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

Abstract

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.