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Waves in composites and metamaterials/Fading memory and waves in layered media

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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.

Viscoelastic MaterialsEdit

In the previous lecture, we discussed viscoelastic materials and wondered why the Maxwell model works even though the effective Young's modulus   for such materials is analytic in the entire complex plane (except for a few isolated points).

Recall that the Maxwell model (see Figure 1) predicts that the frequency dependent Young's modulus of the system is given by

 
 
Figure 1. A generalized Maxwell model for viscoelasticity.

This implies that the function   is analytic in the entire imaginary   plane except for poles at  . On the other hand, for the frequency dependent metamaterials that we have discussed earlier, the effective modulus is generally analytic only in the upper half   plane (see Figure~2). Also, for such materials,  , where   indicates the complex conjugate. Note that we do not consider the mass when we derive the modulus of the Maxwell model. The relation between viscoelastic models of the Maxwell type and general frequency dependent materials continues to be an open question.

 
Figure 2. Poles for a general frequency-dependent material vs. poles for a generalized Maxwell model.

A justification of the Maxwell model can be provided by considering the behavior of viscoelastic materials (Christ03). Consider an experiment where a bar of viscoelastic material of length   is deformed by a fixed amount. We want to see how the stress changes with time. Recall, that if the bar is extended by an amount   where   at one end of the bar, then the one-dimensional strain is defined as

 

Therefore, the displacement in the bar can be expressed in terms of the strain as

 

Also, if   is the applied force on the bar and   is its cross-sectional area, then the stress is given by

 

Let us now apply a strain to the bar at time   and hold the strain fixed. Due to the initial application of the strain, the stress reaches a value   and then relaxes at time increase (due to the relaxation of polymer chains for instance). Figure 3 shows a schematic of this situation.

 
Figure 3. One-dimensional stress relaxation of a viscoelastic material.

If the strain is applied by the superposition of a two step strains as shown in the figure, we have

 

The stress is then given by

 

If the strain is applied by a series of infinitesimal steps, then we get a more general form for the stress:

 

where the integral should be interpreted in the distributional sense. Integrating by parts (and assuming that   at  ), we get

 

Now,   clearly depends on past values of  . We expect   should have a stronger dependence on   in the recent past than in the distant past. More precisely, the dependence should decrease monotonically as   increases. This implies that   should decrease at   increases, i.e.,

 

This is the assumption of fading memmory.

From equation (1) the rate of change of   is given by

 

Again, we expect   to have a stronger dependence on   in the recent past than in the far past, i.e,.

 

Now,

 

Such functions are said to be completely monotonic. An example is

 

More generally,

 

is completely monotonic if   for all   and  . The function   is called the {\bf relaxation spectrum.}

Conversely, any completely monotonic function can be written in this form (Bernstein28).

Specifically, if

 

then

 

Therefore,

 

Let  . Then

 

Define

 

Then we have

 

If we define

 

we get

 

Now, let   be a completely monotonic function of the form

 

Then from equation (2) we get

 

Assume that   has a very small poistive imaginary part (which implies that   increases very slowly as   goes to  ). Then

 

or,

 

This is the generalized Maxwell model.

This brings up the question: Is the assumption of fading memory always correct?

Recall the model of the Helmholtz resonator shown in Figure~4.

 
Figure 4. A model of the Helmholtz resonator.
[hb]

If we apply a strain in the form of a step function to this model, the resulting stress response is not a monotonically decreasing function of time. Rather if oscillates around a certain value and may damp out over time. A similar oscillatory behavior is expected in other spring-mass systems and   will, in general, not be monotonic.

A short interlude: Maxwell's equations in Elasticity FormEdit

In this section, we discuss how Maxwell's equation can be reduced to the form of the elasticity equations. Recall that, at a fixed  , Maxwell's equation take the form

 

Therefore,

 

Recall that, in index notation and using the summation convention, we have

 

where   is the permutation tensor defined as

 

Therefore,

 

or,

 

This is very similar to the elasticity equation

 

The permittivity is similar to a negative density and the electric field is similar to the displacement. The equations also hint at a tensorial density. However, continuity conditions are different for the two equations, i.e., at an interface,   is continuous while only the tangential component of   is continuous. Also, the tensor   has different symmetries for the two situations. Interestingly, for Maxwell's equations

 

Waves in Layered MediaEdit

A detail exposition of waves in layer media can be found in Chew95. In this section we examine a few features of electromagnetic waves in layered media.

Assume that the permittivity and permeability are scalars and are locally isotropic though not globally so. Then we may write

 

The TE (transverse electric field) equations are given by

 

where   represents the two-dimensional gradient operator.

Multiplying (3) by  , we have

 

Equation (4) admits solutions of the form

 

and equation (4) then becomes an ODE:

 

The quantity

 

can be less than zero, implying that   may be complex. Also, at the boundary, both   and   must be continuous.

Similarly, for TM (transverse magnetic) waves, we have

 

and the ODE is

 

ReferencesEdit

  • S. Bernstein. Sur les fonctions absolument monotones. Acta Mathematica, 52:1--66, 1928.
  • W. C. Chew. Waves and field in inhomogeneous media. IEEE Press, New York, 1995.
  • R. M. Christensen. Theory of viscoelasticity: 2nd Edition. Courier Dover Publications, London, 2003.