Closed‐form matrix exponential and its application in finite‐strain plasticity

Abstract

SUMMARY A new method to compute numerically efficient closed‐form representation of matrix exponential and its derivative is developed for 3 × 3 matrices with real eigenvalues. The matrix exponential is obtained by automatic differentiation of an appropriate scalar generating function in a general case, and highly accurate asymptotic expansions are derived for special cases in which the general formulation exhibits ill‐conditioning, for instance, for almost equal eigenvalues. Accuracy and numerical efficiency of the closed‐form matrix exponential as compared with the truncated series approximation are studied. The application of the closed‐form matrix exponential in the finite‐strain elastoplasticity is also presented. To this end, several time‐discrete evolution laws employing the exponential map are discussed for J 2 plasticity with isotropic hardening and nonlinear kinematic hardening of Armstrong–Frederick type. The discussion is restricted to the case of elastic isotropy and implicit time integration schemes. In this part, the focus is on a general automatic differentiation‐based formulation of finite‐strain plasticity models. Numerical efficiency of the corresponding incremental schemes is studied in the context of the FEM. Copyright © 2014 John Wiley & Sons, Ltd.