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.

erratum
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 (18v). The deduction of (18v) 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 (18v) = (6S), is therefore supplemented here.

Upon normal force (FN) application the spherical indentation couples the volume formation (V) with pressure formation to the surrounding material + pressure loss by plasticizing (ptotal). 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 “hptotal” is lost for the volume formation. It remains (5S) with m = 2/3 on FNv or the exponent 3/2 on hv.

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 kv (mN/µm3/2) 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 kv the π ( R / h − 1 / 3 ) factor is separately multiplied with h3/2 for every data pair.

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