The Physical Foundation of F\(_{N}\)=kh\(^{(3/2)}\) for Conical/Pyramidal Indentation Loading Curves: Scientific Explanation

Abstract

On the basis of fundamental mathematics, it has been possible to physically deduce the \(F_{\mathrm{N}}=k h^{3 / 2}\) relation for conical/pyramidal indentation loading curves (where \(F_{\mathrm{N}}\) is normal force, \(k\) penetration resistance, and \(h\) penetration depth) for conical/pyramidal indentation loading curves. It has been achieved on the basis of elementary mathematics. The displacement of material, which frequently partially plasticizes as a result of such pressure, is coupled with the productions of volume and pressure by the indentation process. As the pressure/plasticizing depends on the indenter volume, it follows that \(F_{\mathrm{N}}=F_{\mathrm{Np}}^{1 / 3} \cdot F_{\mathrm{Nv}}^{2 / 3}\), where the index \(\mathrm{p}\) stands for pressure/plasticizing and \(\mathrm{V}\) for indentation volume. \(F_{\mathrm{Np}}\) does not contribute to the penetration only \(F_{\mathrm{NV}}\). The exponent \(2 / 3\) on \(F_{\mathrm{NV}}\) shows that while \(F_{\mathrm{N}}\) is experimentally applied; only \(F_{\mathrm{N}}^{2 / 3}\) is responsible for the penetration depth \(h\). Thus, \(F_{\mathrm{N}}=k h^{3 / 2}\) is deduced and the physical reason is the loss of \(F_N^{1 / 3}\) for the depth. Unfortunately, when the Love/Sneddon deductions of an exponent 2 on \(h\) were accepted and applied, this was not taken into account in instruction, textbooks, or the earlier deduction of a number of common mechanical parameters. The author mentions and cites several unexpected experimental verifications and applications of the correct exponent \(3 / 2\).