Real and Fitted Spherical Indentations

Spherical indentations that rely on original date are analyzed with the physically correct mathematical formula and its integration that take into account the radius over depth changes upon penetration. Linear plots, phase-transition onsets, energies, and pressures are algebraically obtained for germanium, zinc-oxide and gallium-nitride. There are low pressure phase-transitions that correspond to, or are not resolved by hydrostatic anvil onset pressures. This enables the attribution of polymorph structures, by comparing with known structures from pulsed laser deposition or molecular beam epitaxy and twinning. The spherical indentation is the easiest way for the synthesis and further characterization of polymorphs, now available in pure form under diamond calotte and in contact with their corresponding less dense polymorph. The unprecedented results and new possibilities require loading curves from experimental data. These are now easily distinguished from data that are “fitted” to make them concur with widely used unphysical Johnson’s formula for spheres (“ ( ) 3 2 1 2 4 3 P h R E = ”) not taking care of the R/h variation. Its challenge is indispensable, because its use involves “fitting equations” for making the data concur. These faked reports (no “experimental” data) provide dangerous false moduli and theories. The fitted spherical indentation reports with radii ranging from 4 to 250 µm are identified for PDMS, GaAs, Al, Si, SiC, MgO, and Steel. The detailed analysis reveals characteristic features.


Introduction
The spherical indentations were not described by H. Hertz, who only deduced that the pressure of a contacting sphere is related to "impact" area 3/2 , but without How to cite this paper: Kaupp [2]. When the penetration depth was also considered, all involved researchers tried with "elastic theory" with respect to the area (but not to the volume). The not practically useable mathematical formalism became extremely complicated, using a multitude of most extended mathematical techniques with always hundreds of complicated equations, apparently only readable by highly trained mathematicians. Also, Sneddon reported in 1965 a still very complicated hardly usable multi-term and multi-exponent equation in [3] that was forgotten, when Johnson in 1985 presented his simple (but false) " ( ) 3 2 1 2 4 3 P h R E * = " formula for spherical indentations [4]. It promised an easy analysis and direct way to the reduced "Young's modulus" and avoidance of very complicated mathematics. This was highly acclaimed and later taken up into the ISO 14577 standard. This formula was thus used by most authors for spherical indentations by citing [1] [2] [4] without checking content and validity. That is rather strange, because the experimental data of spherical indentations exclude the validity of this formula and every simple circle directly shows its incorrectness: it does not consider the enormous R/h changes during the penetration. For example, the R/h ratios in [5] vary from >1500 to 20 (for h max = 20 nm) or from >480 to 50 (for h max = 3.6 µm). Every researcher should have immediately seen that any R/h term is missing in Johnson's formula and that a cone or a pyramid behaves different from a sphere. The second obvious error is claiming "Young's modulus" that is a unidirectional property, totally different from an indentation modulus. A very complex "equation for fitting" of the depth values is published as equation (9) in (6) The δ in [6] is penetration depth; P is "load" (force). The "fit parameters" are a 0 and P adh . This data-falsification is published in [6] together with a series of crazy modulus values, relying on them. Related other fitting equations might also be in use. Only few authors published true experimental data of spherical indentations. These publications will be analyzed and the wealth of their loading curves is used in Section 3 on the basis of the physically and mathematically correct Equation (1), as deduced in [5] [7]. These honest papers concentrate on Bradby's group of 2002 (e.g. [8]). Correct data for Ge, ZnO, and GaN will be analyzed and compared with hydrostatic anvil pressurizations. Several by fitting falsified spherical indentations from peer reviewed publications will be analyzed in Section 4. They are largely prevailing and all of them are worthless in all respects. But we present easy and strict methods to sort them out. Furthermore, the use of unidirectional "Young's modulus" for the simulation of the loading curve to result in the false Johnson equation is incorrect. The data fitting for concurring with it are severe falsification. The therefrom determined "Young's moduli" are dangerously misleading. The enormous trouble when Young's moduli are equalized with indentation moduli has been amply exemplified in [9]. The publication of moduli from "fitted" data [6] adds another aggravation to these problems. The unsound theories that emerged from falsified data are, of course, also completely worthless. The purpose of this work is to ask Authors, Peer Reviewers, and Editors to reject papers that use data that are falsified by fitting.
Unfortunately, such publications are the basis of certificates for industries, from which these must not deviate. Therefore, numerous false materials' properties create a daily risk for failure upon mechanical stress [5] [10]. Our sorting out techniques below for recent false reports, are clearly developed in this paper.
They shall help to repeat the mechanical characterization with true experimental data, when these are unavailable from the recent authors for genuine publications. Falsified results must no longer be used. The below cited and further authors of falsified reports on spherical indentations obtain the possibility to revisit their published data and the connected non-physical theories. The publication of their experimental data for obtaining correct important materials' properties will be highly welcomed. The false theories and the severe risks from incorrect mechanical properties of technical materials in daily life must be removed.

