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 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 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

$$
\mu(x) = \mu\_0 \mathbf{e}^{\mathrm{ax}} \tag{4}
$$

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

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

96

relations, and the equilibrium equations in the radial and axial directions disregarding the body forces are given by the following relations [7]:

For the present axisymmetric problem, the strain components, stress-strain

<sup>ε</sup>rr <sup>¼</sup> <sup>∂</sup><sup>u</sup> ∂r

, (1a)

two-dimensional adhesive contact problem.

Mechanics of Functionally Graded Materials and Structures

#### Figure 1.

The linear mutli-layered model for the functionally graded coating (a) and the cylindrical coordinate system (b).

where <sup>α</sup> <sup>¼</sup> <sup>h</sup>�<sup>1</sup> <sup>0</sup> log ð Þ μ0=μ ∗ is a constant characterizing the material inhomogeneity with μ<sup>0</sup> being the value of μð Þz at the surface, i.e., μ<sup>0</sup> ¼ μð Þ h<sup>0</sup> . μ<sup>0</sup> and μ ∗ are related by

$$
\mu\_0 = \mu \ast \mathbf{e}^{ah\_0} \tag{5}
$$

(6a)

f g <sup>2</sup>sD <sup>þ</sup> <sup>α</sup>ð Þ <sup>3</sup> � <sup>k</sup> <sup>s</sup> 〈u~〉<sup>1</sup> <sup>þ</sup> ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>D</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>D</sup> � ð Þ <sup>k</sup> � <sup>1</sup> <sup>s</sup>

Axisymmetric Indentation Response of Functionally Graded Material Coating

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

ai ¼ � <sup>2</sup>smi <sup>þ</sup> <sup>s</sup>αð Þ <sup>3</sup> � <sup>k</sup>

2 þ 1

2 � 1

2 þ 1

〈σ~zz〉<sup>0</sup> <sup>¼</sup> <sup>X</sup>

1 μð Þz

ð Þ¼ z cj 1 þ z=bj

� � <sup>¼</sup> cj

� � <sup>¼</sup> <sup>μ</sup> hj

2 � 1

4

i¼1

〈σ~rz〉<sup>1</sup> <sup>¼</sup> <sup>X</sup>

4

i¼1

For a homogeneous layer without the gradient, the gradient index α in Eqs. (9)

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 z, μð Þz , with boundary values μð Þ¼ h<sup>0</sup> μ0. Poisson's ratio v is taken as 1/3. The linear multilayered model divides functionally graded coating into N sub-layers. The shear modulus μð Þz in each sub-layer is assumed to take the following form:

> z ∗ b j

where z ∗ ¼ z þ b and μ<sup>j</sup> is equal to the real value of the shear modulus at the

� �, which lead to

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

ð Þ k þ 1 mi

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

<sup>m</sup><sup>1</sup> ¼ � <sup>α</sup>

<sup>m</sup><sup>2</sup> ¼ � <sup>α</sup>

<sup>m</sup><sup>3</sup> ¼ � <sup>α</sup>

<sup>m</sup><sup>4</sup> ¼ � <sup>α</sup>

κ � 1 μð Þz

where

and

may be expressed as

and (10) equals to 0.

μð Þz ≈μ<sup>j</sup>

99

sub-interfaces, z ¼ h <sup>j</sup>, i.e., μ<sup>j</sup> h <sup>j</sup>

3.2 Linear multi-layered model

<sup>2</sup> � �〈w~〉<sup>0</sup> <sup>¼</sup> 0, (8b)

〈u~〉<sup>1</sup> <sup>¼</sup> <sup>A</sup>3ð Þ<sup>s</sup> <sup>e</sup>m1<sup>z</sup> <sup>þ</sup> <sup>A</sup>4ð Þ<sup>s</sup> <sup>e</sup>m2<sup>z</sup> <sup>þ</sup> <sup>A</sup>5ð Þ<sup>s</sup> <sup>e</sup>m3<sup>z</sup> <sup>þ</sup> <sup>A</sup>6ð Þ<sup>s</sup> <sup>e</sup>m4<sup>z</sup> (9a)

<sup>2</sup> <sup>þ</sup> <sup>α</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> mi � ð Þ <sup>k</sup> � <sup>1</sup> <sup>s</sup><sup>2</sup> ,ð Þ <sup>i</sup> <sup>¼</sup> 1, ::…, 4

r

r

r

r

( )<sup>1</sup>=<sup>2</sup>

( )<sup>1</sup>=<sup>2</sup>

( )<sup>1</sup>=<sup>2</sup>

( )<sup>1</sup>=<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffi 3 � k k þ 1

ffiffiffiffiffiffiffiffiffiffiffi 3 � k k þ 1

ffiffiffiffiffiffiffiffiffiffiffi 3 � k k þ 1

ffiffiffiffiffiffiffiffiffiffiffi 3 � k k þ 1

αs

αs

αs

αs

f g ð Þ <sup>κ</sup> <sup>þ</sup> <sup>1</sup> miai <sup>þ</sup> ð Þ <sup>3</sup> � <sup>k</sup> <sup>s</sup> Aiþ2emiz (10a)

f g mi � ais Aiþ2emiz (10b)

� �, hj <sup>≤</sup> <sup>z</sup><sup>≤</sup> hj�1, <sup>j</sup> <sup>¼</sup> 1, 2…<sup>N</sup> (11)

,

,

,

:

〈w~〉<sup>0</sup> <sup>¼</sup> <sup>A</sup>3ð Þ<sup>s</sup> <sup>a</sup>1e<sup>m</sup>1<sup>z</sup> <sup>þ</sup> <sup>A</sup>4ð Þ<sup>s</sup> <sup>a</sup>2e<sup>m</sup>2<sup>z</sup> <sup>þ</sup> <sup>A</sup>5ð Þ<sup>s</sup> <sup>a</sup>3e<sup>m</sup>3<sup>z</sup> <sup>þ</sup> <sup>A</sup>6ð Þ<sup>s</sup> <sup>a</sup>4e<sup>m</sup>4<sup>z</sup> (9b)

<sup>2</sup> <sup>þ</sup> <sup>i</sup><sup>4</sup>

<sup>2</sup> <sup>þ</sup> <sup>i</sup><sup>4</sup>

<sup>2</sup> � <sup>i</sup><sup>4</sup>

<sup>2</sup> � <sup>i</sup><sup>4</sup>

According to Hooke's law and strain-displacement relations, stress components

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

$$\begin{aligned} &(k+1)\left\{\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{1}{r^2}u + \frac{\partial^2 w}{\partial r \partial z}\right\} + (k-1)a\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}\right) + (k-1)\left\{\frac{\partial^2 u}{\partial z^2} - \frac{\partial^2 w}{\partial r \partial z}\right\} \\ &= 0, \end{aligned}$$

$$\begin{split} &(k+1)\left\{\frac{\partial^2 u}{\partial r \partial \mathbf{z}} + \frac{1}{r}\frac{\partial u}{\partial \mathbf{z}} + \frac{\partial^2 w}{\partial \mathbf{z}^2}\right\} - (k-1)\left(\frac{\partial^2 u}{\partial r \partial \mathbf{z}} - \frac{\partial^2 w}{\partial r^2}\right) - \frac{(k-1)}{r}\left\{\frac{\partial u}{\partial \mathbf{z}} - \frac{\partial w}{\partial r}\right\} \\ &+(3-k)a\left(\frac{\partial u}{\partial r} + \frac{u}{r}\right) + (k+1)a\frac{\partial w}{\partial \mathbf{z}} = 0 \end{split} \tag{6b}$$

where k ¼ 3 � 4ν and ν is Poisson's ratio.

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

$$\left< \tilde{\boldsymbol{\phi}}(\boldsymbol{s}, \boldsymbol{z}) \right>\_{\mathcal{P}} = \bigcap\_{\mathbf{0}}^{\infty} \boldsymbol{\alpha}(\boldsymbol{s}, \boldsymbol{z}) \boldsymbol{r} \boldsymbol{J}\_{\mathcal{P}}(\boldsymbol{s} \boldsymbol{r}) d\boldsymbol{r}, \tag{7a}$$

$$\rho\_p(r,z) = \int\_0^\infty \langle \ddot{\alpha}(s,z) \rangle\_p s I\_p(sr) ds \tag{7b}$$

where the bar � indicates Hankel transform; 〈〉<sup>p</sup> is the pth-order Hankel transform; and Jp is the pth-order Bessel function of the first kind.

By using the Hankel transform and defining D ¼ d=dz, Eq. (6) can be expressed as

$$\left\{ (k-1)D^2 + a(k-1)D - (k+1)s^2 \right\} \langle \ddot{u} \rangle\_1 - \left\{ 2\varepsilon D + a\varepsilon (k-1) \right\} \langle \ddot{w} \rangle\_0 = 0,\tag{8a}$$

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

$$\left\{2\mathfrak{s}D + a(\mathfrak{z} - k)s\right\}\langle\ddot{u}\rangle\_1 + \left\{(k+1)D^2 + a(k+1)D - (k-1)s^2\right\}\langle\ddot{u}\rangle\_0 = 0,\tag{8b}$$

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

$$\langle \hat{u} \rangle\_1 = A\_3(\mathfrak{s}) \mathbf{e}^{m\_1 x} + A\_4(\mathfrak{s}) \mathbf{e}^{m\_2 x} + A\_5(\mathfrak{s}) \mathbf{e}^{m\_3 x} + A\_6(\mathfrak{s}) \mathbf{e}^{m\_4 x} \tag{9a}$$

$$\langle \hat{w} \rangle\_0 = A\_3(s) a\_1 \mathbf{e}^{m\_1 x} + A\_4(s) a\_2 \mathbf{e}^{m\_2 x} + A\_5(s) a\_3 \mathbf{e}^{m\_3 x} + A\_6(s) a\_4 \mathbf{e}^{m\_4 x} \tag{9b}$$

where

$$a\_i = -\frac{2sm\_i + sa(3-k)}{(k+1)m\_i^2 + a(k+1)m\_i - (k-1)s^2}, (i = 1, \dots, 4)$$

and

where <sup>α</sup> <sup>¼</sup> <sup>h</sup>�<sup>1</sup>

related by

Figure 1.

system (b).

ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>∂</sup><sup>2</sup>

¼ 0,

ð Þ k þ 1

98

u ∂r<sup>2</sup> þ 1 r ∂u <sup>∂</sup><sup>r</sup> � <sup>1</sup>

∂2 u ∂r∂z þ 1 r ∂u ∂z þ ∂2 w ∂z<sup>2</sup>

∂r þ u r � �

þð Þ <sup>3</sup> � <sup>k</sup> <sup>α</sup> <sup>∂</sup><sup>u</sup>

<sup>0</sup> log ð Þ μ0=μ ∗ is a constant characterizing the material inhomoge-

∂z þ ∂w ∂r � �

� ∂2 w ∂r2

� �

<sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>μ</sup> <sup>∗</sup> <sup>e</sup><sup>α</sup>h<sup>0</sup> (5)

� ð Þ <sup>k</sup> � <sup>1</sup> r

ωð Þ s, z rJpð Þ sr dr, (7a)

〈ω~ð Þ s, z 〉psJpð Þ sr ds (7b)

<sup>þ</sup> ð Þ <sup>k</sup> � <sup>1</sup> <sup>∂</sup><sup>2</sup>

∂u <sup>∂</sup><sup>z</sup> � <sup>∂</sup><sup>w</sup> ∂r � �

u <sup>∂</sup>z<sup>2</sup> � <sup>∂</sup><sup>2</sup>

w ∂r∂z � �

(6a)

(6b)

neity with μ<sup>0</sup> being the value of μð Þz at the surface, i.e., μ<sup>0</sup> ¼ μð Þ h<sup>0</sup> . μ<sup>0</sup> and μ ∗ are

The linear mutli-layered model for the functionally graded coating (a) and the cylindrical coordinate

<sup>þ</sup> ð Þ <sup>k</sup> � <sup>1</sup> <sup>α</sup> <sup>∂</sup><sup>u</sup>

∂2 u ∂r∂z

In order to solve Eq. (6), we use the technique of Hankel integral transform.

∞ð

0

∞ð

0

where the bar � indicates Hankel transform; 〈〉<sup>p</sup> is the pth-order Hankel

By using the Hankel transform and defining D ¼ d=dz, Eq. (6) can be expressed as

<sup>2</sup> � �〈u~〉<sup>1</sup> � f g <sup>2</sup>sD <sup>þ</sup> <sup>α</sup>s kð Þ � <sup>1</sup> 〈w~〉<sup>0</sup> <sup>¼</sup> 0, (8a)

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

<sup>þ</sup> ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>α</sup> <sup>∂</sup><sup>w</sup>

where k ¼ 3 � 4ν and ν is Poisson's ratio.

ð Þ <sup>k</sup> � <sup>1</sup> <sup>D</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup>ð Þ <sup>k</sup> � <sup>1</sup> <sup>D</sup> � ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>s</sup>

The Hankel transform and its inversion are defined as

〈ω~ð Þ s, z 〉<sup>p</sup> ¼

ωpð Þ¼ r, z

transform; and Jp is the pth-order Bessel function of the first kind.

∂2 w ∂r∂z

� ð Þ k � 1

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup>

<sup>r</sup><sup>2</sup> <sup>u</sup> <sup>þ</sup>

Mechanics of Functionally Graded Materials and Structures

� �

� �

$$\begin{aligned} m\_1 &= -\frac{a}{2} + \frac{1}{2} \left\{ \alpha^2 + 4s^2 + i4\sqrt{\frac{3-k}{k+1}} \alpha \right\}^{1/2}, \\ m\_2 &= -\frac{a}{2} - \frac{1}{2} \left\{ \alpha^2 + 4s^2 + i4\sqrt{\frac{3-k}{k+1}} \alpha \right\}^{1/2}, \\ m\_3 &= -\frac{a}{2} + \frac{1}{2} \left\{ \alpha^2 + 4s^2 - i4\sqrt{\frac{3-k}{k+1}} \alpha \right\}^{1/2}, \\ m\_4 &= -\frac{a}{2} - \frac{1}{2} \left\{ \alpha^2 + 4s^2 - i4\sqrt{\frac{3-k}{k+1}} \alpha \right\}^{1/2}. \end{aligned}$$

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

$$\frac{\kappa - 1}{\mu(z)} \langle \tilde{\sigma}\_{\text{xx}} \rangle\_0 = \sum\_{i=1}^{4} \{ (\kappa + 1)m\_i a\_i + (3 - k)s \} A\_{i+2} \mathbf{e}^{m\_i x} \tag{10a}$$

$$\frac{1}{\mu(z)} \langle \tilde{\sigma}\_{rz} \rangle\_1 = \sum\_{i=1}^{4} \{m\_i - a\_i s\} A\_{i+2} \mathbf{e}^{m\_i x} \tag{10b}$$

For a homogeneous layer without the gradient, the gradient index α in Eqs. (9) and (10) equals to 0.

#### 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 z, μð Þz , with boundary values μð Þ¼ h<sup>0</sup> μ0. Poisson's ratio v is taken as 1/3. The linear multilayered model divides functionally graded coating into N sub-layers. The shear modulus μð Þz in each sub-layer is assumed to take the following form:

$$\mu(\boldsymbol{z}) \approx \mu\_j(\boldsymbol{z}) = c\_j(\mathbb{1} + \boldsymbol{z}/b\_j) = c\_j\left(\frac{\boldsymbol{z} \ast}{b\_j}\right), h\_j \le \boldsymbol{z} \le h\_{j-1}, j = \mathbf{1}, 2...N \tag{11}$$

where z ∗ ¼ z þ b and μ<sup>j</sup> is equal to the real value of the shear modulus at the sub-interfaces, z ¼ h <sup>j</sup>, i.e., μ<sup>j</sup> h <sup>j</sup> � � <sup>¼</sup> <sup>μ</sup> hj � �, which lead to

Mechanics of Functionally Graded Materials and Structures

$$b\_j = \frac{\mu\_{j-1}h\_j - \mu\_j h\_{j-1}}{\mu\_j - \mu\_{j-1}},\tag{12a}$$

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

Axisymmetric Indentation Response of Functionally Graded Material Coating

〉<sup>0</sup> <sup>þ</sup> <sup>d</sup>〈ϕ~<sup>j</sup>

s 2

〉0=dz ∗

s 3

I1ð Þ sz ∗

2

Piece wise exponential multi-layered model divides functionally graded coatings into N sub-layers as shown in Figure 2. The shear modulus μð Þz in each sub-layer is

