Hardness measures and their test dependence

An indentation is a defined experiment

A hardness test presses a specified indenter into a surface under a prescribed load and procedure. The reported scale depends on geometry and how size or depth is interpreted. Scratch hardness probes another interaction; different scales are not interchangeable numbers.

A tip concentrates force into a small contact. The surrounding solid restrains material beneath it, so the stress state differs from uniform uniaxial tension. In a ductile solid, dislocation motion or other plastic flow can leave an impression; elastic deformation recovers during unloading. In a brittle solid, cracking may accompany the impression. Polymers can continue deforming during a hold. These responses explain why tip shape, time history and the chosen reading matter.

Hardness therefore answers a local, procedural question: how much resistance did this specified contact encounter? Elastic modulus describes recoverable stiffness; yield strength describes the onset of a defined plastic response in a stated loading test; fracture toughness concerns resistance to crack extension. A small indentation alone does not supply all three. A harder surface can still be brittle, and a coating can be hard while the underlying part bends readily.

From a square impression to a Vickers result

Consider conventional Vickers indentation. A square-based diamond pyramid has a nominal angle of 136° between opposite faces. Apply the stated force history, unload, and measure the two corner-to-corner diagonals d₁ and d₂ of the residual impression. Use their arithmetic mean d = (d₁ + d₂)/2. The corner of the impression, rather than the end of a crack issuing from it, defines the diagonal.

For the ideal square pyramid, the sloping four-face area inferred from that impression is Aₛ = d²/[2 sin(68°)] ≈ d²/1.8544. The square’s side is d/√2. Dividing its projected area d²/2 by sin(68°) accounts for the faces’ inclination and gives Aₛ. Thus a force divided by the Vickers face area scales as F/d²: the numerical coefficient comes from geometry, not from an alloy constant.

Keep the unit convention explicit. In this lesson Hᵥ,SI = 1.8544 F/d² is the Vickers face-area value in MPa when F is in N and d in mm, since 1 N/mm² = 1 MPa. The traditional HV number uses F in kilogram-force (kgf) and d in mm instead. One kgf is 9.80665 N, so Hᵥ,SI in MPa = 9.80665 × the traditional HV number. A force in kgf is not a mass in kg. Values in GPa and traditional HV numbers must not be compared as though their digits use one scale.

This is a defined area inferred after unloading, not a direct measurement of the actual contact surface under peak force. Record the method, force, loading/hold history, surface condition and measurement location with the result. Distorted boundaries, chips or cracks can make a diagonal ambiguous; averaging two readings does not cure an invalid impression. The geometry here explains the reading, rather than specifying a complete testing or acceptance procedure.

The denominator can change the conclusion

A projected area is the contact’s footprint viewed along the loading direction; it differs from the sloping faces. Instrumented indentation records force and penetration through loading and unloading. Its commonly used hardness Hᵢ = Fmax/A꜀ uses the projected contact area A꜀ at maximum force Fmax, estimated with a calibrated tip-area function and contact analysis. Contact depth is the height of the tip actually in contact; it need not equal the total measured penetration because the surface can sink in or pile up around the tip.

Consequently, neither the residual optical footprint nor an uncorrected maximum-depth reading can automatically replace A꜀. Pile-up means displaced material rises beside the tip and changes the contact. If the true area is larger than the analysis assumed, Fmax divided by the underestimated area overstates hardness. Force, displacement and tip-area calibration have distinct roles; good repeatability alone does not remove a shared area bias.

For an optical Vickers result at fixed F, H ∝ 1/d². If the measured d is 10% too small, the reported hardness is multiplied by 1/0.9² = 1.2346, about 23.5% too high. A 10% length error is not a 10% hardness error. This is a deterministic error example, not a confidence interval. Small uncertain edges can matter greatly when the whole impression is small.

Change the load and you change the experiment

For an ideal self-similar pyramid in a uniform bulk material with load-independent hardness, F/d² stays constant. Hence d₂/d₁ = √(F₂/F₁). Four times the force would double the diagonal, not quadruple it. This is a useful consistency prediction, not a promise that every real surface obeys it.

