(RVE) for the Simulation of Heterogeneous Polymers at Finite Strains

Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at finite strains

Citation for published version (APA): Mirkhalaf, S. M., Andrade Pires, F. M., & Simoes, R. (2016). Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at finite strains. Finite Elements in Analysis and Design, 119, 30-44. https://doi.org/10.1016/j.finel.2016.05.004

DOI: 10.1016/j.finel.2016.05.004

Document status and date: Published: 01/01/2016

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1 Contents lists available at ScienceDirect 2 3 4 Finite Elements in Analysis and Design 5 6 journal homepage: www.elsevier.com/locate/finel 7 8 9 10 11 Q2 12 Determination of the size of the Representative Volume Element (RVE) 13 for the simulation of heterogeneous polymers at finite strains 14 a a,n b,c 15 Q1 S.M. Mirkhalaf , F.M. Andrade Pires , R. Simoes 16 a 17 Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias 4200–465, Porto, Portugal b – 18 Institute for Polymers and Composites (IPC/I3N), University of Minho, 4800 058 Guimarães, Portugal c School of Technology, Polytechnic Institute of Cávado and Ave, Campus do IPCA, 4750–810 Barcelos, Portugal 19 20 21 article info abstract 22 23 Article history: The definition of the size of the Representative Volume Element (RVE) is extremely important for the 24 Received 11 November 2015 mechanics and physics of heterogeneous materials since it should statistically represent the micro- 25 Received in revised form structure of the material. In the present contribution, a methodology based on statistical analysis and 26 29 March 2016 numerical experiments is proposed to determine the size of the RVE for heterogeneous amorphous Accepted 23 May 2016 27 polymers subjected to finite deformations. The approach is applied to Rubber Toughened Polystyrene 28 (RT-PS) composed by a two phase random micro-structure. Different micro-structural samples with two 29 Keywords: different percentages of rubbery particles, namely 10% and 15%, inside the micro-structure are studied. 30 RVE size Periodic boundary conditions (PBC) are enforced to the RVE due to their fast convergence to the theore- Heterogeneous tical/effective solution when the RVE size increases. The Finite Element Method (FEM) is used in combi- 31 Polymers nation with mathematical homogenization to obtain the macro-stress. Two criteria are proposed for the 32 Finite deformations 33 RVE size determination. The proposed statistical-numerical approach is general and easy to use, when compared to the previously proposed approaches, and covers other criteria available in the literature. 34 & 2016 Elsevier B.V. All rights reserved. 35 36 67 37 68 38 69 39 1. Introduction based composite materials have been the focus of a large number 70 40 of studies during the last decade. 71 41 Polymers have attracted a considerable attention from both The most widely used strategy to improve the mechanical 72 42 academic and industrial communities due to their interesting properties of materials consists in directly modifying the micro- 73 43 characteristics such as good thermal and electrical insulation, lower structure of the material, see e.g. [3], which usually promotes a 74 44 density and also usually higher yield strain when compared to more heterogeneous nature including different phases in the 75 45 metals [1]. Besides the properties mentioned above, one of the micro-structure. Therefore, it becomes vital to establish relation- 76 46 relevant features of polymers is that complicated shapes can be ships between the micro-structure of the material and the 77 47 easily fabricated through processes such as extrusion, injection engendered macroscopic properties. There are different approa- 78 48 molding and casting. For these reasons, polymers have a variety of ches to characterize the behaviour of materials with hetero- 79 49 applications, which are far more than any other class of materials, geneous micro-structures. One possible approach is to fit material 80 50 such as electronic products, automotive engineering, bio-mechan- properties to experimental results for a continuum level con- 81 51 ics, civil and building engineering, among others. stitutive model (e.g. [4]). Another approach consists in changing a 82 52 In spite of very interesting properties, polymeric materials also continuum level constitutive model by including some informa- 83 53 have some undesirable characteristics. For instance, glassy poly- tion from the micro-structure of the material into the constitutive 84 54 mers show brittleness under specific conditions. Polystyrene (PS) equations of the model so that the model could characterize the 85 55 and Polymethylmethacrylate (PMMA), among other glassy poly- behaviour of the material with modified micro-structure (e.g. [2]). 86 56 mers, are considered brittle since they show brittle failure under The most recent strategy entails the use of micro-mechanical or 87 57 low stress triaxiality such as uniaxial tension [2]. Therefore, in coupled multi-scale models based on homogenization techniques 88 58 order to take advantage of the mentioned qualities of polymers to transfer information between different scales (e.g. [5,6]). In fact, 89 59 and also improving the unfavourable characteristics, polymer 90 60 different homogenization techniques are used by different authors 91 61 in order to model the mechanical behaviour of heterogeneous n – 92 62 Corresponding author. materials (see e.g. [7 10]). The reader is referred to [11] for a 93 63 E-mail address: [email protected] (F.M. Andrade Pires). comprehensive review on homogenization techniques. 94 64 http://dx.doi.org/10.1016/j.finel.2016.05.004 95 65 0168-874X/& 2016 Elsevier B.V. All rights reserved. 96 66

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i 2 S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 In order to work within the framework of micro-mechanical determination of the RVES which is illustrated with numerical 67 2 and coupled multi-scale approaches, it is required to have the examples. Finally, Section 6 is devoted to some conclusions and 68 3 Representative Volume Element (RVE) of the heterogeneous concluding remarks. 69 4 material defined. In published literature, there are different defi- 70 5 nitions for the RVE (see e.g. [12–18]). What is common in the 71 6 variety of definitions is that the RVE should accommodate enough 2. Kinematics of the micro-scale problem 72 7 information about the micro-structure, while its dimensions are 73 8 smaller than the macroscopic dimensions, so that the concept of In this section, the mathematical formulation and main kine- 74 9 separation of scales is satisfied. matical assumptions governing this study will be introduced. If x is 75 10 During the last two decades, there have been different methods an infinitesimal point at the macro-scale, y is an infinitesimal point 76 11 employed by the scientific community to determine the RVE on the RVE domain and TðyÞ is a generic field defined over the RVE 77 12 size (RVES) from image processing approaches (e.g. [19–21]), domain, the homogenized response of the field, T, for the mac- 78 13 experimental-image processing methods (e.g. [22]), analytical roscopic point to which the RVE is linked, is obtained by the fol- 79 14 approaches (e.g. [13,23]) and statistical-numerical methods (e.g. lowing volume average relation: 80 15 [24–29]). Grimal et al. [19] have derived the field of elastic coef- Z 81 1 16 ficients from one acoustic microscopy image of a human femur TðxÞ¼ TðyÞdV; ð1Þ 82 V Ω 17 cortical bone sample with an overall porosity of 8.5%. They have 83 18 also used FEM to obtain the homogenized properties of RVEs. Shan where V is the RVE volume and Ω is the RVE domain. It must be 84 19 and Gokhale [20] have introduced a methodology involving image emphasized that the above relation can be applied to both unde- 85 20 analysis techniques for determination of the RVES for Ceramic formed and deformed configurations of the RVE. In this study, all 86 21 Matrix Composites (CMC) with different fiber sizes. Graham and the homogenized variables are defined as the volume average of 87 22 Yang [21] determined the RVES for HY-100 steel using image the respective quantities over the initial configuration of the RVE. 88 23 analysis of the polished cross section of the material. Romero and The deformation of the RVE is driven by the macroscopic 89 24 Masad [22] have obtained the RVES for an asphalt concrete spe- deformation gradient, which is the volume average of the micro- 90 25 cimen utilizing an image analysis approach and they have verified scopic deformation gradient: 91 Z 26 their results with some mechanical tests. Drugan and Willis [13] 92 ð ; Þ¼ 1 ð ; Þ : ð Þ 27 employed a variational formulation to derive a nonlocal con- F x t F y t dV 2 93 V 0 Ω 28 stitutive equation for a class of random linear elastic composite 94 29 materials. They have estimated the RVES by comparing the mag- The microscopic deformation gradient is given by: 95

