Interfacial Contact Model in a Dense Network of Elastic Materials

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 $D(0,1)$.