Review of measures and hardness tests
The concept of hardness is included in the science of material strength, but it is also associated with a branch of mechanics called contact mechanics. Heinrich Hertz made pioneering progress in this area [1]. His achievements were continued, among others, by Maximilian Huber [2]. An outline of the history of hardness studies can be found in the study by Walley [3].
The hardness of a material is not as unambiguous as, for example, strength or the modulus of elasticity. There are, therefore, several types of hardness as well as many tests and hardness scales. There are basically three types of hardness as resistance: scratch resistance, static indentation and dynamic indentation [4, 5]. Moreover, permanent (plastic) indentation differs significantly from elastic indentation. Different shapes of indenters are used in the case of static permanent indentation. For hard materials, indenters with a sharp end such as a cone (Rockwell test), a regular pyramid (Vickers test) or an extended pyramid (Knoop test) work well. However, rounded indenters, such as a ball (Brinell, Janka, Krippel and Meyer tests) and a cylinder (Monnin test), are much better suited for wood. Due to the specific structure of wood, we basically distinguish the hardness at the longitudinal sections and the significantly higher hardness at the cross section.
Regardless of the indenter’s shape and the load applied, three basic indenter measures are used alternatively: indenter projection area, total indenter area or indenter depth [the width (diameter) of indentation d is the easiest to measure indentation parameter, but is not a direct measure of its hardness]. The first measure correctly reflecting the average stress was used by Meyer [6,7,8]:
$$H_{{\text{M}}} = \frac{F}{S},$$
(1)
where F is the load force; S is the indentation projection area. Meyer referred this measure to the ball test, but it can also be tried on other indenters. The ball or cylinder projection area is determined by elementary formulae:
$$S = \pi d^{2} /4, \;S = d \cdot L,$$
(2)
where d is the average width of indentation; L is the length of the cylinder contact with the wood. However, it turns out that formula (1) is not widely used and, contrary to appearances, does not experimentally reflect the concept of hardness. In other words, this measure significantly depends on the load used and the size of the indenter. However, for wood, the traditional Janka hardness test [9,10,11,12,13,14] (the values in kg/cm2 used by Janka need to be multiplied by 0.0980665 ≈ 1/10 to change to the value expressed in MPa = N/mm2) is based on the above simple principle of determining hardness [15]:
$$H_{{\text{J}}} = \frac{{F_{\max } }}{{S_{\max } }},$$
(3)
where HJ is the Janka hardness (in MPa) at a full indentation, Fmax is the force inducing full indentation and Smax is the full indentation projection area. In the Janka method, there is no freedom in selecting the load, as it is measured when the ball enters the depth of its radius (diameter section). In standards [9, 10], the indenter speed is 6 mm/min (except for measuring the hardness module at a speed of 1.3 mm/min in standard [9]), which means that the measurement time is about 1 min (as in standard [16]). And the standards [15, 17, 18] recommend a measurement time of 1–2 min, which also effectively results from the recommendations of the standard [19]. A half-depth indentation is often allowed with the same definition of measure [15] (in this standard the hardness is expressed in N/mm2 = MPa and the formula (4) is implicitly written by the expansion of the area S, leading to a factor of 4/3):
$$H_{{{\text{J}}1/2}} = \left. \frac{F}{S} \right|_{ h = R/2} ,$$
(4)
where HJ1/2 is the Janka hardness (in MPa) at half-depth indentation, h is the depth of indentation and R is the indenter radius. A ball with a projection area of 1 cm2 = 100 mm2, less frequently 25π mm2, is normally used. The standards [15, 18, 19] recommend minimum wood sample sizes of 50 × 50 × 50 mm. In modern Janka-type standards [17,18,19], the division of force by any surface area is abandoned. In this case, the hardness characteristics are the force itself (for indenter diameter D = 11.28 mm):
$$F_{{\text{J}}} \left[ {\text{N}} \right] = 100 \cdot H_{{\text{J}}} \left[ {{\text{MPa}}} \right] , \;F_{{\text{J}}}^{*} \left[ {\text{N}} \right] = 100 \cdot H_{{{\text{J}}1/2}} \left[ {{\text{MPa}}} \right] = 4/3 \cdot F \left[ {\text{N}} \right].$$
(5)
Janka hardness is very comfortable for machine use and has been continuously improved [20]. However, accurate measurement of indentation force is somehow problematic with respect to controlling indentation depth, and the use of a type (1) measure transfers its imperfections to (3) or (4). Another example of using the indentation projection area (i.e. the Meyer measure) is the Knoop test (hardness HKN) [6]. The indenter in this test has the shape of a pyramid with a diamond base, but such indenters are not used for wood.
