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

Abstract

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=εB_m+z_m$, of arbitrary shapes with $ 0<\textcolor{black}ε<<1$ and with constant Lamé coefficients $λ$ and $μ$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $Ω$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}ε):=O(\textcolor{black}ε^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}ε)\approx \textcolor{black}ε^{t}$ with $t$ in some appropriate range, as $\textcolor{black}ε \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $ω^2ρI_3-(K+1)\mathbf{C}_0 $ where $ρ$ 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 $\overlineΩ$) 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 $ρ$ is variable in $Ω$ and $ρ=1$ in $\mathbb{R}^3\setminus{\overlineΩ}$, then if we remove from $Ω$ appropriately distributed small bodies then the equivalent medium will be equal to $ω^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $Ω$ characterized by $ρ$ is approximately cloaked at the given and fixed frequency $ω$.