Erratum to “Valid Geometric Solutions for Indentations with Algebraic Calculations”, [Advances in Pure Mathematics, Vol. 10 (2020) 322-336]

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The original online version of this article (Gerd Kaupp 2020) Valid Geometric Solutions for Indentations with Algebraic Calculations, (Volume, 10, 322-336, https://doi.org/10.4236/apm.2020.105019) needs some further amendments and clarification.

The Deduction Details for the Spherical Indentations Equation

The incorrect proportionalities (16) and (17) in the published main-text are useless and we apologize for their being printed. They were not part of the deduction of the Equation (18_{v}). The deduction of (18_{v}) follows the one for the pyramidal or conical indentations (4) through (8). The only difference is a dimensionless correction factor
$\pi \left(R/h-1/3\right)$ that must be applied to every data pair due to the calotte volume. The detailed deduction of (18_{v}) = (6S), is therefore supplemented here.

Upon normal force (F_{N}) application the spherical indentation couples the volume formation (V) with pressure formation to the surrounding material + pressure loss by plasticizing (p_{total}). One writes therefore Equation (1S) (with m + n = 1)

${F}_{\text{N}}={F}_{\text{N}v}^{m}{F}_{\text{N}p\text{total}}^{n}$ (1S)

There can be no doubt that the total pressure depends on the inserted calotte volume that is $V={h}^{\text{2}}\pi \left(R-h/3\right)$ . It is multiplied on the right-hand side with 1 = h/h to obtain (2S). We thus obtain (3S) and (4S) with n = 1/3.

$V={h}^{\text{3}}\pi \left(R/h-1/3\right)$ (2S)

${F}_{\text{N}p\text{total}}\propto {h}^{\text{3}}$ (3S)

${F}_{\text{N}p\text{total}}^{1/3}\propto {h}_{p}{}_{\text{total}}$ (4S)

(4S) with pseudo depth “h_{p}_{total}” is lost for the volume formation. It remains (5S) with m = 2/3 on F_{Nv} or the exponent 3/2 on h_{v}.

${F}_{\text{N}v}^{2/3}\propto {h}_{v}$ or ${F}_{\text{N}v}\propto {h}_{v}^{3/2}$ (5S)

The proportionality (5S) must now result in an equation by multiplication with the dimensionless correction factor
$\pi \left(R/h-1/3\right)$ and with a materials' factor k_{v} (mN/µm^{3/2}) to obtain Equation (6S) that is Equation (18) in the main paper.

${F}_{\text{N}v}={k}_{v}{h}^{3/2}\pi \left(R/h-1/3\right)$ (6S)

For plotting of (6S) for obtaining *k _{v}* the
$\pi \left(R/h-1/3\right)$ factor is separately multiplied with

An additive term F_{a} can be necessary for the axis cut correction if not zero due to initial surface effects of the material.

References