Multiscale Damage

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Multi-scale boundary element modelling of materialdegradation and fracture

G.K. Sfantos, M.H. Aliabadi *

Department of Aeronautics, Faculty of Engineering, Imperial College, University of London, South Kensington Campus, London SW7 2AZ, United Kingdom

Received 2 June 2006; received in revised form 16 September 2006; accepted 18 September 2006

Abstract

In the present paper a multi-scale boundary element method for modelling damage is proposed. The constitutive behaviour of a poly-crystalline macro-continuum is described by micromechanics simulations using averaging theorems. An integral non-local approach isemployed to avoid the pathological localization of micro-damage at the macro-scale. At the micro-scale, multiple intergranular crackinitiation and propagation under mixed mode failure conditions is considered. Moreover, a non-linear frictional contact analysis isemployed for modelling the cohesive-frictional grain boundary interfaces. Both micro- and macro-scales are being modelled with theboundary element method. Additionally, a scheme for coupling the micro-BEM with a macro-FEM is also proposed. To demonstratethe accuracy of the proposed method, the mesh independency is investigated and comparisons with two macro-FEM models are madeto validate the different modelling approaches. Finally, microstructural variability of the macro-continuum is considered to investigatepossible applications to heterogeneous materials.

2006 Elsevier B.V. All rights reserved.

Keywords: Multi-scale modelling; Damage; Non-local approach; Polycrystalline; Microfracture; Finite element method

1. Introduction

In every day engineering, many failures are due to thepre-existence of various types of defects in the materialsmicro-scale [1]. The propagation and coalescence of micro-cracks, microvoids and similar defects in the micro-scaleleads eventually to the complete rapture of the component[2]. However, from a modelling perspective, the transitionof a microcrack to the macro-scale is still not very clear.

Continuum damage mechanics aims to ll that gap. Fromthe early work of Kachanov [3], continuum damagemechanics, in its simplest form, introduces an isotropicscalar multiplier that reduces the initial elastic stiffness of the material over a specic region of the macro-continuum,in order to describe the local loss of the material integritydue to the formation and propagation of microcracks. A

macrocrack is subsequently represented by the regionwhere the damage is so extensive that the material cannotsustain more load [4,5]. Even though continuum damagemechanics can actually deal with initiation of macrocracks,it does not provide sufficient details about the actual initi-ation and behaviour of cracks at micro-scale. Therefore, itis evident that there is a need for modelling materials in dif-ferent scales and actually monitoring their behavioursimultaneously.

Multi-scale modelling is receiving much attention nowa-days due to the increasing need for better modelling andunderstanding of materials behaviour. Engineering materi-als are in general heterogeneous at a certain scale. Textilecomposites, concrete, ceramic composites, etc. are all natu-rally heterogeneous. Even classic metallic materials areheterogeneous at the micro and grain scale. Multi-scalehomogenization methods provide the advantage of model-ling a specic material at different scales simultaneously[69]. At scales where the mechanical behaviour isunknown due to the complexity of the material structure,

0045-7825/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.09.004

* Corresponding author. Tel.: +44 20 7594 5077; fax: +44 20 7594 5078.E-mail address: [email protected] (M.H. Aliabadi).

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no constitutive law is required since this can be dened atsmaller scales where the behaviour may be known. There-fore, different classes of materials can be modelled withthe same principles and monitor simultaneously theirbehaviour at several scales, providing a better understand-ing of their actual behaviour [10].

Multi-scale methods can also provide valuable infor-mations of the damage evolution in a material throughoutdifferent scales [1114]. The macro-continuum can be mod-elled as in the case of continuum damage mechanics, butwithout considering a priori any constitutive law for themechanical behaviour of the material or any damage lawfor the degradation of the materials integrity. Both consti-tutive laws can be deduced from the micromechanicsin situ . Hence, any heterogeneities of the material in themicro-scale will affect directly the macro-continuumresponse and moreover, microcracking initiation and prop-agation in the micro-scale and their affect on the macro-scale will be monitored simultaneously as the micro-scalewill pass information to the macro-scale and vice versa.A schematic representation of the aforementioned method-ology is illustrated in Fig. 1. The macro-continuum feedsthe micro-scale model with boundary conditions that areevaluated by the macro-stress/strain elds of the zoomedarea, while on the other hand, the micro-scale model feedsthe macro-continuum with an updated constitutive law,encountering any possible micro-damage.

To date, multi-scale modelling is mainly carried outwithin the context of the nite element method (FEM)[69,1214]. The boundary element method (BEM), analternative method to the FEM, nowadays provides a pow-

erful tool for solving a wide range of fracture problems[15,16]. The main advantage of BEM, the reduction of the dimensionality of a problem, becomes very attractivein cases of large scale problems that are computationallyexpensive as the multi-scale modelling. In this paper, a par-allel processing multi-scale boundary element method is

proposed for the rst time, for modelling damage initiationand progression in the micro- and macro-scale. Bothmicro- and macro-mechanics are being formulated by theproposed method, nevertheless a link for coupling themicro-BEM with a macro-FEM solution scheme is alsopresented.

Multi-scale modelling of intergranular microfracture inpolycrystalline brittle materials is the problem in consider-ation. Grain boundaries of polycrystalline materials, asappears in the majority of engineering metallic alloys (fer-rous, nonferrous) and ceramics, are often characterised bythe presence of deleterious features and increased surfacefree energy that makes them more susceptible to aggressiveenvironmental conditions. These conditions often lead tobrittle intergranular failure [17,18] and stress-corrosioncracking [19,20], respectively. The cohesive surfacesapproach inside the FEM remains the most popularapproach for modelling such micromechanics failures.Among the proposed cohesive failure models, the linearlaw proposed by Ortiz and Pandol [21] for mixed modefailure initiation and propagation, and the potential-basedlaws proposed by Tvergaard [22] and Xu and Needleman[23] are among the most popular. In the present work,the recently proposed boundary cohesive grain elementmethod by Sfantos and Aliabadi [24] is used for modellingthe micro-scale. Multiple intergranular microfracture initi-ation, propagation, branching and arresting, under mixedmode failure conditions is being modelled in a polycrystal-line material, by incorporating a linear cohesive law [21].Moreover, the random grain morphology, distributionand orientation is taken into consideration.

The macro-continuum is also modelled using the BEM.To monitor the material behaviour in the micro-scale andto pass information to the macro-scale, representative vol-ume elements (RVE) are assigned to points in the domainof the macro-continuum. These RVEs represent the micro-structure, at the grain level, of the macro-continuum at theinnitesimal material neighbourhood of that point. Theformation and propagation of intergranular microcracksis monitored individually to each RVE. Since this micro-damage reduces the elastic stiffness of the RVE, conse-quently the material integrity of the local macro-elementis also reduced. Therefore a non-linear boundary elementformulation is presented for the macro-continuum. Never-theless, there are still advantages for using the macro-BEMas will be discussed later. Moreover, the RVEs are assignedin a random manner in order to encounter stochastic effectsof the microstructure and of possible defects.

Microcracking initiation and propagation in the micro-scale results in strain softening at the macro-scale. Thisstrain softening causes the loss of positive deniteness of the elastic stiffness resulting in an ill-posed problem[25,26]. In the FEM, this loss of ellipticity results in meshsensitivity, where as much as the nite element discretiza-tion is rened, the numerical solution does not convergeto a physically meaningful solution [27,28]. To overcome

this pathological localization of damage, the so-calledFig. 1. Schematic representation of a multi-scale damage model.

