Spatial behaviour in plane incompressible elasticity on a half-strip
Authors:
R. J. Knops and P. Villaggio
Journal:
Quart. Appl. Math. 58 (2000), 355-367
MSC:
Primary 74G55; Secondary 74B05, 74G50
DOI:
https://doi.org/10.1090/qam/1753404
MathSciNet review:
MR1753404
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Abstract: Growth and decay estimates are derived for an incompressible homogeneous isotropic elastic material occupying a plane semi-infinite strip in equilibrium under self-equilibrated loads on the base and zero traction along the lateral sides. The estimates depend upon a pair of differential inequalities for two cross-sectional line integrals related to different kinds of energy fluxes. A comparison with the exact solution shows that the estimates are somewhat conservative. The method, however, is applicable to non-rectangular plane regions.
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V. L. Berdichevskii, On the proof of the Saint-Venant principle for bodies of arbitrary shape, Prikl. Mat. Mech. 38, 851–864 (1974). [Jl. Appl. Math. Mech. 38, 799–813 (1975)]
J. H. Bramble and L. E. Payne, Some inequalities for vector functions and applications in elasticity, Arch. Rational Mech. Anal. 11, 16–26 (1962)
R. D. Gregory, The traction boundary value problem for the elastostatic semi-infinite strip; existence of solution, and completeness of the Papkovich-Fadle eigenfunctions, J. of Elasticity 10, 295–327 (1980)
C. O. Horgan, Eigenvalue estimates and the trace theorem, J. Math. Anal. Appl. 69, 231–242 (1979)
C. O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Review 37, 491–511 (1995)
C. O. Horgan and J. K. Knowles, Recent developments concerning Saint- Venant’s principle, T. Y. Wu and J. W. Hutchinson, eds., Advances in Applied Mechanics 23, Academic Press, New York, 1983, pp. 179–269
C. O. Horgan and L. E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz, Arch. Rational Mech. Anal. 82, 165–179 (1983)
C. O. Horgan and L. E. Payne, Saint-Venant’s principle in linear isotropic elasticity for incompressible and nearly incompressible materials, J. of Elasticity 46, 43–52 (1997)
G. Horvay, The end problem of rectangular strips, J. Appl. Mech. 75, 87–94 (1953)
R. J. Knops, End effects in fluid flows along a pipe, Proceedings of the 7th Conference on Waves and Stability in Continuous Media, Ed. by S. Rionero and T. Ruggeri, World Scientific, 1994, pp. 224–235
R. J. Knops and C. Lupoli, End effects for plane Stokes flow along a semi-infinite strip, Z. Angew. Math. Phys. 48, 905–920 (1997)
R. J. Knops and C. Lupoli, End effects for non-steady Stokes flow along a pipe (to appear)
J. R. Kuttler and V. G. Sigillito, Inequalities for membrane and Stekloff eigenvalues, J. Math. Anal. Appl. 23, 148–160 (1968)
C. Lupoli, Some problems of spatial behaviour in continuum mechanics, Ph.D. Thesis, Department of Mathematics, Heriot-Watt University, June 1995
P. Maremonti and R. Russo, On Saint-Venant’s principle and related results in incompressible linear elastostatics, Ricerche di Matematica 42, 361–375 (1993)
B. S. Mittelman and A. P. Hillman, Zeros of $z + \sin z$, Math. Tables and other Aids to Computation 2, 60 (1946)
N. Week, An explicit Saint-Venant’s principle in three-dimensional elasticity. Ordinary and Partial Differential Equations, Dundee (1976), Lecture Notes in Mathematics, vol. 564, Springer-Verlag, New York, 1976, pp. 518–526
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© Copyright 2000
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