_{1}

The Nanoindentation is a precise technique for the elucidation of mechanical properties. But such elucidation requires physically based interpretation of the loading curves that is widely still not practiced. The use of indentation hardness
*H* and indentation modulus
*E _{r}* is unphysical and cannot detect the most important phase-transitions under load that very often occur. The claim that

*H*versus

*E*plots relate linearly for all different materials is neither empirically found nor correctly deduced. It is most dangerous by producing incorrect materials properties and misleading. The use of

*H/E*(that is also called “elasticity index”) in complicated formulas for brittle parameter, yield strength, toughness, and so-called “true hardness” is also in error. The use of

*H/E*cannot reveal the true qualities of materials without considering phase-transitions under load that require the correct exponent 3/2 on

*h*for the loading curves (instead of disproved 2). This is exemplified with the physical data of different mollusk shells that experience phase-transitions, a new bionics model, and different contributions for their strengthening. The data are compared to the ones of aragonite and calcite and vaterite.

The recent papers of Labonte, Lenz and Oyen [_{N}) vs depth curves, the slope of which is the penetration resistance, that is the physical hardness when calibrated with the indenter cone or effective cone. It was deduced in 2013 [_{r} since 2017 [_{N} vs h^{3/2} relation [_{N} vs h^{3/2} plots detect important phase transitions upon indentations in [

But all of that has been disregarded by the criticized authors who refer to Oliver-Pharr [_{r} are fake-values and so are there from created theories of [

The published Berkovich indentation loading curves from the Limacina Helicina Antarctica and Haliotis rufescens mollusks were scanned and enlarged to A4 size. 25 to 42 data points were taken for the Excel calculation of F_{N} vs h^{3/2} diagrams. The regression lines of the linear branches provide the slopes as k-values (the physical hardness) for the precise calculation of the kink position (phase-transition onset). These are the basis for the calculation of indentation work (W_{indent}), applied work (W_{applied}), full applied work (full W_{applied}) and phase-transition energy (W_{conversion}), using a pocket calculator (10 digits before final rounding). We normalize them per µN to make them comparable. The necessary equations are well known as repeatedly published in [_{r} world.

The methods of [_{r} (here for Berkovich). The according to [_{N} per contact area A_{hc} that is geometrically 27.15h_{c}^{2}. The dimension is (force/depth^{2}) usually reported as GPa. The elastic property is experimentally measured as stiffness S = ΔF_{Nmax}/Δh with the different dimension (force/depth). To obtain an ISO-E_{r} with the same dimension as the ISO-H, one multiplies the stiffness S with 0.5 π^{1/2} A_{hc}^{−}^{1/2}. This provides the further h_{c}^{−}^{1} for the dimension of ISO-E_{r} as force per area (GPa). But the A_{hc} value requires one iteration with 3 and another iteration with up to 8 free parameters (also + or − sign selection) according to [_{r} numbers, suggesting a “≈0.05 ratio of H/E_{r}” for uncountable published Berkovich indentations would at best indicate very poor worldwide measurements of indentations over the years if that would be reality. It cannot be used for the calculation of E_{r} from H numbers with a “statistical confidence of 95%” and R^{2} = 0.96. For example a hardness number H of 0.6 GPa in figure 1 of [_{r} of the densely overlapping entries. Or an entry at H = 7 × 10^{−}^{5} GPa has a data triangle value of about 1.05 × 10^{−}^{2} GPa for E_{r}, while the corresponding H value on the H/E_{r}line for that E_{r} is at 4.8 × 10^{−}^{4} GPa, which is an about 6.9-fold higher hardness number. These examples show drastically that the claimed linear relation between the ISO hardness H and ISO modulus E_{r} numbers is not correct. And it will be shown in Section 3.2 why it cannot be correct. The claimed statistic confidence of 95% for the log-log plot is useless and dangerous. In the figures 1 and 4a, 4b of [_{r} plot, straight linear plots are imaged. These are for spherical and for Berkovich indentations without any visible deviations. Beware from the risk of H vs E_{r} plots in view of figure 1 of [

Particularly risky and dangerous are the use of H/E_{r} plots or values for the evaluation of brittleness characterizations, critical load ratios, strengthening, toughness, and “true hardness”. For example figure 4a in [_{r})^{2} numbers, and the normalized characteristic indentation dimension vs critical normalized cracking load ratios are plotted in figure 4b of [_{I} and E_{I}, even though they repeatedly invoked the “Oliver-Pharr analysis” or “-model”. And only a few entries are directly cited, but almost none of these disclosed published original loading curves that could be checked and used for the calculation of real properties like physical hardness, iteration-free elastic moduli and phase-transitions under load.

Furthermore, the authors of [_{true} = H/{1 − (H/E)^{1/2} (2/tan ß)^{1/2}}^{2}, where ß is the cone angle of the indenter. This shall be the “resistance to plastic deformation” or “resistance to irreversible deformation”, which “depends on the ratio between indentation hardness and indentation modulus”. It is strangely claimed that “a large indentation hardness does not imply a large resistance to irreversible deformation per se”. This hard to understand basis by using the H/E_{r} fraction is exemplified in [_{r} = 10.5 GPa shall imply “true hardness” of 17.8 GPa”; or H = 3.12 GPa and E_{r} = 87.02 GPa shall imply “true hardness” of 10.8 GPa”. The H/E_{r} fraction is contained in such calculations. And the other formulas are in the appendix of [

The proportionality claims of ISO-H with ISO-E_{r} in [^{3/2}] from the slope of the so named Kaupp-plot F_{N} vs h^{3/2}. None of the cited and used H_{I} and E_{I} values tells which polymorph of the material was probed, because their onset forces cannot be found with the wrong exponent 2 on h. One needs such linear plots for the detection of polymorph formation onsets [

All of these H and E_{r} values are unphysical and so are their ratios, because they rely on the disproved exponent 2 on h (instead of the F_{N} vs h^{3/2} relation) and require data-fitting iterations [_{r} numbers are not well defined and depend strongly on the details of their detection, as outlined in [_{r} when comparing different methods.

The Oliver-Pharr technique and the still present ISO 14577 standard assume the physically disproved exponent 2 on h for the loading curves instead of undeniably physically deduced h^{3/2} [_{r} values that are entirely unphysical parameters. If indentation hardness have to be compared with indentation modulus one should only take physically sound values from the so named Kaupp plots (F_{N} vs h^{3/2}) that most easily provide penetration resistance onsets and differentiate the properties of every polymorph under load. And it provides directly measured indentation moduli (E_{phys}) without any iteration. All of the trouble in Section 3.2 originates from the widespread refusal to check the exponent of their loading curves. These F_{N} vs h loading curves follow always the physically correct relation F_{N} = kh^{3/2}. All of the respective authors stay with the physically disproved h^{2}, apparently until ISO and the authors of [

The claims of linear relations between ISO-hardness H and ISO-modulus E_{r} include nacre, eggshell, aragonite, calcite, hydroxyapatite, enamel, dentine, bone, etc in the unphysical and incorrect log/log plots in [

We show now that hitherto unthinkable materials’ properties are straightforwardly obtained on the physical analysis of indentations from correctly cited publications. This will be exemplified for the case of two mollusk varieties with their aragonite shells, including the distribution of the organic materials. The physical analysis of the indentation loading curve onto the polar pteropod Limacina Helicina Antarctica shell, as recently published with figure 4 in [_{N} vs h^{3/2} diagram of _{N} vs h curve and it does never show up in the unphysical F_{N} vs h^{2} relation with its false exponent 2 on h. The authors of [_{r} values [^{2} = 0.9994, 0.9993, and 0.9993, respectively) have the equations that are inserted in

are three different polymorphs up to a loading range of 5 mN load with a Berkovich. Interestingly these values are similar to the ones of calcite in table 1 of [

