Shear & Bulk Deformation

Tensile loading pulls perpendicular to the area it acts on; shear loading drags parallel to it. Clamp the bottom of a block, push its top face sideways with a force F∥, and the square cross-section leans into a parallelogram. The stress is still force per area, τ = F∥/A, but A is now the face the force slides along. The strain is the sideways displacement of the top face divided by the block's height, γ = Δx/h. That ratio is tan θ for the tilt angle θ, and at the strains a solid survives, tan θ ≈ θ, so shear strain is simply the tilt in radians. Both quantities keep the units of their tensile cousins: pascals for stress, nothing at all for strain.

Divide one by the other and you have the shear modulus, G = τ/γ. Rearranged, Δx = F∥h/(AG): a taller block shears further, a wider one less, exactly as Young modulus predicts for stretching. In pure shear the volume hardly changes at all — the block leans, it does not swell — so G measures resistance to change of shape alone. That makes G the sharpest test of what counts as a solid: a fluid at rest has G = 0, because any tangential stress, however small, keeps it moving. It is also why seismic S-waves, which shear the rock they pass through, stop at the Earth's liquid outer core, while P-waves, which compress it, carry on.

Now apply the same normal stress to all faces. The load is a pressure increase Δp, and the response is a fractional volume change ΔV/V₀ called volume strain — the only one of the three strains that is not a ratio of lengths. The bulk modulus is B = −Δp/(ΔV/V₀); the minus sign is there because raising the pressure shrinks the volume, and it keeps B positive. Its reciprocal, the compressibility k = 1/B, states the same fact from the other end. Water has B ≈ 2.2 GPa, so k ≈ 4.5 × 10⁻¹⁰ Pa⁻¹. At 1000 m depth the water pressure is about ρgh = 1025 × 9.81 × 1000 ≈ 1.0 × 10⁷ Pa, giving ΔV/V₀ ≈ −(1.0 × 10⁷)/(2.2 × 10⁹) ≈ −0.46 %. Water is nearly, but not quite, incompressible.

The three moduli are not three theories. They are one definition — stress over strain — applied to three loadings. Pull or push along a single axis and use E with ε = ΔL/L₀. Drag one face across another and use G with γ = Δx/h; twisting a shaft is that same shear wrapped around an axis, so G governs it too. Squeeze from all directions and use B with ΔV/V₀. Choosing wrongly costs a whole factor: for steel, E ≈ 200 GPa, G ≈ 79 GPa and B ≈ 160 GPa, so using E where B belongs understates the volume strain by about 20 %, and using G overstates it by a factor of two. A fluid at rest has only B, since G = 0 and there is no way to pull it along one axis.

For an isotropic material — one with no built-in grain direction — linear elasticity has only two independent constants. Stretching a bar lengthways also thins it sideways, and Poisson's ratio ν is the fractional sideways contraction divided by the fractional lengthways extension: a ratio of two strains, not of two distances. Once E and ν are known the rest follow: G = E/[2(1 + ν)] and B = E/[3(1 − 2ν)]. Take steel with E = 200 GPa and ν = 0.30. The relations give G = 200/2.6 = 77 GPa and B = 200/1.2 = 167 GPa, close to the measured 79 GPa and 160 GPa. They also explain the extremes: as ν → 0.5 the factor 3(1 − 2ν) → 0 and B runs away. That is rubber, which changes shape freely (G of order 1 MPa) while resisting compression almost like water (B of order 2 GPa).

Every relation here assumes small strains and a response that is proportional and recoverable — the same restriction that limits Hooke's law. Push past the elastic limit and no single modulus describes the material. Three further limits matter. Anisotropic materials such as wood, fibre composites and single crystals need more than two constants, because their stiffness depends on direction. A gas has no one bulk modulus: for an ideal gas held at constant temperature B = p itself, about 10⁵ Pa at sea level, some four orders of magnitude below water's, and it climbs as the gas is compressed. And even for a liquid the number depends on how fast you squeeze it, since the adiabatic bulk modulus is larger than the isothermal one.