30 nitude of the non-local term and the portion of the equation that Fðy; tÞ¼Iþ∇puðy; tÞ; ð3Þ 96 31 relates ensemble average stresses and strains through a constant 97 where I is the second order identity tensor, uðy; tÞ is the dis- 32 ”overall” modulus tensor. Sebsadji and Chouicha [23] have pro- 98 placement field at micro-scale and ∇ ðnÞ is the material gradient 33 posed an analytical approach, using fractal analysis, to obtain the p 99 operator. Using relations (2) and (3), the macroscopic deformation 34 RVES for concrete mixtures. Kanit et al. [24] have performed a 100 gradient is given by: 35 statistical analysis of the numerical examples in order to quantify Z 101 36 the RVES for two-phase three-dimensional Voronoii mosaic in the 1 102 Fðx; tÞ¼Iþ ∇puðy; tÞdV: ð4Þ 37 case of linear elasticity and thermal conductivity. Kanit et al. [25] V 0 Ω 103 38 have extended the statistical-numerical approach introduced in The displacement field at the micro-scale can be split into two 104 39 the previous work by Kanit et al. [24] to the case of real micro- terms: 105 40 structures of two materials from the food industry for RVES 106 ð ; Þ¼½ ð ; Þ þ ~ ð ; Þ; ð Þ 41 determination. Pelissou et al. [26] have started from the work of u y t F x t I Y u y t 5 107 42 Kanit et al. [24] and introduced a new approach for RVES deter- where the first term is the linear displacement varying linearly 108 43 mination for a metal matrix composite with randomly distributed with Y, which represents the RVE coordinates in the reference 109 44 aligned brittle inclusions. Skarzynski and Tejchman [27] have used configuration, and the second term denoted by u~ ðy; tÞ is the dis- 110 45 two different approaches, within a statistical analysis, to deter- placement fluctuation field. Using Eqs. (3) and (5), the microscopic 111 46 mine the RVES for softening quasi-brittle materials. The first deformation gradient, Fðy; tÞ, can be expressed as: 112 47 approach, failure zone averaging approach, was previously intro- 113 Fðy; tÞ¼Fðx; tÞþ∇ u~ ðy; tÞ: ð6Þ 48 duced by Nguyen et al. [30]. Gitman et al. [28] investigated RVE p 114 49 existence in different stages of loading, elastic-hardening-soft- Doing some algebraic manipulations, using divergence theorem 115 50 ening, and also proposed a method to determine the RVES for and relations (2) and (6), results in the minimal kinematical 116 51 random three-phase (matrix, inclusion and ITZ) heterogeneous admissible constraint: 117 52 materials. Stroeven et al. [29] have used different criteria (peak Z 118 53 load, dissipated energy, strain concentration factor), within a u~ ðy; tÞNðYÞdA ¼ 0; ð7Þ 119 ∂Ω 54 statistical-numerical framework to determine RVES for materials 120 55 with particles in a matrix material. where N (Y) denotes the outward unit vector to the undeformed 121 56 In this contribution, a statistical-numerical approach is pro- boundary of the RVE. The virtual work equilibrium of the RVE 122 57 posed to determine the RVES for heterogeneous amorphous states that: 123 Z Z Z 58 polymers subjected to finite deformations. Polystyrene (PS) is used 124 Pðy; tÞ : ∇ηdV Bref ðy; tÞηdV Tref ðy; tÞ 59 as the matrix material toughened with rubbery particles. Two Ω Ω ∂Ω 125 60 different volume fraction of inclusions are considered. The paper is 126 ηdA ¼ 0 8ηAν; ð8Þ 61 structured as follows. Section 2 presents the kinematics of the 127 62 micro-scale problem. The material model used for the polymeric where Bref ðy; tÞ and Tref ðy; tÞ are respectively the reference body 128 63 matrix material is presented in Section 3. Section 4 presents some force and the reference boundary traction force of the RVE. The 129 64 numerical examples at the micro-structural level with different symbol ν denotes the space of the virtual admissible displacement 130 65 percentage of inclusion volume fraction of rubbery particles. fluctuation field η. The first Piola–Kirchhoff stress of the infinite- 131 66 Section 5 details the proposed statistical-numerical approach for simal point at the macro-scale is the volume average of 132

