功能梯度板中Griffith裂纹尖端应力场的三维解析研究

3D Analytical Solutions of Stress Fields at Griffith Crack Tips in Functionally Graded Plates

基于推广后的England⁃Spencer板理论,研究了横观各向同性功能梯度板中Griffith裂纹尖端的三维应力场.假定材料参数沿板厚方向可以任意连续变化,利用复变函数解法和保角变换技术分别给出了受无穷远处荷载作用和受均匀内压时裂纹尖端应力的三维解析解.当材料退化为各向同性均匀材料时,将该解答与经典二维解进行了比较,进而验证了该解答的有效性.通过数值算例,进一步讨论了材料梯度因子和荷载条件等因素对裂纹尖端三维应力场的影响.

Based on the generalized EnglandSpencer plate theory,the 3D stress field at the Griffith crack tip in a transversely isotropic functionally graded plate was investigated.With the material parameters assumed to vary continuously and arbitrarily along the thickness direction,by means of the complex function theory and the conformal mapping technology,the analytical results of the 3D stress field at the Griffith crack tip under loading at infinity and uniform internal pressure were obtained respectively.With the material reduced to an isotropic homogeneous material,the validity of the solution was verified through comparison with the classical 2D solution.Numerical examples were given to demonstrate the effects of material gradient factors and load conditions on the stress field.