The physical foundation of F_N = kh^3/2 for conical/pyramidal indentation loading curves

Abstract

SummaryA physical deduction of the F N = kh 3/2 relation (where F N is normal force, k penetration resistance, and h penetration depth) for conical/pyramidal indentation loading curves has been achieved on the basis of elementary mathematics. The indentation process couples the productions of volume and pressure to the displaced material that often partly plasticizes due to such pressure. As the pressure/plasticizing depends on the indenter volume, it follows that F N = F Np 1/3 · F NV 2/3, where the index p stands for pressure/plasticizing and V for indentation volume. F Np does not contribute to the penetration, only F NV. The exponent 2/3 on F NV shows that while F N is experimentally applied; only F N 2/3 is responsible for the penetration depth h. Thus, F N = kh 3/2 is deduced and the physical reason is the loss of F N 1/3 for the depth. Unfortunately, this has not been considered in teaching, textbooks, and the previous deduction of numerous common mechanical parameters, when the Love/Sneddon deductions of an exponent 2 on h were accepted and applied. The various unexpected experimental verifications and applications of the correct exponent 3/2 are mentioned and cited. Undue mechanical parameters require correction not only for safety reasons. SCANNING 38:177–179, 2016. © 2015 The Authors. Scanning published by Wiley Periodicals, Inc.

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

“…(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].…”

“…However, Kaupp et al do not fit treated data but are analyzing experimental loading curves according to the physically deduced universal Eq. (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].…”

“…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. The polynomial or FE correlations (Table 1) [11] are not helpful (for example phase changes are unavoidable for the partially crystallized POM and PEEK thermoplastics or compacted Al).…”

“…Clearly, one was not willing to recognize specific properties of materials under loading stress and strangely strived for concurring with the disproved Sneddon theory. The exponent on h is with mathematical precision 3/2 and the dimension of k is (force/length 3/2 ) [10]. Minimal deviations are experimental errors.…”

“…After the recent physical foundation of the exponent 3/2, giving the explanation why it must be so, by considering both the simultaneous volume-formation and the thereby created total pressure with elementary mathematics [10], the physical law (1) is additionally enforced beyond any doubt and cannot be denied any more. We must herewith point out the necessity of using this new state of the art for removal of the dilemma between ISO 14577 and physics (for undue NIST tutorial from 2009) for all mechanical parameters that rely on h 2 .…”