TITLE:
The Loading Curve of Spherical Indentions Is Not a Parabola and Flat Punch Is Linear
AUTHORS:
Gerd Kaupp
KEYWORDS:
Closed Formula for Spherical Indentation, Challenge of ISO14577, Mathematical Proofs, Volume Instead of Area, Correct Flat Indentations, Physical Indentation Hardness, Hardness Dependence on Indenter Shape, Data Treatment Detection
JOURNAL NAME:
Advances in Materials Physics and Chemistry,
Vol.9 No.8,
August
30,
2019
ABSTRACT: The purpose of this paper is the physical deduction
of the loading curves for spherical and flat punch indentations, in particular
as the parabola assumption for not self-similar spherical impressions appears
impossible. These deductions avoid the still common first energy law violations
of ISO 14577 by consideration of the work done by elastic and plastic pressure
work. The hitherto generally accepted “parabolas” exponents on the depth h (“2 for cone, 3/2 for spheres, and 1 for flat punches”) are still the unchanged
basis of ISO 14577 standards that also enforce the up to 3 + 8 free iteration
parameters for ISO hardness and ISO elastic indentation modulus. Almost all of
these common practices are now challenged by physical mathematical proof of
exponent 3/2 for cones by removing the misconceptions with indentation against a
projected surface (contact) area with violation of the first energy law,
because the elastic and inelastic pressure work cannot be obtained from
nothing. Physically correct is the impression of a volume that is coupled with
pressure formation that creates elastic deformation and numerous types of
plastic deformations. It follows the exponent 3/2 only for the
cones/pyramids/wedges loading parabola. It appears impossible that the
geometrically not self-similar sphere loading curve is an h3/2 parabola. Hertz did only deduce the touching of the sphere and Sneddon did not
get a parabola for the sphere. The radius over depth
ratio is not constant with the sphere. The apparently good correlation of such parabola
plots at large R/h ratios and low h-values does not withstand against the deduced physical equation for the spherical
indentation loading curve. Such plots are unphysical for the sphere and so tried
regression results indicate data-treatments. The closed physical deduction
result consists of the exponential factor h3/2 and a
dimensionless correction factor that is depth dependent. The non-parabola
against force plot using published data is concavely bent even for large
radius/depth-ratios at the shallow indents. The capabilities of
conical/pyramidal/wedged indentations are thus lost. These facts are outlined
for experimental nano- and micro-indentations. Spherical indentations reveal
that linear data regression is suspicious and worthless if it does not
correspond with physical reality. This stresses the necessity of the
straightforward deductions of the correct relations on the basis of
iteration-less and fitting-less undeniable calculation rules on an undeniable basic physical understanding. The straightforward physical
deduction of the flat punch indentation is therefore also presented, together
with formulas for the physical indentation hardness, indentation work, and
applied work for these geometrically self-similar indentations. It is
exemplified with a macroindentation.