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Interfacial contact model in a dense network of elastic materials. (English. Russian original) Zbl 1482.74125

Funct. Anal. Appl. 55, No. 1, 1-14 (2021); translation from Funkts. Anal. Prilozh. 55, No. 1, 3-19 (2021).
The authors consider an Apollonian packing of the closed disk \(\overline{ \Omega }=D(0,1)\subset \mathbb{R}^{2}\) through a network \( (D_{k})_{k=1,...,\infty }\) of disjoint elastic disks, \(\Lambda =\overline{ \Omega }\setminus \cup _{k}D_{k}\), and \(\Sigma =\Lambda \setminus \Gamma _{0} \), where \(\Gamma _{0}\) is the boundary of \(\Omega \). They also consider a decreasing sequence \((\rho _{h})_{h}\) of positive numbers which satisfies \( \lim_{h\rightarrow \infty }\rho _{h}=0\) and \(\lim_{h\rightarrow \infty } \mathcal{H}^{2}(\mathcal{V}_{h})=0\), where \(\mathcal{V}_{h}\) is the union of disks of radius less than or equal to \(\rho _{h}\) and \(\mathcal{H}^{2}\) is the 2D Hausdorff measure. They finally consider the linear elastic equilibrium problem in the union of indented disks \((D_{k}^{\ast })_{k=1,...,\infty }\) assuming perfect adhesion between part of their boundary. The purpose of the paper is to describe the asymptotic behavior of the elastic energy associated with this problem.The extra term added in the limit energy functional is \(\frac{4\mu _{0}\pi c}{\mathcal{H} ^{d}(\Lambda )(1+\kappa _{0})(\ln 2)^{2}}\int_{\Sigma }\left\vert [u]_{\Sigma }\right\vert ^{2}d\mathcal{H}^{d}(s)\), where \(\mu _{0}\) and \( \kappa _{0}\) are material coefficients, \(d\) is the fractal dimension of the fractal set \(\Lambda \), \(\mathcal{H}^{d}\) is the \(d\)-dimensional Hausdorff measure and \(c\) is a positive constant associated with the set of perfect adhesion. The authors use the \(\Gamma \)-convergence tools. They first introduce a small parameter \(h>0\) and modifications of the disks, whence of the domain. They prove Poincaré and Korn inequalities within this context, from which they deduce the appropriate topology to be considered. For the proof of the \(\lim \sup \) property, the authors consider boundary layer problems posed in \(\mathbb{R}^{2+}\), and they use the Kosolov-Muskhelishvili relation between the components of the solution to this local problem. For the proof of the \(\lim \inf \) property, they use a subdifferential inequality.

MSC:

74M15 Contact in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74E20 Granularity
28A80 Fractals
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