Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion

@article{Milton2017BoundsOC,
  title={Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion},
  author={Graeme W. Milton},
  journal={arXiv: Mathematical Physics},
  year={2017}
}
  • G. Milton
  • Published 22 April 2017
  • Physics, Mathematics
  • arXiv: Mathematical Physics
Here we obtain explicit formulae for bounds on the complex electrical polarizability at a given frequency of an inclusion with known volume that follow directly from the quasistatic bounds of Bergman and Milton on the effective complex dielectric constant of a two-phase medium. We also describe how analogous bounds on the orientationally averaged bulk and shear polarizabilities at a given frequency can be obtained from bounds on the effective complex bulk and shear moduli of a two-phase medium… Expand

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References

SHOWING 1-10 OF 71 REFERENCES
Bounds on Herglotz functions and fundamental limits of broadband passive quasistatic cloaking
Using a sum rule, we derive new bounds on Herglotz functions that generalize those given in Bernland et al. [J. Phys. A: Math. Theor. 44(14), 145205 (2011)] and Gustafsson and Sjoberg [New J. Phys.Expand
Inequalities for electric and elastic polarization tensors with applications to random composites
Abstract N ew bounds on the elastic and electric polarization tensors are found for grains of arbitrary shape or connectivity. For a grain shape specified by the characteristic function χ(x), theExpand
The dielectric constant of a composite material—A problem in classical physics
Abstract The problem of calculating the effective dielectric constant of a composite material ee is analyzed by separating out the dependence of ee on the microscopic geometry. A set ofExpand
Bounds on the transport and optical properties of a two‐component composite material
An infinite set of bounds on the effective permittivity ee of two‐component composite materials is derived. All the bounds can be expressed in terms of a single function g. Analogous bounds apply toExpand
On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus
  • L. Gibiansky, G. Milton
  • Mathematics
  • Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1993
The dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in fixed proportions is considered in the frequency range where the acoustic wavelength is much larger than theExpand
Multicomponent composites, electrical networks and new types of continued fraction I
The development of bounds on the complex effective conductivity tensor σ* (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of aExpand
Properties of a periodically stratified acoustic half‐space and its relation to a Biot fluid
The problem of the reflection of acoustic waves from a periodically layered acoustic half‐space is solved exactly at all frequencies. Pass and stop bands and the associated complex slowness surfacesExpand
Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization
Abstract Wave propagation in periodic elastic composites whose phases may have not only highly contrasting but possibly also (in particular) highly anisotropic stiffnesses and moderately contrastingExpand
Elastostatic resonances—a new approach to the calculation of the effective elastic constants of composites☆
Abstract A new method is presented for a systematic evaluation of the effective elastic tensor C (e) in a two-component composite. Both C (e) and local strain field are expanded in terms of aExpand
Variational bounds on the effective moduli of anisotropic composites
Abstract The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ∗ of an anisotropic composite madeExpand
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