XUE Yun, QU Jia-le, CHEN Li-qun. Gauss Principle of Least Constraint for Cosserat Growing Elastic Rod Dynamics[J]. Applied Mathematics and Mechanics, 2015, 36(7): 700-709. doi: 10.3879/j.issn.1000-0887.2015.07.003
 Citation: XUE Yun, QU Jia-le, CHEN Li-qun. Gauss Principle of Least Constraint for Cosserat Growing Elastic Rod Dynamics[J]. Applied Mathematics and Mechanics, 2015, 36(7): 700-709.

# Gauss Principle of Least Constraint for Cosserat Growing Elastic Rod Dynamics

##### doi: 10.3879/j.issn.1000-0887.2015.07.003
Funds:  The National Natural Science Foundation of China(11372195; 10972143)
• Rev Recd Date: 2015-06-09
• Publish Date: 2015-07-15
• The dynamic modeling of growing elastic rods, with the background of a kind of growing, deforming and moving slender bodies in nature and engineering, was studied based on the Gauss principle of least constraint in the classical mechanics. This provides a new method for the dynamic modeling of growing elastic rods, and meanwhile expands the application scope of the Gauss principle of least constraint. With the Cosserat growing elastic rod as the object, the geometric rules for growth and deformation of the rod were analyzed, which show that the growing strain and elastic strain are in a nonlinear coupling relation. The constitutive equations were given as a linear relationship between the internal forces and elastic deformations of the rod’s cross section; through definition of the inverse of dyad, the Gauss principle of least constraint was used to model the growing elastic rod dynamics and get 2 equivalent forms of the Gauss variation, which reflect the symmetry between time and arc coordinates in the expression of rod dynamics. The closedform dynamic differential equations were derived. 2 forms of constraint functions were given, which indicate that the actual motion of an elastic rod made the function at a stationary value, and also the minimum value. Finally, some problems about the constraints and conditional extremums of the growing elastic rod dynamics were discussed.
•  [1] 刘延柱. 弹性细杆的非线性力学——DNA 力学模型的理论基础[M]. 北京: 清华大学出版社, Springer, 2006: 14, 32.(LIU Yan-zhu. Nonlinear Mechanics of Thin Elastic Rod— Theoretical Basis of Mechanical Model of DNA[M]. Beijing: Tsinghua University Press, Springer, 2006: 14, 32.(in Chinese)) [2] 刘延柱. 弹性杆基因模型的力学问题[J]. 力学与实践, 2003,25(1): 1-5.(LIU Yan-zhu. Mechanical problems on elastic rod model of DNA[J]. Mechanics in Engineering, 2003,25(1): 1-5.(in Chinese)) [3] 马拉森斯基 乔治 M. 分子生物学精要[M]. 第4版. 魏群 译. 北京: 化学工业出版社, 2005: 59.(Malacinski George M. Essentials of Molecular Biology[M]. 4th ed. WEI Qun transl. Bejing: Chemical Industry Press, 2005: 59.(Chinese version)) [4] CAO Deng-qing, Tucker R W. Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation[J]. International Journal of Solids and Structure,2008,45(2): 460-470. [5] 薛纭, 刘延柱, 陈立群. 超细长弹性杆的分析力学问题[J]. 力学学报, 2005,37(4): 485-493.(XUE Yun, LIU Yan-zhu, CHEN Li-qun. On analytical mechanics for a super-thin elastic rod[J]. Chinese Journal of Theoretical and Applied Mechanics,2005,37(4): 485-493.(in Chinese)) [6] 薛纭, 刘延柱, 陈立群. Kirchhoff弹性杆动力学建模的分析力学方法[J]. 物理学报, 2006,55(8): 3845-3851.