Asymptotic analysis of an elastic material reinforced with thin fractal strips

Mustapha El Jarroudi; Youness Filali; Aadil Lahrouz; Mustapha Er-Riani; Adel Settati; Mustapha El Jarroudi; Youness Filali; Aadil Lahrouz; Mustapha Er-Riani; Adel Settati

doi:10.3934/nhm.2021023

Networks and Heterogeneous Media

2022, Volume 17, Issue 1: 47-72. doi: 10.3934/nhm.2021023

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Asymptotic analysis of an elastic material reinforced with thin fractal strips

Laboratory of Mathematics and Applications, Abdelmalek Essaâdi University, FST Tangier, B.P. 416 Tangier, Morocco

Received: 01 February 2021 Revised: 01 August 2021
35B40, 28A80, 35J20

We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use $ \Gamma $-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.
- Three-dimensional elastic material,
- highly contrasted fractal strips,
- boundary layers,
- asymptotic analysis,
- effective behavior
Citation: Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips[J]. Networks and Heterogeneous Media, 2022, 17(1): 47-72. doi: 10.3934/nhm.2021023

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Abstract

We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use $ \Gamma $-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.

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