| Generally, cavity formation, growth and run-through of the adjacent cavities in solid materials are considered as important mechanisms of failure and fracture. Thus, prediction of cavity formation and growth in materials has long attracted much attention. In this dissertation, by means of the basic viewpoints, methods and conclusions of bifurcation theory and dynamical theory, the quasi-static problems of cavity formation and growth, motion rules of the formed cavity along with time in the interior of the spheres that are composed of imperfect hyper-elastic materials are discussed systemically. Some new results are obtained and the corresponding numerical analyses are carried out. The main works are as follows:1. In Chapter 3, problems of cavitated bifurcation for a solid sphere composed of imperfect incompressible hyper-elastic materials, subjected to a surface tensile dead load, are examined. The cavitated bifurcation equation that describes cavity formation and growth in the interior of the sphere is obtained. It is proved that there exists a unique bifurcation point on the trivial solution of the cavitated bifurcation equation. And the distinguished conditions of left-bifurcation and right-bifurcation are carried out. It is presented that secondary turning point can be occurred on the cavitated bifurcation solution which bifurcates locally to the left. Equivalent normal forms of the cavitated bifurcation equation at the bifurcation point were presented by using singularity theory. The imperfect parameters' plane is divided into several regions, and it is proved that the critical dead load for the sphere composed of imperfect hyper-elastic materials is less than that for perfect hyper-elastic materials as the imperfect parameter belongs to some regions. Stability of solutions of the cavitated bifurcation equation is discussed in each region by using the minimal potential principle, and the catastrophic phenomenon of cavity formation is explained.2. In Chapter 4, problems of cavitated bifurcation for two spheres, which are composed of two different kinds of imperfect compressible hyper-elastic materials, were examined, respectively. The parametric analytic solutions of the radially deformed function and the parametric cavitated bifurcation solutions are all obtained for the two spheres. It is pointed out that the critical stretch of the sphere composed of the first kind of materials given in this chapter is less than that for the perfect materials. Finally, stability of solutions of the cavitated bifurcation equation is discussed.3. In Chapter 5, problem of radial motion for a solid sphere composed of an imperfect incompressible hyper-elastic material, subjected to a surface tensile dead load, is examined. And a cavity motion equation with respect to time is obtained. As an example that the sphere is composed of an imperfect incompressible hyper-elastic material, effects of material imperfection on the existence and the qualitative properties of the nontrivial solutions of the cavity motion equation were studied. It is proved that a cavity forms in the interior of the sphere as the surface tensile dead load exceeds certain critical value, and the motion of the formed cavity alongwith time is periodic motion which the derivative of the solution has the first kind of discontinuous point.4. In Chapter 6, problem of radial motion for a spherical shell composed of an imperfect incompressible hyper-elastic material under sudden dead loads at the inner- and outer-surface respectively, is examined. And a second-order nonlinear differential equation that describes radial motion of the inner-surface of the sphere is obtained. Dynamical behavior of the differential equation is analyzed, and effects of each parameter on the qualitative properties of the solutions of the equation are discussed in detail. After that, numerical examples of period vibration and destroy of the spherical shell are presented. The effects of material imperfection on period vibration of the spherical shell are discussed.
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