平面偶应力问题的辛求解方法
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摘要
平面偶应力理论虽然早在上世纪初就出现了，但是其分析求解一直没有得到很好的解决。现有的求解手段主要采用数值方法——如有限元法。而能给出其解析解的只限于某些特殊的偶应力问题。辛方法作为一种崭新的理论求解体系已成功应用于板、梁等弹性力学问题的求解，与经典的弹性力学求解体系相比有着其独特的优越性。本文目的在于把这种解析方法应用到平面偶应力问题的求解。
本文借助于Reissner板与平面偶应力的模拟关系，在平面偶应力问题的类Hellinger-Reissner变分原理的基础上，以应力函数为原变量，部分应变为其对偶变量，推导出力法形式的平面偶应力问题的Hamilton对偶方程组。于是把平面偶应力问题引入到Hamilton体系，从而利用辛空间的分离变量和本征函数向量展开法获得其解。本文讨论了两种典型边界条件——对边自由和对边固支矩形域问题的解析求解。首先求解出由于用应变代替位移作为基本变量而带来的对边自由矩形域问题的所有非齐次特解，这些解均是有特殊物理意义的解。然后，推导出这两类边界条件各自的本征值超越方程，并进一步给出其对应的非零本征值的本征解。从而依据叠加原理，获得这两种典型边界条件问题的解。最后，本文求解了一半无穷矩形域单向拉伸问题，数值结果证明了微尺寸下经典弹性力学的求解方法得出的结果不再适用，由于偶应力的影响，单向拉伸问题在固定端角点处的奇异性消失。
本文将辛方法成功应用于矩形域平面偶应力问题的求解，为这一类问题提供了一条崭新的解决途径。算例结果也很好地证明了辛方法的有效性和优越性。
Plane couple stress problem appeared at the beginning of last century, but it has not been well analytically solved before. There are mainly some numerical methods such as FEM etc. at present, and there are only few analytic solutions for some special problems. Symplectic method has been applied in some plate and beam bending problems successfully. Compared with classical method, it has some unique advantages. This paper aims to apply this analytic method in plane couple stress problem.
Based on the Pro-Hellinger-Reissner variational principle of plane couple stress problem, the dual PDEs are proposed corresponding to the force method extension. The duality solution methodology is thus extended to plane couple stress problem, and then the method of separation of variables and eigenfunction expansion in the symplectic space is used to find the analytical solutions. A long strip domain plate with both lateral edges free, fixed at one end and under simple tension at the far end, is solved analytically. The solution is composed of the inhomogeneous boundary condition solutions (which are induced from that the stress functions are selected as primary unknowns) and the superposition of eigensolutions of homogeneous lateral boundary conditions. The method of separation of variables is used for the dual PDEs, from which the eigen-root transcendental equation is solved and the corresponding eigenvector functions are obtained. The boundary conditions at the fixed end are derived via' the variational method. The superposition of these eigensolutions gives the stress distribution at the fixed end. Numerical results show that due to the effect of couple stress, the stress distribution is no longer infinity as given by the classical theory of elasticity at the corner of fixed end.
Results improve that in micro scale the couple stress effect should not be neglect, and symplectic method is effective and advantageous to handle with plane couple stress problem in rectangle field.
本文借助于Reissner板与平面偶应力的模拟关系，在平面偶应力问题的类Hellinger-Reissner变分原理的基础上，以应力函数为原变量，部分应变为其对偶变量，推导出力法形式的平面偶应力问题的Hamilton对偶方程组。于是把平面偶应力问题引入到Hamilton体系，从而利用辛空间的分离变量和本征函数向量展开法获得其解。本文讨论了两种典型边界条件——对边自由和对边固支矩形域问题的解析求解。首先求解出由于用应变代替位移作为基本变量而带来的对边自由矩形域问题的所有非齐次特解，这些解均是有特殊物理意义的解。然后，推导出这两类边界条件各自的本征值超越方程，并进一步给出其对应的非零本征值的本征解。从而依据叠加原理，获得这两种典型边界条件问题的解。最后，本文求解了一半无穷矩形域单向拉伸问题，数值结果证明了微尺寸下经典弹性力学的求解方法得出的结果不再适用，由于偶应力的影响，单向拉伸问题在固定端角点处的奇异性消失。
本文将辛方法成功应用于矩形域平面偶应力问题的求解，为这一类问题提供了一条崭新的解决途径。算例结果也很好地证明了辛方法的有效性和优越性。
Plane couple stress problem appeared at the beginning of last century, but it has not been well analytically solved before. There are mainly some numerical methods such as FEM etc. at present, and there are only few analytic solutions for some special problems. Symplectic method has been applied in some plate and beam bending problems successfully. Compared with classical method, it has some unique advantages. This paper aims to apply this analytic method in plane couple stress problem.
Based on the Pro-Hellinger-Reissner variational principle of plane couple stress problem, the dual PDEs are proposed corresponding to the force method extension. The duality solution methodology is thus extended to plane couple stress problem, and then the method of separation of variables and eigenfunction expansion in the symplectic space is used to find the analytical solutions. A long strip domain plate with both lateral edges free, fixed at one end and under simple tension at the far end, is solved analytically. The solution is composed of the inhomogeneous boundary condition solutions (which are induced from that the stress functions are selected as primary unknowns) and the superposition of eigensolutions of homogeneous lateral boundary conditions. The method of separation of variables is used for the dual PDEs, from which the eigen-root transcendental equation is solved and the corresponding eigenvector functions are obtained. The boundary conditions at the fixed end are derived via' the variational method. The superposition of these eigensolutions gives the stress distribution at the fixed end. Numerical results show that due to the effect of couple stress, the stress distribution is no longer infinity as given by the classical theory of elasticity at the corner of fixed end.
Results improve that in micro scale the couple stress effect should not be neglect, and symplectic method is effective and advantageous to handle with plane couple stress problem in rectangle field.
引文
[1] A.R. Boresi and Ken.P. Chong, Elasticity in Engineering Mechanics, 1999, appendix 5A. 2nd ed. J. Wiley & Sons.
[2] Koiter, Couple Stresses in the Theory of Elasticity. Ⅰ and Ⅱ Proc.K.Ned.Akad. 1964, 67: 17-44
[3] P. Tong, D.C.C. Lam, F. Yang, A.C.M. Chong and J.Wang, Size dependence in nanometer and micron-scaled structures, Solid Mechanics Conference. 2002
[4] R.D. Mindlin & H.F.Tiersten, Effects of couple-stresses in linear elascticity. Arch. Ration. Mech. Anal., 1962, 11: 415-448.
[5] Fleck N A, Muller G M,Ashaby M F, etal, Strain gradient plasticity:theory and experiment.Acta Metall Mater, 1994,42:475-487
[6] Stolken J S, Evans A G., A microbend test method for measuring the plasticity length scale. Harvard University Report Mech 320 Division of Engineering and Applied Sciences,Harvard University, Cambridge. 1997.
[7] 陈少华，王自强，应变梯度理论进展，力学进展，2003，33(2)：207-216
[8] Toupin, R.A., Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 1962,11:385-414
[9] Toupin, R.A., Theories of Elasticity with couple -stress. Arch. Ration. Mech. Anal 1964, 17(2): 85-112.
[10] R.D. Mindlin, Influence of couple-stresses on stress concentrations. Exp. Mech. 1963, 3(1); 1-7.
[11] 黄克智，邱信明，姜汉卿，应变梯度理论的新进展(一)——偶应力理论和SG理论，机械强度，1999，21(2)：81-87
[12] L. Zhang, Y. Huang, J.Y. Chen and K.C. Hwang, The mode Ⅲ full-field solution in elastic material with strain gradient effects, Int.J.Fracture, 1998,92:325-348
[13] Y. Huang, J.Y. Chen, T.F. Guo, L. Zhang and K.C. Hwang, Analytic and numerical studies on mode Ⅰ and mode Ⅱ fracture in elastic-plastic materials with strain gradient effects, Int.J.Fracture, 1999, 100:1-27
[14] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong,P., Couple stress based strain gradient theory for elasticity. Int.J.Solids&Struct,, 2002,39:2731-2743
[15] 肖其林，凌中，吴永利，姚文慧，偶应力问题的非协调元分析，中国科学院研究生院学报，2003，20(2)：223-231
[16] 郑长良，张志峰，应用离散偶应力单元分析弹性Cosserat介质，中国计算力学大会——工程和科学中的计算力学，2003(上)：321-326
[17] Zhong Wanxie and Yao weian, New solution System for plate bending and its application, ACTA mechanics Sinica, 1999, 31(2): 173-184 (in Chinese)
[18] Yao Weian and Sui Yongfeng, Symplectic solution system for Reissner plate bending, Applied Mathematics and Mechanics.2004, 25(2): 178-185
[19] Zhong Wanxie, Yao Weian, Zheng Changliang, Analogy between Reissner plate bending and plane couple-stress, Journal of Dalian University of technology, 2002, 42(5): 519-521 (in Chinese)
[20] Yao Weian and Zhong Wanxie, Symplectic Elasticity, Beijing: Higher Education Press (in Chinese), 2002.
[21] Zhong Wanxie, A new systematic methodology for theory of elasticity, Dalian: Dalian University of Technology press (in Chinese), 1995.
[22] 龙云亮，文希理，谢处方，复平面上超越函数零点的数值计算，数值计算与计算机应用，1994，第二期：88-92
[23] G.K.Hu,B.Han and L.Liao, A Note on Microstructural Interpretation of the Material Constants for Couple Stress Theory. Mechanics Research Communications, 1999, Vol.26, No.5, pp.541-545.
[24] S.H.Chen, T.C.Wang, A new deformation theory with strain gradient effects. J.Inter.Plas. 2002,18:971-995.
[25] 胡海昌，各向同性夹层板反对小挠度的若干问题，力学学报，1963，第六卷第一期：53-59
[26] Yechiel Weitsman, Couple-stress Effects on Stress Concentration Around a Cylindrical Inclusion in a Field of Uniaxial Tension, J.App.Mech., 1965, Jnue:424-428
[27] 刘俊，黄铭，葛修润，考虑偶应力影响的应力集中问题求解，上海交通大学学报，2001，35(10)：1481-1485
[28] 赵俭斌，李子国，应力偶对孔洞附近应力集中的影响，应用数学和力学，2000，21卷第8期：803-808
[29] 佘成学，陈胜宏，张玉珍，偶应力理论的工程适用性分析，武汉水利电力大学学报，1996，29(4)：60-64
[30] 黄克智，邱信明，姜汉卿，应变梯度理论的新进展(二)，机械强度，1999，21(3)：161-165
[31] 王则柯，Kuhn算法的程序实施及数值实验，数值计算与计算机应用，1981，第三期：175-181
[32] 龙云亮，蒋鸿雁，龙晓莉，电磁理论中复超越方程数值解法研究，微波学报，1999，15卷第2期：179-184
[33] 程锦松，多项式根的分布理论与代数方程解法，应用数学与计算数学，1966，3：225-236
[34] J.Y. Chen, Y. Wei, Y. Huang, J.W. Hutchinson, K.C. Hwang, The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses, Engineering Fracture Mechanics, 1999,64:625-648
[35] N.A. Fleck, J.W. Hutchinson, A reformulation of strain gradient plasticity, Journal of the Mechanics and Physics of Solids, 2001,49: 2245-2271
[36] Masanobu Oda, Kazuyoshi Iwashita, Study on couple stress and shear band development in granular media based on numerical simulation analyses, International Journal of Engineering Science, 2000,38:1713-1740
[37] Chien H. Wu, Ming L. Wang, The effect of crack-tip point loads on fracture,Journal of the Mechanics and Physics of Solids, 2000,48:2283-2296
[38] S.H. Chen, T.C. Wang, A new deformation theory with strain gradient effects,International Journal of Plasticity, 2002,18:971-995
