Finite deformations of a generalized Blatz-Ko material
In this dissertation, the finite deformations of a certain class of compressible, isotropic elastic materials are investigated. The class is characterized by a two-parameter family of strain energy functions which includes the well-known Blatz-Ko material model for foam rubbers. The Blatz-Ko material, which has been arrived at by experiment and whose deformations have been studied previously, is obtained from the considered class of materials by specifying one of the two parameters involved in the definition of the class. On employing the semi-inverse method, according to which the form of the solution is given at the outset in terms of functions which are then determined from the equilibrium equations and boundary conditions, closed-form solutions to the equilibrium equations are obtained for the non-homogeneous deformations describing the straightening of a sector of a circular tube, the bending of a rectangular block into a sector of a circular tube, the eversion of cylindrical and spherical shells, and the cylindrical and spherical expansions, and a number of associated boundary value problems are investigated using both analytical and numerical methods. Certain situations in which solutions of the pre-assigned form cannot exist are identified and cases of non-uniqueness are dealt with by discriminating between the different solutions on physical grounds. The homogeneous deformations of the materials in this class are also examined and, throughout, comparison is being made with the behaviour of the Blatz-Ko material. For the whole range of deformations examined, it is found that the materials for which one of the parameters is greater than, or equal to, two (the case when this parameter equals to two corresponds to the Blatz-Ko material) become harder as this parameter increases, but that otherwise they all behave in a similar manner. Consequently, it is concluded that the materials in this particular subclass will also represent foam rubbers of the type described by the Blatz-Ko material. In order to describe the situations in which the solutions become unstable, the conditions for the strong ellipticity of the equilibrium equations for non-linearly elastic materials are reformulated so as to be expressible in terms of the derivatives of the strain-energy function regarded as a function of the principal stretches. Use of these conditions reveals that the solutions to the considered boundary value problems become unstable at certain critical values of the applied loads.