Medical, Pharma, Engineering, Science, Technology and Business

University of Oldenburg, Germany

- *Corresponding Author:
- Kaupp G

University of Oldenburg

Diekweg 15 D-26188

Edewecht, Germany

**Tel:**4944868386

**Fax:**494486920704

**E-mail:**gerd.kaupp@uni-oldenburg.de

**Received Date:** January 24, 2017; **Accepted Date:** February 08, 2017; **Published Date:** February 18, 2017

**Citation: **Kaupp G (2017) The ISO Standard 14577 for Mechanics Violates the
First Energy Law and Denies Physical Dimensions. J Material Sci Eng 6:321. doi:
10.4172/2169-0022.1000321

**Copyright:** © 2017 Kaupp G. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Material Sciences & Engineering

The basis of the quantitative conical/pyramidal (nano) indentation, without fittings, iterations, or simulations, is the physically founded FN=k h3/2 relation. The constant k (penetration resistance, mN/μm3/2) from linear plot with excellent regression discards initial surface effects, identifies important phase transformation onsets, conversion and activation energies, and reveals errors. The failing Sneddon theory of ISO with unphysical exponent 2 on h lacks these possibilities, disregards shear-force work, and violates the first energy law since 50 years. The denied but strictly quantified loss of energy (20% for physical h3/2; 33.33% at believed h2) violates the first energy law and disregards the force remaining for penetration. Straightforward correction is performed for the dimensions, by replacing unphysical exponent 2. The correction factors hmax 1/2 and 0.8 are applied via joint maximal force to the universal, FE-simulated, (approximately) ISO hardness, and ISO modulus that unduly rely on h2, to give the physically founded values with their correct dimensions. Previous corrected k-values obtain Hphys directly from the loading curve regression. Previous incomplete corrections are rectified. The new dimensions and daily risk liabilities from ISO versus physics dilemma are discussed, considering the influence on all mechanical parameters from hardness and modulus, regarding technique, biology, medicine, daily life.

Correction of ISO hardness and modulus; Energy law violations; Failure risks; False materials parameters; False ISO-standards; Indentation exponent; New hardness and modulus definitions; Penetration resistance; Physical consequences; Physical hardness from loading curve

The most basic natural and technical law that can never be dismissed, but must be strictly obeyed, is the energy conservation law. All worlds work on it and must trust in its validity, and that must not be dismissed by any organization. However, ISO and its subsidiary NIST in USA still violate against with ISO standard 14577, claiming the exhaustively complicated mathematical deductions of Sneddon and Love [1,2]. However, these authors obviously missed taking into account the shear-force work, when a rigid indenter is forced to penetrate vertically into a solid. It must be very clear that the pressure (and or plastic deformation) from the rigid indenter against its displaced solid material requires work. Nevertheless, the whole applied force and thus the whole applied energy is still falsely considered to be only acting in vertical direction of the impact. Unfortunately, there was no protest from physics. Rather the work of Oliver and Pharr [3] on the indentation of cones or pyramids was highly acclaimed and adapted by ISO/NIST for ISO 14577. It thus became undisclosed that their assumed relation between force and depth is incorrect and that the hardness and elastic modulus determinations violate the first principle of energy conservation. Such disregard has still been retained till now, even though the unphysical exponent 2 on the depth h had been experimentally demonstrated to be replaced by 3/2 from the present author since 2000 with convincing evidence.

In 1939 and 1965 two mathematicians solved the long standing
Boussinesq problem using very complicated mathematics and came
(with different constant) to the same exponent 2 on the depth *h* in
relation to the normally applied force in conical indentation when the
indenter remains stiff (**Figure 1**). The Sneddon/Love exponent [1,2]
has also been used for partly plastic response (it is a consequence of
pressure!) by Oliver and Pharr in 1992 [3], the ISO standard 14577, and
finite element (FE) simulations (e.g. ANSYS or ABACUS software), even though the shear force of the conical (similarly effective cone of
pyramids) indenter to the environment did apparently not find any
concern in physics. Rather numerous fitting procedures were put
forward over the years for the excuse, that the exponent 2 on *h* could not
be found experimentally but only with FE-simulations converging to
such exponent. Thus, these mathematical deductions (**Figure 1**) cannot
be correct. It did apparently not help that the energetics of the (pseudo)
conical indentation was for the first time quantitatively clarified in a
publication from 2013 [4] because the experimental exponent on *h* was
consistently found as 3/2 instead of 2 [5]. The thoughtful convincing physical foundation of exponent 3/2 in Equation (1) that followed prepublished
since 2015 [6] requires only first grade mathematics.

