功能梯度矩形板的三维弹性分析

将功能梯度三维矩形板的位移变量按双三角级数展开，以弹性力学的平衡方程为基础．导出位移形式的平衡方程。引入状态空间方法，以三个位移分量及位移分量的一阶导数为状态变量，建立状态方程。考虑四边简支的边界条件，由状态方程得到了功能梯度三维矩形板的静力弯曲问题和自由振动问题的精确解。由给出的均匀矩形板自由振动问题的计算结果表明．与已有的理论解以及有限元方法的计算结果相吻合。假设功能梯度三维矩形板的材料常数沿板的厚度方向按照指数函数的规律变化．进一步给出了功能梯度三维矩形板的自由振动问题和静力弯曲问题的算例分析，并讨论了材料性质的梯度变化对板的动力响应和静力响应的影响。

Three Dimensional Elasticity Analysis of A Functionally Graded Rectangular Plate

The displacement components of a three dimensional functionally graded rectangular plate were expanded in the forms of double trigonometric series. The state equation were established based on the governing equations, in which the displacement components and their first derivatives were chosen as state variables. Given simply supported boundary conditions, the exact solutions of bending and free vibration problem of a three dimensional functionally graded rectangular plate were derived by using the state space equation. It is noted that the present method gives results in good agreement with those obtained from other analytical methods and FEM. Examples in which the material coefficients were assumed to have exponent law dependence on the thickness-coordinate were presented for free vibration and bending problems. Discussions on the effect of the different functionally gradient material properties on the dynamical and static response were given.