Effect of Inhomogeneity of FGM Coating on Contact with Adhesion under the Axisymmetric Indenter

The contact with adhesion between the FGM coating and axisymmetric indenter is probed into in the present paper. Coating is made of FGM with linear varying modulus. Poisson’s ratios of coating and substrate are chosen to 1/3. The Maugis-type adhesive model is used to describe the adhesive interaction in the adhesion zone. According to the basic equations for axisymmetric problem, the controlling singular integral equations of the contact problem with adhesion are gained by making use of Hankel integral transformation. These integral equations are solved by applying a numerical collocation technique. The results show that the contact damage and deformation on the coating surface can be reduced by adjusting the inhomogeneity of the coating.


Presentation
Functionally graded material (FGM) is often used as coating or interface layers to resist contact damage or delete the stress concentrating [1,2]. With the development of information technology, FGM is considered to be applied in the micro-nano field. The influences of adhesion are more obvious with the contact size of materials decreases [3]. During these years, many scholars have done lots of research on contact problem for functionally graded materials. Ke and Wang [4,5] solved the contact problem of FGM coating with any function by using the multi-layered model which requires the elastic modulus of FGM in each layer vary according to the linear function. Liu and wang [6,7] developed the model to explore the contact problems of functionally graded coating under axisymmetric indenter. Recently, Balci and Dag [8] consider the dynamic frictional contact mechanics of a FGM coating loaded by a rigid moving cylindrical punch based on the analytical method. By considering the adhesion effect which presented in JKR adhesion model, Jin et al [9] solved the axisymmetric contact problem for power-law gradient materials by considering the adhesion effect. Li and Liu [10,11] applied the Maugis model to investigate the impact of adhesion on the contact problem between a functionally graded coating in which material parameter is characterized by exponentially varying modulus and a cylindrical indenter. Meanwhile, Liu et al [12] solved full adhesive contact problem for FGM coating by assuming the assuming the interfacial friction sufficient to prevent any slip taking place.
In this paper, the contact with adhesion for FGM coating with linear varying modulus under a cylindrical indenter is considered. The singular integral equations which describe the adhesive contact problem are gained by using the Hankel integral transformation. The Maugis' adhesion model is adopted to depict the adhesive interaction in the adhesion zone. An iteration method is applied to solve the singular integral equations. The effect of inhomogeneity of FGM coating on the contact behavior is analyzed.

Formulation
The contact geometry is shown in Figure 1a, in which a rigid spherical indenter is loaded on the surface of functionally graded coating. The homogenous materials substrate coated by FGM film with thickness h . The ratio of Poisson is chosen to 0.33. The shear modulus of the substrate is an invariant constant. The shear modulus of the graded coating is assumed to vary as linear function: Where 0 μ refer to the elastic parameter of the coating surface and μ * represent the elastic parameter of the substrate. According to reference [6,12], the non-dimension normal displacement can be expressed as: indicate respectively the first and second kinds of the complete elliptic integral [13], and With p J is the first kind of the th p order Bessel functions [14].  In order to obtain the solution of the singular integral equation (2), the geometric relationship for the gap of the contacting bodies at r c = and a is needed [15]. Using the normalizations, the additional equation is given by Where ( ) The non-dimension normal load is expressed as ( ) The non-dimension penetration depth can be determined 3

Numerical Analysis and Results
Base on the above analysis, the approximate expression of integral equations (2), (7), (10) and (11) can be solved by using the Gauss-Chebyshev formula [14]. The effect of the shear modulus ratio on load vs. contact radius, load vs. maximum indentation depth, surface stress distribution of coating and the ratio of adhesion zone to contact zone are analyzed in this section.  It can be found from figure 2a that the pull-off force determined by the vertical tangents increase with the decrease of the shear modulus ratio. The similar results can be gained from figure 2b in which the horizon tangents determine the pull-off force. In figure 2b, the indentation depth increases with the shear modulus ratio increases for the fixed pressure. Figure 2c demonstrates that the surface stress σ gradually transitions from compressive stress to tensile stress along the radial direction because of the effect of adhesion. It can also be seen from figure 2c that the shear modulus ratio has a great influence on the coating surface stress distribution. This behavior indicates that the contact behavior can be improved by adjusting the stiffness of the coating. In figure 2d, the ratio of adhesion zone to contact zone m increase with the decrease of the shear modulus ratio.

Conclusions
The impacts of shear modulus ratio on the contact behavior of between a graded coating and a spherical indenter are analyzed. The numerical results demonstrated that the gradient index (denoted by 0 μ μ * ) of graded coating can alter the pull-off force and distribution of surface stress. The results in this paper provide a method to suppress contact deformation and damage by using the FGM coating.