Phase-transitions upon indentations cover all types of materials and the penetration resistances = physical hardnesses) of the two branches in their Kaupp-plots (they profit from the physical and mathematical proved Equations (1)-(7). Typical examples are collected in Table 1. All units are made equal by using Equation (2) with the corresponding units. It contains salts, oxides, metal, semiconductor, metal organic framework (MOF), polymer composite (wood), organic crystals, and the complicated mineral Mica. They cover nano- and micro-indentations, using Berkovich or for the organics cube corner indenters. The maximal forces are given to indicate the experimentally studied ranges. This does not exclude further phase-transitions at higher loads as has been shown with NaCl, where 4 phase-transitions (6 polymorphs) had been located [17] . Further polymorphs are even likely for calcite, where already two phase-transitions (4 polymorphs) are energetically analyzed in Table 1 (further examples are in Tables 3-5). The k-, F_N- and h_kink-values do not systematically relate to their unprecedented transition-works that are normalized per mN of their ranges. We did not transform the mNµm or µNnm units into Joules for easier transformation possibility into different units. These quantities relate to the chemistry of the materials and the transitions can be mostly endothermic but also exothermic. It is seen that the less stable aragonite has a lower endothermic W_transition than the calcite polymorph. The second transition of calcite affords about 7 times more energy than its first one. Surprisingly, the W_transition of the very hard sapphire is only 8.4 times higher (at 94 times higher transition onset) than the first one of calcite. Furthermore, W_transition of sapphire is only 1.6 and1.7 times higher than the ones of the organic crystals in this Table 1. The latter are however extended molecules requiring energy consuming solid-state migrations into other polymorph structures, whereas the cooperative transformation of Al₂O₃ from its trigonal space group (R-3c) does not require considerable molecular migrations. The proportion of transition energies between Al₂O₃ (m.p. 2050˚C) and ZrO₂ (m.p. 2715˚C) (4.75 fold) is much higher than the one of physical hardnesses k (1.25 fold). This underlines the independence of transition energies from such qualities as hardness, or melting point, etc. The high pressure polymorphs of sapphire are probably either the orthorhombic (Pbcn) or the (Pbnm) polymorph of Al₂O₃, both with a volume decrease of 3.1% [21] , even now at room temperature, because of the shearing upon pyramidal indentation. Similarly, monoclinic ZrO₂ (P2₁/c) transforms probably into the denser monoclinic polymorph (still P2₁/c) or the orthorhombic polymorph with (Pbcm) structure, both with higher density. The hard metal tungsten (m.p. 3400˚C) has a lower hardness than the two oxides, but its transition energy is 2.5 times higher than the one of ZrO₂, which is quite remarkable.

Surprisingly, the semiconductor InGaAs₂ exhibits a significantly exothermic phase-transition, and mica (Muscovite) has a very low normalized endothermic transition energy, smaller than the two organics. Both of them experience very easy phase-transitions and they can only be detected by our iterative-free precise technique.

The metallorganic MOF, wood, and organics exhibit phase-transition energies that are not very different from the ones of the inorganics. The anisotropy of radial or circular pinewood is remarkable. Clearly, the non-iterative technique is very sensitive and avoids all of the strange recent techniques that are wiping out all of the important information from phase-transitions with incredible data treatments.

The phase-transitions of organic polymers under mechanical load are initial bond-breakings into radicals. This requires relatively high energy if these primary cleavage products are not stabilized by substituents. Therefore, their phase-transition onset must relate to the bond energies of the weakest chain bond. Further reactions of the so formed radical pairs are manifold, but overall there is hardening by cross-linking, which is well known for the technical treatments of polymers, e.g. upon extrusions, etc. Table 2 collects the data of some linear polymers upon Berkovich indentation. Most of these are highly amorphous and their hardnesses (k-values) are low. An exception is isotactic polypropylene it-PP with high crystallinity and much higher k-values. This is however not so important for the C-C bond strength. It is nicely seen in Table 2 that the phase change onset (chain breaking) relates to the strength of the carbon-carbon bonds. In the case of high density polyethylene (hd-PE) without chain substitution and isotactic polypropylene (it-PP) with slightly less efficient methyl-substituents, the strong unsubstituted C-C bonds exhibit similar bond energies and thus also similar phase-transition energies, much higher than those in Table 1. The details with kink forces and depths depend only slightly on further properties of the materials. Substitution of the polymer chains with phenyl groups as in polystyrene

^a)86th Handbook of Chemistry and Physics, CRC Press, 2006; ^b)not available, probably loss of the side group rather than chain breakage.

