A novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity

A novel approach to computational almost inextensible transversely isotropic and nearly incompressible finite hyperelastic fibre mechanics is introduced. It relies on using an equivalent generalised right Cauchy–Green stretch tensor which is volume preserving and simply-stretch free in the limit of incompressibility and inextensibility. In other words, its third and fourth principal invariants become trivial. Otherwise it represents volume change and fibre stretch with the aid of point-wise equivalent auxiliary measures in the continuous case. The generalised kinematics implies the usual orthogonal spherical–deviatoric decomposition of the stresses. The deviatoric stresses are further orthogonally decomposed into axial fibre- and ground substance-stresses. The novel approach implies that the deviatoric ground substance stresses are trivial in the fibre direction as opposed to the current standard formulation. The approach is also able to represent exact inextensible fibres which is a problem recently addressed in the literature using an additive volumetric–isochoric decoupled strain energy density function, relying on volume preserving stretch. The formulation is corroborated by a couple of numerical examples using a preliminary finite element setting. The basis for the implementation is provided.