[39] D.J. Unger, E.C. Aifantis, Strain gradient elasticity theory for antiplane shear cracks.Part Ⅰ: Oscillatory displacements, Theoretical and Applied Fracture Mechanics, 2000,34: 243-252
[40] V.A. Lubarcla, The effects of couple stresses on dislocation strain energy,,Internationai Journal of Solids and Structures, 2003,40:3807-3826
[41] Mohan K. Kadalbajoo, Kailash C. Patidar, Singularly perturbed problems in partial differential equations: a survey Applied Mathematics and Computation, 2003,134:371-429
[42] Weishi Liu, Exchange Lemmas for Singular Perturbation Problems with Certain Turning Points, Journal of Differential Equations, 2000,167:134-180
[43] Mami Suzuki, Singular perturbation for difference equations, Nonlinear Analysis 2001,47:4083-4093
[44] 柳春图，蒋持平，板壳断裂力学，国防工业出版社，北京，2000
[2] Koiter, Couple Stresses in the Theory of Elasticity. Ⅰ and Ⅱ Proc.K.Ned.Akad. 1964, 67: 17-44
[3] P. Tong, D.C.C. Lam, F. Yang, A.C.M. Chong and J.Wang, Size dependence in nanometer and micron-scaled structures, Solid Mechanics Conference. 2002
[4] R.D. Mindlin & H.F.Tiersten, Effects of couple-stresses in linear elascticity. Arch. Ration. Mech. Anal., 1962, 11: 415-448.
[5] Fleck N A, Muller G M,Ashaby M F, etal, Strain gradient plasticity:theory and experiment.Acta Metall Mater, 1994,42:475-487
[6] Stolken J S, Evans A G., A microbend test method for measuring the plasticity length scale. Harvard University Report Mech 320 Division of Engineering and Applied Sciences,Harvard University, Cambridge. 1997.
[7] 陈少华，王自强，应变梯度理论进展，力学进展，2003，33(2)：207-216
[8] Toupin, R.A., Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 1962,11:385-414
[9] Toupin, R.A., Theories of Elasticity with couple -stress. Arch. Ration. Mech. Anal 1964, 17(2): 85-112.
[10] R.D. Mindlin, Influence of couple-stresses on stress concentrations. Exp. Mech. 1963, 3(1); 1-7.
[11] 黄克智，邱信明，姜汉卿，应变梯度理论的新进展(一)——偶应力理论和SG理论，机械强度，1999，21(2)：81-87
[12] L. Zhang, Y. Huang, J.Y. Chen and K.C. Hwang, The mode Ⅲ full-field solution in elastic material with strain gradient effects, Int.J.Fracture, 1998,92:325-348
[13] Y. Huang, J.Y. Chen, T.F. Guo, L. Zhang and K.C. Hwang, Analytic and numerical studies on mode Ⅰ and mode Ⅱ fracture in elastic-plastic materials with strain gradient effects, Int.J.Fracture, 1999, 100:1-27
[14] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong,P., Couple stress based strain gradient theory for elasticity. Int.J.Solids&Struct,, 2002,39:2731-2743
[15] 肖其林，凌中，吴永利，姚文慧，偶应力问题的非协调元分析，中国科学院研究生院学报，2003，20(2)：223-231
[16] 郑长良，张志峰，应用离散偶应力单元分析弹性Cosserat介质，中国计算力学大会——工程和科学中的计算力学，2003(上)：321-326
[17] Zhong Wanxie and Yao weian, New solution System for plate bending and its application, ACTA mechanics Sinica, 1999, 31(2): 173-184 (in Chinese)
[18] Yao Weian and Sui Yongfeng, Symplectic solution system for Reissner plate bending, Applied Mathematics and Mechanics.2004, 25(2): 178-185
[19] Zhong Wanxie, Yao Weian, Zheng Changliang, Analogy between Reissner plate bending and plane couple-stress, Journal of Dalian University of technology, 2002, 42(5): 519-521 (in Chinese)
[20] Yao Weian and Zhong Wanxie, Symplectic Elasticity, Beijing: Higher Education Press (in Chinese), 2002.