The author`s nanoindentations used a fully calibrated Hysitron
Inc. TriboScope^{(R)} Nanomechanical Test Instrument with a twodimensional
transducer and leveling device in force control mode after
due calibration, including instrument compliance. The samples were
glued to magnetically hold plates and leveled at slopes of ±1° in x and
y directions under AFM control with disabled plain-fit, and loading
times were 10-30 s for 400-500 or 3000 data pairs [5,7]. The radii of
the cube corner (55 nm) and Berkovich (110 nm) diamond indenters
were directly measured by AFM in tapping mode. Three-dimensional
microscopic inspection of the indenter tips secured smooth side faces
of the diamonds for at least 2 μm from the (not resolved) apex. The
whole data set of the loading curve was used for analysis, using Excel^{(R)}.
Most analyses were however with published loading curves from the
literature, as rapid sketches with pencil, paper, and calculator (10-20
data pairs), but for linear regressions always by digitization to give 50-
70 almost uniformly arranged data pairs using the Plot Digitizer 2.5.1
program (www.Softpedia.com), unless complete original data sets
could be obtained from the scientists with 400-500 or 3000 data points.
They were handled with Excel^{(R)}. The distinction of experimental and
simulated loading curves succeeded by performing the "Kaupp-plot" (1)
revealing F_{N} ∝ h^{3/2} (experimental), surface effects and most important
phase changes' onset [8]. The necessary force correction to comply with
the energy law is made with the physical k-value (0.8 times the slope).
Only FE-simulated or iterated curves gave linear unphysical F_{N} ∝ h^{3/2} plots. The linear regressions were calculated with Excel^{(R)}. In the case of
phase changes the kink positions were precisely calculated by equating
the regression lines before and after the kink. Initial surface effects were,
of course, exempt from the linear regressions. Previous penetration
resistance values *k* was corrected for complying with the energy/force/
depth loss in **Figure 2**. A 10-figures pocket-calculator was used for the
physical calculations, but the final results are reasonably rounded. It
was tried to cover all different materials types, all different indentation
modes, equipments, response mechanisms, depth ranges, penetration
resistance sizes, from numerous authors from all around the globe, in
order to show their universal obeying to basic mathematics.

**The mathematical clarification of the energetics upon
(pseudo)conical indentation**

We proceed analogous to the deduction in Kaupp [4]. In force
controlled indentations the total force *F _{N}* is linearly applied. This can
provisionally be imaged together with an assumed normal parabola
(with exponent 2) as is used by ISO etc. in a force versus depth
diagram, as obtained by a FE-calculation from the literature (

**Figure 3:** FE-simulated force displacement curve for 500 nm thick gold
assuming *h*^{2} with an ideal Berkovich and our comparison with the linearly
applied total work (straight line from zero to F_{Nmax} [4]); the force-corrected
parabola would end at the (2/3) F_{Nmax} point; evidently a large part of the applied
work would be lost for the indentation; the dotted simulated force curve would
precisely follow *h*^{2}; the simulated data points were taken from Reference [8]
(their Figure 3b).

*F*_{N-phys}=*k h*^{3/2} (1)

*F*_{N-simul}=const *h*^{2} (2)

*W*_{simul-indent}=1/3 const *h*^{3} (3)

*W*_{applied}=0.5 *F*_{N-max} *h*_{max} (4)

*W*_{simul-applied}=0.5 const' *h*_{max}^{3} (5)

In order to clarify the unlikely objection that the applied force
would be parabolic, we plot here in **Figure 4** both applied force and
depth side by side against the time as these develop. It is, of course,
seen that these develop simultaneously with total *F*_{N} linearly but depth *h* parabolic. We can thus safely calculate the total applied work from
the triangle as in **Figure 3** (or **Figure 5**). Different ways of normal force
applications (force controlled, displacement controlled, continuous
stiffness, squared progression of the load increments) cannot decrease
this applied work. Furthermore, the analysis of strongly creeping
loadings (e.g. PMMA data in **Figure 2**) also gives the unfitted *h*^{3/2} parabolas (1) with excellent correlation [5] excluding chances to
improve the ISO- or FE-indentation efficiency. The formerly forgotten
and not considered decreased energy for the indentation and thus also
for the actual indentation load part is a striking violation of the first
energy law. Only the fraction of the full applied work depends on the
exponent on *h*.

**Figure 5:** Experimental force displacement curve of aluminum (following the
physical exponent 3/2 on the depth *h* [6]) and the comparison with the linearly
applied force line, showing the loss of force (and energy) for the indentation
depth; the measurement was with a Hysitron Nanoindenter ^{(R)}; the forcecorrected
parabola would end at the 0.8 *F*_{Nmax} point.

But unfortunately we have to respond against continuing
strange attacks on the quantitative treatment of conical or pyramidal
indentations without any approximations simulations or fittings,
despite the publications [4,5]. The probably last denial of the wellestablished
experimental evidence of the exponent 3/2 on *h* [9] repeats
the offence of Troyon (advocating depth dependent broken exponents
such as 1.64533 or 1.75285 on *h* without discussing the incredibly
changing dimensions) [10], which is combined with the violation of
the first energy law (not considering [4]). Furthermore, Merle [9] tries
to invoke the undisputed self-similarity of cones and pyramids as a
theoretical argument. But Merle [9] incorrectly claims that this should
be in favor of exponent 2. Self-similarity can by no means decide between the exponents in question. The exponent 3/2 is physically founded [6],
and all data relying on the false exponent 2 require correction with the
dimensional factor *h*^{1/2}. Furthermore, these unduly opposing authors
tried to discredit the successful Kaupp-plot (*F*_{N} versus *h*^{3/2}) by calling
it "Kaupp's double *P*-*h*^{3/2} fit" [9] (P means force, the same as *F*_{N} here),
even though the "Kaupp-plot" does not fit at all. They pretend that the
kink (phase transformation) in the fused quartz example would have
been claimed by intersecting an initial surface effect extrapolation line
with the second linear branch, instead of equating the first and second
linear branches (more of it in the Discussion). Kaupp has always been
identifying surface effects and removing them from the regression.

