Shear modulus and bulk modulus
Shear modulus
Clamp the bottom of a metal block and push its top face sideways with a force F. The top slides by Δx, and the block leans over by a small angle θ.
The shearing stress is F over A, where A is the area of the face the force acts on. The shearing strain is Δx over L, which equals the angle θ in radians.
Within the elastic limit, shearing stress is proportional to shearing strain. Their ratio is the shear modulus, G, also called the modulus of rigidity.
G equals F over A, divided by Δx over L. That is F L over A Δx, or simply F over A θ. Its unit is the newton per square metre.
Look at the table of shear moduli. Steel is about 8.4 × 10¹⁰, copper 4.2 × 10¹⁰, aluminium 2.5 × 10¹⁰, and soft lead only 5.6 × 10⁹ newtons per square metre.
Compare these with Young’s moduli. For most materials, G is roughly one third of Y. It is easier to shear a solid than to stretch it.
Worked example. A square lead slab, 50 centimetres on each side and 10 centimetres thick, has its lower edge riveted to the floor. A shearing force of 9.0 × 10⁴ newtons acts on its narrow top face.
The top face is 50 centimetres by 10 centimetres, an area of 0.05 square metres. So the stress is 9.0 × 10⁴ divided by 0.05, which is 1.8 × 10⁶ newtons per square metre.
Δx equals F L over A G: 1.8 × 10⁶ times 0.5 metres, divided by 5.6 × 10⁹. The top edge moves by about 1.6 × 10⁻⁴ metres, just 0.16 millimetres.
Bulk modulus and compressibility
Now lower a body deep into a fluid. The fluid presses on it from every side, at right angles to its surface. Its shape stays the same, but its volume shrinks by ΔV.
The hydraulic stress is the pressure, p. The volume strain is ΔV over V. Within the elastic limit, they are proportional.
The bulk modulus B equals minus p divided by ΔV over V. Why the minus sign? An increase in pressure causes a decrease in volume, so ΔV is negative. The minus sign keeps B positive.
The reciprocal of the bulk modulus is called compressibility, k. It is one over B: the fractional change in volume per unit increase in pressure.
Look at the values. Steel has a bulk modulus of about 1.6 × 10¹¹, water 2.2 × 10⁹, and air at normal pressure only about 1.0 × 10⁵ newtons per square metre.
So solids are the least compressible, liquids a little more, and gases are by far the most compressible: roughly a million times more than solids.
Worked example. The average depth of the Indian Ocean is about 3000 metres. At the bottom, the extra pressure is h ρ g: 3000 times 1000 times 9.8, which is 2.94 × 10⁷ newtons per square metre.
The fractional compression of water there is p over B: 2.94 × 10⁷ divided by 2.2 × 10⁹, which is about 1.34 × 10⁻². Water at the bottom of the ocean is squeezed by only about 1.3 percent.
One last point. Young’s modulus and the shear modulus exist only for solids, because only solids keep their shape. Liquids and gases flow, so they have only a bulk modulus.
Try it: shear a block
- This starts with NCERT’s lead slab: h = 0.5 m, A = 0.05 m², F = 90 kN. The top moves about 0.16 mm.
- Swap lead for steel. Which material is hardest to shear?
- Double the force: the angle θ doubles too, as long as we stay in the elastic region.
- The drawing exaggerates the tilt. The real angle is only a hundredth of a degree or so.
Try it: squeeze it in the ocean
- Water at 3000 m (the Indian Ocean) is squeezed by about 1.3 %.
- Try steel and then glycerine at the same depth. Which has the smallest compressibility?
- Choose air: even a few metres of water squeezes a gas a lot. Gases are the most compressible.
- The deepest point of the ocean, the Mariana Trench, is about 11 km deep.
Which moduli for a fluid?
Why do liquids and gases have a bulk modulus, but no Young’s modulus or shear modulus?
- They are too light to be stretched
- Their molecules do not exert any forces on each other
- They cannot keep their shape: they flow under tensile or shearing stress, but resist being squeezed
- Their Young’s and shear moduli are infinite
Answer
C. A fluid has no fixed shape. Under a tangential or stretching stress it simply flows, so there is no restoring force and no Y or G. But squeeze it from all sides and its volume resists the change, so it does have a bulk modulus. Only solids have all three moduli.
Shearing an aluminium cube
An aluminium cube of edge 0.10 m has its lower face fixed. A tangential force of 2.5 × 10⁴ N acts on its upper face. The shear modulus of aluminium is 2.5 × 10¹⁰ N/m². How far does the upper face move?
- 0.001 mm
- 0.01 mm
- 0.1 mm
- 1 mm
Answer
B. Area A = 0.10 × 0.10 = 0.01 m², so the shearing stress is 2.5 × 10⁴ ÷ 0.01 = 2.5 × 10⁶ N/m². Shearing strain θ = stress ÷ G = 2.5 × 10⁶ ÷ 2.5 × 10¹⁰ = 1 × 10⁻⁴. Then Δx = θ × L = 10⁻⁴ × 0.10 m = 1 × 10⁻⁵ m = 0.01 mm.
Summary
- Shear modulus:
G = (F/A) / (Δx/L) = F / (Aθ) - Bulk modulus:
B = −p / (ΔV/V) - Compressibility:
k = 1/B
- The shear modulus (modulus of rigidity) is shearing stress ÷ shearing strain. For most materials G ≈ Y/3.
- The bulk modulus is hydraulic stress ÷ volume strain. The minus sign makes B positive, because pressure reduces volume.
- Solids are the least compressible and gases the most. Water at 3000 m depth is compressed by only about 1.3 %.
- Y and G exist only for solids. Liquids and gases have only a bulk modulus.