Phenomenological hyperelastic strain-stiffening constitutive models for rubber

Many rubber-like materials and soft biological tissues exhibit a significant stiffening or hardening in their stress-strain curves at large strains. The accurate modeling of this phenomenon is a key issue for a better understanding of the thermomechanics of rubber and the biomechanics of soft tissues. In this paper, we provide a review of some phenomenological hyperelastic constitutive models that have been proposed to model this strain stiffening effect and summarize recent advances in the solution of boundary-value problems that illustrate the utility of such models. The emphasis is on constitutive models that reflect limiting chain extensibility at the molecular level. A remarkably simple phenomenological model of this type has been proposed by Gent. The Gent model has a molecular basis related to the inverse Langevin function compact support non-Gaussian statistics for the end-to-end distance function. The mathematical simplicity of the Gent model. which contains just two constitutive parameters, has facilitated the analytic solution of a variety of specific boundary-value problems that are relevant to the rubber industry and we summarize the main results here. These problems include those of torsion, axial, azimuthal and helical shear, anti-plane shear, mode III crack problems, rotation induced deformation of circular cylinders and fracture problems. It is shown that the results are radically different from those obtained in the literature for classical models such as the neo-Hookean and Mooney-Rivlin models for incompressible rubber. Extensions to include thermoelasticity, material compressibility, anisotropy and stress softening are also briefly described.