Asymptotic analysis of an elastic material reinforced with thin fractal strips

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

doi:10.3934/nhm.2021023

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

This issue Previous Article $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices Next Article

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

^* Corresponding author: M. El Jarroudi
^* Corresponding author: M. El Jarroudi

Received: January 31, 2021

Revised: July 31, 2021

Published: February 2022

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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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Mathematics Subject Classification: 35B40, 28A80, 35J20.

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Figure 1. The graph $ \Sigma _{h} $ for $ h = 0,1,2,3 $

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Figure 2. The network $ \left\{ \mathcal{T}_{m}\right\} _{m\in \mathbb{N}} $ where $ \sigma _{\alpha 3}|_{\Sigma \times \left\{ 0^{+}\right\} } $ is the outward normal stress on $ \Sigma \cap \partial \mathcal{T}_{m} $ and $ -\sigma _{\alpha 3}|_{\Sigma \times \left\{ 0^{-}\right\} } $ is the inward normal stress

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