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non-local models, either in integral form [25,29,30] or ingradient form [3133] have been proposed. In the contextof BEM, non-local approaches can be found in [34,35].In the present study, an integral non-local approach isenforced to ensure macro-mesh independency and objectiv-ity of the results.

The macromicro interface is being constructed in termsof averaging theorems [36,37]. All quantities transferredfrom the micro to the macro are being volume averagedover the RVE. A brief discussion on possible RVE bound-ary conditions is given and the implementation of the peri-odic boundary conditions in the context of the proposedBEM is explained in detail. The rst order computationalhomogenization is being used in the present work [8,9].

Finally, several numerical examples are presented forsimulating damage and fracture in a polycrystalline brittlematerial. Intergranular cracking evolution at the micro-scale and the resulting damage progression and fractureat the macro-scale are illustrated. The mesh independencyof the proposed formulation is discussed and comparisonswith the FEM for the different damage modellingapproaches concludes the present study.

2. Macromechanics

2.1. Modelling the continuum

In the case of multi-scale damage modelling, the propa-gation and coalescence of microcracks and other defectsthat pre-exist due to manufacturing or are formed due toloading, leads to a progressive degradation of the materialstiffness, in the macro-scale, that introduces a non-linearbehaviour to the problem [4,5]. Therefore, a non-linear for-mulation for the macro-continuum is required, in order toexploit the local non-linear material behaviour that themicromechanics pose to the macro-scale.

Nowadays, it is well established that for linear contin-uum mechanics problems, the BEM can be considered asa powerful alternative tool to the FEM [15]. In cases of non-linear problems, some kind of domain discretizationis usually required to accommodate the non-linear behav-iour of the eld unknowns into BEM. In the present work,an initial stress approach is proposed to include for

micromechanics material non-linearities.In terms of continuum mechanics, the macroscopicallyobserved degradation of the material stiffness due to thepropagation and coalescence of various microdefects inthe micro-scale, suggests the reduction of the local elasticitystiffness tensor. In the present work, the non-linear materialdegradation is introduced in terms of initial decrementalstresses, that soften locally the material. For this initial stressapproach , the boundary integral equation can be written as

C ij x 0_u jx 0 Z --S T ij x 0; x_u jxdS

Z S

U ij x0; x_t jxdS

Z V

E ijk x 0; X_r D jk XdV ; 1

where _u j ; _t j denote the displacement and tractions onboundary S , respectively, T ij , U ij , E ijk are fundamentalsolutions given in Appendix , _r D jk denotes the decrementalcomponent of stress, that is introduced by the micro-scalesolution to soften locally the material in the macro-scaleand C ij is the so-called free term [15]. Even though the

problem in consideration is time-independent, due to theincremental formulation and to maintain a general nota-tion with respect to other time-dependent inelastic phe-nomena, it is regarded as a rate problem where the eldunknowns are denoted by an upper dot. Moreover, withX 2 V a domain point is denoted while with x 2 S a bound-ary point. The source point is denoted by x 0 while the eldpoint is without the dash.

To solve the above equation, the boundary S of themacro-continuum is discretized into N quadratic isopara-metric boundary elements while the expected non-lineardomain V is discretized into M constant subparametricquadrilateral cells. For each cell the eld unknowns areevaluated at its geometrical center and is assumed to beuniformly distributed over its area. In other words, thenon-linear domain is assigned M points, in which themicromechanics response will be evaluated, and thisresponse will be uniformly distributed over the neighbour-hood of the point that is limited by the neighbourhood of the adjacent points. For each point, a representative vol-ume element (RVE) is assigned that would give all theinformation about the micromechanics state in the inni-tesimal material neighbourhood.

After the discretization and using the point collocationmethod for solution, the nal system of equations can be

written, in matrix form asA _x _f E _r D ; 2

where the matrices A, E contain known integrals of theproduct of shape functions, Jacobians and the fundamentalelds, the vector _f contains contributions of the prescribedboundary values and the vector _x contains the unknownboundary values.

The size of the domain that must be discretized is limitedby the distribution of the micro-damage during the loadingprocess, that would introduce non-linear material behav-iour at the macro-scale. However, in cases of non-homoge-neous materials of which behaviour depends on the locationeven in the elastic regime, the whole domain must be disret-ized. A great advantage of the proposed boundary elementformulation is that even if all the macro-continuum domainwas discretized and an RVE was assigned to each domainpoint, as long as the material remains locally undamaged,the micromechanics simulations are linear and the contri-bution to the computational effort is negligible. On theother hand, for the completely damaged zones, the RVEssimulations are stopped and computational storage andtime is saved yet again. Therefore, it should be mentionedhere that even in cases where the macro-damage patternis unknown, discretizing the whole domain would not

increase the computational effort substantially.

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Another advantage of the proposed formulation is thatthe size of the nal system of equations, for the macro-con-tinuum, remains unchanged irrespective of the number of the domain points and therefore RVEs they are considered.From the nal system of equations that must be solved, Eq.(2), it can be seen that the material non-linearities due to the

micro-damage are acting as a right-hand side vector thatdoes not increase the system size. Hence, at every incrementthis right-hand vector is evaluated and the new solution isgiven by forward and back substitution with the L and Udecomposed matrices of the coefficient matrix A [38].

After solving the macro-continuum, the internal strainson every domain point must be evaluated, in order todene the boundary conditions on the correspondingRVE, in the micro-scale, for the next increment. Consider-ing Somiglianas identity for the internal displacements [15]and the Cauchy strain tensor for small deformations_eij 12 _ui; j _u j ;i, the boundary integral equation for theinternal strains can be obtained by differentiating Eq. (1)with respect to the source point X0 and gives

_eij X0 Z S Deijk X0; x_t k xdS Z S S eijk X0; x_uk xdS Z --V W eijkl X0; X_r Dkl XdV _ g eij X0; 3

where Deijk and S eijk are fundamental solutions produced by

the derivatives of the U ij and T ij fundamental solutions,respectively. The fourth order fundamental solution W eijklhas been evaluated by the derivative of the domain integral,Eq. (1), using the Leibniz formula and the free term _ g eij isdue to the treatment of the O( r 2) singularity in the senseof Cauchy principal value [15]. All the fundamental solu-tions can be found in Appendix .

Finally, the boundary integral equation for the internalstresses at the macro-continuum is derived through theapplication of Hookes law and Eq. (3). i.e.,

_r ij X0 Z S Drijk X0; x_t k xdS Z S S rijk X0; x_uk xdS Z --V W rijkl X0; X_r Dkl XdV _ g rij X0: 4

2.2. Non-local approach

To ensure mesh independency and reproducibility of the numerical results, a non-local approach must beintroduced in order to avoid the pathological localizationof micro-damage at the macro-scale. Generally, a non-localapproach consists of replacing a specic variable by itsnon-local weighted volume averaged counterpart [25,29,30]. The choice of the variable to be averaged is arbi-trary, in some extent. However the new non-local modelmust exactly agree with the standard modelling approach,as long as the material behaviour remains elastic.

In the proposed multi-scale boundary element formula-

tion, the local degradation of the material stiffness due to

the micro-damage evolution is modelled by introducing atthe macro-scale the decremental stress, _r D , which resultsfrom the initiation and propagation of microcracks insideeach RVE, at the micro-scale. However this stress compo-nent cannot be replaced directly by its non-local counter-part. To overcome this, the following technique is

introduced. For every domain point, i = 1, M , that has beenassigned an RVE for monitoring the microscopic behav-iour, the non-local macro-strain _eMX0 is evaluated afterevery macroscopic solution, by considering the macro-strains in the neighbourhood of this point, as follows:

_eMX0 Z V aX0; X_eMXdV X; 5where aX0; X a0X0; X R V a0X0; ndV n 1 anda0(X 0, X) in the present work is taken to be the Gauss dis-tribution function, given for the two-dimensional case as

a0X0

; X exp 2 j X X0j2

l 2 !; 6where l denotes the material characteristic length, whichmeasures the heterogeneity scale of the material [29].

