In this chapter we provide an overview of the basic equations governing the mechanical response of rubberlike materials capable of finite deformations. We use the term rubberlike materials to refer to highly deformable, nonlinear elastic continua, which respond differently in their reference state and their deformed configuration. An elastic material, by definition, will always return to its original state after the application and removal of forces. Nonlinear elastic materials, in particular, exhibit a change of material behavior during deformation, but assume the original shape after removal of all applied forces. Therefore, we do not include the effects of time and rate dependence, nor do we assume that the applied forces are of such magnitude to induce damage to the material.
We first introduce the mathematics of deformation and different stress tensors for the analysis of rubberlike materials. Subsequently, we summarize the general constitutive theory and specialize its use to isotropic compressible and incompressible materials. A large number of constitutive functions are available in the literature to describe the nonlinear elastic response. We provide an overview of the most frequently used formulations.
The last section of this chapter covers the solution of some boundary value problems. In particular, we consider incompressible, isotropic and elastic materials to illustrate the mechanical response when different strain energy functions are used. We apply the theories to circular, cylindrical thick-walled tubes subjected first to a pure azimuthal shear deformation, followed by combined axial extension and radial inflation.