The smooth transition zones rather than sharp kinks that are here only seen by the intersecting regression lines reveals a gradual change of the strengthening organic material between the different polymorphs. This is certainly a bionics model for avoiding the crack increasing risk when unavoidable polymorph interfaces contact smoothly [

The precise distribution of the about 5 wt% of organic material is certainly worth further studies. Conversely, the unphysical H and E_{r} measurements led to the claim of “essentially homogenous distribution throughout the shell for “strengthening the cell” [_{r} values in figure 5 of [^{3/2}) of the different polymorphs up to the same load maximum would tell, whether there are zones with more or less organic material also laterally distributed. In the case of micro-caverns empty or filled with water, these would show-up as spurious pop-ins [

It is particularly unsuitable that the authors of [_{r} that are against physics.

The averaged Berkovich indentation curve of the red abalone Haliotis rufescens shell from Baja, California [_{first-aragonite-shell} = 0.9058 µN/nm^{3/2} is sharply distinguished by the organic layer with k_{organic} = 0.274 µN/nm^{3/2} and the following inner apatite layer with k _{inner-aroganite-shell} = 1.1495 µN/nm^{3/2}. The stepwise behavior is also shown in the original F_{N} vs h plot from figure 7 in [_{N} vs h^{3/2} plot in ^{3/2}) with the inserted regression lines is totally different from ^{3/2}. Therefore at least some of this k-value increase represents the shift relative to the end of the first aragonite layer. The thinner strengthening organic layer is considerably softer. The 16.8% difference between the k-values of the two aragonite layers cannot solely be responsible for their displacement. The second aragonite layer should also be phase-transformed at the increased force with an onset right above the organic layer at 4312 µN. The regression line values of the hard branches correlate both with R^{2} = 0.9997.

However, there are difficulties for the calculation of the phase-transition

energy, because we do not have a kink-point between the displaced aragonite layers. We must try to secure that the inner aragonite layer is a polymorph by a phase-transition. ^{3/2}). The second aragonite branch starts directly at the upper end of the organic layer at 4312 µN. This data point is already part of the regression line for the steeper (harder) branch. The 4000 µN load would thus be the phase transition onset point if the organic layer was not there. Unlike the repair of pop-ins [_{N} and by higher h^{3/2} values at its start. This influence on the steepness cannot be undoubtedly judged and also minor corrections in that sense would strongly influence our precise and highly sensitive calculations. A phase transition part from about 4500 µN load of the hard nacre shell is however most likely. That judgment is in view of the k_{1} and k_{2} values of the softer Limacina Helicina Antarctica shell that experiences the phase transition and the k_{1} and k_{2} values of the Haliotis rufescens shell that are in the same order of magnitude even though the shells of Limacina Helicina Antarctica are softened by the embedded organic layers. Final proof would require comparison with an indentation of pure aragonite at forces up to about 7000 µN load. Unfortunately we did not find accessible reliable Berkovich indentation curves of pure aragonite at such a loading range with smooth loading curves that are not interrupted by continuously repeated unloads. There is however a phase-transition within a 500 µN loading range of pelletized aragonite from [

We must report here how one can identify and exclude experimentally false reported data by the calculation of transition energies. The reported Berkovich indentation onto (001) of aragonite up to 1000 µN load from the figure 1 of [_{Nkink} = 408.6 µN when physically analyzed. Thus, figure 15b in [

The behavior of aragonite must now be compared with the other ambient modifications of CaCO_{3}. Hexagonal calcite, orthorhombic aragonite, and hexagonal vaterite crystallize in the respective space groups R3−c, Pbm6n, and P6_{3}/mmc. Their X-ray densities are 2.71, 2.93, and 2.93 g/cm^{3}, respectively. The most frequent twins of calcite occur along (10 - 11) by mechanical stress on (01 - 12) and those of aragonite on (110) by mechanical stress on (010) [

