Brinell hardness from indentation diameter

A Brinell test presses a tungsten carbide ball of diameter D into the surface under a force F and holds it, then the operator measures the diameter of the permanent impression. Hardness is defined as the test force divided by the curved surface area of the spherical cap left behind, not by its projected area. That surface area works out to pi*D*(D - sqrt(D^2 - d^2))/2, which gives the classic expression HBW = 2F / (pi*D*(D - sqrt(D^2 - d^2))) with F in kgf and both diameters in millimetres.

Because the indentation is large — typically 2 to 6 mm across — Brinell averages over many grains, second phases and banding. That is exactly why it is the scale of choice for castings, forgings, thick plate and coarse-grained weld metal, and why it correlates better with bulk tensile strength than the micro-scales do. It is unsuitable for thin sections, hardfacing overlays, case-hardened surfaces or anything where the indentation would be closer than 2.5d to an edge or would show through the back face.

Two constraints govern whether the result is valid. The load-diameter ratio 0.102*F/D^2 (or simply F/D^2 with F in kgf) must be held constant for results on the same material to be comparable — 30 for steel and cast iron, 10 for copper alloys, 5 or 2.5 for aluminium and soft alloys. The indentation ratio d/D must fall between 0.24 and 0.60; outside that band the ball is either barely marking the surface or is being pushed in so far that the geometry departs from the assumed spherical cap.

The indicative tensile strength shown alongside comes from the long-standing empirical relation UTS[MPa] ≈ 3.45 * HBW for wrought carbon and low-alloy steel. It is a screening figure only. It has no validity for austenitic stainless steel, cast iron, non-ferrous alloys, cold-worked material or weld metal, and it can never substitute for a tensile test where a specification calls for one.

d = (d₁ + d₂) / 2
HBW = 2F / (π·D·(D − √(D² − d²)))    F in kgf, D and d in mm
Load-diameter ratio = F / D²
Indentation ratio = d / D    (0.24 … 0.60 per ISO 6506-1)
UTS ≈ 3.45 × HBW  [MPa]  — indicative, wrought carbon steel only