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.

Recap edit

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.

References edit

  • [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.