压电材料平面裂纹尖端场的杂交应力有限元分析

ANALYSES OF CRACK-TIP FIELDS OF PLANE PIEZOELECTRIC MATERIALS BY THE HYBRID STRESS FINITE ELEMENT METHOD

基于复势理论和杂交变分原理建立了一种适用于力电耦合分析的杂交应力有限元模型.给出了建立刚度矩阵的主要公式和推导过程,单元内的位移场和应力场采用满足平衡方程的复变函数级数解,假设的复变函数级数解事先精确满足裂纹的无应力和电位移法向分量为零的条件,单元外边界的位移场假设按抛物线变化,单元的刚度矩阵采用Gauss积分的方法得出.通过对力电耦合裂尖场的数值计算验证了程序的正确性和单元的有效性,同时也用所得结果校验了理论解.

A hybrid stress finite element based on the complex potential theory and hybrid variational principle is proposed for mechanical-electrical coupling analyses. The formulations are given in detail in this paper. The complex series solutions satisfying the equilibrium equations and compatibility equations are chosen as the displacement and stress fields in the element domain. Assume that the series solutions satisfy exactly the traction free and the zero normal electrical displacement boundary conditions along the crack surface in advance. While the displacements along the element outer boundaries vary parabolically. The element stiffness matrix is then obtained by using the Gauss quadrature method. Numerical examples verify the accuracy of the program and the efficiency of the proposed element. Meanwhile, the theoretical results are verified by the finite element results.