17th International Workshop on Computational Mechanics of Materials August 2007 - IWCMM17-Paris BOOK...

Wednesday

Thursday

Friday

Plenary

session

Heterog.Materials

Multiphysics

Heterog.Mat.

Crystalplasticity

830-9

00

G.Cailletaud

M.Apel

T.Ricken

G.Laschet

K.Jöchen

900-9

30

T.Sadowski

P.Cartraud

O.Goy

J.-M

.Pipard

S.Naamane

930-100

0S.

Schm

auder,W.Lutz

T.Antretter

K.P.Jayachandran

R.Piat

A.Prakash

1000-103

0A.Bertram

C.Barbier

S.Meftah

M.M

.Aghdam

M.Razandrazaka

1030-110

0Co

eebreak

Com

posites

Macrosc.behaviour

SizeDepEn-Plenary

session

Multiphysics

Interfaces

1100-113

0I.Scheider

S.Benke

T.Hochrainer

N.Marchal

M.Bäker

1130-120

0B.Latha

Shankar

A.Kovacs

M.Rodriguez

K.Weinberg

L.Marsavina

1200-123

0C.Jochum

Z.Hruby

I.Groma

B.X.Xu

E.-M.Craciun

1230-140

0Lunch

Com

posites

Macrosc.behaviour

SizeDepEn

Localizationanddamage

Plenary

session

1400-143

0T.Seelig

A.M

.Karlsson

Y.Cheng

S.Dimitrov

S.Lurie

1430-150

0L.Figiel

S.K.Panthi

F.F.Csikor

R.Bargellini

L.Delannay

1500-153

0I.Ivanov

V.V.Silberschm

idt

S.Limkumnerd

G.Franz

B.Fedelich

1530-160

0Co

eebreak

U.Prahl

1600-163

0Postersession

H.Yuan

1630-180

0

1800-200

0Welcomecocktail

Reception/banquet

1 Wednesday, 22nd

1.1 Plenary session - 900 − 1030

T. Sadowski : Non-symmetric thermal shock in layered FGM cylindrical plates -comparison of two applied methods for problem solution, T. Sadowski, S.Ataya andK. NakoniecznyW. Lutz : Damage development in short ber reinforced injection molded thermo-plastic composites, W. Lutz, J. Herrmann, M. Kockelmann, A. Jäckel, S. Schmauder,S. Predak and G. BusseA. Bertram : Micro macro modeling of metal forming processes, A. Bertram, T.Böhlke, G. Risy and V. Schulze

1.2 Composites - 1100 − 1230

I. Scheider : Modelling of the damage behaviour of bre reinforced metal matrixcomposites using a representative unit cell, I. ScheiderB. Latha Shankar : Experimental study on mechanical fastening of FRP compositelaminates for structural use, B. Latha Shankar, S.M. Shashidhara and G.S.ShivaShankarC. Jochum : A FEM coupling model for properties prediction during the curing ofan epoxy matrix, N. Rabearison, Ch. Jochum and JC. Grandidier

1.3 Macroscopic behaviour - 1100 − 1230

S. Benke : Modeling of the tensile and compression behavior of semi-solid A356alloy, S. Benke, S. Dziallach, G. Laschet, U. Prahl and W. BleckA. Kovacs : Eective material properties of multi-layered porous nanolters, A.Kovacs, A. Kovacs and U. MeschederZ. Hruby : On critical plane criteria assessments in case of combined tensile andtorsion load, Z. Hruby, J. Papuga, K. Doubrava and M. Ruzicka

1.4 Composites - 1400 − 1530

T. Seelig : Modeling the interaction of crazing and matrix plasticity in rubber-toughened glassy polymers, T. Seelig and E. Van der GiessenL. Figiel : Physically-based modelling of large deformations of clay/polymer nano-composites, L. Figiel, F.P.E. Dunne, C.P. BuckleyI. Ivanov : Experimental investigation and modelling of crack propagation in ply-wood by CT test, T. Sadowski, I.Ivanov and M.Filipiak

iii

1.5 Macroscopic behaviour - 1400 − 1530

A.M. Karlsson : Modeling and simulations of interfacial crack growth of cyclicallyloaded structures, D. Cojocaru and A. M. KarlssonS.K. Panthi : An analytical modeling and FE simulation of arc bending process topredict the springback, S. K. Panthi and N. RamakrishnanV.V. Silberschmidt : Creep damage study at power cycling of lead-free surfacemount device, P. Hegde, D. Whalley and V. V. Silberschmidt

iv

2 Thursday, 23rd

2.1 Heterogeneous materials - 830 − 1030

M. Apel : Eective elastic constants of a 3D two phase structure calculated by amultiphase-eld approach, M. Apel, S. Benke and I. SteinbachP. Cartraud : Image-based Micromechanics Analysis Using Level Sets and the Ex-tended Finite Elements Method, I. Ionescu, N. Moës, P. Cartraud, N. Chevaugeonand M. BéringhierT. Antretter : An Energy Approach to Determine the Martensite Morphology inNano-Structured NiTi Alloys, T. Antretter, W. Pranger, T. Waitz and F.D. FischerC. Barbier : Computational approach to the mechanics of entangled materials, C.Barbier, R. Dendievel and D. Rodney

2.2 Multiphysics - 830 − 1030

T. Ricken : Remodeling and growth of living tissue - A Multiphase Theory, T.Ricken, J. BluhmO. Goy : Modelling of Point Defects in anisotropic media with an application toFerroelectrics, O. Goy, R. Mueller, D. GrossK.P. Jayachandran : Homogenization of textured as well as randomly orientedferroelectric polycrystals, K P Jayachandran, J M Guedes and H C RodriguesS. Meftah : A numerical modelling of the plasticity induced by martensitic trans-formation in a 16MND5 steel. A multigrain approach, S. Meftah, F. Barbe, L. Taleb,F. Sidoro

2.3 Plenary session Size-dependent mechanical properties -

1100 − 1130

T. Hochrainer : A non-linear theory of multiple slip in continuum dislocation dy-namics, T. Hochrainer, P. Gumbsch, M. Zaiser and S. SandfeldM. Rodriguez : Simulation of the strain path change eect on drawn tungstenwires, M. Rodriguez Ripoll, H. Riedel and S. Forest.I. Groma : Stress screening by dislocations, I. Groma, G. Gyorgyi and P. D. Ispa-novity

2.4 Size-dependent mechanical properties - 1400 − 1530

Y. Cheng : Simulated low-angle tilt grain boundaries and their response to stress,Y. Cheng, D. Weygand and P. GumbschF.F. Csikor : Pair correlations in 3D dislocation systems, F. F. Csikor, I. Groma,T. Hochrainer and M. Zaiser

v

S. Limkumnerd : Formulation of a multiple-slip nonlocal plasticity theory ba-sed on a statistical approach, S. Limkumnerd and E. van der Giessen

2.5 Localization and damage - 1400 − 1530

S. Dimitrov : Incremental modeling of damage in brittle materials, S. DimitrovR. Bargellini : Towards a robust non local numerical analysis of ductile fracture,R. Bargellini, J. Besson, E. Lorentz and S. Michel-PonnelleG. Franz : Strain localization analysis using a multiscale model, G. Franz, F.Abed-Meraim, T. Ben Zineb, X. Lemoine and M. Berveiller

vi

3 Friday, 24th

3.1 Heterogeneous materials - 830 − 1030

G. Laschet : Eective permeability and thermal conductivity of open-cell metallicfoams via homogenization on a microstructure model, G. Laschet, O. Reutter, J.Sauerhering, J. Scheele and R. Pitz-PaalJ.-M. Pipard : Mean eld approach with internal length scale accounting for nonlocal eects due to geometrically necessary dislocations, J.-M. Pipard, S. Berbenni,O. Bouaziz, N. Nicaise and M. BerveillerR. Piat : Material parameter identication of heterogeneous materials with inclu-sions on dierent length scales, R. Piat, P. Gumbsch, T. BöhlkeM.M. Aghdam : Overall properties of randomly distributed particle reinforcedcomposites using simplied unit cell model, M. M. Aghdam, M. Shakeri and M. J.Mahmoodi

3.2 Crystal plasticity - 830 − 1030

K. Jöchen : Approximation of crystallographic texture as a mixed integer quadraticprogramming problem, T. Böhlke, K. Jöchen, U.-U. Haus and V. SchulzeS. Naamane : Dislocations mobility laws and precipitation strengthening in BCCcrystals at low temperatures by Dislocation Dynamics simulations, S. Naamane, G.Monnet and B. DevincreA. Prakash : Modeling the evolution of texture and grain shape in magnesium alloyAZ31 under plane strain compression, A. Prakash, H. Riedel and S. WeygandM. Razandrazaka : Discrete Dislocation Dynamics Simulations of dislocationchannelling eects on Inter granular fracture, M. Razandrazaka, D. Delafosse andD. Tanguy

3.3 Multiphysics - 1100 − 1230

N. Marchal : Finite element simulation of Pellet-Cladding Interaction (PCI) innuclear fuel rods, N. Marchal, C. Campos and C. GarnierK. Weinberg : Kirkendall voids in the intermetallic layers of solder joints in MEMS,K. Weinberg, T. Böhme and W. H. MüllerB.X. Xu : Micromechanical analysis of ferroelectric structures by a phase eldmodel, D. Schrade, B.X. Xu, R. Mueller, and D. Gross

3.4 Interfaces - 1100 − 1230

M. Bäker : Simulation of crack propagation at interfaces using energy release rates,M. BäkerL. Marsavina : Fracture parameters at bi-material ceramic interfaces under bi-axialstate of stress, L. Marsavina and T. SadowskiE.-M. Craciun : Mathematical modelling of the interface crack propagation in a

vii

pre-stressed ber reinforced elastic composite, N. Peride, A. Carabineanu and E.-M.Craciun

3.5 Plenary session - 1400 − 1630

S. Lurie : Advanced theoretical and numerical multiscale modeling of cohesion/adhesioninteractions in continuum mechanics and its applications for lled nanocomposites,S. Lurie , D. Volkov-Bogorodskii, V.Zubov and N. TuchkovaL. Delannay : Multisite modelling of grain interactions in single- and multiphasepolycrystalline aggregates, L. Delannay, M. A. Melchior and J. W. SignorelliB. Fedelich : A simplied modelling of rafting in Ni-base superalloys, B. Fedelich,U. Brückner, A. Epishin, G. Künecke, T. Link, T. May and P. PortellaU. Prahl : Deformation induced phase transformation and mechanical properties ofdual phase steels using combined phase eld and nite element modelling, U. Prahl,S. Benke, N. Warnken, V. Uthaisangsuk and W. BleckH. Yuan : Prediction of fretting fatigue crack propagation using extended niteelement methods, Y. Xu and H. Yuan

viii

4 Posters

4.1 Heterogeneous materials

• The Mori-Tanaka homogenization scheme for particulate composites in nite strainwith interface debonding, L. Brassart, I. Doghri, L. Delannay and P. Geubelle• Stretch-angeability characterisation of multi phase steel using a microstructurebased failure modelling, V. Uthaisangsuk, U. Prahl and W. Bleck• Rubber lled with carbon black from the nanoscopic structure to the macrosco-pic behaviour, A. Jean, S. Cantournet, S. Forest, D. Jeulin, V. Mounoury and F.N'Guyen• On tunability amplication factor of composite dielectrics, Kolpakov, A.G.

4.2 Macroscopic behaviour

• Modelling of semi-crystalline polymers behaviour during ECAE process, B. Aour,F. Zaïri, M. Naït-Abdelaziz, J.M. Gloaguen and J.M. Lefebvre• Quantication of the inuence of vertex singularities on crack behaviour, P. Hutaø,L. Náhlík and Z. Knésl• Computational assessments of residual stress relaxation induced by shot peening,J. Liu, H. Yuan and M. Becker• Glass ber reinforced plastic leaf springs for light weight vehicle suspension,G.S.Shiva Shankar, B. Latha Shankar and S.Vijayarangan• Dynamic hysteresis behavior of TMCP steel(SM490-TMC) and its modeling, G.-C. Jang, K.-H. Chang• Analysis of structural performance of aluminium sandwich plates with foam-lledhexagonal cores, T. Sadowski and V. Burlayenko• Modelling of temperature shocks in composite materials using a meshfree FEM,K. Nakonieczny and T. Sadowski• Improved beam nite element for analysis of slender transversely cracked beam-columns, M. Skrinar• Calculus of the cylindric spring with an uniformly vertical load by transfer-matrixmethod, M. Suciu

4.3 Interfaces

• Numerical failure analysis of dual phase steel using cohesive modelling on realmicrostructures, U. Prahl, B. v. Binsbergen, C. Thomser, V. Uthaisangsuk and W.Bleck• Interface damage of SiC/Ti metal matrix composites subjected to axial shear loa-ding, M. M. Aghdam, M. Gorji and S.R. Falahatgar• A 3D nite element modelling of crack propagation under fretting fatigue condi-tions, H. Benzaama, E. Giner, Aour B., F.J. Fuenmayor and S.M. Elachachi• Prediction of failure metal/composite bonded joints, Derewoñko Agnieszka

ix

4.4 Composites

• Fracture mechanics assessment of singular stress concentrations caused by bresat a free surface, J. Klusak and Z. Knésl• Fatigue damage simulation in hybrid Titanium-PEEK /AS4 composite laminates,P. Naghipour, J. Hausmann, M. Bartsch, H. Voggenreiter• Analytical elasticity solution for general laminated long rotating cylinder undervarious loading conditions, M.M. Aghdam, M. Shakeri, M.J. Mahmoodi• Numerical analysis of the energy localization in high-contrast dense-packed com-posites, A.A. Kolpakov

4.5 Crystal plasticity

• Crystal plasticity modelling of texture development in TWIP steel, M.A. Melchior,P.J. Jacques and L. Delannay• Modelling of creep damage in polycrystals : the combination of crystal plasticitywith cohesive zone models for grain boundary cavitation, S. Kassbohm, B. Fedelich,M. Budnitzki and D. Noack• Numerical Implementation of a 3D-Continuum Theory of Dislocations, S. Sand-feld, T. Hochrainer, M. Zaiser and P. Gumbsch• Comparison of DDD and continuum approach in simulations of tensile test of thinlms, F. Siska, S. Forest, P. Gumbsch and D. Weygand• Multi-scale approach to the analysis of indentation experiments in polycrystallinecoatings, O.Casals and S.Forest• Study of non-proportionnal loading paths : comparison between experimental re-sults and simulations performed by nite element and homogenized models, C. Gé-rard, B. Bacroix, M. Bornert, R. Brenner, G. Cailletaud, D. Caldemaison, T. Chau-veau, J. Crépin, S. Leclercq and V. Mounoury.• Free meshing of microstructures based on modied Voronoi tessellations, R. Queyand F. Barbe

4.6 Localization and damage

• Crack growth modelling of single crystals for higher-order continua, O. Aslan andS. Forest• Simulation of generation of mesobands of inelastic deformation in surface layersof deformable solid : the stochastic approach of excitable cellular automata, D.D.Moiseenko, A.L. Zhevlakov, P.V. Maksimov and V.E. Panin

4.7 Size-dependent mechanical properties

• Role of elastic nonlinearity in dislocation patterning, P.D. Ispanovity and I. Groma• Time evolution and numerical analysis of dislocation-dislocation correlation func-tions, R. Deak and I. Groma

x

• Microscale patterning of plastic deformation : self ane surfaces and scale freestatistics of deformation-induced surface steps, J. Schwerdtfeger, E. Nadgorny, V.Koutsos and M. Zaiser

4.8 Multiphysics

• Diusion induced phase transformations in mechanical stressed lead-free alloys, T.Böhme, W.H. Müller and K. Weinberg• Finite element formulation of phase els models based on the concept of generali-zed stresses, K. Ammar, B. Appolaire, G. Cailletaud, F. Feyel and S. Forest• The study of exposure time to defend on reectance of interface at holographiclithography, J.-B. Yeo, S.-D. Yun and H.-Y. Lee• FE analysis of the eect of the kinetics of nucleation on the plasticity inducedduring diusive transformations, H. Hoang, F. Barbe, R. Quey and L. Taleb

4.9 Late posters

• Numerical determination of heat distribution and castability simulations of as castMg-Al alloys, S. S. Khan, N. Hort, I. Steinbach, S. Schmauder and U. Weber

xi

!"$# %&('*),+-),.0/21)435)4/65798;:+-),.0<=:>?7 @ACBDA(EGFIHKJMLONDPCHQESRUTCV"LONUJXWYCZVLOT[Z,L\V]Y[^_Ta`Cbdc5eYgfh2TCVHXi,LOjOkJ\LOkjOHXJKT[RUJlkRUT0LmH-npo4qrTgs*kR LONDth2TCVHluwv_NDHKRUnyxzttjmY,TCJ\h|o4qrsBxzt2HKR ~ACB wgT[CHluwoTCVH-nas*NUJljOYCH-JOh2T[ZNUJKVxzZ2T[RDqVNV$VmNDZHKPCHKR|i4HlLOV]T[ZnaLmhHEG4LOHKZnH-npv_NDZN LOHESRDHKHKZ,LOV!sHlLmhYn|o4qaf_B2WT[jmLmj\T[knACB `xzZyHKZHKjOCqaT[ttjOY,TCJ\haLOYgnHlLmHKjONDZHQLmhHQaT[jLOHKZVN LOHYCjOthYCRDY,CqNDZyZ2T[ZYgVLmjOkJMLmkjmH-npzNDuwc]NT[RDRDY0qVK4o4qpcBxzZ,LOjmHlLmLmHKjQBBACAACB WYCtkL\T0LmNDY,ZT[RIT[ttjOY,TCJ\haLOY9LOhHQH-J\hT[ZNUJKV]Y[^_HKZ,L\T[ZCRDH-naaT0LmHKjONUT[RUVlo4qyW$B2T[jOoNDHKjBBXBQBBQBBQBBQBQBBXBBQBBQBBA-ÀCB EGFIHKJMLONDPCHSt2HKjmgHKT[oNDRDN L"qT[Z2n LmhHKjOgT[RCJlYCZnkJMLONDP4N L"q Y[^YCt2HKZuwJlHKRDRCHlL\T[RDRDNUJG^Y,T[gVIPNUT!hYCYCCHKZND-T0LONDYCZ YCZXT!NUJljOY,V"LOjmkJMLOkjOH

YnHKR?o4qy9B2TCVOJOhHML BBQBBXBBQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBAACB s*H-T[ZHKRUnT[ttjOY,TCJ\he]NDLmhNDZ,LOHKjmZT[RRDHKZ[LOhVmJKT[RDHyTCJlJlYCkZ4LmNDZ^YCjZYCZRDYJKT[RHlFIHKJML\VnkHpLmYCHKYCHlLOjmNUJKT[RDRDqZH-JlHKVOVOT[jmq

nNUVmRDYJKT0LmNDYCZVK4o4qp B uwsBfSNDtT[j\n¡BQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBA0~ACB ~sT0LOHKjmNUT[RItT[j\T[HlLmHKj]NUnHKZ,LmND2JKT0LmNDYCZ*Y[^5hHlLOHKjmYCCHKZHKYCkVgT0LOHKjmNUT[RUVe!N LOhyNDZ2JMRDk2VNDYCZ2VYCZ*nN FIHljOHKZ,LzRDHKZ[LmhVmJKT[RDH-VK4o4qp¢BfSNUT0L A-ACB £¥¤$PCHKjOT[RDR,tjmYCt2HKjmLmNDH-V_Y[^2j\T[ZnYCRUqXnNUVLmjONDokLOHKn9tT[jmLmNUJlRDHjmHKNDZ^=YCjOJlH-n¦JlYCt2Y,VmN LOHKV_kVmNDZQVmNDtRDNDH-n¦kZN LJlHlRURYnHKR?[oq¦s§B sB

xzChnT[¨BBQBBQBBQBQBBQBBXBBQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBA©ª8«:¬[.0/27-¬C/G<=¬y®S)¯5:C°<=/265. ©±BDA²zYCZuwVmq4gHMLq4jmNJ_LOhHKjmaT[RVhYJO³NDZ9RUTqCHKjmH-nQv5Qs´Jlq4RDNDZ2njmNUJKT[R,tRUT0LmH-V|uJlYCtT[jONUVYCZXY[^L"eY T[ttRDNUHKn¦HlLmhYnV^YCjGtjOYCoRDHK

VYCRDkLmNDYCZ|o4qrcBiTCnY0ezV³Nµ¶BXBBQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBCB ·s*YnHlRUNDZY[^5LmhHQLmHKZVmNDRDHXT[ZnyJMYCgtjOHKVOVNUYCZro2HKhTP4NDYCj!Y[^GVHKNDuwVmYCRDNUnpx$`,[9T[RURDY0q[o4qpiIBHKZ³CHXBBQBBQBBQBQBBXBBQBBQBB,~B `·EGFIHKJMLONDPCHQgT0LOHKjmNUT[RItjmYCtHKjLONDH-VY[^_¦kR LON u¸RT-qCHKjOH-nat2YCjOYCkV]ZT[ZY[2R LmHKj\Vlo4qpxXB2¹$YP0TCJKV9BBQBBXBQBBQBBQBBQBQBBXBBQBBQBB[B ¤$ZyJljON LmNUJKT[R|tRUT[ZHQJljON LmHKjONUTTCVOVH-VmVmHKZ,L\VNDZ*JKTCVmHY[^GJlYC¦oNUZHKnrLOHlZVmNDRDHQT[Z2nrLOYCjOVmNDYCZrRDY4TCnIo4q*º5B2»!jOko4q¼BQBQBBXBBQBBQBB½ÀB ·xzZ*T[ZT[RUq,LmNUJKT[RgYnHKRDNDZaT[Znyv_E¾VmND¦kRUT0LONDYCZpY[^T[jOJ$oHKZnNDZatjmYJlH-VOVSLOYgtjmH-nNUJML]LmhHXVtjONDZCo2TCJO³o4qpiB ¹BfT[Z,LmhNpBQBB½`C`B ¥WjmHKHKtynT[aT[CHV"LOknqyT0L!t2Y0eHlj!JlqJlRDNDZgY[^_RDH-TCn4u¸^jmHKHQVmkjm^=TCJlH YCkZ,L$nHKP4NUJlHCo4qr¿¦B ¿¦Bi4NURDo2HKjOVOJ\hNUn4LÀBQBQBBXBBQBBQBB½`,B ~s*YnHlRUNDZgT[ZnyVNU¦kRUT0LONDYCZV]Y[^_NDZ4LmHKj^?TCJMNT[RIJljOTCJ\³gCjOY0e]LmhpY[^GJlqJlRDNUJKT[RDRDqgRUY,TCnH-npVLmjOkJMLmkjOH-Vlo4qrxXB sB¹XT[jmRUVOVYCZBBQBBQBB½`4~

±ÁM3I+K)4.Âw:¬C)47 ±2Ã`BDA(fjOH-nNUJMLmNDYCZyY[^_^jmHlLmLmNDZ^=T0LONDCkHXJljOTCJ\³gtjmYCtT[,T0LONDYCZak2VNDZHl4LmHKZnH-npZN LOHQHKRDHKHKZ,LzHlLmhYnVlo4qr»XBÄk2T[ZµBXBBQBBQBBA`B i4ND¦kRT0LmNDYCZyY[^_Jlj\TCJO³atjmYCt2T[,T0LmNDYCZyT0L!NDZ,LmHKjm^=TCJlHKV]k2VNDZgHlZHljOCqgjOHKRDH-TCVHj\T0LmH-VK4o4qps§B2]Å[³CHljpBQBBQBBQBBQBQBBXBBQBBQBB,``B `vj\TCJ\LOkjOH tT[jOT[gHlLmHKjOV]T0L!oN uwgT0LOHljONUT[R|JlHKj\T[NUJ$NDZ,LOHKj^=TCJlH-V]kZnHKj!oN u"T0NUT[R|VLOT0LOHY[^V"LOjmH-VOVlo4qpSBsT[jOVOTP4NDZTÆBXBBQBBQBB4`B ÇsT0LOhHKgT0LONUJKT[RYnHlRDRUNDZ9Y[^5LmhHNUZ,LmHKjm^=TCJlHJMj\TCJ\³tjOYCtT[,T0LmNUYCZgNDZ*T9tjOHMu"V"LOjmH-VOVH-ngo2HKj]jmHKNDZ^=YCjOJlH-naHKRUTCVLmNUJ JlYCtY,VN LOHCo4q

E!B u"s§BIWjOTCJlNDkZ*BQBBQBQBBQBBXBBQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBB~ÈÇÉ¦/2Ê/27-<+-)47 ÈÃBDA²b T[gT[CHnHKPCHKRDYCtHKZ,L!NDZVhYCjL]oHlj!jOHKNDZ^YCj\JMH-npNDZËH-JMLmNDYCZ*YCRUnHKnrLmhHljOYCtRTCV"LONUJJlYCt2Y,VmN LmH-VKo4qrÌdBkLOµaBBQBBQBBAB ·x!nP0T[ZJlHKnLmhHKYCjOHMLONUJKT[R,T[ZnZ4kgHKjmNUJKT[RC¦kRDLmNUVOJKT[RDHgYnHKRDNDZ$Y[^JlYChH-VmNDYCZÍ0TCnhH-VmNDYCZ NDZ,LOHlj\TCJMLmNDYCZ2V|NDZ¦JlYCZ,LONDZ4kkÎH-JOh2T[ZNUJKV

T[ZnrNDLOVzT[ttRUNUJlT0LONDYCZV^YCj]RDRUHKnyZT[ZYJlYCgt2Y,VNDLmH-Vlo4qpiIB|kjmNDH0µQBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBB[`B `s*YnHlRURDNDZpY[^SLOhHgnT[aT[CH¦o2HKhTP4NDYCkj$Y[^SojmHjmHKNDZ^YCj\JlH-n*HlL\T[R_aT0LmjON ÏJMYCgt2Y,VmN LmH-V$kVmNDZ*TrjOHltjmH-VmHlZ,L\T0LONDPCHXkZN LXJlHKRDR?

o4qaMB2iJ\hHKNUnHKjBQBBQBQBBQBBXBBQBQBBQBBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBCB EGtHKjmNDgHlZ,L\T[R|VLmk2nqrYCZyH-J\hT[ZNUJKT[R^?TCV"LOHlZNDZY[^Gv_¢!fJlYCtY,VN LOHRUT[NUZT0LmH-V^YCjzVLmjOkJMLmkj\T[RIk2VHCoqr B5T0LmhTai4hT[Z³0T[j½,~B ·xdv_ESs¼JlYCktRUNDZYnHKRI^YCj!tjOYCt2HKjmLmNDH-VtjOHKnNUJ\LONDYCZ*nkjONDZLOhHQJlkjONDZY[^T[ZpHKt2YqrgT0LOjmN Io4qyW$B2 CYJ\h4k BBQBBQBB[B s*YnHlRUNDZ9LOhHQNDZ,LOHKjOTCJMLmNUYCZrY[^JMj\T[KNDZT[ZnpgT0LOjmNDrtRUTCVLmNUJlN LqrNDZyjmkoo2HKju¸LmYCkChHKZH-nrCRUTCVOVqat2YCRDq4gHlj\Vlo4qrcBi4HKHlRUND;BQBB½AB ~ªfh4qVNUJKT[RDRDq,uwoTCVmHKnrYnHKRDRDNDZY[^5RT[jmCHQnHl^YCjOgT0LONDYCZVY[^JMRUTqÍtYCRDq4HKjZT[ZYJlYCgt2Y,VNDLmH-Vlo4qpSBv_NDCNDHKRÐBQBQBBXBBQBBQBB½C`B £·EGtHKjmNDgHlZ,L\T[RINDZPCHKVLmNU,T0LmNDYCZyT[ZnrgYnHKRDRDNDZaY[^_Jlj\TCJO³atjmYCt2T[,T0LmNDYCZpNDZytRDq4eY4Ynro4q*Wc¾LmH-VLKo4qrMB"PT[ZY0PBXBBQBBQBB½,

@·ÉX.Ñ|7K+-:>]ÒQ>=:7K+-<=¬C<=+OÑ ÓÔBDA²s*NUJljOY9gTCJljOYYnHlRUNDZY[^_HlLOT[R|^YCjONDZtjmYJlH-VmVmH-Vlo4qrxXB2HljmLmj\T[rµÕBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBB½CB ·s*kR LmNUVmN LOHQYnHKRDRDNDZY[^_Cj\T[NDZpNDZ,LmHKj\TCJ\LONDYCZVNDZVNDZCRDHluST[Znp¦kR LONDthTCVmHQt2YCRUqJMjOqVLOT[RDRDNUZHT[CCjmHK,T0LOH-Vl,oqrBb$HKRUT[ZZTqµªBÖ~4AB `xdVmNDtRDN 2HKnyYnHlRDRUNDZY[^_jOT0^LONDZgNDZ*zN uwoTCVHXVktHlj\T[RDRDY0qVK4o4qp$B2vH-nHKRDNUJ\hµ¼BQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBÖ~0`B ÇxzttjOYNDgT0LONDYCZrY[^SWjmqVLOT[RURDYCCjOT[thNUJ$cHl4LOkjmHXTCV]Tgs*N HKnywZ,LOHKCHKj$× kTCnj\T0LmNUJ$fSjmYC,jOT[gNDZ¦fSjmY,oRDHKyo4qp¹B CØJOhHKZÙBÖ~[B ·b$NUVRUYJlT0LONDYCZVQgYCoNDRDN LqRUTezVXT[Z2ntjmH-JlNDtN L\T0LmNDYCZVLmjOHKZ[LmhHlZNDZNDZzW!WÕJMjOqVLOT[RUVQT0L9RDY0e;LOHlgt2HKjOT0LOkjOH-VQo4q§nNUVRUYJlT0LONDYCZ

nq4ZT[gNUJlV!VmND¦kRUT[LmNDYCZVK4o4q*iB$TCT[aT[ZHÚBQBBQBXBBQBBQBBQBBQBQBBXBBQBBQBQBBQBBQBBXBQBBQBBQBBQBQBBXBBQBBQBBÖ~C~

Û

ÜÝ Þß*àáâlãUäDåæçOèâXâKéCàCãDêçOäDàCåyà[ëGçOâMì4çOêíOâ¦î[åá*æCíOî[äDå*ïmèî[ð2âXäDåñgî[æCåâ-ïmäDêñî[ãDãDà0òpóôIõöQêå2áâlízðãUî[åâ¦ïçmí\î[äDå*÷làCñgðímâ-ïOïäDàCåøù4òóXÝúSíOî[û0îCïmèÇÝÝQÝÝQÝQÝÝQÝÝXÝÝQÝQÝÝQÝÝQÝÝQÝXÝÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝÖü0ýÜÝ üþ$äUïm÷líOâlçmâQþ$äUïmãDà÷Kî0çmäDàCå*þ$ò4åî[ñgäU÷Kï!ÿäDñ¦êãUî0çOäDàCåï]à[ëáäUïãUà÷lî0çOäDàCå*÷Oèî[ååâKãDãDäDåæ9ââ-÷Mç\ï]àCådëí\îC÷\çOêíOâCøù4òrßÝ!î-îå2áíOî-î[û0î ö

! #"$"#%&'( )*ÞÝDö+wå÷líOâlñgâKå,çOî[ã|ß*àáâlãDäDåægà[ëGþ î[ñgî[æCâäDå-,íOä çmçmãDâQßî0çOâlíOäUî[ãUïKøù4òyÿÝþ$äDñgä çmíOà0éÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ.,ÜÞÝ /10à2]î[í\áïî9ímàCùêï"ç!åàCåyãDà÷Kî[ãIå4êñâKímäU÷Kî[ãî[åî[ãDòïmäUïà[ëGáê2÷\çOäDãDâëí\îC÷MçmêíOâCøù4ò3QÝ4,]î[íOæCâlãDãUäDåä ÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ.4üÞÝ õÇÿ,çmí\î[äDåyãDà÷Kî[ãDä5-î0çmäDàCåyî[åî[ãDòïmäUïêïäUåæaî9ñ9êã çmäïm÷Kî[ãDâñàáâKã?øù4ò69Ý47í\î[åÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ.Cý

819 (;:<"#(;=(> #"#(> ?@%&(>BA# CD=CE4=D(;EB(>F GIHü4ÝDö²óÎåàCåJwãDäDåâ-î[íçmèâlàCíOògà[ëGñ¦êã çOäDðãDâXïmãDäDð*äDå*÷làCå,çOäDå4êêñ´áäUïmãDà÷Kî0çmäUàCåyáò4å2î[ñäU÷Kïløù4òK0ÝLzà÷OèíOî[äDåâKíMXÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ½ýCõü4Ý /ÿ4äDñ¦êãî0çmäDàCåyà[ëçOèâXïçmí\î[äDåpð2î0çmè÷Oè2î[åæCâ ââ-÷\çzàCåáí\îB2]årçmêåæ,ï"çOâKå62!äDíOâ-ïløù4òrßÝ4!àáíOäDæCêâNO!äDð2àCãUãPMpÝÝQÝQÝÝXÝÝQÝÝQÝÝ½ý,Üü4Ý õÇÿ,çmíOâ-ïmï!ïO÷límâKâKåäDåæù4òráäUïmãDà÷Kî0çmäUàCåïløù4òQMÝ íOàCñgî>MpÝÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ½ý4üü4Ý R¥ÿ4äDñ¦êãî0çmâ-árãDà2SJ"î[åæCãDâ$çmäUã çzæCíOî[äDåyù2àCêåáî[ímäDâ-ï]î[åápçmèâKäDí!íOâ-ïð2àCå2ïâ çOàgïçmíOâ-ïmïKøù4òQT9Ý?UèâKåæpÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ½ýCýü4Ý ÜúGî[äDíz÷làCíOímâKãUî0çOäDàCåïäDå*õCþ«áäUïmãDà÷Kî0çmäDàCå*ïòïçOâlñaïløù4òK7Ý 7ÝVUïäDûCàCízÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öWöü4Ý ÞX7àCíOñ¦êãUî0çOäDàCårà[ëîßêã çOäDðãDâJ"ÿ4ãDäDð-YzàCåãDà÷Kî[ã|úãUîCïçOäU÷Mä çòK0]èâKàCíOòK,]îCïmâKápàCåyîaÿ,çOî0çOäUïçmäU÷Kî[ã|ózððímà,îC÷\è|øù4òpÿÝZ|äUñû4êñåâlí\á Ý;öWCõ

)[]\CP=#AVÎFF H_?`ÝDö²þ$âMë=àCímñaî0çmäDàCåyäDåáê÷lâ-á*ðèîCïmâçmí\î[åïëàCímñaî0çmäDàCå*î[åáyñâ-÷Oè2î[åäU÷Kî[ãIðíOàCð2âKímçmäDâ-ï!à[ëGáêî[ãðèîCïâ¦ï"çOâlâKãUï!êïmäDåær÷MàCñ9ùäDåâ-ápðèîCïâ

âKãUápî[åá3åäDçmâQâKãDâKñâKå,çzñàáâlãDãUäDåæøù4ò3aXÝúSíOî[èãPMÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öW4üÝ /1!âlñgàáâKãDäDåæaî[åá6 íOà2]çmèyà[ëbZäDé4äDåæQ0äïmïmêâdcró«ß*êã çmäDðè2îCïâe0]èâKàCíOòCø4ù4ò30Ý4!äU÷OûCâKåÎÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öWCýÝ õß*àáâlãUãDäDåæà[ë_ú_àCäDå4çzþ$âlëâ-÷Mç\ï]äUå*î[åäUïmà[çOímàCðäU÷$ñgâ-áäUîf2!ä çOè*î[åî[ððãDäU÷Kî0çOäDàCåaçOàQ7âKíOímà4âKãDâ-÷MçmíOäU÷Kïløù4ò6g¦Ý à0ò ÝQÝÝXÝÝQÝÝQÝÝ;öCöCöÝ RhLzàCñàCæCâKåä5-î0çmäUàCårà[ëçOâlì4çmêímâ-á*îCïi2âKãDã|î,ïí\î[åáàCñgãDògàCíOäDâKå,çmâ-árëâKímíOà4âKãDâK÷MçOímäU÷$ðàCãDò÷límòïç\î[ãUïløù4òKjÝ úGÝ4k,îò,îC÷\èî[åáí\î[å ÝÝQÝÝ;öCö-õÝ Ü·ódå4êñgâlíOäU÷Kî[ã|ñgàáâKãDãDäDåæaà[ë_çOèâXðãUîCïçOäU÷MäDç"òräUåáê÷lâ-áyù4òyñgî[ímçmâKåïmä çOäU÷ çmí\î[åïëàCímñaî0çmäDàCåyäDåî*ö-ÞCßlY$þÜïçmâKâKã?Ý]ódñ¦êã çmäDæCí\î[äDå

î[ððíOà,îC÷Oèø,ù4òyÿIÝßâMëçOî[è ÝÝXÝÝQÝQÝÝQÝÝQÝÝQÝXÝÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öCöÜÝ ÞX75äDåä çmâQâKãDâKñâKå,ç$ïmäDñ¦êãUî0çOäDàCårà[ëGúGâlãDãDâlçmJUãUîCááäDåæ@"å,çmâKí\îC÷\çOäDàCåonúSUDpäDå*å4ê÷lãDâ-î[íë=êâKãíOàáïKø4ù4ò3YXÝ2ßî[íO÷\èî[ãÐÝÝXÝÝQÝÝQÝÝ;öCö0üÝ üjäDímû,âlåáî[ãDãéCàCäUáïäDåyçOèâQäDå,çOâlíOñâlç\î[ãDãDäU÷ ãUîòCâKíOïà[ëGïàCãUáâlí#qàCäUå,çOï]äDåßrßÏÿøù4ò6jÝsâKäDå4ù2âKíOæÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öCö-ýÝ ß*äU÷líOàCñâ-÷\èî[åäU÷Kî[ãî[åî[ãUòïäUïà[ë_ëâKímíOà4âKãDâ-÷\çOímä÷ ï"çOímê÷MçOêíOâKï]ù4òpî9ðèîCïmâtâKãUáyñàáâKã?øù4òK, Ý u¦Ý4u$ê ÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öB/ö

vv wxyzI|~?xDDz|4tzxDdy C

G(;(;E'?(> C(>?\CFf[(;EF H`ýÝDö0]èâQß*à,ímä J05î[åî[û0î¦èàCñàCæCâKåä5-î0çmäDàCåpïm÷\èâKñâëàCízðî[íçOäU÷lêãUî0çOâ÷làCñð2à,ïmä çmâ-ï]äDå6åä çOâXï"çOí\î[äDå32!ä çOèyäDå,çOâKíë=îC÷lâQáâlùàCåáäDåæ(ÝÝ;öB/,üýÝ /ÿ,çmíOâlçO÷\èJ2î[åæCâ-î[ùäDãDä çòg÷\èî[í\îC÷\çOâKímäUïOî0çmäUàCåaà[ë_ñ9êã çOä|ðèîCïmâQïçmâKâKãIêïmäDåæaîñäU÷líOà,ï"çOímê2÷\çOêíOâ ùîCïâ-árë=î[äDãDêíOâñàáâlãUãDäDåæ ÝQÝÝQÝÝ;öB/[ýýÝ õX!êùù2âKíSãDãDâ-á62!ä çmè÷Kî[ímù2àCåyùãUîC÷\ûgë=ímàCñÐçmèâXåî[åà,ïO÷MàCðäU÷ ï"çOímê2÷\çOêíOâ çmàçmèâQñaîC÷límà,ïO÷làCðäU÷$ù2âKèîé4äDàCêí¡ÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-õöýÝ Rg$årçOêåî[ùäUãDä ç"òyî[ñðãDä ÷lî0çOäDàCåpë=îC÷MçmàCí]à[ëG÷làCñgð2à,ïäDçmâQáäDâKãDâ-÷MçmíOäU÷KïÕÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-õCõ

H_l[]EF4=b6(AC\CE H*?`öWÝDöß*àáâlãUãDäDåæà[ëGïâKñä5Jw÷límòïçOî[ãUãDäDåâð2àCãDòñâKíOïù2âKèîé4äDàCêízáêíOäDåæ@rSUór¾ðíOà÷Mâ-ïOïÚÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-õ4üöWÝ /$êî[å,çOä 2÷Kî0çmäDàCåpà[ë5çOèâQäDåêâKå÷lâà[ë_éCâKímçmâlìrïmäDåæCêãUî[ímä çOäDâ-ï]àCåy÷líOîC÷\ûaù2âKèîé4äDàCêíÙÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-õCýöWÝ õ$UàCñðêç\î0çmäDà,åî[ãIîCïOïâ-ïOïñâKå,ç\ïà[ë_íOâ-ïäUáêî[ãIïçmíOâ-ïmï]íOâlãUî0ìî0çmäDàCåpäDåáê÷lâ-áyù4òrïmèà[ç!ð2âKâKåäDåæ ÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öNRööWÝ R$ ãUîCïOï2ù2âKízímâKäDåëàCí\÷lâ-árðãUîCïçmä÷ ãDâ-î0ë_ïmðímäUåæ,ïëàCí!ãDäDæCè,ç2âläDæ,è,ç]éCâKèä÷MãDâQïmêïmð2âKåïmäDàCåÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öNR,õöWÝ Üþ$ò4åî[ñäU÷è4òï"çOâKímâ-ïmäUïùâKèîé4äDàCí!à[ë#0!ßUú¾ï"çOâKâlãn¸ÿß-R,ýWJ0!ßUSpî[å2árä ç\ï!ñàáâKãDäDåæÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öNR4ÜöWÝ ÞÙózåî[ãDòïmäUï]à[ë_ïçmíOê÷MçmêíOî[ã|ð2âKíëàCíOñgî[å2÷Mâ à[ëî[ãDêñgäDåäDêñïmî[å2á2!äU÷OèyðãUî0çOâ-ïi2!ä çOèyëà,î[ñfJ2ãDãDâ-árèâlìî[æCàCåî[ã|÷làCímâ-ï ÝÝXÝÝQÝÝQÝÝ;öNRüöWÝ üß*àáâlãUãDäDåæà[ë5çmâKñðâlí\î0çOêímâXïmèà÷\ûïäDå÷MàCñgð2à,ïmä çmâñaî0çmâKíOäUî[ãUïêïäDåæaî9ñgâKïmèëíOâKâ7Crß ÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öNR,ýöWÝ ~wñðímà0éCâ-áaùâKî[ñåäDçmâXâKãDâKñâKå,ç]ëàCízî[åî[ãDòïmäUïà[ëïãDâKåáâKí]çOíOî[åïméCâKíOïmâKãDòg÷líOîC÷\ûCâ-áaùâKî[ña÷làCãDêñåï ÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öÜööWÝ ý$Uî[ãU÷MêãDêï]à[ëçOèâX÷lò4ãDäDå2áímäU÷ïmðíOäDåæ@2!ä çOèî[åpêåäDëàCímñgãDòréCâKímçmäU÷Kî[ããDà,îCápù4ògçOí\î[åï"ë=âlímJ¸ñaî0çmíOä ìañâlçOèàáÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öÜ[õ

HH6 VN(>EB(>F H``öCöCÝDöYzêñâKíOäU÷lî[ãIë=î[äDãDêíOâQî[åî[ãDòïäUïà[ëáêî[ãIðèîCïâQïçOâlâKãêïäDåæg÷làCèâ-ïäDéCâ ñgàáâKãDãDäDåægàCåyíOâ-î[ãñäU÷líOà,ï"çOímê2÷\çOêíOâ-ïSÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öÜ,üöCöCÝ /wå,çOâKíë=îC÷lâáî[ñaî[æCâà[ëGÿ4äUS0]äñâlçOî[ã|ñgî0çOímä ìy÷làCñð2à,ïmä çmâ-ï!ïmêù>qâ-÷MçOâKápçmàaî0ìäUî[ã|ïmèâ-î[í]ãDà,îCáäDåæpÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;öÜ[ýöCöCÝ õÙódõCþ2åä çmâXâlãDâKñâKå,çzñgàáâKãDãDäDåæà[ë÷líOîC÷\ûgðímàCðî[æ,î0çOäDàCåaêåáâKí!ëíOâMçmçmäUåæ9ë=î0çOäDæCêâX÷làCåáäDçmäDàCåïÕÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-ÞööCöCÝ RÙúíOâ-áäU÷MçmäDàCåyà[ë_çmèâë?î[äDãDêíOâñâlçOî[ãP0÷làCñðà,ïä çOâ ù2àCå2áâ-áeqàCäDå,ç\ïÝÝQÝQÝÝXÝÝQÝÝQÝQÝÝQÝÝQÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-ÞCõ

H ¡%=DFP(K%&(;E4F H?`öB/ÝDö7í\îC÷\çOêíOâ ñâ-÷Oè2î[åäU÷Kï!îCïmïmâ-ïmïmñâKå,ç]à[ëGïmäDåæCêãUî[í!ïçmíOâ-ïmï]÷làCå÷lâKå,çmí\î0çmäUàCåï]÷lî[ê2ïâ-árùò¢ùíOâ-ï!î0ç$î¦ëíOâlâQïmêímë=îC÷lâ´ÝQÝÝXÝÝQÝÝQÝÝ;ö-Þ4üöB/Ý /~7î0çmäUæCêâQáî[ñgî[æCâïmäDñ¦êãUî0çmäDàCåyäDåyèò4ùímäá30]ä ç\î[åäDêñfJwú£rrDj0ó ÿRg÷làCñð2à,ïmä çmâQãUî[ñäDåî0çOâ-ïÝÝXÝQÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö-ÞCýöB/Ý õÙózåî[ãDò,çOäU÷Kî[ãIâKãUîCï"çOäU÷lä ç"òpïàCãDêçmäDàCåpëàCí!æCâKåâKíOî[ãIãUî[ñäDåî0çOâ-árãDàCåægímà[ç\î0çmäUåæ÷lò4ãDäDåáâlízêå2áâKí]é0î[íOäDàCêïãDà,îCáäUåæg÷làCåáä çOäDàCåïÝÝQÝÝ;ö0ü4ööB/Ý RYzêñâKíOäU÷lî[ãî[åî[ãDòïmäUïà[ëçOèâQâKåâKímæCòrãDà÷Kî[ãDä5-î0çmäUàCåräDå*èäDæCèJw÷làCå,çOí\îC÷\ç!áâKåïmâJ¸ðîC÷\ûCâ-ár÷làCñð2à,ïmä çOâKïrÝÝQÝÝQÝÝQÝQÝÝXÝÝQÝÝQÝÝ;ö0ü0õ

¤

¥¦§e¨B©IªN«¬i®#¬ªN«N¯°¯P«±© ¥²³´µ¶5´·D¸m¹º»<¼½V¾½¿¼º»mÀ¿ÁÀ »¹QÂ@ÃÄÅ½5½¿À5ÆÇ¢ÃÈC»mÅÉ>»<Ê¸<ÅeÄÅNËÅ½5Ã¾4Â@ÅNÆ;»À5Æ6ÌSÍÎÏ$º»<ÅNÅ½Ð¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´ÑÑ´µ¶ Ò~ÓÃÄÅ½¿½5À5ÆÇKÃÈS·D¸mÅNÅN¾ Ôt¼Â¢¼ÇÅeÀ5ÆlÏbÃ½5¹Á¸m¹º»<¼½¿ºNÕÌSÖÅ¢·DÃÂf×À5Æ4¼»<À5ÃÆ-ÃÈS·D¸m¹º»±¼½Ï½¿¼º»mÀ¿ÁÀ »¹ØÀ »<Öo·DÃÖÅºmÀ5ËÅ@ÙÃÆ4Å¡ÓÃÄÅN½¿º

