Gradient plasticity modeling of geomaterials in a meshfree environment. Part I: Theory and variational formulation

Deformation and strength behavior of geomaterials in the pre- and post-failure regimes are of significant interest in various geomechanics applications. To address the need for development of a realistic constitutive framework, which allows for an accurate simulation of pre-failure response as well as an objective and meaningful post-failure response, a strain gradient plasticity model is formulated by incorporating the spatial gradients of elastic strain in the evolution of stress and gradients of plastic strain in the evolution of the internal variables. In turn, gradients of only kinematic variables are included in the constitutive equations. The resulting constitutive equations along with the balance of linear momentum for the continuum are cast as a coupled system of equations, with displacements and plastic multiplier appearing as the primary unknowns in the final governing integral equations. To avoid singular stress fields along element boundaries, a finite element discretization of the governing equations would require C² continuous displacements and C¹ continuous plastic multiplier, which is undesirable from a numerical implementation point of view. This issue is naturally resolved when a meshfree discretization is used. Hence the developed model is formulated within the framework of a meshfree environment. The new constitutive model allows an analysis of grain size effects on strength and dilatancy of rocks. The role and effectiveness of the new gradient terms on regularizing the underlying boundary value problems of geomechanics beyond the initiation of strain localization will be assessed in a future paper.