Hollomon Flow Curve

Past the yield point a metal does not simply flow at constant stress; it hardens as it deforms, and the stress needed to keep it moving climbs. J. H. Hollomon's 1945 paper proposed describing that climb with a power law — true stress equals a coefficient times true plastic strain raised to a small exponent — and eighty years later it is still the first thing anyone fits to a flow curve. It is empirical. Nothing derives it. It survives because over the strain range that matters in sheet forming it is remarkably close to right, and because its exponent turns out to mean something.

The physical reason a metal hardens is that dislocations get in each other's way. Deformation multiplies them; a higher density means shorter free paths between obstacles; and a shorter free path means a higher stress to keep moving. Taylor's relation puts the flow stress as proportional to the square root of dislocation density, and combining that with a dislocation storage rate gives a power law with n near 0.5 in a pure annealed metal — which is about what annealed brass and austenitic stainless actually measure. Everything that stores dislocations in advance drives n down: cold work, solutes, precipitates. A heavily worked alloy has n near 0.05 and hardens hardly at all, because most of the hardening has already been spent.

This equation is written in true stress and true strain, and feeding it engineering values is the classic way to fit a bad curve. True stress divides load by the section the bar actually has, which is smaller than the original; true strain is the logarithm of the length ratio. The conversion is on this site's true-stress page. Mix the two — a true stress against an engineering strain — and you will get an n and a K that fit nothing and predict less.

K is an extrapolation, not a measurement, and this is worth being blunt about because K values scatter badly between sources for the same alloy. K is the true stress the metal would be flowing at if it reached a true strain of exactly 1.0 — 172 % engineering elongation, which most metals never see and none see uniformly. A fit made over strains of 0.05 to 0.20 is being projected five times past its own data to name K. Quote it with the strain range it was fitted over, or it is not usable.

What n is worth is stretchability. A high n spreads deformation out: any spot that starts to thin immediately hardens, becomes locally stronger than its neighbours, and hands the strain to them. That self-levelling is exactly what lets a sheet draw into a deep shape rather than tearing at the first thin patch, and it is why deep-drawing grades of sheet steel are sold on their n value. By Considère's argument n is also the true strain at which necking begins, so it puts a hard ceiling on uniform elongation. Between those two facts, n is arguably the single most useful number on a sheet-metal certificate.

Where Hollomon fails is worth knowing. It predicts zero stress at zero plastic strain, which is nonsense — real metals start plastic flow at the yield strength — so it has nothing to say near the yield point and cannot represent the elastic region at all. That is what the Ludwik form (σ = σ₀ + Kεⁿ, adding a yield offset) and the Swift form (σ = K(ε₀ + ε)ⁿ, offsetting the strain instead) exist to fix. Steels with a yield plateau and Lüders bands are badly served by all three, because the plateau is a propagating instability rather than hardening. And n itself drifts with strain in most real alloys, so the value you fit depends on the range you fit it over — which is one more reason to record that range beside the number.