Abstract

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 \(D(0,1)\).

中文翻译：

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弹性材料密集网络中的界面接触模型

摘要

我们考虑由弹性磁盘的密集网络建模的弹性材料的密集网络。更具体地说，我们考虑了从弹性圆盘的阿波罗包装获得的\(\mathbb{R}^{2}\)的单位盘\(D(0,1)\)中的弹性盘的密集网络，通过删除小尺寸磁盘。我们假设圆盘相互挤压以形成小的直线接触区，其中在较薄的区域上发生完美的粘附。我们使用\(\Gamma\)-收敛方法，以研究结构相对于描述材料之间小的完美接触线厚度的消失参数的渐近行为。我们在通过从\(D(0,1)\) 中去除阿波罗盘网络获得的残余分形界面上推导出有效边界条件。