## Abstract

In this chapter, the indentation response of the functionally graded material (FGM) coating is considered due to the contact between the coating and axisymmetric indenter. The mechanical properties of FGM coating is assumed to vary along the thickness direction. Three kinds of models are applied to simulate the variation of elastic parameter in the FGM coating based on the cylindrical coordinate system. The axisymmetric frictionless and partial slip contact problems are reduced to a set of Cauchy singular integral equations that can be numerically calculated by using the Hankel integral transform technique and the transfer matrix method. The effect of gradient of coating on the distribution of contact stress is presented. The present investigation will provide the guidance for the indentation experiment of coating.

### Keywords

- functionally graded material
- coating
- indentation
- axisymmetric contact

## 1. Introduction

Functionally graded material (FGM) [1] which is new kind of nonhomogeneous composite material has many predominant properties, so it has been widely used in many fields. In recent years, many researchers have conducted the experiment to prove that FGM used as coatings can resist the contact deformation and reduce the interface damage [2], so it is very important to study the indentation response of FGM coating. Because FGM are composites whose material properties vary gradually along a coordinate axis, the governing equations which represent the mechanical behaviors of the materials are very difficult to solve. Researchers usually describe the properties of FGM according to some specific functional forms such as exponential functions and power law functions of elastic modulus [3, 4]. By assuming the elastic modulus of FGM varying as exponential function form, Guler and Erdogan [5, 6] studied the two-dimensional contact problem of functionally graded coatings. Liu et al. [7, 8] investigated the axisymmetric contact problem of FGM coating and interfacial layer with exponentially varying modulus by using the singular integral equation. The axisymmetric problems for a nonhomogeneous elastic layer in which the shear modulus follows the power law function are taken into account by Jeon et al. [9]. Because solving the controlling equations of FGM is difficult, the contact problem of FGM is limited to assume the elastic modulus varying as some specific functional forms. To eliminate this disadvantage, Ke and Wang [10, 11] applied the linear multilayered (LML) model to simulate the FGM with arbitrarily varying elastic parameter. Based on the model, some two-dimensional contact problems are studied. The axisymmetric contact problem of FGM coating with arbitrary spatial variation of material properties is considered by making use of the extended linear multilayered model [12, 13]. Recently, a piecewise exponential multilayered (PWEML) model [14] is presented to solve the frictionless contact problem of FGM with the shear modulus of the coating varying in the power law form. Subsequently, Liu and Li [15] applied the model to solve the two-dimensional adhesive contact problem.

When two bodies are brought together under the applied force, contact occurs at interface. Hertz [16] first considers the frictionless contact problem between elastic bodies. Researchers obtained the classical solution to the indentation problem under the flat, cylindrical, and cone punch based on Hertz’s theory [17]. The contact tractions and displacement field can be given to characterize the mechanical properties of various materials. Liu et al. [7, 12, 14] solved the axisymmetric frictionless contact problem for FGM coating by using the singular integral equation. They discussed the effect of the gradient of FGM coating on the indentation response. Because the materials of the two contact solids are dissimilar, the slip will take place at the contact surface. If slip is opposed by friction, the contact region is divided into two parts: the stick region and the slip region. Spence [18] gives the contact stress fields in homogeneous materials by assuming a self-similarity at each stage of finite friction contact when the normal load monotonically increases. Ke and Wang [19] solved the two-dimensional contact problem with finite friction for FGM coating. Liu et al. [13] considered the axisymmetric partial slip contact problem of a graded coating. When the coefficient of friction is sufficiently large, slip might be prevented entirely. The self-similar solution to nonslip contact problems with incremental loading was considered by Spence [20]. Goodman [21] investigated the axisymmetric contact problem with full stick when elastically dissimilar spheres are pressed together. Mossakovski [22] studied contact with adhesion for the elastic bodies under condition of adhesion. Norwell et al. [23] adopt an iteration method to solve the coupled equations which can describe the partial slip contact problem.

In this chapter, the axisymmetric frictionless and partial slip contact problems for FGM coating are considered. The basic formulation for nonhomogeneous material layer with elastic parameter varying along the thickness direction is given in Section 2. Based on the basic formulations for nonhomogeneous layer, three types of computational model for FGM coating are introduced in Section 3 for axisymmetric contact problem. The displacement and stress components in the transform domain are gained by using the Hankel transform technology and transfer matrix method. In Section 4, we will investigate the solution for the axisymmetric frictionless and partial slip contact problems. The indentation response of FGM coating under frictionless and frictional condition will be discussed in Section 5. Finally, we will depict some conclusions on the axisymmetric indentation response of FGM coating.

## 2. Basic formulations for nonhomogeneous material layer

For the present axisymmetric problem, the strain components, stress-strain relations, and the equilibrium equations in the radial and axial directions disregarding the body forces are given by the following relations [7]:

in which

## 3. Computational models for FGM coating

The properties of nonnonhomogeneous material may vary arbitrarily along a certain spatial direction, which makes the solution of contact problem very difficult in mathematics. In the present work, we adopt three methods to model the axisymmetric FGM layer based on the cylindrical coordinate system. First, exponential function (EF) model [7] is used to assume the elastic modulus of the FGM layer that varies as the exponential function. Second, the linear multi-layered (LML) model [12] is applied to simulate the FGM layer with arbitrarily varying material modulus, and Poisson’s ratio is chosen as 1/3. The model divided FGM layer into a series of sub-layers in which the shear modulus varies as linear function form. The shear modulus is taken to be continuous at the sub-interfaces and equal to their real values. Third, the piecewise exponential multilayered (PWEML) model [14] is employed in modeling the functionally graded material layer with arbitrary spatial variation of material properties. In this model, the functionally graded layer is cut into several sub-layers where the elastic parameter varies according to the exponential function form. Three types of computational model for FGM coating are the following.

