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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: June 21, 2001
MathSciNet review: 1852096
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Hans-Dieter Alber, Materials with memory, Lecture Notes in Mathematics, vol. 1682, Springer-Verlag, Berlin, 1998. Initial-boundary value problems for constitutive equations with internal variables. MR 1619546 (99i:73040)
  • 2. J.-P. Aubin, Optima and Equilibria, An Introduction to Nonlinear Analysis, 2nd ed., Graduate Texts in Mathematics 140, Springer, Berlin, Heidelberg, New York 1998. CMP 2000:06
  • 3. M. Brokate and P. Krejc, Wellposedness of kinematic hardening models in elastoplasticity, Christian-Albrechts-Universität Kiel, Berichtsreihe des Mathematischen Seminars Kiel, Bericht 96-4, Februar 1996.
  • 4. Ioan R. Ionescu and Mircea Sofonea, Functional and numerical methods in viscoplasticity, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1993. MR 1244578 (95e:73025)
  • 5. M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. MR 0126722 (23 #A4016)
  • 6. P. Krejc, Hysteresis, Convexity, and Dissipation in Hyperbolic Equations, Gakkotosho, Tokyo, 1996.
  • 7. P. Laborde and Q. S. Nguyen, Étude de l’équation d’évolution des systèmes dissipatifs standards, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 1, 67–84 (French, with English summary). MR 1034899 (90m:58025)
  • 8. J. Lemaitre and J. L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge 1994.
  • 9. Paul Germain and Bernard Nayroles (eds.), Applications of methods of functional analysis to problems in mechanics, Lecture Notes in Mathematics, vol. 503, Springer-Verlag, Berlin, 1976. Joint Symposium, IUTAM/IMU, held in Marseille, September 1–6, 1975. MR 0521351 (58 #25196)
  • 10. M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. MR 1113700 (92e:46059)
  • 11. R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 (1968), 525–539. MR 0236689 (38 #4984)
  • 12. R. T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math. 39 (1971), 439–469. MR 0310612 (46 #9710)
  • 13. R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362 (98m:49001)
  • 14. E. S. Suhubi, Thermoelastic solids, in Continuum Physics II: Continuum Mechanics of Single Substance Bodies, C. Eringen, ed., Academic Press, New York, San Francisco, London 1975.
  • 15. K. Takamizawa and K. Hayashi, Strain energy density function and uniform strain hypothesis for arterial mechanics, J. Biomechanics 20 (1987), 7-17.

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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: http://dx.doi.org/10.1090/S0002-9947-01-02847-1
PII: S 0002-9947(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