Hybrid-Trefftz stress elements for three-dimensional elastoplasticity

Abstract

The stress model of the hybrid-Trefftz finite element formulation is applied to the elastoplastic analysis of solids. The stresses and the plastic multipliers in the domain of the element and the displacements on its boundary are approximated. Harmonic and orthogonal hierarchical polynomials are used to approximate the stresses, constrained to solve locally the Beltrami governing differential equation. They are derived from the associated Papkovitch-Neuber elastic displacement solution. The plastic multipliers are approximated by Dirac functions defined at Gauss points. The finite element equations are derived directly from the structural conditions of equilibrium, compatibility and elastoplasticity. The non-linear governing system is solved by the Newton method. The resulting Hessian matrices are symmetric and highly sparse. All the intervening arrays are defined by boundary integral expressions or by direct collocation. Numerical applications are presented to illustrate the performance of the model.