_{1}

^{*}

This paper analyzes the force vs depth loading curves of conical, pyramidal, wedged and for spherical indentations on a strict mathematical basis by explicit use of the indenter geometries rather than on still world-wide used iterated “contact depths” with elastic theory and violation of the energy law. The now correctly analyzed loading curves provide as yet undetectable phase-transition. For the spherical indentations, this includes an obvious correction for the varying depth/radius ratio, which had previously been disregarded. Only algebraic formulas are now used for the calculation of material’s properties without data-fittings, or simplifications, or false simulations. Penetration resistance differences of materials’ polymorphs provide precise intersection values as kink unsteadiness by equalization of linear regression lines from mathematically linearized loading curves. These intersections indicate phase transition onset values for depth and force. The precise and correct determination of phase-transition onsets allows for energy and phase-transition energy calculations. The unprecedented algebraic equations are most simply and mathematically reproducibly deduced. There are no restrictions for elastic and/or plastic behavior and no use of different formulas for different force ranges. The novel indentation formulas reveal unprecedented access to the onset, energy and transition energy of phase-transitions. This is now also achieved for spherical indentations. Their formula as deduced for plotting is reformulated for integrations. The distinction of applied work (
*W _{applied}*) and indentation work (

*W*) allows now for comparing spherical with pyramidal indentation phase-transitions. Only low energy phase-transitions from pyramidal indentation may be missed in spherical indentations. The rather low penetration depths of sphere calottes calculate very close for cap and flat area values. This allows for the calculation of the indentation phase-transition onset pressure and thus the successful comparison with hydrostatic anvil pressurizing results. This is very helpful for their interpretations, as low energy phase-transitions are often missed under the anvil, and it further strengthens the unparalleled ease of the indentation techniques. Exemplification is reported for pyramidal, spherical, and hydrostatic anvil stressing by the numerical analysis of published germanium data. The previous widely accepted historical indentation theories and standards are challenged. Falsely simulated and even published so-called “experimental” indentation data from the literature can most easily be checked. They are mathematically unsound and their correction is urgently necessary for scientific reasons and daily safety with stressed materials. The motivation for this paper is the challenge of worldwide incorrect ISO 14577 standards for false and incomplete characterization of materials. The minimization of catastrophic failures e.g. in aviation requires the strengthening and the advancements of the mathematical truth by using our closed formulas that are based on undeniable geometric and algebraic calculation rules.

_{indent}Normal indentations onto flat surfaces are a long-term mathematical problem initially posed by Boussinesq in 1882 [^{3/2} where α describes the impact area. He repeated that he did not describe penetration in [^{3/2}). Much effort was put to the refinement in the early 1900ies, but advancements became over complicated, and also conical indentation had to be mathematically described. Really useful equations for practical use had to wait until 1939, when Love in [_{N} for normal force and do not violate the energy law). Thus, the depth was squared for cones and pyramids in [^{*} is reduced elastic modulus) for the sphere. This again did not consider the particular geometry of the sphere calotte. Interestingly, Oliver and Pharr cited in [^{m} for spheres “with m = 1.5”, arguing that the sphere can be described as a solid of revolution giving “a smooth function”. But they did not tell that this was Johnson’s “summarized” formula in [_{N} vs h^{3/2} plot from the present author’s group. This plot (later from believers disdainfully termed as “Kaupp-plot”) disproved and disproves the more than 1000-fold falsely claimed exponent “2” on h (as enforced by ISO) for all conical, pyramidal and wedged indentations. It also discloses whether published spherical indents were truly spherical. However, several researchers continued to simulate spherical indentations as one exponent parabola with h^{3/2} according to Johnson’s formula in [_{N} vs h^{3/2} plot (we now also call it “Kaupp-plot”), which, of course, cannot give straight lines neither for the simulation and nor for the published so called “experimental” curves. Such publications are disastrous and some examples for such clearly manipulated data are published in [

All mathematical deductions in the Introduction started with the indented surface area, and by applying the elasticity theory. They ran into enormous mathematical problems that could for a long time not be solved since 1882 with practically useable formulas. Only the still incorrect formulas of Love in [_{N} vs h^{3/2} parabola for cones was geometrically deduced in 2016 using basic algebra in [

