Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-31T03:27:20.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Saint-Venant's principle on unbounded regions

Published online by Cambridge University Press:  14 November 2011

R. J. Knops ,
S. Rionero  and
L. E. Payne
R. J. Knops
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, U.K.
S. Rionero
Affiliation:
Istituto di Matematica, Università di Napoli, Cap 80134, Napoli, Italy
L. E. Payne
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 319 - 336
Copyright
Copyright © Royal Society of Edinburgh 1990

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.)

Article purchase

Temporarily unavailable

References

1Boussinesq, J.. Application des potentials à I'étude de I'equilibre et des mouvements des solides étastiques Paris: Gautier-Villars, (1885).Google Scholar
2Bramble, J. H. and Payne, L. E..Some inequalities for vector functions with applications in elasticity. Arch. Rational Mech. Anal. 11 (1962), 16–26.CrossRefGoogle Scholar
3Bramble, J. H. and Payne, L. E..Bounds for solutions of second-order elliptic partial differential equations. Contrib. Differential Equations 1 (1963), 95–127.Google Scholar
4Fichera, G..Sull'esistenza e sul calcolo della soluzioni dei problemi al contorno relativi all'equilibrio di un corpo elastico. Ann. Scuolo Norm. Sup. Pisa 4 (1950), 35–99.Google Scholar
5Flavin, J. N.Knops, R. J. and Payne, L. E..Energy bounds in dynamical problems for a semi-infinite elastic beam. In Elasticity: Mathematical Methods and Applications, eds Eason, G. and Ogden, R. W. (Ellis Horwood: Chicester, 1990), 101–111.Google Scholar
6Flavin, J. N., Knops, R. J. and Payne, L. E.Decay estimates for the constrained elastic cylinder of variable cross section. Quart. Appl. Maths. 47 (1989), 325–350.CrossRefGoogle Scholar
7Galdi, G. P. and Rionero, S..Continuous dependence theorems in linear elasticity on exterior domains. Int. J. Eng. Sci. 17 (5) (1979), 521–526.CrossRefGoogle Scholar
8Galdi, G. P. and Rionero, A..On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions. J. Elast. 10 (1980), 333–340.Google Scholar
9Galdi, G. P. and Rionero, S.. Weighted Energy Methods in Fluid Dynamics and Elasticity. Lecture Notes in Mathematics 1134 (Berlin: Springer, 1985).CrossRefGoogle Scholar
10Gurtin, M. E. and Sternberg, E..Theorems in linear elastostatics for exterior domains. Arch. Rational Mech. Anal. 8 (1961), 99–119.CrossRefGoogle Scholar
11Horgan, C. O. and Knowles, J. K..Recent developments concerning Saint-Venant's principle.In Advances in Applied Mechanics 23 ed Hutchinson, J. W., (New York: Academic Press, 1983), 179–269.Google Scholar
12Horgan, C. O. and Payne, L. E.Decay estimates for a class of nonlinear boundary value problems in two dimensions. SIAM J. Math. Anal. 20 (1989), 782–788.CrossRefGoogle Scholar
13Horvay, G..Some aspects of Saint-Venant's principle. J Mech. Phys. Sols. 5 (1957), 77–94.CrossRefGoogle Scholar
14Horvay, G..Saint-Venant's principle: a biharmonic eigenvalue problem. J. Appl. Mech. 24 (1957) 381–386.CrossRefGoogle Scholar
15Howell, K. B..Uniqueness in linear elasticity for problems involving unbounded regions. J Elast. 10 (1980), 407–428.CrossRefGoogle Scholar
16Howell, K. B., Periodic and ‘slightly’ periodic boundary value problems in elastostatics on bodies unbounded in several directions. Int. J. Eng. Sci. 20 (1982) 455–481.CrossRefGoogle Scholar
17Huilgol, R. R..On Liouville's Theorem for biharmonic functions. SIAM J. Appl. Maths. 20 (1971), 37–39.Google Scholar
18Keller, H. B.,Saint-Venant's procedure and Saint-Venant's principle. Quart. Appl. Math. 22 (1965), 293–304.CrossRefGoogle Scholar
19Knops, R. J. and Payne, L. E.. Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy 19 (Berlin: Springer, 1971).CrossRefGoogle Scholar
20Kondratiev, V. A. and Oleinik, O. A..On the asymptotics at infinity of solutions of elliptic systems with constant coefficients. Uspekhi Mat. Nauk. 40 (1985), 233.Google Scholar
21Kondratiev, V. A. and Oleinik, O. A.On the asymptotic behaviour of solutions of systems of differential equations. Uspekhi Mat. Nauk. 40 (1985), 306.Google Scholar
22Kondratiev, V. A. and Oleinik, O. A.Asymptotic properties of solutions of the elasticity system.In Applications of Multiple Scaling in Mechanics, eds Ciarlet, P. G. and Sanchez-Palencia, E.,188–205 (Paris: Masson, 1987).Google Scholar
23Kondratiev, V. A. and Oleinik, O. A..On the behaviour of solutions of elliptic systems with finite energy integral. Arch. Rational Mech. Anal. 99 (1987), 75–89.Google Scholar
24Kondratiev, V. A. and Oleinik, O. A.Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities. Uspekhi Mat. Nauk. 43 (1988), 65–119; English translation: Russian Math. Surveys 43 (1988), 65–119.Google Scholar
25Kondratiev, V. A., Kopachek, I. and Oleinik, O. A..On the behaviour of weak solutions of second order elliptic equations and the elasticity system in a neighbourhood of a boundary point. Trudy Sent. Petrovsky 8 (1982), 135–152.Google Scholar
26Payne, L. E. and Weinberger, H. F..Note on a lemma of Finn and Gilbarg. Ada Math. 98 (1957), 297–299.Google Scholar
27Payne, L. E. and Weinberger, H. F..New bounds for solutions of second order elliptic partial differential equations. Pacific J. Math. 8 (1958), 551–573.CrossRefGoogle Scholar
28Russo, R..Continuous data dependence and uniqueness in homogeneous linear elastostatics. Boll. Un. Mat. Ital. A (6) 5 (1986), 227–234.Google Scholar
29Russo, R..An extension of the basic theorems of linear elastostatics to exterior domains. Ann. Univ. Ferrara 34 (1988), 101–119.Google Scholar
30Russo, R.. On traction problem in linear elastostatics (to appear).Google Scholar
31Sternberg, E.On Saint-Venant's Principle. Quart. Appl. Math. 11 (1954), 393–402.CrossRefGoogle Scholar
32von Mises, R..On Saint-Venant's Principle. Bull. Amer. Math. Soc. 51 (1945), 555–562.CrossRefGoogle Scholar
33Wilcox, C. A..Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics. J Elast. 9 (1979) 221–243Google Scholar