含有非圆形双孔的无限平板中应力的解析解研究
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摘要
无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.
The analytical stress solution for an infinite plate containing a single hole of arbitrary shape can be obtained by complex variable method. As to the doubly-connected domain problem that an infinite plate contains two round holes or two elliptical holes, it can also be solved using a variety of methods, such as the bi-polar coordinate method, the stress function method, the complex variable method, and the Schwarz alternating method. The complex variable method combined with conformal mapping is one of importance methods which can be used to obtain analytical stress solution,but it is not yet used to solve the problem of an infinite plate containing a square hole and an elliptical hole. Taking advantage of the conformal mapping method, the problem that an infinite plate contains two specific holes, which far-field uniform stress is applied at infinity and the boundaries of the two holes are subjected to uniform vertical compression, can be solved. The key step of this method is to find the corresponding mapping function with which the considered region can be mapped onto a ring in the image plane. Based on the Riemann mapping theorem, we propose a general form of the mapping function and figure out the concrete mapping function for the specific problem using optimization method. Thebasic equation set for solving the two analytical functions is established through the stress boundary condition of the twholes. Then the analytical stress solution can be obtained according to the two analytical functions. The analytical stressolution is compared with numerical stress solution of ANSYS finite element method. Effects of separation distancesize of elliptical hole, and the orientation of holes on tangential stress of the boundary is investigated using the newlderived solution. The stress distributions on the line that connects the centers of the two holes under different loadare presented.
The analytical stress solution for an infinite plate containing a single hole of arbitrary shape can be obtained by complex variable method. As to the doubly-connected domain problem that an infinite plate contains two round holes or two elliptical holes, it can also be solved using a variety of methods, such as the bi-polar coordinate method, the stress function method, the complex variable method, and the Schwarz alternating method. The complex variable method combined with conformal mapping is one of importance methods which can be used to obtain analytical stress solution,but it is not yet used to solve the problem of an infinite plate containing a square hole and an elliptical hole. Taking advantage of the conformal mapping method, the problem that an infinite plate contains two specific holes, which far-field uniform stress is applied at infinity and the boundaries of the two holes are subjected to uniform vertical compression, can be solved. The key step of this method is to find the corresponding mapping function with which the considered region can be mapped onto a ring in the image plane. Based on the Riemann mapping theorem, we propose a general form of the mapping function and figure out the concrete mapping function for the specific problem using optimization method. Thebasic equation set for solving the two analytical functions is established through the stress boundary condition of the twholes. Then the analytical stress solution can be obtained according to the two analytical functions. The analytical stressolution is compared with numerical stress solution of ANSYS finite element method. Effects of separation distancesize of elliptical hole, and the orientation of holes on tangential stress of the boundary is investigated using the newlderived solution. The stress distributions on the line that connects the centers of the two holes under different loadare presented.
引文
1西田正孝,李安定.应力集中.北京:机械工业出版社,1986(Nishida Masataka,Li Anding.Stress Concertrations.Beijing:Machinery Industry Press,1986(in Chinese))
