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Substantiation of two-scale homogenization of the equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material

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Abstract

Initial-boundary value problems for the system of quasilinear operator-differential equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material with nonsmooth rapidly oscillating coefficients and initial data are investigated. The system involves the hysteresis Prandtl-Ishlinskii operator. Passage to the limit to initial-boundary value problems for the corresponding system of two-scale homogenized operator integro-differential equations is strictly substantiated globally in time without assuming that the data are small.

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References

  1. N. S. Bakhvalov and M. E. Eglit, “Processes in Periodic Media not Described in Terms of Mean Characteristics,” Dokl. Akad. Nauk SSSR 268, 836–840 (1983).

    MathSciNet  Google Scholar 

  2. N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media (Nauka, Moscow, 1984; Kluwer, Dordrecht, 1989).

    Google Scholar 

  3. M. E. Eglit, “Homogenized Description Processes in Periodic Elastoplastic Media,” Mekh. Kompoz. Mater., No. 5, 825–831 (1984).

  4. G. Allaire, “Homogenization and Two-Scale Convergence,” SIAM J. Math. Anal. 23, 1482–1518 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Yu. Ishlinskii, “Some Applications of Statistics to Describing the Laws Governing Strains in Bodies,” Izv. Akad. Nauk SSSR. Otd. Tekh. Nauk, No. 9, 580–590 (1944).

  6. M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis (Nauka, Moscow, 1983; Berlin, Springer-Verlag, 1989).

    Google Scholar 

  7. A. A. Amosov and A. A. Zlotnik, “Substantiation of Quasi-Homogenized One-Dimensional Equations of Motion for a Viscous Barotropic Medium with Rapidly Oscillating Properties,” Dokl. Akad. Nauk 342, 295–299 (1995).

    MathSciNet  Google Scholar 

  8. A. A. Amosov and A. A. Zlotnik, “On Quasi-Homogenized One-Dimensional Equations of Motion for a Viscous Barotropic Medium with Rapidly Oscillating Data,” Zh. Vychisl. Mat. Mat. Fiz. 36, 87–110 (1996).

    MathSciNet  Google Scholar 

  9. A. A. Amosov and A. A. Zlotnik, “Quasi-Averaging of the System of Equations of One-Dimensional Motion of a Viscous Heat-Conducting Gas with Rapidly Oscillating Data,” Zh. Vychisl. Mat. Mat. Fiz. 38, 1204–1219 (1998) [Comput. Math. Math. Phys. 38, 1152–1167 (1998)].

    MathSciNet  Google Scholar 

  10. A. Amosov and A. Zlotnik, “On Two-Scale Homogenized Equations of One-Dimensional Nonlinear Thermoviscoelasticity with Rapidly Oscillating Nonsmooth Data,” C. R. Acad. Sci. Paris, Ser. IIb 329, 169–174 (2001).

    Google Scholar 

  11. A. A. Amosov and A. A. Zlotnik, “Substantiation of Two-Scale Homogenization of One-Dimensional Nonlinear Thermoviscoelasticity Equations with Nonsmooth Data,” Zh. Vychisl. Mat. Mat. Fiz. 41, 1713–1733 (2001) [Comput. Math. Math. Phys. 39, 1630–1650 (2001)].

    MathSciNet  Google Scholar 

  12. A. A. Amosov and I. A. Goshev, “Existence and Uniqueness of Global Weak Solutions to the Equations Describing the Longitudinal Vibrations of a Viscoelastoplastic Ishlinskii Material,” Dokl. Akad. Nauk 410, 7–11 (2006) [Dokl. Math. 74, 623–627 (2006)].

    Google Scholar 

  13. A. A. Amosov and I. A. Goshev, “Global Unique Solvability of the System of Equations Describing the Longitudinal Vibrations of a Viscoelastoplastic Ishlinskii Material,” Differ. Uravn. 43 (2007).

  14. J. Franků and P. Krejčí, “Homogenization of Scalar Wave Equations with Hysteresis,” Cont. Mech. Thermodyn. 11, 371–391 (1999).

    Article  Google Scholar 

  15. K. Yoshida, Functional Analysis (Mir, Moscow, 1967; Springer-Verlag, Berlin, 1980).

    Google Scholar 

  16. H. Gajewski, K. GrÖger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978).

    MATH  Google Scholar 

  17. I. P. Natanson, Theory of Functions of a Real Variable (Ungar, New York, 1995; Nauka, Moscow, 1974).

    Google Scholar 

  18. S. Saks, Theory of the Integral (Stechert, New York, 1937; Inostrannaya Literatura, Moscow, 1949).

    Google Scholar 

  19. A. A. Amosov, “On Weak Convergence of a Class of Rapidly Oscillating Functions,” Mat. Zametki 62(1), 145–150 (1997).

    MathSciNet  Google Scholar 

  20. A. A. Amosov and A. A. Zlotnik, “Global Solvability of a Class of Quasilinear Systems of Composite-Type Equations with Nonsmooth Data,” Differ. Uravn. 30, 596–609 (1994).

    MathSciNet  Google Scholar 

  21. A. Visintin, Differential Models of Hysteresis (Springer-Verlag, Berlin, 1994).

    MATH  Google Scholar 

  22. M. Brokate and J. Sprekels, Hysteresis and Phase Transitions (Springer-Verlag, New York, 1996).

    MATH  Google Scholar 

  23. P. Krejčí, Hysteresis, Convexity, and Dissipation in Hyperbolic Equations. Gakuto Int. Ser. Math. Sci. & Appl. (Gakkotosho, Tokyo, 1996).

    Google Scholar 

  24. L. Prandtl, “Ein Gedankenmodell zur knetischen Theorie der festen Körper,” Z. Angew. Math. Mech. 8, 85–106 (1928).

    Article  Google Scholar 

  25. A. A. Amosov and A. A. Zlotnik, “Uniqueness and Stability of Weak Solutions to Quasi-Homogenized One-Dimensional Equations of Motion of a Viscous Barotropic Medium,” Differ. Uravn. 31, 1123–1131 (1995).

    MathSciNet  Google Scholar 

  26. I. A. Goshev, “Global Unique Solvability of the Two-Scale Homogenized Problem of Longitudinal Vibrations of a Viscoelastoplastic Ishlinskii Material,” Vestn. Mosk. Energ. Inst., No. 6 (2006).

  27. G. Andrews, “On the Existence of Solutions to the Equation u tt = u xxt + σ(u x(x),” J. Differ. Equations 35, 200–231 (1980).

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Modeling, Moscow Power Engineering Institute (Technical University), ul. Krasnokazarmennaya 14, Moscow, 111250, Russia

    A. A. Amosov & I. A. Goshev

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  1. A. A. Amosov
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  2. I. A. Goshev
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Original Russian Text © A.A. Amosov, I.A. Goshev, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 6, pp. 988–1006.

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Amosov, A.A., Goshev, I.A. Substantiation of two-scale homogenization of the equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material. Comput. Math. and Math. Phys. 47, 943–961 (2007). https://doi.org/10.1134/S0965542507060061

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  • DOI: https://doi.org/10.1134/S0965542507060061

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