|
On the wellposedness of constitutive laws involving dissipation potentials
Author(s):
Wolfgang
Desch;
Ronald
Grimmer
Journal:
Trans. Amer. Math. Soc.
353
(2001),
5095-5120.
MSC (1991):
Primary 73B05;
Secondary 46E30
Posted:
June 21, 2001
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
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:
-
- 1.
- H.-D. Alber, Materials with Memory, Lecture Notes in Mathematics 1682, Springer, Berlin, Heidelberg, New York 1998. MR 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.
- J. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford, New York, Tokyo 1993. MR 95e:73025
- 5.
- M. A. Krasnosel'skii and Y. B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen 1961. MR 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 systémes dissipatifs standards, Mathematical Modelling and Numerical Analysis 24 (1990), 67-84. MR 90m:58025
- 8.
- J. Lemaitre and J. L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge 1994.
- 9.
- J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in Applications of Methods of Functional Analysis to Problems in Mechanics, P. Germain and B. Nayroles, eds., Lecture Notes in Mathematics 503, Springer, Belin, Heidelberg, New York 1975, pp. 57-89. MR 58:25196
- 10.
- M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Pure and Applied Mathematics 146, Marcel Dekker, New York 1991.MR 92e:46059
- 11.
- R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
- 12.
- R. T. Rockafellar, Integrals which are convex functionals II, Pacific J. Math. 39 (1971), 439-469. MR 46:9710
- 13.
- R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin, Heidelberg, New York 1998.MR 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.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
73B05,
46E30
Retrieve articles in all Journals with MSC
(1991):
73B05,
46E30
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:
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
Posted:
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
Copyright of article:
Copyright
2001,
American Mathematical Society
|
Comments: webmaster@ams.org