This non-local macro-strain is used to evaluate the peri-odic boundary conditions to be assigned to the correspond-ing to point X

0

, RVE, as it will be described in Section 3.After solution of the specic micromechanics problem withthe prescribed boundary values dened by _eM , the volumeaverage total stress _r t is evaluated using averaging theoremspresented in Section 3. From this stress, the decrementalcomponent that is used as initial stress in the boundary

element formulation is evaluated as _r D

_r e _r t

, where_r e is the elastic stress in case of no-microdamage; the upperhat denotes that these stresses resulted by the non-localmacro-strain that corresponded to the specic point X0 atthe macro-continuum the RVE was assigned.

However, the aforementioned decremental componentof stress, _r D , cannot be directly implemented in the bound-ary integral Eq. (1), since it corresponds to the non-localstrain eld and not to the local one. At this point amacro-damage coefficient is introduced, denoted by D ij ,given by the subdivision of the decremental stress by thenon-local elastic stress, resulting in

Dij X0

1 ^_r

tij X

0

^_r

eij X

0

1

; 7where no summations are implied for the repeated indices i , j and D ij = D ji due to the symmetry of the strain and stresstensors. In the case where D ij = 0, no damage has takenplace, where in cases of Dij = 1 the macro-continuum iscompletely damaged and a macrocrack (fracture) must beintroduced.

In the context of the proposed boundary elementmethod for the macro-continuum, to implement the afore-mentioned damage, a local decremental stress is evaluatedby

_r

Dij X

0

D ij C Mijkl

_e

Mkl X

0

; 8

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where C Mijkl denotes the fourth order elasticity stiffness ten-sor of the macro-continuum and, again, no summation isimplied for the repeated indices ij .

At this stage it should be noted that some attention mustbe paid to cases of loading an RVE by a strain tensor of theform {e11 , e22 , e12 } = {0, a , 0}, where a 2 R . In this case,

damage is expected to appear along the 11- direction (foran isotropic material without defects). Therefore, the afore-mentioned damage coefficient D ij should describe the devel-oped damage due to loading on the 22- direction ; that is0 < D22 6 1. However, due to the Poisson effect, the devel-oped average stress component on the 11- direction will alsobe lower than the undamaged (linear elastic) componenton the same direction. Consequently, Eq. (7) would givea damage coefficient D 11 , which is articial, since no dam-age has occurred on this direction and is due to the Poissoneffect. For the case of a perfect homogeneous isotropicmaterial, this articial damage is always equal to the actualone. For the general case of a polycrystalline material com-posed of randomly orientated anisotropic grains, as in thepresent work, this articial damage appears to be lower, orin the range of the actual one.

3. Denitions, averaging theorems

As pointed out before, an RVE represents the micro-structure of an innitesimal material neighbourhood for apoint in a macro-continuum mass. Hence the stress andstrain elds corresponding to the macro-scale will bereferred to as macro-stress/strain and will be denoted bya superscript M, as r M and eM , respectively. On the other

hand, the stress, strain elds corresponding to the RVEs(that is the micro-scale), will be referred as micro-stress/strain and denoted by a superscript m, as r m and em ,respectively. In multi-scale mechanics, averaging theoremsand quantities are required in order to transfer informationthrough the different scales [36,37]. Therefore, every aver-aged quantity referring to the RVEs will be denoted byan upper bar; that is r m, em for the volume averagemicro-stress and micro-strain, respectively. As pointedout before, a rate problem is regarded here, where the eldunknowns are denoted by an upper dot; that is _r ; _e for thestresses and strains respectively. Moreover, as innitesimaldeformations are considered in the present work, it shouldbe noted that the average micro-stress/strain rates, equalsthe rate of change of the average micro-stress/strain [36];that is: _r m _r m; _em _em.

As a benchmark problem in the present work, a poly-crystalline brittle material is considered, that is susceptibleto intergranular fracture. Assume now the RVE illustratedin Fig. 2. This RVE represents the microstructure of a poly-crystalline brittle material and is composed of randomlydistributed and orientated single crystal anisotropic elasticgrains. It was produced by the PoissonVoronoi tessella-tion method, which is extensively used in the literaturefor modelling polycrystalline materials in a random man-

ner [17,39]. Each grain is assumed to have a randomly

assigned material orientation, dened by an angle h sub-tended from the x geometrical axis, where 0 6 h < 360(non-directional solidication is assumed). Since the pres-ent study considers two-dimensional problems, to maintainthe random character of the generated microstructure andthe stochastic effects of each grain on the overall behaviourof the system, three different cases are considered for eachgrain in view of which material axis is normal to the plane[40], i.e. Case 1: 1 z, Case 2: 2 z and Case 3: 3 z(working plane is assumed the xy).

Since every grain is assumed to have a general aniso-tropic mechanical behaviour, the RVE would behave ina linear elastic manner as long as the interfaces are stillintact. Each grain H : H = 1, N g, where N g denotes thetotal number of grains in the RVE, has a volume denoted

by V H

and a surface denoted by S H

. Therefore the volumeof the RVE, V m , is given by

V m [ N g

H 1V H : 9

The boundary of each grain is divided into the contactboundary S H c , indicating the contact with a neighbour grainboundary and into the free boundary S H nc , indicating thegrain boundaries that coincide with the boundary of theRVE, S m . Hence for every grain

S H S H nc [ S H c : 10

For the internal grains S H nc and thus S

H

S H c . There-fore S H nc exists only on the RVE boundary grains resulting

to

S m [ N g

H 1S H nc; 11

where S m denotes the boundary of an RVE.Let us assume the overall collection of all grain bound-

ary interfaces within an RVE to be denoted by S mpc, andgiven as

S mpc 1

2 [ N g

H 1

S H c : 12

Fig. 2. Articial microstructure with randomly distributed materialorientation for each grain.



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Along this path, potential intergranular microcracksmay be initiated and propagated. In general, the overallproperties of this RVE are strongly affected by the mor-phology and the material orientation of its grains and thecondition of all its grain boundary interfaces S mpc. Thesegrain boundary interfaces may be undamaged, partially

damaged and completely damaged. The latter deectsintergranular cracks that can propagate along the grainboundaries. This type of debonding consumes mechanicalenergy and leads to greater toughness. Therefore, its effecton the overall behaviour of the RVE must be consideredwhen averaging theorems are used. The overall volumeaverage micro-stress _r m;t ij of an RVE composed by grainscan be given as

_r m;t ij 1V m Z V m _r mij dV m 1V m X

N g

H 1 Z V H _r H ij dV H 13and since the stress tensor is divergence-free [36], using thedivergence theorem and considering Eq. (10)

_r m;t ij 1V m X

N g

H 1 Z S H nc x H i _t H j dS H nc Z S H c x H i _t H j dS H c( ); 14where _t j _r ij ni denotes the surface tractions.

Consider now that the debonding of the grain bound-aries can be modelled as displacements discontinuities,d _u I , and tractions jumps, d_t I . However, to ensure equilib-rium, traction jumps must always vanish. In other words,in cases of partially damaged boundaries or closed cracks,the local tractions must cancel each other and in cases of completely formed opened cracks their surfaces must betraction free. Hence, by using the denition of the RVEboundary, Eq. (11), the overall volume average stress,_r m;t ij , can be evaluated by

_r m;t ij 1V m Z S m xmi _t m j dS m; 15

where xmi ; _t m j represents the position vectors of the pointslying on the RVE boundary and their tractions,respectively.

