Materials with negative Poisson's ratios are a striking example that
composites can exhibit behavior unlike that of the constituent materials.
Are there anisotropic materials with similar extreme elastic behavior? It
turns out there is a very rich family of such materials, each of which is
stiff to a certain set of loadings, and compliant to an orthogonal set of
loadings. We review the theory of these extremal materials and their
synthesis using appropriate microstructures. In the dynamic setting
composites can exhibit even stranger behaviors. We show that the effective
mass density is in many respects similar to the effective dielectric
constant in electromagnetism. Recent (and not so recent) research has shown
it is frequency dependent, can be negative over a narrow band of
frequencies, can be anisotropic, and can have an imaginary part similar to
the way the effective stiffness can have an imaginary part corresponding
to viscous losses. We also show how one can construct composites which,
over a narrow band of frequencies, have a Willis type constitutive law, where the
stress depends not only on the strain but also on the velocity, and the momentum depends
not only on the velocity but also on the strain. Such materials may be useful
for providing a cloak to shield an object from elastic waves. This is joint work with Marc Briane,
Andrej Cherkaev, and John Willis.
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