Finer grains make stronger metal. Every grain boundary is a wall that pins the slip — dislocations pile up against it and cannot pass, so the more boundaries you pack in, the higher the yield point climbs. The law is exact and inverse-root: σy = σ0 + k·d-1/2. Rendered, not quoted.
sourceHall, E.O. (1951), "The Deformation and Ageing of Mild Steel: III", Proc. Phys. Soc. B 64, 747-753; Petch, N.J. (1953), "The Cleavage Strength of Polycrystals", J. Iron Steel Inst. 174, 25-28. AMBER — paywalled/print-era, no stable open link; cited by author, journal, volume, year.
Yield strength is a lattice-friction floor plus a grain-boundary term that grows as grains shrink:
Plot σy against d-1/2 and the points fall on a straight line: intercept σ0, slope k. Two grain sizes are enough to recover both constants exactly.
Finer grains, stronger metal — the boundaries pin slip and raise the yield point. That raised yield point is precisely the elastic-to-plastic knee on the-stress-strain curve: Hall-Petch is the microstructural knob that slides that knee upward without changing Young's modulus (the elastic slope is set by bonding, not grain size).
Live re-check: does the finer grain still out-yield the coarser one?
CHECKING…
Re-runs the finer-is-stronger law against the engine on every render. If window 6 flips the sign, this badge goes red within one frame.
Live engine — σy(d) computed each frame from pure functions. Curve = strength vs grain size; the boundary term is the inverse root.
computing…
wall"Just make every grain enormous — one big single crystal — and it will be the strongest of all."
Wrong direction. As d→∞ the boundary term k/√d → 0 and σy→σ0, the bare lattice-friction floor — the weakest Hall-Petch state, not the strongest. Single crystals win only in creep/fatigue at high temperature, where boundaries slide; that is a different regime, not this law.
"Strength scales as 1/d." → It scales as 1/√d. Halving the grain multiplies the boundary term by √2, not 2.
"Refine grains forever and strength rises without limit." → Below ~10-20 nm the law inverts (inverse Hall-Petch): grain-boundary sliding takes over and finer becomes weaker. amber
"k is a universal constant." → k depends on the alloy, slip system and temperature; it is measured, not assumed.
Swap the boundary term to +k·√d (grain size itself, not its inverse root). Now bigger grains wrongly test as stronger — the witness in 7 must catch it.
law intact — d-1/2