Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann–Ogden form

Abstract

In this paper, the surface elasticity in the form proposed by Steigmann and Ogden is applied to study a plane problem of frictionless contact of a rigid stamp with an elastic upper semi-plane. The results of this work generalize the results for contact problems with Gurtin–Murdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integro-differential equations, which is further regularized using the Fourier transform. The size dependency of the solutions of the problem is highlighted. It is observed that the curvature dependence of the surface energy is increasingly important at small scales. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and numerical results are obtained for different values of the mechanical parameters.

Flamant solution of a half-plane with surface flexural resistibility and its applications to nanocontact mechanics

“…For brevity, their derivation procedure is not detailed here. Interested readers may refer to [15,33,35,40] for more details.…”

“…where p m represents the average value of the contact pressure and a is the half-width of the contact. Under the application of the surface traction (33), the normal stress component can be determined by integrating Equation (31b): Figure 9 shows the distribution of the normal stress component along the half-plane boundary and two horizontal interfaces inside the half-plane. The surface traction ( 33) is identically revealed by the classical normal stress along the half-plane boundary.…”

“…Until very recently, the curvature-dependent nature of a half-plane boundary under contact loads had not been considered. In [33] the boundary conditions of Steigmann-Ogden type on a half-plane boundary were first derived . More details of the Steigmann-Ogden boundary conditions were later provided in [15].…”

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

“…For this purpose, Eremeyev and Lebedev [39] and Zemlyanova and Mogilevskaya [40] rederived the boundary conditions of the Steigmann-Ogden model, using the variational approach. Since then, this model has been successfully employed in a few research areas, including nanocontact mechanics [41][42][43][44][45][46], fracture mechanics [47], dislocation mechanics [48,49], and fibrously reinforced nanocomposites [40,[50][51][52]. Until very recently, the elastic states and effective material properties of particulately reinforced nanocomposites have not been addressed.…”

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

“…As a result, the Gurtin–Murdoch theory enables the surface effects to be expressed in terms of surface constitutive parameters and surface residual stress. At present, although the generalization of the Gurtin–Murdoch theory by incorporating the curvature dependence of the surface energy starts to attract some investigations, 7 the classical Gurtin–Murdoch theory can yield satisfactorily accurate results in most analyses about nanomaterials and nanostructures. 8 Besides of the phenological Gurtin–Murdoch theory, a few different approaches, such as surface roughness, 9 molecular simulation 10 and surface energy density method, 11 can be found in literatures to model the surface effects.…”