Modifications to Newton’s second law and continuum
elastodynamics
with John WillisCambridge University
and supported by the National Science Foundation
Happy Birthday Luc, and thanks for your greatcontributions to mathematics andin particular homogenizationand for your inspiration
Newton’s second lawF=ma
has withstood the centuries but itis a microscopic law (holding at the atomiclevel) which is frequently applied as a macroscopic law without modification.
Is this correct?
We are not saying Newton’s law is wrongbut presenting a different perspective whereit seems advantageous to generalize it
Of course (following Einstein) we can rewriteNewton’s second law as
F=dp/dt, p=mv
At the macroscopic scale p can be definedas the sum of momenta of the microscopicmasses, and Newton’s law can be triviallymade to hold at a macroscopic level if we define the macroscopic velocity to be
v=total momentum/total mass
But is it always sensible to define the macroscopic velocity in this way?
The “macroscopic velocity” depends on themass of the cores which are unconnected withthe object: more sensible to take the velocity ofthe matrix as the macroscopic velocity
Maxwell’s Equations:
Continuum Elastodynamics:
Suggests that andmight have similar properties
Specifically a similar dependence on frequency
Sheng, Zhang, Liu, and Chan (2003) foundthat materials could exhibit a negativeeffective density over a range of frequencies
Red=Rubber, Black=Lead, Blue=Stiff Matrix
A simplified onedimensional model:
Avila, Griso and Miara (2005) extending work ofBouchitte and Felbacq (2004) essentially proved that materials could have an anisotropic density
Simplified Model:
General properties ofIf the velocity was not harmonically varying
Causality implies is analytic for
Also since H is real
We expect
At fixed frequency, with
the average work done in a cycle of oscillation is
So must be positive semidefinite forfor all real
Generalization of the model whichcan mimic any response function
If each of the springs behaves according toa Maxwell model of a dashpot in series witha purely elastic spring then:
and the material will have a mass:
One could argue that the failure of Newton’slaw in these models is due to the failure to takeinto account the internal “hidden mass” which isnot moving in phase with the motion of the rest of the body. But there is always some internal mass vibrating at microscopic levels.
Key point: Rather than look to deeper and deeperlevels of the microstructure (perhaps far beyondthe reach of experiments) to explain everything using Newton’s law, it seems much more sensibleto allow modifications of Newton’s second law toexplain the response of bodies.
Why would one want anisotropic density? One reason: Transformation based cloakingThis was first discovered by Greenleaf, Lassasand Uhlmann (2003) for conductivity.
It was later extended to 2dimensional geometricoptics by Leonhardt (2006) and to the time harmonicMaxwell equations by Pendry, Schurig and Smith (2006)The conductivity or Maxwell equations retain their form under coordinate transformations and the keyidea of Greenleaf, Lasses and Uhlmann is to use a transformation which maps a point to a sphere (the cloaking region) and which is the identity mapping outside a larger sphere.
From Pendry, Schurig and Smith (2006)
Putting an object in the orange cloaking regionIs equivalent to disturbing the moduli inside apoint in the equivalent problem, which will notdisturb the surrounding fields, in particular outsidethe blue shell where the mapping is the identity.
Cloaking for elasticity (joint work withMarc Briane and John Willis)
Idea: Apply the Greenleaf, Lassas and Uhlmann/ Pendry, Schurig, Smith method of cloaking to elastodynamics
Requires new materials with new behavior,in particular, materials with anisotropicdensityCummer and Schurig found that materials with anisotropic density are also needed fortwodimensional acoustic cloaking
The Black circles have positive massThe White circles have negative mass
Origin atcenter
Displacements at vertices of unit cellare assumed to derive from somesmooth complex displacement field in the limit h to 0:
where
Displacements at interior points are internalhidden variables
From the geometry:
Let the masses at the interior points be:
Then the physical momentum density is:
So the associated complex momentumdensity is
Depends on the displacement gradientthrough s
Acceleration of the material will generate stress
To leading order in h the masses at E and Fmust be accelerated by complex forces
andThis must be provided by tension in the rods.Forgetting for the moment the springs and surrounding material a force
needs to act on the vertex D to maintainthe motions. Similarly a force needsto act on the vertex B
To compensate for the inertial stresscaused by the acceleration a traction oft per unit length needs to be applied tothe top boundary of a small sample, anda traction of –t to the bottom boundary, andno tractions to the side boundaries. Equating these tractions with gives:
Total stress: with
Constitutive law:
in a basis with
the tensors S and D are
Generalization:
One gets:
When the stress is symmetricand the momentum only depends on thestrain, not on local rotations. The consititutivelaw then has the general form required forelasticity cloaking.
Besides transformation based cloaking thereis also cloaking based on anomalous localizedresonance (joint work with Nicorovici, McPhedranand Botton)
Clusters of polarizable dipoles near superlensesget cloaked. In contrast to transformation basedcloaking, the cloaking region is outside thecloaking device.
Effective equations for a random media:the ensemble averaging approach of Willis.
Summary of the approach initiated by Willis(1981)For each realization:
Ensemble averaging:
The equations of motion could be solved if therewere constitutive laws:
Introduce a comparison medium and define“polarization fields”
The equation of motion then implies:
which has the solution
where G is the relevant Green’s function. Then we have:
and
in the sense
Expressed in terms of the polarization fields:
and by ensemble averaging:
which has the formal solution
from which it follows that
Extension allowing a weighted average.
If there are voids then u is not defined thereand <u> has no meaning. To allow for thisand the presence of “hidden regions” it makessense to introduce a weighted displacementfield uw(x)=w(x)u(x) and find the equation governing its ensemble averaged behavior
H is the Heaviside function and we assume
Then
so
where
Giving
which with and determine
This analysis breaks down if voids are presentbut we still expect the final equations to hold.
Of course this only gives the expected (ensembleaveraged behavior, not the behavior in a particularrealization.
For more details see:Milton and Willis, On modifications to Newton’s second law and linear continuum elastodynamics Proc. Roy. Soc A, 463 855880 (2007)
Milton, Briane and Willis, On cloaking for elasticity and physical equations witha transformation invariant form, New J. Phys. 8, 248 (2006)
Milton, New metamaterials with macroscopic behavior outside that ofContinuum elastodynamics, http://arxiv.org/abs/0706.2202
and references therein.
Thankyou for your attention!
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