“A Two Point Boundary Value Problem for the Axial Shear of Isotropic Incompressible Nonlinearly Elastic Materials”

The purpose of this research is to investigate a pure axial shear problem for the annular region between two concentric rigid cylinders occupied by an incompressible isotropic nonlinearly elastic material. Our main concern is with the subclass of these materials that exhibits hardening at large deformations. Such hardening at large strains is observed experimentally but is often not predicted by commonly used strain-energy densities. The particular axial shear problem that we investigate arises when the elastic material is perfectly bonded to an inner rigid circular core and to an outer rigid circular container. The deformation is driven by an axial pressure gradient. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ODE. For a broad class of incompressible isotropic hyperelastic materials, existence of smooth solutions is established on conversion to an initial value problem. We then specialize the material class to be of generalized neo-Hookean type, that is, where the strain-energy density depends only on the first invariant of the strain tensor. Two classes of such materials that exhibit hardening at large deformations are considered. The first class models limiting chain extensibility at the molecular level, and the second class is of power-law type. An interesting limiting case of the latter leads to an exponential strain-energy function commonly used in modeling biological materials. Some specific strain-energy densities from each class are examined in detail. Numerical boundary-value methods with quasilinearization for nonlinear second-order ODEs are used to illustrate a special feature of the solutions. It was previously shown by other authors that, for softening power-law materials, a boundary layer behavior is exhibited near the bonded surfaces: the axial displacement undergoes a sharp increase from zero at the endpoints and is slowly varying in the interior. We provide further numerical results regarding this phenomenon here. Our main concern is, however, with hardening materials. For such materials, the numerical results exhibit an interior localization at a location which is almost the midpoint between the boundaries. The axial displacement has a sharp change of slope which, in the limit of infinitely large pressure gradients, gives rise to a cusp in the displacement profile. The results highlight the contrasting behavior between softening and hardening rubber-like or biological materials.