Boundary Element Analysis for the Plane Elasticity Problems of Finite Icosahedral Quasicrystal Plates Containing Elliptical Holes
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摘要: 基于扩展的Stroh方法, 对含椭圆孔有限大二十面体准晶板平面弹性问题进行边界元分析.首先利用扩展的Stroh方法, 研究了二十面体准晶的Green函数, 得到了含椭圆孔无限大二十面体准晶平面弹性问题位移和应力的基本解.利用该基本解, 通过加权余量法建立了区域内积分方程和边界积分方程, 并采用线性插值函数及Gauss积分对含未知量的边界积分方程和区域内积分方程分别进行离散，得到了离散格式.进一步, 对椭圆孔的孔边应力进行了数值求解, 并将有限大板的数值结果与无限大板的解析解进行了对比验证, 说明当板与椭圆孔尺寸之比小于某下限值时, 不能用无限大板的解析解对有限大板进行分析.最后, 分析了在垂向拉伸作用下, 板的大小、孔口尺寸及倾斜角度对孔边应力的影响.结果表明: 板的尺寸沿垂直拉伸方向变化对孔边应力的影响更明显; 随着椭圆孔尺寸的增加, 孔边应力集中现象越明显; 若长轴垂直拉伸方向, 椭圆孔倾斜会减缓孔边应力集中程度.Abstract: Based on the extended Stroh method, a boundary element analysis was conducted for the plane elasticity problem of finite-sized icosahedral quasicrystal plates with elliptical holes. Firstly, the extended Stroh method was used to study Green's function for the icosahedral quasicrystal, to obtain the fundamental solutions of displacements and stresses of the plane elasticity problem about infinite-sized icosahedral quasicrystal plates with elliptical holes. With these fundamental solutions, the weighted residual method was employed to establish the integral equations within the domain and on the boundary, and the linear interpolation functions and the Gaussian integration were used to discretize the boundary integral equations and the domain integral equations with unknown variables, respectively. Furthermore, the stress at the hole boundary was numerically solved, and the numerical results of the finite-sized plate were compared with the analytical solution of the infinite-sized plate to demonstrate that, the analytical solution of the infinite-sized plate cannot be used for the analysis of the finite-sized plate with the ratio of the plate size to the hole size below a certain threshold. Finally, the effects of the plate size, the hole size, and the inclination angle on the stress at the hole boundary were analyzed under tensile loading in the vertical direction. The results show that, the variation of the plate size along the vertical tensile direction has a more significant effect on the stress at the hole boundary. As the elliptical hole size increases, the stress concentration phenomenon becomes more pronounced. If the major axis is perpendicular to the vertical tensile direction, the inclination of the elliptical hole will mitigate the degree of stress concentration at the hole boundary.
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Key words:
- icosahedral quasicrystal /
- Stroh method /
- elliptical hole /
- finite size /
- stress concentration factor
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图 3 声子场应力集中系数随板尺寸的变化
Figure 3. The variations of phonon stress concentration coefficients with plate sizes
图 4 边界元解与无限大板解析解对比(H/a=200, W/a=100)
Figure 4. Comparison of the boundary element solution and the infinite plate analytical solution(H/a=200, W/a=100)
图 5 边界元解与无限大板解析解对比(H/a=40, W/a=8)
Figure 5. Comparison of the boundary element solution and the infinite plate analytical solution (H/a=40, W/a=8)
图 6 取a=0.2 m，孔边应力值随b的变化情况
Figure 6. For a=0.2 m, the changes of hole edge stresses with b
图 7 取b=0.1 m，孔边应力值随a的变化情况
Figure 7. For b=0.1 m, the changes of hole edge stresses with a
表 1 应力集中系数随α的变化情况
Table 1. The change of the stress concentration coefficient with α
α/(°) 0 15 45 75 90 stress concentration coefficient 5.098 2 4.767 9 3.651 6 2.258 2 2.018 6 
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