New Results in Dynamics Stability Problems of Elastic Rods

Abstract:

This article is about the nonlinear dynamic stability problems of the exact (Cosserat) theory of elastic rods. There is examined the general geometrically nonlinear theory with no restrictions on displacements and rotations being imposed. In this article, it is shown that the variational problem can be defined as the search for the stationary point of the Hamilton’s functional. The new exact solutions of the stability problems for different types of the end fixities of the rod were obtained taking into account bending, shear and longitudinal stiffness.

• Periodical: Applied Mechanics and Materials (Volume 617)
• Pages: 181-186
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.617.181
• Online since: August 2014
• Authors: Vladimir Vladimirovich Lalin*, Daria Aleksandrovna Kushova
• Keywords: Conjugate Pairs of Strain, Dynamic Approach to Study the Stability Problem, Dynamic Stability, Dynamic Stability Criterion, Functional of Variational Formulation, Large Displacements, Rotations, Stability Equations, Stress Vectors

References

[1] D.P. Goloskokov, P.А. Zhilin, Obshaja nelinejnaja teorija uprugih sterzhnej s prilozheniem k opisaniju jeffekta Pojntinga, Deposited VINITI No. 1912-V87, Dep., 20 p. (Russian).

[2] V.V. Eliseev, Mehanika uprugih sterzhnej, Saint-Petersburg, SPbSPU, 1994, 88 p. (Russian).

[3] P.A. Zhilin, A.D. Sergeev, Ravnovesie i ustojchivost' tonkogo sterzhnja, nagruzhennogo konservativnym momentum. Mehanika i processy upravlenija, Trudy SPbSPU, 1994, No. 448, pp.47-56. (Russian).

[4] P.A. Zhilin, A.D. Sergeev, T.P. Tovstik, Nelinejnaja teorija sterzhnej i ee prilozhenija, Trudy XXIV letnej shkoly: Analiz i sintez nelinejnyh mehanicheskih kolebatel'nyh sistem, Sankt-Peterburg, 1997, p.313 – 337 (Russian).

[5] P.A. Zhilin, Prikladnaja mehanika, Teorija tonkih uprugih sterzhnej, SPb, Izd-vo SPbSPU, 2007, 102 p. (Russian).

[6] V.V. Eliseev, T.V. Zinoveva, Mehanika tonkostennyh konstrukcij, Teorija sterzhnej, SPb, Izd-vo SPbSPU, 2008, 96 p. (Russian).

[7] G. Jelenic, M.A. Crisfield, Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for static and dynamics. Comp. Meths. Appl. Mech. Engng, 1999, No. 171, pp.141-171.

DOI: 10.1016/s0045-7825(98)00249-7

[8] A. A. Shabana and R.Y. Yakoub, Three dimensional absolute nodal coordinate formulation for beam elements theory. ASME Journal of Mechanical Design, 123 4 (2001) 606–613.

DOI: 10.1115/1.1410100

[9] J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, 2004, 482 p.

[10] S.S. Antman, Nonlinear problems of elasticity, Springer, Berlin Heidelberg New York, 2005, 835 p.

[11] J. Gerstmayr, A.A. Shabana, Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45 1-2 (2006) 109-130.

DOI: 10.1007/s11071-006-1856-1

[12] A.A. Shabana, Computational continuum mechanics. Cambridge University Press, 2008, 349 p.

[13] P. Wriggers, Nonlinear finite element methods. Springer-Verlag Berlin Heidelberg, 2008, 566 p.

[14] S. Krenk, Non-linear modelling and analysis of solids and structures. Cambridge University Press, 2009, 361 р.

[15] А. Ibrahimbegovic, Nonlinear Solid Mechanics. Springer Science+Business Media B.V. 2009, 585 p.

[16] D. Bigoni, Nonlinear solid mechanics: bifurcation theory and material instability. Cambridge University Press, 2012, 550p.

[17] S. Coskun, B. Öztürk, Elastic Stability Analysis of Euler Columns Using Analytical Approximate Techniques, Advances in Computational Stability Analysis, Dr. Safa Bozkurt Coşkun (Ed. ), 2012, DOI 10. 5772/45940.

DOI: 10.5772/45940

[18] V.P. Bagmutov, A.A. Belov, A.S. Stoljarchuk, Jelementy raschetov na ustojchivost'. Ucheb. Posobie. V. P. Bagmutov, A. A. Belov, A. S. Stoljarchuk. – Volgograd: IUNL VolgGTU, 2010, 56 p. (Russian).

[19] V.V. Lalin, Razlichnye formy uravnenij nelinejnoj dinamiki uprugih sterzhnej. Trudy SPbGTU, 2004, No. 489, pp.121-128, (Russian).

[20] V.V. Lalin, D.A. Kushova, Geometricheski nelinejnoe deformirovanie i ustojchivost' ploskih uprugih sterzhnej s uchetom zhestkostej na rastjazhenie-szhatie, sdvig i izgib. International Journal for Computational Civil and Structural Engineering, 9 4 (2013).

[21] I.M. Gelfand, S.V. Fomin, Variacionnoe ischislenie. M.: GIFML, 1961, 228 p. (Russian).

[22] V.V. Lalin, L.A. Rozin, D.A. Kushova, Variacionnaja postanovka ploskoj zadachi geometricheski nelinejnogo deformirovanija i ustijchivosti uprugih sterzhnej. Inzhenerno – stroitelnyj zhurnal, 36 1 (2013) 87-96, (Russian).

[23] A.V. Perelmuter, V.I. Slivker, Ustojchivost' ravnovesija i rodstvennye problem. Vol. 1, Мoscow, SKAD SOFT, 2010, 647 p. (Russian).