ÈPÃ¸dÚt¸<¼À¿Æ6ÛiÃÊÆ4Ä4¼¸m¹6·D¼BË>À5»<¼»<À5ÃÆ ¶¶O¶¶O¶¶O¶¶e¶O¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´ÑÜ´µ¶ µ!ÝÊÂ@ÅN¸<À¿Á¼½IÎÂ¢¾½5ÅNÂ@ÅNÆ;»<¼»<À5ÃÆ6ÃÈb¼@µÔÞm·DÃÆ;»mÀ5ÆÊÊÂßÌiÖ4ÅÃ¸<¹¢ÃÈ£ÔdÀ¿ºm½5ÃÁN¼»mÀ5ÃÆ4º¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´à´´µ¶ áâ·DÃÂ@¾4¼¸<À¿ºÃÆKÃÈbÔtÔtÔÐ¼Æ4Ä6ÁÃÆ;»<À5Æ>ÊÊ4Âß¼¾¾4¸mÃ;¼Á±ÖQÀ5Æ-ºmÀ5Â¡Ê½¿¼»<À5ÃÆ4ºSÃÈ»<ÅNÆ4ºmÀ5½5ÅO»mÅº»ÃÈ»<ÖÀ5Æã4½5Â¢ºä¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´àµ´µ¶ å~ÓÊ4½ »mÀ ÞºmÁN¼½5Å¼¾¾4¸mÃ;¼Á±Ö¢»<Ã@»<ÖÅe¼Æ4¼½5¹ºmÀ¿ºiÃÈCÀ¿Æ4ÄÅNÆ;»<¼»<À5ÃÆÅÉ¾?ÅN¸mÀ5Â¢ÅÆ;»±ºSÀ5Æ¾Ã½5¹Á¸m¹º»±¼½5½5À5ÆÅOÁÃ;¼»mÀ¿ÆÇ;ºf¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´à;å´µ¶ æç;»mÊÄ¹3ÃÈ£ÆÃÆÞ¾¸mÃ;¾Ã¸»<À5ÃÆÆ4¼½V½5Ã;¼ÄÀ5ÆÇQ¾4¼»<Ö4ºÕSÁÃ;Â@¾4¼¸<À¿ºÃÆ3×Å»ØiÅNÅNÆÅÉ¾ÅN¸<À5Â@ÅNÆ;»±¼½I¸<ÅNºmÊ½ »±º¼Æ4Ä-ºÀ5Â¡Ê4½¿¼»mÀ5ÃÆº¾ÅN¸ÈPÃ¸<Â@ÅÄ

×>¹¢ã4Æ4À »mÅÅN½5ÅNÂ@ÅNÆ;»d¼Æ4Ä3ÖÃÂ@ÃÇÅNÆÀ5èNÅÄQÂ¢ÃÄÅN½¿º6¶¶e¶O¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´à>Ñ´µ¶ Ñé¸<ÅÅOÂ¢ÅNºmÖÀ5Æ4Ç@ÃÈ#Â¢À¿Á¸mÃ;º»m¸<Ê4Á»mÊ4¸mÅºD×4¼ºmÅNÄ3ÃÆ6Â¢ÃÄÀ ã4ÅÄ6êbÃ¸<ÃÆÃÀ»<ÅºmºmÅN½5½¿¼»mÀ¿ÃÆ4º]¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´àÜ

¥>ë ìí°¬¯î¬«¯íï!¬ï#ð$ñ6¬ò&¬óô ¥õI¥Ńá4¶5´·D¸<¼Á±ö¢Ç¸<ÃØS»mÖ3Â@ÃÄÅN½5½5À5ÆÇ¢ÃÈbºmÀ5ÆÇ½5ÅÁ¸<¹º»±¼½¿ºDÈPÃ¸ÖÀ5ÇÖ4Å¸mÞÃ¸±ÄÅN¸SÁÃÆ;»<À5Æ>Ê4¼¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´ÜµŃá4¶ Ò!ç>À5Â¡Ê½÷¼»mÀ5ÃÆfÃÈVÇÅNÆÅN¸<¼»<À5ÃÆ¡ÃÈ?Â¢ÅNºmÃ×4¼Æ4Ä4ºCÃÈVÀ5ÆÅN½¿¼º»<À¿ÁSÄÅÈPÃ¸<Â¢¼»mÀ5ÃÆ@À5Æ¢ºmÊ¸mÈ¼ÁÅi½¿¼B¹ÅN¸±ºCÃÈVÄÅÈPÃ¸mÂQ¼×½5ÅºÃ½5À÷ÄVÕ»mÖÅº»mÃÁ±Ö4¼º»mÀ¿Á

¼¾¾¸<Ã;¼Á<ÖKÃÈ#ÅÉÁÀ »±¼×½5ÅÁÅN½5½5Ê½¿¼¸¼Ê»<ÃÂ¢¼»±¼ø¶O¶¶e¶O¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´Ü;å

¥³ ùI¯îô;ú<ð#ô;®ô>ï#ð#ô>ï?«@ò&ô>°Bû#¬ïC¯°¬D®C¨í4®Dô;¨B«¯ô>ª ¥õ?²´Bå¶5´üÃ½5ÅOÃÈCÅN½¿¼º»mÀ¿ÁtÆ4ÃÆ½5À5ÆÅ¼¸<À »¹KÀ5ÆÄÀ÷º½5ÃÁN¼»<À5ÃÆK¾4¼»m»<Å¸<ÆÀ5ÆÇý¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶´ÜÜ´Bå¶ Ò~ÌSÀ5Â@ÅOÅNËÃ½5Ê»<À5ÃÆ6¼ÆÄKÆ>ÊÂ¢Å¸<À¿ÁN¼½I¼Æ¼½5¹ºÀ¿ºiÃÈ£ÄÀ¿ºm½5ÃÁN¼»mÀ5ÃÆÞÄÀ¿ºm½5ÃÁN¼»mÀ5ÃÆ3ÁÃ¸m¸<ÅN½¿¼»mÀ5ÃÆKÈPÊÆ4Á»<À5ÃÆ4ºi¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶þÒÿ´´Bå¶ µ!ÓÀ¿Á¸<Ã;ºmÁN¼½5Åe¾¼»»<ÅN¸mÆÀ5Æ4ÇKÃÈD¾½¿¼º»mÀ¿Á¡ÄÅÈÃ¸mÂQ¼»mÀ5ÃÆIÕtºÅN½ Èi¼¢ÆÅ¢ºÊ¸mÈ¼ÁÅºd¼Æ4Ä ºmÁN¼½5ÅeÈP¸<ÅNÅ@º»<¼»<À¿º»<À¿ÁNºdÃÈiÄÅÈÃ¸mÂQ¼»mÀ5ÃÆÞÀ5Æ4ÄÊÁÅÄ

ºÊ¸mÈ¼ÁÅOº»mÅN¾4º+¶¶O¶¶¶O¶¶O¶e¶O¶¶¶O¶¶O¶¶O¶¶e¶O¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶¶O¶¶O¶¶O¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶þÒÿµ

¥C«¯P®#ûV©Iª¯°ª ?³´æ¶5´ÔdÀ ?Ê4ºmÀ5ÃÆÎÆ4ÄÊÁÅÄ6Ï£Ö¼ºÅÌÇ<¼ÆºÈPÃ¸<Â¢¼»<À5ÃÆ4ºDÀ5Æ-ÓÅÞÁ±Ö4¼ÆÀ÷Á¼½ç;»m¸<ÅºmºmÅÄ ÅN¼ÄÞé¸<ÅNÅ½5½5Ã¹º¶e¶¶O¶¶O¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶þÒÿ>Ñ´æ¶ Ò~éCÀ5Æ4À »mÅ½5ÅNÂ@ÅNÆ;»ÈPÃ¸<Â¡Ê½¿¼»<À5ÃÆ3ÃÈ#ÏÖ4¼ºÅ¡éCÀ5ÅN½¿ÄÓÃÄÅ½¿ºÛS¼ºmÅÄKÃÆ3»mÖÅf·DÃÆÁÅN¾»ÃÈÚtÅNÆÅN¸<¼½5À5èNÅÄ3ç>»m¸<ÅNº<ºmÅNº ¶¶O¶e¶O¶¶O¶¶O¶þÒÿÜ´æ¶ µ!é¼Æ¼½5¹ºÀ¿ºiÃÈ#»mÖ4ÅÅVÅÁ±»ÃÈC»mÖÅeö>À5ÆÅ»mÀ¿ÁNºÃÈ#Æ>Ê4Á½5Å¼»mÀ5ÃÆ6ÃÆK»<ÖÅ¾½¿¼º»mÀ÷ÁÀ »¹QÀ5ÆÄÊ4ÁÅÄ6ÄÊ4¸mÀ5ÆÇQÄÀ ?ÊºÀ5ËÅO»<¸<¼ÆºÈPÃ¸<Â¢¼»<À5ÃÆ$¶O¶þÒ´´´æ¶ áÌSÖÅº»<Ê4Ä¹3ÃÈCÅÉ¾?Ã;ºÊ¸<Åt»mÀ¿Â@ÅO»<ÃQÄÅÈPÅNÆ4Ä6ÃÆ3¸<ÅÅNÁ»±¼Æ4ÁÅOÃÈCÀ5Æ>»mÅN¸È¼ÁÅ¼»Ö4Ã½5ÃÇ¸<¼¾Ö4À¿Ád½5À »<ÖÃÇ¸<¼¾4Ö>¹ ¶¶O¶¶¶O¶e¶O¶¶O¶¶O¶þÒ´µ

! "$#&%('$)+*,%.- /

0

1325476 8

9 :<;>= ?A@CBEDF:<GIHKJLD<GI@>BNM

O

PRQTSUSWVYX[Z \

] ^ D ^ :F@>_ ^ B ^ @>JNM ;`D ^ :<Ga;>=aM

b

c

dfed

Effective Elastic Constants of a 3D Two Phase Structure calculated by a Multiphase-Field Approach

Markus Apel, Stefan Benke, Ingo Steinbach

The multiphase-field method allows for the calculation of 3D grain structures in multicomponent and multiphase materials. As all geometrical information about the individual grains, their phase properties and their crystallographic orientations are known one can apply these structures directly in a kind of virtual testing to calculate e. g. the effective Youngs-modulus. In the multiphase-field framework the elastic stress and strain fields within the grain structure can be calculated. We now expand the model to further take into account external stresses. Using this model we calculate the internal stress and strain distribution and the volume response on external load for various grain structures with different grain sizes and phase fractions. From these calculations we derive the effective elastic constants which can be used as effective parameters for e.g. finite element calculations on a larger scale. References: “Multiphase-field model for solid state transformation with elastic strain”, I. Steinbach, M. Apel, Physica D 217 (2006) 153-160 “Multiphase-field approach for multicomponent alloys with extrapolation scheme for numerical application”, J. Eiken, B. Böttger and I. Steinbach, Physical Review E 73, 066122 (2006)

g

h

dfeji

!

!" #

$ $#%# &'&(")*+ # #),+

! # - ! ' # $ . # ! # " /

# # ! #)0+# ## # ' # !#*

*121"# #"

!"

" !#3'#"

"! #"#"

!"#$%& "

k

& ' ' # #4# '#

& # & $ .5 "#)6+# 7# 8*998 1 # # # ! : ;: 0#< ## 5! 0

01 " 5 ! = 7 #" ;:

& # ! # # ' !&#$- ## ')*+ &#>42>?@0AAB"#$"%&'()0AAB(&

)0+ !C2?!?D.@0AA6 E% E?!%! *F06*966*,,

)6+ (2G4@H.!440AA9E% 1 . E ? ! %! *FB6F9A6FI0

%

4J0"

;:)!

D+

6FI*BF896,90BBA8

Ûl

dfenm

AN ENERGY APPROACH TO DETERMINE THE MARTENSITE MORPHOLOGY IN NANO-STRUCTURED NI-TI ALLOYS

T. Antretter1, W. Pranger1, T. Waitz2, F.D. Fischer1

1Institute of Mechanics, Montanuniversität Leoben, 8700 Leoben, Austria 2Institute of Materials Physics, University of Vienna, 1090 Vienna, Austria

[email protected] Nano-structured NiTi alloys are produced by high pressure torsion up to complete amorphization and subsequent recrystallization. Its morphology is essentially influenced by the energies introduced into the material when an austenitic grain instantaneously transforms into martensite. For grain sizes in the order of 50nm one frequently observes typical martensite laminates composed of an alternating sequence of twin-related Bain correspondence variants. For larger grains in the order of 100nm it becomes more likely to observe two such martensite laminates whose arrangement relative to each other, as it appears in a micrograph, gives the impression of a “herring-bone pattern”. Such a configuration prevails if it minimizes the energy that needs to be overcome in order to be able to create the new phase. Consequently, this paper focuses on the evaluation of the energies involved in the phenomenon of martensitic transformation. As one of the predominant contributions the strain energy is computed by both the finite element method and, where available, analytical means. The total energy is then obtained by additionally taking into account the chemical interface energies at the twin- as well as the grain-boundaries. All relevant material data are listed in [1] and [2]. Obviously the total energy strongly depends on geometrical parameters like the grain diameter G, the twin-band width d and the width of the central laminate B (see Fig. 1). An extensive parameter study allows to find the configuration minimizing the total energy barrier Eb (see Fig. 2), which enables to predict what eventually appears in a grain of a given diameter expressed in terms of the ratios B/G and d/G. The resulting morphology is in good agreement with the experimental evidence obtained by means of high-resolution transmission electron microscopy which also indicates a change from the single laminate to the herring-bone microstructure at a diameter range of 70 to 90 nm. Fig.1.: Parameters defining the geometry of a martensitically transformed grain

Fig.2.: Comparison of the total energy as a function of the grain diameter.

[1] Waitz, T., Karnthaler, H.P., 2004. Martensitic transformation of NiTi nanocrystals embedded in

an amorphous matrix,” Acta Materialia 52, 5461-5469. [2] Waitz, T., Antretter, T., Fischer, F.D., Simha, N., Karnthaler, H.-P. „Size Effects on the

Martensitic Martensitic Phase Transformation of NiTi Nanograins“, J. Mech. Phys. Solids 55 (2007) 419-444.

Û2Û

Û>¤

dfepo

Computational approach to the mechanics

of entangled materials

C. Barbier, R. Dendievel, D. RodneySIMAP-GPM2

101, rue de la Physique, Domaine Universitaire, BP 46, 38 402 St-Martin d’Hères Cedex

[email protected]

Abstract

We employ a discrete simulation adapted from molecular dynamics techniques in order tostudy the mechanics of entangled semiflexible fibres. A previous model represents a fibre as a"pearl necklace" [1]. Fibres are discretised by nodes such that the number of nodes per fibreis equal to the fibre aspect ratio (ratio of the length to the diameter of the fibres). The nodalpositions are the degrees of freedom of the model. So the main restriction in based on the factthat the computational load increases rapidly with the fibre aspect ratio that was consequentlylimited to a value of 100, lower than in most real materials.

To reach larger aspect ratios, we develop here a new model ,the "spring" model in whichthe fibers are discretised by springs, the number of springs per fibre depending on the accuracywished for the simulation. As proposed by Rodney et al. [1], traction and bending stiffnesses,as well as non-penetrability between fibres are modeled by means of a potential energy. It isminimized to obtain equilibrium configurations. The distance between two springs is computedaccording to the method of Kumar et al. [2]. Depending on the configuration, the vector ofclosest approach is either perpendicular to the two springs or links one spring to one end ofthe other spring, or it may link one end of a spring to an end of the other spring. The mainadvantage of discretising the fibres by springs is to decrease the number of degrees of freedomper fibre. The drawback of this method is to lead to spring crossings between two simulationsteps. Determining whether the motion of a given spring intersects any other spring is thereforeof crucial importance for the integrity of the simulation.In order to validate this model, we compared its predictions with that of the node model. We

Figure 1: Example of simulation box of 450 fibresof aspect ratio equal to 100 (viewed with softwareAmira 3.1).

first studied the bending behavior of a single fibre clamped at one end and subjected to a forceF applied to its other end. We performed simulations for a fibre with an aspect ratio of 20 andcompared these results with Finite Element simulations. The most important result is that the

Û0

fibre follows a power law well-known in beam theory, δ ∝ FL3. We also simulated incremental

isostatic compressions of fibre systems. Different assemblies of homogeneous aspect ratio fibresare initially placed in the cell and oriented at random (fig. 1). At each compression increment, anenergy relaxation is performed to obtain an equilibrium configuration. The shape of the curvesobtained with the both models are nearly identical. For all the range of aspect ratios studied(20 nodes or 4 springs, 100 nodes or 15 springs, and 20 springs corresponding to an aspect ratioof 200), three states could be identified. At low density, the fibres do not interact at equilibrium("free-fiber state"). At higher densities, the fibres strongly interact and lock each other, witha finite number of contacts and an energy that increases rapidly with the density ("entangledstate"). Between these two states, there is a "mechanical transition" region where no equilibriumis reached. We can conclude that for fibres of aspect ratio equal to 20, only 4 springs can beused to model a fibre, instead of 20 nodes. In the same manner, fibres with a aspect ratio of100 can be modeled by only 15 springs, leading to a large decrease in the number of degrees offreedom in the simulation. We can also see that it is possible with the spring model, to performsimulations of fibres of aspect ratio up to 200, which is close to the aspect ratio of 250 measuredby Masse et al. [3] for steel wools.

We also implemented a Coulomb-like friction at contacts points, in order to study the influenceof sliding between fibres during isostatic compressions. A first result is that applying frictionforces shifts the transition (i.e. the region of rapid increase of contact number) to lower densitiesand lower contact numbers. Another result is that after the transition, either with or withoutfriction, fibres are arranged to hold the same number of contacts per fibre. We are also interestedin the hysteresis that is observed experimentally during compression/relaxation cycles. Thisphenomenon may be attributed either to the friction between fibres and/or irreversible fibrerearrangements. We performed cycles of compressions followed by relaxations, going to largerdeformations at each cycle. Relaxations were performed by incrementally increasing the cellvolume until it was back to its initial value. The curves (either number of contacts per fibre orstress versus relative density) follow a master curve that corresponds to a one-way compression, asmentioned by Masse et al. [3] in experimental compressions of steel wools. But most importantly,there is hysteresis even without sliding friction. We can therefore conclude that hysteresis is atleast partly due to irreversible rearrangements of the fibres.

References

[1] Rodney D., Fivel M., Dendievel R. "Discrete Modeling of the Mechanics of Entangled Mate-rials" Phys. Rev. Lett. 95 108004 (2005)

[2] Kumar S., Larson R.G. "Brownian dynamics simulations of flexible polymers with spring-spring repulsions" J. Chem. Phys. 114 6937-6941 (2001)

[3] Masse J.P., Salvo L., Rodney D., Bréchet Y., Bouaziz, O. "Influence of relative density onarchitecture and mechanical behaviour of steel metallic wool" Scripta Materialia 54 1379-1383(2006)

Û O

dfejq

Effective permeability and thermal conductivity of open-cell metallic foams via homogenization on a microstructure model

G. Laschet, O. Reutter1, J: Sauerhering1, J. Scheele and R. Pitz-Paal1

ACCESS e.V, Intzestrasse 5, D-52072 Aachen, Germany 1German Aerospace Center, Linder Höhe, 51143

In order to increase the efficiency of combined cycle power plants, open-cell metallic foams with fine pores, suitable for transpiration cooling, are manufactured by a SlipReactionFoamSintering (SRFS)-process [1] and characterized. This process allows to generate a great variety of open-cell foams like iron and superalloy foams by a chemical reaction in the powder metallurgy process route. Due to the complexity of their random microstructure, with small secondary pores and large primary pores (see fig: 1), a specific procedure has been developed recently to define a realistic 3-D micromodel of such SlipReaction (SR)-foams [2]. Starting from a detailed foam description by tomographic images, a spectral analysis of the foam microstructure is achieved. This method provides the characterictic lengths and specifies the dimensions of a geometric unit cell model. To reduce the mesh size of the corresponding finite element model several CT voxels are joined together to one hexahedron element (see fig 2). In order to specify a RVE model suitable for further Monte-Carlo simulations, different FE discretizations of the unit cell are built and compared: fine and coarse meshes, full and reduced integrated ones.

Figure 1: Pore structure of SR-foam sample analyzed by computer tomography (a) and micrography

with primary and secondary pores.

Figure 2: Coarse finite element mesh of the unit cell where 4*4*4 voxels have been joined to one

hexahedron element. For metallic open-cell foams, dedicated to serve as basic material of future combustion chamber wall-shingles, the prediction of their effective permeability and thermal conductivity is essential. For this prediction, a multi-scale approach based on the homogenization method has been adopted. The implementation of the homogenisation method for advective heat transfer leads to solve numerically, for each component of the conductivity tensor, an additional problem on the unit cell.

Ûrb

In order to evaluate the effective Darcy permeability tensor, special Stokes flow problems, under unitary pressure gradient in X, Y and Z direction respectively, are solved numerically on the unit cell. The well-known analogy of Stokes flow with the incompressible elasticity is used to solve these microscopic problems. Effective Darcy permeabilities are finally expressed by averaging the resulting velocity fields over the unit cell.

The numerical investigation is realized for a representative Inconel 625 based SR-foam. At first, the influence of some micromodel parameters like the diameter of closed secondary wall pores and the position of the unit cell on the foam porosity is investigated. Effective orthotropic conductivities and permeabilities are then evaluated for the different defined unit cell models and discussed. In order to validate the numerical predictions, experiments have been conducted. At first, the effective thermal conductivity is determined by the transient plane source technique. For the sintered Inconel 625 powder material (foam without primary pores) also Laser Flash measurements have been accomplished up to 700°C and taken into account in the numerical homogenization procedure as the equivalent wall thermal conductivity. A good agreement between experimental and numerical effective thermal conductivity is observed. Moreover, pressure drop measurements on Inconel foam samples of 80 mm diameter have been realized. In these experiments the mean inlet velocity of the air is about 0.3 m/s. Nevertheless, the pressure drop measurements present a slight parabolic dependence with the inlet velocity, indicating that a Forchheimer law is more indicated than the Darcy law, the experimental and numerical permeabilities are in a relative good agreement.

References

[1] S. Angel, W. Bleck, P.-F Scholz and Th. Fend: “Influence of powder morphology and chemical composition on metallic foams produced by SRFS-process”, Steel Research Int., vol. 75, pp. 483-488, 2004.

[2] G. Laschet, T. Kashko, S. Angel, J. Scheele, R. Nickel, W. Bleck and K. Bobzin: „Microstructure based model for permeability predictions of open-cell metallic foams via homogenization“, accepted for publishing in Material Science & Engineering: A and will appear online in April 2007.

Û c

dfens

!!"

#$%&! ' ()%

*#+,& -.)--/0$ %&! 0(1)2&&)%

*#+34-5)). - /

!" # $ $ %!& #

' 4)6+

*/ 6

*/

*6786/ 9:-

0 0.05 0.1 0.150

100

200

300

400

500

600

700

Macroscopic Strain

Mac

rosc

opic

Stre

ss (M

Pa)

'%; 4*

/4-< 4* /+

6=0'>*/?6=!>* /?6=(-!>

*/?6=!-!@*/-

8) ? ?86?& ?A%-

!"$$-96--*BBB/- &

-()" !B &C-

5-%4D-E-54-*0''"/- &

4 44 -)*(1 00(&1C-

5)) 3- *0''C/ F F< FF2 GF < F-

+ '&,-.),()($/0%1%2%334('

!# ##

$

%

!#

#

Û g

Û h

dfet

MATERIAL PARAMETER IDENTIFICATION OF HETEROGENEOUS MATERIALS WITH INCLUSIONS ON DIFFERENT LENGTH SCALES

R. Piat1, P. Gumbsch2 , T. Böhlke1

1) Institute of Engineering Mechanics, University of Karlsruhe 2) Institute for Reliability of Components and Systems, University of Karlsruhe

Classical homogenization procedures for the identification of effective material parameters deal with two different phases: the inclusion and the matrix [1]. Materials studied in this paper are carbon/carbon composites (CFCs) which are produced through chemical-vapour infiltration [2]. These materials consist of three different phases: pyrolytic carbon matrix, carbon fibers and pores (Fig.1). The volume fractions of fibers and pores are small in comparison to the volume fraction of the matrix.

Fig.1: Typical microstructures of CFCs with 2D preforms and infiltrated felts.

Fig. 2: Model I: First step homogenization of the fibers and pyrolytic carbon matrix; second step embedding of the pores in the effective material; Model II: first step homogenization of the pores and pyrolytic carbon matrix; second step embedding fibers in the effective material.

In this procedure, the first step is to obtain the homogenized pyrolytic carbon matrix along with one of the two phases, after that, in the second step, we embed the remaining phase in the effective material that has the material parameters obtained from the first step. The essential question is the decision about the sequence of these homogenization steps, which of the proposed approximation procedures is more accurate, i.e. which phase will first be homogenized with the matrix and which phase will be embedded later in the resulting material with effective parameters from the first step (Model I or Model II on Fig. 2)? To answer this question, numerical studies were performed and effective parameters of the

Ûk

material with random distribution of fibers and pores in isotropic matrix using Model I and Model II have been calculated. For this material the result of each homogenization step is an isotropic material. Effective parameters of the isotropic matrix and pores are calculated using a non-interaction approximation. The effective compliance tensor can be calculated as

ii

eff HDD ISO += 1 where 1ISOD is the effective compliance of the matrix and iH is the

compliance tensor of pore i . The effective compliance MTD of random distributed fibers with the volume fraction If and the isotropic matrix with the compliance 2ISOD the Mori-Tanaka

approximation [3]:

−+++= klijjkiljlikklijI

ijklijkl hhf

DD δδδδδδδδ32

100 432ISOMT can be used,

whereby 3h and 4h are the functions of the material parameters of the fibers. Results of these calculations for both models are presented in Fig. 3 for different volume fractions of the fibers and pores.

Fig. 3: Numerical studies of the influence of the sequence of homogenization steps on

material parameters of the resulting material for different volume fractions of the microconstituents.

Numerical results show that the decision on the sequence of the homogenization steps has an influence on the result. In the studied materials, the inhomogeneities have their characteristic lengths on different length scales. Fiber diameters are ~10 m and pores are with diameters ~200-300 m. As a result, Model I is expected to provide better approximation of the effective properties. The reason for this hides in the order which we follow embedding the inclusions in the matrix. Accordingly we homogenize first the matrix with embedded smaller inclusions and after that perform subsequent homogenization for the larger ones. Statistical studies of the porosity of infiltrated felt and of the material with 2D preforms (Fig.1) were provided using polarized light microscopy and micro-computed tomography accordingly. Using Model I, a two-step homogenization-procedure considering statistical data of real material porosity was provided and effective material parameters of both materials were calculated. The obtained numerical results are close to the experimental results as in [3]. This model gives good results for small volume fractions of fibers and pores. References [1] S. Nemat-Nasser and M. Hori: Micromechanics: Overall properties of heterogeneous materials, Elselier, 1999. [2] W. Benzinger and K.J. Hüttinger: Carbon, Vol. 37 (1999), p. 941-946. [3] R. Piat, I. Tsukrov , N. Mladenov, V.Verijenko, M. Guellali, E. Schnack and MJ.

Hoffmann: Comp. Sci. and Tech. Vol.66 (2006), p. 2997-3003.

¤l

dfenu

Overall properties of randomly distributed particle reinforced

composites using simplified unit cell model

M. M. Aghdam∗ , M. Shakeri, M. J. Mahmoodi

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

Abstract

A three dimensional micromechanical model is presented to determine various overall

thermo-mechanical properties of randomly distributed particle reinforced composite

materials. The geometry of the model is based on the extension of the previous [1-2] 2D

model known as the Simplified Unit Cell (SUC) model. The presented representative

volume element (RVE) consists of r×c×h unit cells in which particle reinforcements are

randomly distributed within the matrix. The system of governing partial differential

equations of the problem is obtained using theory of elasticity for heterogeneous

materials. With proper simplification assumptions, the governing equations are converted

to a system of r×c+r×h+c×h linear algebraic equations with the same number of

unknowns. The presented model is general and can also be used to predict the behavior of

randomly distributed particles reinforced composites subjected to thermal and mechanical

loading. Results revealed that while implementation of elasticity equations to the sub-

cells is very simple, the model provides reasonably good predictions by dividing the RVE

into several meshes. It is shown that predictions of various overall properties of the

composite system show (see Figure 1) good agreement with experimental data and other

analytical models.

∗ Corresponding author: [email protected] (M.M. Aghdam), Fax: +98-21-6641 9736 Tel. +98-21-64543429.

¤Û

Keywords: Micromechanics, Particulate composite, Simplified unit cell model, Overall

thermo-mechanical properties, Representative volume element.

Figure 1- Overall Young modulus of Tungsten Carbide/Cobalt particulate composite.

References

[1] M.M. Aghdam, A. Dezhsetan, 2005, “Micromechanics based analysis of randomly

distributed fiber reinforced composites using simplified unit cell model.” Comp

Struct, 71, 327–332.

[2] Aghdam M.M., Smith, D.J., Pavier, M.J., 2000, ‘Finite element micromechanical

modeling of yield collapse behavior of metal matrix composites’, Journal of

Mechanics and Physics of Solids, 48, 499-528.

¤¤

PRQTSUSWVYX[Z v

;>?A:F@CMw?A@>xKGa? H ^zy ;|GI@>J~:

¤0

¤ O

ied

IWCMM’17 2007 22-24 August 2007 Paris, France

Non-symmetyric thermal shock in layered FGM cylindrical plates – comparison of two applied methods for problem solution

T. Sadowski 1), S.Ataya2), K.Nakonieczny 3)

1) Lublin University of Technology, Faculty of Civil and Sanitary Engineering 20-618 Lublin, Nadbystrzycka 40 str., Poland e-mail: [email protected] 2) Department of Materials Science, Aachen University, Augustinerbach 4, 52062-Aachen, Germany e-mail: [email protected] 3) Lublin University of Technology, Faculty of Mechanical Engineering 20-618 Lublin, Nadbystrzycka 36 str., Poland e-mail: [email protected]

The modern FGM composite materials have a complex internal structure due to the fact

that they consist of several different phases. In this paper the thermal shock problem in FGM cylindrical plates having discrete variation of the composite features was analysed. The

samples of dimension: thickness w = 2.5 mm and diameter d = 30 mm were made of five ceramic layers (each of them 0.5mm thick): purely Al2O3 layer and composite layers made of

Al2O3 matrix and 5, 10, 15, 20 wt% content of ZrO2. The plates were subjected to non-

symmetric thermal shock for different temperature levels ∆T.

The theoretical solution of the problem consists of several steps:

• description of the “position – dependent” properties of the analysed material based

on experimental data. Step distribution functions were constructed for the thermal expansion coefficient, thermal conductivity, thermal diffusivity and Young’s

modulus,

• solution of the non-stationary heat conduction equation for arbitrary smooth or

step variation of functions describing properties of the analysed material. The

considered boundary conditions obey: perfect cooling and real cooling process of the material by introduction of heat transfer coefficient at the cooled surface. The

¤b

Fourier-Kirchhoff equation for the cylindrical specimen is solved using two methods. The first one is application of numerical code by Euler method with use

of finite difference. In the second method the considered problem was solved with the help of ABAQUS code. The heat transfer coefficient at the cooled surface was

estimated by a linear monotonic model and by a linear model with application of step function. The solution gives the distribution of the temperature for different

times.

• calculation of the thermal stress and estimation of the residual stress influence on

the thermal shock behaviour with FEM approach (ABAQUS). Obtained experimental results confirm qualitatively and quantitatively theoretical predictions

of the temperature and thermal stress distributions under thermal shock.

¤ c

ieji

Modeling of the tensile and compression behavior of semi-solid A356 alloy

S. Benke1, S. Dziallach2, G. Laschet1, U. Prahl2, W. Bleck2

1ACCESS e. V. Materials and Processes,RWTH Aachen, D-52072 Aachen, Germany

2Department and Chair of Ferrous MetallurgyRWTH Aachen University, D-52072 Aachen, Germany

ABSTRACT

During the solidification in casting processes, the volume fraction of the solidified metal graduallyincreases. The material changes from liquid via a slurry to a mush and finally to a solid. In the slurrythe solid germs and the continuous liquid forms a suspension. The isolated germs are free to move andthe mixture behaves like a non-Newton fluid. The mush is characterized by a continuous solid networkwith liquid metal in between. The transition from a slurry to a mush depends on the volume fractionand the morphology of the solid which is determined by the solidification conditions.In order to determine the coherency temperature, hot tensile tests were performed using a hot formingsimulator. Starting at the liquidus temperature tensile tests are performed at different temperatures in thesemi-solid range. By this procedure, the temperature where the specimen is able to sustain a tensile loadis identified. At various temperatures between the coherency temperature and the solidus temperaturetensile and compression tests are performed under varying strain rates and the local deformation in thepartially solidified region of the specimen is measured by laser speckle extensometer.The continuum-mechanical model for the mush is based on the theory of porous media and distinguishestwo constituents, namely the elasto-viscoplastic solid skeleton and the viscous liquid metal, and takestheir interaction in terms of mass, momentum and energy exchange into account. Using the experimentalfindings a modified two-parameter creep law for the solid network is deduces, which takes into accountthe coherency of the solid and provides a continuous description of thermo-mechanical behavior of thealuminium alloy from the semi-solid state into the creep regime of the fully solidified metal.

¤ g

¤ h

ienm

EFFECTIVE MATERIAL PROPERTIES OF MULTI-LAYERED POROUS NANOFILTERS

Ádám Kovács1, András Kovács2, Ulrich Mescheder2

1Budapest University of Technology and Economics Department of Applied Mechanics

1111 Budapest, Muegyetem rkp. 3, Hungary E-mail: [email protected]

2Hochschule Furtwangen University of Applied Sciences Institute for Applied Research

78120 Furtwangen, Robert-Gerwig-Platz 1, Germany E-mail: [email protected]

ABSTRACT

Perforated and porous membranes are often used in various engineering applications. As an example, they can serve for micro- and nanofiltering purposes in micro-electromechanical systems (MEMS). Investigated membranes are made from very thin perforated silicon-nitride on a porous polysilicon (PS) layer, which is supported and reinforced by single-crystal silicon (Fig. 1).

Porous silicon

c-silicon columnsSiN or c-Si

Fig. 1: Schematic view of a multi-layer filter membrane

The investigated porous silicon layer was formed in a thin p-type c-Si membrane from the frontside of the wafer by electrochemical etching. Measured porosity was 75% with a mean pore size of 4.85 nm (pore size range varied between 28 nm). To achieve sufficient stability an upper perforated silicon nitride layer was used as masking structure during electrochemical anodization. Due to this special masking structure, anodization current was mainly directed in vertical direction and therefore, residual crystallized silicon (c-Si) columns were formed between the etched SiN-structures. RIE-technique was used to remove the residual c-Si at the backside of the wafer after the anodization. The crystalline silicon support layer can be controlled by the anodization conditions, RIE etching time and distance between the perforations. The performance of filters highly depends on the porosity. In order to obtain higher filtration rate the porosity should be as high as possible [1] which diminishes the strength, and consequently, the load capacity of the device. This effect can be compensated by the application of SiN support layer and c-Si columns as a support grid within the porous Si layer. In order to estimate the load capacity of the membrane the elastic and fracture material properties (E, Q and VB) of the structure should be known. Since elastic modulus E and fracture strength VB are both dependent not only on the porosity but also on the thickness of layers [2], the use of effective material properties could simplify the numerical estimation of bursting pressure in the design phase. Poisson ratio has been kept as constant. Different filter-configurations have been analyzed. They differ in the thicknesses of SiN and PS layers as well as the etching rate and column-width of crystallized silicon (cSi). The elastic modulus and the fracture strength for full structures have been determined by experiments for different layer-thicknesses (Figs. 3a-b). The same constants for perforated and porous layers were then calculated by the following estimates:

¤k

Ep = (1P)eE, VBp = (1P)f

VB .

In the above equations P is the porosity and e, f > 1 are material constants. Their values can be determined by simulations [3].

0,00

0,50

1,00

1,50

1,00 2,42 3,84 5,26 6,68 8,11 9,53

h_SiN / h_min

w_m

ax / h

_S

iN

0

20

40

60

80

100

120

0,02 0,06 0,10 0,14 0,18 0,22

p (Pa)

E_e

xp

(G

Pa)

Fig. 3a: Measured relative maximum deflections 3b: Measured elastic modulus

Once the material constants had been determined for SiN- and PS-layers, c-Si reinforcement was replaced by an equivalent third layer. The thickness of this layer has been calculated by the equivalence of bulk volumes. Effective material parameters (Eeff, VBeff) were then defined as an average of the layers parameters weighted by the thickness [4]. The mechanical response has been determined numerically with the use of the commercial FE-code COSMOS/M. Simulated results show good accordance with measured values (Fig. 4).

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

0,00 0,05 0,09 0,13 0,17 0,21 0,25

p (Pa)

w_m

ax (P

m)

measured

analytical

FEM

Fig. 4: Measured and estimated maximum deflection of a three-layer nanofilter membrane

References [1] van Rijn, C.J.M.: Nano and Micro Engineered Membrane Technology, Elsevier, 2004. [2] Lange, V., Higelen, G.: Static and dynamic characterization of Si-membranes. Sensors and

Actuators A, 1995, Vol. 46-47, pp. 47-50. [3] Á. Kovács, A. Kovács, U. Mescheder, Estimation of Elasticity Modulus and Fracture Strength of

Thin Perforated SiN Membranes with Finite Element Simulations, Comp. Materials Science, to appear.

[4] van Rijn, C.J.M., van der Wekken, M., Nijdam, W., Elwenspook, M.: Deflection and maximum load of microfiltration membrane sieves made with silicon micromachining. J. of Microelectromechanical Systems, 1997, Vol. 6, No. 1, pp. 48-54.

0l

iepo

On critical plane criteria assessments in case of combined tensile and torsion load

Zbynk HRUBÝ, Jan PAPUGA, Karel DOUBRAVA, Milan RŽIKA1

The number of papers concerning multiaxial fatigue has decreased in comparison to the last decade, but there is still no generally accepted criterion for multiaxial fatigue in practical use. Nevertheless, there is a market demand still strong enough that the leading fatigue code producers are forced to incorporate multiaxial modules into their fatigue post-processor codes. There is great risk involved, thus these manufacturers usually apply criteria, which are more or less dated, believing apparently that their old age together with casual appearance as comparative criteria in the literature is a sufficient cover for their implementation. There is an underlying conservativeness hidden in such a way of thinking as discussed by Papuga [1]. The multiaxial criteria mapping only either the normal or shear component of loading belong here. The critical plane criterion takes the state of stress at the plane most indisposed as the decisive one for the fatigue durability. There are a few basic load modes – tension, torsion and bending. Nevertheless, simple load cases occur very rarely. One can usually find some combination of them. In expectation of elastic response to the loading, these local load states induced by separate load channels can be superposed, but not in the case of elasto-plastic response, where principle of superposition is not valid anymore and so-called transient analysis has to be performed. The fatigue analyses performed by authors consisted of two parts – the finite element method elasto-plastic response determination by FE-code ABAQUS [2] and fatigue post-processing by the developed code PragTic [1]. However, two reasons why the results are still not reliable in comparison to experimental results [3], [4]. First, the constitutive model much relies on the combined non-linear kinematic and isotropic hardening for analysing bodies subjected to cyclic loading at the state of saturation is not suitable for the tube samples tested. Second, the implemented low-cycle multiaxial criteria [1] in PragTic code yield unconservative results for load regimes given combinations of tension and torsion as schematically depicted in the Figure 1.

Figure 1: Experimentally performed load regimes (tension – horizontal axis; torsion – vertical axis)

Figure 2: Globe analogy concept towards the plane normal direction

1 Ing. Zbynk Hrubý, Ing. Jan Papuga, Ph.D., Ing. Karel Doubrava, Ph.D., Prof. Ing. Milan Ržika, CSc.: Department of Mechanics, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Czech Technical University in Prague; Technická 4; 166 07 Praha 6; CZECH REPUBLIC; tel.: +420 224 352 519, fax: +420 233 322 482; e-mail: [email protected], [email protected].

05Û

The study of state of stress at the critical plane has been performed. According to the Figure 2, number of planes was checked to signify the critical one. It was ascertained that not only the shear stress vector loops as mentioned above, but also the normal and the resultant stress vector loops at the critical plane affect the durability. I has been figured out that for different load regimes the damage variables vary even in the same planes. It is supposed, this is the reason, why different durabilities were embodied for samples subjected to various load regimes with the same amplitudes of the tensile force and the torque. All these items will be discussed in detail in the text of the contribution. [1] Papuga, J. Mapping of Fatigue Damages – Program Shell of FE-Calculation

[Ph.D. Thesis]. Praha: CTU, 2005. http://www.pragtic.com [2] ABAQUS Online Documentation. Version 6.5. ABAQUS, Inc. 2004.

http://man.fsid.cvut.cz:2080/v6.5/ [3] Papuga, J.; Ržika, M.; Hrubý, Z.; Balda, M. Searching for multiaxial fatigue

solution. In Fatigue 2006 – Proceeding of the 9th International Fatigue Congress. Atlanta, May 14–19th, 2006. Oxford: Elsevier, 2006. pp 152-153.

[4] Hrubý, Z.; Papuga, J.; Ržika, M.; Balda, M.; Svoboda, J. Prediction and verification of the lifetime for the various combined tensile and torsion load [in Czech]. In Engineering Mechanics 2006. Svratka, May 15-18th, 2006. Prague: ITAM AS CR, 2006. pp. 116-117.

.

0?¤

iejq

An analytical modeling and FE simulation of arc bending process to predict the springback

S. K. Panthi, N. Ramakrishnan

Advanced Materials and Processes Research Institute (AMPRI), Bhopal-26, India

One of the most widely used sheet metal forming process is bending. This is employed in

automobile industry, construction of large spherical & cylindrical pressure vessels, curved structural

components in aerospace industry etc. Bending is a process in which a planer sheet is plastically

deformed to a curved one [1]. The precision in dimension is a major concern in sheet metal bending

process because of the considerable elastic recovery during unloading leading to springback.

This investigation pertains to the analysis of springback in arc bending of sheet metal, which

can serve as an excellent design support. The amount of springback is considerably influenced by the

geometrical and the material parameters associated with the arc bending. In addition, the applied load

during bending also has a significant influence on the springback. This investigation covers analytical

modeling as well as Finite Element (FE) simulation of arc bending carried out as numerical

experiment. An analytical model is proposed to predict the springback in arc bending, considering the

effect of load, based on the strain-based approach and energy-based approach. FE simulation has been

carried out by an indigenous software, based on Total-Elastic-Incremental-Plastic Strain (TEIP) [2], to

compare the results of presented model. The results of analytical model are also compared with the published experimental one [3]. It shows a considerable agreement with the FE simulation and

published experimental one.

Figure 1 shows the initial undeformed sheet, bent sheet after loading and final configuration

after springback. A plot between the springback ratio rf/ri (ratio of final radius to die radius) and ratio of

rf/t (t: thickness of the sheet) is shown in figure 2 by analytical model (strain based approach and

energy based approach), published experimental and FE simulation for the copper sheet. The strain-

based approach shows a better agreement with the published experimental as well as FE simulation.

The effect of load on springback is shown in figure 3 by analytical model as well as by FE simulation

at the different value of d/t ratio, where ‘d’ is the reduction in thickness of the sheet due to higher load.

It shows that the springback ratio increases with increase in ratio of ‘rf/t’ and it decreases with increase

in ratio of ‘d/t’.

References: [1] G.E. Dieter, Mechanical Metallurgy, 3rd ed., McGraw Hill, London, 1988.

[2] N. Ramakrishnan, K. M. Singh, R. K. V. Suresh, N. Srinivasan, An algorithm based on total-

elastic-incremental-plastic strain for large deformation plasticity. J. of Materials Processing

Technol. (86) (1999) 190-199.

[3] K. Lange, Handbook of metal forming. Mc-Graw Hill Book Company.1985. 00

Figure 1: Bending process shows the initial, loaded

and unloaded sheet

Figure 2: Comparison of analytical, FE simulation and

published experimental results for copper sheet

Figure 3: Comparison of effect of load on springback by analytical model and FE

simulation for different value of ratio compression depth (d)/ thickness (t) for

copper sheet

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

0 50 100 150 200 250rf/t

r f/r i

AnalyticalFEMPublished ExperimentSeries4

Strain Based Approach

Energy Based Approach

FEM

Sheet before unloading

Sheet after unloading

Undeformed sheet

Die

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

0 50 100 150 200 250rf/t

r f/r i

d/t=0.0004d/t=0.0

d/t=0.001d/t=0.0014

d/t=0.002

d/t=0.01

d/t=0.004

d/t=0.006 d/t=0.008

0 O

iens

Creep damage study at power cycling of lead-free surface mount device

Pradeep Hegde, David Whalley and Vadim V. Silberschmidt

Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University,

Loughborough, Leicestershire, LE11 3TU, UK

KEYWORDS: power cycling, lead-free solder, finite element analysis

Soldering is extensively used to assemble electronic components to printed circuit boards or

chips to a substrate in microelectronic devices. Solder joints serve as mechanical, thermal

and electrical interconnections, therefore, their integrity is a key reliability concern.

However, newly introduced lead-free solders do not have a long history of applications in the

industry and there is a lack of established material models of their behaviour over the wide

temperature range experienced by electronics systems.

The use of accelerated tests to characterise the reliability of electronic packages is well

established in the microelectronic industry. At the present time, a widely used accelerated life

test is the accelerated temperature cycle test where the components or printed circuit assemblies are placed inside a chamber and cycled between the extreme temperatures.

Popularity of the thermal cycling test is due to simplicity of its concept and operation.

However, there is a growing debate on the validity of thermal cycling tests since under the

service conditions, a powered electronic device generates the heat resulting in non-isothermal

conditions for the package. Thus, it is believed that power-cycling accelerated test, in which

the package is differentially heated, is more representative of the real life conditions. Typical

temperature and power-cycling tests are carried out in the range from -55oC to 125oC and

0oC to 100oC making both creep and fatigue potential microscopic failure mechanisms due to

high homologous temperatures.

This paper focuses on finite element thermal analysis for establishing temperature

distributions in the surface mount chip resistor during maximum rated powered conditions.

These temperature distributions are later used in finite-element creep analysis to study creep

damage and fatigue life of lead-free solder joints. A surface mount chip resistor, which was

used in [1], is simulated in the finite element analysis. This paper continues our previous

study [1] with an aim to elucidate the effect of a non-uniform temperature distribution on

stress levels in surface mount devices. The chip resistor is modelled in the 2D plane strain

0b

formulation and material properties of lead-free SnAgCu are used to simulate solder joints.

[1] P. Hegde, D.C. Whalley, V.V. Silberschmidt,, Finite Element Analysis of Lead-free

Surface Mount Devices. Computational Materials Science (in press).

0 c

iet

1(2)

Modeling and simulations of interfacial crack growth of cyclically loaded structures

A.M. Karlsson1 and D. Cojocaru, Department of Mechanical Engineering

University of Delaware, Newark, DE 19716, USA

Abstract Cyclically load structures commonly fail due to incremental crack growth, fatigue. Even

though experimental investigations are critical to understand the evolution of crack growth, the experiments must be complemented with mechanics based theoretical approaches. The finite element method (FEM) is commonly employed to this end [1, 2]. Finite element analyses (FEA) typically give “snap-shots” of the stress/strain fields along with how these fields change during one load cycle. A proposed technique to reduce the computational time associated with long term evolution is the so called cycle-jump technique, e.g. [3-5]. The spatial modeling of the degradation phenomena, such as crack growth, must also be addressed, which is the focus of the current work. In general, two approaches are used for simulating crack propagation in FEA [6]: (i) a geometric representation of the cracks or voids in the model, utilizing node release techniques for propagation, e.g. [7]; or (ii) an appropriate constitutive response accounts for discontinuities and damage, e.g. [8].

We investigate here a modeling frame based on a “node release” approach to simulate the cyclic crack propagation, where the propagation occurs when a propagation criterion is fulfilled. In the current frame work, the fracturing path is assumed to be known beforehand, which is commonly the case for interfacial decohesion.

