Rate theory of nonlocal gradient damage-gradient viscoinelasticity

A general concept for the analysis of damage evolution in heterogeneous media is proposed. Since macroscopic failure is governed by physical mechanisms on two different length-scale levels, the macro- and mesolevel, we introduce a 6-dimensional kinematical model with manifold structure accounting for discontinuous fields of microcracks, microvoids and microshear bands. As point of departure a variational functional is introduced with a Lagrangian density depending on macro- and microdeformation gradients and of a damage variable representing scalar-, vector- and/or tensor-type quantities. To derive the equations of motion for viscoinelastic damage evolution on macro- and mesolevel, we introduce into the Lagrangian the macro- and microdeformation gradients, damage variable and also their gradients and time rates. The equations of motion on macro- and mesolevel are derived for non-equilibrium states. We assume that the Lagrangian can be split into two contributions, a time-independent and a time-dependent one which can be identified with the Helmholtz free energy and a dissipation potential. This split of the Lagrangian can be used to decompose the stresses and forces into reversible and irreversible ones. The latter can be considered as dissipative driving stresses and driving forces, respectively, on defects. The model presented in this paper can be considered as a framework, which enables to derive various nonlocal and gradient, respectively, damage theories by introducing simplifying assumptions. As special cases a scalar damage and a solid-void model are considered.