Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T02:04:34.714Z Has data issue: false hasContentIssue false
  • English
  • Français

On Saint-Venant's problem for an elastic strip*

Published online by Cambridge University Press:  14 November 2011

Alexander Mielke
Alexander Mielke
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D7000 Stuttgart 80, West Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

The equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 1-2 , 1988 , pp. 161 - 181
Copyright
Copyright © Royal Society of Edinburgh 1988

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.. Large Lateral Buckling of Nonlinearity Elastic Beams. Arch. Rational Mech. Anal. 84 (1984), 293–305.CrossRefGoogle Scholar
2Ball, J. M.. Constitutive Inequalities and Existence Theorems in Nonlinear Elastostatics. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I, ed. Knops, R. J., pp. 187–241 (London: Pitman, 1977).Google Scholar
3Breuer, S. and Roseman, J. J.. On SV's Principle in 3-dimensional nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 191–203.CrossRefGoogle Scholar
4Ciarlet, P. G.. Lectures on Three Dimensional Elasticity. Tata Institute Lectures onMath (Berlin: Springer, 1983).CrossRefGoogle Scholar
5Ericksen, J. L.. On the Formulation of St.-Venant's Problem. In Nonlinear Analysis and Mechancs: Heriot-Watt Symposium, Vol. I, ed. Knops, R. J., pp. 158–186 (London: Pitman, 1977).Google Scholar
6Ericksen, J. L.. Periodic Solutions for Elastic Prisms. Quart. Appl. Math. 37 (1980), 443–446.CrossRefGoogle Scholar
7Ericksen, J. L.. Problems for Infinite Elastic Prisms—St.-Venant's Problem for Elastic Prisms. In Systems of Nonlinear Partial Differential Equations, ed. Ball, J. M., NATO ASI Series C 111 (Boston: Reidel, 1983).Google Scholar
8Gusein-Zade, M. I.. On Necessary and Sufficient Conditions for the Existence of Decaying Solutions of the Plane Problem of the Theory of Elasticity for a Semistrip. J. Appl. Math. Mech. 29 (1965), 892–901.CrossRefGoogle Scholar
9Iesan, D.. On Saint-Venant's Problem. Arch. Rational Mech. Anal. 91 (1986), 363–373.CrossRefGoogle Scholar
10Kirchgässner, K.. Wave Solutions in Reversible Systems and Applications. J. Differential Equations 45 (1982), 113–127.CrossRefGoogle Scholar
11Knops, R. J. and Payne, L. E.. A Saint-Venant Principle for Nonlinear Elasticity. Arch. Rational Mech. Anal. 81 (1983), 1–12.CrossRefGoogle Scholar
12Kohn, R. V.. New Integral Estimates for Deformations in Terms of their Nonlinear Strains. Arch. Rational Mech. Anal. 78 (1982), 131–172.CrossRefGoogle Scholar
13Love, A. E. H.. A Treatise on the Mathematical Theory of Elasticity, (4th edn.) (Cambridge: University Press, 1927).Google Scholar
14Mielke, A.. Reduction of Quasilinear Elliptic Equations in Cylindrical Domains with Applications. Math. Meth. Appl. Sci. 10 (1988), 51–66.CrossRefGoogle Scholar
15Muncaster, R. G.. Saint-Venant's Problem in Nonlinear Elasticity: A Study of Cross-Sections. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, ed. Knops, R. J., pp. 17–75 (London: pitman, 1979).Google Scholar
16Oleinik, O. A. and Yosiflan, G. A.. On the Asymptotic Behavior at Infinity of Solutions in Linear Elasticity. Arch. Rational Mech. Anal. 78 (1982), 29–53.CrossRefGoogle Scholar
17Roseman, J. J.. The Principle of Saint-Venant in Linear and Nonlinear Plane Elasticity. Arch. Rational Mech. Anal. 26 (1967), 142–162.CrossRefGoogle Scholar
18Saint-Venant, B. de. Memoire sur la torsion des prismes, Mémoires présentes par divers savant à l'academie de sciences de l'institut imperial de France, 2. Ser. 14 (1856), 233–560.Google Scholar
19Sternberg, E. and Knowles, J. K.. Minimum Energy Characterizations of Saint-Venant's solution to the Relaxed Saint-Venant Problem. Arch. Rational Mech. Anal. 21 (1966), 89–107.CrossRefGoogle Scholar