nUKhina

Keywords:B. Delamination growthB. Cohesive elements

dapas an p

the onset damage is not changed in the proposed model. In addition, the critical energy release rate of the

of lamads. It

age law. In each model the viscosity parameter (g) is used to varythe degree of rate dependence. Kubair et al. [11] thoroughly sum-marized the evolution of these rate-dependant models and pro-vided the solution to the mode III steady-state crack growthproblem as well as spontaneous propagation conditions.

A main advantage of the use of cohesive elements is the capabil-ity to predict both onset and propagation of delamination without

of loading history in the loading stage before the happening ofdamages. Furthermore, various generally oriented methodologiescan be used to remove this numerical instability, e.g. Riks method[18] which can follow the equilibrium path after instability. Also,Gustafson andWaas [19] have used a discrete cohesive zone meth-od nite element to evaluate traction law efciency and robustnessin predicting decohesion in a nite element model. They provideda sinusoidal traction law which found to be robust and efcientdue to the elimination of the stiffness discontinuities associatedwith the generalized trapezoidal traction law.

* Corresponding author.

Composites Science and Technology 69 (2009) 23832391

Contents lists availab

ce

evE-mail address: ahmed.elmarakbi@sunderland.ac.uk (A.M. Elmarakbi).reduction in the compressive load-carrying capacity of a structure.Cohesive elements are widely used, in both forms of continuousinterface elements and point cohesive elements [17], at the inter-face between solid nite elements to predict and to understand thedamage behaviour in the interfaces of different layers in compositelaminates. Many models have been introduced including: perfectlyplastic, linear softening, progressive softening, and regressive soft-ening [8]. Several rate-dependent models have also been intro-duced [913]. A rate-dependent cohesive zone model was rstintroduced by Glennie [9], where the traction in the cohesive zoneis a function of the crack opening displacement time derivative. Xuet al. [10] extended this model by adding a linearly decaying dam-

lem as pointed out by Mi et al. [14], Goncalves et al. [15], Gao andBower [16] and Hu et al. [17]. This problem is caused by a well-known elastic snap-back instability, which occurs just after thestress reaches the peak strength of the interface. Especially forthose interfaces with high strength and high initial stiffness, thisproblem becomes more obvious when using comparatively coarsemeshes [17]. Traditionally, this problem can be controlled usingsome direct techniques. For instance, a very ne mesh can alleviatethis numerical instability, however, which leads to very high com-putational cost. Also, very low interface strength and the initialinterface stiffness in the whole cohesive area can partially removethis convergence problem, which, however, lead to the lower slopeC. Finite element analysisC. Quasi-static and dynamic analysis

1. Introduction

Delamination is a mode of failurerials when subjected to transverse lo0266-3538/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.compscitech.2009.01.036materials is kept constant. Moreover, the constitutive equation of the new cohesive model is developed tobe dependent on the opening velocity of the displacement jump. The traction based model includes acohesive zone viscosity parameter (g) to vary the degree of rate dependence and to adjust the maximumtraction. The numerical simulation results of DCB in Mode-I is presented to illustrate the validity of thenew model. It is shown that the proposed model brings stable simulations, overcoming the numericalinstability and can be widely used in quasi-static, dynamic and impact problems.

2009 Elsevier Ltd. All rights reserved.

inated composite mate-can cause a signicant

previous knowledge of the crack location and propagation direc-tion. However, when using cohesive elements to simulate interfacedamage propagations, such as delamination propagation, there aretwo main problems. The rst one is the numerical instability prob-Accepted 30 January 2009Available online 10 February 2009

a pre-softening zone is proposed ahead of the existing softening zone. In this pre-softening zone, the ini-tial stiffness and the interface strength are gradually decreased. The onset displacement corresponding toFinite element simulation of delaminatioin composite materials using LS-DYNA

