A review of analytical micromechanics models on composite elastoplastic behaviour

A Review of Analytical Micromechanics Models on Composite Elastoplastic Behaviour Procedia Engineering 173 ( 2017 ) 1283 – 1290 Available online at www sciencedirect com 1877 7058 © 2017 Published by[.]

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 173 (2017) 1283 – 1290 11th International Symposium on Plasticity and Impact Mechanics, Implast 2016 A review of analytical micromechanics models on composite elastoplastic behaviour Yanchao Wanga,b, Zhengming Huanga* a Department of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China b Department of Mechanical Engineering, Northwestern University, Evanston, ILˈUSA ῐ bstract Elasto-plastic behavior of composite structures is a key problem in light-weight application, which has been investigated vastly for decades Most approaches for composites elasto-plastic simulation can be split into three categories: 1) numerical methods also known as computational micromechanics methods, 2) variational approaches for upper and lower bounds, and 3) analytical homogenization models In this study only focuses on the analytical models Two steps are essential to establish an analytical model, i.e choosing a suitable linear elastic homogenization models and a rational linearization theory so that linear elastic models can be extended to nonlinear elasto-plastic regime Besides, considering the situation that isotropic materials may become anisotropic due to plastic deformation, three approaches are found in the literature to deal with the anisotropic properties and corresponding Eshelby’s tensor In summary, a comparative study is made among nine different models to evaluate their capabilities on elasto-plastic behavior simulation Experiment data of a series of unidirectional composites under different loading cases are used for the purpose of comparison © 2017 2016Published The Authors Published by Elsevier Ltd © by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: analytical models; micromechanics; elasto-plastic behavior; fibrous composites Introduction How to evaluate elasto-plastic behaviour of a fibrous composite structure accurately and efficiently is an ongoing important and difficult point in the community of composite mechanics As the demand of light-weight design * Corresponding author Tel.:+86-021-65985373 E-mail address: huangzm@tongji.edu.cn 1877-7058 © 2017 Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 doi:10.1016/j.proeng.2016.12.159 1284 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 increasing, this problem becomes more urgent Numerous work regarding composites’ elasto-plastic behaviour have been done for decades Mostly, analytical models are established on basis of homogenization theories The critical problem of composites’ homogenization theories is to obtain the stress/strain concentration tensor to connect averaged stress/strain tensors in constituent matrix and fiber Many elastic homogenization theories have been established in literature The existed elastic theories can be applied to elasto-plastic problems with the help of linearization theories An incremental method was presented by Hill[1] that the matrix behaviour can be seen as linear elastic in each loading step as long as the load step is small enough In that case, elastic theories are applicable in each loading step However, since the isotropic matrix behaves anisotropic when it undergoes plastic deformation, the original Eshelby’s solution for isotropic materials should be modified It has been proved that when a composite subjected to monotonous proportional loads the matrix can be seen as isotropic approximately in each loading step where the instantaneous stiffness tensor of matrix and the instantaneous Eshelby’s tensor can be obtained by replacing the elastic quantities such as Young’s modulus and Poisson ratio with tangent or secant ones[2] This method is named as “isotropic matrix method” in this work The other way to deal with the Eshelby tensor in this work is named as “anisotropic Eshelby’s tensor method” In this way, the instantaneous anisotropic stiffness of matrix is calculated by a chosen plastic flow rule while the instantaneous Eshelby’s tensor is given by a complex integration in terms of Green’s function[3] Additionally, another approximate way was proposed by Doghri[4] that the instantaneous Eshelby’s tensor is obtained by replacing the elastic Poisson in the original Eshelby’s tensor for isotropic materials with tangent or secant one whereas the matrix instantaneous stiffness remains anisotropic calculated from a chosen plastic flow rule Apart from the three methods in regard to the Eshelby’s tensor, Huang[5] presented a different approach for composites elasto-plastic problems on basis of tangent increment method In Huang’s bridging model, the elements of the instantaneous bridging tensor are divided into dependent and independent variables The dependent variables can be derived from the symmetric condition of the effective compliance tensor of a composite