Stress integration procedures for inelastic material models within the finite element method

A review of numerical procedures for stress calculation in the inelastic finite element analysis is presented. The role of stress integration within a time (load) step in the incremental-iterative scheme for the displacements based FE formulation is first given briefly. Then, the basic relations of the explicit algorithms, as the first ones developed in the 70s, are presented. The shortcomings of these algorithms are pointed out. The implicit procedures are presented in some detail, with the emphasis on a general return mapping procedure and the governing parameter method (GPM). Derivation of the consistent tangent moduli represents an important task in the inelastic FE analysis because the overall equilibrium iteration rate depends on these moduli. The basic concepts of this derivation are presented. An important field, very challenging in today's stage of design and technology, is the large strain deformation of material. A review of the approaches in the large strain domain that includes the rate and the total formulations is given in some detail. Special attention is devoted to the multiplicative decomposition of deformation gradient concept, since that concept is generally favored today. Some unresolved issues, such as the use of the stress and strain measures, are discussed briefly. A number of selected numerical examples illustrate the main topics in the stress integration task, as well as the applications of the stress integration algorithms to various material models. Some concluding remarks and an outline of further research topics are given at the end of the paper. This review article includes 205 references.