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Uniqueness theorems for linearized theories of interacting continua
Author(s) -
Atkin R. J.,
Chadwick P.,
Steel T. R.
Publication year - 1967
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300007981
Subject(s) - uniqueness , mathematics , helmholtz free energy , boundary value problem , mathematical analysis , uniqueness theorem for poisson's equation , constitutive equation , entropy (arrow of time) , physics , thermodynamics , finite element method
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.