On Brinell and Boussinesq indentation of creeping solids

Abstract: As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.