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

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

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,




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,




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


If we choose the   s so that




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

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,




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






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




If we now take a volume average, we get


Also, from the definition of   we have






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




Linear Elastic laminates edit

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)




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

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

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

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

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






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,




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

Footnotes edit

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

References edit

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