Analysis of axisymmetric indentation of functionally graded piezoelectric coating or substrate systems under an insulator indenter

Abstract

Axisymmetric indentation problem of a rigid perfect electrical insulator indenter on a functionally graded piezoelectric coating bonded to a piezoelectric substrate is investigated. The electromechanical properties of the functionally graded piezoelectric coating are assumed to vary as an exponential function along the thickness direction. The technique of the Hankel integral transform is applied to reduce the indentation problem to a singular integral equation. The numerical methods are developed and applied to compute the contact pressure for a cylindrical indenter and a spherical indenter. The effects of the material property gradient on the contact pressure, contact region, indentation depth, and electrical potential are analyzed. The numerical results are also obtained for the indentation responses of three different piezoelectric substrates.

Keywords

functionally graded piezoelectric materials coating indentation

Introduction

Piezoelectric materials are a kind of smart materials which are widely used as sensors, actuators, and transducers in many engineering fields. In order to gain the piezoelectric material with high strength, high toughness, and low thermal expansion, the functionally graded piezoelectric material (FGPM; Wu et al., 1996) was developed based on the concept of functionally graded material (FGM) which requires the continuous and gradual change in the material properties. Recently, Suresh (2001) proved that an FGM coating can alter the damage and failure of a surface to normal and sliding contact. The results offer opportunities for the design of novel surfaces with high resistances to contact deformation and damage. Using the analytical method, Guler and Erdogan (2004) proved that the appropriate gradual variation of the shear modulus can significantly alter the contact tractions. The contact problems of FGM coating with the different variations in the material properties were considered by Ke and Wang (2006) and Liu et al. (2008). Chen and Chen (2013) and Chen et al. (2015) solved the frictional and partial sliding contact problem with shear modulus gradient variation in an arbitrary direction. These articles which related to the FGM will provide guidance for studying the contact problems of FGPMs.

Many researchers have focused on the indentation problem of piezoelectric materials in the past few years. Giannakopoulos and Suresh (1999) developed the general theory for electromechanical response of a piezoelectric half-space indented by an axisymmetric indenter of spherical, conical, and cylindrical punch geometries. Experimental studies of mechanical and electrical responses of piezoelectric materials subjected to shallow cone indentation were conducted by Sridhar et al. (2000). Giannakopoulos (2000) considered the indentation of a transversely isotropic and linear piezoelectric half-space for a rigid spherical indenter using the theoretical and computational methods. Chen et al. (1999) and Chen (2000) applied the potential theory to solve the contact problem of piezoelectric materials. Wang (2004) and Wang and Han (2006) investigated the piezoelectric layer indented by conducting and insulating indenters. They established the expressions for the singular mechanical and electric fields near the indenter front. Using the technique of Hankel transformation, Wang et al. (2008) and Wang and Chen (2011) obtained the closed-form solutions for the frictionless indentation response of a piezoelectric film or rigid substrate system under three kinds of axisymmetric indenters. Most of the previous works did not consider the indentation problem of FGPM. Recently, Ke et al. (2008a, 2008b, 2010) dealt with the two-dimensional (2D) contact problem of an FGPM layered half-plane with an exponential variation of the material properties along the thickness direction using the technique of singular integral equation. To the authors’ knowledge, no three-dimensional (3D) axisymmetric indentation analysis of the FGPM coating or substrate system has been yet reported in the literature.

In this study, the axisymmetric indentation problem of a FGPM coating or substrate system which is indented by a rigid perfect electrical insulator indenter is considered. The elastic, piezoelectric, and dielectric constants of the FGPM coating are assumed to vary continuously along the thickness direction. In order to solve the problem in an analytical way, the gradients for all material constants are assumed to follow the same exponential rule. The contact problem is reduced to a singular integral equation by the method of Hankel integral transform. Numerical calculations are carried out to compute the contact pressure for a cylindrical indenter and a spherical indenter. The effects of the material gradation on the contact pressure, contact region, indentation depth, and electric potential are investigated. The electrical and mechanical responses for three different piezoelectric substrates are analyzed.

