Using null strain energy functions in compressible finite elasticity to generate exact solutions

Abstract

In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain-energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler-Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.

Linear Equations of Motion in Finite Elasticity

“…It can be shown that there are no other null energies [1] and so we may now write down a full SNL (spherical-null-linear) strain-energy function…”

“…There are several ways to determine the null energies for this problem, see [1] for details. One illustrative method is to start with the total (partial) energy E which can be written as…”

“…Details of the deformation can be found in Ogden [1,[256][257][258][259][260]. The principal stretches are given by…”

“…In a recent paper [1] it was shown how partial null strain energies may be useful in obtaining exact solutions to problems in non-linear elasticity or in the general modelling process for a particular material. The basic idea is to express the total strain-energy function W as a sum of partial energies W = n i=1 ω i , say.…”

A generalized Ogden model for the compressibility of rubber-like solids

“…[4,5]). The strain-energy functions constructed with this method have successfully been used to characterize the mechanical behaviour of rubber-like solids in various loading cases such as uniaxial tension, biaxial tension and hydrostatic tension/compression [5,[11][12][13][14][15][16]. The cavitation damage and elastic wave propagation in compressible rubber-like solids have also been analysed based on this two-terms strain-energy function, which includes contributions due to the volume change [8,17,18].…”