Waves in composites and metamaterials/Backus formula for laminates and rank-1 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.

Recap edit

Recall the material laminated in the   direction as shown in Figure 1. [1]

Figure 1. A laminate with direction of lamination  .

We showed that we could write






We also showed that




Therefore, we got the relation


This provides the relations




When the off diagonal elements vanish, we get


The harmonic average corresponds to a situation in which each layer may be thought of as a capacitor in series while the arithmetic average corresponds to a situation where the capacitors are in parallel.

Effective Elastic Properties of a Layered Medium edit

In this section we use the approach of arranging constant fields to find the effective elastic properties of a layered medium in which each layer is anisotropic. The approach used is that of Backus (Backus62).

Consider the layered medium shown in Figure 2. In this case the displacement field is continuous across the interfaces between the layers.

Figure 2. An elastic layered medium with direction of lamination  .

The strain field ( ) is given by


In a Cartesian basis ( ) with coordinates ( ) we can write


Therefore the strain components   are also continuous across the interfaces. Moreover, these components of strain are also constant in each layer. Since a piecewise constant field that is also continuous must be constant, the strain components   must be constant throughout the laminate.

The tractions (normal components of the stress) at each interface are given by


where   is the stress and   is the normal at the interface. Now the tractions must be continuous at the interfaces. Since the normal components of the stress are piecewise constant in each layer, this implies that the normal components of the stress must also be constant throughout the laminate.

We have chosen   such that  . Therefore the stress components   must be constant.

Recall that the constitutive relation for an anisotropic elastic material is given by


Following the approach that we used for the permittivity, we now write the constitutive relation in the form






From the major symmetry of  , we see that  . Also,   and   are symmetric.

Writing the first row out, we get


From the second row we get


Substituting the expression for   from the first row, we get


Collecting (1) and (2) we get


Taking a volume average gives


If the effective stiffness of the material is given by


we can also show that


Comparing (3) and (4) we can show that


Isotropic layers edit

If the material in each layer is isotropic, then the constitutive relation is


where   is the Lame modulus and   is the shear modulus. In that case the effective properties of the laminate are


Laminates with Arbitrary Direction of Lamination n edit

So far we have dealt with laminates with a single direction of lamination that was oriented with the   axis. In this section we generalize our approach to deal with laminates with a normal   which is not parallel to the   axis.

Recall that the normal component of  , i.e.,  , is constant and the tangential components of   are constant throughout the entire laminate (if there is only one direction of lamination).

Let us introduce the second-order tensor basis


These are useful because




Let us now introduce a polarization field


where   is an arbitrary constant.

The volume averaged polarization field is given by






Applying the projection   to (7), we get


Using equations (5)  and (6), we have


From the definitions (9) we can then write




Then we have




Also, form equations (10) and (8) we have




Inverting (11) and (12) we have


Also, taking the volume average of (13) , we have


Therefore, comparing (13)  and (14) and invoking the arbitrary nature  , we have


This relation provides us with a means of computing the effective permittivity of a layered medium oriented at a random angle (given by the normal  ) with respect to the coordinate basis.

Case 1: Simple or Rank-1 Laminate edit

Consider the Rank-1 laminate shown in Figure 3. The layers have permittivities alternating between   and  . The volume fraction of phase   is   while that of phase   is   such that  .

Figure 3. A Rank-1 laminate.

Recall that


Let us take the limit as  . Since   in phase  , we have




Hence, right hand side of


reduces to an average only over phase  . If we define


we get


Taking the inverse of both sides of (15) gives






we then have


This is the formula of Tartar, Murat, Lurie, and Cherkaev and can be shown to be equivalent to the Backus formula.

Footnotes edit

  1. The discussion in this lecture is based on Milton02. Please consult that book for more details and references. The method of Backus (Backus62) (see also Postma55 and Tartar76) has been used.

References edit

  • [Backus62]     G. E. Backus. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res., 67:4427--4440, 1962.
  • [Milton02]     G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.
  • [Postma55]     G. W. Postma. Wave propagation in a stratified medium. Geophysics, 20:780--806, 1955.
  • [Tartar76]     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.