A small impression may mainly probe one grain or phase, while a larger one encounters grain boundaries, pores or several phases. Surface preparation can introduce a deformed layer; roughness and tip rounding become relatively important at shallow depth. Even carefully measured homogeneous crystalline materials can show an indentation size effect, a systematic hardness change with indentation size. A change with load needs investigation before it is labelled either a material improvement or an instrument fault.

Choose sampling to match the question. To compare a local hard phase, locate it with microscopy and keep the affected region suitably confined. To characterise a representative heterogeneous surface, distribute indents across its structure and retain the location information and spread. A few targeted indents in the hardest particles do not estimate the whole surface’s typical response. Nearby impressions, edges and supports can also alter the surrounding constraint.

The surrounding material participates

Surface preparation, coating thickness, indentation size and nearby structure can affect a result. An indentation in a thin coating can sample the substrate. Correlations with strength require evidence for the relevant material and state rather than a universal conversion.

Penetration depth is not the boundary of the mechanically affected material. Stresses and deformation extend laterally and beneath the tip. A hard film on a softer substrate can therefore bend or transmit load into the substrate before the tip itself reaches their interface. The measured value can describe the coupled coating–substrate system, with a depth dependence that changes as the sampled region grows.

Use a depth series, film-thickness measurement, surface examination and a suitable substrate comparison to investigate that coupling. A shallow apparent plateau is useful evidence, but it does not alone certify a film-only value: surface effects, area calibration and genuine size dependence may also matter. No fixed depth/thickness percentage guarantees negligible substrate influence for every material pair and property. A criterion established for hardness must not automatically be transferred to modulus.

The original illustrative dataset below represents a 1.00 µm coating on a substrate with a separately measured hardness of 3.0 GPa across the same depth range. Here h is the maximum penetration into the specimen after instrument corrections, not contact depth. Hᵢ was obtained from the appropriate projected-contact analysis at each depth. These chosen points have no uncertainty bars because no measurement uncertainty has been supplied; their small differences do not establish statistical significance.

A scratch ranking asks a different question

A scratch introduces lateral motion as well as normal contact. The Mohs mineral scale ranks relative resistance to scratching against reference minerals: orthoclase is 6 and quartz is 7. A mineral that scratches orthoclase but is scratched by quartz is bracketed between those references by that evidence. The observation needs a real scratch rather than loose transferred powder, and surface direction or condition can matter.

The ranks are ordinal: they express an order, not equal intervals of force, energy or indentation pressure. Quartz at 7 is not established to require 7/6 times the force to scratch as orthoclase at 6. There is no unit conversion from a Mohs rank to a Vickers number. A controlled scratch test reporting force and groove dimensions is a different quantitative experiment; its tip, motion, normal force and damage criterion must also be defined.

Scratch resistance alone is not wear life. Repeated sliding can introduce debris, cracking, adhesion and changes to the contacting surfaces. The next lessons develop contact, friction and wear as interacting-system behaviour; a single hardness rank cannot replace that evidence.

Make a comparison that answers the material question

For a useful comparison, state what should differ and what should be matched. If the intended change is heat treatment, compare material and location, method, force/time history, preparation and sampling before attributing a hardness difference to the treatment. Repeated results and an uncertainty estimate help decide whether the observed separation is resolved. A calibrated reference checks measurement performance; it does not make two different methods measure one identical quantity.

For many ductile metals, hardness can correlate with flow resistance because indentation requires plastic deformation. But the conversion depends on the material family, plastic strain and hardening response, processing state and test convention. A relation checked for one family is evidence for that domain, not a universal equation turning H into tensile strength. A hard surface layer also need not represent the tensile response of the whole section.

Separate the evidence from the desired application. A coating result of 12 GPa, a Mohs rank of 7 and a tensile strength in MPa describe different observations. First identify the quantity, sampled material and test conditions; then ask whether the relevant deformation, cracking or wear evidence supports the proposed use.

Worked examples

Try the reasoning

1. A surface has high indentation hardness, but a precracked specimen from that material fractures at a relatively low crack-driving force. Is either observation necessarily inconsistent, and which response does each test probe?

   Show the answer and reasoning

   No necessary inconsistency. The indentation measures resistance in a confined local contact; the precracked test probes resistance to crack extension under its stated conditions.

   Hardness is not fracture toughness. Cracking may also occur around an indent, but an ordinary hardness number does not supply a validated crack-resistance measurement.

