This article was originally published on July 31, 2014.

Microindentation hardness testing is a very powerful tool for characterizing materials and diagnosing problems. It is a complicated process, however, and the subject is rarely taught in schools. We must decide which test – Knoop or Vickers – is best for the problem at hand and then choose the most appropriate test force to evaluate the problem. This is not a trivial task.


The Vickers test has an advantage in that the indent cavity is geometrically identical in shape as a function of the test force and the cavity depth. Therefore, HV values are statistically identical over a very wide range of test loads as long as we use the proper optics to measure the diagonals with precision. This becomes much more difficult when the diagonals are ≤20 µm in mean length. As will be shown, part of this problem is due to the equation used to calculate the hardness. A small measurement imprecision leads to greater and greater potential error as the diagonal becomes smaller.

    The Knoop test has an advantage in that the long diagonal is 2.7 times greater than the mean Vickers diagonal for the same test force. So, the imprecision problem becomes more acute at lower test loads than for the Vickers test. Also, due to its shape, the Knoop indenter is better for measuring the hardness of coatings, segregation (such as banding) and the hardness of surface treatments that produce a rapid change in hardness over a small distance. Its weakness is that the cavity produced is not geometrically constant with test force and indent cavity depth.

    Consequently, the Knoop hardness does vary with test force, and this becomes more acute at test forces of 100 gf and below. Specimen preparation is also very important and becomes more important as the applied force is decreased. All damage from sectioning, grinding and polishing must be removed in the preparation process while keeping the specimen flat and producing excellent edge retention when surface gradients are to be assessed.

    There are many applications where hardness determination is very important, but it must be determined with a small indent. Bulk hardness tests are generally conducted to obtain an average value, often to determine if an annealing treatment meets the required maximum hardness permitted so that the alloy can be successfully machined. It is also used to determine if a heat-treatment process yielded the correct strength range because hardness correlates very well with tensile strength.

    Applications for microindentation hardness testing are quite different. Instead of determining a mean hardness, averaging out any minor inhomogeneity present, the tests are used to reveal variations in hardness – desirable or undesirable. For example, the depth of decarburization can be assessed using Knoop microindentation tests at known depths from the surface by determining the depth where the hardness increases and becomes reasonably constant. Conversely, the depth affected by surface-hardening processes can be determined by making very small indents from the surface inward until the core hardness is obtained.

    Segregation can also be evaluated by using small indents that are within normal versus segregated regions. Very low-load tests are also made to aid in identifying various phases or constituents that may be present in alloys that are known to be either harder or softer than the matrix. In each of these studies, specimen preparation deficiencies that will bias the hardness test values must not be present. The test force must be chosen so that the indent size is small enough to give sensitivity in revealing the hardness variations present. Performing these tasks properly is not a trivial exercise.


Precision Influence of the Equations Defining HV and HK

The basic equations (Eq. 1 and 2) defining the Knoop (HK) and Vickers (HV) hardness, where the applied force (in gf) is multiplied by a geometric constant and then divided by the long diagonal squared or the mean diagonal squared (in µm) respectively, cause an inherent problem in measuring small indents (i.e. indents with diagonals ≤20 µm in length). Figure 1 shows the calculated relationship between the diagonal and load and the resulting hardness for Knoop indents. Figure 2 shows this relationship for Vickers indents. As the test load decreases and the hardness rises, the slope of the curves for diagonal versus hardness becomes nearly vertical. Hence, small variations in diagonal measurements in this region, even those well within the precision of the test, will result in large hardness variations and produce imprecise data.


HK = 14229 L/d2  (1)

HV = 1854.4 L/d2 (2)


    If we assume that the repeatability of diagonal measurements by the average user is about ±0.5 µm, which is quite reasonable, and we add and subtract this value from the long diagonal length or the mean diagonal length, we can then calculate two hardness values. The difference between these values is ΔHK and ΔHV (Figs. 3 and 4). From these two figures, we can see how the steepness of the slopes shown in Figures 1 and 2 will affect the possible range of obtainable hardness values as a function of the diagonal length and test force for a relatively small measurement imprecision, ±0.5 µm. These figures show that the problem is significantly greater for the Vickers indenter than for the Knoop indenter for the same diagonal length and test force. However, for the same specimen hardness and the same applied test force, the long diagonal of the Knoop indent is 2.7 times greater than the mean of the Vickers’ diagonals (Fig. 5).


Consistency in HV for Specimens as a Function of Test Force

Numerous studies of Knoop and Vickers tests made on metals over a range of hardness and test forces have shown an inconsistency in the hardness values, or the so-called “load-hardness” problem. For the Knoop indenter, because the indent cavity is not geometrically identical as a function of indent depth, the hardness should vary somewhat with test force. Because of the difficulty in measuring small indents and the influence of small variations in measurement, this inconsistency would be expected to be greater for high-hardness materials than for soft materials and will be greater as the test force decreases. On the other hand, the Vickers indenter does produce geometrically identically shaped indent cavities as a function of depth, so the Vickers hardness should be constant, within statistical precision, with variations in the test force. However, many studies conducted using test forces ≤1,000 gf
have shown deviations from constancy at test loads ≤100 gf. In almost all cases, this problem has been attributed to interactions between dislocations and the indenter at these low loads.[1-3]

    A review of more than 60 publications about such studies has revealed four different load-HV trends. They are (from most common to least common):

1) At test loads ≤100 gf, HV decreases.

