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

In a companion paper Perez-Foguet, A., Rodriguez-Ferran, A. and Huerta, A. 'Numerical differentiation for local and global tangent operators in comput ational plasticity'. Computer Methods in Applied Mechanics and Engineering, 2000, in press, the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of c onsistent tangent matrices. Relatively simple models were treated in that r eference. The approach is extended here to a complex model: the MRS-Lade mo del. This plastic model has a cone-cap yield surface and exhibits strong co upling between the flow vector and the hardening moduli. Because of this, d ifferentiating these quantities with respect to stresses and internal varia bles-the crucial step in obtaining consistent tangent matrices-is rather in volved. Numerical differentiation is used here to approximate these derivat ives. The approximated derivatives are then used to (1)compute consistent t angent matrices (global problem) and (2) integrate the constitutive equatio n at each Gauss point (local problem) with the Newton-Raphson method. The c hoice of the stepsize (i.e. the perturbation in the approximation schemes), based on the concept of relative stepsize, poses no difficulties. In contr ast to previous approaches for the MRS-Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational ef ficiency (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. Copyright (C) 2000 John Wiley & Sons, Ltd.