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.

**[1]**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)] MR**0373425****[2]**J. H. Bramble and L. E. Payne,*Some inequalities for vector functions and applications in elasticity*, Arch. Rational Mech. Anal.**11**, 16-26 (1962) MR**0170516****[3]**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) MR**589248****[4]**C. O. Horgan,*Eigenvalue estimates and the trace theorem*, J. Math. Anal. Appl.**69**, 231-242 (1979) MR**535293****[5]**C. O. Horgan,*Korn's inequalities and their applications in continuum mechanics*, SIAM Review**37**, 491-511 (1995) MR**1368384****[6]**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 MR**889288****[7]**C. O. Horgan and L. E. Payne,*On inequalities of Korn, Friedrichs and Babuska-Aziz*, Arch. Rational Mech. Anal.**82**, 165-179 (1983) MR**687553****[8]**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) MR**1458835****[9]**G. Horvay,*The end problem of rectangular strips*, J. Appl. Mech.**75**, 87-94 (1953) MR**0053735****[10]**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 MR**1320082****[11]**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) MR**1488687****[12]**R. J. Knops and C. Lupoli,*End effects for non-steady Stokes flow along a pipe*(to appear)**[13]**J. R. Kuttler and V. G. Sigillito,*Inequalities for membrane and Stekloff eigenvalues*, J. Math. Anal. Appl.**23**, 148-160 (1968) MR**0226226****[14]**C. Lupoli,*Some problems of spatial behaviour in continuum mechanics*, Ph.D. Thesis, Department of Mathematics, Heriot-Watt University, June 1995**[15]**P. Maremonti and R. Russo,*On Saint-Venant's principle and related results in incompressible linear elastostatics*, Ricerche di Matematica**42**, 361-375 (1993) MR**1283366****[16]**B. S. Mittelman and A. P. Hillman,*Zeros of*, Math. Tables and other Aids to Computation**2**, 60 (1946)**[17]**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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DOI:
https://doi.org/10.1090/qam/1753404

Article copyright:
© Copyright 2000
American Mathematical Society