Formulation of the problem

As shown in Figure 1, an FGPM coating of thickness $h_{0}$ , which is bonded to a piezoelectric substrate, is loaded by a rigid axisymmetric indenter under an applied normal load P. Because the thickness of the substrate is much thicker compared to the FGPM coating, the substrate is treated as a half-space. The cylindrical coordinate system ( $r, θ, z$ ), where the origin o is located at the center of the interface between the coating and the half-space, is employed with the z-axis being along the thickness direction and pointing upward. The axisymmetric indenter is assumed to be a perfect insulating body with no electric charge distribution. The radius of the contact region is denoted by a. The substrate (half-space) is a transversely isotropic piezoelectric material with the z-axis being parallel to the poling direction. The material property of the FGPM coating is assumed to vary exponentially along the z-axis. Then, the elastic constants $c_{ij} (z)$ , the piezoelectric constants $e_{ij} (z)$ , and the dielectric constants $\in_{ii} (z)$ have the following form

{c_{kl} (z), e_{kl} (z), \in_{kl} (z)} = {c_{kl 0}, e_{kl 0}, \in_{kk 0}} e^{β z} (0 \leq z \leq h)

(1)

where $β$ is the gradient index, and $c_{kl 0}$ , $e_{kl 0}$ , and $\in_{kl 0}$ are the material properties of the piezoelectric substrate.

Figure 1.

Functionally graded piezoelectric coating or substrate system indented by a perfect electrical insulator indenter.

General solution of axisymmetric indentation of FGPM coating or substrate system

The equilibrium equations in the absence of body forces and the geometric equations for the axisymmetric problem can be expressed as (Liu et al., 2008)

\frac{\partial σ_{rrj}}{\partial r} + \frac{\partial σ_{rzj}}{\partial z} + \frac{σ_{rrj} - σ_{θ θ j}}{r} = 0

(2a)

\frac{\partial σ_{rzj}}{\partial r} + \frac{\partial σ_{zzj}}{\partial z} + \frac{σ_{rzj}}{r} = 0

(2b)

ε_{rrj} = \frac{\partial u_{rj}}{\partial r}

(2c)

ε_{θ θ j} = \frac{u_{rj}}{r}

(2d)

ε_{zzj} = \frac{\partial u_{zj}}{\partial z}

(2e)

γ_{rzj} = \frac{\partial u_{rj}}{\partial z} + \frac{\partial u_{zj}}{\partial r} (j = 1, 2)

(2f)

where $σ_{klj} (r, z)$ and $ε_{klj} (r, z)$ with $(k, l) = (r, θ, z)$ are the stress and strain components, $u_{rj}$ and $u_{zj}$ are the radial and axial components of the displacement vector, $j = 1$ refers to the FGPM coating, and $j = 2$ with β = 0 to the piezoelectric half-space, respectively.

The Maxwell electrostatic equation in the absence of body charge is given by (Giannakopoulos and Suresh, 1999)

\frac{\partial D_{rj}}{\partial r} + \frac{D_{rj}}{r} + \frac{\partial D_{zj}}{\partial z} = 0

(3)

where $D_{rj} (r, z)$ and $D_{zj} (r, z)$ are the electric displacement components, respectively.

The Gauss equations which describe the relation between the electric field and the electric potential are given by

E_{rj} = - \frac{\partial ϕ_{j}}{\partial r}

(4a)

E_{zj} = - \frac{\partial ϕ_{j}}{\partial z}

(4b)

where $E_{rj}$ and $E_{zj}$ are the electric flux components and $ϕ_{j} (r, z)$ is the electric potential.

The constitutive equations for the piezoelectric materials can be written as (Ueda, 2008)

σ_{rrj} = c_{11} (z) ε_{rrj} + c_{12} (z) ε_{θ θ j} + c_{13} (z) ε_{zzj} - e_{31} (z) E_{zj}

(5a)

σ_{θ θ j} = c_{12} (z) ε_{rrj} + c_{11} (z) ε_{θ θ j} + c_{13} (z) ε_{zzj} - e_{31} (z) E_{zj}

(5b)

σ_{zzj} = c_{13} (z) (ε_{rrj} + ε_{θ θ j}) + c_{33} (z) ε_{zzj} - e_{33} (z) E_{zj}

(5c)

σ_{rzj} = c_{44} (z) γ_{rzj} - e_{15} (z) E_{rj}

(5d)

The relations between the electric displacements and the electric fluxes are

D_{rj} = e_{15} (z) γ_{rzj} + \in_{11} (z) E_{rj}

(6a)

D_{zj} = e_{31} (z) (ε_{rrj} + ε_{θ θ j}) + e_{33} (z) ε_{zzj} + \in_{33} (z) E_{zj}

