非均匀热载荷作用下功能梯度梁的非线性静态响应

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。

NONLINEAR STATIC RESPONSES OF FGM BEAMS UNDER NON-UNIFORM THERMAL LOADING

Due to the variation in material properties through the thickness of functionally graded material (FGM) structures, an FGM beam simply supported at both ends exhibits characteristics quite different from those of a FGM beam clamped at both ends. An exact, closed form solution is obtained for nonlinear static responses of FGM beams subjected to non-uniform in-plane thermal loadings. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear classical beam theory and the physical neutral surface concept. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformation. For an FGM beam clamped at both ends, the equation and the corresponding boundary conditions lead to a differential eigenvalue problem, whereas for an FGM beam simply supported at both ends, an eigenvalue problem does not arise due to the inhomogeneous boundary conditions. This consequently results in quite different behavior between a clamped and a simply supported FGM beams. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal bending deformation is obtained as a function of the applied thermal load. By using the exact solutions, the nonlinear deformation problems for buckling, postbuckling and nonlinear bending of the beam can be investigated. Finally, the numerical analyses are carried out to investigate the effects of material gradient properties and thermal loads on the nonlinear static responses of FGM beams.