Surface effect in axisymmetric Hertzian contact problems

Abstract

Surface effect in three different axisymmetric Hertzian contact models is investigated in this paper with a recently developed elastic theory for nanostructured materials, including a Boussinesq problem, contact problem between a rigid flat-ended cylindrical indenter and an elastic half space as well as contact problem between a rigid spherical indenter and an elastic half space. With the help of the Love's strain function method and Hankel integral transformation, closed-form solutions of the stress and displacement fields at the surface of an elastic half space subjected to a concentrated force are achieved, based on which the interface tractions and displacements in the three different axisymmetric contact problems can be further obtained. It is found that surface effect in these contact problems can be characterized only by an intrinsic length, i.e., the ratio of the bulk surface energy density to the bulk shear modulus of the indented material. When the contact radius is comparable with the intrinsic length, surface effect is much obvious, leading to a serious deviation between the two solutions predicted respectively by the theoretical model developed for nanomaterials and the classical contact model. A more interesting phenomenon is about surface effect on the indentation hardness, which is found to increase with the reduction of the indenter radius when the external load is fixed, or to increase with the decrease of the external load when the indenter radius keeps unchanged. All the results in this paper should be helpful not only for deep understanding of the surface effect on nano-contact behaviors but also for further revealing the nature of surface effect of nano-indentation hardness.

Axisymmetric contact analysis of piezoelectric materials with surface effect

“…So, we extend this idea to the surface piezoelectric theory in which the surface stresses are related to both surface strain and electric field. On the basis of this theory, the equilibrium equations can be represented as (Jia et al, 2017(Jia et al, , 2018Li and Pan, 2016;Long et al, 2012)…”

“…According to the assumption of small deformation, the high order terms of the partial derivatives of u r0 in equations ( 34) to (37) can be reasonably neglected (Jia et al, 2018). Therefore, applying the Hankel integral transform, and substituting equations ( 26), ( 30), (31), and (33) into equations ( 34) to (37), the total boundary conditions at Z = 0 in terms of the stresses and electric displacement can be rewritten as…”

“…In this theory, the surface behavior was governed by the constitutive relation of surface, and all material constants were generally obtained from either experimental measurements (Huang et al, 2008) or atomistic simulations (Grigoras et al, 2002). Furthermore, compared to the classical continuum theory, the presence of the surface stress in the surface elasticity theory leads to the non-classical boundary conditions (Gao et al, 2013;Jia et al, 2017Jia et al, , 2018.…”

“…Subsequently, Long and Wang (2013) studied the contact problem of a spherical punch indenting on an elastic half-space by considering the residual surface tension. A Boussinesq contact model or the contact problem of a half-space were presented by Jia et al (2017Jia et al ( , 2018 considering the surface effect. Their analysis showed that the surface effect leaded to a significant deviation between the nonclassical and classical solutions as the contact radius was comparable with the material intrinsic length.…”

Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects

“…Pharr et al [28] showed that the slope of the unloading P − δ curve, normald P ∕ normald δ , can be used to represent the contact stiffness, where P and δ are the indentation load and depth, respectively. However, at present, the contact pressure considering surface effects can only be numerically obtained by using special integration schemes [16, 29]. To avoid the heavy mathematical efforts in numerical calculation, for an isotropic elastic half space, Gao et al [3] investigated the influence of surface effects on the contact stiffness in detail by using the classical contact pressures.…”