(6b)

The governing equations for the electromechanical fields of the FGPM coating can be written as (Ueda, 2010)

\begin{matrix} c_{110} (\frac{\partial^{2} u_{rj}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{rj}}{\partial r} - \frac{u_{rj}}{r^{2}}) + c_{440} \frac{\partial^{2} u_{rj}}{\partial z^{2}} \\ + (c_{130} + c_{440}) \frac{\partial^{2} u_{zj}}{\partial r \partial z} + (e_{310} + e_{150}) \frac{\partial^{2} ϕ_{j}}{\partial r \partial z} \\ + β {c_{440} (\frac{\partial u_{rj}}{\partial z} + \frac{\partial u_{zj}}{\partial r}) + e_{150} \frac{\partial ϕ_{j}}{\partial r}} = 0 \end{matrix}

(7a)

\begin{matrix} c_{440} (\frac{\partial^{2} u_{zj}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{zj}}{\partial r}) + c_{330} \frac{\partial^{2} u_{zj}}{\partial z^{2}} + (c_{130} + c_{440}) \\ \frac{\partial}{\partial z} (\frac{\partial u_{rj}}{\partial r} + \frac{u_{rj}}{r}) + e_{150} (\frac{\partial^{2} ϕ_{j}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{j}}{\partial r}) + e_{330} \frac{\partial^{2} ϕ_{j}}{\partial z^{2}} \\ + β {c_{130} (\frac{\partial u_{rj}}{\partial r} + \frac{u_{rj}}{r}) + c_{330} \frac{\partial u_{zj}}{\partial z} + e_{330} \frac{\partial ϕ_{j}}{\partial z}} = 0 \end{matrix}

(7b)

\begin{matrix} e_{150} (\frac{\partial^{2} u_{zj}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{zj}}{\partial r}) + e_{330} \frac{\partial^{2} u_{zj}}{\partial z^{2}} \\ + (e_{150} + e_{310}) \frac{\partial}{\partial z} (\frac{\partial u_{rj}}{\partial r} + \frac{u_{rj}}{r}) \\ - \in_{110} (\frac{\partial^{2} ϕ_{j}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{j}}{\partial r}) - \in_{330} \frac{\partial^{2} ϕ_{j}}{\partial z^{2}} \\ + β {e_{310} (\frac{\partial u_{rj}}{\partial r} + \frac{u_{rj}}{r}) + e_{330} \frac{\partial u_{zj}}{\partial z} - \in_{330} \frac{\partial ϕ_{j}}{\partial z}} = 0 \end{matrix}

(7c)

Using the technique of Hankel integral transform, the solution of the differential equation (7) for the FGPM coating ( $j = 1$ ) is given by (Ueda, 2010)

{〈 σ_{zz 1} (r, z) 〉}_{0} = \exp (β z) \sum_{i = 1}^{6} [p_{1 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8a)

{〈 σ_{zr 1} (r, z) 〉}_{1} = \exp (β z) \sum_{i = 1}^{6} [p_{2 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8b)

{〈 D_{z 1} (r, z) 〉}_{0} = \exp (β z) \sum_{i = 1}^{6} [p_{3 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8c)

{〈 u_{z 1} (r, z) 〉}_{0} = \sum_{i = 1}^{6} [p_{4 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8d)

{〈 u_{r 1} (r, z) 〉}_{1} = \sum_{i = 1}^{6} [p_{5 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8e)

{〈 ϕ_{1} (r, z) 〉}_{0} = - \sum_{i = 1}^{6} [p_{6 i 1} (s) A_{i 1} (s) \exp (s γ_{i 1} z)]

(8f)

where $A_{i 1} (s)$ $(i = 1, 2, \dots, 6)$ are the unknown functions to be determined, the functions $γ_{i 1} = γ_{i 1} (s)$ and $p_{ki 1} (s)$ $(i, k = 1, 2, \dots, 6)$ are given in Appendix 1, and the symbols ${〈〉}_{0}$ and ${〈〉}_{1}$ denote the zeroth- and first-order Hankel transforms.

