This article defines and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type defined in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case.
@article{bwmeta1.element.doi-10_2478_BF02475187,
author = {Krzysztof Che\l mi\'nski},
title = {On quasistatic inelastic models of gradient type with convex composite constitutive equations},
journal = {Open Mathematics},
volume = {1},
year = {2003},
pages = {670-689},
zbl = {1038.35135},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475187}
}
Krzysztof Chełmiński. On quasistatic inelastic models of gradient type with convex composite constitutive equations. Open Mathematics, Tome 1 (2003) pp. 670-689. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475187/
[1] H.-D. Alber: Materials with memory, Lecture Notes in Math., Vol. 1682, Springer, Berlin Heidelberg New York, 1998.
[2] H.-D. Alber and K. Chełmiński: “Quasistatic problems in viscoplasticity theory I: Models with linear hardening”, A part of the monograph: I. Gohberg et al.: Operator theoretical methods and applications to mathematical physics. The Erhard Meister memorial volume, Birkhäuser, Basel, in print.
[3] G. Anzellotti and S. Luckhaus: “Dynamical evolution of elasto-perfectly plastic bodies”, Appl. Math. Optim., Vol. 15, (1987), pp. 121–140. http://dx.doi.org/10.1007/BF01442650 | Zbl 0616.73047
[4] J.P. Aubin and A. Cellina: Differential inclusions. Springer, Berlin Heidelberg New York, 1984.
[5] M. Brokate: “Elastoplastic constitutive laws of nonlinear kinematic hardening type”, Technical Report 97-14, Berichtsreihe des Mathematischen Seminars, Kiel, 1997.
[6] K. Chełmiński: “Coercive limits for a subclass of monotone constitutive equations in the theory of inelastic material behaviour of metals”, Mat. Stos., Vol. 40, (1997), pp. 41–81. | Zbl 1071.74582
[7] K. Chełmiński: “On self-controlling models in the theory of inelastic material behaviour of metals”, Contin. Mech. Thermodyn., Vol. 10, (1998), pp. 121–133. http://dx.doi.org/10.1007/s001610050085 | Zbl 0911.73023
[8] K. Chełmiński: “Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations”, Math. Meth. Appl. Sci., Vol. 25, (2002), pp. 1195–1230. http://dx.doi.org/10.1002/mma.336 | Zbl 1099.74029
[9] K. Chełmiński: “Coercive approximation of viscoplasticity and plasticity”, Asymptotic Analysis, Vol. 26, (2001), pp. 105–133. | Zbl 1013.74010
[10] K. Chełmiński: “On noncoercive models in the theory of inelastic deformations with internal variables”, ZAMM, Vol. 81, Suppl. 3, (2001), pp. 595–596. | Zbl 1051.74548
[11] K. Chełmiński: “On quasistatic models in the theory of inelastic deformations”, Proceedings in Applied Mathematics, Vol. 1, (2002), pp. 401–402. http://dx.doi.org/10.1002/1617-7061(200203)1:1<401::AID-PAMM401>3.0.CO;2-Z | Zbl 05296956
[12] K. Chełmiński and P. Gwiazda: “Monotonicity of operators of viscoplastic response: application to the model of Bodner-Partom”, Bull. Polish Acad. Sci.: Tech. Sci., Vol. 47, (1999), pp. 191–208. | Zbl 0987.74019
[13] K. Chełmiński and P. Gwiazda: “Nonhomogeneous initial-boundary value problems for coercive and self-controlling models of monotone type”, Contin. Mech. Thermodyn., Vol. 12, (2000), pp. 217–234. http://dx.doi.org/10.1007/s001610050136 | Zbl 1003.74033
[14] K. Chełmiński and Z. Naniewicz: “Coercive limits for constitutive equations of monotone-gradient type”, Nonlinear Anal., Vol. 48, (2002), pp. 1197–1214. http://dx.doi.org/10.1016/S0362-546X(01)00096-7 | Zbl 1056.35146
[15] F.H. Clarke: Optimization and nonsmooth analysis, CMR Université de Montréal, Montréal, 1989. | Zbl 0727.90045
[16] S. Guillaume: “Subdifferential evolution inclusion in nonconvex analysis”, Positivity, Vol. 4, (2000), pp. 357–395. http://dx.doi.org/10.1023/A:1009821417484 | Zbl 0968.35068
[17] P. Gwiazda: “Non-homogeneous boundary value problem for the Chan-Bodner-Linholm model”, Math. Methods Appl. Sci., Vol. 23:11, (2000), pp. 1011–1022. http://dx.doi.org/10.1002/1099-1476(20000725)23:11<1011::AID-MMA148>3.0.CO;2-# | Zbl 0976.35090
[18] B. Halphen and Nguyen Quoc Son: “Sur les matèriaux standards généralisés”, J. Méc., Vol. 14, (1975), pp. 39–63. | Zbl 0308.73017
[19] P.-M. Suquet: “Evolution problems for a class of dissipative materials”, Quart. Appl. Math., Vol. 38, (1980), pp. 391–414. | Zbl 0501.73030
[20] R. Temam: “A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity”, Arch. Rational Mech. Anal., Vol. 95, (1986), pp. 137–183. http://dx.doi.org/10.1007/BF00281085 | Zbl 0615.73035