Asymptotic behaviour in linear viscoelasticity
Author:
Jaime E. Muñoz Rivera
Journal:
Quart. Appl. Math. 52 (1994), 628-648
MSC:
Primary 73F15; Secondary 35B40, 35Q72, 45K05
DOI:
https://doi.org/10.1090/qam/1306041
MathSciNet review:
MR1306041
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We study the asymptotic behaviour of the solution of the viscoelastic equation, and we prove for a bounded domain that the energy associated to this system approaches zero exponentially as time goes to infinity. Moreover, for the whole space ${\mathbb {R}^n}$ we will prove that the displacement vector field can be decomposed into two parts, solenoidal and irrotational, whose corresponding energies decay to zero uniformly as time goes to infinity with rates that depend on the regularity of the initial data.
G. Creus, Viscoelasticity basic theory and applications to concrete structures, (C. A. Brebbia and S. A. Orszag, eds.), Lecture Notes in Engineering, vol. 16, Springer-Verlag, New York, 1986
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations (3), 178–219 (1979)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non lineares, Dunod Gauthier Villars, Paris, 1969
G. Creus, Viscoelasticity basic theory and applications to concrete structures, (C. A. Brebbia and S. A. Orszag, eds.), Lecture Notes in Engineering, vol. 16, Springer-Verlag, New York, 1986
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations (3), 178–219 (1979)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non lineares, Dunod Gauthier Villars, Paris, 1969
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73F15,
35B40,
35Q72,
45K05
Retrieve articles in all journals
with MSC:
73F15,
35B40,
35Q72,
45K05
Additional Information
Article copyright:
© Copyright 1994
American Mathematical Society
