Jm. Hill et Dj. Arrigo, ON AXIALLY-SYMMETRICAL DEFORMATIONS OF PERFECTLY ELASTIC COMPRESSIBLEMATERIALS, Quarterly Journal of Mechanics and Applied Mathematics, 49, 1996, pp. 19-28
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.