A.M. Elmarakbi a,*, N. Hu b,c, H. Fukunaga c

a School of Computing and Technology, University of Sunderland, Sunderland SR6 0DD,bDepartment of Engineering Mechanics, Chongqing University, Chongqing 400011, PR CcDepartment of Aerospace Engineering, Tohoku University, Sendai 980-8579, Japan

a r t i c l e i n f o

Article history:Received 25 June 2008Received in revised form 20 January 2009

a b s t r a c t

In this paper, a modied aimplemented in LS-DYNA,simulations of delaminatio

Composites Scien

journal homepage: www.elsll rights reserved.growth

tive cohesive element is presented. The new elements are developed anduser dened material subroutine (UMAT), to stabilize the nite element

ropagation in composite laminates under transverse loads. In this model,

le at ScienceDirect

and Technology

ier .com/ locate/compsci tech

placement increments of nodes in the cohesive zone of laminatesafter delaminations occurred. Therefore, similar to the articial

of delamination, an 8-node cohesive element shown in Fig. 1 isdeveloped to overcome the numerical instabilities.

The need for an appropriate constitutive equation in the formu-lation of the interface element is fundamental for an accurate sim-ulation of the interlaminar cracking process. A constitutiveequation is used to relate the traction to the relative displacementat the interface. The bilinear model, as shown in Fig. 2, is the sim-plest model to be used among many strain softening models.Moreover, it has been successfully used by several authors in im-plicit analyses [2326]. However, using the bilinear model leadsto numerical instabilities in an explicit implementation. To over-come this numerical instability, a new adaptive model is proposedand presented in this paper.

The adaptive interfacial constitutive response shown in Fig. 3 isimplemented as follows:

1. In pre-softening zone,adom < dmaxm < d

om, the constitutive equation

is given by

r rm g _dm dmdom1

and rm Kdom 2where r is the traction, K is the penalty stiffness and can be writtenas

ce and Technology 69 (2009) 23832391damping method [16], the move-limit method introduces the arti-cial external work to stabilize the computational process. Asshown later, although these methods [16,17] can remove thenumerical instability when using comparatively coarse meshes,the second problem occurs, which is the error of peak load in theloaddisplacement curve. The numerical peak load is usually high-er than the real one as observed by Goncalves et al. [15] and Huet al. [17].

Similar work has also been conducted by De Xie and Waas [20].They have implemented discrete cohesive zone model (DCZM)using the nite element (FE) method to simulate fracture initiationand subsequent growth when material non-linear effects are sig-nicant. In their work, they used the nodal forces of the rod ele-ments to remove the mesh size effect, dealt with a 2D study anddid not consider viscosity parameter. However, in the presentedpaper, the authors used the interface stiffness and strength in acontinuum element, tackled a full 3D study and considered the vis-cosity parameter in their model.

With the previous background in mind, the objective of this pa-per is to propose a new cohesive model named as adaptive cohe-sive model (ACM), for stably and accurately simulatingdelamination propagations in composite laminates under trans-verse loads. In this model, a pre-softening zone is proposed aheadof the existing softening zone. In this pre-softening zone, with theincrease of effective relative displacements at the integrationpoints of cohesive elements on interfaces, the initial stiffnessesand interface strengths at these points are reduced gradually. How-ever, the onset displacement for starting the real softening processis not changed in this model. The critical energy release rate orfracture toughness of materials for determining the nal displace-ment of complete decohesion is kept constant. Also, the tractionbased model includes a cohesive zone viscosity parameter (g) tovary the degree of rate dependence and to adjust the peak or max-imum traction.

In this paper, this cohesive model is formulated and imple-mented in LS-DYNA [21] as a user dened materials (UMAT). LS-DYNA is one of the explicit FE codes most widely used by the auto-mobile and aerospace industries. It has a large library of materialoptions; however, continuous cohesive elements are not availablewithin the code. The formulation of this model is fully three-dimensional and can simulate mixed-mode delamination. How-ever, the objective of this study is to develop new adaptive cohe-sive elements able to capture delamination onset and growthunder quasi-static and dynamic Mode-I loading conditions. Thecapabilities of the proposed elements are proven by comparingthe numerical simulations and the experimental results of DCB inMode-I.

