球面各向同性功能梯度球壳的自由振动  

FREE VIBRATIONS OF A FUNCTIONALLY GRADED SPHERICALLY ISOTROPIC HOLLOW SPHERE

作　　者：陈伟球[1] 叶贵如[1] 蔡金标[1] 丁皓江[1] 

机构地区：[1]浙江大学土木系,杭州310027

出　　处：《力学学报》2001年第6期768-775,共8页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金资助项目(10002016).

摘　　要：通过引入位移函数和应力函数，针对沿径向为非均匀的球面各向同性弹性动力学的基本方程建立了两个解耦的变系数状态方程．利用层合近似理论，将变系数状态方程转化为常系数状态方程，并给出了相应的解．进而利用层间连续条件，得到关于内外球壳表面边界量的相应线性方程．由自由表面条件导出对应两类独立振动的频率方程．最后给出了算例，并讨论了材料梯度指标对球壳两类固有频率的影响．在地球动力学分析中有一定的应用前景．By introducing displacement functions as well as stress functions, this paper establishes two independent state equations with variable coefficients from the three-dimensional dynamic equations of a radially inhomogeneous spherically isotropic elastic medium. Using the laminated approximation theory, one can transfer the state equations to the ones with constant variables in each divided sub-layers, of which the solutions can be easily obtained using available matrix theory. The continuity conditions at each interface then lead to linear algebraic equations about the state variables only at the inner and outer spherical surfaces. It is found that there exist two independent classes of vibrations. Considering the free conditions at the boundaries, frequency equations are then derived. No matter how many sub-layers there are, it is shown that for the first class of vibration, the frequency equation is a first-order determinant, while for the second class, it is first-order when n = 0 and second-order when n > 0 . Numerical calculations are finally presented and discussion on the effect of the material gradient index is also given. Obviously, it is rather straightforward to apply the present method to analyze spherically isotropic laminated spheres. It may be useful in the geodynamic study because the latest geophysical results have shown that the Earth in fact should be modelled as a laminated hollow sphere with spherical isotropy. Since the present method is completely based on the three-dimensional elasticity without introducing any assumptions on deformation, the solution could be accurate enough when the shell is divided into a certain number of sub-layers. Therefore, the solution can be a benchmark to verify any two-dimensional shell theories and/or numerical methods.

关 键 词：球面各向同性 球壳 自由振动 状态空间法 层合近似理论 功能梯度材料 

分 类 号：TB34[一般工业技术—材料科学与工程]