Some Results in the Dynamics of Viscoelastic Mixtures

Abstract

The present article studies some qualitative properties of solutions in the dynamics of viscoelastic mixtures made by two constituents: a porous elastic solid and a viscous fluid. In this sense, some results concerning the theory of semigroups of linear operators are used to establish the existence and uniqueness of the weak solutions of the initial boundary value problems associated with the linear theory. The continuous dependence problem upon initial data and supply terms is also investigated. Then, the temporal behavior is studied in terms of the Cesáro means of various parts of the total energy. The relations that describe the asymptotic behavior of mean energies are established by using some Lagrange—Brun identities.

Keywords

Viscoelastic mixture existence and uniqueness of weak solutions asymptotic partition of energy

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References

Bowen, R.M. Theory of mixtures, in Continuum Physics, vol. III, ed. A. C. Eringen , Academic, New York, 1976.

Atkin, R.J. and Craine, R.E. Continuum theories of mixtures: Basic theory and historical development. The Quarterly Journal of Mechanics and Applied Mathematics, 29, 209—245 ( 1976).

Atkin, R.J. and Craine, R.E. Continuum theories of mixtures: applications . Journal of the Institute of Mathematics and its Applications , 17, 153—207 (1976).

Bedford, A. and Drumheller, D.S. Theory of immiscible and structured mixtures . International Journal of Engineering Science, 21, 863—960 (1983).

Samohyl, I. Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner Verlag, Leibzig, 1987.

Rajagopal, K.R. and Tao, L. Mechanics of Mixtures, World Scientific, Singapore, 1995 .

Ieşan, D.

On the theory of viscoelastic mixtures

Journal of Thermal Stresses, 27, 1125—1148 ( 2004).

Goodman, M.A. and Cowin, S.C. A continuum theory for granular materials . Archive for Rational Mechanics and Analysis, 44, 249—266 (1972).

Nunziato, J.W. and Cowin, S.C. A nonlinear theory of elastic materials with voids. Archive for Rational Mechanics and Analysis, 72, 175—201 (1979).

10.

Galeş, C.

On the spatial behavior in the theory of viscoelastic mixtures

Journal of Thermal Stresses, 30, 1—24 (2007).

11.

Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer, Berlin, 1983.

12.

Day, W.A.

Means and autocorrelations in elastodynamics

Archive for Rational Mechanics and Analysis, 73, 243—256 (1980).

13.

Levine, H.A.

An equipartition of energy theorem for weak solutions of evolutionary equations in Hilbert space: the Lagrange identity method

Journal of Differential Equations, 24, 197—210 (1977).

14.

Chiriţă S. On the asymptotic partition of energy in linear thermoelasticity. Quarterly of Applied Mathematics, 45, 327—340 (1987).

15.

Bedford, A. and Stern, M. A multi-continuum theory for composite elastic materials. Acta Mechanica, 14, 85—102 (1972).

16.

Adams, R.A. Sobolev Spaces, Academic, New York , 1975.

17.

Hlavàcek, I. and Necas, J. On inequalities of Korn's type. Archive for Rational Mechanics and Analysis, 36, 305—334 (1970).

18.

Ieşan, D. and Quintanilla , R. Existence and continuous dependence results in the theory of interacting continua. Journal of Elasticity, 35, 85—98 (1994 ).

19.

Martìnez, F. and Quintanilla, R. Some qualitative results for the linear theory of binary mixtures of thermoelastic solids. Collectanea Mathematica, 46, 263—277 (1995).

20.

Galeş, C.

Existence and uniqueness results in the theory of swelling porous elastic soils

Analele Ştiinţif ice ale Univ. Al. I. Cuza Iaşi Matematica, 49, 161—174 (2003).

21.

Brun, L.

Mètodès ènergetiques dans les systémes linèaires

Journal de Mècanique, 8, 125—192 (1969).

22.

Ciarletta, M. and Passarella, F. Some results in the dynamics of porous elastic mixtures. International Journal of Engineering Science, 41, 995—1018 (2003).