Methods
The loading data of the materials are taken from the published curves that were enlarged to A4 size. The cone depths were checked with h cone = R(1 − sinβ) where β is the half angle of the cone and R the sphere radius. When pop-ins was present these were repaired [10]. 20 data points were used and pocket calculator with10 decimals. The results are suitably rounded in text and The indices in the Equations (1) (2) (3) are N for normal, s for spherical, a for axis cut when not zero, and py for pyramidal or conical.
The area of the immersed calotte for the onset of the phase-transition pressure calculations is given by its flat surface (πr 2 ) or by its cap surface (2πRh). The radius r is easily obtained by the combination of sinα = (R − h)/R and cosα = R/r when looking at the geometric situation for the penetration of the calotte from the sphere with radius R [5].

Germanium Spherically Indented
The spherical indentation analysis of germanium [7] [8] is reported here for comparison with the further examples in Table 1. Figure 1 provides the basic data from the F N vs πh 3/2 (R/h − 1/3) plot and by using Equations (1) and (2). It shows the penetration resistances k 1 and k 2 as the slopes for the two phases up to 50 mN load.
The phase-transition onset of Figure 1

Zinc Oxide Spherically Indented
The spherical indentation onto ZnO with wurtzite structure follows Equation (1) and it reveals two phase-transitions. The analysis had to be performed after repair [10] of the published pop-ins in [12]. The plotted data are in Figure 2, where the included regression data are the basis for the calculations.
The spherical indentation onto ZnO exhibits two phase-transitions, distinguishing 3 polymorphs in the force range up to 100 mN load, as revealed with the plot by application of Equation (1) to the published original load-depth data.
The inserted linear regression equations are the basis for the calculation of the energetic terms and the pressure data in Table 1.

GaN Spherically Indented
The spherical indentation onto a GaN epilayer (R = 4.2 µm) was reported in 2002 [13] and the analysis (after pop-in repair [10]) with Equation (1) gives the linear plot with two phase-transitions (three linear branches) as shown in Figure   3. The inserted regression formulas allow for the calculation of the onset forces, the energetic terms, and onset pressures in Table 1.
The more recent spherical indentation onto a single crystal of GaN [14] cannot be compared. Unfortunately, the analysis of these data with Equations (1) and ( Figure 1 in [13], showing the required linearity and the kinks for two phase-transitions, ready for the calculation of the results in Table 1; kink positions in Table 1.

Comparison of the Results from Ge, ZnO, and GaN
The results with Ge, ZnO, and GaN are compared in Table 1. These materials cover maximal loads that are 50 mN for Ge, 100 mN for ZnO, and 250 mN for GaN. More phase-transition onsets are to be expected at higher loads. The force for the first phase-transition onset describes the sensitivity of the materials for their stability with respect to mechanical interactions. It is equally reflected by the sequence of the penetration resistance values k 1 (physical hardness) in Figures 1-3. The penetration depth values are not in the same sequence and neither so the first indentation work that are required for reaching the transition onset. But the transition work values are for the first and second transition of ZnO and GaN in the same sequence as the F Nkink values. The data reflect the situation of spherical indentations with the same radius covering one or two phase-transitions per sample. The depths are always very low. The results are calculated from 0 to kink, from kink to kink, and from kink to the maximal force. The transition-energy and the onset pressure values can also be calculated for every force of interest, but they cannot be normalized per force as in the pyramidal case. The energy law requires multiplication of the phase-transition F Nkink values with W indent /W applied = F Nindent /F Napplied [7] [15] when only the penetration is addressed, for obeying the energy law. The advantage of experimental spherical indentations is the reliable pressure calculations at the transition onsets. The close similarity of flat surface and cap surface for the low depths is very favorable. It allows the comparison with hydrostatic pressurizing and new insights are therewith achieved.

Comparison of the Spherical Indentation with Hydrostatic Pressurizing Data and with Synthesized Polymorphs
The pressure data of germanium has already been compared with anvil pressure as combined with X-ray diffraction date. The low anvil pressure of 2.5 GPa required enormous effort for being detected and was long disregarded, as outlined and discussed in [7]. It corresponds with the onset pressure of 2.3 GPa in Table   1. The present analysis of the spherical indentation is by far easier than anvil experiments but we profit from the X-ray proof for the low-pressure GeI to GeIII phase-transition in a highly rewarding manner. The pressure values for ZnO in Table 1 up to 100 mN load are not large enough for the B4 to B1 (NaCl-type) transition, because the hydrostatic pressure for that phase-transition are reported to start at 9 GPa [16] or 9.8 GPa [17]. There are numerous reports on theoretically calculated caged ZnO polymorphs [18] that are not better space-filling than the wurtzite structure (B4) of these materials and need not to be considered here. The theoretical calculations of 7 bulk polymorph structures of ZnO with respect to expected optical properties is also not helpful, because these lack density predictions [19] and  [21]. The fact that the growing of the film succeeded indicate that the pressure for its formation must be rather low. We can thus confidently claim having detected the first synthesis technique for the bulk ZnO B3 polymorph under the diamond calotte cap for further investigation. It is provisionally attributed to the 4 GPa onset pressure. The spacious wurtzite tetrapod with its complicated twinned shape including legs, as obtainable from thermal "vapor deposition on a polymer decorated silicon substrate" [22], must be excluded. The high tendency of ZnO to form twin structures is known [23]. A ZnO twin had already been synthesized by pulsed laser deposition [23]. It can also be obtained by electro-deposition on indium doped tin oxide (ITO) [24].
Again, spherical indentation appears to be the easiest synthesis of this species.
These attributions must be confirmed by on-site X-ray diffraction at a synchrotron, or spectroscopically. This will include their further characterization. The spherical indentation results of ZnO reveal that the hydrostatic anvil pressurizing experiments did not resolve these lower pressure phase-transitions. The almost uniform pressure distribution in the low depth spherical indentations is particularly suitable for the suggested investigations.
The GaN B4 phase (wurtzite) transforms upon hydrostatic pressurizing at 47 GPa into the GaN B1 phase (rock-salt) [25]. Another report found this transition pressure at 37 GPa [26]. These publications do not report on transitions at lower Advances in Materials Physics and Chemistry pressures that are again not remarked under anvil pressurizing. Thus, the B4 to B3 (zinc-blende = sphalerite) transition pressure was not known. The ab-initio calculations of [27] predict a pressure of 11.45 GPa for the B4 into B3 transition. This value is close to our 12 GPa value in Table 1 and we assign accordingly.
Very interesting is our 10 GPa value in Table 1. When compared to ZnO this phase-transition occurs at 0.65 times the depth at 1.75 times the force and at 2.75 times the pressure. This may at first glance indicate twinning of GaN, but why should that require so much force and pressure? Another attribution comes to mind: it could be the formation of the rhombohedral phase (R3m) of GaN that has recently be found in the B3 (zinc-blende, F4-3m) phase of GaN epilayers, as grown on sapphire by molecular beam epitaxy [28]. This would prevent the twinning by pressurizing upon spherical indentation. Unfortunately, no energetics is available for the rhombohedral phase. The rhombohedral GaN (R3m) forms also via migration enhanced encapsulation growth by encapsulation between silicon and graphene [29]. It is thus easily formed. This would also support our attribution of the first phase-transition onset pressure of GaN to the rhombohedral polymorph with space group R3m. Clearly, experimental spherical indentation extends the knowledge from hydrostatic anvil pressurizing. Both assignments should be checked by Synchrotron X-ray diffraction together with the now far easier further analyses under the diamond calotte of these elusive and now easily available polymorphs.