For the piezoelectric half-space ( $j = 2$ with β = 0), the stresses, electric displacements, mechanical displacements, and electric potential are given by

{〈 σ_{zz 2} (r, z) 〉}_{0} = \sum_{i = 4}^{6} [p_{1 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9a)

{〈 σ_{zr 2} (r, z) 〉}_{1} = \sum_{i = 4}^{6} [p_{2 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9b)

{〈 D_{z 2} (r, z) 〉}_{0} = \sum_{i = 4}^{6} [p_{3 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9c)

{〈 u_{z 2} (r, z) 〉}_{0} = \sum_{i = 4}^{6} [p_{4 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9d)

{〈 u_{r 2} (r, z) 〉}_{1} = \sum_{i = 4}^{6} [p_{5 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9e)

{〈 ϕ_{2} (r, z) 〉}_{0} = - \sum_{i = 4}^{6} [p_{6 i 2} (s) A_{i 2} (s) \exp (s γ_{i 2} z)]

(9f)

where $A_{i 2} (s)$ $(i = 4, 5, 6)$ are the unknown functions to be calculated. Considering the regularity conditions at infinity, that is, $u_{z 2} (r, z)$ , $u_{r 2} (r, z)$ , and $ϕ_{2} (r, z)$ $\to 0$ when $z \to - \infty$ , the characteristic roots $γ_{i 2} = γ_{i 2} (s)$ (given in Appendix 1) with negative real parts should be removed (Wang and Han, 2006). The functions $p_{ki 2} (s)$ $(i = 4, \dots, 6, k = 1, 2, \dots, 6)$ are given in Appendix 1.

Because the FGPM coating is perfectly bonded to the piezoelectric half-space (substrate), the continuity conditions at the interfaces $z = 0$ are given by

σ_{zz 1} (r, 0) = σ_{zz 2} (r, 0)

(10a)

σ_{zr 1} (r, 0) = σ_{zr 2} (r, 0)

(10b)

D_{z 1} (r, 0) = D_{z 2} (r, 0)

(10c)

u_{z 1} (r, 0) = u_{z 2} (r, 0)

(10d)

u_{r 1} (r, 0) = u_{r 2} (r, 0)

(10e)

ϕ_{1} (r, 0) = ϕ_{2} (r, 0)

(10f)

And along the coating surface $z = h_{0}$ , we have

σ_{zz 1} (r, h_{0}) = p (r) (0 \leq r \leq a)

(11a)

σ_{rz 1} (r, h) = 0 (r > 0)

(11b)

D_{z 1} (r, h) = 0

(11c)

where $p (r)$ is the unknown stress distribution and the indenter is electrically insulated with the coating.

In the transformed domain, the boundary conditions (10) and (11) may be written as

T_{1} A_{1} = T_{2} A_{2}

(12)

S = T_{3} A_{1}

(13)

where

\begin{matrix} T_{1} = \\ [\begin{matrix} p_{111} (s) & p_{121} (s) & p_{131} (s) & p_{141} (s) & p_{151} (s) & p_{161} (s) \\ p_{211} (s) & p_{221} (s) & p_{231} (s) & p_{241} (s) & p_{251} (s) & p_{261} (s) \\ p_{311} (s) & p_{321} (s) & p_{331} (s) & p_{341} (s) & p_{351} (s) & p_{361} (s) \\ p_{411} (s) & p_{421} (s) & p_{431} (s) & p_{441} (s) & p_{451} (s) & p_{461} (s) \\ p_{511} (s) & p_{521} (s) & p_{531} (s) & p_{541} (s) & p_{551} (s) & p_{561} (s) \\ - p_{611} (s) & - p_{621} (s) & - p_{631} (s) & - p_{641} (s) & - p_{651} (s) & - p_{661} (s) \end{matrix}] \end{matrix}

(14a)

T_{2} = [\begin{matrix} p_{142} (s) & p_{152} (s) & p_{162} (s) \\ p_{242} (s) & p_{252} (s) & p_{262} (s) \\ p_{342} (s) & p_{352} (s) & p_{362} (s) \\ p_{442} (s) & p_{452} (s) & p_{462} (s) \\ p_{542} (s) & p_{552} (s) & p_{562} (s) \\ - p_{642} (s) & - p_{652} (s) & - p_{662} (s) \end{matrix}]

(14b)

A_{1} = [\begin{matrix} A_{11} & A_{21} & A_{31} & A_{41} & A_{51} & A_{61} \end{matrix}]

(14c)

A_{2} = [\begin{matrix} A_{42} & A_{52} & A_{62} \end{matrix}]

(14d)

S = {[\begin{matrix} g (s) & 0 & 0 \end{matrix}]}^{T} = g (s) S_{0}

(14e)

