Williams单元分析I-II混合型裂纹应力强度因子

Stress Intensity Factor for mixed mode cracks by Williams element

为了建立高效、精确的混合型裂纹应力强度因子分析的裂尖奇异区单元,文章在改进Williams级数的基础上建立了裂尖应力奇异域单元的整体位移场,基于普通有限元形函数建立了奇异域子单元的局部位移场,利用整体位移场控制局部子单元的节点位移,结合有限项等比级数求和公式建立了I-II混合型Williams单元刚度方程。根据该单元模型中与应力强度因子相关的参数,可以直接计算裂尖I型、II型应力强度因子,克服了普通单元和奇异单元需要通过中间物理量回归分析、外推计算裂尖处应力强度因子的缺陷,并能取得很高的计算精度和计算效率。结合算例,分析了裂纹长度和倾斜角等参数对应力强度因子的影响,确定了Williams单元的径向比例因子、子单元数、级数项等三个重要参数的取值。

An element discretizing the singular region around the crack tip for stress intensity factor of mixed mode cracks is developed, the improved Williams series is applied to define the global displacement field of element in singular region around crack tip, while the local displacement field of subelement is ap- proximated by employing the shape function of common finite element method. The global displacement field governs nodal displacement of subelement, so that the stiffness equation of Williams element for mixed mode is developed by using the theorem for summation of the finite geometric proportion series. The stress intensity factor can be evaluated analytically for mixed mode cracks by the corresponding undetermined constant in the model of Williams element, which overcomes the disadvantage of the singular element in de- termination of the stress intensity factor via extrapolation and regression analysis of intermediate physical quantity. Examples are considered to demonstrate the high accuracy and efficiency of proposed Williams element. The parametric study is implemented to illustrate the stress intensity factor versus the length and inclination angle of the crack, and determine the values of three important parameters for the Williams element, including the radial scale factor, the number of subelement and the series term.