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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[1] YANG W G, DING D H, WANG R H, et al. Thermodynamics of equilibrium properties of quasicrystals[J]. Zeitschrift für Physik B: Condensed Matter, 1997, 100: 447-454. doi: 10.1007/s002570050146 [2] 范天佑. 准晶数学弹性理论及应用[M]. 北京: 北京理工大学出版社, 1999.FAN Tianyou. Quasicrystal Mathematical Elasticity Theory and Its Application[M]. Beijing: Beijing Institute of Technology Press, 1999. (in Chinese) [3] 祝爱玉. 三维准晶弹性高阶偏微分方程的解析解和准晶的弹性动力学研究[D]. 北京: 北京理工大学, 2009.ZHU Aiyu. Analysis solutions of high order partial differential equations of three-dimentional quasicrystals in elasticity and elasto-hydro dynamics of quasicrystals[D]. Beijing: Beijing Institute of Technology, 2009. (in Chinese) [4] HWU C, YEN W J. Green's functions of two-dimensional anisotropic plates containing an elliptic hole[J]. International Journal of Solids and Structures, 1991, 27(13): 1705-1719. doi: 10.1016/0020-7683(91)90070-V [5] 舒小敏, 李坚. 基于边界元法求解三维弹性摩擦接触问题[J]. 计算力学学报, 2022, 39(5): 557-565. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG202205004.htmSHU Xiaomin, LI Jian. Solving 3D elastic friction contact problem by boundary element method[J]. Chinese Journal of Computational Mechanics, 2022, 39(5): 557-565. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG202205004.htm [6] XU X L, RAJAPAKSE R. Boundary element analysis of piezoelectric solids with defects[J]. Composites Part B: Engineering, 1998, 29(5): 655-669. doi: 10.1016/S1359-8368(98)00022-5 [7] LIANG Y C, HWU C. Electromechanical analysis of defects in piezoelectric materials[J]. Smart Materials and Structures, 1996, 5(3): 314-320. doi: 10.1088/0964-1726/5/3/009 [8] 袁彦鹏. 准晶材料平面断裂问题分析[D]. 郑州: 郑州大学, 2018.YUAN Yanpeng. Analysis of plane fracture problem of quasicrystal[D]. Zhengzhou: Zhengzhou University, 2018. (in Chinese) [9] 陈帅. 一维六方准晶复合材料平面断裂问题研究[D]. 郑州: 郑州大学, 2019.CHEN Shuai. Study on plane fracture problem of one-dimensional hexagonal quasicrystal bi-material[D]. Zhengzhou: Zhengzhou University, 2019. (in Chinese) [10] 潘先云, 余江鸿, 周枫林. 非齐次弹性力学问题双互易边界元方法研究[J]. 应用数学和力学, 2022, 43(9): 1004-1015. doi: 10.21656/1000-0887.420208PAN Xianyun, YU Jianghong, ZHOU Fenglin. Research on the dual reciprocity boundary element method for non-homogeneous elasticity problems[J]. Applied Mathematics and Mechanics, 2022, 43(9): 1004-1015. (in Chinese) doi: 10.21656/1000-0887.420208 [11] 翟婷, 马园园, 赵雪芬. 三维二十面体准晶圆弧形界面刚性线的平面弹性问题[J/OL]. 应用力学学报, 2023[2023-12-12]. https://kns.cnki.net/kcms/detail//61.1112.O3.20230223.1117.004.html .ZHAI Ting, MA Yuanyuan, ZHAO Xuefen. The plane elasticity problem of circular arc interface rigid lines in three-dimensional icosahedral quasicrystals[J/OL]. Chinese Journal of Applied Mechanics, 2023[2023-12-12].https://kns.cnki.net/kcms/detail//61.1112.O3.20230223.1117.004.html . (in Chinese)[12] WANG J B, MANCINI L, WANG R, et al. Phonon-and phason-type spherical inclusions in icosahedral quasicrystals[J]. Journal of Physics: Condensed Matter, 2003, 15(24): L363- L370. doi: 10.1088/0953-8984/15/24/102 [13] FAN T Y, GUO L H. The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals[J]. Physics letters A, 2005, 341 (1/4): 235-239. [14] DING D H, YANG W G, HU C Z, et al. Generalized elasticity theory of quasicrystals[J]. Physical Review B, 1993, 48(10): 7003-7010. doi: 10.1103/PhysRevB.48.7003 [15] GAO Y, RICOEUR A, ZHANG L. Plane problems of cubic quasicrystal media with an elliptic hole or a crack[J]. Physics Letters A, 2011, 375(28/29): 2775-2781. [16] LI L H, LIU G T. Stroh formalism for icosahedral quasicrystal and its application[J]. Physics Letters A, 2012, 376(28/29): 987-990. [17] YANG L Z, ANDREAS R, HE F M, et al. Finite size specimens with cracks of icosahedral Al-Pd-Mn quasicrystals[J]. Chinese Physics B, 2014, 23(5): 056102. doi: 10.1088/1674-1056/23/5/056102 [18] LI L H. Generalized 2D problem of icosahedral quasicrystals containing an elliptic hole[J]. Chinese Physics B, 2013, 22(11): 116101. doi: 10.1088/1674-1056/22/11/116101 [19] QIN Q H. Thermoelectroelastic Green's function for a piezoelectric plate containing an elliptic hole[J]. Mechanics of Materials, 1998, 30(1): 21-29. doi: 10.1016/S0167-6636(98)00022-2 [20] 张炳彩, 丁生虎, 张来萍. 