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This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.

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 [

To study the mechanical behavior of nano-materials, the most celebrated continuum-based surface/interface model was first established by Gurtin, Murdoch and coworkers [

Now we consider a material occupying the upper half-plane

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.

In the absence of body force, the equilibrium equations, constitutive law, and geometry relations in the bulk are as follows

where G and

The strain tensor is related to the displacement vector

On the surface, the generalized Young-Laplace equation [

where

Based on previous work by Wang [

where A and B are generally functions of

As a particular example, let us consider the effect of a normal triangle distribution force

(6)

Due to the surface tension mostly influences the normal stress [

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),

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

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

It is seen, when

where

On the contact surface (

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

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

As show in

Due to the different surface tension value, the horizontal displacement is displayed in

Now, let us consider the effect of a tangential triangle distribution force

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

where

(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

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

It is seen, when

On the contact surface

Based on the previous assumption,

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.

It can be seen from

singularity predicted by classical elasticity.

Due to the different surface elasticity value, the horizontal displacement is displayed in

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

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