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

1 corresponding stress at the micro-scale: 3. Material model 67 2 Z 68 1 3 Pðx; tÞ¼ Pðy; tÞdV: ð9Þ Over the last century, a considerable effort has been devoted to 69 V Ω 4 0 the development of new material models capable of reproducing 70 5 In order to relate the micro-scale and macro-scale, the Hill– experimentally observed data. It can be said that the initiation of 71 6 Mandel principle is used. This principle is expressed as follows: the efforts to determine the behaviour of polymers dates back to 72 7 Z 1930s. Eyring [36] proposed a molecular theory for the yield stress 73 _ 1 _ 8 Pðx; tÞ : Fðx; tÞ¼ Pðy; tÞ : Fðy; tÞdV; ð10Þ of amorphous polymers. Later, Mooney [37] proposed a strain 74 V Ω 9 0 energy function for rubber elastic materials. Haward and Tackray 75 10 which states equality between the macroscopic stress power and [38] developed a one dimensional constitutive model for glassy 76 11 volume average of microscopic stress power over the RVE domain. polymers and Boyce et al. [39] established a three dimensional 77 12 The Hill–Mandel principle can be equivalently written as: version of Haward and Tackray model. Another constitutive 78 13 Z approach that has been introduced to characterize the deforma- 79 _ 14 Tref ðy; tÞu~ ðy; tÞdA ¼ 0; ð11Þ tion behaviour of polymers is the generalized compressive Leonov 80 15 ∂Ω 81 model (currently known as EGP model). This model has been 16 82 Z proposed by Baaijens [40] and extended by Tervoort et al. [41] and 17 ref _ 83 B ðy; tÞu~ ðy; tÞdV ¼ 0: ð12Þ Govaert et al [42]. In addition to the models developed to char- 18 Ω 84 acterize the behaviour of polymers at continuum level, some 19 85 authors have devoted their work to predict the deformation 20 86 behaviour of polymers using molecular dynamics simulations (see 21 87 e.g. [43,44]) or through multi-scale models from the molecular 22 2.1. Periodic boundary condition 88 23 structure to the macro-scale (see e.g. [45]). 89 fl 24 The periodic boundary condition (PBC) is the most commonly This section brie y describes the elasto-viscoplastic con- 90 25 adopted kinematical boundary constraint for the solution of RVE stitutive model which is used in this study. The total deformation 91 26 equilibrium problem. It has been demonstrated that PBC con- gradient, F, is multiplicatively composed of the elastic deformation 92 e p 27 verges faster to the effective (or theoretical) as the RVE size gradient, F , and the plastic deformation gradient, F , [46]: 93 28 increases [24,31]. In other words, for the same RVE size, PBC gives e p 94 F ¼ F F : ð17Þ 29 a better estimation of the effective properties in comparison to the 95 30 other boundary conditions. This boundary condition is particularly It should be emphasized that the aforementioned decomposition 96 31 suitable for the analysis of the mechanical behaviour of materials of the deformation gradient has been mainly attributed to Lee [47] 97 32 with periodic or even quasi-periodic microstructure. This bound- but recently Sadik and Yavari [48] showed that the decomposition 98 33 ary condition assumes that the boundary of the RVE is composed was first introduced by Bilby et al. [46]. To measure the elastic 99 34 of negative and positive parts: deformations, the logarithmic (or natural) Eulerian (or spatial) 100 35 þ strain is adopted. The plastic flow rule of the model is character- 101 ∂Ω ¼ðΓ [ Γ Þ; ð13Þ 36 i i ized by the generalized Eyring equation, which is given by: 102 37 þ AΓ þ AΓ 103 where each point y i has a counterpart point y i . Fur- 38 p ∂Ψ 104 thermore, the following condition must be satisfied: d ¼ γ_ eq ¼ γ_ eqN; ð18Þ 39 ∂τ 105 þ 40 n ¼n ; ð14Þ 106 where γ_ eq is the equivalent rate of shear strain, τ is the Kirchhof 41 þ 107 where n þ and n are the unit normal vectors of Γ and Γ at stress tensor, Ψ is the dissipation potential, N is the flow vector 42 i i 108 points y þ and y , respectively. The displacement fluctuation field and dp is the spatial plastic stretching tensor. The dissipation 43 109 for periodic boundary condition is periodic: potential of the model is assumed to be given by: 44 110 rffiffiffiffiffiffiffiffiffiffiffiffi ~ þ ~ 45 uðy ; tÞ¼uðy ; tÞ; ð15Þ 1 111 46 Ψ ¼ s : s: ð19Þ 112 fi 2 47 and the traction eld is anti-periodic: 113 fl 48 ref þ ref Consequently, the ow vector is obtained as: 114 T y ; t ¼T ðÞy ; t : ð16Þ rffiffiffi 49 115 ∂Ψ 1 s 50 N ¼ ¼ ; ð20Þ 116 Therefore, the application of PBC on the RVE consists in the ∂τ 2JsJ 51 enforcement of an identical fluctuation field for each pair of cor- 117 52 responding boundaries of the RVE. This forces the RVE to be dis- where JsJ is the norm of s which is given by: 118 53 cretized with a conform mesh. Nevertheless, in order to apply pffiffiffiffiffiffiffiffiffi 119 JsJ ¼ s : s: ð21Þ 54 periodic boundary conditions on arbitrary finite element mesh 120 55 discretization, several strategies have been proposed (e.g. [32–34]). Combining relations (18) and (20), the multi-dimensional plastic 121 56 122 A comprehensive account of the numerical aspects required for flow rule of the model is obtained as: 57 the computational implementation of multi-scale constitutive rffiffiffi 123 58 models within a generic non-linear implicit finite element fra- 1 s 124 dp ¼ γ_ eq : ð22Þ 59 mework at finite strains can be found in [35], among others. 2JsJ 125 60 126

61 Table 1 127 62 Q6 Material properties for PS required by the constitutive model. 128 63 129 ν Δ ð = Þ ð Þ τ μ 64 E (MPa) H J mol A0 s 0 (MPa) D1 h RH(MPa) 130 65 131 PS 3300 0.37 1.7Eþ51.11E20 2.559 9 60 0.14 8.3143 11 66 132

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i 4 S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 μ 19 Fig. 1. Two different samples with a size of 10 m and volume fraction of inclusions equal to (a) 10% and (b) 15%. 85 20 86 21 87 22 88 23 5,0E+07 The rate of equivalent plastic shear can be expressed in terms of 89 24 multi-dimensional plastic flow rule: 90 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25 4,0E+07 91 γ_ eq ¼ p : p: ð Þ 26 2d d 29 92

27 3,0E+07 The scalar parameter A is given by: 93 28 94 ΔH μP 29 A ¼ A exp þ D ; ð30Þ 95 2,0E+07 0 RT τ 30 0 96

31 where the scalar A0 is a constant or pre-exponential factor invol- 97 32 1,0E+07 ving the fundamental vibration energy; the parameter ΔH is the 98

