Waves in composites and metamaterials/Willis equations for elastodynamics

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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.


In the previous lecture introduced the Willis equations (Willis81,Willis81a,Willis83,Willis97,Milton07). In this lecture we will discuss how those equations are derived.

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[edit]

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[edit]

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[edit]

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.

Figure 1. A composite consisting of a rigid matrix and deformable phases.

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.