Abstract
We consider a homogenization Winkler–Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler–Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order \(\varepsilon \). For fixed \(\varepsilon \), the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues \(\{\beta _k^\varepsilon \}_{k=1}^{\infty }\) as \(\varepsilon \rightarrow 0\). We show that \(\beta _k^\varepsilon =O(\varepsilon ^{-1})\) for each fixed k, and we observe a common limit point for all the rescaled eigenvalues \(\varepsilon \beta _k^\varepsilon \) while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain “groups” of eigenmodes.
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References
Bertram, A.: Elasticity and Plasticity of Large Deformations. Springer, Berlin (2005)
Brillard, A., Lobo, M., Pérez, E.: Homogénéisation de frontières par épi-convergence en élasticité linéaire. RAIRO Modél. Math. Anal. Numér. 24(1), 5–26 (1990)
Brillard, A., Lobo, M., Pérez, E.: Un problème d’homogénéisation de frontiére en élasticité linéaire pour un corps cylindrique. (French) [A boundary homogenization problem in linear elasticity for a cylindrical body] C. R. Acad. Sci. Paris Sr. II Mc. Phys. Chim. Sci. Univers. Sci. Terre 311(1), 15–20 (1990)
Bucur, D., Ionescu, I.: Asymptotic analysis and scaling of friction parameters. Z. Angew. Math. Phys. (ZAMP) 57(6), 1–15 (2006)
Campillo, M., Dascalu, C., Ionescu, I.: Instability of a periodic systems of faults. Geophys. J. Int. 159, 212–222 (2004)
Cao, L., Zhang, L., Allegretto, W., Lin, Y.: Multiscale asymptotic method for Steklov eigenvalue equations in composite media. SIAM J. Numer. Anal. 51(1), 273–296 (2013)
Chechkina, A.G.: Homogenization of spectral problems with singular perturbation of the Steklov condition, Izvestiya RAN: Ser. Mat. 81:1 203–240, (2017). (English transl. Izv. Math. 81(1):199–236, 2017)
Conca, C., Planchard, J., Vanninathan, M.: Fluids and Periodic Structures, RAM: Research in Applied Mathematics 38. Wiley, Chichester (1995)
Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
El Jarroudi, M., Addou, A., Brillard, A.: Homogénéisation de frontière en élasticité linéaire. (French) [Boundary homogenization in linear elasticity] C. R. Acad. Sci. Paris Sr. I Math. 328(8):725–730, (1999)
Gelfand, I.M.: Expansions in eigenfunctions of an equation with periodic coefficients. Dokl. Acad. Nauk SSSR 73, 1117–1120 (1950). (in Russian)
Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem. J. Spectr. Theory 7(2), 321–359 (2017)
Gómez, D., Nazarov, S.A., Pérez, E.: Dependence on the Floquet parameter of the spectrum of the elasticity problem in an infinite prism (in preparation)
Ionescu, I., Onofrei, D., Vernescu, B.: \(\Gamma \)-convergence for a fault model with slip-weakening friction and periodic barriers. Quart. Appl. Math. 63(4), 747–778 (2005)
Jayawardana, K.: Mathematical theory of shells on elastic foundations: an analysis of boundary forms, constraints, and applications to friction and skin abrasion, dissertation of doctor of philosophy in mathematics, Department of Mathematics, University College London (2016)
Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)
Kondratiev, V.A.: Boundary problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313 (1967)
Kondratiev, V.A., Oleinik, O.A.: Boundary value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities, Uspekhi Mat. Nauk 43:55–98, (1988). (English transl. Russian Math. Surveys 43, 65–119 1988)
Kuchment, P.: Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37:3–52, 1982. (English transl. Russ. Math. Surveys 37:1–60, 1982)
Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)
Kuznetsov, N., Kulczycki, T., Kwasnicki, M., Nazarov, A., Poborchi, S., Polterovich, I., Siudeja, B.: The legacy of Vladimir Andreevich Steklov. Notices Am. Math. Soc. 61(1), 9–22 (2014)
León Baldelli, A.A., Bourdin, B.: On the asymptotic derivation of Winkler-type energies from 3D elasticity. J. Elast. 121(2), 275–301 (2015)
