Precipitation (age) hardening strengthens a metal by seeding it with a fine dispersion of a second-phase particle that a gliding dislocation must get past. Two competing mechanisms set the resistance, and whichever needs less stress wins:
Cutting (shearing): τ_shear ≈ k·√r
Bypass (Orowan): τ_Orowan = G·b / L
Particle spacing: L = r·√(2π / 3f)
Governing strength: τ_y = min(τ_shear, τ_Orowan)
For small, coherent precipitates it is cheaper for the dislocation to cut straight through — but cutting gets harder as r grows (more antiphase-boundary / coherency-strain energy to pay). For large, widely-spaced precipitates it is cheaper to bow around them, leaving a residual dislocation loop (Orowan looping) — but that gets easier as particles coarsen at fixed volume fraction, since L grows with r. The crossover between the two curves is exactly the peak strength seen in real age-hardening curves: alloys are heat-treated to land near that peak radius, and "over-ageing" (coarsening the precipitates further) softens the metal again.
The bow itself follows a line-tension force balance. With line tension T = Gb²/2, an applied resolved shear stress τ pushes the pinned dislocation into a circular arc of radius:
R = G·b / (2·τ)
Bypass condition: R ≤ L/2 ⇔ τ ≥ τ_Orowan
- Particle radius / volume fraction — reshapes the precipitate row and moves the cutting-vs-bypass crossover.
- Applied stress — drives the live bow of the dislocation; below τy it stays elastically pinned, above τy it breaks through.
- Alloy system — swaps in typical shear modulus G, Burgers vector b, and cutting-resistance constant k for an Al–Cu (θ′) versus a Ni-base superalloy (γ′) system.
Real-world relevance: this is the mechanism behind heat-treatable aluminium alloys (2xxx/6xxx/7xxx series, aircraft structure) and nickel superalloys strengthened by γ′ (turbine blades) — the same competing-mechanism model used in undergraduate and graduate mechanical-behavior-of-materials courses (after Orowan, 1948, and standard texts such as Courtney and Ashby).