Irrotational Deformations in Finite Compressible Elasticity

This paper examines nonhomogeneous deformations of homogeneous, isotropic, compressible, nonlinearly elastic solids. It is assumed that the deformation is the gradient of a scalar field, and then determine the equation satisfied by this field from the equation of equilibrium. The strain energy function assumed is a generalization of the harmonic material and the generalized Varga material. For the special case of the generalized Varga material, a similarity solution is found for the scalar field. This solution describes the deformation of regions bounded by a family of surfaces which includes circles, ellipses, and hyperbolas in two and three dimensions. The solution is then partially extended to the more general elastic material.