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Effective elastic wave transmission through a periodically voided interface

Markus Gahn, Tanja Lochner, Malte A. Peter

Abstract

Effective interface conditions for a periodically voided thin layer separating two homogeneous bulk regions are derived for the elastic wave equation by taking the simultaneous limit of vanishing layer periodicity and layer thickness. The limit problems are obtained using the unfolding method for thin perforated domains. We consider three different scalings of the material parameters in the layer that characterise its stiffness, each leading to a distinct type of interface condition and requiring the solution of scaling-dependent cell problems. Depending on the scaling, the resulting effective model yields either a membrane equation or a Kirchhoff-Love plate equation. In the critical regime of reduced stiffness, the interface equation additionally depends on the microscopic variable. By selecting appropriate cell problems, this equation can be reformulated as an effective interface condition between the bulk domains.

Effective elastic wave transmission through a periodically voided interface

Abstract

Effective interface conditions for a periodically voided thin layer separating two homogeneous bulk regions are derived for the elastic wave equation by taking the simultaneous limit of vanishing layer periodicity and layer thickness. The limit problems are obtained using the unfolding method for thin perforated domains. We consider three different scalings of the material parameters in the layer that characterise its stiffness, each leading to a distinct type of interface condition and requiring the solution of scaling-dependent cell problems. Depending on the scaling, the resulting effective model yields either a membrane equation or a Kirchhoff-Love plate equation. In the critical regime of reduced stiffness, the interface equation additionally depends on the microscopic variable. By selecting appropriate cell problems, this equation can be reformulated as an effective interface condition between the bulk domains.

Paper Structure

This paper contains 15 sections, 19 theorems, 143 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Statement of the problem
  4. Main results and the macroscopic model
  5. The case $\gamma = 1$
  6. The case $\gamma = -1$
  7. The case $\gamma = -3$
  8. Existence and uniform boundedness
  9. Compactness results
  10. Identification of the limit problems
  11. The case $\gamma = 1$
  12. Effective interface conditions
  13. The case $\gamma = -1$
  14. The case $\gamma = -3$
  15. Weak solutions for wave equation in the reference cell

Key Result

Lemma 1

There exists a constant $C$ independent of ${\varepsilon}$ such that for every $u\in H^1_{\Gamma_\mathrm{D}}(\Omega_\varepsilon)^3$ Moreover, the following estimate holds:

Figures (3)

  • Figure 1: Illustration of the domain $\Omega$
  • Figure 2: Subdomains of $\Omega$ depending on ${\varepsilon}$
  • Figure 3: Reference cell $Y$

Theorems & Definitions (40)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Lemma 1: Korn inequality I
  • Lemma 2: Korn inequality II
  • Theorem 4.1
  • proof
  • Remark 4
  • ...and 30 more