**Experimental and physical basis of pyramidal and conical
instrumental indentation**

The violation of the basic energy law is connected with the use of
unphysical exponent 2 on *h* with implied assumption that the one third
loss of the applied energy ∝ force (**Figure 3**) would not count for the
peak load in the hardness *H* and modulus *E*_{r} calculations that use *F*_{Nmax} for the start of the unloading curve. The connection is quite simple
and direct with the definition of universal hardness for indentations *H*_{universal}=*F*_{Nmax}/*A*_{proj} (where *A*_{proj} is the projected area of the indenter). This has been worked out in Kaupp [7] with the formula sequence (6)
leading to a disproved ISO *F*_{N} ∝ *h*^{2} relation:

*F*_{Nmax}=π*R*^{2}*H*_{universal} and *R*/*h*=tanα gives *F*_{Nmax}=π*h*_{max}^{2}tanα *H*_{universal} (6)

The ISO *F*_{N} ∝ *h*_{c}^{2} relation is also obtained for the ISO-hardness *H*_{ISO}=*F*_{Nmax}/*A*_{hc}, where the so called contact height *h*_{c} must be adjusted
to a standard material in a complicated procedure, including two
multiparameter iteration steps [7]. Clearly there are three undisputable
flaws against physics with these hardness determinations: 1. the
violation of the basic energy law, 2. the use of unphysical exponent and
3. the non-considering of the often occurring phase transformations
under load before the chosen peak load is reached, which can only be
detected with the Kaupp-plot of (1). The energy law correction will
be discussed in the next Section after presenting further support. The
dimensional correction will be exemplified in the Sections dealing with
the correction of hardness and modulus into physical values.

The convincing physical foundation of exponent 3/2 in the
force depth relation (1) [6] (pre-published in 2015) leaves no doubt
whatsoever with respect to the present author's analysis of his own
and published loading curves from others who wrongly trusted and
used the Sneddon/ISO/Oliver-Pharr exponent 2. All details of the
loading curves can only be detected when the correct exponent 3/2 on *h* is used for the analysis. The details are lost with unphysical plots
and more so with data fitting, iterations, or present FE-simulations.
Conversely, the physically founded linear *F*_{N} versus *h*^{3/2} Kaupp-plots, as
first introduced in lectures since 2000, correct for initial surface effects,
reveal phase transformation if they occur within the chosen force range.
Furthermore, they detect alternating layers, gradients, pores, defective
tips, tilted impressions, and edge interface or too close-by impressions.
For example, fused quartz Berkovich indents exhibit the well-known
amorphous to amorphous phase transformation [11,12] at about 2.50
or 2.25 mN applied work and 113 or 107 nm depth (analyzed loading
curve of Triboscope or CSIRO-UMIS manual, respectively) [11]. This
is indicated by a sharp kink in the Kaupp-plot, as it occurs in the chosen
loading range [5,11,13].

The force *F*N is linearly applied in force controlled experimental
indentations. This can again be imaged together with the exponent
3/2 parabola, which is physically founded [6] and experimentally
found (**Figure 2** [5,11]) and (1). Similar to Equations (2)-(5) deducing *W*_{applied}/*W*_{indent} for the wrongly assumed ISO exponent 2 on *h*, the
energetic deduction for the physical exponent 3/2 on *h* is given by
the formulas (7)-(9). The physical ratio is thus *W*_{applied}/*W*_{indent}=5:4. The
difference 5−4=1 is for the shear force component exerting pressure and plasticization on the adjacent material. That means: precisely 80%
of the applied work and (as *W* ∝ *F*) also applied force *F*_{N} is left for the
penetration. Thus, 20% is for exerting the sum of pressure and plastic
deformation energies to the solid environment. This is considerably
less loss for the indentation than if the assumed unphysical exponent
2 would apply (33.33%, see above). The new knowledge is expressively
supported with **Figure 5** that shows the difference in relation to the **Figure 3** for the false exponent.

*F*_{N}=*k h*^{3/2}

*W*_{indent}=0.4 const *h*^{5/2} (7)

*W*_{applied}=0.5 *F*_{Nmax} *h*_{max} (8)

*W*_{applied}=0.5 const *h*_{max}^{5/2} (9)

We have now *W*_{indent}=0.8 *W*_{applied}. The basic energy law is thus no
longer violated when the applied force *F*_{N} (and thus also *k*) is corrected
with the factor 0.8. Furthermore the definition of all physical parameters
that are related to the indentation force must also not violate the first energy law and require the factor 0.8, provided the exponent correction
(2 giving 3/2) has also be performed. Importantly, the now deduced
universal 5/4/1 ratio (applied/indent/long-range work) for pyramids
and cones is valid for all uniform materials, be they elastic, plastic,
migrating, viscous, sinking in, piling up, and flowing. Particular cases
are surface effects, gradients, tilted or too tight or edge indentations,
pores, micro-voids, cracks, defective tips' effects, and most important
kink indicating phase transformation onset. It is valid for all differently
angled smooth pyramids or cones with mathematical precision. Any
deviations are experimental errors. Surface effects include water
layers, gradients, oxides, hydroxides, surface compaction, tip rounding
(sometimes compensating other surface effects), and the like. They do
not belong to the bulk material and must therefore be eliminated from
regressions.

