This paper analyzes the force vs depth loading curves of conical, pyramidal,\nwedged and for spherical indentations on a strict mathematical basis by explicit\nuse of the indenter geometries rather than on still world-wide used iterated\nâ??contact depthsâ? with elastic theory and violation of the energy law. The\nnow correctly analyzed loading curves provide as yet undetectable phasetransition.\nFor the spherical indentations, this includes an obvious correction\nfor the varying depth/radius ratio, which had previously been disregarded.\nOnly algebraic formulas are now used for the calculation of materialâ??s properties\nwithout data-fittings, or simplifications, or false simulations. Penetration\nresistance differences of materialsâ?? polymorphs provide precise intersection\nvalues as kink unsteadiness by equalization of linear regression lines from\nmathematically linearized loading curves. These intersections indicate phase\ntransition onset values for depth and force. The precise and correct determination\nof phase-transition onsets allows for energy and phase-transition energy\ncalculations. The unprecedented algebraic equations are most simply and\nmathematically reproducibly deduced. There are no restrictions for elastic\nand/or plastic behavior and no use of different formulas for different force\nranges. The novel indentation formulas reveal unprecedented access to the\nonset, energy and transition energy of phase-transitions. This is now also\nachieved for spherical indentations. Their formula as deduced for plotting is\nreformulated for integrations. The distinction of applied work (Wapplied) and\nindentation work (Windent ) allows now for comparing spherical with pyramidal\nindentation phase-transitions. Only low energy phase-transitions from\npyramidal indentation may be missed in spherical indentations. The rather\nlow penetration depths of sphere calottes calculate very close for cap and flat\narea values. This allows for the calculation of the indentation phase-transition\nonset pressure and thus the successful comparison with hydrostatic anvil pressurizing\nresults. This is very helpful for their interpretations, as low energy\nphase-transitions are often missed under the anvil, and it further strengthens\nthe unparalleled ease of the indentation techniques. Exemplification is reported\nfor pyramidal, spherical, and hydrostatic anvil stressing by the numerical\nanalysis of published germanium data. The previous widely accepted\nhistorical indentation theories and standards are challenged. Falsely simulated\nand even published so-called â??experimentalâ? indentation data from the\nliterature can most easily be checked. They are mathematically unsound and\ntheir correction is urgently necessary for scientific reasons and daily safety\nwith stressed materials. The motivation for this paper is the challenge of\nworldwide incorrect ISO 14577 standards for false and incomplete characterization\nof materials. The minimization of catastrophic failures e.g. in aviation\nrequires the strengthening and the advancements of the mathematical\ntruth by using our closed formulas that are based on undeniable geometric\nand algebraic calculation rules.
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