(PS, 1.04 g/cm³) decreases the bond energy, due to stabilization of the radicals, and thus also the phase-transition energy is decreased. The methylester groups in the side chain of polymethylmethacrylate PMMA (1.19 g/cm³) are 13 times more effective with decreasing the transition energy when compared with polystyrene PS. The h_kink of PMMA is 7.7 times lower even though the required onset force is 6.2 fold higher. But it appears that the elimination of methylformate (HCO₂CH₃) is energetically easier than the chain breakage. Polycarbonate Macrolon^(R) (PC) contains no C-C breakage chain. But the ArO-C(O)OAr bond (Ar = aryl) is cleaved, starting the energetically favorable loss of CO₂. This is the largest decrease of phase-transition energy of the polymers in Table 2: 145 fold decrease of W_transition when compared with the C-C breakage of PE. The analyses of cross-linked polymer indentations promise further interesting insights.

Table 1 contains already the double phase-transition of calcite and it is well-known that several phase-transitions can occur consecutively at higher and higher loads. This has been shown with the depth-sensing macroindentation of sodium chloride up to 50 N loads with a Vickers indenter when 4 phase-transitions were detected. The thus experimentally detected 6 polymorphs formed their polymorph interfaces under load with the facilitated production of seeds for catastrophic breakages [17] . Further examples have been found by analyses of published macroindentations [22] . For example, sapphire has its second phase-transition at 12.8 N and 5.82 µm depth (not included in Table 1) and soda-lime glass has the third transition at 14.0 N and 11.7 µm depth (not included in Table 3) [1] . Importantly, phase-transitions are, of course, not achievable by technical one-point

^a)h_end (µm); ^b)final W_applied/mN; ^c)k₂ = 124.80 from a separate measurement up to 11 mN after cut-off of the initial effect at 1 mN, including the hydrated oxide layer.

Vickers, Brinell, Rockwell, etc. hardness measurements. Interestingly, already microindentations (mN to low N-ranges) are a rich but hitherto undetected field of consecutive phase-transitions. Table 3 shows it for the standard materials soda lime glass, aluminium, and silicon, the latter on two different faces (another example is calcite in Table 1). The h_end and W_applied/mN values indicate only the experimental range. The undoubtedly amorphous soda lime glass exhibits two endothermic phase-transitions at the microindentation range (sharp kinks with linear branches in the Kaupp-plot) and exists thus in 4 or together with the macroindentation 5 polymorphic forms, the structures of which are subject to further studies. We can only conclude from the absence of cracks or pop-ins that there are different degrees of density under load that increases the penetration resistance from state to state. The transition energies of the first two transitions of soda lime glass vary by a factor of roughly 4, which is quite remarkable. Aluminium requires about 12.1 and 60.4 mN load for its first and second phase-transition and its proper use as a calibration standard ends already below the values of the first one. Some onset-depths are still in a range of nanoindentations. Also the transition work for both transitions is similar to the one of much harder sapphire and smaller than the ones of pinewood (Table 1) and PE, PP, and PS (Table 2). This reflects the weak metallic bonds of aluminium that facilitate rearrangements in the crystal lattice.

The transitions of cubic silicon (Ed3m), exhibiting an exceptional pop-out of the unloading curve, found more theoretical interest. Several polymorphs (including an amorphous phase) were detected by electron diffraction, Raman spectroscopy, and electric conductance onset from the loading curve [10] . The obtained values from the papers, as cited in [10] , are similar to the ones found more easily directly from the Kaupp-plot of the loading curve in Table 3. Particularly helpful in that respect are the in-situ current flow onsets at about 5 mN of the metallic Si II [23] , which agrees with the first kink discontinuity in our analysis. This electrical unsteadiness confirms again the first phase-transition onset under the (001) face of silicon. We observe however 3 phase-transitions (5 polymorphs) from the physical Kaupp-plots (Table 3). And the data from the two different faces exhibit considerable differences. The first transition is hidden at the scales of 50 and 100 mN. It is thus helpful that also a nanoindentation was published in [24] for the (001) face. The variation of the 3 endothermic transition energies is by a factor of 28 for the (001) indention and there were no pop-ins but only “fine cracks emanating from the corners” at 50 mN load/displacement. It can be assumed that the boron-doping does not make much difference to the mechanical properties of silicon. We can, therefore, compare the (001) with the (100) phase. The five polymorphs of the silicon in the loading curve up to 50 mN load have been located for the first time together with the energetic information. The striking differences in the endothermic phase-transition energies at the different faces require a crystallographic understanding. This can be obtained by considering the crystal structure, similar to the procedure, as developed in [16] . Figure 3 compares the different surface structures. It shows the two probed surfaces of silicon. The channels under (001) are slightly smaller than the ones under (100). That explains the deeper onset penetration under (100), but it cannot explain the differences in the corresponding transition onset forces F_N-kink and the phase-transition energies W_transition. We have thus to consider how the Berkovich indenter with its skew faces at the face angle (semi angle θ = 65.3˚) interacts orthogonally with the interior of the crystal at the opposite angle, due to the penetrated Berkovich. The equally skew face structures around the Berkovich are calculated by rotations from the indented face by rx ± 65˚ and ry ± 65˚. This models 8 opposing skew faces because the corresponding rotations around 245˚ and 115˚ are mirrored or doubly mirrored. Figure 4 and Figure 5 image the striking differences. Figure 4 under (001) exhibits large channels for migrations that facilitate