Challenge of Simulations and Data-Fitting for Spherical Indentations
It appears that after publication of the false Johnson's formula requiring an relation, the ISO standard 14,577, several textbooks, and publications believed in it (Section 1). A first glance on that Formula (from the beginning in 1985) should have evidenced that it does not take into account the self-evident change of the R/h ratio during penetration. As the experimental data did not concur with the assumed F N − h 3/2 relation for spheres, Authors did not hesitate to simulate spherical loading curves by using Young's modulus and Poisson's ratio of the material with e.g. the JKR procedures [30] to produce such relation. Thereafter, the experimental depths have been fitted to concur with it by using published "fitting formulas" (e.g. Equation (9) with "fitting parameters" in [6], as is written out in Section 1). That is in fact severe data-treatment. Related other fit-equations might also be in use. But most of the involved scientists stopped with publishing their experimental spherical indentation depths in favor of publishing "fitted depths".
They so avoided the inefficient formula for spheres from [3] and credulously thought to have a simple means for direct determination of the reduced elastic modulus. When doing so they agreed with data manipulation, not recognizing that it was against scientific ethics or practical value. However, anonymous Referees and Editors of books and papers did not stop such data treatments. There were though the important correct publications with Germanium, ZnO, and GaN G. Kaupp from 2002 that are cited and successfully analyzed in Section 3. These pioneering papers contain experimental spherical load-depth curves that were apparently neglected, but they rightfully did not agree with Johnson's formula. So, we urge on their revival here.
The "fitting" of depth data is the creation of fake data for making them obey the falsely prescribed F N vs h 3/2 relation. That relation is however only valid for cones and pyramids [5] [11]. It is more than misleading to unpardonably call the depth values in [6] "experimental data" instead of fitted ones, as in their Figure 4 for a spherical (R = 200 or 192 µm) indentation onto PDMS. The exponent analysis according to Equation (3) gave a perfect straight line with a correlation R 2 = 0.9999 (no phase-transition) [31]. This was already complained against in [5] (with apology for not expressively having criticized the false F N ∝ h 3/2 relation for spherical indentations in [31]). When the correct F N vs πh 3/2 (R/h − 1/3) plot according to Equation (1) is applied to fitted spherical indentations onto materials, one obtains concave curves. That is imaged below for GaAs, Al, and Si. It is also typical for all the further analyzed materials in this Section 4 (including the fitted curve for GaN from [14] in Section 3.  It appears that most published spherical indentations were "fitted" to obey the incorrect Johnson's formula with its false promise to obtain "Young's moduli" values that are however incorrect fake values, not to speak of the fact that Young's moduli are unidirectional moduli. Importantly, we can easily distinguish valid from fitted invalid spherical indentation reports by simply checking their loading curves with plots according to the Equations (1) and (3).
We do not further deal here with the details of the JKR simulations [30] and the data-fitting techniques concerning the spherical indentations, because that is unscientifically false. Such data treating is dangerous and against scientific ethics. The falsified outcome will now be further exemplified with some unbelievably manipulated published loading curves of varied types. It is hoped that this somehow difficult task will help to positively develop this important field of materials' analyses.

