Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T19:47:43.332Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness theorems for linearized theories of interacting continua

Published online by Cambridge University Press:  26 February 2010

R. J. Atkin,
P. Chadwick and
T. R. Steel
R. J. Atkin
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich.
P. Chadwick
Affiliation:
School of Mathematics, University of Newcastle upon Tyne.
T. R. Steel
Affiliation:
School of Mathematics, University of Newcastle upon Tyne.
Get access

Summary

Recent work on the mechanics of interacting continua has led to the formulation of linearized theories governing thermo-mechanical disturbances of small amplitude in mixtures of an elastic solid and a viscous fluid and of two elastic solids. These theories are well posed in the crude sense that the number of field and constitutive equations equals the number of field quantities to be determined, but the proper posing of initial and initial-boundary-value problems has not been studied. In this paper we prove, for both types of mixtures, the uniqueness of sufficiently smooth solutions of the field and constitutive equations subject to initial and boundary data which include conditions of direct physical significance. Sufficient conditions for uniqueness are given in the form of inequalities on the material constants appearing in the constitutive equations. These restrictions fall into two categories, one arising from the application of the entropy-production inequality, and therefore intrinsic to the theory, and the other representing constraints on the Helmholtz free energy of the mixture. Our results include as special cases uniqueness theorems for unsteady linearized compressible flow of a heat-conducting viscous fluid and for the linear theory of thermoelasticity.

Information

Type
Research Article
Information
Mathematika , Volume 14 , Issue 1 , June 1967, pp. 27 - 42
Copyright
Copyright © University College London 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Green, A. E. and Naghdi, P. M., Int. J. Engng. Sci., 3 (1965), 231–241.CrossRefGoogle Scholar
2.Mills, N., Int. J. Engng. Sci., 4 (1966), 97–112.Google Scholar
3.Green, A. E. and Steel, T. R., Int. J. Engng. Sci., 4 (1966), 483–500.Google Scholar
4.Steel, T. R., Quart. J. Mech. Appl. Math, 20 (1967), 57–72.CrossRefGoogle Scholar
5.Atkin, R. J.. To be published.Google Scholar
6.Weiner, J. H., Quart. Appl. Math., 15 (1957), 102–105.CrossRefGoogle Scholar
See also Boley, B. A. and Weiner, J. H., Theory of thermal stresses (Wiley: New York, 1960), pp. 37–40).Google Scholar
7.Truesdell, C. and Toupin, R. A., The classical field theories. Handbuch der Physik (ed. Flügge, S.), Band III/1, 226–793 (Springer: Berlin, 1960).Google Scholar
8.Kellogg, O. D., Foundations of potential theory (Ungar: New York, 1929).CrossRefGoogle Scholar
9.Chadwick, P. and Powdrill, B., Int. J. Engng. Sci., 3 (1965), 561–595.CrossRefGoogle Scholar
10.McNamee, J. and Gibson, R. E., Quart. J. Mech. Appl. Math., 13 (1960), 210–227.CrossRefGoogle Scholar
11.Pai, S. I., Viscous flow theory. I-Laminar flow (Van Nostrand: Princeton, 1956).Google Scholar
12.Serrin, J., Arch. Rat. Mech. Analysis, 3 (1959), 271–288.CrossRefGoogle Scholar
13.Chadwick, P., Progress in solid mechanics (ed. Sneddon, I. N. and Hill, R.), Vol. 1, 265–328 (North Holland: Amsterdam, 1960).Google Scholar