广义参数Williams单元分析受弯裂纹梁的应力强度因子

Williams element with generalized degrees of freedom for stress intensity factor of crack in bending beam

鉴于普通有限元和奇异单元分析三点或四点弯曲裂纹梁时存在精度不高的问题,研究建立了裂尖应力强度因子分析的广义参数Williams单元。首先对梁中裂尖奇异区子单元建立普通有限元刚度方程,然后利用总体位移场和单元位移场之间的关系,将普通单元刚度方程转换成广义参数单元刚度方程,并利用等比级数求和公式获得Williams单元刚度方程的显性表达式,进而分析了影响Williams单元的3个重要参数——径向离散比例因子、子单元数以及级数项数对应力强度因子的影响,并确定了3个参数的取值,α=0.9,m=6,n=100,同时分析了裂纹长度对裂纹梁跨中位移的影响。结果表明,Williams单元比ANSYS商业软件具有更高的计算精度,能够快速稳定收敛,可以较好地解决梁的断裂问题。

In order to circumvent the problem arising from the ordinary and singular element in three-point and four-point bending beams with crack, a Williams element with generalized degrees of freedom （GDOFs） is presented for the calculation of stress intensity factor （SIF）. The stiffness e- quation of ordinary sub-elements is developed in the singular region, and the stiffness equation of Williams element can be obtained by means of geometric series summation based on the relationship between local and global displacement field. The influence of three important parameters, including discretization ratio, number of series terms and sub-element number, is investigated and the values are determined to be α=0. 9, m=6, and n= 100. Displacements at mid-span of the beam are ana- lysed when crack grows. Example analyses show that Williams element is applicable to tacklecracked beams and has a higher accuracy and faster convergence compared to singular elements in commercial FEM code.