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 twodimensional 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.

εθθ <sup>¼</sup> <sup>u</sup> r

<sup>ε</sup>zz <sup>¼</sup> <sup>∂</sup><sup>w</sup>

∂u ∂r

u r þ λð Þz

∂w ∂z

> ∂u ∂z þ ∂w r

in which r and z are the variables of the cylindrical coordinate system; εrr, εθθ, εzz, and εrz are the strain components; u and w are the displacement components in the radial and axial directions; σrr, σθθ, σzz, and σrz are the stress components; λð Þz

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 sublayers 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

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

<sup>μ</sup>ð Þ¼ <sup>z</sup> <sup>μ</sup>0e<sup>α</sup><sup>z</sup> (4)

þ λð Þz

þ λð Þz

u r þ ∂w ∂z

∂u ∂r þ ∂w ∂z

> ∂u ∂r þ u r

<sup>2</sup>εrz <sup>¼</sup> <sup>∂</sup><sup>u</sup> ∂z þ ∂w

σrr ¼ ð Þ λð Þþ z 2μð Þz

Axisymmetric Indentation Response of Functionally Graded Material Coating

DOI: http://dx.doi.org/10.5772/intechopen.89312

σθθ ¼ ð Þ λð Þþ z 2μð Þz

σzz ¼ ð Þ λð Þþ z 2μð Þz

∂σrz ∂r þ ∂σzz ∂z þ 1 r

and μð Þz are Lame's constants which vary along the z-axis direction.

types of computational model for FGM coating are the following.

3.1 Exponential function model

described by

97

∂σrr ∂r þ ∂σrz ∂z þ 1 r

3. Computational models for FGM coating

σrz ¼ μð Þz

, (1b)

<sup>∂</sup><sup>z</sup> , (1c)

<sup>∂</sup><sup>r</sup> : (1d)

, (2a)

, (2b)

, (2c)

: (2d)

ð Þ¼ σrr � σθθ 0, (3a)

σrz ¼ 0: (3b)

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]:

$$
\varepsilon\_{rr} = \frac{\partial u}{\partial r},
\tag{1a}
$$

Axisymmetric Indentation Response of Functionally Graded Material Coating DOI: http://dx.doi.org/10.5772/intechopen.89312

$$
\varepsilon\_{\theta\theta} = \frac{u}{r},
\tag{1b}
$$

$$
\varepsilon\_{xx} = \frac{\partial w}{\partial \mathbf{z}},
\tag{1c}
$$

$$
\Delta \varepsilon\_{\rm rx} = \frac{\partial u}{\partial \mathbf{z}} + \frac{\partial w}{\partial r} \,. \tag{1d}
$$

$$
\sigma\_{rr} = \left(\lambda(x) + 2\mu(x)\right) \frac{\partial u}{\partial r} + \lambda(z) \left(\frac{u}{r} + \frac{\partial w}{\partial z}\right), \tag{2a}
$$

$$
\sigma\_{\theta\theta} = \left(\lambda(x) + 2\mu(x)\right)\frac{u}{r} + \lambda(x)\left(\frac{\partial u}{\partial r} + \frac{\partial w}{\partial z}\right), \tag{2b}
$$

$$
\sigma\_{\rm zx} = \left(\lambda(z) + 2\mu(z)\right) \frac{\partial w}{\partial z} + \lambda(z) \left(\frac{\partial u}{\partial r} + \frac{u}{r}\right), \tag{2c}
$$

$$
\sigma\_{rz} = \mu(z) \left( \frac{\partial u}{\partial z} + \frac{\partial w}{r} \right). \tag{2d}
$$

$$\frac{\partial \sigma\_{rr}}{\partial r} + \frac{\partial \sigma\_{rz}}{\partial \mathbf{z}} + \frac{1}{r}(\sigma\_{rr} - \sigma\_{\theta\theta}) = \mathbf{0},\tag{3a}$$

$$\frac{\partial \sigma\_{rx}}{\partial r} + \frac{\partial \sigma\_{xx}}{\partial z} + \frac{1}{r} \sigma\_{rx} = 0. \tag{3b}$$

in which r and z are the variables of the cylindrical coordinate system; εrr, εθθ, εzz, and εrz are the strain components; u and w are the displacement components in the radial and axial directions; σrr, σθθ, σzz, and σrz are the stress components; λð Þz and μð Þz are Lame's constants which vary along the z-axis direction.
