Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T22:51:21.635Z Has data issue: false hasContentIssue false
  • English
  • Français

5.—Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods.§

Published online by Cambridge University Press:  14 February 2012

Stuart S. Antman  and
Kathleen B. Jordan
Get access Rights & Permissions [Opens in a new window]

Extract

In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 85 - 105
Copyright
Copyright © Royal Society of Edinburgh 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Antman, S. S., The theory of rods. Handbuch der Physik, VIa/2, (Ed.: Truesdell, C.). Berlin: Springer Verlag, 641–703, 1972.Google Scholar
2Antman, S. S., Qualitative theory of the ordinary differential equations of nonlinear elasticity. Mechanics Today, 1, 1972. (Ed.: Nemat-Nasser, S.). New York: Pergamon, 58–101, 1974.Google Scholar
3Antman, S. S., Monotonicity and invertibility conditions in one-dimensional nonlinear elasticity, In Nonlinear elasticity (Ed.: Dickey, R. W.). New York: Academic Press, 57–92, 1973.Google Scholar
4Antman, S. S., Kirchhoff's problem for nonlinearly elastic rods. Quart. Appl. Math, 32, 221–240, 1974.CrossRefGoogle Scholar
5Antman, S. S., Boundary value problems of one-dimensional nonlinear elasticity. (To appear.)Google Scholar
6Cosserat, E. and F., , Théorie des corps déformables. Paris: Hermann, 1909.Google Scholar
7Ericksen, J. L., Simpler static problems in nonlinear theories of rods. Internal. J. Solids and Structures, 6, 371–377, 1970.CrossRefGoogle Scholar
8Ericksen, J. L. and Truesdell, C, Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal, 1, 295–323, 1958.CrossRefGoogle Scholar
9Euler, L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne, 1744.CrossRefGoogle Scholar
10Green, A. E. and Laws, N., A general theory of rods. Proc. Roy. Soc. London, 293A, 145–155, 1966.Google Scholar
11Kirchhoff, G., Über das Gleichgewicht und die Bewegung eines unendlich diinnenelastischen Stabes. J.Reine Angew. Math., 56, 285–313, 1859.Google Scholar
12Kovári, K., Räumliche Verzweigungsprobleme des dünnen elastischen Stabes mit endlichen Verformungen. Ing.-Arch., 37, 393–416, 1969.CrossRefGoogle Scholar
13Love, A. E. H., A treatise on the mathematical theory of elasticity (4th edn.). New York: Dover, 1944.Google Scholar
14Whitman, A. B. and DeSilva, C. N., An exact solution in a nonlinear theory of rods. (To appear.)Google Scholar