Abstract
In this chapter, nonlinear theories for rods and beams will be discussed in the Cartesian coordinate frame and the curvilinear frame of the initial configuration. Without torsion, the theory for in-plane beams will be presented. The traditional treatises of nonlinear rods were based on the Cosserat’s theory (e.g., E. and F. Cosserat, 1896) or the Kirchhoff assumptions (e.g., Kirchhoff, 1859; Love, 1944). This chapter will extend the ideas of (1915), and the nonlinear theory of rods and beams will be developed from the general theory of the 3-dimensional deformable body. The definitions for beams and rods are given as follows.
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References
Cosserat, E. and Cosserat, F., 1896, Sur la théorie de l’élasticitié. Premier Mémoire, Annals de la Faculté des Sciences de Toulouse, 10, 1–116.
Cosserat, E. and Cosserat, F. 1909, Theorie des corps deformables, Hermann, Paris.
Ericksen, J.L. and Truesdell, C., 1958, Exact theory of stress and strain in rods and shells, Archive Rational Mechanics Anal, 1, 295–323.
Frenet, F., 1847, Sur les courbes a double courbure, Thèse, Toulouse.
Galerkin, B.G., 1915, Series solutions of some problems of elastic equilibrium of rods and plates, Vestnik Inzhenerov, 1, 879–908.
Goldstein, H., Poole, C. and Safko, J., 2002, Classic Mechanics (3rd edition), Addison Wesley, San Francisco.
Kresyszig, E.,1968, Introduction to Differential Geometry and Riemannian Geometry, University of Toronto Press, Toronto.
Kirchhoff, G., 1859, Ueberdas Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, Journal für die reine und angewandte Mathematik, 56, 285–313.
Love, A.E.H., 1944, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York.
Luo, A.C.J. and Han, R.P.S., 1999, Analytical predictions of chaos in a nonlinear rod, Journal of Sound and Vibration, 227, 523–544.
Novozhilov, V.V., 1953, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, New York.
Reissner, E., 1972, On one-dimensional finite-strain beam theory: the plane beam, Journal of Applied Mathematics and Physics (ZAMP), 23, 759–804.
Verma, G.R., 1972, Nonlinear Vibrations of beam and membranes, Studies in Applied Mathematics, LII, 805–814.
Whitman, A.B. and DeSilva, C.N., 1969, A dynamical theory of elastic directed curves, Journal of Applied Mathematics and Physics (ZAMP), 20, 200–212.
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© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
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Luo, A.C.J. (2010). Nonlinear Beams and Rods. In: Nonlinear Deformable-body Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12136-4_8
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DOI: https://doi.org/10.1007/978-3-642-12136-4_8
Publisher Name: Springer, Berlin, Heidelberg
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