  A long-standing question is why Poisson's ratio v nearly always exceeds 0.2
for isotropic materials, whereas classical elasticity predicts v to be between
-1 to 1/2. We show that the roots of quadratic relations from classical
elasticity divide v into three possible ranges: -1 < v <= 0, 0 <= v <= 1/5, and
1/5 <= v < 1/2. Since elastic properties are unique there can be only one valid
set of roots, which must be 1/5 <= v < 1/2 for consistency with the behavior of
real materials. Materials with Poisson's ratio outside of this range are rare,
and tend to be either very hard (e.g., diamond, beryllium) or porous (e.g.,
auxetic foams); such substances have more complex behavior than can be
described by classical elasticity. Thus, classical elasticity is inapplicable
whenever v < 1/5, and the use of the equations from classical elasticity for
such materials is inappropriate.
