An analysis of fully plastic Brinell indentation

Abstract

Indentation of a hard sphere into inelastic solids, Brinell indentation, is examined theoretically and numerically by aid of classical plastic flow theory. With the main interest focused on fully plastic behaviour at indentation the mechanical analysis is carried out for power-law hardening rigid-plastic materials where self-similarity features play a dominant role. It is explained in detail how the problem of a moving contact boundary may be reduced to a stationary one by an appropriate transformation of field variables. Within this setting classical empirical findings by Meyer (1908) and O'Neill (1944) are established on a rigorous theoretical ground. In particular, it is shown to advantage also for nonlinear materials how intermediate solutions for a flat die may by cumulative superposition generate solutions for a class of curved indenters. In the case of perfect plasticity it turns out in the present context that indentation hardness is independent of die profiles. For hardening solids when the material behaviour is history dependent, reduction to a stationary geometry is achieved also by expressing the accumulated strain by cumulative superposition. The intermediate flat die problem is then solved for a variety of hardening exponents by a finite element procedure designed to account for material incompressibility. With finite element computations as a basis desired solutions are obtained by straightforward numerical superposition procedures. Detailed results are then given for bulk quantities such as the mean contact pressure as well as relevant field variables. The influence of hardening characteristics on sinking-in and piling-up of indented surfaces and contact pressure distributions are discussed in the light of earlier findings based on deformation theory of plasticity and available discriminating experiments. Correlation is particularly sought with the celebrated universal hardness parameters proposed by Tabor (1951) and the existence of representative strain measures. Attention is also given to the elastic-plastic transition region of Brinell indentation in search for loading levels sufficiently high that the results tend to an asymptotic fully plastic state. A standard finite element technique employing contact elements for a moving boundary is used to analyse with tolerable accuracy the influence of elasticity and more elaborate hardening behaviour. Some relevant features are shown for a sequence of solutions from elastic Hertzian to fully plastic behaviour.