Study on Non-Equivalence of Pyramids and Their Pseudo-Cones

Abstract

It is challenged that simulating indentations using ostensibly "equivalent" pseudo-cones will take less computer time. Basic trigonometry and arithmetic rules out the mimicking of pseudo-cones with equal basal surface and depth with pyramidal indenters. The widely recognised angles of supposedly "identical" pseudo-cones cannot also assert that their depth is equal. The historical values of the often-employed half-opening angles of pseudo-cones are biassed, as evidenced by the answers provided for the problems to be answered. On that basis, it invalidates all simulations or findings. Not just for artificial intelligence, the large inaccuracies in the resulting hardness HISO and elastic modulus Er-_ISO values are devastating. For equal basal surface and equal volume, the straightforward deduction for potentially \(\psi\)-cones (\(\psi\) for pseudo) without biassed depths' errors is presented. These \(\psi\)-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). Additionally, the more sideways and their resulting downhill directions' opposite forces' direction angles are indicated. They are reflected by the parallelograms' diameters that are long and and their angles to the vertical axis. It is essential to have experimental loading curves before and after the phase-transition onsets. Quantitatively, imitation of pyramids and \(\psi\)-cones is likewise disallowed. For industrial and solid pharmaceutical materials, every simulation based on their assumptions would also be dangerously invalid.