A finite strain mixed J2 − u − p low-order tetrahedron

Abstract

With the goal of improving upon the accuracy of D. Arnold's MINI element for finite strain plasticity, and more precisely calculate the elastic/plastic interface, we extend this element formulation to include, as nodal degrees-of-freedom, a function of the second invariant of the deviatoric stress, J2. A finite-strain J2 − u − p mixed formulation of the classical low-order tetrahedron element is introduced. We therefore have continuous displacements, pressures and J2. This element contains an internal displacement bubble that is not condensed out. For hyperelastic materials, we adopt a relative Green-Lagrange formulation whose conjugate stress approximates the Cauchy stress. For the elasto-plastic case, we combine this formulation with the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. In contrast with nodally integrated and variational multiscale methods, there are no additional parameters. High accuracy is obtained for four-node tetrahedra with three incompressibility and bending benchmarks being solved. Accuracy similar to the F¯ hexahedron are obtained. Although the ad-hoc factor is removed and performance is competitive, computational cost is higher than MINI's, with each tetrahedron containing 23 degrees-of-freedom.