In terms of strains, the volume average strain, _emij , can beevaluated in a similar manner as

_emij 12V m Z V m _umi; j _um j;idV m

12V m X

N g

H 1 Z V H _u H i; j _u H j ;idV H 16and by using again the divergence theorem and Eq. (10),leads to

_emij 12V m X

N g

H 1 Z S H nc _u H i n H j _u H j n H i dS H nc(

Z S

H c

_u H i n H j _u

H j n

H i dS

H c

): 17

Considering now small deformations, for two adjacentgrains A and B over an interface, the displacements discon-tinuities are dened as d _u I _u A _u B, in global coordinates,and the outward normal unit vectors of each grain are nA

and nB = nA , respectively. The volume average straincan be evaluated after using Eqs. (11) and (12) by

_emij 12V m Z S m _umi nm j _um j nmi dS m Z S mpc d _u I i n A j d _u I j n Ai dS mpc( ):

18

Transforming the displacements discontinuities fromglobal, d _u I , to local, d~_u I , coordinates, the opening gapd~_u I n and the sliding gap d~_u

I t along the damaged interfaces

can be used directly for evaluating the volume averagestrain. The transformation of the displacements discontinu-ities is given by

d _u I i Rik d~_u I k ; 19

where Rik denotes the transformation tensor. Finally, Eq.(18) can be written as

_emij 12V m Z S m _umi nm j _um j nmi dS m Z S mpc Rik d~_u I k n A j R jk d~_u I k n Ai dS mpc( ):

20

In the case of perfect grain boundary interfaces, the dis-placement discontinuities vanishes, i.e. , d~_u I 0 and there-fore the last term of the above equation vanishes too. Onthe other hand, when the interfaces are imperfect, partiallydamaged and/or completely damaged (cracked), displace-

ment discontinuities exist, i.e. d~_u I

6 0, and therefore anadditional strain appears due to the presence of micro-cracks and partially damaged interfaces. This additionalstrain is represented by the last term in Eq. (20) and pro-vides a correction to the effective volume average straindue to the possible discontinuity of the displacements ona grain boundary interface that has been partially damagedor cracked [14,36,41]. It should be noted that for the slidingcomponent of the displacements discontinuities, d~_u I t , bothpositive and negative values may be considered to modelthe two way sliding of the grain boundary interfaces. How-ever, for the normal opening component, d~_u I n , only openingis considered, that is negative values for convection withthe denition of the outward normal unit vectors of thegrains. This is because the impenetrability conditions areenforced in the contact detection algorithm to ensure thenon-penetration of the cracked grain boundaries [24].Moreover, the detailed contact history of every interfacecrack is being recorded throughout the incremental pro-cess, in order for the internal friction effect on the slidingand the sticking of the crack interfaces to be consideredin evaluating the volume average strain.

Generally speaking, in a multi-scale method, the macro-stress _r M and macro-strain _eM tensors corresponding to apoint XM in the macro-continuum, can be evaluated

directly by the volume average micro-stress _r m and



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micro-strain _em over the RVE, which represents the micro-structure of the innitesimal material neighbourhood atpoint XM . On the contrary, the macro-stress/strain canprovide the boundary conditions for the RVE [36]. Adetailed discussion on the possible boundary conditionsand the one used here is given in Section 4.2.

4. Micromechanics

4.1. Microstructure modelling

For each domain point in the macro-continuum, anRVE is assigned to represent the microstructure of themacro-continuum at the innitesimal material neighbour-hood of that point. In the present work, the macro-contin-uum is assumed to be made from a polycrystalline brittlematerial. Therefore, each RVE represents the microstruc-ture of a polycrystalline material, that is subjected to brittleintergranular fracture. In this context, the newly proposedboundary cohesive grain element method by Sfantos andAliabadi [24] is being used here to model multiple micro-fracture initiation, propagation, branching and arresting,under mixed mode failure conditions in polycrystallinebrittle materials. Here only the basics will be outlined forcompleteness. For more details the readers are referred toRef. [24].

Fig. 2 illustrates an RVE composed by randomly dis-tributed and orientated single crystal anisotropic grains,as pointed out in Section 3. For each grain interface, thatis a boundary of two neighbour grains, say A and B , trac-tions equilibrium and displacements compatibility are

directly imposed; that is~_t I ~_t Ac

~_t Bc and d~_u I ~_u Ac

~_u Bc 0; 21

where ~_t I and d~_u I denote the interface tractions and relativedisplacements jump and the upper bar ~ denotes values inthe local coordinate system. The local coordinate system isdenoted by the outward normal vector to the grain bound-ary (i.e. ~_t I n and d

~_u I n) and the tangential vector to the bound-ary (i.e. ~_t I t and d

~_u I t ). The transformation from the global tothe local is given by ~_t I R _t I and d~_u I Rd _u I , where R de-notes the local rotation matrix [24].

The displacements integral equation [24] for each grain,

can now be written as

C H ij z 0k ~_u

H j z

0k Z --S H nceT

H ij z

0k ; z k ~_u

H j z k dS

H nc

Z --S H c eT H ij z

0k ; z k ~_u

H j z k dS

H c

Z S H nc eU H ij z

0k ; z k

~_t H j z k dS H nc

Z S H c eU H ij z

0k ; z k

~_t H j z k dS H c ; 22

where

eT H ij ,

eU H ij denote the anisotropic fundamental solu-

tions and z 0k x01 l k x02, zk = x1 + l k x2, for k = 1,2, de-

note the source and the eld points in a complex plane,respectively, and l k are the roots of the characteristic equa-tion [15]. All components in Eq. (22) refer to the local coor-dinate system. In the case of internal grains, the rstintegral on the left and the right-hand side of Eq. (22) van-ishes since for these grains S H nc .

The boundaries S H c and S

H nc of each grain H = 1, N g are

discretized into N H c and N H nc constant sub-parametric

elements respectively. The motivation for using constantelements is that all eld unknowns, these are interface trac-tions and displacements discontinuities, are located at thecenter of these elements and not at the edges; thus prob-lems at triple points (points where three grains meet) areautomatically avoided. After the discretization and apply-ing the interface boundary conditions Eq. (21), the nalsystem of equations can be written, in matrix form as

" A

0 BC # ~_x

d~_u I ~_t I 8>: 9>=>;

R~_y

F( ); 23

where the submatrices A and R are sparsed containingknown integrals of the product of the shape functions,the Jacobians and the fundamental elds. Submatrix A alsocontains the interface boundary conditions Eq. (21). Thevectors ~_x and ~_y denotes the unknown boundary conditionsand the prescribed boundary values along the domainboundary S m , respectively. The submatrix BC contains allthe interface conditions for the grain facets, correspondingto d~_u I and ~_t I , while the submatrix F contains the right-handsides of these interface conditions.

To ensure mesh independency and reproducibility of thesolution in the present study, the grain boundary elementssize was always LCZ2 Le > 15, where LCZ denotes the cohesivezone size at the crack tip, given by [42]

LCZ p2

K ICT max 2; 24

where K IC denotes the fracture toughness of the material inMode I for plane strain conditions and T max denotes thestrength of the cohesive grain boundary pair under purenormal separation [42].

4.2. RVE boundary conditions

The accurate estimation of the overall response of anRVE is of great importance in a multi-scale modelling,and is directly related to the applied type of boundary con-ditions. In order to be able to use the averaging theoremspresented in Section 3, for transferring informationthrough the scales, four types of boundary conditions canbe used; these are uniform tractions, uniform displace-ments, mixed boundary conditions and periodic boundaryconditions [7,36,43,44].

