Constitutive modelling of rubber-like and biological materials with limiting chain extensibility

Many rubber-like materials and soft tissues exhibit a significant stiffening or hardening in their stress-strain curves at large strains. The accurate modelling of this phenomenon is a key issue for a better understanding of the thermomechanics of rubber and the biomechanics of arteries, blood vessels, tendons and other biological tissues. In this paper, we consider several strain-energy functions that have been proposed to model this strain hardening effect in the framework of the theory of isotropic hyperelasticity. These models may be divided in two main families: power law models and limiting chain extensibility models, both of which are based on non-Gaussian statistics for their molecular structure. The known limiting chain extensibility models involve a constraint on the strain invariants. Here we consider alternative models which limit the maximum stretch. The two homogeneous deformations of uniaxial extension and pure shear are examined in detail. It is shown that simple modifications of the incompressible neo-Hookean or Varga materials to reflect limiting chain extensibility can be used to match well with the classic experimental data of Treloar. Some remarks on empirical inequalities and ellipticity are also given.