2 2

Aj1ð Þ<sup>s</sup>

<sup>2</sup> <sup>z</sup> <sup>∗</sup>K1ð Þ� sz <sup>∗</sup> sK0ð Þ sz <sup>∗</sup> Aj2ð Þ<sup>s</sup>

<sup>K</sup>0ð Þþ sz <sup>∗</sup> <sup>2</sup>sK1ð Þ sz <sup>∗</sup> Aj4ð Þg <sup>s</sup>

2

<sup>2</sup> <sup>z</sup> <sup>∗</sup>K0ð Þ sz <sup>∗</sup> Aj2ð Þþ<sup>s</sup> f g <sup>z</sup> <sup>∗</sup> sI1ð Þ� sz <sup>∗</sup> <sup>I</sup>0ð Þ sz <sup>∗</sup> Aj3ð Þ<sup>s</sup>

<sup>2</sup> <sup>z</sup> <sup>∗</sup>K1ð Þ sz <sup>∗</sup> Aj2ð Þ<sup>s</sup>

z ∗ I1ð Þ sz ∗ Aj3ðÞþs s

, hj ≤ z≤ hj�1, j ¼ 1, 2, ……N (19a)

, (19b)

2

(18a)

(18b)

(18c)

(18d)

z ∗K1ð Þ sz ∗ Aj4ð Þg s

〉0

�sz ∗ I0ð Þ sz ∗ Aj3ðÞ�s sz ∗K0ð Þ sz ∗ Aj4ð Þs

s 2

<sup>2</sup> <sup>z</sup> <sup>∗</sup> <sup>I</sup>1ð Þ sz <sup>∗</sup> Aj1ðÞ�<sup>s</sup>

<sup>I</sup>0ð Þ� sz <sup>∗</sup> <sup>2</sup>sI1ð Þ sz <sup>∗</sup> Aj3ðÞþ<sup>s</sup> <sup>z</sup> <sup>∗</sup> <sup>s</sup>

<sup>2</sup> <sup>z</sup> <sup>∗</sup> <sup>I</sup>0ð Þ� sz <sup>∗</sup> <sup>s</sup>

K1ð Þ sz ∗

〉<sup>0</sup> � <sup>s</sup>〈ϕ~<sup>j</sup>

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

<sup>2</sup> <sup>z</sup> <sup>∗</sup> <sup>I</sup>1ð Þ� sz <sup>∗</sup> sI0ð Þ sz <sup>∗</sup> Aj1ð Þþ<sup>s</sup>

〉0=dz <sup>∗</sup> � 〈 <sup>~</sup><sup>f</sup> <sup>j</sup>

þ �f g z ∗ sK1ð Þ� sz ∗ K0ð Þ sz ∗ Aj4ð Þs

<sup>2</sup> <sup>z</sup> <sup>∗</sup> <sup>I</sup>0ð Þ sz <sup>∗</sup> Aj1ðÞþ<sup>s</sup>

z ∗ bj f s 3

〈u~rj〉<sup>1</sup> ¼ �sz <sup>∗</sup> 〈 <sup>~</sup><sup>f</sup> <sup>j</sup>

¼ � <sup>s</sup> 2

〈w<sup>~</sup> <sup>j</sup>〉<sup>0</sup> <sup>¼</sup> <sup>z</sup> <sup>∗</sup> <sup>d</sup>〈 <sup>~</sup><sup>f</sup> <sup>j</sup>

¼ s 2

〈σ~zzj〉<sup>0</sup> ¼ 2cj

〈σ~rzj〉<sup>1</sup> ¼ 2cj

þ � s 3

Figure 2.

101

þ z ∗ s 2

> z ∗ b j

<sup>2</sup> <sup>z</sup> <sup>∗</sup>K0ð Þþ sz <sup>∗</sup> <sup>s</sup>

f � <sup>s</sup> 3

> 2 2

3.3 Piece wise exponential multi-layered model

assumed to vary as an exponential function form:

Piece wise exponential multi-layered model for the graded coating.

ð Þ¼ z aje

b jz

μ hj

<sup>¼</sup> <sup>μ</sup><sup>j</sup> <sup>h</sup> <sup>j</sup>

μð Þz ≈μ<sup>j</sup>

Aj2ðÞ�<sup>s</sup> <sup>s</sup>

$$c\_j = \frac{\mu\_j}{1 + h\_j/b\_j}.\tag{12b}$$

As in [12], introduces two potential functions to write the displacement components uj and wj in each sub-layer:

$$
\Delta u\_j = \left(\mathbf{z} + b\_j\right) \partial \mathbf{f}\_j / \partial r + \partial \phi\_j / \partial r \tag{13a}
$$

$$\mathcal{L}w\_j = \left(\mathbf{z} + b\_j\right) \partial \!\!/ \left. \not\!/ \partial \!\!z - f\_j + \partial \phi\_j / \partial \!\!z , \ \left. \left(h\_j \le \mathbf{z} \le h\_{j-1}\right) . \right. \tag{13b}$$

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

$$
\nabla^2 \phi\_j + \frac{1}{\mathbb{Z} \ast} \partial \phi\_j / \partial \mathbf{z} \ast = \mathbf{0}, \tag{14a}
$$