2. The constitutive model

Cohesive elements are used to model the interface betweensublaminates. The elements consists of a near zero-thickness volu-metric element in which the interpolation shape functions for thetop and bottom faces are compatible with the kinematics of theRecently, the articial damping method with additional energydissipations has been proposed by Gao and Bower [16]. Also, thepresent authors proposed a kind of move-limit method [17] to re-move the numerical instability using cohesive model for delamina-tion propagation. In this technique, the move-limit in the cohesivezone provided by articial rigid walls is built up to restrict the dis-

2384 A.M. Elmarakbi et al. / Composites Scienelements that are being connected to it [22]. Cohesive elementsare typically formulated in terms of traction vs. relative displace-ment relationship. In order to predict the initiation and growthelements element

Fig. 1. Eight-node cohesive element.

mfmom

o

Closed crack K Ko dm 0Ki d

maxm < d

om

Kn dom dmaxm < dfm

8>:

3

dm is the relative displacement in the interface between the top andbottom surfaces (in this study, it equals the normal relative dis-placement for Mode-I), dom is the onset displacement and it is re-mained constant in the simulation and can be determined asfollows:

dom roKo

riKi

rminKmin

4

where ro is the initial interface strength, ri is the updated interfacestrength in the pre-softening zone, rmin is the minimum limit of the

Solid Cohesive Fig. 2. Normal (bilinear) constitutive model.

n ,

ituti

nce ainterface strength, Ko is the initial stiffness, Ki is the updated stiff-ness in the pre-softening zone, and Kmin is the minimum value ofthe stiffness.

For each increment and for time t + 1, dm is updated as follows:

dt1m tcet1 tc 5where tc is the thickness of the cohesive element and et + 1 is thenormal strain of the cohesive element for time t + 1, et + 1=et + De,where De is the normal strain increment.

The dmaxm t is the max relative displacement of the cohesive ele-ment occurs in the deformation history. For each increment and fortime t + 1, dmaxm is updated as follows:

dmaxm t1 dt1m if dt1m dmaxm t and; 6dmaxm t1 dmaxm t if dt1m < dmaxm t 7Using the max value of the relative displacement dmaxm rather than

Closed crack

nfmm

o

m rmin and adom < dmaxm < dom 8

Ki dmaxm

domKmin Ko Ko; Ko > Kmin and adom < dmaxm < dom 9

It should be noted that a in Eqs. (8) and (9) is a parameter to denethe size of pre-softening zone. When a = 1, the present adaptivecohesive mode degenerates into the traditional cohesive model.

In our computations, we set a = 0. From our numerical experi-ences, the size of pre-softening zone has some inuences on theinitial stiffness of loadingdisplacement curves, but not so signi-cant. The reason is that for the region far always from the crack tip,the interface decrease or update according to Eqs. (8) and (9) is notobvious since dmaxm is very small.

The energy release rate for Mode-I GIC also remains constant.Therefore, the nal displacements associated to the complete dec-ohesion dfim are adjusted as shown in Fig. 3 as

dfim 2GICri

10

Once the max relative displacement of an element located at thecrack front satises the following conditions; dmaxm > d

om, this ele-r 1 drm g _dm dmdom11

where d is the damage variable and can be dened as

d dfmdmaxm dom

dmaxm dfm dom; d 2 0;1 12

The above adaptive cohesive mode is of the engineering meaningment enters into the real softening process. Where, as shown inFig. 3, the real softening process denotes a stiffness decreasing pro-cess caused by accumulated damages. Then, the current strength rnand stiffness Kn, which are almost equal to rmin and Kmin, respec-tively, will be used in the softening zone.

2. In softening zone, dom dmaxm < dfm, the constitutive equation isgiven by

o

mm

o

m

3. Finite element implementation

The proposed cohesive element is implemented in LS-DYNA -nite element code as a user dened material (UMAT) using thestandard library 8-node solid brick element. This approach forthe implementation requires modelling the resin rich layer as anon-zero thickness medium. In fact, this layer has a nite thicknessand the volume associated with the cohesive element can in factset to be very small by using a very small thickness (e.g.0.01 mm). To verify these procedures, the crack growth along theinterface of a double cantilever beam (DCB) is studied. The twoarms are modelled using standard LS-DYNA 8-node solid brick ele-

ments and the interface elements are developed in a FORTRAN sub-routine using the algorithm shown in Fig. 4.

