Open access peer-reviewed chapter - ONLINE FIRST

Axisymmetric Indentation Response of Functionally Graded Material Coating

By Tie-Jun Liu

Submitted: May 20th 2019Reviewed: August 22nd 2019Published: September 30th 2019

DOI: 10.5772/intechopen.89312

Downloaded: 30

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]:

εrr=ur,E1a
εθθ=ur,E1b
εzz=wz,E1c
2εrz=uz+wr.E1d
σrr=λz+2μzur+λzur+wz,E2a
σθθ=λz+2μzur+λzur+wz,E2b
σzz=λz+2μzwz+λzur+ur,E2c
σrz=μzuz+wr.E2d
σrrr+σrzz+1rσrrσθθ=0,E3a
σrzr+σzzz+1rσrz=0.E3b

in which rand zare the variables of the cylindrical coordinate system; εrr, εθθ, εzz, and εrzare the strain components; uand ware the displacement components in the radial and axial directions; σrr, σθθ, σzz, and σrzare the stress components; λzand μzare Lame’s constants which vary along the z-axis direction.

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

Figure 1.

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

μz=μ0eαzE4

where α=h01logμ0/μis a constant characterizing the material inhomogeneity with μ0being the value of μzat the surface, i.e., μ0=μh0. μ0and μare related by

μ0=μeαh0E5

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

k+12ur2+1rur1r2u+2wrz+k1αuz+wr+k12uz22wrz=0,E6a
k+12urz+1ruz+2wz2k12urz2wr2k1ruzwr+3kαur+ur+k+1αwz=0E6b

where k=34νand νis 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

ω˜szp=0ωszrJpsrdr,E7a
ωprz=0ω˜szpsJpsrdsE7b

where the bar ∼ indicates Hankel transform; pis the pth-order Hankel transform; and Jpis the pth-order Bessel function of the first kind.

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

k1D2+αk1Dk+1s2u˜12sD+αsk1w˜0=0,E8a
2sD+α3ksu˜1+k+1D2+αk+1Dk1s2w˜0=0,E8b

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

u˜1=A3sem1z+A4sem2z+A5sem3z+A6sem4zE9a
w˜0=A3sa1em1z+A4sa2em2z+A5sa3em3z+A6sa4em4zE9b

where

ai=2smi+3kk+1mi2+αk+1mik1s2,i=1..4

and

m1=α2+12α2+4s2+i43kk+1αs1/2,m2=α212α2+4s2+i43kk+1αs1/2,
m3=α2+12α2+4s2i43kk+1αs1/2,m4=α212α2+4s2i43kk+1αs1/2.

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

κ1μzσ˜zz0=i=14κ+1miai+3ksAi+2emizE10a
1μzσ˜rz1=i=14miaisAi+2emizE10b

For a homogeneous layer without the gradient, the gradient index αin 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, μz, with boundary values μh0=μ0. Poisson’s ratio vis taken as 1/3. The linear multilayered model divides functionally graded coating into Nsub-layers. The shear modulus μzin each sub-layer is assumed to take the following form:

μzμjz=cj1+z/bj=cjzbj,hjzhj1,j=1,2NE11

where z=z+band μjis equal to the real value of the shear modulus at the sub-interfaces, z=hj, i.e., μjhj=μhj, which lead to

bj=μj1hjμjhj1μjμj1,E12a
cj=μj1+hj/bj.E12b

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

uj=z+bjfj/r+ϕj/rE13a
wj=z+bjfj/zfj+ϕj/z,hjzhj1.E13b

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

2ϕj+1zϕj/z=0,E14a
2fj+1zfj/z=12z2ϕj/z.E14b

where

2=1rr/r/r+2/z2.

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

uj=zfj/r+ϕj/r,hjzhj1E15a
wj=zfj/zfj+ϕj/z,hjzhj1E15b
σrrj=2cjzbjz2fj/r2+2ϕj/r2fj/z12zϕj/z,E16a
σθθj=2cjzbjz1rfj/r+1rϕj/rfj/z12zϕj/z,E16b
σzzj=2cjzbjz2fj/r2+2ϕj/z2fj/z12zϕj/z,E16c
σrzj=2cjzbjz2fj/rz+2ϕj/rz.E16d

Applying Hankel transformation Eqs. (7a)–(14), we obtain the solutions for displacement functions ϕjand fjin each sub-layer:

ϕ˜j0=Aj1sI0sz+Aj2sK0sz,E17a
f˜j0=Aj3sI0sz+Aj4sK0sz+s2Aj1sI1szAj2sK1szE17b

where I0, I1, K0, and K1are modified Bessel functions of the 0th and 1th order.

