Influence of Surface Energy Effects on Elastic Fields of a Layered Elastic Medium under Surface Loading

Abstract

This paper presents the analysis of a layered elastic half space under the action of axisymmetric surface loading and the influence of the surface energy effects. The boundary value problems for the bulk and the surface are formulated based on classical linear elasticity and a complete Gurtin-Murdoch constitutive relation. An analytical technique using Love’s representation and the Hankel integral transform is employed to derive an integral-form solution for both displacement and stress fields. An efficient numerical quadrature is then applied to accurately evaluate all involved integrals. Selected numerical results are presented to portray the influence of various parameters on elastic fields. Numerical results indicate that the surface stress displays a significant influence on both displacement and stress fields. It is also found that the layered half space becomes stiffer with the presence of surface stresses. In addition, unlike the classical elasticity solution, size-dependent behavior of elastic fields is noted. The present analytical solutions provide fundamental understanding of the influence of surface energy on layered elastic materials. It can also be used as a benchmark solution for the development of numerical techniques such as FEM and BEM, for analysis of more complex problems involving a layered medium under the influence of surface energy effects.

Axisymmetric indentation problem of a transversely isotropic elastic medium with surface stresses

“…For example, Vasu and Bhandakkar investigated the cylindrical indentation of a layer-substrate system, 21 and Rungamornrat research group analyzed an elastic layer under the action of different axisymmetric surface loads. 22–24 All researches reveal that the bulk material becomes stiffer with surface effects, and the surface residue stresses dominate in the normal deformation while the surface elasticity in the shear deformation. Since this paper concentrates on the frictionless normal contact problem, hence only the effect of surface residue stresses will be taken into account.…”

Surface effect in axisymmetric Hertzian contact problems

“…Recently, surface effect in contact mechanics attracts many interests, including the Boussinesq problem Zhou and Gao, 2013 ), the Cerruti problem Zhou and Gao, 2014 ) and the plane strain or axisymmetric model of indenters with different shapes contacting an elastic substrate or a layer-substrate system ( Long et al, 2012;Long and Wang, 2013;Pinyochotiwong et al, 2013;Zhou and Gao, 2013;Rungamornrat et al, 2016;Tirapat et al, 2017 ). Compared with the classical contact model ( Johnson, 1987 ), additional surface-induced tractions as a function of surface stress are introduced at the contact surface, which depends not only on the surface residual stress resulted from surface relaxation but also on the surface deformation yielded by the external load, i.e., σ s = τ 0 I + C s : ε s .…”

Variational Formulation of Interaction between Elastic Plate and Elastic Medium under the Influence of Surface Energy

“…The Gurtin-Murdoch continuum-based model has been employed to study a variety of problems with the consideration of surface energy effects in the past (e.g. see [4][5][6][7][8][9][10][11][12][13][14]).…”