Shear modulus

Shear modulus (modulus of rigidity) G = shearing stress σs ÷ shearing strain = (F/A)/(Δx/L) = FL/(AΔx).

Because shearing strain ≈ θ, G = (F/A)/θ = F/(Aθ), and so σs = Gθ.

G is in N m⁻² or Pa. It is generally smaller than Young's modulus; for most materials G ≈ Y/3.

Typical G values: tungsten has the stiffest at 150 GPa, steel 84, nickel 77 and iron 70; copper 42, brass 36, aluminium 25 and glass 23; wood 10 and lead only 5.6 GPa.

Lead slab 50 cm square and 10 cm thick, lower edge riveted to the floor, sheared by 9.0 × 10⁴ N on its narrow face: A = 0.5 m × 0.1 m = 0.05 m², stress = 1.8 × 10⁶ N m⁻², and with G = 5.6 × 10⁹ N m⁻² the upper edge moves Δx = stress × L/G = 1.6 × 10⁻⁴ m = 0.16 mm.

Shear, like stretching, needs a body with a definite shape, so Young's modulus and shear modulus apply only to solids.