A geometric approach to the algorithmic tangent stiffness

Abstract

The elastoplastic tangent stiffness is the linear operator which provides the stress rate corresponding to a prescribed strain rate. As such, it plays a central role in the computational aspects of elastoplastic problems. According to the usual approach to the nonlinear evolutive analysis of elastoplastic models, a finite time-step is considered and the evolution law describing the constitutive behavior is reformulated as a finite step flow rule. The algorithmic tangent stiffness was first introduced by Simo and Taylor in [1]. They showed that the adoption of the algorithmic tangent stiffness leads to a significant improvement of the asympthotic convergence rate. The expression of the algorithmic tangent stiffness provided in [1], and in all the subsequent references to their contribution, was based on an explicit formulation of the elastoplastic constitutive law in terms of a plastic scalar multiplier. The geometrical analysis developed in the present paper is based on a formulation of the constitutive problem in terms of the nonlinear projector, in complementary elastic energy, on the convex elastic domain. The algorithmic tangent stiffness is evaluated as the composition between the derivative of the nonlinear projector and the elastic stiffness. The key point consists in the evaluation of the derivative of the nonlinear projector. A direct geometric argument, based on hypersurface theory, shows that the derivative can be expressed as the difference between the linear projector on the hyperplane tangent at the trial stress point and the shape operator of the parallel hypersurface passing thru the trial stress-state, multiplied by the distance between the trial stress and the projected stress, evaluated in the complementary elastic norm. The composition of the linear projector with the elastic stiffness is in fact the rate elastoplastic tangent stiffness. Since the analytic expression of the parallel hypersurface thru the trial stress point is available only in special cases, an effective procedure consists in substituting it with the level set of the yield function passing thru the trial stress point. In this way, the exact expression of the algorithmic stiffness is got when the level sets of the yield function are homothetic hypersurfaces, as in von Mises plasticity criterion, and a simple useful approximation is obtained in the general case. Indeed the proposed procedure greatly simplifies the computations while preserving the benefit of an improved convergence rate, since it takes effectively into account the curvature of the yield hypersurface, thus leading to a reduced tangent stiffness, in comparison with the rate tangent stiffness.