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

Gerd Kaupp^{}

University of Oldenburg, Oldenburg, Germany.

**DOI: **10.4236/apm.2020.109034
PDF HTML XML
201
Downloads
541
Views
Citations

University of Oldenburg, Oldenburg, Germany.

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.

Keywords

Share and Cite:

Kaupp, G. (2020) Erratum to “Valid Geometric Solutions for Indentations with Algebraic Calculations”, [Advances in Pure Mathematics, Vol. 10 (2020) 322-336]. *Advances in Pure Mathematics*, **10**, 545-546. doi: 10.4236/apm.2020.109034.

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.

Conflicts of Interest

The author declares no conflicts of interest.

Journals Menu

Contact us

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2023 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.