ZAMM · Z. Angew. Math. Mech. 89, No. 8, 687 – 697 (2009) / DOI 10.1002/zamm.200800171 On constitutive and configurational aspects of models for gradient continua with microstructure B. Svendsen 1, ∗ , P. Neff 2 , and A. Menzel 1,3 1 Institute for Mechanics, Dortmund University of Technology, 44227 Dortmund, Germany 2 Fachbereich Mathematik, Technische Universit¨ at Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany 3 Division of Solid Mechanics, Lund University, P.O. Box 118, 22100 Lund, Sweden Received 2 October 2008, accepted 11 November 2008 Published online 20 February 2009 Key words Inelastic gradient microstructure, gradient elastoplastic decomposition, incremental variational approach. MSC (2000) 74A30 The purpose of this work is the investigation of some constitutive and configurational aspects of phenomenological model formulations for a class of materials with history-dependent gradient microstructure. The assumption that the behavior of a material point is affected by history-dependent processes in a finite neighbor of this point yields an extended con- tinuum characterized by non-simple material behavior and by additional degrees-of-freedom. This includes both standard micromorphic materials as well as inelastic gradient materials as special cases. As in the case of simple materials, the cor- responding constitutive relations are subject to restrictions imposed by material frame-indifference and material symmetry. In the latter case, both direct and differential restrictions are obtained in the case of assuming that the free energy density is an isotropic function of its arguments. In addtion, the concept of material isomorphism is shown to extend to inelastic gra- dient continua, resulting in a gradient generalization of the well-known elastoplastic multiplicative decomposition of the deformation gradient. Finally, we examine the consequences of gradient extension for the formulation of configurational field and balance relations, and in particular for the Eshelby stress. This is carried out with the help of an incremental stress potential formulation as based on a continuum thermodynamic approach to the coupled field problem involved. c  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The behavior of many materials of engineering interest (e.g., metals, alloys, granular materials, composites, liquid crystals, polycrystals) is often influenced by an existing or emergent microstructure (e.g., phases in multiphase materials, phase tran- sitions, voids, microcracks, dislocation substructures, texture). In general, the components of such a microstructure have different material properties, resulting in a macroscopic material behavior which is highly anisotropic and inhomogeneous. Attempts to account for these effects in the modeling of such materials have lead to a number of approaches to and view- points on the issue depending in particular on the nature of the microstructure in question (e.g. [8, 37, 45]). One class of models idealizing the behavior of such systems phenomenologically is that of gradient continua, including in particular micromorphic materials (e.g. [19–21, 33]) and strain-gradient materials (e.g. [15, 34]) as special cases. As a foundation for the second part of the current work, our first purpose here is to examine some basic constitutive issues for such materials, in particular those of material frame-indifference, material isomorphism and material symmetry. In this, we follow and extend the earlier work of [32]. Beyond anisotropic and heterogeneous material properties, processes associated with the microstructure which are rep- resented in the model by continuum fields (e.g., damage and order parameter fields, director field) also contribute to con- figurational fields and processes. Such fields represent additional continuum degrees-of-freedom for which corresponding field relations must be formulated. Contingent on the premise that the corresponding processes contribute to energy flux and energy supply in the material, field relations for such degrees-of-freedom result from the Euclidean frame-indifference of the total rate-of-work (e.g. [8]), or more generally from that of the total energy balance (e.g. [9,46,48]), or even from the exploitation of the dissipation principle (e.g. [24]). Once thermodynamically-consistent field relations and reduced constitu- tive relations have been obtained, one is in a position to formulate and solve initial-boundary-valueproblems. In the context of elastic material behavior and thermodynamic equilibrium, such initial-boundary-value problems are often formulated variationally. Examples here include elastic phase transitions (e.g. [2]), elastic liquid crystals (e.g. [53]), configurational ∗ Corresponding author E-mail: bob.svendsen@udo.edu, Phone: +0049 231 755 2686, Fax: +0049 231 755 2688 c  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim