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Dimension reduction analysis of a three-dimensional thin elastic plate reinforced with fractal ribbons

Part of: Elliptic equations and systems Partial differential equations Classical measure theory

Published online by Cambridge University Press:  02 March 2023

Mustapha El Jarroudi Open the ORCID record for Mustapha El Jarroudi [Opens in a new window] ,
Mhamed El Merzguioui ,
Mustapha Er-Riani ,
Aadil Lahrouz  and
Jamal El Amrani
Mustapha El Jarroudi*
Affiliation:
Abdelmalek Essaâdi University, LMA, FST Tanger, B.P. 416, Tangier, Morocco
Mhamed El Merzguioui
Affiliation:
Abdelmalek Essaâdi University, LMA, FST Tanger, B.P. 416, Tangier, Morocco
Mustapha Er-Riani
Affiliation:
Abdelmalek Essaâdi University, LMA, FST Tanger, B.P. 416, Tangier, Morocco
Aadil Lahrouz
Affiliation:
Abdelmalek Essaâdi University, LMA, FST Tanger, B.P. 416, Tangier, Morocco
Jamal El Amrani
Affiliation:
Abdelmalek Essaâdi University, LMA, FST Tanger, B.P. 416, Tangier, Morocco
*
*Correspondence author. Email: [email protected]
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Abstract

The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the $\Gamma $-convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.

Keywords

Thin elastic platesfractal ribbonsdimension reduction analysismicroscopic interactionseffective material