33 (Pa),stress First Piola-Kirchhof activation energy; the parameter R is the universal gas constant; 99 34 0,0E+00 the scalar T is the absolute temperature; the parameter μ is a 100 1 1,03 1,06 1,09 1,12 1,15 35 pressure coefficient related to the shear activation volume, V, and 101 Deformation gradient, 36 the pressure activation volume, Ω, according to: 102 37 Fig. 2. Stress-deformation of the samples shown in Fig. 1, solid line: 10% and Ω 103 38 dashed line: 15%. μ ¼ : ð31Þ 104 V 39 105 40 The deformation behaviour of polymers is also dependent on the 106 41 hydrostatic pressure and thus the hydrostatic pressure should take 107 The equivalent rate of shear strain is given by: part in the constitutive relations of the material model. In relation 42 108 τeq (30), P represents the total hydrostatic pressure of the analysis. 43 γ_ eq ¼ 1 ; ð Þ 109 sinh τ 23 The parameter D, in relation (30), is a softening parameter. The 44 A 0 110 evolution of the softening parameter is modelled according to 45 τeq fi 111 where is an equivalent stress de ned by: [49]: 46 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 112 47 eq 1 1 _ D 113 τ ¼ trðssÞ ¼ s : s; ð24Þ D ¼ h 1 γ_ p; ð32Þ 48 2 2 D1 114 49 where s is the deviatoric stress. Using relation (23) and also Eq. 115 where D1 is the saturation value of the softening parameter and 50 fl 116 (24), the plastic ow rule can be expressed in another form: γ_ p is an equivalent plastic strain rate defined by [42]: 51 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 117 1 τeq s 52 dp ¼ sinh ; ð25Þ γ_ p ¼ dp : dp: ð33Þ 118 τ τeq 53 A 0 2 119 Using relations (24), (28)–(30) and (32) together with some alge- 54 or equivalently, 120 braic manipulations results in the following viscosity function: 55 121 p ¼ "#s ; ð Þ "# pffiffiffi ! 56 d 26 ΔH μP h 3 εp τeq 122 τeq η ¼ þ þ pffiffiffi τeq= ; ð Þ A0 exp τ D1 D1 exp sinh τ 34 57 2A RT 0 2D1 0 123 sinh τeq=τ 58 0 124 where εp is the accumulated plastic strain. The accumulated 59 125 which can be represented as: plastic strain rate is defined by: 60 rffiffiffiffiffiffiffiffi 126 p ¼ s : ð Þ 61 d eq 27 _ p 2 p p 127 2ηðτ Þ ε ¼ ε_ : ε_ : ð35Þ 62 3 128 eq 63 The viscosity function, ηðτ Þ, is given by: total 129 "# The total stress, τ , is additively composed of driving stress, 64 driving hardening 130 τeq τ , and hardening stress τ : 65 ηðτeqÞ¼A : ð28Þ 131 sinh τeq=τ total driving hardening 66 0 τ ¼ τ þτ : ð36Þ 132

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 Fig. 3. Morphologies of the samples with size of 10 μm and volume fraction of inclusions equal to (a) 10% and (b) 15%. 85 20 86 21 5,0E+07 4,0E+07 87 22 88 23 4,0E+07 89 24 3,0E+07 90

25 3,0E+07 91 26 2,0E+07 92 27 93 2,0E+07 28 94 29 1,0E+07 95 30 1,0E+07 96

31 (Pa), stress Piola-Kirchhof First First Piola-Kirchhof stress (Pa), stress Piola-Kirchhof First 97 32 0,0E+00 0,0E+00 98 1 1,03 1,06 1,09 1,12 1,15 1 1,03 1,06 1,09 1,12 1,15 33 99 Deformation gradient, Deformation gradient, 34 100 35 Fig. 4. Stress-deformation of the samples shown in Fig. 3, solid line: 10% and Fig. 6. Stress-deformation of the samples shown in Figs. 1(b) and 3(b), both with 101 36 dashed line: 15%. inclusion volume fraction equal to 15%: Fig. 1(b) and red line: Fig. 3(b). (For 102 interpretation of the references to colour in this ﬁgure caption, the reader is 37 103 referred to the web version of this paper.) 38 5,0E+07 104

39 For a more elaborate discussion on the constitutive model and 105 40 4,0E+07 also the integration algorithm and finite element implementation, 106 41 the reader is referred to [50]. 107 42 3,0E+07 108 43 109 44 110 2,0E+07 4. Numerical examples 45 111 46 112 In this section, some numerical examples are performed on 47 1,0E+07 113 samples of rubber toughened Polystyrene composed by two pha- 48 First Piola-Kirchhof stress (Pa), 114 ses: Polystyrene (PS) and rubber. The material properties for PS are 49 0,0E+00 115 1 1,03 1,06 1,09 1,12 1,15 taken from van Melick et al. [52] and are given in Table 1. It should 50 116 Deformation gradient, be mentioned that due to the deformation induced cavitation early 51 117 during deformation [2], the rubbery particles are modelled as 52 Fig. 5. Stress-deformation of the samples shown in Figs. 1(a) and 3(a), both with 118 53 inclusion volume fraction equal to 10%: blue line Fig. 1(a) and red line Fig. 3(a). (For voids. Two levels of inclusion volume fraction, namely 10% and 119 interpretation of the references to colour in this figure caption, the reader is 54 15%, are considered and the inclusions size range is assumed to be 120 Q5 referred to the web version of this paper.) μ μ 55 from 0.7 mto1 m. 121 56 In this model, the hardening stress is given as the deviatoric strain Remark 1. The rubbery particles are chosen to be within the 122 57 123 multiplied by a hardening modulus: mentioned size range due to the fact that the rubber toughening 58 process is normally performed using sub-micron sized rubbery 124 59 τhardening ¼ ε ; ð Þ 125 H d 37 particles [2,51]. 60 126 ε 61 where d gives the total deviatoric strain and H is the hardening The first sample size is 10 μm. The samples are analysed with a 127 62 modulus, which is one of the material parameters. The deviatoric non-linear finite element framework at finite strains which con- 128 63 strain is given by: tains the constitutive model described in Section 3. The numerical 129 64 ε ¼ I : ε; ð Þ approach employed to analyse each micro-structural sample 130 d d 38 65 consists in enforcing a macroscopic deformation gradient, F, to the 131

66 where Id is the fourth order deviatoric identity tensor. RVE and solving the micro-scale equilibrium problem and then 132