the phase changes. Clearly, that makes them less endothermic. Conversely, in the correspondingly calculated Figure 5, under (100) one observes 3 blocking (a, c, d) opposing faces and only one (b) with channels. Clearly, the migrational possibilities are more impeding under (100). The phase-transition and the endothermic result are thus considerably higher, as has to be expected.

The multiple phase transitions of silicon (Berkovich) are also face specific and this is described in detail in Section 3.3, Table 3 and Figures 3-5. Also, the anisotropic single exothermic phase-transition energies of α-quartz (cube corner) on four different faces have already been analyzed for crystallographic reasons and reported in [16] . All crystalline materials must have different phase-transition energies under different faces. Further typical examples are with bcc α-iron (ferrite, Im3m) under two faces and with strontium titanate perovskite type (Pm−3m) under three faces (Table 4). These measurements require the high precision that is only available with the physical regression analysis without iterations or data fittings.

The crystal of α-iron on (100) and (110) exhibits different exothermic phase-transition values. For strontium titanate with three endothermic phase-transition energies, two of them have closer together values, but the lowest and largest values are far apart. This is again not consistently reflected by the different onset forces and onset depths (Table 4). The reasons are the different crystal packing, similar to the reasoning with silicon (Table 3) and α-quartz in [16] . The understanding of the exothermic phase-transition of α-iron rests again on the fact that the penetrating pyramid interacts orthogonally with its opposing

skew face at the semi-angle, which is θ = 35.26˚ for the cube corner. A rotation from the surface faces around x and y by ±35˚ yield these skew surfaces. The eight by 35˚inclined faces around the three-sided indenter pyramid are completely represented with only two images for symmetry reasons of the bcc cubic crystal under (100) (Figure 6) and under (110) (Figure 7). This facilitates the understanding of the differences for crystallographic reasons.

The crystal face on (100) and the skew faces under it in Figure 6 under (100) stand for the higher exothermic phase-transition of α-iron. The cube corner opposes all of the eight faces almost orthogonally that are devoid of channels. That means, there is no energy lost by the migration of phase transformed material. The gained exothermic energy remains highly negative.

Conversely, Figure 7 with Fe on (110) with the lower exothermic phase-transition energy shows the skew faces by rotations of ±35˚ from it. Four of the eight exhibit open channels orthogonally to the indenter. Therefore, part of the gained transition-energy is used up for migration of the phase transformed material, leaving less of the gained energy by phase-transition. The channel face is here fortuitously the {111} faces across the crystal. Only the other four faces impede migrations.

The analogous analysis has been successful with the exothermic phase-transition of α-SiO₂ with cube corner in [16] , where 4 different faces were studied: the absence of channels retains exothermic negative transition energy. Everything is thus well understood by the straightforward analysis in the case of exothermic phase-transition.

As already shown for the endothermic phase-transitions of silicon in Section 3.3, the analysis of the strontiumtitanate using Berkovich with θ = 65.3˚ is equally successful (not imaged here, due to many images that are required). The minimal endothermic work (6.5 µNnm/µN) is required under (100), where channels are available on the skew faces to facilitate the conversion. Conversely, under (111) the strongest endothermic energy (14.8 µNnm/µN) is observed, as there are no channels at the skew faces that enforce the transition to stay blocked

in space, which increases the endothermic work. Consistently, under (110) the value (7.0 µNnm/µN) stays between the extremes, as there are some smaller channels allowing for minor migration of the transformed material. Again, channels facilitate migrations for the endothermic phase-transitions and decrease the endothermic work. When migrations are blocked more work is required for the phase-transition. The face-dependency of phase-transition energies is thus well understood by crystal structure analysis.

