A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

A novel Hu–Washizu type 5-field virtual work principle for nearly inextensible and almost incompressible finite hyperelasticity is developed and implemented. The formulation is implemented in an hp-adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation. The novel implementation includes residual based error estimation and mesh adaptivity. In the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraints of inextensibility and incompressibility. A study using a semi-inverse analytical solution and h-refinements and p-enrichments corroborates the convergence characteristics of the 5-field implementation. A new closed form solution for pure torsion of a circular cylindrical tube with inextensible fibres is derived and used for verification. As typical applications are found in e.g. soft tissue biomechanics, passive pressurisation of an ellipsoidal geometry resembling the left ventricle of a rabbit heart is analysed to demonstrate the capability of modelling transversely isotropic materials.