含等边三角形孔孔边裂纹横观各向同性压电弹性体的反平面问题研究

The anti-plane solution for the cracked equilateral triangle hole in transverse isotropic piezoelectric materials

利用Stroh公式,研究了含等边三角形孔孔边裂纹的横观各向同性压电弹性体的反平面问题。首先,基于Savin给出的三角形近似保角变换函数,导出了带直线裂纹三角形孔的近似保角变换函数。其次,假设边界条件为电不可穿透裂纹且孔周及裂纹表面为自由表面,结合柯西积分,导出了力-电耦合作用下复势函数的近似解,从而得到了裂尖的场强度因子和能量释放率的表达式。最后,考虑面内电载荷和面外机械载荷共同作用,分析了三角形孔尺寸、裂纹尺寸、外载变化对裂尖场强度因子和能量释放率的影响。结果表明:随着三角形孔洞尺寸的增加,场强度因子的增加非常明显,三角形及裂纹的尺寸对裂尖场的影响不容忽视,且正的机械载荷总是促进裂纹的扩展,而纯电载荷则可以抑制裂纹的扩展。

The Stroh formula is used to study the anti-plane problem of a cracked equilateral triangle hole in the transversely isotropic piezoelectric materials. Firstly, the mapping function which maps the outside of the equilateral triangle hole and the crack into the outside of a circular hole is derived based on the equilateral triangle's conformal mapping function which was given by Savin. Secondly, the approximate expressions for the complex potential with coupling effect by electric and mechanics is derived by Cauchy integral combine with the assumption that the surface of the crack and hole is electrically impermeable and traction free, then we can obtain the expression for the field factor of crack tip and the energy release rates. Finally, combined in-plane electric load with the mechanical load, effects of the triangle size, the crack length and changes of applied loading on the field intensity factors and energy release rates are analyzed. Result shows that the shape of the hole and the crack cannot be ignored, the positive mechanical loads always promote crack growth, while the pure electric loads can retard crack growth.