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in this work we seek to characterize the conditions under which an elastic-plastic stress update algorithm preserves the symmetries inherent to the material response. From a numerical standpoint, the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality. For the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving. For hardcning plasticity, symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations. We show that these restrictions amount to the existence of a pseudo-internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli. In view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and, hence, in symmetric tangents, we address the question of whether any connections exist between the two approaches. We show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.

This work surveys a broad range of related issues in quasistatic elastoplasticity, beginning with a development of an internal variable constitutive theory. The initial-boundary value problem is then considered, and the remainder of the work is concerned with the properties of the time-discrete problem. It is shown how this discrete problem has associated with it a holonomic constitutive law (that is, one relating stress to strain or strain increment), and this holonomic law in turn forms the basis of a solution algorithm. Conditions for the convergence of the algorithm are discussed. The entire treatment applies to the spatially continuous problem.

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