**Implementation of the first energy law in instrumented
indentation**

The energetics of the instrumented depth sensing indentation with
pyramids or cones has first been published in 2013 [4] for the *F*_{N}=*k h*^{3/2} relation. 20% of the applied work is lost for the indentation with
mathematical precision due to the shear-force elastic and plastic work,
including sink-in or pile-up. This is universal for all different shapes
and materials.

As deduced above, the applied force *F*_{N} with the directly
proportional otherwise physically correct published parameters
(including *H*_{phys} in [7]) must be corrected with the factor 0.8 (5/4 ratio,
80%) (similarly for *E*_{r-phys}, see below). Thus, **Figure 2** (all with correct
exponent 3/2) corrects now the data from the originals in Kaupp
[5,7]. Considering the advanced knowledge, this includes all the
penetration resistance values *k* and phase-transformation conversion
energies *W*_{conv} (both correction with the factor 0.8) that were published
up to 2016. Not affected are the activation energies and the phasetransformation
onsets at characteristic depth, because of cancellation.
Also most of the other mechanical parameters from indentations in
the literature including ISO-hardness and ISO-modulus are affected.
The new knowledge that requires a further specification also for the
hardness and modulus definitions requires separate treatment in the
next Section below, because these require also the above mentioned
dimensional correction.

The tip influence on the *k*-values (**Figure 2**) and their conversion
between different tips has been demonstrated and can be normalized
[13]. Creep depends on force and temperature. It is a materials property
but does not change the exponent on *h* of the loading curve, only the
penetration resistance *k*. Loading times should thus not exceed 30 s to
avoid such influence. Independent creep measurements and corrections
must only be performed for most precise rankings of materials. But it
is usually much less severe than with the viscoelastic PMMA (strongly
diverging from different authors) and certainly for the PDMS values of **Figure 2**. Indentation times are in fact generally very fast (10-30 s) and
creep is mostly slow even at high temperatures, so that a rating along
the *k*-values is a good choice already without creep corrections. Creep
is mostly not corrected for or published, while thermal drift can be
easily corrected for. Creep has however great importance for long-term
pressure under heat and for the properties of viscoelastic materials with
time dependent behavior. Importantly, the exponent on *h* remains 3/2
also at indentations of organic crystals with lattice guided anisotropic migrations [13,14].

**Basic energy law and dimensional corrections of indentation hardness**

A quantitative foundation of conical or pyramidal nanoindentation
results as for hardness (and modulus) has to obey the first energy law.
All world suffers from such violation that requires correction. The *F*_{N}=*k h*^{3/2} relation (1) corrects the fact that only 80% of *F*_{N} is used for the
indentation with the adjusted *k*-value in accordance with the energy
law. The correction of *H*_{phys}=*k*/π tanα^{2} (mN/μm^{3/2}), as taken from the
correct loading curve, where the factor 0.8 is included in the *k*-values,
is exhaustive and complies with the first energy law. The physical
indentation hardness has unavoidably the dimension (μN/nm^{3/2}) or
(mN/μm^{3/2}) (11). The loading curve provides the easiest, most precise,
most rapid and cheap way to obtain the correct physical hardness *H*_{phys}. The deduction of (11) starts with the definition of the universal
hardness (*F*_{N}/*A*_{proj}) relying on unphysical *h*^{2} and violating the energy
law. This has to be corrected with the dimensional factor *h*^{1/2} [that is
also required to make Equation (6) concur with physics] for exponent
correction to concur with the correct exponent 3/2 on *h* (1) [7] and the
factor 0.8 to concur with the first energy law that is already contained
in the penetration resistance *k*. This leads via (10) to (11) after
expression of the projected area and insertion of (1) with cancellation
of *h*^{3/2}. Importantly, the physical hardness *H*_{phys} is thus independent
of projected area, depth, *F*_{Nmax}, and standard material. It avoids all
iterations or fittings or approximations but is experimentally obtained
by linear regression and it becomes a genuine physical quantity for the
first time. It is also not falsified by undetected phase transformations,
because these would show-up in the linear regression. A sharp kink
before *F*_{Nmax} must be absent! The applications of *H*_{phys} should be very
welcome. It is nothing else than a normalized penetration resistance.
For example the physical hardness values can be directly obtained from
the examples in **Figure 2** by using the α-values of the corresponding
indenters (Berkovich is ISO-standard).

(10)

(11)

The odd appearing dimension mN/μm^{3/2} (also GPa μm^{1/2}) of the
physical indentation hardness, which does only resemble to a pressure
is unavoidable, due to the mathematically fixed shear force component
of indentations that cannot be avoided. Nevertheless, indentation
remains a very useful particularly precise technique.

Universal hardness, ISO hardness, and FE-simulated hardness
would require a factor 2/3 for correction of *F*_{N} to give the force for
the indentation in order to accept the energy law (**Figure 3**). But after
the necessary multiplication with *h*_{max}^{1/2} for dimensional correction the
force correction becomes 0.8 (**Figure 5**). However, such corrections of
the ISO hardness can only be approximate, because the *h*_{c} and thus *A*_{hc} iterations with respect to a standard material cannot be reverted.
Force induced phase-transformations must always be excluded with a
Kaupp-plot that at the same time obtains the physical hardness more safely and directly (11).