The LS-DYNA code calculates the strain increments for a timestep and passes them to the UMAT subroutine at the beginningof each time step. The material constants, such as the stiffnessand strength, are read from the LS-DYNA input le by the subrou-tine. The current and maximum relative displacements are savedas history variables which can be read in by the subroutine. Usingthe history variables, material constants, and strain increments,the subroutine is able to calculate the stresses at the end of thetime step by using the constitutive equations. The subroutine thenupdates and saves the history variables for use in the next time

Update i and iK

Compute the traction Eq. (1)

Material constants

ICoo GKK &,,,, minmin

Compute om

and fm

Compute the normal relative displacement

m

History variables Calculate strain

max

m = history variable 1

max

m = { }

mm ,max

No

Yesomm

step and outputs the calculated stresses. Note that the *DATA-BASE_EXTENT_BINARY command is required to specify the storageof history variables in the output le.

It is worth noting that the stable explicit time step is inverselyproportional to the maximum natural frequency in the analysis.The small thickness elements drive up the highest natural fre-quency, therefore, it drives down the stable time step. Hence, massscaling is used to obtain faster solutions by achieving a larger ex-plicit time step when applying the cohesive element to quasi-staticsituations. Note that the volume associated with the cohesive ele-ment would be small by using a small thickness and the elementskinetic energy arising from this be still several orders of magnitudebelow its internal energy, which is an important consideration forquasi-static analyses to minimize the inertial effects.

4. Numerical simulations

4.1. Quasi-static analysis

The DCB specimen is made of a unidirectional bre-reinforced

using different mesh sizes. The aim of the rst ve cases is to studythe effect of the element size with constant values of interfacestrength and stiffness on the loaddisplacement relationship. Dif-ferent element sizes are used along the interface spanning fromvery small size of 0.5 mm to coarse mesh of 2 mm. Moreover, Cases3, 6, and 7 are to study the effect of the value of minimum interfacestrength on the results. Finally, Cases 6 and 8 are to nd out the ef-fect of the high interfacial strength.

Figs. 8 and 9 show the loaddisplacement curves for both nor-mal (bilinear) and adaptive cohesive elements in Cases 1 and 5,respectively, with different element sizes. Fig. 8 clearly shows thatthe bilinear formulation results in a severe instability once thecrack starts propagating. However, the adaptive constitutive lawis able to model the smooth, progressive crack propagation. It isworth mentioning that the bilinear formulation brings smooth re-sults by decreasing the element size. And it is clearly noticeablefrom Fig. 9 that both bilinear and adaptive formulations are foundto be stable in Case 5 with very small element size. This indicatesthat elements with very small sizes need to be used in the soften-ing zone to obtain high accuracy using bilinear formulation. How-ever, this leads to large computational costs compare to Case 1. On

0

20

40

60

80

0 2 4 6 8 10 12 14

Adaptive-Mesh size=1 mm- Case BAdaptive-Mesh size=1 mm- Case A

Opening displacement (mm)

Load

(N)

Fig. 7. Loaddisplacement curves for a DCB specimen in both Cases A and B.

A.M. Elmarakbi et al. / Composites Science alaminate containing a thin insert at the mid-plane near the loadedend. A 150 mm long specimen (L), 20 mm wide (w) and composedof two thick plies of unidirectional material (2 h = 2 1.98 mm)shown in Fig. 5 was tested by Morais [27]. The initial crack length(lc) is 55 mm. A displacement rate of 10 mm/s is applied to theappropriate points of the model. The properties of both carbon -bre-reinforced epoxymaterial and the interface are given in Table 1.

The LS-DYNA nite element model, which is shown deformed inFig. 6, consists of two layers of fully integrated S/R 8-noded solidelements, with three elements across the thickness. Two caseswith different mesh sizes are used in the initial analysis, namely:Case A, which includes eight elements across the width, and CaseB, which includes one element across the width, respectively. Thetwo cases are compared using the new cohesive elements withmesh size of 1 mm to gure out the anticlastic effects.

A plot of a reaction force as a function of the applied end dis-placement is shown in Fig. 7. It is clearly shown that both casesbring similar results with peak load value of 64 N. Therefore, theanticlastic effects are neglected and only one element (Case B) isused across the width in the following analyses.

