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A new triangular element for crack front and the application in analysis of crack problems

DOI：

 作者 单位 E-mail 周枫林 湖南大学机械与运载工程学院 zhoufl@hnu.edu.cn 谢贵重 湖南大学机械与运载工程学院 张见明 湖南大学机械与运载工程学院 zhangjm@hnu.edu.cn 李落星 湖南大学机械与运载工程学院

构造了一种适合边界元分析裂纹问题的三角形单元，该单元中的形函数包含两部分，其主要部分用于捕捉裂纹尖端上位移分布的陡峭特性（性质），另一部分为常规的拟合函数，体现裂纹尖端位置附近的物理量在其他方向上的连续分布。形函数主要部分的构造充分利用了已有理论研究获得的结论，在裂纹表面上，随着距离远离尖端，位移分布与sqrt(r)函数保持同阶变化。在传统形函数的基础上，通过先乘以一项同阶于sqrt(r)的变量项，再在系数中将其在形函数所在点上的值除去，便得到新型的用于拟合裂纹尖端附近位移和面力分布的形函数。新的形函数能够满足形函数的delta性质，但归一性不再满足，因此，新的形函数只用于物理量的拟合，而几何量的拟合依然采用传统方案。在通过对偶边界元方法计算裂纹尖端的张开位移之后，利用一种位移外插方法计算获得应力强度因子。数值算例关注了一种限域内的圆盘裂纹，应用新构造的三角形单元于对偶边界元中计算结构在受到斜拉力时裂纹尖端的三种应力强度因子。通过与参考解进行对比，验证了该插值方案用于对偶边界元分析裂纹问题时的正确性和高精度。

A new type of triangular element one edge of which lies on the crack front is developed for boundary element analysis of crack problems. The shape function in this type of elements consisted of two parts. The major part of the shape function is constructed to approximate the sharp variation of the physical quantities including displacements and stresses in the places near the crack front in the crack surfaces. The other part is the traditional type of shape functions that are defined for triangular elements. In the major part of the shape function, a variation similar to sqrt(r) is involved since it has been theoretically proved that the distribution of the displacement around the crack front is of the same order as that of sqrt(r). In the newly defined shape function, a specially constructed term, which is of the same order as that of , is multiplied by the traditional shape function. In order to guarantee the delta property of the shape function, however, the value of the specially constructed term at the interpolation points should be divided by the corresponding coefficients of the shape function. The proposed elements should only be applied for physical quantities approximation near the crack front. The geometric quantities in the same area should be approximated by a traditional method. In the application of this elements in the dual boundary element method for crack problems, the crack of displacements (CODs) in the crack surfaces is first computed. Then an extrapolation method is applied to compute the stress intensity factors at the crack front. As the illustration example, a circle shaped crack in an infinite domain is considered. Three types of SIFs at crack front are computed by the dual boundary element method with employing the new element. Comparisons with the analytical results have been made to demonstrate the validity and accuracy of the element.
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