The sudden first order sharp phase-transitions form polymorph interfaces that are shifted away from the indenter after their onset when the load increases. These polymorph interfaces are preferential sites for the nucleation of large far-distant cracks. This has already been imaged in connection with the multiple consecutive phase-transitions of sodium chloride in [17] . These circumstances are most important for the occurrence of catastrophic failures of superalloys at work. Clearly, polymorph interfaces must be avoided with materials under mechanical and thermal stress. We must therefore urgently look for phase-transition onsets and transition energies of published loading curves of some superalloys (also called entropic alloys). This shall convince industries that they must complement their Vickers, Brinell, Rockwell, etc. hardness measurements with physical depth-sensing macroindentations’ control. Only these provide the necessary phase-transition information. The problems might occur primarily after multi-1000-fold load and unload both mechanically and thermally. The thermal stress calculation requires the activation energy of the phase-transitions that can also be obtained by physically correct temperature dependent indentations [10] . The transition energies W_trans in Table 5 are purposely given in µNnm/µN units, in order to show that such phase transformations can also here occur at comparatively low loads and depths. There were no “pop-ins”, which could disqualify

^a)Cube corner; ^b)Berkovich; ^c)Second transition.

the alloy’s use beforehand. However, these published examples do certainly not represent alloys that are in practical use, as the actual alloys are secret and only available to the industrial personnel.

Table 5 starts with the 68/32 γTiAl alloy, because a propeller blade close to the turbine each out of a “titanium-aluminium alloy” from two identical airliners broke away, during flight within one year in 2017 and 2018. Both blades hit the body of the airliners, unfortunately killing a woman passenger. This “titanium aluminium alloy” of publicly unknown composition contained certainly several other components. The data of Table 5 show however comparably low force and energy values for the phase-transition of the alloy base TiAl. It is seen: the way for obtaining high enough phase-transition onset and endothermic transition energy for a reasonable TiAl alloy must probably have been too long. The other alloys of Table 5 exhibit much higher values, but these alloys are certainly not the secret alloys for airplanes. There are means to engineer alloys by trial and error with ductilizers, for example by oxide dispersion strengthening (ODS-superalloys) with Y₂O₃, ThO₂, La₂O₃, Al₂O₃, etc. [25] . Also, carbon and other materials have been widely used as additives. Here is now the still not industrially used (legal ISO-ASTM standard certification!) more systematic way to improve superalloy compositions for best performance. All what’s required is the easy and reliable physical indentation analysis with respect to phase-transitions. The hardness of materials is again a poor guide for judging the phase-transition onset force ranges and the phase-transition energies. For example, the by far best alloy in Table 5 is Vileroy-105 with a comparably low hardness (k₁= 0.90332 µN/nm^3/2 or with other units 29.652 mN/µm^3/2) that are much smaller than those of Fe₄₃Cr₁₆Mg₁₆C₁₅B₁₀ (k₁= 7.21378 µN/nm^3/2 or with other units 228.12 mN/µm^3/2), while the W_transition ratio is 291/87, respectively. The Newton-range is almost reached with Vileroy-105. Its first transition is at highest force and also the transition energy is highest, whereas the well-known iron-based superalloy surmounts the kink force only at its second phase-transition but with much lower transition energy. The magnesium based alloy is inferior and the copper-based alloy has two phase-transitions at still lower forces and transition energies.

The here achieved maximal loading forces (0.5 N, corresponding to HV 0.05) are at or below the lower level of industrial Vickers, Brinell, Rockwell, etc. hardness measurements. These are certainly highly standardized by ASTM, but unable to detect phase-transitions. It is to be expected that further as yet undetected phase-transitions will occur at the much higher loads (for example from Vickers hardness HV 0.5 to HV 10) in all cases.

The field of superalloys for technical constructions is widespread and extremely important, and so is the improvement of the stability of superalloys when being at work. Any alloy must only be admitted to forces below its first phase-transition onset. Both the onset force and the transition energies must be as high as possible. We present here a simple systematic way to improve the engineers’ efficiency, but this depends on the profound change of the present ISO and ASTM standards that are the basis of the manufacturer’s certification. Physical mathematically proved standards must be urgently created, edited, and enforced. Present ISO-ASTM hardness or indentation moduli are not physical quantities and they are unsuitable for the safety. Their biggest flaw is their inability to detect or even know of phase-transitions under load. The here described improvements are, of course, worldwide indispensable, but unfortunately not easily achieved for various unscientific reasons.