This article provides answers to the following questions, among others:

• How is the hardness of materials defined?
• On which principles are all hardness testing methods based?
• Which indenter is used for Brinell hardness testing?
• What is the load factor and what is it used for?
• For which materials is Brinell testing particularly suitable?
• Which indenter is used for the Vickers hardness test?
• Under what conditions can Vickers hardness values only be compared with each other?
• For which materials is Vickers testing particularly suitable?
• Which indenters are used for Rockwell hardness testing and for which materials are they used?
• What is the purpose of applying the preload?
• What are the advantages and disadvantages of Rockwell testing?

## Introduction

In many applications, components should have not only a high strength but also a high wear resistance. This generally applies whenever two or more components are in moving contact with each other. These include, for example, gears, shafts, bolts, pins, etc.

High wear resistance ultimately means a hard surface, so that the surface is not damaged in contact with adjacent components and thus wear is kept to a minimum. For this reason, characteristic values are required to characterize the hardness of a material. In order to obtain such parameters, hardness must first be defined:

Indentation hardness is the resistance of a material to penetration by an indenter (indentation resistance)!

According to this definition, all hardness testing methods are ultimately based on the same principle. An indenter (e.g. ball, cone, pyramid, etc.) is pressed with a certain force into the material surface to be tested. The indentation hardness value is determined from the indentation left behind.

Depending on the material to be tested and the given boundary conditions, different hardness tests have developed, whose respective measured values generally cannot be converted into one another. Therefore, hardness values can only be compared if they have been obtained by identical test procedures. The most important procedures and their advantages and disadvantages are explained in more detail below:

• Brinell hardness test
• Vickers hardness test
• Rockwell hardness test

Specially prepared specimens or real components can be used for hardness testing, provided that their functionality is not impaired due to the indentation left behind.

## Brinell hardness test

In Brinell hardness testing, a hard metal ball (carbide ball) is pressed into the material surface to be tested within approximately 10 seconds as the force increases. The applied test force is maintained for 15 to 20 seconds so that the material can settle during this time and the measurement provides reproducible and comparable test results. The indentation left behind on the material surface is then determined under a light microscope. The ratio of testing force $$F$$ and the indentation surface $$A$$ (spherical segment) serves as a measure for the Brinell hardness value HBW:

\begin{align}
\label{brinellhaerte}
&HBW=\frac{0.102 \cdot F}{A}  \\[5px]
\end{align}

With the Brinell hardness test, a carbide ball is pressed into the material. The indentation surface left behind serves as a measure of the indentation resistance!

The factor 0.102 in the equation is due to the unit “kilopond” or “kilogram-force” (1 kp ≙ 9.807 N), which was used in the past but is no longer permissible today. Therefore, the unit kilopond was replaced by the physically correct unit “Newton” with the corresponding conversion factor of 0.102 (=1/9.807).

The indentation surface $$A$$ can be determined by the diameter $$D$$ of the penetrator ball and by the indentation diameter $$d$$ left behind using the following formula:

\begin{align}
\label{kugelsegment}
&A=\frac{\pi}{2} \cdot D \cdot \left(D-\sqrt{D^2-d^2} \right)  \\[5px]
\end{align}

By combining equation (\ref{kugelsegment}) and equation (\ref{brinellhaerte}), the unit-less Brinell hardness HBW is calculated as a function of the applied penetration force $$F$$ (in N) and the ball diameter $$D$$ (in mm) and the indentation diameter $$d$$ (in mm) as follows:

\begin{align}
\label{brinellhaertewert}
&\boxed{HBW =\frac{0.204 \cdot F}{\pi \cdot D \cdot \left(D-\sqrt{D^2-d^2} \right)}}  ~~~~~\text{Brinell hardness} \\[5px]
\end{align}

