Comptes Rendus
On two-scale homogenized equations of the Ishlinskii type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth data
Comptes Rendus. Mécanique, Volume 334 (2006) no. 12, pp. 713-718.

We study the initial-boundary value problems for a system of operator-differential equations describing Ishlinskii type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth coefficients and initial data. The main feature is an presence of hysteresis Prandtl–Ishlinskii operator. We rigorously justify the passage to the corresponding limit initial-boundary value problems for a system of two-scale homogenized operator-integro-differential equations, including the existence theorem for the limit problems. The results are global with respect to the time interval and the data.

Nous étudions l'homogénéisation du système d'équations décrivant les oscillations longitudinales du matériau viscoélastoplastique d'Ishlinskii, avec des données rapidement oscillantes. La propriété principale est la présence de l'operateur de l'hystérèse de Prandl–Ishlinskii. Nous justifions rigoureusement la convergence vers un problème limite pour un système d'équations operateur-intégro-différentielles homogéneisées à deux échelles, pour lequel nous prouvons un théorème d'existence. Les résultats sont globaux par rapport au temps et aux données.

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Published online:
DOI: 10.1016/j.crme.2006.10.007
Keywords: Solids and structures
Mot clés : Solides et structures

Andrey Amosov 1; Ivan Goshev 1

1 Department of Mathematical Modelling, Moscow Power Engineering Institute (Technical University), Krasnokazarmennaja 14, 111250 Moscow, Russia
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Andrey Amosov; Ivan Goshev. On two-scale homogenized equations of the Ishlinskii type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth data. Comptes Rendus. Mécanique, Volume 334 (2006) no. 12, pp. 713-718. doi : 10.1016/j.crme.2006.10.007. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.10.007/

[1] N. Bakhvalov; G. Panasenko Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht/Norwell, MA, 1990

[2] E. Sanchez-Palencia Non-Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin/New York, 1980

[3] N.S. Bakhvalov; M.E. Eglit Processes in periodic media which are not describable by averaged characteristics, Soviet Phys. Dokl., Volume 28 (1983), pp. 125-127

[4] G. Allaire Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992), pp. 1482-1518

[5] A.A. Amosov; A.A. Zlotnik Quasi-averaged equations of the one-dimensional motion of a viscous barotropic medium with rapidly oscillating data, Comp. Math. Math. Phys., Volume 36 (1996), pp. 203-220

[6] D. Serre Variations de grande amplitude pour la densite d'un fluide visqueux compressible, Physica D, Volume 48 (1991), pp. 113-128

[7] A.A. Amosov; A.A. Zlotnik Quasi-averaging of the system of equations of one-dimensional motion of a viscous heat-conducting gas with rapidly oscillating data, Comp. Math. Math. Phys., Volume 38 (1998), pp. 1152-1167

[8] A. Amosov; A. Zlotnik On two-scale homogenized equations of one-dimensional nonlinear thermoviscoelasticity with rapidly oscillating nonsmooth data, C. R. Acad. Sci. Paris, Ser. IIb, Volume 329 (2001), pp. 169-174

[9] A.A. Amosov; A.A. Zlotnik Substantiation of two-scale homogenization of one-dimensional nonlinear thermoviscoelasticity equations with nonsmooth data, Comp. Math. Math. Phys., Volume 41 (2001), pp. 1713-1733

[10] J. Franku; P. Krejci Homogenization of scalar wave equations with hysteresis, Cont. Mech. & Ther., Volume 11 (1999), pp. 371-391

[11] L.C. Evans; R.F. Gariepi Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992

[12] A.A. Amosov Weak convergence for a class of rapidly oscillating functions, Math. Notes, Volume 62 (1997), pp. 122-126

[13] M. Brokate; J. Sprekels Hysteresis and Phase Transitions, Appl. Math. Sci., vol. 121, Springer-Verlag, New York, 1996

[14] M.A. Krasnoselskii; A.V. Pokrovskii Systems with Hysteresis, Springer-Verlag, Berlin, 1989 (Russian edition:, 1983, Nauka, Moscow)

[15] P. Krejci Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Int. Series Math. Sci. Appl., vol. 8, Gakkotosho, Tokyo, 1996

[16] A. Visintin Differential Models of Hysteresis, Springer-Verlag, Berlin/Heidelberg, 1994

[17] L. Prandtl Ein Gedankenmodell zur knetischen Theorie der festen Körper, Z. Angew. Math. Mech., Volume 8 (1928), pp. 85-106

[18] A.Yu. Ishlinskii Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR. Tech. Ser., Volume 9 (1944), pp. 580-590

[19] A.A. Amosov; I.A. Goshev Existence and uniqueness of global weak solutions to the equations describing the longitudinal oscillations of a viscoelastoplastic Ishlinskii material, Dokl. Math., Volume 74 (2006), pp. 623-627

[20] A.A. Amosov, I.A. Goshev, Global one-valued solvability to the system of equations describing Ishlinskii body longitudinal vibrations, Differential Equations, submitted for publication

[21] A.A. Amosov, I.A. Goshev, Substantiation of two-scale homogenization of the system of equation of viscoelastoplastic Ishlinskii body longitudinal vibrations, Comp. Math. Math. Phys., submitted for publication

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