Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation
Abstract
We consider a homogenization problem for the elasticity operator posed in a bounded domain of the half-space, a part of its boundary being in contact with the plane. This surface is traction-free out of “small regions”, where we impose nonlinear Winkler-Robin boundary conditions containing “large reaction parameters”. Non-periodical distribution of these regions is allowed provided that they have the same area. We show the convergence of solutions towards those of the homogenized problems depending on the relations between the parameters distance, sizes, and reaction.
Keyword : boundary homogenization, elasticity operator, nonlinear Winkler foundations
How to Cite
Gómez, D., & Pérez-Martínez, M.-E. (2024). Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation. Mathematical Modelling and Analysis, 29(4), 694–713. https://doi.org/10.3846/mma.2024.20137
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Brillard, M. Lobo and E. Pérez. Homogénéisation de frontières par epiconvergence en élasticité linéare. RAIRO Modél. Math. Anal. Numér., 24(1):5–26, 1990. https://doi.org/10.1051/m2an/1990240100051
M. Dalla Riva, G. Mishuris and P. Musolino. A degenerating Robin-type traction problem in a periodic domain. Math. Model. Anal., 28(3):509–521, 2023. https://doi.org/10.3846/mma.2023.17681
A. Gaudiello and T.A. Mel’nyk. Homogenization of a nonlinear monotone problem with non-linear Signorini boundary conditions in a domain with highly rough boundary. J. Differential Equations, 265(10):5419–5454, 2018. https://doi.org/10.1016/j.jde.2018.07.002
A. Gaudiello and T.A. Mel’nyk. Homogenization of a nonlinear monotone problem with a big nonlinear Signorini boundary interaction in a domain with highly rough boundary. Nonlinearity, 32:5150–5169, 2019. https://doi.org/10.1088/1361-6544/ab46e9
D. Gómez, M. Lobo, M.E. Pérez, T.A. Shaposhnikova and M.N. Zubova. On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems. Math. Methods Appl. Sci., 38(12):2606– 2629, 2015. https://doi.org/10.1002/mma.3246
D. Gómez, S.A. Nazarov and E. Pérez. Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys, 69(2):35, 2018. https://doi.org/10.1007/s00033-018-0927-8
D. Gómez, S.A Nazarov and M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elast., 142(1):89–120, 2020. https://doi.org/10.1007/s10659-020-09791-8
D. Gómez, S.A. Nazarov and M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In Computational and Analytic Methods in Science and Engineering, pp. 127–150. Springer Nature Switzerland AG, Cham, 2020. https://doi.org/10.1007/978-3-030-48186-5_7
D. Gómez and M.-E. Pérez-Martínez. Boundary homogenization with large reaction terms on a strainer-type wall. Z. Angew. Math. Phys., 73(6):234, 2022. https://doi.org/10.1007/s00033-022-01869-8
D. Gómez and M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity: the averaged Robin reaction matrix. In Integral Methods in Science and Engineering: Applications in Theoretical and Practical Research, pp. 145– 154. Springer Nature Switzerland AG, Cham, 2023.
D. G Gómez and M.-E. Pérez-Martínez. Extreme cases in boundary homogenization for the linear elasticity system. In Exact and Approximate Solutions for Mathematical Models in Science and Engineering, pp. 17–34. Springer Nature Switzerland AG, Cham, 2024.
G. Griso, A. Migunova and J. Orlik. Homogenization via unfolding in periodic layer with contact. Asymptot. Anal., 99:23–52, 2016. https://doi.org/10.3233/ASY-161374
I. Ionescu, D. Onofrei and B. Vernescu. γ-convergence for a fault model with slip-weakening friction and periodic barriers. Quart. Appl. Math., 63(4):747–778, 2005. https://doi.org/10.1090/S0033-569X-05-00981-7
N. Kikuchi and J.T. Oden. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, PA, 1988.
J.L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris, 1969.
M. Lobo, O.A. Oleinik, M.E. Pérez and T.A. Shaposhnikova. On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4e série, 25(3–4):611–629, 1997. Available on Internet: http://www.numdam.org/item/ASNSP_1997_4_25_3-4_611_0
M. Lobo and E. Pérez. Asymptotic behaviour of an elastic body with a surface having small stuck regions. RAIRO Modél. Math. Anal. Numér., 22(4):609–624, 1988. https://doi.org/10.1051/m2an/1988220406091
V.A. Marchenko and E.Ya. Khruslov. Boundary Value Problems in Domains with a Fine-grained Boundary. Naukova Dumka, Kiev, 1974.
S.A. Nazarov and J. Taskinen. A model of a plane deformation state of a twodimensional plate with small almost periodic clamped parts of the edge. Zap. Nauchn. Sem. (POMI), 506:130–174, 2021. Available on Internet: https://www.mathnet.ru/eng/znsl/v506/p130
G. Nguetseng and E. Sanchez-Palencia. Stress concentration for defects distributed near a surface. In Local Effects in the Analysis of Structures, Stud. Appl. Mech., pp. 55–74. Elsevier, Amsterdam, 1985. https://doi.org/10.1016/B978-0-444-42520-1.50007-4
O.A. Oleinik, A.S. Shamaev and G.A Yosifian. Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and its Applications, 26. NorthHolland, Amsterdam, 1992.