T_{3} = [\begin{matrix} e^{β h_{0}} p_{111} (s) e^{s γ_{11} h_{0}} & e^{β h_{0}} p_{121} (s) e^{s γ_{21} h_{0}} & e^{β h_{0}} p_{131} (s) e^{s γ_{31} h_{0}} \\ e^{β h_{0}} p_{211} (s) e^{s γ_{11} h_{0}} & e^{β h_{0}} p_{221} (s) e^{s γ_{21} h_{0}} & e^{β h_{0}} p_{231} (s) e^{s γ_{31} h_{0}} \\ e^{β h_{0}} p_{311} (s) e^{s γ_{11} h_{0}} & e^{β h_{0}} p_{321} (s) e^{s γ_{21} h_{0}} & e^{β h_{0}} p_{331} (s) e^{s γ_{31} h_{0}} \end{matrix} \begin{matrix} e^{β h_{0}} p_{141} (s) e^{s γ_{41} h_{0}} & e^{β h_{0}} p_{151} (s) e^{s γ_{51} h_{0}} & e^{β h_{0}} p_{161} (s) e^{s γ_{61} h_{0}} \\ e^{β h_{0}} p_{241} (s) e^{s γ_{41} h_{0}} & e^{β h_{0}} p_{251} (s) e^{s γ_{51} h_{0}} & e^{β h_{0}} p_{261} (s) e^{s γ_{61} h_{0}} \\ e^{β h_{0}} p_{341} (s) e^{s γ_{41} h_{0}} & e^{β h_{0}} p_{351} (s) e^{s γ_{51} h_{0}} & e^{β h_{0}} p_{361} (s) e^{s γ_{61} h_{0}} \end{matrix}]

(14f)

Then, we can obtain the displacement component and the electrical potential in the transformed domain as

{〈 u_{z 1} (r, z) 〉}_{0} = U ({T_{1}}^{- 1} T_{2}) (T_{3} {T_{1}}^{- 1} T_{2}) S = G (s, z) g (s)

(15a)

{〈 ϕ_{1} (r, z) 〉}_{0} = W ({T_{1}}^{- 1} T_{2}) (T_{3} {T_{1}}^{- 1} T_{2}) S = H (s, z) g (s)

(15b)

where $g (s) = \int_{0}^{\infty} p (r) r J_{0} (sr) ds$ and

U = [\begin{matrix} p_{411} (s) e^{s γ_{11} z} & p_{421} (s) e^{s γ_{21} z} & p_{431} (s) e^{s γ_{31} z} & p_{441} (s) e^{s γ_{41} z} & p_{451} (s) e^{s γ_{51} z} & p_{461} (s) e^{s γ_{61} z} \end{matrix}]

(16a)

W = [\begin{matrix} - p_{611} (s) e^{s γ_{11} z} & - p_{621} (s) e^{s γ_{21} z} & - p_{631} (s) e^{s γ_{31} z} & - p_{641} (s) e^{s γ_{41} z} & - p_{651} (s) e^{s γ_{51} z} & - p_{661} (s) e^{s γ_{61} z} \end{matrix}]

(16b)

G (s, z) = U ({T_{1}}^{- 1} T_{2}) (T_{3} {T_{1}}^{- 1} T_{2}) S_{0}

(16c)

H (s, z) = W ({T_{1}}^{- 1} T_{2}) (T_{3} {T_{1}}^{- 1} T_{2}) S_{0}

(16d)

Applying the inverse Hankel transform (Liu et al., 2008) to equation (15), we obtain

\begin{matrix} u_{z 1} (r, z) = \int_{0}^{\infty} {〈 u_{z 1} (r, z) 〉}_{0} s J_{0} (sr) ds \\ = \int_{0}^{a} p (t) tdt \int_{0}^{\infty} sG (s, z) J_{0} (st) J_{0} (sr) ds \end{matrix}

(17a)

\begin{matrix} ϕ_{1} (r, z) = \int_{0}^{\infty} {〈 ϕ_{1} (r, z) 〉}_{0} s J_{0} (sr) ds \\ = \int_{0}^{a} p (t) tdt \int_{0}^{\infty} sH (s, z) J_{0} (st) J_{0} (sr) ds \end{matrix}

(17b)

At the coating surface $z = h_{0}$ , we have

lim_{s \to \infty} sG (s, h_{0}) = G^{\infty}

(18)

