Waves in composites and metamaterials/Willis equations for elastodynamics
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.
Recall that by ensemble averaging the governing equations of elastodynamics we get
where is the ensemble average over realizations and not a volume average.
We need to derive the effective constitutive relations
where the operator represents a convolution over time, i.e.,
and the adjoint operator (represented by the superscript ) is defined via
for all vector fields and second order tensor fields and at time . Note that the quantities and are third-order tensors. In the above definition the convolutions are defined as
where are vectors and are second-order tensors.
Derivation of Willis' equations 
Let us introduce a homogeneous reference medium with properties and (constant). The polarization fields are defined as
Taking the divergence of the equation (2), we get
Also, taking the time derivative of equation (2), we have
Recall that the equation of motion is
Plugging (3) and (4) into (5) gives
In the reference medium, and . Let be the solution in the reference medium in the presence of the body force and with the same boundary conditions and initial conditions. For example, if the actual body has as , then as . Then, in the reference medium, we have
Remember that we want our effective stress-strain relations to be independent of the body force . So all we have to do is subtract (7) from (6). Then we get
If we assume that is fixed, then (8) can be written as
where is a linear operator. The solution of this equation is
where is the Green's function associated with the operator . Plugging back our definitions of and , we get
The strain-displacement relation is
Plugging the solution (9) into the strain-displacement relation gives
Define and via
Then we can write (10) as
Also, taking the time derivative of (9), we get
Define and via
Then we can write (12) as
Willis (Willis81a) has shown that and are formal adjoints, i.e., , in the sense that
From (11) and (13), eliminating and via equations (1), we have
Also, ensemble averaging equations (11) and (13), we have
From (14) and (15), eliminating and , we get
Equations (16) are linear in and . Therefore, formally these equations have the form
That such an argument can be made has been rigorously shown for low contrast media but not for high contrast media. Hence, these ideas work for composites that are close to homogeneous.
From the definition of and , taking the ensemble average gives us
Also, from (17), taking the ensemble average leads to
Plugging in the relations (18) in these equations gives us
These are the Willis equations.
Willis equations for electromagnetism 
For electromagnetism, we can use similar arguments to obtain
where is a coupling term.
In particular, if the fields are time harmonic with non-local operators being approximated by local ones, then
If the operators are local, then will just be matrices that depend on the frequency .
If the composite material is isotropic, then
Under reflection, reflects like a normal vector. However, reflects like an axial vector (i.e., it changes direction). Hence would have to change sign under a reflection. Therefore, with fixed, the constitutive relations are not invariant with respect to reflections! This means that if the medium has a certain handedness and is called a chiral medium.
Extension of the Willis approach to composites with voids 
Sometimes the quantity is not an appropriate macroscopic variable. For example, in materials with voids is undefined inside the voids. Even if the voids are filled with an elastic material with modulus tending to zero, the value of will depend on the way this limit is taken. Also, for materials such as the rigid matrix filled with rubber and lead (see Figure 1), it makes senses to average only over the deformable material phase.
Therefore it makes sense to look for equations for where
where is a weight which could be zero in the region where there are voids. Also, the weights could vary from realization to realization.
Also, if we have we can recover by integrating over time, i.e.,
Hence we can write
So, from the definitions of and and using the relation (22), we have
Form the Willis equations (17) we have
Now, if the weighted strain is defined as
then, taking the ensemble average, we have
Using equation (21) we can show that
Using (23) we can express (24) in terms of and , and hence also in terms of . After some algebra (see Milton07 for details), we can show that
where when .
Taking the inverse, we can express the Willis equations (20) in terms of and as
These equations have the same form as the Willis equations. However, . We now have a means of using the Willis equations even in the case where there are voids.
- [Milton07] G. W. Milton and J. R. Willis. On modifications of Newton's second law and linear continuum elastodynamics. Proc. R. Soc. London A, 463:855--880, 2007.
- [Willis81] J. R. Willis. Variational and related methods for the overall properties of composites. Advanced in Applied Mechanics, 21:1--78, 1981.
- [Willis81a] J. R. Willis. Variational principles for dynamics problems in inhomogenous elastic media. Wave Motion, 3:1--11, 1981.
- [Willis83] J. R. Willis. The overall elastic response of composite materials. J. Appl. Mech., 50:1202--1209, 1983.
- [Willis97] J. R. Willis. Dynamics of composites. In Suquet P., editor, Continuum Micromechanics: CISM Courses and Lectures No. 377, pages 265--290. Springer-Verlag-Wien, New York, 1997.