Lobo, M., Pérez, E.: Asymptotic behaviour of an elastic body with a surface having small stuck regions. RAIRO Modél. Math. Anal. Numér. 22(4), 609–624 (1988)
Lobo, M., Pérez, E.: Boundary homogenization of certain elliptic problems for cylindrical bodies. Bull. Sci. Math. 116(3), 399–426 (1992)
Lobo, M., Pérez, E.: On the local vibrations for systems with many concentrated masses near the boundary. C.R. Acad. Sci. Paris, Sér. IIb 324(5), 323–329 (1997)
Lobo, M., Pérez, E.: Local problems for vibrating systems with concentrated masses: a review. C.R. Mécanique 331, 303–317 (2003)
Lobo, M., Pérez, E.: Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems. Math. Methods Appl. Sci. 33(11), 1356–1371 (2010)
Mazja, V.G., Plamenevskii, B.A.: Estimates in \(L_p\) and in Hölder classes and the Miranda–Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary. Amer. Math. Soc. Trans. 123, 1–56 (1984)
Nazarov, S.A.: Elliptic boundary value problems with periodic coefficients in a cylinder. Izv. Akad. Nauk SSSR Ser. Mat. 45(1):101–112, (1981). (English transl. Math. USSR Izvestija 18(1):89–98, 1982)
Nazarov, S.A.: The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Uspehi Mat. Nauk. 54(5), 77–142 (1999). (English transl. Russ. Math. Surveys. 54(5), 947–1014, 1999)
Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods, vol. 1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirck (2002)
Nazarov, S.A.: Eigenoscillations of an elastic body with a rough surface, Prikl. Mekh. Techn. Fizika. 48(6), 103–114, (2007). (English transl. J. Appl. Mech. Techn. Physics. 48(6), 861–870 2007)
Nazarov, S.A.: Asymptotics of solutions and modeling of the elasticity problems in a domain with the rapidly oscillating boundary, Izv. Ross. Akad. Nauk. Ser. Mat. 72(3), 103–158 (2008). (English transl. Math. Izvestiya. 72(3), 509–564, 2008)
Nazarov, S.A., Pérez, E.: New asymptotic effects for the spectrum of problems on concentrated masses near the boundary. C.R. Mécanique 337(8), 585–590 (2009)
Nazarov, S.A., Pérez, E.: On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary. Rev. Mat. Complut. 31(1), 1–62 (2018)
Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)
Nazarov, S.A., Ruotsalainen, K., Taskinen, J.: Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps. Appl. Anal. 89(1), 109–124 (2010)
Nguetseng, G., Sanchez-Palencia, E.: Stress concentration for defects distributed near a surface. In: Local Effects in the Analysis of Structures. Studies in Applied Mechanics, vol. 12, pp. 55–74. Elsevier, Amsterdam (1985)
Pérez, E.: On periodic Steklov type eigenvalue problems on half-band and the spectral homogenization problem. Discrete Contin. Dyn. Syst. Ser. B 7(4), 859–883 (2007)
Pérez, E.: Long time approximations for solutions of evolution equations from quasimodes: perturbation problems. Math. Balkanica (N.S.) 25(1–2), 95–130 (2011)
Sanchez-Hubert, J., Sanchez-Palencia, E.: Vibration and Coupling of Continuous Systems, Asymptotic Methods. Springer, Berlin (1989)
Sivaji Ganesh, S., Vanninathan, M.: Bloch wave homogenization of linear elasticity system. ESAIM Control Optim. Calc. Var. 11(4), 542–573 (2005)
Stekloff, W.: Sur l\(\acute{}\)existence des fonctions fondamentales. C.R. Acad. Sci. Paris, Phys. Math. 128(1), 808–810 (1899)
Turbé, N.: Applications of Bloch expansion to periodic elastic and viscoelastic media. Math. Methods Appl. Sci. 4(1), 433–449 (1982)
Visik, M.I., Lusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Am. Math. Soc. Trans. 20, 239–364 (1962)
Acknowledgements
This research work has been partially supported by the Spanish MINECO, MTM2013-44883-P and by Russian Foundation of Basic Research 18-01-00325.
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Gómez, D., Nazarov, S.A. & Pérez, M.E. Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys. 69, 35 (2018). https://doi.org/10.1007/s00033-018-0927-8
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DOI: https://doi.org/10.1007/s00033-018-0927-8
Keywords
- Homogenization
- Linear elasticity
- Steklov problem
- Winkler foundation
- Spectral perturbation theory
- Floquet–Bloch–Gelfand transform