Valid Geometric Solutions for Indentations with Algebraic Calculations

This chapter analyzes the force vs depth loading curves of conical, pyramidal, wedged and for spherical indentations on a strict mathematical basis by explicit use of the indenter geometries rather than on still world-wide used iterated “contact depths” with elastic theory and violation of the energy law. The loading curves, which have now been accurately examined, offer undetected phase-transition. This contains a clear correction for the variable depth/radius ratio, which was previously ignored, for the spherical indentations. Fortunately, no data-fitting, simplification, or fake simulations must be utilised in the determination of a material's properties; only algebraic formulas. By equating the linear regression lines from the mathematically linearized loading curves, the penetration resistance differences of the materials' polymorphs offer accurate intersection values as kink unsteadiness. The phase transition onset values for depth and force are indicated by these intersections. Energy and phase-transition energy calculations are made possible by the accurate and precise estimation of phase-transition onsets. The derivation of the novel algebraic equations is most straightforward and reproducible mathematically. There are no limitations on the behaviour of elastic or plastic materials, and no need to utilise distinct formulas for various force ranges. This is now also achieved for spherical indentations. Their formula as deduced for plotting is reformulated for integrations. The unique indentation formulas provide previously unattainable access to the phase-transitions' onset, energy, and transition energy. (Windent) allows now for comparing spherical with pyramidal indentation phase-transitions. Only low energy phase-transitions from pyramidal indentation may be missed in spherical indentations. The spherical calottes' relatively shallow penetration depths are calculated extremely closely for both cap and flat area values. As a result, the indentation phase-transition onset pressure may be calculated and successfully compared to the results of hydrostatic anvil pressurising. Low energy phase-transitions are frequently overlooked beneath the anvil; therefore, this is highly beneficial for their interpretations and further supports the unparalleled simplicity of the indentation experiments. The numerical examination of previously published germanium data has compared pyramidal, spherical, and hydrostatic anvil stresses. The preexisting, commonly accepted theories and guidelines for historical indentation are contested. It is easiest to check falsely generated and even published so-called "experimental" indentation data from the literature. Due to their mathematical inadequacies and the pressing need for daily safety with stressed materials, they must be corrected. The issue of faulty ISO 14577 standards for inaccurate and inadequate material categorization is what spurred the author to write this essay. By employing our closed formulas, which are founded on indisputable geometric and algebraic calculation laws, we can reinforce and advance the mathematical truth while minimising catastrophic failures, such as in aviation.