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, z ¼ 0, which state.

$$
u\_2(r, \mathbf{0}) - \boldsymbol{u}\_1(r, \mathbf{0}) = \mathbf{0},\tag{23a}$$

$$w\_2(r, \mathbf{0}) - w\_1(r, \mathbf{0}) = \mathbf{0},\tag{23b}$$

$$
\sigma\_{2xx}(r,0) - \sigma\_{1xx}(r,0) = 0,\tag{23c}
$$

$$
\sigma\_{2\text{rz}}(r, \mathbf{0}) - \sigma\_{1\text{rz}}(r, \mathbf{0}) = \mathbf{0}.\tag{23d}
$$

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

$$
\sigma\_{1\varpi}(r, h\_0) = p(r) \ (0 \le r \le a),
\tag{24a}
$$

$$
\sigma\_{1xx}(r, h\_0) = 0 \quad (a < r < \infty), \tag{24b}
$$

$$
\sigma\_{1rx}(r, h\_0) = q(r) \ (0 \le r \le a),
\tag{24c}
$$

$$
\sigma\_{1\pi}(r, h\_0) = 0 \quad (a < r < \infty) \tag{24d}
$$

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, the surface displacement components can be expressed as

in which:

as [14]

kj <sup>þ</sup> <sup>1</sup> � � <sup>∂</sup><sup>2</sup>

þð Þ 3 � k bj

a <sup>j</sup> ¼ μ hj � �e

assumed to be a constant v <sup>j</sup>.

kj <sup>þ</sup> <sup>1</sup> � � <sup>∂</sup><sup>2</sup>

uj ∂r∂z þ 1 r ∂uj ∂z þ ∂2 wj ∂z<sup>2</sup>

> ∂uj ∂r þ uj r

� �

directions in layer j and kj ¼ 3 � 4v <sup>j</sup>.

<sup>þ</sup> kj � <sup>1</sup> � � <sup>∂</sup><sup>2</sup>

� �

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

Mechanics of Functionally Graded Materials and Structures

� �

<sup>þ</sup> kj <sup>þ</sup> <sup>1</sup> � �bj

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

� ln ½ � <sup>μ</sup>ð Þ <sup>h</sup> <sup>j</sup>þ<sup>1</sup> <sup>=</sup><sup>μ</sup>ð Þ <sup>h</sup> <sup>j</sup> <sup>h</sup> <sup>j</sup>=ð Þ <sup>h</sup> <sup>j</sup>þ1�<sup>h</sup> <sup>j</sup> , bj <sup>¼</sup> ln <sup>μ</sup> hjþ<sup>1</sup>

<sup>r</sup><sup>2</sup> uj <sup>þ</sup>

� <sup>k</sup> <sup>j</sup> � <sup>1</sup> � �bj

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

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

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

cji ¼ � <sup>2</sup>smji <sup>þ</sup> sbj <sup>3</sup> � <sup>k</sup> <sup>j</sup>

2 þ 1 2 b j <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

2 � 1 2 bj <sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup>

2 þ 1 2 b2 <sup>j</sup> þ 4s

2 � 1 2 b2 <sup>j</sup> þ 4s

4

i¼1

mj<sup>1</sup> ¼ � <sup>b</sup> <sup>j</sup>

mj<sup>2</sup> ¼ � bj

mj<sup>3</sup> ¼ � bj

mj<sup>4</sup> ¼ � bj

〈σ~zzjð Þ <sup>s</sup>, <sup>z</sup> 〉<sup>0</sup> <sup>¼</sup> <sup>X</sup>

may be expressed as

102

k <sup>j</sup> � 1 μj ð Þz

where uj and wj are the displacement components in the radial and z axial

〈u~jð Þ <sup>s</sup>, <sup>z</sup> 〉<sup>1</sup> <sup>¼</sup> Aj1ð Þ<sup>s</sup> <sup>e</sup><sup>m</sup> <sup>j</sup>1<sup>z</sup> <sup>þ</sup> Aj2ð Þ<sup>s</sup> <sup>e</sup><sup>m</sup> <sup>j</sup>2<sup>z</sup> <sup>þ</sup> Aj3ð Þ<sup>s</sup> <sup>e</sup><sup>m</sup> <sup>j</sup>3<sup>z</sup> <sup>þ</sup> Aj4ð Þ<sup>s</sup> <sup>e</sup><sup>m</sup> <sup>j</sup>4<sup>z</sup> (21a)

