XFEM for Deformation Theory of Plasticity

Abstract

In this work, an approach is presented to improve global accuracy properties for physically non-linear problems in the frame of elastoplasticity. The work is motivated by the fact that convergence characteristics of a finite element solution are dominated by the regularity of the exact solution. For a material undergoing inelastic deformations, however, very few analytical solutions for the field variables are known, especially for the displacement field. Considering a simple, one-dimensional example, it is shown that the convergence rates are far from optimal. The reason is explained by establishing ties to a familiar, but more demonstrative problem. In the next section two remedies for the problem, based on the Extended Finite Element Method (XFEM) are presented and discussed, in the last section a 2D-problem is considered. Elasticity in 1DAs shown in [1], it is possible to make a priori estimates on convergence rates of a finite element solution. Depending on the regularity properties of the exact solution and the polynomial degree of the elements (further referred to as m), the optimal convergence rate (k) for the displacement field using a uniform refinement in a L 2 -norm, is given by k = m + 1. This result however, is valid for a smooth distribution of the field variables only, therefore the exact solution u ex is required to be at leastConsider a one-dimensional elastic bar, consisting of two non-overlapping domains Ω = Ω 1 ∪ Ω 2 with the intersection Γ = Ω 1 ∩ Ω 2 being the interface. The properties associated with the two domains are the cross-section A and the Young's moduli E 1 and E 2 . Now, in case that E 1 = E 2 = E, E ∈ C ∞ , a finite element analysis gives optimal convergence rates, as pictured in Fig. 1 by the continuous lines. In this case, u ex ∈ C ∞ (Ω), therefore clearly fulfilling the minimum requirement of C m (Ω)-continuity. However, altering the material properties, let e.g. E 1 > E 2 , E ∈ C −1 , changes the behaviour drastically. The convergence rates drop, settling at a rate of 1, this is depicted in Fig. 1 by the dashed lines. This is explained by a kink in the analytical displacement field u ex , right at the material interface Γ. The analytical solution is therefore C 0 (Ω)-continuous, but still C ∞ in Ω 1 and Ω 2 .