A thermodynamics- and mechanism-based framework for constitutive models with evolving thickness of localisation band

Abstract

Localised failure in geomaterials invalidates the assumption of homogeneous deformation that constitutive models based on continuum mechanics rest on. In such cases, the deformation and nonlinear processes inside the localisation zone dominate the inelastic response of the material, while the material outside this zone usually undergoes negligible inelastic or even elastic deformation. As a consequence, internal variables representing micromechanical failure processes should better be defined inside the localisation zone, not averaged over the whole volume element containing it. In this study, we propose a thermodynamics-based framework for constitutive models that take into account the transition from homogenous to localised deformation. Two spatial scales involved in the mechanisms of localised failure, macro scale of the considered volume element and smaller scale of the localisation zone, are included in the formulation and derived constitutive models. This separation of spatial scales is combined with enrichments of the constitutive kinematics for the integration of three constitutive relationships describing the behaviour of the materials inside and outside the localisation zone, and the evolving size of this zone. As a result, the internal variables are associated with their own spatial zones, instead of being averaged over the whole volume element like in classical continuum approaches. The gradual transition from homogenous to localised deformation is represented by the onset and evolution of the thickness of the localisation band, both of which appear naturally in the proposed formulation. The obtained model therefore consists of both size and orientation of the localisation band, and three constitutive relationships connected through the equilibrium condition across the boundary of the localisation zone. They help provide a smooth transition from homogeneous to localised failure. Numerical examples show promising features of the proposed approach in connecting the macro behaviour with the underlying evolution of the localisation zone.