MICROMECHANICAL FAILURE ANALYSIS OF UNIDIRECTIONAL FIBER-REINFORCED COMPOSITES UNDER IN-PLANE AND TRANSVERSE SHEAR

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1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS MICROMECHANICAL FAILURE ANALYSIS OF UNIDIRECTIONAL FIBER-REINFORCED COMPOSITES UNDER IN-PLANE AND TRANSVERSE SHEAR Lei Yang*, Ying Yan, Zhiguo Ran School o Aeronautic Science and Engineering, Beihang University, Beijing, China * Corresponding author (leiyang.buaa@gmail.com) Keywords: Fiber-reinorced polymer (FRP), In-ane shear, Transverse shear, Representative volume element (RVE), Failure mechanism, Stress-strain curve 1 Introduction With the increasing use o iber-reinorced polymer (FRP) composites in aerospace and other industries, the ailure mechanisms o them should be well understood in order to take ull advantage o their excellent perormance and to develop even higher perormance materials. However, as FRPs consist o heterogeneous materials, the interactions between the reinorcements and matrix as well as the interace make the ailure process very comicated. Up to now the ailure mechanisms o FRPs are not well revealed, and their strength cannot be accurately predicted. Generally, a ull experimental approach is used to obtain their strengths, which requires a considerable amount o human, inancial and time resources. Even so, as compared with the longitudinal properties, the experimental characterization o the properties in the transverse direction is subjected to more uncertainties and experimental data are actually scarcer. Thereore, an accurate and cost-eective method to reveal the damage initiation and evolution process o the composites and then to predict their strengths is urgently required. Previously, the authors [1] developed a micromechanical damage model by inite element method (FEM) to reveal the ailure mechanisms and predict the strength o unidirectional FRP composites under transverse tension and compression, with some valuable conclusions obtained. However, the shear load has not been considered in the investigation, which may cause more comicated ailure behavior. Though some researchers have carried out investigations on the micromechanical ailure behavior o iber reinorced composites under in-ane shear [2] and transverse shear [3][4], the ailure mechanisms were still not well revealed. Thus, this paper uses the presented micromechanical damage model to investigate the ailure behavior o FRPs subjected to in-ane and transverse shear load. The stress-strain curves are obtained and compared with the experiment. The damage initiation and evolution process is discussed in detail, giving reasonable exanations to the ailure mechanisms o the composite under both in-ane and transverse shear loads. 2 Simulation Strategies 2.1 Generation o RVE To perorm micromechanical analysis o composite material, a representative volume element (RVE) large enough to represent the bulk material should be constructed. LLorca et al [5] have proved that an RVE including 30 ibres is adequate to represent the macroscopic material, and this suggestion is adopted in our investigation. The RVE consists o three phases: the ibers, the matrix and the interace. It is well known that or FRPs the ibers are randomly distributed in the matrix, thus the RVE should take into account this characteristic. In our previous work [6], a novelty method named random sequential expansion (RSE) algorithm was proposed, which can overcome the jamming limit o hard-core model [7] to generate random iber distributions or high value o iber volume raction (FVF). The RSE algorithm is used in this paper to generate the RVE o the composite material. The representative RVEs used or micromechanical analysis are shown in Fig.1, which contains 30 integrated ibers o 10 μm diameter, with FVF o 50% and 60%, respectively. The RVEs with 50% FVF are used or the transverse shear simulation, and those with 60% FVF or in-

2 Micromechanical Failure Analysis o Unidirectional Fiber-Reinorced Composites under In-ane and Transverse Shear ane shear simulation. In each case, ive separate RVEs with dierent iber distributions are generated to take into account the eect o microscopic coniguration. Fig.1. The RVEs used or micromechanical analysis with iber volume raction o (a) 50% and (b) 60% 2.2 FEM model FEM models are generated in ABAQUS/Exicit, which uses an exicit direct-integration procedure to overcome the convergence diiculty o numerical analysis. Plain strain models are used or transverse shear simulation, since the mechanical behavior in any transverse ane is basically the same or long iber reinorced composite. While or in-ane shear simulation, three dimensional models must be adopted, as the load is along the iber direction. The detailed description about the ain strain model can be ound in [1], thus is not repeated here. For the three dimensional model, the ibers and matrix are meshed with 8-node linear brick element with reduced integration and hourglass control (C3D8R). A layer o 8-node three-dimensional cohesive elements (COH3D8) with very small thickness (0.01 μm) are inserted between each iber and the surrounding matrix to simulate the interace. Shown in Fig.2 is the FEM model o one representative RVE with FVF o 60%. The whole model contains 228,200 elements. Periodic boundary conditions are apied to the RVE to ensure continuity between neighboring RVEs, which are expressed by the ollowing equations: U u( a, y, z) u(0, y, z) 1 U u( x, b, z) u( x,0, z) 2 (1) U3 u( x, y, c) u( x, y,0) where a, b and c are dimensions o the RVE along the x, y and z axis direction, respectively. Fig.2. 3-D FEM model o RVE 2.3 Material Models The material is chosen rom the World Wide Failure Exercise as E-glass/MY750/HY917/DY063 [8], with the properties o the iber and matrix listed in Table 1.

