Constitutive Assumptions and the Second Law
Let a fixed reference configuration R be assigned for the body B under consideration, and identify each of the material points X of B with the place in space that X occupies when B has the configuration R. A thermodynamic process of B is a collection of f
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LECTURES
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No. 73
BERNARD D. COLEMAN CAR...-.EGIE - MEILON UNIVERSITY, PITTSBURGH
THERMODYNAMICS OF MATERIALS WITH MEMORY
COURSE HELD AT THE DEPARTMENT OF MECHANICS OF SOLIDS JULY 1971
UDINE 1971
SPRINGER-VERLAG WIEN GMBH
This worlt is suiject to copyright. All rights are reserved, witether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data hanks.
©
1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81125-2 DOI 10.1007/978-3-7091-2951-7
ISBN 978-3-7091-2951-7 (eBook)
This text has been submitted as being "ready for camera" by professor B.D. Coleman and has been reproduced without any corrections or additions, except for the titles at the top of pages. For the CISM. Dr. Franeo Buttazzoni Responsable for the Editorial Board
1. Introduction
In theories of the dynamical behavior of continua, there are several ways of describing the dissipative effects which, in addition to heat conduction, accompany deformation.
The oldest way is to employ a
viscous stress which depends on the rate of strain, as is done in the theory of Navier-Stokes fluids.
In another description of dissipation,
one postulates the existence of internal state variables which influence the stress and obey differential equations in which the strain appears. A third approach is to assume that the entire past history of the strain influences the stress in a manner compatible with a general postulate of smoothness or "principle of fading memory". Experience in high-polymer physics shows that the mechanical behavior of many materials, including polymer melts and solutions, as well as amorphous, cross-linked solids and semi-crystalline plastics, is more easily described within the theory of materials with fading memory than by
theories of the viscous-stress type, which do not account for gradual stress-relaxation, or by theories which rest on a finite number of internal state variables and which, therefore, give rise to discrete relaxation spectra when linearized. Some years ago, Walter Noll and I proposed a systematic procedure for rendering explicit the restrictions which the second law places on constitutive relations.
1
The procedure was easily applied in theories of
Thermodynamics of Materials with Memory
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materials of the viscous-stress type 1 ' 2 and in theories which employ evolution equations for internal state variables 3. These applications, which did not yield results a physicist would consider surprising, were presented as attempts at clarification, with the emphasis laid upon logical relations.
Implementation of the procedure in the theory of materials with
memorywas a different matter, however, for it there led to conclusions 4 which, although not anticipated by other arguments, have recently been shown to have important bearing on wave propagation5 and dynamical stability6 ' 7 In this course we shal
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