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

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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.

Recap[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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 | edit source]

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[edit | edit source]

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[edit | edit source]

  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 | edit source]

  • [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.