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Greenhill formula for Kirchhoff elastic rod is extended to that of exact model of the rod. Under the assumption of the plane cross section, the configuration of an extensible and shearable elastic rod is expressed as a history of the cross section with arc coordinate. A special solution which describes equilibrium in straight line state of the rod is obtained from a differential equilibrium equation. A linear perturbation equation is derived and its general solution is obtained in which the integral constants are determined by constrained conditions at two ends of the rod. The condition for a non zero solution of the integral constants to exist leads to the Greenhill formula of exact elastic rod model, which shows that the boundary of stable area of the force screw is a closed curve and of symmetry and the inference of extensible and shearable to stability of the rod is dependent on three factors: the difference in flexibility between shear and extension of a section of the rod, the bending stiffness, and the length of the rod.
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Keywords:
- Greenhill formula /
- exact model of elastic rod /
- force screw /
- Euler stability
[1] Travers1 A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265
[2] Liu Y Z 2003 Mech. Eng. 25(1) 1 (in Chinese) [刘延柱 2003 力学与实践 25(1) 1]
[3] Science Press) p103 (in Chinese)[武际可、 苏先樾 1994 弹性系统的稳定性 (北京: 科学出版社) 第103页]
[4] Press) p96 (in Chinese) [刘延柱 2001 高等动力学 (北京: 高等教育出版社) 第96页]
[5] Liu Y Z 2009 Chin. Phys. B 18 1
[6] Healey T J, Mehta P G 2005 Int. J. Bifur. Chaos 15 949
[7] Cao D Q, Tucker R W 2008 Int. J. Sol. Struc. 45 460
[8] He X S 2010 Acta Phys. Sin. 59 1428 (in Chinese) [和兴锁 2010 物理学报 59 1428]
[9] Liu Y Z, Xue Y, Chen L Q 2004 Acta Phys. Sin. 53 2424 (in Chinese) [刘延柱、 薛 纭、 陈立群 2004 物理学报 53 2424]
[10] Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛 纭、 陈立群、 刘延柱 2004 物理学报 53 4029]
[11] Liu Y Z 2002 Mech. Eng. 24(4) 56 (in Chinese) [刘延柱 2002 力学与实践 24(4) 56]
[12] Xue Y, Chen L Q 2004 Mech. Eng. 26(5)71 (in Chinese)[薛纭、 陈立群 2004 力学与实践 26(5) 71]
[13] Timoshenko S P, Gere J M 1965 Theory of Elastic Stability (2nd ed) (Beijing: Science Press ) p169 (in Chinese)[铁摩辛柯 S P、盖莱J M 1965 弹性稳定理论(第二版)(中译本)(北京: 科学出版社) 第169页]
[14] Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod: Theoritical Basis of Mechanical Model of DNA (Beijing: Tsinghua University Press, Springer) pp14, 61, 89 (in Chinese) [刘延柱 2006 弹性细杆的非线性力学: DNA力学模型的理论基础 (北京:清华大学出版社, Springer) 第14, 61, 89页]
[15] Wu J K, Su X Y 1994 Stability of Elastic System (Beijing:
[16] Liu Y Z, Xue Y 2005 Mech. Eng. 27(1) 64 (in Chinese)[刘延柱、 薛 纭 2005 力学与实践 27(1) 64]
[17] Xue Y, Chen L Q 2008 J. Dyn. Contr. 6 198 (in Chinese)[薛纭、 陈立群 2008 动力学与控制学报 6 198]
[18] Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 61 (in Chinese)[薛纭、刘延柱 2009 物理学报 58 61]
[19] Liu Y Z 2001 Advanced Dynamics (Beijing: Higher Education
[20] Xue Y, Weng D W, Chen L Q 2009 Chin. Quart. Mech. 30 116 (in Chinese)[薛 纭、 翁德玮、 陈立群 2009 力学季刊 30 116]
[21] Hu H C 1981 Variational Principles of Elastic Mechanics and Their Applications(Beijing: Science Press)p145 (in Chinese) [胡海昌 1981 弹性力学的变分原理及其应用 (北京: 科学出版社) 第145页]
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[1] Travers1 A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265
[2] Liu Y Z 2003 Mech. Eng. 25(1) 1 (in Chinese) [刘延柱 2003 力学与实践 25(1) 1]
[3] Science Press) p103 (in Chinese)[武际可、 苏先樾 1994 弹性系统的稳定性 (北京: 科学出版社) 第103页]
[4] Press) p96 (in Chinese) [刘延柱 2001 高等动力学 (北京: 高等教育出版社) 第96页]
[5] Liu Y Z 2009 Chin. Phys. B 18 1
[6] Healey T J, Mehta P G 2005 Int. J. Bifur. Chaos 15 949
[7] Cao D Q, Tucker R W 2008 Int. J. Sol. Struc. 45 460
[8] He X S 2010 Acta Phys. Sin. 59 1428 (in Chinese) [和兴锁 2010 物理学报 59 1428]
[9] Liu Y Z, Xue Y, Chen L Q 2004 Acta Phys. Sin. 53 2424 (in Chinese) [刘延柱、 薛 纭、 陈立群 2004 物理学报 53 2424]
[10] Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛 纭、 陈立群、 刘延柱 2004 物理学报 53 4029]
[11] Liu Y Z 2002 Mech. Eng. 24(4) 56 (in Chinese) [刘延柱 2002 力学与实践 24(4) 56]
[12] Xue Y, Chen L Q 2004 Mech. Eng. 26(5)71 (in Chinese)[薛纭、 陈立群 2004 力学与实践 26(5) 71]
[13] Timoshenko S P, Gere J M 1965 Theory of Elastic Stability (2nd ed) (Beijing: Science Press ) p169 (in Chinese)[铁摩辛柯 S P、盖莱J M 1965 弹性稳定理论(第二版)(中译本)(北京: 科学出版社) 第169页]
[14] Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod: Theoritical Basis of Mechanical Model of DNA (Beijing: Tsinghua University Press, Springer) pp14, 61, 89 (in Chinese) [刘延柱 2006 弹性细杆的非线性力学: DNA力学模型的理论基础 (北京:清华大学出版社, Springer) 第14, 61, 89页]
[15] Wu J K, Su X Y 1994 Stability of Elastic System (Beijing:
[16] Liu Y Z, Xue Y 2005 Mech. Eng. 27(1) 64 (in Chinese)[刘延柱、 薛 纭 2005 力学与实践 27(1) 64]
[17] Xue Y, Chen L Q 2008 J. Dyn. Contr. 6 198 (in Chinese)[薛纭、 陈立群 2008 动力学与控制学报 6 198]
[18] Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 61 (in Chinese)[薛纭、刘延柱 2009 物理学报 58 61]
[19] Liu Y Z 2001 Advanced Dynamics (Beijing: Higher Education
[20] Xue Y, Weng D W, Chen L Q 2009 Chin. Quart. Mech. 30 116 (in Chinese)[薛 纭、 翁德玮、 陈立群 2009 力学季刊 30 116]
[21] Hu H C 1981 Variational Principles of Elastic Mechanics and Their Applications(Beijing: Science Press)p145 (in Chinese) [胡海昌 1981 弹性力学的变分原理及其应用 (北京: 科学出版社) 第145页]
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