r-pou

.M

.22812.2

19 August 2011

sticur.posi

models were formulated and implemented into ABAQUS/Explicit nite element code. Comparisons

compt force

becomes more important. The rst attempt to simulate armour wasmade by Tate [1], using an analytical model to study thephenomena of impact. The impact onto ceramic/composite armourwas rst studied by Woodward [2]. Later Benloulo and Snchez-Glvez [3] used the Woodward model as a base to a more

Another way to study the problem is through nite elementmethod. In the last few decades, with the advance in computers,this method had a large progress. The rst constitutive model forbrittle materials specic for ballistic was developed by Espinosaet al. [7], modied later by Espinosa et al. [8] and by Zavattieri et al.[9]. The model was based on micro cracks in multi planes, so thedamage could evolve to any direction. Johnson and Holmquistdeveloped one of the most used constitutive models for brittlematerials, the JH-1 [10]. This model uses two yield strengths, an

* Corresponding author. Tel.: 55 12 3947 6433; fax: 55 12 3947 6405.

Contents lists available at

International Journal o

lse

International Journal of Impact Engineering 43 (2012) 63e77E-mail address: bur.daniel@gmail.com (D. Brger).heavier metal armours. Instead, new materials were introducedand combined to improved armours performance, such as bercomposites and ceramics. Recently, themost used bers for ballisticprotection are aramid and Ultra High Molecular Weight Poly-ethylene (UHMWPE). But the armours composed only of compos-ites are inefcient against Armour Piercing (AP) projectiles. For APprojectiles protection a ceramic plate is used to break projectiletips. So the composite can hold both ceramic and projectile frag-ments. The most used ceramics are alumina, silicon carbide andboron carbide.

As the prices of the materials involved in the experimentsincrease, the need for developing accurate simulation tools

study did not present a larger number of experimental results.Vaziri et al. [5] proposed a model to predict the transient responseof composite plates subjected to non-penetrating impacts. Themodel uses 2-D elements to simulate the composite, so the internalphenomena cannot be predicted, but it can predict in a simple andefcient mean the structural impact response. The study of impact,using an analytical model, provides a very limited solution, sinceevery geometry or type of material requires a new model. Thisresults in a very large number of analytical models. As an example,Ben-dor et al. [6] compile several analytical models and classied20 of them as the most cited models. The compilation had 280models.Keywords:ArmourBallistic impactConstitutive modelsFinite elements

1. Introduction

Until World War II, armours wereHowever, the projectile developmen0734-743X/$ e see front matter 2011 Elsevier Ltd.doi:10.1016/j.ijimpeng.2011.12.001osed basically of metal.d the use of larger and

comprehensive model. Parga-Landa and Hernndez-Olivares [4]developed an analytical model for composite armours, taking intoaccount the fabric architecture. With dynamic properties thesimulations had a very good agreement with experiments, but theAvailable online 13 December 2011

also presented and discussed in the paper.

2011 Elsevier Ltd. All rights reserved.Accepted 3 December 2011between numerical predictions and experimental results in terms of damage shape/extent and V50 areBallistic impact simulation of an armouceramic/ber reinforced composite arm

Daniel Brger a,*, Alfredo Rocha de Faria b, Srgio FMaurcio V. Donadon b

a Instituto de Aeronutica e Espao, Praa Mal. Eduardo Gomes, 50 Vila das Accias, 12b Instituto Tecnolgico de Aeronutica, Praa Mal. Eduardo Gomes, 50 Vila das Accias,

a r t i c l e i n f o

Article history:Received 20 December 2010Received in revised form

a b s t r a c t

This paper presents a ballireinforced composite armoweight polyethylene com

journal homepage: www.eAll rights reserved.iercing projectile on hybridrs

. de Almeida b, Francisco C.L. de Melo a,

-904 So Jos dos Campos - SP, Brazil28-900 So Jos dos Campos - SP, Brazil

impact simulation of an armour-piercing projectile in hybrid ceramic/berThe armour is composed by an alumina plate and an ultra high molecularte. In order to model the armour behavior three different constitutive

SciVerse ScienceDirect

f Impact Engineering

vier.com/locate/ i j impeng

2. Constitutive models

2.1. JH-2 model

2.1.1. Description of the JH-2 modelThe JH-2 model is a constitutive model suitable to predict the

behaviour of brittle materials subjected to extreme loading. Themain features of the model include pressure-dependent strength,damage and fracture, signicant strength after fracture, bulkingand strain rate effects. A general overview of the JH-2 model interms of strength is shown in Fig. 1. The idea behind the modelformulation is that the material begins to soften when damagebegins to accumulate (D > 0). This allows for gradual softening ofthe material under increasing plastic strain. The strength generallyis a smoothly varying function of the intact strength, fracturestrength, strain rate and damage [11].

