Mixed variational principles for the evolution problem of gradient–extended dissipative solids

Abstract

AbstractThis work develops minimization and saddle point principles for a class of gradient–extended dissipative solids, incorporating micro–structural fields (micro–displacements, order parameters or generalized internal variables), whose gradients enter the energy storage and dissipation potential functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro–structural fields are governed by additional balance–type partial differential equations including micro–structural boundary conditions. Typical examples are theories of phase field evolution, gradient damage or strain gradient plasticity. Such models incorporate non–local effects based on length scales, which reflect properties of the material micro–structure. We outline a unified variational framework for the evolution problem of first–order gradient–extended standard dissipative solids. Particular emphasis is put on mixed multi–field representations, where both the microstructural variable itself as well as the local driving forces are present (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)