2王从曾,刘会亭.材料性能学.北京:北京工业大学出版社,2004(Wang Congzeng,Liu Huiting.Material Properties.Beijing:Beijing University of Technology Press,2004(in Chinese))
3孙可明,张树翠.含层理页岩气藏水力压裂裂纹扩展规律解析分析.力学学报,2016,48(5):1229-1237(Sun Keming,Zhang Shucui.Hydraulic fracture propagation in shale gas bedding reservoir analytical analysis.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1229-1237(in Chinese))
4陈建霖,励争,储鹏程.大开口复合材料层合板强度破坏研究.力学学报,2016,48(6):1326-1333(Chen Jianlin,Li Zheng,Chu Pengcheng.Strength analysis of fiber reinforced composite laminates with big cutouts.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1326-1333(in Chinese))
5付云伟,倪新华,刘协权等.颗粒缺陷相互作用下复合材料的细观损伤模型.力学学报,2016,48(6):1334-1342(Fu Yunwei,Ni Xinhua,Liu Xiequan,et al.Micro-damage model of composite materials with particle and defect interaction.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1334-1342.(in Chinese))
6徐芝纶.弹性力学简明教程(第三版).北京:高等教育出版社,1983(Xu Zhilun.A Concise Course in Elasticity(Third Edition).Beijing:Higher Education Press,1983(in Chinese))
7 Gao XL.A general solution of an infinite elastic plate with an elliptic hole under biaxial loading.International Journal of Pressure Vessels and Piping,1996,67(1):95-104
8 Sharma DS.Stress distribution around polygonal holes.International Journal of Mechanical Sciences,2012,65(1):115-124
9 Jafari M,Ardalani E.Stress concentration in finite metallic plates with regular holes.International Journal of Mechanical Sciences,2016,106:220-230
10侯祥林,王家祥,贾连光.正六边形孔角裂纹应力强度因子复变函数解.应用力学学报,2018,35(3):483-488(Hou Xianglin,Wang Jiaxiang,Jia Lianguang.Complex variable solution for stress intensity factor of hexagon angle crack.Chinese Journal of Applied mechanics,2018,35(3):483-488(in Chinese))
11 Sharma DS.Stresses around hypotrochoidal hole in infinite isotropic plate.International Journal of Mechanical Sciences,2016,105:32-40
12朱大勇,蔡永祥.对称载荷正多边形孔洞弹性应力解析逼近方法.岩土力学,2016(s2):60-66(Zhu Dayong,Cai Yongxiang.Analytical approximation method for elastic stress analysis of regular polygonal holes under symmetrical loads.Rock and Soil Mechanics,2016(s2):60-66(in Chinese))
13朱大勇,蔡永祥.含单孔洞无限平面体弹性应力场解析逼近方法.岩石力学与工程学报,2017,36(3):705-715(Zhu Dayong,Cai Yongxiang.Approaching analytical solution to elastic stress field of an infinite plane containing a single hole.Chinese Journal of Rock Mechanics and Engineering,2017,36(3):705-715(in Chinese))
14 Muskhelishvili NI.Some Basic Problems of Mathematical Theory of Elasticity.Groningen:Noordhoff,1953
15陈子荫.围岩力学分析中的解析方法.北京:煤炭工业出版社,1994(Chen Ziying.Analytical Method for Mechanics Analysis of Surrounding Rock.Beijing:Coal Industry Press,1994(in Chinese))
16 Jeffery GB.Plane stress and plane strain in bipolar co-ordinates.Philosophical Transactions of the Royal Society A,1921,221:265-293
17 Ling CB.On the stresses in a plate containing two circular holes.Journal of Applied Physics,1948,19:77-82
18 Iwaki T,Miyao K.Stress concentration in a plate with two unequal circular holes.International Journal of Engineering Science,1980,18(8):1077-1090
19 Kumari S,Uphadhyay AK,Shukla KK.Stress analysis for an infinite plate with circular holes.Materials Today:Proceedings,2017,4(2):2323-2332
20 Green AE.General biharmonic analysis for a plate containing circular holes.Proceedings of the Royal Society of London A,1940,37(1):29-33
21 Hoang SK,Abousleiman YN.Extended Green’s solution for the stresses in an infinite plate with two equal or unequal circular holes.Journal of Applied Mechanics,2008,75(3):031016
22 Haddon RAW.Stresses in an infinite plate with two unequal circular holes.Quarterly Journal of Mechanics and Applied Mathematics,1967,203:277-291
23王林江,林佳铿.用多复变量应力函数计算任意多连通弹性平面问题.应用数学和力学,1994,15(7):657-664(Wang Linjiang,Lin Jiakeng.Computation of arbitrary multiply connected elastic plane problems by multiple complex variables stress functions.Applied Mathematics and Mechanics,1994,15(7):657-664(in Chinese))
24 Lü AZ,Xu Z,Zhang N.Stress analytical solution for an infinite plane containing two holes.International Journal of Mechanical Sciences,2017,128:224-234
