# Waves in composites and metamaterials/Anisotropic mass and generalization

**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.**

## Rigid Bar with Frequency Dependent Mass edit

Recall the a rigid bar with cavities, each containing a spring-mass system with mass and complex spring constant (see Figure 1).

The momentum of the bar ( ) is related to its velocity ( ) by the relation

where is the frequency and is the ** effective mass**
which is given by

where is the mass of the rigid bar~.

Let us now consider a specific model for the springs. A simple model is the one-dimensional Maxwell model shown in Figure 2. In this case instead of a spring with a complex spring constant we have an elastic spring (with real spring constant ) and a dashpot (with a real viscosity ) in series. Let be the displacement of the elastic spring and let be that of the dashpot under the action of a force .

For the elastic spring, we have

For the dashpot, we have

Once again, assuming that the functions , , and can be expressed as harmonic functions, we have

Plugging equations (3) into equations (1) and (2), we get

Recall that the displacement of the sphere of mass inside the cavity is related to the applied force by the relation

Now, . Hence,

or,

So, with the Maxwell model, the effective mass is

or,

This model is remarkably similar to a simple model for the frequency dependent dielectric constant .

### Comparison with a simple model for edit

Consider the following simple model of an electron bound to an atom by a harmonic force under the influence of a slowly varying electric field (a detailed description can in found in Jackson75, Sec. 7.5). A schematic of the situation is shown in Figure 3.

Let the electric field be and assume that it varies slowly in space over the distance that the electron moves, i.e., . Let be the mass of the electron and let be its charge. Let be the frequency of the harmonic force binding the electron to the atom and let be a damping coefficient due to interactions with obstacles.

Then, the equation of motion of the electron is given by

Let

Plugging equations (7) into equation (6), we get

or

Let be the charge density of the electron-atom system, i.e.,

where is the Dirac delta function and is the position of the atom. Then, the dipole moment (the first moment of the charge) contributed by the electron-atom system is given by

or,

Suppose now that instead of a binding frequency for all the atoms and all the electrons in a volume, there are atoms per unit volume with binding frequencies and damping constants and. Then, we can write the polarization as

Recall that the electric displacement is related to the electric field and the polarization by

Therefore, from equations (8) and (9) we get

Comparing equation (5) with equation (10) we observe that they have the same form which implies that the effective permittivity is analogous to the effective mass. This also implies that the electric field is analogous to the velocity. Similarly, the polarization is analogous to the momentum.

### More on frequency dependence of the mass edit

More generally, we get a frequency dependent density if all the constituents do not move in lock step. Lock step motion almost never occurs because there are thermal vibrations at the microscale. There are also many macroscopic situations in which lock step motion does not occur. Consider the example of a porous rock containing some water (see Figure 4). Both the rock grains and the water are connected, though this is not obvious from the figure

In this case, the water will move with a different frequency that the rock and the density of the composite will be dependent on the frequency.

At a molecular level, we can have a crystal with lead atoms attached by single bonds to the structure (see Figure 5). Presumably, the resonant frequency of such molecules is very high. so we will see the frequency dependence of the mass only at very high frequencies.

In fact, Sheng et al. Sheng03 have shown in experiments that materials indeed have frequency dependent masses. An example of such a material is shown in Figure 6.

## Generalizations of the Rigid Bar Model edit

Consider the rigid body containing cavities shown in Figure~7. This is just a two-dimensional extension of the model shown in Figure~1. Here and are complex spring constants in the and directions.

In this case, the effective mass along the -direction is given by

while that along the -direction is given by

In matrix form, we then have

Hence, the effective mass is clearly ** anisotropic**. Note however that
from a macroscopic perspective it is not the average velocity in the
matrix which is important. In fact, such a quantity does not even make
sense because the velocity is not defined in the void phase. Rather it is
the velocity of the matrix that is the relevant quantity in this model.