The procedure [7] is implemented in the commercially available software ABAQUS [9], using ABAQUS Scripting Interface (ASI) and through an object oriented programming approach. Based on user input, the FE-model is generated automatically. The two-dimensional (2D) multi-material system is described by two sets: (i) a set of continua and (ii) a set of continuum interfaces, fig. 1A. This decomposition aims to take advantage of the object-oriented programming featured by ABAQUS Scripting Interface (ASI) [9]. Here, a continuum represents a 2D sub-domain of the complete model. Each continuum has its own constitutive description and meshing related parameters (e.g., element type, meshing algorithm). A continuum interface describes either: (i) how two continua interact with each other (CC interface); or (ii) how one continuum interacts with the exterior (CE interface). A continuum interface is described by a sequence of interface segments. Two types of segments are necessary for the CC interface (fig. 1A), type D for modeling interface discontinuities (i.e., cracks) and type C for modeling continuous portions (e.g., crack ligaments). The CE continuum interface is described by the interface segment, type S, capturing the interface between the continuum and the exterior.

A critical part to the modeling frame is the crack-propagation criteria, which can be based on any quantity available in the result database. Here, we will limit the discussion to the case to investigate the accumulation of dissipated energy (which can be directly related to accumulated plastic strain) as a propagation criterion. To this end, a region where the dissipated energy is evaluated must be defined, illustrated with just four elements in fig. 1B. Selected benchmark problems will be presented, e.g., fig 1B, to elucidate the effectiveness of the modeling frame and 1 Corresponding Author: A.M. Karlsson. phone: +1-302-831-6437, fax+1-302-831-3619, e-mail: [email protected]

0 g

2(2)

the influence of parameters such as the critical value of the propagation criterion and load conditions. We will show that, depending on the parameters used, the modeling frame is capable of capturing a stationary crack, intermitted crack growth, as well as accelerated crack growth associated with each cycle.

(A) (B) Fig. 1: Using symmetry conditions, a cyclically loaded center crack (A) schematic of the modeling frame and (B) FE-mesh.

References 1. Karlsson, A.M., Evans, A.G., A numerical model for the cyclic instability of thermally grown oxides in thermal

barrier systems. Acta Materialia, 2001. 49: p. 1793-1804. 2. Shi, J., Darzens S., Karlsson A.M., Aspects of the morphological evolution in thermal barrier coatings and the

intrinsic thermal mismatch therein. Materials Science and Engineering, 2005. A392: p. 301-312. 3. Cojocaru, D., Karlsson, A.M., A simple numerical method of cycle jumps for cyclically loaded structures.

International Journal of Fatigue, 2006. 28(12): p. 1677-1689. 4. Kiewel, H., Aktaa, J., Munz, D., Application of an extrapolation method in thermocyclic failure analysis.

Computer Methods in Applied Mechanics and Engineering, 2000. 182: p. 55-71. 5. Oskay, C., Fish, J., Fatigue life prediction using 2-scale temporal asymptotic homogenization. International

Journal for Numerical Methods in Engineering, 2004. 61: p. 329-359. 6. Ingraffea, A.R., Computational Fracture Mechanics, in Encyclopedia of Computational Mechanics, Volume 2:

Solids and Structures (edited by E. Stein, R. de Borst and T.J.R. Hughes). 2004: John Wiley & Sons. 7. Cojocaru, D. and Karlsson, A.M., An object-oriented approach for modeling and simulation of interfacial crack

growth of cyclically loaded structures. Submitted, 2006. 8. De Borst, R., Some recent developments in computational modelling of concrete fracture. International Journal

of Fracture, 1997. 86: p. 5-36. 9. ABAQUS, ABAQUS 6.5. 2004, Pawtucket, Rhode Island: ABAQUS Inc.

0 h

PRQTSUSWVYX[Z

BED ^ :<;? ^ M

0k

O l

med

Prediction of fretting fatigue crack propagation using extended finite element methods Y. Xu & H. Yuan* Dpt. of Mechanical Engineering, Bergische University of Wuppertal, Wuppertal, Germany

Abstract: In the present paper we are using the extended finite element methods (XFEM) combined with an irreversible cohesive zone model to predict fatigue crack propagation due to fretting loading. As known, fretting fatigue reduces component life significantly. One of main reasons is high stress concentrations near the friction pad corner which cannot accurately be considered by common fatigue life models. With help of the XFEM fretting fatigue damage can be computed with additional assumption about stress distributions and crack propagation. The computational prediction is based on an irreversible cohesive zone model which considers accumulative material damage under mixed mode loading conditions. The numerical computations give us a realistic picture of fretting fatigue.

* Corresponding author. Email: [email protected]

O Û

O ¤

meji

Simulation of crack propagation at interfacesusing energy release rates

Martin Baker

Institut fur Werkstoffe, Langer Kamp 8, 38106 Braunschweig, [email protected]

Finite element calculations of crack propagation problems are of growingimportance. One of the main problems is the determination of the crack propa-gation direction using criteria that are appropriate even in systems comprisingdifferent materials or complicated geometries. Recently, a method for solvingsuch problems directly using the energy release rate during crack opening hasbeen presented [1]. In this method, trial cracks are propagated from the cur-rent crack tip position in different directions, and the energy release per createdcrack surface is calculated in the simulation. The direction with maximum en-ergy release rate is then chosen as new direction of the crack. To implement thismethod, automatic remeshing of the problem domain is performed for each trialcrack configuration. The method is suitable not only for linear-elastic prob-lems, but also for elasto-plastic materials and materials containing interfaces.Its main disadvantage is the computing time required, which is several timeshigher than in other methods.

The method is used to predict the angle of crack propagation in mode II ina homogeneous material, comparing the results with other criteria. The mainapplication of the method is in heterogeneous materials. As an example, aninterface crack at the interface of a bi-material is consideed and the direction ofcrack propagation is calculated under different circumstances. Finally, possiblefuture applications of the method to the prediction of failure in thermal barriercoatings and other heterogeneous materials are discussed.

References

[1] Martin Baker, Finite element crack propagation calculation using trial

cracks, Talk at IWCMM16, to appear in Computational Materials Science

O 0

OO

menm


Fracture parameters at bi-material ceramic interfaces under bi-axial state of stress

L. Marsavina, T. Sadowski

Lublin University of Technology, Faculty of Civil and Sanitary Engineering 20-618 Lublin, Nadbystrzycka 40 str., Poland e-mail: [email protected]

The presence of cracks has a major impact on the reliability of advanced materials, like fiber or particle reinforced ceramic composites, ceramic interfaces, laminated ceramics. The understanding of the failure mechanisms is very important, as much as the estimation of fracture parameters (energy release rate, stress intensity factors) at the tip of the crack approaching an interface normal or inclined. Different researchers have investigated the interaction between an interface and a perpendicular or inclined crack. Zak and Williams [1] showed that the stress field singularity at the tip of a crack perpendicular to an interface or terminating at the interface is of the form:

( )θσ λ ijij frK

−= 1 (1)

where (r, θ) represents the polar coordinates, K is the stress intensity factor, fij(θ) represents the angular distribution of the stress field, and λ is the real part of the eigenvalue, depends on the elastic properties of the bi-material, and can be obtained as a root of the following characteristic equation:

0)cos()1()1)((2 222 =−++−+− λπββαββαλ , (2) with α and β the Dundur’s bi-material parameters [2]:

*2

*1

*2

*1

EEEE

+−=α and ( ) ( )

( ) ( )1221

1221

112121

21

νµνµνµνµβ

−+−−−−= (3)

where µj and νj are the shear modulus and Poisson’s ratio, jj EE =* for plane stress and )1/(*

jjj EE ν−= for plane strain, j=1, 2 (1 and 2 represent the material) . The fracture parameters at the tip of an inclined crack approaching a ceramic interface

were numerically calculated under bi-axial loading state. An initial study for determining the fracture parameters for the homogeneous case was carried out. Then the bi-axial specimen with an interface was numerically investigated using FRANC2DL code. Eight node isoparametric elements were used to model a quarter of the biaxial specimen, Fig. 1. Eight singular elements were placed around the crack tip as a common technique to model the stress singularity. The model was loaded with different combinations of stresses σx and σy in order to produce mixed modes from pure Mode I (KII/KI=0) to pure Mode II (KII/KI=). The symmetric boundary conditions were imposed. The crack was extended in 13 increments of 5 mm starting from 5 mm to 65 mm. Then problem was analyzed by considering the two parts

O b

of the model from ceramic materials (Al2O3, respectively ZrO2). The stress intensity factors results for bi-material are normalized to the stress intensity factors for homogeneous case, and presented in Fig. 2 for predominantly mode I loading.

It can be observed that the Mode I SIF in the presence of interface has values above the homogeneous case for all considered mixed modes. For the Mode II SIF the presence of the interface produces an increase in the KII only when the crack is close to the interface (for a/b > 0.5).

A study of crack deflection or penetration at the interface will be presented taking into account the energy release rate ratio for the penetrated crack versus the deflected crack (Gp/Gd), [3].

Fig. 1 Bi-axial model for numerical analysis

0.00.20.40.60.81.01.21.41.6

0.0 0.2 0.4 0.6 0.8 1.0a/b [-]

KI_b

i/KI_

hom

[-]

KII/KI=0.00KII/KI=0.25KII/KI=0.33KII/KI=0.50KII/KI=0.66KII/KI=0.75

00.20.40.60.8

11.21.41.61.8

0.0 0.2 0.4 0.6 0.8 1.0a/b [-]

KII_

bi/K

II_ho

m [-

]

KII/KI=0.00KII/KI=0.25KII/KI=0.33KII/KI=0.50KII/KI=0.66KII/KI=0.75

a. KI_bimaterial/KI_homogeneous b. KII_bimaterial/KII_homogeneous

Fig. 2 The normalized stress intensity factor versus a/b for predominantly Mode I loads 1. Zak A.R., Williams M.L., Crack point stress singularities at a bi-material interface, J.

Appl. Mech, Volume: 30, (1963), pp. 142 – 143 2. Dundurs J., Effect of elastic constants on stress in a composite under plane

deformation, J. Compos. Mater., Volume: 1, (1969), pp. 310 – 322 3. Kaddouri K., Belhouari M., Bachir Bouiadjra B., Serier B., Finite element analysis of

crack perpendicular to bi-material interface: Case of couple ceramic-metal, Comput. Mater. Sci., 35, (2006), pp. 53 – 60

w w

2w

σx

σy

b a

Al2O3 ZrO2

O c

mepo

Mathematical modelling of the interface crack propagation in apre-stressed fiber reinforced elastic composite

Niculae Peride*, Adrian Carabineanu**, Eduard-Marius Craciun***

Abstract: A pre-stressed fiber elastic composite containing an interface crack in themode three of classical fracture is considered. The boundary conditions of our problem areleading to a homogeneous and a non-homogeneous problem. Using the theory of Cauchy’sintegral ([1]-[4]) and the numerical analysis we determine the incremental fields in thevecinity of the crack tips. Generalising Sih’s ([5]) criterion we obtain the critical values of theincremental stresses which are producing crack propagation and the crack propagationdirection. A numerical example regarding the propagation of an antiplane interface crack ingraphite epoxy is studied.

References

[1] Guz, A.N., Mechanics of brittle fracture of materials,with initial stresses, Naukova Dumka, Kiev,1983 (in Russian). [2] Muskhelishvili N.I., Some Basic Problems of Mathematical Theory of Elasticity, Nordhoff,Groningen, 1953.[3] Cristescu, N., Craciun, E.M., Soos, E., Mechanics of Elastic Composites, CRC Press, Chapman &Hall, 2003. [4] Lekhnitski, S.G., Theory of elasticity of aniosotropic elastic body. Holden Day, San Francisco,1963.[5] Sih, G.C., Leibowitz, H., Mathematical theories of brittle fracture, in Fracture - An advancedtreatise, Vol.II, Mathematical fundamentals, Editor H. Lebowitz, pp 68-191, Academic Press, NewYork, 1968.

............................................................................................................................ * Faculty of Mechanical Engineering, Industry and Maritime, “OVIDIUS” University of Constanta ** Faculty of Mathematics and Informatics, University of Bucharest *** Faculty of Mathematics and Informatics, “OVIDIUS” University of Constanta,corresponding author, email: [email protected]

Og

O h

PRQTSUSWVYX[Z

@C x @CMwGD ^ M

O k

bl

o7ed

Damage development in short fiber reinforced injection molded thermoplastic

composites

W. Lutz1, J. Herrmann2, M. Kockelmann1, A. Jäckel1, S. Schmauder1

S. Predak3, G. Busse3

1 University of Stuttgart, Institute for Materials Testing, Materials Science and

Strength of Materials (IMWF), Pfaffenwaldring 32, 70569 Stuttgart, Germany

2 University of Stuttgart, Institute of Applied and Experimental Mechanics (IAM)

Allmandring 5b, 70569 Stuttgart, Germany

3 University of Stuttgart, Institute for Polymer Technology (IKT), Dept. of Non-

destructive Testing, Pfaffenwaldring 32, 70569 Stuttgart, Germany

The demand for injection molded products increases because of the capability of high-volume-production, suitable properties and high geometrical freedom of design. To improvethe properties such as stiffness and the heat distortion temperature of thermoplastic materials,reinforcing fibers are embedded in the thermoplastic matrix. Due to the flow conditionsduring processing the resulting components show a complex fiber orientation distributionwhich is exhibited by a layered skin and core structure. Depending on the fiber orientationrelative to the loading direction, various damage processes and damage propagation appear onthe fiber-matrix interface. In the case of fibers loaded perpendicular to their fiber axis, thedebonding of the matrix starts from the outer fiber ends. Fibers which are loaded in theirlongitudinal axis are cracked or pulled out. To consider local fiber orientations as well as thelocal damage behavior at the fiber-matrix interface, the fiber orientation is simulated applyinginjection molding simulations. The damage behavior is investigated by using a statisticalapproach together with a combined cell model. The damage development of the fiber-matrixinterface is captured through considering two extreme cases which are a totally damagedfiber-matrix interface and a perfect or intact interface within the unit cell. The process ofdamage evolution of the fiber-matrix interface during loading of the specimen is realized by aWeibull statistic which describes the transition from the intact interface to the totally damagedinterface. The parameters of this Weibull law are determined using inverse modeling bycomparing simulation and experiment. The results show the different extent of damagedevelopment depending on the local fiber orientation as well as the poor mechanicalproperties of weak areas of a component such as a weld line where two melt fronts coalesce.

bÛ

b¤

o7eji

ADVANCED THEORETICAL AND NUMERICAL MULTISCALE MODELING OF COHESION/ADHESION INTERACTIONS IN CONTINUUM MECHANICS AND ITS APPLICATIONS

FOR FILLED NANOCOMPOSITES.

S. Lurieab , D. Volkov-Bogorodskya, V. Zubovb, N. Tuchkovab

aInstitute of Applied Mechanics Russian Academy of Scienses , Leninskii pr. 32 a, 119991, Moscow Russia bDorodnicyn Computer Centre of the Russian Academy of Sciences, Vavilov st. 40, 119991 Moscow GSP-1, Russia

Fax number: +7 (495) 135-61-90. E-mail: [email protected], and [email protected]

The correct consistent mechanical model of the media with spectrum of cohesion and adhesion interactions is

constructed using the variant of the kinematics variation principle. It is assumed that kinematics restrains models of

the media under consideration in full measure define the corresponding spectrum of internal interactions. The

sequential description of kinematics of media with possible internal structures (field of defects) of the various scales

is introduced, to investigate defective media with kept defects-dislocations, to describe of the defects generation

conditions [1]. This approach allows introducing the set of internal interactions of various types consequently,

which correspond to various types of microstructures in the considered media. In result it was received the various

complexity media models: model of porous media, turbulence and twinning models, model of an interphase layer

and model of disperse composites with spectrum of adhesion and cohesion interactions. As the most simple,

particular case the continual model of interphase layer with cohesion interactions was constructed (scale effect) [2-

4]. With its help it was possible to explain many known nonclassical effects of the mechanics of the continuous

mediums: -in the mechanics of fracture it was shown, that the nonsingular stresses are modeled in top of a crack; the

theoretical substantiation of Barenblatt hypothesis about existence of the cohesion fields was given first by the

modeling way; -it was established on connection of length of the cohesion interaction zone with parameters of the

fracture mechanics; -in the theory filled composites it was modeled effect of strengthening with reduction of radius

micro and nanoparticles at the constant volumetric contents of inclusions (nonclassical effect); the substantiation

of known hypotheses given: an effective matrix, effective inclusion, an effective continuum.

The further development of the offered approach to modeling a new materials demands the account of the scale

factors determined by superficial effects that is connected to modelling and the detailed analysis of adhesive

properties, development of methods of definition of the appropriate characteristics of materials. In the present work

the continuum model of media is developed in which known superficial effects such as a superficial tension, friction

of rest of bodies with ideally smooth surface of contact, the meniscus, wettability and capillarity are modeled within

the framework of the unified continuous description as particular scale effects in the mediums. So, for the first time

use the theory of media with microstructures with common description of the all spectrum of the superficial

phenomena in contrast to other variants of theories (of Mindlin types) of the media with microstructures. The

formulation and treatment of the constitutive equations on the a surface is given, and the test problems for definition

of new physical constants are formulated. Generalized Pascal equation for a surface tension pressure, generalized

Young law for the description of wettability will be received theoretically whereas earlier corresponding

dependences were known as empirical.

As application the dispersed composites with micro- and nano- inclusions were considered. An approach and the

model has been validated to predict some basic mechanical properties of a polymeric matrix reinforced with

b0

nanoscale particles/fibres/tubes (including carbon nanotubes) as a function of size and also dispersion of

nanoparticles. Using interphase layer model the prediction methodology and modeling tools have been developed

by numerical simulations and analysis of the mechanical properties across the length scales. In according with a

technique of asymptotic homogenization for periodic media, the infinite media with periodic micro/nano inclusions

it was considered. The problem of homogenization was solved numerically with the aid of the special analytical-

numerical method [2] and effective mechanical properties of the filled composite were found across the length

scales.

On the figure the influence of adhesion properties on the effective longitudinal elastic modulus for filled

composite with the ellipsoidal inclusions with semi-axis a is shown.

1K

2.0,8.0 == b

Figure.

The following adhesion characteristics are used: 0.03A B C= = for good-quality adhesion (thin line on the top);

and 0.01A B C= = −

0== B

for low (damage) adhesion (thin line below), thick line respond to case of adhesion neglect

( ). Parameter C is non-classical parameter of model that defines the width of cohesion type

interaction near the bound of inclusion-matrix contact;

A 2/1−

Dµ , Mµ are shear modulus of the inclusion and matrix. (The

work was supported by RFBR Grant N-6-01-00051 and Int. Grant EOARD N 2154p.).

References.

1. Lurie S., Kalamkarov A. (2005). Int. J. Solids & Str., 43 (1):91-111 2. Lurie, S. Belov, P. Volkov-Bogorodsky, D. and Tuchkova, N. (2003). Int J Comp Mater Scs, l.(3-4):529-539. 3. Lurie, S. Belov, P. and Volkov-Bogorodsky, D. (2003) Analysis and Simulation of Multifield Problems,

Springer, 12: 101-110. 4. Lurie S., Belov P. & Tuchkova N. (2005). Int. J. Comp. Mater. Scs. A., 36(2):145-152 5. Lurie S., Hui D., Zubov V., Tomlinson G., and Williams. (2006). Int. J. of Comp. Scs. and Eng. V2. (3/4),

2006: 228-241 6. Lurie S., Belov P., Volkov-Bogorodsky D., Tuchkova N.( 2006). J. of Mat. Scs, v.41, 20, pp. 6693-6707. 7. Volkov-Bogorodsky D., Evtushenko Y, Zubov V., Lurie S.(2006). Comput. Math. and Math. Phys.,

46(7):13181337.

b O

o7enm

Modelling of the damage behaviour of fibre reinforced metal matrix composites using a representative unit cell.

Ingo Scheider

GKSS Research Centre Geesthacht, Institute of Materials Research

Abstract:

For the understanding of the damage of fibre reinforced material under monotonic loading, it is advantageous to use micromechanical modelling. “Micro” in this case means that the single fibres are modelled explicitly, whereas the bonding between fibre and matrix is not treated as an additional material layer but simply as an interface. On that scale, different damage mechanisms can be distinguished: ductile matrix failure, which can be modelled by various continuum damage models, brittle fibre breaking, usually simulated by a maximum stress criterion (possibly combined with a stochastic approach), and fibre debonding, for which the cohesive model is most suitable. During the past years, many research groups have been worked on that issue, who first modelled a single fibre with surrounding matrix material. If loading and fibre direction are coincident, an axisymmetric cell can be used, for which fibre breaking and debonding has been modelled first by Tvergaard in 1993 [1]. If the loading is perpendicular to the fibre direction, a plane strain model (often containing several fibres) can be applied, see e.g. [2-4]. However, if a random orientation of fibres is investigated, 2D simulations are not sufficient. In such cases, three dimensional simulations with complex microstructure representation have to be utilized. This technique has become popular in recent years for particle reinforced materials, as shown e.g. in [5], but for fibre reinforced material, this kind of modelling is rather new, see e.g. the arrangement of fibres with single orientation in a metal matrix simulated by Peters et al. [6].

The current investigation is aimed at describing the failure of a metal matrix composite with arbitrary oriented fibres by a representative volume element containing several fibres in different directions. With such a model, also the interaction between fibres and the localization of damage in the matrix can be taken into account. The material under consideration is a Titanium composite as described in [6]. Due to the complexity of such a unit cell, a simplified 3D finite element model containing twelve fibres in the three principal directions is generated, see Figure 1. The structure is loaded by uniaxial straining.

As a result of this simulation, the mesoscopic stress-strain behaviour of the cell is used to develop a simplified failure law, which can then be embedded as a traction-separation law in a cohesive model on the macroscopic scale. This approach is already well established for the determination of traction-separation laws for ductile metals, see e.g. [7], where the stress-strain behaviour of a voided unit cell is used as constitutive behaviour of a cohesive law on the macroscale.

At the present state, experiments are missing to validate the model with engineering structures, but a numerical study shows the principle applicability of the scale bridging to assess the structural integrity of engineering structures.

bb

Figure 1: Representative unit cell of a fibre reinforced metal composite containing 12 fibres

in the three principal directions.

References:

1. V. Tvergaard, Model studies of fibre breakage and debonding in a metal reinforced by short fibres, J. Mech. Phys. Solids 41, 1993, 1309-1326.

2. C.J. Lissenden, C.T. Herakovich, Numerical modelling of damage development and viscoplasticity in metal matrix composites, Comput. Methods Appl Mech. Engrg. 126, 1995, 289-303.

3. H. Ismar, F. Schröter and F. Streicher, Effects of interfacial debonding on the transverse loading behaviour of continuous fibre-reinforced metal matrix composites, Comput. Struct. 79, 2001, 1713-1722.

4. N. Bonora, A. Ruggiero, Micromechanical modelling of composites with mechanical interface – Part II: Damage mechanics assessment, Comp. Sci. Techn. 66, 2006, 323-332.

5. W. Han, A. Eckschlager, H.J. Böhm The effects of three-dimensional multi-particle arrangements on the mechanical behavior and damage initiation of particle-reinforced MMCs, Comp. Sci. Techn. 61, 2001, 1581–1590

6. P.W.M. Peters, Z. Xia, J. Hemptenmacher, H. Assler, Influence of interfacial stress transfer on fatigue crack growth in SiC-fibre reinforced titanium alloys, Composites Part A 32, 2001, 561–567.

7. T. Siegmund, W. Brocks, The role of cohesive strength and separation energy for modelling of ductile fracture, Fatigue and Fracture Mechanics: 30th Volume, ASTM STP 1360, P.C. Paris and K.L. Jerina, Eds. ASTM, 2000, 139-151.

b c

o7epo

EXPERIMENTAL STUDY ON MECHANICAL FASTENING OF FRP COMPOSITE LAMINATES FOR STRUCTURAL USE

B. Latha Shankar1, S.M.Shashidhara2, G.S.Shiva Shankar3 1Corresponding Author and research scholar, IEM Department, 2professor and Head, 3 professor, Mechanical Engineering Department SIT, Tumkur -572103, Karnataka State, INDIA Phone:0816-2214001,2282696,2214000. Fax: 0816-2282994 E-Mail: [email protected], [email protected]

Abstract: The work presented in this paper covers an experimental study on finding the

effect of fiber volume fraction, width to diameter (W/D) ratio and clamping pressure on single bolted lap joint of woven jute fabric reinforced epoxy and woven glass fibre reinforced epoxy (GFRP) composites. Composite laminates were prepared by hand lay up technique and were made with two different fiber volume fractions. GFRP laminates were made with three staking sequence. For each volume fraction, four different W/D ratios were studied. In each group unnotched member was also fabricated and tested to determine the efficiency of the joint. All the tests were conducted on Zwick universal testing machine. The result of the study indicates that the joint efficiency increases continuously with the increase in the fiber volume fraction but decreases with the increase in W/D ratio. Tension and cleavage type of failure modes were observed on the specimens. The effect of clamping pressure and clearance on joint strength of GFRP laminates is also presented. EXPERIMENTAL ANALYSIS: Figure 1: Test set-up for GFRP joints

DISCUSSIONS: All types of GFRP test specimens failed without much yielding .All types of

joints with low thickness constraints (with less clamping pressure) showed larger displacement to failure and higher energy absorption during damage processes and

b g

resulted in initial bearing damage followed by final net-section failure. Load–displacement curves gave important information concerning properties of joints. Beyond the initial linear ranges, all load-displacement curves seemed to exhibit two different types of profiles that could be categorized as ductile and brittle type. In all types of GFRP laminates failed in brittle-type curve. Since ratio w/d = 6 and ratio e/d = 3 were imposed to all joints in order to have bearing damage initially, accordingly all composite joints investigated in the present study started failure with initial bearing damage prior to ultimate net– section failure.

The hole deformation behaviour was investigated for laminates failing by bearing

failure mode. The hole deformation was found to be slightly larger for clearance fit laminates in comparison to neat fit laminates for a given load level. The stiffness of the joint was also shown to decrease in clearance fit laminates as a result of the reduced contact area and larger hole deformation. Bolt-hole clearance should be minimized in order to achieve maximum bearing strength of the joint. The deformation of the bolt-hole in the laminate influences the stiffness and strength of a mechanically fastened joint. Permanent deformation of the hole results in slackness in the joint, which can results in significant strength reduction.

The experimental results show that, 15N-m tightening torque has the maximum strength. As a result of increasing the tightening torque (contact pressure), the slope of the load-displacement curve (stiffness) increases with increasing the tightening torque. The bearing strength of bolted joint increases with increasing tightening torque. The load-displacement curve of bolted joint specimen with 5 N-m tightening torque has the less slope/stiffness.

It is clear that from the study the ultimate loads increases with increasing the clamping torque from lower level to higher level (5 to 15 N-m). At the applied clamping torque of 20 N-m, on the surface of top ply around and below the washer area matrix cracking damage was observed. So analysis has been limited up to a clamping torque of 15 N-m.

REFERENCES [1] HART-SMITH.J,: Mechanically fastened joints for advanced composites, phenomenological considerations and simple analyses, Fibrous composites in structural design, Plenium Press, 1993. [2] WHITNEY.J and NUISMER.R,: Stress fracture criteria for laminated composites containing stress concentrations, J. Composite Materials, 1974,Vol.8, pp.253-265 [3] McCarthy M. A., Padhi G.S., W. Stanley, C.McCarthy and V. Lawlor, (December 8-9, 2000), ‘Three-dimensional Stress Analysis of Composite Bolted Joints’, National Seminar on Aerospace Structures, Indian Institute of Technology, Kanpur, India, pp.153-167 [4] Collings T.A., (1977), ‘The strength of bolted joints in multi-directional CFRP laminates’, Composites, Vol. 8, pp.43-54. [5] Chang F.K., Scott R.A., and Springer G.S., (1982), ‘Strength of Mechanically Fastened Composite Joints’, Journal of Composite Materials, Vol.16, pp. 470-494.

b h

o7ejq 17TH INTERNATIONAL WORKSHOP ON COMPUTATIONAL MECHANICS OF MATERIALS

The paper presents a modelling and simulation approach of material properties evolution during

the curing of a thermosetting epoxy matrix. Indeed, a highly accurate knowledge of material properties and

of the internal state obtained at the end of the manufacturing process is becoming more and more necessary for

the optimization of composites structures. This crucial need has recently emerged in naval construction for thick composites designs. The manufacturing of such a class of composites is a complex and delicate process where the thermal, the chemistry, the physics, the viscosity and the mechanical phenomena are

involved.

Therefore, the first part of the paper contains experimental results about material quality description to

take the attention to the reader to the quality problems generated by the curing. Especially, gradients of properties (degree of cure, cure shrinkage, viscoelasticity) are clearly established and their dependency to the cure is presented. Unfortunately, traditional know-how of manufacturers is less and less able to face quality problems encountered during the curing. For example, during the curing thermal gradients and internal stresses are generated leading to several imperfections (bubbles, fiber waviness…) decreasing therefore the quality of the composite. This complexity is increased by the thermo activated and exothermal behaviors of the thermosetting chemical reaction of the epoxy resin. Moreover throughout the cure, mechanical characteristics are evolving with temperature and degree of conversion of the chemical reaction.

However, tools using advanced finite elements made their appearance but these tools are established on formulations where all the couplings are not taken into account. A complete modelling of these phenomena coupled with the temperature and with the mechanics seems to never have been approached previously (Blesta et al. 1999; Golay 1991). Thus, the second part of the paper concentrates on a simulation tool for the curing of the epoxy matrix, based on coupling models taking into account the mechanics, the thermal and the chemistry. The presentation exposes the numerical setting of the coupling model developed with the finite elements modeling software Abaqus. The couplings taken into consideration concern the shrinkage induced by the thermosetting reaction and the corresponding heat generated. The mechanical behaviour of the resin is described by a viscoelastic constitutive law depending on the degree of cure. From a technical point of view, three Abaqus user subroutines were considered.

A FEM COUPLING MODEL FOR PROPERTIES PREDICTION DURING THE CURING OF AN EPOXY

MATRIX N. Rabearison *, Ch. Jochum *, JC. Grandidier ** : [email protected] *ENSIETA, 2 rue François Verny, F-29806 Brest cedex 9, FRANCE

** ENSMA, BP 40109, F-86961 Futuroscope Chasseneuil Cedex, FRANCE

bk

Fem Simulation of a Thermosetting Matrix: Application to internal stresses: Rabearison et al.

A comparison of simulation results with experimental data was performed. Results are in a very good agreement concerning the temperature field. Elastic stress level calculated by the FEM model seems to be realistic in comparison with experimental observations done on the epoxy mould..

0

50

100

150

200

250

300

0 200 400 600 800 1000 1200 1400 1600 1800time (s)

Tem

pera

ture

(°C)

HeatingtemperatureLocal resintemperatureModel

0.E+001.E+062.E+063.E+064.E+065.E+066.E+067.E+068.E+069.E+061.E+07

0 200 400 600 800 1000 1200 1400 1600 1800time (s)

Von

Mise

s (P

a.)

Element 73Element 77Element 80Verre

8077

73

Glass

807773

0.E+001.E+062.E+063.E+064.E+065.E+066.E+067.E+068.E+069.E+061.E+07

0 200 400 600 800 1000 1200 1400 1600 1800time (s)

Von

Mise

s (P

a.)

Element 73Element 77Element 80Verre

8077

73

Glass

807773

Verre

8077

73

Glass

807773

Verre

8077

73

Verre

8077

73

Glass

807773

Glass

807773

Figure 1: Local temperature simulation. (Element 73, 120°C isothermal curing).

Figure 2: Von Mises elastic stress gradient prediction within the epoxy matrix.

First Internal stress estimation by taking into account the viscoelasticity of the matter in formation is under development at this time and should be presented at the conference. This is a strategic way to understand material properties growth during the cure. It should lead to a better description of material characteristics gradients present at the end of cure and namely the internal residual stress state obtained. Moreover, the taken into account of viscoelastic behaviour should lead to material relaxation prediction after the curing.

References Blesta D.C, Duffyb B.R., McKeeb S., Zulkifleb A.K. "Curing simulation of thermoset composites"; Composites: Part A 30 (1999), pp.1289–1309. Golay F., "Contribution à la modélisation par éléments finis des phénomènes thermomécaniques apparaissant lors de l’élaboration de matériaux. composites applications industrielles", Thèse de l’université d’Aix-Marseille. (1991).

c l

o7ens

MODELING THE INTERACTION OF CRAZING AND MATRIX PLASTICITY IN RUBBER-TOUGHENED GLASSY POLYMERS

Thomas Seelig1, Erik Van der Giessen

2

1Fraunhofer-Institute for Mechanics of Materials, Freiburg, Germany

2University of Groningen, Department of Applied Physics, Groningen, The Netherlands

Abstract The ductility and fracture toughness of originally brittle glassy polymers is known to be enhanced when

these materials are blended with small, finely dispersed rubber particles. Generally, the rubber particles

act as initiation sites for dissipative micromechanisms distributed throughout the material. In so-called

ABS materials consisting of an amorphous thermoplastic matrix of SAN (styrene-acrylonitrile) and sub-

micron sized rubber particles, the prevailing micromechanisms are crazing as well as plastic flow (shear

yielding) of the glassy matrix. The latter is often accompanied by void growth upon cavitation of the

rubber particles, especially in situations of high stress triaxiality as found ahead of crack tips or notches.

Despite numerous experimental investigations, details about the conditions for the coexistence or

competition of crazing and matrix flow, in dependence on microstructural parameters (e.g. rubber

particle size and volume fraction) and overall loading (e.g. loading rate), are not yet fully understood.

In contrast to previous numerical studies which have mostly focused solely on matrix yielding and

void growth, e.g. [1],[2], the present work particularly aims at investigating the interaction between both,

matrix plasticity and crazing. Finite element analyses are performed on micromechanical models where

the rubber particles are explicitly resolved and the rate-dependent finite strain deformation behavior of the

glassy matrix, displaying intrinsic softening and rehardening, is described by an appropriate constitutive

model [3],[4]. Owing to their appearance as highly localized crack-like zones, yet bridged by stress-

carrying polymer fibrils, the crazes are modeled as cohesive surfaces. They are governed by a stress-

based initiation criterion, a rate-dependent traction-separation law describing craze widening, and a

criterion for craze breakdown, i.e. microcracking [5]. An issue of special interest is whether and how the

occurrence of crazes —which accommodate overall dilatation and at the same time sustain stress— may

prevent the unrealistic overall softening inevitably obtained from previous models where only void

growth in the incompressible matrix is considered.

Further topics discussed are model extensions to account for experimentally observed size effects and

critical aspects of the cohesive surface modeling of crazing in the vicinity of small rubber particles.

Moreover, due to the different intrinsic time scales of matrix flow and crazing, the overall loading rate is

expected to have an influence on the predominance of either of the two deformations mechanisms.

References 1. Steenbrink, A.C. and Van der Giessen, E., J. Mech. Phys. Solids, vol. 47, 843-876, 1999.

2. Socrate, S. and Boyce, M.C., J. Mech. Phys. Solids, vol. 48, 233-273, 2000.

3. Boyce, M.C., Parks, D.M. and Argon, A.S., Mech. Mater., vol. 7, 15-33, 1988.

4. Wu, P.D. and Van der Giessen, E., Eur. J. Mech. A/Solids, vol. 15, 799-823, 1996.

5. Tijssens, M.G.A., Van der Giessen, E. and Sluys, L.J., Mech. Mater., vol. 32, 19-35, 2000.

c Û

c ¤

o7et

PHYSICALLY-BASED MODELLING OF LARGE DEFORMATIONS OFCLAY/POLYMER NANOCOMPOSITES

Ł. Figiel, F.P.E. Dunne, and C.P. BuckleyDepartment of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, UK

The layered silicate (clay)/polymer nanocomposites have been the most extensivelyinvestigated type of nanocomposites because of their application potential (in e.g. packagingor automobile industry) at relatively low production cost. Depending on the polymer andorgano-silicate thermodynamic compatibility, intercalation kinetics and the processingconditions, the nanocomposites can have different morphologies. Moreover, it is well-knownthat the processing-induced morphologies have different impacts on final properties (e.g.mechanical) of nanocomposites. However, it is still unclear what the influence of thosemorphologies is on end-product properties, and how they arise during processing. Betterunderstanding and predictive capability is required, to take full advantage of the extraordinarypotential of polymer nanocomposites. The task is complex, because it involves a relationshipbetween variables of material composition, process parameters and end-product properties.However, the authors believe the complexity involved can be handled by exploitation ofadvanced numerical modelling of the material.

The present work aims to develop a multi-scale, physically-based model of the shapingprocess for layered silicate nanocomposites based on poly(ethylene terephtalate) (PET)matrices, as might be used for packaging such as bottles. Here the matrix is PET in thevicinity of the narrow time/temperature window just above the glass transition, where itbehaves approximately as hyperelastic. During a process such as injection-stretch blow-moulding, the material undergoes finite deformations under largely biaxial macroscopicstress-states. This is modelled at the level of assemblages of particles, by representing theentire material by a repeating Representative Volume Element (RVE) with periodic boundaryconditions. The RVE is designed to mimic structures as observed by electron microscopy, andcontains a variable number of individual clay platelet particles, or incompletely exfoliatedgroups of particles with given aspect ratio and initial orientation distribution. They areassumed to be linear elastic, while the PET is represented by a finite deformation 3-Dconstitutive model as proposed and justified experimentally for PET by our laboratory inprevious work. Finite deformation of the RVE is simulated numerically using Finite Elementanalysis, treating particles and matrix as continua, and invoking these material models.Results of the simulations yield predictions of the overall rheology of the nanocomposite, asneeded for macroscopic simulations of processing, and also of deformation and re-orientationof the particles.

c 0

c O

o7enu


Experimental investigation and modelling of crack propagation in plywood by CT test

T. Sadowski, I.Ivanov, M.Filipiak


Plywood is made from thin sheets of wood veneer, called plies, which are stacked together symmetrically with the grain direction of each ply perpendicular to the grain direction of its adjacent ones. The veneers are bonded under heat and pressure with strong adhesives, which makes plywood a type of composite material. Plywood is used in many applications as a timber of a higher quality because of its resistance to cracking, breaking, twisting and warping.

In this paper the problem of crack propagation in plywood material is investigated by Compact Tension (CT) tests and Finite Element (FE) analysis. The plywood specimens are made from five veneers of beech or pine. The gradual degradation and fracture process of this material is very complicated. It obeys both description of damage growth in plies and delamination of composite layers. The theoretical modelling of the problem was done by application FE technique and ABAQUS code. The plywood is treated as orthogonal layered composite and in order to model the material behaviour, all mechanical data concerning single ply and adhesive joint were incorporated into the algorithm. Even in case of early stage of deformation process of CT specimen one can observe different stress concentration in different layers of the plywood (Fig. 1).

Fig. 1 Stress concentration in layers of the plywood for two different sequences of plies

The Hashin (1980) failure criteria are used for damage initiation determining four damage modes. Three damage variable degrade the material stiffness with linear softening evolution functions to simulate the progressive failure of one play. The orthotropic strength of layers is justified from the single ply tests. The adhesive inter-layers between the plies are modelled with cohesive elements in ABAQUS FE code. When the overall behaviour of plywood specimen in CT tests (Fig. 2) is simulated successfully by FE analysis, the history of damage accumulation and internal crack propagation is revealed including delamination of plies.

c b

Fig. 2. Compact Tension (CT) test of plywood specimen

REFERENCES 1. Myriam Chaplain, Gerard Valentin. Fracture mechanics models applied to delayed failure of LVL beams. Holz Roh Werkst 65 (2007): 7–16. 2. Jernkvist, Lars Olof. Fracture of wood under mixed mode loading: I Derivation of fracture criteria. Engineering fracture mechanics 68 (2001): 546–563. 3. R H Leicester. Application of linear fracture mechanics to notched timber elements. Prog. Struct. Engng Mater. 8 (2006):29–37 4. A. Reiterer, S.E. Stanzl-Tschegg, E.K. Tschegg. Mode I fracture and acoustic emission of softwood and hardwood. Wood Science and Technology 34 (2000): 417–430. 5. M.A.L. Silva, M.F.S.F. de Moura b, J.J.L. Morais. Numerical analysis of the ENF test for mode II wood fracture. Composites: Part A 37 (2006): 1334–1344. 6. Eric N. Landis, Svetlana Vasic, William G. Davids, and Perrine Parrod. Coupled Experiments and Simulations of Microstructural Damage in Wood. Experimental Mechanics 42(4) (2002): 389–394. 7. A. Reiterer, I. Burgert, G. Sinn, S. Tschegg. The radial reinforcement of the wood structure and its implication on mechanical and fracture mechanical properties—A comparison between two tree species. Journal of Materials Science 37 (2002): 935– 940. 8. Erik Serrano. A numerical study of the shear-strength-predicting capabilities of test specimens for wood–adhesive bonds. International Journal of Adhesion & Adhesives 24 (2004): 23–35. 9. E. K. Tschegg, A. Reiterer, T. Pleschberger, S. E. Stanzl-Tschegg. Mixed mode fracture energy of sprucewood. Journal of Materials Science 36 (2001): 3531 – 3537. 10. Jørgen Lauritzen Jensen. Quasi-non-linear fracture mechanics analysis of the double cantilever beam specimen. J Wood Sci 51 (2005): 566–571. 11. Jean-Luc Coureau, Per Johan Gustafsson, Kent Persson. Elastic layer model for application to crack propagation problems in timber engineering. Wood Sci Technol 40 (2006): 275–290. 12. Shigehiko Suzuki, Hideto Miyagawa. Effect of element type on the internal bond quality of wood-based panels determined by three methods. J Wood Sci 49 (2003):513–518. 13. Aleksandra Sretenovic, Ulrich Muller, Wolfgang Gindl. Comparison of the in-plane shear strength of OSB and plywood using five point bending and EN 789 steel plate test methods. Holz als Roh- und Werkstoff 63 (2005): 160–164.

0 2 4 6Displacement ∆l [mm]

0

0.2

0.4

0.6

0.8

1

Forc

e P

[kN]

P

P

∆ l

cc

PRQTSUSWVYX[Z

:wMDF;>= =a;>MD<Ga?GD(

c g

ch

qed

Micro macro modeling of metal forming processes

A. Bertram1, T. Bohlke2, G. Risy, V. Schulze

1Otto-von-Guericke-Universitat Magdeburg, Institut fur Mechanik, PSF 4120, 39016 Magdeburg, Germany,

email: [email protected], 2Institut fur Technische Mechanik, Universitat Karlsruhe (TH), PSF

6980, 76128 Karlsruhe, Germany, email: [email protected]

The elastic and plastic anisotropy of metals strongly depends on the crystallographic texture.Such anisotropies become important in typical industrial applications, like, e.g., the earing be-haviour and the springback phenomenon which are observed in deep drawing processes. Thesimulation of these industrial applications is usually based on large-scale FE models. If thetexture has to be taken into consideration, then often phenomenological approaches (see, e.g.Hill, 1948; Barlat et al., 1997) are implemented into the FE code. The drawback of these meth-ods is that the texture evolution and therefore the evolution of the material anisotropy duringthe deformation process is neclected. In contrast to phenomenological approaches, polycrystalplasticity models allow for a description of an evolving crystallographic texture. The main disad-vantage of this kind of models is the high numerical effort, in particular if an implementation intoan FE code is carried out. This problem also occurs, if homogenization methods like the Taylormodel (Taylor, 1938) are applied at the integration points of an FE model (see, e.g. Bronkhorstet al., 1992; Miehe et al., 1999). If one wants to reduce the numerical costs to an acceptable levelthen the number of crystals has to be minimized. As a consequence, however, the quality of thenumerical results is generally not acceptable or the induced anisotropy is strongly overestimated(Bohlke et al., 2006).

In the present work different means to reduce the numerical effort are introduced which still allowfor an acceptable quality of the numerical results. Two of the models are based on a low numberof discrete crystals (Bohlke et al., 2006; Schulze, 2006). In order to reduce the amount of inducedanisotropy which is overpredicted by the Taylor assumption, an isotropic background is addedin both approaches. The third model (CT model) is particularly suitable for crystallographictexures consisting of a small number of components (Bohlke et al., 2005, 2006). It is basedon continuous model functions on the orientation space. The applied Mises-Fisher distributionfunctions (see, e.g. Eschner, 1993) permit an explicit consideration of the scattering aroundtexture components. The Mises-Fisher distribution is a central distribution. The scattering ofa texture component can be described by the half-width parameter. In contrast to the model ofRaabe and Roters (2004), the CT model directly uses this parameter for the calculation of themacroscopic stresses. Furthermore, the degree of anisotropy can be adjusted by appropriatelychoosing this parameter.

The models are applied to the simulation of the deep drawing process in aluminum and steel.The numerical results for the earing profiles and for springback are compared with experimentaldata. In Fig. 1 the earing profiles calculated from the CT model and the numerical results takenfrom Engler and Kalz (2004) are shown. If the half-widths of the components are increased bya factor β = 2, then a good agreement of the numerical and experimental results is performed.

References

Barlat, F., Becker, R., Hayashida, Y., Maeda, Y., Yanagawa, M., Chung, K., Brem, J., Lege, D.,Matsui, K., Murtha, S., and Hattori, S. (1997). Yielding description for solution strengthenedaluminum alloys. Int. J. Plast., 13(4), 385–401.

c k

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 10 20 30 40 50 60 70 80 90α

Experiment

Nor

mal

ized

cup

hei

ght

CT (β = 1)CT (β = 2)

Figure 1: Comparison of the earing profile of a deep drawn cup calculated by the CT modelwith experimental data Engler and Kalz (2004)

Bohlke, T., Risy, G., and Bertram, A. (2005). A texture component model for anisotropicpolycrystal plasticity. Comp. Mat. Science, 32, 284–293.

Bohlke, T., Risy, G., and Bertram, A. (2006). Finite element simulation of metal formingoperations with texture based material models. Modelling Simul. Mater. Sci. Eng., 14, 1–23.

Bronkhorst, C., Kalidindi, S., and Anand, L. (1992). Polycrystalline plasticity and the evolutionof crystallographic texture in fcc metals. R. Soc. Lond., A341, 443–477.

Engler, O. and Kalz, S. (2004). Simulation of earing profiles from texture data by means of avisco-plastic self-consistent polycrystal plasticity approach. Materials Science and Engineering

A, 373, 350–362.

Eschner, T. (1993). Texture analysis by means of modelfunctions. Textures and Microstructures,21, 139–146.

Hill, R. (1948). A theory of yielding and plastic flow of anisotropic materials. Proc. Phys. Soc.

Lond., A 193, 281–297.

Matthies, S. (1980). Standard functions in texture analysis. Phys. Stat. Sol. B, 101, K111–K115.

Miehe, C., Schroder, J., and Schotte, J. (1999). Computational homogenization in finite plastic-ity. Simulation of texture development in polycrystalline materials. Comp. Meth. Appl. Mech.

Engng., 171, 387–418.

Raabe, D. and Roters, F. (2004). Using texture components in crystal plasticity finite elementsimulations. Int. J. Plast, 20, 339–361.

Schulze, V. (2006). Anwendung eines kristallplastischen Materialmodells in der Umformsimula-

tion. Ph.D. thesis, Otto von Guericke Universitat Magdeburg.

