Poisson's ratio

Remarks

Example: for $\sigma = s \, e_1 \otimes e_1$ one finds $\varepsilon_{11} = \frac{s(\lambda + \mu)}{\mu(3\lambda + 2\mu)}$ and $\varepsilon_{22} = \varepsilon_{33} = -\frac{s \lambda}{2\mu(3\lambda + 2\mu)}$, and the quotient gives the formula. For $\lambda = 120$ and $\mu = 80$ (in GPa, steel) it gives $\nu = 120/400 = 0.3$.

Non-example: in uniaxial strain the sides are held fixed, $\varepsilon_{22} = 0$, and the quotient $-\varepsilon_{22}/\varepsilon_{11} = 0$ is not Poisson's ratio; the lateral faces must be free of stress.

Positivity of the elasticity tensor, $\mu > 0$ and $3\lambda + 2\mu > 0$, is equivalent to $E > 0$ and $-1 < \nu < 1/2$. The value $\nu = 1/2$ is the incompressible limit, and materials with $\nu < 0$ are called auxetic. The volume change in uniaxial tension is $\tr \varepsilon = (1 - 2\nu) \varepsilon_{11}$.