Waves in composites and metamaterials/Elastodynamics and electrodynamics

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

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

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




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

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








Plugging into the wave equation (11) we get


or (using the two-dimensional gradient operator  )


TM and TE modes in electromagnetism edit

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.,  .



This implies that






Plugging into equation (13) we get the TE equation


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

More generally, if




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

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

  • 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.