Taylor, G. (1938). Plastic strain in metals. J. Inst. Metals, 62, 307–324.

g l

qeji

Multisite modelling of grain interactions

in single- and multiphase polycrystalline aggregates

Laurent Delannay (1), Maxime A. Melchior (1), Javier W. Signorelli (2)

(1) Department of Mechanical Engineering, Université catholique de Louvain (UCL), av. G. Lemaître 4, 1348 Louvain-la-Neuve (Belgium)

(2) Instituto de Fisica Rosario, Fac. de Ciencias Exactas, Ingenieria y Agrimensura,CONICET-UNR, Bv. 27 de febrero 210 bis, 2000 Rosario, Santa Fe, Argentina

An original modelling approach is proposed for the prediction of heterogeneous strain fieldsthrough polycrystalline aggregates. The so-called “multisite” model assumes that close-rangeinteractions of adjacent grains have a dominant influence on the micromechanical response.Hence, the approach is different of self-consistent homogenization schemes in which onefocuses instead on long-range interactions, by considering that each grain is embedded in auniform matrix. The computational cost of the multisite model is several orders of magnitudeless than that of crystal plasticity based finite element modelling (CPFEM), yet the accuracyof the predictions is in many circumstances similar.

The first part of the study considers the complementary effects of grain shape and texture onthe plastic anisotropy of single- and multiphase polycrystalline aggregates. Predictions of themultisite model are compared to the predictions of CPFEM and of a self-consistent scheme.

In the second part of the study, the multisite model is used as a user-defined constitutive lawfor the full-scale simulation of a forming operation. In order to reduce the computational cost(and to facilitate the transport of state variables when remeshing is required), everyintegration point of the FE mesh is represented by a different set of grains. The statisticalrepresentation of texture is incomplete at the level of individual integration points, but it isensured on average over 5-10 adjacent elements.

g Û

g ¤

qenm

A simplified modelling of rafting in Ni-base superalloys

Authors: Bernard Fedelich*, Udo Brückner*, Alexander Epishin**, Georgia Künecke*,

Thomas Link*, Thomas May**, Pedro Portella*

* Bundesanstalt für Materialprüfung und -forschung, (BAM) Berlin, Germany

** Technical University Berlin, Germany

Rafting, i.e. directional coarsening of the γ' precipitates, usually occurs in Ni-base superalloys

under high temperature creep loading and has been often observed in turbine blades after

service. This instability of the microstructure can also lead to a significant degradation of the

monotonous and the cyclic strength of these alloys. Hence, modelling tools to predict the

occurrence of rafting in blades under service condition are highly desirable.

Recent results suggest that the directional coarsening of the γ' precipitates is mostly due to the

anisotropy of the dislocation structure generated in the γ /γ' interfaces. Indeed, the dissimilar

superposition of external stresses and misfit stresses in each channel type lead to different

dislocation densities in the three types of interfaces. It is argued that the driving force for

rafting can be best described by the internal stresses, i.e. the back stresses, which can be

identified by the usual testing methods at the macroscopic level.

A constitutive model that takes into account some of the recent advances in the understanding

of the deformation behaviour of Ni-base superalloys at high temperature has been developed.

The kinetics of rafting is assumed to be driven by the macroscopic back-stress. The model has

been validated for the alloys SRR99 and CMSX-4. It is implemented as a user-subroutine in a

commercial FE code and can be used for structural analysis.

The Figure 1 shows a typical model prediction for the rafting kinetics, where the advancement

degree of rafting 10),( ≤λ≤λ t is defined by the variation of the γ channel width )(tw between

the initial width 0w and a final value ∞w , with 0)1( www λ−+λ= ∞ .

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.002 0.004 0.006 0.008 0.010

creep strain

λλλλ

near [001]t =2h

t =5h

t =20h

t =45h

t =115h

Fig. 1. Comparison between the computed rafting advancement degree in the alloy SRR99

during creep at 980°C, σ=200MPa and SEM micrographs of several crept specimens.

g 0

gO

qepo

Approximation of Crystallographic Texture as a Mixed Integer

Quadratic Programming Problem

T. Bohlkea K. Jochena U.-U. Hausb V. Schulzec

a Institut fur Technische Mechanik, Universitat Karlsruhe (TH), Germanyb Institut fur Mathematische Optimierung, Otto-von-Guericke-Universitat Magdeburg, Germany

c Volkswagen AG, EZVT Methodenplanung/Umformsimulation, Germany

Single phase polycrystals are composed of grains of the same material which differ with respectto their lattice orientations. The simplest statistical description of such microstructures isbased on the crystallite orientation distribution function (codf) which specifies the volumefraction of a material having a specific lattice orientation. The codf is the one-point correlationfunction of lattice orientation and describes the crystallographic texture in the material.Higher-order correlation functions allow for a description of the morphological texture. Thecorrelation functions can be estimated based on orientation data determined experimentallyfor example by X-ray diffraction or by automated electron back-scatter diffraction orientationmeasurements. For a review concerning the representation of microstructures of polycrystalsand the experimental determination of their mesoscale microstructure see Adams and Olson[1]. Due to the general complexity of a codf it is often necessary to look for simplified, butphysical, descriptions which are essentially low dimensional.In the context of crystallographic textures, so called texture components have been used todescribe textures peculiar to specific processing histories. A large amount of work has beendone to formulate isotropic and anisotropic model functions and to identify the dominantcomponents in experimental textures see Helming and Eschner [6]. Texture components areused on the one hand to obtain a physical interpretation of experimental codfs and on theother hand to homogenize the mechanical behavior with an acceptable numerical effort.The problem of finding approximations for a given texture has been considered in detail byKocks et al. [8], Toth and Van Houtte [10], and Helming et al. [7]. A special approach forthe approximation of steel textures was developed by Delannay et al. [5] characterizing thetexture by a set of parameters and describing typical features of industrial steel sheets byprescribed fibers.Taylor type material models [11] allow for a description of the macroscopic mechanical behav-ior due to a specific slip system geometry and orientation distribution. From the numericalpoint of view, large scale finite element simulations of metal forming operations based onthe Taylor model are very time-intensive and storage-consuming if the crystallographic tex-ture is approximated by several hundred discrete crystals. Therefore, Raabe and Roters[9] introduced the so called texture component crystal plasticity method which describescrystallographic textures by small sets of discrete orientations. Due to the discontinuousapproximation of the codf the approach by Raabe and Roters [9] requires a random variationof the discrete crystal orientation through the sample. In the approach by Bohlke et al. [3]texture components are described by continuous model functions. The effective stress is ob-tained by integrating the crystal stress – weighted by the codf – over the orientation space.The two material models mentioned before require an approximation of the codf by discreteor continuous texture components. The method suggested in this paper yields both a discrete

g b

(taking only the main orientations into account) and a continuous approximation, in the lastcase also giving a rigorous error bound. In some applications, the codf is already discretizedinto a finite set of weighted orientations. If the number of orientations needs to be reduced,the proposed method can be used to find an optimal set of components to approximate theinitial distribution.In this paper we restrict our attention to the formulation as a mixed integer quadratic pro-gramming problem (MIQP) problem and its approximate solution without considering me-chanical properties. These could only be predicted based on several simplifications and as-sumptions which themselves are still a subject under discussion. For the case of a continuousapproximation of the codf a discussion of effective mechanical properties can be found forexample in Refs. [3, 4].Special emphasis is given to the generation of a class of approximations with an increasingnumber of texture components [2]. Furthermore, the constraints resulting from the non-negativity, the normalization, and the symmetry of the codf are analyzed. Finally, a set ofapproximations of three different experimental textures determined with this solution schemeis presented and discussed. Based on these hierarchical solutions, the engineer can decide inwhat detail the microstructure is considered.

References

[1] Adams B, Olson T. The mesostructure-properties linkage in polycrystals. Progr MaterSci 43, 1–88 (1998)

[2] Bohlke T, Haus, U-U, Schulze V. Crystallographic texture approximation by quadraticprogramming. Acta Materialia 54, 1359–1368 (2006)

[3] Bohlke T, Risy G, Bertram A. A texture component model for anisotropic polycrystalplasticity. Computat Mater Sci 32, 284–293 (2005)

[4] Bohlke T, Risy T, Bertram A. A texture based model for polycrystal plasticity. Proceed-ings of the 14th international conference on textures of materials (ICOTOM-14) (2005).

[5] Delannay R, Van Houtte P, Van Bael A, Vanderschueren D. Application of a textureparameter model to study planar anisotropy of rolled steel sheets. Model Simul MaterSci Eng 8, 413–422 (2000)

[6] Helming K, Eschner T. A new approach to texture analysis of multiphase materials usinga texture component model. Cryst Res Technol 25, K203–208 (1990)

[7] Helming K, Schwarzer R, Rauschenbach B, Geier S, Leiss LB, Wenk H-R, et al. Textureestimates by means of components. Z Metallkd 85, 545–553 (1994)

[8] Kocks U, Kallend J, Biondo A. Accurate representation of general textures by a set ofweighted grains. Text Microstruct 14, 199–204 (1991)

[9] Raabe D, Roters F. Using texture components in crystal plasticity finite element simu-lations. Int J Plast 20, 339–361 (2004)

[10] Toth L, Van Houtte P. Discretization techniques for orientation distribution functions.Text Microstruct 19, 229–244 (1992)

[11] Van Houtte P. A comprehensive mathematical formulation of an extended Taylor-Bishop-Hill model featuring relaxed constraints, the Renouard-Winterberger theory and a strainrate sensitive model. Text Microstruct 8, 313–350 (1988)

g c

qejq

DISLOCATIONS MOBILITY LAWS AND PRECIPITATION STRENGTHENING

IN BCC CRYSTALS AT LOW TEMPERATURES

BY DISLOCATION DYNAMICS SIMULATIONS

Sanae Naamane*, Ghiath Monnet

*, Benoit Devincre

**

* EDF-R&D, MMC, Avenue des Renardières, 77818 Moret sur Loing, France,

[email protected], [email protected] ** LEM, CNRS-ONERA, 29 av. de la division Leclerc, 92230 Châtillon, France,

[email protected]

This study is dedicated to the problem of precipitation strengthening in α-iron and in the low

temperature regime. In this case, a thermally activated process controls the motion of

dislocation lines and the mobility is strongly dependant on dislocation character. For this

reason, usual analytical models used for the prediction of precipitation strengthening may not

be valid and 3D simulations [1] of dislocations gliding through a random distribution of

precipitates are of strong interest.

Velocity laws for both screw and non-screw dislocations were first determined in self-

consistency with the theory of activated process [2]. The agreement between such laws and

the experimentally observed dependencies on temperature and strain rate was examined.

These dislocation velocity laws have been validated using dislocation dynamics simulations

by modeling the evolution of microstructures and stress-strain curves in α-iron as a function

of temperature. This study shows that the effects of temperature and strain rate are well

reproduced by the simulations (see figure below).

In a second step, the question of precipitation strengthening at low temperature was

considered. We first looked at the simple geometry of dislocations gliding through a periodic

row of particles. More precisely, the interaction of an infinite dislocations with spherical

precipitates was examined as a function of the dislocation character, the size and the spacing

of precipitates, temperature and strain rate. Such simulations based on a simple geometry are

of interest since they can be easily compared with the static prediction of Bacon et al. [3].

Finally, the strengthening associated with a random distribution of particles was also

estimated.

[1] B. Devincre, L. P. Kubin, C. Lemarchand, R. Madec. Mater. Sci. Eng. A 309, 211 (2001).

[2] U. F. Kocks, A. S. Argon, M. F. Ashby. Prog. Mater. Sci. 19 (1975).

[3] D. J. Bacon, U. F. Kocks, and R. O. Scattergood, Phil. Mag. 13, 911 (1973).

gg

Simulated dislocation microstructures after a tensile test with a constant imposed strain rate of

10-4

s-1

. Thin foils of thickness 1 µm were extracted from the simulation cell along the (110)

slip plane. The temperature considered is 50 K in (a) and 250 K in (b). Dislocations of the

active slip system are shown in gray, dislocations of the inactive systems in white and

junctions in black.

1 µm

(b)

1 µm ]111[

(a)

g h

qensModeling the evolution of texture and grain shape in magnesium

alloy AZ31 under plane strain compression

Aruna Prakasha,1, Hermann Riedela, Sabine Weygandb

aFraunhofer Institute for Mechanics of Materials IWM, Woehlerstraße 11, 79108 Freiburg, Germany bForschungszentrum Karlsruhe GmbH, Institute for Materials Research II, 76021, Karlsruhe, Germany

1Corresponding Author: Aruna Prakash<[email protected]>

Summary: This contribution deals with the micro-structural modeling of Magnesium. A crystal plasticity finite element model

(CPFEM), which accounts for twinning reorientation, is used in combination with representative volume elements (RVEs) to

identify bands of strain localization which occur during rolling of magnesium. Texture evolution and grain shape change are

additional areas of interest in this work.

Introduction Magnesium, with its high strength to weight ratio and high

impact strength, is a suitable candidate for use in the aerospace

and automobile industry. Its usage has however been limited

due to its low formability. Magnesium, commonly used as

alloys, has an HCP crystallographic structure, which forms a

strong texture upon processing. For instance, in wire drawing,

the hexagonal planes tend to orient themselves parallel to the

wire axis. In rolling, the hexagonal planes orient themselves in

the rolling plane. This preferred orientation is referred to as a

basal texture. At low temperatures, rolling of magnesium is

practically impossible. The developed texture combined with

the lack of sufficient number of active slip systems causes

bands of high stress localization, ultimately leading to failure.

In this work, we use a Crystal Plasticity based Finite Element

Model (CPFEM) to simulate these shear bands. The CPFEM

presents a major advantage in comparison to the well-known

Self-Consistent (SC) models. The SC models fail to accurately

predict grain shape change; the shape is always restricted to be

an ellipsoid. Needless to say, this is insufficient to describe

complex deformation behavior such as grain curling in BCC

materials [1], when subjected to wire drawing.

Crystal plasticity model Grains in the polycrystalline aggregate are modelled within the

framework of crystal plasticity [2,3]. The kinematics of

deformation is based on Rice [4] and Hill & Rice [5]. In the

original model, plastic deformation is assumed to be entirely

due to dislocation movement (slip).

In the framework of large deformations, this model assumes a

multiplicative decomposition of the deformation gradient Fij.

pkjikij FFF *= (1)

where, Fpij denotes plastic shear and F*

ij accounts for elastic

stretching and rotation of the lattice.

The rate of change of Fpij

is related to the slipping rate )(αγ of

the slip system by

( ) ( ) ( )∑=

−

α

αααγ jip

kjp

ik msFF 1

, (2)

where ( ) ( )αα

ji ms , , are the slip direction and the slip plane

normal respectively.

It is convenient to define si*( )

, lying along the slip direction of

the system and mj*( )

, normal to the slip plane, in the deformed

configuration, by

)(*)*( ααjiji sFs = ,

1*)()*( −= ijjj Fmm αα . (3)

Based on the Schmid law, the shear rate )(αγ is determined by

the resolved shear stress τ( ) on the slip system in form of a

power law,

)*()*()( ααα στ jiji ms= ,

n

ga

/1

)(

)()()( sign α

ααα ττγ = ,

(4)

where a ( ) denotes a reference shear rate on the system , n

denotes the material rate sensitivity and g( ) describes the

current strength of the system.

The strain hardening is defined through a modified VOCE

hardening law:

)()(

)( ˆ β

βαβ

αα γτ

∑Γ= h

ddg , (5)

with the hardening matrix h , where the diagonal elements

describe self-hardening and off-diagonal elements describe

latent hardening. The hardening function )(ˆ ατ is defined as:

Γ−−Γ++= α

ααααα

τθθτττ

1

0110 exp1)(ˆ , (6)

where ατ0 ,

ατ1 ,αθ0 ,

αθ1 are the hardening parameters and Γ is

the cumulative shear on all slip systems.

This entire constitutive structure has been implemented as a

user material routine UMAT for cubic symmetry in the finite

element code ABAQUS Standard® [6].

g k

Twinning reorientation The constitutive model described above is, however,

insufficient to describe deformation in magnesium, which

shows a lot of twinning activity. To this end, we extend this

model by implementing a twinning model which also accounts

for reorientation.

The proposed model for twinning reorientation is based on Van

Houtte [7] and the Predominant Twin Reorientation scheme [8].

The rate at which the twinned volume fraction changes is given

by:

Sg

tntn

,,

γ = (7)

Heretn,γ is the shear rate on the twin system t and S is the

characteristic twin shear. At any particular increment, the

twinned volume fraction is compared with the threshold volume

fraction FT, which is a random number between 0 and 1. If the

twinned volume fraction gn is greater than the threshold volume

fraction FT, then the material point is reoriented along the most

predominant twin system.

Simulation Plane strain compression was simulated with a representative

volume element (RVE) with periodic boundary conditions.

Plane strain compression is an idealized model for the

deformation during rolling. The RVEs were generated through a

Voronoi tessellation with random seeds. Periodic boundary

conditions are used in order to minimize constraint effects;

displacement vectors of two equivalent points a and b are

coupled by the macroscopic deformation gradient ijF

( ) ( )a

0

b

0

a

0

b

0

ab

iijjijii xxxxFuu −−−=− , (8)

where xai0, x

bi0 indicate the position of a point pair in the non-

deformed configuration.

A finite element mesh of 90x90 elements with 100 grains was

used for the simulation. Each grain was given an initial

orientation so as to reflect a random texture.

Two cases were tested: a) Absence of pyramidal slip, b)

Presence of pyramidal slip. Case a) produces inhomogeneous

deformation with stress and strain localization in the

polycrystal. Case b) develops a more homogeneous deformation

with lower stresses. Sub-graining and grain curling are also

observed in case a). Texture evolution shows a basal texture

evolution in both cases, although case a) seems to produce a

much sharper texture than case b). The results of the

simulations are compared with experimental results from

literature [9].

i) ii)

Figure 1: Stress profile from simulation. i) Absence of pyramidal slip. ii) Presence of pyramidal slip References

[1] J. O enášek, M. Rodriguez Ripoll, S. M. Weygand and H.

Riedel: Multi-grain finite element model for studying the

wire drawing process, Computational Materials Science,

(In Press).

[2] Asaro, R.J. (1983): Crystal Plasticity, Journal of Applied Mech., 50, 921

[3] R.J. Asaro (1983): Micromechanics of Crystals and

Polycrystals, Adv. Appl. Mech., 23, 1

[4] Rice, J.R. (1971): Inelastic constitutive relations for solids,

J. Mech Phys. Solids, 19, 433

[5] Hill, R. and Rice, J.R. (1972): Constitutive analysis of

elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20, 401

[6] Y.G. Huang (1991): A User-material Subroutine Incorporating Single Crystal Plasticity in the ABAQUS Finite Element Program, Harvard Univ., Rep. MECH-178.

[7] Van Houtte, P. (1978): Simulation of rolling and shear

texture of brass, Acta metallurgica, 26, 591

[8] Tomé, C. N., Lebensohn, R. A. and Kocks, U. F. (1991): A

model for texture development dominated by deformation

twinning, Acta metallurgica, 39, 2667

[9] Gottstein, G. and Al. Samman, T. (2005): Texture

development in Mg and Mg alloy AZ31, Mat. Sci. Forum,

495-497, 623

h l

qet

Discrete Dislocation Dynamics Simulations of dislocation channelling effects on IG fracture

M.Razafindrazaka, D.Delafosse, [email protected]

Ecole Nationale Supérieure des Mines de Saint Etienne, Centre Science des Matériaux et des Structures, CNRS LPECM(UMR 5146), 158 cours Fauriel 42023 Saint Etienne, France

Core internal components made of austenitic stainless steel (e.g.: 316L) are liable to Irradiation-Assited Stress Corrosion Cracking (IASCC), which is a complex damage phenomenon involving irradiation effects on materials as well as corrosion/deformation/metallurgy interactions. Among other consequences of irradiation, irradiation defects such as faulted dislocation loops induce a strain localization in narrow slip bands, usually called “dislocation channels”. The specific contribution of dislocation channelling to the IASCC process is neither well modelled today, nor even well characterised experimentally. The interaction between grain boundaries and dislocation channels may cause a stress concentration promoting inter-granular fracture. In order to evaluate the way dislocation channelling influences inter-granular fracture, we develop a two-dimensional discrete dislocation dynamics simulation of crack tip plasticity that includes a model of dislocation channelling. The simulated configuration comprises two adjacent grains with a crack running at their interface. Each 10 to 20µm-wide grain has its own crystal lattice, simulated in 2D by three different slip systems. Dislocation motion takes place in a linear elastic medium, where interaction forces between defects is calculated using the theory of the linear elasticity. The faulted loops we focus on are simulated by obstacles randomly distributed throughout the grains and characterised by a stress opposed to dislocation motion. From atomistic simulations performed in the physics modelling sub-project, the interaction stress lies in the 0 to 1000 MPa range. However, the configurations that can be represented in 2D limit the magnitude of the interactions around 600 MPa. Nevertheless, first simulation results show that such interactions strongly impedes the dynamics of crack-tip shielding, resulting in an increased crack-tip Stress Intensity Factor (SIF). Following the approach of Roberts [1], the fracture criterion is met when this local SIF reaches the theoretical toughness in the absence of dislocation. Investigating various types of grain boundaries (symmetric, anti-symmetric tilt, general grain boundaries...) we observe that the grain orientation affects the time evolution of the local SIF through the spatial distribution of active slip lines at the crack tip.A suitable reproduction of the work-hardening in unirradiated 316L is a prerequisite for the introduction of irradiation effects on dislocation multiplication and motion. However, most of the elementary dislocation mechanisms responsible for work-hardening are inherently 3-dimensional. Following the approach of Gomez-Garcia et al. [2, 3] and Benzerga et al. [4], we add some elementary rules in our 2D simulation that allow to reproduce a realistic behaviour of non irradiated materials, up to strains of several per cents (this approach is often referred to as “2.5D”). Finally, by means of a simple model of obstacle sweeping, the behaviour of an irradiated 316L will be compared to that of a unirradiated 316L, thus enabling the assessment of the specific contribution of dislocation channelling to inter-granular fracture. The simulation tools (codes) are developed in a way that will allow for the introduction of physico-chemical models of corrosion-deformation interactions in future developments.

References

h Û

1. Roberts, S. G. 2000. Modelling brittle - ductile transitions. In Multiscale Phenomena in Plasticity : From Experiments to Phenomenology, Modelling and Materials Engineering, J. Lépinoux, D. Mazière, P. V, and G. Saada eds. Kluwer Academic Publisher, 349-364.

2. Gomez-Garcia, D., B. Devincre, and L. P. Kubin. 2000. Forest hardening and boundary conditions in 2-D simulations of dislocations dynamics. Materials Research Society Symposium - Proceedings 578: 131-136. 3. Gomez-Garcia, D., B. Devincre, and L. P. Kubin. 2006. Dislocation patterns and the similitude principle: 2.5D mesoscale simulations. Physical Review Letters 96: 1-4. 4. Benzerga, A. A., A. Needleman, Y. Bréchet, and E. Van der Giessen. 2004. Incorporating three-dimensional mechanisms into two-dimensional dislocation dynamics. Modelling and Simulation in Materials Science and

Engineering 12: 159-196.

h ¤

PRQTSUSWVYX[Z

@ ?;>=aGI¡¢;£DFGI@CB ;BN¤ ¤N; ;¥_ ^

h 0

h O

sed

Incremental Modeling of Damage in Brittle Materials

Slav Dimitrova

aChair of Continuum Mechanics, Institute of Engineering Mechanics, Department of

Mechanical Engineering, University of Karlsruhe, Kaiserstrasse 10/Wilhelm-Nusselt-Weg 4,

D-76131 Karlsruhe, Germany; E-mail: [email protected]

Abstract. The study proposes a novel approach for definition and analysis of damage in brittleand quasibrittle materials at infinitesimal strains based on finite-step-size reduced energy mini-mization principle (a concept introduced in Mielke et al. 2002, Carstensen et al. 2002). Thepoint of departure is continuum description of the strain driven evolution of material microstruc-ture in terms of internal variables. In variance with (Miehe et al. 2002, Miehe et al. 2004)the local constitutive response then is defined in terms of incremental minimization principlefor internal variables with dissipation accounted by an exact penalty function depending onthe time-rate of microstructural evolution. In a finite-step-sized incremental setting this varia-tional problem defines an inelastic potential for stresses. The existence of the inelastic potentialallows further to recast the incremental boundary value problem of damage mechanics into avariational principle for minimization of the reduced incremental energy in standard dissipa-tive solids. This conceptual line is applied to the numerical modeling of scalar and orthotropicdamage. For these canonical model problems we outline details of the constitutive incrementalvariational formulation, and integration algorithms for stress update. To ensure the numericalstability of the integration algorithm we introduce an intrinsic material length scale motivatedby Γ-convergence result in (Ambrosio & Tortorelli 1990).Key words: inelasticity, energy-relaxation methods, damage, Γ-convergence, brittle fracture.

REFERENCES

Ambrosio, L. & Tortorelli, V. (1990), ‘Approximation of functionals depending on jumps byelliptic functionals via Γ-convergence’, Communications on Pure and Applied Mathematics

43(8), 999–1036.Carstensen, C., Hackl, K. & Mielke, A. (2002), ‘Non-convex potentials and microstructures

in finite-strain plasticity’, Proceedings of the Royal Society of London. Series A. Mathemat-

ical, Physical and Engineering Sciences 458(2018), 299–317.Miehe, C., Lambrecht, M. & Guerses, E. (2004), ‘Analysis of material instabilities in inelastic

solids by incremental energy minimization and relaxation methods: evolving deformationmicrostructures in finite plasticity’, Journal of the Mechanics and Physics of Solids 52, 2725–2769.

Miehe, C., Schotte, J. & Lambrecht, M. (2002), ‘Homogenization of inelastic solid materialsat finite strains based on incremental minimization principles. application to the textureanalysis of polycrystals’, Journal of the Mechanics and Physics of Solids 50(10), 2123–2167.

Mielke, A., Theil, F. & Levitas, V. (2002), ‘A variational formulation of rate-independentphase transformations using an extremum principle’, Archive for Rational Mechanics and

Analysis 162, 137–177.

h b

hc

seji17th International Workshop on Computational Mechanics of Materials

IWCMM 17, August 22-24, 2007 Paris, France

TOWARDS A ROBUST NON LOCAL NUMERICAL ANALYSIS OF DUCTILE FRACTURE

R. Bargellini a, J. Besson b, E. Lorentz a and S. Michel-Ponnelle a a Laboratoire de Mécanique des Structures Industrielles Durables, UMR CNRS/EDF 2832,

1 av. Général de Gaulle, 92141 Clamart Cedex, France. b Mines Paris ParisTech, Centre des Matériaux Pierre-Marie Fourt, UMR CNRS 7633,

BP87, 91003 Evry Cedex, France.

Keywords: Enhanced Finite Element, Damage Mechanics, Ductile Fracture, Non local model, Plastic Incompressibility

1 Introduction

During the last decade, finite element based tools have been developed and used for the predictive simulation of a wide range of industrial situations, for which full-scale experimental approaches are not practicable. Ductile fracture remains a pivotal topic, due to its numerous applications. The aim of this note is to present a finite element method to predict crack nucleation and growth in a ductile material, namely A508 steel. As resistance approaches (see e.g. [1]) are restricted to the extension of a pre-existing crack and cohesive zone models (see e.g. [2] and [3]) often lead to an over-estimation of the cracked area, our approach enters the field of Continuum Damage Mechanics (see e.g. [4] and [5]), which relies on specific constitutive models to describe damage processes.

Among the main difficulties encountered when simulating ductile fracture, one may mention: (i) volumetric locking phenomena, caused by the constraint on the volumetric strain (guiding the damage growth); (ii) spurious mesh dependency, due to the strain localization in softening constitutive laws [6]. Consequently, a specific finite element formulation, adapted to quasi-incompressible behavior in finite strain, is first considered. Then, the use of a non-local damage model permits to unravel mesh dependency. 2 An quasi-incompressible finite element formulation The first crucial task is the development of appropriate finite elements which can model the quasi-incompressible nature of the plastic flow without exhibiting volumetric locking behavior. This phenomenon is classically observed with von Mises plasticity where the strain should remain isochoric, and more generally when a constraint exists on the plastic volumetric flow. In ductile damage behavior, this volumetric strain is related to voids nucleation and growth, and is constrained by constitutive models (see e.g. [7] and [8]). Volumetric locking phenomena is consequently expected. As it is thought to be caused by kinematics constraints, two major solutions rise; the first one consists in relaxing these kinematics constraints by reducing the number of integration points (low-order elements); however, it is often restricted to quadrilateral and hexahedral elements; the second one is to enrich the kinematics (enhanced finite elements). Due to the high strain level often encountered in ductile fracture, the possibility of remeshing should be kept in mind to maintain an appropriate quality throughout the entire simulation. The lack of robustness of currently available hexahedral remeshing tools leads to prefer tetrahedral (and consequently enhanced) finite elements.

Following [5], the volumetric strain is here treated as a new unknown. Its relation to the displacement field is weakly enforced by means of a Lagrange multiplier, namely the pressure P. A three field formulation of the Hu-Washizu type is thus obtained. Calling J the volumetric strain,

Lδ the Eulerian deformation variation, extW the potential of external forces, the Kirchhoff stress and D

its deviatoric part, the variational formulation of the problem reads:

( ) ( ) ( ) extWdGJPPtrGPPG δδδδδδδ =Ω

−+

−++∀ Ω0

0ln3

:,,, ILL dD (1)

h g

17th International Workshop on Computational Mechanics of Materials IWCMM 17, August 22-24, 2007

Paris, France Note that G actually corresponds to the logarithmic volumetric strain instead of the volumetric strain itself in [5]. This formulation is validated on a severely notched CT specimen in a von Mises material. 3 Non local damage formulation The starting constitutive model used is the Rousselier model [8]. As the considered material can exhibit large strain, it is first rewritten in a finite strain formulation ([9]); in addition, it is modified to enter the General Standard Material framework (GSM, [10]) to ensure robustness. The porosity growth law now depends on the total volumetric strain instead on the plastic volumetric strain in its initial formulation; this constraint on the volumetric strain justifies the use of the formulation presented in Section 2.

As it represents a softening behavior, strain localization, which classically leads to a spurious mesh dependency, is expected to occur. To control this phenomenon, a coupling between material points is introduced. As the volumetric strain drives the damage mechanism, a control of the volumetric strain gradient seems sufficient to avoid spurious localization. Variational formulation (1) is consequently modified into:

( ) ( ) ( ) extWdGGcGJPPtrGPPG δδδδδδδδ =Ω

∂∂

∂∂+−+

−++∀ Ω0

0.ln3

:,,,XX

ILL dD (2)

where c is a material parameter, introducing implicitly an internal length. Thanks to the finite elements introduced in Section 2, a spatial discretisation of the field G is

already available; the non local model does consequently not introduce higher complexity into the computing procedure, which represents an advantage of the proposed approach.

The presented formulation is tested, for both quadrilateral and triangular elements, in different classical situations, such as axisymmetric tensile or CT severely notched specimens. References [1] Xia L., Shih C.F., Hutchinson J.W., A computational approach to ductile crack growth under large scale yielding conditions, J. Mech. Phys. Solids 43 (3), 389-413, 1995. [2] Tvergaard V., Crack growth predictions by cohesive zone model for ductile fracture, J. Mech. Phys. Solids 49, 2191-2207, 2001. [3] Anvari M., Scheider I., Thaulow C., Simulation of dynamic ductile crack growth using strain-rate and triaxiality-dependent cohesive elements, Engineering Fracture Mechanics 73, 2210-2228, 2006. [4] Andrade Pires F.M., de Souza Neto E.A., Owen D.R.J., On the finite element prediction of damage growth and fracture initiation in finitely deforming ductile materials, Comput. Methods Appl. Mech. Engrg. 193, 5223-5226, 2004. [5] Taylor R.L., A mixed-enhanced formulation for tetrahedral finite elements, Int. J. Num. Engng. 47, 205-227, 2000. [6] Forest S., Lorentz E., Localization Phenomena and regularization Methods, In Local Approach to Fracture, Ed. J. Besson, Presse de l’Ecole des Mines, Paris. [7] Gurson A.L., Continuum Theory of Ductile Rupture of Void Growth: Part I – Yield Criterion and Flow Rules for Porous Ductile Media, J. Eng. Mater. Tech. 99, 2-15, 1977. [8] Rousselier G., Dissipation in porous metal plasticity and ductile fracture, J. Mech. Phys. Solids 49, 1727-1746, 2001. [9] Simo J.C., Miehe C., Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation, Comp. Meth. Appl. Mech. Eng. 98, 416104, 1992. [10] Germain P., Nguyen Q.S., Suquet P., Continuum Thermodynamics, J. Appl. Mech. 50, 1010-1020, 1983.

hh

senm

STRAIN LOCALIZATION ANALYSIS USING A MULTISCALE MODEL Gérald Franz1, Farid Abed-Meraim1, Tarak Ben Zineb2, Xavier Lemoine3, Marcel Berveiller1 1 LPMM, Laboratoire de Physique et Mécanique des Matériaux, UMR CNRS 7554, ENSAM 4 rue Augustin Fresnel, 57078 Metz Cedex 3, France 2 LEMTA, Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée, UMR CNRS 7563, ESSTIN UHP 2 rue Jean Lamour, 54519 Vandoeuvre-lès-Nancy, France 3 Centre Automobile Produit, Arcelor Research S.A., voie Romaine B.P. 30320, 57283 Maizières-lès-Metz, France

Abstract

The development of a relevant constitutive model adapted to sheet metal forming simulations requires an accurate description of the most important sources of anisotropy, i.e. the slip processes, the intragranular substructure changes and the texture development. During plastic deformation of thin metallic sheets, strain-path changes often occur in the material resulting in macroscopic effects. These softening/hardening effects must be correctly predicted because they can significantly influence the strain distribution and may lead to flow localization, shear bands and even material failure. The main origin of these effects is related to the intragranular microstructure evolution. This implies that an accurate description of the dislocation patterning during monotonic or complex strain-paths is needed to lead to a reliable constitutive model. First, the behaviour at the mesoscopic scale (which is the one of the grain or a single crystal) is modelled by the micromechanical law written within the large strain framework [1]. Hardening is taking into account by a matrix whose internal variables are the mean dislocation densities on each slip system. This crystal plasticity based model is implemented into a large strain self-consistent scheme, leading to the multiscale model which achieves, for each grain, the calculation of plastic slip activity, with help of regularized formulation drawn form viscoplasticity [2]. An improvement of this model is suggested with the introduction of intragranular microstructure description. The substructure of a grain is described taking into account the experimental observations as stress-strain curves and TEM micrographs. Following Peeters approach [3], three local dislocations densities, introduced as internal variables in the multiscale model, allow representing the spatially heterogeneous distributions of dislocations inside the grain (Figure 1). Rate equations, based on the consideration of associated creation, storage and annihilation, are used to describe the dislocation cells evolution. The coupling of the substructure to the critical shear stresses is performed thanks to the concepts of isotropic hardening, latent hardening and polarity.

Dislocation wall ρ(wd)

ρ(wd)

Cells ρ

Dislocation wall ρ(wd)

ρ(wd)

Cells ρ

FIGURE 1. Schematic representation of the intragranular microstructure [3].

h k

Moreover, a ductility loss criterion, first introduced by Rice [4], based on the ellipticity loss of the elastic-plastic tangent modulus, is used in these two models to plot Ellipticity Loss Diagrams (ELD). Qualitative comparisons are made with experimental Forming Limit Diagrams (FLD) for ferritic steel involving simple and complex loading paths. In particular, it is shown that numerical ELD have a shape close to experimental FLD and reproduce qualitatively the effects due to complex loading paths. Figure 2 gives an example of ELD plotted with the first model, without taking account of intragranular microstructure. This ELD is compared with experimental data and a FLD plotted with an Arcelor model [5]. The impact of intragranular microstructure on strain localization is studied thanks to comparisons between ELD plotted with the two models.

FIGURE 2. Comparison between direct experimental FLD and numericals ELD for a ferritic steel.

Reference

1. J. P. Lorrain, " Critère de ductilité basé sur la perte d’ellipticité du module tangent élastoplastique déduit d’un modèle autocohérent ", PhD. Thesis, ENSAM Metz, 2005

2. J. P. Lorrain, T. Ben Zineb, F. Abed-Meraim, M. Berveiller, "BCC single crystal ductility loss modeling", International Journal of Forming Processes, 8(2), pp. 135-158, 2005

3. B. Peeters, "Multiscale modelling of the induced plastic anisotropy in IF steel during sheet forming", Ph.D. Thesis, Katholieke Universiteit Leuven, 2002

4. J. R. Rice, "The localization of plastic deformation", 14th International Congress of Theorical and Applied Mechanics, pp. 207-220, 1976.

5. X. Lemoine, F. Cayssials, "Predictive model of FLC (Arcelor model) upgraded to UHSS Steels", International Deep-Drawing Research Group (IDDRG) Congress, Besançon, 20-22 June 2005.

kl

PRQTSUSWVYX[Z ¦

§ GI¡ ^©¨ ¤ ^ x ^ BN¤ ^ BED ^ ? y ;>BNGa?;>= xN:F@>x ^ :TD<G ^ M

k5Û

k?¤

tªed

A non-linear theory of multiple slip in continuum

dislocation dynamics

Thomas Hochrainer1,2, Peter Gumbsch1,2, Stefan Sandfeld1,

Michael Zaiser3

1Universitat Karlsruhe, Institut fur die Zuverlassigkeit von Bauteilen und

Systemen, Kaiserstr. 12, 76131 Karlsruhe, Germany

2Fraunhofer Institut fur Werkstoffmechanik IWM, Wohlerstr. 11, 79108

Freiburg, Germany

3The University of Edinburgh, Centre for Materials Science and

Engineering, King’s Buildings, Sanderson Building, Edinburgh EH93JL,

United Kingdom

Abstract

Crystal plasticity is nowadays recognised as the result of the motion and

complex and effectively non-linear interactions of dislocations. Strain hard-

ening, for example, results from the increasing dislocation density, in that the

dislocations mutually obstruct each others motion. The individual strength

of dislocations acting as obstacles to dislocations on other slip systems has

therefore been the subject of intensive research both theoretically and by

means of atomistic and discrete dislocation simulations. There is, however,

still a major gap between the results obtained by the microscopic and meso-

scopic simulations and current continuum crystal plasticity models. Espe-

cially the character (line direction) of the discloations and their average ve-

locity which both affect the frequency of dislocation cutting events are not

taken into account in any available model of crystal plasticity.

A recently defined higher dimensional dislocation density measure ac-

counts for the line like nature of all dislocations in reasonably general dislo-

cation distributions [1]. This measure is called the dislocation density tensor

of second order [2] and is in fact a direct generalisation of Kroner’s dislo-

cation density tensor. The evolution equation for this tensor is given by a

conservation law obtained by rigorous averaging procedures and thus is a

continuum version of discrete dislocation dynamics.

In the current work we use the second order dislocation density tensor

to obtain the rate of dislocation cutting events per volume. Furthermore we

introduce the concept of a mean free surface area swept by dislocations cross-

k0

ing a possibly moving field of forest dislocations. The mean free surface area

and the closely related mean free segment length are shown to be dynamic

quantities depending on the relative orientation, velocity and direction of

motion of the dislocations. If the mean dislocation velocity itself is taken to

be dependend on the mean free segment length this leads to a non-linear the-

ory of dynamic forest hardening in multiple slip configurations. This theory

anticipates that multiple slip deformation, as a sort of dynamic equilibrium

resulting from the dislocation distributions and velocities on the active and

passive slip systems, might be effectively unpredictable.

Acknowledgements: We acknowledge support of the of the European

Commission under contract MRTN-CT-2003-504634 (SizeDepEn).

References

[1] T. Hochrainer, M. Zaiser, P, Gumbsch. A three-dimensional continuum

theory of dislocation systems: kinematics and mean field formulation.

Phil. Mag., 87 (8-9):1261-1282, 2007

[2] T. Hochrainer. Evolving systems of curved dislocations: Mathematical

foundations of a statistical theory. Thesis, Universitat Karlsruhe (TH),

2006

k O

tªeji

Simulation of the strain path change effect on drawn tungsten wires

Manel Rodriguez Ripolla*, Hermann Riedela, Samuel Forestb

a Fraunhofer Institute for Mechanics of Materials IWM, Woehlerstraße 11, 79108 Freiburg, Germany.

b Ecole des Mines de Paris/CNRS, Centre des Matériaux/UMR 7633, BP 87, 91003 Evry, France.

* Corresponding Author. E-mail address: [email protected]

Summary: The aim of this work is to study the hardening behaviour of tungsten wires tested under uniaxial loading. The severe

plastic deformation produced during the drawing process results in a strong asymmetry between tension and compression. During

compression of a drawn wire, the effect of the strain path change produces a transient strain softening. Simulations using a texture

self-consistent model can successfully pick up this effect. In addition, the deformation of a representative volume element

containing typically hundred grains is modelled using the finite element method. Each of the grains obeys the constitutive

equations of crystal plasticity. The causes of the softening are analyzed and the results of both models are discussed.

Introduction

Tungsten wires used as bulb filaments are obtained by

reducing their diameter in successive wire drawing steps.

During the drawing process the wires undergo severe plastic

deformation that dramatically modifies their microstructure.

Grains elongate in the drawing direction and establish a sharp

<110> fibre texture. In the cross section, initially equiaxed

grains bend around each other, a process which is called

curling [1] leading to a particular pattern known as Van Gogh sky structures.

One of the consequences of these modifications at the

macroscopic level is the effect of loading path changes on the

hardening behaviour. When a material deformed by wire

drawing (axisymmetric elongation) is tested under simple

tension, the strain path is maintained. Combining wire

drawing with tensile tests is a common practice for obtaining

the hardening behaviour of materials under large strains.

However, when the mode of deformation is switched from

elongation to compression, the hardening behaviour of the

material radically changes. A transient softening stage is

observed in drawn wires tested under compression. This

phenomenon is referred to as strain path change in the

literature [2, 3] and is attributed to the formation of dislocation

substructures and the activation of new slip systems on load

reversal.

Experimental results

Figure 1a shows the hardening behaviour of three tungsten

wires with different diameters tested under tension at a

temperature comparable to the one used during drawing (T «

0.3Tm, being Tm the melting temperature). The flow curves of

the single wires provide information only for small strains due

to early necking of the specimens. The flow curve of tungsten

is obtained by taking the envelope of the individual curves.

The flow curve shows a characteristic, approximately linear

hardening behaviour observed for bcc metals subjected to

axisymmetric elongation [4].

a)

b)

Figure 1: Uniaxial tests on tungsten wires tested at T « 0.3Tm.

a) Tension, b) compression.

Compression tests on tungsten wires (see fig. 1b) exhibit the

asymmetric behaviour of heavily deformed metals. The wires

kb

do not show a constant hardening rate. Rather, a transient

softening that increases with increasing pre-deformation is

observed.

Simulation using a self-consistent model

Our first attempt to reproduce the experimental results uses a

viscoplastic self-consistent (VPSC) model proposed by

Lebensohn and Tomé [5]. In this model, each grain is treated

as an ellipsoidal inclusion embedded in a matrix representing

the average properties of the remaining grains. The interaction

between the grain and the matrix is solved using the Eshelby

inclusion formalism. The model is able to predict the

macroscopic material response and texture development. It

also accounts partially for grain shape evolution by keeping

track of the aspect ratio of the ellipsoid during the deformation

process. Homogenization of the effective properties of the

aggregate allows for calculations using a large number of

grains.

Recently, the model has been successfully applied by other

authors [6, 7] for modelling the behaviour of materials after

severe plastic deformation by equal channel angular extrusion.

Our simulations show that VPSC is also able to qualitatively

reproduce the uniaxial hardening behaviour of drawn tungsten

wires (fig. 2). This implies that hardening of certain slip

systems during forming and the activity of new slip systems

during compression plays a role in the softening observed

during strain path change.

Figure 2: Simulation of tension and compression tests of

drawn tungsten wires using VPSC.

Finite element simulation using a representative volume element

Finite element simulations using representative volume

elements (RVEs) provide a detailed insight into the evolution

of the microstructure during deformation. Along with crystal

plasticity as a constitutive model [8], they are a powerful tool

for predicting the macroscopic response of the material,

texture evolution and morphological changes at the grain

level.

A novel approach is to use the RVE technique to study the

effect of strain path changes. The simulation of the drawing

process using a RVE accounts for texture development and

grain curling (fig. 3) [9] and also keeps track of the inter- und

intra-granular stresses [10].

a) b)

Figure 3: Grain curling in a tungsten wire after drawing.

a) Structure observed in the cross section, b) simulated

microstructure using a RVE

Special attention is paid to the activity of the slip systems

during the forming process and the subsequent uniaxial

deformation. The importance of latent hardening, as well as

the influence of the material parameters on the amount of

softening produced during strain path change is investigated. References

[1] W.F. Hosford, Trans. AIME, 230 (1964) 12.

[2] E.F. Rauch, J.-H. Schmitt, Mater. Sci. Eng. A 113 (1989)

441.

[3] N.A. Sakharova, J.V. Fernandes, Mater. Chem. Phys. 98

(2006) 44.

[4] J. Gil Sevillano, P. Van Houtte and E. Aernoudt, Prog. Mater. Sci. 25 (1981) 69.

[5] R.A. Lebensohn, C.N. Tomé, Acta Metall. Mater. 41

(1993) 2611.

[6] I.J. Beyerlein, S. Li, D.J. Alexander, Mater. Sci. Eng. A

410-411 (2005) 201.

[7] S. Li, Scripta Mater. 56 (2007) 445.

[8] R.J. Asaro, Adv. Appl. Mech. 23 (1983) 1.

[9] J. O ¬ enášek, M. Rodriguez Ripoll, S. M. Weygand and H.

Riedel, Comp. Mat. Sci. 39, 1, (2007) 23.

