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On spatial behavior in linear viscoelasticity
Author(s):
Catalin
Gales;
Stan
Chirita
Journal:
Quart. Appl. Math.
MSC (2000):
Primary 74D05, 74G50;
Secondary 74H45, 74E10
Posted:
May 12, 2009
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Abstract:
Within the framework of linear viscoelasticity this paper deals with the study of spatial behavior of solutions describing harmonic vibrations in a right cylinder of finite extent. Some exponential decay estimates of Saint-Venant type, in terms of the distance from the excited end of the cylinder are obtained from a first-order differential inequality concerning an appropriate measure associated with the amplitude of the steady-state vibration. The dissipative mechanism guarantees the validity of the result for every value of the frequency of vibration and for the class of viscoelastic materials compatible with thermodynamics whose relaxation tensor is supposed to be symmetric and sufficiently regular. The case of a semi-infinite cylinder is also studied, and some alternatives of Phragmé n-Lindelöf type are established.
References:
-
- 1.
- J. N. Flavin and R. J. Knops, Some spatial decay estimates in continuum dynamics, J. Elasticity, 17 (1987), 249-264. MR 888318 (88g:73014)
- 2.
- J. N. Flavin, R. J. Knops and L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross section, Quart. Appl. Math., 47 (1989), 325-350. MR 998106 (90g:73026)
- 3.
- J. N. Flavin, R. J. Knops and L. E. Payne, Energy bounds in dynamical problems for a semi-infinite elastic beam, In Elasticity: Mathematical methods and applications (G. Eason and R.W. Ogden, eds.) The Ian N. Sneddon 70th birthday volume, Ellis Horwood Limited, Chichester, 1990, pp. 101-112.
- 4.
- R. J. Knops, Spatial decay estimates in the vibrating anisotropic elastic beam, In Waves and Stability in Continuous Media (S. Rionero ed.), World Scientific, Singapore, 1991, pp. 192-203. MR 1193270
- 5.
- S. Chiriţă, Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder, J. Thermal Stresses, 18 (1995), 421-436. MR 1423300 (97k:73005)
- 6.
- M. Aron and S. Chiriţă, Decay and continuous dependence estimates for harmonic vibrations of micropolar elastic cylinders, Arch. Mech., 49 (1997), 665-675. MR 1482613 (98j:73006)
- 7.
- W. A. Day, The thermodynamics of simple materials with fading memory, Springer, Berlin, 1972. MR 0366234 (51:2482)
- 8.
- M. J. Leitman and G. M. Fischer, The linear theory of viscoelasticity, Handbuch der Physik, Band VIa/3, Springer, Berlin, 1973.
- 9.
- R. M. Christensen, Theory of viscoelasticity: An introduction, 2, Academic Press, New York, 1982.
- 10.
- M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. MR 1153021 (93a:73034)
- 11.
- M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity, 19 (1988), 63-75. MR 928727 (89m:73019)
- 12.
- R. A. Toupin, Saint-Venant's Principle, Arch. Rational Mech. Anal., 18 (1965), 83-96. MR 0172506 (30:2725)
- 13.
- J. K. Knowles, On Saint-Venant's principle in the two-dimensional linear theory of elasticity, Arch. Rational Mech. Anal., 21 (1966), 1-22. MR 0187480 (32:4930)
- 14.
- R. S. Lakes and A. Wineman, On Poisson's ratio in linearly viscoelastic solids, J. Elasticity, 85 (2006), 45-63. MR 2254005 (2007e:74015)
- 15.
- R. S. Lakes, Viscoelastic solids, CRC Press, Boca Raton, FL, 1998.
- 16.
- S. Chiriţă, M. Ciarletta and M. Fabrizio, Saint-Venant's principle in linear viscoelasticity, Internat. J. Engrg. Sci., 35 (1997), 1221-1236. MR 1488530 (99b:73022)
- 17.
- S. Chiriţă and M. Ciarletta, Time-weighted surface power function method for the study of spatial behaviour in dynamics of continua, Eur. J. Mech., A/Solids, 18 (1999), 915-933. MR 1723228 (2000i:74044)
- 18.
- W. A. Day, Restrictions on relaxation functions in linear viscoelasticity, Quart. J. Mech. Appl. Math., 24 (1971), 487-497. MR 0317631 (47:6178)
- 19.
- N. S. Wilkes, Thermodynamic restrictions on viscoelastic materials, Quart. J. Mech. Appl. Math., 30 (1977), 209-221. MR 0495550 (58:14220)
- 20.
- M. A. Gurtin, The linear theory of elasticity, Handbuch der Physik, Band VIa/2, Springer, Berlin, 1972.
- 21.
- J. Merodio and R. W. Ogden, A note on strong ellipticity for transversely isotropic linearly elastic solids, Quart. J. Mech. Appl. Math., 56 (2003), 589-591. MR 2026873
- 22.
- S. Chiriţă, On the strong ellipticity condition for transversely isotropic linearly elastic solids, An. St. Univ. Iasi, Matematica, 52 (2006), 245-250. MR 2341092 (2008f:74010)
- 23.
- S. Chiriţă, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27. MR 2310629 (2007m:74013)
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Additional Information:
Catalin
Gales
Affiliation:
Faculty of Mathematics, Al. I. Cuza University of Iasi, Blvd. Carol I, no. 11, 700506 - Iasi, Romania
Email:
cgales@uaic.ro
Stan
Chirita
Affiliation:
Faculty of Mathematics, Al. I. Cuza University of Iasi, Blvd. Carol I, no. 11, 700506 - Iasi, Romania
Email:
schirita@uaic.ro
PII:
S0033-569X-09-01149-0
Keywords:
Viscoelastic cylinder,
harmonic vibrations,
spatial behavior,
dissipative effects
Received by editor(s):
May 16, 2008.
Posted:
May 12, 2009
Additional Notes:
The authors are very grateful to the reviewer for useful observations which have led to the improvement of this paper. The authors were supported by the Romanian Ministry of Education and Research, CNCSIS Grant code ID-401, Contract no. 15/28.09.2007.
Copyright of article:
Copyright
2009,
Brown University
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