Shear Modulus
G = F/(Aφ) — block sheared into a parallelogram. Watch angle φ grow linearly with τ.
Key Notes
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).
Formulas
Shear modulus
θ ≈ Δx/L for small shear angles.
Torsion of a wire
Used in torsion-balance experiments.
Transverse wave speed (solid)
S-waves in seismology.
Relation to Y, B, σ (isotropic)
σ_p = Poisson's ratio. Y, B, G all linked.
Important Points
Fluids have G = 0 — they shear easily and continuously under any tangential force.
Shear modulus is typically 1/3 to 1/2 of Young's modulus for metals.
Earthquakes: P-waves (longitudinal) travel through everything; S-waves (shear) travel only through solids. This is how we know Earth's outer core is LIQUID — S-waves don't pass.
Torsion in mechanical engineering (drive shafts, springs) is governed by G.
G, B, and Y for an isotropic material are related — only two are independent.
Soft tissues and biological materials often have very low G — leading to specialized 'elastography' imaging.
Shear Modulus notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.