Determination of the Eshelby tensor in mean-field schemes for evaluation of mechanical properties of elastoplastic composites

The conventional Eshelby solution based mean-field schemes may dramatically over-estimate the stress in composites consisting of elastoplastic matrix and elastic reinforcement particles. A new and rational approach is presented to determine the Eshelby tensor of elastoplastic medium to solve this problem. We assume that, during an increment of loading in the Eshelby problem for an elastoplastic medium (EPM), the mechanical property is homogeneous, isotropic and incrementally linearized, and introduce a reference elastic medium (REM) with the identical configuration and elastic property of the EPM studied. For a purely traction boundary value problem, the distribution of the stress increment as well as the constrained elastic strain increment in the EPM can be approximated with that in the REM if both of them are subjected to exactly identical traction boundary condition. As the surface traction increment induced by the eigenstrain increment in an ellipsoidal subdomain Ω in the EPM is applied to the REM, the Eshelby tensor of the REM, Se, can be easily derived with the regular procedure of the traditional Eshelby problem. Then, the Eshelby tensor for the EPM can easily be obtained as S=[(Le)-1:L]-1:Se:[(Le)-1:L], where Le and L are the elastic and the tangent elastoplastic moduli of the EPM respectively. The developed Eshelby tensor is embedded in the conventional Mori-Tanaka scheme to predict the responses of particulate composites subjected respectively to tensile, shear, combined biaxial tensile and shear, and cyclic tensile-compressive, and non-proportional tensile and shear loadings. The comparison with the results obtained with the approaches using isotropic approximation of the tangent elastic modulus, and with the reference ones obtained with full-field FE analyses showed satisfactory agreement, demonstrating the validity of the proposed approach. The approach to determine the Eshelby tensor for elastoplastic media has the following distinct advantages: (1) it employs the regular anisotropic tangent modulus and has a unified form in both elastic and elastoplastic cases; (2) it can easily be embedded in the conventional mean-field homogenization schemes (MFHs) and used for the analyses of effective properties of composites subjected to complex loading histories; (3) it possesses high computational efficiency, since the Eshelby tensor can be simply computed using the elastic and elastoplastic properties of composites with no need of time-consuming numerical ellipsoidal integral in each increment.