The Non-Equivalence of Pyramids and Their Pseudo-Cones: Important New Insights  

作　　者：Gerd Kaupp Gerd Kaupp(University of Oldenburg, Oldenburg, Germany)

机构地区：[1]University of Oldenburg, Oldenburg, Germany

出　　处：《Journal of Applied Mathematics and Physics》2022年第4期1158-1166,共9页应用数学与应用物理（英文）

摘　　要：The simulation of indentations with so called “equivalent” pseudo-cones for decreasing computer time is challenged. The mimicry of pseudo-cones having equal basal surface and depth with pyramidal indenters is excluded by basic arithmetic and trigonometric calculations. The commonly accepted angles of so called “equivalent” pseudo-cones must not also claim equal depth. Such bias (answers put into the questions to be solved) in the historical values of the generally used half-opening angles of pseudo-cones is revealed. It falsifies all simulations or conclusions on that basis. The enormous errors in the resulting hardness H_ISO and elastic modulus E_r-ISO values are disastrous not only for the artificial intelligence. The straightforward deduction for possibly ψ-cones (ψ for pseudo) without biased depths’ errors for equal basal surface and equal volume is reported. These ψ-cones would of course penetrate much more deeply than the three-sided Berkovich and cube corner pyramids (r a/2), and their half-opening angles would be smaller than those of the respective pyramids (reverse with r > a/2 for four-sided Vickers). Also the unlike forces’ direction angles are reported for the more sideward and the resulting downward directions. They are reflected by the diameter of the parallelograms with length and off-angle from the vertical axis. Experimental loading curves before and after the phase-transition onsets are indispensable. Mimicry of ψ-cones and pyramids is also quantitatively excluded. All simulations on their bases would also be dangerously invalid for industrial and solid pharmaceutical materials.The simulation of indentations with so called “equivalent” pseudo-cones for decreasing computer time is challenged. The mimicry of pseudo-cones having equal basal surface and depth with pyramidal indenters is excluded by basic arithmetic and trigonometric calculations. The commonly accepted angles of so called “equivalent” pseudo-cones must not also claim equal depth. Such bias (answers put into the questions to be solved) in the historical values of the generally used half-opening angles of pseudo-cones is revealed. It falsifies all simulations or conclusions on that basis. The enormous errors in the resulting hardness H_ISO and elastic modulus E_r-ISO values are disastrous not only for the artificial intelligence. The straightforward deduction for possibly ψ-cones (ψ for pseudo) without biased depths’ errors for equal basal surface and equal volume is reported. These ψ-cones would of course penetrate much more deeply than the three-sided Berkovich and cube corner pyramids (r a/2), and their half-opening angles would be smaller than those of the respective pyramids (reverse with r > a/2 for four-sided Vickers). Also the unlike forces’ direction angles are reported for the more sideward and the resulting downward directions. They are reflected by the diameter of the parallelograms with length and off-angle from the vertical axis. Experimental loading curves before and after the phase-transition onsets are indispensable. Mimicry of ψ-cones and pyramids is also quantitatively excluded. All simulations on their bases would also be dangerously invalid for industrial and solid pharmaceutical materials.

关 键 词：Basic Mathematics Extreme Errors False Cone Angles INDENTATION Unphysical Cone Models for Pyramids Undue Simulations 

分 类 号：O17[理学—数学]