The normalized equivalent stress shown in Fig. 1 is dened as

s* s*i Ds*i s*f

(1)

l of Impact Engineering 43 (2012) 63e77intact, and a failed. This model was upgraded by their authors,leading to the JH-2 [11] model, which allows a gradual stiffnessreduction, as the damage increases. Also, the model was lesssensitive to model parameters, besides, some numerical problemsin Eulerian codes when the material reaches complete failure(damage 1.0) were xed. However, major advances of JH-1 werekept, such as: pressure-dependent strength, damage and fracture,signicant strength after fracture, bulking, and strain rate effects.Rajendran and Grove [12] proposed a model for silicon carbide,boron carbide and titanium diboride, the main achievement of thismodel is the property loss under tension and compression. Simhaet al. [13] proposed a model based on impact experiments inalumina, the model shows a great agreement with experiments atimpact velocities from 1.5 km/s to 3.5 km/s.

Recently, a large number of constitutive models for compositematerials have been proposed. Lim et al. [14] proposed a modelbased on membrane elements, suitable for aramid armours. Themodel presented a good prediction for the projectile residualvelocity for high speeds. But for velocities close to V50 the errorbecame larger. The main restriction to the model is the membraneelement, which cannot represent the internal compositephenomena. Grujicic et al. [15] develop a model to study impact oncarbonecarbon composites. Later, Grujicic et al. [16] used thismodel to simulate a projectile impact in a panel composed bya ceramic plate and a composite base, obtaining a good descriptionof the phenomena involved. However, the model does not accountfor ceramic-composite interfacial bonding. Iannucci [17] proposeda model suitable for thin woven carbon composites. The maincharacteristics of this model are: a stress threshold for damage tocommence; damage cannot grow faster than the crack velocity;gradual reduction in the properties and gradual energy dissipation(with increasing damage). The model presents good predictions forseveral parameters such as force-time history and damage extend,but the author highlights the difcult to obtain some of theconstants for the model.

One of the rst attempts for modeling the adhesive layer wasmade by Zaera et al. [18], where the authors assumed that thearmour is composed by an alumina plate and an aluminum base inan analytical model. The solution usually adopted to simulate theadhesive is to use a CONTACT_TIED_SURFACE_TO_SURFACE (LS-DYNA 3D) [19], or an equivalent, which allows to connect ceramicand composite layers. This solution uses two failure criteria todetect damage initiation: maximum normal tensile stress criterion,and maximum shear stress criterion. Despite of giving reasonablepredictions for damage initiation, the approach aforementioned isnot able to predict damage extent accurately for problems havinga variable mixed normal to shear failure interaction. Moreover, thesolution is also mesh dependent.

This work presents a numerical model for ballistic impactsimulations in hybrid ceramic/ber reinforced composite armours.The simulations were carried out using ABAQUS/Explicit niteelement code. Four different material models have been used forthis purpose: (i) Johnson-Cook model to predict the materialbehaviour of the projectile; (ii) JH-2 model to predict the materialbehaviour of the ceramic; (iii) a new 3-D progressive failure modelto predict the structural response of the composite base; (iv)a contact-logic to predict debonding between the ceramic plate andthe composite base. Thematerial models (ii), (iii) and (iv) have beenimplemented into ABAQUS as user dened material models withinsolid elements.

The aim of the model is to achieve an accurate V50, that is, thevelocity at which there is a 50% probability of specimen completeperforation in the armour. Different models were run at differentvelocities, and then the results were compared to experimental

D. Brger et al. / International Journa64results.where si* and sf* are the normalized intact and fractured equivalentstress, respectively; and D is the damage (0 < D < 1) [11].

The normalized intact and fractured strengths are respectivelygiven by:

s*i AP* T*

N1 Cln _* (2)

s*f BP*M

1 Cln _* (3)The material constants are A, B, C, M, and N. P* and T* are the

normalized pressure and maximum tensile hydrostatic stress. Thedimensionless strain rate is _* _= _0, where _ is the actual strainrate and _0 is the reference strain rate. The damage for fracture isaccumulated in a manner similar to that used in JohnsoneCookfracture model [20], and it is expressed as,

D XDp

pf

(4)

where Dp is the equivalent plastic strain increment during a cycleof integration and pf f(P) is the plastic strain to fracture underconstant pressure. The expression for pf is given as follows,Fig. 1. Description of the JH-2 model.

compressibility factor.After damage begins to accumulate (D > 0), bulking can occur.