Due to the anisotropy in the deformation behavior, it can happen that there is no exactly circular imprint on the material surface. Then the indentation diameter $$d$$ is determined from the mean of two indentation diameters $$d_1$$ and $$d_2$$ at right angles to each other:

\begin{align}
\label{durchmesser}
&\boxed{d=\frac{d_1+d_2}{2}}  \\[5px]
\end{align}

### Validity

To prevent the material from being pushed over the edge of the specimen during testing and therefore pretending a lower hardness value, the center of the indentation should be at least as far from the edge as 2.5 times the diameter of the indentation.

\begin{align}
\label{mindestabstand}
&\boxed{a \ge 2.5 \cdot d}  \\[5px]
\end{align}

If several hardness tests are carried out on one single specimen, care must be taken to ensure that the indentations do not fall below a minimum distance from each other. Otherwise, the measurement result would be influenced by hardening phenomena that occur around the respective indentations. This distance should not be less than 3 times the indentation diameter.

\begin{align}
\label{mindestabstand_proben}
&\boxed{\Delta a \ge 3 \cdot d} \\[5px]
\end{align}

In order to obtain comparable results, the indentation diameter $$d$$ should not be smaller than 24 % and not larger than 60 % of the indenter diameter $$D$$:

\begin{align}
\label{mindestdurchmesser}
&\boxed{0.24 \cdot D \le d \le 0.6 \cdot D} \\[5px]
\end{align}

If the indentation diameters are too large and lie in the range of the test ball diameter, the test ball is pressed too deeply into the material. A further penetration then hardly produces a larger indentation diameter, which then leads to no longer reproducible hardness values due to measurement inaccuracies in the diameter determination.

If, on the other hand, the indentation diameter is too small compared to the test ball diameter used, however, the ball is hardly pressed into the material. Blurred edges are the result, from which it is very difficult to determine the indentation diameter left behind. Due to the low deformation, elastic portions are particularly high, so that the indentation diameter decreases relatively strongly when the ball is lifted off. The hardness values obtained from small indentation diameters are no longer valid, as well as those from large diameter values.

For the above mentioned reasons of too much or too little penetration, the surface pressure between the ball and material sample must therefore not be too high and not too low. Comparable results for different materials are only given if the test was carried out with the same stress intensity. Due to the larger surface area, larger test balls also require higher test forces compared to smaller test balls, in which the forces are distributed over a smaller surface.

In order to do justice to this fact, the so-called load factor $$B$$ is defined. The load factor is ultimately defined by the ratio of test load to test ball surface and can be regarded as a kind of “surface pressure”:

\begin{align}
&\boxed{B =\frac{0.102 \cdot F}{D^2}} ~~~~~\text{load factor} \\[5px]
\end{align}

For comparability of the hardness values obtained with different test balls on different materials, the load factor $$B$$ must have the same value in all cases!

The factor 0.102 results again from the obsolete unit “kilopond”. In contrast to softer materials, hard materials must be tested with a higher load and thus with a higher load factor in order to maintain the diameter range according to the equation (\ref{mindestdurchmesser}).

The load factor is standardized to the values 1 – 2.5 – 5 – 10 – 15 – 30. Depending on the expected hardness, reference values for the load factor used can be found in the table books. The test force $$F$$ (in N) to be set can then be determined with equation (\ref{beanspruchungsgrad}) depending on the dimensionless load factor $$B$$ and the selected ball diameter $$D$$ (in mm).

### Test balls

Sintered carbide balls with a standardized diameter of 10 mm, 5 mm, 2.5 mm, 2 mm or 1 mm are available as test balls for Brinell hardness testing. Small diameters are necessary for thinner sheets, as balls that are too large would only bulge out the material on the opposite side of the sheet. In principle, the sample thickness $$s$$ should be at least 8 times the penetration depth $$h$$:

\begin{align}
\label{mindestprobendicke}
&\boxed{s \ge 8 \cdot h} ~~~~~\text{minimum thickness of the sample} \\[5px]
\end{align}

Large test balls are also not suitable for determining the hardness of thin surface layers. In such cases, there is a risk that the surface layer will only be pressed into the underlying base material.