:

$$
\nabla^2 \mathcal{f}\_j + \frac{1}{\mathbb{Z}\*} \partial \!\!\!/ \_j / \partial \!\!\!/ \* = \frac{1}{2\mathbb{Z}\*^2} \partial \phi\_j / \partial \!\!\!/ \_j \text{.}\tag{14b}
$$

where

$$\nabla^2 = \frac{1}{r}\partial(r\partial/\partial r)/\partial r + \partial^2/\partial \mathbf{z} \ast \mathbf{z}^2$$

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

uj <sup>¼</sup> <sup>z</sup> <sup>∗</sup> <sup>∂</sup><sup>f</sup> <sup>j</sup> <sup>=</sup>∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup>ϕ<sup>j</sup> <sup>=</sup>∂r, <sup>h</sup> <sup>j</sup> <sup>≤</sup> <sup>z</sup><sup>≤</sup> <sup>h</sup> <sup>j</sup>�<sup>1</sup> (15a)

$$\boldsymbol{w}\_{j} = \boldsymbol{z} \ast \partial \boldsymbol{f}\_{j} / \partial \boldsymbol{z} \ast \ -\boldsymbol{f}\_{j} + \partial \phi\_{j} / \partial \boldsymbol{z} \ast \ , \ (\boldsymbol{h}\_{j} \le \boldsymbol{z} \le \boldsymbol{h}\_{j-1}) \tag{15b}$$

$$\sigma\_{\eta\eta} = 2c\_j \left( \frac{z \ast}{b\_j} \right) \left( z \ast \partial^2 f\_j / \partial r^2 + \partial^2 \phi\_j / \partial r^2 - \partial f\_j / \partial z \ast - \frac{1}{2z \ast} \partial \phi\_j / \partial z \ast \right), \tag{16a}$$

$$\sigma\_{\theta\theta\overline{\eta}} = 2c\_j \left( \frac{\mathbf{z} \cdot \mathbf{s}}{b\_j} \right) \left( \mathbf{z} \ast \frac{\mathbf{1}}{r} \partial \!\!f \!/ \slash \!r + \frac{\mathbf{1}}{r} \partial \!\!\phi\_j \slash \!r - \partial \!\!f \!\/\_j \slash \!\mathbf{z} \ast \ - \frac{\mathbf{1}}{2\mathbf{z} \ast} \partial \!\!\phi\_j \slash \!\mathbf{z} \ast \right), \tag{16b}$$

$$\sigma\_{\text{zz}j} = 2c\_j \left( \frac{\mathbf{z} \ast}{b\_j} \right) \left( \mathbf{z} \ast \partial^2 f\_j / \partial r^2 + \partial^2 \phi\_j / \partial \mathbf{z} \ast^2 - \partial f\_j / \partial \mathbf{z} \ast - \frac{\mathbf{1}}{2\mathbf{z} \ast} \partial \phi\_j / \partial \mathbf{z} \ast \right), \tag{16c}$$

$$
\sigma\_{\rm rzj} = 2c\_j \left( \frac{\mathbf{z} \cdot \mathbf{s}}{b\_j} \right) \left( \mathbf{z} \ast \partial^2 f\_j / \partial r \partial \mathbf{z} \, \ast + \partial^2 \phi\_j / \partial r \partial \mathbf{z} \, \ast \right). \tag{16d}
$$

Applying Hankel transformation Eqs. (7a)–(14), we obtain the solutions for displacement functions ϕ<sup>j</sup> and f <sup>j</sup> in each sub-layer:

$$
\langle \tilde{\phi}\_j \rangle\_0 = A\_{j1}(\mathfrak{s}) I\_0(\mathfrak{s} \mathfrak{x} \ast) + A\_{j2}(\mathfrak{s}) K\_0(\mathfrak{s} \mathfrak{x} \ast), \tag{17a}
$$

$$
\langle \bar{f}\_j \rangle\_0 = A\_{j3}(s)I\_0(\text{sz}\*) + A\_{j4}(s)K\_0(\text{sz}\*) + \frac{s}{2} \left\{ A\_{j1}(s)I\_1(\text{sz}\*) - A\_{j2}(s)K\_1(\text{sz}\*) \right\} \tag{17b}
$$

where I0, I1, K0, and K<sup>1</sup> are modified Bessel functions of the 0th and 1th order.

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

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

$$\begin{aligned} \langle \tilde{u}\_{\tilde{f}} \rangle\_1 &= -\text{sz} \ast \langle \dot{f}\_{\tilde{f}} \rangle\_0 - s \langle \dot{\phi}\_{\tilde{f}} \rangle\_0 \\ &= \left\{ -\frac{s^2}{2} \text{z} \ast I\_1(\text{sz} \ast) - s I\_0(\text{sz} \ast) \right\} A\_{j1}(\text{s}) + \left\{ \frac{s^2}{2} \text{z} \ast K\_1(\text{sz} \ast) - s K\_0(\text{sz} \ast) \right\} A\_{j2}(\text{s}) \\ &- \text{sz} \ast I\_0(\text{sz} \ast) A\_{j3}(\text{s}) - \text{sz} \ast K\_0(\text{sz} \ast) A\_{j4}(\text{s}) \end{aligned} \tag{18a}$$

$$\begin{aligned} \langle \check{w}\_j \rangle\_0 &= z \ast d(\check{f}\_j)\_0 / dz \ast - \langle \check{f}\_j \rangle\_0 + d \langle \check{\phi}\_j \rangle\_0 / dx \ast \\ &= \frac{s^2}{2} z \ast I\_0(\mathfrak{s} z \ast) A\_{j1}(\mathfrak{s}) + \frac{s^2}{2} z \ast K\_0(\mathfrak{s} z \ast) A\_{j2}(\mathfrak{s}) + \{ z \ast s I\_1(\mathfrak{s} z \ast) - I\_0(\mathfrak{s} z \ast) \} A\_{j3}(\mathfrak{s}) \\ &+ \{ -z \ast s K\_1(\mathfrak{s} z \ast) - K\_0(\mathfrak{s} z \ast) \} A\_{j4}(\mathfrak{s}) \end{aligned}$$