### 3.1 Exponential function model

In Figure 1(a), the shear modulus of the functionally graded coating can be described by

where

Substituting Eqs. (2) and (5) into Eq. (3), we obtain

where

In order to solve Eq. (6), we use the technique of Hankel integral transform. The Hankel transform and its inversion are defined as

where the bar ∼ indicates Hankel transform; * p*th-order Hankel transform; and

*th-order Bessel function of the first kind.*p

By using the Hankel transform and defining

The solution of the differential Eqs. (8) is given by [7]

where

and

According to Hooke’s law and strain-displacement relations, stress components may be expressed as

For a homogeneous layer without the gradient, the gradient index

### 3.2 Linear multi-layered model

Consider the linear multi-layered model shown in Figure 1. The shear modulus of the functionally graded coating can be described by an arbitrary continuous function of

where

As in [12], introduces two potential functions to write the displacement components

By making use of Eqs. (1), (2), and (13), the equilibrium equations (3) are represented as [12].

where

Then the displacement and stress components given by Eqs. (13) and (2) are given by

Applying Hankel transformation Eqs. (7a)–(14), we obtain the solutions for displacement functions

where

Applying Hankel transform to Eqs. (15) and (16), we get

### 3.3 Piece wise exponential multi-layered model

Piece wise exponential multi-layered model divides functionally graded coatings into

in which:

and

In each sub-layer (

where

The solution of differential equations (20) may be expressed as [7]

where

According to Hooke’s law and strain-displacement relations, stress components may be expressed as

## 4. Solution for the axisymmetric frictionless and partial slip contact problem

In this section, we will solve axisymmetric contact and fretting problem for the functionally graded coating bonded to the homogeneous half-space under the spherical indenter. A functionally graded coated half-space subjected to normal and radical distributed external loads is shown in Figure 3. The stresses and displacements are continuous at the interfaces,

And along the coating surface,

in which

By using the Hankel integral transform technique and transfer matrix method, the surface displacement components can be expressed as

where

where

Considering the asymptotic behavior of Bessel functions for large arguments [13], one may prove

Differentiation of Eq. (5) with respect to

where

with

The system of the singular integrals, Eqs. (27a) and (27b), must be solved subjected to the following condition:

### 4.1 Frictionless contact problem of FGM coating

In this section, the axisymmetric frictionless contact problem between FGM coatings and a rigid spherical punch is studied. As shown in Figure 4, an applied force

Because the frictionless contact is considered, the shear traction

The Gauss-Chebyshev integration formula [24] is applied to solve Eqs. (28) and (30) with the consideration of Eq. (29).

### 4.2 Partial slip contact problem with finite friction for FGM coating

Consider the axisymmetric partial slip contact problem as shown in Figure 5. The normal surface displacement, * z* =

The inner stick region,

where

where

Finally, the partial slip contact problem with consideration of the boundary conditions (31), (32), and (33) can be expressed according to the singular integral equations:

The Goodman approximate method (uncoupled solution) [21] and the iteration method (coupled solution) [13] are used to solve coupled singular integral equation (34).

## 5. Indentation response of FGM coating under a spherical indenter

The indentation response of FGM coating under frictionless and frictional condition will be presented in this section.

Firstly, the effects of the stiffness ratio

Secondly, the linear multi-layered model is used to model the shear modulus of the coating varying in the following power law form:

where

Figure 8 shows the distributions of the contact pressure for some selected values of n with * n*.

Thirdly, the effect of the variation of Poisson’s ratio on the frictionless contact problem is considered by using piece wise exponential multi-layered model. The shear modulus of FGM coating varies as power law form according to Eq. (35). Poisson’s ratio of FGM coating is assumed to vary as the linear function along the thickness as follows

where

Finally, the axisymmetric contact problem of a functionally graded coated half-space is indented by a rigid spherical punch in the case of the partial slip. The linear multi-layered model is used to solve the problem.

The normal contact traction and radial tangential traction for some selected values of the shear modulus ratio

## 6. Conclusions

In this chapter, we introduced the axisymmetric indentation response for FGM coating under frictionless and partial slip condition by using the three types of computational models. The exponential function model can solve the axisymmetric contact problem for FGM coating whose elastic modulus continuously varies, but it cannot simulate FGM with arbitrarily varying properties. The linear multi-layered model allows arbitrarily the variation of the material properties of FGM, but it requires Poisson’s ratio which is 1/3. The Piece wise exponential multi-layered model can simulate functionally graded coating with arbitrarily varying material modulus with no limit to Poisson’s ratio, but numbers of sub-layers are larger. In practice, the computational model is chosen according to properties of the problem. Hankel integral transformation technology and transfer matrix method are used to solve the axisymmetric contact problem of FGM coating based on the cylindrical coordinate system. The results show that the contact behavior can be improved by adjusting the gradient of FGM coating. The present investigation will be expected to provide a guidance for design considerations and applications of FGM coating.

## Acknowledgments

The support from the National Natural Science Foundation of China under Grant Nos. 11662011 and 11811530067 are gratefully acknowledged.