Limits to Poisson's RATIO

We present answers to two longstanding questions on Poisson’s ratio v: first, what happens to the bulk modulus B in a material as it approaches the softening zone, where v --> 1/2 and the substance becomes “incompressible,” and second, why v nearly always exceeds 0.2 for isotropic materials. For the first question, mathematically because the ratio of the bulk to the shear modulus B/G becomes infinite when v=1/2, it is often assumed that as B becomes very large as v approaches 1/2. However, experimental results show that B is nearly constant in the softening zone. The analysis clarifies the apparent conflict between the classic equations of elasticity and the experimental results. For the second question, we show that the range of v is divided by the roots of quadratic elasticity relations 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 v outside of this range are rare, and tend to be either very hard (diamond, beryllium) or porous (auxetic foams); such substances have more complex behavior than can be described by classical elasticity.