Gallium Arsenide "Spherically" and Pyramidally Indented
The spherical indentation (R = 10 µm) of GaAs in [32] does not follow the equation for spherical indentations (1) and the Authors cited paper [4]. The analysis of the published loading curve-after repair of the "pop-in" in their Figure   1 Conversely, the application of Equation (3) for cones and pyramids as falsely claimed by Johnson to the data of [32] leads, after a short initial effect, to two straight lines, the first steeper as the second by forming a kink. This is shown in   Figure 4 from the spherical indentation (R = 10 µm) onto GaAs (100) with the clearly fitted data as taken from [32], showing after an initial effect two straight lines with trial regression lines of F N1 = 433.6h 3/2 + 6.6 and F N2 = 325h 3/2 + 57.9 mN that would falsely simulate an exothermic transition event with an onset at 213 mN and a here provisionally calculated impossible transition energy per mN of −0.101 mNµm/mN. This exothermicity is dangerously in error and the slopes in Figure 5 have no physical meaning.  [11]. Thus, the spherical indentation with R = 10 µm onto GaAs is a misleading fitting artifact showing again the data treatment in [32]. The importance of the argument deserves the printing of the Berkovich indentation analysis of the experimental result from [34] in Figure 6.

Aluminum "Spherically" Indented
The analysis of the spherical indentation onto aluminum [35] reveals a similar outcome as with GaAs. The sphero-conical indenters are precisely described with a cone half angle of 45˚ and nominal radii of 10 µm and 5 µm. Scanning electron microscopic images indicated "well formed" radii of 8.5 and 5 µm down to depths of 2.5 and 1.5 µm, respectively. The nominal radius of 10 µm corresponds to h cone = 2.93 µm. This is considerably larger than the maximal depth of <1 µm) for the indentation data of pure Al. Again, the analysis with Equation (1) for spheres does not give a straight line but the concave curve of Figure 7, which clearly indicates a data fitting for concurring with Johnson's formula and unfortunately ISO standard.
The trial plot with Equation (3) to the spherical indentation in Figure 8, using the same fitted data as for Figure 7 is linear with after the extended initial effect two straight lines. These simulate an exothermic event, because the second branch has the lower slope. This is the false and dangerous result of the fitting error as above with GaAs ( Figure 5): the phase transition of aluminum under load is endothermic, as was already shown with Berkovich indentations onto aluminum in [31].
Again, the calculations of elastic moduli according to false Johnson's formula and its discussion are useless and misleading. The data fitting is again safely confirmed with Figure 7 and Figure 8. The previous branch is again steeper than the following one, simulating a false exothermic event. The known phase-transition of aluminium upon Berkovich indention has the endothermic phase-transition onset at close to 40 mN load (two different sources) [31]. The similarity with the above GaAs case repeats the falsifications by the fittings. Again, the falsified data for an h 3/2 loading parabola retain only the information that there must be a phase-transition at a similar force, as characterized with a Berkovich indentation. Figure 7. Analysis of the published fitted data from the spherical indentation with a nominal radius of 10 µm onto pure aluminum according to Equation (1); data are taken from the upper data points of the multiple partial loading Figure 12(b) in [35]; the concave bending does not agree with Equation (1) and proves the data-manipulation. branches with unsteadiness at about 33 mN loads that would simulate an exothermic event, but not the endothermic phase-transitions of aluminium; the "averaged" data crosses are taken from Figure 12(b) in [35]; the slopes in Figure 8 have no physical meaning.
Data fitting destroys the value of spherical indentation and excludes any use of them. "Fitted" data points must not be told or suggested as being "experimental" ones.

Silicon "Spherically" Indented
Spherical indentations (R = 8.5 µm) of silicon were published in [36]. Photos of a sphere and a description of their indenter setup as "All of the force-displacement measurements were made on a UMIS-2000 instrument", and also the copious description how "The indentations were carried out" are indicating that they might have been without data-fitting. Only the cone angle of the conoidal indenter was not disclosed. However, the authors relied in their text on the incorrect Johnson's formula with the impossible F N vs h 3/2 relation for spherical indentations. This must again be severely challenged. Our analysis of the published "experimental" data pair crosses from their Figure 4(a) (similarly in the further images) give again no straight line when the Equation (1) for spheres is applied to the published data. The concave curve in Figure 9 is obtained instead.
The data-fitting is again additionally secured with the trial Kaupp-plot (F N vs h 3/2 ) in Figure 10, using the so called "observed" but in reality, fitted loading curve data pairs (the data crosses next to the simulated curve in [36]). This second proof is evident by the linearity with exothermic unsteadiness at about 30 mN and very pronouncedly at 80 mN load.
As shown with GaAs and aluminium, the residual information of exothermic unsteadiness tells only that phase-transitions will be found by Berkovich indentation. The steepness of the preceding lines in Figure 10 is again higher than Advances in Materials Physics and Chemistry  (1); the concave form instead of the required linearity proves the data-manipulation in Figure 4(a) of [36], from where these were taken. and 81 mN (data taken from [38]) were analyzed in [39]). They reveal endothermic phase-transitions.
Unfortunately, the more recent spherical indentation with R = 5 µm of [40] also used Johnson's formula and the Authors do not disclose their cone angle.
We therefore do not discuss it here.
They used their SiC-N brand "that is similar to (0001) of the single crystal".
With their sphere radii of 25 µm and 7.5 µm they reached fully elastic or elastic and plastic indentations, respectively. The data were "fitted" to "Hertz spherical contact solution", which means: the disproved Johnson's equation was again the fitting target. These published load-depth curves with fitted depths do again not follow the Equation (1)  The phase-transition pressures of SiC have been calculated to 102 and105 GPa (hexagonal 6H to 1B) or (cubic 3C to 1B = sodium chloride) phase, respectively, and the experimental shock data have them at about 100 GPa [42]. This is far from being reached with the maximal force of 500 mN in [41].
There seems to be the same data fitting techniques in all of the here analyzed cases, but we still need further analyses with a crystalline oxide.

Magnesium Oxide "Spherically" Indented
The authors of [43] used a polished and rinsed (001) surface of MgO for a "spherical indenter in diamond" with an iterated radius of 9.5 µm. Their published Figure 1(c) in [43] is completely reversible up to 300 mN load and 0.4 µm depths. The data fitting is clearly revealed by their reference to [4] and again by the concave plot that results by application of Equation (1) to the published force-depth curve. We need thus not deal any further with this report.

