Numerical differentiation for non-trivial consistent tangent matrices: an application to the MRS-Lade model

In a companion paper P erez-Foguet, A., Rodr guez-Ferran, A. and Huerta, A. `Numerical di erentiation for local and global tangent operators in computational plasticityʹ. Computer Methods in Applied Mechanics and Engineering, 2000, in press, the authors have shown that numerical di erentiation is a competitive alter- native to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS-Lade model. This plastic model has a cone-cap yield surface and exhibits strong coupling between the ow vector and the hardening moduli. Because of this, di erentiating these quantities with respect to stresses and internal variables|the crucial step in obtaining consistent tangent matrices|is rather involved. Numerical di erentia- tion is used here to approximate these derivatives. The approximated derivatives are then used to (1) compute consistent tangent matrices (global problem) and (2) integrate the constitutive equation at each Gauss point (local problem) with the Newton{Raphson method. The choice of the stepsize (i.e. the perturbation in the ap- proximation schemes), based on the concept of relative stepsize, poses no di culties. In contrast to previous approaches for the MRS-Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational e ciency (CPU time) and robustness of the proposed approach is illustrated by means of several numerical examples, where the major relevant topics are discussed in detail