Shear Modulus

Shear modulus G (also called rigidity modulus η or μ) measures resistance to SHAPE change at constant volume.

Apply tangential force F to top face of a solid block (area A) while bottom is fixed: produces angular shear θ.

Shear stress = F/A; shear strain = θ ≈ Δx/L (for small angles). G = shear stress / shear strain.

Units: Pa. Typical values: steel ≈ 80 GPa, copper ≈ 45 GPa, rubber ~ MPa.

Liquids and gases have G = 0 — they have no rigidity and cannot sustain shear stress at equilibrium.

Shear modulus governs torsion of rods, propagation of TRANSVERSE waves in solids, etc.

Torsion of a wire: torque τ = (πGr⁴/2L)·θ, where r = radius, L = length, θ = twist angle.

Speed of transverse (shear) waves in solid: v_s = √(G/ρ).