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… 

Figures from this paper

On the Range of Complex Effective Permittivities of Isotropic Two-Phase Composites and Related Problems
Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to
Tight Bounds on the Effective Complex Permittivity of Isotropic Composites and Related Problems
Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to
A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems
We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for
A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents, their rapid computation; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint
We obtain rapidly convergent series expansions of operators taking the form A = Γ1BΓ1 where Γ1(k) is a projection that acts locally in Fourier space and B(x) is an operator that acts locally in real
Approximating the Effective Tensor as a Function of the Component Tensors in Two-Dimensional Composites of Two Anisotropic Phases
  • G. Milton
  • Mathematics, Computer Science
    SIAM J. Math. Anal.
  • 2018
TLDR
Using the approximations for the relevant operators one can also directly obtain approxIMations, with the same geometry, for the effective tensors of coupled field problems, including elasticity, piezoelectricity, and thermoelectrics.
A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint
We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an
A new route to finding bounds on the generalized spectrum of many physical operators
  • G. Milton
  • Physics, Mathematics
    Journal of Mathematical Physics
  • 2018
Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a
Some open problems in the theory of composites
  • G. Milton
  • Mathematics, Medicine
    Philosophical Transactions of the Royal Society A
  • 2021
A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase composites with general geometries have the
A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better
A unifying perspective on linear continuum equations prevalent in science. Part I: Canonical forms for static, steady, and quasistatic equations
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better

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.
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), the
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 of
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 to
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 the
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 a
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 surfaces
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 contrasting
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 a
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 made
...
1
2
3
4
5
...