Generalized radial‐return mapping algorithm for anisotropic von Mises plasticity framed in material eigenspace

Abstract

A computationally efficient integration algorithm for anisotropic plasticity is proposed, which is identified as a generalization of the radial-return mapping algorithm to anisotropy. The algorithm is based upon formulation within the eigenspace of a material anisotropy tensor associated with anisotropic quadratic von Mises (J 2 ) plasticity (also called Hill plasticity), for which it is shown to ensure that the flow rule remains associative, ie, the normality condition is satisfied. Extension of the algorithm to include anisotropic elasticity (anisotropic elastoplasticity) is further provided, made possible by the identification of a certain fourth-order material tensor dependent on both the elastic and plastic anisotropy. The derivation of the fully elastoplastically anisotropic algorithm involves further complexity, but the resulting algorithm is shown to closely resemble the purely plastically anisotropic one once the appropriate eigenspace is identified. The proposed generalized radial-return algorithm is compared to a classical closest-point projection algorithm, for which it is shown to provide considerable advantage in computational cost. The efficiency, accuracy, and robustness of the algorithm are demonstrated through various illustrative test cases and in the finite element simulation of Taylor impact tests on tantalum. KEYWORDS algorithm efficiency, anisotropy, elastoplasticity, finite element method, return mapping, von Mises 202 /journal/nme Int J Numer Methods Eng. 2018;116:202-222.VERSINO AND BENNETT 203 models can take advantage of the J 2 yield surface being a hypersphere in the five-dimensional space of the deviatoric stress. In this case (barring other complications, such as nonlinear elasticity), the normal to the updated yield surface is identical to the normal calculated from the trial stress (predictor) value. This simplification gives rise to the well-known radial-return type return mapping algorithms, 7,8 which can save considerable computational effort by avoiding iterative computation of the yield surface normal, ie, the flow "direction" in associative plasticity. 3,9 Many crystalline and polycrystalline solids exhibit anisotropic plasticity. Anisotropic plasticity, however, typically precludes the use of radial-return like algorithms for associative plasticity because the predictor and corrector flow directions (ie, the trial and actual surface normals) are not equivalent due to the yield surface no longer being a hyper-sphere. Ensuring associative flow, ie, enforcing the normality condition, 10,11 can be achieved by simultaneously solving for the amount and direction of the plastic flow within the return mapping algorithm, ie, the so-called closest-point projection (CPP) type algorithms 12 ; however, the additional computational expense of this approach 3,13 can make it an unrealistic option for large scale computations where efficiency of numerical algorithms is paramount. [14][15][16] Various alternative approaches for minimizing the computational cost of computing the flow directi...

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