Larger ball diameters are necessary when testing coarse-grained, heterogeneous microstructures (e.g. cast iron). Due to the large sphere, as many individual (heterogeneous) structural components as possible are involved in the deformation, resulting in a hardness value that covers the entire microstructure and not just individual phases. This testing of heterogeneous microstructures is a big advantage of Brinell hardness testing. In principle, however, it is only suitable for soft to medium-hard materials.

Brinell hardness testing is particularly suitable for thicker, heterogeneous materials in the low to medium hardness range! Thin sheets cannot be tested with the Brinell hardness test!

The Brinell hardness test is not suitable for very hard materials or hardened surface layers because the ball does not penetrate sufficiently into the material. Higher test loads are not the solution at this point, as this leads to deformation of the carbide ball. The flattening of the ball results in a larger indentation diameter and thus pretends a softer material.

Even very thin sheets cannot be tested due to the aforementioned bulging of the material on the opposite side of the sheet. In order to close this gap, a new hardness test method was developed by Vickers, which is explained in the next section.

### Indication of the hardness value

The standard-compliant specification of Brinell hardness consists of the hardness value (HBW), the ball diameter (in millimeters), the test force (in kiloponds) and the application time (in seconds). These values are given without units and separated by slashes. The indication of the time can be omitted if the test was performed with the standard application time of 10 to 15 seconds.

### Empirical relationship between tensile strength and hardness for non-alloy steels

For unalloyed and low-alloyed steels there is an empirical relationship between the Brinell hardness HBW and the tensile strength $$\sigma_u$$. This relationship means that the tensile strength (in N/mm²) corresponds approximately to 3.5 times the Brinell hardness value:

\begin{align}
\label{zugfestigkeit_brinell}
&\boxed{R_m \approx 3.5 \cdot \text{HBW}} \\[5px]
\end{align}

## Vickers hardness test

For the Vickers hardness test, a square base pyramid with a opening angle of 136° is used as the indenter (opening angle = angle between two opposite surfaces of the pyramid). The angle was chosen so that the Vickers hardness values are comparable to a certain degree with the Brinell hardness values (applies to approx. 400 HBW or 400 HV). The diamond pyramid is pressed into the material surface with increasing force and maintained for about 10 to 15 seconds when the desired test force is reached. As with the Brinell hardness test, the ratio of test force $$F$$ and indentation surface $$A$$ (pyramid surface area) serves as hardness value for the Vickers method:

\begin{align}
\label{vickershaerte}
&HV=\frac{0,102 \cdot F}{A} \\[5px]
\end{align}

In the Vickers hardness test, a four-sided diamond pyramid is pressed into the material to be tested. The indentation surface left behind serves as a measure of the hardness value!

The factor 0.102 again comes from the no longer used unit “kilopond” (see Brinell hardness test). The indentation surface can be determined from the diagonals of the indentation left behind. With this indentation diagonal $$d$$ (in mm) and the test force $$F$$ (in N), the Vickers hardness value HV is then determined as follows:

\begin{align}
\label{vickershaertewert}
&\boxed{HV =\frac{0.1891 \cdot F}{d^2}} ~~~~~\text{Vickers hardness} \\[5px]
\end{align}

The indentation diagonal $$d$$ is determined by the mean value of the two diagonals $$d_1$$ and $$d_2$$ at right angles to each other:

\begin{align}
\label{durchmesserdiagonale}
&\boxed{d=\frac{d_1+d_2}{2}} \\[5px]
\end{align}

### Validity

To avoid the risk of material bulging on the opposite side of the sample, the thickness should not fall below a certain minimum value. The minimum thickness depends on the expected hardness of the material and the test load.