3 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS As iber ailure is unlikely to happen under the current loadings, no damage model has been imemented or the ibres, which are modelled as linear elastic and transverse isotropic solids. On the other hand, experimental studies have shown that the dominant damage mechanisms involved in Table 1 Mechanical properties o the iber and matrix transverse y cracking o composites are matrix astic deormation and interacial debonding [9]. Thereore the ailure behaviour o the polymeric matrix and the interace are introduced and discussed in detail as ollows. Fiber Matrix Longitudinal modulus, E 1 (GPa) 74 Modulus, E m (GPa) 3.35 Transverse modulus, E 2 (GPa) 74 Shear modulus, G m (GPa) 1.24 In-ane shear modulus, G 12 (GPa) 30.8 Poisson's ratio, ν m 0.35 Major Poisson's ratio, ν Tensile strength, σ mt (MPa) 80 Transverse shear modulus, G Compressive strength, σ mc (MPa) Polymeric matrix It is oten observed that the behavior o polymers is sensitive to the hydrostatic stress [10][11]. To capture these characteristics, the extended linear Drucker-Prager yield criterion is emoyed, which is expressed by the ollowing equation [12]: r F t p tan d 0, t q 1 1 (2) 2 K K q where p is the hydrostatic stress, q is the Mises equivalent stress, r is the third invariant o deviatoric stress, β is the slope o the linear yield surace in the p t stress ane, d is the cohesion o the material, and k is the ratio o the yield stress in triaxial tension to the yield stress in triaxial compression and, thus, introduces dierent yield behaviours between tension and compression. The parameters needed or the numerical simulation, β, d and k, can be determined by the ollowing equations [12]: 6sin tan (3) 3 sin 1 d 1 tan mc (4) 3 3 sin k (5) 3 sin where, is the internal riction angle o the Mohr- Coulomb yield criterion, which can be obtained rom the tensile and compressive strength o the matrix [13]: mc mt sin (6) mc mt Thus, it gives 11.5, and then we have 23.2, d 103 MPa, k Besides the yield criterion, there also should be a criterion or predicting the onset o damage. Experimental results indicate that the polymer shows rather brittle racture behaviour at a very low strain in uniaxial tension, but it yields and shows considerable astic deormation in uniaxial compression and pure shear [14]. Thus the criterion should be able to predict the damage onset discriminating between dierent triaxial stress states. This is achieved by the ductile criterion which assumes the equivalent astic strain at the onset o damage D to be a unction o stress triaxiality η, where p/ q. The stress triaxiality is a measurement o the triaxial stress state, or exame, η takes the value o zero under pure shear, and the values o ±1/3, ±2/3 and ± or uniaxial, equibiaxial and equitriaxial tension and compression, respectively. In this paper, the equivalent astic strains at the onset o damage or uniaxial tension and compression are set as 0.05 and 0.5, respectively, the reasonable values extracted rom experimental results [14]. Ater the onset o ailure, the damage evolution is controlled by a progressive ailure procedure, the stress-strain behavior o which is illustrated in Fig.3. The dashed curve in the igure is the stress-strain response in the absence o damage, while the solid curve represents the damaged response. The damage maniests itsel in two orms: sotening o the yield stress and degradation o the elasticity, both o which are related to the damage variable D, which

