Some exact solutions of the equations of finite elasticity
Abstract
Finite deformations of special, compressible, elastic materials are considered. Attention is restricted to elastic materials having the following separable form of the strain energy function: W = f(i(sub 1)) + g(i(sub 2)) + h(i(sub 3)), where f, g, and h are twice continuously differentiable functions of the appropriate principal invariant of the stretch tensor. Three special cases are then considered; for each class, two of the functions f, g, and h are linear functions and the third is an arbitrary function. The effect of several constitutive restrictions on the form of the strain energy functions of the materials will be examined. Controllable deformations have a form which is independent of the specific form of the strain energy function. Necessary and sufficient conditions for a deformation to be controllable for each of the materials described above will be obtained and then used to show that azimuthal shearing deformation is controllable for these materials. Qualitative features of such azimuthal shearing are then studied. Spherical and cylindrical expansion deformations are also controllable for the materials of interest and the qualitative features of these deformations for the socalled materials of type 2 will be considered. The steady rotation of a cylinder or tube and the eversion of both cylindrical and spherical shells are not controllable deformations for all three types of elastic material. However, for some of the representative materials chosen, a number of closed form solutions to the equations of equilibrium or motion, as the case may be, will be obtained which describe the abovementioned deformations. These solutions provide the basis for the discussion of the qualitative features of such deformations for compressible materials.
 Publication:

Ph.D. Thesis
 Pub Date:
 1990
 Bibcode:
 1990PhDT........69M
 Keywords:

 Deformation;
 Elastic Properties;
 Equations Of Motion;
 Numerical Analysis;
 Shearing;
 Strain Energy Methods;
 Stress Tensors;
 Compressibility;
 Constraints;
 Cylindrical Shells;
 Elastic Shells;
 Rotating Cylinders;
 Rotating Spheres;
 Spherical Shells;
 Structural Mechanics