Waves in composites and metamaterials/Hierarchical laminates and Hilbert space formalism

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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.

Hierarchical Laminates[edit | edit source]

In the previous lecture we found that, for rank-1 laminates, the effective permittivity can be calculated using the formula of Tartar-Murat-Lurie-Cherkaev. In this lecture we extend the ideas used to arrive at that formula to hierarchical laminates. [1]

An example of a hierarchical laminate is shown in Figure 1. The idea of such materials goes back to Maxwell. In the rank-2 laminate shown in the figure there are two length scales which are assumed to be sufficiently separated so that the ideas in the previous lecture can be exploited. There has to be a separation of length scales so that the layer material can be replaced by its effective tensor.

Figure 1. A rank-2 hierarchical laminate.

Recall the Tartar-Murat-Lurie-Cherkaev formula for the effective permittivity of a rank-1 laminate:

By iterating this formula one gets, for a rank- laminate,

where

and is the number of laminates in the hierarchy, is the proportion of phase in a rank- laminate, and is the orientation of the -th laminate.

In particular,

Then,

For a rank-3 laminate, if the normals , , and are three orthogonal vectors, then

If we choose the s so that

then

In this case, equation (1) coincides with the solution for the Hashin sphere assemblage!

This implies that different geometries can have the same .

Is there a formula as simple as the Tartar-Murat-Lurie-Cherkaev formula when ε1 and ε2 are both anisotropic?[edit | edit source]

The answer is yes.

In this case we use an anisotropic reference material and define the polarization as

The volume average of this field is given by

Therefore, the difference between the field and its volume average is

Let us introduce a new matrix defined through its action on a vector , i.e.,

where and projects parallel to . Therefore,

where . Also,

Therefore,

From the definition of we then have

Taking the projection of both sides of equation (3) we get

Now continuity of the normal component of and the piecewise constant nature of the field implies that the normal component of is constant. Therefore,

Hence we have,

Recall from the previous lecture that

Since the conditions in (4) are satisfied with

from the definition of we then have

Now, from equation (2) we have

Plugging this into (8) gives

or,

Define

and note that this quantity is constant throughout the laminate. Therefore we can write

or

If we now take a volume average, we get

Also, from the definition of we have

Therefore,

or,

Comparing equation (9) and (10) and invoking the arbitrariness of , we get

This relation has a simple form and can be used when the phases are anisotropic.

For a simple (rank-1) laminate where , equation (11) reduces to

where

Linear Elastic laminates[edit | edit source]

For elasticity, exactly the same analysis can be applied. In this case we introduce a reference stiffness tensor and define the second order polarization tensor as

where the strain is given by

Following the same process as before, we can show that the effective elastic stiffness of a hierarchical laminate can be determined from the formula

where (the components are in a rectangular Cartesian basis)

and

Note that has the same form as the acoustic tensor.

If is isotropic, i.e.,

where is the Lame modulus and is the shear modulus, simplifies to

Hilbert Space Formulation[edit | edit source]

The methods discussed above can be generalized if we think in terms of a Hilbert space formalism. Recall that our goal is to find a general formula for and .

Let us consider a periodic material with unit cell . We will call such materials -periodic.

Electromagentism[edit | edit source]

Consider the Hilbert space of square-integrable, -periodic, complex vector fields with the inner product

where and are vector fields and denotes the complex conjugate. We can use Parseval's theorem to express the inner product in Fourier space as

where is the phase vector.

The Hilbert space can be decomposed into three orthogonal subspaces.

  1. The subspace of uniform fields, i.e., is independent of , or in Fourier space, unless .
  2. The subspace of zero divergence, zero average value fields, i.e., and , or in Fourier space, and .
  3. The subspace of zero curl, zero average value fields, i.e., and , or in Fourier space, and .

Thus we can write

In Fourier space, we can clearly see that

if we choose from any one of and from a a different subspace. Therefore the three subspaces are orthogonal.

Elasticity[edit | edit source]

Similarly, for elasticity, is the Hilbert space of square-integrable, -periodic, complex valued, symmetric matrix valued fields with inner product

In Fourier space, we have

Again we decompose the space into three orthogonal subspaces , , and where

  1. is the subspace of uniform fields, i.e., is independent of , or in Fourier space, unless .
  2. is the subspace of zero divergence, zero average value fields, i.e., and , or in Fourier space, and .
  3. is the subspace of zero average "strain" fields, i.e., , or in Fourier space, and .

Problem of determining the effective tensor in an abstract setting[edit | edit source]

Let us first consider the problem of determining the effective permittivity. The approach will be to split relevant fields into components that belong to orthogonal subpaces of .

Since , we can split into two parts

where and .

Also, since , we can split into two parts

where and .

The constitutive relation linking and is

where can be thought of as an operator which is local in real space and maps to . Therefore, we can write

The effective permittivity is defined through the relation

Let denote the projection operator that effects the projection of any vector in onto the subspace . This projection is local in Fourier space. We can show that, if

then

where

More generally, if we choose some reference matrix , we can define an operator which is local in Fourier space via the relation

if and only if

In Fourier space,

where

In the next lecture we will derive relations for the effective tensors using these ideas.

Footnotes[edit | edit source]

  1. The discussion in this lecture is based on Milton02. Please consult that book for more details and references.

References[edit | edit source]

  • [Milton02]     G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.