Analytical Solutions to Problems of Elliptical Holes With 4 Edge Cracks in 1D Orthorhombic Quasicrystals

GAO Yuanyuan; LIU Guanting

doi:10.21656/1000-0887.390032

Volume 40 Issue 2

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GAO Yuanyuan, LIU Guanting. Analytical Solutions to Problems of Elliptical Holes With 4 Edge Cracks in 1D Orthorhombic Quasicrystals[J]. Applied Mathematics and Mechanics, 2019, 40(2): 210-222. doi: 10.21656/1000-0887.390032

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Analytical Solutions to Problems of Elliptical Holes With 4 Edge Cracks in 1D Orthorhombic Quasicrystals

doi: 10.21656/1000-0887.390032

College of Mathematical Science, Inner Mongolia Normal University, Hohhot 010022, P.R.China

Funds: The National Key R&D Program of China(2017YFC1405605)

Received Date: 2018-01-22
Rev Recd Date: 2018-06-24
Publish Date: 2019-02-01

Abstract

Abstract

The plane elastic problems of elliptical holes with 4 cracks in 1D orthorhombic quasicrystals were investigated through introduction of a new generalized conformal mapping with the generalized complex variable method. With the stress functions, the basic elasticity equations were reduced to 4th-order partial differential equations, the complex expression of stress components was derived in the image plane and the analytical solution of stress intensity factors (SIFs) at the crack tips was found out. With the change of parameters describing the defects, the results can not only reduce to the conclusions in previous literatures, but also give the SIFs of a variety of common defect configurations, which provides a theoretical basis for engineering mechanics analysis.
- generalized conformal mapping,
- 1D orthorhombic quasicrystal,
- elliptical hole with 4 cracks,
- stress intensity factor

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References(23)

