An Introduction to Metamaterials and Waves in Composites by Biswajit Banerjee

By Biswajit Banerjee

Requiring no complicated wisdom of wave propagation, An creation to Metamaterials and Waves in Composites specializes in theoretical points of metamaterials, periodic composites, and layered composites. The publication offers newcomers a platform from which they could begin exploring the topic in additional detail.

After introducing thoughts concerning elasticity, acoustics, and electrodynamics in media, the textual content provides airplane wave suggestions to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the airplane wave growth of assets in addition to scattering from curved interfaces, particularly spheres and cylinders. the writer then covers electrodynamic, acoustic, and elastodynamic metamaterials. He additionally describes examples of modifications, elements of acoustic cloaking, and purposes of pentamode fabrics to acoustic cloaking. With a spotlight on periodic composites, the textual content makes use of the Bloch-Floquet theorem to discover the potent habit of composites within the quasistatic restrict, provides the quasistatic equations of elastodynamic and electromagnetic waves, and investigates Brillouin zones and band gaps in periodic constructions. the ultimate bankruptcy discusses wave propagation in easily various layered media, anisotropic density of a periodic layered medium, and quasistatic homogenization of laminates.

This booklet presents a release pad for learn into elastic and acoustic metamaterials. some of the principles provided have not begun to be discovered experimentally―the e-book encourages readers to discover those principles and convey them to technological maturity.

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33) to get ∂p ∂p + κ0 ∇ · v = 0 ⇔ + ρ0 c20 ∇ · v = 0 . 33) as p(x,t) = − Z ∞ −∞ κ(x,t − t ) ∇ · u(x,t ) dt . 34) Causality implies that the pressure at time t depends only on velocities from previous times and hence κ(x, τ) = 0 if τ = t − t < 0. We can write the above in terms of velocity as Z ∞ ∂p(x,t) =− κ(x,t − t ) ∇ · v(x,t ) dt . , ∇ · σ + ρ(x) b(x) = ρ(x) Dv ; Dt Dv ∂v ∇v)T . 36) as ∇ p + ∇ · s + ρ(x) b(x) = ρ(x) −∇ ∂v ∇v)T . , the deviatoric stress depends on the symmetric part of the velocity gradient which is defined as ε˙ := 1 2 ∇ v + ∇ vT .

The free current density (J f ) arises only from conduction currents arising from the response of the medium and not from beams of charged particles. 2. In the far distant past (t = −∞) all fields are zero. 3. The media are at rest. Define Z t D(x,t) := D(x,t) + −∞ J f (x, τ) dτ . 81) and using Maxwell’s equation connecting the magnetic field to the total current, we have ∂D ∂D = +Jf = ∇ ×H . 67) and Maxwell’s equation relating the electric displacement to the free charge density, we get ∇ ·D = ∇ ·D+ Z t −∞ ∇ · J f dτ = ∇ · D − Z t ∂ρ f −∞ ∂t dτ = ∇ · D − ρ f = 0 .

55), the average acoustic power density in a cycle of oscillation will be ω Z π/ω ∇ p · v + p∇ ∇ · v) dt (∇ Pden = 2π −π/ω = ω κi ( p2 + p2r ) − ρi (vr · vr + vi · vi ) . 2 κ2i + κ2r i This quadratic form will be non-negative for all choices of p and v if and only if κi = Im(κ) > 0 and ρi = Im(ρ) < 0 for all real ω > 0. Note that the quadratic form does not contain the real part of ρ. Since the work done in a cycle should be zero in the absence of dissipation, this implies that the imaginary parts of the bulk modulus and the density are connected to energy dissipation.

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