The normal force vs depth curves are empirically described since 2004 in [_{N} is the normal force (we do not use “P” as in the formulas of the Introduction), h (µm) is the depth, and k (mN/µm^{3/2}) is the material’s penetration resistance. The indentation work W_{indent} (mNµm) in Formula (2) is obtained by integration. The constantly increased normal force from zero to the same force gives the applied work W_{applied} (mNµm) in Formula (3). For the maximal force F_{Nmax} we substitute F_{N} of Formula (1) in Formula (3) and obtain the W_{applied}/W_{indent} ratio of 5/4, which is universally valid for all materials upon conical, pyramidal and wedged indentations. Clearly 20% of W_{applied} (and thus F_{N}) is not used for the penetration with a cone, pyramid or wedge. The non-consideration is the already mentioned violation of the energy law that led to a false exponent on h, which is still enforced by ISO, but urgent subject to change. This is the reason why we use F_{N} and not “P” for the normal force.

F N = k h 3 / 2 (1)

W i n d e n t = 0.4 k h 5 / 2 (2)

W a p p l i e d = 0.5 F N h (3)

The geometric deduction of the correct exponent (3/2 but not 2) in [_{Nv} with the force for total pressure formation F_{Np} in Equation (4). Now one considers that the total pressure must be proportional to the immersed volume of the cone in Equation (5), so that F_{Np} is proportional to h^{3} and h proportional to F N p 1 / 3 . When the exponent n is 1/3 the exponent m must be 2/3 and Equation (4) becomes Equation (6). As F N p 1 / 3 is lost for the indentation only F N v 2 / 3 is proportional to the depth in Equation (7). The deduction is completed with inclusion of the materials property factor, which is the penetration resistance or the physical hardness k_{v} or k = 1.25k_{v} (mN/µm^{3/2}) as the proportional constant to give Equation (8_{v}) and by its multiplication of both sides with the W_{applied}/W_{indent} ratio = 1.25 that is herewith also deduced. Equation (8_{v}) is used when only the volume formation must be considered as e.g. in Section 5. Equation (1) thus describes the whole indentation.

F N = F N v m F N p n (4)

V c o n e = h 3 π tan α 2 / 3 (5)

F N = F N v 2 / 3 F N p 1 / 3 (6)

F N v 2 / 3 ∝ h or F N v ∝ h 2 / 3 (7)

F N v = k v h 3 / 2 (8v)

Equation (1) is generally valid for all materials with respect to the used cone angle and for pyramids and wedges with their effective cone angles (e.g. α = 70.3˚ for the most used Berkovich pyramid). Normalization of these with tanα^{2} is possible when different indenter geometries among these must be compared. The k_{v} value of Equation (8_{v}) is the physical hardness with respect to the force F_{Nv} that is responsible for the volume formation. The technically more important k = 1.25k_{v} value of Equation (1) is the physical hardness for the whole indentation force with F_{N} = 1.25F_{Nv}. We thus distinguish two different physical hardness values. This has to be taken into account for the comparison of e.g. pyramidal indentations with spherical ones with different W_{applied}/W_{indent} = F_{N}/F_{Nv} ratios. Unlike iterations for false “ISO-hardness” we obtain from the correct analysis of Equation (1) by plotting F_{N} vs h^{3/2} (the “Kaupp-plot”) linear regression lines via Excel^{(R)} calculation and detect the phase-transition onsets at the kink unsteadiness of intersecting regression lines, because different polymorphs exhibit different k-values. The calculated intersection point by equalization of the regression lines at F_{Nkink} and h k i n k 3 / 2 reveals also the practically important indentation energies [cf Equation (2) and Equation (3)]. Furthermore, the phase-transition energy can be calculated by using the Equation (9) through (13). The application Equation (9) and Equation (11) contain the corrections for axis-cut F_{a} when not zero. All of these have been deduced and are repeated here in modified form [

W a p p l i e d 1 = 0.5 h k i n k ( F N k i n k + F a 1 ) (9)