A very attractive concept of hardness measure was proposed by Brinell [21, 22], in which he used the total indentation area A:
$$H_{{\text{B}}} = \frac{F}{A}.$$
(6)
In the SI system, force F is expressed in newtons (N), so the hardness is consistently expressed in MPa = N/mm2. Brinell (like Janka) gave the force in units of load mass, i.e. in kilograms (kg). This was later clarified by the introduction of a unit of kilogram-force (kG or kgf). Unfortunately, the turbulence associated with changing units continues to lead to inaccuracies, and even to one formal error repeated in some standards. In the metal hardness standards [23, 24] and in the Brinell hardness standards for wood-based flooring materials [25, 26], a factor of 0.102 or 1/g appears in formula (6). The purpose of this coefficient is to refer to the traditional unit kg/mm2 instead of N/mm2 = MPa. However, calculating the hardness in kg/mm2 and writing it in N/mm2 is an error in the wooden standards [25, 26] because these units are incomparable, and using an N/mm2 unit requires a conversion rate of 0.102 to kG/mm2 (and not vice versa: 1kG = g·1 kg = 9.80665 N, 1 N ≈ 0.102 kG). Metal hardness standards [23, 24] due to the use of dimensionless of the factor 0.102 do not contain any formal error, but only artificial normalization. Work [27] is proof that this is not just an academic problem. The author quotes the formula from standard [25] containing effectively the factor 0.102, and in the table of wood hardness does not take this factor into account (which results from the order of hardness values given in MPa). Also factor 0.102 does not take into account work [4], while different normalization is used in studies [5, 28]. A more correct and consistent use of the unit kG = kgf is included in the standard for wood-based flooring [29] and in the work [30] on lignofol. Additional information on Brinell hardness can be found in numerous metal standards [31,32,33,34] and in conversion values of different scales [35].
Brinell referred his measure (6) to a ball (usually with a diameter of 10 mm), but the same hardness formula idea is used for a regular square pyramid in the Vickers test (hardness HV) [6, 8]. The phenomenon of Brinell measure for the sphere is based on a few interesting properties of a fragment of the sphere. First of all, the area of the sphere section is directly proportional to its height h (the depth of indentation):
$$A = \pi Dh = {{\pi D\left( {D - \sqrt {D^{2} - d^{2} } } \right)} \mathord{\left/ {\vphantom {{\pi D\left( {D - \sqrt {D^{2} - d^{2} } } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}.$$
(7)
As a result, Brinell hardness is based on the depth of penetration and its total area at the same time, which makes this measure universal. It is worth noting at this point that there are measures based only on the depth of indentation. An example for conical and spherical indenters here is the Rockwell scale (hardness HR) and for cylindrical indenters the Monnin measure (hardness HMO) described below. The Rockwell scale is based on the arbitrary linear function of the depth of indentation h with a negative slope [6].
The surface area (7) in the form of a central expression may also be interpreted as a side area of a cylinder of ball diameter and height h. Moreover, field A can be treated as area of a circle (disk) with a radius l equal to the chord of half an arc of indentation (Fig. 1):
$$A = \pi l^{2} ,\;l = \sqrt {Dh} .$$
(8)
There is also an interesting approximate mathematical relationship between Brinell measure and Meyer measure for small indentations (it follows from the Laurent series):
$$H_{{\text{B}}} \approx H_{{\text{M}}} - F/A_{D} ,$$
(9)
where AD = πD2 is the total sphere area. The Brinell () and (7) hardness test of wood is carried out using a ball with a diameter of D = 10 mm loaded in with a force of 6F = 1 kN (or in practice F = 100 kg = 980.7 N). For floor boards is recommended a square sample of 50 × 50 mm with the original thickness [25, 26]. The application of force should last about 15 s, and its sustaining about 25 s [25, 26]. The paper [36] compares the results of wood indentations made at different speeds of 5, 1 and 0.5 mm/min. In work [37], five different loads of 10, 20, 30, 40 and 50 kg were used for wood with oak hardness.