� � <sup>k</sup> <sup>j</sup> <sup>þ</sup> <sup>1</sup> � �mji<sup>2</sup> <sup>þ</sup> bjð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> mji � <sup>k</sup> <sup>j</sup> � <sup>1</sup> � �s<sup>2</sup> ,ð Þ <sup>i</sup> <sup>¼</sup> 1, 2, 3, 4

<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

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 � k <sup>j</sup> k <sup>j</sup> þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 � k <sup>j</sup> kj þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 � k <sup>j</sup> kj þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 � k <sup>j</sup> kj þ 1 bjs

bjs

bjs

bjs

kj <sup>þ</sup> <sup>1</sup> � �mjicji <sup>þ</sup> ð Þ <sup>3</sup> � <sup>k</sup> <sup>s</sup> � �Ajiemjiz (22a)

,

,

,

:

s

s

s

s

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

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

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

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

〈w<sup>~</sup> <sup>j</sup>ð Þ <sup>s</sup>, <sup>z</sup> 〉<sup>0</sup> <sup>¼</sup> Aj1ð Þ<sup>s</sup> cj1e<sup>m</sup> <sup>j</sup>1<sup>z</sup> <sup>þ</sup> Aj2ð Þ<sup>s</sup> cj2e<sup>m</sup> <sup>j</sup>2<sup>z</sup> <sup>þ</sup> Aj3ð Þ<sup>s</sup> cj3e<sup>m</sup> <sup>j</sup>3<sup>z</sup> <sup>þ</sup> Aj4ð Þ<sup>s</sup> cj4e<sup>m</sup> <sup>j</sup>4<sup>z</sup> (21b)

� �

wj ∂r∂z

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

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

∂2 wj ∂r∂z

> ∂2 uj ∂r∂z

� �=μ hj

<sup>þ</sup> <sup>k</sup> <sup>j</sup> � <sup>1</sup> � �bj

� ∂2 wj ∂r2

� �

¼ 0, (20a)

� � � � <sup>=</sup> <sup>h</sup> <sup>j</sup>þ<sup>1</sup> � hj

∂uj ∂z þ ∂wj ∂r � �

� kj � <sup>1</sup> � � r

� �

∂uj <sup>∂</sup><sup>z</sup> � <sup>∂</sup>wj ∂r � �

(20b)

$$w\_0(r) = \int\_0^a p(t)t \int\_0^\infty s M\_{11}(s, h\_0) f\_0(st) f\_0(sr) ds dt + \int\_0^a q(t)t \int\_0^\infty s M\_{12}(s, h\_0) f\_1(st) f\_0(sr) ds dt \tag{25a}$$

Iijð Þ¼ � <sup>r</sup>, <sup>t</sup> ð Þ<sup>1</sup> <sup>i</sup>

h2ð Þ¼ r, t

subjected to the following condition:

and second kinds.

∞ð

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

sMijð Þ� <sup>s</sup>, <sup>h</sup><sup>0</sup> <sup>α</sup>ij � �sJ2�ið Þ sr <sup>J</sup> <sup>j</sup>�1ð Þ st ds,ð Þ <sup>i</sup> <sup>¼</sup> 1, 2, <sup>j</sup> <sup>¼</sup> 1, 2 ,

<sup>2</sup> � <sup>r</sup><sup>2</sup> ð Þ=j j tr � �K tð Þþ <sup>=</sup><sup>r</sup> ð Þ <sup>r</sup>=<sup>t</sup> E tð Þ <sup>=</sup><sup>r</sup> , ð Þ j j<sup>t</sup> <sup>&</sup>lt;j j<sup>r</sup> E rð Þ =t , ð Þ j jt >j jr