Now an additional incremental pressure, DP is added, such as,

P K1m K2m2 K3m3 DP (7)The pressure increment is determined from energy consider-

ations: it varies from DP 0 at D 0 to DP DPmax at D 1.

4) Compute current equivalent stress:

sn1 3

p 12

trialSn1xx

2trialSn1yy 2trialSn1zz 2

trialSn1xy

2trialSn1yz 2trialSn1zx 2

12

(13)

Sny

l of I3) Based on the current time step Dt and current strain incre-based on the updated Lagrangian formulation which is used inconjunctionwith the central difference time integration scheme forintegrating the resultant set of nonlinear dynamic equations. Themethod assumes a linear interpolation for velocities between twosubsequent time steps and no stiffness matrix inversions arerequired during the analysis. The drawback of the explicit methodis that it is conditionally stable for nonlinear dynamic problems andthe stability for its explicit operator is based on a critical value ofthe smallest time increment for a dilatational wave to cross anyelement in the mesh. A detailed description on the model imple-mentation into ABAQUS VUMAT user-dened material modelsubroutine is given step-by-step as follows.

1) Based on the current time step compute strain, stress incre-ments and update strain and trial stresses at current time:

fgn1 fgnfDg (8)trailfsgn1 fsgnCfDg (9)

where [C] andfDgT fDxx Dyy Dzz Dxy Dyz Dzx g arethe material stiffness matrix and strain increment vector, respec-tively. The superscripts n and n 1 refer to previous and currenttime, respectively.

2) Compute the current deviatoric trial stresses trialfSgn1 basedon the decomposition of the total trial stresses trialfsgn1 intodeviatoric trialfSgn1 and hydrostatic sn1H stresses:

trialfSgn1 trial fsgn1sn1H dij (10)where dij is the Dirac delta function.

n1

Dp fagT C_Dt2

pfagT Cfag

Sn1xx Sn1yy

2Sn1xx Sn1zz 2Sn1zz rA detailed description on the model formulation can be found inRef. [11].

2.1.2. Numerical implementationThe JH-2 model has been implemented into ABAQUS Explicit

nite element code within brick elements. The code formulation isD1 and D2 are material constants. The hydrostatic stresses aredened in terms of the pressure given by the following equation ofstate (EOS),

P K1m K2m2 K3m3 (6)K1 is the bulk modulus, K2 and K3 are material constants. m is thepf D1

P* T*

D2(5)

D. Brger et al. / International Journaments {D}, compute the current strain rate vector f_g andcurrent effective strain rate _n1eff :else, Dp 0 and Dn1 Dnpf

n1 D1P*n1T*D2 (20)

Dn1 Dn Dppf

n1 (21)

1y

26hSn1xy 2Sn1yz 2Sn1zx 2i (19)fSgn1

s*n1

sHEL=sn1

trialfSgn1

(18)

and compute the equivalent plastic strain increment Dp, currentplastic strain to failure

pf

n1 and update the damage variableDn1,

s*n1 s*i n1Ds*i n1s*f n1 (16)

with P*n1 sn1H =PHEL

6) Check for Yielding at current time n 1:

Fn1s sn1 sHELs*n1

(17)

If Fn1s > 0, return the deviatoric stresses to the Yield surfaceusing the Radial Return Algorithm,5) Compute current normalized Yield stress s*n1 with _*n1 _n1eff =_0;

s*i

n1 AP*n1T*N1 Cln _*n1 (14)s*f

n1 BP*n1M1 Cln _*n1 (15) _gn1yz _gn1zx 12_n1 1

DtfDg (11)

_n1eff 23

r _n1xx

2_n1yy

2_n1zz

20:5

_gn1xy

2 2 2!1=2

mpact Engineering 43 (2012) 63e77 65where fag vFn1s; sHEL; s*=vfSg

{D} using the central difference method integration schemeand return to step 1).

Case C incorporated both fractured material strength and the

Table 1Material model constants.