W i n d e n t 1 = 0.8 W a p p l i e d 1 (10)

W i n d e n t 2 = 0.4 ( h 5 / 2 − h k i n k 5 / 2 ) + F a 2 ( h 2 − h k i n k ) (11)

full W a p p l i e d = 0.5 F N 2 h 2 (12)

W t r a n s i t i o n = full W a p p l i e d − Σ W a p p l i e d (13)

The fast calculation of W_{applied}_{1} for cones, pyramids and wedges with Equation (10) avoids the integration of Equation (1). Higher phase-transition W_{indents} must be integrated from kink to the next kink etc. The F_{N}_{2}-h_{2} pair can be freely chosen above the kink_{1}. In the case of several phase-transitions one proceeds from kink to further kink and after the last phase-transition there is free choice for the F_{Nn}-h_{n} pair. A practical example will be calculated in Section 3.

A one member “parabola with exponent 3/2” for spheres in [_{N} vs h curve of sphere calottes (with sphere radius R and calotte radius r) is more involved, because the depth related R/h ratio is changing during the penetration as shown in [^{3} but with the varying dimensionless R/h term for its π ( R / h − 1 / 3 ) correction. This correction term can be treated like a variable factor that has to be separately applied for every force point of the plot, according to Equation (18) that is not at all a one-member parabola. The sequence of the deduction is now similar to the one for cones. We consider again that the total pressure must be proportional to the immersed volume of Equation (14) and get the Formulas (15) for F_{N}, (16) for F N p 1 / 3 and (17) for F N v 2 / 3 . Equation (18_{v}) is obtained for F_{Nv} after multiplication with the material’s proportionality factor k_{sv} (_{s} for sphere; _{v} for volume). Equation (18_{v}) describes only the volume formation and it is used when the penetration part F_{Nv} must be exclusively considered. Equation (18) for the whole indentation is obtained by multiplication of both F_{Nv} and k_{v} with the W_{applied}/W_{indent} ratio. Equation (18) is used for plotting F_{N} vs { h 3 / 2 π ( R / h − 1 / 3 ) } . One obtains the penetration resistance values k_{s}_{1} and k_{s}_{2} (mN/µm^{3/2}), the phase-transition onset with F_{Nkink}, the { h 3 / 2 π ( R / h − 1 / 3 ) } value, and the W_{applied}_{1} at the kink position from the intersection of the regression lines. The necessary h_{kink} must not be calculated. It is available from the F_{N}-h for F_{N} = F_{Nkink} curve. For the now necessary calculation of W_{indent} we reformulate Equation (18) and add the axis cut F_{a} (+, 0, or −) to give Equation (19). Its integration gives Equation (20) for W_{indent}. The W_{applied}_{2} is calculated from F_{Nkink} to F_{N}_{2} (here chosen at F_{Nmax}). The balance of full W_{applied} − SW_{applied} is the phase-transition energy W_{transition} as in Equation (13). A practical example will be calculated in Section 4. Equation (18_{v}) will be used in Sections 4, 5, and 6.

V = h 3 π ( R / h − 1 / 3 ) (14)

F N ∝ h 3 π ( R / h − 1 / 3 ) (15)

F N p 1 / 3 ∝ h 1 / 3 [ π ( R / h − 1 / 3 ) ] 1 / 3 (16)

F N v 2 / 3 ∝ h 2 / 3 [ π ( R / h − 1 / 3 ) ] 2 / 3 (17)

F N v = k s v h 3 / 2 π ( R / h − 1 / 3 ) (18v)

F N = k s h 3 / 2 π ( R / h − 1 / 3 ) (18)

F N = k s π R h 1 / 2 − k s π h 3 / 2 / 3 + F a (19)

W i n d e n t = ∫ F N k i n k d h = 2 / 3 ⋅ k s π R h 3 / 2 − 2 / 15 ⋅ k s π h 5 / 2 + Δ F h (20)