[10] J. Gil Sevillano, D. González, JM. Martínez-Esnaola,

Heterogeneous deformation and internal stresses

developed in BCC wires by axisymmetric elongation,

Materials Science Forum (2006).

k c

tªenm

¯®±°³²WÍ´µá¶f°³²²·¹¸·¹º »¼ ½¹¸Í¾¿À¶ÂÁÃ®a¸¿Ä·T´ÅÆÇpÈÉËÊ[ÌªÍÎ&Ï¯ÐÑ7ÒÓÔ&Õ,Ör×¯ÒØ+ÖÙËÚ ÛFÔÖrÒÕ,ÜÝÔËÞàßªáãâ+äåß,ÜàæßçéèãêËÖ,ëìêìßíØÜîëËÒÕ,ßÜîÖjäTïðñÔËÓÒßòÖáfóôïÙõTö+÷øç$ù.úåùûïÂð ñÔ&ÓÒß,Öçéüð ØýìÔËÕ,äÌþrÿÐ<Ì&ÍØ ß,ÖrÜîÖð ÖÒ Ú ÙìÕªâÒÙËÕÒÖÜàæÔ&Þaáãâ+äåß,ÜàæßçéèãêËÖ,ëìêËß7íØÜîëËÒÕ,ßÜîÖjäTïðñÔËÓÒß,ÖáfóôïÂÙõTö+÷øç$ù.úåùûïÂð ñÔ&ÓÒß,Öçéüð ØýìÔËÕ,äþÇ ÍIÅrÆ ÉËÊÎ&È ÇÑUÒÓÔ&Õ,Ör×¯ÒØ+ÖÙËÚ ÛFÔÖrÒÕ,ÜÝÔËÞàßªáãâ+äåß,ÜàæßçéèãêËÖ,ëìêìßíØÜîëËÒÕ,ßÜîÖjäTïðñÔËÓÒßòÖáfóôïÙõTö+÷øç$ù.úåùûïÂð ñÔ&ÓÒß,Öçéüð ØýìÔËÕ,äjÖÃÜàßÃÔÀÞàÙìØýß,ÖÔ&ØñÜîØý¹æâÔ&ÞàÞàÒØ ýìÒÄÙ&Ú ñ ÜàßÞàÙåæÔÖrÜàÙìØwÖrâ ÒÙìÕòä ÖÙ¹ñÒÖrÒÕ,×ÀÜàØ Ò Örâ ÒßòÖrÕÒßßß,ÖÔ&ÖÒÙËÚÂÔ¹ñÜàß,ÞàÙåæÔ&ÖÒñÙåñ äFßð,ÒæÖÄÙ&ÚÒõåÖrÒÕØÔËÞ ÖÕrÔËæÖrÜàÙËØÙËÕÃÖâÒ ýìÒØÒÕrÔËÞãöìÑæÔËß,Ò Örâ Ò Ó ÕÙ ÞàÒ×µÜàßßòÖrÜàÞàÞ$ðØ ßÙìÞîëËÒñð ÖÚ ÙìÕ×¯ÙåñÒÞIß,äåßòÖrÒ×¯ß7æÙËØ+ÖÔËÜàØ ÜàØýß,ÖÕrÔËÜàýËâ+ÖÓÔËÕÔËÞàÞàÒÞãÒñ ýìÒÀñÜàß,ÞàÙøæÔ&ÖrÜîÙìØß ÔwæÙìØ+ÖÜàØYðð × ÖâÒÙËÕ,ä ÙËÚªñ ÜàßÞàÙåæÔÖrÜàÙìØ Üàß ÓÕ,ÙìÓÙìß,Òñ ÖÙâÔËØ ñÞàÒ ÖâÒTß,ÖÕÒß,ß¯ß,æÕÒÒØ ÜàØý æÔ&ðßÒñ +ä Örâ Ò¹ÜàØñ ðæÒñAýìÒÙì×¯ÒÖrÕ,ÜàæÔËÞàÞîä5ØÒæÒß,ßrÔËÕòäñÜàßÞàÙåæÔ&ÖrÜàÙËØß!"$#7Ñ&%' ÖãÜàß³ß,âÙ(Ø¯ÖrâÔ&ÖãÔËØ Ò')ÒæÖÜîëìÒÂÚ ÕÒÒªÒØ ÒÕ,ýËäÄÚ ðØ æÖrÜîÙìØÔ&ÞåÙËÚéÖrâ Òß,ÖrÕÒßßWÚ ð ØæÖÜàÙìØ

χçøÖrâ ÒUÖrÙËÖrÔËÞ$ÔËØñ¹ýËÒÙË×ÀÒÖrÕÜàæÔËÞàÞîäØÒæÒßßrÔ&Õ,äTñÜàß,ÞàÙåæÔ&ÖÜàÙìØñ ÒØ ßÜîÖrÜàÒßæÔËØ*ÒÃÜàØ+ÖrÕÙåñ ðæÒñ+(Wâ Üàæâ ÜàßÂ×¯ÜàØÜà×¯Ü-,Òñ.(ÜîÖrâTÕÒßÓÒæÖÖrÙ Örâ Ò/"0#7Ñ ñ ÒØ ßÜîÖjäTÔËØ ñ

χÔ&ÖÒ21Yð ÜàÞàÜ-Õ,Üàð×CæÙìØ43ýËðÕrÔÖrÜàÙìØ5

0

0.005

0.01

0.015

0.02

0.025

0.03

0 5 10 15 20

κ(0,

y)/k

02

k0 y ÜàýËðÕÒ5ù6*"$#7Ñ ÔËÞàÙËØý<Örâ Ò

yÜàØ ñð æÒñ +äAÔ5ÒõåÖrÕÔ ñ ÜàßÞàÙåæÔ&ÖrÜàÙËØ7ªâÒTÚð ÞàÞÂÞàÜàØÒæÙìÕ,ÕÒß,ÓÙìØñ ß ÖrÙ5Örâ Ò ÖrâÒÙìÕÒÖrÜàæÔËÞÓÕ,Òñ ÜàæÖrÜàÙËØ8(âÜàÞàÒ¯ÖrâÒ¹æÜàÕæÞàÒßÔËÕ,Ò ÙøÖÔËÜàØ Òñ9+äñÜàßæÕÒÖÒ ñ ÜàßÞàÙåæÔÖrÜàÙìØÀñ äåØÔË×¯ÜàæßßÜà×(ðÞÝÔÖrÜàÙËØß2

:|ÜîÖrâ¯ÖrâÜîßÓâÔ&ßÒ;3ÒÞàñ Ô&ÓÓ ÕÙ+Ô&æâ ÖrâÒ7ß,æÕÒÒØ ÜàØý Ù&Ú±Örâ ÒUß,ÖrÕ,Òßß<3ÒÞàñ Ù&Ú Ô ß,ÜàØýËÞàÒñÜàßÞàÙåæÔ&ÖrÜàÙËØ=!³Üàý>Âù%ªÔ&Øñ Ôñ ÜàßÞàÙåæÔÖrÜàÙËØ?(ÂÔËÞàÞ±ÔËÕ,Ò ÔËØÔËÞîä,Òß2

k g

@BACDAEFAHG+I<AKJLNM2O/P0QSRT.UWVYXZP&[RQS\[]<PX_^KR4`bac]-a$d;Xfe;g2hi[gWac`bQcgbg2j>]kj>\*Rlm>]kacn-RòUpc]-Rja2Xqsri[a2tu g'vwtyx_g'pcpzgbQca|y~M+i

LDO/P&QcRT.UVYtP[RQc\[]PXy^|R`bac]-a;d;Xwe|[j>UT]-`2a|Rls`bRiUQcacg/\QzU]-jg2m*m>]kacn-RòUpc]-Rjmg2j>aS]kpz]-gbalQcRTUj+gbg2`'pz]kvg$lQcgbg$gbj>g2QS\[Xqsr]-ntyU\>t=MMiXit

k h

tªepo

Simulated low-angle tilt grain boundaries and their response to

stress

Y. CHENG†, D. WEYGAND†, P. GUMBSCH†‡ ∗

†IZBS, Universitat Karlsruhe (TH), 76131 Karlsruhe, Germany‡Fraunhofer-Institut fur Werkstoffmechanik, 79108 Freiburg, Germany

March 14, 2007 abstract to IWCMM17-Paris

The atomistic structures and energies of the low-angle tilt grain boundaries in bcc tungsten and

fcc aluminum are simulated by molecular statics. The results show that the characters of dislocations

constructing grain boundaries and the grain boundary energies are determined by the tilt axis, the

misfit angle between the two grains and the orientation of the grain boundary plane. Comparisons

are made with the high resolution electron microscopic observations and the dislocation model for

grain boundaries which is restricted to simple cubic lattices. The response of the grain boundaries to

applied stresses are simulated. It is find that the grain boundary may move under stress, depending

on the characters and the arrangement of the grain boundary dislocations.

∗Email: [email protected]

kk

Ûll

tªejq

! "#$#%$& ' $% % $ (#%) %*# $ %'$# $ %$ (#%+,-)

! "#$#%

.#/ #012 2/113 4+ 5". (#%6

& ' $% % $ (#%) %*# $ %'$# $ %$ (#%+,-)

' #$ %$ $6$72 ( $ ##$( $% % $ $ 36*$2 $# 8# $ $$ $ 9 6): ;7 $(# $< =3> $7 $$%$ 7 # # $$# ? # %#$ <=3/ # $ % # $ %7 ( % % $# 8# 3

># $ $ %($ $ ((#$ % #$$ <=3/$$$ $ % 3/ 27#%#($ $ $ # 7$ 7 %$ $$$ # 3 ( # $ %7 ($ #$3

< =/36 !!3@3!"549 ++4;3<=/36.3.3& 2$3A " ##!$B 4 955;3<=>3 3A $36#(% ##$!$& !!# 3%3 C 9554;3

Ûl5Û

. %# D> # $ %# $ %B#(3@$%(#$$$ # 7 ( -23

. %#D# $ $ # #$$ $ $# %# % 3E# % $ $#$ # 3

Ûl?¤

tªens

Formulation of a Multiple-Slip Nonlocal

Plasticity Theory Based on a Statistical

Approach

Surachate Limkumnerd, Erik van der Giessen

Abstract

Due to recent successes of a statistical–based nonlocal continuumcrystal plasticity theory for single-glide in explaining various aspectssuch as dislocation patterning and dependent plasticity, several at-tempts have been made to extend the theory to describe crystalswith multiple slips using ad-hoc phenomenological considerations. Wepresent here a mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations. The transport equations fortotal dislocation densities and geometrically necessary dislocation den-sities are derived by systematically incorporating couplings betweenslips driven by Peach–Koehler interactions due to both single and pairdislocation correlations. The evolution law is applied to the problem ofstress relaxation in single crystalline thin films on substrates subjectedto thermal loading, where results between the former single-slip con-tinuum theory and discrete dislocation simulations deviated the most.Implementation of finite element simulations and the results will bediscussed.

Ûl0

Ûl O

PRQTSUSWVYX[Z

JN=D<GIx y MwGa?M

Ûlb

Ûl c

ued

Deformation induced phase transformation and mechanical properties of dual phase steels

using combined phase field and finite element modelling

U. Prahl1, S. Benke2, N. Warnken2, V. Uthaisangsuk2, W. Bleck2

1Department of Ferrous Metallurgy, RWTH Aachen University, D-52072 Aachen

2ACCESS e. V. Materials and Processes, RWTH Aachen University, D-52072 Aachen

ABSTRACT

In order to achieve low cost solutions for crash resistant and lightweight structures inautomotive applications, modern multiphase steels will be increasingly applied innext future. Dual steels consisting of hard martensitic particles in a ductile ferriticmatrix offer high strength and deformability in parallel, and are a cost effective steelconcept using low alloying compositions. For the industrial production, twocontinuous concepts are implemented: hot rolling with controlled and acceleratedcooling or intercritical annealing with accelerated cooling of cold rolled sheet.

The current work deals with the dual phase steel production using hot rollingscheme, where the last hot rolling pass is performed in the temperature range of norecristallisation. Thus, stored dislocations and deformed austenite grain structurehave a strong influence on the ferrite transformation kinetics. In addition, after ferriteformation, the martensitic transformation of austenite during quenching causesplastic strain gradients in the ferritic grains.

A combined approach based on representative volume elements using finite elementcalculations for the deformation and phase field modelling for the transformation isused in order to model the hot rolling processing of dual phase steels. Mechanicalproperties can be directly deduced using numerical tension tests of thesemicrostructures based on empirical flow curves for the participating phases.

Ûl g

Ûl h

ueji

Remodeling and Growth of Living Tissue – A Multiphase TheoryT. Ricken1, J. Bluhm2∗,

1Computational Mechanics, 2Institute of Mechanics, University of Duisburg-Essen, D-45117 [email protected], [email protected]

Summary: A continuum triphasic model (solid with interstices filled with water containing nutrients) which is based on theTheory of Porous Media (TPM) is proposed for the phenomenological description of porous transversely isotropic biologicaltissue. Particular attention is directed to the description of stress, strain or nutrient driven growth and remodeling phenomena.Finally, we gain a coupled set of equations determining the solid motion, mixture temperature, inner pressure as well as solid andnutrient volume fractions. After presenting the developed framework, a representative numerical example is examined.

Basic Model

The investigated porous body consists of ϕS (solid) which issaturated by a fluid which also is composed of ϕL (liquid) andϕN (nutrients), see Fig.1. The volume fraction nα refer the vol-

Organic Solid (S)

Fluid (F)=Liquid (L) + Nutrient (N)

homogenization

of theconstituents

true structure smeared model

Figure 1: Homogenization

ume elements dvα of the constituents ϕα to the bulk volumeelement dv, viz.

nα(x, t) =dvα

dv,

κ∑

α=1

nα(x, t) =

κ∑

α=1

ρα

ραR= 1, (1)

where x is the position vector of the spatial point x in the actualplacement and t is the time. The partial density ρα = nα ραR

is related to the real density of the materials ραR involved viathe volume fractions nα, see (1)2. Moreover, we define the Ja-cobian Jα = detFα, where Fα = (∂xα)/(∂Xα) = Gradα χ

α

is the deformation gradient. The differential operator “Gradα”denotes a partial differentiation with respect to the referenceposition Xα. The local statements of the balance equations ofmass and momentum are given for the constituents ϕα by

(ρα)′α

+ ρα div x′

α= ρα ,

div Tα + ρα (b − x′′

α) + pα − ρα x′

α= o .

(2)

Therein,“div” denotes the spatial divergence operator, x′

αis the

velocity of the constituent ϕα, ρα represents the mass supplybetween the phases which has to conform to ρS + ρL + ρN = 0,Tα is the partial symmetric CAUCHY stress tensor, ρα b speci-fies the volume force while pα describes the interactions of theconstituents ϕα which are restricted to pS + pL + pN = o.The system is investigated under the condition of a materialincompressible components. The nutrient phase is assumed tobe contained in the liquid phase so that both phases are assignedwith the same velocity x′

Fand to the same pore pressure λ.

Moreover, we assume that the liquid phase is not involved inthe mass transition and we exclude accelerations.

Constitutive Modeling

The entropy inequality for the mixture yields the followingconstitutive relations for the partial Cauchy stress tensors

TS = − nS λ I − (nS)2 ρSR∂ψS

∂nSI + 2 ρS FS

∂ψS

∂CS

FT

S

= − nS λ I − (nS)2 ρSR∂ψS

∂nSI + TS

E,

TF = −(nL + nN )λ I = − nF λ I , nF = nL + nN

(3)of the constituents solid and fluid (ϕF = ϕL + ϕN) with therealistic fluid pressure λ, the tensor of identity I and the rightCauchy-Green tensor CS = FT

SFS which is related to the solid

phase, see [4].In many living tissues an anisotropic strain response whichis caused by the inner structure of the tissue can be ob-served. A convenient and common avenue to formulateanisotropic constitutive relations is the usage of the conceptof integrity bases which allows a coordinate-invariant formu-lation. Therefore, we introduce the so-called structural tensorM = a0 ⊗ a0 where a0 denotes the vector of the preferreddirection. The functional dependency of the stored energy issuggested with the usage of the principle invariants I1,3 andthe basic invariant J4 := tr[CS M] of the argument tensors(CS,M). And now the stored energy function can be writtenas ψS = [nS/(nS

0S)]n ψS

iso, neo (I1, I2, I3) + ψS

ti (J4). Therein,the term connected with the solid volume fraction nS describesthe change of solid rigidity relating to the reference solidvolume fraction nS

0Sat t = t0. The isotropic part of ψS

iso, neois

of Neo-Hookean type, viz.

ψSiso, neo

= ψSiso, neo

(I1, JS =√

I3)

=1

ρS

0S

λS1

2(log JS)2 − µS log JS +

1

2µS(I1 − 3),

(4)

where µS and λS are the macoscopic Lame constants. For thetransversely isotropic part of the Helmholtz free energy func-tion we choose

ψS

ti = ψS

ti(J4) =

1

2 ρS

0S

α1 (J4 − 1)α2 for J4 ≥ 1

0 for J4 < 1,(5)

where α1,2 ≥ 1 are parameters due to the stiffness of thepreferred direction a, see e.g. [1] or [3]. Due to the structureÛlk

of ψS both the invariance condition and the polyconvexitycondition are satisfied, see [2] and thence the solid effectiveCauchy stress tensor is

TS

E= (

nS

nS

0S

)n JS

ρS

ρS

0S

TS

E, iso, neo + JS

ρS

ρS

0S

TS

E, ti ,

TS

E, iso, neo=

1

JS

[ 2µS KS + λS ( log JS ) I ],

TS

E, ti=

1

JS

α1 α2 [ tr(CS M) − 1 ]α2−1 FS MFT

S .

(6)

Herein, the Karni-Reiner strain KS = 1/2 (BS − I) with theleft Cauchy-Green tensor BS = FS FT

Shas been used.

The motions of both the solid and the fluid are connected by theinteraction forces pF = −pS with pF = λ grad nF −SF wFS,where SF is obtained with

SF = αF0 [αF1 I + αF2 M ]−1,

(nF)2

αF0

= kS

0/µFR [nF

0S/nF]m

(7)neglecting the influence of the mass supply due to the momen-tum of the body. The permeability tensor SF describes the re-action between the fluid and solid phase in connection to theseepage velocity wFS = x′

F− x′

S, αF1,2 are the parameters

for the isotropic and transversely isotropic ratio of porosity re-spectively, m denotes a dimensionless material parameter, kS

0

is the initial intrinsic permeability and µFR denotes the shearviscosity of the fluid, see [4].We assume a mass exchange which acts between the solid andnutrient phase only. Using the ansatz ρS = −δL

ρ(ΨS − ΨN)

where Ψα = ψα + nα (λ/ρα + ∂ψα/∂nα) denotes the chemi-cal potential neglecting the influence of the velocity square, theentropy inequality is fulfilled if δL

ρ≥ 0, see [4]. We postulate

ρS as a function of the norm of the total Kirchhoff stresses τvMi,the solid Jacobian JS and the nutrient content nN viz.

ρS = ρSmax ρS

nN ρS

JSρS

τvMi,

ρS

nN = − exp [−κnN (nN)2 ] + 1 ,

ρS

JS= − exp [−κJS

(JS − 1)2 ] + 1 ,

ρSτvMi

= −2 exp [− log(2) τvMi/τvMi0] + 1 .

(8)

Under consideration of the above given constitutive relations aclosed calculation concept has been developed by using of thefinite element approximation, see [4] which enables the calcu-lation of the following example.

Example: Optimized organic structures

Biological material is able to adapt its internal structure for thegiven load case. A substantial characteristic is the optimizationof weight reduction, where only in those places structures aretrained which are directly relevant to the stability, for exam-ple the structure of our skeleton which was developed by theevolution to a system optimal for our purposes. This evolution-ary process is subjected to the following simulation. In order tostudy this phenomena more further, a simple cantilever with agiven load is regarded, see Fig. 2. In our example the materialdistribution tends to the state of optimal stress and low strainstate. As shown in the left column of Fig. 2 the red areas mark

Figure 2: Structure optimization of organic material, left: mate-rial, right: effective stress, load = 2 N, height = 7 mm, width =14 mm, µS = 1 ·104 N/mm2, λS = 0.0 N/mm2, kF

0S= 8.3 · 102

mm/d, γFR = 1.0 ·104 = N/mm3, nS

0S= 0.5, nL

0S= 0.05, nN

0S=

0.45, ρSR

0S= 0.2 · 10−3 g/mm

3, ρLR

0S= 1.0 · 10−3 g/mm

3, ρNR

0S

= 0.2 · 10−3 = g/mm3, αF1 = 1.0, αF2 = 0.0, m = 0, ρS

max = 1

kg/d mm3, κnN 5.0, κJS

= 2.0·106, τvMi0 = 10.0 N/mm2, d =virtual time step

the places with a high material content and the blue ranges witha low one. In the right column of Fig. 2 the pertinent change ofthe effective stress is represented. It can be recognized that theinner structure of the cantilever adjusts itself to the given loadso that an optimal structure is finally formed. In the case ofa change of the load case the trained structure would changeagain as well. With this simulation it is possible to optimizetechnical products, e.g. the basic structure of an automobile.

References[1] REESE, S., RAIBLE, T., WRIGGERS, P. [2001], “Finite

element modelling of orthotropic material behaviour inpneumatic membranes.”, International Jornal of Solidsand Structures 38, 9525-9544.

[2] SCHRODER, J., NEFF, P. [2003], “Invariant Formulationof Hyperelastic Transverse Isotropy Based on PolyconvexFree Energy Functions.”, International Journal of Solidsand Structures 40, 401-445.

[3] BALZANI, D., NEFF, P., SCHRODER, J., HOLZAPFEL,G.A. [2006], “A Polyconvex Framework for Soft Biolog-ical Tissues. Adjustment to Experimental Data.”, Interna-tional Journal of Solids and Structures 43, 20, 6052-6070.

[4] RICKEN, T., SCHWARZ, A., BLUHM, J. [2007], Compu-tational Materials Science 39, 124-136.

Û2Ûl

uenm

Modelling of Point Defects in anisotropic media with an applica-

tion to Ferroelectrics

Oliver Goy, Ralf Mueller, Dietmar Gross

Solid Mechanics, Department of Civil Engineering & GeodesyTU-Darmstadt, D-64289 Darmstadt, Germany

Elastic Interaction between point defects in isotropic and anisotropic mediahave been investigated in many studies. Solutions for the Green’s function ofpoint sources in anisotropic media are given e.g. in [1, 5] and for anisotropicpiezoelectric solids in e.g. [4, 6]. These solutions can be applied within theboundary element method (BEM) as well as when simulating concentrateddefects like foreign atoms, vacancies or point charges.

x1 [mm]

x2[m

m]

0 2E-06 4E-06 6E-06 8E-06 1E-05 1.2E-050

2E-06

4E-06

6E-06

8E-06

1E-05

lambda2E-201.8E-201.6E-201.4E-201.2E-201E-208E-216E-214E-212E-210

-2E-21-4E-21-6E-21-8E-21-1E-20-1.2E-20-1.4E-20-1.6E-20-1.8E-20-2E-20

Figure 1: Driving forces on three defects due to interaction energies.

In this work a dilatation centre is represented as an eigenstrain in the con-stitutive equations, which formally leads to the same field equations asthe description with force multipoles. The fields produced by mechanicallyisotropic and anisotropic defects will be investigated and their interactionwill be simulated by means of material forces. Transversally isotropic ma-terial with linear electromechanical coupling will be used.The defect parameters used in the continuum model are obtained by fittingthe results of molecular dynamics (MD) simulations to continuous spatialfields. Transferring data from the atomic to the continuum level is a fieldof active research and no unique solution can be provided. On the atomic

Û2Û2Û

level, Coulomb–interaction causes a displacement field incompatible to anelastic solution. In order to solve this difficulty and motivated by [2], thetrace of the so called Formation Volume Tensor, which denotes the volumechange around the defect is used to determine the defect parameters.The model is applied to get a better insight in processes leading to electricfatigue in ferroelectric materials. Ferroelectrics are used in a wide field ofapplications and can be found in actuators, sensors and in electronic devices.During the high number of mechanical and electrical load cycles to whichthe material often is exposed, fatigue phenomena may occur. This involvesdegradation of the material and a decrease of the electromechanical couplingcapability. The causes are assumed to be ionic and electronic charge carriersmodelled as point defects, which interact with each other, with microstruc-tural elements in the bulk and with interfaces. Accumulation of defects canprimarily lead to degradation, or finally to mechanical damage and dissoci-ation reactions.In order to get a better understanding of the defect accumulation processes,the interaction of defects in periodic and in infinite cells is simulated. Ap-plying reasonable kinetic laws, defect migration is studied in a deterministicway in order to obtain a general tendency of defect migration, for detailssee [3].

References

[1] D. M. Barnett. The precise evaluation of derivatives of the anisotropicgreen’s functions. Phys. stat. sol. (b), 49:741–748, 1972.

[2] K. Garikipati, M. Falk, M. Bouville, B. Puchala and H. Narayanan. Thecontinuum elastic and atomistic viewpoints on the formation volume andstrain energy of a point defect. J. Mech. Phys. Sol., 54:1929–1951, 2006.

[3] O. Goy, R. Mueller and D. Gross. Interactions of point defects in piezo-electric materials – numerical simulation in the context of electric fatigue.

[4] E. Pan and F. Tonon. Three-dimensional green’s functions in anisotropicpiezoelectric solids. Int. Journal of Solids and Structures, 37:943–958,2000.

[5] C.-Y. Wang. Elastic fields produced by apoint source in solids of generalanisotropy.

[6] M. Wang X. Li. Three-dimensional green’s functions for infiniteanisotropic piezoelectric media. Int. Journal of Solids and Structures,44:1680–1684, 2007.

Û2Û>¤

uepo

Homogenization of textured as well as randomly oriented ferroelectric polycrystals

K P Jayachandran, J M Guedes and H C Rodrigues

IDMEC Instituto de Engenharia Mecânica, Instituto Superior Técnico, Technical University

of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Abstract

Microstructural features such as spatial correlation of crystallographic orientations and

morphological texture in piezoelectrics is modelled computationally. The mathematical

homogenization of piezoelectric material is implemented using the finite element

method (FEM) by solving the coupled equilibrium electrical and mechanical fields. A

three-dimensional (3D) FEM model is adapted to polycrystalline medium. The

dependence of macroscopic electromechanical properties on domain orientation and

texture of single-crystalline as well as polycrystalline ferroelectric BaTiO3 and PbTiO3

are demonstrated based on this model. The homogenized piezoelectric moduli of the

polycrystals and single-crystals of these materials exhibit remarkable but contrasting

behaviours. An optimum texture at which the piezoelectric response of these

polycrystals can be maximized is demonstrated.

0.01 0.1 0.5 1 2 5 10

0

100

200

300

400

500

600

Piez

olel

ectri

c st

rain

mod

uli d

jQ (p

C/N)

V

d15

d31

d33

Figure 1. Dependence of the piezoelectric strain moduli jKd of polycrystalline BaTiO3

on texture parameter 1.

Results The aggregate texture for polycrystalline ferroelectrics is assumed to have a Gaussian

(normal) distribution [1, 2] defined by the normal probability distribution function

2

2

1 ( )( | , ) exp

22f T PT PV

VV S

ª º « »

¬ ¼ (1)

Û2Û0

where 1 is an increasing parameter describing worsening texture for grain orientation T

and µ is the mean of the distribution. We are showing some of the important results on

the polycrystalline BaTiO3 and PbTiO3. The longitudinal piezoelectric strain coefficient

d33 assumes a higher value than the corresponding single-crystalline value of BaTiO3 at

a distribution characterised by 1 between 0.5 and 0.7 (figure 1) however, and for

PbTiO3 the maximum is shown at the perfect 001 texture (figure 2). Therefore,

polycrystalline materials can be improved in their global piezoelectric performance by

engineering the textures.

0.01 0.1 0.5 1 2 5 10-30

-15

0

15

30

45

60

75

90Pi

ezoe

lect

ric s

train

mod

uli d

iQ(p

C/N)

V

d15

d31

d33

Figure 2. Dependence of the piezoelectric strain moduli jKd of polycrystalline PbTiO3

on texture parameter 1.

References [1] Ruglovsky J L, Li J, Bhattacharya K and Atwater H A 2006 The effect of biaxial

texture on the effective electromechanical constants of polycrystalline barium titanate

and lead titanate thin films Acta Mater. 54 3657- 3663

[2] Jayachandran K P, Guedes J M and Rodrigues H C 2007 Homogenized

electromechanical properties of crystalline and ceramic relaxor ferroelectric

0.58Pb(Mg1/3Nb2/3)O30.42PbTiO3 Smart Mater. Struct. (under review)

Û2Û O

uejq

A numerical modelling of theplasticity induced by martensitic

transformation in a 16MND5 steel. Amultigrain approach

S. Meftah(1), F. Barbe(1,∗), L. Taleb(1), F. Sidoroff(2)

(1) Laboratoire de Mecanique de Rouen, EA 3828,INSA Rouen, BP 08, 76801 Saint Etienne du Rouvray, France

(2) LTDS, UMR 5513,

Ecole Centrale Lyon - 69134 Ecully, France

(∗) Corresponding author: Fabrice Barbe

e-mail: [email protected] - tel: 33 2 32 95 97 60, fax: 33 2 32 95 97 04

Keywords: Martensitic transformation, TRIP, micromechanical modelling, finite

elements

AbstractIt has been shown in (Grostabussiat et al., 2001; Grostabussiat et al., 2004) that 16MND5

steel, when subjected to phase transformation, can exhibit transformation induced plasticity

(TRIP) even though no stress is imposed during the transformation: this happens when the

parent phase has been strain-hardened. This peculiar feature can be reproduced with the

model of Leblond (Leblond, 1989), one of the most currently used micromechanical models,

but with a TRIP in the opposite direction to that observed experimentally (Taleb and Petit,

2006).

The finite elements (FE) micromechanical modelling of martensitic transformation of

Ganghoffer (Ganghoffer and Simonsson, 1998) has been improved (Meftah et al., 2006) so

to test the abilities of FE analysis to predict the phenomenon of TRIP resulting from pre-

hardening. In this modelling, plates of martensite appear according to a given criterion

(energetic or based on the driving force) in any of the possible variant directions. The

appearance is accompanied by the shear transformation strain of the variant and by the

variation of the mechanical properties of the transforming plate from those of austenite to those

of martensite. The resulting interaction between austenite and martensite being computed

locally, without adopting any particular assumption on the distribution of stress and strain,

the mechanisms involved in the transformation are described more accuratly with the FE

micromechanical modelling than with the Leblond model. This may explain why the quality

of the prediction is much better with FE, at least from the qualitative point of view.

The first comparisons have dealt with a single grain of austenite in 2D. The sensitivity to

mesh size have been tested, along with other numerical parameters, in order to determine a

configuration of modelling devoid of mesh size effect. Besides, several criteria for the selection

of plates to be transformed have been studied. The object of this work is to present the

extension of the modelling to a domain of study containing several grains of austenite seperated

by a grain boundary -or interacting zone-. Using an adequate criterion, the first plate to be

transformed is determined according to the load and then martensite propagates plate afterÛ2Ûrb

plate from a grain of austenite to its neighbours. One of the main improvements due to this

extension concerns the kinetics of transformation: it is more continuous with several grains,

which corresponds better to physics. At the current state of the modelling, all the grains have

the same crystallographic orientation (same set of variants); it is under purpose to have these

crystallographic orientations taken at random so that the domain of computation be more

representative of a real crystalline austenite subjected to martensitic transformation.

References

Ganghoffer, J. F. and Simonsson, K. (1998). A micromechanical model of the martensitic

transformation. Mechanics of Materials, 27:125–144.

Grostabussiat, S., Taleb, L., and Jullien, J. F. (2004). Experimental results on classical

plasticity of steels subjected to structural transformations. Int J Plasticity, 20(8-9):1371–

1386.

Grostabussiat, S., Taleb, L., Jullien, J. F., and Sidoroff, F. (2001). Transformation induced

plasticity in martensitic transformation of ferrous alloys. J Phys IV, 11:173–180.

Leblond, J. B. (1989). Mathematical modelling of transformation plasticity in steels II:

coupling with strain hardening phenomena. Int J Plasticity, 5:573–591.

Meftah, S., Barbe, F., Taleb, L., and Sidoroff, F. (2006). Parametric Numerical Simulations

of TRIP and its Interaction with Classical Plasticity in Martensitic Transformation. Eur

J Mech, A/Solids. In press, doi:10.1016/j.euromechsol.2006.10.004.

Taleb, L. and Petit, S. (2006). New investigations on transformation induced plasticity and

its interaction with classical plasticity. Int J Plasticity, 22:110–130.

Û2Û c

uens

Finite element simulation of Pellet-Cladding Interaction (PCI) in nuclear fuel rods N. Marchal*˜, C. Campos*, C. Garnier*

*AREVA NP, Fuel sector, 10 rue Juliette Récamier, 69456 Lyon cedex 06, FRANCE

Abstract Introduction The core of nuclear Light Water Reactors (LWRs) holds fuel rods. They consist in zirconium alloy tubes containing uranium dioxide pellets. The zirconium alloy cladding is the first containment barrier for fission products. Due to water pressure, cladding creeps down until contact with pellet after a few operating cycles. In case of power increase, this Pellet-Cladding Interaction (PCI) induces large stresses in the cladding that might lead to fuel rod failure. It is therefore necessary to be able to prevent the occurrence of PCI failure. Thus, tests are carried out to determine the PCI resistance and improve the understanding of this phenomenon. Finite element simulations shown here are used to help interpreting experiments and finally develop efficient core protections and PCI resistant designs. The Pellet-Cladding Interaction phenomenon Nuclear reactions within the fuel material generate heat that is transferred to the water. A large thermal gradient is created within the pellet resulting in fuel cracking and the formation of pellet fragments. The pellet gets an hour-glass shape (figure 1). A gap exists initially between the pellet and the clad. In reactor, irradiation-induced phenomena cause the external diameter of the pellet to increase in the first stages of irradiation. On the other hand, the cladding creeps down because of compressive stresses due to water pressure. Consequently, the gap size decreases up to gap closure. In case of power rise, the thermal expansion of the pellet results in an increase in pellet diameter accompanied by pellet crack opening. This causes enhanced stresses in the cladding that might trigger Stress-Corrosion Cracking (SCC) because of the presence of corrosive fission products such as iodine, and finally lead to clad failure. PCI failure is the combination of mechanical interaction (PCMI) and iodine-induced SCC. Modelling of Pellet-Cladding Mechanical Interaction As PCI is a complex phenomenon occurring under irradiation, measurements are difficult. Thus, finite element simulations are useful. TOUTATIS is an application of CEA’s finite element code CAST3M [1], dedicated to the simulation of thermomechanical behaviour of LWR fuel rods at the pellet scale [2]. This code is currently being developed by CEA in the framework of the PCI research project, common to AREVA NP, EDF and CEA. Its main computational features consist in a coupled thermal-mechanical computing scheme, 2D and 3D friction between fuel and clad. Complex material behaviours are modelled, such as fuel densification and swelling due to irradiation. Fuel cracking and creep are computed using the “UO2 model”, developed at CEA Saclay DEN/DM2S/SEMT. It couples a smeared crack model with an elastic-viscoplastic material behaviour. An anisotropic elasto-viscoplastic constitutive model is used for cladding. Although this code is devoted mainly to 3D simulations, it allows also 2D(r,z) and 2D(r,T) simulations. Results on 2D(r,T) simulations AREVA NP mainly focused on 2D(r,T) simulations because this representation allows to simulate fuel crack opening and offers limited computing time, which facilitates sensitivity studies. Beside the improvement of our understanding of PCMI, these computations are part of a worldwide

˜Corresponding author: [email protected]

Û2Û g

Figure 3: Distribution of VTT in a clad portion at peak power

fuel crack

Symmetry plane (half fuel fragment)

Figure 2: Pattern of secondary cracks appeared during a power ramp

effort to develop cutting-edge mechanical modelling in COPERNIC3, AREVA’s future global fuel rod performance code [3]. PCI modelling is an essential part of this mechanical modelling. In 2D(r,T), the hypothesis of plane strains is considered. This is equivalent to a mid-pellet plane computation. Several interesting results arise. The most important ones are summarized below. At first, if 8 initial fuel fragments are assumed, it appears that secondary cracking of the pellet is due to the friction between pellet and clad. At peak power, extended secondary fuel cracking occurs, as shown on figure 2. A stress concentration can be observed in the cladding in front of the fuel crack (figure 3). Pellet-clad friction is a first-order parameter for this stress distribution. The effect of fuel cracking properties can be easily investigated. Actually, the pattern of secondary cracks appearing during the power increase strongly influences the distribution of shear stresses along the pellet-clad interface. Conclusions Finite elements simulations are useful to get a better understanding of PCMI. They revealed for example that pellet-clad friction controls the apparition of peripheral radial cracks in the pellet. This influences the stress distribution in the cladding. These improvements in the understanding of PCMI will lead to new PCI criteria that will be included in AREVA’s future global fuel rod performance code COPERNIC3. Acknowledgments The authors want to thank their partners of EDF and CEA in the PCI project for interesting and useful discussions. References [1] CAST3M code, Commissariat à l’Energie Atomique CEA DEN/DM2S/SEMT. http://www-

cast3m.cea.fr [2] Toutatis, an application of the CAST3M finite element code for PCI three-dimensionnal

modelling. F. Bentejac, N. Hourdequin. Proceedings of the International Seminar on PCI with Water Reactor Fuel, Aix-en-Provence, France. 2004.

[3] The COPERNIC3 project : how AREVA is successfully developing an advanced global fuel rod performance code. Ch. Garnier, F. Sontheimer, P. Mailhé, H. Landskron, D. Deuble, V.I. Arimescu, M. Billaux. Accepted for the 2007 international LWR fuel performance meeting, September 30 – October 3, 2007, San Francisco, USA. Available online at: http://www.inspi.ufl.edu/fuel07/

Figure 1: fragments and pellet hourglass shape

(after Gittus)

Û2Û h

uet

KIRKENDALL VOIDS IN THE INTERMETALLIC

LAYERS OF SOLDER JOINTS IN MEMS

Kerstin Weinberg, Thomas Böhme and Wolfgang H. Müller

Technische Universität Berlin, Fakultät V, Verkehrs- und Maschinensysteme,

Institut für Mechanik, LKM, Sekr. MS 2, Einsteinufer 5, 10587 Berlin, Germany

ABSTRACT

Microelectronic systems consist of the functional chip unit itself and its packaging, which includes several electro-mechanical connections, e.g., solder joints between different metal layers. Experimental observation shows that aging of the solder alloy as well as the formation and growth of so-called Kirkendall voids significantly contribute to the degradation of the joining capability and drop impact reliability. For a continuum mechanical approach to such processes the question is: How can the additional information about the micro-structural development be included in a constitutive model? To this end we introduce here a general mesoscopic concept which enables us to describe materials with a certain microstructure by means of a continuum mechanical framework. A mesoscopic distribution function provides a statistical addition to the classical continuum theory. In this contribution we model a typical intermetallic layer of Cu-Sn solder joints and link the mechanisms of diffusion and deformation induced growth of voids to parameters which describe the macroscopic material response. Numerical simulations on the material point level provide inside into the (not yet completely understood) mechanisms of failure by formation and growth of Kirkendall voids.

Û2Ûk

Û>¤l

uenuMicromechanical analysis of ferroelectric structures by

a p hase fi eld mod el

D. Schrade, B.X. Xu, R. Mueller, and D. Gross

Division of Solid Mechanics, Department of Civil Engineering and Geodesy, TU Darm-

stadt, H ochschu lstr. 1 , D-6 4 2 8 9 Darmstadt, Germany

A continuum p hase fi eld m odel has b een dev elop ed to sim ulate the m icrostructureof ferroelectric m aterials, see Schrade et.al (2 0 0 6 ). T he m odel is characterized b y thetreatm ent of the sp ontaneous p olarization as an order p aram eter and the ex tension of theclassic electric enthalp y b y a p hase sep arating p otential and an interface energ y . F rom thisp hase fi eld p otential, the coup led constitutiv e eq uations and the Ginzb urg -L andau ty p eev olution eq uation can b e deriv ed. Im p lem ented in an im p licit F inite E lem ent Method,the m odel is used to determ ine the electric and m echanical fi elds as w ell as the ev olutionof the dom ain structures.

In the p resent w ork , the p hase fi eld m odel is ap p lied to inv estig ate v arious ferroelectricdev ices w ith p oint and surface defects and crack s. F ig ure.1 show s the interaction of thecrack w ith the dom ain structure and the induced electric disp lacem ent concentrationaround the crack tip .

4.36E-02 9.63E-02 1.49E-01 2.02E-01 2.54E-01 3.07E-01 3.60E-01 4.12E-01 4.65E-01 5.18E-01 5.70E-01

-9.09E-03

6.23E-01

_________________ S T R E S S 5

Time = 4.00E-10Time = 4.00E-10

F ig ure 1 : Interaction of a crack w ith the dom ain structure of ferroelectrics

Û>¤Û

Another example is the local analysis of a cofired stack actuator, which finds a widerang e of applications in practice. O ne drawb ack of this dev ice lies in the fact that at theinternal electrode tips the activ e ferroelectric material may fail, due to hig h electrome-chanical fields. U p to now, inv estig ations on diff erent aspects of this prob lem hav e b eendone solely b y macroscopic models, see e.g ., H om and S hankar (1 9 9 6 ), G ong and S uo(1 9 9 6 ), S hindo et.al (2 0 0 4 ) and E lhadrouz et.al (2 0 0 6 ). A more detailed analysis can b eprov ided b y the phase field model. D iscussed and compared are the electroelastic fieldand the domain structure ev olution around the electrode tip for diff erent loading stag es.F ig ure 2 shows the unit cell of the actuator and the F E M mesh.

F ig ure 2 : U nit cell of an actuator and F E M mesh

References

[1 ] E lhadrouz M ., B en Z ineb T ., and P atoor E ., F inite element analysis of a multilayerpiezoelectric actuator taking into account the ferroelectric and ferroelastic b ehav iors.Int. J . E ng ing . S ci. 4 4 [1 5 -1 6 ], pp. 9 9 6 -1 0 0 6 , 2 0 0 6 .

[2 ] G ong X .Y . and S uo Z ., R eliab ility of C eramic M ultilayer Actuators: a N onlinearF inite E lement S imulation. J . M ech. P hys. S olids, 4 4 [5 ], pp. 7 5 1 -7 6 9 , 1 9 9 6

[3 ] H om C .L . and S hankar N ., A F inite E lement M ethod for E lectrostrictiv e C eramicD ev ices. Int. J . S olids S truct., 3 3 [1 2 ], pp. 1 7 5 7 -1 7 7 9 , 1 9 9 6 .

[4 ] S chrade D ., M ueller R . and G ross D ., P hase field simulations in ferroelectric materi-als, P AM M 6 [1 ], pp. 4 5 5 -4 5 6 , 2 0 0 6 .

[5 ] S hindo Y ., Y oshida M ., N arita F . and H orig uchi K ., E lectroelastic field concentra-tions ahead of electrodes in multilayer piezoelectric actuators: experiment and finiteelement simulation. J . M ech. P hys. S olids, 5 2 [5 ], pp. 1 1 0 9 -1 1 2 4 , 2 0 0 4 .

Û>¤¤

132 476 878

@CMD ^ : ?A@CB DF:5GIHKJLDFGI@CBNM

Û>¤0

Û>¤ O

PRQTSUSWVYX[Z

] ^ D ^ :F@>_ ^ B ^ @>JNM ;`D ^ :<Ga;>=aM

Û>¤b

Û>¤ c

ed

!"#

$% % % & '()* # + , -(( + +% . + # $ %

/012&0%1 2 *#012 #&3 *&4 *% 5 + #0%% + 6%7 +$ % + 8 9-: % % 012 ##

%# #% #" 93: %

9-: !/5;3((4 < %; ,#'=->?@-'(493: " !"A""B3((' ;% < % #1 4C'D3((@>?(>C>

Û>¤ g

Û>¤ h

eji

Stretch-flangeability characterisation of multi phase steel using amicrostructure based failure modelling

Vitoon Uthaisangsuk, Ulrich Prahl, Wolfgang Bleck

Department of Ferrous Metallurgy, RWTH Aachen University

The demand on modern high strength steel with excellent strength and ductility has increasedin the automotive industry due to their potential in terms of weight reduction, improvedpassive safety features, energy saving considerations and environmental protection. For thecar body construction, a reliable prediction of material formability is required. Stretch-flangeability has become an especially important formability parameter in addition to tensileproperties, particularly for complicated auto body parts or parts under heavy deformationconditions. The hole expansion ratio is strongly influenced by microstructures and theirdistribution in steel. Therefore, failure behaviour by the stretch-flangeability analysis has tobe described depending on microstructure.

In this work, hole expansion tests are carried out for steel sheet with different microstructuresand their local fracture surfaces are afterwards investigated. For the failure modelling, a FEapproach will be presented by means of Representative Volume Elements (RVE) incombination with Continuum Damage Mechanics (CDM) in order to consider influence ofmulti phase microstructure on mechanical properties and complex failure mechanisms [1, 2].The local deformation fields from macroscopic FE simulation of the hole expansion test areused as boundary conditions for the RVE simulation. At the failure moment, local straindistributions between different phases are studied, and correlated with macroscopicdeformability results. The influence of material properties of individual phases and the localstates of stress on material formability as well as failure behaviour were examined.Additionally, damage mechanisms including crack initiation and propagation of multiphasesteel will be investigated and a local microstructure based failure model will be formulated. Inconclusion, a better understanding of the stretch-formability analysis is allowed on amicrostructure basis for sheet metal characterisation of multi phase steels.

References:

[1] Tvergaard, V. and Needleman, A.: Analysis of the cup-cone fracture in an round tensilebar. Acta metall., 32(1): 157-169, 1984.

[2] U. Prahl, S. Papaefthymiou; V. Uthaisangsuk; W. Bleck; J. Sietsma; S. van der Zwaag:Micromechanical based modelling of properties and failure of multi phase steels,Computational Materials Science, Volume 39, Issue 1, 2007, p. 17-22.

Û>¤k

Û0l

enm

Rubber filled with carbon black

from the nanoscopic structure to the macroscopic behaviour

A. Jean, S. Cantournet, S. Forest, D. Jeulin, V. Mounoury, F. N’Guyen

Centre des Materiaux Pierre Marie Fourt,CNRS-UMR 7633, Evry cedex, France

Centre de Morphologie Mathematique, Fontainebleau cedex, France.

Michelin Research - 63040 Clermont Ferrand Cedex 09 (France)

Keywords : carbon black, homogenization, RVE, morphology

Many materials, assumed homogeneous at a scale, can be heterogenous at smaller ones.

Homogenization approach consists in linking the material properties through different scales.

In this work, we apply such methods to rubber filled with carbon black. Indeed, at nanoscopic

scale, compare to macrosopic one, the filled rubber is heterogenous about microstructure and

mechanical properties.

First, we propose to describe the morphology of the material at nanoscopic scale to model

it with mathematical morphology approach (Jeulin, 1991), (Savary et al., 1999), (Delarue,

2001), (Pecastaings, 2005), (Moreaud, 2006). The observation and the characterization of the

microstructure is based on using of TEM (Transmission Electronic Microscope) and image

analysis.

(a) (b)

Fig. 1 – (a) : TEM Image - (b) : Simulation 16003nm

3, 20% of Carbon Black ' 60000 particles

After being able to simulate nanoscopic structure of filled rubber, the purpose is to find a

RVE (Representative Volume Element) which is a main actor in homogenization methods to

predict the effective properties. A statistical approach is used to determine the RVE size (Kanit

Û05Û

et al., 2003) with a given precision of the estimation of the wanted overall property. A RVE

is to be found for morphological properties like volume fraction, a RVE will be found also for

mechanical properties like Young’s modulus in small deformations.

To conduct properly this study, many numerical tools are needed like microstructure finite

element meshing and parallel computation methods.

Fig. 2 – Finite Element Meshing

We gratefully aknowledge the financial and scientific support of Michelin for this research

project.

References

Delarue, A. (2001). Prevision du comportement electromagnetique de materiaux composites a

partir de leur mode d’elaboration et de leur morphologie. PhD thesis, Ecole des Mines de

Paris.

Jeulin, D. (1991). Modeles mporphologiques de structures aleatoires et de changement d’echelle,

These de Doctorat d’Etat. PhD thesis, Universite de Caen.

Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D. (2003). Determination of the

size of the representative volume element for random composites : statistical and numerical

approach. International Journal of Solids and Structures, 40 :3647–3679.

Moreaud, M. (2006). Proprietes multi-echelles et prevision du comportement dielectrique de

nanocomposites. PhD thesis, Ecole des Mines de Paris.

Pecastaings, G. (2005). Contribution a l’etude et a la modelisation de la mesostructure de

composites polymeres-noir de carbone. PhD thesis, Universite Bordeaux I.

Savary, L., Jeulin, D., and Thorel, A. (1999). Morphological analysis of carbon-polymer

composite materials from thick section. Acta Stereologica, 18(3) :297–303.

Û0?¤

epo

ON TUNABILITY AMPLIFICATION FACTOR OF COMPOSITE DIELECTRICS

A. G. Kolpakov

Siberian State University of Telecommunications and Informatics

324, Bld.95, 9th November str., Novosibirsk 630009 Russia

e-mail: [email protected]

Tunability of nonlinear composite material

Ferroelectric-dielectric composite material are considered as prospective materials for electronic industry

[1]. Ferroelectric is nonlinear material. The characteristics of the nonlinearity is the relative tunability

[2] )0(

)()0(H

HH ET , where H(E) is the permittivity of the material corresponding to the field || I E

applied to the material. It is the property that can be used for fabrication of electronic devised with

parameters that can be tuned by the application of a controlling DC voltage. The progress in the field is

restricted by obscurity of possible values of the overall relative tunability and overall electric loss of

ferroelectric-dielectric composite material (pure ferroelectric cannot be used in electronic devices).

In [2] the question was formulated: Can a ferroelectric-dielectric composite material have relative

tunability greater relative tunability of its ferroelectric component? (tunability of dielectric is 0).

