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On the wellposedness of constitutive laws involving dissipation potentials


Authors: Wolfgang Desch and Ronald Grimmer
Journal: Trans. Amer. Math. Soc. 353 (2001), 5095-5120
MSC (1991): Primary 73B05; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9947-01-02847-1
Published electronically: June 21, 2001
MathSciNet review: 1852096
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Abstract:

We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic.

As a technical tool we generalize the notion of an Orlicz space to a cone ``normed'' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.


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Additional Information

Wolfgang Desch
Affiliation: Institut für Mathematik, Universität Graz, Heinrichstraße 36, A-8010 Graz, Austria
Email: georg.desch@kfunigraz.ac.at

Ronald Grimmer
Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901

DOI: https://doi.org/10.1090/S0002-9947-01-02847-1
Keywords: Dissipation potential, viscoplastic material, constitutive equation, Orlicz space
Received by editor(s): March 9, 2000
Received by editor(s) in revised form: February 16, 2001
Published electronically: June 21, 2001
Additional Notes: Supported by Spezialforschungsbereich F 003 “Optimierung und Kontrolle” at the Karl-Franzens-Universität Graz, grant GAUK 19/1997. W. D. acknowledges the kind hospitality of Southern Illinois University, Carbondale
Article copyright: © Copyright 2001 American Mathematical Society

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