Questions: Hardness Testing and Strength Correlations

Question 1

A materials engineer receives a batch of steel and measures a Brinell Hardness Number of 180 BHN. Using the approximation UTS ≈ 3.45 × BHN (MPa), she needs to verify a specification of UTS ≥ 600 MPa. What should she conclude?

• A. The material fails — hardness tests always underestimate tensile strength for steels
• B. The material likely meets the specification: 3.45 × 180 ≈ 621 MPa
• C. No conclusion is possible — hardness and tensile strength are unrelated properties
• D. The material passes only if confirmed by a full tensile test; the hardness correlation is unreliable

Answer: (B)

The hardness-UTS correlation works because both properties reflect resistance to plastic deformation. The approximation UTS ≈ 3.45 × BHN is accurate enough for production screening of steel — 3.45 × 180 = 621 MPa, which exceeds the 600 MPa target. The correlation is approximate (it assumes similar strain-hardening behavior to the steel database it was derived from), but it is standard industry practice for rapid material certification without destructive tensile testing.

Question 2

A failure analyst wants to map the property gradient across the heat-affected zone of a weld in a thin steel plate. Which hardness test is most appropriate, and why?

• A. Brinell — it uses a large ball that averages across microstructural heterogeneity, giving a representative measurement
• B. Vickers — the small, geometrically self-similar diamond pyramid can resolve fine local variations and is load-independent
• C. Rockwell C — it reads hardness directly off a dial, making it fastest for production-floor screening
• D. Knoop — it produces the deepest indent, reaching through surface oxidation from welding

Answer: (B)

The Vickers test is ideal for fine-scale mapping. Its small diamond pyramid indent can be placed at intervals of ~50–100 μm, resolving the sharp property gradients in the heat-affected zone. The geometrically self-similar pyramid means the hardness number is independent of applied load, so measurements are comparable across the traverse. The Brinell test uses a 10 mm ball that leaves a large indent, averaging over millimeters of microstructure — far too coarse to resolve HAZ gradients. Rockwell is good for production screening but not spatial mapping.

Question 3

The factor-of-three relationship between indentation hardness and yield strength (H ≈ 3σ_y) is a purely empirical fitting result with no theoretical derivation.

• T. True
• F. False

Answer: False

The factor of three has a theoretical basis in slip-line field theory, which analyzes the plastic flow field under an indenter in an ideally plastic material. The mean contact pressure required to plastically indent a material is approximately three times the uniaxial yield stress — a result derived analytically from plasticity theory, not just fitted to data. This is why the correlation works reasonably well across different steel alloys, not just for the specific alloys used to calibrate it.

Question 4

A hardness traverse — a line of indentations across a cross-section — can reveal the depth of a hardened case on a carburized gear without machining the part into a tensile specimen.

• T. True
• F. False

Answer: True

This is one of the principal advantages of hardness testing over tensile testing: spatial resolution on the actual part. A hardness traverse across a carburized gear cross-section will show high hardness (~700–800 HV) in the case layer dropping to lower hardness (~200–300 HV) in the core, directly measuring case depth. Tensile testing requires machining a dedicated dog-bone specimen, destroys a section of the part, and gives only a bulk average over the gauge length — incapable of resolving the case-core gradient.

Question 5

Why does pressing a hard indenter into a metal surface under controlled load measure a property closely related to yield strength, and what theoretical result underlies the approximate factor-of-three relationship?

Model answer: When an indenter presses into the surface, the material directly beneath it must yield and flow plastically to accommodate the indenter volume — the same physical process that governs tensile yield. The mean contact pressure required to drive this plastic flow is related to yield strength by slip-line field theory: for a rigid, perfectly plastic material, the mean indentation pressure is approximately 3σ_y (where σ_y is the uniaxial yield stress). This factor of three comes from the triaxial stress state under the indenter — the hydrostatic constraint means plastic flow requires much higher stress than simple uniaxial tension. Since hardness is defined as load divided by indent area (i.e., mean pressure), hardness directly reflects yield strength through this factor of three.

The practical consequence: a quick, non-destructive hardness number predicts tensile strength because both probe the same underlying resistance to dislocation motion and slip. The correlation breaks down when materials have very different strain-hardening behavior from the steel database (e.g., work-hardened copper vs. annealed copper have the same yield strength but different hardness-UTS ratios), which is why the approximation is called empirical even though it has a theoretical foundation.