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Spectral Homogenization Problems in Linear Elasticity with Large Reaction Terms Concentrated in Small Regions of the Boundary

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Computational and Analytic Methods in Science and Engineering

Abstract

We consider a homogenization problem for the elasticity operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. We assume that this surface is traction-free out of “small regions,” where we impose Winkler–Robin boundary conditions. These “reaction regions” are periodically placed along the plane while its size is much smaller than the period. The Winkler–Robin condition links stresses and displacements in the small reaction regions by means of a symmetric and definite positive matrix and a “reaction parameter” that can be large as the period tends to zero. Using the technique of matched asymptotic expansions, we provide all the possible homogenized problems, depending on the relations between the three parameters: period, size of the small regions, and reaction parameter.

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References

  1. Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II. Non critical size of the holes for a volume distribution of holes and a surface distribution of holes. Arch. Ration. Mech. Anal. 113, 261–298 (1983)

    MATH  Google Scholar 

  2. Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman, London (1984)

    Google Scholar 

  3. Brillard, A.: Asymptotic flow of a viscous and incompressible fluid through a plane sieve. In: Progress in Partial Differential Equations: Calculus of Variations, Applications. Pitman Research Notes in Mathematics Series, vol. 267, pp. 158–172. Longman Scientific & Technical, Harlow (1992)

    Google Scholar 

  4. Brillard, A., Lobo, M., Pérez, E.: Homogénéisation de Frontières par epi-convergence en élasticité linéare. RAIRO Modél. Math. Anal. Numér. 24, 5–26 (1990)

    Article  MathSciNet  Google Scholar 

  5. Brillard, A., Gómez, D., Lobo, M., Pérez, E., Shaposhnikova, T.A.: Boundary homogenization in perforated domains for adsorption problems with an advection term. Appl. Anal. 95, 218–237 (2016)

    Article  MathSciNet  Google Scholar 

  6. Brillard, A., Lobo, M., Pérez, E.: Un probléme d’homogénéisation de frontière en élasticité linéare pour un corps cylindrique. C.R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 311, 15–20 (1990)

    Google Scholar 

  7. Cioranescu, D., Donato, P., Ene, H.: Homogenization of the Stokes problem with non-homogeneous slip boundary conditions. Math. Methods Appl. Sci. 19, 857–881 (1996)

    Article  MathSciNet  Google Scholar 

  8. Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 45–93. Birkäuser, Boston (1997)

    Google Scholar 

  9. Conca, C.: On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64, 31–75 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Conca, C.: Étude d’un fluide traversant une paroi perforée. I. Comportement limite près de la paroi. J. Math. Pures Appl. 66, 1–43 (1987)

    MATH  Google Scholar 

  11. Del Vecchio, T.: The thick Neumann’s sieve. Ann. Mat. Pura Appl. 147, 363–402 (1987)

    Article  MathSciNet  Google Scholar 

  12. Gómez, D., Pérez, E., Shaposhnikova, T.A.: Spectral boundary homogenization problems in perforated domains with Robin boundary conditions and large parameters. In: Integral Methods in Science and Engineering, pp. 155–174. Birkhäuser/Springer, New York (2013)

    Google Scholar 

  13. Gómez, D., Lobo, L., Pérez, E., Sanchez-Palencia, E.: Homogenization in perforated domains: a Stokes grill and an adsorption process. Appl. Anal. 97, 2893–2919 (2018)

    Article  MathSciNet  Google Scholar 

  14. Gómez, D., Pérez, E., Podolskiy, A.V., Shaposhnikova, T.A.: Homogenization of variational inequalities for the p-Laplace operator in perforated media along manifolds. Appl. Math. Optim. 79, 695–713 (2019)

    Article  MathSciNet  Google Scholar 

  15. Gómez, D., Lobo, M., Pérez-Martínez, M.-E.: Asymptotics for models of non-stationary diffusion in domains with a surface distribution of obstacles. Math. Methods Appl. Sci. 42, 403–413 (2019)

    Article  MathSciNet  Google Scholar 

  16. Gómez, D., Nazarov, S.A., Pérez, E.: Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys. 69(2), article 35, 23 pp. (2018)

    Google Scholar 

  17. Gómez, D., Nazarov, S.A., Pérez, E.: Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. Submitted, (2020)

    Google Scholar 

  18. Gómez, D., Pérez, E., Shaposhnikova, T.A.: On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems. Asymptot. Anal. 80, 289–322 (2012)

    Article  MathSciNet  Google Scholar 

  19. Gómez, D., Pérez, E., Shaposhnikova, T.A.: On correctors for spectral problems in the homogenization of Robin boundary conditions with very large parameters. Int. J. Appl. Math. 26, 309–320 (2013)

    Article  MathSciNet  Google Scholar 

  20. Griso, G., Migunova, A., Orlik, J.: Homogenization via unfolding in periodic layer with contact. Asymptot. Anal. 99, 23–52 (2015)

    Article  MathSciNet  Google Scholar 

  21. Ionescu, I., Onofrei, D., Vernescu, B.: Γ-Convergence for a fault model with slip-weakening friction and periodic barriers. Quart. Appl. Math. 63(4), 747–778 (2005)

    Article  MathSciNet  Google Scholar 

  22. El Jarroudi, M., Addou, A., Brillard, A.: Asymptotic analysis and boundary homogenization in linear elasticity. Math. Methods Appl. Sci. 23, 655–683 (2000)

