Poisson's ratio

A tensile stress in a particular direction causes extension ($\epsilon_z$) in that direction and contraction in the other 2 directions ($\epsilon_x$, $\epsilon_y$). For isotropic materials:

For metals, $\nu = 0.34$ if not given. Maximum possible value is $\nu = 0.5$.

Isotropic Materials

Homogeneous materials where properties are direction-independent. $\epsilon_x = \epsilon_y$ under uniaxial stress.

Relation to Elastic Moduli

Here:

• $G$: shear modulus
• $K$: bulk modulus
• $E$: Young’s modulus

Biaxial Stress State

When stresses $\sigma_x$ and $\sigma_y$ both act in the $xy$-plane, the strain in $z$ is:

Why Rubber has $\nu = 0.5$

Rubber is nearly incompressible: volume does not change under stress. Volume conservation requires $\nu = 0.5$ exactly. From the bulk modulus relation, $K \to \infty$ as $\nu \to 0.5$, which corresponds to a perfectly incompressible material.