A mixed higher order FEM for fully coupled compressible transversely isotropic finite hyperelasticity

A higher order mixed finite element method is presented for compressible transversely isotropic finite hyperelasticity. The independent variables of the three-field formulation are; displacement, fibre tension and fibre stretch. The formulation admits the description of a fully coupled stress response which evolves from an almost isotropic one into a hyper-anisotropic simply nearly inextensible response with increasing fibre tension, as for example observed in soft tissue biomechanics. Standard displacement approximation of order p in H1(Ω) is used while the auxiliary variables are approximated element wise by square integrable functions of order p−1 in L2(Ω). For finite extensibility the auxiliary variables are statically condensed out yielding a pure displacement based method. It is implemented in an hp-adaptive finite element code. A matching residual based error estimation capability is added and exercised. Numerical evidence indicating stability of approximation is supplied resolving a boundary layer caused by almost inextensible fibres. Coupled and uncoupled stress responses are compared. A generalised compressible transversely isotropic Holzapfel-Gasser-Ogden model is developed for the formulation that intentionally avoids the volumetric-isochoric split. Solutions obtained with the mixed method compare favourably to the corresponding ones obtained with pure displacement formulation. The latter fails in certain strongly anisotropic cases.