Under condition that the volume fraction q of inclusions is small, the following formulas for

computation of overall permittivity

ferroT

effH and overall relative tunability of composite material filled

with spherical inclusions were derived [1]

effT

)5.11( qeff HH , )2.01( qTT ferroeff , (1) The second formula proclaims the possible increasing of the relative tunability. Usually, the formulas

derived under condition q <<1 (so called dilute composite material). In [3] the increasing of tunability

was documented for not dilute in ferroelectric-dielectric composites. At the same time, the increasing of

the relative tunability predicted by formula (1) does not exceed 10% (the volume fraction q<0.5).

The ratio ferro

eff

TT

k is called tunability amplification factor

The experiments [2] do not provide us information about a composite material having the relative

tunability greater the relative tunability of its components. Thus, in the present time, there is no

agreement on the relative tunability as a function of q. The problem of bounds (the possible values) on

the relative tunability, to the best knowledge of the author, is "terra incognita", although the

Û00

2corresponding problem for the overall permittivity was intensively investigated. Note that there is no

good reason to believe that the volume fraction determines the relative tunability solely. At the same

time, to the best knowledge of the author, the specialists in electroceramics do not pay proper attention

to the influence of microgeometry and microtopology on the overall property of composite dielectrics.

Probably, the reason for this is the using in electronic industry the technology, which can be briefly

described as "blending and baking" and represents a technology of low level as compared with usual

technologies of composite material industry.

This paper presents theoretical and numerical analysis of tunability of composite materials the

mathematically rigor method known as the homogenization theory. The results of the analysis

combined with the current results on the high-contrast composite materials allow to estimate the

effectiveness of the existence technological solutions and indicated the possible ways to improve the of

composite electroceramics.

Our computations demonstrate that composite material can have tunability greater the tunability

of its components. The tunability amplification factor takes values from 3 to 30 (against 0.1 maximum

predicted in [1]) for designs, which can be characterized as practically realized. Here "practically

realized" means that the designs obtained do not exceed the complexity of designs usual for the modern

composite materials. At the same time, the designs obtained cannot be realized using simple

technologies like "blending and baking" used in the modern electronic industry.

The electrical characteristics of ferroelectric-dielectric composite materials are determined by the

microgeometry and topology of components of composite rather then volume fraction of the

ferroelectric and dielectric components.

The theoretical analysis predicts that the overall loss tangent of composite material lies between

the loss tangents of its components. For high-contrast ferroelectric (matrix)-dielectric (inclusion)

composite material, the theory predicts overall loss tangent to be equal to the loss tangent of

ferroelectric.

References 1. L.C. Sengupta, S. Stowell, E. Ngo et al Barium strontium titanate and non-ferroelectric oxide

ceramic composites for use in phased array antennas. Integrated Ferroelectrics. 1995, 8. 77-78.

2. A. K. Tagantsev, V. O. Sherman, K. F. Astafiev et al Ferroelectric materials for microwave

tunable applications. J. Electroceramics. 2003. 11. 5-66.

3 A. G. Kolpakov, A. K. Tagantsev, L. Berlyand et al Nonlinear dielectric response of periodic

composite materials J. Electroceramics (accepted).

Û0 O

PRQTSUSWVYX[Z \+

;>?A:F@CMw?A@>xKGa? H ^zy ;|GI@>J~:

Û0b

Û0 c

dFe,d

Modelling of semi-crystalline polymers behaviour during ECAE process

B. Aour(1), F. Zaïri(2), M. Naït-Abdelaziz(2) J.M. Gloaguen3, J.M. Lefebvre3

(1) Laboratoire de Technologie de l’Environnement, Département de Mécanique, ENSET d'Oran, BP 1523El'Mnaour 31000, E-mail : [email protected]

(2)Laboratoire de Mécanique de Lille (UMR CNRS 8107), Polytech’Lille, USTL (3)Laboratoire de Structure et Propriétés de l’Etat Solide (UMR CNRS 8008), USTL,

59655 Villeneuve d’Ascq, Cedex, France

AbstractThe study that we present here has two principal objectives. The first consists on the numericalinvestigation of the behaviour of semi-crystalline polymers during a super plastic deformationprocess commonly named “Equal Channel Angular Extrusion (ECAE)”. This process, which wasdeveloped initially for metals, is used more recently for polymers in order to improve their physicaland mechanical properties. The second includes the fabrication and experimental study of the effectof the principal process parameters on the development of the plastic strain and the force requiredfor the extrusion. In the present work, the process has been carried out for two typical semi-crystalline polymers:high-density polyethylene (HDPE) and polypropylene (PP). Therefore, a characterization bycompressive tests under different temperatures and strain rates was adopted for the identification ofthe selected viscoplastic models. Then, numerical simulations were carried out using the finiteelement software MSC.Marc to highlight and optimize the various effects of the geometricalparameters and the processing conditions on the material behaviour during ECAE process. Anexperimental study was then carried out and consolidated by a detailed numerical analysis. Keywords: ECAE, semi-crystalline polymers, FEM, FEM-BEM, severe plastic deformation.

IntroductionEqual channel angular extrusion (ECAE) has attracted a great deal of interest primarily due to thesubstantial microstructural changes it imparts on the extruded samples [1]. Many papers haveshown that the properties of the ECAE products depend on processing conditions such as number ofpasses and route, material properties, friction effect, temperature, back pressure and die geometry[2,3]. In order to optimize the process conditions, the knowledge of the strain distribution in agiven material is fundamental. Several analyses of ECAE process have been made to investigate theplastic strain in the extruded material. These include geometric analyses [4,5], slip lines method [6],upper bound theory [7] and finite element method (FEM) [8]. However, these theoretical,experimental and numerical investigations have been carried out for a large number of metallicmaterials. In contrast, in polymeric materials, little work is available to address the mechanicalbehaviour during ECAE process [9,10]. It is interesting to note that the ability of polymers toachieve high toughness and ductility is linked to its microstructure. In this respect, ECAE process isa potential way to trigger a significant morphological changes and hence in profound modificationsof the mechanical properties of the extruded materials.

Finite element modellingA comparison between analytical solutions and finite element results for different channel angles Φwith θ=0°, was carried out for the ideal case of elastic perfectly plastic behaviour, which neglectsany hardening or viscous effects of the material and makes the comparison perfectly geometrical.The values of the equivalent plastic strain at the middle of the workpiece (Fig.1.a) were plotted as afunction of the corner angle θ for representative values of the channel angles Φ=90°, 105°, 120°,135° and 150°.

Û0 g

0

0,2

0,4

0,6

0,8

1

1,2

1,4

80 100 120 140 160Channel angle (deg)

Equi

vale

nt p

last

ic st

rain

Analytical sol. Eq.(7)

FEM Perfectly-plastic

(a) (b)Fig. 1. (a) Selected cross-section (b) Comparison between FEM and analytical solution of the equivalent

plastic strain according to various die angles Φ for θ=0°.

The results are shown in Fig. 1.b, where the solid lines correspond to analytical solution and thesolid points correspond to the finite element solutions. It is apparent from this plot that niceagreement is highlighted between the two types of solutions. Furthermore, it can be seen that thechannel angle of 90° produces the high level of deformation and the magnitude of the equivalentplastic strain decreases with the increase of the channel angle Φ.

Experimental considerationsA polypropylene plate was machined into 10×10×70mm3 samples. The samples were extrudedthrough a die having the geometrical features presented in the numerical part under various ramspeeds. Experimental results in terms of the applied load evolution are shown in figure 2a and aquite good agreement is obtained when compared with finite element results in figure 2b. Is alsoshown the friction effect between the tool and the sample.

Figure 2: (a) Experimental load-ram displacement curves for polypropylene. (b) Comparison betweenexperimental and finite element results for 0.001/s (f is the friction coefficient).

References[1] Segal, V.M.; Reznikov, V.I.; Drobyshevskiy, A.E.; Kopylov, V.I., Russ. Metall. 1981, 1, 99-105.[2] Aour, B.; Zairi, F.; Gloaguen, J.M.; Nait-Abdelaziz, M.; Lefebvre, J.M.; Comp. Mater. Sci. 2006, 37, 491. [3] Aour B., Thèse de Doctorat, Oran, 2007.[4] Iwahashi, Y.; Wang, J.; Horita, Z.; Nemoto, M.; Langdon, T.G., Scripta Mater. 1996, 35, 143-146.[5] Segal, V.M., Mater. Sci. Eng. A, 1995, 197, 157-164.[6] Segal, V.M., Mater. Sci. Eng. A, 2003, A345, 36-46.[7] Lee, D.N., Scripta Mater., 43, 115-118.[8] Zairi, F.; Aour, B.; Gloaguen, J.M.; Nait-Abdelaziz, M.; Lefebvre, J.M.; Comp. Mater. Sci. 2006, 38, 202.[9] Sue, H.J.; Li, C.K.Y., J. Mater. Sci. Letter 1998, 17, 853-856.[10] Weon, J.I.; Creasy, S.T.; Sue, H.J.; Hsieh, A.J., Polym. Eng. Sci. 2005, 45, 314-324.

Û0 h

dFe i

Quantification of the influence of vertex singularities on crack behaviour

P. Huta, L. Náhlík, Z. Knésl1 Institute of Physics of Materials, Academy of Sciences of the Czech Republic

Žižkova 22, 616 62 Brno, Czech Republic [email protected]

Due to the existence of vertex singularity at the point where the crack intersects the free

surface, stress distribution around the crack tip and the type of the singularity is changed [1,2,3]. In the interior of the specimen the classical singular behaviour of the crack is dominant and can be described using analytic equations of two-dimensional stress distribution around the sharp crack. Contrary to this, at the free surface or in the boundary layer close to free surface the vertex singularity is significant. Over the last 20 years a large number of papers which address various aspects of the influence of vertex singularity on crack behaviour have been written eg.[1-6]. The three-dimensional singular stress field near the point where the crack front intersecting the free surface (vertex point) of an elastic body was investigated by Bažant and Estenssoro [4], Benthem [5], Picu and Gupta [6]. They found that the singularity exponent induced by the free surface differs from 0.5 and is dependent on Poisson’s ratio ν .

The influence of corner singularity on crack behaviour for a three-dimensional structure is described in this paper. The singularity exponents in corner point vary between 0.5 (corresponding to 0=ν ) and 0.33 (corresponding to 5.0=ν ), so the singularity of the crack inside the boundary layer is changed from 0.5 in the interior to 0.33 on a free surface. To estimate fatigue crack growth rate influenced by a free surface the model of a general singular stress concentrator was applied. After comparison between shape functions for stress distribution in the three-dimensional model of the crack intersecting free surface and a general singular stress concentrator (V-notch or crack on bi-material interface), the two-dimensional singular stress concentrator with corresponding stress singularity was used as a model of a crack influenced by a corner singular field. The methodology suggested describes the behaviour of a crack with a different singularity exponent than 0.5.

The results presented make it possible to estimate the behaviour of a crack influenced by another singular stress field, using the known concept of the generalized stress intensity factor [7]. The estimated fatigue crack growth rate of the threshold values can help to provide a more reliable estimation of the fatigue life of the structures considered.

Acknowledgement: This investigation was supported by grants no.106/06/P239 and 101/05/0320 of the Czech Science Foundation. [1] M. Heyder, G. Kuhn : International Journal of Fatigue Vol. 28 (2006), p. 627-634 [2] L.P. Pook : Engineering Fracture Mechanics Vol. 48 (1994), p. 367-378 [3] K.N. Shivakumar, I.S Raju : International Journal of Fracture Vol. 45 (1990), p. 159-178 [4] Z.P. Bažant, L.F. Estenssoro : Int. J. Solids Structures Vol. 15 (1979), p. 405-426 [5] J.P. Benthem : Int. J. Solids Structures Vol. 13 (1977), p. 479-492 [6] C.R. Pictu, V. Gupta : J. Mech. Phys. Solids Vol. 45 (1997), p. 1495-1520 [7] Z.Knésl, L.Náhlík, J.C.Radon : Computational Mat. Science Vol. 28 (2003), p.620-627

Û0k

Û O l

dFepm

Computational Assessments of Residual Stress Relaxation induced by Shot Peening J. Liu, Beijing Institute of Technology, Peking, PR China H. Yuan*, Bergische University of Wuppertal, Wuppertal, Germany M. Becker, MTU Aero Engines AG, Munich, Germany

Abstract: Shot peening is a wide spread manufacture process method to arise component fatigue strength. Generally it is assumed that the life extension is realized due to high residual stresses near component surfaces after shot peening. To quantify life improvement it is necessary to figure out interaction of residual stresses and applied mechanical loading. Experimental observation implies that the residual stresses vary with applied mechanical loading significantly which cannot be described by a common cyclic plasticity model. Based on experimental results the present paper suggests a multi-layer material model to simulate over-proportional relaxation of residual stresses. The computational results are in a good agreement with experiments. It implies strong strain-hardening occurring near the peened surface.

* Corresponding author. Email: [email protected]

Û O Û

Û O ¤

dFeo

GLASS FIBER REINFORCED PLASTIC LEAF SPRINGS FOR LIGHT WEIGHT VEHICLE SUSPENSION

G.S.Shiva Shankar1, B. Latha Shankar2, S.Vijayarangan3 1 professor, Mechanical Engineering Department, 2Research scholar, IEM Department, SIT, Tumkur -572103, Karnataka State, INDIA, 3 Director (Academic), Dr. Mahalingam College of Engineering, Pollachi, TN Phone:0816-2214001,2282696,2214000. Fax: 0816-2282994 E-Mail: [email protected]

ABSTRACT: One of the vital parts during the design of an automobile is the Suspension

system. The primary function of the suspension system is to minimize the transmission of road shocks. The role of composites in weight reduction and fuel saving in automobiles is highly significant. A combination of better material and better design approach will further enhance the weight reduction process. In the present work, multi leaf steel leaf spring is replaced by the optimally designed composite mono and multileaf leaf spring. A single leaf with variable thickness and width for constant cross sectional area of unidirectional glass fiber reinforced plastic with similar mechanical and geometrical properties to the multileaf spring, was designed, fabricated and tested for static strength and fatigue life. Design is also done by analytical method and values were compared with values obtained by using genetic algorithm (GA). The objective was to obtain a spring with minimum weight that is capable of carrying external forces with out failure. The finite element and experimental results has provided the necessary link between material properties, component design and fabrication to achieve performance optimization. Compare to steel springs, the composite spring has stresses that are much lower, the natural frequency is higher and the spring weight is reduced considerably. 1.0 FABRICATIONS OF COMPOSITE LEAF SPRINGS

. Fabricated composite mono leaf spring with and with out end joints and multi leaf springs are shown in Figure .

Figure: Fabricated composite multi leaf springs

Û O 0

2.0. RESULTS

The optimum values for the design variables, constraints and leaf spring weight obtained through the GA process. The obtained GA results were compared with experimental data. Results comparison of load, deflection, weight and stresses are shown in Table.

Table: Comparison results of load, deflection, weight and stresses Maximum deflection (mm) Maximum Stress (MPa) Material Static

load (N)

Experimental FEA GA Experimental FEA GA

Weight (N)

Steel 3980

107.5

90

- 503.3

511

- 260

COMPOSITE 1.Mono leaf 2.Multi leaf

4250

4050

105.0

98.5

94

-

119.8

-

473.0

487.8

466

-

386.8

-

38.8(Analytical)

18.04(GA) 82.3

4. CONCLUDING REMARKS

x The weight of the composite mono leaf spring can be reduced by 53.5% from 38.8

N to 18.04 N by applying the GA optimization technique. x Composite mono leaf spring reduces the weight by 85% for E-Glass/Epoxy over

conventional leaf spring. The reduction of 91%weight is achieved by replacing conventional steel spring with an optimally designed composite mono-leaf spring.

x Composite multi leaf spring reduces the weight by 68.3% for E-Glass/Epoxy over conventional leaf spring.

x From the results, it is observed that the composite leaf spring is lighter and more economical than the conventional steel spring with similar design specifications

x The study demonstrated that composites can be used for leaf springs for light weight vehicles and meet the requirements, together with substantial weight savings.

References

[1] Al-Qureshi, H.A. “Automobile leaf spring from composite materials”, Material processing Technology, 2001,Vol.118, pp.58-61. [2] Rajendran, I.and Vijayarangan.S. “Optimal design of a composite leaf spring using

genetic algorithm”, Int.J.Computers and structures, 2001,Vol. 79, pp. 1121-1129. [3] Daugherty, R. L. “Composite Leaf Springs in Heavy Truck Applications,” Proc.

Japan-US Conference, Kawata, K., Akasaka, T., eds., Composite Materials, 1981,Tokyo, pp.529-538.

[4] Dharam, C. K “Composite Materials Design and Processes for Automotive Applications,” The ASME Winter Annual Meeting, San Francisco, 1978, December 10-15, pp. 19-30.

Û OO

dFe q

'<1$0,&+<67(5(6,6%(+$9,252)70&367((/6070&$1',76

02'(/,1*

*DE&KXO-DQJ .\RQJ+R&KDQJ

,QVWLWXWHRI7HFKQRORJ\DQG6FLHQFH&KXQJ$QJ8QLYHUVLW\6HRXO.RUHD

'HSDUWPHQWRI&LYLODQG(QYLURQPHQWDO(QJLQHHULQJ&KXQJ$QJ8QLYHUVLW\6HRXO.RUHD

JDEFKXOMDQJ#JPDLOFRP FKDQJNRU#FDXDFNU

5HFHQWO\XVHRI70&37KHUPR0HFKDQLFDO&RQWURO3URFHVVVWHHOV LVJUDGXDOO\ LQFUHDVLQJGXHWR

LWVDGYDQWDJHVKLJKZHOGDELOLW\IRUORZFDUERQHTXLYDOHQW&HTKLJKVWUHQJWKDQGKLJKWRXQJKQHVV

70&3VWHHOVGXULQJG\QDPLFGHIRUPDWLRQVKRZGLIIHUHQWPHFKDQLFDOFKDUDFWHULVWLFVDQGK\VWHUHWLF

EHKDYLRUWKDQWKDWGXULQJVWDWLFGHIRUPDWLRQGXHWRWKHFRXSOHGHIIHFWRIWHPSHUDWXUHULVHDQGVWUDLQ

UDWHKDUGHQLQJ7RSUHGLFWK\VWHUHWLFEHKDYLRURI 6070&70&3VWHHODFFRUGLQJ WR.RUHDQG

6WDQGDUGXQGHUG\QDPLFF\FOLF ORDGLQJ LW LVQHFHVVDU\ WRGHYHORSD VWUDLQ UDWHDQG WHPSHUDWXUH

GHSHQGHQW K\VWHUHVLVPRGHOZKLFK FDQ GHVFULEH WKH VWDWLFG\QDPLFPHFKDQLFDO SURSHUWLHV DQG WKH

QRQOLQHDU K\VWHUHWLF VWUHVVVWUDLQ UHODWLRQVKLS 7KHUHIRUH WKH G\QDPLF K\VWHUHVLV EHKDYLRU RI

6070& LV LQYHVWLJDWHG E\ SHUIRUPLQJ WKHPDWHULDO WHVWVWHQVLOH DQG F\FOLF WHVW XQGHU VWDWLF

G\QDPLFORDGLQJ$QGEDVHGRQWKHWHVWUHVXOWVUDWHGHSHQGHQWK\VWHUHVLVPRGHOLVIRUPXODWHGDQG

DSSOLHGWRWKUHHGLPHQVLRQDOHODVWLFSODVWLFILQLWHHOHPHQWDQDO\VLV

6WUDLQ

6WUHVV0

3D

V

V

V

V

V

V

D

6WUDLQ

6WUHVV0

3D

V

V

V

F

7HPSHUDWXUH

R&

7HQVLOHDQG\LHOGVWUHVV0

3D

<LHOGVWUHVV

7HQVLOHVWUHVV

)LJXUHV7HVWUHVXOWVRI6070& DPRQRWRQLF ORDGLQJ IRU VWUDLQ UDWH EF\FOLF ORDGLQJ IRU

VWUDLQ UDWH FGLVWULEXWLRQRIPHFKDQLFDOSURSHUWLHV\LHOG VWUHVVDQG WHQVLOH VWUHVV IRU WHPSHUDWXUH

ULVH

Û O b

Û O c

dFeps


Analysis of structural performance of aluminium sandwich plates with foam-filled hexagonal cores

T. Sadowski, V. Burlayenko


The role of low-density structural polymeric foams filling the interstices of the cores of

metal sandwich plates is studied to ascertain the strengthening of the cores and the enhancement of plate performance under crushing and impulsive loads [1, 2]. Hexagonal

aluminium honeycomb and folded aluminium plate cores filled with different densities polymeric foams are investigated. The foam makes direct contributions to core strength and

stiffness, but its main contribution is in supplying lateral support to core members thereby enhancing the buckling strength of these members.

The aim of the paper was to estimate of the polymeric foam influence on the free vibration and buckling characteristics of the sandwich plates. The paper starts with the specification of

the sandwich plate geometries and material properties. The material comprising sandwich plate core and face sheet is aluminium. The details of the continuum constitutive model for

metallic sandwich core can be found in [3]. The foam material is a closed-cell PVC foam material, commercially available. Two different densities of foam were applied to the

analysis. The commercial finite element ABAQUS code [4] was used to simulate the dynamic and

buckling response of sandwich plates. In defining core and face sheet, the four-node, doubly curved layered shell elements S4 and S4R that are available in ABAQUS were utilized. These

elements are designed for both “thin” and “thick” problems of composite shells. Thus, core shear effect in finite element model was modeled by the first-order transverse shear flexible

theory. A limited investigation of the buckling response of the core is carried out to provide insight

into the effect of filling the core interstices with foam. The paper presents comparative results

Û Og

of load carrying capacity and imperfection sensitivity for sandwich plates without and with the foam for different boundary conditions. The problem of free vibration of the aluminium

sandwich plates was particularly analysed to assess the role of foam core on the natural frequencies and mode shapes of the plates. The calculated results for sandwich plates were

compared with those for plates made of convectional metallic construction having the same weight. Several eigenfrequencies and corresponding eigenmodes of the plates were estimated

and plotted. The obtained results lead to the conclusion that polymeric foam filler strongly influences the dynamic and stability responses of the sandwich plates. REFERENCES 1. Yasui Y. Dynamic axial crushing of multi-layer honeycomb panels and impact tensile behaviour of the components member. Int. J. of Impact Eng 24 (2000): 659-671. 2. Zhenyu Xue, John Hutchinson. Crush dynamics of square honeycomb sandwich cores. Int. J. Num Meth Eng 65 (2006): 2221–2245. 3. Zhenyu Xue, John Hutchinson. Constitutive model for quasi-static deformation of metallic sandwich cores Int. J. Num Meth Eng 61 (2004): 2205–2238. 4. ABAQUS User’s Manual, Version 6.6. Hibbit, Karlsson and Sorensen Inc.: 2001.

Û O h

dFejt


Modelling of temperature shocks in composite materials using a meshfree FEM

K.Nakonieczny 1), T. Sadowski 2)

1) Lublin University of Technology, Faculty of Mechanical Engineering 20-618 Lublin, Nadbystrzycka 36 str., Poland e-mail: [email protected]

2) Lublin University of Technology, Faculty of Civil and Sanitary Engineering 20-618 Lublin, Nadbystrzycka 40 str., Poland e-mail: [email protected]

This work discusses an application of a meshfree Galerkin finite element method to solving the temperature shock problem for a thin cylindrical plate. Shape functions used in the discretisation of the heat conduction equation are the Shepard interpolation functions. Various weights are used in the construction of the shape functions to find an efficient approach to the discretisation. Among them are the Swartz function and the splines. The domains of influence are chosen to be rectangles, so the weight functions are defined as tensor products of the reference 1D weights in r and z directions. The governing equation is discretised in time by using the explicit forward time stepping scheme.

The cmputations are performed on two model samples: ¦ the homogenous sample made of the pure Al203 material, and ¦ the composite plate made of the alumina/zirconium layers with variating weight content of

Zr02.

The description of the “position – dependent” properties of the analysed material is based on experimental data. Step distribution functions were constructed for the thermal conductivity and thermal diffusivity.

The heat transfer coefficient on the surface subjected to the thermal shock is modelled with the theoretical distribution function based on the experimental findings.

Û O k

The jet impingement cooling experimental data obtained for homogenous and composite plates are used for the verification of the computations results. The comparison shows good accordance between the theoretical predictions and the experimental findings. The computational costs of the applied meshfee method are not competitive with the costs of other methodologies in case of the regular domain analysed. However, the applications to complex geometry domains, which is natural field of the implementation of FEM are expected to be more promising.

Ûrbl

dFepu

Improved beam finite element for analysis of slender transversely cracked beam-

columns

0DWMDå6NULQDU

Faculty of Civil Engineering, University of Maribor

Smetanova 17

2000 Maribor

Slovenia

Tel.: +386 40 26 25 61, Fax: + 386 2 25 24 179, E-mail: [email protected]

Summary

This paper presents new geometrical stiffness matrix that is implemented in the

prediction of buckling load Pcr for slender beam-type structures with a transverse crack.

The matrix is derived on the basis of a simplified model, presented by Okamura, where

the crack is modeled as a rotational spring. This model is commonly used for the

computation of transverse displacements regarding transversely cracked slender beams.

Φ1 Φ2

Kr x,u

Y2 Y1

L

L1

y,v

1 2

Figure 1: Beam finite element with a crack represented by a linear rotational spring

ÛrbÛ

Although the first geometrical stiffness matrix for these problems has already been

presented previously, together with a stiffness matrix of a beam-column finite element

(both matrixes were derived by implementing Okamura’s model), this new matrix allows

modeling of compressive axial force that has a linear change over the length of the finite

element.

The non-uniform distribution of axial forces now increases the accuracy of the

computational model in situations where the axial force has a linear distribution over the

element’s length (for example due to self weight). Consequently, due to the improved

reliability of the model less finite elements are required. Therefore, the compact form of

the complete computational model, which is essential in inverse identification problems,

where, for example, the crack should be identified, is preserved.

Numerical examples covering several structures with different boundary conditions are

briefly presented in order to support the presented matrix. The results obtained using the

presented matrix are further compared with those values from more complex models,

thus clearly proving the quality of the presented compact FE model.

Some references

1. H. Okamura, H.W. Liu, C. Chorng-Shin, H. Liebowitz, A cracked column under

compression, Engineering Fracture Mechanics 1 (1969) 547-564

2. M. Skrinar, On the application of a simple computational model for slender

transversely cracked beams in buckling problems, Computational Materials Science,

(2007) Volume 39, Issue 1, 242-249

Ûrb¤

dFe

CALCULUS OF THE CYLINDRIC SPRING WITH AN UNIFORMLY VERTICAL LOAD BY TRANSFER-MATRIX METHOD

Mihaela SUCIU -¡ -¢¡£¤ ¥'¦2 -§¡¥ ¨¤©Dª«>-z¢¡£¬ ¥'®¢¡£ ¯¡-°²±S£ ¨©s³z´¡µ¥ §¶·N¸y -¹¡³z ªzº³»¼ -¢¡£¤ ½ªE-mail: [email protected]

ABSTRACT «¡±¨¾ ¿©À³zÝ¨¤¡±¹z°£ ¢zÁz±£ ±i¯¡-°N©À£ »_¹¡³°N¨- -¢z¨´²³°4 ¥¤³¨³z´£ ¢¡¿§z±¨¬°£¬>¿z³»< £ ¢z±ÃÂY®i±Ã£ ¢zÁ¨¤¡«° -¢z±S´²-°· -¨¤°£ ÄK-¨¬¡³z¿zªSÅ -¢Åw°£ ¨¾4¨¤D4Æ¡ k±S£¬Ç§¡ -¨¾£¬³¢z±Z³zó¨¤D4±¹z°£ ¢zÁ¡È ±¨¬¡³°N©ÉÅ£ ¨¬HÊw£ ° È ±Z -¢¡¿ÉËy -¯¡£ ±S£¤¿YÈ ±Ń³¢¡-¨-£¬³¢z±$ -¢¡¿/³¹¡-° -¨-³°N±$ -¢¡¿KÅ ¥¤-§¡¥¬ -¨¾¼¨¤¡<±Ã£ Ä+¥¤-»<-¢z¨¤±$³z´¨¤¡&³°£ Á¡£ ¢/±¨- -¨¾¼¯¡-¨-³°Â4Ì¼¡ -¯¡¿z¿§¡-¨-¿¨¬¡iÁ¡-¢¡-° ¥o-Äz¹z°-±±S£¬³¢<Ń³°¨¬¡Í«° k¢z±SŃ-°·² -¨¬°£ Ä³z´ w-©D¥¬£ ¢¡¿°£¬i±¹z°£ ¢zÁ¡ªÅ£ ¨¤ -¢< -¹z¹¡¥¬£¤ -¨¾£¤³¢Ń³° f°³§z¢¡¿s±¹z°£ ¢zÁ|Å£ ¨¬Î -¢|§z¢¡£¬Ń³°N»À¥ ©;¯¡-°N¨-£¬ ¥¥¤³z ¿;£ ¢Î£ ¨¤±¹¡¥¬ -¢¡fÏF´ ¨--°ª¡Å< k¢ ¥¬k§D¥¤ -¨¾ª4£ ¢Î ¥¤¥±¹z°£ ¢zÁ±S-¨¾£¤³¢z±Sª¨¬¡±¨- -¨¾¯¡k¨¾³°N±ÃÂÌ|¡ -¯¡;±¨¬§¡¿z¿W Î£ °k§¡¥¬ -°H±¹z°£ ¢zÁ?Å£ ¨¤= Î³¢z±¨- -¢Y¨;£ ¢¡-°N¨¾£¤ Î -¢¡¿¡ -¯¡ÎÐz-¹¡¿¨¤¡;±Ã£ Áz¢³¢z¯¡-¢z¨-£¬³¢z±· -±FÅF¥¬¥> -±F¨-³É¨¬¡_Æ¡ -»_±·5Ń³°¨¬¡¼£ ¢z¨¾-°N¢¡ ¥>´²Ń³°N¨¤±Sª4´²³°5¨¬¡¼¿z-¹¡¥¬ k»¼-¢z¨¬± -¢¡¿KŃ³°5¨¤¡-Äz¨--°£¬³°¥¬³z ¿±SÂc«¡_¿z-¹¡¥¬ -»¼-¢z¨¬±F³ÅF£ ¢zÁs¨-³5¨¤¡_-§z¨¬¨--°5Ń³°_£ ±y¢¡-Á¡¥¤¾¨-¿ÉŃ >¨-³5¨¤¡_¿z-¹¡¥¤ -»<-¢z¨¬±F¿§DÆz©_¨¬¡yŃ¥¤-Ä¡£¬³¢_»¼³»<-¢z¨wÑ²ÒÓ²ÂÌ¡ -¯¡y³±Ã£¬¿z-°¿f ¹¡ k°Nï³z´ y£ °-§¡¥¤ -°±Ô¹z°£ ¢zÁ¼Õ Ö ªÅF£ ¨¤¼ i»¼³Æ¡£¬¥¤°Ńk°-¢¡½±©z±¨¾-»*Õ ¨¾ª¢ Ö ªz³¢z¢¡k¨¾¿5 -¨ >k§z°N°-¢z¨¹¡³z£ ¢z¨×Õ²¦2£ Áz§z°fÒªz Â ª¡ÑNÒÓ Ö Â

y

xM

Ø

n t

Ù2Ú

Ù2ÛÜ Ý

Ü Ý

ÝHÞ Ù Ý

Ý?Þ Ù Ý

ß

ß

ß

Ú ÚÚ

y

xØ Øfà

á

xââ

M

¦2£ Áz§z°fÒÂÌi¡ -¯¡£ ±Ã³z¥¬ -¨¾¿f ¥¬£ ¢¡£¬w¥¬-»¼-¢z¨Í¿ÄÕN¦2£ ÁY§z°ÉÒÂ ª½Æ¡Â Ö · w-§z°N¯¡¿Z¥¬-»¼-¢z¨Í¿±_·-¡ ¥¤¥¬-¢zÁ¡¿>Æz©F¨¤¡-Äz¨--°£¤³°¥¤³z ¿±y¹¡Õ Ä Ö ¿ÄH· ´ ¨¾-°ãiÄH -Ä¡¥¬ -¢¡¿ÉÇzÕ Ä Ö ¿ÄH·Í ´ ¨--°5ã©K -Ä¡¥¬ÂSÌ¡ -¯¡Ń³°Í¨¤¡±¹z°£ ¢zÁH ±Ô¨¾ -¨-¯¡k¨¾³°5 -¨D¨¤¡-§z°N°-¢z¨'¹¡³z£ ¢z¨ª äå ªÅ£ ¨¬5±S£ Äs¥¬-»¼-°²¢z¨¬±Sæ

çèè ,,,,, éêëëìå íää ÕÒ Ö¦2³°¡¨¤¡±¹z°£ ¢zÁs¥¤¾»À-¢z¨¿Ä¡ªÅ> k¢Åy°£ ¨¾¨¬¡>-Äz¹z°k±±S£¬³¢z±wÕ²î Ö æ

rhfEI

vffhfEI

TghfEI

ThfEI

Mv

rhgEI

ugfhgEI

TghgEI

ThgEI

Mu

hrEI

fhEI

TghEI

ThEI

M

qTT

pTT

rfTgTMM

ïð

ïð

ïðïïðð

ïð

'1

'1

'1

'1

'1

'1

'1

'1

1111

ñññññ

ñññññ

ññññòññòññ

ñññ

Õ²î Ö

Ì¼ -¢HÅw°£ ¨-¼ _»¼ -¨¬°£ ÄH°¥¬ k¨¾£¬³¢HÆ¡-¨¬ÅF-¢K¨¤¡£ ±Z±¨- -¨¾¯¡-¨-³°ª´²³°5¨¤¡±S-¨-£¬³¢KÄ -¢¡¿É¨¬¡±¨- -¨¾¯¡-¨-³°4Ń³°¡¨¤DD±S-¨-£¬³¢sózæ

Ûrb0

ôõ ö ÷ø÷÷ õõùõ ô ú²ûüýyþ¡ÿÿ ö þ¡ÿ ÿ þ¡ÿ ¡ÿ¡ÿ ýFÿÿ þ¡ÿ ÿ þ¡ÿ ÿ ! "õ þ¡ÿ ÿ$#¡ÿ þ¡ÿ ÿÿ ÿ% & þ¡ÿ ÿ '( ' ýFÿ)5ý* ÿ ö

+-,./0

+-,1/20

+3,/0

45

+

67

889

.. ,./01:,./20,./0

1. ,1/01:,1/0,1/20

. ,/01,/0,/20

.1

67

889

;<

;<

'1

'1

!

10'1

'1

'1

01'1

'1

'1

001111

000100

000010

0001

==

>>>>>>

ú?ü

@ ÿþ#¡ÿ$ - ÿ! ö

AA

BDCCEE

BDCCFF

GHGH

úJIüK ÿ þ¡ÿ)L ML M !ý þ NPO ' ýÿþ#¡ÿ þ¡ÿ$¡ÿ¡ÿQ-L - ÿ- RJ þ¡ÿ S ( úTü ö

10sin2cos2sin21

2sin2cos1sin2

0102sin2cos122cos2sin23

cossin2

001sin2

cossin22

000100

000010

000sin201

UUV

UUV

VV

WX2YW

XYW

XYW

X2YW

X2YW

XYW

XYW

XYW

XYW

W

Z)[ úTü

\ þ¡ÿ$#¡ÿ []^ (_ þ¡ÿ$ÿL úOü ý L L¡ÿ)3L ML ÿ! (( þ¡ÿ>ÿ ( ÿ !zÿ ` L ! a@ ÿ)5ý* ÿ þ¡ÿ)!zÿ ÿ ' ý þcb -&d !fewÿQ# !zÿgd h ! ¡ÿ- Ãö

i0

jjkmljkjn

úoü@ ÿþ#¡ÿp(L ÿ!zÿ! þ (( -þ p ` L ! ( ( ' ÿq¡ÿQ!zÿ!r m s ý ÿ! ' -þL L QS¡ÿ3!ýyþ $#¡ÿ L3M( RL ` L ! L ¡ÿt!ýwþ uNv u ! O Nswx u a\ þ¡ÿ \ ÿ yz yÿ þ ! #¡ÿ ` ÿ `| (L ÿ! þ¡ÿ ( (LML M a @ ÿ~Q

( ( þ LML M !yý þ þ !zÿ ' ÿ!* ` þ¡ÿfyz \ - ÿQhyÿ þ ! ' ýÿfQ3L ML ÿ!-Q !(L `'$ þ¡ÿ ÿ |! þ¡ÿFÿ! ÿ ($ þ¡ÿ ( ( '& þ¡ÿ wO ÿL¬ÿ¼ÿ c($ þ¡ÿ ý ÿ#¡ÿ a @ þ þ ÿ ML ' ýFÿ3L ML ÿ3!3L L þ¡ÿ ÿ#¡ÿ LL ÿ J þ¡ÿ rL¬ÿ ÿ ( ( aq(*( ÿ#¡ÿ ' \ ÿ yz yÿ þ ! ' M(! ( ( ' b -gd ' eyÿ# !zÿgd a*22h22&2 t¡¢ £¤¥¦ §¥¨©¢ ª«2¢ ¬ ®¡¥¯°± ²®Qª³¥´®µ¯¶¥´®·¦ m²¥¦ ²¸D¦¶®·®¯ ¸²¯°¸®Q£¹¶± ¯°± ¨´®·º ¨¦ ¦ ®£»(¥± ®£g¼½¾ ¿g³± À¥m¡¢ ¬pÁS¯ ´§Ât¨µr¡¥¯ ± ¥¦ ®£ºÃ2»ÅÄ¸²Â¥®¯ £Æ¨DÇp¥´± ¥£g¼È½ ¾&É¥´t¤$¢Ês¢ ¬*ÆËs«2ÆÌ2ÍÁÅÎ(¨Ç)¸D¦ ¥®µ¨&ÁS¯ ®®·ÏÁS¯ ¥± ´£Èª¯ ÂtÐ¶D± ¯ ± ¨´£¡²s¥Ñ2Òs± ¦°¦Ä¨¨Ót¤$¨Ç)À¥´ £g¼Ô¼

Ûrb O

PRQTSUSWVYX[Z \ \

BED ^ :<;? ^ M

Ûrbb

Ûrb c

ddfe,d

Numerical failure analysis of dual phase steel using cohesive modelling on real microstructures

U. Prahl1, B. v. Binsbergen2, C. Thomser1, V. Uthaisangsuk1, W. Bleck1

1Department of Ferrous Metallurgy, RWTH Aachen University,D-52056 Aachen

2CORUS Research Development & Technology, NL-1970 IJmuiden

Abstract

Modern modern multiphase steels gain interest in automotive application due to theirattractive mechanical properties that are characterised by highstrength and gooddeformability in parallel. The reason of this property characteristic can be found inthe distribution of hard martensitic particles in a ductile ferritic matrix.

The current work aims the prediction of damage and failure in dual phase steels. Amicrostructural based approach by means of representative volume elements is usedtaking into account carbon partitioning for the flow curve description of ferrite andmartensite. Real microstructures are investigated in a twodimensional approach andcohesive zone modelling has been used for the debonding of martensite ferriteinterfaces. The parameter identification for the cohesive model is based onmetallographic investigations combined with calculated stress strain distribution inthe heterogeneous microstructure.

The investigations give a physical based correlation between multiphasemicrostructure and mechanical properties in terms of flow curve and failure of dualphase steel for automotive applications.

Ûrb g

Ûrb h

ddfe i

Interface damage of SiC/Ti metal matrix composites subjected toaxial shear loading

M. M. Aghdam∗, M. Gorji, S.R. Falahatgar

Department of Mechanical Engineering, Amirkabir University of Technology,

Hafez Ave., Tehran, Iran

Abstract

A three-dimensional micromechanical finite element model is developed to studyeffects of interface damage on the behavior of a unidirectional SiC/Ti-6Al-4V metalmatrix composite (MMC) subjected to axial shear loading. Effects of variousimportant parameters such as manufacturing process thermal residual stress, fibercoating, interface bonding and fiber volume fraction are included in the model. Themodel includes a representative volume element (RVE) consists of a quarter of SiCfiber covered by interface and coatings which are all surrounded by Ti-6Al-4Vmatrix. A suitable failure criterion for interface damage is introduced to predictinitiation and propagation of interface de-bonding during shear loading. It is shownthat while predictions based on perfectly bonded and de-bonded interface are far fromreality, the predicted stress-strain curve for damaged interface demonstrate very goodagreement with experimental data. It is also revealed that inclusion of thermal residualstress and fiber coatings in the model are necessary to correctly predict shear strengthof the SiC/Ti composite material.

Keywords: Micro-mechanics; Metal matrix composites; Axial shear loading; Interface

damage; Finite element; Thermal residual stress

∗ Corresponding author. [email protected] (M.M. Aghdam), Fax: +98-21-641 9736 Tel. +98-21-6454 3429

Ûrbk

Û c l

ddfepm

A 3D finite element modelling of crack propagation under fretting fatigue conditions

H. Benzaama, E Giner *, Aour B, F.J. Fuenmayor, S.M. Elachachi,

Département de génie mécanique, ENSET. Oran BP 1523 El-Mnaouer, Oran 31000, [email protected], Tel. 041-58-20-64, Fax: 041-58-20-66

ABSTRACT This work is focused on the evolution of the propagation of circular and elliptic crack form, inclined or not using finite element analysis in three dimensions under conditions of fretting fatigue with complete contact. A structured mesh has been designed with a discretized crack like an independent module, in which the geometry can parametrically be varied. The initial crack is supposed to be nucleated at the end of the complete contact zone between the two bodies in contact (specimen and bulk). We simulate the propagation of the crack according to the loads applied to the specimen, by taking into account the influence of the strong gradients of tensions in contact and their effects corresponding to the orientation of the crack with a possible propagation under conditions of mixed mode loading (I, II and III). The form of the crack and its dimensions can be varied parametrically to model the orientation according to the values of the stress intensity factors SIF (KI, KII, KIII). The criterion of orientation of crack in 3D based on the maximum principal tension (criterion of Schöllmann) and his approximate equivalent criterion of Richard have been used. The study was applied under conditions of axial loading with variable amplitude to the specimen by solving the non linear contact problem by finite element using the commercial code ABAQUS 6.5. The bulk is in complete contact with the specimen under the action of normal loading, and the conditions of fretting fatigue are due to the variable extension of the specimen along the loading cycle. Preliminary examples of the MFEL with known solutions have been used to validate the suggested methodology. Keywords: Finite Element, crack propagation3D, mixed mode, fretting fatigue, complete contact. INTRODUCTION Fretting fatigue is a widespread phenomenon that occurs in a vast number of engineering components where two bodies are in contact and undergo a small oscillatory slip and at least one of the bodies is experiencing a bulk cyclic loading. Fretting fatigue is often the root cause of fatigue crack nucleation. This implies that an initial crack has propagated due to cyclic loading. These initial cracks will grow as two-dimensional surfaces in a three-dimensional body. Generally, an initially plane crack can warp due to mixed-mode conditions. Therefore, a procedure to calculate crack propagation must be capable of dealing with arbitrarily curved crack shapes in three-dimensional space. One of the first techniques used the boundary element method [Wawrzynek, Martha, and Ingraffea (1988), Aliabadi (1997), Wen, Aliabadi, and Young (2004)]. The principal advantage of this method is that only the boundary of the structure has to be meshed. The BEM requires only 2D meshing of the crack surface and the outer boundary. Several examples of such an approach are provided in Bonnet (1994). Methods based on integral equations are especially attractive for infinite bodies since the sole meshing of the crack surface is then necessary; see for instance the examples provided [ Fares (1989) and Xu and Ortiz (1993). If, in addition, the crack propagates along a plane, compelling methods requiring the sole 1D meshing of the crack front are envisageable. Using an earlier theoretical work [Rice (1989) and Bower and Ortiz (1990, 1991, 1993) proposed such an approach and applied it to various problems of practical interest. [Lazarus (2003) later defined a simplified variant of this method which resulted in no significant loss of numerical accuracy. Alternatively, one can use extended finite elements [Gravouil, Möes, and Belytschko (2002), Sukumar et al. (2003)]. This is a recent technique, in which the crack propagation is independent of the mesh. Therefore, the method has similar advantages as the boundary element method, and this to an even higher degree: the problem of remeshing does not arise. In another, the hexahedral technique developed at MTU, This technique has a long history: the first articles describing the procedure appeared in 1998 [Dhondt (1998)]. It has been applied to aero engine components on a regular basis since 1995. The technique has been proven to yield excellent results for Mode-I applications, exhibiting amazing capabilities describing crack propagation across corners and other geometric discontinuities [Dhondt (2005)]. An extension to real three-dimensional mixed-mode

Û c Û

crack propagation calculations has been proposed and applied to a three-point bending specimen. However, the artificial extension of the triangulation of the crack shape always remained a problem and been finally presented as new technique to solve that problem. An overview is given of all necessary steps to apply the procedure in [Dhondt (2005)]. Finally, more traditional approach the finite element method is used en two and three dimensions. A nontrivial example involving a complex, non-planar crack shape is provided in the work of Xu et al. (1994). [Richard, Fulland, Buchholz, and Schöllmann (2002),Schöllmann, Fulland, and Richard (2003), Buchholz, Chergui, and Richard (2004)]. There, the crack front is surrounded by a flexible cylinder consisting of hexahedral elements, whereas the remaining space is filled with tetrahedral elements. In fact, the structure without crack is meshed with tetrahedra and only the immediate neighbour hood of the crack front is remeshed. In this work a finite element method using a parametrical mesh with a crack as an independent module is presented. The two bodies (bulk and specimen) are in complete contact under conditions of fretting fatigue. The semi elliptic crack plane can be inclined as shown in fig.1. The use of linear quadratics elements is recommended for complete contact in fretting fatigue. The Abaqus 6.5 code is used to FE analysis. The criterion of Richard is used for 3D crack propagation in mixed mode.

Figure 1. Specimen and bulk. Figure 2. FE mesh under fretting fatigue conditions Abaqus 6.5

Figure 3. Crack propagation for a number of cycle (N1=1000, N2=500, N3=200, N4=100).

References : [1] Shivakumar,K.N. y Raju,I.S., "An equivalent Domain Integral Method for Three-Dimensional Mixed Mode Fracture Problemes", Engineering Fracture Mechanics, vol.42,Nº6,1992,935,959 [2] Nikishkov, G.P. y Atluri, S.N., "Calculation of Fracture Mechanics Parameters for an Arbitrary Three-Dimensional Crack by the Equivalent Doamin Integral Method", International Journal for Numerical Methods in Engineering, vol.24, 1987, 1801-1821. [3] Gerstle W.H., Martha L., Ingraffea A.R., "Finite and Bondary Element Modeling of Crack Propagation in Two and Three Dimensions". Engineering with Computers 1987; 2:167-183. [4] Xu Y., Moran B., Belytshko T. "Self-semilar carck expension method for three dimension analysis". ASME, Journal of Applied Mechanics 1997; 64:729-737. [5] H. A. Richard M. Fulland, M. Sander” Theoretical crack path prediction”, Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 28, 3–12, 2005 [6]Buchholz, F.-G.; Chergui, A.; Richard, H. A.” Fracture analyses and experimental results of crack growth under general mixed mode loading conditions”. Engineering Fracture Mechanics, vol. 71, pp. 455–468. 2004 [7]Wen, P. H.; Aliabadi,M. H.; Young, A.” Crack growth analysis for multi-layered airframe structures by boundary element method”. Engng. Fract.Mech., vol. 71, pp. 619–631. 2004 [8]Dhondt, G. (1998): Automatic 3-D Mode I crack propagation calculations with finite elements. Int. J. Num. Meth. Engng., vol. 41, pp. 739–757. [9]Dhondt, G. (2005): Cyclic crack propagation at corners and holes. Fatigue Fract. Engng. Mater. Struct., vol. 28, pp. 25–30.