The equations (12) and (13) show how easy it is to calculate *H*_{phys} from published *H*_{univ} or *H*_{simul} values, provided the *h*_{max} values for *F*_{Nmax} are available, and when phase changes are excluded before *F*_{Nmax} is
reached. The corrections are multiplications with *h*_{max}^{1/2} for the correct
exponent 3/2 and factor 0.8 for the force loss.

(12)

(13)

This is exemplified in **Table 1** with a numerical example from a
published indentation onto aluminum, where *H*_{ISO} and both the FE-simulated
H_{simul} (ANSYS software) with exponent 2 on h and the
experimental Berkovich loading curves are published (falsely claimed
exponent 2 but according to the Kaupp-plot determined with exponent
3/2 on *h*) [15]. Any universal hardness (*H*_{univ}) treatment would be
the same as the one for H_{simul}. The published loading curve was also
provisionally analyzed as FN-h^{2} plot but only used for numerical
achievement of the conversions.

Number | Technique | hmax^{n} |
k or h_{ma}^{(a)} |
Hardness calculations and corrections |
---|---|---|---|---|

1 | Experimental linear regression | h_{max}^{3/2} |
k=5.9540(mN/µm^{3/2}) (energy corrected)(b) |
H_{phys}=k/πtanα^{2}=0.24295(mN/µm^{3/2}) independent on FN andh_{max}(no phase trans.) |

2 | Experimental with 2/3 factor | h_{max}^{2} |
- - | H_{ISO}=0.716 (GPa) x (2/3) ≈ 0.477 (mN/µm^{2})(unphysical dimension) h not known_{max} |

3 | FE-simul.h_{ma}^{1/2} no energycorr. |
h_{max}^{2} |
h_{max}=251.984 nm |
H_{simul-corr1} (as Huniv)=FN_{max}/πtanα2h_{max}3/2=0.2977 (mN/µm^{3/2}) (energy law violation!) |

4 | FE-simul.2/3; no exponent corr. |
h_{max}^{2} |
h_{max}=251.984 nm |
H_{simul-corr2}=2FN_{max}/3πtanα^{2}h_{max}^{3/2}=0.4011(mN/µm ^{2})(wrong exponent) |

5 | FE-simul, 0.8, and h_{max}^{1/2} |
h_{ma}^{2} |
h_{max}=251.984 nm |
Hsimul-phys=0.8 FN_{max}/πtanα2h_{max}^{3/2}=0.2382 (mN/µm^{3/2}) |

**Note:** ^{(a)}Simulated parameters are not italicized; ^{(b)}correction factor 0.8.

**Table 1:** Comparison and correction of unloading *H*_{ISO} and FE-simulated H_{simul} loading curves of Al on Si [15] with the physical *H*_{phys}, which is in accordance with the energy law.

Entry 1 in **Table 1** gives the *H*_{phys} from the analyzed loading curve
(11), which is certainly the most reliable value. It does not rely on *F*_{Nmax}, *h*_{max}, any *h*_{c} or *A*_{hc} and it secures the absence of a phase change up to the
maximal force. And it compares with *H*_{ISO} and the hardness values that
derive from H_{simul} with various stages of correction.

Entry 2 shows that *H*_{ISO} exhibits a far too high value and an
unphysical dimension. The energy correction for leaving exponent
2, removing only the energy law violation, decreases the value
insufficiently, still with the unphysical dimension mN/μm^{2}. A value for *h*_{max} is not available for a final correction. When exceptionally a guess
were tried that it might be in a 0.25 μm region one would guess a further
decrease that would look like 0.239 with the changed dimension mN/
μm^{3//2}. This would be in the region of *H*_{phys} although with all reservation,
because it is only a free guess only indicating the direction. This show
the difficulties for the conversion when *h*_{max} for the used *F*_{Nmax} is not
reported. It is thus much easier to apply the Kaupp-plot to the loading
curve (1). We renounce of including the uncorrected simulated value
(0.6016 mN/μm^{2}).

Entry 3 gives only the exponent correction of FE H_{simul} (ANSYS-software)
that was probably obtained by using Young's modulus E
(either known or iterated) input, with converging criterion to exponent
2 on h.

Entry 4 gives only the energy correction with a rather high value. **Table 1** show that neither the exponent correction for exponent 2
alone nor the energy correction (**Figure 3**) alone (removing energy law
violation) is sufficient.

Entry 5, finally with both exponent correction and then smaller
energy correction factor for h ^{3/2} (**Figure 5**) provides H_{simul-phys}, with surprisingly good match (2%) with *H*_{phys}. The surprisingly close
coincidence of *H*_{phys} and H_{simul-phys} supports the numerical correctness
of the non-fitting (!) straightforward deduction and it also reminds the
unbeatable precision of the Kaupp-plot's linear regression (**Figure 2**).
The close correspondence with H_{simul-phys} in this case should however
be tested for generality, because this single example could be fortuitous
when considering the parameterizations and iteration procedures at FE
simulations.

Importantly, the striking dilemma of ISO with physics persists with
the false dimension of too large *H*_{ISO} and unphysical dimension. All of
the values and dimensions of the mechanical parameters that depend
on it are severely wrong, also those that depend on wrong ISO elastic
modulus *E*_{r} (see next Section). Clearly, **Table 1** and Equations (12)
with (13) show an easy way for straightforward corrections of *H*_{univers},
probably H_{simul}, and with reservation *H*_{ISO}, provided the *h*_{max} values
are known. However, despite the straightforward corrections none
of them can handle the very often occurring and so important phase
transformations under load (here they were experimentally excluded
with Kaupp-plot).