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

u˜rj1=szf˜j0sϕ˜j0=s22zI1szsI0szAj1s+s22zK1szsK0szAj2sszI0szAj3sszK0szAj4sE18a
w˜j0=zdf˜j0/dzf˜j0+dϕ˜j0/dz=s22zI0szAj1s+s22zK0szAj2s+zsI1szI0szAj3s+zsK1szK0szAj4sE18b
σ˜zzj0=2cjzbj{s32zI1szAj1ss32zK1szAj2s+zs2I0sz2sI1szAj3s+zs2K0sz+2sK1szAj4s}E18c
σ˜rzj1=2cjzbj{s32zI0szs22I1szAj1s+s32zK0sz+s22K1szAj2ss2zI1szAj3s+s2zK1szAj4s}E18d

3.3 Piece wise exponential multi-layered model

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

Figure 2.

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

μzμjz=ajebjz,hjzhj1,j=1,2,NE19a
μhj=μjhj,E19b

in which:

aj=μhjelnμhj+1/μhjhj/hj+1hj,bj=lnμhj+1/μhj/hj+1hj

and hjis the zcoordinate at the end of layer j. Poisson’s ratio in each sub-layer is assumed to be a constant vj.

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

kj+12ujr2+1rujr1r2uj+2wjrz+kj1bjujz+wjr+kj12ujz22wjrz=0,E20a
kj+12ujrz+1rujz+2wjz2kj1bj2ujrz2wjr2kj1rujzwjr+3kbjujr+ujr+kj+1bjwjz=0E20b

where ujand wjare the displacement components in the radial and zaxial directions in layer jand kj=34vj.

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

u˜jsz1=Aj1semj1z+Aj2semj2z+Aj3semj3z+Aj4semj4zE21a
w˜jsz0=Aj1scj1emj1z+Aj2scj2emj2z+Aj3scj3emj3z+Aj4scj4emj4zE21b

where Aj1Aj4are unknown constants to be solved in layer j.

cji=2smji+sbj3kjkj+1mji2+bjk+1mjikj1s2,i=1234
mj1=bj2+12bj2+4s2+i43kjkj+1bjs1/2,mj2=bj212bj2+4s2+i43kjkj+1bjs1/2,
mj3=bj2+12bj2+4s2i43kjkj+1bjs1/2,mj4=bj212bj2+4s2i43kjkj+1bjs1/2.

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

kj1μjzσ˜zzjsz0=i=14kj+1mjicji+3ksAjiemjizE22a
1μjzσ˜rzjsz1=i=14mjicjisAjiemjizE22b

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=0, which state.

Figure 3.

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

u2r0u1r0=0,E23a
w2r0w1r0=0,E23b
σ2zzr0σ1zzr0=0,E23c
σ2rzr0σ1rzr0=0.E23d

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

σ1zzrh0=pr0ra,E24a
σ1zzrh0=0a<r<,E24b
σ1rzrh0=qr0ra,E24c
σ1rzrh0=0a<r<E24d

in which i=1refers to the graded coating and i=2refers to the homogeneous half-space. prand qrare 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

w0r=0aptt0sM11sh0J0stJ0srdsdt+0aqtt0sM12sh0J1stJ0srdsdtE25a
u0r=0aptt0sM21sh0J0stJ1srdsdt+0aqtt0sM22sh0J1stJ1srdsdtE25b

where J0.and J1.are Bessel functions and

M11sh0M12sh0M21sh0M22sh0=12μ0B3M,M=T1h0+b1V¯1BT1h0+b1V¯11,
B3=01001000,C=1001.

where Mijsh0is the kernel function (see Ref. [13]).

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

limssM11sh0sM12sh0sM21sh0sM22sh0=α11α12α21α22=1vμ012v2μ012v2μ01vμ0.E26

Differentiation of Eq. (5) with respect to rand extension of the definition of the unknown functions, prand qr, into the range ar0yields.

m1r=12aapttI11rt+qttI12rtdt+α1πaapttrdt+α1πaaptH1rtdtα2qrE27a
m2r=12aaqttI22rt+pttI21rtdt+α1πaaqttrdt+α1πaaqtH2rtdt+α2prE27b

where

m1r=uz0rh0r,m2r=1rrur0rh0r,Hirt=hirt1/tr,
Iijrt=1i0sMijsh0αijsJ2isrJj1stds,i=12j=12,
h1rt=t/rEt/r,(t<r)t2/r2Er/tt2r2/r2Kr/t,(t>r),
h2rt=t2r2/trKt/r+r/tEt/r,(t<r)Er/t,(t>r),

with K.and E.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=πaapttdtE28

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 Pis acted on the rigid spherical punch along the z-direction to form an indent depth δ0and a circular contact region with a radius a. The displacement boundary condition in the contact region is expressed as

Figure 4.

FGM coating indented by a spherical indenter.

wrh0=δ0r2/2R0raE29

Because the frictionless contact is considered, the shear traction qris zero, and the controlling equation is

m1r=12aapttI11rtdt+α1πaapttrdt+α1πaaptH1rtdtE30

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

Figure 5.

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

uz0r=δ0r2/2RE31

The inner stick region, rb, and outer slip annulus, bra, 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

ur0r=Cr2,rbE32

where Cdenotes 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

qr=qrfpbrb,rb,E33a
qr=fpr,bra.E33b

where fdenotes 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:

α2qr+α1πaapttrdt+α1πaaptH1rtdt+12aapttI11rt+qttI12rtdt=r/R,E34a
α2pr+α1πaaqttrdt+α1πaaqtH2rtdt+12aaqttI22rt+pttI21rtdt=3CrE34b

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 μ0/μon 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 pr(a) and radial stress σrrr(b) on the surface of FGM coating indented by a rigid spherical indenter for various stiffness ratio μ0/μwhen R/h0=10and a/h0=0.2is shown in Figure 6. With the increase of μ0/μ, the contact pressure prdecreases. It can be observed that the tensile spike in the distribution of σrrras rahas clearly some implications regarding the initiation and subcritical growth of surface cracks. Figure 7 presented the relation of Pvs. aand Pvs. 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 δ0(b). The results give an indentation testing method to measure the stiffness of the coating surface and the gradient of the coating.

Figure 6.

Distribution of the dimensionless contact pressure p r (a) and radial stress σ rr r (b) on the surface of the graded coating loaded by a rigid spherical indenter for some selected values of the stiffness ratio μ 0 / μ ∗ with R / h 0 = 10 and a / h 0 = 0.2 .

Figure 7.

Relations of P vs. a (a) and P vs. δ 0 (b) for some selected values of the stiffness ratio μ 0 / μ ∗ with R / h 0 = 10 .

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

μz=μ+μ0μz/h0n,E35

where nis a gradient index characterizing the gradual variation of the shear modulus. In the following calculation, the LML model divided the FGM coating into 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/h0=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 Pvs. a(a) and Pvs. δ0(b) for some selected values of nwith μ0/μ = 1/8 are shown in Figure 9 [12]. To create the same contact region and the same maximum indentation depth δ0(b), the larger applied normal load is needed for larger values of n.

Figure 8.

Distribution of the dimensionless contact pressure σ zz r (a) and radial stress σ rr r (b) for some selected values n with a / h = 0.1 and R / h 0 = 10 .

Figure 9.

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

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

vj=v1+j1N1vv1,j=1,2,NE36

where v1and vare Poisson’s ratio for the first layer and homogeneous half-space. vjdenotes Poisson’s ratio in layer j. The contact pressure (a) and the relations of Pvs. a(b) for the different variation forms of Poisson’s ratio when v1 = 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 Pvs. a(b) when Poisson’s ratio in the coating-substrate structure is a constant (vj=v) as shown in [14]. We can observe that the value of Poisson’s ratio has a significant effect on the contact pressure. While the values of vobviously 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 δ0for larger values of v.

Figure 10.

The contact pressure (a) and the relations of P vs. a (b) for the different variation form of the Poisson’s ratio with v 1  = 1/3 and μ 0 / 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 μ 0 / μ ∗  = 1/5 while n  = 0.2.

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 μ0/μwith P/μh02=4×104and f=0.16are 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 non the contact traction distributions with P/μh02=4×104and f=0.16[14]. With the increase of n, 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 contact traction by adjusting the gradient of the coating while remaining the shear modulus of the coating surface unchanged.

Figure 12.

Contact traction distributions for selected values of the shear modulus ratio μ 0 / μ ∗ with P / μ ∗ h 0 2 = 4 × 10 − 4 and f = 0.16 : (a) p r and (b) q r .

Figure 13.

Contact traction distributions for selected values of n with P / μ ∗ h 0 2   =   4 × 10 − 4 and f = 0.16 : (a) p r and (b) q r .

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.

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Tie-Jun Liu (September 30th 2019). Axisymmetric Indentation Response of Functionally Graded Material Coating [Online First], IntechOpen, DOI: 10.5772/intechopen.89312. Available from:

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