Case A Case B Case C

Density (kg/m3) 3700 3700 3700Shear modulus (Pa) 9.016 e 10 9.016 e 10 9.016 e 10Strength constantsA 0.93 0.93 0.93B 0 0 0.31C 0 0 0M 0 0 0.6N 0.6 0.6 0.6Ref. strain rate 1.0 1.0 1.0T (Pa) 2 e 8 2 e 8 2 e 8HEL (Pa) 2.79 e 9 2.79 e 9 2.79 e 9PHEL (Pa) 1.46 e 9 1.46 e 9 1.46 e 9D1 0.0 0.005 0.005D2 0.0 1.0 1.0K1 (Pa) 1.3095 e 11 1.3095 e 11 1.3095 e 11K2 (Pa) 0.0 0.0 0.0K3 (Pa) 0.0 0.0 0.0b 1.0 1.0 1.0

l of Impact Engineering 43 (2012) 63e77strength but was allowed to accumulate plastic strain so thatcomplete damage did not occur instantaneously. In this case thebulking pressure increased with damage to a maximum value of0.72 GPa when the material was completely damaged. A compar-ison between predictions obtained using the actual JH-2 imple-mented into ABAQUS and the results published by Johnson andled to an instantaneous increase in bulking pressure of 0.56 GPa. Acomparison between predictions obtained using the actual JH-2implemented into ABAQUS and the results published by Johnsonand Holmquist [11] for the case A is depicted in Fig. 2.

In case B, the material was dened as having no fracturethis constitutive model. All cases involve the conned compres-sion and release of a ceramic material with variation of damagerepresentation to demonstrate the response of the model. For allthree cases the model consists of a cube with sides 1.0 m in lengthmodeled using a single three-dimensional element. Thedisplacements on ve faces of the element were constrained inrespect to the faces normal direction, and loaded under normaldisplacement control on the sixth face (top). For each test, thematerial was displaced vertically downwards by 0.05 m and thenreleased until a zero stress-state was reached. Due to bulking, thenal volume of the material was larger than the original volumeresulting in a non-zero displacement corresponding to zero stress.Table 1 presents the material properties for the three validationcases.

For case A, the material was dened as having no fracturestrength and was not allowed to accumulate plastic strain. As such,the material is fully damaged once the strength was exceeded. This2.1.3. ValidationJohnson and Holmquist [11] present three validation cases for7) Compute compressibility factor mn1 based on the currentvolumetric strain n1v :

mn1 lnn1v 1

(22)

8) Update pressure using EOS:

Pn1

K1mn1 K2mn1

2K3mn1 3K1mn1

if mn1>0otherwise

(23)

If Dn1>0 compute energy loss due to damage DU and update totalpressure Pn1 using additional pressure increment DPn1:

DU s2HEL6G

s*n2s*n12 (24)

DPn1 K1mn1 K1mn1 DPn 2bK1DUq (25)

else, DPn1 0

Pn1 Pn1 DPn1 (26)

9) Update total stresses:

fsgn1 fSgn1Pn1dij (27)

10) End of one-direct integration cycle.11) Compute new stable time increment Dt and strain increments

D. Brger et al. / International Journa66Holmquist [11] for the case B is depicted in Fig. 3.accumulation of plastic strain. A comparison between predictionsobtained using the actual JH-2 implemented into ABAQUS and theresults published by Johnson and Holmquist [11] for the case C isdepicted in Fig. 4.

2.2. JohnsoneCook model

The JohnsoneCook model [20] is a phenomenological modelthat is commonly used to predict the material response of metalssubjected to impact and penetration, since it can reproduce strainhardening, strain-rate effects, and thermal softening. These prop-erties are coupled in amultiplicative manner by using the followingexpression,

sy hC1C2

peff

Ni 1C3ln

_peff_0

!!1

TTRTMTR

M

(28)

where peff is the effective plastic strain, TM is the melting temper-ature; TR is the reference temperature when determining C1, C2, C3,M and N; _0 is the reference strain rate; C1, C2, C3, N and M arematerial constants. The fracture in the JohnsoneCook model [20] isFig. 2. Stress versus pressure histories for a single element validation based on case A.

Fig. 3. Stress versus pressure histories for a single element validation based on case B.

D. Brger et al. / International Journal of Ibased on the value of the equivalent plastic strain. Failure isassumed to occur when damage exceeds 1. The cumulative damageis given by

D XDpeff

F (29)

with

F

"D1 D2exp

D3

Pseff

!# 1 D4ln

_peff_0

!!

1 D5

T TRTM TR

M

30

where P is the pressure, seff is the Mises stress; D1, D2, D3, D4, D5 arefailure parameters. The Johnson-Cook model used in this work iscurrently available in the ABAQUS Explicit material model library

for both shell and solid elements.

