Poisson's Ratio

The Story Behind Poisson's Ratio

When you stretch a rubber band lengthwise, you notice it also becomes thinner. This isn't a coincidence — it's a fundamental property of how solids deform. When a material is stretched in one direction, it contracts in the perpendicular directions. Similarly, when compressed lengthwise, it bulges out sideways.

This coupling between longitudinal and lateral strain is quantified by Poisson's ratio, named after the French mathematician Siméon Denis Poisson. It tells us how much a material "squeezes in" when pulled, or "puffs out" when pushed.

Defining the Ratio

Consider a rectangular bar of length $L$ and diameter $D$. When you apply a tensile force along its length:

• The longitudinal strain is the fractional change in length: $\frac{\Delta L}{L}$.
• The lateral strain is the fractional change in diameter: $\frac{\Delta D}{D}$.

For most materials, these two strains have opposite signs — when length increases ($\Delta L > 0$), diameter decreases ($\Delta D < 0$), and vice versa.

Poisson's ratio, denoted by the Greek letter $\sigma$ (sigma) or sometimes $\nu$ (nu), is defined as:

The negative sign is deliberately inserted to make $\sigma$ a positive number for ordinary materials. Since lateral strain is opposite in sign to longitudinal strain, the ratio of their magnitudes is positive.

Physical Range and Meaning

Poisson's ratio is a dimensionless quantity — it has no units. For most common engineering materials, its value lies between 0 and 0.5.

• $\sigma = 0.5$: The material is perfectly incompressible. Volume remains constant during deformation. Rubber comes very close to this ideal.
• $\sigma = 0$: The material shows no lateral contraction when stretched. Cork is a classic example — you can compress a cork stopper lengthwise without it expanding sideways noticeably.
• Typical metals: Steel has $\sigma \approx 0.3$, aluminium $\sigma \approx 0.33$, copper $\sigma \approx 0.35$.

Why the Range is Bounded

The limits $0 < \sigma < 0.5$ for ordinary materials arise from energy considerations. If $\sigma$ were greater than 0.5, a material could be stretched and would actually increase in volume — requiring energy to be created from nothing. If $\sigma$ were negative, the material would expand sideways when stretched, which would also violate certain stability conditions for isotropic solids.

Typical Values Table

The NCERT textbook provides a table of Poisson's ratio for common materials. Here it is reproduced:

Notice how rubber is nearly 0.5 (almost incompressible) while cork is nearly 0 (no lateral deformation). This is why cork is used for bottle stoppers — you can push it into the neck without it bulging out and getting stuck.

Relation to Other Elastic Moduli

Poisson's ratio is not an independent elastic constant. For an isotropic material, it is related to Young's modulus $Y$, shear modulus $G$, and bulk modulus $B$ through the following relations:

$Y = 3B(1 - 2\sigma)$ …