The rst case of the aforementioned boundary condi-

tions, uniform tractions, do not provide all the required



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information for a numerical analysis, since rigid bodymotion will be inevitable.

The uniform displacements boundary conditions can beapplied directly on the RVE boundary, considering themacro-strain _eM at the domain point X0 in the macro-continuum [36]

_um;oi _eMij xm j ; 25

where x m denotes the position vector of every point on thedomain boundary S m of an RVE, i.e., xm 2 S m . By apply-ing uniform displacements boundary conditions on theRVE an underestimation of the mechanical properties of the RVE is achieved [7]. However, in the present case whereintergranular cracks may run up to the RVE boundaries,uniform displacements boundary conditions are overcon-straining the response of the RVE in excess loading thatwould result in excess micro-damage. This is due to the factthat the applied displacements are always a linear transla-tion of the square boundaries of the RVE and thereforethey overconstraint crack propagation close to the RVEsboundaries.

The mixed boundary conditions would not overcon-straint crack propagation, however are not applicable inthe present case since they require the RVE to have at leastorthotropic behaviour and the mixed uniform boundarydata must exclude shear stresses or strains [43].

To date, the Periodic Boundary Conditions (PBC) areusually preferred since they provide the most reasonableestimates of mechanical properties of heterogeneous mate-rials, even in cases where the microstructure is not periodic

[7,8]. To apply the PBC, the RVE boundary S m is separatedinto left, right, top and bottom parts, as Fig. 3 illustrates,and for the two-dimensional case the following conditionsare applied:

_u Ri _u Li _e

Mij x

2 j x

1 j ; and _u

T i _u

Bi _e

Mij x

4 j x

1 j ; 26

_t Ri _t Li ; and _t

T i _t

Bi ; 27

where us and ts, for s = {T , B , R , L} represents the applieddisplacements and tractions, respectively, on the top, bot-tom, right and left side of the RVE boundary. The position

vectors of the vertices 1, 2 and 4, as Fig. 3 illustrates, aredenoted by x i , i = {1,2,4}. In the present case where alleld unknowns in the micro-scale are referred to the localcoordinates, Eq. (22), the PBCs take the following form:~_u Ri ~_u

Li d _ x

R Li and

~_uT i ~_u Bi d _ x

T Bi ; 28

~_t Ri ~_t Li and

~_t T i ~_t Bi ; 29

where d_ x R Li R Rij

1 _eM jk x2k x1k , d_ xT Bi RT ij

1 _eM jk x4k x1k and R

R and RT are the right and top side rotationmatrices.

However, closer examination of Eqs. (28) and (29)

shows that these boundary conditions cannot be directlyimplemented into the BEM, as they are constraint equa-tions instead of prescribed boundary values as in the caseof uniform displacements, Eq. (25). In other words, theprescribed boundary conditions are obtained from the nalsolution of the RVE. Hence, there are no initial prescribedconditions but boundary constraints that increase the sizeof the nal system of Eq. (23). In order to implement theaforementioned periodic boundary conditions in the pre-sented boundary cohesive grain element formulation, with-out increasing the nal system of equations, the PBC, Eqs.(28) and (29), are directly implemented in the coefficientsubmatrix [ A], Eq. (23), and the unknown boundary valuesare now the displacements and tractions of the right andtop RVE boundary sides. To be more precise, consideringEq. (23), the part of submatrix [ A] that corresponds to theRVE boundary unknown values would take the followingform:

H T H B H R H L G T G B G T G L _uT

_u R

_tT

_t R

8>>>>>:9>>>=>>>; f H Bd _x T B H Ld _x R Lg; 30

where the submatrices H s, G s, for s = {T , B , R , L}, containknown integrals of the products of the Jacobian and theanisotropic tractions and displacements fundamental solu-tions, respectively, corresponding to the RVE boundarynodes.

The general condition for applying the aforementionedPBC is that the discretization of the RVE boundary onopposite sides must coincide. Therefore the grain boundarymesh generator must place the same number of elements atsame locations on opposite sides, for the PBC to be directlyimplemented. Fortunately, in the framework of boundaryelement methods, such implementations of the mesh are

relatively easy to achieve. Moreover, considering Fig. 3,Fig. 3. Schematic representation of a typical RVE under periodic

boundary conditions.



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rigid body motions can be eliminated by requiring _uk 0,for either k = {1, 2,4} [45].

4.3. Grain boundaries interface

Cohesive modelling is suitable for interfaces where

materials with different properties join, since it avoids thesingular crack elds close to the crack tip. In the presentformulation, the displacements compatibility conditions(21) are directly implemented in the BEM resulting in thecancellation of any penetration or separation of the grainboundary interfaces. In this way, difficulties with the initialslope of the bilinear cohesive law (extensively used in theFEM) are avoided [42,40]. However, to initiate damagein the BEM formulation, considering mixed mode failurecriteria, all the information must be gathered by the inter-face tractions. Therefore an effective traction is introduced_t I ;eff , over all grain boundary interface node pairsi 1; M

c : i 2 PC , where PC denotes the potential crack

zone. Once damage has initiated on a specic grain bound-ary node pair, say i 0, it is assumed that this pair enters thecohesive zone; that is i0 2 C Z . Following Ortiz and Pan-dol [21], an effective opening displacement is introduced,that accounts for both opening (Mode I) and sliding (ModeII) separation. The effective traction and opening displace-ment are given as

_t I ;eff h_t I ni2

ba

_t I t 2" #12

and

_d

d_u I ndu I ;crn

2

b2 d_u

I t

du I ;crt 2

" #12

; 31

where _t I n; _t I t are the normal and tangential components of

the interface traction ~_t I ; b and a assign different weightsto the sliding and opening mode and hi denotes the Mc-Cauley bracket dened as hx i = max{0, x} x 2 R . Damageis initiated once the effective traction, _t I ;eff , exceeds a max-imum traction, denoted as T max ; hence: _t I ;eff P T max . Theterms d_u I n , d_u

I t denote the normal and tangential relative

displacements of the interface and du I ;crn ; du I ;crt are critical

values at which interface failure takes place in the case of pure Mode I and pure Mode II, respectively.

The normal and tangential components of the tractionacting on the interface in the fracture process zone aregiven by

_t 1 _d

_d Kd _u; 32

where K T max=du I ;crn 0

0 aT max =du I ;crt and a b2 du I ;crndu I ;crt .Due to the irreversibility of the interface cohesive law,unloadingreloading in the range 0 6 _d < d is given byEq. (32) where _d is replaced by d *, which denotes the lasteffective opening displacement where unloading took place.

Once a microcrack has formed, that is _d 1, the two

free surfaces of the microcrack can come into contact, slide

or separate. Upon interface failure, the equivalent nodaltangential tractions are computed using the Coulombsfrictional law. Therefore a fully frictional contact analysisis introduced in the proposed formulation to encountersuch effects [24].

It worth noting that all the aforementioned interface

laws can be implemented directly into the submatrix BCof the nal system of Eq. (23). This is a signicant advan-tage of the proposed boundary element formulation, sincethe introduction of the cohesive elements and later of thefree microcracks do not affect the size of the nal system.This is due to the fact that all the interface laws can bedirectly implemented as local boundary conditions alongthe grain boundaries of the microstructure, by couplingthe local tractions and relative displacements discontinu-ities through the interface laws. The system becomes non-linear only when interface elements exist along grainboundaries that are in the loading case (not unloading/reloading), since the interpretation of Eq. (32) is required.For all other cases the system is fully linear.

