Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

Abstract

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Axisymmetry in Mechanical Engineering

“…In their analysis of the limiting no-wear profile for multi-mode fretting, Dmitriev et al [41] conducted experiments on torsional fretting of a rubber half-sphere on a flat covered by abrasive paper. The radius of the sphere is (approximately) 19 mm (the paper wrongly says 14.5 mm, but from the respective figure, the correct value can be estimated), the indentation depth is 2 mm, and the radius of the permanent stick area is 3.25 mm [41]. The theoretical prediction will only depend on the number of normalized fretting cycles, N/N 0 .…”

“…It is necessary to note here that the general solutions outlined above hold true in a more general case of a transversely isotropic elastic half-space, provided that the plane of isotropy is parallel to the half-space surface (see, for example, [9]). The potential for further generalization and development of the general solutions presented above relies on the fact that the problem of elastic contact is a core issue in similar contact problems with a circular contact region for functionally graded [29,30], viscoelastic [31,32], thermoelastic [33,34], poroelastic [35,36], magneto-electro-elastic [37,38], multiferroic [39,40] semi-infinite media as well for elastic semi-infinite media with surface effects [41,42].…”

“…It is of interest to note that, in contrast to formulas due to Borodachev (40) and Fabrikant (41), Formula (44) separates the contributions to the contact SIF from the punch displacement δ 0 and the punch shape function Φ(r, ϕ).…”

Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

“…For example, it was established that due to surface elasticity a nanoporous material could be even stiffer than its solid counterpart (Duan et al, 2008). Surface stresses affect a material response during nanoindetation that could be useful for determination of surface elastic moduli (Li & Mi, 2019;Argatov, 2022;Zemlyanova & White, 2022). Since surface energy results in non-classic boundary conditions often called the generalized Young-Laplace equation, they may essentially change stress singularity in the vicinity of defects such as crack tips (Kim et al, 2013;Gorbushin et al, 2020) or dislocations (Dai & Schiavone, 2019;Grekov & Sergeeva, 2020).…”

Three-dimensional analytical solutions to the half infinite thermo-elastic body pressed by a rigid cylindrical structure in thermal environment

“…Argatov [23] compared general solutions of contact pressure in literatures for non-axisymmetric circular contact problems. Based on Hankel integral transformation, Zemlyanova et al [24] discussed the indentation of a half-space with surface energy. Furthermore, using the finite element method (FEM), Abbas and Abd-alla [25], Abbas [26], and Abbas and Kumar [27] studied thermo-elastic interactions in different elastic half-spaces and full space.…”