Question 2 of 7: True Strain Derivations & the Brinell Hardness Test

Question 2: True Strain Derivations & the Brinell Hardness Test (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Definition of true strain as the integral of incremental length change over current length; constant-volume assumption for plastic deformation ($A_0l_0=Al$).

Find. (a) Derive Eqs. (1) and (2); identify which monitoring method is valid through necking. (b) Describe the Brinell test and explain its correlation (and lack thereof) with tensile strength.

Approach

True strain is defined incrementally, $d\varepsilon_T=dl/l$, because engineering strain (which normalizes by the fixed original length) understates strain once the gauge length itself has grown substantially; integrating that incremental definition gives Eq. (1) directly. Eq. (2) follows by substituting the constant-volume relation between length and diameter into Eq. (1). The necking question and the Brinell correlation question are both, at heart, about whether a measurement reflects the true local state of the material or only an average over a region that may include material that has already stopped deforming.

1. (a)(1) True strain from gauge length. By definition, true strain accumulates as $d\varepsilon_T=\dfrac{dl}{l}$ (each increment of stretch is normalized by the current, not the original, length). Integrating from the initial length $l_0$ to the current length $l$, $$\varepsilon_T=\int_{l_0}^{l}\frac{dl}{l}=\ln l-\ln l_0=\boxed{\ln\!\left(\frac{l}{l_0}\right)}.$$
2. (a)(2) True strain from diameter. For a cylindrical specimen, volume conservation during plastic deformation ($A_0l_0=Al$, with $A=\tfrac{\pi}{4}d^2$) gives $$\frac{l}{l_0}=\frac{A_0}{A}=\frac{d_0^2}{d^2}.$$ Substituting into Eq. (1), $$\varepsilon_T=\ln\!\left(\frac{d_0^2}{d^2}\right)=\boxed{2\ln\!\left(\frac{d_0}{d}\right)}.$$
3. Which monitoring survives necking? Once necking begins, deformation localizes — the neck cross-section thins rapidly while the rest of the gauge length barely changes further. An extensometer measuring $l$ over the full gauge length reports an average strain diluted by all the non-necked material, understating the true strain at the neck. Monitoring the minimum diameter at the neck itself tracks the local cross-section directly, and volume conservation still holds locally at that cross-section even though it no longer holds uniformly along the whole gauge length. Diameter monitoring (Eq. 2) should therefore be used once necking starts, since it measures the true local strain rather than a gauge-length average that becomes increasingly wrong as deformation localizes.
4. (b) The Brinell hardness test. A hardened steel (or, for softer materials, tungsten carbide) ball — standard diameter $D=10$ mm — is pressed into the flat, polished surface under a fixed load $P$ (typically 3000 kgf for steels) for a standard dwell time, and the diameter $d$ of the resulting permanent indentation is measured optically. The Brinell hardness number is $$\text{HB}=\frac{2P}{\pi D\left(D-\sqrt{D^2-d^2}\right)}$$ (load divided by the indent's curved surface area).
5. Why HB correlates with tensile strength for structural steels. Both hardness (resistance to localized plastic indentation) and tensile strength (resistance to bulk plastic flow to fracture) are, at bottom, measures of the same underlying quantity — the material's resistance to plastic deformation. For as-rolled or normalized structural steels, the indentation zone deforms plastically in a manner that scales consistently with the material's bulk flow stress, giving the well-known empirical relation $\sigma_{ts}(\text{psi})\approx 500\times \text{HB}$ over a broad hardness range — a correlation calibrated on, and valid for, a reasonably uniform, ductile microstructure.
6. Why the correlation breaks down for heat-treated steel, and a better test. The $500\times\text{HB}$ constant is empirical, calibrated on annealed/normalized microstructures; a quenched-and-tempered (largely martensitic) steel has a different work-hardening response and much lower ductility, so the relationship between the highly localized indentation flow under the ball and the bulk bulk-tensile flow-to-fracture behaviour no longer tracks the same proportionality — and at very high hardness the ball itself can begin to flatten, corrupting the diameter reading. The Rockwell C (HRC) test — a conical diamond (Brale) indenter under a high load, reading hardness directly from indentation depth — is the standard, better-suited test for hardened steels: it doesn't deform under the hardness levels typical of heat-treated steel, needs no optical measurement, is fast enough for routine QC, and its correlations to tensile properties are established specifically for that hardness range.