On axially symmetric deformations of perfectly elastic compressible materials

The governing partial differential equations for static deformations of homogeneous isotropic compressible hyperelastic materials (sometimes referred to simply as perfectly elastic materials) are highly nonlinear and consequently only a few exact solutions are known. For these materials, only one general solution is known which is for plane deformations and is applicable to the so-called harmonic materials originally introduced by John. In this paper we extend this result to axially symmetric deformations of perfectly elastic harmonic materials. The results presented hinge on a reformulation of the equilibrium equations and a similar procedure can be exploited to derive the known solution due to John. It is shown that axially symmetric deformations of a harmonic material can be reduced to two linear equations whose coefficients involve the partial derivatives of an arbitrary harmonic function omega(R, Z). For any harmonic material, this linear system admits a simple general solution for the two special cases omega = omega(R) and omega = omega(Z). For the 'linear-elastic' strain-energy function, the linear system is shown to admit a general solution for the two harmonic functions which, in spherical polar coordinates, arise from the assumptions of spherical and radial symmetry.