功能梯度与均匀圆板弯曲解的线性转换关系

LINEAR TRANSFORMATION BETWEEN THE BENDING SOLUTIONS OF FUNCTIONALLY GRADED AND HOMOGENOUS CIRCULAR PLATS

基于一阶剪切理论,研究了功能梯度材料圆板与均匀圆板轴对称弯曲解之间的线性转换关系.通过理论分析和比较功能梯度材料圆板和均匀圆板在一阶剪切理论下的位移形式的轴对称弯曲控制方程,发现了功能梯度材料圆板的转角与均匀圆板的转角之间的相似转换关系.在假设材料性质沿板厚连续变化的情况下,给出了相似转换系数的解析表达式.在此基础上,进一步导出了一阶剪切理论下功能梯度圆板的挠度与经典理论下,均匀圆板的挠度之间的线性关系.从而,可将功能梯度材料圆板在一阶剪切理论下的弯曲问题求解,转化为相应均匀薄圆板在经典理论下的弯曲问题求解,以及转换系数的计算问题.这一方法为功能梯度非均匀中厚度圆板的求解提供了简捷途径,而且更便于工程应用.作为例子,采用上述方法分别求得了周边简支和夹紧条件下,梯度圆板在均布横向载荷作用下的弯曲解析解,该解答与Reddy得到的结果完全吻合.

Based on the first-order shear deformation theory （FSDT）, linear transformation relationship be- tween bending solutions of functionally graded material （FGM） and homogenous circular plates was studied. Firstly, through theoretically analyzing and comparing the displacement-type governing equations for axially symmetrically bending of FGM and homogenous circular plates based on the first-order shear deformation theory, linearly dependent relationship between the rotationM angles of FGM circular plate and those of homogenous one was found. By giving the material properties of FGM circular plates changing as continuous functions in the thickness direction, the corresponding transition factor between the solutions of the two kind plates were derived in analytical forms. Furthermore, a linear relationship between the deflections of FGM circular plate based on FSDT and those of homogenous one based on the classical plate theory were derived. As a result, solutions for static bending based on the first-order shear deformation plate theory of a non-homogenous circular plate can be reduced to that based on classical plate theory of a homogenous one and the calculation of the transformation factors. This approach provides a simple and facile procedure for solutions of the non-homogenous moderately thick FGM circular plates, which can be very easily and conveniently used in engineering. By using the above mentioned approach, analytical bending solutions of FGM circular plates with simply supported as well as clamped boundary conditions under uniformly distributed lateral force were presented, which show a very good agreement with the results given by Reddy.