Abstract
Abstract Diametric compression testing can be used to measure the hoop strength of spherical particles if one uses platens whose deformation behaviour is tailored in a way that allows the particles to sink sufficiently far into the platens before particle fracture takes place. To obtain the hoop stress of a compressed spherical particle at the moment of failure, and thus interpret the strength tests, an analytical solution derived by Hiramatsu and Oka is generally used. In deriving their analytical equations, Hiramatsu and Oka assumed that the contact stress between the spherical particle and the platens is radial and uniform along two equal diametrically opposed spherical caps. Here we revisit, by means of the finite element method, the mechanics that underlie the diametrical compression of a linear elastic sphere between two parallel and planar elasto-plastic platens. We show that the Hiramatsu and Oka equations give valid hoop stress values only when the contact area between the compressed sphere and the platens is less than roughly 5 % of the equatorial sphere cross-sectional area. At higher contact areas, when the spheres are more deeply embedded in the platen material and the test can be used to measure the particle hoop strength, the Hiramatsu and Oka solution underestimates the hoop stress. By conducting a parametric study, we provide simple expressions that can be used to quickly, yet accurately to within the ≈ 5 % , compute the hoop stress of compressed spheres in diametric compression tests as a function of the contact area knowing only two parameters, namely the Poisson's ratio of the material making the sphere and the sphere/platen contact friction coefficient.
Highlights
The diametric compression of a sphere between two parallel platens is a mechanical problem that is frequently encountered, notably in the context of particle compression testing
By performing a large number of finite element simulations under a range of conditions, we propose a set of simple expressions that can replace the Hiramatsu and Oka equations and be used to quickly, yet accurately, estimate the hoop strength of compressed spherical particles
Finite element modeling of an isotropic elastic sphere of radius R subjected to a diametric compression by a pair of elasto–plastic platens was implemented in Abaqus FEA v6.11 (Dassault Systèmes Simulia Corp., Providence, RI, USA) software as a Python script, which is made freely available on the website of our laboratory
Summary
The diametric compression of a sphere between two parallel platens is a mechanical problem that is frequently encountered, notably in the context of particle compression testing. The problem is far from new: the mechanics of the embedding of various objects, e.g. spheres, discs or cylinders, as they are pushed against a flat deformable surface, has been studied both analytically and numerically in many settings, for example due to its connection with the Brinell hardness tests or instrumented indentation tests used to assess various material mechanical properties (Hill et al, 1989; Bower et al, 1993; Biwa and Storkers, 1995; Huber and Tsakmakis, 1998; Mesarovic and Fleck, 1999; Kucharski and Mrz, 2001; Sakai et al, 2003; Kogut and Komvopoulos, 2004; Niederkofler et al, 2009; Ghaednia et al, 2016; Clausner and Richter, 2016) In these studies, it is shown that the indentation laws, i.e., the dependence of the indentation load on the indentation depth (or equivalently on the contact radius), depend strongly on the elasto-plastic parameters of the indented material. We examine here in detail, with a goal to complement analytical studies such as that of Hiramatsu and Oka, the evolution of stress within an elastic sphere that is diametrically compressed between two elasto-plastic platens up to large contact radii, with focus on (tensile) stresses within the sphere that might cause it to fracture
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