In addition, the distance $$a$$ from the center of the indentation to the edge of the sample should be at least 2.5 times the value of the indentation diagonal $$d$$ to prevent the material from flowing sideways:

\begin{align}
\label{mindestrandabstand}
&\boxed{a \ge 2.5 \cdot d} \\[5px]
\end{align}

Furthermore, the distance between two adjacent indentations for steel and copper samples should be at least as far apart as three times the diagonal length of an indentation (six times for aluminum samples). This is to eliminate the influence of work hardening phenomena around the area of the indentation.

\begin{align}
\label{mindestprobenabstand}
&\boxed{\Delta a \ge 3 \cdot d} \\[5px]
\end{align}

### Comparability of hardness values

In contrast to a ball (as in Brinell hardness test), a pyramid always provides to a certain extent geometrically similar indentations even with different test loads. Thus, with identical samples, the double force also leads to a double indentation surface. As a ratio of force and indentation surface, the hardness value is therefore always identical despite different test loads*. However, the independence of the hardness value from the test load does not apply to low test loads. In this case, the elastic deformation accounts for a larger proportion of the total deformation. As a result, the remaining pyramid indentation is smaller and thus pretends a higher hardness value.

*) This is not the case with Brinell hardness test. There the double force (higher load factor) would lead to a different hardness value for the same ball used.

Therefore, Vickers hardness values should only be compared with each other if they were determined with the same test loads. A harder material always requires higher test loads than a softer material. Depending on the expected hardness of the material, different test load ranges are prescribed. A distinction is made between three ranges of loads.

On the one hand, the so-called macro test range with test loads between 49.03 N (5 kp) and 980.7 N (100 kp), within which the hardness values are practically independent of the test load (“macrohardness”).

On the other hand, the micro test range is differentiated between 1.961 N (0.2 kp) and 29.42 N (3 kp). Such a load range is used for thin surface layers and sheet metals as well as for finished parts in order not to damage the components too much (“microhardness”).

In special cases, the nano test range between 0.098 N (0.01 kp) and 1.961 N (0.2 kp) is also used (“nanohardness”). The pyramid tip used offers an additional advantage over the ball in the Brinell process, since the pyramid-shaped indentation leaves sharper edges even at low indentation depths and can thus be better measured. At low indentation depths, therefore, the accuracy of the Vickers test increases compared to the Brinell hardness test.

In contrast to the Brinell hardness test, the Vickers test method is suitable for all hardness ranges, i.e. from very soft to very hard materials. In addition, this method can also be used for thin sheets or thin surface layers, which makes it a universal hardness testing method.

The Vickers hardness test is suitable for soft to very hard materials and especially for thin sheets!

### Indication of the hardness value

The standard-compliant specification of Vickers hardness consists of the hardness value, the test force and the application time. The latter can be omitted with the standard time of 10 to 15 seconds.

Both the Brinell and Vickers hardness test use the indentation surface left behind as a hardness measure. The indentation geometry is determined by a light microscope. This usually requires a glossy surface so that the indentation is clearly visible. It may be necessary to polish the sample before testing. Therefore, these processes are generally not suitable for automated production. For this reason, the Rockwell hardness test described below was developed.

## Rockwell hardness test

In the Rockwell hardness test, the measure of the hardness is not an indentation surface but an indentation depth. Either a carbide ball or a rounded diamond cone with a tip angle of 120° and a tip radius of 0.2 mm serves as the indenter. The indentation depth can be read directly from a dial gauge via the traverse path of the testing machine.

In the Rockwell hardness test, an indenter is pressed into the material to be tested. The indentation depth serves as a measure of the hardness!

The measuring process of the Rockwell test is carried out in three steps. First, the indenter is placed on the surface to be tested with a so-called preload $$F_0$$ of 98 N. In this way, the influences of possible setting processes in the sample and any clearance in the measuring instrument can be compensated. After the preliminary test force has been applied for a short time, the dial gauge is set to zero (reference level). The actual hardness value can then be determined.

