Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory

Abstract

This study develops a general consistent and systematic framework for the analysis of heterogeneous media that assesses a strong coupling between rate-dependent plasticity and anisotropic rate-dependent damage for dynamic problems within the framework of thermodynamic laws and gradient theories. The proposed formulation includes thermo-elasto-viscoplasticity (rate-dependent plasticity) with anisotropic thermo-viscodamage (rate-dependent damage); a dynamic yield criterion of a von Mises type and a dynamic damage growth criterion; the associated flow rules; thermal softening; non-linear strain hardening; strain-rate hardening; strain hardening gradients; and strain-rate hardening gradients. Since the material macroscopic thermomechanical response under dynamic loading is governed by different physical mechanisms on the meso- and macroscale levels, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic flow) and crack and void interactions (damage growth). The gradient theory of rate-independent plasticity and rate-independent damage that incorporates macroscale 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 viscodamage models, which are considered here physically and mathematically related to their local counterparts. Furthermore, the constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material; that is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses and strains are replaced by their effective ones. In addition, computational issues concerned with the current gradient-dependent formulation of initial-boundary value problems are introduced in a finite element context. A weak (virtual work) formulation of the non-local dynamic viscoplastic and viscodamage conditions is derived, which can serve as a basis for the numerical solution of initial boundary value problems in the sense of the finite element method. Explicit expressions for the generalized tangent stiffness matrix and the generalized nodal forces are given. The model presented in this paper can be considered as a feasible thermodynamic approach that enables one to derive various coupled gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.