混合Trefftz有限元法反平面断裂问题的探讨

DISCUSSION ON ANTI-PLANE CRACK PROBLEMS BY HYBRID TREFFTZ FINITE ELEMENT APPROACH

基于修正变分原理,采用满足控制微分方程的应力和位移混合Trefftz函数,满足裂纹单元断裂性质的特殊Trefftz函数以及满足裂纹尖端条件的附加试函数,推导出混合Trefftz有限元法反平面断裂问题公式,给出应力集中因子解析表达式.同时,给出单个边裂纹、单个曲折裂纹和三个边裂纹反平面裂纹问题三个算例,探讨特殊Trefftz函数个数、破裂单元个数、高斯点数以及不同附加试函数对结果的影响.最后,将计算结果与一般有限元算法或其它方法结果进行对比,分析了混合Trefftz有限元法的精确性和高效性.

This paper presents basic formulations and applications of method to anti-plane crack problems. Based on a modified variational princ satisfying governing differential equations for intraelement stresses and displ hybrid Trefftz finite element iple, hybrid Trefftz functions acements,a special purpose T-functions meeting the frcture behavior for crack elements, and uxiliary functions agreeing with singular properties for crack tips, the process are deduced. After that, the stress intensity factor（SIF） is directly evaluated from the coefficients of the special purpose T-functions. The performance of the proposed element formultions is assessed by three examples of a plane with a side crack, kinked crack and triple-crack in anti-plane shear problems. The sensitivity of number of special T-complete functions, number of crack elements,number of Gauss polnts,and different type of auxiliary functions to the accuracy of SIF are discussed. Finally,the results of this paper are compared with those obtained by conventional FEM or other approaches.