功能梯度材料板壳静动态宏观力学行为的均匀化经典化表示

On the assumption that the material properties vary arbitrarily continuously in the thickness direction, relationship between the static and dynamic mechanical responses of functionally graded material plates/shells and those of the corresponding homogenous structures will be investigated by using analytical approach in conjunction with numerical simulation. By using load equivalence condition and similarity theory of differential equations, transition relations of the bending, buckling and free vibration solutions of the functionally graded plates/shells based on the classical and shear deformation theories, respectively, and those of the corresponding homogenous structures based on the classical theory will be established, the related transition parameters will be presented and the regulation of the effects of the boundary conditions on the transition relations will be examined. Consequently, solution for the static and dynamic problem of FGM plates/shells can be simplified as the solution of that of homogenous thin plates/shells with the same geometry, the same loading conditions and constraints and calculation of the related transition parameters. Accordingly, we can obtain the deflections, the critical buckling loads, the natural frequencies and other interesting solutions from the familiar classical theory solutions for homogenous structures. In other words, we can realize homogenized and classical treatment to the problem of FGM structures. Through investigation of this project, it may be hopeful to explore new approches and find simplified method to analyze the macroscopic behaviors of FGM structures. The research results will be helpful to us to further understand the characteristics of the static and dynamic responses of FGM structures, to reveal the correlation and difference between the responses of inhomogenous and homogenous structures and to enrich the theory and methodology of mechanics of inhomogenous structures.

假设材料性质沿厚度任意非均匀变化，采用理论分析和数值计算相结合的方法研究功能梯度材料板壳与相应均匀材料结构的静动态响应之间的关系。利用载荷等效条件和微分方程的相似理论，分别建立在经典理论和剪切变形理论下功能梯度板壳的弯曲、屈曲和振动解与参考均匀结构在经典理论下的解答之间的解析转换关系，给出转换系数的计算公式，分析边界条件的影响规律。将FGM板(壳)的静动态问题求解简化为具有相同几何尺寸、载荷工况和边界约束均匀薄板(壳)问题的求解以及转换系数的计算问题。从而可利用我们熟悉的均匀材料结构的经典理论解答获得FGM结构的挠度、临界载荷和固有频率等感兴趣的解答，实现对非均匀结构静动态问题的均匀化、经典化处理。本项目研究有望能为FGM结构宏观力学行为分析开辟新的途径，探索简便方法。研究成果有助于进一步认识功能梯度材料结构宏观力学行为的特性，揭示其与均匀结构的联系与区别，丰富非均匀结构力学的理论和方法。

功能梯度材料梁板壳结构的宏观力学行为研究是固体力学的一个活跃分支。现有研究成果更多地关注了这种特殊非均匀材料结构的宏观力学行为与均匀结构的力学行为之间的差异，而很少去研究功能梯度结构与相应的均匀材料结构的静动态响应之间的联系，以及不同理论之间功能梯度结构的静动态问题解答与参考均匀结构的解答之间的相互关系。本项目采用理论分析与数值计算相结合的方法，假设材料性质沿厚度任意非均匀变化，主要研究了功能梯度梁板壳与相应均匀材料结构的静动态响应之间的关系。建立了各阶剪切变形理论下功能梯度材料梁的弯曲、屈曲和振动解与相应均匀梁的经典理论解之间的转换关系，推导了经典理论下功能梯度薄板与参考均匀薄板的挠度、屈曲载荷、频率之间的转换关系以及Levinson高阶理论下功能梯度圆板弯曲解的均匀化经典化表示。给出了转换系数的解析计算公式，这一系数综合反映了材料非均匀性和剪切变形效应的影响。将功能梯度梁板的静动态问题求解简化为具有相同几何尺寸、荷载工况和边界条件的均匀梁板问题的求解以及转换系数的计算问题。从而能够利用已有的均匀材料结构的经典理论解答获得功能梯度结构的挠度、临界载荷和固有频率等解答，实现了对非均匀的功能梯度结构静动态相关问题解答的均匀化、经典化表示。