4 Micromechanical Failure Analysis o Unidirectional Fiber-Reinorced Composites under In-ane and Transverse Shear increases with the evolution o damage according to [12]: L u D (7) u u where L is the characteristic length, and u respectively are the equivalent astic strain and equivalent astic disacement. Beore damage initiation u 0 ; ater damage initiation u L. And u is the equivalent astic disacement at ailure computed as u 2 G / y0, in which, y 0 is the yield stress at the time when the ailure criterion is reached, and G is the racture energy per unit area: u G L yd ydu (8) 0 0 in which, 0 and respectively are the equivalent astic strains at the onset o damage (D=0) and at inal ailure (D=1). The experimental stress-strain curves show that the orm o inal ailure is brittle instead o ductile no matter the epoxy is under uniaxial tension or compression, thus the racture energy is given as a very small value o 0.5 J/m 2, since a value smaller than this may induce numerical instability. tn K n, ts K s, tt K t (9) where t n, t s and t t are the normal and the two shear tractions, respectively, δ n, δ s and δ t are the corresponding separations. The elastic stiness is selected as K = 10 8 GPa/m, large enough to ensure the disacement continuity at the interace and to avoid any modiication o the stress ields around the ibers in the absence o damage [4]. The maximum nominal stress criterion is used to predict the damage initiation o the interace, which is represented as: tn ts tt max,, 1 (10) t t t n s t where t n, t s and t t represent the peak values o the nominal stress when the deormation is either purely normal to the interace or purely in the irst or the second shear direction, respectively. is the Macaulay brackets, which return the argument i positive and zero otherwise, to impede the development o damage when the interace is under compression. As the strength o interace can hardly be measured by experiment, or simicity, it is assumed to have similar mechanical properties with the matrix: the normal and the two shear strength o the interace are assumed to be equal to the tensile strength and the cohesion o the matrix, respectively, i.e., tn 80 MPa, ts tt 103 MPa. Once the damage begins, the traction stress is reduced depending on the interace damage parameter D, which evolves rom 0 (in the absence o damage) to 1 (at ultimate ailure), as shown in Fig.4. The disacement at ailure ( n or s ) is determined by the racture energy G, which corresponds to the area under the traction-separation curve. The interace racture energy is taken rom Re. [3] as 100 J/m 2. Fig.3. Stress-strain behavior o the matrix Interace model Interacial debonding is included in the simulation by cohesive elements, with the constitutive response deined in terms o a bi-linear traction-separation law which relates the separation disacement between the top and bottom aces o the element to the traction vector acting upon it. The initial response is linear in absence o damage with an elastic stiness o K: Fig.4. Traction-separation law o the interace

5 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS 3 Results and Discussion 3.1 In-ane shear Stress-strain curves The simulated in-ane shear stress-strain curves o the RVEs are shown in Fig.5, compared with the experimental result rom [8]. As can be seen, the simulated results o all RVEs have very good consistency, with the elastic regimes o all the curves almost superposed. The dierent iber distributions o RVEs cause very little dierence in the simulated shear strength and ailure strain, thus in return proves the RVE is large enough to represent the bulk material. The predicted in-ane shear strengths are a little smaller than the experimental value, but generally the errors are within 15%. The predicted in-ane shear ailure strain also agrees well with the experimental result. Considering the uncertainties o the interace properties, the accuracy o the numerical model is acceptable. Fig.5. The simulated in-ane shear stress-strain curves and comparison with experiment [8] Fig.6. Ultimate damage state o RVE under in-ane shear Failure mechanism Take RVE-1 or exame, Fig.6 illustrates the ultimate damage state o the RVE under in-ane shear. As can be seen, a comete racture surace is generated in the model, which runs through the iber direction. To get a clearer understanding o the damage orm, the model is broken up along the racture surace, showing the details o the racture surace. It is interesting that the racture surace is not parallel to the ibers, but has certain angle o inclination with the iber direction. The actual value o the inclination angle is about 11, which is approximately equal to the internal riction angle o the matrix (11.5 ), indicating a leading role o the matrix astic damage in the ailure behaviour o the

6 Micromechanical Failure Analysis o Unidirectional Fiber-Reinorced Composites under In-ane and Transverse Shear composite. However, this exact inclination angle is not necessarily to orm, as the crack propagation path will be aected by the distribution o ibers. It can also be seen that interacial debonding only happens at the locales near the ront and back surace, leaving the most o the racture surace being matrix astic damage. Thus it is concluded that damage irst occurs in the orm o interacial debonding near the outer surace, and then matrix astic damage is induced at the vicinity o the interace damage and propagates inward, inally, the matrix damage links as a comete surace that runs through the RVE, causing the ultimate racture o the RVE. 3.2 Transverse shear Stress-strain curves The simulated transverse shear stress-strain curves are illustrated in Fig.7. As can be seen, the dierence in iber distribution causes more divergence in the stress-strain relation than in-ane shear, especially in the ailure strain. And the average transverse shear ailure strain is smaller than the in-ane shear ailure strain. This is probably because when the loading is along the longitudinal direction, more interace slipping is allowed to happen, or instance, in the orm o iber-matrix debonding or iber pull-out. Fig.7. The simulated transverse shear stressstrain curves Fig.8. Damage initiation and evolution o RVE-1 under transverse shear