One can generalize the model one step further by having the springs be oriented at different angles to each other as shown in Figure 8. Also let the springs in each cavity have different spring constants and the masses in each cavity are different.

In this case, let be the rotation that is needed to orient each set of springs with the and axes. Then, if is the rotation matrix for cavity containing a mass , and if and are the complex spring constants for that cavity, the effective mass can be written as

The eigenvalues of can therefore depend on .

## General form of edit

Take any body with a rigid matrix. Suppose, instead of applying harmonically varying velocities, we apply a time varying velocity and observe what the momentum is. There will be some linear constitutive relation

The kernel is second-order tensor valued and may possibly be singular. Also, since both and are physical and real, must be real. Causality implies that when (or ) since the inertial force cannot depend on velocities in the future.

Taking the Fourier transform of equation (13) and using the convolution theorem, we get

where

The quantity can be shown to satisfy the Cauchy-Riemann equations only if . This is a consequence of causality and the fact that the integral in equation (15) only converges in the upper half of the complex plane. \footnote{ Note the positive sign in the power of in equation (15). This occurs because we have chosen the inverse Fourier transform to be of the form

The effect of such a choice is that the imaginary part of is positive instead of negative.} Hence, is analytic in when . The quantity is real.

Now, the complex conjugate of is given by

where denotes the complex conjugate of (for any complex ).

Also assume that, for large enough frequencies, the dynamic mass tends toward the static mass, i.e.,

Equation (15) can be used to establish the Kramers-Kronig equations for the material. To do that, recall Cauchy's formula for a function which is analytic on a domain that is enclosed in a piecewise smooth curve :

Since the function is analytic on the upper-half plane, for any point in a closed contour in the upper-half plane, we have

Let us now choose the contour such that it consists of the real axis and a great semicircle at infinity in the upper half plane (see Figure~9).

Also, from equation (17) we observe that

Hence, there is no contribution to the integral in equation (18) due to the semicircular part of the contour and we just have to perform an integration only over the real line:

Now, let us consider the integral

There is a pole at the point (in the figure, the contour is shown as a semicircle of radius centered at the pole). In the limit , this integral may be interpreted to mean the Cauchy principal value (not the same as principal values in complex analysis)

From Figure~9 we observe that this integral can be broken up into the integral over the path minus the sum of the integrals over the paths and . From the Cauchy-Goursat theorem, the integral over the closed path is zero. We have also seen that since as , the integral over is zero. The integral over the path around the pole is obtained from the Residue Theorem (where the value is divided by two because the integral is over a semicircle), i.e.,

The negative sign arises because the curve is traversed in the counter-clockwise direction.

Therefore,

or,

Letting and taking the limit as , we get

Note that, in the above equation, both and are real. Expanding equation (20) into real and imaginary parts and collecting terms, we get the first form of the Kramers-Kronig relations

Therefore, the real part of the frequency dependent mass can be determined if we know the imaginary part and vice versa.

We can also eliminate the negative frequencies from equations (21). Recall from equation (16) that

Since, in equations (21), and are real, we have

This implies that

Consider the first of equations (21). We can write this relation as

Similarly, the second of equations (21) may be written as

Therefore, the alternative form of the Kramers-Kronig relations is

### Significance of edit

Consider harmonically varying and given by

Alternatively, we may write these as

Then,

which implies that

The average rate of work done on the system in a cycle of oscillation will be

This quadratic form will be non-negative for all choices of if and only if is positive semidefinite for all real . Note that the quadratic form does not contain . Since the work done in a cycle should be zero in the absence of dissipation, this implies that the imaginary part of the mass is connected to the energy dissipation (for instance, into heat).

## References edit

J. D. Jackson. * Classical Electrodynamics: 2nd Edition*. John Wiley and Sons, New York, 1975.

P. Sheng, X. X. Zhang, Z. Liu, and C. T. Chan. Locally resonant sonic materials. * Physica B. Condensed Matter*, 338:201--205, 2003.