The flow stress of a work-hardened metal is estimated with the Taylor equation σ = MαGb√ρ (ρ is the dislocation density). For cold-worked copper with M = 3.06, α = 0.3, G = 48 GPa, b = 0.256 nm and ρ = 1.0×10¹⁴ m⁻², which is closest to σ?

Answer: C. About 113 MPa

Since √ρ = 1.0×10⁷ m⁻¹, σ = 3.06 × 0.3 × 48×10⁹ × 0.256×10⁻⁹ × 1.0×10⁷ ≈ 1.13×10⁸ Pa, about 113 MPa. About 37 MPa is the resolved shear stress obtained by forgetting to multiply by the Taylor factor M, and about 376 MPa forgets the coefficient α. The Taylor equation comes from the fact that moving dislocations are caught where they intersect other dislocations (forest dislocations), whose spacing is of the order of 1/√ρ. The dislocation density of annealed metals is about 10¹⁰ to 10¹² m⁻², rising to about 10¹⁵ m⁻² after heavy cold work. A hundredfold increase in dislocation density raises the strength only tenfold, so work hardening gradually slows as the strain grows.