It appears more than surprising that a paper like [

An H/E_{r} ratio (also called “elasticity index”) is unphysical, as are ISO-H and ISO-E_{r}. Physical hardness is H_{phys}= k/π tanα^{2} (k in force/depth^{3/2}, α effective cone angle) for conical and pyramidal indenters. And the not iterated physical modulus is E_{rphys} = S/2h_{max} tanαas deduced in [_{Nmax} [_{r} has not been demonstrated by the log-log plot in figure 1 of [^{2} = 0.96 for the selected materials by not considering their undetermined phase-transitions that are however most frequent for all kinds of materials upon load. Even under these unsuitable conditions the actual deviations are very often enormous. However, most materials have to be again indented when neither original data, nor published F_{N} vs h loading curves had been published. ISO-H and ISO-E_{r} values cannot reinstall the physical indentation results, due to the exhaustive data fitting iterations. It has to be rejected that the H/E_{r} ratio is used for defining a so-called “true hardness” with extremely high useless values of hardness and moduli. Also the revival of the complicated formulas, using H/E_{r} ratios for brittle parameter, yield strength, and toughness, is misleadingly incorrect and useless. Correct unprecedented qualities of materials (as exemplified in Section 3.3.) are to be deduced from a physically sound basis. The easily obtained experimental achievements on the basis of the physical analyses of indentations are withheld in [

The exemplarily analysis of the indentations onto seawater mollusks shows that ISO-H and ISO-E values are unable to differentiate between the construction principles of different mollusk shells. The aims to solve important biological questions are not attained and so are the theoretical speculations. It requires the so named Kaupp-plot (F_{N} vs h^{3/2}) for most easily and rewardingly revealing the striking differences. In the case of Limacina Helicina Antarctica the linearized loading curve undergoes two phase-transitions in the load ranges up to 5 mN. The within aragonite distributed material cushions shocks so that the shells are protected. We quantified the phase-transition onsets and energies that reflect the details for the lattice conversion in Section 3.3. A new bionics model is extracted from the shell behavior. Its cell strength is achieved by mitigation of the dangerous effects of polymorph interfaces by softening with gradual approach to polymorphs interface from the unavoidable phase-transitions. This bionics model should become most useful for technical materials that are exposed to mechanical forces that induce phase-transition onsets in e.g. ballistics or earthquakes etc. Furthermore, these results open up new technical and biological insights. Further indentations onto Limacina Helicina Antarctica, as requested in Section 3.3 will in the future facilitate the crystallographic understanding of these phase-transitions with eventually further bionics models.

Totally different is the already known bionics model of Haliotis rufescens. It uses alternating layers of the thin soft organic material between thicker aragonite layers for cushioning. Also further studies with the numerous further mollusks become worthwhile and promising now. Variations of the layer thickness and detailed structures in the not layered varieties with respect to environmental conditions will provide biological answers. Also snail-shell indentations should be physically analyzed, but not with the disproved and unable techniques in figures 1 and 2 of [

Further advances of the physical analyses, in addition to the precise detection of phase-transition onset forces and energies for explaining and avoiding catastrophic failures, are the sorting out of initial surface effects, the detection and elimination of experimentally false reports with the calculation of phase-transition energies and the distinction of phase-transition onsets from those of different material layers. When measured at various temperatures one can also calculate the activation energies of phase-transitions [_{r}, H/E_{r} and there from deduced obsolete techniques, because they are all unphysical.

The future will detect and understand further phase-transition onsets and energies from all kinds of materials. This opens their discussion for the widening of their understanding and applications, and the search for further bionics models appears promising.

It is hoped that not only biologists, pharmacists, physicians, and also industrial Engineers take their chance to increase and revise their knowledge for preventing dangerous disasters by using the penetration resistance instead of ISO-H and ISO-E_{r}. The revision of ISO-14577 must also be accelerated. It is strongly hoped that ISO 14577 and the authors of [

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

Kaupp, G. (2021) Undue Hardness/Modulus Ratio Claims instead of Physical Penetration Resistance and Applications with Mollusk Shells. Advances in Materials Physics and Chemistry, 11, 45-57. https://doi.org/10.4236/ampc.2021.112005