For the numerical exemplification, the published data of the semiconductor germanium are chosen from the literature. This covers pyramidal (Berkovich diamond), spherical (diamond) indentations, and hydrostatic anvil compression. The analysis of the Berkovich indentation onto cubic germanium from [

equations after a short initial surface effect (including the inevitable tip rounding). The proportionality of W_{indent} = 0.8W_{applied} (Equation (2) and Equation (3)) for a loading parabola with exponent 3/2 in [_{indent} for every chosen work so that normalization per force unit provides comparable values for different materials. The phase-transition kink position by equalization of the regression lines is at 4.149 mN and 0.151 µm. The pristine polymorph withstands a phase-transition up to 0.1342 mNµm applied work and W_{indent}_{1} is thus 0.1074 mNµm. With W_{applied}_{2} = 1.7927 and full W_{applied} = 2.3059 mNµm the phase transition energy into the second polymorph calculates easily as full W_{applied} − SW_{applied} (Equation (13)) to give W_{transition} = 0.37898 mNµm from kink to 15.25 mN load. These are after normalization per mN 0.03414 mNµm/mN. Such transition energies based on physically valid application of geometry and arithmetic calculation rules are not available by any other means. Unfortunately, we could not search for further phase-transitions of germanium, due the smoothness lack by the repeated load-unload sequences at higher loads in [

The spherical indentation onto germanium at a rate of 7 mN/s follows Equation (18) and Equation (19), but not Johnson’s equation of a one exponent F_{N}-h^{3/2} parabola for spheres in [_{N} vs h curve far away from the phase-transition position are not corrected for, because there is no force hold interruptions and there are no “discontinuities” in the F_{N} vs h curve of [

Only our physically correct two-exponent parabola plot for spherical indentations in _{N} vs { h 3 / 2 π ( R / h − 1 / 3 ) } for germanium, according to Equation (18). It visualizes the linearity for obtaining the penetration resistance values k (mN/µm^{3/2}) with respect to the spherical tip with radius 4.2 µm. The phase-transition onset is clearly seen by the kink and the very different k-values of the polymorphs. The regression line results are inserted. The slightly steeper data above the horizontal shut-off line are not included in the regression.

We do not dare to claim a “second kink” at about 46 mN load due to the short penetration length, but did not include the data pairs above 46 mN load in the regression. There is also a risk of spheres’ quality at higher depths. The 4.2 µm

diamond sphere was however in good shape, at least up to about 0.4 µm depths. The inserted regression lines give the materials’ dependent penetration resistance values k_{s} (mN/m^{3/2}) of the respective polymorphs, because the uninterruptedly varying geometric factor is taken care of by the R/h ratios in Equation (18). The equalization of the (not drawn) regression lines provides the sharp phase transition onset at the kink position at 5.197 { h 3 / 2 π ( R / h − 1 / 3 ) } µm^{3/2}. One obtains F_{Nkink} = 10.703 mN by insertion in any one of the two regression line equations. With the F_{Nkink} value one obtains h_{kink}= 0.124 µm from the F_{N} vs h loading curve and according to Equation (9) also W_{applied}_{1} = 0.67704 mNµm. The integrated Equation (20) provides W_{indent}_{1} = 0.4756 mNµm.

The W_{inden}_{t}/W_{applied} ratio is here not 0.8 as in the case of conical, pyramidal, and wedged indentations (cf Section 3). It changes for every point at spherical indentations as in _{indent}_{2} and W_{applied}_{2}. In the absence of a second kink within the loading range we integrated from F_{Nkink} to F_{Nmax} at linearly interpolated 50 mN and the interpolated depth of h_{max} = 0.4444 µm. Equation (20) provides W_{indent}_{2} = 7.07438 mNµm for the sphere with radius 4.2 µm. Equation (9) is correspondingly used for (50 − F_{Nkink}_{1}) and (h_{max} − h_{kink}) to give W_{applied}_{2} = 8.8219 mNµm up to 50 mN load for the sphere of radius 4.2 µm. The sum (W_{applied}_{1} + W_{applied}_{2}) is 9.4989. As the full applied work (Equation (12)) from 0 to F_{Nmax} and h_{max} is 11.1111 mNµm one obtains the endothermic balance as full W_{applied} – SW_{applied} = W_{transition} = 1.6122 mNµm, according to Equation (13) for the phase transition energy of germanium at the kink onset position for the whole indentation force.

