Modified Mixed Least-Squares Finite Element Formulations For Small And Finite Strain Plasticity

In this contribution, we propose mixed least-squares finite element formulations for elastoplastic material behavior. The resulting two-field formulations depending on displacements and stresses are given through the L2(B)-norm minimization of the residuals of the first-order system of differential equations. The residuals are the balance of momentum and the constitutive equation. The advantage of using mixed methods for an elastoplastic material description lies in the direct approximation of the stresses as an unknown variable. In addition to the standard least-squares formulation, an extension of the least-squares functional as well as a modified formulation is done. The modification by means of a varied first variation of the functional is necessary to guarantee a continuous weak form, which is not automatically given within the elastoplastic least-squares approach. For the stress approximation, vector-valued Raviart-Thomas functions are chosen. On the other hand, standard Lagrange polynomials are taken into account for the approximation of the displacements. We consider classical J(2) plasticity for a small and a large deformation model for the proposed formulations. For the description of the elastic material response, we choose for the small strain model Hooke's law and for finite deformations a hyperelastic model of Neo-Hookean type. The underlying plastic material response is defined by an isotropic von Mises yield criterion with linear hardening.