International Journal o f Plasticity, Vol. 9, pp. 813-831, 1993 0749-6419/93 $6.00 + .00 Printed in the USA Copyright © 1993Pergamon Press Ltd. MICROPOLAR ELASTOPLASTICITY AND ITS ROLE IN LOCALIZATION ANDREAS DIETSCHE,* PAUL STEINMANN,* and KASPAR WILLAM'~ *University of Karlsruhe and ~University of Colorado (Communicated by Romesh Batra, University of Missouri-Rolla) Abstract- In this article we focus on micropolar elastoplasticity and the conditions for the for- mation of spatial discontinuities. After a brief review of micropolar kinematics and balance equa- tions, an augmented localization tensor is developed for elastoplastic constitutive behavior that describes discontinuous bifurcation at the constitutive level. In this context an auxiliary condi- tion is encountered that restricts the jump of the stress rate to remain symmetric across discon- tinuity surfaces of second order. The expanded localization conditions are studied with two constitutive models-micropolar RANKINE and micropolar VON MISES J2-flow theories of elas- toplasticity-for which definite statements are derived with regard to their regularization properties. 1. INTRODUCTION The classical description of local continua reveals loss of ellipticity of the governing partial differential equations as soon as localization occurs. This causes strong mesh dependence of numerical solutions that require special provision to capture localized deformations in failure bands of narrow width. To remedy this pathological behavior, nonlocal constitutive descriptions have been recently proposed, see Bxa-a~ [1987], BaP_~T [1988], and ERIr~EN [1992], which introduce an internal length scale to regularize the postbifurcation behavior. Micropolar continua like the CossERm-formulation [1909] provide a natural length measure in the elastoplastic material description which tends to regularize the impending loss of ellipticity in the vicinity of localization. The present study will focus on the localization properties of micropolar elastoplasticity and will address possible distortions of the underlying failure mechanisms assuming linear- ized kinematics. II. COSSERAT STATICS AND KINEMATICS Let 6] C ~3 denote the domain occupied by a body with particles labeled by x E 6]. Omitting body forces and inertia effects, the local balance of linear momentum results in the traditional divergence statement div(o) = O. (1) If an asymmetric stress tensor is admitted, the balance of angular momentum will not be satisfied by the shear stresses, so that the couple-stresses m enter the description. In 813