Different cases are considered in this study and given in Table 2to investigate the inuence of the new adaptive cohesive element

Fig. 5. Model of DCB specimen.

Table 1Properties of both carbon bre-reinforced epoxy material and specimen interface.

Carbon bre-reinforced epoxy material DCB specimen interface

q = 1444 kg/m3 GIC = 0.378 kJ/m2

E = 150 GPa, E = E = 11 GPa K = 3 104 N/mm311 22 33 ot12 = t13 = 0.25, t23 = 0.45 ro = 45 MPa Case IG12 = G13 = 6.0 MPa, G23 = 3.7 MPa ro = 60 MPa Case IIFig. 6. LS-DYNA nite element model of the deformed DCB specimen.

nd Technology 69 (2009) 23832391 2387the other hand, Fig. 10, which presents the loaddisplacementcurves, obtained with the use of the adaptive formulation in therst ve cases, show a great agreement of the results regardless

the mesh size. Adaptive cohesive model (ACM) can yield very goodresults from the aspects of the peak load and the slope of loadingcurve if rmin is properly dened. From this gure, it can be foundthat the different mesh sizes result in almost the same loadingcurves. Even, with 2 mm mesh size, which considerable large size,

Table 2Different cases of analyses.

Case 1 Mesh size = 2 mm ro = 45 MPa, rmin = 15 MPa Ko = 3 104 N/mm3, Kmin = 1 104 N/mm3Case 2 Mesh size = 1.25 mmCase 3 Mesh size = 1 mmCase 4 Mesh size = 0.75 mmCase 5 Mesh size = 0.5 mmCase 6 Mesh size = 1 mm ro = 45 MPa, rmin = 22.5 MPa Ko = 3 104 N/mm3, Kmin = 1.5 104 N/mm3Case 7 Mesh size = 1 mm ro = 45 MPa, rmin = 10 MPa Ko = 3 104 N/mm3, Kmin = 0.667 104 N/mm3Case 8 Mesh size = 1 mm ro = 60 MPa, rmin = 30 MPa Ko = 3 104 N/mm3, Kmin = 1.5 104 N/mm3

-20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14

Bilinear - Case 1Adaptive - Case 1

Lo

ad (N

)

Opening Displacement (mm) Fig. 8. Loaddisplacement curves obtained using both bilinear and adaptiveformulations Case 1.

0

20

40

60

80

0 2 4 6 8 10 12 14

Bilinear - Case 5Adaptive - Case 5

Lo

ad (N

)

Opening displacement (mm) Fig. 9. Loaddisplacement curves obtained using both bilinear and adaptiveformulations Case 5.

0

20

40

60

80

0 2 4 6 8 10 12 14

Adaptive - Case 1Adaptive - Case 2Adaptive - Case 3Adaptive - Case 4Adaptive - Case 5

Opening displacement (mm)

Lo

ad (N

)

Fig. 10. Loaddisplacement curves obtained using the adaptive formulation Cases15.

2388 A.M. Elmarakbi et al. / Composites Science and Technology 69 (2009) 23832391although the oscillation is higher compared with those of nemesh size, ACM still models the propagation in stable manner.The oscillation of the curve once the crack starts propagates be-came less by decreasing the mesh size. Therefore, the new adaptivemodel can be used with considerably larger mesh size and thecomputational cost will be greatly minimized.

The loaddisplacement curves obtained from the numericalsimulation of Cases 3, 6 and 7 are presented in Fig. 11 togetherwith experimental data [28]. It can be seen that the average max-imum load obtained in the experiments is 62.5 N, whereas theaverage maximum load predicted form the three cases is 65 N. Itcan be observed that numerical curves slightly overestimate theload. It is worth noting that with the decrease of interface strength,the result is stable, very good result can be obtained by comparingwith the experimental ones, however, the slope of loading curvebefore the peak load is obviously lower than those of experimentalones (Case 7; rmin = 10.0 MPa). In Case 6 (rmin = 22.5 MPa) andCase 3 (rmin = 15 MPa), excellent agreements between the experi-mental data and the numerical predictions is obtained althoughthe oscillation in Case 6 is higher compared with those of Case 3.Also, the slope of loading curve in Case 3 is closer to the experi-mental results compared with that in Case 6.

