Nowadays, nanometer material and technology have been widely used in industrial and engineering fields. Many new nano-materials have been developed by utilizing the fact that materials begin to exhibit unique mechanical properties at nano-scale, which significantly differ from those at larger scale.

Nano-indention tests have been widely used to measure such mechanical properties of materials. For micro-nano solids with large surface-to-bulk ratio the significance of surfaces is likely to be important. Form the viewpoint of continuum mechanics, this difference can be described by such concepts as surface effects [1] . This is especially true for nano-scale materials or structures. In such cases, the surface tension and the surface elasticity play a critical role and thus have been adding its appeal to many researchers. For example, Miller and Shenoy [2] probed the size-dependent elastic properties of nano-plates and beams. Hang et al. [3] explained the size dependent phenomenon by the strain gradient continuum theory. Dingreville et al. [4] investigated the surface free energy and its effect on elastic behavior of the nano-sized particles, wires, and films. Yang [5] studied the size-dependent effective modulus of spherical nano- cavities at dilute concentrations. Gao et al. [6] built a simple model to describe the influence of surface stress on the nanoscale adhesive contact. There are a lot of work regarding the surface/interface energy effects on the nanostructures and solids, and we can only include a small part of them here. For more recent developments in this field, the readers can refer to a review article by Wang et al. [7] .

To study the mechanical behavior of nano-materials, the most celebrated continuum-based surface/interface model was first established by Gurtin, Murdoch and coworkers [8] [9] [10] , which known as the theory of surface elasticity. In the study of nano-scale problems, all material constants appearing in that constitutive model were commonly calibrated with data obtained from either experimental measurements [11] or atomistic simulations [2] [12] . Therefore, the surface effect has been widely adopted to investigate the mechanical phenomena at nano-scale. Cammarata et al. [13] considered the size-dependent deformation in thin film with surface effects. Gao et al. [14] developed a finite-element method to account for the effect of surface elasticity. Wang et al. [15] studied to a half-plane subjected to normal pressures with surface tension. Zhao and Rajapakse [16] proposed a continuum-based model to study the influence of surface stresses on mechanical responses of an elastic half-space compressed by an axisymmetric, rigid, frictionless nano-indentor. Long and Wang [17] studied the effect of the residual surface stress on the two dimensional Hertz contact problem, and later Long et al. [18] generalized their work to the three dimensional case. Wang [19] derived the general analytical solution of nano-contact problem with surface effects by using the complex variable function method. Gao et al. [20] [21] derived the influence of the surface stress on the JKR adhesive contact, which is investigated by employing the non-classical Boussinesq fundamental solutions. In this paper, Fourier integral transform method is used to solve the non-classical boundary value problems with surface effects.

Now we consider a material occupying the upper half-plane, we refer to a Cartesian coordinate system (o-xyz), as shown in Figure 1, where the x axis is along the surface and the z axis perpendicular to the surface. It is assumed that the material is subjected to triangle distribution force and over the region. While the normal and shear force form zero (and) uniformly increased to maximum and at the point O. The plane-strain conditions are assumed to, and

the contact is assumed to be frictionless.

The problem statement is to determine the triangle distribution force exerted by the elastic field (e.g., displacement and stresses) with the half-plane for the influence of surface effects.

In surface elasticity theory, the equilibrium and constitutive equations in the bulk of material are the same as those in classical elastic theory, but the presence of surface stresses gives rise to a non-classical boundary condition.

As a particular example, let us consider the effect of a normal triangle distribution force over the region, and the normal force form zero (and) uniformly increased to maximum (O), while remainder of the boundary being unstressed as shown in Figure 1.

(6)

Due to the surface tension mostly influences the normal stress [20] . Therefore, we keep only the first term in Equation (4). Then, the surface stresses are given by

On the surface, the boundary conditions (3) can be written by

Substituting Equation (8) into Equation (5), one obtains

Due to deformation the radius of curvature of the surface is given by

By substituting Equations (9) and (10) into the surface condition Equation (8), is determined by

where

where s is a length parameter depending on the surface property and material elastic constants. It should be pointed out that this parameter indicates the thickness size of the zone where the surface effect is significant, and plays a critical role in the surface elasticity. For metals, s is estimated on the order of nanometers.

Therefore is given by

Substituting Equation (14) into Equation (5), the stresses component and displaces component are obtained as

It is seen, when, that is, the surface influence is ignored in Equation (15), the stresses of the half-plane are consistent with those in the classical elastic contact results which is the same result [23] , respectively.

where

.

On the contact surface (), the normal stress is given by

According to the Saint-Venant’s Principle, we assume that the normal displacement is w specified to be zero at a distance on the contact surface, that is, , the displacement on the surface is derived as

Assuming that the origin has no displacement in the x direction, that is, one obtains

As show in Figure 2, the results indicated that the normal stress is a smooth distribution when the surface tension is considered by at the loading boundary (), However, it is just the reverse with classical results that is the surface tension is ignored (s/a = 0). In addition, the actual normal stress is smaller than the classical value in the loading zone and is larger outside of the zone.

Due to the different surface tension value, the horizontal displacement is displayed in Figure 3, where we set, and. It is seen that the horizontal displacement is continuous everywhere on the deformed surface. However, the classical elasticity theory predicted unreasonably that the horizontal displacement is discontinuously at the load boundary, as seen from the curve of s/a = 0. The indent depth is plotted in Figure 4 with, which also shows that the slope of the deformed surface is continuous everywhere. It is also found the indent depth decreases with the increase of surface tension.

Now, let us consider the effect of a tangential triangle distribution force over the region, while the shears force form zero (and) uniformly increased to maximum at the point O, while remainder of the boundary being unstressed as shown in Figure 1.

At this moment, the boundary conditions (3) on the contact surface () are simplified to

where is the tangential triangle distribution force applied on the materials surface, and is a surface constant.

(21)

Substituting Equation (21) into Equation (20), one can be obtained

where

where b is a length parameter depending on the material surface property. It should be pointed out that this parameter plays a critical role in the surface elasticity.

Substituting Equation (24) into the surface condition Equations (12) and (13), the solution of stresses and displaces under pure shear load were obtained

Substituting Equation (24) into Equations (22), one obtains

Therefore is given by

Substituting Equation (27) into Equations (25), the stresses component and displaces component are obtained as

It is seen, when, that is, the surface influence is ignored in Equation (28), the stresses and displacements of the half-plane are consistent with those in the classical elastic results [24] , respectively.

On the contact surface, the stresses is given by

Based on the previous assumption, , the displacement on the surface component is derived as

It is instructive to examine the influence of the surface elasticity on the stresses and displacements of the contact surface and compare them with those in classical contact problem. Figure 5 and Figure 6 show the distribution of the stresses and on the contact surface, where the solution of b/a = 0 is consistent with the classical elastic result.

It can be seen from Figure 7 that the normal stress transits continuously across the loading boundary, which is increasing monotonically with respect to x in the loading region (), and the inverse is observed outside the loading region (). It means that the analytical expressions can give approximate results in the region that is far from the loading boundary. It is also found Figure 5 that shear stress changes smoothly across the loading boundary, which is different from a

singularity predicted by classical elasticity.

Due to the different surface elasticity value, the horizontal displacement is displayed in Figure 6, where we set, and. It is seen that the slope of

the deformed surface for a > 0 is continuous everywhere. It is also found the horizontal displacement decreases with the increase of surface elasticity. The indent depth is plotted in Figure 8 with, which also shows that the normal displacement is continuous everywhere on the deformed surface. In addition, the indent depth decreases continuously with the increase of surface elasticity.

In this paper, we consider the two-dimensional contact problem in the light of surface elasticity theory. Fourier integral transform method is adopted solving general analytical solution. For two particular loading cases of triangle distribution forces, the results are analyzed in detail and compared with the classical linear elastic solutions. A series of theoretical and numerical results show that the influences of the surface tension and the surface elasticity on the stresses and displacements are not always equal. It is found that the surface elasticity theory illuminates some interesting characteristics of contact problems at nano-scale, which are distinctly different from the classical solutions of elasticity without surface effects. Therefore, the influence of surface effects should be considered for nano-contact problems.

Wang, L.Y., Han, W., Wang, S.L., Wang, L.H. and Xin, Y.P. (2016) Nano-Contact Problem with Surface Effects on Triangle Distribution Loading. Journal of Applied Mathematics and Physics, 4, 2047-2060. http://dx.doi.org/10.4236/jamp.2016.411204