Û c ¤

ddfeo

!

"#$ %&

$'())* *'+)),)+ -()) "", -.-() "-,/& ",0012034, * 567830//93:209;(<67830//93:209;1' 65*4=4'545*+5 >

>** ? @)' .- + )(, +'. *' )5*"(*@-**)()A,:,8B5().-**'*+' *'(@**?ACB5

'D' *+@ ? ."*5 '*)' +)+ '+()+*@-)"'+ E.F -("5)'*' ?*@-*.5" @ '* (*+@ ?*E"5F5

"5)'(."*?

. " - * ' () '* * )'+@5 )()'+@5C9''54*' * +* '* " ( *. @*" )' * ' +@5 *. -050C'')5*.D+@* '()'* *4(*A/B5)E*F*'*()"( *"5' (+)( *")"4"5/5

!"#$

%& '&

"5/' .5)"( *"

Û c 0

+') '* * *' '* ' () (*)(* ')() *"5*) 4'+ ( +(?*)'+@5

G+ ) ) +* -*) .- +* *)@*+) ()+ +) .**.5

:$ )' '* ) +* ) ' 4 ' )6H+"I '*+ , , '*+ *)+ *), -)* - " AB5)) ( )'4@* ('<' ."5

4 @ '* @ +" () + ( )() )''E&>F+*)"8/00/5

!"

AB 5 5>+",55),5 55G4, 5 -) @- ,&/00/5

A/B 5 $4, 5 #"*, J5 G*#', :$ +') ." ( **>**J()(+)+,:/* )()"4* &/009,41+@ 1 )#4, *,01:'@/0095

A:B J5G*#',''*)(+)+ @**?E F,,/00/5A8B J5 G*#', 5 )#+, +') '* " ( @** ? +@?)*

*"E F,>+ -8,/00C,#45ACB J5G*#',5)#+,$'"')) (@**?E

F,) "+'-#)?K+:,8D/0085A9B $5 5 +, G5 J5 &-, 5 5 > ),

,*> 1@-""(),* "*'+ , ,,/005

AB !"#$% ,L/00C5

Û c O

PRQTSUSWVYX[Z \ v

@C x @CMwGD ^ ;£D ^ :<Ga;>=aM

Û c b

Û cc

d ie,dFracture mechanics assessment of singular stress concentrations

caused by fibres at a free surface

J. Klusák, Z. Knésl1 Institute of Physics of Materials, Academy of Sciences of the Czech Republic

Žižkova 22, 616 62 Brno, Czech Republic [email protected]

Failure in composite structures frequently starts at singular stress concentrators at the surface of the structures. Typically, in fibre composites failure initiation of the composite may be induced by singular stress distributions caused by the ends of fibres touching the free surface. The geometry of the stress concentrator leads us to model such a configuration as a bi-material notch. The basic feature of the solution in this case is the existence of singular stress distribution with a type of the singularity different from those of a crack. Consequently the approach of standard linear elastic fracture mechanics cannot be applied directly. In the previous papers, see e.g. [1], the singular stress distribution around the free end of a fibre has been analysed and the generalized stress intensity factors have been evaluated for various geometrical and material configurations. The present study follows it by applying the stability criteria suggested in [2], [3]. The objective of the presentation is to express the conditions under which a crack is initiated in the vicinity of a free fibre end, and to estimate the presumed crack initiation direction. On the basis of the results the question of interfacial debonding between the fibre and matrix is discussed. Generally the stress distribution around the free fibre end represents an inherently combined mode of normal and shear loading and the angles of the presumed crack initiation directions have to be estimated first. From the basic assumption of the generalized strain energy density factor (GSEDF) it follows that the direction of propagation coincides with the direction of its local minimum. Therefore, the GSEDF was calculated and the angles of the presumed crack initiation directions were estimated for varying ratios of elastic moduli of a matrix EM and fibres EF. Then the conditions for crack nucleation were applied and the corresponding critical stress for crack initiation was evaluated. The critical stress for crack initiation in the matrix and the conditions for interfacial matrix fibre debonding are estimated for various geometrical configurations and different values of material properties of the matrix and the fibre and their adhesion. The results facilitate to predict whether a crack initiated due to a free fibre end will propagate into the matrix or along the interface. Acknowledgements The authors would like to thank the Czech Science Foundation (grant 101/05/0320) for financial support. [1] Noda, N.-A., Shirao, R., Li, J., Sugimoto, J.-S.: Intensity of singular stress fields causing interfacial debonding at the end of a fiber under pullout force and transverse tension., Int. J. Solids Struct. (2007), doi10.1016/j.ijsolstr.2006,11,034 [2] Klusák, J., Knésl, Z.: Determination of crack initiation direction from a bi-material notch based on the strain energy density concept, Computational Materials Science (2007), Volume 39, issue 1, pp. 214-218. [3] Klusák, J., Knésl, Z., Náhlík, L.: Crack initiation criteria for singular stress concentrations, Part II: Stability of sharp and bi-material notches, Engineering mechanics (2007), to be published.

Û c g

Û ch

d ie i

Fatigue damage simulation in hybrid Titanium-PEEK /AS4 composite laminates

Parya Naghipour, Joachim Hausmann, Marion Bartsch, Heinz Voggenreiter

German Aerospace Center (DLR), Institute of materials research, Cologne-Germany

Abstract Hybrid laminates made of carbon fiber reinforced plastic plies (CFRP) with metal foil plies are being developed

to satisfy the requirements for future stiffer aerospace applications. The titanium protects the CFRP core from

environmental and temperature-dependent effects such as oxidation and moisture, as well as potentially

providing improved impact resistance and bearing properties. The composite core has higher strength and

stiffness to weight ratios than the monolithic titanium and is presumed to be less sensitive to fatigue effects [1].

The combination of the two materials as a hybrid composite is beneficial, because it provides the ability to utilize

the attractive aspects of the two component materials whereas their weaknesses are avoided. A schematic of the

laminate is shown in Fig. 1.

Fig. 1: Schematic view of Ti- CF/PEEK layup.

Hybrid laminates typically exhibit complex damage states as a result of mixed mode failure and fatigue loading.

Often different interlaminar and intralaminar damage modes interact with one another to cause structural failure.

Currently hybrid materials undergo extensive fracture and fatigue experimental programs to verify their

performance before they are certified for use in aircraft design. Nevertheless, the time and high cost of

conducting these experiments has led to a significant volume of numerical research, on fracture and fatigue

damage modelling. There are two principal objectives for this study. The first objective is to analyze the fracture

failure of unidirectional orthotropic CFRP (PEEK/AS-4 with 40% fiber volume fraction) core with a numerical

delamination meso-model, which is composed of individual laminas plus interface elements, under mixed mode

loading. The model will then be validated through mixed mode bending experimental results. The second goal is

the addition of titanium layers and investigation on the fatigue damage evolution in titanium facesheets and the

composite core numerically and experimentally. The broader goal is to improve the structural durability

according to design parameters such as: layup configuration, total thickness, and metal volume fraction.

The particular critical failure mechanism is supposed to be the coupled growth of cracks in the titanium

facesheets and interlaminar delamination between facesheet and the composite core that occurs in the wake of

the crack. In order to model the coupled damage growth, the behaviour of the individual damage modes, i.e.

crack propagation in titanium and interlaminar or intralaminar fracture of the CFRP core, will be studied

experimentally and numerically. In the CFRP core, assuming the lamina as an orthotropic homogenized

continuum in plane stress permits the modelling of damage by three independent non-negative damage

parameters, df, dm, and ds which reduce the stiffness numerically in fiber, transverse and shear directions

Û c k

respectively. This is caused by initiation, growth, and coalescence of micro-cracks. Therefore the compliance

matrix degrades as [2]:

21

11 22

12

11 22

12

10

(1 )

1( ) 0

(1 )

10 0

(1 )

f

m

s

d E E

H wE d E

d G

Q

Q

§ ·¨ ¸¨ ¸¨

¨ ¸¨ ¸

¨ ¸¨ ¸¨ ¸© ¹

¸ (1)

where E11, E22 and G12 are the longitudinal, transverse elastic and in plane shear modulus of the orthotropic

lamina. The constitutive law for the interface, between titanium facesheet and CFRP-core or in between CFRP

laminas, is the general relation, which connects the traction vector, 2, to the vector of displacement

discontinuities, /. In the interlaminar cohesive elements an isotropic damage growth as a function of mixed mode

relative displacement, , with a unique damage variable, d, degrades the stiffness [3]. m

maxG

)(

)(

0max

0max

mfm

m

mmmfd

GGG

GGG

Where 0< d <1 (2)

Fig. 2: Schematic of mixed mode softening law [2]

Titanium facesheets will absorb energy undergoing plastic deformation and ductile damage with a similar

methodology to the abovementioned stiffness reduction.

Propagation of the abovementioned damages results in a stiffness reduction in the laminate that will lead to

catastrophic failure. At the end, the procedures used to develop the models, will be analyzed to determine their

applicability to similar problems in hybrid laminates. Results of mixed mode bending tests of CFRP panels and

their usage for modelling of hybrid laminate will be presented.

References: [1] De Vries, T. J., Blunt and Sharp Notch Behavior of Glare Laminates, Ph.D dissertation, Delft University

Press, 2001.

[2] Davila, C. G., and P. P. Camanho, Analysis of the Effects of Residual Strains and Defects on Skin/Stiffener

Debonding using Decohesion Elements, SDM Conference, Norfolk, VA, April 710, 2003.

[3] O. Allix, L.Blanchard, Meso modelling of delamination: towards industrial application, Composites

science and Technology 66., 2006, pp. 731-744.

Û g l

d iepm

Analytical elasticity solution for general laminated long rotating

cylinder under various loading conditions

I. Ahmadi, M. M. Aghdam∗

Mechanical Engineering Department, Amirkabir University of Technology,

Hafez Ave., Tehran, Iran

ABSTRACT

A displacement based discrete-layer analytical solution is presented for a thin/thicklaminated composite cylinder subjected to various loading conditions. Loading includesaxially uniform extension, torsion, thermal load, internal and external pressure.Centrifugal forces due to a constant speed about longitudinal axis are also considered. Ageneral form of stacking sequence for layers of the laminate can be considered. It isassumed that the cylinder is long enough to ignore edge effects. A reduced displacementfield is determined for each layer of a long general laminated composite cylinder usingthree-dimensional elasticity theory. The governing equations of each layer are solvedanalytically while continuity conditions are imposed for out of plane stresses and radialdisplacement within the interfaces of layers. It is analytically observed that inter-laminar shear stresses vanish in the long composite cylinders. Numerical results arethen presented for the various stress and displacement components through thethickness of the laminated composite cylinder under rotation, torsion, pressure andthermal loads. Results show close agreement with other analytical and numerical dataavailable in the literature. The presented analytical solution can be used for verificationof other numerical techniques for the same problem.

Keywords: Laminated composite cylinder; Discrete-layer theory; Analytical solution;

Elasticity formulation.

∗ Corresponding author. [email protected] (M.M. Aghdam), Fax: +98-21-641 9736 Tel. +98-21-64543429

Û g Û

Û g ¤

d ieoNUMERICAL ANALYSIS OF THE ENERGY LOCALIZATION

IN HIGH-CONTRAST DENSE-PACKED COMPOSITES

Kolpakov A. A.

Mechanics and Mathematical Department, Novosibirsk State University

324, Bld.95, 9-th November str., Novosibirsk, 630009 Russian


The effect of energy concentration between the closely placed inclusions in high-contrast composite

material attracts attention of numerous researchers. The papers [1-7] (the list is not complete) tackle this

effect in the analytical context. The paper [6] demonstrates that the high-contrastness and close placing

are necessary but not sufficient conditions for the energy concentration and approximation of the

continuum problem with so-called "network" models widely used in natural sciences [8]. That is why the

verification of the effect of the energy concentration is a problem of both scientific and practical value.

The most mathematical results were obtained for perfectly conducting inclusions under condition

that the distances between the neighboring inclusions tend to zero. In practice, the conductivity of

inclusions is large but finite and the distances between the neighbor inclusions are small but do not

vanish. There is no reason to expect exact solution of the thermoconductivity problem for the composite

material of random structure. That is why the author analyses the problem with numerical (finite element)

method.

Another problem analyzed in the paper is the problem of the approximation of the temperature of

inclusions with network model. This problem is new.

Analyzing specific problem formulated above, the author used standard finite element approach and

the general purpose finite element computer programs. The high accuracy solutions were obtained due to

the use of modern computer, relatively simple geometry of domains and the special mesh refinement.

Numerical methods developed for high-contrast composites of complex structure can be found in [9-12].

The methods [9,12]can be applied to analysis of both linear and nonlinear composites.

The author obtained following results:

1) Numerical analysis of the thermoconductivity problem for high contrast dense packed composite

materials predicts the transformation of the phenomenon of the energy concentration to the phenomenon

of the energy localization for high-contrast high-filled composites and arising of the "energy channels".

The "energy channels" start to take specific shape for the value of contrastness and the relative

distance between the neighbor inclusions

10tc

1.0dG . However, the distribution of the density of the energy

in the "energy channel" becomes stable (depends only on the relative distance between the inclusions and

negligibly depends on the contrastness) for the contrastness . 1000tc2) The network models start to approximate the corresponding continuum problem with accuracy 10%

(this accuracy can be accepted in material science) the relative distance between inclusions 015.0dG

and the contrastness . 1000tc

Û g 0

3) The phenomenon of the energy concentration in domains between the neighbor inclusions takes place

for nonlinear high contrast dense packed composite materials, although the structure of the "energy

channels" is not identical to the structure of the "energy channels" in linear composite materials.

4) The author formulates a new problem of approximation of temperatures of high-conducting filling

particles with solution of network model and demonstrates numerically that solution of the network

problem approximates the temperatures of the inclusions. The author assumes such kind of approximation

to be the general property of network approximation.

5) The phenomenon of energy concentration looks like a prospective method for developing of non-

linear composites possessing the required properties. We introduce the notion of the coefficient of the

amplification of tunability in the composite material and demonstrated that it has the order of G

R for

high-contrast dense-packed composite materials.

The results obtained in the paper for thermoconductivity problem can be expanded to a wide range of

transport problems (diffusion, electrostatics and so on).

References 1. Flaherty J.E. and Keller, J.B. (1973). Comm. Pure Appl. Math. 26. 2. Keller J.B. (1963). J. Appl. Phys. V. 34. N 4. 3. McPhedran R., Poladian L. and Milton G.W. (1988), Proc. R. Soc. London. A, 415. 4. Borcea L. and Papanicolaou G., (1998) SIAM J. Appl. Math. V 58. N 2.. 5. Berlyand L. and Novikov A. (2002) SIAM J. Math. Anal. 34(2). 6. Kolpakov, A.G. (2005) J. Appl. Mech. Thech. Phys. N3. . 7. Berlyand L. and Kolpakov A. (2001). Arch. Rational Mech. Anal. 3. 8. Sahimi, M. (2003) Heterogeneous Materials. Springer, New York. 9. Moulinec H. and Suquet P. (1994) Comt. Rend. Seances Acad. Sci. Ser.II. 318. 10. Michel J.C., Moulinec H. and Suquet P. (2000). Comp. Modeling Engng Sci. 1(2). 11. Michel J.C., Moulinec H. and Suquet P. (2001) Int. J. Numer. Meth. Engng. 52. . 12. Vinogradov V. and Milton G.W. (2005) Advances Comp. Experim. Enging Sci. An example of numerical computations: field distribution between two densely-paced inclusions in high-contrast composite

Û gO

PRQTSUSWVYX[Z \T

:wMDF;>= xK=a;MD<Ga?GD

Û g b

Û g c

d³me,d

Crystal plasticity modelling of texture

development in TWIP steel

M.A. Melchior*, P.J. Jacques** and L. Delannay*

*Department of Mechanical Engineering (CESAME-MEMA),

Universite catholique de Louvain (UCL),

batiment Euler, av. G. Lemaitre 4, 1348 Louvain-la-Neuve (Belgium)

**Department of Materials Sciences and Processes (IMAP),

Universite catholique de Louvain (UCL),

Place Sainte Barbe 2, 1348 Louvain-la-Neuve (Belgium)

The steel grade studied here presents an excellent combination of ductilityand strength. It is called TWIP (TWinning-Induced Plasticity) steel becausethe increased hardening capacity is attributed to the formation of very thintwin lamellae inside the grains. Twin boundaries act as barriers to dislocationmotion, leading to anisotropic hardening of the slip systems in the parent grain.The twinned regions themselves are probably too thin for plasticity to be achie-ved by conventional dislocation slip. Hence, twin lamellae can be regarded ashard inclusions embedded in a soft matrix.

Kalidindi [1] proposed an efficient mathematical formalism for the incorpora-tion of mechanical twinning in crystal-plasticity models. His assumption of ahomogeneous deformation of the parent grain and the twin tends, however, tocontradict the fact that the two regions have a very different strength in thecase of TWIP steels.

In the present study, it is assumed that the heterogeneity of strain betweenthe parent grain and the twin amounts to a simple shear parallel to the twin-ning plane. Hence, twin boundaries act as flat, coherent interfaces. Such partialrelaxation of the geometrical constraints reduces the deformation of twinned re-gions (due to their hardness) whereas the macroscopic deformation is achievedon average over one or several adjacent grains. The amplitude of the relaxationis computed by minimising the collective plastic work [2].

Simulations of cold rolling and uniaxial tensile tests have been performed withthis modelling of the twin-grain interaction and also under the hypothesis ofa uniform strain within each grain. In the latter case, several models of graininteraction have been considered (Taylor FC, ALAMEL and FEM). Textureevolutions have been compared to experimental measurements carried out onFe-Mn-C alloys in which the occurrence of mechanical twinning is controlled

Û gg

both by the chemical composition and the testing temperature.

References

[1] S.R. Kalindindi, ”Incorporation of deformation twinning in crystal plasticitymodels”, J. Mech. Phys. Solids, 46, (1998), 267–290,

[2] L. Delannay, R.E. Loge, J.W. Signorelli, Y. Chastel, ”Evaluation of a multi-site model for the prediction of rolling textures in hcp metals”, International

Journal of Forming Processes, 8, (2005), 2-3, 131–149

Û g h

d³me i

Abstract for an oral presentation at IWCMM17

Authors:

S. Kaßbohm1, B. Fedelich, M. Budnitzki, D. Noack

Address:

Federal Institute for Materials Research and TestingBundesanstalt fur Materialforschung und -prufungFachgruppe V.2, WerkstoffmechanikUnter den Eichen 87, 12205 Berlin

Title:

Modelling of Creep Damage in Polycrystals: The Combination of Crystal Plasticity withCohesive Zone Models for Grain Boundary Cavitation

Abstract:

The creep behaviour of polycrystals comprising grains and grain boundaries is studied.At high temperatures, creep in the grains is coupled with intergranular damage. Thelatter being controlled mainly by cavity nucleation [Riedel 87], cavity growth (throughgrain boundary diffusion and surface diffusion) and power-law creep [Weertman 74,Beere 78, Chen 81, Svensson 81, Cocks 82]. During cavity growth the grain boundariesbecome thicker, which leads to a stress redistribution in the grains.

As an introduction, the crystal plasticity model for the grains as well as a new cohesivezone interface model for the grain boundaries are described. The crystal plasticitymodel takes into account primary and secondary creep. The cohesive zone model takesinto account cavity growth by diffusion as well as dislocation creep. Both models havebeen implemented as user defined material subroutines for use in the finite elementprogram Abaqus.

To demonstrate the capabilities of the two models, the evolution of creep damage ina representative volume element of a polycrystal is simulated.

1corresponding author, e-mail: [email protected], phone: +49-(0)30-8104-3147

Û g k

[Beere 78] W. Beere & M. V. Speight. Metal Science, vol. 12, page 172, 1978.

[Chen 81] I.-W. Chen & A. Argon. Acta Metallurgica, vol. 29, page 1759, 1981.

[Cocks 82] A. C. F. Cocks & M. F. Ashby. Progress in Materials Science, vol. 27,page 189, 1982.

[Riedel 87] H. Riedel. Fracture at high temperatures. Springer-Verlag Berlin,Heidelberg, 1987.

[Svensson 81] L.-E. Svensson & G. L. Dunlop. International Metals Reviews, vol. 2,page 109, 1981.

[Weertman 74] J. Weertman. Metallurgical Transactions, vol. 5, page 1743, 1974.

Û h l

d³mepm

Õ×Ö|ØÚÙfÛtÜDÝÅÞRßàmØâá|ßÙØâÙqãåäÞäÜDæcãçæRèéÞêcëíìî æRãåäÜDã~ÖÖ|Ø ïñðÙræcÛòóæRè ë Üô2ßæÝÅÞõäÜæRãô

ö÷)øsùúÅûüösúÅû*ýhùmøsþý1 ÿ ú úû*ø 1,2 ÿ úÅøsþhú:ø 3 ÿ ø÷)ø 1,2

1 !#"%$'&(*),+-).0/1$(23&($54768"59:5;&%)**)*48) <(=7$>-?"@A4B!#"($(2C+#&5&5CD#E-"(*)F!HG-JI/148)K"(2L"%M47N>OQP &%)K"(R"%;9.0/1S&T"($5&()?$VUXW#Y-9ZH[\W1]7W.0/-$523&($(4^68"59`_a"%$5Rb/- P2 c $d/14B;6BGe<T"($f:5;&()?*)*4B);<(=B$hgi"($5ET&()KGejkR"dl(6B/1 SE:Bgnmo9pgJq16^2L"($5&()?$VUrW\WH9ZTs7W\t1u

c $d"%,v(47$?D#9w_"($(Rb/1 P3 x 68"h S!1"($5&5*) P Ge<yk>1v%47$?D#6\9wzX"%Q)*$"r<VG1$wm|/-)K"($(?/123& O lV,"(Nl'"0/1N>

y7D-""($(7D#9.7D~ &Ik4B2C>17D#&59 O /1N>^"%$'&VG1oIk472C>17D19yk>1v(4B$?D#6nyrsn]^#9 *)K"d>i.p7D\>\G1R

F -'TL;;%# 0?1;7BdT#CL0b1dC-#ViV17dC^;;d V1dLTX#Q;1dL%CVf#iL %T1Vb(1dX#V1^dL^;;;V'%dC;d1;XQ1Vb1iBd7T#d1; ¡ 1Ci;CC¢QVMd8d%bfHd'#C-7V8\i8LL7T#L1 wLMnC;-LiCL|T#8¢ -8'#d1£fK#BLb% d1QHAd V1dbLV-#CS#8-¤B;%C\Vb'1¥d a'#dLdLT#Nb(Q#;LV¦#C^%51%dC;pQ#ddL%Bd'%bT1§`T%\%T¤^#LQ#d%i;Xfd'# e?%Xd;Lb%d Bifd;V(K8C% dL1Q#B%br#%;d\VBd7T#d1;X,#LLVi8 kwd 1'¨£Hr08Ld7T#d1h;% dLKnbT18'w%L;#;Lf'0(©QV%`HLBd7T#d1;1CL fCL¨\18ªdV% F bT11MLfdQ#d b«w¬1 %'%;'1,® Q¯ û*ø X°-± ²B³µ´L¶%·¸ ¤1nb #;?1\V1b%ddT#CLh VVV''#do8LLBT#dL1 ,¹fº`» ¸ V1i;dL'Vw1;Chi*'1%dL1|#r#L8Ld7T#L1 T¤Tk;CLd ¦T##;% dLK#;'#dLdLT#LCd1Vp8LL7T#L1 F,¼;¼B» ¸V1^'#L ;bBLV%d1 #L8?1d#d#½¾n(¿B;L1'n#µ(¿B% ;VÀV1^dL^;;Ád V1d úû*ø ÂBÃ;Ã ÄÅ´ÇÆH·*¸ ¤;;dCLCÈTV|À\% %'#CVÉ;(¢ ;Cd1Ê#id ¾8LLBT#dL1Ê;% dLKË% 1 ½ÌCd;Cd;Lf(¿B% ;VMV1^dL^;;Í;V1od 8ÎV%^dS#dL1¾%KV%Ï¹fº`»Ð# |¼;¼B»VV1Va8d% 5#8-

½¾£1d;bnÑdC;-b-C;ÒB%ÓCµd b¿B,7,;# ÒLdÀo #1\%;1 i8eddL;8L1#r-C;;S#;TChÈ08C'V%dL1M# ÑV1;'#^fÔF;d\%TÕ \V%'# N¡;;ed %db1'f1'd;bfdQHa8LL7T#L1 k; 1kC^'%V%krd %£#fbTBC;b18ÎA%%^;; T#d^JQ#'#LL%-C;f;# V Ö8FT1;`8dLBT#dL1£;% dLK% 1F#d `(¿B'% 8VnV1^C^;;×d V1dL0;(¢ VÀ#ÀhV18¢ -;'#d#Ø 1V-¤;L'ÀT# Ld'p#ad Q#d#1C^#d f%Bd5H

r = (x, y)# £d wCL 1d%^'#d1h;(¢ VJ^nd #;-L

ϕ¤8;

L?1dbVo^nd 8LLBT#dL1oLC VaLdo;Ô;d\% Õ8\V%'1TF J(¿8% ;VÅ8Ld7T#d1Å;%;CÀ% 1ii;(¢; VØ7Ïd J;% dLK

ρ(r,ϕ)

H%-%^#BLV%d1

ϕ# d b Hn%;d-#d;

k(r,ϕ)

#d V'k'%-b%^ ^F

Ù h Ù

Ú%Û\Ü1ÝCÞ;ßdàÜ1áÑÚVâ7Þ ã#ßàLÜ1á äaß'ã#å\Úfßdæ Ú`ç?Ü1èé

∂tρ(r,ϕ) = −

( ê àCÛ (ρ(r,ϕ)v(r,ϕ)) + ∂ϕ(ρ(r,ϕ)ϑ(r,ϕ)))

+ ρ(r,ϕ)v(r,ϕ)k(r,ϕ)

∂tk(r,ϕ) = −v(r,ϕ)k2

(r,ϕ) + L(r,ϕ)(−ϑ(r,ϕ)) − V (r,ϕ)(k(r,ϕ)),

ë æ Ú%èÚvàäwßdæ ÚiäìQãHßàã#ÝXÛ\Ú%ÝLÜ7íVàCßî|ã#á ê ϑ = −L(v) àLä`ßdæ Úè'Ü1ß'ã#ßdàLÜ1áQã#ÝrÛ\Ú%ÝLÜBí%àCßîÜ#ç

ßdæ ÚwÝLàCá ÚwÚ%ÝÚ%ébÚ%á^ßTïQðánßdæ Ú`ç?Ü1ÝLÝLÜ ë àCá;ñßdæ Ú`ìÜ1àCá7ßkÜ#çßdæ ÚíVÜ1áBò ñ-Þ;è'ã#ßdàÜ1áhädìQã1íVÚwÞ;á ê Ú%èíVÜ1á ädà ê Ú%è5ãHßàLÜ1á ë àCÝLÝ óÚ ë èdàCßßÚ%áÒã1äXädÞ;ó äí%èdàLì;ßTï

Lê Ú%á Ü1ßÚVäXßdæ Ú ê àLèÚVí%ßdàLÜ1áQã#Ý ê Ú%èdàLÛ1ã#ßdàCô

Û\Úaã#ÝÜ1á;ñßdæ Úkñ\Ú%á Ú%è'ã#ÝCàäÚ ê ÝCàCá Ú ê àCèÚVí%ßàLÜ1á£ãHá êV

ßdæ Ú ê àCèÚVí%ßdàÜ1áQã#Ý ê Ú%èdàCÛ-ã#ßdàCÛ\Ú`ã#ÝLÜ1á;ñßdæ Úfñ\Ú%á Ú%è'ã#ÝLàLä'Ú ê Û\Ú%ÝLÜBí%àCßî

L(r,ϕ) := cos ϕ∂x − sin ϕ∂y + k(r,ϕ)∂ϕ

V (r,ϕ) := v(r,ϕ) sin ϕ∂x − v(r,ϕ) cos ϕ∂y + ϑ(r,ϕ)∂ϕ.

ðáßdæ Úhí%Þ;èdè'Ú%á^ß ë Ü1èdå ë ÚoàLéiì;ÝLÚ%éÚ%á^ß£ßdæ ÚVäÚÚVâ7Þ ã#ßdàÜ1á äá7Þ;ébÚ%èdàLíTã#ÝLÝCî\ïõ¾Úhädæ Ü ëßdæQã#ßßdæ ÚfßÚ%ébìÜ1è'ã#ÝÚ%Û\Ü1ÝCÞ;ßdàÜ1áÑÜ#çXÛ\Ú%èdî£ñ\Ú%á Ú%è'ã#Ý ê àLädÝÜ7íTã#ßdàÜ1á ê àädßdèdàLó8Þ;ßdàÜ1á ä`íTã#áoóÚì;èÚ ê àLí%ßÚ ê ó^îJãiäàCépÞ;Ýã#ßdàÜ1áhóQã1äÚ ê Ü1áhßdæ ÚÚ(ö8ßÚ%á ê Ú ê íVÜ1á^ßdàLá^Þ;Þ;é÷ßdæ ÚVÜ1èdî\ïø äfãóÚ%á í'æ8éÒã#èdå ë Úiäæ Ü ëù ßdæQã#ß`ßdæ Ú ê àLädßèdàCó;Þ;ßÚ ê ìÚVá ê ã#á^ßÜ#çãÒädàLá;ñ-ÝLÚ ê àLä'í%èÚ%ßÚÝLÜBÜ1ìoíTã#áoóÚfæQã#á ê ÝÚ ê ì;èÜ1ìÚVèdÝCî\ïú àLáQã#ÝCÝCîÌßdæ ÚMÚ(ûÚVí%ßoÜ#ç ê àLäÝLÜ7í ãHßàLÜ1áüì;àCÝÚ(ô,Þ;ì ähã#ßhã#áËàCéiìÚ%á Ú%ßdè'ã#ó;ÝLÚMóÜ1Þ8á ê ã#èdîÌàLäädßdÞ ê àÚ ê ë àLßdæ;àCáoßæ;ÚàCébì8ÝÚ%ébÚ%á7ß'ã#ßdàLÜ1áï

ýþ#ÿ õ¾Úfã1íåBá Ü ë ÝLÚ ê ñ\ÚäÞ8ì;ìÜ1èdßÜ#çßdæ ÚXÞ;èÜ1ìÚTã#áFÜ1éiàLä'ädàLÜ1áÞ;á ê Ú%èíVÜ1á^ßè'ã1í%ß! aô"#ô"$%%'&Hô"(%*)+'&*)-,/.7à10TÚ32wÚ%ì4á65(ï

798;:6<=>!:6?@=ACBEDGF@HJILK MNPO4K , BQ (RS5TVU'W XZYW4[[\V]^X_a`bU'cdY/`fe`^chgi`^cj]`^Xlk3[WmÈWn[W6epoqYCm`^Wi]/rasW4W4[Wm`^W ù ,/.7ì8èàCá;ñ\Ú%èt Ú%èÝã#ñ ù4u Ú%èdÝLàCá ù BQ (RS5A $ DwvxHJyVz|PK~JNPO4KHJ~O4K@HG6| ,$%%+S5 X_cb`b`^"e'Y\x`^Wi]dYU'W6s'PbU'W XYW4[[\X_a`bU'cdU;e'Y]dCUbsXZYUW]d*]^XÈ\V]YWÈ\LsXYE]

s'W6ex\x`s'W4#ÈCe3U'cE\w[CsXZYU'W ùa æ;àLÝ,ï |ã#ñ ïi ¡

Ù h*¢

d³meo

Comparison of DDD and continuum approach in simulations oftensile test of thin films

Filip SISKA1, Samuel FOREST1, Peter GUMBSCH2,3, Daniel WEYGAND2

1. Centre des Materiaux, Mines Paris, Paristech, CNRS UMR 7633, BP 87, 91003 Evry Cedex,France

2. Institut fur Zuverlassigkeit von Bauteilen und Systemen (izbs), Universitat Karlsruhe, Kaiserstr.12, 76131 Karlsruhe, Germany

3. Fraunhofer-Institut fur Werkstoffmechanik IWM / Woehlerstr. 11, 79108 Freiburg, [email protected]

Thin film structures are in the center of interest, because they play important role in presenttechnologies. Their behavior is different from the bulk material and new type of effects is observed.Their mechanical properties are related to the size of these structures and therefore are called sizeeffects. Investigation of mechanical behavior and size effect can be done by the numerical simulations.Two type of approaches are chosen and compared. The discrete dislocation dynamics approach andcontinuum mechanics one. Simulations of tensile test of multicrystalline aggregates are made withDDD theory [1, 2]. These aggregates contain 9 grains with crystallographic orientation 111, 001and random perpendicular to the plane of the film. The grain size was chosen within the interval0.5 - 1 µm. The There is also included the shape of the grains since different value of ratio of in-plane grain size d and film thickness h is considered. This d/h ration varies between 0.5 - 1. Grainboundaries are impenetrable for dislocation and they remains stored at these boundaries. Analysisis performed on macroscopic and microscopic level. Macroscopic level is based on the comparison ofglobal stress/strain curves for different realiztaions of aggregate (see fig 1(a)). The microscopic levelconsits of analysis of evolution and distribution of stress and strain fields and dislocation density(seefig 1(b). These quantities are correlated to the grain size and initial dislocation density and sourcelengths. Continuum mechanics simulations are made with the same grain shape and orientations tocompare the results between the two approaches. First the model based on classical crystal plasticity isused [3]. Within these theory it is possible to compare the similarity of stress/strain field distributionsin grains of aggregates. Further comparison is furnished with the simulations done by the Cosseratcrystal plasticity which allows to input the intrinsic length scale into simulations [4]. This allows tocompare also the influence of the grain size on the stress/strain fields. The dislocation density tensorwhich is obtained from DDD simulations is related to parameters of Cosserat theory.

References

1. D. Weygand, L. H. Friedman, E. van der Giessen, and A. Needleman. Discrete dislocation modelingin three-dimensional confined volumes. Materials Science and Engineering A, 309–310:420–424,2001.

2. D. Weygand and P. Gumbsch. Study of dislocation reactions and rearrangements under differentloading conditions. Materials Science and Engineering, A 400–401:158–161, 2005.

3. L. Meric, P. Poubanne, and G. Cailletaud. Single crystal modeling for structural calculations. Part1: Model presentation. J. Engng. Mat. Technol., 113:162–170, 1991.

4. Forest S., Barbe F., and G. Cailletaud. Cosserat modelling of size effects in the mechanical behaviourof polycrystals and multiphase materials. International Journal of Solids and Structures, 37:7105–7126, 2000.

Ù h 0

(a)

(b)

Figure 1: Results of DDD simulations of 9 grain polycrystalline aggregates: (a) comparison ofstress/strain curves for different initial source length, (b) distribution of σ22 (direction of tension)and dislocation structure.

Ù h O

d³me q

Multi–scale approach to the analysis of

indentation experiments in polycrystalline coatings

O. Casals and S. Forest

Centre des Materiaux / Mines Paris, CNRS

UMR 7633, B. P. 87, 91003 Evry Cedex, France


Abstract

The development of advanced materials and components has acquired increasing impor-

tance in high technological fields as for instance aeronautics, biomechanics or microelectron-

ics. Applications within these areas often deal with local mechanical solicitations of the

materials, as is the case for instance of coatings, thin films, advanced composite materials,

thermal barrier coatings or welded joints [1]. In addition, due to the increasing tendency to

miniaturization, small volumes of material are usually involved in the mechanical response of

such micro-systems. As a consequence, the actual behavior of the components results from

a complex interaction between the intrinsic deformation mechanisms of the solid and a large

number of relevant microstructural details. In particular, the mechanical properties of thin

films significantly differ from that of the corresponding bulk material [2,3,4]. Within this

context, hardness experiments become relevant to the characterization of small volumes of

materials and to the analysis of the mechanical response of individual grains and microcon-

stituents. Instrumented indentation techniques have gained popularity as well, since they

provide the opportunity to systematically characterize the local response of materials over

the length scales. In this sense, great efforts have been directed over the years to determine

the viability of new methodologies to evaluate mechanical properties from contact param-

eters assessed through the aforementioned indentation techniques [5]. Nevertheless, such

methodologies are usually based on macroscopic interpretations of the contact response. In

order to extend their use to the micro– and the nano–scales, a detailed study of indenta-

tion experiments conducted on few grains or isolated micro–constituents is required [6]. To

this aim, crystal plasticity models, that account for the crystallographic nature of grains,

become pertinent. This modelling approach provides the possibility of analyzing the me-

chanical behavior of polycrystalline aggregates at the mesoscopic level. For instance, large

scale computations to investigate the development of plastic flow heterogeneities and sur-

face roughness in polycrystalline films under cyclic loading solicitations have been recently

undertaken following this modelling strategy [4].

The present work concerns the simulation of indentation experiments in single crystals,

(see Fig.1), as well as indentation of metallic coatings on hard substrates to a penetration

depth at which few grains are encompassed by the indentation imprint. In this case, a

spherical indenter is brought into contact with a finite element mesh resembling a true 3D

polycrystalline microstructure with columnar grains. The main objective is to compare the

Ù h b

plastic zone within a single crystal and that developed in a polycrystalline aggregate. We

also seek to establish the micromechanics of plastic deformation of polycrystalline coatings

and its interaction with the substrate. The size of the plastic zone beneath the indenter,

its complex interaction with the grain boundaries and the influence of film thickness are

analyzed in light of the results obtained using a classical continuum crystal plasticity theory

[7]. The analysis of microstructure–related size effects and their role on the local contact

response of crystalline metals is also addressed within such a multi-scale framework.

0.0 0.001equivalent plastic strain (octgeq)

Figure 1: Cross section of the anisotropic plastic zone developing in a copper single crystalindented in the (001) crystallographic plane using a spherical indenter. A sector encompass-ing 90 of the full 3D model is presented in the figure, showing the 4–fold symmetry of theplastic zone for this particular orientation.

References

[1] S.P.Baker, A.Kretschmann, E.Arzt, Thermomechanical behavior of different texture com-

ponents in Cu thin films, Acta Mater. 49 (2001) 2145–2160

[2] M.Hommel, O.Kraft, Deformation behaviour of thin copper films on deformable sub-

strates, Acta Mater. 49 (2001) 3935–3947

[3] H.Huang, F.Spaepen, Tensile testing of free–standing Cu, Ag and Al thin films and

Ag/Cu multilayers, Acta Mater. 48 (2000) 3261–3269

[4] F. Siska, S. Forest, P. Gumbsch and D. Weygand, Finite element simulations of the cyclic

elastoplastic behavior of copper thin films, Modelling Simul. Mater. Sci. 14 (2006) 1–22

[5] O.Casals, J.Alcala, The duality in mechanical property extractions from Vickers and

Berkovich instrumented indentation experiments, Acta Mater. 53 (2005) 3545–3561

[6] O.Casals, J.Ocenasek and J.Alcala, Crystal plasticity finite element simulations of pyra-

midal indentation in copper single crystals, Acta Mater 55 (2006) 55–68

[7] R. J. Asaro, Micromechanics of crystals and polycrystals. Adv. Appl. Mech. 23 (1983)

1–115

Ù hc

d³meps

IWCMM-17 : Abstract submitted.

Study of non-proportionnal loading paths : comparisonbetween experimental results and simulations

performed by finite element and homogenized models.

C. Gérard (∗), B. Bacroix, M. Bornert,R. Brenner, G. Cailletaud, D. Caldemaison,

T. Chauveau, J. Crépin, S. Leclercq, V. Mounoury.

3 avril 2007

The plastic behavior of fcc materials is studied under complex loading path at room tempe-rature. Multi-scale experimental results are compared to finite element computations of poly-crystalline aggregates, and to simulation by homogenization techniques.Combinations of sequences of simple loading paths such as tension, simple shear, rolling, cyclictension-compression, with different orientations with regard to rolling direction, are considered.The material used in the study has been cold rolled and annealed.Specimen are marked with several gold microgrids to measure local strain field by digital corre-lation of scanning electron microscope images, whereas global strains are measured by classicalextensometry.A polycrystalline aggregate taking into account the material microstructure is used to performfinite element simulations corresponding to the experiments. The texture measured by X-raydiffraction is represented. Several single crystal models, using the 12 slip systems of the octa-hedral families, are introduced. Various assumptions are considered for the description of theself and latent hardening.The comparisons between the simulation and the experiments are made on three types of va-riables : on a global level, with the description of the macroscopic stress-strain curve, on a locallevel, by considering the local fields, and in terms of phase averages. Informations about theconsistency of the transition rules are deduced.

(*) Fédération Francilienne de Mécanique des Matériaux Structures et Procédés (F2Mmsp)CNRS, FRANCE.Tel : +33 6 14 99 23 93. Mail : [email protected]

Ù h g

Ù hh

d³mejt

Free meshing of microstructures basedon modified Voronoi tessellations

R. Quey(1), F. Barbe(2,∗)

(1) Laboratoire PECM, UMR 5146,Ecole des Mines de Saint Etienne - 42023 Saint Etienne, France

(2) Laboratoire de Mecanique de Rouen, EA 3828,

INSA Rouen, BP 08, 76801 Saint Etienne du Rouvray, France

(∗) Corresponding author: Fabrice Barbee-mail: [email protected] - tel: 33 2 32 95 97 60, fax: 33 2 32 95 97 04

Keywords: Polycrystals, finite elements, free meshing, free software

AbstractVoronoi tessellations are commonly used for the representation of the microstructure ofpolycrystals, especially when generated by a diffusive transformation (e.g. solidification orrecrystallization). Indeed, the process of construction of a Voronoi tessellation is quite similarto the one of such polycrystals: starting from a set of nuclei –usually described by a Poissondistribution, the Voronoi polyhedra grow diffusively at a uniform and constant rate. At theend of the process, they fill the space with no overlaps and no gaps. The resulting intersectionsof 2/3/4 polyhedra are planes (representing grain boundaries), straight edges (representingtriple lines) and vertices (representing quadruple points). An important feature of Voronoitessellations is that they account for the random feature of polycrystals, as opposed to regularrepresentations, e.g. through truncated octahedra.

Numerous studies of the plasticity of polycrystals have been made by resorting to theFE computation of a polycrystalline domain described by Voronoi tessellation (Barbe et al.,2001a; Diard et al., 2005; Zhegadi et al., 2006a). One of the main advantages of this approachis that it gives access to detailed information about the local interactions between grains andintra-granular inhomogeneities, as long as the discretization inside each grain is fine enough.

The simplest ways for meshing Voronoi tessellations is to rely on a regular mesh madeof brick elements, and to represent each grain by a set of elements or integration points(Barbe et al., 2001a; Diard et al., 2005). However, the intersections between grains are poorlydescribed, so that the study of local interactions with grain boundaries can be intrisicallylinked with mesh effects. Only the use of free meshing (e.g. using tetrahedral elements)enable to conform with the morphology of Voronoi polyhedra. But to automatically producesuch a mesh, with a good quality, is far from being an easy task because of the numerousconstraints of Voronoi tessellations. It has been achieved in the mesher of the FE softwareZset (Zset 8.4, 2006) calling upon the meshing algorithms of Tetmesh-GHS3D of INRIA.

As an alternative solution, we propose an algorithm integrated in the NePeR software(NePeR 1.5, 2006), whose general purpose is to generate microstructures of diffusively phasetransforming solids and/or of polycrystals. NePeR takes advantage of the free software Gmsh(Gmsh 1.60, 2005) to generate free meshes. The algorithm consists mainly in modifyingslightly the morphology of Voronoi tessellations so to eliminate the small edges, which can beconsidered to have a neglectable influence on the local behaviour. As can be seen on fig. 1,Ù h k

this slight morphology modification allows to get a half-size mesh –meanwhile maintaininga good quality. Polycrystals containing 1000 grains can be automatically obtained using astandard PC; for larger polycrystals, more computing ressources are required.

x

y

z

x

y

z

Figure 1: A multicrystal containing 100 grains discretized freely, as generated by (NePeR 1.5,2006). (Left) Pure Voronoi tessellation. (Right) Slightly modified Voronoi tessellation.

References

Barbe, F., Decker, L., Jeulin, D., and Cailletaud, G. (2001a). Intergranular and intragranularbehavior of polycrystalline aggregates. Part 1: F.E. model. Int J Plasticity, 17:513–536.

Diard, O., Leclercq, S., Rousselier, G., and Cailletaud, G. (2005). Evaluation of finiteelement based analysis of 3D multicrystalline aggregates plasticity. Application to crystalplasticity model identification and the study of stress and strain fields near grainboundaries. Int J Plasticity, 21:691–722.

Gmsh 1.60 (2005). User Manual. http://www.geuz.org/gmsh.

NePeR 1.5 (2006). Reference Manual. Ecole des Mines de Saint Etienne (R. Quey), INSARouen (F. Barbe).

Zhegadi, A., Forest, S., Gourgues, A., and Bouaziz, O. (2006a). Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure – Part2: crystal plasticity. Phil Mag. In press, doi:10.1080/14786430601009517.

Zset 8.4 (2006). Ecole des Mines de Paris, ONERA, www.nwnumerics.com

Ùkl

PRQTSUSWVYX[Z \

@ ?;>=aGI¡¢;£DFGI@CB ;BN¤ £ ;> ;¥_ ^

Ùk^Ù

Ùk ¢

d o7e,d

Crack growth modelling of single crystalsfor higher-order continua

O.Aslan and S.Forest

Centre des Materiaux / Mines Paris, CNRS

UMR 7633, B.P. 87, 91003 Evry Cedex, France

e-mail : [email protected]

Abstract

Due to its remarkable material properties, single crystals have been used in

many advanced engineering applications, especially in integrated circuits, high-tech

wires and turbine blades of gas turbines in jet engines. Research in these fields

of application generally necessitates proper life-time prediction modeling of single

crystals operating under thermo-mechanical creep-fatigue conditions.

In previous works, stress and strain fields have been analysed at the tip of a

static crack subjected to creep-fatigue loading, assuming an elasto-viscoplastic single

crystal behavior and a crack growth model based on local approach to fracture

in fatigue of single crystals has been proposed in order to enhance the life-time

prediction [1]. However, due to the mesh dependent nature of the model, the crack

path predictions were valid only for a certain size and type of mesh as in many other

damage models [2].

Fig. 1 – Results of a crack growth simulation in a (001)[110] single crystal CT

specimen at 650oC, ∆ K = 40 MPa

√

m, f=0.1. Hz. (a) after 119 fatigue cycles

(b) after 78 fatigue cycles with another set of parameters [2].

Ùk0

The present work aims the regularization of the model by introducing elasto-

viscoplastic constitutive equations for weakly nonlocal theories in the frame of

micromorphic continuum which adds to the displacement degrees of freedom ~u

a full, non-symmetric, micro-deformation tensor χ∼

[3]. As being one of the most

general higher order continuum theory, micromorphic theory is able to encompass

several other generalized models like Cosserat, second gradient or a microstrain

model which makes it an attractive framework to explore the regularization

capabilities of enhanced continua. Therefore, taking the computational cost and

the material behaviour into account, to choose a well-suited higher order continuum

becomes necessary. For instance, the microstrain theory is a good candidate for the

modelling of strain localization phenomena in metallic foams and the Cosserat is not

appropriate, due to the fact that the rotation of the cells is not the main deformation

mechanism [4]. In the same sense, regarding the regularization of the previous model,

microstrain theory is chosen as a basis for the modelling of crack growth in single

crystals, sparing 3 degrees of freedom compared to the full micromorphic approach.

References

[1] S. Flouriot, S. Forest, G. Cailletaud, et al. Strain localization at the crack tip in

single crystal CT specimens under monotonous loading : 3D Finite Element analyses and

application to nickel-base superalloys. Int. J. of Fracture, 124 : 43-77, 2003.

[2] N. Marchal, S. Forest, L. Remy, S. Duvinage. Simulation of fatigue crack growth in

single crystal superalloys using local approach to fracture.