Fig. 4. Stress versus pressure histories for a single element validation based on case C.2.3. Contact-logic

A typical failure mode experimentally observed in hybridceramic-ber reinforced composite armors is delamination.Delamination occurs mainly due to high interlaminar stressesdeveloped at the bonding interface between layers of dissimilarorientation and/or materials. The delamination failure modes areusually classied into mode I, mode II, mode III and mixed-modedelamination modes according to the predominant stresses actingon the interface. For instance for Mode I, also dened as openingmode, the delamination is exclusively due to the through-thicknesstensile normal stress which leads to layer debonding in the direc-tion normal to the interface. Mode II and III are related to the out-of-plane shear stresses which results in relative sliding betweenupper and lower layers. Mixed-mode delamination is a combina-tion of modes I, II and III.

Different techniques for delamination modelling have beenproposed by many researchers in recent years. Approaches basedon stress criteria like those proposed by Lee [21], Kim and Soni [22],Brewer and Lagace [23], Liu and co-workers [24] and Jen et al. [25]are suitable to model the initiation of delamination. However, theydo not predict delamination growth realistically. Moreover, theyrequire a precise calculation of stresses and usually the stresses aresingular at the crack tip or free edge. Therefore, the determinationof stresses using nite element models becomes mesh dependent.Also, as discussed before, they do not give any information aboutthe delamination mode involved in the failure process. Numericalapproaches based on fracture mechanics require an initial aw andthey are used in conjunction with techniques such as the VirtualCrack Closure (VCC) method for the determination of the strainenergy release rate. The VCC method is based on Irwins assump-tion that when a crack extends by a small amount, the energyrelease in the process is equal to work required to close the crack toits original length. The energy release rates can then be computedfrom the nodal forces and displacements obtained from the solu-tion of the nite element model and crack propagation is simulatedby advancing the crack front when the local energy release raterises to a critical value [26]. The method predicts well the delami-nation growth, however as aforementioned the structure must bepre-cracked and different meshes are required for each delamina-tion front as the crack advances.

An alternative and efcient way for delamination modellingwhich has been widely reported in the literature is by usinginterface elements. Interface elements offer the possibility ofcoupling stress based criteria and fracture mechanics based criteriawithin a unied way. Therefore, they enable the model to predictboth initiation and growth of delamination. For bi-dimensionalproblems interface elements can be dened as a one-dimensionalentity inserted between two adjacent layers. In a similar waythey can be extended to three-dimensional problems which theone dimensional element is replaced by two dimensional elementconnecting adjacent layers. In elastic cases the interface elementsare very stiff in order to ensure the transference of displacementand traction between the adjacent layers. To model delaminationgrowth an interfacial material behaviour is assumed to control therelative displacements and traction between layers and as soon ascertain failure criteria are fullled, the delamination is allowed toinitiate and propagate. Mi, Criseld and Davies [27] proposeda continuous interface element for delamination modelling in brecomposites. The interface element was embedded between twoeight-noded isoparametric plane strain elements. A bi-linear soft-ening stress-relative displacement relationship was assumed forthe interface material model and linear and quadratic interactioncriteria were used for mixed-mode prediction. For unloading

mpact Engineering 43 (2012) 63e77 67conditions a simple elastic damage model was adopted in which

constitutive law without membrane effects for the contactelement can be written as follows

Fig. 5. Three-dimensional contact element.

l of Impact Engineering 43 (2012) 63e77the material is assumed to unload directly towards the origin.Excellent agreement was obtained between simulations, experi-mental and closed form solutions for mode I, mode II and mixed-mode delamination.

Daudeville and Ladeveze [28] proposed a delamination modelbased on a damage mechanics approach. In their model connectinglayers were used to represent the resin rich interface between twoadjacent layers. Three internal damage variables were used in orderto describe delamination associated with modes I, II and III. Theauthors studied the delamination in the vicinity of a straight edgeof a specimen under static tension or compression. Good correla-tion between numerical simulations and experimental results wasobtained for the prediction of damaged areas and onset strains.Based on the works proposed by Daudeville and Ladeveze [28] andCriseld and Davies [27], Camanho and co-workers [29] proposeda mixed-mode decohesion interface elements to model delamina-tions in composite laminates. The authors obtained a good corre-lation between numerical predictions and experimental results forDCB, ENF and MMB specimens. Their interface element is currentlyavailable in ABAQUS FE code and it was later implemented into LS-DYNA 3D explicit nite element code via user-dened materialmodels within brick elements by Pinho et al. [30]. An alternativeversion of their model for dynamic delamination modelling incomposites was also proposed by Iannucci [31].

