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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。
The development of 3D printing technology makes the manufacturing of complex gradient structure easier and it is necessary to study the solution of complex gradient problem. At present, there are more researches about the gradient of beam structure modulus in axial direction or thickness direction. However, There are few studies on the simultaneous changes of the amount along two directions. Therefore, the problem of exponentially laminar composite beams with varying gradient in thickness and axial direction is solved by the complex Fourier decomposition method. First, the general solution to the fourth-order variable coefficient partial differential equations governing the governing equations is obtained by using the semi-inverse elastic mechanics. Then, by using the series expansion, the special solution of the beam under the symmetrical load is solved. Finally, In contrast, the correctness of series solution is illustrated. The results show that the stress distribution and deformation of the beam structure are more complicated when the gradient changes bidirectionally, the position stress is higher at higher modulus and less at lower modulus. The proposed series solution can also be generalized to other related studies of two-way gradient non-uniform planar and semi-planar problems.