Extension of the unsymmetric 8-node hexahedral solid element US-ATFH8 to 3D hyper-elastic finite deformation analysis

This work extends the recent US-ATFH8 element to 3D hyper-elastic finite deformation analysis. Using two sets of shape functions, the new 3D element comprises of 8 nodes and 24 DOFs. The first set of shape functions represent the test functions that come from the conventional isoparametric interpolation, and the second set, representing the trial functions, are constructed from the homogenous solutions for linear elasticity governing equations, termed analytical trial functions (ATFs). This study considers finite deformation for hyper-elastic materials, but it is assumed that the analytical solutions associated with hyper-elastic materials can be updated to hold approximately in each incremental step. Moreover, the deformation information required for stress computation is updated by using the incremental deformation gradient, which is constructed from the updated ATFs. Numerical examples show that without additional pressure DOF, the element US-ATFH8 still behaves well in nearly incompressible hyper-elastic 3D problems with finite deformation, even when the meshes are extremely distorted.