A mixed finite element formulation for compressible finite hyperelasticity with two fibre family reinforcement

Abstract

A mixed finite element formulation for compressible finite hyperelasticity, possibly with strongly deformation dependent stiffening fibre reinforcement, is developed and verified. Reinforcement by two, mechanically equivalent, in general oblique, fibre families is considered. In the provided numerical experiments, residual based error estimation is essential for a simple h-adaptive mesh refinement strategy. It is shown to be effective to resolve stress concentrations, also in combination with stress oscillations (checker boarding) in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used. The novel mixed generalised (Lagrangian) metric based formulation is illustrated using a simple but fully coupled material model. The ground substance is the common so-called compressible neo-Hooke model and the standard reinforcement part is augmented by second order anisotropic invariants. The theoretical formulation admits approaching the inextensible limit seamlessly, by-passing the use of a separate decoupled material model. The Hu–Washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables, approximated in L2 at the element level. The condensation yields a purely H1-based displacement formulation at the global level.