[3] S. Forest, E. Lorentz. Localization and regularization. In J. Besson (Ed.), Local

Approach to Fracture. Ecole des Mines de Paris-Les Presses, 311-373, 2004.

[4] S. Forest, R. Sievert. Nonlinear microstrain theories. Int. j. of Solids and Structures,

43 : 7224-7245, 2006.

Ùk O

d o7e iSimulation of generation of mesobands of inelastic deformation

in surface layers of deformable solid: the stochastic approach of excitable cellular automata

D.D. Moiseenko, A.L. Zhevlakov, P.V. Maksimov, V.E. Panin

(Institute of Strength Physics and Materials Science,

Akademichesky 2/1, Tomsk, Russia)

Stochastic excitable cellular automata method (SECAM), which allows taking into account the self-organization of configuration perturbations of various scales in the system “surface layer – its interface with the substrate”, is developed. It is shown that any plastic shear can be generated in the zones of normal tensile stresses that give rise to inelastic configuration disturbances in a crystal as a plastic shear precursor.

In a last time methods of cellular automata [1] are widely used for simulation of various

dynamic processes in a solid under external influence. They are especially effective for modeling of such processes as generation of new phase, recrystallization, generation of cracks etc.

Since influence of external field is always accompanied with persistent energy influx, its transformation and dissipation into environment, the character of the behaviour of the chosen type of cellular automaton has to have a possibility to describe all stages of excitation of active element. The class of excitable cellular automata obeys this principle. Moreover, as the inner structure and distribution of energy levels on the elements of this structure in a real solid have more or less probabilistic character, the cellular automata have to be stochastic.

In this study the approach basing on thermodynamic law dE = –dA + dQ on the one hand, and on the other hand the notion of 2-well potential of multiparticle interaction in medium with stochastic distribution of atomic clusters with various configurations is proposed.

It is well-known that in strongly deformed solids mesobands of localized plastic deformation (MLPD) are developed intensively. This is accompanied with deformation weakening of material, its fragmentation on mesoscopic level and is finished by fracture on macroscopic level. It is experimentally shown that MLPD are generated in surface layers of deformable solid and are propagated in material volume in conjugated lines of maximal tangent stresses. In this study the simulation of the process of generation and propagation of macrobands of localized plastic deformation in two-level system «surface layer – substrate» was implemented with the help of the method of stochastic excitable cellular automata.

Surface layer in deformable solid is considered as independent subsystem. To study the development of MLPD in strongly deformed material larger density of defects in this material is introduced, and it is classified as highly non-equilibrium structure. In such materials on initial stage the quasi-periodical corrugation in the form of «chess-board» appears in a surface layer where the areas of tensile and compressing normal stresses are formed [2]. For clear description of tensile and compressing normal stresses at the interface the simulation was implemented using the Murnaghan theory [3].

The uniaxial tension of the composition «surface layer – basic material» was implemented. The specimen was represented by the cube of steel-3 with investigated plane in the form of square with side length of 1 mm. The cellular automata dividing the specimen were represenred by cubes with side length of 5 mcm. At zero time step of the algorithm for every cellular automaton the value of entropy was determined in a stochastic way (with the help of lognormal distribution of probabilities).

Thus, the surface of the specimen was uniformly «rough». The values of temperature of every automaton were defined equal between each other. Also it was supposed that initially the potential energy of elastic deformation of each cellular automaton equals zero, i.e. that in unloaded specimen the stresses are absent.

The process of the specimen tension is simulated so that at every time period equaling to 1 second the influx of mechanical energy into each layer of automata located on two opposite front planes of the specimen equals 1,2*10-6 joule (∆A = 1,2*10-6 joule). Starting from the fact

Ùkb

that the plane of the specimen consists of 40000 automata for 1 second 1 automaton receive the energy 1110*3 − joule. It corresponds to the power of energy influx in automaton 1110*3 −=CAq watt. If all this energy comes into the mechanical part of the energy of each cellular automaton then the deformation velocity υ equals 1,1*10-4 sec-1.

When loading of such specimen at the interface «relaxed surface layer – substrate» in the areas of tensile normal stresses the embryos of macrobands of plastic deformation appears (see pic. 1a). One can see from the picture 1b that at further stage of loading the macrobands of localized deformation begin to growth from these embryos in the line of maximal tangent stresses.

a) b)

c) d)

Pic. 1. Propagation of the macrobands of localized deformation on various stages of the specimen deformation: a) ε=0.01%; b) ε=0.1%; c) ε=0.2%; d) ε=0.5%. In case of extremely heterogeneous distribution of entropy and correspondingly of free

energy on mesovolumes of the basic material defects in substrate are concentrated in macrobands and are subjected to dynamic recrystallization. Comparing pictures 1a–1d one can see that when growing of the macrobands in the areas those are not subjected to plastic deformation the spatial distribution of levels of the entropy related to the stochastic distribution of defects in crystal becomes more and more homogeneous, and the total density of defects is decreased.

The macrobands of plastic deformation are formed by the way of segregation of nanoconfigurational disturbances with excess «non-equilibrium» volume. This process of appearance of embryos of inelastic deformation in the areas of tensile normal stresses in surface layer and their directional growth in material volume is similar to interrupted breakup in supersaturated solid solutions. However in deformable solid this process is conditioned by mass transfer in mechanical field of internal stresses, and not by diffusion.

The variation of the characteristics of surface layer (its thickness, modulus of elasticity, number and kind of defects) allows to embody wide change of the distribution of tensile and compressing normal stresses at the interface and in that way to have an influence on plasticity and strength of deformable solid.

References

1. Theory and application of Cellular Automata. Wolfram S. Singapore: World Scientific, 1986. 2. The Chess-Board Effect in the Stress–Strain Distribution at Interfaces of a Loaded Solid. V.E. Panin,

A.V. Panin, D.D. Moiseenko, et al. // Doklady Physics, Vol. 51, No. 8, 2006, pp. 408-412. 3. Finite deformation of an elastic solid. Francis D. Murnaghan. New York, Wiley, 1951

Ùk c

PRQTSUSWVYX[Z \

§ GI¡ ^©¨ ¤ ^ x ^ BN¤ ^ BED ^ ? y ;>BNGa?;>= xN:F@>x ^ :TD<G ^ M

Ùk g

Ùk h

d qe,d

Role of elastic nonlinearity in dislocationpatterning

Peter D. Ispanovity

Department of Materials Physics, Eotvos UniversityPO Box 32, 1518 Budapest, Hungary

Istvan Groma

Department of Materials Physics, Eotvos UniversityPO Box 32, 1518 Budapest, Hungary

Starting from the equation of motion of individual dislocations, a continuumdescription of a 2D dislocation system was derived by Groma and his co-workers for single slip [1,2], which is, according to earlier investigations [3],able to predict dislocation and GND densities under many different boundaryconditions. By applying the model, however, one cannot get a dislocationarrangement with a length parameter in the order of µm. So it is unable todescribe many observed patterns, like PSB, where the distance of the bandsdefines the caracteristic length parameter.

The models proposed so far are based on linear elasticity. The goal of theinvestigations presented in this talk is to study how nonlinear behaviouraffects the dislocation dynamics. After examining the different sources ofnonlinearity, a modification of the original model is outlined. It is shownthat, contrary to the linear case, it has spatially periodical solutions, whichis necessary for formation of patterns with mentioned properties. We alsoanalyze the results of numerical discrete dislocation dynamics simulations,when relaxations from different random initial states have been performed,and prove the appearance of the desired length parameter.

[1] I. Groma Link between the microscopic and mesoscopic length-scale

description of the collective behavior of dislocations Phys. Rev. B 56, 5807(1997)[2] I. Groma, F. F. Csikor and M. Zaiser Spatial correlations and

higher-order gradient terms in a continuum description of dislocation

dinamics Acta Mat. 51, 1271 (2003)[3] S. Yefimov, I. Groma and E. van der Giessen A comparison of a

statistical-mechanics based plasticity model with discrete dislocation

plasticity calculations J. Mech. Phys. Sol. 52, 279 (2004)

Ùkk

Figure 1: A relaxed dislocation configuration in the linear (left side), and thenonlinear (right side) case.

1/λx [ρ0.5]

1/λ y

[ρ0.

5 ]

(a)

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

1/λx [ρ0.5]

1/λ y

[ρ0.

5 ]

(b)

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Figure 2: The statistically averaged Fourier transform of the autocorrelationfunctions of relaxed dislocation configurations. While there is no character-istic wavelength in the linear case (a), after introducing nonlinearity, a sharppeak appears (b).

¢ ll

d qe i

Time evolution and numerical analysis of

dislocation-dislocation correlation functions

Robert Deak and Istvan Groma The dislocation-dislocation correlation functions play a crucial role in collective dislocation behaviour like pattern formation or size dependent plastic deformation [1]. In the present talk a simple case is studied, the 2D system of parallel straight edge dislocations in single glide, where the interesting quantities are the pair-correlation functions of the same and opposite signed dislocations. The correlation functions was analyzed earlier by discrete dislocation dynamics simulations, by solving the equation of motion of the above 2D dislocation system [2]. Another way to determine these pair-correlation functions of dislocations is to write and solve numerically the coupled evolution equations for two dislocation densities [2]. In the present approach we cut the hierarchy of many-dislocation densities by applying the Mayer cluster expansion. By solving these continuum equations using a finite-difference method, the results for relaxation of dislocations from random configurations are compared with results from discrete dislocation dynamics (see Fig. 1).

Fig. 1: Correlation functions for opposite signed dislocations, from discrete dislocation dynamics (left) and numerical simulations (right). References [1] I. Groma, F.F. Csikor, M. Zaiser. Acta Materialia 51 (2003) 1271–1281 [2] M. Zaiser, M. Carmen Miguel, I. Groma. Physical Review B, VOLUME 64, 224102

¢ l^Ù

¢ l ¢

d qepm

Microscale patterning of plastic deformation: self affine surfaces and scale free statistics of deformation-induced surface steps. Jan Schwerdtfeger, Edward Nadgorny, Vasileios Koutsos and Michael Zaiser The University of Edinburgh, Institute for Materials and Processes, The Kings Buildings, Edinburgh EH9 3JL, UK Abstract: We use atomic force microscopy and scanning white light interferometry to investigate the spatial intermittency of plastic flow in crystalline materials. To this end we analyze one-dimensional surface profiles obtained from surface maps of alkali halide single crystals deformed in uni-axial compression. The profiles of samples deforming in dual slip show self affine behaviour over many orders of magnitude. For the special case of single slip a different behaviour is observed. In this case, height distributions of surface steps extracted from the profiles show scale-free behaviour, similar to the strain bursts observed in compression of micron sized samples and in dislocation dynamics simulations. We extend our experimental investigation to micron sized Cu-bending beams for which simulations indicate a similar behaviour. References:

x Evolution of self affine surface roughness in plastically deforming KCL single crystals, Edward M. Nadgorny, Jan Schwerdtfeger, Frederic Madani-Grasset, Vasileios Koutsos, Elias C. Aifantis and Michael Zaiser. Proceedings in Science, International

Conference on Statistical Mechanics of Plasticity and Related Instabilities, Bangalore,

India 2005 x Scale-free statistics of plasticity-induced surface steps on KCL single crystals, Jan

Schwerdtfeger, Edward Nadgorny, Vasileios Koutsos, Jane Blackford and Michael Zaiser Journal of Statistical Mechanics (April 2007) L04001

Figures:

Left: AFM image (10 µmx10 µm) taken on deformed KCl (0§ 0.5%) Right: SWLI image (130 µm X 350µm) taken on deformed KCl (0§ 4%)

¢ l0

Roughness plot for 0§ 0.5% (stage I) deduced from AFM and SWLI image spanning several orders of magnitude

0.01

0.1

1

10

100

1000

1 10 100 1000 10000 100000 1000000 10000000

displacement along surface L [nm]

<|y

(x)-

y(x

+L

)|>

[n

m]

2micron scan

150micron scan

Zygo scan

slope of 0.7 to compare

H= 0.7

¢ l O

PRQTSUSWVYX[Z \T

JN=D<GIx y MwGa?M

¢ lb

¢ l c

d³se,d

Diffusion Induced Phase Transformations in Me-chanical Stressed Lead-Free Alloys

Thomas Böhme, Wolfgang H. Müller and Kerstin Weinberg

Technische Universität Berlin, Fakultät V,

Verkehrs- und Maschinensysteme, Institut für Mechanik, LKM, Sekr. MS 2, Einsteinufer 5,

10587 Berlin, Germany

ABSTRACT This contribution is subjected to diffusion phenomena in solid mixtures. As an example

we turn the attention to lead-free alloys which are of special interest for microelectronics due to environmental concerns. Such materials guarantee the electrical and mechanical bonding between the Chip and the circuit board, and, consequently, their reliable behaviour must be predictable.

One aspect that considerably affects strength and lifetime of solder alloys represents diffu-sion-induced phase separation, e.g., nucleation, spinodal decomposition and subsequent OST-WALD ripening (cf., Figure 1). The theoretical quantitative description of these processes is still a strongly discussed and partially open question. There are mainly three reasons for these shortcomings: (1) it is not sufficiently solved, how the different effects, such as external me-chanical loadings or surface tensions between the different phases, enter the diffusion equa-tion; (2) the occurring materials parameters, e.g., the surface energy, are often unknown and must be estimated and (3) the resulting diffusion equation represents a nonlinear PDE of fourth order and their numerical solution is non-trivial.

According to these problems we present a general phenomenological theory allowing for the theoretical quantification of diffusion in inhomogeneous solids. It is based on a novel method for the exploitation of the sec-ond law of thermodynamics and incorporates effects of concentration gradients, surface tensions and mechanical fields. Furthermore we illustrate how all occurring material pa-rameters can exactly be determined by means of databases or atomistic theories (EAM). Finally we present selected numeri-cal simulations of the phase evolution and compare the results with performed experi-ments.

Figure 1: Spinodal decomposition in Ag-Cu after 20 h heat treatment (1000 K).

REFERENCES [1] Dreyer, W.; Müller, W.H.: A study of the coarsening in tin/lead solders, International

Journal of solids and structures, Vol. 37, pp. 3841–3871 (2000). [2] Böhme, Th., Dreyer, W., Müller, W.H.: Determination of stiffness and higher gradient

coefficients by means of the embedded atom method, An approach for binary alloys, Continuum Mechanics and Thermodynamics, Vol. 18 (7-8), pp. 411-441 (2007).

¢ l g

¢ l h

d³se iFinite Element formulation of Phase Field Models Based on the

Concept of Generalized Stresses

Kais Ammara, Benoıt Appolaire

b, Georges Cailletaud

a, Frederic Feyel

a, Samuel

Foresta

a Centre des Materiaux, Ecole des Mines de Paris, BP 87, 91003 Evry, Franceb LSG2M, Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy Cedex, France

(Dated : April 26, 2007)

In this work, we propose a recent phenomenological approach, developed within the general

framework of thermodynamics, that provides a variational formulation for the phase-field/diffusion

model. It is based on the generalized continuum approach proposed by Gurtin [3]. The formulation

involves an additional balance equation for generalized stresses associated with the order parameter

and its gradient. Boundary conditions are clearly stated for the concentration and order parameter

and their dual quantities. The method can be applied to finite size non periodic samples and

heterogeneous materials.

This formulation is employed to simulate the morphological and micro structural evolution

and to describe the diffuse interface regions between two phases. The local state of heterogeneous

microstructures is described by conservative fields of concentration and non-conservative fields of

order parameter. Using the principles of continuum thermodynamics, we determine the state laws

and the evolution equations of these variables. Balance laws and constitutive equations are clearly

separated in the formulation so that nonlinear and strongly coupled models can be more easily

implemented in the finite element program.

The set of coupled evolution equations, which are the phase-field equation and the balance of

mass, is solved using a finite element method to discretize space and a finite difference method to

discretize time. Numerical results show the concentration and the order parameter profiles across

the interface. These results are presented for different values of interface thickness to validate

the numerical method developed and to illustrate its usefulness and its convergence. These test

cases are compared to simulations based on more standard finite volume techniques to assess the

efficiency of the finite element method.

Keywords : Phase field model ; Balance law for microforces ; Variational Formulation ; Finite element ; Diffuse interfacemodel.

References

[1] Q. Bronchart. Developpement de methodes de champs de phase quantitatives et applications a la precipitions homogenedans les alliages binaires. These de doctorat, Universite de Cergy-Pontoise, 2006.

[2] P. Germain. La methode des puissances virtuelles en mecanique des milieux continus, premiere partie : theorie du secondgradient. J. de Mecanique, 12 :235–274, 1973.

[3] M.E. Gurtin. Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D,92 :178–192, 1996.

[4] O. Kruger. Modelisation et analyse numerique de problemes de reaction-diffusion provenant de la solidification d’alliagesbinaires. These de doctorat, Ecole Polytechnique Federale de Lausanne, 1999.

[5] Y. Le Bouar. Phenomenological phase field models for solid-solid phase transformations. Centre de Physique, LesHouches, 2006.

¢ lk

Concentrationφ(δ =

√

0.004)

x(mm)

φ,C

(kg/m

m3)

10.50-0.5-1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

y

z

0.27 0.3 0.33 0.36 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.6

Cd map:69.000000 time:5000 min:0.300000 max:0.700000

x

y

z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

PHId map:69.000000 time:5000 min:0.000000 max:1.000000

Fig. 1 – Finite element simulation og the formation of a diffuse interface between two phases based ona polynomial free energy function

¢ Ùl

d³sepm

FE analysis of the effect of the kineticsof nucleation on the plasticity induced

during diffusive transformation

H. Hoang(1), F. Barbe(1,∗), R. Quey(2), L. Taleb(1)

(1) Laboratoire de Mecanique de Rouen, EA 3828,INSA Rouen, BP 08, 76801 Saint Etienne du Rouvray, France

(2) Laboratoire PECM, UMR 5146

Ecole des Mines de St Etienne - 42023 St Etienne, France

(∗) Corresponding author: Fabrice Barbe

e-mail: [email protected] - tel: 33 2 32 95 97 60, fax: 33 2 32 95 97 04

Keywords: Diffusive transformation, TRIP, micromechanical modelling, finite

elements

AbstractDuring the cooling diffusive transformation of a steel from the austenitic domain, the

product phase is usually the harder phase and also the less compact. These two features

can be sufficient to explain the appearance of a transformation induced plasticity (TRIP)

while a loading stress is exerted on the phase transforming medium; they constitute the

two main features introduced in several micromechanical modellings dedicated to TRIP

prediction (Leblond, 1989; Taleb and Sidoroff, 2003). These modellings do have a predictive

character in simplest cases of phase transformation under uniaxial or biaxial loadings but they

do not when the parent phase has been strain-hardened (Taleb and Petit, 2006).

The use of finite elements (FE) representing the evolution of the microstructure during

the transformation is particularly convenient to elicit information about the local interactions

at the origin of the discrepancies between micromechanical modelling and experiments. This

has been the first motivation for the development of a FE based modelling which has lead to a

thorough analysis in (Barbe et al., 2006) and to extensions towards more physically consistent

modelling in (Barbe et al., 2007; Hoang et al., 2006): the improvements have consisted mainly

in determining the effective properties of a phase transforming representative volume element

by ensemble averaging over domains extracted from a quasi-infinite domain where nuclei

appear randomly in space as well as in time. This approach allows to reproduce kinetics of

transformation as it is described in the JMAK theory. Further, it has been proved that, for

large enough domains of computation, results were not sensitive to mesh size or type of brick

elements.

At the current state of our modelling, there is still a small quantitative discrepancy with

experimental results in a simple uniaxial TRIP test. Nevertheless, it has been observed that

TRIP results were dependent upon the kinetics of nucleation and growth. We propose here

to study the effect of the kinetics of nucleation on the effective TRIP from two points of

view: (i) by varying the spatial distribution from a Poisson distribution to the distribution of

the vertices of a Voronoi tesselation, a distribution which corresponds better to the physics

of diffusive transformation; (ii) by varying the laws governing nucleation according to time

¢ ÙHÙ

(or volume fraction of product phase). The objective is to determine the configurations of

kinetics modelling which are the most suitable for reproducing experimental TRIP results. On

a longer term, computations will be performed on a particular case of phase transformation

where experimental TRIP results as well as microscopic kinetics are known.

References

Barbe, F., Quey, R., and Taleb, L. (2006). Numerical modelling of the plasticity induced

during diffusive transformation. Case of a cubic array of nuclei. Eur J Mech, A/Solids,

26:611–625.

Barbe, F., Quey, R., Taleb, L., and Souza de Cursi, E. (2007). Numerical modelling of the

plasticity induced during diffusive transformation. Case of a random instantaneous array

of nuclei. Submitted.

Hoang, H., Barbe, F., Quey, R., and Taleb, L. (2006). FE determination of the plasticity

induced during diffusive transformation in the case of nucleation at random locations

and instants. Comput Mat Sc. to be published.

Leblond, J. B. (1989). Mathematical modelling of transformation plasticity in steels II:

coupling with strain hardening phenomena. Int J Plasticity, 5:573–591.

Taleb, L. and Petit, S. (2006). New investigations on transformation induced plasticity and

its interaction with classical plasticity. Int J Plasticity, 22:110–130.

Taleb, L. and Sidoroff, F. (2003). A micromechanical modeling of Greenwood-Johnson

mechanism in transformation induced plasticity. Int J Plasticity, 19(10):1821–1842.

¢ Ù ¢

d³seo

G G G G G G G G G G G G G

G

G

Jong-Bin YeoQ, Sang-Don Yun, Hyun –Yong Lee,

Center for Functional Nano Fine Chemicals, Faculty of Applied Chemical Engineering., Chonnam National

University, 300 Yongbong-dong, Gwangju 500-757, Korea G

G

G

We could make periodic nano or micro size pattern by holographic lithography (HL) in 442 nm

wavelength light source. Then exposure dose is proportionate exposure time. The enough exposure

time depend on change of incident angle. Because transmittance of incident light is chosen by

incident angle. Also incident angel is important parameters of pattern size in HL, in that pattern size

depend on incident angle. So we needs condition of exposure time for desired pattern size and enough

exposure dose.

This experiment is holographic lithography of rotated double exposure methode for PCs pattern.

The method use only one beam. Result data is shown by optical micrograph, atomic force

microscope(AFM), scanning electron microscope(SEM) and 3D contour image.

In this work, we confirmed enough exposure time condition for pattern generation. We calculated

transmittance of exposure dose by computing simulation for example MATLAB, MathCAD, Rsoft,

MBP, etc. And we demonstrated calculation result.

Holographic lithography (HL) is simple and easy method to periodic pattern generation (i.e.,

photonic crystals, etc.) using holographic phenomenon. Holography recodes the interference patterns

between object beam and reference beam on a substrate. To recorde the patterns, a special photo resist

(PR) with standard treatment is used. This interference patterns are transferred onto the PR layer in the

form of standing wave. Using this technique with He-Cd laser but no other special equipments, we

fabricated gratings and periodic dot, hole to reduce the size to about 2 Ã ~ theoretically down to

the /2. Also, we could change only some parameters to vary the period of PCs. For example laser

power, exposure time, post exposure baking (PEB) time, develop time, and PR thickness, etc. In this

work, we tried to the exposure time changing for incident angle to make periodic grating or arrayed

¢ Ù0

dot or hole structure. It is applied to the fabrication of one dimensional photonic crystals (periodic

multi layer or grating structure), two dimensional photonic crystals (periodic dot or hole structure), etc.

We simulated HL system by using ourselves manufactured computer calculation tool. Program is

organized Snells law and Maxwell equation. We designed sample structure like Figure 1. We used

Fresnells equation solution. We calculate transmittance in PR. Result of calculation is Figure 2.

We studied exposure time condition for nano or micro size pattern generation at changing incident

angle in HL. Figure 3 is schematics of exposure system.

Used photo resist (PR) was DMI-150, developer was TMAH 2.38%, exposure system was

organized components, develop system was using ultra-sonication, wavelength of light source was

laser of 442nm. We fixed that develop time is 15 sec.

We had demonstrated exposure time for enough dose. We confirmed that exposure time to make

enough light intensity is increased by changing incident angle. Because transition intensity was

decrease by reflectance followed incident angle in boundary. These results will be applied various

lithography system.

pGOBSGP

W `W

X

WU\

XW YW ZW [W \W ]W ^W _W

pGOBSGP

W `W

X

WU\

XW YW ZW [W \W ]W ^W _W

pGG

wyG

~G

Figure 1. Exposure model of computing

simulation.

Figure 2. Result of calculation() and

experiment(|)

oTj sGOEd[[YP

ukGz

t sOGP

wG

jz

oTj sGOEd[[YP

ukGz

t sOGP

wG

jz

Figure 3. Schematics of exposure system.

¢ Ù O

BN¤ ^¥¤

¦q§¨^©ª«¨d¬®°¯²±³µ´ h k¦q¶·©¯¸«¹³ «¹³°´¢ Ù´ Ùrbk ´ Ù g Ù¦q·¯º©S® »b³°´ Ù g Ù¦q¯¼¯º¬;¾½¾³°´

¢ lk¦q¿*À;¬"¨dÀÁÀ;¨d¬qÂÃ³°´ ÙHÙ¦Ä*Å¬ÇÆ!³µ´ Ùr0 g ´ Ù c Ù¦qÈËÉ«¹³°´ g¦qÈÈÄ*É°®µ¬;¨ÊÆ!³µ´¢ lk¦#ËÁÉC¿ÍÌ³°´ Ùk0¦À3Î*¾Ïi³µ´

¢ bÆÐd¬"Ä*®ÑVÆ!³°´ Ù h gÆÒÓ*¨d¬Ô«¹³°´ O 0Æ¬;§a¨Ã±³°´ ÙHÙb ´ Ù h k ´ ¢ Ù#ÙÆ¬;§®°¨d¬qÕ#³°´ Ù0Æ¬;¶ËÉµÉ°®°¿®4ÖÊ³µ´ h gÆ¬"À;Ë;Ð·w«³µ´ Ù c kÆËÐÓ*¨d¬Ô«¹³°´ Ù O ÙÆ¨d¿Í×a®°¿Ë§GÂÃ³°´ h kÆ¨d¿Ó¨Ïi³°´ g ´ ¢ g ´ Ùl gÆ¨d¿ØE¯ºÙ³µ´ Ù c ÙÆ¨d¬;§a¨d¿¿®Ïi³µ´ Ù gÆÚd¬;®°¿¶·®°¨d¬Ô«¹³°´ kÆ¨d¬"À"¬¯¸¦³°´ c kÆ¨d¬;ÛË®µÉ°É°¨d¬Ô«¹³°´ Ù g ´ h kÆËË;Ë"Ä¿LÜ³°´ h gÆ®°¿ËÁ§a¨d¬;¶Ë¿GÆ!³ Ý³°´ Ùrb gÆÉ°ËÐÓ@Þh³°´¢ g ´ Ùl g ´ Ù ¢ k ´ Ùrb gÆÉ°Å·¯¸Ü³°´ ÙlkÆß·ÉµÓ*¨ÃÂÃ³°´ Ùk ´ c k ´ g bÆß·¯¼¨ÊÂÃ³°´ ÙHÙrk ´ ¢ l gÆÄ¬;¿Ë¬ÁÀÔ«¹³°´ Ù h gÆÄÅØd®°ØÌ³µ´ Ù gÆ¬Ë;Ë"¬ÁÀàJ³°´ Ù ¢ gÆ¬;¨d¿¿Ë¬ÇÖÊ³°´ Ù h gÆ¬;áÐÓ¿¨d¬Ôâ³°´ g 0ÆÅÐÓÉ°¨dÎGÕ#³ ã³µ´ c 0ÆÅ©S¿®À;ØdÓ®6«¹³°´ Ù g kÆÅ¬;É°3Î*¨d¿Ó*ÄÝ³µ´ Ù OgÆÅË;ËÁ¨ä¼³°´ b\Ù

Õ®µÉ°Éµ¨dÀ;Å©Gä¼³°´ Ù h g ´ ¢ lkÕÉ°©SË¯º®CËÁÄ*¿wå³µ´ Ù h gÕ¯¼ÈaÄ*ËqÕ#³°´ Ù#Ù gÕ¿*À;ÄÅ¬;¿¨dÀÇÏ4³µ´ Ùr0^ÙÕ¬;§®µ¿Ë¿ÅG¦³°´ OgÕ¬ÁÀ;¬;Å©@ã³°´ kÕ*Ë"É°ËÇÌ³°´ Ù h bÕ·¿¶º½¾³ Ù³µ´ Ù O bÕ·ÅÛ¨^ÅLÂÃ³µ´ Ù h gÕ·Ë¿¶¼æ³°´ kkÕ·ËÛÅ¶*¨dÄ¿Lç³°´ kÕÄèÄÐE¬;Åwå³µ´ 0 g

Õ¬Ðj®°Å¿GéÇ³ ª«¹³°´ O.gÕ¬;ÚdÈ®°¿xÜ³°´ Ù h gÕË"®µÓ*Ä¬Ô±³ ±³°´ Ùl7Ùå#ËÓGÖÊ³°´

¢ l^Ùå#¨dÉCêlÄ*Ë;Ë"¨Ãå³°´ h Ùå#¨dÉC¿¿3ÎwàJ³°´ g Ù ´ Ù ¢ g ´ Ù ggå#¨d¿©S®°¨dÛ*¨dÉ6Ö³µ´ Ùr0å#¨d¬;¨dëÄ¿Ó*Ä¼¦³µ´ Ù c 0å#¨dÛ®µ¿Ðj¬;¨Æ#³°´ ggå#®µ¯¾®µÀ"¬;ÄÛwÏi³°´ h bå#Ä¶·¬"®4»b³°´ Ù ¢ gå#ÄÅ§¬3Û½¾³°´ 07Ùå#Å¿¿¨±³ ã³ éÇ³°´ c 0å#Ød®CÉ°É°*Ð·LÏi³°´¢ géJÉCÐ·*Ð·®6Ïi³ «¹³°´ Ù c ÙéJÈ®CË"·®µ¿¦³µ´ g 0

±É°·À;¶*¬ÔÏ4³ ÖÊ³°´ Ùbk±¨^©S¨dÉ°®CÐ·LÆ!³°´ g 0 ´ Ù g k±ËÎËÉ6±³µ´¢ lk±®µ¶*®µËÉ6àJ³°´ c 0±®µÉ°®°È®°Ów«¹³°´ c b±®°Ë;Ð·Ë¬Ç±³ å³µ´ Ù#Ù±Ä*¬"¨^ËÀqÏi³°´ kb ´ Ù07Ù ´ Ù h 0 ´ Ù h b ´ Ùk0 ´ ¢ lk±¬¿ØÊä¼³°´ h k±Å¨d¿¯º3ÎÄ*¬Ô±³ Ü³µ´ Ù c ÙäÃ¬;¿®°¨d¬#Õ#³°´ ÙHÙ gä!Úd¬¬©xÕ#³µ´ Ù h gä!¨dÅ§ËÉµÉ°¨Êã³ Ù³µ´ Ù ¢ gä!®°¿¨d¬Çéq³µ´ Ù c Ùä!É°Ä*¶*Å¨d¿LÜ³ «¹³°´ Ù0 gä!Ä¬è®6«¹³°´ Ùrbkä!ÄÎGÌ³°´ ÙHÙ#Ùä!¬¿©S®C©S®°¨d¬ÇÜ³ìÕ#³°´ bkä!¬;Ä¯º¼»b³µ´ k g ´ Ùl7Ù ´ Ùkk ´ ¢ l^Ùä!¬;Ä*Ë;ËÔå³°´ ÙHÙ#Ù ´ Ù ¢ Ùä!Å¨^©SËËÇÜ³ «¹³µ´ Ù#Ù0ä!Å¯§Ë"Ð·Lã³°´ Ùk ´ k0 ´ kk ´ Ù h Ù ´ Ù h 0ä!Îß*¬"¶*Î®4ä¼³°´ k g

ÙqÅËÇâ³ â³µ´ g bÙqÅË"¯¾¿¿LÜ³°´ Ù c kÙÇË¶*©S¨Êã³°´ 0bÙÇË¬"¬;¯º¿¿LÜ³µ´ b^ÙÙÇÄ'¿¶ºÙ³µ´¢ ÙHÙÙÇÄSÐ·¬®°¿Ë¬ÂÃ³°´ k0 ´ Ùl7Ù ´ Ù h ÙÙÇ¬;Å§ÎG×³°´ 07ÙÙÇÅÀ;¬qã³°´ Ùr0k

»/Ä¿ËË;ÐjÅG»b³µ´ k»Ë"È¿ÄÛ®µÀÎºã³ å³µ´ k g ´ Ùkk

¢ Ùrb

»/Û¿ÄÛ@»b³µ´ c bÜ'ÒÐÓ*¨dÉ4¦³°´ b\ÙÜ'ÐEí*ÅËËã³ Ü³µ´ Ù ggÜ'¿¶@ä¼³îÕ#³µ´ Ù O bÜ'3Î**Ð·¿©¬;¿@½¾³ ã³°´ ÙHÙr0Ü*Ë¿G¦³µ´ Ùr0^ÙÜ*¨dÅÉ°®°¿Lå³°´ Ù07ÙÜ*ßSÐ·¨d¿L½¾³µ´ g bÜ*ÄSÐ·'Å¯ïÕ· ³°´ bk½Ê¬"ÉCË"Ë"Ä¿G¦³ «¹³°´ 0 g½Ê*Ë"Ë"§Ä*·¯¸Ïi³°´ Ù g k½ÃÉ°ÅË;ÓVÜ³µ´ Ù c g½Ã¿Ú^ËÁÉ×³µ´ Ùr0k ´ Ù c g½#ÄSÐÓ¨dÉ°¯º¿¿G«¹³°´ b\Ù½#ÄÉ°ÈÓ*ÄÛº¦³ ¦³°´ Ù g 0½#ÄÉ°ÈÓ*ÄÛº¦³ìä¼³µ´ Ùr00½#ÄÅSÀËÁÄ'ËÝ³°´

¢ l0½#ÄÛðÐdË¦³µ´¢ k½Ãá¿ËÐÓ¨ä¼³°´ g 0

àË;Ð·¨jÀqä¼³°´ Ùrb ´ ¢ gàÀ"·¾ÏS·¿Ó¬ÇÆ#³°´ b g ´ Ù O 0à6ËÐjÉ°¨d¬ÐdíwÏi³µ´ Ù h gà6¨d¨Ù³ æ³°´¢ Ù0à6¨jêl¨d§Û¬;¨ÃÜ³ «¹³µ´ Ùr0 gà6¨d¯¾Ä®°¿¨Ãñ³°´ h kà6®µ¯¾ÓÅ¯¾¿¨d¬©xÏi³°´ Ùl0à6®µ¿ÓGÂÃ³µ´ g 0à6®µÅxÜ³µ´ Ù O Ùà6Ä¬;¨d¿'À"ØÃéÇ³°´ h gà6Å¬"®°¨Ïi³°´ b0à6ÅSÀ"ØÊÞh³°´ b\Ù

«·¯¾ÄÄ©®6«¹³ Ü³°´¢ Ù«ÓSËÁ®°¯¾ÄÛºã³ Ý³°´ Ùrkb«¬;Ð·É4ç³°´ ÙHÙ g«¬;Ë;3Û®µ¿àJ³°´ O b«3ÎwÂÃ³µ´ g 0«x¨dê1À;·xÏ4³µ´ Ù#Ùrb«xËÉ°Ð·®µÄ*¬Ç«¹³ ¦³µ´ g Ù ´ Ù gg«x¨^Ë"Ð·Ë©SË¬Çâ³°´

¢ k«x®CÐ·ËÉªãÄ¿¿ËÉµÉ°¨Ïi³°´ h g«xÄò^ËÔç³°´ k«xÄ*®°Ë"¨dË¿ÓÄ¾å³ å³µ´ Ùkb«xÄ*¿¿¨dÀ#ä¼³µ´ gg«xÄ*Å¿Ä*Å¬"ÎVÝ³°´ Ù07Ù ´ Ù h g«xÅ¨dÉ°ÉµË¬ÇÖÊ³µ´ ÙHÙHÙ ´ Ù ¢ Ù«xáÉµÉ°¨d¬ÇÞh³ Ù³°´ Ù#Ùk ´ ¢ l gçó ä!ÅÎ*¨d¿x±³µ´ Ù0^Ùç#¯º¿¨Ïi³µ´ ggç#©S¶*Ä¬;¿'Îºéq³µ´¢ l0ç#¶·®µÈaÄÅ¬Ôã³°´ Ù c kç#ð·É°ôµÓGàJ³°´ Ù0kç#õÀÁª/¦Ç§i©S¨dÉCØE®µØÊ«¹³°´ Ù0 gç#ÓÄ*¿®°ËÐjØE¿Î@½¾³µ´¢ b ´ Ù O kçq®°ÐE®CËÁ¨Êç³°´ Ù gçqÄ*ÐÓwå³°´ Ù g k

ãP¿®µ¿LÝ³ éÇ³°´ ÙkbãP¿'À;·®Ïi³ ½¾³µ´ 00

ãPÈÅ¶*¾Ü³°´ 0^Ùã¨d¬;®°©¨ç³°´ OgãJ®CÀqÖÊ³°´ ÙrkãJ®°È¬©LÜ³ ª/«¹³µ´ Ù gãJ®µÀ"ØdªZãPÉ6ÖÊ³°´ ÙrbãÄ¬"À"ËÉµÉC¼ã³°´ g 0ãJ¬·Éâ³°´¢ g ´ Ùrl g ´ Ù ¢ k ´ Ùrb gãJ¬ÓË"·L¦³°´ g kãJ¬¿¶*¨d¬Þh³°´ ÙHÙãJ¬;Ë©ÓwÏi³°´ b\Ù

ö ÅËÎVÖÊ³°´ Ù h k ´ ¢ Ù#ÙÖÇ§¨^¬;®°Ë"Ä¿Gç³°´ bkÖÇ¯¾Ó¬"®CËÁ·¿¿Gç³°´ 00ÖÇØE÷¿©S¬ØEÓ«¹³°´ h ÙÖÔËÅSÀÁÀ;¨d¬!Ì³°´ ÙrbÖÔ®CÐÓË¿GÂÃ³°´ ÙlkÖÔ®°Ë©SËÉ6Ù³°´ g k ´ kbÖÔ®CËÁÎLä¼³µ´ c kÖÔÄS©S¿ËÎwå³µ´ Ù0ÖÔÄS©S¬;®µ¶*ÅËËÔÙ³ìÕ#³°´ Ù#Ù0ÖÔÄS©S¬;®µ¶*Å¨dØÊÖÔ®°ÈaÄÉ°É6«¹³°´ kbÖÔÅØE®°ÐÓ¾«¹³°´ 0^ÙÏS*©SÄëÇËÁÓ®4ÂÃ³°´

¢ b ´ O b ´ c b ´ Ù O.g ´ Ù O kÏS¿©êl¨dÉC©xÏi³°´ k0 ´ Ù h ÙÏSÅ¨d¬;·¨d¬;®°¿¶¼Ü³°´ ÙrbÏSÐ·Ë¨dÉ°¨ÊÜ³°´ ÙrbÏSÐ·Ë®°©¨d¬Ô»b³°´ bbÏSÐ·¯ºÅ©S¨d¬qÏ4³µ´ b^ÙÏSÐ·¬©¨Ãå³°´ Ù ¢ ÙÏSÐ·ÅÉ°Ød¨ÃÝ³°´ c k ´ g bÏSÐ·ë¨d¬©ÀÁêlË¶¨d¬ÔÜ³µ´¢ l0ÏË¨dÉ°®µ¶ºÂÃ³°´ c ÙÏ·ÓË¬"® «¹³°´

¢ ÙÏ·*ËÁ·®°©S·¬¼Ïi³ «¹³°´ b gÏ·®°Û@Ï·¿Ó¬qä¼³ìÏi³µ´ b g ´ Ù O 0Ï®C©SÄ¬;ÄøÍ±³°´ ÙHÙbÏ®°¶¿Ä*¬"ËÉµÉ°® Ü³ Þh³°´ g ÙÏ®°Éµ§a¨d¬Ë;Ð·¯¾®°©ÀÇÝ³ Ý³µ´ 0bÏ®CËÁÓº±³°´ Ù h 0ÏÓ¬;®µ¿¬Ç«¹³°´ Ùrb\ÙÏ'À;¨d®°¿'§Ð·G»b³µ´ gÏÅÐd®µÅx«¹³°´ Ùb0ÂÉ°¨d§xàJ³°´ ÙHÙb ´ ¢ Ù#ÙÂ¿¶*ÅÎVå³µ´ h ÙÂ·Ä*¯¾Ë"¨d¬#Õ#³°´ Ùb gÂÅÐ·ÓÄÛ¼ç³°´ b0âÔÀ;·®CË;¿¶'ËÁÅÓVÝ³°´ Ùl g ´ Ù ¢ k ´ Ùrb gÝJ¿x©SË¬qä!®µ¨^Ë"Ë"¨d¿GéÇ³°´ c Ù ´ Ùl0Ýq® èÁ3Î*¬;¿¶*¿wÏi³°´ Ù O 0ÝPÄ¶*¶Ë¿¬"Ë®À;¨d¬ÔÙ³°´ Ù c kÝPÄÉ°ÓÄÛ'ª/ÆÄ¶Ä*¬"ÄS©Ë"ÓÎå³µ´ b0Þù®µÀ"ØÂÃ³°´ ÙHÙÞù¬;¿ÓË¿Gç³µ´ Ùrl gÞ¨d®°¿'§a¨d¬;¶º½¾³µ´ ÙHÙk ´ ¢ l gÞ¨dÎ¶*¿©wå³°´ kk ´ Ù h 0Þ¨dÎ¶*¿©GÏ4³µ´ g k

¢ Ù c

Þú·ÉµÉ°¨dÎGå³°´ 0bñ#ÅLÆ!³ ñ³°´ Ù ¢ Ùñ#Åwæ¼³µ´ O ÙæJËÄ@Ü³ Æ!³µ´

¢ Ù0æJÅ¿xÙ³°´ O Ù ´ Ù O ÙæJÅ¿Ïi³ å³µ´¢ Ù0×iõ¬;®4±³µ´ Ùr0 g×i®°Ë"¨d¬Ô«¹³°´ k0 ´ Ùl7Ù ´ Ù h Ù ´ ¢ l0×a·¨dÛÉ°ÓÄÛº¦³ àJ³°´ Ùrkb×aÅ§ÄÛ@Ý³µ´ b0

¢ Ù g

13254U6 87878

;£D ^ x @>MD ^ :

¢ Ù h

Numerical Determination of Heat Distribution and Castability Simulations of as cast Mg-Al alloys.

Shehzad Saleem Khan1 , Norbert Hort1 , Ingo Steinbach2 , Siegfried Schmauder3, and Ulrich Weber3

1 GKSS Forschungszentrum GmbH ,Max Planck Strasse 1, 21502 Geesthacht 2 RWTH Aachen ACCESS e.V. Intzestraße 5 52072 Aachen

3 Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), Universität Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart

Research and development of magnesium alloys depends largely on the metallurgist’s understanding and ability to control the microstructure of the as-cast part. Currently only little information about the solidification of magnesium and about the as-cast microstructures exist. So far, there has not been any subsequent progress in the field of casting simulations for magnesium fluidity. Therefore, the goal of this paper is to increase the general knowledge base of magnesium fluidity behavior with two different mold geometries and to simulate the resultant microstructures. It is a well known fact that cast Mg alloys possesses better fluidity and higher specific mechanical properties in comparison with other mainstream alloys. Application of Mg is able to increase structural integrity and to maximize weight saving of the castings.

Various thermodynamic equipments have been used to study the thermo-physical properties of different magnesium alloys including the (CALPHAD) calculation of phase diagram analysis. These analyses have been performed on 13 binary magnesium- aluminum alloys, the resultant microstructures have been simulated and then have been compared with the experimental output. The temperature distribution and the heat dissipation during casting has been simulated with the finite difference method and compared with experiments for different geometries.

This paper presents an overview of a range of ideas that have been undertaken to improve our understanding on the gravity die-casting behavior and solidification characteristics of Mg-Al alloys. Furthermore, the solidification process of binary Mg-Al alloys, beginning with the nucleation was investigated and a detailed discussion about the design and the constraints evolved in the fabrication of an optimized mould system is undertaken. The whole casting process using (using the finite difference based Magmasoft®, fig 1) as well as the microstructure composition using MICRESS (Micro Structure Evolution Simulation Software ,fig 2) are simulated. Closing, the validation of the results is performed by comparison of the simulated and the experimental results.

Figure 1: The numerical fluidity results from Magmasoft®.

Figure 2: The working mechanism of MICRESS®.

Salient References:

Bottger, B. E.-F.-P. (2006). Controlling microstructure in magnesium alloys: A combined thermodynamic, experimental and simulation approach. Advanced Engineering Materials 8 (4) , 241-247.

J.Eiken, B. I. (2007). Role of solute in grain refinement of Magnesium alloys. Physical Review E 73 , 66-122.

Experimental and computational analysis of toughness anisotropy in

2139 Al-alloy sheet

T.F. Morgeneyer1,2,3a,

,J. Besson1,b

, M.J. Starink2,c

and I.Sinclair2,d

1 Centre des Matériaux, CNRS UMR 7633, Mines Paris, ParisTech, BP 87, 91003 Evry Cedex,

France

2 Materials Research Group, School of Engineering Sciences, University of Southampton,

Southampton, SO17 1BJ, United Kingdom

3 Alcan Centre de Recherches de Voreppe, BP 27, 38341 Voreppe Cedex, France

a [email protected],

b [email protected],

c [email protected],

d [email protected]

Keywords: AA2139, plastic anisotropy, anisotropic yield function, fracture toughness, Gurson

model

Abstract The subject of the present study is the experimental assessment and analysis of toughness anisotropy

of AA2139 Al-alloy sheet material in T3 condition and its prediction using a finite element model.

For this purpose plastic and fracture behaviour of the material was assessed experimentally via

tensile testing in rolling direction (L), long-transverse direction (T) and diagonal direction (45º).

Tests on notched tensile specimens (NT2) [1] and Kahn tear tests [2] are carried out in rolling (L)

and long-transverse direction (T). The Lankford coefficient has been found to be about equal to 1.

The moderate plastic anisotropy of the material is mainly attributed to the prestraining of 2% of the

material in rolling direction. Notched specimens and smooth tensile specimens fail under slant

fracture whilst a flat horizontal triangle region is formed near the notch in the Kahn tear test samples

that is turning into a slant stationary crack. Fracture surfaces are investigated via scanning electron

microscopy. In a previous synchrotron radiation computed tomography study [3] it has been

identified that toughness anisotropy may also be linked to anisotropy of the initial void shape. For

the modelling of toughness a Gurson-type [4] void nucleation and growth model is used that is

implemented in a finite element code Zébulon [5]. The plastic anisotropy is taken into account using

the yield function for anisotropic materials suggested by Bron and Besson [1]. The toughness

anisotropy may be attributed to plastic anisotropy and microstructural factors like the identified

anisotropic initial void shape. It is accounted for via introducing direction dependent weighting

factors in the expression of the Gurson yield surface.

References

[1] Bron F, Besson J, Int. J. of Plasticity, 2004 ; 20 : 937-963

[2] ASTM-international, Standard B 871 – 01. 2001.

[3] Morgeneyer TF, Starink MJ, Sinclair I, article under submission

[4] Gurson A, J. Engng Mater. Technology, 1977 ; 99 : 2-15

[5] Besson J, Foerch R., Comput. Methods Appl. Mech. Eng., 1997 ; 142 : 165-187

Date post:	28-Jul-2015
Category:	Documents
Upload:	sreesh-p-somarajan
View:	259 times
Download:	29 times

17th International Workshop on Computational Mechanics of Materials August 2007 - IWCMM17-Paris BOOK...

Documents