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i 6 S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 obtain the homogenized response of the stress field. The numer- generated. The standard six-noded plane strain triangular ele- 67 2 ical simulations are performed under tensile loading conditions ment, with three Gauss integration points, is adopted for the 68 3 with the following deformation gradient: spatial discretization of the samples. The mesh discretization for 69 4 both cases is defined such that the relevant features of the geo- 70 1:15 0 5 F ¼ : ð39Þ metry are well captured without an excessive level of refinement 71 01 6 with PBC enforced to the relevant edges. Fig. 2 shows the stress 72 7 Fig. 1 shows two samples with dimension of 10 μm and volume deformation for both samples subjected to tensile loading under 73 8 fraction of inclusions equal to 10% and 15% that were randomly periodic boundary conditions. As expected, the increase in the 74 9 inclusion volume fraction from 10% to 15% leads to a significant 75 10 76 decrease of the material response quantified by P11 F11 curve. In 11 order to analyse the impact of changing the morphology of the 77 12 micro-structure on the overall deformation behaviour, one more 78 13 sample of each inclusion volume fraction is generated. The sam- 79 14 ples are shown in Fig. 3. Fig. 4 depicts the stress-deformation 80 15 graph for the samples shown in Fig. 3. In order to directly compare 81 16 the difference between different realizations of the same inclusion 82 17 volume fraction, the stress-deformation graph for samples of the 83 18 size of 10 μm and inclusion volume fraction of 10%, shown in 84 19 Figs. 1(a) and 3(a), is shown in Fig. 5. Similar to Fig. 5, the stress- 85 20 86 deformation graph for samples of the size of 10 μm and inclusion 21 87 volume fraction of 15%, depicted in Figs. 1(b) and 3(b), is presented 22 88 in Fig. 6. As can be seen in Figs. 5 and 6, changing the morphology 23 89 of the micro-structure clearly affects the overall behaviour of the 24 90 micro-structural samples. Therefore, it is necessary to define what 25 91 is the appropriate RVE size for each volume fraction of inclusions, 26 92 such that the results obtained can be regarded as representative. 27 93 To achieve this, a combined numerical-statistical analysis is pro- 28 94 posed in Section 5, where two criteria are introduced for the 29 95 determination of the RVE size. 30 96 31 97 32 98 33 5. RVE size 99 34 100 35 As mentioned before, working within the framework of micro- 101 36 mechanical modelling and coupled multi-scale modelling approa- 102 37 ches requires an appropriate definition of the RVE. An RVE should 103 38 be large enough to accommodate enough information from the 104 39 material micro-structure and at the same time, small enough to 105 40 comply with the separation of scales. In this section, a statistical 106 41 procedure combined with numerical simulations to determine the 107 42 RVES is presented. Again, Rubber Toughened Polystyrene (RT-PS) is 108 43 considered for the numerical examples. Two different volume 109 44 Fig. 7. Flow chart for RVES determination. fraction of inclusions are considered namely 10% and 15%. The initial 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 Fig. 8. Different realizations for inclusion volume fraction equal to 10% and different volume element sizes, (a): 10 μm, (b): 15 μm, (c): 20 μm. 132

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1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 Fig. 9. Different realizations for inclusion volume fraction equal to 10% and volume element size equal to 10 μm. 78 13 79 14 1,0E+08 1. The standard deviation of the deformation behaviour, which is 80 15 represented by a specific deformation measure, should not be 81 fi 16 8,0E+07 more than a prede ned percentage of average deformation 82 17 behaviour. In this work, 10% is considered for the variation of 83 18 the first invariant of the first Piola–Kirchhof stress tensor, I1ðPÞ . 84 19 6,0E+07 The criterion is given by: 85 20 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 86 u 21 4,0E+07 u Xn 2 87 t 1 i 22 SI ðPÞ ¼ I ðPÞI1ðPÞ rð0:1 I1ðPÞÞ ð40Þ 88 1 n1 1 i ¼ 1 23 2,0E+07 89 24 90 ð Þ 25 where SI1 P is the standard deviation, n is the number of sam- 91 First Invariant of the stress tensor (Pa) 0,0E+00 i ð Þ fi fi – 26 ples, I1 P is the ith value of the rst invariant of the rst Piola 92 1 1,03 1,06 1,09 1,12 1,15 ð Þ 27 Kirchhof stress tensorand I1 P is the mean value which is given 93 Deformation gradient, 28 by: 94 29 Fig. 10. Stress-deformation of the samples shown in Fig. 9: blue line: Fig. 9(a), 95 orange line: Fig. 9(b), green line: Fig. 9(c), red line: Fig. 9(d), black line: Fig. 9(e). Xn i ð Þ 30 I1 P 96 (For interpretation of the references to colour in this figure caption, the reader is I ðPÞ¼ : ð41Þ 1 n 31 referred to the web version of this paper.) i ¼ 1 97 32 98

33 1,0E+08 Checking this criterion ensures that changing the realization, for 99 34 the same sample size and same inclusion volume fraction, does 100 35 not remarkably change the deformation behaviour. 101 8,0E+07 36 2. The average behaviour of all the realisations of the same volume 102 37 k 103 element size, k, denoted by I ðPÞ is within a desirable pre- 38 1 104 6,0E+07 defined error with the average response of the next volume 39 105 k þ 1 40 element size, kþ1, denoted by I1ðPÞ : 106 4,0E+07 41 107 k þ 1 k 42 I1ðPÞ I1ðPÞ 108 ð%Þ¼ r %: ð Þ 43 2,0E+07 Error þ 10 42 109 ð Þk 1 44 First invariant of the stress tensor (Pa) I1 P 110 45 111 0,0E+00 46 1,00 1,03 1,06 1,09 1,12 1,15 In this work, 10% is considered for the variation of the average 112 first invariant of the first Piola–Kirchhof stress tensor of two 47 Deformation gradient, 113 48 consecutive volume element size. The second criterion checks if 114 Fig. 11. Average stress-deformation of the samples shown in Fig. 9 with shaded by increasing the size of the micro-structural sample, the 49 bounds of standard deviation. 115 50 deformation behaviour changes. If no considerable change is 116 51 observed, while increasing the sample size, this criterion is 117 Table 2 satisfied. 52 The percentage of (standard deviation/average) for some representative values of 118 53 deformation gradient for the samples shown in Fig. 9. 119 fi 54 The next stage is to check the rst criterion to see whether it is 120 Deformation gradient, F 1,01 1,02 1,05 1,08 1,12 1,15 55 11 satisfied. If the first criterion is satisfied, different distributions of 121 56 Standard deviation/average (%) 1.26 4.90 12.85 13.55 14.25 14.42 the next volume element size with the same inclusion volume 122 57 fraction are to be generated and analysed and the second criterion 123 58 is checked. In case the second criterion is satisfied as well, the 124 μ 59 volume element size is assumed to be 10 m and the inclusion size previous volume element size is RVES. If the first criterion is not 125 μ μ 60 range is from 0.7 mto1 m. satisfied, the next volume element size would be considered as the 126 61 The procedure for RVES determination is as follows. First, dif- initial volume element size and the same process, as mentioned 127 62 ferent distributions of one volume element size and inclusion above, would be done. This procedure is represented in Fig. 7. 128 63 volume fraction are generated. Then, the samples are loaded and 129 64 130 the stress-deformation graphs are extracted. Two criteria are Remark 2. The procedure suggested is basically a method for 65 131 introduced in order to determine the RVES: determining a numerical RVE size for a given heterogeneous 66 132

