Theoretical framework for coupling of nonlocal damage and viscoplasticity for dynamic localization problems

Conventional continuum mechanics models of inelastic deformation processes are size scale independent. In contrast, there is considerable experimental evidence that inelastic flow in crystalline materials is size-dependent. At present there is no generally accepted framework for analyzing the size-dependent response of an inelastically deforming material. This is due to the fact that very limited quantitative numerical comparisons with experimental results were conducted, particularly, in localization problems. As soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computation is considerably affected by the mesh resolution. Several localization limiters that incorporate length scale measures in the constitutive relations have been successfully used in the literature to remove the inherent mesh sensitivity of the numerical failure predictions and to solve size scale dependency. This study develops a general consistent and systematic framework for the analysis of heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic visco-damage for dynamic problems. Since the material macroscopic thermo-mechanical response under dynamic loading is governed by different physical mechanisms on the meso- and macro-scale levels, the proposed model is introduced with manifold structure accounting for discontinuous fields of dislocation, crack, and void interactions. The gradient theory of rate-independent plasticity and rate-independent damage that incorporates macro-scale interstate variables and their higher-order gradients is generalized here for rate-dependent plasticity and rate-dependent damage to properly describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related rate- and temperature dependent materials. The idea of bridging length-scales is made more general and complete by introducing spatial higher-order gradients in the temporal evolution equations of the internal state variables that describe hardening in coupled viscoplasticity and visco-damage models, which are considered here dependent on their local counterparts. Computational issues concerned with the current gradient-dependent formulation of initial-boundary value problems are introduced in a finite element context. This framework is used to study among others the effect of material length scales on the localization of inelastic flow in shear bands.