<sup>2</sup> � <sup>r</sup><sup>2</sup> ð Þ=r<sup>2</sup> � �K rð Þ <sup>=</sup><sup>t</sup> , ð Þ j j<sup>t</sup> <sup>&</sup>gt;j j<sup>r</sup>

,

,

p tð Þtdt (28)

=2R ð Þ 0≤ r≤ a (29)

p tð ÞH1ð Þ r, t dt (30)

<sup>h</sup>1ð Þ¼ <sup>r</sup>, <sup>t</sup> j j <sup>t</sup>=<sup>r</sup> E tð Þ <sup>=</sup><sup>r</sup> , ðj j<sup>t</sup> <sup>&</sup>lt; j jÞ <sup>r</sup>

with Kð Þ: and Eð Þ: being, respectively, the complete elliptic integrals of the first

The system of the singular integrals, Eqs. (27a) and (27b), must be solved

ð a

�a

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 P is acted on the rigid spherical punch along the z-direction to form an indent depth δ<sup>0</sup> and a circular contact region with a radius a. The displacement boundary

2

Because the frictionless contact is considered, the shear traction q rð Þ is zero, and

p tð Þ t � r dt <sup>þ</sup> <sup>α</sup><sup>1</sup> π ð a

�a

P ¼ π

w rð Þ¼ , h<sup>0</sup> δ<sup>0</sup> � r

π ð a

�a

p tð Þj j<sup>t</sup> <sup>I</sup>11ð Þ <sup>r</sup>, <sup>t</sup> dt <sup>þ</sup> <sup>α</sup><sup>1</sup>

0

�

�

t

t

4.1 Frictionless contact problem of FGM coating

condition in the contact region is expressed as

the controlling equation is

1 2 ð a

�a

FGM coating indented by a spherical indenter.

m1ð Þ¼ r

Figure 4.

105

<sup>2</sup>=r<sup>2</sup> ð ÞE rð Þ� <sup>=</sup><sup>t</sup> <sup>t</sup>

Axisymmetric Indentation Response of Functionally Graded Material Coating

$$u\_0(r) = \int\_0^a p(t)t \int\_0^\infty s M\_{21}(s, h\_0) f\_0(st) f\_1(sr) ds dt + \int\_0^a q(t)t \int\_0^\infty s M\_{22}(s, h\_0) f\_1(st) f\_1(sr) ds dt \tag{25b}$$

where J0ð Þ: and J1ð Þ: are Bessel functions and

$$
\begin{bmatrix} M\_{11}(s, h\_0) & M\_{12}(s, h\_0) \\ M\_{21}(s, h\_0) & M\_{22}(s, h\_0) \end{bmatrix} = \frac{1}{2\mu\_0} \mathbf{B}\_3 \mathbf{M},
$$

$$
\mathbf{M} = \left[ T\_1(h\_0 + b\_1) \right] \left[ \overline{V}\_1 \right] \left\{ \left[ B \right] \left[ T\_1(h\_0 + b\_1) \right] \left[ \overline{V}\_1 \right] \right\}^{-1},
$$

$$
\mathbf{B}\_3 = \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix}, \mathbf{C} = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{bmatrix}.
$$

where Mijð Þ s, h<sup>0</sup> is the kernel function (see Ref. [13]).

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

$$\lim\_{s\to\infty} \begin{bmatrix} s\mathcal{M}\_{11}(s, h\_0) & s\mathcal{M}\_{12}(s, h\_0) \\ s\mathcal{M}\_{21}(s, h\_0) & s\mathcal{M}\_{22}(s, h\_0) \end{bmatrix} = \begin{bmatrix} a\_{11} & a\_{12} \\ a\_{21} & a\_{22} \end{bmatrix} = \begin{bmatrix} \frac{1-v}{\mu\_0} & \frac{1-2v}{2\mu\_0} \\ \frac{1-2v}{2\mu\_0} & \frac{1-v}{\mu\_0} \end{bmatrix}.\tag{26}$$

Differentiation of Eq. (5) with respect to r and extension of the definition of the unknown functions, p rð Þ and q rð Þ, into the range �a≤ r≤ 0 yields.