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1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 Fig. 12. Different realizations for inclusion volume fraction equal to 10% and volume element size equal to 15 μm. 98 33 99 34 1,0E+08 1,0E+08 100 35 101

36 8,0E+07 8,0E+07 102 37 103 38 104 6,0E+07 39 6,0E+07 105 40 106 41 4,0E+07 4,0E+07 107 42 108 43 109 2,0E+07 2,0E+07 44 110

45 (Pa)tensor stress the of invariant First 111 First invariant of the stress tensor (Pa) 46 0,0E+00 0,0E+00 112 1 1,03 1,06 1,09 1,12 1,15 47 1,00 1,03 1,06 1,09 1,12 1,15 113 48 Deformation gradient, Deformation gradient, 114 49 Fig. 13. Stress-deformation of the samples shown in Fig. 12: blue line: Fig. 12(a), Fig. 14. Average stress-deformation of the samples shown in Fig. 12 with shaded 115 50 orange line: Fig. 12(b), green line: Fig. 12(c), red line: Fig. 12(d), black line: Fig. 12(e). bounds of standard deviation. 116 ﬁ 51 (For interpretation of the references to colour in this gure caption, the reader is 117 referred to the web version of this paper.) 52 Table 3 118 53 The percentage of (standard deviation/average) for some representative values of 119 54 deformation gradient for samples shown in Fig. 12. 120 55 material when the macroscopic stress and strain measures 121 Deformation gradient, F11 1,01 1,02 1,05 1,08 1,12 1,15 56 converge. 122 57 Standard deviation/average (%) 0.73 1.41 4.70 4.99 4.99 4.95 123 58 124 59 5.1. RT-PS with 10% of rubbery particles 125 60 126 Fig. 8 shows three micro-structural samples with 10% of 61 than 10 μm according to the introduced criteria. Thus, the initial 127 inclusion volume fraction and different sizes namely, 10 μm, 62 volume element size to start the analysis is chosen to be 10 μm. 128 63 15 μm and 20 μm. 129 64 Five different morphologies for inclusion volume fraction equal 130 65 Remark 3. Preliminary simulations with micro-structural samples to 10% and volume element size equal to 10 μm are shown in 131 66 smaller than 10 μm revealed that the RVE size could not be smaller Fig. 9. For each volume size, a trial and error algorithm is employed 132

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1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 Fig. 15. Different realizations for inclusion volume fraction equal to 10% and volume element size equal to 20 μm. 98 33 99 34 100 35 1,0E+08 1,0E+08 101 36 102 37 103 8,0E+07 8,0E+07 38 104 39 105 6,0E+07 40 6,0E+07 106 41 107 42 4,0E+07 108 4,0E+07 43 109 44 2,0E+07 110 45 2,0E+07 111 First invariant of the stress tensor (Pa)

46 0,0E+00 First invariant of the stress tensor (Pa) 112 47 1 1,03 1,06 1,09 1,12 1,15 113 0,0E+00 48 Deformation gradient, 1,00 1,03 1,06 1,09 1,12 1,15 114 49 Fig. 16. Stress-deformation of the samples shown in Fig. 15: blue line: Fig. 15(a), Deformation gradient, 115 50 orange line: Fig. 15(b), green line: Fig. 15(c), red line: Fig. 15(d), black line: Fig. 15(e). 116 Fig. 17. Average stress-deformation of the samples shown in Fig. 15 with shaded (For interpretation of the references to colour in this figure caption, the reader is 51 bounds of standard deviation. 117 52 referred to the web version of this paper.) 118 53 119 54 120 55 121 56 to compute different radii of inclusions, within the range Table 4 122 57 0.7–1.0 μm, such that the desired inclusion volume fraction is The percentage of (standard deviation/average) for some representative values of 123 deformation gradient for samples shown in Fig. 15. 58 achieved. This procedure attempted to mimic the process of rub- 124 59 125 ber toughening, which is usually performed using sub-micron Deformation gradient, F11 1,01 1,02 1,05 1,08 1,12 1,15 60 126 sized rubbery particles with different radii. Once the radii of all 61 Standard deviation/average (%) 0.85 2.28 6.76 7.86 8.99 9.33 127 inclusions for each volume size and volume fraction are defined, 62 128 five realizations are generated using a random sequential addition 63 129 64 algorithm that places the same inclusions within the domain. This Remark 4. As pointed out by Kanit et al. [24], the RVE size could 130 65 strategy guarantees that all realizations have exactly the same be associated with the defined precision and also the number of 131 66 inclusion volume fraction. realizations. We considered five realizations for each inclusion 132

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1 1,0E+08 Five different morphologies for inclusion volume fraction equal 67 2 to 10% and volume element size equal to 15 μm are shown in 68

3 8,0E+07 Fig. 12. The same loading is applied to these realizations and the 69 4 stress-deformation curves are given in Fig. 13. Fig. 14 shows the 70 5 average response of the samples depicted in Fig. 12 with the upper 71 6 6,0E+07 and lower bounds of standard deviation. As expected, making a 72 7 comparison between Figs. 10 and 13 and also between Figs. 11 and 73 8 4,0E+07 14 shows that by increasing the volume element size from 10 μm 74 9 to 15 μm, the deformation behaviour of different realizations are 75 10 closer to each other. 76 2,0E+07 11 For the samples in Fig. 12, the ﬁrst criterion is satisﬁed, i.e. the 77