Fig. 12 show the loaddisplacement curves of the numericalsimulations obtained using the bilinear formulation in both cases,i.e., Cases 6 and 8. The bilinear formulations results in a severeinstabilities once the crack starts propagation. It is also shown thata higher maximum traction (Case 8) resulted in a more severeinstability compared to a lower maximum traction (Case 6). How-ever, as shown in Fig. 13, the loaddisplacement curves of thenumerical simulations obtained using the adaptive formulations

60

800

20

40

0 2 4 6 8 10 12 14

Adaptive - Case 6Adaptive - Case 3Adaptive - Case 7Experiment

Opening displacement (mm)

Lo

ad (N

)

Fig. 11. Comparison of experimental and numerical simulations using the adaptiveformulation Cases 3, 6 and 7.

rial parameter depending on deformation rate, which appears inEqs. (1) and (11). When g = 0, it would be a traditional model with-out rate dependence. By observing Eq. (1), g determines the ratiobetween viscosity stress g _dm and interface strength rm sincerm = ri if we consider Eqs. (1) and (4) by setting K = Ki. For example,if we assume _dm =6.5 mm/s on the interface here (i.e., 1% of loadingrate). g = 0.01 N s/mm3 corresponds to a low viscosity stress of0.065 MPa, which is much lower than the initial interface strengthof 50 MPa. However, g = 1.0 N s/mm3 corresponds to a viscositystress of 6.5 MPa, which is around 13% of the interface strength,and denotes a higher rate dependence. In addition, two sets of sim-ulations are performed here. The rst set involves simulations ofnormal (bilinear) cohesive model. The second set involves simula-tions of the new adaptive rate-dependent model.

A plot of a reaction force as a function of the applied end dis-placement of the DCB specimen using cohesive elements with vis-cosity value of 0.01 N s/mm3 is shown in Fig. 14. It is clearly shownfrom Fig. 14 that the bilinear formulation results in a severe insta-bility once the crack starts propagating. However, the adaptiveconstitutive law is able to model the smooth, progressive crackpropagation. It is worth mentioning that the bilinear formulationmight bring smooth results by decreasing the element size.

The loaddisplacement curves obtained from the numericalsimulation of both bilinear and adaptive cohesive model using vis-cosity parameter of 1.0 N s/mm3 is presented in Fig. 15. It can be

nce and Technology 69 (2009) 23832391 23890

20

40

60

80

100

0 2 4 6 8 10 12 14

Bilinear - Case 8Bilinear - Case 6

Opening displacement (mm)

Lo

ad (N

)

Fig. 12. Loaddisplacement curves obtained using the bilinear formulation Cases6 and 8.

60

80

) A.M. Elmarakbi et al. / Composites Scieare very similar in both cases. The maximum load obtained fromCase 8 is found to be 69 N while in Case 6, the maximum load ob-tained is 66 N. The adaptive formulation is able to model thesmooth, progressive crack propagation and also to produce closeresults compared with the experimental ones.

4.2. Dynamic analysis

The DCB specimen, as shown in Fig. 5, is made of an isotropicbre-reinforced laminate containing a thin insert at the mid-planenear the loaded end, L = 250 mm, w = 25 mm and h = 1.5 mm, wasanalyzed by Moshier [29]. The initial crack length (lc) is 34 mm. Adisplacement rate of 650 mm/s is applied to the appropriate pointsof the model. Youngs modulus, density and Poissons ratio of car-bon bre-reinforced epoxy material are given as E = 115 GPa,q = 1566 Kg/m3, and t = 0.27, respectively. The properties of theDCB specimen interface are given as following:

GIC 0:7 kJ=m2;Ko 1 105 N=mm3;Kmin 0:333 105 N=mm3;ro 50 MPa; and rmin 16:67 MPa:

Similarly, the LS-DYNA nite element model consists of two lay-ers of fully integrated S/R 8-noded solid elements, with three ele-ments across the thickness.

