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)
Defining
we then have
If we assume that is symmetric, we have (in two dimensions)
with respect to the basis ()
Then
The dual system of equations is then given by
where
So and solve Maxwell's equations for electricity in a
dual medium of permittivity .
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
Therefore,
As before, defining
gives
Therefore the relation between the effective permittivity of the original
and the dual material is
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.
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
Now
since
Therefore,
Let us choose the constant such that
Then
Recall that
If
we have
Hence
But we also have
Therefore,
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
Consider the composite of two isotropic phases shown in Figure 2. Define an indicator function
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
Hence,
This is an useful result that can be used to test numerical codes.
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.
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 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.
To find the relation between and we cannot
average the constitutive relation
because
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.,
where
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
or,
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.
Define
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
where
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.
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.