Decay and Other Estimates for an Annular Elastic Cylinder in an Axisymmetric State of Stress

Abstract

This paper considers an annular circular cylinder, consisting of homogeneous isotropic linear elastic material, in a state of axisymmetric (torsionless) stress corresponding to traction-free lateral boundaries and such that the traction on every cross-section is self-equilibrated, body forces beign absent; in the main case considered, one plane end is traction-free, the other being self-equilibrated. A positive-definite cross-sectional measure of stress is defined and shown to be a convex function of the axial coordinate, and, in the main case considered, to be a function of “generalized convexity”. Interences are drawn, including a spatial decay estimate for a semi-infinite cylinder reflecting Saint-Venant's principle. The paper concludes with a discussion of the estimate obtained.

Get full access to this article

View all access options for this article.

References

Knowles, J. K. and Horgan, C. O. On the exponential decay of stresses in circular elastic cylinder subject to axisymmetric self-equilibrated end loads . International Journal of Solids and Structures, 5, 33-50 (1969).

Horgan, C. O. and Knowles, J. K. Recent developments concerning Saint-Venant's principle, in Advances in Applied Mechanics, pp. 179-269, ed. T. Y. Wu & J. W. Hutchison , Vol. 23, Academic, New York , 1983.

Flavin, J. N. and Knops, R. J. Some convexity considerations for a two-dimensional traction problem . Zeitschrift für Angewandte Mathematik und Physik, 39, 166-176 (1988).

Horgan, C. O. Recent developments concerning Saint-Venant's principle: an update . Applied Mechanics Review, 42, 295-303 (1989).

Horgan, C. O. Recent developments concerning Saint-Venant's principle: a second update . Applied Mechanics Reviews, 49, 101-111 (1996).

Love, A. E. H. The Mathematical Theory of Elasticity, 4th edn, Dover, New York (1944).

Gleeson, B. Qualitative estimates for partial differential equations in elasticity. Ph.D. Thesis, Chapter 5, NUI Galway (2001).

Flavin, J. N. and Rionero, S. Qualitative Estimates for Partial Differential Equations: An Introduction, CRC Press, Boca Raton, FL , 1995.

Flavin, J. N. , Knops, R. J. and Payne, L. E. Decay estimates for the constrained elastic cylinder of variable cross-section . Quarterly Journal of Applied Mechanics, 47, 325-350 (1989).

10.

Knowles, J. K. On Saint-Venant's principle in the two-dimensional linear theory of elasticity . Archive for Rational Mechanics and Analysis, 21, 1-22 (1966).

11.

Flavin, J. N. On Knowles' version of Saint-Venant's principle in two-dimensional elastostatics . Archive for Rational Mechanics and Analysis, 53, 366-375 (1974).

12.

Stephen, N. G. and Wang, M. Z. Decay rates for the hollow circular cylinder . Journal of Applied Mechanics, 59, 747-753 (1992).

13.

Mitrinović, D. S. , Pečarić, J. E. and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer, Dordrecht , 1991.