$$\begin{split} \langle \bar{\sigma}\_{\text{zx}\sharp} \rangle\_0 &= 2c\_j \Big( \frac{\text{z} \ast}{b\_j} \Big) \{ \frac{\text{s}^3}{2} \text{z} \ast I\_1(\text{sz} \ast) A\_{j1}(\text{s}) - \frac{\text{s}^3}{2} \text{z} \ast K\_1(\text{sz} \ast) A\_{j2}(\text{s}) \\ &+ \big( \text{z} \ast \text{s}^2 I\_0(\text{sz} \ast) - 2\text{s} I\_1(\text{sz} \ast) \big) A\_{j3}(\text{s}) + \big( \text{z} \ast \text{s}^2 K\_0(\text{sz} \ast) + 2\text{s} K\_1(\text{sz} \ast) \big) A\_{j4}(\text{s}) \} . \end{split} \tag{18c}$$

$$\begin{split} \langle \bar{\sigma}\_{\text{rij}} \rangle\_1 &= 2c\_j \left( \frac{\mathbf{z} \ast}{b\_j} \right) \{ \left( -\frac{\mathbf{s}^3}{2} \mathbf{z} \ast I\_0(\mathbf{z} \mathbf{z} \ast \mathbf{}) - \frac{\mathbf{s}^2}{2} I\_1(\mathbf{z} \mathbf{z} \ast \mathbf{}) \right) A\_{j1}(\mathbf{s}) \\ &+ \left( -\frac{\mathbf{s}^3}{2} \mathbf{z} \ast K\_0(\mathbf{z} \mathbf{z} \ast \mathbf{}) + \frac{\mathbf{s}^2}{2} K\_1(\mathbf{z} \mathbf{z} \ast \mathbf{}) \right) A\_{j2}(\mathbf{s}) - \mathbf{s}^2 \mathbf{z} \ast I\_1(\mathbf{z} \mathbf{z} \ast) A\_{j3}(\mathbf{s}) + \mathbf{s}^2 \mathbf{z} \ast K\_1(\mathbf{z} \mathbf{z} \ast) A\_{j4}(\mathbf{s}) \}. \end{split} \tag{18d}$$

#### 3.3 Piece wise exponential multi-layered model

Piece wise exponential multi-layered model divides functionally graded coatings into N sub-layers as shown in Figure 2. The shear modulus μð Þz in each sub-layer is assumed to vary as an exponential function form:

$$
\mu(\mathbf{z}) \approx \mu\_j(\mathbf{z}) = a\_j e^{b\_j \mathbf{z}}, \\
h\_j \le \mathbf{z} \le h\_{j-1}, j = \mathbf{1}, \mathbf{2}, ..., N \tag{19a}
$$

$$
\mu(h\_j) = \mu\_j(h\_j), \tag{19b}
$$

(18b)

Figure 2. Piece wise exponential multi-layered model for the graded coating.

bj <sup>¼</sup> <sup>μ</sup>j�1<sup>h</sup> <sup>j</sup> � <sup>μ</sup><sup>j</sup>

cj <sup>¼</sup> <sup>μ</sup> <sup>j</sup>

<sup>=</sup>∂<sup>z</sup> � <sup>f</sup> <sup>j</sup> <sup>þ</sup> <sup>∂</sup>ϕ<sup>j</sup>

By making use of Eqs. (1), (2), and (13), the equilibrium equations (3) are

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> <sup>¼</sup> <sup>1</sup>

<sup>∂</sup>ð Þ <sup>r</sup>∂=∂<sup>r</sup> <sup>=</sup>∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

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

<sup>=</sup>∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup>ϕ<sup>j</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> � <sup>f</sup> <sup>j</sup> <sup>þ</sup> <sup>∂</sup>ϕ<sup>j</sup>

z ∗ ∂<sup>2</sup> f j

ϕj=∂r

<sup>2</sup> � <sup>∂</sup><sup>f</sup> <sup>j</sup>

<sup>=</sup>∂r∂<sup>z</sup> <sup>∗</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

2

where I0, I1, K0, and K<sup>1</sup> are modified Bessel functions of the 0th and 1th order.

<sup>=</sup>∂<sup>r</sup> � <sup>∂</sup><sup>f</sup> <sup>j</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> <sup>2</sup> � <sup>∂</sup><sup>f</sup> <sup>j</sup>

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

<sup>2</sup><sup>z</sup> <sup>∗</sup> <sup>2</sup> <sup>∂</sup>ϕ<sup>j</sup>

=∂z ∗ <sup>2</sup> :

<sup>=</sup>∂r, <sup>h</sup> <sup>j</sup> <sup>≤</sup> <sup>z</sup><sup>≤</sup> <sup>h</sup> <sup>j</sup>�<sup>1</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> , hj <sup>≤</sup> <sup>z</sup><sup>≤</sup> hj�<sup>1</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> � <sup>1</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> � <sup>1</sup>

<sup>=</sup>∂<sup>z</sup> <sup>∗</sup> � <sup>1</sup>

〉<sup>0</sup> ¼ Aj1ð Þs I0ð Þþ sz ∗ Aj2ð Þs K0ð Þ sz ∗ , (17a)

