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

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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

 

or,

 

where

 

We also showed that

 

where

 

Therefore, we got the relation

 

This provides the relations

 

where

 

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

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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

 

where

 

and

 
 

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

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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

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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

 

Therefore,

 

Let us now introduce a polarization field

 

where   is an arbitrary constant.

The volume averaged polarization field is given by

 

Define

 

Then,

 

Applying the projection   to (7), we get

 

Using equations (5)  and (6), we have

 

From the definitions (9) we can then write

 

Define

 

Then we have

 

or

 

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

 

or,

 

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

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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

 

Therefore,

 

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

 

or,

 

Since

 

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

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  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

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  • [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.