5. Micromacro interface

5.1. Coupling with macro-BEM

Considering now the case where the RVE boundaryconditions are dened by a macro-strain _eM . In the absenceof any partially damaged, cracked grain boundary inter-face, the corresponding overall volume average stress _r m;t ijassociated with the prescribed macro-strain would be equalto

_r m;elij C mijkl _eMkl ; 33

where the term _r m;elij denotes the corresponding averageelastic stress, related to the prescribed macro-strain andC mijkl is the fourth order elasticity tensor corresponding tothe RVE. If the RVE is sufficiently large so that eventhough is composed of randomly distributed and orien-tated single crystal anisotropic grains, its overall mechani-cal behaviour is isotropic due to the homogenization [7,24]and equal to the macro-continuum (if the macro-contin-uum is assumed to be isotropic). In this case, Eq. (33)can be used directly by replacing the RVE elasticity tensor

with the macro-continuum elasticity tensor C Mijkl . Neverthe-less, the elastic average stress can always be computed by

the averaging theorem, Eq. (15), for each RVE by consid-ering no damage at the grain boundary interfaces.

Due to the presence of partially damaged and crackedgrain boundary interfaces, the volume average micro-stressis not in general equal to Eq. (33). Nevertheless the totalvolume average micro-stress is dened by

_r m;t ij _rm;elij _r

m;Dij ; 34

where _r m;D denotes the decrement in the overall stress, dueto the presence of cracked and damaged grain boundary

interfaces.



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Taking into account Eq. (15) for the evaluation of theoverall volume average stress over an RVE, the additionalstress term in the above equation can be evaluated as

_r m;Dij _rm;elij

1V m Z S m xmi _t m j dS m: 35

This component of stress is considered as initial stressfor the macro-continuum boundary element formulationpresented in Section 2.1. When no microdamage has takenplace, the last term in Eq. (35) is equal to _r m;el and thereforethe initial stress component vanishes. Hence, the macro-continuum is still in the elastic regime without any damage.On the other hand, when the RVE is completely brokenand cannot carry anymore load, the last term in Eq. (35)vanishes and the decremental component of stress equalsthe fully elastic. In the macro-continuum BE formulationthis initial stress completely cancels the elastic and there-fore the macro-material stiffness has completely degradedat that point.

5.2. Coupling with macro-FEM

In the case where the macro-continuum is being mod-elled with a domain numerical method, like the niteelement method, an RVE can be assigned at every integra-tion point or centroid of an element. Degradation of theRVE stiffness due to possible initiation and propagationof microcracks can be modelled directly by assuming anew stiffness tensor, C Dijkl , that correlates the total volumeaverage micro-stress with the prescribed macro-strain; i.e._r m;t ij C Dijkl _e

Mkl : 36

To this extent and considering Eq. (34), the overall averagestress over an RVE can be evaluated in terms of strains as

_r m;t ij C mijkl _eMkl C

mijkl

_em;Dkl ; 37

where _r m;Dij C mijkl _em;Dkl , and _e

m;Dkl denotes the additional

strain component due to the presence of microcracks [36].Considering now Eq. (20), this additional volume aver-

age strain component can be evaluated by

_em;Dij 12V m Z S mpc Rik d~_u I k n A j R jk d~_u I k n Ai dS mpc( ): 38

Following Kouznetsova [44], and considering the peri-odic boundary conditions for an RVE presented in Section4.2, Eq. (30), the nal system of the proposed microme-chanics BEM, Eq. (23), can be rearranged in terms of thedisplacements discontinuities as

K1 K2

K3 K4" # ~_xd~_u I ( ) P _eM0 ; 39where P = H B (R T ) 1dx 4 1 H L (R R ) 1dx2 1 and K1 R m m, K2 R m n , K3 R n m, K4 R n n denote sub-matrices.

At the end of a microstructural increment, where a con-

verged state has been achieved, a third order tensor L ijk can

be evaluated, that relates directly the displacement discon-tinuities with the prescribed macro-strains, i.e. d~_u I i X Lijk X_eM jk , where Lijk K

3il K

1lp

1 K 2 pn K 4in

1 K 3nm K 1ms

1

P sjk and Lijk = L ikj .Using now the relation between the displacements dis-

continuities and the prescribed macro-strain, Eq. (38) takes

the form_em;Dij J ijkl _eMkl ; 40

where J ijkl is a fourth order tensor with symmetriesJ ijkl = J jikl = J ijlk given by

J ijkl 12V m Z S mpc Rim Lmkl n A j R jm Lmkl n Ai dS mpc( ): 41

Finally, the damaged stiffness tensor is obtained bysubstituting Eqs. (41) and (40) into Eq. (37) and consider-ing Eq. (36). The resulting expression must be valid for anyconstant symmetric macro-strain [36], given by

C Dijkl C ijkl C ijmn J mnkl : 42

From the above expression, the damaged stiffness matri-ces, in the context of the FEM, are evaluated, depending if the specic RVE is assigned to an integration point or thecentroid of a macro-nite element.

6. Multi-processing algorithm

The proposed multi-scale boundary element method is aparallel processing formulation that requires special atten-tion during the implementation, in order to be efficient

and robust. Each micromechanics simulation, that is eachRVE, is assumed to be an individual sub-program that runsseparately and in parallel with all the other micromechanicsprograms and the macromechanics main program. Sincethe proposed formulation is an incremental solutionmethod, for every micromechanics simulation the inversecoefficient matrix of the nal system of equations, Eq.(23), must be stored. As micro-damage progresses andtherefore the interface boundary conditions are changing,the coefficient matrix of each micromechanics simulationwould dynamically change. Therefore, throughout the sim-ulation only updates of the inversed matrix should be madein order to reduce the computational effort of repeatedinversion of the coefficient matrix. For more details onthe implementation of the micromechanics the readers arereferred to [24]. The macromechanics main program con-trols all the micromechanics programs. The macro-programstarts all the micro-programs and gives them the green agfor reading its output. Once all the micro-programs havenished, the macro-program reads their outputs and pro-cesses them. When the micro-programs are running, themacro-program is placed on pause and vice versa.

The main algorithm is illustrated in Fig. 4. Once all theRVE sub-programs have started, built the BE mesh andinvert their main coefficient matrix, the critical macro-loadk where micro-damage will be initiated in the 1st RVE, for



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the 1st time, is evaluated. This is done directly since thewhole system remains fully linear elastic and saves compu-tational effort of incrementing the macro-load in the linearelastic regime. The incremental scheme starts by increasingstep-by-step the applied macro-load. The macro-contin-uum is being solved and the macro-strains are evaluatedfor every domain point that has assigned an RVE to repre-sent the corresponding microstructure. Parallel processingof every RVE micro-mechanics starts by applying thenew periodic boundary conditions. When all of them havenished, the main program reads their outputs, i.e. the dec-remental component of stress, and evaluates the right-handvector to encounter the possible microdamage. Afterresolving the macro-continuum, the convergence is checkedby evaluating the macro internal energy at each internal

loop k , by U M ;k

R V M r

Mij e

Mij dV

M

, and enforcing the follow-

ing tolerance: 100 j U M ;k U M ;k 1

U M ;k j6 0:1% . If the prescribedtolerance has not been reached, the macro-strains are re-evaluated considering the previous macromicrodamagestate and the micromechanics sub-programs resolve theRVEs for the new boundary conditions. When convergenceis achieved, the intermediate results are printed andanother macro-load increment is applied.

