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Continuous dependence and uniqueness theorems in boundary-initial value problems for a class of porous bodies occupying bounded or unbounded domains

Part of: Flows in porous media; filtration; seepage

Published online by Cambridge University Press:  26 February 2010

Alessandra Borrelli  and
Maria Cristina Patria
Alessandra Borrelli
Affiliation:
Istituto Matematico dell'Università, via Machiavelli 35, 44100, Ferrara, Italy.
Maria Cristina Patria
Affiliation:
Istituto Matematico dell'Università, via Machiavelli 35, 44100, Ferrara, Italy.
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Summary

In this paper the authors formulate a boundary-initial value problem for a linear elastic porous body saturated with an inviscid fluid and establish a continuous dependence theorem (Theorem 2) and two uniqueness theorems (Theorems 3, 4) for a particular class of such continua. Theorems 2, 3 are proved without hypotheses on the sign of the constants and, if the domain is unbounded, under mild assumptions on the spatial asymptotic behaviour of the field variables. Theorem 4 holds for body-forces not equal to zero and, if the domain is unbounded, without restrictions upon the behaviour of the unknown fields at infinity, but under suitable conditions on the sign of the constants.

MSC classification

Secondary: 76S05: Flows in porous media; filtration; seepage
Type
Research Article
Information
Mathematika , Volume 31 , Issue 2 , December 1984 , pp. 196 - 213
Copyright
Copyright © University College London 1984

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References

1.. Szefer, G.. Non-linear problems of consolidation theory. Symposium Franco-Polonais, Problémes non lineaires de Mecanique (Cracovie, 1977).Google Scholar
2.. Biot, M. A.. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 33 (1962), 1482–1498.CrossRefGoogle Scholar
3.. Biot, M. A.. Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am., 34 (1962), 1254–1264.CrossRefGoogle Scholar
4.. Green, A. E. and Naghdi, P. M.. A dynamical theory of interacting continua. Int. J. Eng. Sci., 3 (1965), 231–241.CrossRefGoogle Scholar
5.. Green, A. E. and Naghdi, P. M.. On basic equations for mixtures. Quart. J. Mech. Appl. Math., 22 (1969), 427–438.CrossRefGoogle Scholar
6.. Atkin, R. J., Chadwick, P. and Steel, T. R.. Uniqueness theorems for linearized theories of interacting continua. Mathematika, 14 (1967), 27–42.CrossRefGoogle Scholar
7.. Green, A. E. and Naghdi, P. M.. The flow of fluid through an elastic solid. Acta Mech., 9 (1970), 329–340.CrossRefGoogle Scholar
8.. Szefer, G. and Pallotti, G.. Non-linear theory of porous media with application to articular cartilage. To appear in Biomechanics.Google Scholar
9.. Knops, R. J. and Payne, L. E.. Uniqueness theorems in linear elasticity (Springer, 1971).CrossRefGoogle Scholar
10.. Galdi, G. P. and Rionero, S.. Continuous dependence in linear elastodynamics on unbounded domains without definiteness conditions on the elasticities. Proc. Roy. Soc. Edin. (A), 93 (1983), 299–306.CrossRefGoogle Scholar
11.. Wheeler, L. T.. On the uniqueness of solutions to the displacement problem in linear elastodynamics. J. Elasticity, 1 (1971), 121–124.CrossRefGoogle Scholar
12.. Ignaczak, J.. Tensorial equation of motion for a fluid saturated porous elastic solid. Bull. Acad. Pol. Sci., Ser. Sci. Tech., 26 (1978), 371–375.Google Scholar
13.. Borrelli, A. and Patria, M. C.. Propagation of discontinuity waves of any order through an elastic solid saturated with an rnviscid fluid. ZAMP, 35 (1984), 373–386.Google Scholar
14.. Knops, R. J. and Payne, L. E.. Some uniqueness and continuous dependence theorems for nonlinear elastodynamics in exterior domains. Appl. Anal., 15 (1983), 33–51.CrossRefGoogle Scholar
15.. Galdi, G. P. and Rionero, S.. On the uniqueness of viscous fluid motions. Arch. Rational Mech. Anal., 62 (1976), 295–311.Google Scholar
16.. Hartman, P.. Ordinary differential equations (J. Wiley and Sons, 1964).Google Scholar