A 3-field formulation for strongly transversely isotropic compressible finite hyperelasticity

A Hu-Washizu type 3-field virtual work principle for strongly transversely isotropic compressible finite hyperelasticity is developed, implemented and verified. The independent variables are: the stretch, its energy conjugate uniaxial tension and the displacement. The formulation is based on a new decomposition of the deformation gradient into a simply stretchless part and a lateral contraction-free uniaxial extension. In the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraint of inextensibility. The finite element formulation is implemented in an hp-adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation. A study using a semi-inverse analytical solution corroborates the convergence characteristics for h-refinements and p-enrichments of the 3-field implementation. The case of isostatic loading of a circular cylinder reinforced with a single family of fibres, and the pressurisation of a saccular aneurysm-like configuration with circumferential fibre reinforcement are used for verification and illustration. A posteriori residual discretisation error estimation is used for making mesh refinements in the latter case. Typical applications are found in soft tissue biomechanics.