In continuum damage models, a macrocrack is repre-sented by a region of completely damaged material. How-ever, this completely damaged region should be excludedfrom the macro-continuum formulation, since the govern-ing equations are meaningless as the material has no stiff-ness there. Moreover, in non-local formulations as theone used here, the large strains due to the complete lossof the material stiffness would lead to wrong estimates of

the non-local averaged strains. Additionally, by excluding

Fig. 4. Incremental solution algorithm for boundary element multi-scale modelling.



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this region from the macro-continuum formulation, theassigned completely damaged RVEs are also excluded,resulting in savings in computational time and storage.By excluding this completely damaged region, a new inter-nal or external boundary is specied and boundary condi-tions are applied. In order to do so, the macro-continuum

is remeshed and the local solution is remapped onto thenew mesh [46]. However this is rather complicated andinterpolation errors will be inevitably introduced. More-over in the case of multi-scale modelling the macro-posi-tions where the RVEs are assigned cannot change duringthe solution process. In this paper, after following Peerlingset al. [33] who proposed the following remeshing method inthe context of the FEM, the completely damaged macro-cells, that is the assigned RVEs macro-points and theirneighbourhood, are removed from the macro-continuumand the additional newly formed macro-boundary is beingdiscretized using quadratic boundary elements. To ensuresmooth transition and crack propagation and on the otherhand to avoid numerical singularities, a critical damagefactor is specied; i.e. D* = 0.999. The criterion for remov-ing a completely damaged cell was chosen to bemax{D11 , D22 , D12 } P D *.

7. Multi-scale damage simulations

Multi-scale damage simulations are performed usingthe proposed method for a polycrystalline Al 2O3 ceramic

material. At the micro-scale, multiple intergranular crackinitiation and propagation under mixed-mode failure con-ditions is considered. Moreover, the random grain distri-bution, morphology and orientation is also taken intoaccount. In cases of fully cracked grain boundary inter-faces, a fully frictional contact analysis is performed to

allow for sliding, sticking and separation of the crackssurfaces. The mesh independency of the proposed formu-lation is addressed. Additionally, comparisons with theFEM are made in order to investigate the different model-ling philosophies. Several examples are illustrated to con-clude the study.

Fig. 5 illustrates a schematic representation of the prob-lem solved here. A polycrystalline Al 2O3 is subjected tothree-point bending, at the macro-scale, by applying dis-placements control. The expected non-linear macro-regionis assigned a number of domain points and on each pointan RVE is handed over. Two cases are investigated: (a) ini-tially the same RVE is considered for every macro-domainpoint and (b) a randomly picked different RVE is assignedto each point to investigate heterogeneous microstructureswith possible defects, randomly distributed in the macro-domain. The RVEs are randomly generated by Voronoitesselations as it was described in Section 3 [24]. The singlecrystal elastic constants of Al 2O3 considered here are:C 11 = 496.8 GPa, C 33 = 498.1 GPa, C 44 = 147.4 GPa, C 12 =163.6 GPa, C 13 = 110.9 GPa, C 14 = 23.5 GPa [47].The fracture toughness of the material K IC 4 MPa m

12,

Fig. 5. Schematic representation of the multi-scale problem.



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T max = 500 MPa, a = b = 1 and plain strain conditionswere assumed. The RVEs were composed by 21 grains ,randomly distributed with random material orientation,of average grain size ASTM G = 10 ( Agr 126 l m2,d gr 11:2 l m [48]). The interface internal friction coeffi-cient was assumed to be l = 0.2. The macro-continuum

elastic properties were E = 415.0 GPa, for the elastic mod-uli and m = 0.24, for the Poisson ratio [49]. The non-localmaterials characteristic length was set to l = 1.5 mm.

The macro-continuum was modelled using 65 quadraticboundary elements and 228 domain points and therefore228 RVEs. The macro-continuum was also modelled usingthe nite element commercial software ABAQUS [50].To compare directly the results from both macro-formula-tions, the expected non-linear region was modelled inexactly the same manner in both numerical methods. TheFEM model was created using quadratic quadrilateral ele-ments in order to match exactly the BEM model in thenon-linear region, and the rest was discretized using qua-dratic triangular elements. In order to investigate the inu-ence of modelling the damage, which the micro feeds themacro, using the initial stress approach in the context of the BEM, two different formulations were considered inthe case of macro-FEM. The rst one is to consider thedamage as an initial decremental stress that softens thematerial locally, as exactly the same as in the case of the proposed boundary element formulation. The second

formulation is to directly implement the new damagedmaterial stiffness, as cracks initiate and propagate in themicro-scale (see Section 5.2). In both cases it was assumedthat the damage is uniformly distributed inside a nite ele-ment in order to make a direct comparison with the BEMand to avoid partially damaged elements [33]. Fig. 6 illus-

trates the different meshes used in the case of macro-BEM and macro-FEM.The results from the macro-BEM/FEM comparison are

illustrated in Fig. 7, where the dimensionless macro-stresscomponent r 22 in front of the hole, along the cross sectionX X 0, Fig. 5, is presented. The rst frame shot, (i), illus-trates the stress state when no-damage has appeared yet;i.e. is still in the fully elastic regime. The next two frameshots illustrate some damage, due to partially damagedand cracked grain boundary interfaces in the micro-scale,which reduce the stiffness of the macro-continuum andtherefore less stress can be sustained over this area. Theelastic BEM stress curve is also presented as a dashed-dotdot line for comparison. The last frame shot is theincrement just before a macrocrack will be initiated. Asdashed line the initial stress FEM approach is denoted,while with dashdot line the damaged stiffness FEMapproach is denoted. It can be seen that both macro-FEM results are very close and moreover the proposedmacro-BEM formulation is in good agreement with bothmacro-FEMs.

Fig. 6. Macro-BEM mesh and macro-FEM mesh, used in the present study.



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Fig. 8 illustrates the two different domain discretizationthat were used in the present study to investigate the meshindependency of the proposed formulation. The same exactregion in front of the hole was assigned 120 points for cellmesh A (61 quadratic boundary elements) and 228 points

for cell mesh B (65 quadratic boundary elements). The

same characteristic length in the integral non-local modelwas used for both cases and the same RVE was assignedat each macro-domain point. Fig. 9 illustrates the resultingdimensionless stress component in front of the hole. It canbe seen that the proposed formulation, with the non-local

approach for the macro-continuum, does not suffer from

Fig. 7. Comparison between a macro-BEM and a macro-FEM formulation, in the context of the proposed multi-scale damage modelling.

Fig. 8. Investigating mesh independency: Comparison of the domain discretisation for the macro-BEM.



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severe localization of the damage that eventually leads tomesh dependent results. In Fig. 9, frame shot (iii) corre-sponds to the last increment just before a macrocrack is ini-tiated, while in the last frame a macrocrack has alreadybeen initiated. The corresponding frame shots of Fig. 9

macro-Damage patterns, due to microcracking evolution,for both mesh cases, are illustrated in Fig. 10. Even thoughthe damage patterns are represented in a discrete manner(uniform damage distribution over each cell), both meshcases give similar macro-damage pattern.

Fig. 9. Dimensionless stress component along X X 0 cross section: comparison between different domain discretisations for the macro-BEM.

Fig. 10. Macro-damage patterns for different domain discretisations.



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Fig. 11 illustrates the macro-damage evolution for thecase of cell mesh A density, Fig. 8, but with additional

domain discretization. The new discretisation is composedof 180 macro-domain points with the same correspondingRVEs. Even though between the previous cell mesh Aexample and the current example there is a difference of +50% more RVEs, until case (iv) in Fig. 10 and case (vi)in Fig. 11, which corresponds at the same macro-loadincrement, the computational effort was only 9% higher.

