A two-dimensional ordinary state-based peridynamic model for plastic deformation based on Drucker-Prager criteria with non-associated flow rule

Abstract

The plastic behaviors of geomaterials often obey non-associated flow rule. In this paper, a two-dimensional (2D) ordinary state-based peridynamic (OSB-PD) increment model with shear deformation based on Drucker-Prager (D-P) criterion with non-associated flow rule is proposed to study the plastic behavior of geomaterials. An incremental form of peridynamic (PD) strain energy density for plane elastic problems is first established. The material parameters are determined when the incremental PD strain energy density is equal to the incremental strain energy density in classical mechanics. The PD force density for the OSB-PD theory with shear deformation in plane elastic cases is derived accordingly. The relationship between the incremental second invariant of the stress tensor dJ2, the first invariant function of stress tensor dI1 in the classical mechanics and PD are established in this paper, and the Drucker-Prager yield surface is expressed in the framework of peridynamics. To avoid the excessive plastic dilation, the non-associated flow rule is proposed, in which isotropic and deviatoric plastic elongation are employed to describe the plastic behaviors of a bond. Also, the equivalent stress and equivalent plastic strain for the proposed PD model are given. An integration algorithm for the proposed increment model is developed. Numerical examples including circular opening in rocks are performed, and the PD results are compared with that obtained by the finite element method or analytical solution. Those numerical simulations have verified the validity of the proposed PD model to predict the elasto-plastic behavior of geomaterials.