**Basic energy law and corrections of indentation elastic
modulus**

Also the correction of unphysical *E*_{r-ISO} into a physical value
is essential, because elasticity is a technically important materials
property. Young's moduli E are required for the deduction of numerous
mechanical qualities and for example increasingly as input parameter
for numerical FE-simulations, often including FE-iterations with E-Y
pairs as free parameters (where Y is yield strength). The determination
of the elastic modulus requires the unloading stiffness *S*=d*F*_{Nmax}/d*h*_{max} from the pressure to the displaced material that must be separated
from the plastic response. This is achieved for *F*_{Nmax}, which is a joint
quantity of loading and unloading curves. Thus, there must again be
corrections for dimension adjustment with *h*^{1/2} [for not violating (1)]
and for shear force loss during the loading (for not violating the energy
law). These are not applied in ISO 14577 that applies the Oliver-Pharr
iterations [3]. Thus, the slope correction for Er-ISO (14) requires again
the exponent correction with *h*^{1/2} and then the force correction factor
0.8 (**Figure 5**) to comply with the energy law for obtaining *E*_{r-phys}. This
gives via (14) *E*_{r-phys} (15) in complete correspondence with the necessary
treatment of *H*_{ISO} (this replaces the incomplete formula 11 in [7]).
Again one must be certain that the unloading was performed at *F*_{max} before any onset of a force derived phase transformation had occurred.
By comparing (14) and (15) the correction factors are found to be 0.8
and *h*_{max}^{1/2}. The corrections for obtaining physical modulus values with
changed dimension is simply by multiplication with 0.8 *h*_{max}^{1/2}. The
unloading stiffness *S* and *h*_{max} (before creep) must be known. This is
another dilemma between ISO and physics.

(14)

The extremely complicated mathematical deductions of Sneddon/
Love ([1,2]; **Figure 1**) for the conical or pyramidal indentations did not
consider the energetics of the process, as illustrated with the **Figures 3** **and 5**. And there was no protest from physicists. Almost all involved
people followed Sneddon [1], Oliver Pharr [3], and ISO 14577 all
with violating the first energy law for more than half a century. The general acceptance for half a century of the implied claim that pressure
formation and plasticization could be workless achieved is hard to
understand. It is apparently the result of hype upon the publication
[1] that unfortunately was believed by ISO/NIST. The simple equations
as derived starting in 2000 ([14] and before in lectures and in refused
manuscripts) and the point by point unraveling of the field until now
against strong impediments did not help. The newcomers had to obey
ISO 14577 and many very complex rules, and they used the software of
the instrument suppliers that had to trust in the ISO/NIST-standards.
By doing so they forgot to think on the physical foundations. Thus,
the basic formulas (3)-(5) and (7)-(9) that essentially rely on the
experimentally (since 2000) (**Figure 2**) and physically founded (since
2015) Equation (1) [6] found much refusal, various excuses for not
experimentally finding exponent 2 with fittings, multi-parameter
iterations, and simulations. The actions against the elementary
algebraic treatment without any fitting/iterating/simulating were
undue repetitive offenses. Rather acknowledgement had to be expected
because everything became much easier and quantitative on a sound
physical basis with simple closed mathematical formulas, proving the
universal validity.

Apparently, nobody else (not even textbook or tutorial writers)
asked themselves why all of the applied normal force with cones or
pyramids is claimed to be used for the indentation depth, even though
the loading curve proceeds not linearly but parabolic. The obvious
answer is that well-known long range effects and pressure formation
to the environmental solid material require energy that is lost for
the indentation depth. When this energy/force/loss was quantified
and finally (after difficulties with anonymous Reviewers) published
in 2013 [4] with the universal loss of 1/5 for the physical (1) and 1/3
for unphysical (2) equations, there was discussion about the validity
for comparing applied work and indentation work. But these proceed
at the same time to the same endpoint *F*_{max}. Surprised about the ease
of the mathematical deduction and the strict and universal result,
requiring difficult necessary changes, there were objections and
much open discussion in plenary lectures from the audience with the
guess that all of the linearly applied force might instead go along the
parabolic curve during the experiment. This prevented the opponents
from recognizing that the first energy law was evidently violated. The
linearity of the applied force is however also evident, simply from the
additional applied force *F*_{N} versus time plot in **Figure 4**.