References

[1]	丁棣华, 王仁卉, 杨文革, 等. 准晶的弹性、塑性与位错[J]. 物理学进展, 1998,18(3): 223-260.(DING Dihua, WANG Renhui, YANG Wenge, et al. Elasticity, plasticity and dislocations of quasicrystals[J]. Progress in Physics,1998,18(3): 223-260.(in Chinese))
[2]	FAN T Y. Mathematical Theory of Elaticity of Quasicrystals and Its Applications [M]. Beijing: Science Press, 2010.
[3]	董闯. 准晶材料[M]. 北京: 国防工业出版社, 1998.(DONG Chuang. The Quasicrystal Material [M]. Beijing: National Defence Industry Press, 1998.(in Chinese))
[4]	LIU G T, FAN T Y. Complex method of the plane elasticity in 2D quasicrystal with point group 10 mm tenfold rotational symmetry and holey problems[J]. Science in China,2003,46(3): 326-336.
[5]	王仁卉, 胡承正, 桂嘉年. 准晶物理学[M]. 北京: 科学出版社, 2004.(WANG Renhui, HU Chengzheng, GUI Jianian. Quasicrystal Physics [M]. Beijing: Science Press, 2004.(in Chinese))
[6]	ZHOU W M, FAN T Y. Contact problem in decagonal two-dimensional quasicrystal[J]. Journal of Beijing Institute of Technology,2001,10(1): 51-55.
[7]	WANG X, PAN E. Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals[J]. Pramana,2008,70(5): 911-933.
[8]	刘官厅. 准晶弹性的复变方法与非线性发展方程的显示解[M]. 呼和浩特: 内蒙古人民出版社, 2005.(LIU Guanting. The Complex Variable Function Method of Quasicrystals Elastiaty and Analytic Solutions of Nonlinear Equations [M]. Hohhot: Inner Mongolia People Press, 2005.(in Chinese))
[9]	郭俊宏, 刘官厅. 一维六方准晶中具有不对称裂纹的圆形孔口问题的解析解[J]. 应用数学学报, 2007,30(6): 1066-1075.(GUO Junhong, LIU Guanting. Analytic solutions of the one-dimensional hexagonal qsicrystals about problem of a circular hole with asymmetry cracks[J]. Acta Mathematicae Applicatae Sinica,2007,30(6): 1066-1075.(in Chinese))
[10]	YANG J, LI X, DING S H. Anti-plane analysis of a circular hole with three unequal cracks in one-dimensional hexagonal piezoelectric quasicrystals[J]. Chinese Journal of Engineering Mathematics, 2016,33(2): 184-198.
[11]	杨丽星, 刘官厅. 一维六方准晶中带三条不对称裂纹的圆形孔口问题的解析解[J]. 数学的实践与认识, 2010,40(2): 148-156.(YANG Lixing, LIU Guanting. Analytic solutions of about a circular hole with three unequal straight edge cracks in one-dimensional hexagonal quasicrystals[J].Mathematics in Practice and Theory,2010,40(2): 148-156.(in Chinese))
[12]	郭俊宏, 刘官厅. 一维六方准晶中带双裂纹的椭圆孔口问题的解析解[J]. 应用数学和力学, 2008,29(4): 439-446.(GUO Junhong, LIU Guanting. Analytical solutions of problem about an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals[J]. Applied Mathematics and Mechanics,2008,29(4): 439-446.(in Chinese)).
[13]	李联和, 刘官厅. 一维六方准晶中螺型位错与楔形裂纹的相互作用[J]. 物理学报, 2012,61(8): 326-330.(LI Lianhe, LIU Guanting. A serew dislocation interacting with a wedge-shaped crack in one-dimensional hexagonal quasicrystals[J]. Acta Physica Sinica,2012,61(8): 326-330.(in Chinese))
[14]	FAN T Y, LI X F, SUN Y F. A moving screw dislocation in a one-dimensional hexagonal quasicrystal[J]. Acta Physica Sinica (Overseas Edition),1999,8(4): 288-295.
[15]	马小丹, 李星. 一维六方准晶的两类周期接触问题[J]. 应用数学和力学, 2016,37(7): 699-709.(MA Xiaodan, LI Xing. Kinds of periodic contact problems of 1D hexagonal quasicrystals[J]. Applied Mathematics and Mechanics,2016,37(7): 699-709.(in Chinese))
[16]	SLADEK J, SLADEK V, PAN E. Bending analyses of 1D orthorhombic quasicrystal plates[J]. International Journal of Solids and Structures,2013,50(24): 3975-3983.
[17]	LI Y, YANG L Z, GAO Y. An exact solution for a functionally graded multiayered one-dimensional orthorhombic quasicrystal plate[J]. Acta Mechanica,2017: 1-17. DOI: 10.1007/s00707-017-2028-8.
[18]	YANG L Z, GAO Y, PAN E, et al. An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate[J]. Acta Mechanica,2015,226(11): 3611-3621.
[19]	ZHANG L L, ZHANG Y M, GAO Y. General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect[J]. Physics Letters A,2014,378(37): 2768-2776.
[20]	于静, 刘官厅. 一维正方准晶椭圆孔口平面弹性问题的解析解[J]. 固体力学学报, 2010,31(4): 411-416.(YU Jing, LIU Guanting. Analytic solution of plane elasticity of one-dimensional orthorhombic quasicrystals with elliptical hole[J]. Chinese Journal of Solid Mechanics,2010,31(4): 411-416.(in Chinese))
[21]	高健, 刘官厅. 一维正方准晶中半无限裂纹问题的解析解[J]. 应用数学和力学, 2015,36(9): 945-955.(GAO Jian, LIU Guanting. Analytical solutions for problems of 1D orthorhombic quasicrystal with semi-infinite crack[J]. Applied Mathematics and Mechanics,2015,36(9): 945-955.(in Chinese))
[22]	匡震邦, 马法尚. 裂纹端部场[M]. 西安: 西安交通大学出版社, 2002.(KUANG Zhenbang, MA Fashang. Crack Tip Fields [M]. Xi’an: Xi’an Jiaotong Univercity Press, 2002.(in Chinese))
[23]	范天佑. 断裂理论基础[M]. 北京: 科学出版社, 2003.(FAN Tianyou. Fracture Theory Basis [M]. Beijing: Science Press,2003.(in Chinese))