Comparison of Additive and Multiplicative Kinematics in Large Deformation Plasticity

Abstract

The most widely used plasticity model in commercial finite element analysis software packages is the additive plasticity model. One of the more well-known alternatives to additive plasticity is multiplicative plasticity. This work brings out the differences between these two plasticity models for problems of large deformation in metal plasticity. Such problems are relevant to engineering processes such as metal forming and machining. Fundamental differences and differences in implementation between the two models are discussed. The additive plasticity model is based on an additive decomposition of the rate of deforamtion tensor into elastic and plastic parts. The integration of this rate equation leads to a whole host of concerns related to objectivity. The solution of such concerns leads to involved objectivity preservation algorithms. In contrast, a multiplicative plasticity model is based on the multiplicative decomposition of the total deformation gradient into elastic and plastic parts. This difference allows one to formulate the stress response as a hyperelastic function in the case of multiplicative plasticity. This difference is crucial since it removes all concerns related to objectivity with regard to the integration of the constitutive model and reduces the complexity of implementation of the multiplicative plasticity model. The multiplicative plasticity model is implemented as part of an explicit dynamics IBVP solver written in MATLAB/C++. The additive plasticity model is accessed through the ABAQUS simulation package. These two platforms are used to compare the effects of the use of additive and multiplicative plasticity models on the numerical solution of IBVP's. The specific problem chosen is the problem of plane strain specimen placed under a monotonic tensile load. Problems of this kind provide a good test for large deformation plasticity codes since the solutions are usually complicated by large stress and strain gradients. These gradients arise as a result of strain localisation phenomena such as necking and shear banding. Large deformation plasticity codes are tested based on their ability to reproduce such phenomena with a reasonable degree of fidelity. The first test of the multiplicative plasticity solver is for problems of a single element subjected to small strains. These are followed by a problem of several elements under small strains. Upto this stage, all verification is done by solving the same problem in ABAQUS. Comparisons are made between hysteresis loops and plots of the external work done, plastic dissipation and strain energy stored. This preliminary verification stage is followed by the study of large deformation metal plasticity problems. The first step at this stage is the verification of convergence. This is done by solving the same plane strain necking problems at two different levels of mesh refinement. Convergence is confirmed by the small difference in the results from the two different meshes. Once the solver passes the convergence check, it may be used to solve large deformation metal plasticity problems. This process is started by making the first comparison for a full scale problem between the additive and multiplicative plasticity models. The plane strain necking problem is solved for the case of a material with hardening rule similar to the Johnson-Cook hardening rule, with the rate dependence and temperature dependence dropped. This comparison reveals that the multiplicative plasticity model predicts higher plastic strains than the additive plasticity model. Several other problems of plane strain necking are solved to test the multiplicative plasticity model. The comparison to the additive plasticity models are followed by parametric studies of the multiplicative plasticity model. The slope of the hardening curve is varied. The yield criterion is changed between a the smooth yield surface of Tresca's yield criterion and the Tresca yield criterion, whose yield surface has sharp corners. It is found that as the hardening slope becomes shallower, the extent of the plastic deformation increases. In extreme cases of low hardening slopes, shear bands show up very early in the simulation procedure. It appears that for low hardening slopes, the Tresca yield criterion is more resistant to shear banding than the von Mises yield criterion.