1

uj ¼ z þ bj <sup>∂</sup><sup>f</sup> <sup>j</sup>

nents uj and wj in each sub-layer:

represented as [12].

where

given by

σrrj ¼ 2cj

σθθ<sup>j</sup> ¼ 2cj

σzzj ¼ 2cj

〈 ~f j

100

z ∗ bj 

z ∗ b j 

z ∗ bj 

wj ¼ z þ b <sup>j</sup>

<sup>∂</sup><sup>f</sup> <sup>j</sup>

Mechanics of Functionally Graded Materials and Structures

∇2 f <sup>j</sup> þ 1 <sup>z</sup> <sup>∗</sup> <sup>∂</sup><sup>f</sup> <sup>j</sup>

∇2 ϕ<sup>j</sup> þ

<sup>∇</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> r

uj <sup>¼</sup> <sup>z</sup> <sup>∗</sup> <sup>∂</sup><sup>f</sup> <sup>j</sup>

wj <sup>¼</sup> <sup>z</sup> <sup>∗</sup> <sup>∂</sup><sup>f</sup> <sup>j</sup>

z ∗ ∂<sup>2</sup> f j =∂r <sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

<sup>z</sup> <sup>∗</sup> <sup>1</sup> r ∂f j <sup>=</sup>∂<sup>r</sup> <sup>þ</sup> 1 r ∂ϕ<sup>j</sup>

z ∗ ∂<sup>2</sup> f j =∂r <sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> ϕj

σrzj ¼ 2cj

displacement functions ϕ<sup>j</sup> and f <sup>j</sup> in each sub-layer:

〉<sup>0</sup> <sup>¼</sup> Aj3ð Þ<sup>s</sup> <sup>I</sup>0ð Þþ sz <sup>∗</sup> Aj4ð Þ<sup>s</sup> <sup>K</sup>0ð Þþ sz <sup>∗</sup> <sup>s</sup>

〈ϕ~j

z ∗ bj  μ<sup>j</sup> � μ <sup>j</sup>�<sup>1</sup>

1 þ hj=bj

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

hj�<sup>1</sup>

, (12a)

: (12b)

<sup>=</sup>∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup>ϕj=∂<sup>r</sup> (13a)

: (13b)

=∂z: (14b)

(15a)

<sup>2</sup><sup>z</sup> <sup>∗</sup> <sup>∂</sup>ϕj=∂<sup>z</sup> <sup>∗</sup>

<sup>2</sup><sup>z</sup> <sup>∗</sup> <sup>∂</sup>ϕj=∂<sup>z</sup> <sup>∗</sup>

Aj1ð Þ<sup>s</sup> <sup>I</sup>1ð Þ� sz <sup>∗</sup> Aj2ð Þ<sup>s</sup> <sup>K</sup>1ð Þ sz <sup>∗</sup>

<sup>2</sup><sup>z</sup> <sup>∗</sup> <sup>∂</sup>ϕ<sup>j</sup>

ϕj=∂r∂z ∗

(15b)

=∂z ∗

, (16a)

, (16b)

, (16c)

(17b)

: (16d)

<sup>=</sup>∂z, hj <sup>≤</sup> <sup>z</sup><sup>≤</sup> hj�<sup>1</sup>

<sup>z</sup> <sup>∗</sup> <sup>∂</sup>ϕj=∂<sup>z</sup> <sup>∗</sup> <sup>¼</sup> 0, (14a)

in which:

$$\mathfrak{a}\_{j} = \mu(h\_{j})e^{-\ln\left[\mu\left(h\_{j+1}\right)/\mu\left(h\_{j}\right)\right]h\_{j}/\left(h\_{j+1} - h\_{j}\right)}, \\ \mathfrak{b}\_{j} = \ln\left[\mu\left(h\_{j+1}\right)/\mu\left(h\_{j}\right)\right]/\left(h\_{j+1} - h\_{j}\right)$$

1 μjð Þz

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

ments are continuous at the interfaces, z ¼ 0, which state.

And along the coating surface, z ¼ h0, we have

the surface displacement components can be expressed as

problem

Figure 3.

103

〈σ~rzjð Þ <sup>s</sup>, <sup>z</sup> 〉<sup>1</sup> <sup>¼</sup> <sup>X</sup>

Axisymmetric Indentation Response of Functionally Graded Material Coating

4

mji � cjis � �Ajiemjiz (22b)

u2ð Þ� r, 0 u1ð Þ¼ r, 0 0, (23a) w2ð Þ� r, 0 w1ð Þ¼ r, 0 0, (23b) σ2zzð Þ� r, 0 σ1zzð Þ¼ r, 0 0, (23c) σ2rzð Þ� r, 0 σ1rzð Þ¼ r, 0 0: (23d)

σ1zzð Þ¼ r, h<sup>0</sup> p rð Þ ð Þ 0≤ r≤ a , (24a)

σ1zzð Þ¼ r, h<sup>0</sup> 0 ð Þ a< r< ∞ , (24b)

σ1rzð Þ¼ r, h<sup>0</sup> q rð Þ ð Þ 0≤ r≤ a , (24c)

σ1rzð Þ¼ r, h<sup>0</sup> 0 ð Þ a<r< ∞ (24d)

i¼1

In this section, we will solve axisymmetric contact and fretting problem for the

in which i ¼ 1 refers to the graded coating and i ¼ 2 refers to the homogeneous half-space. p rð Þ and q rð Þ are normal contact tractions and shear stress, respectively. By using the Hankel integral transform technique and transfer matrix method,

A functionally graded coated half-space subjected to normal and radical distributed external loads.