12 First invariant of the stress tensor (Pa) values of the standard deviation during deformation are less than 78 13 0,0E+00 10% of the average behaviour. The values of the standard devia- 79 1 1,03 1,06 1,09 1,12 1,15 14 tion/average versus deformation gradient are given in Table 3. 80 Deformation gradient, 15 The next stage consists in generating bigger samples in order to 81 16 Fig. 18. Average stress-deformation of the samples shown in Figs. 12 and 15. Solid proceed with the determination of the RVES. Five different reali- 82 ¼ μ ¼ μ 17 line: volume element size 20 m and dashed line: volume element size 15 m. zations with volume element size equal to 20 μm are shown in 83 18 Fig. 15. The respective stress-deformation curves are shown in 84 19 8 Fig. 16. Again, it is required to check if the first criterion is satisfied 85 20 7 for the samples with volume element size equal to 20 μm. The 86 21 87 6 average response of the samples shown in Fig. 15 with the bounds 22 88 5 of standard deviation are given in Fig. 17. Table 4 shows that the 23 values of standard deviation are within the bounds defined, 10% of 89 24 4 90 the average. Hence, the second criterion should be checked. In this 25 91 Error (%) 3 study, 10% difference between the average behaviour of two 26 92 2 volume element size is allowed. In other words, if the average 27 93 behaviour of samples with 15 μm of volume element size, shown 28 1 94 in Fig. 12, are within 10% difference to the average behaviour of 29 0 95 samples with 20 μm of volume element size, the second criterion 30 1,00 1,02 1,03 1,05 1,06 1,08 1,09 1,11 1,12 1,14 1,15 96 is satisfied too. Fig. 18 shows the average responses of both series 31 Deformation gradient, 97 of samples. In order to appreciate the evolution of the difference 32 Fig. 19. Difference between the behaviour of samples shown in Figs. 12 and 15. 98 between the two averages, Fig. 19 shows the measure of the dif- 33 99 ference, i.e. error, between the average behaviour of samples 34 100 shown in Figs. 12 and 15 versus deformation gradient. In Fig. 19, 35 101 volume fraction and volume element size, as considered by other the error is computed by: 36 authors (see e.g. [28]). Apparently, by increasing the number of 102 37 20 μm 15 μm 103 samples and also reducing the errors defined above (in the two I1ðPÞ I1ðPÞ 38 ð%Þ¼ : ð Þ 104 criteria), more accurate RVE size will be obtained. Error 20 μm 100 44 39 I1ðPÞ 105 40 Fig. 10 shows stress-deformation curves for different realiza- 106 According to the values of the average deformation behaviours of 41 tions of samples with 10% of rubbery particles and volume ele- 107 the two series of samples and also the defined criteria and algo- 42 ment size equal to 10 μm. In order to proceed with the determi- 108 rithm, it can be concluded that for Rubber Toughened Polystyrene 43 nation of the RVES, it is required to check whether or not the first 109 (RT-PS) with 10% of rubbery particles, the size of the Representa- 44 criterion is satisfied. Fig. 11 shows the average response of the 110 tive Volume Element (RVES) can be considered as 15 μm. 45 samples shown in Fig. 9 together with upper and lower bounds of 111 46 standard deviation, defined respectively by: Remark 5. The rather time consuming and cumbersome RVE size 112 47 determination process is due to the finite deformations. From 113 48 Upper Bound ¼ I ðPÞþS ð Þ; Lower Bound ¼ I ðPÞS ð Þ: ð43Þ 114 1 I1 P 1 I1 P Fig. 10, it can be obviously realized that within small deformation, 49 still in the elastic regime, all realizations depict the same beha- 115 50 In this study, if the standard deviation is within 10% of the 116 average response, it is considered acceptable. In other words, if viour. It can also be perceived that even smaller volume element 51 sizes could be satisfactory for small deformations. 117 52 the standard deviation of the deformation behaviour, through the 118 53 whole deformation, is less than 10% of the average deformation As stated before, the rubbery particles are considered as voids 119 54 behaviour, the first criterion is satisfied. For the first series of in this study due to the internal cavitation of the particles during 120 55 samples with inclusion volume fraction equal to 10% and volume tensile deformation. Thus, the material considered here is, in fact, 121 56 element size equal to 10 μm, the first criterion is not satisfied. porous PS. It is known that porosity magnifies the post-yield 122 57 This means that during the deformation process, the standard softening of polymers. Considering the stress-deformation curves 123 58 deviation is not always less than 10% of the average deformation of the samples with different sizes, Figs. 10, 13 and 16, it is possible 124 59 behaviour. Table 2 gives the values of the percentage of the to conclude that the differences appear mainly after the elastic 125 60 standard deviation to the mean value as a function of the domain, during softening. Due to the aforementioned reasons, the 126 61 deformation gradient at some representative deformation gra- RVE size determined is, most likely, a representative size for other 127 62 dients. According to the proposed algorithm, shown in Fig. 7,the loading conditions as well. Therefore, the same methodology could 128 63 next step is to generate samples with the same inclusion volume be used to define such size using eventually a more appropriate 129 64 fraction and a bigger size. deformation measure. 130 65 131 66 132

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1 Remark 6. In this study the effect of inclusion size range on the in a narrow band, while the rest of the material experiences 67 2 RVES is not studied. Stroeven et al. [29] investigated the effect of unloading, is attenuated and pathological dependence on mesh 68 3 particle size distribution on the RVES. They concluded, considering discretization is alleviated. To confirm this, some of the micro- 69 4 peak load and dissipated energy as driving parameters, that structural samples in this study were spatially discretized with 70 5 changing particle sizes does not significantly change the RVES. 71 6 72 7 5.2. RT-PS with 15% of rubbery particles 73 8 8,0E+07 74 9 The next volume fraction of rubbery particles to be con- 75 10 sidered is 15%. Fig. 20 shows five different realizations for 15% 76 11 inclusion volume fraction and 10 μmvolumeelementsize.The 6,0E+07 77 12 respective stress-deformation curves are given in Fig. 21.In 78 13 order to check the first criterion, if the standard deviation is 79 14 within the 10% of the average response, Fig. 22 presents the 4,0E+07 80 15 average response of the samples given in Fig. 20 together with 81 16 the bounds of standard deviation of the samples responses. 82 17 Table 5 shows the percentage of standard deviation/average 2,0E+07 83 18 through the deformation. The criterion is checked and it is not 84 19 85 satisfied. The next stage is to generate samples with the same First of the Invariantstress tensor (Pa) 20 fi 0,0E+00 86 inclusion volume fraction and a bigger size. Fig. 23 shows ve 1 1,03 1,06 1,09 1,12 1,15 21 87 different morphologies for 15% inclusion volume fraction and Deformation gradient, 22 15 μmvolumeelementsize.Fig. 24 depicts stress-deformation 88 23 curves for different realizations of samples with 15% inclusion Fig. 21. Stress-deformation of the samples shown in Fig. 20: blue line: Fig. 20(a), 89 24 μ orange line: Fig. 20(b), green line: Fig. 20(c), red line: Fig. 20(d), black line: Fig. 20 90 volume fraction and 15 m volume element size. The average fi 25 (e). (For interpretation of the references to colour in this gure caption, the reader 91 response of the samples presented in Fig. 23 with the bounds of is referred to the web version of this paper.) 26 standard deviation is given in Fig. 25. Table 6 gives the values of 92 27 standard deviation/average through the deformation. The stan- 93 8,0E+07 28 dard deviation of the behaviour of the samples is within the 94 29 defined 10% of the average. Samples with volume element size 95 30 equal to 20 μm and inclusion volume fraction equal to 15% are to 96 31 be generated in order to check the first and second criterion. Five 6,0E+07 97 32 different morphologies for inclusion volume fraction equal to 98 33 15% and volume element size equal to 20 μm are shown in 99 34 Fig. 26. Fig. 27 depicts the respective stress-deformation curves. 4,0E+07 100 35 Fig. 28 shows the average response of the samples presented in 101 36 Fig. 26 with the bounds of standard deviation. Table 7 depicts the 102 37 103 standard deviation/average for different values of deformation 2,0E+07 38 104 gradient. The first criterion is satisfied for this series of samples

