An arbitrary Lagrangian-Eulerian finite element method for incompressible hyperelasticity

Abstract

In this paper, we present an arbitrary Lagrangian-Eulerian (ALE) formulation for large deformation analysis of incompressible hyperelastic materials, which are often employed for modeling rubber-like materials. Conventional ALE formulations, which have been successfully applied to not only fluids but also solid materials with rate constitutive equations, are usually described in velocity-based variables. Thus, in the application of such formulations to hyperelasticity, the velocity needs to be integrated in the time variable to obtain the deformation gradient necessarily appearing in hyperelastic material modeling. However, such integration is expensive and difficult to evaluate precisely. From such aspects, we propose a new ALE finite element formulation which directly incorporates the total deformation to avoid integration of velocity-based variables and can deal with arbitrary mesh motion by considering the variation of geometric mappings. Some basic numerical results are also given to illustrate the validity of our formulation.