ZHANO Xuefen, LI Xing. A Frictionless Contact Problem of 2D Decagonal Quasicrystal Plane Elasticity With Cracks[J]. Applied Mathematics and Mechanics, 2019, 40(2): 223-236. doi: 10.21656/1000-0887.390127
 Citation: ZHANO Xuefen, LI Xing. A Frictionless Contact Problem of 2D Decagonal Quasicrystal Plane Elasticity With Cracks[J]. Applied Mathematics and Mechanics, 2019, 40(2): 223-236.

# A Frictionless Contact Problem of 2D Decagonal Quasicrystal Plane Elasticity With Cracks

##### doi: 10.21656/1000-0887.390127
Funds:  The National Natural Science Foundation of China(11762017)
• Rev Recd Date: 2018-06-24
• Publish Date: 2019-02-01
• With the classical complex function method, a frictionless contact problem of 2D decagonal quasicrystal semiplane elasticity with arbitraryform cracks was addressed under the action of a rigid convex basal punch. Based on complex expressions of stresses and displacements of 2D decagonal quasicrystals, the problem was converted into solvable boundary value problems with analytic functions, and then reduced to a class of Riemann boundary problems. Solutions to the Riemann boundary problems give the stress functions in closed form, the explicit expressions of the stress intensity factors at crack tips and the contact stress distribution under the punch. The expression of the contact stress shows that, it has singularity at the edge of the contact zone and the crack tips. Without the effect of the phason field, the obtained results match well with those classical conclusions for elastic materials. Numerical examples illustrated the solutions to the frictionless contact problem in 2D decagonal quasicrystal semiplane elasticity with a vertical crack and a horizontal straight crack under a rigid punch. The work provides a theoretical reference for the application of quasicrystalline materials.
•  [1] 杨臻, 牛莉莎. 弹性半平面中边界裂纹研究的新方法[J］. 计算力学学报, 2007,24(1): 35-39.(YANG Zhen, NIU Lisha. A new method studying the edge crack problem on an elastic half plane[J］. Chinese Journal of Computational Mechanics,2007,24(1): 35-39.(in Chinese)) [2] 李星, 庞明军. 含裂纹功能梯度材料热接触的奇异积分方程方法[J］. 应用数学进展, 2012,1(2): 49-58.(LI Xing, PANG Mingjun. Singular integral equation method for thermal contact problem of FGM with crack[J］. Advances in Applied Mathematics,2012,1(2): 49-58.(in Chinese)) [3] CHEN Y Z, LIN X Y, WANG Z X, et al. Solution of contact problem for an arc crack using hypersingular integral equation[J］. International Journal of Computational Methods,2008,5(1): 119-133. [4] CHEN W Q. Some recent advances in 3D crack and contact analysis of elastic solids with transverse isotropy and multifield coupling[J］. Acta Mechanica Sinica,2015,31(5): 601-626. [5] SHECHTMAN D, BLECH I, GRATIAS D, et al. Metallic phason with long-range orientational order and no translational symmetry[J］. Physical Review Letters,1984,53: 195l-1953. [6] DUBOIS J M, BRUNET P, COSTIN W,et al. Friction and fretting on quasicrystals under vacuum[J］. Journal of Non-Crystalline Solids,2004,334/335: 475-480. [7] FAN T Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications [M］. Beijing: Science Press, 2010. [8] 徐文帅, 杨连枝, 高阳. 二维十次对称压电准晶含Griffith裂纹的平面问题[J］. 浙江大学学报(工学版), 2018,52(3): 487-496.(XU Wenshuai, YANG Lianzhi, GAO Yang. Plane problems of 2D decagonal quasicrystals of piezoelectric effect with Griffith crack[J］. Journal of Zhejiang University(Engineering Science),2018,52(3): 487-496.(in Chinese)) [9] TUPHOLME G E. An antiplane shear crack moving in one-dimensional hexagonal quasicrystals[J］. International Journal of Solids and Structures,2015,71: 255-261. [10] ZHONG L J, LIU G T. The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals[J］. Chinese Physics B,2017,26(4): 245-251. [11] 赵雪芬, 李星. 一维六方准晶非周期半平面的有限摩擦接触问题[J］. 应用力学学报, 2018,35(1): 8-14.(ZHAO Xuefen, LI Xing. The frictional contact problem for aperiodical half-plane in one-dimensional hexagonal quasicrystals[J］. Chinese Journal of Applied Mechanics,2018,35(1): 8-14.(in Chinese)) [12] 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): 699-707.(MA Xiaodan, LI Xing. 2 kinds of periodic contact problems of 1D hexagonal quasicrystals[J］. Applied Mathematics and Mechanics,2016,37(7): 699-707.(in Chinese)) [13] LI X Y, WU Y F, CHEN W Q. Indentation on two-dimensional hexagonal quasicrystals[J］. Mechanics of Materials,2014,76: 121-136. [14] 叶玉娇, 李星, 赵雪芬. 十二次对称二维准晶中的无摩擦接触问题[J］. 固体力学学报, 2016,37(1): 83-89.(YE Yujiao, LI Xing, ZHAO Xuefen. Frictionless contact problem of dodecagonal system in two-dimensional quasicrystals[J］. Chinese Journal of Solid Mechanics,2016,37(1): 83-89.(in Chinese)) [15] ZHAO X F, LI X, DING S H. Two kinds of contact problems in dodecagonal quasicrystals of point group 12 mm[J］. Acta Mechanica Solida Sinica,2016,29(2): 167-177. [16] ZHOU W M, FAN T Y, YIN S Y. Axisymmetric contact problem of cubic quasicrystalline materials[J］. Acta Mechanica Solidarity Sinica,2002,15(1): 68-74. [17] 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. [18] 胡承正, 丁棣华, 杨文革. 五次八次十次和十二次对称准晶的弹性性质[J］. 武汉大学学报(自然科学版), 1993,3: 21-28.(HU Chengzheng, DING Dihua, YANG Wenge. Elastic properties of pentagonal, octagonal, decagonal and dodecagonal quasicrystals[J］. Journal of Wuhan University(Natural Sciences),1993,3: 21-28.(in Chinese)) [19] 路见可. 解析函数边值问题教程[M］. 武汉: 武汉大学出版社, 2009.(LU Jianke. A Tutorial on the Boundary Value Problem of Analytic Functions [M］. Wuhan: Wuhan University Press, 2009.(in Chinese)) [20] MUSKHELISHIVILI N I. Some Basic Problems of the Mathematical Theory of Elasticity [M］. Cambridge: Cambridge University Press, 1953. [21] 黄民海, 姜永. 带裂纹的弹性半平面接触问题[J］. 厦门大学学报(自然科学版), 2000,39(4): 437-440.(HUANG Minhai, JIANG Yong. Contact problem of elastic half-plane with cracks[J］. Journal of Xiamen University(Natural Science),2000,〖STHZ〗 39(4): 437-440.(in Chinese)) [22] PANASYUK V V, DATSYSHYN O P, MARCHENKO H P. Stress state of a half-plane with cracks under rigid punch action[J］. International Journal of Fracture,2000,101(4): 347-363.

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