Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme

We present a new cell-centered Lagrangian scheme on unstructured mesh for hyperelasticity. It is based on the recently proposed Glace scheme [11] for compressible gas dynamics. We show how to use the multiplicative decomposition of the gradient of deformation and the entropy property to derive the new scheme. We also prove the compatibility of this discretization with usual calculations of mass. Our motivation is to use hyperelasticity models for the study of finite plasticity, which is an extension of hypoelasticity to finite deformations. Hyperelasticity is a natural choice for extended models in solid mechanics, because of its mathematical structure which is a system of conservation laws with full rotational invariance. We study these properties for the Lagrangian system, and detail the various Eulerian formulations.