具有缺陷的不可压缩neo-Hookean球壳的动力学行为的定性分析  

Qualitative Analysis of Dynamical Behavior for an Imperfect Incompressible Neo-Hookean Spherical Shell

作　　者：袁学刚[1,2] 朱正佑[1] 程昌钧[1] 

机构地区：[1]上海大学,上海市应用数学和力学研究所 [2]烟台大学,数学与信息科学系,山东烟台264005

出　　处：《应用数学和力学》2005年第8期892-898,共7页Applied Mathematics and Mechanics

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

摘　　要：研究了一类具有缺陷的不可压缩超弹性材料球壳的径向对称运动问题,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缩的neo_Hookean材料.得到了描述球壳内表面运动的二阶非线性常微分方程,并给出了方程的首次积分.通过对微分方程的解的动力学行为的分析,讨论了材料的缺陷参数和球壳变形前的内外半径的比值对解的定性性质的影响,并给出了相应的数值算例.特别地,对于一些给定的参数,证明了存在一个正的临界值,当内压与外压之差小于临界值时,球壳内表面随时间的演化是非线性周期振动;当内压与外压之差大于临界值时,球壳的内表面半径随时间的演化将无限增大,即球壳最终将被破坏.The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained.And the first integral of the equation was then carried out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carried out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely.That is to say, the shell will be destroyed ultimately.

关 键 词：具有缺陷的不可压neo-Hookean材料 动力学行为 临界值 非线性周期振动 

分 类 号：O175[理学—数学] O343[理学—基础数学]