The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials

The classical phenomenological constitutive modeling of the mechanical behavior of isotropic incompressible rubber-like hyperelastic materials involves strain-energy densities that depend on the first two principal invariants of the strain tensor. For rubber, the most well-known of these is the Mooney–Rivlin model which has a linear dependence on the two principal invariants and its specialization to the neo-Hookean form which is independent of the second invariant. While each of these models provides a reasonably accurate prediction for the mechanical behavior of rubber-like materials at small stretches, they fail to reflect the strain-stiffening that is observed as the stretch increases. For soft biomaterials, an exponential dependence on the strain-invariants is well known to capture this predominant stiffening effect. The most celebrated of these models is that of Fung and Demiray which depends only on the first strain invariant. In the limit as the strain-stiffening parameter tends to zero, one recovers the neo-Hookean model. A modification of the Fung–Demiray (FD) model that also depends on the second invariant was proposed by Vito. For the Vito model, one recovers the Mooney–Rivlin model as the strain-stiffening parameter tends to zero. It is well known that, in general, inclusion of a dependence on the second invariant models the stress response of rubber-like materials more accurately. More importantly, in the solution of some basic problems involving homogeneous and inhomogeneous deformations, constitutive models that do not include a dependence on the second invariant fail to capture some significant physical effects. These issues for elastomers and soft biomaterials are addressed here and explicitly illustrated by using the FD and Vito models.

► The inclusion of the second strain-invariant in modeling of isotropic materials.
► For soft biomaterials, the Fung strain-energy depends only on the first invariant.
► A modification to include dependence on the second invariant is due to Vito.
► The stress responses for homogeneous and inhomogeneous deformations are compared.