Mechanical behavior of strain-stiffening rubber-like nonlinearly elastic materials

It is well-known that rubber-like materials and soft biological tissues undergo significant strain-stiffening upon loading. In this dissertation our focus is on obtaining explicit analytic results for the stress response for special classes of constitutive models for incompressible isotropic nonlinearly elastic materials that give rise to such severe strain-stiffening in their stress-stretch curves at large strains. We investigate the effects of strain-stiffening for the classical problems of plane strain bending of a rectangular bar, axial and azimuthal shearing of a long circular cylindrical tube, and torsion superimposed on axial extension of a circular cylinder. We consider each problem first for a general strain-energy density for homogeneous isotropic incompressible nonlinearly elastic materials. The results are then specialized for two particular constitutive models that are tractable analytically, involve just two constitutive parameters, and capture severe strain-stiffening so that they agree well with experimental data at large values of stretch. The solutions that are obtained are then compared with results for classical models and with those for a less abruptly strain-stiffening model widely used in biomechanics. The closed-form analytic solutions obtained here are valuable as benchmarks if these models are to be implemented in finite element computational packages. This work leads to a better understanding of the effects of strain-stiffening on the deformations of rubber-like materials and soft biological tissues.