_{1}

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 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 π ( R / h − 1 / 3 ) 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 N = F N v m F N p total n (1S)

There can be no doubt that the total pressure depends on the inserted calotte volume that is V = h 2 π ( R − h / 3 ) . 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 3 π ( R / h − 1 / 3 ) (2S)

F N p total ∝ h 3 (3S)

F N p total 1 / 3 ∝ h p 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 N v 2 / 3 ∝ h v or F N v ∝ h v 3 / 2 (5S)

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

F N v = k v h 3 / 2 π ( R / h − 1 / 3 ) (6S)

For plotting of (6S) for obtaining k_{v} the π ( R / h − 1 / 3 ) factor is separately multiplied with h^{3/2} for every data pair.

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