The independent variables in instantaneous bridging tensor have the same form as the ones in elastic bridging tensor as long as the elastic quantities are replaced with tangent ones In elastic range, it has been proved that the bridging tensor can also be derived from the two-phase CCA model[6-8] Besides, different from the models established basing on incremental method, Dvorak[9] presented a method of transformation field analysis In Dvorak’s work, the plastic strains of constituent materials are considered as eigenstains In this way, the elasto-plastic problems can be solved in the framework of elasticity Moreover, with the concept of pre-stress, Masson et al[10] proposed a non-incremental tangent formulation named as affine formulation to predict the nonlinear properties of composites As a semi-empirical model, the Chamis formulations[11] were modified to solve the composites’ nonlinear problems In this work, nine different elasto-plastic models are presented and compared The first six models are MoriTanaka model based, i.e the isotropic matrix method, the anisotropic Eshelby tensor method, the isotropic Eshelby tensor method, secant formulations, the transformation fields analysis and the affine formulations The other three models are respectively, self-consistent model, bridging model and Chamis model All the models are incremental method based except the transformation fields analysis, the affine formulations and the secant formulations The nonlinear stress-strain curves for several UD composites under different loading cases are predicted by the nine different elasto-plastic models Comparison between prediction results and experiment data are made General Framework Fig Schematic of RVE for UD composites 1285 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 For a two-phase composite RVE (representative volume element) shown in Fig 1, there exist: f f m m d  V f d f  V m d m , d  V d  V d (1) In this work, the f, m used in superscripts denote fiber and matrix Quantities without superscripts denote quantities for composites For a given composite, there exists a non-singular tensor to link stress/strain in each phase to external ones, like d r  Ar d , d r  B r d , r  f , m (2) r r where A ˈ B ˈr=f, m are, respectively, the instantaneous strain and stress concentration tensor for fiber and matrix Let M and L denote the instantaneous compliance and stiffness tensor, the effective stiffness and compliance tensor of a composite are expressed as: L  Lm  V f ( Lf  Lm ) A f , M  M m  V f ( M f  M m ) B f (3) From equation (3), it is found that once the instantaneous strain or stress concentration tensor is given, the effective properties of a composite can be determined easily Therefore, the general critical point of many homogenization theories such as the Mori-Tanaka model, self-consistent model and the generalized self-consistent model etc is how to determine the instantaneous stress/strain concentration tensor when extended to elasto-plastic range with the help of linearization theory Elasto-plastic theories 3.1 Tangent incremental Mori-Tanaka method The effective stiffness and compliance tensor is given by M tan  M m  tan  V f ( M T f  tan  [I  S M tan m  tan f  tan (L f  tan f  tan tan  M m  tan ) L f  tanT f  tan M m  tan (V m I  V f L f  tanT f  tan M m  tan ) 1 m  tan L 1 )] (4) (5) m  tan and M are, respectively, tangent form compliance tensors of composites, fiber and where M , M tan matrix The tangent compliance tensor of matrix depends on stress state and flow rule S is the instantaneous Eshelby’s tensor According to the solution to the instantaneous Eshelby tensor, three different methods are shown below (a) Isotropic Matrix method When a composite subjected to monotonic proportional load, the matrix can be considered as isotropic in each loading step Non-zero elements of the approximate compliance tensor are shown below m 1 , mtan  T , m  tan M iijj  m M ijij  m , i, j  1,2,3 m GT ET ET The corresponding Eshelby’s tensor follows m  tan  M iiii (6) T   2Tm  tan tan ,  , S 2211  S3311   m  2(1  Tm ) 2(1  T )   m tan tan  S3333  S 2222 S tan 2233 S tan 3322  1  2Tm  tan  1  2Tm  1 tan      , S S , 2323 3322 2(1  Tm )   2(1  Tm )   (7) (b) Anisotropic Eshelby tensor method When a composite subjected to complex load, the elasto-plastic material behaves anisotropic whose compliance tensor should be calculated from J2 flow rule The corresponding Eshelby tensor have to be calculated from 1286 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 integration of Green functions due to the anisotropy Expressions for the Green functions involved in the integration can be found in Ref[3] 1 2 m  tan Lmnkl  d  {Gimjn ( )  G j ( )}d 8 1 tan S ijkl  (8) (c) Isotropic Eshelby tensor method Doghri proposed that a better result can be obtained if the elasto-plastic material remains anisotropic and calculated from the J2 flow rule while the Eshelby tensor is given as equation (7) 3.2 Secant Mori-Tanaka method Formulations of secant Mori-Tanaka method can be easily obtained by rewriting equations (4) - (5) with secant quantities instead of tangent ones One thing should be noted that the Eshelby’s tensor involved in the secant MoriTanaka method is given by replacing the Poisson ration in the original Eshelby’s tensor used for isotropic materials with the secant one 3.3 Transformation field analysis Dvorak[9] presented a transformation field analysis in which the plastic strain is seen as eigenstrain The stressstrain relationship is give as   L    ˈ  M    p (9)  r  Lr  r  r ,  r  M r r   r  p , r  f , m r (10) r where L and M are the elastic stiffness and compliance tensor of constituent materials The effective elastic stiffness and compliance tensor of a composite L and M are calculated based on the Mori-Tanaka model The relationship between phase strain/stress, composite’s averaged strain/stress and the plastic strain in each phase is shown as followings r r rs s s  p  r  Ar    D rs s p ˈ   B   F L  , s  f , m, (11)  s s  p  V r [ B r ]T  r  p ,   V r [ Ar ]T r r r r (12) r where the elastic strain and stress concentration tensor A and B are calculated from elastic Mori-Tanaka rs rs model and keep unchanged throughout the loading procedure The influence tensor D and F can be found in Ref[9] 3.4 Affine formulations The affine formulations are proposed by Masson[10] to predict the nonlinear behaviour of composites The constitutive equation is given as   Ltan   ˈ   M tan   (13)  r  Lr tan r   r ˈ  r  M r  tan r   r , r  f , m (14) where  ˈ  , r  f , m are the so called pre-stress for a composite and its constituent material  ˈ r  r , r  f , m are the corresponding pre-strain The Ltan and M tan are calculated from the Mori-Tanaka model in each loading step The relationship of pre-stress/prestrain in each phase is given by   V r [ Ar tan ]T r ˈ   V r [ B r tan ]T r , r  f , m  r  r (15) 1287 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290  r  Ar tan  Ar tan P tan (   r ), r  f , m (16) It should be noted that the Eshelby’s tensor involved in the calculation of instantaneous stress or strain concentration tensor should be calculated with equation (8) 3.5 Tangent incremental self-consistent model A model of a single inclusion embedded into an infinite medium is employed in the self-consistent model The properties of the infinite medium are set to be effective properties of the composite Rewrite the homogenization equations in the incremental form: M tan  M mtan  V f ( M f tan  M mtan ) L f T tan M tan (17) T tan  [ I  S tan M tan ( Lf  tan  Ltan )]1 (18) tan tan on the right side of equation (74) depend on the effective properties of It should be noted that the T and S tan the composite Hence, self-consistent is implicit Besides, the instantaneous Eshelby’s tensor S in each loading step can be obtained from equation (8) 3.6 Tangent incremental bridging model The effective compliance tenor of a composite was expressed by Huang[5] in terms of bridging tensor [ M ij ]  (V f [ M ijf ]  Vm [ M ijmtan ][aijtan ])(V f [ I ]  Vm [aijtan ]) 1 , i, j  1,2,3 a    [aijtan ]       tan 11 tan 12 tan 22 a a a a 0 a 0 a11tan tan 13 tan 23 tan 33 tan 14 tan 24 tan 34 tan 44 tan 15 tan 25 tan 35 tan 45 tan 55 a a a a a a a a 0 a 0 a   a  a   a  a   a  tan 16 tan 26 tan 36 tan 46 tan 56 tan 66 (20) Em  Tf E11 tan tan tan a22  a22  a22    (1   ) (19) (21) ETm , (0    1) E22f (22) GTm , (0    1) (23) G12f The bridging parameters  and  can be determined by simple tension or shear test Empirically, the bridging parameters can be set as   0.4 ˈ   0.45 if experiment test is not available The off diagonal elements can be derived by symmetric condition of effective compliance tensor tan a55tan  a66    (1   ) M ijtan  M tan ji (24) 3.7 Chamis formulations The Chamis formulations are semi-empirical The equations for effective properties are given below E11  V f E11f  V m ETm , E22  E33  ETm /(1  V f (1  EE )) m T f 22 G23  GTm /(1  V f (1  GTm f G 23 )) , G12  G13  GTm /(1  V f (1  (25) GTm f G 22 )) 12  13  Tm  V f (12f  Tm ) , 23  V f 23f  V m (2Tm  12 E22 / E11 ) (26) (27) 1288 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 Comparison and discussion Experiment data of two kinds of UD composites subjected to various loading case are used to evaluate the prediction capability of different elasto-plastic models Detail information of these two composites can be found in Ref[12] Figure show the stress-strain curves predicted by different models for the two kinds of composites under longitudinal tension When the stress-strain curves along fiber direction are focused on, it can be found that all the prediction results agree well with experiment data and show nearly no nonlinearity However, when it comes to the transverse direction, the stress-strain curve given by the Mori-Tanaka model with isotropic Eshelby’s tensor shows great nonlinearity while results provided by other models show nearly no nonlinearity The reason to this phenomenon can be revealed by stresses predicted in phases shown in Table 1-2 The nonlinearity of stress-strain in transverse direction indicates that the absolute value of averaged stress predicted