Hardness to Tensile Strength⚠ unverified

Understanding the Parameters

• Brinell hardness $H_B$ — the ratio of test force to the spherical indentation area (kgf/mm²), measured by pressing a hardened ball into the surface. Because both hardness and strength reflect resistance to plastic flow, they correlate tightly for a given class of steel.

• The factor $3.45$ — the constant that converts $H_B$ to MPa for carbon and low-alloy steels ($S_{ut}\,[\text{MPa}] \approx 3.45\,H_B$; equivalently $\approx 0.5\,H_B$ in ksi). It varies by alloy family — stainless, aluminium, and copper alloys need different constants — and the correlation degrades at very high or very low hardness.

• Output $S_{ut}$ — an estimate, not a measurement; treat it as ±10 % and not a substitute for qualification testing on critical parts.

• Why it works — indentation hardness is essentially a constrained compression yield test; for work-hardening metals the mean indentation pressure scales with the flow stress, which in turn tracks the ultimate strength.

Derivation (Approaching a Proof)

Hardness is not derived from strength but correlated with it. Tabor's analysis of indentation showed that the mean pressure under a hardness indenter is about three times the material's flow (yield) stress, $H \approx 3\,\sigma_y$ — the constraint factor from the surrounding elastic material. For a work-hardening steel the ultimate strength bears a roughly fixed ratio to the flow stress at the strain the indentation imposes. Combining the constraint factor with unit conversions (kgf/mm² → MPa) and the steel-specific strength/flow ratio collapses to a single empirical constant:

$$S_{ut}\,[\text{MPa}] \approx 3.45\,H_B.$$

The constant is fitted to tensile-vs-hardness data across many steels; the physical basis (Tabor's constraint factor) explains why a linear relation exists, while the precise slope is empirical.

Dimensional check. $H_B$ is reported as a pure number (kgf/mm² with the units suppressed by convention), so the constant $3.45$ carries the MPa units: $S_{ut}\,[\text{MPa}] = 3.45\,[\text{MPa}]\times H_B$.

History and Development

Johan August Brinell introduced his hardness test in 1900, and the strength–hardness correlation was established empirically soon after as tensile data accumulated. David Tabor's mid-20th-century work gave it a mechanistic footing (the $H \approx 3\sigma_y$ constraint factor). The relation remains a workshop staple — a Rockwell or Brinell reading gives an immediate strength estimate — while codes still require actual tensile testing for design allowables.

Related Concepts: Specific Strength, Endurance Limit Unmodified, Endurance Limit steel, Fatigue Strength Coefficient, Material Selection Index Strength

Notes: Empirical, for carbon/low-alloy steels (~±10 %); different constants for stainless/aluminium/ copper. $S_{ut}\,[\text{MPa}] \approx 3.45\,H_B$ ($\approx 0.5\,H_B$ ksi). Rooted in Tabor's $H\approx3\sigma_y$ indentation constraint. Not a substitute for qualification testing.