The Loading Curve of Spherical Indentions is Not a Parabola and Flat Punch is Linear: Scientific Explanation

The goal of this paper is to physically deduce the loading curves for spherical and flat punch indentations, especially since the one exponent parabola assumption appears to be impossible for not self-similar spherical impressions. By taking into account the work done by elastic and plastic pressure work, these deductions avoid the still common first energy law violations of ISO 14577. The misleading though 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 (false for cone and sphere) up to 3 + 8 free iteration parameters for ISO hardness and ISO elastic indentation modulus. Since the elastic and inelastic pressure work cannot be created out of nothing, almost all of these common practices are now refuted by physical mathematical proof of exponent 3/2 for cones, which also dispels the myths about indentation against a flat projected surface (contact) area with violation of the first energy law. The impression of a volume that is associated with pressure formation, which results in elastic deformation and a variety of plastic deformations, is physically accurate. Only the loading parabola for cones, pyramids, and wedges follows the exponent 3/2. 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 one-exponent parabola for the sphere. The radius over depth ratio is not constant with the sphere. The sole exponent 3/2 on h (Johnson’s formula) for the spherical indentation loading curve does not hold up against the ostensibly strong correlation of such parabola graphs at large R/h ratios and low h-values. Such graphs do not represent the sphere physically, and so, attempted regression results point to data manipulations using at least one published data fitting formula. Unbiased algebraically we now deduce with the exponential factor h3/2 and a depth-dependent dimensionless correction factor containing the R/h ratio as result the closed physical formula for the spherical indentations. Even with huge radius/depth-ratios at the shallow indents, the h3/2 against force plot utilizing published data is concavely bent. Thus, the benefits of conical, pyramidal, and wedged indentations are negated. For experimental nano- and micro-indentations, these details are listed. Spherical indentations showing linear data regression for force versus h3/2 plots do not agree with physical reality. They are useless and reveal the data-fitting. This emphasizes the need for simple deductions of the right relationships based on indisputable calculation procedures and indisputable fundamental physical knowledge that don't require fitting or iteration. Therefore, in addition to formulas for the physical indentation hardness, indentation work, and applied work for these indentations, the simple physical deduction of the flat punch indentation is also supplied. A macroindentation serves as an illustration.