A Lie Group Analysis of the Axisymmetric Equations of Finite Elastostatics for Compressible Materials

Many boundary-value problems in nonlinear elastostatics for incompressible isotropic materials have been solved analytically. The situation for compressible materials is quite different. The main difficulty is the absence of the simplified kinematics arising from the incompressibility constraint. Considerable progress in the analytic solution of boundary-value problems for classes of compressible materials has however taken place in recent years and further developments in this direction are considered in this paper. A very special topic in the equilibrium theory for finite deformations of homogeneous isotropic compressible materials is investigated, namely that of spherically symmetric or cylindrically symmetric deformations. Since only axisymmetric deformations are considered, the equilibrium equations reduce to a single second-order nonlinear ordinary differential equation for the radial deformation field. In previous work, two main methods for obtaining analytic solutions for these equations have been developed, the first based on a change of variable that allows one to transform the second-order ODE to a pair of first-order ODEs that may be solved in parametric form. In some cases, the parameter can be eliminated to yield an explicit closed form expression for the deformation. The second method is based on the observation that the second-order ODE is invariant under a stretching transformation of the independent and dependent variables and thus may always be transformed to a first-order ODE. The purpose of this paper is to show how the powerful techniques of Lie group analysis (LGA) for ODEs can be used to unify both the above approaches and how LGA is an invaluable guide in the development of analytic solution methods. It is shown that both of the approaches can be derived from Lie algebra invariance criteria and that the second approach is, in fact, a special case of the first. Results from LGA are then applied to investigate the structure and solution set of the governing ODEs. A systematic procedure for solving the axisymmetric equations of compressible finite elastostatics is thereby proposed. The results are illustrated by application to several specific strain-energy density functions.