含椭圆孔有限大二十面体准晶板平面弹性问题的边界元分析  

Boundary Element Analysis for the Plane Elasticity Problems of Finite Icosahedral Quasicrystal Plates Containing Elliptical Holes

作　　者：王会苹 王桂霞[1,2] 陈德财 WANG Huiping;WANG Guixia;CHEN Decai(College of Mathematics Science,Inner Mongolia Normal University,Hohhot 010022,P.R.China;Inner Mongolia Center of Applied Mathematics,Hohhot 010022,P.R.China)

机构地区：[1]内蒙古师范大学数学科学学院,呼和浩特010022 [2]内蒙古自治区应用数学中心,呼和浩特010022

出　　处：《应用数学和力学》2024年第4期400-415,共16页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11962026);内蒙古自治区自然科学基金重点项目(2022ZD05);内蒙古自治区高校科研项目(NJZZ21003)。

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.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.

关 键 词：二十面体准晶 Stroh方法 椭圆孔 有限尺寸 应力集中系数 

分 类 号：O343.1[理学—固体力学] O343.4[理学—力学] O241.82