2. One report gives 62 HRC and another 700 HV. Can subtracting their displayed numbers establish a hardness difference of 638, or can either number be ranked directly against Mohs 7?

   Show the answer and reasoning

   Neither operation is justified. HRC, HV and Mohs use different definitions and conventions; a justified material-specific relation would be needed for any proposed conversion.

   Scale symbols belong to the result. A number’s apparent size is not a common physical distance between these quantities.

3. At F = 4.903325 N (0.5 kgf), another valid Vickers impression has diagonals 38 µm and 42 µm. Calculate the traditional HV number and the SI face-area value. Use the stated ideal coefficient 1.8544.

   Show the answer and reasoning

   d = 40 µm = 0.040 mm; HV = 1.8544 × 0.5/0.040² = 579.5. The SI value is 579.5 × 9.80665 = 5682.95 MPa ≈ 5.68 GPa.

   Average the diagonals, convert µm to mm, and keep the force convention consistent. Inserting 40 as though it were mm would introduce a millionfold area error.

4. At fixed force, a reading based on d = 50 µm gives hardness H₀. Suppose only the possible diagonal interval 49–51 µm is supplied. Express the corresponding hardness interval as multiples of H₀. Does the interval itself define a 95% confidence statement?

   Show the answer and reasoning

   H/H₀ lies from (50/51)² = 0.96117 to (50/49)² = 1.04123, approximately −3.88% to +4.12%. No confidence level follows from a possible-value interval alone.

   The upper length limit gives the lower hardness limit because H decreases with d². Squaring the reciprocal produces slightly asymmetric hardness bounds.

5. A constant-hardness ideal pyramid gives d = 30 µm at force F₁. What diagonal should result at force 2F₁? What force ratio would give d = 90 µm under that same model?

   Show the answer and reasoning

   At 2F₁, d = 30√2 ≈ 42.43 µm. A 90 µm diagonal is three times larger and requires F₂/F₁ = 3² = 9.

   For the stipulated constant-hardness self-similar model, area grows as d² and force grows in proportion to area. Real size dependence must still be checked.

6. Careful comparison shows an unknown mineral scratches orthoclase (Mohs 6), while quartz (Mohs 7) scratches it. What ranking is supported? A colleague assigns it 6.5 and predicts a force exactly halfway between those references. Assess both claims.

   Show the answer and reasoning

   The evidence brackets it between the 6 and 7 references under those comparisons. It supplies neither a measured exact rank of 6.5 nor a halfway scratching force.

   Ordinal references establish order. An intermediate numerical assignment and a quantitative force law need additional definitions and evidence.

7. Two multiphase specimens have the same hard-particle fraction. For A, every indent is deliberately centred in a hard particle. For B, positions are distributed across particles and matrix. A has a larger mean. Does this establish that A has the harder representative surface? How would you redesign the comparison?

   Show the answer and reasoning

   No. Different spatial sampling confounds the mean. Use matched distributed locations for a representative-surface question, or matched particle-targeted locations for a phase question, retaining phase/location information and spread.

   A correct local reading can answer the wrong population question. Matching the number of indents does not repair a systematic difference in where they were placed.

8. For a different coating, t = 2.0 µm and maximum penetration h = 0.10 µm. A report asserts that h/t is below 10%, so both hardness and modulus must be independent of the substrate. What follows from the arithmetic and what remains unproved?

   Show the answer and reasoning

   h/t = 0.05 = 5%. That ratio alone proves neither a film-only hardness nor a film-only modulus, and a criterion justified for one property cannot automatically be assigned to the other.

   The depth is a geometric comparison, not the edge of the elastic or plastic field. Material contrast, contact analysis, surface condition and property-specific validation still matter.

What to carry forward

• Keep the indenter, method, force/time history, location and surface state with a hardness result.
• Vickers residual diagonals, peak projected-contact area and Mohs scratch ranks define different readings.
• Area and length errors can strongly bias a force/area result; repeatability does not exclude shared bias.
• A load or depth series helps reveal sampling and size dependence without identifying the cause by itself.
• Indentation depth does not bound the affected material, and no universal coating-depth percentage isolates every property.
• Correlations with tensile strength or wear performance need evidence for the intended material and interacting system.