一种模量双向变化的梯度复合平面梁的级数解
Series solutions of graded composite plane beam whose modulus varying in bi-directions

3D打印技术的发展使复杂梯度结构的制造更加容易，有必要对复杂梯度问题的求解开展研究；目前，关于梁结构模量沿轴向或厚度方向梯度变化问题的研究已经较多，但对模量沿2个方向同时变化的研究较少。因此，通过复数形式傅里叶分解的方法对模量以指数形式沿厚度方向和轴向同时变化梯度平面复合梁问题进行了求解。首先，采用弹性力学半逆解法得到了问题的四阶变系数偏微分控制方程的通解；然后，利用级数展开，求解了对称载荷作用下该梁的特解；最后，通过与有限元结果进行对比，说明了级数解的正确性。结果表明：当梯度双向变化时，梁结构的应力分布和变形情况更加复杂，模量较高的位置应力较大，而模量较低的位置应力较小。提出的级数解还可推广至其他相关的梯度双向变化非均匀平面和半平面问题的研究。 As the development of 3D print technology makes the manufacturing of complex graded structures more easily, it is necessary to carrying out the investigations with respect to the solving for complex graded problems. Now, the investigations about the problems that the modulus of beam structures varying along the axial direction or thickness direction grade are already more, while the investigations about the modulus varying along two directions synchronously are less. Thus a graded composite plane beam with modulus changed in exponential form synchronously along thickness direction and axial direction was investigated by using Fourier decomposition in complex form. The general solutions for four-order differential governing equation with variable coefficients were solved firstly by using semi-inverse method of elasticity. Then, the particular solution of the beam subjected to symmetric load effects was derived by series expansions. Finally, the correctness of series solutions was verified by comparing with finite element results. The results show that when the gradient varying in bi-directions, the stress distribution and deformation situations of the beam structure become more complex, that the stress is larger at the region with higher modulus, while the stress is smaller at the region with lower modulus. The series solutions proposed could be also extended to the investigation of other relevant plane or semi-plane inhomogeneous problem with graded varying in bi-directions. 中国博士后科学基金（2015M581681）；中国商飞中欧合作专项（SAMC14-JS-13-012）