(XUE Yun, LIU Yan-zhu，CHEN Li-qun. Methods of analytical mechanics for dynamics of the Kirchhoff elastic rod[J]. Acta Physica Sinica,2006,55(8): 3845-3851.(in Chinese)) [7] 薛纭, 翁德玮. 超细长弹性杆动力学的Gauss原理[J]. 物理学报, 2009,58(1): 34.(XUE Yun, WENG De-wei. Gauss principle for a super-thin elastic rod dynamics[J]. Acta Physica Sinica,2009,58(1): 34.(in Chinese)) [8] 薛纭, 翁德玮, 陈立群. 精确Cosserat弹性杆动力学的分析力学方法[J]. 物理学报, 2013,62(4): 044601.(XUE Yun, WENG De-wei, CHEN Li-qun. Methods of analytical mechanics for exact Cosserat elastic rod dynamic[J]. Acta Physica Sinica,2013,62(4): 044601.(in Chinese)) [9] 梅凤翔. 分析力学（下卷）[M]. 北京: 北京理工大学出版社, 2013: 621.(MEI Feng-xiang. Analytical Mechanics[M]. Bejing: Bejing Institute of Technology Press, 2013: 621.(in Chinese)) [10] 陈滨. 分析动力学[M]. 北京: 北京大学出版社, 2012: 288.(CHEN Bin. Analytical Dynamics[M]. Beijing: Peking University Press, 2012: 288.(in Chinese)) [11] 刘延柱. 高等动力学[M]. 北京: 高等教育出版社, 2001: 50.(LIU Yan-zhu. Advanced Dynamics[M]. Beijing: Higher Educational Press, 2001: 50.(in Chinese)) [12] Kalaba R E, Udwadia F E. Equations of motion for nonholonomic, constrained dynamical systems via Gauss’s principle[J]. Journal of Applied Mechanics,1993,60(3): 662-668. [13] Kalaba R, Natsuyama H, Udwadia F. An extension of Gauss’s principle of least constraint[J]. International Journal of General Systems, 2004,33(1): 63-69. [14] 董龙雷, 闫桂荣, 杜彦亭, 余建军, 牛宝良, 李荣林. 高斯最小拘束原理在一类刚柔耦合系统分析中的应用[J]. 兵工学报, 2001,22(3): 347-351.(DONG Long-lei, YAN Gui-rong, DU Yan-ting, YU Jian-jun, NIU Bao-liang, LI Rong-lin. Application of the Gauss minimum constraint theory in a rigid-flexible coupled system[J]. Acta Armamentarii,2001,〖STHZ〗 22(3): 347-351.(in Chinese)) [15] 刘延柱, 薛纭. 基于高斯原理的Cosserat弹性杆动力学模型[J]. 物理学报, 2014,64(4): 044601.(LIU Yan-zhu, XUE Yun. Dynamical model of Cosserat elastic rod based on Gauss principle[J] . Acta Physica Sinica, 2014,64(4): 044601.(in Chinese)) [16] Moulton D E, Lessinnes T, Goriely A. Morphoelastic rods—part Ⅰ: a single growing elastic rod[J]. Journal of the Mechanics and Physics of Solids, 2013,61(2): 398-427. [17] Cao D Q, Song M T, Tucker R W, Zhu W D, Liu D S, Huang W H. Dynamic equations of thermoelastic Cosserat rods[J]. Communications in Nonlinear Science and Numerical Simulation, 2013,18(7): 1880-1887. [18] Wolgemuth C W, Goldstein R E, Powers T R. Dynamic supercoiling bifurcations of growing elastic filaments[J]. Physica D: Nonlinear Phenomena, 2004,190(3/4): 266-289. [19] Goriely A, Neukirch S. Mechanics of climbing and attachment in twining plants[J]. Phys Rev Lett,2006,97(18): 184302. [20] Lockhart J A. An analysis of irreversible plant cell elongation[J]. Journal of Theoretical Biology,1965,8(2): 264-275. [21] Cosgrove D J. Cell wall yield properties of growing tissue: evaluation by in vivo stress relaxation[J]. Plant Physiol,1985,78(2): 347-356. [22] Goodwin B C, Briére C. A mathematical model of cytoskeletal dynamics and morphogenesis in acetabularia[C]//Menzel D ed. The Cytoskeleton of the Algae. Boca Raton: CRC Press, 1992: 219-233. [23] Stein A A. The deformation of a rod of growing biological material under longitudinal compression[J]. Journal of Applied Mathematics and Mechanics,1995,59(1):139-146.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142