Homogenization via unfolding in periodic layer with contact

TL;DR

This paper analyzes the asymptotic behavior of elasticity problems in a layered medium with micro-contact and cracks, using unfolding methods to derive a limit problem as the layer's periodicity vanishes.

Contribution

It introduces a homogenization approach for elasticity with micro-contact and cracks in a periodic layer, utilizing unfolding techniques and modified Korn inequalities.

Findings

Derived the limit transmission problem as the periodicity parameter tends to zero.

Established a modified Korn inequality for the periodic layer.

Applied unfolding method to analyze the micro-contact and crack effects.

Abstract

In this work we consider the elasticity problem for two domains separated by a heterogeneous layer. The layer has an $\varepsilon-$periodic structure, $\varepsilon\ll1$, including a multiple micro-contact between the structural components. The components are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates modification of the Korn inequality for the $\varepsilon-$dependent periodic layer is performed. An asymptotic analysis with respect to $\varepsilon \to 0$ is provided and the limit problem is obtained, which consists of the elasticity problem together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior.