Waves in composites and metamaterials/Bloch waves in elastodynamics and bubbly fluids

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.

Recap edit

In the previous lecture we discussed Bloch wave solutions for Maxwell's equations in periodic media in the quasistatic limit. [1] Recall the periodic medium shown in Figure 1. The lattice spacing is  .

Figure 1. Periodic medium.

The permittivity and permeability of the medium are periodic functions of the form


where   are the primitive lattice vectors. We defined periodic functions


We then wrote Maxwell's equations (at fixed frequency) as


We also found that the constitutive relations could be expressed as


Next, we examined solutions to (1) that correspond to Bloch waves with wavevector   (possibly complex), i.e.,


The wavelength of the Bloch waves is


For the quasistatic limit, we look for solutions that satisfy


Such a condition is satisfied if  . Following standard multiple scale analysis, we assumed that the periodic complex fields have the expansions


and that dependence of   on   and   has an expansion of the form




we then showed that we got the equations in the quasistatic limit


and the constitutive equations


Note that the quasistatic equations (5), (6) have the same form as the Maxwell equations (1) and (2).

We also found that the volume averaged fields ( ) satisfy the equations


along with the necessary conditions


Effective Permittivity and Permeability edit

Let   be an effective permittivity tensor associated with the field   and let   be an effective permeability tensor associated with the field  . These effective tensors are defined through


Plugging (9) into (8) gives


Eliminate   from (10) to get




Define the matrix   such that for all vectors  


Then (11) can be written as


The eigenvalue problem (13) may be used to determine the dispersion relations   and also possible values of   associated with each Block wave mode. Once we know   we can then find  .

Note that the eigenvalue of   is zero when  . So, in practice, it is necessary to examine only the other two eigenvalues.


If the average field   is to be a Bloch wave solution of wavevector   and frequency  , then   must be an eigenvector of the matrix   and   must be the associated eigenvalue. More general solutions can be obtained by superposing different Block wave solutions.

Isotropic constituents edit

When the permittivity and permeability of the medium are isotropic, equation (12) becomes




Since  , equation (13) becomes


If we assume that


where   is the wavelength,   is the attenuation length, and   is a real unit vector, equation (15) gives


Therefore for a given frequency, the wavelength and attenuation length are determined by the complex permittivity   and the complex permeability  .

Elastic wave propagation in the quasistatic limit edit

Let us now perform the exercise for linear elastic materials. Recall that the momentum equation at fixed frequency is given by


where   is the stress,   is the mass density, and   is the displacement. For infinitesimal deformations, the strain is related to the displacement by


Also, the stress is related to the strain for linear elastic materials by


For a periodic medium, define


Then the governing equations can be written as


Let us examine Bloch wave solutions to these equations of the form


Plug (17) into (16) to get






Equations (18) have solutions if the frequency   takes one of a discrete set of values,  ,  . Let us assume that the fields have perturbation expansions in powers of  , i.e,


where the functions  ,  ,   are periodic. Let us also assume that


Plugging these expansions into (18) and using the definition


we get


Collecting terms containing   from (20) we get


Collecting terms of order   from (20) we get


From (22) we see that   must be linear in  . We also know that (from our definition),   is periodic in  . A function that is both linear and periodic must be constant. Therefore,


Then we can write (24) in the form




Note that equations (21), (26), and (25) have a form similar to that of the elasticity equations in the absence of inertial and body forces.

The complex effective elasticity tensor may be defined via the relation


Taking the average of (26) over the periodic cell, we get




Also, taking the average of (23) we get




Plugging (28) into (27) gives


Plugging (30) into (29) gives


From the minor symmetry of the tensor  , i.e.,


we have






where, for any vector  ,


The quantity   is the effective acoustic tensor. The dispersion relations may then be calculated in a manner similar to that for electromagnetism.

Effective properties of bubbly fluid edit

So far we have found that, in the quasistatic limit, the Bloch solutions satisfy equations which are directly analogous to the electromagnetic or elasticity equations but with complex fields and complex effective tensors. This implies that if we have a formula for the effective tensor which is valid for real tensors, then (by analytic continuation) we can use the same formula when the tensors take complex values.

Let us examine one such situation which arises in bubbly fluids. We can assume bubbly fluids to be an assembly of coated spheres. For such a geometry, we have Hashin's relations Hashin62,Hashin62a when the material properties are real. Let us now discuss how the Hashin relations are obtained.

Hashin's relations for assemblages of coated spheres edit

Consider the coated sphere shown in Figure~2. The sphere has a permittivity   and the coating has a permittivity  . Let the coated sphere be embedded in a matrix with permittivity  .

Figure 2. Coated sphere in a matrix.

For certain choices of the permittivities, current will tends to flow around the coated sphere while for other choices current will be attracted toward the coated sphere. Therefore we expect that there might be a situation in which the permittivities are such that the coated sphere will have no effect on a current flowing through the matrix. In such a situation, we could continue to add spheres and completely fill space (except for a set of measure zero) without affecting the field as can be seen in Figure 3. Clearly, in that case, the effective permittivity of the assemblage of coated spheres is equal to that of the matrix.

Figure 3. Assemblage of coated spheres.

In determining the effective properties of an assemblage of coated spheres we make the following assumptions:

  1. the coated spheres do not overlap the boundary of the unit cell.
  2. fluxes and potentials at the boundary remain unaltered due to addition of coated spheres.

The goal is to find a matrix whose properties are such that when a coated sphere is added to it, the fields in the matrix remain unaltered.

Effective Permittivity edit

Consider the single coated sphere shown in Figure 2. Let us assume that the sphere is centered at the origin. We look for a solution to the time-independent Maxwell's equations


with potentials


Then the electric field is given by


The potentials satisfy Laplace's equations


and we only need to match the boundary conditions at the interfaces to get a solution, i.e.,


Continuity of the tangential component of the electric field at the interface implies that


Combining (31) and (32) gives


Defining the volume fraction   as


leads to the following expression for  :


Effective Bulk Modulus edit

In this case, consider the coated sphere shown in Figure~2 with the difference that each material now has two properties- the bulk and shear moduli. Let the bulk and shear moduli of the sphere be   and  , let those of the coating be   and  , and let the moduli of the effective medium (the matrix in the figure) be   and  , respectively. As before, the radius of the sphere is   and the outer boundary of the coating has a radius  .

The governing equations are


Let the matrix be subject to a hydrostatic state of stress, i.e.,


Let us look for a solution that does not perturb this field when the coated sphere is added to the matrix.

Therefore, we look for a solution with a radial displacement field


From the continuity of displacements at the interfaces, we have


And the continuity of radial tractions   across the interface implies that


From the continuity relations and using the definition


we can show that the effective bulk modulus is given by


Bubbly Fluid edit

For the assemblage of coated spheres, when the bulk and shear moduli of the two phases are real, the effective bulk modulus is given by equation (33). We can then use the Bloch wave solutions to show that the same result holds when the moduli are complex.

Returning to the bubbly fluid problem, suppose that phase 1 is gas and phase 2 is water. The mixture is then a bubbly fluid. If we assume that water is incompressible, then  . Hence from (33) we have


Also, assume that   (air) is independent of frequency.

Now consider a plane shear wave propagating into a bubbly fluid. Let the frequency of the wave be   and let it be real. Let the wave be spatially attenuated with a complex wavevector  . Then the associated strain field is given by


The stress field is given by (assuming a Newtonian fluid)


where   is the shear viscosity of water and is assume to be independent of frequency. Therefore, we can think of water as having a complex shear modulus, i.e.,


Plugging this in (34) gives


where the effective bulk viscosity is defined as


When  , we get


Therefore, the shear viscosity of water has been converted into the bulk viscosity of the bubbly fluid. This is the reason that sound is damped strongly in bubbly fluids.

Footnotes edit

  1. The following discussion is based on Milton02

References edit

  • Z. Hashin. The elastic moduli of heterogeneous materials. J. Appl. Mech., 29:143--150, 1962.
  • Z. Hashin and S. Shtrikman. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys., 33(10):3125--3131, 1962.
  • G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.
  • M. I. Hussein. Reduced Bloch mode expansion for periodic media band structure calculations. Proc. R. Soc. A, 465:2825-2848, 2009.