一维六方准晶双材料中圆孔边共线界面裂纹的反平面问题[J]. 应用数学和力学, 2022, 43(6): 639-647. doi: 10.21656/1000-0887.420202ZHANG Bingcai, DING Shenghu, ZHANG Laiping. The anti-plane problem of collinear interface cracks emanating from a circular hole in 1D hexagonal quasicrystal bi-materials[J]. Applied Mathematics and Mechanics, 2022, 43(6): 639-647. (in Chinese) doi: 10.21656/1000-0887.420202 [21] 卢绍楠, 赵雪芬, 马园园. 一维六方压电准晶双材料界面共线裂纹问题[J]. 应用数学和力学, 2023, 44(7): 809-824. doi: 10.21656/1000-0887.430111LU Shaonan, ZHAO Xuefen, MA Yuanyuan. Research on interfacial collinear cracks between 1D hexagonal piezoelectric quasicrystal bimaterials[J]. Applied Mathematics and Mechanics, 2023, 44(7): 809-824. (in Chinese) doi: 10.21656/1000-0887.430111 [22] 尹姝媛, 周旺民, 范天佑. 八次对称二维准晶中的Ⅱ型裂纹[J]. 应用数学和力学, 2002, 23(4): 376-380. http://www.applmathmech.cn/article/id/1803YIN Shuyuan, ZHOU Wangmin, FAN Tianyou. A mode Ⅱ crack in a two-dimensional octagonal quasicrystals[J]. Applied Mathematics and Mechanics, 2002, 23(4): 376-380. (in Chinese) http://www.applmathmech.cn/article/id/1803 [23] LU P, WILLIAMS W. Green functions of piezoelectric material with an elliptic hole or inclusion[J]. International Journal of Solids and Structures, 1998, 35(7/8): 651-664. [24] ZHOU Y B, LI X F. Fracture analysis of an infinite 1D hexagonal piezoelectric quasicrystal plate with a penny-shaped dielectric crack[J]. European Journal of Mechanics A: Solids, 2019, 76: 224-234. doi: 10.1016/j.euromechsol.2019.04.011 [25] KAMEL M, LIAW M. Green's functions due to concentrated moments applied in an anisotropic plane with an elliptic hole or a crack[J]. Mechanics Research Communications, 1989, 16(5): 311-319. doi: 10.1016/0093-6413(89)90071-2 [26] GUO J H, YU J, XING Y M, et al. Thermoelastic analysis of a two-dimensional decagonal quasicrystal with a conductive elliptic hole[J]. Acta Mechanica, 2016, 227(9): 2595-2607. doi: 10.1007/s00707-016-1657-7 [27] ZHAO X F, LI X, DING S H. Two kinds of contact problems in three-dimensional icosahedral quasicrystals[J]. Applied Mathematics and Mechanics(English Edition), 2015, 36(12): 1569-1580. doi: 10.1007/s10483-015-2006-6 [28] WANG X F, FAN T Y, ZHU A Y. Dynamic behaviour of the icosahedral Al-Pd-Mn quasicrystal with a Griffith crack[J]. Chinese Physics B, 2009, 18(2): 709-714. doi: 10.1088/1674-1056/18/2/050 [29] 吴祥法, 范天佑, 安冬梅. 用路径守恒积分计算平面准晶裂纹扩展的能量释放率[J]. 计算力学学报, 2000, 17(1): 34-42. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG200001005.htmWU Xiangfa, FAN Tianyou, AN Dongmei. Energy release rate of plane quasicrystals with crack determined by path-independent E-integral[J]. Chinese Journal of Computational Mechanics, 2000, 17(1): 34-42. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG200001005.htm [30] 吴云龙. 十面体准晶和二十面体准晶动态断裂的有限差分分析[D]. 北京: 北京理工大学, 2013.WU Yunlong. Finite difference analysis of dynamic fracture in decagonal and icosahedral quasicrystals[D]. Beijing: Beijing Institute of Technology, 2013. (in Chinese) [31] 杨连枝, 张亮亮, 孙振东, 等. 二十面体准晶的有限元模拟方法[C]//中国力学大会: 2013论文摘要集. 工程科技I辑, 2013.YANG Lianzhi, ZHANG Liangliang, SUN Zhendong, et al. Finite element simulation of icosahedral quasicrystals[C]//Chinese Mechanics Congress: 2013 Abstracts Collection. Engineering Technology Series I, 2013. (in Chinese) [32] 杨连枝, 何蕃民, 高阳. 二十面体Al-Pd-Mn准晶板的断裂行为研究[C]//北京力学会第20届学术年会论文集. 北京: 中国学术期刊电子出版社, 2014: 2.YANG Lianzhi, HE Fanmin, GAO Yang. Study of fracture behavior of icosahedral Al-Pd-Mn quasicrystal slab[C]//The Proceedings of the 20th Annual Conference of Beijing Society of Theoretical and Applied Mechanics. Beijing: China Academic Journal Electronic Publishing House, 2014: 2. (in Chinese) [33] GAUL L, KÖGL M, WAGNER M. Boundary Element Methods for Engineers and Scientists: an Introductory Course With Advanced Topics[M]. New York: Springer-Verlag, 2003. [34] 航空工业部科学技术委员会. 应力集中系数手册[M]. 北京: 高等教育出版社, 1990.Science and Technology Committee of the Ministry of Aeronautics Industry. Stress Concentration Factors Handbook[M]. Beijing: Higher Education Press, 1990. (in Chinese) -
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