Unified hexahedral finite elements for large strains analysis of compressible and nearly incompressible materials

Numerous mechanical and aeronautical engineering applications, soft tissues modeling and bio-inspired material deal with hyperelastic materials. During the years, many efficient models based on strain energy functions have been proposed for the simulation of hyperelastic media, each of them based on different phenomenological assumptions. Dealing with physical and material nonlinear analysis of materials and structures, finite element models must deal with mathematical limitation that arise, like instabilities, shear and membrane locking but in particular volumetric locking, coming from the superimposition of mass conservation equation in compressible and slightly-compressible materials like rubbers or elastomers. This work proposes a displacement-based finite element model for large strains analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high order hexahedral finite elements are derived adopting a three-dimensional formalism based on the Carrera Unified Formulation, to obtain nonlinear governing equation independent of the chosen polynomial expansion for the displacement field. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells and curved structures.