关于超弹性材料球体中空穴分岔问题的研究(Ⅰ)  

Cavitated Bifurcation Problems for Spheres Composed of Hyper-elastic Materials (Ⅰ)

作　　者：袁学刚[1] 

机构地区：[1]烟台大学数学与信息科学系,山东烟台264005

出　　处：《烟台大学学报（自然科学与工程版）》2005年第2期79-90,共12页Journal of Yantai University(Natural Science and Engineering Edition)

基　　金：国家自然科学基金资助项目(10272069);烟台大学博士基金资助项目(SX04B24).

摘　　要：研究了由一类关于径向各向异性不可压缩超弹性材料组成的球体在给定的表面径向拉伸死载荷作用下的空穴分岔问题.得到了描述球体内部有空穴生成和增长的空穴分岔方程;讨论了在由表征径向各向异性参数划分的不同区域内空穴分岔方程解的定性性质;用奇点理论方法证明了空穴分岔方程等价于一类具有单边约束条件的正规形;用最小势能原理分析了解的稳定性和突变现象,并讨论了实际稳定的平衡状态.最后给出了球体内部有空穴生成时的径向和环向应力分布,指出了应力的集中和突变现象是导致空穴生成的基本原因.The spherical cavitated bifurcation problems is examined for a solid sphere composed of a class of incompressible hyper-elastic materials which are transversely isotropic about the radial direction. The sphere is subjected to a uniform radial tensile dead-load. A cavitated bifurcation equation that describes cavity formation and growth in the interior of the sphere is obtained. Some qualitative properties of the solutions of the cavitated bifurcation equation are discussed in detail in the different regions of the plane partitioned by material parameters indicating the degree of radial anisotropy. It is shown that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraint conditions at the critical point by using singularity theory. Stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed by using the minimal potential energy principle and the actual stable equilibrium state is then discussed. Finally, analyses of distribution of radial and circumference stresses as a cavity forms in the sphere are carried out, and then the basic reason that the concentration and catastrophe of stresses causes cavity formation is pointed out.

关 键 词：超弹性材料 空穴分岔 临界死载荷 奇点理论 二次转向分岔点 

分 类 号：O175[理学—数学] O343[理学—基础数学]