39 First invariant of the stress tensor (Pa) 105 too. The second criterion should be checked. Fig. 29 depicts the 40 106 average response of the samples shown in Figs. 23 and 26. 0,0E+00 41 1,00 1,03 1,06 1,09 1,12 1,15 107 Fig. 30 shows the difference between the average behaviour of 42 Deformation gradient, 108 samples depicted in Figs. 23 and 26. Based on the deﬁned factors 43 109 for determination of the RVES and also considering the values of Fig. 22. Average stress-deformation of the samples shown in Fig. 20 with shaded 44 bounds of standard deviation. 110 the average deformation behaviours of the two series of samples, 45 111 the RVES for Rubber Toughened Polystyrene (RT-PS) with 15% of 46 112 rubbery particles could be chosen as 15 μm. 47 113 Table 5 ﬁ 48 Remark 7. In this work a nite strain elasto-viscoplastic con- The percentage of (standard deviation/average) for some representative values of 114 49 stitutive model, described in Section 3, is employed to model the deformation gradient for samples shown in Fig. 20. 115 50 non-linear behaviour of Polystyrene (PS). Since the viscoplastic 116 51 formulation includes rate effects, which implicitly introduce a Deformation gradient, F11 1,01 1,02 1,05 1,08 1,12 1,15 117 52 118 characteristic length through the viscosity, it acts as a regulariza- Standard deviation/average (%) 3.50 7.36 12.48 13.87 14.61 15.17 53 tion method. Therefore, the phenomenon of localization of strains 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 Fig. 20. Different realizations for inclusion volume fraction equal to 15% and volume element size equal to 10 μm. 132

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1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 Fig. 23. Different realizations for inclusion volume fraction equal to 15% and volume element size equal to 15 μm. 98 33 99 34 8,0E+07 8,0E+07 100 35 101 36 102 37 6,0E+07 6,0E+07 103 38 104 39 105 40 4,0E+07 4,0E+07 106 41 107 42 108 43 2,0E+07 2,0E+07 109 44 110

45 First invariant of the stress tensor (Pa) 111 First Invariant of the stress tensor (Pa) 0,0E+00 46 0,0E+00 112 47 1 1,03 1,06 1,09 1,12 1,15 1,00 1,03 1,06 1,09 1,12 1,15 113 48 Deformation gradient, Deformation gradient, 114

49 Fig. 24. Stress-deformation of the samples shown in Fig. 23: blue line: Fig. 23(a), Fig. 25. Average stress-deformation of the samples shown in Fig. 23 with shaded 115 50 orange line: Fig. 23(b), green line: Fig. 23(c), red line: Fig. 23(d), black line: Fig. 23 bounds of standard deviation. 116 51 (e). (For interpretation of the references to colour in this figure caption, the reader 117 is referred to the web version of this paper.) 52 Table 6 118 53 The percentage of (standard deviation/average) for some representative values of 119 54 finer finite element meshes and the difference in the results was deformation gradient for samples shown in Fig. 23. 120 55 negligible. 121 Deformation gradient, F 1,01 1,02 1,05 1,08 1,12 1,15 56 11 122 57 Remark 8. One may consider more intermediate dimensions, Standard deviation/average (%) 1.29 3.48 6.34 7.96 9.14 9.62 123 58 between the considered sizes in this study, to obtain a more pre- 124 59 cise RVE size. 125 60 126 61 Remark 9. The authors also applied the two criteria for RVES stress tensor) were obtained for both cases of RT-PS with 10% and 127 62 determination, introduced at the beginning of this section, to the 15% of rubbery particles. 128 fi fi – 63 rst component of the rst Piola Kirchhof stress tensor (P11), 129 64 which is basically the stress component in the loading direction, It should be emphasized that the statistical-numerical approach 130 65 and similar results (as obtained using the first invariant of the for RVES determination, introduced in this study, has certain 131 66 132

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1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 Fig. 26. Different realizations for 15% inclusion volume fraction and 20 μm volume element size. 98 33 99 34 8,0E+07 8,0E+07 100 35 101 36 102 37 6,0E+07 6,0E+07 103 38 104 39 105 40 4,0E+07 4,0E+07 106 41 107 42 108 43 2,0E+07 2,0E+07 109 44 110

45 First invariant of the stress tensor (Pa) 111 First Invariant of the stress tensor (Pa) tensor the stress of First Invariant 0,0E+00 46 0,0E+00 112 1 1,03 1,06 1,09 1,12 1,15 47 1,00 1,03 1,06 1,09 1,12 1,15 113 Deformation gradient, 48 Deformation gradient, 114 49 Fig. 27. Stress-deformation of the samples shown in Fig. 26: blue line: Fig. 26(a), Fig. 28. Average stress-deformation of the samples shown in Fig. 26 with shaded 115 50 orange line: Fig. 26(b), green line: Fig. 26(c), red line: Fig. 26(d), black line: Fig. 26 bounds of standard deviation. 116 (e). (For interpretation of the references to colour in this ﬁgure caption, the reader 51 117 is referred to the web version of this paper.) 52 Table 7 118 53 The percentage of (standard deviation/average) for some representative values of 119 54 deformation gradient for samples shown in Fig. 26. 120 55 advantages compared to the previously introduced methods: 121 Deformation gradient, F11 1,01 1,02 1,05 1,08 1,12 1,15 56 122 57 Using this approach it is not required to investigate different Standard deviation/average (%) 1.46 2.33 3.06 3.79 4.56 6.27 123 58 regimes of the deformation, namely elastic, yield and softening, 124 separately and thus it facilitates the statistical study. 59 125 60 All possible criteria (e.g. slope of the graph in different regimes 6. Conclusions 126 61 of the deformation, the strain at which the material yields and 127 62 dissipated energy) are covered using this approach. In this paper, some numerical examples on Rubber Toughened 128 63 Polystyrene (RT-PS), with two different levels of Inclusion Volume 129 64 Based on the analysis conducted, we conclude that for materials which Fraction, were presented. Having different morphologies showed 130 65 show a moderate softening response, an RVE size can be determined. the necessity of determination of the RVE Size. Due to moderate 131 66 132

Please cite this article as: S.M. Mirkhalaf, et al., Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at ﬁnite strains, Finite Elem. Anal. Des. (2016), http://dx.doi.org/10.1016/j.ﬁnel.2016.05.004i 14 S.M. Mirkhalaf et al. / Finite Elements in Analysis and Design ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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