2) At test forces ≤100-200 gf, HV rises slightly and then decreases.

3) At test loads ≤100 gf, HV increases.

4) At test forces from 1,000 gf to either 10 or 25 gf, HV was constant within statistical precision.


    Trend number 1 was by far the most commonly observed. An example of such a study[1] is shown in Figure 6. In this experiment, five indents at each test force, from 5-500 gf, were measured at 500X magnification using specimens representing five HRC test blocks covering a wide hardness range. Note that trends 1 and 2 are observed in this data. In an ASTM E-4 interlaboratory study,[2,3] all four trends were observed for the same indents measured by different laboratories, clearly showing that this is a measurement problem and has nothing to do with dislocation interactions with the indenter.

    The literature claimed that macro-Vickers testers were immune from this problem. However, no published examples of work like that shown in Figure 6 could be found in the literature for macro-Vickers testers using test loads from 1-120 kgf or from 1-50 kgf. Consequently, the same HRC test blocks were evaluated as a function of applied test force from 1-50 kgf, revealing results[1] shown in Figure 7 (trends 1 and 2). These indents were measured using 100X magnification, typically used with macro-Vickers systems. These results clearly suggest that the inconsistency in HV at low test loads is a visual perception problem due to inadequate resolution and perhaps inadequate image contrast.

    If higher-magnification optics with high numerical aperture ratings are utilized, can the load-hardness problem be overcome? To test this, Vickers indents (six at each test load) were made over a range of test forces varying from 10 gf to 10 kgf, and the indents were measured with objectives varying from 10-100X using four HRC test blocks with a range of hardness.[4]

    Modern microindentation testers, such as the DuraScan 70 (Fig. 8), can be set up with a turret that holds both indenters (Knoop and Vickers) and four objectives. Figure 9 shows that the results using the machine and increasing the magnification with decreasing indent size were much better than those shown in Figures 6 (5-500 gf) and 7 (1-50 kgf).

    All indents made with a 10-gf load were <8 µm in length; all of the indents made with a 25-gf load were <12.5 µm in length; and all indents made with a 50-gf load were <17.5 µm in length. Despite these very small sizes due to the light loads, measurements using a 100X objective (0.95 NA) gave reasonably good data. The overall results are much better than the results in Figures 6 and 7, as well as those reported in the references.[2,3]


Variability in HK as a Function of Applied Force

Unlike the Vickers square-based indenter, the rhombohedral-shaped Knoop indenter does not produce geometrically identical indent cavities as a function of depth, and the hardness should vary with the applied force. In general, this variation is small over the range of 200-1,000 gf. As the force decreases below 200 gf, the increase in HK becomes greater.

    Unlike the Vickers indents, the Knoop indents are more likely to be undersized than oversized due to the difficulty in seeing the indent tips when they are smaller. This error also increases the HK value and adds to the usual upward trend observed as the test force decreases.

    The literature contains many examples of this trend. However, in the first interlaboratory test[2,3] a few laboratories actually showed the opposite trend, decreasing HK with decreasing test force below 200 gf, which was never previously published. The four test blocks evaluated for Vickers hardness (Fig. 9) were also evaluated with the DuraScan 70 for Knoop hardness, using six indents at each test force and a range of 10-1,000 gf. The results are shown in Figure 10.

    All of the indents made with a 10-gf load were <20 µm in length; all of the indents made with a 25-gf load were <31 µm in length; all of the indents made with a 50-gf load were <45 µm in length; and all of the indents made with a 100-gf load were <65 µm in length.

    For the same specimen and test load, the Knoop long diagonal is ~2.7 times longer than the Vickers mean diagonal (Fig. 5), which improves the precision in determining HK values versus HV values at low loads. However, the variation in HK with test force is a constraint to using the Knoop test at varying test loads and then trying to compare that data to results from other hardness scales. Knoop data only at a 500-gf load is correlated to other test scales in ASTM E 140. Being able to correct for this deviation would be advantageous.


Next: We conclude this discussion in the second and final installment by comparing and contrasting Knoop and Vickers indents for different test conditions and look at the influence etching has on a sample’s microindentation hardness results. IH


For more information:  Contact George F. Vander Voort, Vander Voort Consulting LLC, Consultant – Struers Inc., 24766 Detroit Rd., Westlake, OH 44145; tel: 847-623-7648; e-mail:; web: and



1. Vander Voort, G. F., Metallography: Principles and Practice, McGraw-Hill Book Co., NY, 1984; ASM International, Materials Park, OH, 1999, pp. 356, 357, 380 and 381.

2. Vander Voort, G.F., “Results of an ASTM E04 Round Robin on the Precision and Bias of Measurements of Microindentation Hardness,” Factors that Affect the Precision of Mechanical Tests, ASTM STP 1025, ASTM, Philadelphia, 1989, pp. 3-39.

3. Vander Voort, G.F., “Operator Errors in the Measurement of Microindentation Hardness,” Accreditation Practices for Inspections, Tests, and Laboratories, ASTM STP 1057, ASTM, Philadelphia, 1989, pp. 47-77.

4. Vander Voort, G. F. and Fowler, R., “Low-Load Vickers Microindentation Hardness Testing,Advanced Materials & Processes, Vol. 170, April 2012, pp, 28-33.