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.
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.
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
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
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).
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.