插值矩阵法分析正交各向异性板切口应力奇异性

Singularity Analysis for Notches in Orthotropic Composite Plates With the Interpolating Matrix Method

基于切口尖端附近区域位移场渐近展开,提出了分析正交各向异性复合材料板切口奇异性的新方法.将位移场的渐近展开式的典型项代入弹性板的基本方程,得到关于正交各向异性板切口奇异性指数的一组非线性常微分方程的特征值问题;再采用变量代换法,将非线性特征问题转化为线性特征问题,用插值矩阵法求解获得的正交各向异性板切口若干阶应力奇异性指数和相应特征函数.该法可由相应的特征角函数对板切口的平面应力和反平面奇异特征值加以区分,并将计算结果与现有结果对照,表明了该文方法的有效性.

Based on asymptotic expansion of generalized displacement field at the V-notch tip, a new method for analyzing the stress singularity exponents of the notches in orthotropic com- posite plates was proposed. Through introduction of the typical terms in asymptotic expansion of the generalized displacement functions into the basic elastic equations of the plate, the eigen- value problem of a set of nonlinear ordinary differential equations（ODEs） about the stress sin- gularity exponents of the notch was obtained, then the nonlinear eigenvalue problem was trans- formed into a linear one by means of variable substitution, and the interpolating matrix method was employed to solve the problem to determine the stress singularity exponents and associated characteristic functions at the notch tip in the orthotropic bi-material plate. With the present method, both the stress singularity exponents and the associated characteristic angle functions can be acquired simultaneously, and the stress singularity exponents can be easily distinguished between plane and anti-plane singularities according to the corresponding characteristic angle functions. Validity of the present method is confirmed in comparison with the existing results through numerical calculation.