The undue opposition against straight forward physics and algebra
is surprising even after it was very clear with Kaupp [4] that the ISOsystem
violates the first energy law (the present author could not
dare to verbally express the energy law violation at that time). The
offenses have been continuing. For example, the opposing manuscript
[9] was received at Scanning on May 27, 2014, whereas the clarifying
manuscript [4] was received at Scanning on October 4, 2012 and
published on February 25, 2013. The content of Kaupp [4] had thus to
be taken up again in Kaupp [7] with more details, because the authors,
reviewers, and editors of Merle [9] continued violating the basic energy
law. And the Merle [9] continued arguing against the most precise
Kaupp-plot that actually was the basis for the quantification of the
violation. The opponents tried with iterated own loading curves of
fused quartz. But when doing it correctly, even the invoked curve in
Troyon [10] would roughly reproduce the well-known transformation
onset, despite its using a blunt tip that gave an unusually long
initial effect. And Merle [9] tries again with a false intersection at its
microindentation "Kaupp-plot" (up to 300 mN and 1600 nm) where
the region with all of the nanoindentation details is almost totally
obscured in a short unstructured part of it. The false intersection with a remote line far away from the plot is useless. But it is used for falsely
criticizing the Kaupp-plots that never used or use such faulty tricks.
When properly looking at this linear plot in Reference [9] with a ruler,
one recognizes an intersection of two straight lines at about 175 mN
and 1225 nm, which the authors do neither trace nor recognize. Four
possibilities exist for this kink very close to the plot: either a new highload
phase transition of fused quartz occurred, or a smoothness defect
of the tip was present at this depth, or a remote crack at such deep
impression was formed, or the impression was too close to an edge/
interface/impression. Furthermore, these authors claim and draw a
straight single line for their unphysical so called "P-*h*^{2} fit with 0.999 fit
quality". However, despite their claimed "three-nines fit", their depicted
unphysical "P-*h*^{2} fit" gives two roughly linear branches, intersecting in
the region of 60-70 mN force (that is far away from surface effects).
This deviation from the claim is easily "overlooked" without a ruler in
a wide pencil stroke representation at totally false depth-square scaling
(better seen when more precisely drawn, the first part steeper and
cutting at small angle). This shall only be a necessary contradiction to
the false claim of linearity for a "P-*h*^{2} fit" trying to discredit our simple
algebraic treatment on a sound physical foundation. Fitted or FE
curves, converging with *h*^{2}, must not be used for denying thoughtful
and repeatable physically founded [6] and experimental Equation (1).
Only untreated experimental loading curves are able to detect surface
effects, the important phase changes, conversion energies, etc., when
using the physically founded exponent on *h*.

A problem might arise when fitted, iterated or FE-simulated curves
and experimental loading curves might be mixed up in publications.
However, when experimental force data are plotted with or fitted to
the non-physical h2, the deviations from a straight line might appear
minor for example as in Merle [9]. Also a minor endothermic phase
change slightly levels the unphysical *F*_{N}-*h*^{2} trial-plot with respect
to the stronger curved appearance without phase change [5,11].
Such leveling behavior of the test material fused quartz might have
strengthened the belief in *h*^{2}, but it reflects the inability to find phase
transformation with the physically wrong exponent 2 on *h*. All of the
important details of nanoindentation are lost with *h*^{2}. But the kink at *F*_{N} ≈ 2.4 mN (Berkovich) and initial surface effects of the fused quartz
standard are easily seen by sharp kinks with the precise Kaupp-plot
(1) in nanoindentations, notwithstanding the cases of later or further
phase changes in microindentations (e.g. NaCl in [5,11]). But there is
no excuse for using the unphysical exponent and thus denial of the
phase transitions if these occur, combined with the violation of the first
energy law.

The readers of Kaupp [4] and the attendants of the present author's
lectures on numerous worldwide conferences were repeatedly urged to
think about the unexpected and surprisingly easy deduced energetic
facts (2)-(6) and (7)-(9) but the expected response of the scientific
establishment is still missing. It appeared unlikely that all of the scientific
Celebrities and their successors including textbook authors, ISO/NIST,
and numerous anonymous referees have, consciously or not, been
violating the first energy law for more than 50 years. Hesitation to use
only the normal force left for the indentation depth was thus advisable,
before any non-apparent compensation effect for saving the energy
law was excluded in the desperate situation. Publications of the truth
should stay as close as possible with the current indentation theory
unless all objections are removed. Clearly, the believers in exponent 2
on *h* could for themselves have easily performed the deductions as in
Equations (1)-(5) and could have tried to change their minds because
of this inexcusable energy law violation. But they did not try to take into account the always occurring energy loss. Based on their believed
exponent 2 on *h* it would have amounted to 1/3 (33.33%) of *F*_{N} due
to the work and force proportionality, as shown above with the trial
Equations (2)-(5). And they would have found that the violation is also
programmed and used in FE-simulations. They refused till now to accept
the undeniable wealth of the Kaupp-plot and the physical deduction for
the correct exponent 3/2 on *h* [6] that finally proves energy/force loss of
1/5 (20%) according to Equations (7)-(9), as only the physical exponent
is correct. Since ISO/NIST have been reluctant to change their minds,
or to announce reconsideration with an alert, there was the urgent
preliminary publication in Kaupp [7] for expressively naming the
incredible claim of workless pressure formation and plasticization
as "violation of the basic first energy law". This is now completed
with valid transformation formulas for obtaining the physical values
and the necessary conditions for that from unphysical publications.
Furthermore, the most easy and precise *H*_{phys} determination by linear
regression of the loading curve (1) (hitherto strongly refuted Kauppplot)
with energy-based correction is now again strongly advocated for.

