ON AXIALLY-SYMMETRICAL DEFORMATIONS OF PERFECTLY ELASTIC COMPRESSIBLEMATERIALS

The governing partial differential equations for static deformations o f homogeneous isotropic compressible hyperelastic materials (sometimes referred to simply as perfectly elastic materials) are highly nonline ar and consequently only a few exact solutions are known. For these ma terials, only one general solution is known which is for plane deforma tions and is applicable to the so-called harmonic materials originally introduced by John. In this paper we extend this result to axially sy mmetric deformations of perfectly elastic harmonic materials. The resu lts presented hinge on a reformulation of the equilibrium equations an d a similar procedure can be exploited to derive the known solution du e to John. It is shown that axially symmetric deformations of a harmon ic material can be reduced to two linear equations whose coefficients involve the partial derivatives of an arbitrary harmonic function omeg a(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 line ar system is shown to admit a general solution for the two harmonic fu nctions which, in spherical polar coordinates, arise from the assumpti ons of spherical and radial symmetry.