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由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。
Due to the non-uniform material properties along the thickness of FGM, the functionally graded beams with clamped and simply supported conditions have quite different behavior characteristics. In this paper, the exact solution of nonlinear static response of FGM under thermal load is given. Based on the nonlinear classical beam theory and the concept of physical medium, the nonlinear governing equations of FGM are derived. The two equations are reduced to a fourth-order integro-differential equation. For functionally graded beams clamped at both ends, the equations and the corresponding boundary conditions form a differential eigenvalue problem; however, for a functionally graded beam with simply supported ends, an eigenvalue problem will not be obtained due to non-homogeneous boundary conditions. The result is a completely different behavior of functionally graded beams that are clamped and simply supported. By solving the integral-differential equation directly, a closed form solution of beam over-buckling and bending deformation is obtained. This solution can be used to analyze nonlinear deformation such as buckling, over-buckling and nonlinear bending of beams. Finally, numerical results are used to study the effect of material gradient and thermal load on the nonlinear static response of FGM.