$$m\_1(r) = \frac{1}{2} \int\_{-a}^{a} \{p(t)|t|I\_{11}(r,t) + q(t)|t|I\_{12}(r,t)\}dt + \frac{\alpha\_1}{\pi} \int\_{-a}^{a} \frac{p(t)}{t-r}dt + \frac{\alpha\_1}{\pi} \int\_{-a}^{a} p(t)H\_1(r,t)dt$$

$$m\_2(r) = \frac{1}{2} \int\_{-a}^{a} \{q(t)|t|I\_{22}(r,t) + p(t)|t|I\_{21}(r,t)\}dt + \frac{\alpha\_1}{\pi} \int\_{-a}^{a} \frac{q(t)}{t-r}dt + \frac{\alpha\_1}{\pi} \int\_{-a}^{a} q(t)H\_2(r,t)dt$$

where

$$m\_1(r) = \frac{\partial u\_{x0}(r, h\_0)}{\partial r}, \\ m\_2(r) = \frac{1}{r} \frac{\partial r u\_{r0}(r, h\_0)}{\partial r}, \\ H\_i(r, t) = (h\_i(r, t) - 1)/(t - r), \text{ a.s.}$$

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

$$\begin{aligned} I\_{\vec{\eta}}(r,t) &= (-1)^i \int\_0^\infty (s\mathcal{M}\_{\vec{\eta}}(s,h\_0) - a\_{\vec{\eta}}) J\_{2-i}(sr) J\_{j-1}(st) ds, \ (i = 1,2, j = 1,2), \\\ h\_1(r,t) &= \begin{cases} |t/r|E(t/r), & (|t| < |r|) \\ (t^2/r^2)E(r/t) - \left\{(t^2 - r^2)/r^2\right\} K(r/t), & (|t| > |r|) \end{cases}, \\\ h\_2(r,t) &= \begin{cases} \left\{(t^2 - r^2)/|tr|\right\} K(t/r) + (r/t)E(t/r), & (|t| < |r|) \\ E(r/t), & (|t| > |r|) \end{cases}, \end{aligned}$$

with Kð Þ: and Eð Þ: being, respectively, the complete elliptic integrals of the first and second kinds.

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

$$P = \pi \int\_{-a}^{a} p(t)t dt\tag{28}$$

#### 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 P is acted on the rigid spherical punch along the z-direction to form an indent depth δ<sup>0</sup> and a circular contact region with a radius a. The displacement boundary condition in the contact region is expressed as

$$w(r, h\_0) = \delta\_0 - r^2 / 2R \ (0 \le r \le a) \tag{29}$$

Because the frictionless contact is considered, the shear traction q rð Þ is zero, and the controlling equation is

$$m\_1(r) = \frac{1}{2} \int\_{-a}^{a} p(t)|t|I\_{11}(r,t)dt + \frac{a\_1}{\pi} \int\_{-a}^{a} \frac{p(t)}{t-r}dt + \frac{a\_1}{\pi} \int\_{-a}^{a} p(t)H\_1(r,t)dt\tag{30}$$

Figure 4. FGM coating indented by a spherical indenter.

w0ð Þ¼ r

u0ð Þ¼ r

ð a p tð Þt ∞ð

p tð Þt ∞ð

0

0

M11ð Þ s, h<sup>0</sup> M12ð Þ s, h<sup>0</sup> M21ð Þ s, h<sup>0</sup> M22ð Þ s, h<sup>0</sup>

" #

where J0ð Þ: and J1ð Þ: are Bessel functions and

Mechanics of Functionally Graded Materials and Structures

sM11ð Þ s, h<sup>0</sup> J0ð Þ st J0ð Þ sr dsdt þ

sM21ð Þ s, h<sup>0</sup> J0ð Þ st J1ð Þ sr dsdt þ

<sup>¼</sup> <sup>1</sup> 2μ<sup>0</sup>

B3 <sup>¼</sup> <sup>0100</sup> 1000 � �

unknown functions, p rð Þ and q rð Þ, into the range �a≤ r≤ 0 yields.