[21] Zhong Wanxie, A new systematic methodology for theory of elasticity, Dalian: Dalian University of Technology press (in Chinese), 1995.
[22] 龙云亮，文希理，谢处方，复平面上超越函数零点的数值计算，数值计算与计算机应用，1994，第二期：88-92
[23] G.K.Hu,B.Han and L.Liao, A Note on Microstructural Interpretation of the Material Constants for Couple Stress Theory. Mechanics Research Communications, 1999, Vol.26, No.5, pp.541-545.
[24] S.H.Chen, T.C.Wang, A new deformation theory with strain gradient effects. J.Inter.Plas. 2002,18:971-995.
[25] 胡海昌，各向同性夹层板反对小挠度的若干问题，力学学报，1963，第六卷第一期：53-59
[26] Yechiel Weitsman, Couple-stress Effects on Stress Concentration Around a Cylindrical Inclusion in a Field of Uniaxial Tension, J.App.Mech., 1965, Jnue:424-428
[27] 刘俊，黄铭，葛修润，考虑偶应力影响的应力集中问题求解，上海交通大学学报，2001，35(10)：1481-1485
[28] 赵俭斌，李子国，应力偶对孔洞附近应力集中的影响，应用数学和力学，2000，21卷第8期：803-808
[29] 佘成学，陈胜宏，张玉珍，偶应力理论的工程适用性分析，武汉水利电力大学学报，1996，29(4)：60-64
[30] 黄克智，邱信明，姜汉卿，应变梯度理论的新进展(二)，机械强度，1999，21(3)：161-165
[31] 王则柯，Kuhn算法的程序实施及数值实验，数值计算与计算机应用，1981，第三期：175-181
[32] 龙云亮，蒋鸿雁，龙晓莉，电磁理论中复超越方程数值解法研究，微波学报，1999，15卷第2期：179-184
[33] 程锦松，多项式根的分布理论与代数方程解法，应用数学与计算数学，1966，3：225-236
[34] J.Y. Chen, Y. Wei, Y. Huang, J.W. Hutchinson, K.C. Hwang, The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses, Engineering Fracture Mechanics, 1999,64:625-648
[35] N.A. Fleck, J.W. Hutchinson, A reformulation of strain gradient plasticity, Journal of the Mechanics and Physics of Solids, 2001,49: 2245-2271
[36] Masanobu Oda, Kazuyoshi Iwashita, Study on couple stress and shear band development in granular media based on numerical simulation analyses, International Journal of Engineering Science, 2000,38:1713-1740
[37] Chien H. Wu, Ming L. Wang, The effect of crack-tip point loads on fracture,Journal of the Mechanics and Physics of Solids, 2000,48:2283-2296
[38] S.H. Chen, T.C. Wang, A new deformation theory with strain gradient effects,International Journal of Plasticity, 2002,18:971-995
[39] D.J. Unger, E.C. Aifantis, Strain gradient elasticity theory for antiplane shear cracks.Part Ⅰ: Oscillatory displacements, Theoretical and Applied Fracture Mechanics, 2000,34: 243-252
[40] V.A. Lubarcla, The effects of couple stresses on dislocation strain energy,,Internationai Journal of Solids and Structures, 2003,40:3807-3826
[41] Mohan K. Kadalbajoo, Kailash C. Patidar, Singularly perturbed problems in partial differential equations: a survey Applied Mathematics and Computation, 2003,134:371-429
[42] Weishi Liu, Exchange Lemmas for Singular Perturbation Problems with Certain Turning Points, Journal of Differential Equations, 2000,167:134-180
[43] Mami Suzuki, Singular perturbation for difference equations, Nonlinear Analysis 2001,47:4083-4093
[44] 柳春图，蒋持平，板壳断裂力学，国防工业出版社，北京，2000