含裂縫異向性彈性板受彎矩作用之破壞力學分析

Abstract

本文發展一種新的積分方程式，用於分析二維含裂縫異向性彈性板彎矩作用之問題上。此法的推導是先利用Stroh理論求得單一旋錯(disclination)的基本解，再利用旋錯會造成旋轉角不連續的特性，將受力矩作用的裂縫視為連續分佈的旋錯，來建構相關的積分方程式。該積分方程式可經由高斯-謝比雪夫積分法轉化為線性代數方程式求解。此法的優點是不論邊界條件、材料常數為何，其裂縫尖端之應力強度因子都可求得；即使在少數的積分點下，也可達到很高的精確性。本文之算例使用之材料有等向性與正交性兩種，計算的模型有單裂縫、雙裂縫、多裂縫、弧形裂縫之無限板受均勻力矩或剪力問題等，其中部分結果參考其他文獻之解析解，驗證其有效性與精確性。

A new integral equation method is developed in this paper for the analysis of two-dimensional general anisotropic cracked elastic plates under bending. Integral equation are constructed by considering cracks as continuous distributions of disclination. Using Gauss-Chebyshev integration formulas, the integral equations can be transformed into the form of algebraic equations, with which the disclination densities and the stress intensity factors associated with each crack tip can be computed. An advantage of the method is that we can get the stress intensity factors regardless of the boundary conditions and material parameters. Another advantage is that accurate results may be obtained with relatively few integration points. Numerical examples are provided for isotropic or orthotropic plates with a single line or arc crack, double line cracks, multiple line cracks, under uniform bending, twisting moments or shearing force. The some results are compared with those in the literature whenever possible to verify their accuracy.