Abstract
Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff's kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler's angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov's stability of the helical equilibrium is discussed in static category. Euler's critical values of axial force and torque are obtained. Lyapunov's and Euler's stability conditions in the space domain are the necessary conditions of Lyapunov's stability of the helical rod in the time domain.
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References
Love, A. E. H. A Treatise on Mathematical Theory of Elasticity, Dover, New York (1927)
Liu, Y. Z. Nonlinear Mechanics of Thin Elastic Rod (in Chinese), Tsinghua University Press, Beijing (2006)
Liu, Y. Z. and Xue, Y. Dynamical stability of helical equilibrium of a thin elastic rod (in Chinese). Chinese Quarterly of Mechanics, 26(1), 1–7 (2005)
Liu, Y. Z. and Sheng, L. W. Dynamical stability and vibration of an helical rod under axial compression (in Chinese). Chinese Quarterly of Mechanics, 27(2), 190–195 (2006)
Liu, Y. Z. and Sheng, L. W. Stability and vibration of an elastic helical rod with circular cross section (in Chinese). Acta Physica Sinica, 56(4), 2305–2310 (2007)
Antman, S. S. Nonlinear Problems of Elasticity, Springer, New York (1995)
Bishop, T. C., Cortez, R., and Zhmudsky, O. O. Investigation of bend and shear waves in a geometrically exact elastic rod model. Journal of Computational Physics, 193, 642–665 (2004)
Schuricht, F. Regularity for shearable nonlinearly elastic rods in obstacle problems. Archive for Rational Mechanics and Analysis, 145, 23–49 (1998)
Tucker, R. W. and Wang, C. Torsional vibration control and Cosserat dynamics of a drill-rig assembly. Meccanica, 38, 143–159 (2003)
Cao, D. Q., Liu, D., and Wang, C. H. T. Nonlinear dynamic modelling of MEMS components via the Cosserat rod element approach. Journal of Micromechanics and Microengineering, 15, 1334–1343 (2005)
Cao, D. Q. and Tucker, R. W. Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. International Journal of Solids and Structures, 45, 460–477 (2008)
Liu, Y. Z. On dynamics of elastic rod based on exact Cosserat model. Chinese Physics B, 18(1), 1–8 (2009)
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Communicated by Li-qun CHEN
Project supported by the National Natural Science Fundation of China (No. 10972143)
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Liu, Yz., Xue, Y. Stability analysis of helical rod based on exact Cosserat model. Appl. Math. Mech.-Engl. Ed. 32, 603–612 (2011). https://doi.org/10.1007/s10483-011-1442-8
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DOI: https://doi.org/10.1007/s10483-011-1442-8