On the computation of design derivatives for Huber–Mises plasticity with non-linear hardening

This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber– Mises deviatoric plasticity with non-linear isotropic=kinematic hardening, one which is applicable to metals. The standard radial return algorithm for linear hardening is generalized to account for non-linear hardening functions. Two generalizations are presented; in both the non-linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design di erentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation. The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of veri cation tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for veri cation of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with nite di erence computations and (b) the e ect of the nite element formulation on the design derivatives for an isochoric plastic ow