25 Zeng XT,Lu AZ,Zhang N.Analytical stress solution for an infinite plate containing two oval holes.European Journal of Mechanics A/Solid,2018,67:291-304
26 Zhang LQ.An analytic algorithm of stresses for any double hole problem in plane elastic-statics.Journal of Applied Mechanics,2001,68(2):350-353
27 Zhang LQ,Yue ZQ,Lee CF,el at.Stress solution of multiple elliptic hole problem in plane elasticity.Journal of Engineering Mechanics ASCE,2003,129(12):1394-1407
28朱大勇,蔡永祥.含多孔洞无限平面体弹性应力场解析逼近方法.岩石力学与工程学报,2017,35(3):705-715(Zhu Dayong,Cai Yongxiang.Analytical approach for elastic stress field of infinite plane containing several holes.Chinese Journal of Rock Mechanics and Engineering,2017,35(3):705-715(in Chinese))
29 Savin GN.Stress Distribution Around Holes.Washington:National Aeronautics and Space Administration,1970
30阿尔福斯.复分析(第三版).北京:机械工业出版社,2005(Ahlfors.Complex Analysis(Third Edition).Beijing:Machinery Industry Press,2005(in Chinese))
31 Papamichael N,Stylianopoulos N.Numerical Conformal Mapping:Domain Decomposition and the Mapping of Quadrilaterals.Singapore,Hackensack:World Scientific,2010
32 Hough DM,Papamichael N.An integral equation method for the numerical conformal mapping of interior,exterior and doublyconnected domains.Numerische Mathematik,1983,41(3):287-307
33 Lai F,Wang Y,Lu Y,et al.Improving the accuracy of the charge simulation method for numerical conformal mapping.Mathematical Problems in Engineering,2017(4):1-9
34 Lo WL,Wu NJ,Chen CS,el at.Exact boundary derivative formulation for numerical conformal mapping method.Mathematical Problems in Engineering,2016(2):1-18
35 Hakula H,Quach T,Rasila A.Conjugate function method for numerical conformal mappings.Journal of Computational and Applied Mathematics,2013,237(1):340-353
36 Korn GA,Korn TM.Mathematical Handbook for Scientists and Engineers(Second Edition).New York:McGraw-Hill,1968
37吕爱钟,张晓莉,王少杰.两向不等压圆形隧洞弹塑性解析分析.岩石力学与工程学报,2018,37(1):14-22(L¨u Aizhong,Zhang Xiaoli,Wang Shaojie.Elastoplastic analysis of circular tunnel with biaxial pressure.Chinese Journal of Rock Mechanics and Engineering,2018,37(1):14-22(in Chinese))
2王从曾,刘会亭.材料性能学.北京:北京工业大学出版社,2004(Wang Congzeng,Liu Huiting.Material Properties.Beijing:Beijing University of Technology Press,2004(in Chinese))
3孙可明,张树翠.含层理页岩气藏水力压裂裂纹扩展规律解析分析.力学学报,2016,48(5):1229-1237(Sun Keming,Zhang Shucui.Hydraulic fracture propagation in shale gas bedding reservoir analytical analysis.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1229-1237(in Chinese))
4陈建霖,励争,储鹏程.大开口复合材料层合板强度破坏研究.力学学报,2016,48(6):1326-1333(Chen Jianlin,Li Zheng,Chu Pengcheng.Strength analysis of fiber reinforced composite laminates with big cutouts.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1326-1333(in Chinese))
5付云伟,倪新华,刘协权等.颗粒缺陷相互作用下复合材料的细观损伤模型.力学学报,2016,48(6):1334-1342(Fu Yunwei,Ni Xinhua,Liu Xiequan,et al.Micro-damage model of composite materials with particle and defect interaction.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1334-1342.(in Chinese))
6徐芝纶.弹性力学简明教程(第三版).北京:高等教育出版社,1983(Xu Zhilun.A Concise Course in Elasticity(Third Edition).Beijing:Higher Education Press,1983(in Chinese))
7 Gao XL.A general solution of an infinite elastic plate with an elliptic hole under biaxial loading.International Journal of Pressure Vessels and Piping,1996,67(1):95-104
8 Sharma DS.Stress distribution around polygonal holes.International Journal of Mechanical Sciences,2012,65(1):115-124
9 Jafari M,Ardalani E.Stress concentration in finite metallic plates with regular holes.International Journal of Mechanical Sciences,2016,106:220-230
10侯祥林,王家祥,贾连光.正六边形孔角裂纹应力强度因子复变函数解.应用力学学报,2018,35(3):483-488(Hou Xianglin,Wang Jiaxiang,Jia Lianguang.Complex variable solution for stress intensity factor of hexagon angle crack.Chinese Journal of Applied mechanics,2018,35(3):483-488(in Chinese))
11 Sharma DS.Stresses around hypotrochoidal hole in infinite isotropic plate.International Journal of Mechanical Sciences,2016,105:32-40
12朱大勇,蔡永祥.对称载荷正多边形孔洞弹性应力解析逼近方法.岩土力学,2016(s2):60-66(Zhu Dayong,Cai Yongxiang.Analytical approximation method for elastic stress analysis of regular polygonal holes under symmetrical loads.Rock and Soil Mechanics,2016(s2):60-66(in Chinese))
