Physical response of hyperelastic models for composite materials and soft tissues

A hyperelastic model must not only characterize the mechanical response of a composite material such as soft tissue, but also ensure numerical stability by a feasible set of material parameters. Apart from the well-known ill-conditioning problem caused by the incompressibility constraint, the paper indicates another ill-conditioning occurring in any general fibre-reinforced material model for tubular organs when unbalance between the fibre strain energy and the matrix strain energy becomes too large. Specifically, although the Holzapfel model is polyconvex, this problem can be observed as an unphysical behaviour in a physiological deformation range of a tissue such as arterial wall and intestine by thickening in the thickness direction associated with a volume growth of a specimen in a tension test. Particularly, the same problem for a polyconvex modified Fung-type model with the matrix characterized by the neo-Hookean model has been discussed for the first time. By investigating the influence of the shear modulus in these two models, we not only prove the cause of the ill-conditioning but also propose a solution to control the unbalance in the strain energy. The numerical results show significant enhancement of the model stability in overcoming the unphysical deformation.