It should be noted that the energy calculations for spheres are with respect to the R/h value at the kink position. The non-constancy of the correction factor in Equation (18) prevents a normalization of the energy values per mN. Every energy value must be separately calculated when compared with the values from conical, pyramidal, or wedged indenters. Only for the latter is it possible to interpolate and even interconvert energy values via (effective) cone half angle. This disadvantage of spherical indentations is outweighed by the pressure distribution over an almost plane area. The sphere calotte radius for R = 4.2 µm and h = 0.124 µm is easily calculated with sin β = ( R − h ) / R and cos β = r / R , to give r = 1.01327 µm. Thus, the flat πr^{2} area is here 3.226 µm^{2}, which is similar to the calotte-cap 2πRh surface area of 3.272 µm^{2}. For the pressure calculation we need the force part for the penetration of Equation (18_{v}), in accordance with the energy law. At the phase transition onset position, the W_{indent}_{1}/W_{applied}_{1} ratio is 0.4756/0.67704 = 0.70247. Therefore, also F_{Nindent}_{1}/F_{Napplied}_{1} = 0.70247. When this factor is multiplied with the whole force of 10.703 mN one obtains the force at the phase-transformation onset that is only responsible for the penetration and calculates to 7.5185 mN. The penetration force/area is thus 2.331 or 2.298 mN/µm^{2} (which is better known as GPa). We do not decide which of the two surfaces give the better value, but upon rounding both read 2.3 GPa. The almost perfect correspondence of this pressure value with much more difficultly obtained hydrostatic anvil pressurization results is discussed in Section 6.

It is certainly more precise to indent with diamond indenters like for example with Berkovich indenter, the smooth diamond faces of which are everywhere uniform. Its inevitable tip rounding ends at the very low h c o n e = R ( 1 − sin 70.3 ) and it is mostly hidden within the diverse surface effects that are abandoned. Conversely, ideal spherical diamond tips require more expertise at their production and control of a constant radius for a certain guarantied height. Such radii are not very precisely known and hardly reproducible. The next difference is the penetration mathematics that is very easy for cones, pyramids, and wedges, but more complicated for spheres. These questions could be answered now with the Equation (18) and Equation (19). Different are the penetration depths with pyramids and spheres. The kink values of the phase-transitions for Berkovich are 4.149 mN at 0.151 ∙ 0.8 µm (Equation (8_{v})) and for the sphere 10.703 mN at 0.124 ∙ 0.70247 µm [Equation (18_{v}), we must here use the penetration force]. The corresponding W_{indent}_{1} values are at 0.10736 mN and 0.4756 µm for Berkovich and sphere, respectively. Also, the comparison of the corresponding phase-transition energies of 0.2487 and 1.6217 mNµm shows that we cannot claim without further data that these phase-transitions did produce the same polymorph. The Berkovich should have reached a much deeper penetration depth than the sphere and the values of W_{indent} and W_{transition} better comparable. It might be twinning of germanium as had been suggested in [

An important advantage of the sphere calotte geometry is its flat πr^{2} area value that is very similar to the one of the 2πRh cap area at low depth (here 3.226 and 3.272 µm^{2}, respectively). One calculates reliable force over area pressure values (mN/µm^{2}, better known as GPa). These do not contain the errors of extensive simulations for ISO hardness for indentations in [_{Nkink} = 10.703 × 0.70247 mN gives 2.321 or 2.298 mN/µm^{2} (GPa) transition pressure when divided by the penetration area at the phase-transition onset (Section 4). This pressure value is smaller than those of the most cited anvil experiments that require about 8 to 11 GPa, depending on the hydrostatic purity of the pressure transfer in [