13朱大勇,蔡永祥.含单孔洞无限平面体弹性应力场解析逼近方法.岩石力学与工程学报,2017,36(3):705-715(Zhu Dayong,Cai Yongxiang.Approaching analytical solution to elastic stress field of an infinite plane containing a single hole.Chinese Journal of Rock Mechanics and Engineering,2017,36(3):705-715(in Chinese))
14 Muskhelishvili NI.Some Basic Problems of Mathematical Theory of Elasticity.Groningen:Noordhoff,1953
15陈子荫.围岩力学分析中的解析方法.北京:煤炭工业出版社,1994(Chen Ziying.Analytical Method for Mechanics Analysis of Surrounding Rock.Beijing:Coal Industry Press,1994(in Chinese))
16 Jeffery GB.Plane stress and plane strain in bipolar co-ordinates.Philosophical Transactions of the Royal Society A,1921,221:265-293
17 Ling CB.On the stresses in a plate containing two circular holes.Journal of Applied Physics,1948,19:77-82
18 Iwaki T,Miyao K.Stress concentration in a plate with two unequal circular holes.International Journal of Engineering Science,1980,18(8):1077-1090
19 Kumari S,Uphadhyay AK,Shukla KK.Stress analysis for an infinite plate with circular holes.Materials Today:Proceedings,2017,4(2):2323-2332
20 Green AE.General biharmonic analysis for a plate containing circular holes.Proceedings of the Royal Society of London A,1940,37(1):29-33
21 Hoang SK,Abousleiman YN.Extended Green’s solution for the stresses in an infinite plate with two equal or unequal circular holes.Journal of Applied Mechanics,2008,75(3):031016
22 Haddon RAW.Stresses in an infinite plate with two unequal circular holes.Quarterly Journal of Mechanics and Applied Mathematics,1967,203:277-291
23王林江,林佳铿.用多复变量应力函数计算任意多连通弹性平面问题.应用数学和力学,1994,15(7):657-664(Wang Linjiang,Lin Jiakeng.Computation of arbitrary multiply connected elastic plane problems by multiple complex variables stress functions.Applied Mathematics and Mechanics,1994,15(7):657-664(in Chinese))
24 Lü AZ,Xu Z,Zhang N.Stress analytical solution for an infinite plane containing two holes.International Journal of Mechanical Sciences,2017,128:224-234
25 Zeng XT,Lu AZ,Zhang N.Analytical stress solution for an infinite plate containing two oval holes.European Journal of Mechanics A/Solid,2018,67:291-304
26 Zhang LQ.An analytic algorithm of stresses for any double hole problem in plane elastic-statics.Journal of Applied Mechanics,2001,68(2):350-353
27 Zhang LQ,Yue ZQ,Lee CF,el at.Stress solution of multiple elliptic hole problem in plane elasticity.Journal of Engineering Mechanics ASCE,2003,129(12):1394-1407
28朱大勇,蔡永祥.含多孔洞无限平面体弹性应力场解析逼近方法.岩石力学与工程学报,2017,35(3):705-715(Zhu Dayong,Cai Yongxiang.Analytical approach for elastic stress field of infinite plane containing several holes.Chinese Journal of Rock Mechanics and Engineering,2017,35(3):705-715(in Chinese))
29 Savin GN.Stress Distribution Around Holes.Washington:National Aeronautics and Space Administration,1970
30阿尔福斯.复分析(第三版).北京:机械工业出版社,2005(Ahlfors.Complex Analysis(Third Edition).Beijing:Machinery Industry Press,2005(in Chinese))
31 Papamichael N,Stylianopoulos N.Numerical Conformal Mapping:Domain Decomposition and the Mapping of Quadrilaterals.Singapore,Hackensack:World Scientific,2010
32 Hough DM,Papamichael N.An integral equation method for the numerical conformal mapping of interior,exterior and doublyconnected domains.Numerische Mathematik,1983,41(3):287-307
33 Lai F,Wang Y,Lu Y,et al.Improving the accuracy of the charge simulation method for numerical conformal mapping.Mathematical Problems in Engineering,2017(4):1-9
34 Lo WL,Wu NJ,Chen CS,el at.Exact boundary derivative formulation for numerical conformal mapping method.Mathematical Problems in Engineering,2016(2):1-18
35 Hakula H,Quach T,Rasila A.Conjugate function method for numerical conformal mappings.Journal of Computational and Applied Mathematics,2013,237(1):340-353
36 Korn GA,Korn TM.Mathematical Handbook for Scientists and Engineers(Second Edition).New York:McGraw-Hill,1968
37吕爱钟,张晓莉,王少杰.两向不等压圆形隧洞弹塑性解析分析.岩石力学与工程学报,2018,37(1):14-22(L¨u Aizhong,Zhang Xiaoli,Wang Shaojie.Elastoplastic analysis of circular tunnel with biaxial pressure.Chinese Journal of Rock Mechanics and Engineering,2018,37(1):14-22(in Chinese))