Year

2000

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

Elastic deformations beyond the range of the classical infinitesimal theory of elasticity are governed by highly non-linear partial differential equations. Usually conventional methods of solution are inapplicable and accordingly only a few exact solutions are known. In this thesis, for some special classes of deformations and for some special materials, we obtain new exact solutions which we apply to certain stress boundary value problems.

perfectly elastic rubber-like materials undergoing large elastic deformations, the non-symmetrical inflation and eversion of circular cylindrical rubber tubes is examined for a family of strain-energy functions, which includes the neo-Hookean and Varga materials as special cases. For both inflation and eversion, new exact solutions are discovered. The exact solutions for inflation are applied to the problem of the lateral compression of a hollow rubber tube. For this problem approximate load deflection relations are obtained and from the limited experimental evidence that is available, they appear to provide reasonable agreement of actual load-deflection curves. The exact solutions corresponding to eversion are examined in the context of the problem of plane strain bending of a sector of a circular cylindrical tube. Two distinct finite elastic exact plane eversion deformations for the Mooney and Varga materials are applied to the problem. By assigning certain resultant forces and moments it is shown that various constants arising in the solution may be determined and a number of numerical examples are given. Each of the above deformations are rendered as approximate solutions of stress boundary value problems. These solutions are approximate in the sense that the pointwise vanishing of the stress vector on a free surface is assumed to be replaceable by the vanishing force and moment resultants.

Finally, for a certain family of non-symmetrical plane strain deformations for hyperelastic materials, we determine every strain-energy function which gives rise to a non-trivial deformation of this type. Previously obtained solutions, such as for the Mooney material and new solutions as obtained in this thesis for special strainenergy functions, are all shown to be deduced from a general formulation. In fact we demonstrate that for physically meaningful response functions, no further exact solutions of this type exist other than the new exact solutions determined in this thesis.

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