The actual test load $$F_1$$ is applied in addition to the preload and the indetor penetrates the material with the total force $$F=F_0+F_1$$. The test load to be set is taken from table books depending on the indenter and the material to be tested.

After the indenter has penetrated the material with a given total force, the test force $$F_1$$ is removed again. Finally, the material is only stressed by the preload $$F_0$$ and the indenter is slightly raised again by the elastic material behavior of the sample. However, contact with the sample remains. The remaining indentation depth $$h$$ (in mm) while maintaining the preload $$F_0$$ is finally measured and used to determine the hardness value.

Depending on the indenter (diamond cone or carbide ball), the unit-less hardness value HR can be determined using the following formulae:

\begin{align}
\label{rockwellhaertewert_1}
&\boxed{HRC, HRA =100-\frac{h}{0,002}} ~~~~~\text{Rockwell hardness for diamond cone} \\[5px]
\label{rockwellhaertewert_2}
&\boxed{HRB, HRF =130-\frac{h}{0,002}} ~~~~~\text{Rockwell hardness for carbide ball} \\[5px]\end{align}

### Testing with diamond cones

For diamond cones, the hardness value is obtained from a reference depth of 0.2 mm. Depending on how far the penetrated indenter reaches this reference depth, a corresponding hardness is assigned to the material. The complete penetration of the indenter to the reference depth obviously means a very soft material; this is assigned a hardness value of 0. If, however, the diamond cone does not penetrate the material at all, an extremely hard material is present, to which the full hardness value 100 is assigned. The scale follows an even subdivision of 0.002 mm (2 µm), so that reaching half the reference depth also corresponds to half the maximum hardness value (Rockwell hardness value 50). When diamond cones are used, the Rockwell scale is thus divided into 100 degrees of hardness.

The testing method with a diamond cone is particularly suitable for very hard materials such as hardened or tempered steels. Apart from special procedures, the preload is 98 N (10 kp). The actual test load can vary depending on the application.

In process variant C, the specimen is subjected to a test load of 1373 N (140 kp). However, especially when testing thin sheets, there is a risk that the material will only bulged out on the opposite side due to the high test force and thus falsify the measurement result.  For this reason, variant A was introduced for diamond cone testing, which operates with a reduced test force of 490 N (50 kp). In addition, there is the less common variant D, in which the hardness value is determined using a test load of 883 N (90 kp). For its determination also equation (\ref{rockwellhaertewert_1}) is used.

Note that in practice Rockwell hardness is not determined by equation (\ref{rockwellhaertewert_1}) and (\ref{rockwellhaertewert_2}) but read directly from a calibrated scale.

### Testing with carbide balls

However, when testing relatively soft materials, the diamond cone would penetrate far too deeply into the material and would lie outside the reference depth of 0.2 mm. Therefore, soft surfaces are tested with carbide balls and the reference depth is extended to 0.26 mm. However, the division of the degrees of hardness in steps of 0.002 mm is maintained. This results in hardness values in the theoretical range of 0 (full indentation depth to 0.26 mm) to 130 (no indentation depth) when using carbide balls.

When using a carbide ball for hardness testing, a main distinction is made between process variants B and F. In contrast to diamond cone testing, they are suitable for softer metals such as construction steels or brass. The ball has a diameter of 1.5875 mm (=1/16 inches). In all process variants the preload is 98 N (10 kp). The procedures differ again only in the actual test load. In variant B the test load is 883 N (90 kp) and in variant F the test load is 490 N (50 kp). Due to its reduced test load, process variant F is particularly suitable for very soft materials such as copper or thin sheets.

### Comparability of hardness values

Hardness values obtained with different process variants cannot be compared with each other. In addition, the hardness value obtained with a certain process method must lie within a certain range. For values outside this range, the method should be changed because the indenter has either penetrated too strongly or too weakly into the material.

• HRC: 20 to 70
• HRA: 20 to 88
• HRB: 20 to 100
• HRF: 60 to 100