一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究  

Qualitative Study of Cavitated Bifurcation for a Class of Incompressible Generalized neo-Hookean Spheres

作　　者：袁学刚[1] 朱正佑[1] 

机构地区：[1]上海大学数学系上海市应用数学和力学研究所

出　　处：《应用数学和力学》2005年第2期169-177,共9页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10272069);上海市重点学科基金资助项目

摘　　要：　研究了一类不可压的广义neo＿Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo＿Hookean材料,得到了球体内部空穴生成的条件·　与均匀各向同性的neo＿Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小·　用奇点理论证明了。The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. The condition of void nucleation for this problem was obtained. In contrast to the situation for a homogeneous isotropic neo-Hookean sphere, it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution, and also the critical load is smaller than that for the material with no perturbations, as the parameters belong to some regions. It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraints near the critical point by using singularity theory. The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.

关 键 词：不可压的广义neo-Hookean材料 空穴分岔 正规形 稳定性和突变 

分 类 号：O175[理学—数学] O343[理学—基础数学]