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- Title
The equivalent medium for the elastic scattering by many small rigid bodies and applications.
- Authors
AL-MUSALLAM, FADHEL; CHALLA, DURGA PRASAD; SINI, MOURAD
- Abstract
We deal with the elastic scattering by a large number M of rigid bodies, Dm := ∊Bm + zm, of arbitrary shapes with 0 < ∊ ⪡ 1 and with constant Lamé coefficients γ and μ. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region Ω of R3 where their number is M := M(∊) := O(∊ -1) and the minimum distance between them is d := d(∊) ≈ ∊t with t in some appropriate range, as ∊ → 0, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by ω2ρI3-(K +1)C0 where ρ is the density of the background medium (with I3 as the unit matrix) and (K +1)C0 is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients K and C0 (which have supports in Ω) modelling 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. C0 might be a matrix). In addition, if the background density ρ is variable in Ω and ρ = 1 in R3 \ Ω, then if we remove from Ω appropriately distributed small bodies then the equivalent medium will be equal to ω2I3 in R3, i.e. the obstacle Ω characterized by ρ is approximately cloaked at the given and fixed frequency ω.
- Subjects
ELASTIC wave scattering; LIPSCHITZ spaces; GEOMETRIC rigidity; LAME'S functions; MASS density gradients; NAVIER-Stokes equations
- Publication
IMA Journal of Applied Mathematics, 2016, Vol 81, Issue 6, p1020
- ISSN
0272-4960
- Publication type
Article
- DOI
10.1093/imamat/hxw042