Geodesic
Interfacial Contact Model in a Dense Network of Elastic Materials
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 3-19.

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We consider a dense network of elastic materials modeled by a dense network of elastic disks. More specifically, we consider a dense network of elastic disks in the unit disk $D(0,1)$ of $\mathbb{R}^{2}$ obtained from an Apollonian packing of elastic circular disks by removing disks of small sizes. We suppose that the disks are pressed against each other to form small rectilinear contact zones where a perfect adhesion occurs on thinner zones. We use $\Gamma$-convergence methods in order to study the asymptotic behavior of the structure with respect to a vanishing parameter describing the thickness of the small perfect contact lines between materials. We derive an effective boundary condition on the residual fractal interface obtained by removing the Apollonian network of disks from $D(0,1)$.
Keywords: Apollonian packing, elastic material, boundary layers, effective boundary condition.
Mots-clés : $\Gamma $-convergence, fractal interface 
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Y. Abouelhanoune; M. El Jarroudi. Interfacial Contact Model in a Dense Network of Elastic Materials. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a0/

[1] S. V. Anishchik, N. N. Medvedev, “Three-dimensional Apollonian packing as a model for dense granular systems”, Phys. Rev. Lett., 75 (1995), 4314–4317 | DOI

[2] H. Attouch, Variational convergence for functions and operators, Appl. Math. Series, Pitman, Boston–London–Melbourn, 1984 | MR | Zbl

[3] Y. Ben-Zion, C. G. Sammis, “Characterization of fault zones”, Pure and Applied Geophysics, 160 (2003), 677–715 | DOI

[4] D. W. Boyd, “The residual set dimension of Apollonian packing”, Mathematika, 20 (1973), 170–174 | DOI | MR | Zbl

[5] R. Capitanelli, M. R. Lancia, M. A. Vivaldi, “Insulating layers of fractal type”, Differential Integral Equations, 26:9/10 (2013), 1055–1076 | MR | Zbl

[6] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and Applications, 8, Birkhäuser, Basel, 1993 | MR | Zbl

[7] M. El Jarroudi, “Asymptotic analysis of contact problems between an elastic material and thin-rigid plates”, Appl. Anal., 89:5 (2010), 693–715 | DOI | MR | Zbl

[8] M. El Jarroudi, M. Er-Riani, “Homogenization of elastic materials containing self-similar microcracks”, Quart. J. Mech. Appl. Math, 72:2 (2019), 131–155 | DOI | MR | Zbl

[9] M. El Jarroudi, A. Brillard, “Asymptotic behaviour of contact problems between two elastic materials through a fractal interface”, J. Math. Pures Appl., 89:5 (2008), 505–521 | DOI | MR | Zbl

[10] K. Falconer, Techniques in Fractal Geometry, J. Wiley Sons and sons, Chichester, 1997 | MR | Zbl

[11] H. J. Herrmann, G. Mantica, D. Bessis, “Space-filling bearings”, Phys. Rev. Lett, 65:26 (1990), 3223–3226 | DOI | MR | Zbl

[12] H. Hertz, “Über die berührung fester elastischer körper”, J. Rein. Angew. Math., 92 (1882), 156–171 | DOI | MR

[13] P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces”, Acta Math., 147:1–2 (1981), 71–88 | DOI | MR | Zbl

[14] A. Jonsson, H. Wallin, “Boundary value problems and Brownian motion on fractals”, Chaos Solitons Fractals, 8:2 (1997), 191–205 | DOI | MR | Zbl

[15] J. C. Kim, K. H. Auh, D. M. Martin, “Multi-level particle packing model of ceramic agglomerates”, Model. Simul. Mater. Sci. Eng., 8 (2000), 159–168 | DOI

[16] M. R. Lancia, “A transmission problem with a fractal interface”, Z. Anal. Anwend., 21:1 (2002), 113–133 | DOI | MR | Zbl

[17] C. Marone, C. B. Raleigh, C. H. Scholz, “Frictional behavior and constitutive modeling of simulated fault gouge”, J. Geoph. Res., 95:B5 (1990), 7007–7025 | DOI

[18] R. D. Mauldin, M. Urbanski, “Dimension and measures for a curvilinear Sierpinski gasket or Apollonian packing”, Adv. Maths, 136:1 (1998), 26–38 | DOI | MR | Zbl

[19] U. Mosco, M. A. Vivaldi, “Fractal reinforcement of elastic membranes”, Arch. Ration. Mech. Anal., 194 (2009), 49–74 | DOI | MR | Zbl

[20] U. Mosco, M. A. Vivaldi, “Thin fractal fibers”, Math. Meth. Appl. Sci., 36:15 (2013), 2048–2068 | MR | Zbl

[21] N. I. Muskhelishvili, Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti, Nauka, M., 1966 | MR

[22] S. Roux, A. Hansen, H. J. Herrmann, “A model for gouge deformation: implication for remanent magnetization”, Geoph. Res. Lett., 20:14 (1993), 1499–1502 | DOI

[23] V. S. Rychkov, “Linear extension operators for restrictions of function spaces to irregular open sets”, Studia Math., 140:2 (2000), 141–162 | DOI | MR | Zbl

[24] C. G. Sammis, R. L. Biegel, “Fractals, fault-gouge, and friction”, Pure and Applied Geophysics, 131:1/2 (1989) | DOI

[25] C. H. Scholz, The Mechanics of Earthquakes and Faulting, Cambridge University Press, Cambridge, 2002

[26] S. J. Steacy, C. G. Sammis, “An automaton for fractal patterns of fragmentation”, Nature, 353 (1991), 250–252 | DOI

[27] D. Sullivan, “Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinan groups”, Acta. Math., 153:3–4 (1984), 259–277 | DOI | MR | Zbl

[28] D. M. Walker, A. Tordesillas, Topological evolution in dense granular materials: A complex networks perspective, 47 (2010), 624–639 | Zbl