On the modelling of incompressibility in linear and non-linear elasticity with the master-slave approach

Abstract

The master-slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well-known stability results for inf-sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non-linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods.