Upper and lower bounds to indentation of rigid ball into semi-infinite solid

A closed-form solution for upper and lower bounds (‘dual bounds’) to a slow indentation of rigid ball (under increasing normal load) into rigid plastic solids are presented. The solutions enlarge the theoretical background to the familiar Brinell hardness tests used in assessing strength of materials and add few new features. w features are mainly: (i) constructing a lower-bound solution to ball indentation, (ii) a closed-form estimation of the dual bounds with any prescribed frictional shear along the indenter/solid contact, (iii) providing dual bounds to two different situations of the material response: (a) the removed material is falling off from the bulk (akin to indentation into brittle-like materials) and (b) the removed material is piled up along the indenter face (akin to indentation into ductile-like materials). al bound solutions of the indenting mean ball pressure for both material cases seem to capture properly well the material constrain to local plastic indentation and the effect of the interfacial friction. Having both bounds simultaneously, despite their unavoidable deviation, provide much more informative results than a single bound. isons between the suggested bounding analysis and experiments (taken from Tabor [Tabor D. Hardness of Metals. Oxford: Clarendon Press; 1951], Johnson [Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985 (new edition, 2001)] and others), numerical FEM solutions (Mesarovic and Fleck [Mesarovic SDj, Fleck N. Spherical indentation of elastic-plastic solids. Proceedings of Royal Society of London 1999; A455: 2707–28] and others) and some self-made measurements are all favorable.