[Sur les équations homogénéisées des oscillations longitudinales du viscoélastoplastique matériau d'Ishlinskii avec données non régulières rapidement oscillantes à deux échelles]
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.
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.
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Mot clés : Solides et structures
@article{CRMECA_2006__334_12_713_0,
author = {Andrey Amosov and Ivan Goshev},
title = {On two-scale homogenized equations of the {Ishlinskii} type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth data},
journal = {Comptes Rendus. M\'ecanique},
pages = {713--718},
publisher = {Elsevier},
volume = {334},
number = {12},
year = {2006},
doi = {10.1016/j.crme.2006.10.007},
language = {en},
}
TY - JOUR AU - Andrey Amosov AU - Ivan Goshev TI - On two-scale homogenized equations of the Ishlinskii type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth data JO - Comptes Rendus. Mécanique PY - 2006 SP - 713 EP - 718 VL - 334 IS - 12 PB - Elsevier DO - 10.1016/j.crme.2006.10.007 LA - en ID - CRMECA_2006__334_12_713_0 ER -
%0 Journal Article %A Andrey Amosov %A Ivan Goshev %T On two-scale homogenized equations of the Ishlinskii type viscoelastoplastic body longitudinal vibrations with rapidly oscillating nonsmooth data %J Comptes Rendus. Mécanique %D 2006 %P 713-718 %V 334 %N 12 %I Elsevier %R 10.1016/j.crme.2006.10.007 %G en %F CRMECA_2006__334_12_713_0
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] Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht/Norwell, MA, 1990
[2] Non-Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin/New York, 1980
[3] Processes in periodic media which are not describable by averaged characteristics, Soviet Phys. Dokl., Volume 28 (1983), pp. 125-127
[4] Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992), pp. 1482-1518
[5] 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] Variations de grande amplitude pour la densite d'un fluide visqueux compressible, Physica D, Volume 48 (1991), pp. 113-128
[7] 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] 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] 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] Homogenization of scalar wave equations with hysteresis, Cont. Mech. & Ther., Volume 11 (1999), pp. 371-391
[11] Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992
[12] Weak convergence for a class of rapidly oscillating functions, Math. Notes, Volume 62 (1997), pp. 122-126
[13] Hysteresis and Phase Transitions, Appl. Math. Sci., vol. 121, Springer-Verlag, New York, 1996
[14] Systems with Hysteresis, Springer-Verlag, Berlin, 1989 (Russian edition:, 1983, Nauka, Moscow)
[15] Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Int. Series Math. Sci. Appl., vol. 8, Gakkotosho, Tokyo, 1996
[16] Differential Models of Hysteresis, Springer-Verlag, Berlin/Heidelberg, 1994
[17] Ein Gedankenmodell zur knetischen Theorie der festen Körper, Z. Angew. Math. Mech., Volume 8 (1928), pp. 85-106
[18] Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR. Tech. Ser., Volume 9 (1944), pp. 580-590
[19] 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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