Graeme W. Milton

On cloaking for elasticity and physical equations with a transformation invariant form

In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density is anisotropic and additional terms appear which couple the stress not only with the strain but also with the velocity, and the momentum gets coupled not only with the velocity but also with the strain. These are a special case of equations which describe the elastodynamic response of composite materials, and which it has been argued should apply to any material which has microstructure below the scale of continuum modelling. If composites could be designed with the required moduli then it could be possible to design elastic cloaking devices where an object is cloaked from elastic waves of a given frequency. To an outside observer it would appear as though the waves were propagating in a homogeneous medium, with the object and surrounding cloaking shell invisible. Other new elastodynamic equations also retain their form under curvilinear transformations. The question is raised as to whether all equations of microstructured continua have a form which is invariant under curvilinear space or space-time coordinate transformations. We show that the non-local bianisotropic electrodynamic equations have this invariance under space-time transformations and that the standard non-local, time-harmonic, electromagnetic equations are invariant under space transformations.

On the cloaking effects associated with anomalous localized resonance

Regions of anomalous localized resonance, such as occurring near superlenses, are shown to lead to cloaking effects. This occurs when the resonant field generated by a polarizable line or point dipole acts back on the polarizable line or point dipole and effectively cancels the field acting on it from outside sources. Cloaking is proved in the quasistatic limit for finite collections of polarizable line dipoles that all lie within a specific distance from a coated cylinder having a shell permittivity where is the permittivity of the surrounding matrix, and is the core permittivity. Cloaking is also shown to extend to the Veselago superlens outside the quasistatic regime: a polarizable line dipole located less than a distance d/2 from the lens, where d is the thickness of the lens, will be cloaked due to the presence of a resonant field in front of the lens. Also a polarizable point dipole near a slab lens will be cloaked in the quasistatic limit.

Bounds on the complex permittivity of a two‐component composite material

A generalization of the bounds obtained by Wiener, Hashin and Shtrikman, and others is derived for complex permittivities. Provided the scale of inhomogeneities in the composite is sufficiently small compared with the wavelength of the applied radiation, the permittivity of the composite is found to lie within a simply constructed region of complex plane. The appropriate region depends on what is known about the composite material. We show that in many cases the region is the most restrictive which can be given, using limited information about the structure of the composite.

On modifications of Newton's second law and linear continuum elastodynamics

In this paper, we suggest a new perspective, where Newtons second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies. We confirm a finding of Willis that the density of a body at a given frequency of oscillation can be anisotropic. The relation between the force and the acceleration is non-local (but causal) in time. Conversely, for every response function satisfying these properties, and having the appropriate high-frequency limit, there is a model which realizes that response function. In many circumstances, the differences between Newtons second law and the new law are small, but there are circumstances, such as in specially designed composite materials, where the difference is enormous. For bodies which are not seemingly rigid, the continuum equations of elastodynamics govern behaviour and also need to be modified. The modified versions of these equations presented here are a generalization of the equations proposed by Willis to describe elastodynamics in composite materials. It is argued that these new sets of equations may apply to all physical materials, not just composites. The Willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the average-weighted displacement field, where the weighted displacement field may attach zero weight to ‘hidden’ areas in the material where the displacement may be unobservable or not defined. From knowledge of the average-weighted displacement field, one obtains an approximate formula for the ensemble averaged energy density. Two other sets of new equations govern the behaviour when the microstructure has microinertia, i.e. where there are internal spinning masses below the chosen scale of continuum modelling. In the first set, the average displacement field is assumed to be observable, while in the second set an average-weighted displacement field is assumed to be observable.

Composite materials with poisson's ratios close to — 1

A family of two-dimensional, two-phase, composite materials with hexagonal symmetry is found with Poissons ratios arbitrarily close to — 1. Letting k∗, k1,k2 and μ∗,μ1,μ2 denote the bulk and shear moduli of one such composite, stiff inclusion phase and compliant matrix phase, respectively, it is rigorously established that when k1 = K2r and μ1 = μ2r there exists a constant c depending only on k2, μ2 and the geometry such that k∗/μ∗ <c√r for all sufficiently small stiffness ratios r (specifically for r <frcase|1/9). This implies that the Poissons ratio approaches — 1 as r → 0 and in this limit it is conjectured that the material deforms conlbrmally on a macroscopic scale. By introducing additional microstructure on a smaller length scale a second family of composites is obtained with substantially lower Poissons ratios, each satisfying k/μ∗ <crThese two families provided conclusive proof that isotropic materials with negative Poissons ratio exist within the framework of continuum elasticity. It is also shown that elastically isotropic two- and three-dimensional composites with Poissons ratio approaching — 1 as r → 0 can be generated simply by layering the component materials together in different directions on widely separated length scales.

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 from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C ∗ of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ∗, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C ∗ for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.

Nonmagnetic cloak with minimized scattering

In an electromagnetic cloak based on a transformation approach, reduced sets of material properties are generally favored due to their easier implementation in reality, although a seemingly inevitable drawback of undesired scattering exists in such cloaks. Here, the authors suggest the use of high-order transformations to create smooth moduli at the outer boundary of the cloak, therefore completely eliminating the detrimental scattering within the limit of geometric optics. The authors apply this scheme to a nonmagnetic cylindrical cloak and demonstrate that the scattered field is reduced substantially in a cloak with optimal quadratic transformation as compared to its linear counterpart.

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 the other transport properties of the composite, such as the thermal and electrical conductivities and the magnetic permeability. The work also applies to the optical properties of the composite, provided the wavelength is sufficiently large compared with the structure of the composite. In all cases we find ee is confined to a region of the complex plane bounded by arcs of circles. The appropriate region is determined by what is known about the composite and as more information is known the region becomes progressively smaller. We show that in many cases the region is the most restrictive which can be found using only the known information about the composite material.

Bounds on the complex dielectric constant of a composite material

A generalization of the bounds obtained by Wiener and by Hashin and Shtrikman is derived for complex dielectric constants.

Which Elasticity Tensors are Realizable

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.