by the Mori-Tanaka model with isotropic Eshelby’s tensor is too high which will bring significant error when used to predict failure behaviour Fig Stress-strain curve of two UD composites under longitudinal tension load Table Averaged stresses in phases predicted by different models for IM7/8551-7 UD composite under longitudinal tension Longitudinal Tension (3200MPa) MT-Anisotropic Eshelby tensor MT-Isotropic Eshelby tensor MT-Isotropic Matrix Bridging Model SC-Anisotropic Eshelby tensor Transformation Fields Analysis Affine Formulations Secant Formulations Chamis Formulations IM7/8551-7 UD Composite Vf=60% σf (MPa) σ11 σ22 5291 -4 5775 518 5292 -3 5295 5288 -8 5292 -4.1 5288 -3.4 5293 -3 5286 -6 σ33 -4 518 -3 -8 -4.1 -3.4 -3 -6 σm(MPa) σ22 -778 12 5.2 4.5 9.2 σ11 64 -662 62 58 68 62 68.7 60.4 70.4 σ33 -778 12 5.2 4.5 9.2 Table Averaged stresses in phases predicted by different models for E-Gass/MY750 UD composite under longitudinal tension E-Glass/MY750 UD Composite Vf=60% σf (MPa) Longitudinal Tension (1300MPa) σ11 σ22 MT-Anisotropic Eshelby tensor 2107 -4 MT-Isotropic Eshelby tensor 2135 31 MT-Isotropic Matrix 2108 -3 Bridging Model 2111 SC-Anisotropic Eshelby tensor 2095 -11 Transformation Fields Analysis 2109 -4.1 Affine Formulations 2114 Secant Formulations 2108 -3 Chamis Formulations 2108 -4 σ33 -4 31 -3 -11 -4.1 -3 -4 σ11 89 48 88 83 107 86 79 87 88 σm(MPa) σ22 -46 16 -10 5 σ33 -46 16 -10 5 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 Fig and Fig show the stress-strain curves predicted by different models for UD composites under transverse compression and in plane shear Firstly, let us focus on the self-consistent model The self-consistent model gives good evaluation on transverse elastic modulus but relatively not accuracy enough for in plane shear modulus Moreover, the stress-strain curves predicted by the self-consistent model tends too stiff In other words, the selfconsistent model underestimates the effect of matrix plastic deformation on composite’s nonlinearity Secondly, let us pay attention to non-incremental models, i.e the secant formulations, transformation field analysis and the affine formulations All the three models give similar results which show basic trend of composites’ nonlinearity Some slightly waviness is observed for the three curves predicted The waviness is induced by the nature of nonincremental method In other words, the strain states calculated by non-incremental models in each loading step are independent of previous loading step, thereby producing strain jumps Eventually, the remaining incremental models give smoothly stress-strain curves but have discrepancy with each other It is hard to identify which model is the best with limit experiment data Besides, the yield stresses predicted by almost all the models are relative too high, particularly for the in-plane shear cases which indicate that stress concentration factors for different loading conditions must be developed and considered Fig Stress-strain curves of two UD composites under transverse compression load Fig.4 Stress-strain curves of two UD composites under in plane shear load 1289 1290 Yanchao Wang and Zhengming Huang / Procedia Engineering 173 (2017) 1283 – 1290 Conclusion Elastic homogenization models can be extended to elasto-plastic range with the help of the plastic flow rule and the linearization methods In this work, nine different elasto-plastic models including incremental models, nonincremental models, and different homogenization theories based models are shown Comparisons are made between prediction results of different models and experimental data The self-consistent model gives too stiff results while the plastic deformation in matrix predicted by Mori-Tanaka method with isotropic Eshelby’s tensor are much overestimated for the case of longitudinal tension What’s more, on the one hand the stress-strain curves predicted by some models in elastic range are still not good enough On the other hand, the yield stresses predicted by most of the models are relative high which indicate that stress concentration factors for different loading cases should be developed Acknowledgements Financial supports from the National Natural Science Foundation of China (Grant No 11272238, 11472192), Doctoral Fund of Ministry of Education of China (Grant No 20120072110036), the Fundamental Research Funds for the Central Universities (Grant No 1330-219-104), and the Chinese Scholarship Council are acknowledged References [1] R Hill, A self-consistent mechanics of composite materials, Journal of the Mechanics and Physics of Solids 13(4) (1965) 213-222 [2] W.G Tandon GP, A theory of particulate reinforced plasticity, Journal of Applied Mechanics (1988) [3] A Gavazzi, D Lagoudas, On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites, Computational Mechanics 7(1) (1990) 13-19 [4] O.A Doghri I, Homogenization of two-phase elasto-plastic composite materials and stes Study of tangent operators, cyclic plasticity and numerical algorithms, International 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