压电复合材料中Ⅲ型唇形裂纹问题的解析解

Analytical solutions of mode-Ⅲ lip-shape crack in piezoelectric composites

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明:唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。

Based on the complex variable function method and the technique of conformal mapping,an anti-plane problem of lip-shape crack in piezoelectric composites was investigated under the anti-plane loading at infinity and inplane electric loads.By using the residue theorem and Cauchy integral formula,the analytical expressions of the field intensity factors and the mechanical strain energy release rate are obtained with the assumption that the crack surfaces were electrically impermeable and electrically permeable.When the height of lip-shape crack approaches to zero,the analytical solutions of an infinitely large piezoelectric solid with a Griffith crack was obtained.Ignoring the effect of the electric field the present results can be reduced to the well-known solutions of classic material.The numerical examples are provided to show the influences of geometrical parameters and applied mechanical/electric loads on the mechanical strain energy release rate.The results indicate that an increase of the height of the lip-shape crack will retard the crack propagation and mechanical loads always accelerate the crack propagation.The influences of electric loads on the crack propagation was about boundary conditions of the lip-shape crack.