This paper compares the mathematical descriptions of conical, pyramidal and wretched indentations with the spherical ones and it numerically exemplifies them with literature data from germanium. The geometrically based mathematical deductions result in arithmetic formulas and application equations. Unprecedented applications are developed. The physically correct formulas are up to replace the false formulas of ISO 14577 that rely on false premises that are still violating the energy law by using elastic theory and iterated projected (contact) area. We therefore urgently ask to abandon historical beliefs. ISO 14577 still standardizes incorrect standards and procedures that do not match with reality. The reasons for the inconsistencies for more than a century are unrepeatable extremely complicated “mathematical deductions”. In addition to that ISO-ASTM use experimentally false standards not only due to not considering phase-transitions under load—that they cannot detect with their false formulas—, but also with poor force linearities and mix-ups of the standards. It is therefore clear that they could not create valid closed equations for indentations. Rather the false belief required numerous iterations, approximations, data-adjustments, and simulations. Any control of experimental data had been impeded and did not occur in the mainstream. It produced false thinking and encouraged various data manipulations. Some striking examples are challenged in [

We complete and extend in this paper the geometric solutions for conical, pyramidal, wedged and spherical indentations by physically sound use of the indenter volume for the coupled pressure and penetration events and we deduce valid formulas and application equations. The calculation of the energies and transition energies of phase transitions upon indentation with the Berkovich indenter is calculated in the usual way as in [_{N} = kh^{3/2} from [

The still worldwide accepted Johnson equation for spherical indentations (“ P = 4 / 3 E * R 1 / 2 h 3 / 2 ”) has again been disproved with a trial Kaupp-plot F_{N} vs h^{3/2} that is not linear.

The spherical loading curve does not at all proceed as a one exponent parabola, because the volume of the sphere calotte V = h 2 π ( R − h / 3 ) can be transformed into V = h 3 π ( R / h − 1 / 3 ) by multiplication with 1 = h/h. In correspondence to the deduction of Equation (4) into Equation (8) [_{v}) and (18) as the plottable equation with a point by point variable dimensionless correction term. It provides the materials constants k_{s}_{1} and k_{s}_{2} (mN/µm^{3/2}) and the onset values of the phase-transition at the intersection of the regression lines. For the energetic terms one transforms Equation (18) into Equation (19) F N = k s π R h 1 / 2 − k s π h 3 / 2 / 3 + Δ F for the integrations by taking care of their ranges to obtain the indentation energies W_{indent}_{1} at the intersection point and W_{indent}_{2} at an arbitrary point. The chosen point must be the same for the full applied work (full W_{applied} = 0.5F_{N}_{2}h_{2}). The unprecedented phase-transition energy W_{transition} is then simply the balance of full W_{applied} minus (W_{applied}_{1} + W_{applied}_{2}).

The unprecedented indentation energy and the phase-transition energy also for spherical indentations enabled the comparison of Berkovich indentations with spherical ones. This was only possible with the penetration forces (not with the full forces from the loading curves!) and thus also with the corresponding W_{indent}_{1} values. Only these are comparable and the spherical W_{applied}/W_{indent} ratios are depth dependent.

The indentations onto germanium were exemplified and the outcome is different with Berkovich at 4.149 mN load and with 42 µm sphere radius at 10.703 mN load. These are different phase-transitions. The low energy transition (most likely twinning) must have been lost in the spherical case where one detects the Ge-I into Ge-III transition.

Importantly, the spherical indentation reveals a reliable pressure calculation for the comparison with published hydrostatic anvil results, because the flat calotte surface at the low penetration depths is only slightly smaller than the cap surface (here 1.4%). The average calculated force/µm^{2} value of the phase-transition pressure amounts to 2.3 mN/µm^{2} (GPa) in excellent correspondence with the published anvil value of 2.5 GPa. These results support the interpretation of the hydrostatic anvil results. Such now possible comparisons are very rewarding.

The presented geometric results are not only comprehensive for academia, but the search for phase-transition onset and energy under load is of immense importance for practical applications and safety, because polymorph interfaces are prominent sites of cracking and crashing as imaged in [

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

Kaupp, G. (2020) Valid Geometric Solutions for Indentations with Algebraic Calculations. Advances in Pure Mathematics, 10, 322-336. https://doi.org/10.4236/apm.2020.105019