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

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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

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Let us introduce a homogeneous reference medium with properties   and   (constant). The polarization fields are defined as

 

Then,

 

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

 

or,

 

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

 

or,

 

Define

 

Then,

 

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

 

or,

 

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

 

or,

 

or,

 

These are the Willis equations.

Willis equations for electromagnetism

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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

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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.,

 

where

 

Hence we can write

 

So, from the definitions of   and   and using the relation (22), we have

 

Form the Willis equations (17) we have

 

Therefore,

 

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

 

or,

 

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

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  • [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.