7 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Damage initiation and evolution The damage initiation and evolution process o the RVE under transverse shear is shown in Fig.8, with each damage state corresponding to one point in the stress-strain curve. Point a is the peak o the stressstrain curve, standing or the transverse shear strength o the composite, and the corresponding damage state is shown in Fig.8a. It is ound that the main damage mechanism at this stage is matrix astic damage, which occurs right near the interaces o two adjacent ibers. At the same time, interacial debonding can also be ound between adjacent ibers, as can be seen more clearly by a partial enlarged drawing. And the connecting line o the two ibers is approximately parallel to the direction o maximum principal stress (i.e. 45 direction), thus the interacial debonding is supposed to be caused by the maximum tensile principal stress. Ater the peak point, the curve shows a slow declining period, at the same time more matrix damages happen at dierent locations where two ibers are close to each other, as shown in Fig.8b. Finally, matrix cracks are linked to orm a main crack, passing through dierent interaces in the path and causing the ultimate racture o the composite, as shown in Fig.8c. Fig.9 shows the ultimate damage orms o the other our RVEs. It is obvious that the iber distributions greatly aect the crack propagation path. In most cases, a main crack that runs through the RVE will orm; but when the iber distribution is too disorderly, the throughout crack may ail to orm, as is the case o RVE-2. This also gives the exanation to Fig.7 that why the stress-strain curve o RVE-2 deviates rom the others. 4 Conclusions The ailure behavior o unidirectional FRPs subjected to in-ane and transverse shear is studied Fig.9. Ultimate damage orms o dierent RVEs using computational micromechanics. RVEs with dierent microscopic structures are constructed to consider the eect o random iber distribution. The predicted in-ane shear stress-strain curves show

8 Micromechanical Failure Analysis o Unidirectional Fiber-Reinorced Composites under In-ane and Transverse Shear reasonable agreement with the experimental results. The microscopic damage mechanisms o the composite under both loadings are clearly revealed, showing comicated interactions between dierent damage orms involving matrix astic deormation and interacial debonding. Acknowledgements The inancial support o the National Basic Research Program (973) o China (under Grant No. 2011CB606105) is acknowledged. Reerences [1] L. Yang, Y. Yan, Y.J. Liu, Z.G. Ran. Microscopic ailure mechanisms o ibre-reinorced polymer composites under transverse tension and compression. Compos Sci Technol, Vol. 72, No. 15, pp , [2] E. Totry, J.M. Molina-Aldareguía, C. González, J. LLorca. Eect o iber, matrix and interace properties on the in-ane shear deormation o carbon-iber reinorced composites. Compos Sci Technol Vol. 70, No. 6, pp , [3] E. Totry, C. González, J. LLorca. Failure locus o iber-reinorced composites under transverse compression and out-o-ane shear. Compos Sci Technol Vol. 68, No. 3-4, pp , [4] L.P. Canal, J. Segurado, J. LLorca. Failure surace o epoxy-modiied iber-reinorced composites under transverse tension and out-o-ane shear. Int J Solids Struct Vol. 46, No , pp , [5] C. González, J. LLorca. Mechanical behavior o unidirectional iber-reinorced polymers under transverse compression: Microscopic mechanisms and modeling. Compos Sci Technol Vol 67, No. 13, pp , [6] L. Yang, Y. Yan, Z.G. Ran, Y.J. Liu. A new method or generating random ibre distributions or ibre reinorced composites. Compos Sci Technol Vol. 76, No. 4, pp 14 20, [7] V.A. Buryachenko, N.J. Pagano, R.Y. Kim, J.E. Spowart. Quantitative description and numerical simulation o random microstructures o composites and their eective elastic moduli. Int J Solids Struct Vol. 40, No.1, pp 47-72, [8] P. Soden, M. Hinton, A. Kaddour. Lamina properties lay-up conigurations and loading conditions or a range o ibre-reinorced composites laminates. Compos Sci Technol Vol. 58, No. 7, pp , [9] T.J. Vaughan, C.T. McCarthy. Micromechanical modelling o the transverse damage behaviour in ibre reinorced composites. Compos Sci Technol Vol. 71, No. 3, pp , [10] L.E. Asp, L.A. Berglund, R. Talreja. A criterion or crack initiation in glassy polymers subjected to a composite-like stress state. Compos Sci Technol Vol. 56, No.11, pp , [11] L.E. Asp, L.A. Berglund, P. Gudmundson. Eects o a composite-like stress state on the racture o epoxies. Compos Sci Technol Vol. 53, No. 1, pp 27-37, [12] ABAQUS theory manual, HKS Inc., [13] T.J. Vaughan, C.T. McCarthy. A micromechanical study on the eect o intra-y properties on transverse shear racture in ibre reinorced composites. Composites Part A Vol. 42, No. 9, pp , [14] B. Fiedler, M. Hojo, S. Ochiai, K. Schulte, M. Ando. Failure behavior o an epoxy matrix under dierent kinds o static loading. Compos Sci Technol Vol. 61, No. 11, pp , 2001.