Rheological model for rock-type materials under large deformations

Abstract

In this paper the behaviour of rock-type materials is described with respect to the current configuration through rate-type constitutive equations, within the finite deformation framework, using the relative motion. The viscoplastic part of the constitutive equations concerns the long-time effects, i.e. the deformation by creep, while the instantaneous elastic response characterizes the short-time effect. The instantaneous elastic response expresses an appropriate objective time derivative of Cauchy stress tensor via the rate of strain, with the aid of fourth-order elastic stiffness tensor. Using mathematical arguments it follows that our new model extends, to finite strains, Cristescu’s rheological model Cristescu (1989)[2]. To solve boundary value problems for rock-type materials described by the proposed model, the appropriate variational equality for the velocity field, at a time t, is derived starting from the weak formulation of the incremental equilibrium problem. The rate form of the constitutive model is coupled with the variational equality, and serves to update the current state of the material.