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

In Reference [1] the authors have shown that numerical differentiation is a competitive alternative 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 [2,3]. This plastic model has a cone-cap yield surface and exhibits strong coupling between the flow vector and the hardening moduli. Because of this, derivating these quantities with respect to stresses and internal variables —the crucial step in obtaining consistent tangent matrices— is rather involved. Numerical differentiation is used here to approximate these derivatives. The approximated derivatives are then used 1) to compute consistent tangent matrices (global problem) and 2) to 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 approximation schemes), based on the concept of relative stepsize, poses no difficulties. In contrast to previous approaches for the MRS-Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational efficiency (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.