Is the Exponent 3/2 Justified in Analysis of Loading Curve of Pyramidal Nanoindentations?

Abstract

Kaupp and Naimi-Jamal (2010) claimed that the analysis of published loading curves reveals the exponent 3/2 to the depth for nanoindentations with sharp pyramidal or conical tips. To demonstrate this, they plotted the load vs. the penetration depth to the power 3/2. We show, through examples, the authors' assertion is not credible because the methodology used is misleading and it cannot be asserted that the exponent 3/2 has a universal validity that applies to all kinds of materials.

Important Consequences of the Exponent 3/2 for Pyramidal/Conical Indentations-New Definitions of Physical Hardness and Modulus

“…This is not too far away from the values for the known sharp Berkovich indents (2-2.5 mN and 120-160 nm) [4,13,14,17] at very minor initial effects. The experimental data printed curves of [16] are therefore supporting but not at all "discrediting" the exponent of Eq. (1), if considering the unusually extended initial effect range (axis cut of the k 1 line at about −1.2 mN; not drawn in [16]) at this indentation.…”

“…The experimental data printed curves of [16] are therefore supporting but not at all "discrediting" the exponent of Eq. (1), if considering the unusually extended initial effect range (axis cut of the k 1 line at about −1.2 mN; not drawn in [16]) at this indentation. Nevertheless, the authors in [16] deny their obvious support of h recalculate these for h 3/2 with the aim to discredit the experimental (now physically founded [10]) exponent 3/2, because such treatment inevitably gives bent curves.…”

“…(1), if considering the unusually extended initial effect range (axis cut of the k 1 line at about −1.2 mN; not drawn in [16]) at this indentation. Nevertheless, the authors in [16] deny their obvious support of h recalculate these for h 3/2 with the aim to discredit the experimental (now physically founded [10]) exponent 3/2, because such treatment inevitably gives bent curves. Such a "treated" curve was used for drawing tangents at the start and the end in Figure 2 of [11] that intersect far away from the plot, for designing a false discrediting term called "Double P-h 3/2 fit after Kaupp et al" [11].…”

“…Unfortunately, leveling equipment for skew surfaces (with AFM precision check) often lack in commercial nanoindentation instruments. Nevertheless, measurements with blunt Berkovich (R ≈ 300 nm) giving unusually long initial effects were tried to "discredit" the exponent 3/2 with the F N versus h 3/2 plot of fused quartz in Figure 1 [16]. However, this plot confuses the 3 initial-effect points with the not considered straight line through the points # 4−17 at the actual kink position where the steeper second linear branch starts.…”

“…(1) [10] to uncover individual properties (e.g. phase change yes or no) that are wiped out by data fittings or simulations as in [11,16]. It is unclear, where the data of [11] in opposition to physics [10] and to the published ones came from, and who did the calculations for fused quartz up to 300 mN load on what assumptions.…”