This is due to the proposed multi-scale boundary elementformulation, where as long as the RVEs remain undam-

aged, only a matrixvector multiplication is performed tonalize the increment. Fig. 12 illustrates the evolution of the dimensionless internal macro-stress along the X X 0cross section at the fracture load. The curves correspondto the damage patterns illustrated in Fig. 11.

Consider now the case that most of the engineeringmaterials are in general heterogeneous at a certain scale.From the denition of the RVE [36], it represents themicrostructure at the innitesimal material neigh-bourhood around a macro-point and moreover it shouldstatistically represents the microstructure of the macro-continuum. Therefore, it could be argued that a materialmay have different microstructure in different areas of the macro-continuum, with certain defects or not. In thiscase, the selected RVE must represents in the same sensethe microstructure of the material at the specic region.For this reason and to demonstrate the capability of theproposed method to deal with such heterogeneous prob-lems, the next examples consist of randomly distributeddifferent RVEs for the macro-domain points. A set of eight RVE-grain morphologies and distributions are pro-duced and assigned randomly to the domain-macropoints. Even though the different RVE-grain morphol-ogies are 8, each RVE has a unique grain material orient-ation, randomly distributed. In this way, a mixture of

microstructure morphologies is randomly distributed at

Fig. 11. Macro-damage evolution.

Fig. 12. Evolution of the dimensionless internal stress r 22 component,

along the X X 0 cross section.



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specic macro-points in the continuum, with the sameaverage grain size, to encounter for cases of microstruc-

tural variation.Two sets of different RVEs were created and simulatedwith the proposed method. Fig. 13 illustrates the damageevolution of the rst set. It can be seen that the damageat the macro-scale, which is due to the intergranular frac-ture evolution in the micro-scale, is not fully symmetric.Moreover at early stages, i.e. frames (ii)(iii), the highestdamage is not exactly at the boundary of the hole butslightly inside of the boundary. Both phenomena are dueto the fact that some RVEs are more susceptible to fracturethan others. Therefore some areas of the macro-continuumare being damaged faster than what it was expected withclassic continuum theory. The capability to model effi-ciently such phenomena is important in terms of modellingmaterials with variable properties through their thickness,such as coated and generally surface treated material.The micro-damage evolution inside the correspondingRVEs is illustrated in Figs. 14 and 15. Fig. 14 illustratesthe microstructural state just at the initiation of the macro-crack, while Fig. 15 at a specic moment after the macro-crack has propagated. In these gures, the progression of microcracking in front of the macrocrack tip is illustrated.This is in agreement with experimental ndings where infront and around the crack tip, microcracks are formed,propagate and coalescence in order to form a macrocrack

[18]. The damage evolution of the second set is illustrated

in Fig. 16 and the corresponding state at the micro-scaleat the initiation and after some propagation of the macro-

crack is illustrated in Figs. 17 and 18, respectively. Com-

Fig. 13. Damage evolution at the macro-continuum, for randomly distributed different RVEs; Set 1.

Fig. 14. Intergranular fracture evolution at the micro-scale for frame shot(v) of Fig. 13.



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paring the damage evolution of the two sets, Figs. 13 and16, a slight difference of the macro-damage response canbe seen. This is due to the microstructural difference that

is illustrated in Figs. 14 and 15 comparing to Figs. 17and 18 . It must be noted, that even though the correspond-ing microstructures of the macro-continuum were different,

Fig. 15. Intergranular fracture evolution at the micro-scale for frame shot(vii) of Fig. 13.

Fig. 16. Damage evolution at the macro-continuum, for randomly distributed different RVEs; Set 2.

Fig. 17. Intergranular fracture evolution at the micro-scale for frame shot(v) of Fig. 16.



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the fracture macro-loads differed by only 1.2% between thetwo random examples.

8. Conclusions

A multi-scale boundary element formulation and itseffective numerical implementation for modelling damageis proposed for the rst time. Information about the consti-tutive behaviour of a polycrystalline material at the macro-continuum are obtained by the micro-scale using averagingtheorems in a multi-processing manner. Both macro-con-tinuum and micro-scale are modelled using the BEM. Anapproach for coupling the micro-BEM with the macro-FEM is also proposed. An integral non-local approach isemployed for avoiding the pathological localization of micro-damage at the macro-scale. At the micro-scale, afterconsidering a random distribution, morphology and orien-tation of the grains, multiple intergranular crack initiationand propagation under mixed-mode failure conditions wasmodelled. A fully frictional contact analysis was used toallow for crack surfaces to come into contact, slide, stickor separate.

Different numerical examples for a polycrystalline Al 2O3were investigated in order to demonstrate the accuracy of the proposed method. Mesh independency of the resultswas achieved due to the non-local approach used at themacro-scale. Comparing the proposed method with twomacro-FEM models, one using an initial stress approachand another with a damaged stiffness tensor approach,

good agreement was also obtained. Cases of not fully

homogeneous materials were also investigated by randomlyassign RVEs with variations in the microstructure.

The analysis demonstrates that the proposed methodcan be considered as a promising tool for future modellingof heterogeneous materials or materials with microstruc-tural variation through their thickness.

Appendix. The fundamental solutions used in the bound-ary integral equations presented in Section 2.1 are given as

U ij x0;x c1 c2 ln 1

r dij r ;ir ; j ;T ij x0;x c3f r ;mnmc4dij 2r ;ir ; j c4r ;in j r ; j n ig;

E ijk x0;X c1r f c4r ; j dik r ;k dij r ;id jk 2r ;ir ; j r ;k g;

Deijk X0;x

c1r f c4r ;id jk r ; j dik r ;k dij 2r ;ir ; j r ;k g;

S eijk X0;x

c3

r 2 f 2r ;mnmf r ;k dij mr ;id jk r ; j dik 4r ;ir ; j r ;k g

nic4d jk 2mr ; j r ;k n j c4dik 2vr ;ir ;k

nk c4dij 2r ;ir ; jg;

W eijkl X0;X

c1r 2

f 2mdli r ; j r ;k dik r ; j r ;l dlj r ;k r ;i d jk r ;l r ;i

2dkl r ;ir ; j c4d jk dli dlj dik dij dkl 2r ;k r ;l 8r ;ir ; j r ;k r ;l g;

_ g eij X0

pc12

f _r DmmX0dij 2c2 _r Dij X

0g;

Drijk X0;x

c3r f c4r ;id jk r ; j dik r ;k dij r ;ir ; j r ;k g;

S rijk X

0

;x c23c1r 2 f 2r ;mnmc4dij r ;k mr ; j dik r ;id jk

4r ;ir ; j r ;k 2mnir ;ir ;k n j r ;ir ;k

c42nk r ;ir ; j n j dik nid jk 1 4mnk dij g;

W rijkl X0;X

c3r 2

f c4dli d jk dik dlj dij dkl 2dij r ;k r ;l

2dkl r ;ir ; j 2mdli r ; j r ;k dik r ; j r ;l dlj r ;k r ;i d jk r ;l r ;i8r ;ir ; j r ;k r ;l g;

_ g rij X0

pc32

f2 _r Dij X0 c2 _r DmmX

0dij g;

where c1 18pl 1 m, c2 = 3 4m, c3 1

4p1 m and c4 =

1 2m. Moreover r ffiffiffiffiffir ir ip , r

i xi x0i, r ;i

r ir ,

r ;mnm or o n and dij 1 if i j ;

0 if i 6 j : denotes the Kronecker

delta function. The Poisson ratio is denoted by m and theshear modulus by l .

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Multiscale Damage

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