An alternative contact logic to predict mixed-mode delamina-tion growth in composites is presented in this section. Theformulation for the interfacial material behaviour is dened interms of a linear-polynomial stress-relative displacement consti-tutive law. Based on fracture mechanics concepts, the area underthe curve dened by the constitutive law is equal to the fractureenergy or energy per unit of area, and once this energy is consumedthe crack propagates. In order to simulate the mixed-modedelamination, a stress-based criterion is used for the failure initi-ation and interactive mixed-mode criteria are used to predictdamage propagation. The advantage of the proposed contact logicformulation over the existing formulations is the use of a singledamage variable to predict interaction between different delami-nation failure modes, without knowing a priori the modes mixityratio. The proposed formulation also uses a high-order damageevolution law which avoids both numerical instabilities and arti-cial stress waves propagation effects commonly observed in thenumerical response of Finite Element Codes based on Explicit TimeIntegration Schemes. These laws compared to the widely usedbilinear law ensure smoothness at damage initiation and fullydamaged stress onsets leading to amore stable numerical response.The numerical predictions obtained using the proposed model wasvalidated against experimental results for Double Cantilever Beam(DCB) Mixed Mode Bending (MMB) specimens.

2.3.1. FormulationThe interfacial material behaviour is dened in terms of trac-

tions and relative displacements between the upper and lowersurfaces of the interface. The relative displacement vector iscomposed of the resultant normal and sliding components deningby the relative movement between upper and lower surfaces of thecontact element (Fig. 5). For a single integration point hexahedronsolid element the relative displacement vector can be written interms of through-thickness normal strain and out-of-plane shearstrains as follows,

fdgT fu v w gT nh*gxz h

*gyz h*zz

oT(31)

where u uT ub, v vT vb, and w wT wb. h* is the elementthickness of the updated geometry Fig. 6. Following the standard

D. Brger et al. / International Journa68interface element formulation, the uncoupled linear-elastic8>>>>>>>>>>>>:GIc

sin2bGIIcGIc24Kuucosa2Kvvsina212

35h

Kwwcos2bsin2bKuucosa2Kvvsina2

12

9>>>>>>>=>>>>>>>;

d0

24Kwwcos2bsin2bKuucosa2Kvvsina212

35(58)

The resultant damage evolution is given by

dmd 1 d0d

h1 k2md2kmd 3

i(59)

where

kIIId d d0df d0

(60)

and the mixed-mode stress-relative displacement relationships aregiven by,fdg :uv; :u

v; :DuDv ; (64)

where the subscripts n and n1 refer to the previous and currenttime step, respectively. h0 is the initial interface thickness associ-ated with the undeformed conguration.

2) Compute total elastic stresses acting on the interface and the

resultant relative displacement dn1

and resultant stress sn1

at current time:

8; 61

fsg Kdmdfdg (62)

The proposed formulation incorporates a consistent singledamage variable dmd for all delamination modes which enablesthe prediction of variable mixed mode delamination growthwithout knowing a priori the mixity ratio between differentdelamination modes.

mpact Engineering 43 (2012) 63e77If Fn1sI ; sII ; sIII 0 compute, bn1, an1, and update de damagevariable:

placed at the midplane of the virtual coupon to represent a resin

in ABAQUS. The mechanical properties for the composite arms aregiven in Table 2 and the mechanical properties for the resin richinterface are presented in Table 3. The simulations were carriedout quasi-statically under displacement control using the dynamicrelaxation method. Comparisons between experimental resultsand numerical predictions are shown in Fig. 9. A very goodagreement between experimental and numerical results obtainedusing the proposed contact-logic was found. Fig. 10 shows thepredicted interfacial damage extent for the DCB and MMBspecimens.

damage variables were dened in order to quantify damageconcentration associated with each possible failure mode andpredict the gradual stiffness reduction during the fracture process.Details about the formulation are given in Donadon et al. [37]. Thedifferent failure modes predicted by this model include ber failurein tension/compression, inter-ber failure (IFF) and in-plane shearfailure.

2.4.1. Fiber failure in tension/compressionThe maximum stress criterion is used to detect ber failure in

tension and compression and is given respectively by

Ft1s1 s1 1 (71)

b

0

0

vv

1 dn1m

dn1

3758