Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: Large‐strain theory for standard dissipative solids

Abstract

Abstract We propose a fundamentally new approach to the treatment of shearband localizations in strain softening elastic–plastic solids at finite strains based on energy minimization principles associated with microstructure developments. The point of departure is a general internal variable formulation that determines the finite inelastic response as a standard dissipative medium. Consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasi‐hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. The existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments of finite elasticity. Furthermore, localization phenomena are interpreted as microstructure developments on multiple scales associated with non‐convex incremental stress potentials in analogy to elastic phase decomposition problems. These microstructures can be resolved by the relaxation of non‐convex energy functionals based on a convexification of the stress potential. The relaxed problem provides a well‐posed formulation for a mesh‐objective analysis of localizations as close as possible to the non‐convex original problem. Based on an approximated rank‐one convexification of the incremental stress potential we develop a computational two‐scale procedure for a mesh‐objective treatment of localization problems at finite strains. It constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the microshearing of a rank‐one laminate aligned to the shear band. This problem is sufficiently robust with regard to applications to large‐scale inhomogeneous deformation processes of elastic–plastic solids. The performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which report on the mesh independence of the results. Copyright © 2003 John Wiley & Sons, Ltd.