Abstract:

Hertzian stress field resulted from a rigid sphere indenting a semi-infinite domain is of great interest to material testing as well as mechanical excavation in rocks. Though both the stress and displacement fields can be expressed in elementary functions, only a handful references in the literature include the full solutions. Surprisingly, among those that report the full solutions, they are actually not the same. Driven by sheer curiosity, we first reviewed the development of the analytical solution for sphere indentation. We verified, to the best of our knowledge, that the internal stress field derived by Love (1929) is the correct version. Next, we developed a numerical scheme that utilizes the Hertzian solution as a far-field asymptotic boundary condition to account for the existence of the semi-infinite domain. Effects of the far-field confining stress on the elasto-plastic deformation and the potential development of the crack systems in both the loading and unloading phases during indentation are investigated.

Introduction

The stress field resulted from a rigid sphere indenting a semi-infinite domain is of great interest to material testing and mechanical excavation in rocks, for example, drilling for oil and gas exploration and tunneling for infrastructure development. The linear elastic solution for the contact pressure distribution on the surface was initially given by Hertz over a hundred years ago (Hertz, 1881; Hertz, 1882). Though both the stress and displacement fields can be expressed in elementary functions, only a handful references in the literature include the full solutions. Generally, only the contact pressure, displacements and stresses on the surface and/or along the contact axis are given (Timoshenko and Goodier, 1969; Frank and Lawn, 1967; Johnson, 1985). Surprisingly, among those that report the full solutions, they are actually not the same (Huber, 1904; Love, 1929; Zeng et al., 1992; Lawn, 1998; Fischer-Cripps, 2007).

Driven by sheer curiosity, we first reviewed the development of the analytical solution for sphere indentation based on the displacement functions. We then derived the full stress and displacement fields by following the method in Love (1929) and corroborated the derivation by applying the method of superposition directly using the Boussinesq displacement and stress fields as the integral kernels. It should be noted that evaluation of the integrals in this case is not a trial exercise. We verified, to the best of our knowledge, that the internal stress field derived by Love (1929) and reported in Fischer-Cripps (2007) is the correct version.

Next, we developed a numerical scheme that utilizes the Hertzian solution as a far-field asymptotic boundary condition and conducted numerical analysis to investigate the effect of the far-field confining stress on the elasto-plastic deformation and potential development of the crack systems in both the loading and unloading phases during indentation. In normal indentation, the plastic deformation that occurs beneath the tool is contained by the surrounding elastic domain. The nonlocal nature of elastic deformation means that the development of plastic deformation and the subsequent crack initiation and propagation in a numerical model could be sensitive to the boundary condition. Using the Hertzian solution as the asymptotic far-field boundary condition could in fact effectively account for the fact that the domain is semi-infinite.

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