Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids

This work outlines a theoretical and computational framework of finite inelasticity with length scales based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to inelasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance-type partial differential equations including micro-structural boundary conditions. This incorporates non-local dissipative effects based on length scales, which reflect properties of the material micro-structure. Typical examples are phase field theories, gradient damage and strain gradient plasticity. We outline unified minimization and saddle point principles for the evolution problem of first-order gradient-type standard dissipative solids. Particular emphasis is put on mixed multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These settings are needed for models with threshold functions formulated in the space of the driving forces, in particular for finite gradient plasticity. The central aim is to define constitutive rate-type and algorithmic incremental potentials, whose variational derivatives govern the coupled macro- and micro-balances in both the continuous as well as the time-discrete setting. Their existence underlines the inherent symmetry of standard dissipative solids. We demonstrate geometrically consistent constructions of these potentials for important classes of finite inelasticity, providing a fresh look on models of multiplicative and additive gradient plasticity of single crystals and amorphous materials. The potentials provide maximum compact representations of those complex material models, are the cornerstones of a subsequent mixed finite element design and should be considered as the primary objects for the theoretical and computational modeling.