压电弹性体中光滑顶点的正三角形孔边裂纹的反平面问题分析
Anti-plane problem analysis of edge crack emanating from regular triangle hole with smooth vertices in piezoelectroelastic solids

通过构造新的保角映射, 利用复变函数的方法, 研究了含光滑顶点的正三角形孔边裂纹的横观各向同性的压电弹性体的反平面问题。在电可穿透和电不可穿透裂纹、 孔周及裂纹面为自由表面的假设下, 充分利用Cauchy积分公式和复变函数方法, 得到了裂纹尖端的场强度因子和能量释放率的表达式。数值算例显示了在不同边界条件下裂纹的几何尺寸、 机电载荷对能量释放率和机械应变能释放率的影响规律。 结果表明: 在电可通和电不可通边界条件下, 裂纹长度和三角形边长的增加会导致能量释放率增加, 机械载荷则总是促进裂纹的扩展。在电不可通边界条件下电位移可以促进或抑制裂纹的扩展, 而在电可通边界条件下电位移对裂纹扩展没有影响。 By constructing new conformal mapping and using complex variable function method, anti-plane problem of isotropic piezoelectroelastic solids containing regular triangle hole with smooth vertices which emanates edge crack was studied. Under the assumption that the surfaces of the crack and hole were electrically permeable and electrically impermeable, respectively, and their surfaces are free traction, combined with Cauchy integral and complex variable function method, the expressions of the field intensity factors and the energy release rates near crack tip were obtained. The numerical examples were conducted to show the influences of the geometrical parameters of crack and applied mechanical loads on energy release rate and mechanical strain energy release rate under different boundary conditions. The results show that, under the electrically permeable and impermeable boundaries, the increases of the length of crack and the size of triangle hole lead to the increase of the energy release rate. The mechanical loads always promote crack growth. Under the electrically impermeable boundary the electric displacements may promote or retard the crack growth, but the electric displacements have no effect on the crack growth under the electrically permeable boundary. 国家自然科学基金(11262012); 高等学校博士学科点专项科研基金(20101102110016); 内蒙古工业大学科学研究项目(ZD201219)