Steel "Spherically" Indented
For rounding up our knowledge of the falsifying effects of data-fitting spherical indentations we also need the analysis of a technical multi-component material.
A spherical indentation onto a standard microhardness steel block (500 HV30; H/E = 0.04), using a sphero-conical tip with radius of 7.2 µm [44] and cone half angle of 45˚ [35] (we calculate h cone = 2.11 µm) appeared appropriate. Actually, the nominal radius of 5 µm ± 6.6 nm (we calculate h cone = 1.46 µm) was in-creased by a "well-fitting simulation" to 7.2 µm. And the "nominal values of 210 GPa and 0.3 were assumed for Young's modulus and Poisson's ratio in all simulations". The load-depth data were taken from Figure 5(a) in [44]. The application of Equation (1) for spherical indentation is valid for every tip radius as long as the h cone value is not surpassed. But the depth values for a certain force are strongly dependent on the tip radius (cf. Section 4.4.). Again, concave curves (not shown here) but not straight lines are obtained both for nominal 5 µm or iterated 7.2 µm radius. These analyses tell that our already multiply complained data-fitting was again performed in [44]. Figure 11  The unsteadiness onsets in Figure 11 cannot be termed phase-transition values, as they would falsely claim "exothermic" behavior and the slopes are worthless. They are equally false as in the cases for GaAs, Al, and Si. Despite the obvious errors of data-fitting for obeying ISO 14577 standards, the authors of [44] relied on the false Johnson formula for "Young's modulus" determinations from a simulated loading curve. These are nothing else than faked results.
Very strange is the publication of Figure 3 in [44]. The tip radius had been shifted from 50 µm ± 1.4 nm to 115 µm, because the data could "only be well fitted" to the simulated Johnson loading curve with R = 115 µm. Such behavior is absurd. The Authors of [44] did not at all recognize that the course of a spherical indentation strongly depends on the radius R. They tried to justify their shifting with the inconceivable claim that "a slight flattening of the spherical shape over the 4.5 µm radius contact region, arising from as little as 0.11 µm at the center, is sufficient to produce an increase in spherical radius of this amount" (from R = 50 µm ± 1.4 nm into R = 115 µm!). Tip radii are mechanical data that cannot be shifted by iterations "as necessary". While all of that already proves the multiple data treatment, we nevertheless checked the fitted data with our test methods.
The application of Equation (1) give the differently sized concave curves (not imaged here) for the data pairs with R = 115 µm and for the simulated curve with nominal R = 50 µm ± 1.4 nm. This confirms the data fitting, as in all of the other tested cases in Section 4.
Furthermore, both simulated curves for the spherical indentation "of a steel standard hardness block (900 HV30 nominal" with H/E = 0.4 in Figure 3 of [44]) give linearity when trial plotted according to Equation (3). The simulated R = 50 µm radius curve gives an exothermic unsteadiness at about 100 mN and 0.150 µm (not imaged here) with our calculated pressure of p = 2.11 GPa. Conversely, the simulated R = 115 µm curve in [44] gives the single straight line in Figure 12 again with the maximal force at 200 mN from the first to the last data point without any unsteadiness. This only tells the high fitting precision that is obtained with the published equations in e.g. [6] (as written down in Section 1).
Our calculated pressure in Figure 12 at its end with 200 mN load gives p = 1.58 GPa. This clearly misses the unsteadiness at p = 2.11 GPa with R = 50 µm, as such a high pressure is not reached with a more than twice as high radius R. It stresses however the enormous influence of the tip radius. In the present case one must be alerted of not mixing up the fitted spherical indentation (that is totally  Figure 12 has no physical meaning. Advances in Materials Physics and Chemistry worthless) with a pyramidal or conical indentation that would give a totally different slope (yet unknown for steel 900 H/E 0.4). It appears rather strange that the paper [44] could pass the Reviewers and Editors for its publication in the Journal of Materials Research. Fortunately, our mathematically sound analyses are now able to detect the invalid simulations, fittings, and iterations, by using Equation (1).
The plot in Figure 12 again underlines the unique power and necessity of checking publication data from spherical indentations with the correctly deduced Formula (1) in [5] and [7]. It reveals: both of the manipulated data in Figure 3 of [44] (R = 115 µm or of nominal 50 µm) are totally worthless due to various simulations, iterations, and fittings for concurring with the disproved Johnson equation that does not take care of the depth-dependent R/h ratio during the penetration. Even worse, the influence of the tip radius to the applied pressure was not acknowledged when nominal tip radii were changed by iterations (5 into 7.2 or 50 into 115 µm).
A very recent report deals with the spherical indentation of several steels with a ball of radius 250 µm, in e.g. Figure 10 of [45]. The Authors used the techniques of [35] and published several "experimental" indentation curves. We analyzed the one for DC01 steel. The test with Formula (1) for spheres gives the concave curve (not imaged here) corresponding to the ones that we always obtained when the spherical indentation data were fitted. This paper [45] covers high loads up to almost 200 N. The trial test with Equation (3) for conical or pyramidal indentations gives three linear branches at very high load with two unsteadiness points, simulating again worthless exothermic behavior. This F N vs h 3/2 plot in Figure 13 proves that the simulation, iteration, and fitting techniques from 1993 [44] and 1995 [35] are unfortunately still in active use. Figure 13. Trial Kaupp-plot according to Equation (3) for the high-load spherical indentation (R approximately 250 µm) onto steel DC 01, using the fitted data as taken from Figure 10 in [45]; it presents after the unresolved initial effect two unsteadiness kink-points that would misleadingly indicate exothermic events; the reasons are the undue simulation, iteration, and data-managing techniques for complying with the incorrect Johnson's equation and with the present ISO-14577 standard; the slopes in Figure 13 have no physical meaning.

Conclusions
A prerequisite for the analysis of spherical indentations is the use of the correctly deduced force-depth relation (Equation (1) that takes into account that the R/h ratio changes strongly during the penetration. Equation (1)

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.