f g p tð Þj j<sup>t</sup> <sup>I</sup>11ð Þþ <sup>r</sup>, <sup>t</sup> q tð Þj j<sup>t</sup> <sup>I</sup>12ð Þ <sup>r</sup>, <sup>t</sup> dt <sup>þ</sup> <sup>α</sup><sup>1</sup>

f g q tð Þj j<sup>t</sup> <sup>I</sup>22ð Þþ <sup>r</sup>, <sup>t</sup> p tð Þj j<sup>t</sup> <sup>I</sup>21ð Þ <sup>r</sup>, <sup>t</sup> dt <sup>þ</sup> <sup>α</sup><sup>1</sup>

1 r

<sup>∂</sup>rur0ð Þ <sup>r</sup>, <sup>h</sup><sup>0</sup>

where Mijð Þ s, h<sup>0</sup> is the kernel function (see Ref. [13]).

sM11ð Þ s, h<sup>0</sup> sM12ð Þ s, h<sup>0</sup> sM21ð Þ s, h<sup>0</sup> sM22ð Þ s, h<sup>0</sup> � � B3M,

M ¼ ½ � T1ð Þ h<sup>0</sup> þ b<sup>1</sup> V<sup>1</sup>

Considering the asymptotic behavior of Bessel functions for large arguments

<sup>¼</sup> <sup>α</sup><sup>11</sup> <sup>α</sup><sup>12</sup> α<sup>21</sup> α<sup>22</sup> � �

Differentiation of Eq. (5) with respect to r and extension of the definition of the

ð a

0 q tð Þt ∞ð

ð a

0 q tð Þt ∞ð

0

0

, <sup>C</sup> <sup>¼</sup> 1 0 0 1 � �

¼

π ð a

π ð a

�a

�a

p tð Þ t � r

q tð Þ t � r

<sup>∂</sup><sup>r</sup> , Hið Þ¼ <sup>r</sup>, <sup>t</sup> ð Þ hið Þ� <sup>r</sup>, <sup>t</sup> <sup>1</sup> <sup>=</sup>ð Þ <sup>t</sup> � <sup>r</sup> ,

dt <sup>þ</sup> <sup>α</sup><sup>1</sup> π ð a

dt <sup>þ</sup> <sup>α</sup><sup>1</sup> π ð a

�a

�a

� � ½ � <sup>B</sup> ½ � <sup>T</sup>1ð Þ <sup>h</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup><sup>1</sup> <sup>V</sup><sup>1</sup> � � � � �<sup>1</sup>

:

1 � v μ0

1 � 2v 2μ<sup>0</sup>

1 � 2v 2μ<sup>0</sup>

3 7 7

5: (26)

p tð ÞH1ð Þ r, t dt

q tð ÞH2ð Þ r, t dt

(27a)

(27b)

1 � v μ0

sM12ð Þ s, h<sup>0</sup> J1ð Þ st J0ð Þ sr dsdt

sM22ð Þ s, h<sup>0</sup> J1ð Þ st J1ð Þ sr dsdt

(25a)

(25b)

,

0

ð a

0

[13], one may prove

lims!∞

m1ð Þ¼ r

m2ð Þ¼ r

where

104

m1ð Þ¼ r

1 2 ð a

1 2 ð a

�a

þ α2p rð Þ

<sup>∂</sup>uz0ð Þ <sup>r</sup>, <sup>h</sup><sup>0</sup>

<sup>∂</sup><sup>r</sup> , <sup>m</sup>2ð Þ¼ <sup>r</sup>

�a

� α2q rð Þ

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, uz0, along the coating interface, z = h0, is given by

$$
\mu\_{x0}(r) = \delta\_0 - r^2 / 2R \tag{31}
$$

The inner stick region, r≤ b, and outer slip annulus, b≤ r≤ a, are shown in Figure 5. According to Spence's work [18], the radial displacement along the coating interface in the stick region may be expressed as

$$u\_{r0}(r) = Cr^2, (r \le b) \tag{32}$$

where C denotes the slop of the relative radial displacement gradient and is an unknown constant. The Coulomb friction law is applied to describe the slip behavior in the slip region. Then, the radial shear traction in the contact region is represented as

$$q(r) = q\*(r) - fp(b)\frac{r}{b}, (r \le b),\tag{33a}$$

$$q(r) = -fp(r), (b \le r \le a). \tag{33b}$$

where f denotes the friction coefficient.

