Waves in composites and metamaterials/Duality relations and phase interchange identity in laminates

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

Duality Relations in Two Dimensions edit

Instead of taking the direct route of the previous lecture, we can determine the effective properties of composites using duality relations.[1]

Let us look at the quasistatic Maxwell's equations first. These equations can be written as


where   and   are periodic.

Let us define the effective permittivity of the medium ( ) using


where   and   denote volume averages, i.e.,


where   is the volume of the region  .

In two dimensions, we have


since  ,  , and   (or constant).

Now, define


where   is the orthogonal tensor that indicates a 90  rotation about the   axis, i.e.,


Therefore, in a rectangular Cartesian basis ( ), we have


Hence, using (3),


and, using (1) ,


Therefore the dual field   and   represent a divergence free electric field (i.e., there are no sources or sinks). Hence the field is the gradient of some potential which has zero curl.

Also, assuming that the permittivity   is invertible, we have from equations (4) and (1) 




we then have


If we assume that   is symmetric, we have (in two dimensions) with respect to the basis ( )




The dual system of equations is then given by




So   and   solve Maxwell's equations for electricity in a dual medium of permittivity  .

Effective permittivity of dual medium edit

The question that arises at this stage is: { what is the effective permittivity ( ) of the dual material in terms of the effective permittivity of the original material ( )?}

Taking volume averages of equations (6)  and (6)  we get


Recall from equation (2) that




As before, defining




Therefore the relation between the effective permittivity of the original and the dual material is


Application to a two-dimensional polycrystal edit

Let us consider a two-dimensional polycrystal as shown in Figure 1. The lattice vectors in each crystal are oriented in a random manner. However, for each crystal, the lattice vector can be determined by a piecewise constant rotation from a reference configuration.

Figure 1. Polycrystal with randomly oriented lattice vectors.

Therefore, for each crystal


where the rotation field


determines the local orientation of the crystal at each point and the rotations   are piecewise constant in each crystal.

If we now consider a medium that is dual to the polycrystal in the sense of equation (6), then the permittivity of the dual medium is given by








Let us choose the constant   such that




Recall that




we have




But we also have




In particular, if the polycrystal is isotropic, i.e.,  , then we have ({\Red Show this!})


In this case only the root with the positive imaginary part is the correct solution unless   is real in which case only the positive root is correct.

Application to a 2-D composite of two isotropic phases edit

Consider the composite of two isotropic phases shown in Figure 2. Define an indicator function

Figure 2. Composite of two isotropic phases.

Then, since the phases are isotropic, we can write


The dual material is defined as one having a permittivity given by


We can write the above as


or alternatively,


If we choose  , we get


Note that the phases are interchanged in the dual material!

Now, recall that the effective permittivities of the original and the dual material are related by


We can use this relation to find the effective permittivities of materials that are invariant with respect to interchange of phases. Examples of such materials are checkerboard material and random polycrystals where each crystal has an equal probability of being of phase 1 or phase 2.

For such a phase interchange invariant material, the effective permittivity of the original material is equal to that of the dual material, i.e.,


If the composite material is isotropic, i.e.,  , then




This is an useful result that can be used to test numerical codes.

Paradox edit

If   and   then both materials are lossless. \footnote{\Red Need to add section here showing that energy dissipation is proportional to  .} But  . So the composite dissipates energy into heat. But where?

To see this we should take   and look at the limit where  . In this limit, the fields lose their square integrability at the corners (in a checkerboard). So an enormous amount of heat per unit volume is dissipated in the vicinity of each corner.

Extensions to 2-D elasticity edit

Duality and phase interchange relations for elasticity were first derived by Berdichevski~Berdi83. In that work, an exact formula for the shear modulus of a checkerboard material (with two incompressible phases) was derived. Further extensions and details of can be found in Sections 3.5, 3.6, 3.7 and 4.7 in Milton02.

We can apply duality transformations to incompressible media or media where the bulk modulus   is equal in both phases and the shear moduli of the two phases are   and  . For example, if we have a phase interchange invariant composite that is isotropic and two-dimensional (such as a checkerboard or a cell material), then the effective elastic moduli are given by


In the case where   we have  .

The Effective Tensors of Laminate Materials edit

In this section we will discuss the method of Backus~Backus62. Similar approaches have also been used by Postma~Postma55 and Tartar~Tartar76.

Consider a material laminated in the   direction as shown in Figure 3.

Figure 3. A laminate with direction of lamination  .

To find the relation between   and   we cannot average the constitutive relation




unless   is constant or   is constant.

However, there are fields which are constant in certain directions and those can be used to simplify things. Since the tangential components (parallel to the layers) of the electric field ( ) are piecewise constant and continuous across the interfaces between the layers, these tangential components must be constant, i.e.,   and   are constant in the laminate. Similarly, the continuity of the normal electric displacement field ( ) across the interfaces and the fact that this field is constant in each layer implies that the component   is constant in the laminate.

Let us rewrite the constitutive relation in matrix form (with respect to the rectangular Cartesian basis ( )) so that constant fields appear on the right hand side, i.e.,




Note that the constant fields are   and  . We want to rewrite equation (7) so that these constant fields appear on the right hand side.

From the first row of (7) we get




From the second row of (7) we get


Substitution of (9) into (10) gives


Collecting (9) and (11) gives


where the negative signs on   and   are used to make sure that the signs of the off diagonal terms are identical.



Then we have,


Since the vector on the right hand side is constant, an volume average of (12) gives


Let us define the effective permittivity of the laminate   via


Since the tangential components of   are constant in the laminate, the average values   and   must also be constant. Similarly, the average value   must be constant. Therefore we can use the same arguments as we used before to write the effective constitutive relation in the form




and   has been decomposed in exactly the same manner as   (see equation (8).

If we compare equations (13) and (14) we get


Thus we have a formula for determining the effective permittivity of the laminate.

Footnotes edit

  1. The following discussion is based on Milton02.

References edit

  • G. E. Backus. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res., 67:4427--4440, 1962.
  • V. L. Berdichevski. Variational Principles in the Mechanics of Continuum Media. Nauka, Moscow, 1983.
  • G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.
  • G. W. Postma. Wave propagation in a stratified medium. Geophysics, 20:780--806, 1955.
  • L. Tartar. Estimation de coefficients homogeneises. In R.~Glowinski and J.~L. Lions, editors, Computer Methods in Applied Sciences and Engineering, pages 136--212. Springer-Verlag, Berlin, 1976.