A semi-analytical integration method for J2 flow theory of plasticity with linear isotropic hardening

In this paper, an exact time integration scheme is presented for von Mises elastoplasticity with linear isotropic hardening at small deformations. The method is based on the constant strain rate assumption, which is widely accepted in displacement based finite element applications. The deviatoric form of the rate equation, using the method proposed by Krieg and Krieg in [R.D. Krieg, D.B. Krieg, Accuracies of numerical solution methods for the elastic-perfectly plastic model, J. Press. Vess. Technol. Trans. ASME 99 (1977) 510–515], is rewritten to a system of nonlinear ODEs with two scalar functions: the radius of the Mises circle and an angle obtained by using the scalar product of the stress deviator and the strain rate. The time integration of these functions gives an exact solution in implicit form using the incomplete beta function. In addition, by the exact linearization of this new semi-analytical solution a stress updating algorithm and the consistent tangent operator are derived. The numerical performance and accuracy of the proposed method are illustrated on numerical examples. Moreover, a comparison of the proposed method with the well-known radial return method is included.