We study the spectrum of one-dimensional eigenoscillations along the $Ox_1$ axis of two-phase layered media with periodic structure occupying the band $0<x_1<L$. The period of the oscillations is a band $0<x_1<\varepsilon$ composed of $2M$ alternating layers of an isotropic elastic or viscoelastic material (the first phase) and a viscous incompressible fluid (the second phase). It is assumed that the number of periods $N=L/ \varepsilon$ is an integer, and the layers are parallel to the $Ox_2 x_3$ plane. The spectrum is denoted by $S_\varepsilon$ and is defined as the set of eigenvalues of a boundary value problem for a homogeneous system of ordinary differential equations with conjugation conditions at the interfaces between the solid and fluid layers. These conditions are derived directly from the initial assumption on the continuity of displacements and normal stresses at the interfaces between the layers. It is shown that the spectrum $S_\varepsilon$ consists of the roots of transcendental equations, the number of which is equal to the number of periods $N$ contained within the band $0<x_1<L$. The roots of these equations can only be found numerically, except for one particular case. In the case of multi-layered media with $N\gg 1$, the finite limits of the sequences $\lambda(\varepsilon)\in S_\varepsilon$ as $\varepsilon\to 0$ are proposed to be used as initial approximations. It is established that the set of all finite limits coincides with the set of roots of rational equations, denoted by $S$. The coefficients of these equations, and hence the points of the set $S$ depend on the volume fraction of the fluid within the layered medium and do not depend on the number $M$ of the fluid layers within the period. It is proved that for any $M\geq 1$ the spectrum $S_\varepsilon$ converges in the sense of Hausdorff to the set $S$ as $\varepsilon\to 0$.
Keywords: spectrum of eigenoscillations, layered medium, two-phase medium, elastic material, viscoelastic material, viscous incompressible fluid
Received August 4, 2022
Revised October 31, 2022
Accepted November 7, 2022
Funding Agency: This study was carried out according a state assignment (state registration no. AAAA-А20-120011690138-6).
Vladlena Valerievna Shumilova, Dr. Phys.-Math. Sci., Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 119526 Russia, e-mail: v.v.shumilova@mail.ru
REFERENCES
1. Oleinik O.A. , Shamaev A.S., Yosifian G.A. Mathematical problems in elasticity and homogenization. Amsterdam: North-Holland, 1992, 397 p. ISBN: 9780080875477 . Original Russian text published in Oleinik O.A., Iosif’yan G.A., Shamaev A.S. Matematicheskie zadachi teorii sil’no neodnorodnykh uprugikh sred. Moscow: MGU Publ., 1990, 311 p.
2. Zhikov V.V. On an extension of the method of two-scale convergence and its applications. Sb. Math., 2000, vol. 191, no. 7, pp. 973–1014. doi: 10.1070/SM2000v191n07ABEH000491
3. Zhikov V.V. On spectrum gaps of some divergent elliptic operators with periodic coefficients. St. Petersb. Math. J., 2005, vol. 16, no. 5, pp. 773–790. doi: 10.1090/S1061-0022-05-00878-2
4. Babych N.O., Kamotski I.V., Smyshlyaev V.P. Homogenization of spectral problems in bounded domains with doubly high contrasts. Netw. Heterog. Media, 2008, vol. 3, no. 3, pp. 413–436. doi: 10.3934/nhm.2008.3.413
5. Kosmodem’yanskii D.A., Shamaev A.S. Spectral properties of some problems in mechanics of strongly inhomogeneous media. Mech. Solids, 2009, vol. 44, no. 6, pp. 874–906. doi: 10.3103/S0025654409060077
6. Vlasov V.V., Gavrikov A.A., Ivanov S.A., Knyaz’kov D.Yu., Samarin V.A., Shamaev A.S. Spectral properties of combined media. J. Math. Sci., 2010, vol. 164, no. 6, pp. 948–963. doi: 10.1007/s10958-010-9776-5
7. Cooper S. Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions. Appl. Anal., 2014, vol. 93, no. 7, pp. 1401–1430. doi: 10.1080/00036811.2013.833327
8. Cherednichenko K.D., Cooper S., Guenneau S. Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients. Multiscale Model. Simul., 2015, vol. 13, no. 1, pp. 72–98. doi: 10.1137/130947106
9. Leugering G., Nazarov S.A., Taskinen J. The band-gap structure of the spectrum in a periodic medium of masonry type. Netw. Heterog. Media, 2020, vol. 15, no. 4, pp. 555–580. doi: 10.3934/nhm.2020014
10. Akulenko L.D., Nesterov S.V. Inertial and dissipative properties of a porous medium saturated with viscous fluid. Mech. Solids, 2005, vol. 40, no. 1, pp. 90–98.
11. Molotkov L.A. Matrichnyi metod v teorii rasprostraneniya voln v sloistykh uprugikh i zhidkikh sredakh [Matrix method in the theory of wave propagation in layered elastic and liquid media]. Leningrad: Nauka Publ., 1984, 201 p.
12. Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow: MGU Publ., 1984, 336 p.
13. Brekhovskikh L.M., Godin O.A. Akustika sloistykh sred [Waves in layered media]. Moscow: Nauka Publ., 1973, 416 p.
14. Shamaev A.S., Shumilova V.V. The spectrum of natural vibrations in a medium composed of layers of elastic material and viscous fluid. Dokl. Phys., 2013, vol. 58, no. 1, pp. 33–36. doi: 10.1134/S1028335813010047
15. Shamaev A.S., Shumilova V.V. Spectrum of one-dimensional natural vibrations of layered medium consisting of elastic material and viscous incompressible fluid, Moscow Univ. Math. Bull.. 2020, vol. 75, no. 4, pp. 172–176. doi: 10.3103/S0027132220040063
16. Shumilova V.V. Spectrum of natural vibrations of a layered medium consisting of a Kelvin–Voigt material and a viscous incompressible fluid. Sib. Elektron. Mat. Izv., 2020, vol. 17, pp. 21–31 (in Russian). doi: 10.33048/semi.2020.17.002
17. Il’yushin A.A., Pobedria B.E. Osnovy matematicheskoi teorii termovyazkouprugosti [Fundamentals of the mathematical theory of thermal viscoelasticity]. Moscow: Nauka Publ., 1970, 280 p.
18. Shamaev A.S., Shumilova V.V. Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials. Proc. Steklov Inst. Math., 2016, vol. 295, pp. 202–212. doi: 10.1134/S0081543816080137
19. Vlasov V.V., Wu J., Kabirova G.R. Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect. J. Math. Sci., 2010, vol. 170, no. 3, pp. 388–404. doi: 10.1007/s10958-010-0093-9
20. Shumilova V.V. Spectral analysis of integro-differential equations in viscoelasticity theory. J. Math. Sci., 2014, vol. 196, no. 3, pp. 434–440. doi: 10.1007/s10958-014-1666-9
Cite this article as: V.V. Shumilova. The spectrum of one-dimensional eigenoscillations of two-phase layered media with periodic structure. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 250–261.
