Equipartition of energy in linearized $3$-D viscoelasticity
Authors:
George Dassios and Filareti Zafiropoulos
Journal:
Quart. Appl. Math. 48 (1990), 715-730
MSC:
Primary 73F15; Secondary 35B40, 35C15, 73B30, 73D99
DOI:
https://doi.org/10.1090/qam/1079915
MathSciNet review:
MR1079915
Full-text PDF Free Access
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Abstract: A regular and compactly supported initial disturbance propagates in a homogeneous isotropic and linearized viscoelastic medium. The elasticities of the medium exhibit a simple exponential decay in time. General expressions for the energies are obtained and the decay and equipartition of the kinetic and strain energies, for both the longitudinal as well as the transverse wave, are demonstrated.
- Alain Bachelot, Équipartition de l’énergie pour les systèmes hyperboliques et formes compatibles, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 45–76 (French, with English summary). MR 877995
- A. R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967), 207–208. MR 212417, DOI https://doi.org/10.1090/S0002-9939-1967-0212417-X
R. M. Christensen, Theory of Viscoelasticity. An Introduction, Academic Press, New York, 1982
- Bernard D. Coleman and Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 0158605, DOI https://doi.org/10.1103/RevModPhys.33.239
- Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308. MR 281400, DOI https://doi.org/10.1007/BF00251609
- George Dassios, Equipartition of energy in elastic wave propagation, Mech. Res. Comm. 6 (1979), no. 1, 45–50. MR 524233, DOI https://doi.org/10.1016/0093-6413%2879%2990078-8
- George Dassios and Manoussos Grillakis, Dissipation rates and partition of energy in thermoelasticity, Arch. Rational Mech. Anal. 87 (1984), no. 1, 49–91. MR 760319, DOI https://doi.org/10.1007/BF00251002
- George Dassios and Manoussos Grillakis, Asymptotic equipartition rate for wave motion in an even number of space dimensions, J. Math. Anal. Appl. 120 (1986), no. 1, 44–52. MR 861907, DOI https://doi.org/10.1016/0022-247X%2886%2990203-9
- W. A. Day, The decay of the energy in a viscoelastic body, Mathematika 27 (1980), no. 2, 268–286 (1981). MR 610712, DOI https://doi.org/10.1112/S0025579300010159
- R. J. Duffin, Equipartition of energy in wave motion, J. Math. Anal. Appl. 32 (1970), 386–391. MR 269190, DOI https://doi.org/10.1016/0022-247X%2870%2990304-5
- A. Cemal Eringen (ed.), Continuum physics. Vol. I, Academic Press, New York-London, 1971. Mathematics. MR 0468443
- Jerome A. Goldstein, An asymptotic property of solutions of wave equations, Proc. Amer. Math. Soc. 23 (1969), 359–363. MR 250125, DOI https://doi.org/10.1090/S0002-9939-1969-0250125-1
J. A. Goldstein, A (More-or-Less) Complete Bibliography of Semigroups of Operators Through 1984, Publication of Tulane University, New Orleans, 1984
- M. E. Gurtin and Eli Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11 (1962), 291–356. MR 147047, DOI https://doi.org/10.1007/BF00253942
- Bernard Hanouzet, Applications bilinéaires compatibles avec un système à coefficients variables. Continuité dans les espaces de Besov, Comm. Partial Differential Equations 10 (1985), no. 4, 433–465 (French). MR 784684, DOI https://doi.org/10.1080/03605308508820384
F. John, Partial Differential Equations, 4th edition, Springer-Verlag, New York, 1982
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
M. J. Leitman, The Linear Theory of Viscoelasticity, Handbuch der Physik (S. Flügge, Ed.), Springer-Verlag, Berlin, 1973
- David W. Reynolds, On the equipartition of energy in a linear viscoelastic body, Quart. Appl. Math. 42 (1984), no. 3, 325–336. MR 757171, DOI https://doi.org/10.1090/S0033-569X-1984-0757171-X
J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948
A. Bachelot, Équipartition de l’Energie pour les Systèmes Hyperboliques et Formes Compatibles, Ann. Inst. Henri Poincaré 46, 45–76 (1987)
R. A. Brodsky, On the asymptotic behaviour of solutions of the wave equation, Proc. Amer. Math. Soc. 18, 207–208 (1967)
R. M. Christensen, Theory of Viscoelasticity. An Introduction, Academic Press, New York, 1982
B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33, 239–249 (1961)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
G. Dassios, Equipartition of energy in elastic wave propagation, Mech. Res Comm. 6, 45–50 (1979)
G. Dassios and M. Grillakis, Dissipation rates and partition of energy in thermoelasticity, Arch. Rational Mech. Anal. 87, 49–91 (1984)
G. Dassios and M. Grillakis, Asymptotic equipartition rate for wave motion in an even number of space dimensions, J. Math. Anal. Appl. 120, 44–52 (1986)
W. A. Day, The decay of the energy in a viscoelastic body, Mathematica 27, 268–286 (1980)
R. J. Duffin, Equipartition of energy in wave motion, J. Math. Anal. Appl. 32, 386–391 (1970)
A. C. Eringen, Continuum Physics II, Academic Press, New York, 1975
J. A. Goldstein, An asymptotic property of solutions of wave equations, Proc. Amer. Math. Soc. 23, 359–363 (1969)
J. A. Goldstein, A (More-or-Less) Complete Bibliography of Semigroups of Operators Through 1984, Publication of Tulane University, New Orleans, 1984
M. E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational. Mech. Anal. 11, 291–356 (1962)
B. Hanouzet, Applications Bilinéaires Compatibles avec un Système à Coefficients Variables. Continuité dans les Espaces de Besov, Comm. Partial Differential Equations 10, 433–465 (1985)
F. John, Partial Differential Equations, 4th edition, Springer-Verlag, New York, 1982
P. Lax and R. Phillips, Scattering Theory, Academic Press, New York, 1967
M. J. Leitman, The Linear Theory of Viscoelasticity, Handbuch der Physik (S. Flügge, Ed.), Springer-Verlag, Berlin, 1973
D. W. Reynolds, On the equipartition of energy in a linear viscoelastic body, Quart. Appl. Math. 41, 325–336 (1984)
J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948
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Article copyright:
© Copyright 1990
American Mathematical Society
