Class Of Nonlinear Elasticity Problem Analytical Solution

At present, the methods for solving nonlinear differential equations have been presented extensively, however, the methods corresponding to different equation are also quite different. Based on the theory of nonlinear elasticity and the variational principle of energy, in this dissertation, cavitation in some axisymmetric structures composed of certain compressible hyper-elastic materials are examined systematically. Firstly, the associated problems are described as a class of boundary-value problems of nonlinear ordinary differential equations; Secondly, the analytic solutions with parametric type of the equations are obtained by using the methods of transformation of variable or the inverse method. Finally, some new results are obtained by using the basic viewpoints, methods and conclusions of the qualitative theory of ordinary differential equations and the bifurcation theory, the numerical simulation are also given. The main works are as follows:1,In chapter 3, under the assumption of spherically symmetric deformation, problems of cavitated bifurcation are presented for spheres composed of several classes of compressible hyper-elastic materials. The analytic solutions with parametric type of the corresponding mathematical models are obtained by different methods.2,In chapter 4, both a circular solid cylinder and a cylinder with a pre-existing micro-void at its center composed of a class of compressible hyper-elastic are examined, where the outer surface of the cylinders is subjected to a radial stretchλ. The critical stretchλcr is obtained for a solid cylinder. And the variation rule of the cavity radius,the stresses distribution and the radial displacement the cylinder with a pre-existing micro-void are discussed by the numerical figures.