Hooke’s law is generalized to the case of arbitrary elastic or plastic indentation \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$\varepsilon = (2/\sqrt \pi )x(w_1 /\sqrt {A)} $$
\end{document}, where ɛ=q/Er is the elastic strain, q is the average pressure over the contact area, Er is the reduced elastic (Young’s) modulus, A is the projected area of the contact, w1 is the deformation in elastic indentation by a flat punch. On this basis a relation is obtained between the reduced hardness H and unreduced hardness Hh, which depends on the ratio ws/w1=ms; ws is the elastic deformation along the perimeter of the indent, and ms≅0.78. It is shown that the correction ΔEr to the elastic modulus Er determined from the condition of linearity of the initial part of the unloading diagram, is ΔEr=0.27(ΔP/Pm), where ΔP is the value used in the calculation of Er for the length of the linear part of the diagram, reckoned from the maximum load Pm. It is shown that for metallic construction materials of medium hardness one has q=HM, where HM is the Meyer hardness. With increasing HM and increasing angle ϕ at the tip of the indenter, the ratio HM/q grows by an exponential law.