The adaptive rate-dependent cohesive zone model is imple-mented using a user dened cohesive material model in LS-DYNA.Two different values of viscosity parameter are used in the simula-tions; g = 0.01 and 1.0 N s/mm3, respectively. Note that g is a mate-

seen that, again, the adaptive constitutive law is able to modelthe smooth, progressive crack propagation while the bilinear for-

0

20

40

0 2 4 6 8 10 12 14

Adaptive - Case 8Adaptive - Case 6Experiment

Opening displacement (mm)

Lo

ad (N

Fig. 13. Comparison of experimental and numerical simulations using the adaptiveformulation Cases 6 and 8.

60

90

120

150Normal-visco =.01Adaptive-visco=.01

d (N

) -30

0

30

0 5 10 15 20 25

Opening displacement (mm)

Lo

a

Fig. 14. Loaddisplacement curves obtained using both bilinear and adaptiveformulations (g = 0.01).

-30

0

30

60

90

120

150

0 5 10 15 20 25

Normal-visco =1Adaptive-visco=1

Opening displacement (mm)

Lo

ad (N

) Fig. 15. Loaddisplacement curves obtained using both bilinear and adaptiveformulations (g = 1).

ce a-30

0

30

60

90

120

150

0 5 10 15 20 25

Normal-visco =.01Normal-visco=1

Opening displacement (mm)

Lo

ad (N

)

Fig. 16. Loaddisplacement curves obtained using bilinear formulations (g = 0.01,1).

120

150

Adaptive-visco =.01Adaptive-visco=1

2390 A.M. Elmarakbi et al. / Composites Scienmulation results in a severe instability once the crack starts prop-agating. The average maximum load obtained using the adaptiverate-dependent model is 110 N, whereas the average maximumload predicted form the bilinear model is 120 N.

Fig. 16 shows the loaddisplacement curves of the numericalsimulations obtained using the bilinear formulation with two dif-ferent viscosity parameters, 0.01 and 1.0 N s/mm3, respectively. Itis noticed from Fig. 16 that, in both cases, the bilinear formulationresults in severe instabilities once the crack starts propagation.There is a very slight improvement to model the smooth, progres-sive crack propagation using bilinear formulations with a high vis-cosity parameter. On the other hand, the loaddisplacement curvesof the numerical simulations obtained using the new adaptive for-mulation with two different viscosity parameters, 0.01 and 1.0 N s/mm3, respectively, is depicted in Fig. 17.

It is clear from Fig. 17 that the adaptive formulation able tomodel the smooth, progressive crack propagation irrespective thevalue of the viscosity parameter. More parametric studies will beperformed in the ongoing research to accurately predict the effectof very high value of viscosity parameter on the results using bothbilinear and adaptive cohesive element formulations.

5. Conclusions

A new adaptive cohesive element is developed and imple-mented in LS-DYNA to overcome the numerical insatiability

-30

0

30

60

90

0 5 10 15 20 25

Opening displacement (mm)

Lo

ad (N

)

Fig. 17. Loaddisplacement curves obtained using adaptive formulations (g = 0.01,1).occurred using the bilinear cohesive model. The formulation isfully three-dimensional, and can be simulating mixed-modedelamination, however, in this study, only DCB test in Mode-I isused as a reference to validate the numerical simulations. Quasi-static and dynamic analyses are carried out in this research tostudy the effect of the new constitutive model. Numerical simula-tions showed that the newmodel is able to model the smooth, pro-gressive crack propagation. Furthermore, the new model can beeffectively used in a range of different element size (reasonablycoarse mesh) and can save a large amount of computation. Thecapability of the new mode is proved by the great agreement ob-tained between the numerical simulations and the experimentalresults.

Acknowledgement

The authors wish to acknowledge the nancial support pro-vided for this ongoing research by Japan Society for the Promotionof Science (JSPS) to A.M.E. and Research Fund for Overseas ChineseYoung Scholars from National Natural Science Foundation of China(No. 50728504) to N.H.

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A.M. Elmarakbi et al. / Composites Science and Technology 69 (2009) 23832391 2391

Finite element simulation of delamination growth in composite materials using LS-DYNAIntroductionThe constitutive modelFinite element implementationNumerical simulationsQuasi-static analysisDynamic analysis

ConclusionsAcknowledgementReferences