By Biswajit Banerjee
Requiring no complex wisdom of wave propagation, An creation to Metamaterials and Waves in Composites specializes in theoretical elements of metamaterials, periodic composites, and layered composites. The ebook offers beginners a platform from which they could commence exploring the topic in additional detail.
After introducing thoughts concerning elasticity, acoustics, and electrodynamics in media, the textual content offers aircraft wave strategies to the equations that describe elastic, acoustic, and electromagnetic waves. It examines the aircraft wave enlargement of assets in addition to scattering from curved interfaces, in particular spheres and cylinders. the writer then covers electrodynamic, acoustic, and elastodynamic metamaterials. He additionally describes examples of variations, features 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 buildings. 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 offers a release pad for examine into elastic and acoustic metamaterials. some of the principles awarded have not begun to be learned experimentally―the booklet encourages readers to discover those rules and convey them to technological maturity.
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Additional resources for An Introduction to Metamaterials and Waves in Composites
The preceding potentials are well known. However, one can go one step further and define superpotentials (see, for example, Bowman et al. (1969)). The most widely used superpotentials are the electric and magnetic Hertz vector potentials Πe and Πm (also known as polarization potentials). In terms of these potentials, the fields E and H can be expressed as Πm ∂Π ∂t Πe ∂Π H = ∇ × ∇ × Πm + ε ∇ × . 71) one sees that the superpotentials lead to symmetric representations of E and H unlike when standard vector and scalar potentials are used.
The deviatoric stress depends on the symmetric part of the velocity gradient which is defined as ε˙ := 1 2 ∇ v + ∇ vT . 37) Then, assuming that the fluid is local in space, we have Z s(x,t) = Kε (x,t − t ) : ε˙ (x,t ) dt . 38) If the fluid is also local in time and isotropic, we have ∇ · v) 1 + µ(x) ∇ v + ∇ vT . s(x,t) = λ(x) (∇ where µ is the shear viscosity and µb = λ + 2/3µ is the bulk viscosity. 10), the balance of momentum for a homogeneous medium becomes ∇ p + (λ + 2µ) ∇ (∇ ∇ · v) − µ ∇ × (∇ ∇ × v) + ρ b(x) = ρ −∇ ∂v ∇v)T .
4 Show that for an isotropic, homogeneous, linear elastic material with stiffness tensor C = λ 1 ⊗ 1 + 2µ I the divergence of the stress can be expressed as ∇ · u) + µ ∇ · (∇ ∇ uT ) . 5 Verify that the volumetric strain e is given by ∇ · u and that the infinitesimal rotation vector is given by ∇ × u. 6 Show that, in the absence of a body force, the displacement potentials φ and ψ satisfy the wave equations ∂2 φ = c2p ∇2 φ ; ∂t 2 ∂2 ψ ∇ψ )T . 7 Show, using spatial curvilinear coordinates, that the gradient of a vector field v can be expressed in spherical coordinates as 1 ∂vr vθ 1 ∂vr vφ ∂vr − − ∂r r sin φ ∂θ r r ∂φ r ∂v cot φ 1 1 v r ∂vθ ∂vθ θ ∇v = + + v φ ∂r r sin φ ∂θ r r r ∂φ ∂v 1 ∂vφ cot φ 1 ∂vφ vr φ − vθ + ∂r r sin φ ∂θ r r ∂φ r where x1 = θ1 cos θ2 sin θ3 , x2 = θ1 sin θ2 sin θ3 , x3 = θ1 cos θ3 , and (θ1 , θ2 , θ3 ) ≡ (r, θ, φ).
An Introduction to Metamaterials and Waves in Composites by Biswajit Banerjee