Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants

• The shear moduli for transversely isotropic soft tissue are typically distinct.
• This is incompatible with the usual simplifying assumption for such materials.
• Both pseudo-invariants are needed to model transversely isotropic soft tissue.

Skeletal muscles, ligaments and tendons are typically assumed to be incompressible, transversely isotropic, non-linearly hyperelastic materials. If one adopts the phenomenological approach to modelling, then the corresponding strain-energy function can be represented as an arbitrary function of two invariants of the Cauchy–Green strain tensors, representing the isotropic contribution, and two pseudo-invariants, representing the anisotropic contribution. For mathematical convenience, dependence on one of these pseudo-invariants is usually dropped. It will be shown here that a necessary consequence of this reduced form of the strain-energy function is that the infinitesimal shear moduli are identical, an assumption that is not supported by experimental data. It will also be shown that a further consequence is that two out of the three shearing modes are identical over the full range of deformation. The conclusion is that transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants.