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 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 coating surface and the gradient of the coating.

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

$$
\mu(\mathbf{z}) = \mu \ast + (\mu\_0 - \mu \ast)(\mathbf{z}/h\_0)^n,\tag{35}
$$

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

Figure 6.

The Gauss-Chebyshev integration formula [24] is applied to solve Eqs. (28) and

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

2

b

1 2 ð a

�a

q rð Þ¼�fp rð Þ,ð Þ b≤ r≤ a : (33b)

=2R (31)

,ð Þ r≤ b (32)

,ð Þ r≤ b , (33a)

f g p tð Þj jt I11ð Þþ r, t q tð Þj jt I12ð Þ r, t dt

(34a)

uz0ð Þ¼ r δ<sup>0</sup> � r

ur0ð Þ¼ <sup>r</sup> Cr<sup>2</sup>

ior in the slip region. Then, the radial shear traction in the contact region is

q rð Þ¼ <sup>q</sup> <sup>∗</sup> ð Þ� <sup>r</sup> fp bð Þ <sup>r</sup>

p tð ÞH1ð Þ r, t dt þ

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

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 behav-

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:

4.2 Partial slip contact problem with finite friction for FGM coating

(30) with the consideration of Eq. (29).

Mechanics of Functionally Graded Materials and Structures

interface in the stick region may be expressed as

where f denotes the friction coefficient.

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

�a

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

represented as

�α2q rð Þþ <sup>α</sup><sup>1</sup>

¼ �r=R,

Figure 5.

106

π ð a

�a

p tð Þ t � r

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

same maximum indentation depth δ<sup>0</sup> (b), the larger applied normal load is needed

Thirdly, the effect of the variation of Poisson's ratio on the frictionless contact

where v<sup>1</sup> and v ∗ are Poisson's ratio for the first layer and homogeneous halfspace. v <sup>j</sup> denotes Poisson's ratio in layer j. The contact pressure (a) and the relations of P vs. a (b) for the different variation forms of Poisson's ratio when v<sup>1</sup> = 1/3 and μ0=μ ∗ = 1/5 are given in Figure 10 [14]. It is assumed that Poisson's ratio for the FGM coating-substrate structure varies from 1/3 to 0.1 and varies from 1/3 to 0.5 according to Eq. (36). The results show that the variation of Poisson's ratio along the thickness has no significant impact on the contact pressure and the relation of force contact region in axisymmetric contact problem when the Poisson's ratio at the upper surface of coating is fixed. Figure 11 presented the effect of value of Poisson's ratio on the contact pressure (a) and the relation of P vs. a (b) when Poisson's ratio in the coating-substrate structure is a constant (v <sup>j</sup> ¼ v) as shown in [14]. We can observe that the value of Poisson's ratio has a significant effect on the contact

The contact pressure (a) and the relations of P vs. a (b) for the different variation form of the Poisson's ratio

The effect of the value of Poisson's ratio on the contact pressure (a) and on the relation of P vs. a (b) with

ð Þ v ∗ � v<sup>1</sup> , j ¼ 1, 2, ……N (36)

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

<sup>v</sup> <sup>j</sup> <sup>¼</sup> <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>j</sup> � <sup>1</sup>

N � 1

Axisymmetric Indentation Response of Functionally Graded Material Coating

for larger values of n.

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

thickness as follows

Figure 10.

Figure 11.

109

μ0=μ ∗ = 1/5 while n = 0.2.

with v1 = 1/3 and μ0=μ ∗ = 1/5.

Figure 7.

Relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for some selected values of the stiffness ratio μ0=μ ∗ with R=h0 ¼ 10.

#### Figure 8.

Distribution of the dimensionless contact pressure σzzð Þr (a) and radial stress σrrð Þr (b) for some selected values n with a=h ¼ 0:1 and R=h0 ¼ 10.

six sub-layers. The axisymmetric indentation response for the frictionless contact under the spherical indenter is considered.

Figure 8 shows the distributions of the contact pressure for some selected values of n with μ0=μ ∗ = 1/8 and a=h<sup>0</sup> ¼ 0:1 [12]. With the increase of n, the contact pressure obviously increases. This behavior shows that the contact traction can be improved by adjusting the gradient of the coating when the stiffness of the coating surface keeps unchanged. When the FGM coating is indented by a conical indenter, the relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for some selected values of n with μ0=μ ∗ = 1/8 are shown in Figure 9 [12]. To create the same contact region and the

Figure 9. Relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for selected values of n with R=h ¼ 10.

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

same maximum indentation depth δ<sup>0</sup> (b), the larger applied normal load is needed for larger values of 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

$$v\_j = v\_1 + \frac{j-1}{N-1}(v\*-v\_1), j = 1,2,...,N\tag{36}$$

where v<sup>1</sup> and v ∗ are Poisson's ratio for the first layer and homogeneous halfspace. v <sup>j</sup> denotes Poisson's ratio in layer j. The contact pressure (a) and the relations of P vs. a (b) for the different variation forms of Poisson's ratio when v<sup>1</sup> = 1/3 and μ0=μ ∗ = 1/5 are given in Figure 10 [14]. It is assumed that Poisson's ratio for the FGM coating-substrate structure varies from 1/3 to 0.1 and varies from 1/3 to 0.5 according to Eq. (36). The results show that the variation of Poisson's ratio along the thickness has no significant impact on the contact pressure and the relation of force contact region in axisymmetric contact problem when the Poisson's ratio at the upper surface of coating is fixed. Figure 11 presented the effect of value of Poisson's ratio on the contact pressure (a) and the relation of P vs. a (b) when Poisson's ratio in the coating-substrate structure is a constant (v <sup>j</sup> ¼ v) as shown in [14]. We can observe that the value of Poisson's ratio has a significant effect on the contact

#### Figure 10.

six sub-layers. The axisymmetric indentation response for the frictionless contact

Distribution of the dimensionless contact pressure σzzð Þr (a) and radial stress σrrð Þr (b) for some selected values

Relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for some selected values of the stiffness ratio μ0=μ ∗ with R=h0 ¼ 10.

of n with μ0=μ ∗ = 1/8 and a=h<sup>0</sup> ¼ 0:1 [12]. With the increase of n, the contact pressure obviously increases. This behavior shows that the contact traction can be improved by adjusting the gradient of the coating when the stiffness of the coating surface keeps unchanged. When the FGM coating is indented by a conical indenter, the relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for some selected values of n with μ0=μ ∗ = 1/8 are shown in Figure 9 [12]. To create the same contact region and the

Relations of P vs. a (a) and P vs. δ<sup>0</sup> (b) for selected values of n with R=h ¼ 10.

Figure 8 shows the distributions of the contact pressure for some selected values

under the spherical indenter is considered.

Mechanics of Functionally Graded Materials and Structures

n with a=h ¼ 0:1 and R=h0 ¼ 10.

Figure 7.

Figure 8.

Figure 9.

108

The contact pressure (a) and the relations of P vs. a (b) for the different variation form of the Poisson's ratio with v1 = 1/3 and μ0=μ ∗ = 1/5.

#### Figure 11.

The effect of the value of Poisson's ratio on the contact pressure (a) and on the relation of P vs. a (b) with μ0=μ ∗ = 1/5 while n = 0.2.

contact traction by adjusting the gradient of the coating while remaining the shear

Axisymmetric Indentation Response of Functionally Graded Material Coating

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.

The support from the National Natural Science Foundation of China under

Department of Mechanics, Inner Mongolia University of Technology, Hohhot,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: liutiejun6204@163.com

provided the original work is properly cited.

Grant Nos. 11662011 and 11811530067 are gratefully acknowledged.

modulus of the coating surface unchanged.

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

6. Conclusions

Acknowledgements

Author details

Tie-Jun Liu

China

111

Figure 12.

Contact traction distributions for selected values of the shear modulus ratio μ0=μ ∗ with P=μ ∗ h0 <sup>2</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>4</sup> and f ¼ 0:16: (a) p rð Þ and (b) q rð Þ.

pressure. While the values of v obviously increase, the contact pressure is observed. The results also show that the larger applied normal load is needed to create the same contact region and the same maximum indentation depth δ<sup>0</sup> for larger values of v.

Finally, the axisymmetric contact problem of a functionally graded coated halfspace 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 μ0=μ ∗ with P=μ ∗ h<sup>0</sup> <sup>2</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>4</sup> and <sup>f</sup> <sup>¼</sup> <sup>0</sup>:16 are shown in Figure 12 [14]. The solid lines correspond to the uncoupled solution, and the scatter symbols correspond to the coupled solution. We can observe that consideration of the coupling between the normal and tangential tractions may result in the increase of the peak contact tractions but slight decrease of the contact tractions near the edges of the contact region for a given shear modulus ratio μ0=μ ∗ . With the increase of μ0=μ ∗ , the peak normal and tangential contact tractions increase. Figure 12b also shows that the stick region and the contact radius decrease with the increase of μ0=μ ∗ . This behavior provides a way for us to change the distribution of the contact pressure by adjusting the stiffness of the coating surface. Figure 13 presents the effects of n on the contact traction distributions with P=μ ∗ h<sup>0</sup> <sup>2</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>4</sup> and <sup>f</sup> <sup>¼</sup> <sup>0</sup>:16 [14]. With the increase of <sup>n</sup>, the peak normal traction (Figure 13a) increases, and the peak tangential traction (Figure 13b) decreases. This behavior provides a way for us to change the distribution of the

Figure 13. Contact traction distributions for selected values of n with P=μ ∗ h0 <sup>2</sup> <sup>¼</sup> <sup>4</sup> � <sup>10</sup>�<sup>4</sup> and f <sup>¼</sup> <sup>0</sup>:16: (a) p rð Þ and (b) q rð Þ.

contact traction by adjusting the gradient of the coating while remaining the shear modulus of the coating surface unchanged.
