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April  1998, 4(2): 339-358. doi: 10.3934/dcds.1998.4.339

Longtime behavior of a homogenized model in viscoelastodynamics

M. Grasselli 1, and  Vittorino Pata 2,

1. 

Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano
2. 

Dipartimento di Elettronica per l'Automazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy

Received  October 1996 Revised  July 1997 Published  February 1998

A material with heterogeneous structure at microscopic level is considered. The microscopic mechanical behavior is described by a stress-strain law of Kelvin-Voigt type. It has been shown that a homogenization process leads to a macroscopic stress-strain relation containing a time convolution term which accounts for memory effects. Consequently, the displacement field $\mathbf{u}$ obeys to a Volterra integrodifferential motion equation. The longtime behavior of $\mathbf{u}$ is here investigated proving the existence of a uniform attractor when the body forces vary in a suitable metric space.
Keywords: Integro-differential equation, translation compact functions, asymptotic behavior, uniform absorbing set, uniform attractor..
Mathematics Subject Classification: Primary: 35B40, 45K05, 73F1.
Citation: M. Grasselli, Vittorino Pata. Longtime behavior of a homogenized model in viscoelastodynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 339-358. doi: 10.3934/dcds.1998.4.339
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