    Article  MathSciNet  Google Scholar 

  23. Landau, L., Lifchitz, E.: Théorie de l’Élasticité. Physique Théorique. Tome 7. Mir, Moscow (1990)

    Google Scholar 

  24. Lobo, M., Oleinik, O.A., Pérez, M.E., Shaposhnikova, T.A.: On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4e série 25, 611–629 (1997)

    Google Scholar 

  25. Lobo, M., Pérez, E.: Comportement asymptotique d’un corps élastique dont une surface présente de petites zones de collage. C.R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers. Sci. Terre 304, 195–198 (1987)

    Google Scholar 

  26. Lobo, M., Pérez, E.: Asymptotic behaviour of an elastic body with a surface having small stuck regions. RAIRO Modél. Math. Anal. Numér. 22, 609–624 (1988)

    Article  MathSciNet  Google Scholar 

  27. Lobo, M., Pérez, E.: Boundary homogenization of certain elliptic problems for cylindrical bodies. Bull. Sci. Math. 116, 399–426 (1992)

    MathSciNet  MATH  Google Scholar 

  28. Marchenko, V.A., Khruslov, E.Ya.: Homogenization of Partial Differential Equations. Birkhäuser, Boston (2006)

    Google Scholar 

  29. Murat, F.: The Neumann sieve. In: Nonlinear Variational Problems (Isola d’Elba, 1983). Research Notes in Mathematics, vol. 127, pp. 24–32. Pitman, Boston (1985)

    Google Scholar 

  30. Nazarov, S.A.: Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes. Uspekhi Mat. Nauk 54, 77–142 (1999). English translation: Russ. Math. Surv. 54, 947–1014 (1999)

    Google Scholar 

  31. Nazarov, S.A.: Asymptotics of solutions and modeling of the elasticity problems in a domain with the rapidly oscillating boundary. Math. Izvestiya 72(3), 509–564 (2008)

    Article  MathSciNet  Google Scholar 

  32. Nazarov, S.A., Sokolowski, J., Specovius-Neugebauer, M.: Polarization matrices in anisotropic heterogeneous elasticity. Asymptot. Anal. 68(4), 189–221 (2010)

    Article  MathSciNet  Google Scholar 

  33. 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)

    Google Scholar 

  34. Oleinik, O.A., Chechkin, G.: On boundary value problems for elliptic equations with rapidly changing type of boundary conditions. Uspekhi Mat. Nauk 48, 163–164 (1993). English translation: Russ. Math. Surv. 48, 173–175 (1993)

    Google Scholar 

  35. Oleinik, O.A., Chechkin, G.: On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7, 5–15 (1996)

    Google Scholar 

  36. Oleinik, O.A., Shamaev, A.S., Yosifian G.A.: Mathematical Problems in Elasticity and Homogenization. North-Holland, London (1992)

    MATH  Google Scholar 

  37. Picard, C.: Analyse limite d’équations variationnelles dans un domaine contenant une grille. RAIRO Modél. Math. Anal. Numér. 21, 293–326 (1987)

    Article  MathSciNet  Google Scholar 

  38. Sanchez-Hubert, J., Sanchez-Palencia, E.: Acoustic fluid flow through holes and permeability of perforated walls. J. Math. Anal. Appl. 87, 427–453 (1982)

    Article  MathSciNet  Google Scholar 

  39. Sanchez-Hubert, J., Sanchez-Palencia, E.: Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer, Heidelberg (1989)

    Google Scholar 

  40. Sanchez-Palencia, E.: Boundary value problems in domains containing perforated walls. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. III. Research Notes in Mathematics, vol. 70, pp. 309–325. Pitman, Boston (1982)

    Google Scholar 

  41. Sanchez-Palencia, E.: Un problème d’ecoulement lent d’un fluide incompressible au travers d’une paroi finement perforée. In: Homogenization Methods: Theory and Applications in Physics. Collect. Dir. Études Rech. Élec. France, vol. 57, pp. 371–400. Eyrolles, Paris (1985)

    Google Scholar 

  42. Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland, Amsterdam (1979)

    Google Scholar 

  43. R. Temam, Problèmes Mathématiques en Plasticité. Gautier Villars, Paris (1983)

    MATH  Google Scholar 

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Acknowledgements

This work has been partially supported by Spanish MICINN grant PGC2018-098178-B-I00, Russian Foundation of Basic Research grant 18-01-00325, and the Convenium Banco Santander—Universidad de Cantabria 2018.

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Authors and Affiliations

  1. Universidad de Cantabria, Santander, Spain

    Delfina Gómez & Maria-Eugenia Pérez-Martínez

  2. Saint-Petersburg State University, St. Petersburg, Russia

    Sergey A. Nazarov

  3. Institute of Problems of Mechanical Engineering RAS, St. Petersburg, Russia

    Sergey A. Nazarov

Authors
  1. Delfina Gómez
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  2. Sergey A. Nazarov
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  3. Maria-Eugenia Pérez-Martínez
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Corresponding author

Correspondence to Maria-Eugenia Pérez-Martínez .

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Editors and Affiliations

  1. The Charles W. Oliphant Professor of Mathematics, The University of Tulsa, Tulsa, OK, USA

    Christian Constanda

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Gómez, D., Nazarov, S.A., Pérez-Martínez, ME. (2020). Spectral Homogenization Problems in Linear Elasticity with Large Reaction Terms Concentrated in Small Regions of the Boundary. In: Constanda, C. (eds) Computational and Analytic Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-48186-5_7

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