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:

$$\begin{aligned} &-a\_2q(r) + \frac{a\_1}{\pi} \int\_{-a}^{a} \frac{p(t)}{t-r} dt + \frac{a\_1}{\pi} \int\_{-a}^{a} p(t)H\_1(r,t)dt + \frac{1}{2} \int\_{-a}^{a} \{p(t)|t|I\_{11}(r,t) + q(t)|t|I\_{12}(r,t)\}dt\\ &= -r/R, \end{aligned}$$

(34a)

<sup>α</sup>2p rð Þþ <sup>α</sup><sup>1</sup>

¼ 3C rj j

equation (34).

Figure 6.

107

R=h<sup>0</sup> ¼ 10 and a=h<sup>0</sup> ¼ 0:2.

π ð a

�a

q tð Þ t � r

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

dition will be presented in this section.

coating surface and the gradient of the coating.

the coating varying in the following power law form:

dt <sup>þ</sup> <sup>α</sup><sup>1</sup> π ð a

�a

q tð ÞH2ð Þ r, t dt þ

Axisymmetric Indentation Response of Functionally Graded Material Coating

method (coupled solution) [13] are used to solve coupled singular integral

5. Indentation response of FGM coating under a spherical indenter

1 2 ð a

The Goodman approximate method (uncoupled solution) [21] and the iteration

The indentation response of FGM coating under frictionless and frictional con-

Firstly, the effects of the stiffness ratio μ0=μ ∗ on the distributions of the contact pressure and the relation between indentation and applied force are investigated for the frictionless contact problem. The exponential function model is applied to obtain the results shown in Figures 6 and 7 [7]. The distribution of the dimensionless contact pressure p rð Þ (a) and radial stress σrrð Þr (b) on the surface of FGM coating indented by a rigid spherical indenter for various stiffness ratio μ0=μ ∗ when R=h<sup>0</sup> ¼ 10 and a=h<sup>0</sup> ¼ 0:2 is shown in Figure 6. With the increase of μ0=μ ∗ , the contact pressure p rð Þ decreases. It can be observed that the tensile spike in the distribution of σrrð Þr as r ! a has clearly some implications regarding the initiation and subcritical growth of surface cracks. Figure 7 presented the relation of P vs. a and P vs. δ0. With the decrease of μ0=μ ∗ , the larger applied normal load is needed to create the same contact region (a) and the same maximum indentation depth δ<sup>0</sup> (b). The results give an indentation testing method to measure the stiffness of the

Secondly, the linear multi-layered model is used to model the shear modulus of

<sup>μ</sup>ð Þ¼ <sup>z</sup> <sup>μ</sup> <sup>∗</sup> <sup>þ</sup> ð Þ <sup>μ</sup><sup>0</sup> � <sup>μ</sup> <sup>∗</sup> ð Þ <sup>z</sup>=h<sup>0</sup> <sup>n</sup>

where n is a gradient index characterizing the gradual variation of the shear modulus. In the following calculation, the LML model divided the FGM coating into

Distribution of the dimensionless contact pressure p rð Þ (a) and radial stress σrrð Þr (b) on the surface of the graded coating loaded by a rigid spherical indenter for some selected values of the stiffness ratio μ0=μ ∗ with

�a

f g q tð Þj jt I22ð Þþ r, t p tð Þj jt I21ð Þ r, t dt

, (35)

(34b)

Figure 5. A functionally graded coated half-space indented by a spherical indenter.

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

$$\begin{split} \frac{1}{2}a\_{2}p(r) + \frac{a\_{1}}{\pi} \int\_{-a}^{a} \frac{q(t)}{t-r} dt + \frac{a\_{1}}{\pi} \int\_{-a}^{a} q(t)H\_{2}(r,t)dt + \frac{1}{2} \int\_{-a}^{a} \{q(t)|t|I\_{22}(r,t) + p(t)|t|I\_{21}(r,t)\}dt \\ = 3C|r| \end{split} \tag{34b}$$

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