# Influence of the Equations Defining HV and HK on Precision

The basic equations defining (see equations 1 and 2) the Knoop (HK) and Vickers (HV) hardness, where the applied force is multiplied by a geometric constant and then divided by the long diagonal squared or the mean diagonal squared, respectively, cause an inherent problem in measuring small indents, that is diagonals ≤20 µm in length.  Figure 1 shows the calculated relationship between the diagonal and load and the resulting hardness for Knoop indents while 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, in this region, small variations in diagonal measurements will result in large hardness variations.

HK = 14229 L/d2    (1)
HV = 1854.4 L/d2    (2)  Figure 1: Relationship between the long diagonal length and the Knoop hardness as a function of the test force.  Note how the slope of the lines becomes more vertical as the test force decreases. Figure 2: Relationship between the mean diagonal length and the Vickers hardness as a function of the test force.  Note how the slope of the lines becomes more vertical as the test force decreases.

If we assume that the repeatability of the diagonal measurement 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, shown in Figures 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 greater for the Vickers indenter than for the Knoop indenter for the same diagonal length and test force. For the same specimen and the same test force, the long diagonal of the Knoop indent is 2.7 times greater than the mean of the Vickers’ diagonals, as shown in Figure 5.   Figure 3: Plot showing the possible range of Knoop hardness due to a ±0.5 µm measurement imprecision as a function of the diagonal length and the applied test force. The results are plotted for materials with a maximum HK of 1100-1200.  Note that the problem for specimens with diagonals ≤20 µm is greatest for 10 and 25 gf test loads. Figure 4: Plot showing the possible range of Vickers hardness due to a ±0.5 µm measurement imprecision as a function of the diagonal length and the applied test force. The results are plotted for materials with a maximum HV of 1100-1200.  Note that the problem for specimens with diagonals ≤20 µm is greatest for 10 to 100 gf test loads. Figure 5: Relationship between the Knoop long diagonal and the mean Vickers diagonal for equivalent hardness (per E 140) at a 500 gf test load. The Knoop diagonal is 2.7 times longter than the Vickers diagonal.

George Vander Voort has a background in physical, process and mechanical metallurgy and has been performing metallographic studies for 47 years. He is a long-time member of ASTM Committee E-4 on metallography and has published extensively in metallography and failure analysis. He regularly teaches MEI courses for ASM International and is now doing webinars. He is a consultant for Struers Inc. and will be teaching courses soon for them. He can be reached at 1-847-623-7648, and through his web site: www.georgevandervoort.com The articles and presentations that can be down-loaded from this web site are based upon work done by GFV while employed at Bethlehem Steel (1967-1983), Carpenter Technology (1983-1996), Buehler Ltd. (1996-2009) and Struers (2009-Present) and from the authors consulting work for companies such as, Latrobe Steel, Scot Forge, etc., and from his litigation work. GFV's bylined articles appearing in various issues of the ASM Handbook series have been listed here courtesy of ASM International, Materials Park, Ohio.