P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions. Birkhauser, Boston, 1985.
M.E. Pérez and T.A. Shaposhnikova. Boundary homogenization of a variational inequality with nonlinear restrictions for the flux on small regions lying on a part of the boundary. Dokl. Math., 85(2):198–203, 2012. https://doi.org/10.1134/S1064562412020135
M.-E. Pérez-Martínez. Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In Emerging Problems in the Homogenization of Partial Differential Equations, SEMA SIMAI Springer Series, pp. 37–57. Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-62030-1_3
M. Dalla Riva, G. Mishuris and P. Musolino. A degenerating Robin-type traction problem in a periodic domain. Math. Model. Anal., 28(3):509–521, 2023. https://doi.org/10.3846/mma.2023.17681
A. Gaudiello and T.A. Mel’nyk. Homogenization of a nonlinear monotone problem with non-linear Signorini boundary conditions in a domain with highly rough boundary. J. Differential Equations, 265(10):5419–5454, 2018. https://doi.org/10.1016/j.jde.2018.07.002
A. Gaudiello and T.A. Mel’nyk. Homogenization of a nonlinear monotone problem with a big nonlinear Signorini boundary interaction in a domain with highly rough boundary. Nonlinearity, 32:5150–5169, 2019. https://doi.org/10.1088/1361-6544/ab46e9
D. Gómez, M. Lobo, M.E. Pérez, T.A. Shaposhnikova and M.N. Zubova. On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems. Math. Methods Appl. Sci., 38(12):2606– 2629, 2015. https://doi.org/10.1002/mma.3246
D. Gómez, S.A. Nazarov and E. Pérez. Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys, 69(2):35, 2018. https://doi.org/10.1007/s00033-018-0927-8
D. Gómez, S.A Nazarov and M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elast., 142(1):89–120, 2020. https://doi.org/10.1007/s10659-020-09791-8
D. Gómez, S.A. Nazarov and M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In Computational and Analytic Methods in Science and Engineering, pp. 127–150. Springer Nature Switzerland AG, Cham, 2020. https://doi.org/10.1007/978-3-030-48186-5_7
D. Gómez and M.-E. Pérez-Martínez. Boundary homogenization with large reaction terms on a strainer-type wall. Z. Angew. Math. Phys., 73(6):234, 2022. https://doi.org/10.1007/s00033-022-01869-8
D. Gómez and M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity: the averaged Robin reaction matrix. In Integral Methods in Science and Engineering: Applications in Theoretical and Practical Research, pp. 145– 154. Springer Nature Switzerland AG, Cham, 2023.
D. G Gómez and M.-E. Pérez-Martínez. Extreme cases in boundary homogenization for the linear elasticity system. In Exact and Approximate Solutions for Mathematical Models in Science and Engineering, pp. 17–34. Springer Nature Switzerland AG, Cham, 2024.
G. Griso, A. Migunova and J. Orlik. Homogenization via unfolding in periodic layer with contact. Asymptot. Anal., 99:23–52, 2016. https://doi.org/10.3233/ASY-161374
I. Ionescu, D. Onofrei and B. Vernescu. γ-convergence for a fault model with slip-weakening friction and periodic barriers. Quart. Appl. Math., 63(4):747–778, 2005. https://doi.org/10.1090/S0033-569X-05-00981-7
N. Kikuchi and J.T. Oden. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, PA, 1988.
J.L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris, 1969.
M. Lobo, O.A. Oleinik, M.E. Pérez and T.A. Shaposhnikova. On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4e série, 25(3–4):611–629, 1997. Available on Internet: http://www.numdam.org/item/ASNSP_1997_4_25_3-4_611_0
M. Lobo and E. Pérez. Asymptotic behaviour of an elastic body with a surface having small stuck regions. RAIRO Modél. Math. Anal. Numér., 22(4):609–624, 1988. https://doi.org/10.1051/m2an/1988220406091
V.A. Marchenko and E.Ya. Khruslov. Boundary Value Problems in Domains with a Fine-grained Boundary. Naukova Dumka, Kiev, 1974.
S.A. Nazarov and J. Taskinen. A model of a plane deformation state of a twodimensional plate with small almost periodic clamped parts of the edge. Zap. Nauchn. Sem. (POMI), 506:130–174, 2021. Available on Internet: https://www.mathnet.ru/eng/znsl/v506/p130
G. Nguetseng and E. Sanchez-Palencia. Stress concentration for defects distributed near a surface. In Local Effects in the Analysis of Structures, Stud. Appl. Mech., pp. 55–74. Elsevier, Amsterdam, 1985. https://doi.org/10.1016/B978-0-444-42520-1.50007-4
O.A. Oleinik, A.S. Shamaev and G.A Yosifian. Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and its Applications, 26. NorthHolland, Amsterdam, 1992.
P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions. Birkhauser, Boston, 1985.
M.E. Pérez and T.A. Shaposhnikova. Boundary homogenization of a variational inequality with nonlinear restrictions for the flux on small regions lying on a part of the boundary. Dokl. Math., 85(2):198–203, 2012. https://doi.org/10.1134/S1064562412020135
M.-E. Pérez-Martínez. Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In Emerging Problems in the Homogenization of Partial Differential Equations, SEMA SIMAI Springer Series, pp. 37–57. Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-62030-1_3