Revisited Simo algorithm for the plane stress state

• The differential type elasto-plastic model with mixed hardening describing the in-plane stress state.
• The update algorithm to solve the model, based on the radial return algorithm adapted to the in-plane stress.
• The discretized weak formulation of the non-linear elastic-type problem; finite element implementation.
• The algorithmic elasto-plastic tangent moduli rebuilt to solve the non-linear system for displacements.
• Comparison of the Simo solution and the in-plane stress solution applied to a trapezoidal plate.

In this paper a new numerical approach of the elasto-plastic problem with mixed hardening in-plane stress state is proposed within the classical constitutive framework of small deformation. In the formalized problem, the non-zero normal component of the strain (namely, the component in the direction perpendicular to the plane of stress) is considered to be compatible with the plane stress. We eliminate the rate of strains which are developed normal to the plane and solve an appropriate plane problem in the strain setting. The integration algorithm for solving the elasto-plastic problem with mixed hardening realizes the coupling of Radial Return Algorithm, namely an update algorithm, with the finite element method applied to the discretized equilibrium balance equation. The solution of the pseudo-elastic nonlinear problem is then solved by Newton’s method. The numerical algorithm (which is an incremental one) consists of the Radial Return Mapping Algorithm, originally proposed by Simo and Hughes (1998) adapted to the stress plane problem and coupled with a pseudo-elastic problem. We rebuild the algorithmic formula giving rise to the plastic factor in terms of the trial stress state and the algorithmic elasto-plastic tangent moduli which is requested to evaluate the Jacobian that is necessary to determine the displacement. The proposed algorithm has been named the revisited Simo algorithm and is applied to exemplify the mode of integration of the bi-dimensional problem.