The equivalent medium for the elastic scattering by many small rigid bodies and applications

TL;DR

This paper demonstrates that a large collection of small rigid bodies can be modeled as an equivalent elastic medium, enabling applications like cloaking and material design by controlling their distribution and shapes.

Contribution

It introduces a rigorous method to approximate the elastic scattering by many small bodies with an equivalent medium, accounting for arbitrary shapes and distributions.

Findings

Far-field patterns approximate those of an equivalent medium as epsilon approaches zero.

The equivalent medium can be isotropic or anisotropic depending on the shapes of the small bodies.

Strategic removal of small bodies can achieve cloaking effects at fixed frequencies.

Abstract

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=\epsilon B_m+z_m$, of arbitrary shapes with $ 0<\textcolor{black}{\epsilon}<<1$ and with constant Lam\'e coefficients $\lambda$ and $\mu$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $\Omega$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}{\epsilon}):=O(\textcolor{black}{\epsilon}^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}{\epsilon})\approx \textcolor{black}{\epsilon}^{t}$ with $t$ in some appropriate range, as $\textcolor{black}{\epsilon} \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $\omega^2\rho I_3-(K+1)\mathbf{C}_0 $ where $\rho$ is the density of the background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients $K$ and $\mathbf{C}_0$ (which have supports in $\overline{\Omega}$) modeling the local distribution of the small bodies and their geometries, respectively. In particular, if the distributed bodies have a uniform spherical shape then the equivalent medium is isotropic while for general shapes it might be anisotropic (i.e. $\mathbf{C}_0$ might be a matrix). In addition, if the background density $\rho$ is variable in $\Omega$ and $\rho =1$ in $\mathbb{R}^3\setminus{\overline{\Omega}}$, then if we remove from $\Omega$ appropriately distributed small bodies then the equivalent medium will be equal to $\omega^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $\Omega$ characterized by $\rho$ is approximately cloaked at the given and fixed frequency $\omega$.