Shear modulus, bulk modulus and Poisson's ratio

Three moduli describe three ways of deforming a body:

Shear modulus (modulus of rigidity): G = (F/A)/(Δx/L) = F/(Aθ). For most materials G ≈ Y/3: sliding layers of atoms over each other is easier than pulling them apart. Only solids resist shear: push a liquid sideways and it simply flows.

Bulk modulus: B = –p/(ΔV/V). The minus sign is there because more pressure means less volume, so ΔV is negative and B comes out positive. Its reciprocal, 1/B, is the compressibility.

Bulk modulus is the one modulus that applies to liquids and gases too. Solids are the hardest to squeeze, liquids next, and gases are easiest: air has B ≈ 10⁵ Pa, while water has 2.2 × 10⁹ Pa.

Example: 1000 m down in the sea the extra pressure is about hρg = 1000 × 1000 × 10 = 10⁷ Pa. Water there is squeezed by only ΔV/V = 10⁷/(2.2 × 10⁹) ≈ 0.45%. That is why we usually treat water as incompressible.

Stretch a wire and it also gets a little thinner. Poisson's ratio compares the two: (sideways strain Δd/d) ÷ (lengthwise strain ΔL/L). It has no unit; for steel it is about 0.28–0.30, for aluminium about 0.33.

A stretched wire stores the work done on it as elastic potential energy: W = ½ × F × ΔL, which works out to ½ × stress × strain × volume. So the energy stored per unit volume is u = ½ × stress × strain.