It turns out that the use of Brinell measure to full indentation gives formally 2 times less HB hardness than HJ hardness. For a half-depth indentation, this ratio is reduced to 4/3 [see below (31), not (5)]––the total area of a half-depth indentation is equal to the cross-sectional area of the ball and both areas are 4/3 times greater than the projection area of the half-depth indentation. This shows that HB measure is not easily proportional to HJ, HJ1/2, i.e. basically to HM. This means that at least one of these measures is not a homogeneous measure, independent of the used load or size of the indentation. The wood standards [19] (strictly, this standard assumes (5), i.e. that for half-depth indentation F = 3/4 Fmax, because S = 3/4 Smax), [18] assume uniformity of HJ and HJ1/2 measures for half-depth and full indentations. However, such an assumption does not have the status of confirmation in experiments or in theory [see power laws (14)–(18) and the implications of the Brinell measure (31)]. In turn, standards for wooden floors [25, 26] recommending a constant load of 1 kN in the Brinell method implicitly assume the homogeneity of the HB measure. The point is that for softwood the indentations will be large and for hardwood the indentations will be small. For example, the research of Karl Huber [38] showed that none of measures HB and HM is sufficiently homogeneous [30]. However, even a greater use of Brinell-type measures (for spheres and regular pyramids) provides that it is a measure that works better than the HM measure. An example of this is the Krippel method (hardness HKR), which combines the idea of Janka and Brinell, because it uses a measure of HB type:
$$H_{{{\text{KR}}}} = \left. \frac{F}{A} \right|_{{ d \approx D/2 \leftarrow h \approx {\text{2 mm}}}} .$$
(10)
However, Meyer measure for this method would only give a value of about 1.07 times greater. The Krippel method is based on a fixed-depth indentation. For the original values h = 2 mm and A = 200 mm2, the indentations are approximately half diameter and the ball should have a diameter D = 100/π mm = 31.831 mm.
The use of cylindrical rather than spherical indenters [39,40,41] has become popular in France. Monnin [42, 43] proposed here a hardness measure HMO inversely proportional to the depth of indentation h [44]:
$$H_{{{\text{MO}}}} = \frac{F}{100 h L} \propto \frac{1}{h}.$$
(11)
For this reason such a definition gave some similarity to Brinell measure. Unfortunately, in the case of a cylinder, the depth of indentation is not proportional to any actual surface area, but to a certain conventional area AMO adopted by Monnin:
$$A_{{{\text{MO}}}} = 100 h L.$$
(12)
Despite the use of the L parameter in the definition, which formally gives a standard stress unit of 1 N/mm2 = 1 MPa, the HMO measure is not comparable to the above measures. The divisor 100 was selected here in such a way that the measure (11) calculated in N/mm2 units for known species ranged from 1 to 10. Such values in order of magnitude corresponded to Brinell measure and Janka measure of wood expressed in kG/mm2. Therefore, a slightly different standardization of Monnin’s measurement (hardness \(H_{{\text{MO}}}^{*}\)) is also justified:
$$H_{{{\text{MO}}}}^{*} = \frac{F}{10 h L},\;A_{{{\text{MO}}}}^{*} = 10 h L,$$
(13)
where \(A_{{{\text{MO}}}}^{*}\) is the conventional area corrected by factor 0.1 ≈ 0.102 in Monnin hardness. According to sources [30, 45] Monnin used a cylinder with a diameter of D = 30 mm and wooden samples with a cross section of 20 × 20 mm (L = 20 mm). The force used was F = 1961 N (200 kg), i.e. 100 kg of load per 1 cm of cylinder length [45]. For soft wood, the load was halved. The time of measurement was to exceed 5 s [45]. According to Ref. [44], the indenter had a speed of 0.5 mm/min for all three used diameters of 10, 30 and 50 mm. More standards concerning Monnin test are given by Starecki [30]. A practical comparison between Monnin test and Janka test was made by Sunley [46].