4. Solution for the axisymmetric frictionless and partial slip contact

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 displace-

and h <sup>j</sup> is the z coordinate at the end of layer j. Poisson's ratio in each sub-layer is assumed to be a constant v <sup>j</sup>.

In each sub-layer ( j ¼ 1, 2, ……N), the equilibrium equations are represented as [14]

$$\begin{split} & \left( (\mathbb{k}\_j + 1) \left\{ \frac{\partial^2 u\_j}{\partial r^2} + \frac{1}{r} \frac{\partial u\_j}{\partial r} - \frac{1}{r^2} u\_j + \frac{\partial^2 w\_j}{\partial r \partial \mathbf{z}} \right\} + (\mathbb{k}\_j - 1) b\_j \left( \frac{\partial u\_j}{\partial \mathbf{z}} + \frac{\partial w\_j}{\partial r} \right) \\ & \left. + (\mathbb{k}\_j - 1) \left\{ \frac{\partial^2 u\_j}{\partial \mathbf{z}^2} - \frac{\partial^2 w\_j}{\partial r \partial \mathbf{z}} \right\} = 0, \end{split} \tag{20a}$$

$$\begin{aligned} \left( (\mathbf{k}\_j + \mathbf{1}) \left\{ \frac{\partial^2 u\_j}{\partial r \partial \mathbf{z}} + \frac{\mathbf{1}}{r} \frac{\partial u\_j}{\partial \mathbf{z}} + \frac{\partial^2 w\_j}{\partial \mathbf{z}^2} \right\} - (\mathbf{k}\_j - \mathbf{1}) b\_j \left( \frac{\partial^2 u\_j}{\partial r \partial \mathbf{z}} - \frac{\partial^2 w\_j}{\partial r^2} \right) - \frac{(\mathbf{k}\_j - \mathbf{1})}{r} \left\{ \frac{\partial u\_j}{\partial \mathbf{z}} - \frac{\partial w\_j}{\partial r} \right\} \\ + (\mathbf{k} + \mathbf{3}) b\_j \left( \frac{\partial u\_j}{\partial r} + \frac{u\_j}{r} \right) + (\mathbf{k}\_j + \mathbf{1}) b\_j \frac{\partial w\_j}{\partial \mathbf{z}} = \mathbf{0} \end{aligned} \tag{20b}$$

where uj and wj are the displacement components in the radial and z axial directions in layer j and k <sup>j</sup> ¼ 3 � 4v <sup>j</sup>.

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

$$\langle \ddot{u}\_{j}(\mathbf{s},\mathbf{z}) \rangle\_{1} = A\_{j1}(\mathbf{s}) \mathbf{e}^{m\_{j1}x} + A\_{j2}(\mathbf{s}) \mathbf{e}^{m\_{j2}x} + A\_{j3}(\mathbf{s}) \mathbf{e}^{m\_{j3}x} + A\_{j4}(\mathbf{s}) \mathbf{e}^{m\_{j4}x} \tag{21a}$$

$$\langle \langle \vec{w}\_j(\mathfrak{s}, z) \rangle\_0 = A\_{j1}(\mathfrak{s}) c\_{j1} \mathbf{e}^{m\_{j1}z} + A\_{j2}(\mathfrak{s}) c\_{j2} \mathbf{e}^{m\_{j2}z} + A\_{j3}(\mathfrak{s}) c\_{j3} \mathbf{e}^{m\_{j3}z} + A\_{j4}(\mathfrak{s}) c\_{j4} \mathbf{e}^{m\_{j4}z} \tag{21b}$$

where Aj1–Aj<sup>4</sup> are unknown constants to be solved in layer j.

$$c\_{ji} = -\frac{2sm\_{ji} + sb\_j(3 - k\_j)}{(k\_j + 1)m\_{ji}^2 + b\_j(k + 1)m\_{ji} - (k\_j - 1)s^2}, (i = 1, 2, 3, 4)$$

$$m\_{j1} = -\frac{b\_j}{2} + \frac{1}{2} \left\{ b\_j^{-2} + 4s^2 + i4\sqrt{\frac{3 - k\_j}{k\_j + 1}}b\_js \right\}^{1/2},$$

$$m\_{j2} = -\frac{b\_j}{2} - \frac{1}{2} \left\{ b\_j^{-2} + 4s^2 + i4\sqrt{\frac{3 - k\_j}{k\_j + 1}}b\_js \right\}^{1/2},$$

$$m\_{j3} = -\frac{b\_j}{2} + \frac{1}{2} \left\{ b\_j^2 + 4s^2 - i4\sqrt{\frac{3 - k\_j}{k\_j + 1}}b\_js \right\}^{1/2},$$

$$m\_{j4} = -\frac{b\_j}{2} - \frac{1}{2} \left\{ b\_j^2 + 4s^2 - i4\sqrt{\frac{3 - k\_j}{k\_j + 1}}b\_js \right\}^{1/2}.$$

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

$$\frac{k\_j - 1}{\mu\_j(z)} \langle \tilde{\sigma}\_{\text{zzj}}(s, z) \rangle\_0 = \sum\_{i=1}^4 \left\{ (k\_j + 1) m\_{ji} c\_{ji} + (3 - k) s \right\} A\_{ji} \mathbf{e}^{m\_{ji}z} \tag{22a}$$

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

$$\frac{1}{\mu\_j(z)} \langle \bar{\sigma}\_{r\bar{z}}(s, z) \rangle\_1 = \sum\_{i=1}^4 \{m\_{ji} - c\_{ji}s\} A\_{ji} \mathbf{e}^{m\_{\bar{j}\bar{z}}} \tag{22b}$$