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 28A80: Fractals 35J20: Variational methods for second-order elliptic equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 4 , August 2023 , pp. 838 - 869
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Adkins, J. E. & Rivlin, R. S. (1955) Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords. Philos. Trans. R. Soc. London A 248, 201–223.Google Scholar
Attouch, H. (1984) Variational Convergence for Functions and Operators , Applied Mathematical Sciences, London, Pitman.Google Scholar
Barlow, M. T. & Bass, R. F. (1990) On the resistance of the Sierpinski carpet. Proc. Math. Phys. Sci. 431(1882), 345–360.Google Scholar
Barlow, M. T., Bass, R. F. & Sherwood, J. D. (1990) Resistance and spectral dimension of the Sierpinski carpets. J. Phys. A 23, L253–L258.10.1088/0305-4470/23/6/004CrossRefGoogle Scholar
Bellieud, M. & Bouchitté, G. (1998) Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(3), 407–436.Google Scholar
Capitanelli, R., Lancia, M. R. & Vivaldi, M. A. (2013) Insulating layers of fractal type. Differ. Integr. Equations 26(9/10), 1055–1076.10.57262/die/1372858561CrossRefGoogle Scholar
Capitanelli, R. & Vivaldi, M. A. (2011) Insulating layers and Robin problems on Koch mixtures. J. Differ. Equations 251(4–5), 1332–1353.10.1016/j.jde.2011.02.003CrossRefGoogle Scholar
Capitanelli, R. & Vivaldi, M. A. (2015) Reinforcement problems for variational inequalities on fractal sets. Calc. Var. 54, 2751–2783.10.1007/s00526-015-0882-6CrossRefGoogle Scholar
Capitanelli, R. & Vivaldi, M. A. (2016) Dynamical quasi-filling fractal layers. SIAM J. Math. Anal. 48(6), 3931–3961.10.1137/15M1043893CrossRefGoogle Scholar
Creo, S. (2021) Singular p-homogenization for highly conductive fractal layers. Z. Anal. Anwend. 40(4), 401–424.CrossRefGoogle Scholar
Dal Maso, G. (1993) An Introduction to $\Gamma $ -Convergence, PNLDEA, Vol. 8, Birkhäuser, Basel.10.1007/978-1-4612-0327-8CrossRefGoogle Scholar
El Jarroudi, M. (2019) Homogenization of an elastic material reinforced with thin rigid von Kármán ribbons. Math. Mech. Solids 24(7), 1965–1991.10.1177/1081286518810757CrossRefGoogle Scholar
El Jarroudi, M. (2022) Homogenization of a quasilinear elliptic problem in a fractal-reinforced structure. SeMA 79, 571–592. https://doi.org/10.1007/s40324-021-00250-5.CrossRefGoogle Scholar
El Jarroudi, M. & Er-Riani, M. (2019) Homogenization of elastic materials containing self-similar microcracks. Q. Jl Mech. Appl. Math. 72(2), 131–155.10.1093/qjmam/hby023CrossRefGoogle Scholar
El Jarroudi, M., Er-Riani, M., Lahrouz, A. & Settati, A. (2018) Homogenization of elastic materials reinforced by rigid notched fibres. Appl. Anal. 97(5), 705–738.10.1080/00036811.2017.1285015CrossRefGoogle Scholar
El Jarroudi, M., Filali, Y., Lahrouz, A., Er-Riani, M. & Settati, A. (2022) Asymptotic analysis of an elastic material reinforced with thin fractal strips. Netw. Heterog. Media 17(1), 47–72.10.3934/nhm.2021023CrossRefGoogle Scholar
Falconer, K. (1997) Techniques in Fractal Geometry , J. Wiley and Sons, Chichester.10.2307/2533585CrossRefGoogle Scholar
Freiberg, U. R. & Lancia, M. R. (2004) Energy form on a closed fractal curve. Z. Anal. Anwendungen 23(1), 115–137.10.4171/ZAA/1190CrossRefGoogle Scholar
Fukushima, M., Oshima, Y. & Takeda, M. (1994) Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, Vol. 19, Eds. Kazdan, Bauer, Zehnder, Berlin.10.1515/9783110889741CrossRefGoogle Scholar
Grigor’yan, A. & Yang, M. (2019) Local and non-local Dirichlet forms on the Sierpi ${\textrm{n}}$ ski carpet. Trans. AMS 372(6), 3985–4030.10.1090/tran/7753CrossRefGoogle Scholar
Izotova, O. V., Nazarov, S. A. & Sweers, G. H. (2008) Asymptotic sharp weight Korn’s inequality for thin-walled elastic structures. J. Math. Sci. (N.Y) 150(1), 1807–1855.10.1007/s10958-008-0098-9CrossRefGoogle Scholar
Jonsson, A. & Wallin, H. (1984) Function Spaces on Subsets of $\mathbb{R}^{n}$ , Mathematical Reports, Vol. 2 (Part 1), Harwood Academic Publisher, London.Google Scholar
Jonsson, A. & Wallin, H. (1995) The dual of Besov spaces on fractals. Stud. Math. 112(3), 285–300.10.4064/sm-112-3-285-300CrossRefGoogle Scholar
Jonsson, A. & Wallin, H. (1997) Boundary value problems and Brownian motion on fractals. Chaos Soli. Frac. 8(2), 191–205.10.1016/S0960-0779(96)00048-3CrossRefGoogle Scholar
Kato, T. (1966) Perturbation Theory for Linear Operators, Springer, Berlin.Google Scholar
Lancia, M. R., Mosco, U. & Vivaldi, M. A. (2008) Homogenization for conductive thin layers of pre-fractal type. J. Math. Anal. Appl. 347(1), 354–369.10.1016/j.jmaa.2008.06.011CrossRefGoogle Scholar
Le Dret, H. (1991) Problèmes Variaonnels dans les Multi-Domaines : Modélisation des Jonctions et Applications, RMA, Vol. 19, Masson, Paris.Google Scholar
Le Jean, Y. (1978) Measures associées á une forme de Dirichlet. Applications. Bull. Soc. Math. France 106, 61–112.10.24033/bsmf.1864CrossRefGoogle Scholar
Lobo, M. & Perez, E. (1992) Boundary homogenization of certain elliptic problems for cylindrical bodies. Bull. Sci. Math. Sér 2 116, 399–426.Google Scholar
Malo, R. J. (2015) Discrete Extremal Lengths of Graph Approximations of Sierpinski Carpets. PhD thesis, Montana State University.Google Scholar
Mosco, U. (2000) Energy functionals on certain fractal structures. J. Conv. Anal. 9, 581–600.Google Scholar
Mosco, U. & Vivaldi, M. A. (2007) An example of fractal singular homogenization. Georgian Math. J. 14(1), 169–194.CrossRefGoogle Scholar
Mosco, U. & Vivaldi, M. A. (2009) Fractal reinforcement of elastic membranes. Arch. Rat. Mech. Anal. 194, 49–74.CrossRefGoogle Scholar
Mosco, U. & Vivaldi, M. A. (2013) Thin fractal fibers. Math. Meth. Appl. Sci. 36, 2048–2068.Google Scholar
Mosco, U. & Vivaldi, M. A. (2015) Layered fractal fibers and potentials. J. Math. Pures Appl. 103, 1198–1227.CrossRefGoogle Scholar