Analytical Solutions to Problems of 1D Orthorhombic Quasicrystal With SemiInfinite Cracks

GAO Jian; LIU Guan-ting

doi:10.3879/j.issn.1000-0887.2015.09.006

Volume 36 Issue 9

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GAO Jian, LIU Guan-ting. Analytical Solutions to Problems of 1D Orthorhombic Quasicrystal With SemiInfinite Cracks[J]. Applied Mathematics and Mechanics, 2015, 36(9): 945-955. doi: 10.3879/j.issn.1000-0887.2015.09.006

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Analytical Solutions to Problems of 1D Orthorhombic Quasicrystal With SemiInfinite Cracks

doi: 10.3879/j.issn.1000-0887.2015.09.006

GAO Jian¹,
LIU Guan-ting²

¹College of Mathematics, Inner Mongolia University for the Nationalities, Tongliao, Neimenggu 028043, P.R.China;²College of Mathematical Science, Inner Mongolia Normal University, Hohhot 010022, P.R.China

Funds: The National Natural Science Foundation of China(11262017)

Received Date: 2015-01-14
Rev Recd Date: 2015-04-26
Publish Date: 2015-09-15

Abstract

Abstract

The anti-plane elasticity problem of the 1D orthorhombic quasicrystal with a semi-infinite crack penetrating along the quasiperiodic direction was investigated through introduction of a new generalized conformal mapping and with the generalized complex variable method. The analytical solutions of the stress fields and the stress intensity factors under the action of the uniform out-of-plane shear load on the partial crack surface were obtained. In addition, this method was applied to solve the plane elasticity problem of the 1D orthorhombic quasicrystal with a semi-infinite crack penetrating perpendicular to the quasiperiodic direction and the analytical solutions were derived. Under the condition of higher symmetry, the analytical solutions to the corresponding problem of the 1D tetragonal quasicrystal were also obtained.
- 1D orthorhombic quasicrystal,
- semi-infinite crack,
- generalized conformal mapping,
- stress field,
- stress intensity factor

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References

[1]	Shechtman D, Blech I, Gratias D, Cahn J W. Metallic phase with long-range orientational order and no translational symmetry[J]. Physical Review Letters,1984,53(20): 1951-1953.
[2]	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))
[3]	DING Di-hua, YANG Wen-ge, HU Cheng-zheng, WANG Ren-hui. Generalized elasticity theory of quasicrystals[J]. Physical Review B,1993,48(10): 7003-7010.
[4]	刘有延, 傅秀军, 董秀清. 一维准晶物理性质[J]. 物理学进展, 1997,17（1）: 1-24.(LIU You-yan, FU Xiu-jun, DONG Xiu-qing. Physical properties of one-dimensional quasicrystals[J]. Progress in Physics,1997,17(1): 1-24.(in Chinese))
[5]	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 Tian-you. Theory and Application of the Mathematic Elasticity About Quasicrystal [M]. Beijing: Beijing Institute of Technology Press, 1999.(in Chinese))
[6]	王仁卉, 胡承正, 桂嘉年. 准晶物理学[M]. 北京: 科学出版社, 2004.(WANG Ren-hui, HU Cheng-zheng, GUI Jia-nian. Quasicrystal Physics [M]. Beijing: Science Press, 2004.(in Chinese))
[7]	丁棣华, 王仁卉, 杨文革, 胡承正. 准晶的弹性、塑性与位错[J]. 物理学进展, 1998,18(3): 228-260.(DING Di-hua, WANG Ren-hui, YANG Wen-ge, HU Cheng-zheng. Elasticity, plasticity and dislocation of quasicrystals[J]. Progress in Physics,1998,18(3): 228-260.(in Chinese))
[8]	PENG Yan-ze, FAN Tian-you. Elastic theory of 1D-quasiperiodic stacking of 2D crystals[J]. Journal of Physics: Condensed Matter,2000,12(45): 9381-9387.
[9]	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 Guan-ting. The Complex Variable Function Method of Quasicrystals Elasticity and Analytic Solutions of Nonlinear Equations [M]. Hohhot: Inner Mongolia People Press, 2005.(in Chinese))
[10]	WANG Ren-hui, YANG Wen-ge, HU Cheng-zheng, DING Di-hua. Point and space groups and elastic behaviours of one-dimensional quasicrystals[J]. Journal of Physics: Condensed Matter,1997,9(11): 2411-2422.
[11]	孟祥敏, 佟百运, 吴玉琨. Al₆₅Cu₂₀Co₁₅准晶体的力学性能[J]. 金属学报, 1994,30（2）: 61-64.（MENG Xiang-min, TONG Bai-yun, WU Yu-kun. Mechanical properties of Al₆₅Cu₂₀Co₁₅ quasicrystal[J]. Acta Metallurgica Sinica,1994,30(2): 61-64.(in Chinese)）
[12]	LI Xian-fang, FAN Tian-you. A straight dislocation in one dimensional hexagonal quasicrystals[J]. Physica Status Solidi (B),1999,212(1): 19-26.
[13]	PENG Yan-ze, FAN Tian-you, JIANG Fu-ru, ZHANG Wei-guo, SUN Ying-fei. Perturbative method for solving elastic problems of one-dimensional hexagonal quasicrystals[J]. Journal of Physics: Condensed Matter,2001,13(18): 4123-4128.
[14]	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 Jun-hong, LIU Guan-ting. 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))
[15]	GUO Jun-hong, LU Zi-xing. Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals[J]. Applied Mathematics and Computation,2011,217(22): 9397-9403.
[16]	GAO Yang, XU Si-peng, ZHAO Bao-sheng. Boundary conditions for plate bending in one-dimensional hexagonal quasicrystals[J]. Journal of Elasticity,2007,86(3): 221-233.
[17]	LIU Guan-ting, GUO Rui-ping, FAN Tian-you. On the interaction between dislocations and cracks in one-dimensional hexagonal quasicrystals[J]. Chinese Physics,2003,12(10): 1149-1155.
[18]	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 Lian-he, LIU Guan-ting. A screw dislocation interacting with a wedge-shaped crack in one-dimensional hexagonal quasicrystals[J]. Acta Physica Sinica,2012,61（8）: 326-330.(in Chinese))
[19]	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 Guan-ting. 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))
[20]	刘官厅, 冯中华. 一维正方准晶椭圆孔口反平面问题的半逆解法[J]. 工程力学, 2013,30（2）: 38-43.(LIU Guan-ting, FENG Zhong-hua. Half-inverse method for the anti-plane problem of one-dimensional orthorhombic quasicrystals with elliptic hole[J]. Engineering Mechanics,2010,30(2): 38-43.(in Chinese))
[21]	刘莹, 刘官厅. 一维正方准晶垂直于准周期方向具有不对称共线裂纹的圆形孔口问题[J]. 数学的实践与认识, 2012,42（4）: 124-132.(LIU Ying, LIU Guan-ting. The problem of a circular hole with asymmetry collinear cracks perpendicular to quasi-periodic direction in one -dimensional orthorhombic quasicrystals[J]. Mathematics in Practice and Theory,2012,42(4): 124-132.(in Chinese))
[22]	范天佑. 断裂理论基础[M]. 北京: 科学出版社, 2003.(FAN Tian-you. Fracture Theory Basis [M]. Beijing: Science Press, 2003.(in Chinese))
[23]	GAO Cun-fa, FAN Wei-xun. Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack[J]. International Journal of Solids and Structures,1999,36(17): 2527-2540.
[24]	Sosa H A. Plane problems in piezoelectric media with defects[J]. International Journal of Solids and Structures,1991,28(4): 491-505.