带裂纹十次对称二维准晶平面弹性的无摩擦接触问题
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摘要
借助经典平面弹性复变函数方法,研究了单个刚性凸基底压头作用下,带任意形状裂纹十次对称二维准晶半平面弹性的无摩擦接触问题.利用十次对称二维准晶位移、应力的复变函数表达式,带任意形状裂纹的准晶半平面弹性无摩擦接触问题被转换为可解的解析函数复合边值问题,进而简化成一类可解的Riemann边值问题.通过求解Riemann边值问题,得到了应力函数的封闭解,并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式.从压头下方接触应力的表达式可以看出,接触应力在压头边缘和裂纹端点处具有奇异性.当忽略相位子场影响时,该文所得结论与弹性材料对应结果一致.数值算例分别给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的十次对称二维准晶下半平面的结果.该文所得结论为准晶材料的应用提供了理论参考.
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 R iemann boundary problems.Solutions to the R iemann 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 show s 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.N umerical 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.
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 R iemann boundary problems.Solutions to the R iemann 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 show s 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.N umerical 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.
引文
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[15]ZHAO X F,LI X,DING S H.Two kinds of contact problems in dodecagonal quasicrystals of point group 12 mm[J].A cta Mechanica Solida Sinica,2016,29(2):167-177.
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[17]ZHAO X F,LI X,DING S H.Two kinds of contact problems in three-dimensional icosahedral quasicrystals[J].A pplied 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 W uhan University(N atural Sciences),1993,3:21-28.(in C hinese))
[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 V alue Problem of A nalytic Functions[M].W uhan:W uhan U niversity Press,2009.(in C hinese))
[20]MUSKHELISHIVILI N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Cambridge:C ambridge U niversity 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.(HU AN G M inhai,JIAN G Yong.C ontact problem of elastic half-plane w ith cracks[J].Journal of Xiamen University(N atural Science),2000,39(4):437-440.(in C hinese))
[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.
[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 w ith crack[J].A dvances in A pplied Mathematics,2012,1(2):49-58.(in C hinese))
[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 C omputational 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].A cta 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 of2D decagonal quasicrystals of piezoelectric effect w ith Griffith crack[J].Journal of Zhejiang University(Engineering Science),2018,52(3):487-496.(in C hinese))
[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 onedimensional hexagonal piezoelectric quasicrystals[J].C hinese 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 onedimensional hexagonal quasicrystals[J].C hinese Journal of A pplied Mechanics,2018,35(1):8-14.(in C hinese))
[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 C hinese))
[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 tw o-dimensional quasicrystals[J].C hinese Journal of Solid Mechanics,2016,37(1):83-89.(in C hinese))
[15]ZHAO X F,LI X,DING S H.Two kinds of contact problems in dodecagonal quasicrystals of point group 12 mm[J].A cta 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].A pplied 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 W uhan University(N atural Sciences),1993,3:21-28.(in C hinese))
[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 V alue Problem of A nalytic Functions[M].W uhan:W uhan U niversity Press,2009.(in C hinese))
[20]MUSKHELISHIVILI N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Cambridge:C ambridge U niversity 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.(HU AN G M inhai,JIAN G Yong.C ontact problem of elastic half-plane w ith cracks[J].Journal of Xiamen University(N atural Science),2000,39(4):437-440.(in C hinese))
[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.