Waves in composites and metamaterials/Elastodynamics and electrodynamics

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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.

Dissipation[edit | edit source]

Recall from the previous lecture that the average rate of work done in a cycle of oscillation of material with frequency dependent mass is

This quadratic form will be non-negative for all choices of if and only if is positive semidefinite for all real . Therefore, a restriction on the behavior of such materials is that

Similarly, for electrodynamics, the average power dissipated into heat is given by

In this case, the quantity is equivalent to the voltage and the quantity rate of change of electrical displacement is equivalent to the current (recall that in electrostatics the power is given by ). In addition, we also have a contribution due to magnetic induction.

Let us assume that the fields can be expressed in harmonic form, i.e.,

or equivalently as

Also, recall that,

Therefore, for real and real , we can write equation (1) as (with the substitution ),

Expanding out, and using the fact that

we have,

Since and the power , the quadratic forms in equation (2) require that

Note that if the permittivity is expressed as

the requirement implies that the conductivity . Therefore, if the conductivity is greater than zero, there will be dissipation.

Brief introduction to elastodynamics[edit | edit source]

A concise introduction to the theory of elasticity can be found in Atkin80. In this section, we consider the linear theory of elasticity for infinitesimal strains and small displacements.

Consider the body () shown in Figure~1. Let be a subpart of the body (in the interior of or sharing a part of the surface of ). Postulate the existence of a force per unit area on the surface of where is the outward unit normal to the surface of . Then is the force exerted on by the material outside or by surface tractions.

Figure 1. Illustration of the concept of stress.

From the balance of forces on a small tetrahedron (), we can show that is linear in . Therefore,

where is a second-order tensor called the stress tensor.

Since the tetrahedron cannot rotate at infinite velocity as its size goes to zero (conservation of angular momentum), we can show that the stress tensor is symmetric, i.e.,

In particular, for a fluid,

where is the pressure.

Let us assume that the stress depends only on the strain (and not on strain gradients or strain rates), where the strain is defined as

Here is the displacement field. Note that a gradient of the displacement field is used to define the strain because rigid body motions should not affect and a rigid body rotation gives zero strains (for small displacements).

Assume that depends linearly on so that

Note that this assumption ignores preexisting internal stresses such as those found in prestressed concrete. If the material can be approximated as being local, then

Taking the Fourier transforms of equation (4), we get

where

In index notation, equation (5) can be written as

Causality implies that stresses at time can only depend on strains of previous times, i.e., if or . Therefore,

This in turn implies that the integral converges only if , i.e., is analytic when .

In the absence of body forces, the equation of motion of the body can be written as

where is the mass density, is the internal force per unit volume, and is the acceleration. Hence, this is just the expression of Newton's second law for continuous systems.

For a material which has a frequency dependent mass, equation (6) may be written as

where causality implies that if then .

Taking the Fourier transform of equation (7), we get

Substituting equation (5) into equation (8) we get

Also, taking the Fourier transform of equation (3), we have

Since and are symmetric, we must have

Because of this symmetry, we can replace by in equation (9) to get

Dropping the hats, we then get the wave equation for elastodynamics

Antiplane shear[edit | edit source]

Let us now consider the case of antiplane shear. Assume that is isotropic, i.e.,

where is the shear modulus and is the Lame modulus.

Let us assume that and are independent of , i.e.,

Let us look for a solution with and independent of , i.e., . This is an out of plane mode of deformation.

Then, noting that , we have

Therefore,

or

Therefore

Plugging into the wave equation (11) we get

or (using the two-dimensional gradient operator )

TM and TE modes in electromagnetism[edit | edit source]

Let us now consider the TM (transverse magnetic field) and TE (transverse electric field) modes in electromagnetism and look for parallels with antiplane shear in elastodynamics.

Recall the Maxwell equations (with hats dropped)

Assume that and are scalars which are independent of , i.e., and .

For the TE case, we look for solutions with and independent of , i.e., .

Then,

This implies that

Therefore,

or,

Plugging into equation (13) we get the TE equation

This equation has the same form as equation (12).

More generally, if

and

we get the TE equation

Similarly, there is a TM equation with of the form

which for the isotropic case reduces to

The general solution independent of is a superposition of the TE and TM solutions. This can be seen by observing that the Maxwell equations decouple under these conditions and a general solution can be written as

where the first term represents the TE solution. We can show that the second term represents the TM solution by observing that

implying that which is the TM solution.

A resonant structure[edit | edit source]

Consider the periodic geometry shown in Figure 2. The matrix material has a high value of shear modulus () while the split-ring shaped region has a low shear modulus or is a void. The material inside the ring has the same shear modulus as the matrix material and is connected to the matrix by a thin ligament. The system is subjected to a displacement in the direction (parallel to the axis of each cylindrical split ring).

Figure 2. A periodic geometry containing split hollow cylinders of soft material in a matrix of stiff material. The direction is parallel to the axis of each cylinder.

Clearly, each periodic component of the system behaves like a mass attached to a spring. This is a resonant structure and the effective density can be negative. A detailed treatment of the problem can be found in Movchan04. Note that the governing equation for this problem is

Let us compare this problem with the TM case where is the out of plane magnetic induction. The governing equation now is

If the value of in the region of the void (ring) is small and hence is large (which implies that the conductivity is large), analogy with the equation of elastodynamics implies that the effective permeability can be negative for this material.

References[edit | edit source]

  • R. J. Atkin and N. Fox. An introduction to the theory of elasticity. Longman, New York, 1980.
  • A. B. Movchan and S. Guenneau. Split-ring resonators and localized modes. Physical Review B, 70:125116, 2004.