It turns out that practically none of the measures (1)–(11) discussed above is strictly compliant with Meyer empirical power law (for n different from 2 and 1):
$$F \propto d^{n} .$$
(14)
Meyer referred to this law mainly to balls, but it can also refer to, at least, cylinders. In case of balls, conformity (14) with the Meyer measure occurs only for the exponent n = 2 associated with Kick [28, 30, 47,48,49,50], true here only for perfectly plastic bodies. For this reason, Meyer law is sometimes incorrectly called the Kick law [51, 52]. In fact, Kick law says that similar stresses cause similar deformations and on this basis derives that for pyramidal shapes of indenters n = 2 (or n = 1 for elongated shapes). The Kick Law is also given in the equivalent form for these shapes in the following forms \(F \propto h^{2}\) [28], and for balls a full square function of the depth of indentation h [28, 53] is considered. The generalization of Kick law to the square function of the d is called the PSR of Li and Bradt or Hays and Kendall model [48, 49, 51, 54]. It can therefore be concluded that the ordinary Kick law applies only to pyramids and cones (n = 2) or wedge-shaped prisms (n = 1), but not to balls or cylinders. On the other hand, the exponent in Meyer law (14) for balls is in the range of 2 ≤ n ≤ 3 [50] (and for cylinders within the range 1 ≤ n ≤ 2). Nevertheless, some sources give a narrower range of exponents 2 ≤ n ≤ 2.5 [6]. Huber [38] determined n ≈ 1.5 or n ≈ 3 depending on the longitudinal or transverse cross section of wood. The case of n < 2 is called indentation size effect (ISE) and n > 2 reverse ISE [50, 54]. The maximum values correspond to a perfectly elastic materials. This fact is confirmed by the contact mechanics formulae for the diameter of the indentation from the ball [55, 56]:
$$d = \sqrt[3]{\frac{3FD}{E}},$$
(15)
and the width of indentation from the cylinder
$$d = \sqrt {\frac{8FD}{{\pi LE}}} ,$$
(16)
where E is the effective surface elasticity modulus. For comparison, the elastic indentation by a loading head in bending is expressed as follows [56]:
$$d = \sqrt[3]{{\frac{8FDT}{{\pi LE}}}},$$
(17)
where T is the sample height. Plastic (or plastic–elastic) indentations from the loading head were tested at work [57]. Note that formulae (15, 16), contrary to Meyer law (14), also take into account the diameter of indenter D. The empirical equivalent (15) for non-elastic indentations is the formula needed to determine International Rubber Hardness Degree (IRHD) [6]:
$$\frac{F}{E} = 0.00242 D^{0.65} h^{1.35} .$$
(18)
The original formula contains increments of forces and indentations, not absolute values, but this is not crucial for this work.
Justification and purpose of the work
The idea is to modify hardness formulae of Brinell HB (6) and Monnin HMO (11) in the direction of Meyer law (14) and relations taking into account the indenter diameter (15)–(18). The modification has the above theoretical basis, but is generally based on experimental research. Meyer law (14) should not be confused with Meyer hardness measure (1). In addition, the hardness measure (mathematical formula) should not be confused with the hardness measurement (experimental test).
The point is that the ideal measure of hardness should reflect some physical law or definition in relation to surface deformation (not just wood). Meanwhile, this criterion is met only by a simple and imperfect Meyer measure HM (1) referring to the definition of pressure. The hardness of Janka is based on this measure. However, everything indicates that for the spherical indenter, the physics of partial indentation is better reflected in the Brinell HB measure (6) than in the Meyer HM measure. Physically, the effectiveness of the Brinell measure is not explained either by the use of the total surface area A of the indentation, nor even the use of the linearly related depth h of the indentation. In view of the above, it is justified to refer to the empirical power law of Meyer (14) with the power n ≠ 2 for balls and n ≠ 1 for cylinders. In contrast to the Brinell measure, Meyer law has a theoretical basis in the form of relationships (15), (16) and (17).
The search for a better measure of hardness is at the same time a search for a more universal and precise hardness law. The law of hardness is understood here as a mathematical formula associating indentation force with parameters describing the indentation size and indenter size. Meyer law still requires a generalization that will take into account the diameter D of the indenter as in formula (18). In addition, Meyer law (14) requires the introduction of a material coefficient that would describe the hardness of the material (and not just the type of material as Meyer index n).
The main unification aspect of the work is the search for similar formulae for measures of hardness for spherical and cylindrical indenters (in Brinell- and Monnin-type tests), which would additionally have similar values. The Brinell measure for half-width indentation was considered the reference measure (see methodological assumptions vii and viii). An additional aspect of unification is the possibility of using the hardness measure for full indentation (or for half depth) without obtaining a significantly different value relative to the hardness obtained with small indentations. Knowledge of a measure having this property would allow a rational conversion of Janka hardness into a new hardness measure or Brinell measure. Rational conversion of hardness measures is already a contribution to unification.