Shear Modulus, Bulk Modulus and Poisson's Ratio: Quick Revision

A brief note on Shear Modulus, Bulk Modulus and Poisson's Ratio. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Shear: a force along a face tilts the sides through an angle $\theta \approx \dfrac{\Delta x}{h}$. $G = \dfrac{F/A}{\theta}$, about $\dfrac{Y}{3}$ for metals.
• Bulk: a pressure on every side shrinks the volume. $B = -\dfrac{p}{\Delta V/V}$; compressibility $= \dfrac{1}{B}$.
• Liquids and gases have a bulk modulus but no shear modulus. A gas at a steady temperature has $B = p$.
• Poisson's ratio: $\sigma = \dfrac{\Delta d/d}{\Delta L/L}$, from $0$ to $0.5$; a stretched rod changes volume by $\varepsilon(1 - 2\sigma)$.
• Links: $Y = 2G(1 + \sigma)$ and $Y = 3B(1 - 2\sigma)$.

The facts at a glance

A worked example

A steel ball $5.0\ \text{cm}$ in radius is lowered to where the extra pressure is $6.0 \times 10^{7}\ \text{Pa}$. ($B = 1.6 \times 10^{11}\ \text{Pa}$.)

1. Fraction. $\dfrac{\Delta V}{V} = \dfrac{6.0 \times 10^{7}\ \text{Pa}}{1.6 \times 10^{11}\ \text{Pa}} = 3.75 \times 10^{-4}$.
2. Volume. $V = \tfrac43\pi(0.050\ \text{m})^3 = 5.24 \times 10^{-4}\ \text{m}^3$, so $\Delta V = 1.96 \times 10^{-7}\ \text{m}^3$, about $0.20\ \text{cm}^3$.
3. Radius. It shrinks by a third of the fraction: $\dfrac{3.75 \times 10^{-4}}{3} \times 5.0\ \text{cm} = 6.25\ \mu\text{m}$.

Practice with new numbers

A program to try

The program works out the steel ball of the worked example.

Mistakes to avoid

• The wrong area in shear. Use the area of the face the force slides along.
• The angle as a distance. Multiply $\theta$ by the height to get the shift.
• A percentage used as a fraction. Divide by $100$.
• The radius shrinking by the whole volume fraction. It shrinks by a third of it.
• $1 - \sigma$ for $1 - 2\sigma$. There are two sideways directions.

Check yourself

1. A sideways force of $800\ \text{N}$ acts on a face of $0.020\ \text{m}^2$. What is the shear stress?

2. Why does a liquid have no shear modulus?

3. What is the bulk modulus of a gas at $2.0 \times 10^{5}\ \text{Pa}$, squeezed slowly?

4. A rod's length grows by $0.20\%$ and its width shrinks by $0.07\%$. What is its Poisson's ratio?

5. A material has $\sigma = 0.25$ and $G = 40\ \text{GPa}$. What is its $Y$?