Poisson's ratio and the relations between the moduli

Longer and thinner

Stretch a rubber band and watch its width: it gets narrower. A metal wire does the same, by a tiny amount.

The longitudinal strain is $\dfrac{\Delta L}{L}$; the lateral strain is $\dfrac{\Delta d}{d}$, the fractional change of width. Poisson's ratio compares them:

taking both as sizes. It has no unit. For most metals $\sigma$ is about $0.3$. For rubber it is very close to $0.5$. For cork it is nearly $0$, which is why cork is good for stoppers: it does not bulge sideways when pushed in.

What happens to the volume

A rod stretched by $\varepsilon$ gets longer by $\varepsilon$ and narrower by $\sigma\varepsilon$ in each of two sideways directions. Its volume changes by

So $\sigma = 0.5$ means the volume stays the same, as for rubber. A real stable material cannot have $\sigma$ above $0.5$: stretching it would shrink its volume.

The moduli are linked

For a material that is the same in every direction, only two of $Y$, $G$, $B$ and $\sigma$ are free. The others follow:

You should find that as $\sigma$ nears $0.5$, $B$ shoots up: a material that keeps its volume is very hard to squeeze from all sides.

Watch out. Poisson's ratio uses the sideways change of one width, not of the area. The area changes by about $2\sigma\varepsilon$.

A worked example

Aluminium has $Y = 70\ \text{GPa}$ and $\sigma = 0.33$. Find its shear and bulk moduli.

1. Shear. $G = \dfrac{70\ \text{GPa}}{2 \times 1.33} = 26.3\ \text{GPa}$.
2. Bulk. $B = \dfrac{70\ \text{GPa}}{3 \times (1 - 0.66)} = \dfrac{70\ \text{GPa}}{1.02} = 68.6\ \text{GPa}$.
3. Check. The tables give $25\ \text{GPa}$ and $75\ \text{GPa}$: close, since real metals are not perfectly the same in every direction.

Practise with new numbers

Try it in Python

The program finds $G$ and $B$ for four metals from their $Y$ and $\sigma$.

Remember these

• $\sigma = \dfrac{\Delta d/d}{\Delta L/L}$, between $0$ and $0.5$; metals about $0.3$.
• $\dfrac{\Delta V}{V} = \varepsilon(1 - 2\sigma)$ for a stretched rod.
• $Y = 2G(1 + \sigma)$ and $Y = 3B(1 - 2\sigma)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A rod's length grows by $0.10\%$ and its width shrinks by $0.032\%$. What is its Poisson's ratio?

2. Why can Poisson's ratio not be more than $0.5$?

3. A metal has $G = 30\ \text{GPa}$ and $\sigma = 0.3$. What is its Young's modulus?

4. Why is cork used for bottle stoppers?