Homogenization of rock-like materials with plastic matrix based on an incremental variational principle

A large class of rock-like materials are composed of a plastic solid matrix in which various inclusions and pores are embedded. This paper is devoted to determine macroscopic inelastic responses of such materials by a nonlinear homogenization procedure. The plastic behavior of solid matrix is described by a plastic model based on the pressure-dependent Drucker-Prager criterion. The plastic strain field of solid matrix is divided into a volumetric part and a shear part. Unlike most mean-field methods previously developed, the strain field in the solid matrix is non-uniform and this nonuniform field is taken into account by using an incremental variational model. The whole loading history is divided into a limit number of increment. For the sake of simplicity, the behavior of solid matrix is first assumed to be elastic-perfectly plastic at each loading increment With the help of a time derivative approximation, the local incremental potential of the solid matrix is deduced. By considering the effect of inclusions and pores, the effective incremental potential of the heterogeneous composite is determined and estimated with the help of a linear comparison material. The macroscopic stress of the composite is finally estimated from the effective incremental potential. The accuracy of the proposed model is assessed by a series of comparisons with reference results obtained from direct finite element simulations respectively for inclusion-reinforced and porous materials. Finally, by assuming that the general form of incremental variational model remains valid when the solid matrix exhibits an isotropic plastic hardening, the proposed model is extended by updating the value of frictional coefficient of solid matrix and keeping it as a constant at each loading increment The proposed model in this case is also well validated by comparisons with finite element reference results. Moreover, as examples of application, the model is used to simulate laboratory tests performed on a cement mortar and a typical porous sandstone.