The still not settled dilemma between ISO and physics with respect
to ISO 14577 (not even an alert has been filed yet) is unbearable due
to its enormous risks for science and daily life's safety. It appears
unbelievable and even desperate that the first energy law was drastically
violated for more than 50 years and none of the physicists protested
against such habit. Everything is easily deduced with first grade
algebra, avoiding fittings, iterations, simulations, and approximations,
making everything much more easy. Hardness is now obtainable by
linear regression, no longer by iterations, fittings, approximations,
and simulations that are not ready for a controlling assessment. The
physical indentation hardness *H*_{phys} (mN/μm^{3/2}) is now for the first time
a genuine physical quantity, obeying Equation (1) and the first energy
law. The same is true for the indentation modulus *E*_{r-phys} (mN/μm^{3/2}).
The complete, more precise deduction than in Kaupp [7] reveals also
the simple conversion from *E*_{r-ISO}. Only the quantitative indentation on
the physical basis reveals numerous otherwise impossible applications.
Examples are phase change [4,5,16], conversion energies [4,16] and
activation energies [16] of materials, all on the basis of the so-called
Kaupp-plot (1) that also checks for correctly performed indentations
and provides extrapolation facility up to recognized phase change
qualities under pressure. Furthermore, it reveals a large number
of special materials' properties and indentation errors that are
named above. But it is still being heavily suppressed by the ignoring
establishment, including ISO and some anonymous Reviewers with
incredible unqualified wording instead of acknowledging this wealth.

The liability with unphysical calculated materials' properties is
totally unclear at the present dilemma between ISO and physics, because
all safety engineers are falsely trained. That means, the issue counts for
every days safety unless ISO files at least an urgent alert. Everybody
knows how many materials fail shortly after the warranty period,
certainly not purposeful but often with falsely calculated materials.
Even worse, falsely calculated components like poorly adjusted
medicinal implants or larger scale composites can produce disasters.
There is good reason why passenger traffic airplanes require frequent
safety checks and complete replacement of all parts within 2 years. For
example *h* goes with *F*_{N} ^{2/3} not with *F*_{N} ^{1/2} [5] with all implications for
fatigue, and wear, to name a few.

Despite the highly comprehensive results of this paper and numerous
worldwide lectures on conferences the ISO versus physics dilemma still
remains. The physical indentation *H*_{phys} and *E*_{r-phys} dimensions that only resemble pressures is perhaps difficult to understand at first glance.
But it is real and the reasons have been discussed. Importantly this
does not detract from indentation as a very precise and reproducible
technique, when properly executed, checked, and algebraic evaluated,
that means without fittings, iterations, and simulations.. Rather the
unavoidable dimensional changes have an enormous bearing for
science and practice. The not fitted and not iterated physical quantities
must be used to redefine the numerous further mechanical parameters
that were deduced from unphysical *H*_{ISO} or *E*_{r-ISO}. Further studies are
necessary and further important insights are to be expected when the
violation of the first and most basic energy law will be removed also for
the deduced parameters. This should help for a better understanding
and open new horizons. Also textbooks must be rewritten for the sake
of physics, compatible materials sciences, and new insights. Since there
was the violation of the first energy law, the new results will prove to be
more compatible with all related techniques that do not violate physical
laws, which is very desirable. The quantitative indentation at the now
physical basis has the indispensable advantages of being precise, and in
accord with basic principles. This is promising and cannot be denied.
The further advancement on the physical basis is a very urgent task
that must be pursued, hopefully soon also with ISO/NIST against all
of the incredible resistance, because violating the first energy law is an
inexcusable fault.

- Sneddon IN (1965) The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Engn Sci 3: 47-57.
- Love AEH (1939) Boussinesq's problem for a rigid cone. Q J Math (Oxford) 10: 161-175.
- Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 7: 1564-1583.
- Kaupp G (2013) Penetration resistance: a new approach to the energetics of indentations. Scanning 35: 392-401.
- Kaupp G (2013) Penetration resistance and penetrability in pyramidal (nano)indentations. Scanning 35: 88-111.
- Kaupp G (2016) The physical foundation of
*F*N=*k h*^{3/2}for conical/pyramidal indentation loading curves. Scanning 38: 177-179. - Kaupp G (2016) Important consequences of the exponent 3/2 for pyramidal/conical indentations-new definitions of physical hardness and modulus. J Mater Sci & Engineering 5: 285-291.
- Wang TH, Fang TH, Lin YC (2008) Finite-element analysis of the mechanical behavior of Au/Cu and Cu/Au multilayers on silicon substrate under nanoindentation. Phys Appl A 90: 457-463.
- Merle E, Maier V, Durst K (2014) Experimental and theoretical confirmation of the scaling exponent 2 in pyramidal load displacement data for depth sensing indentation. Scanning 36: 526-529.
- Troyon M, Abbes F, Garcia Guzman JA (2012) Is the exponent 3/2 justified in analysis of loading curve of pyramidal nanoindentations? Scanning 34: 410-417.
- Kaupp G, Naimi-Jamal MR (2010) The exponent 3/2 at pyramidal nanoindentations. Scanning 32: 265-281.
- Trachenko K, Dove M (2003) Intermediate state in pressurized silica glass: reversibility window analogue. Phys Rev B 67: 212203/1-212203/3.
- Kaupp G (2006) Atomic force microscopy, scanning nearfield optical microscopy and nanoscratching - application to rough and natural surfaces. Springer, Berlin.
- Kaupp G, Schmeyers J, Hangen UD (2002) Anisotropic molecular movements in organic crystals by mechanical stress. J Phys Org Chem 15: 307-313.
- Soare S, Bull SJ, Oila A, O'Neill AG, Wright NG, et al. (2005) Obtaining mechanical parameters for metallization stress sensor design using nanoindentation
*.*Int J Mater Res 96: 1262-1266. - Kaupp G (2014) Activation energy of the low-load NaCl transition from nanoindentation loading curves. Scanning 36: 582-589.

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