Asymptotic analysis of an elastic material reinforced with thin fractal strips

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-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.</p>

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

“…The homogenised problems obtained at the limit generally consist of singular forms containing fractal terms. The asymptotic analysis of a three-dimensional elastic material reinforced with thin vertical strips constructed on horizontal iterated Sierpinski gasket curves was studied in [16]. The problem considered in this work is quite different as we deal here with a threedimensional plate with varying thickness reinforced with vertical strips disposed on iterated Sierpinski carpet curves.…”