Using equation (18), we can obtain

\begin{matrix} u_{z 1} (r, h_{0}) = \int_{0}^{a} p (t) tdt \int_{0}^{\infty} (sG (s, h_{0}) - G^{\infty}) J_{0} (st) J_{0} (sr) ds + G^{\infty} \\ \int_{0}^{a} p (t) tdt \int_{0}^{\infty} J_{0} (st) J_{0} (sr) ds \\ = \int_{0}^{a} p (t) tI (r, t) dt + G^{\infty} \int_{0}^{a} p (t) t H_{00} (r, t) dt \end{matrix}

(19)

where

H_{00} (r, t) = \int_{0}^{\infty} J_{0} (st) J_{0} (sr) ds = {\begin{matrix} \frac{2}{π r} K (\frac{t}{r}), (t < r) \\ \frac{2}{π t} K (\frac{r}{t}), (t > r) \end{matrix}

(20a)

I (r, t) = \int_{0}^{\infty} (sG (s, h_{0}) - G^{\infty}) J_{0} (st) J_{0} (sr) ds

(20b)

and $K (\cdot)$ is the complete elliptic integral of the first kind.

Differentiating equation (19) with respect to r and extending the definition of the unknown function $p (t)$ to the range of $- a \leq r \leq 0$ , we can obtain (Erdogan, 1965)

\begin{matrix} m (r) = \frac{1}{2} \int_{- a}^{a} p (t) | t | L (r, t) dt \\ + \frac{G^{\infty}}{π} \int_{- a}^{a} p (t) \frac{1}{t - r} dt + \frac{G^{\infty}}{π} \int_{- a}^{a} p (t) H_{1} (r, t) dt \end{matrix}

(21)

where

m (r) = \frac{\partial u_{z 1} (r, h_{0})}{\partial r}

(22a)

L (r, t) = \frac{\partial I (r, t)}{\partial r}

(22b)

h_{1} (r, t) = {\begin{matrix} | \frac{t}{r} | E (\frac{t}{r}), (| t | < | r |) \\ \frac{t^{2}}{r^{2}} E (\frac{r}{t}) - \frac{t^{2} - r^{2}}{r^{2}} K (\frac{r}{t}), (| t | > | r |) \end{matrix}

(22c)

H_{1} (r, t) = \frac{h_{1} (r, t) - 1}{t - r}

(22d)

h_{1} (r, r) = 1

(22e)

If the following substitutions are employed

x = \frac{t}{a}

(23a)

y = \frac{r}{a}

(23b)

then equation (21) can be rewritten as

\begin{matrix} m (y) = \frac{1}{2} \int_{- 1}^{1} a^{2} p (x) | x | L (y, x) dx \\ + \frac{G^{\infty}}{π} \int_{- 1}^{1} p (x) \frac{1}{x - y} dx + \frac{G^{\infty}}{π} \int_{- 1}^{1} p (x) H_{1} (y, x) dx \end{matrix}

(24)

Similarly, the equilibrium condition is determined by

P = 2 π \int_{0}^{a} p (t) tdt

(25)

which may be rewritten as

P = π a^{2} \int_{- 1}^{1} p (x) | x | dx

(26)

Examples for cylindrical and spherical indenters

Cylindrical indenter

The displacement $u_{z 1}$ for an FGPM coating indented by a rigid cylindrical indenter (Figure 2(a)) can be stated as

u_{z 1} (r, h_{0}) = constant (0 \leq r \leq a)

(27)

Figure 2.

Geometry of a functionally graded piezoelectric coating indented by (a) a rigid cylindrical indenter and (b) a rigid spherical indenter.

Then, we can obtain

m (y) = 0

(28)

Because the function $p (x)$ in equation (24) has integrable singularities at $x = \pm 1$ , we can obtain the expression as follows

p (x) = \frac{f (x)}{\sqrt{1 - x^{2}}}, (- 1 \leq x \leq 1), (- 1 \leq x \leq 1) .

(29)

The numerical collocation technique developed by Civelek in Appendix 2 is used to solve the singular integral equations (24) and (26) with the consideration of equations (28) and (29).

Spherical indenter

Figure 2(b) shows the indentation problem for an FGPM coating indented by a rigid spherical indenter with

u_{z 1} (r, h_{0}) = δ_{0} - \frac{r^{2}}{2 R} (0 \leq r \leq a)

(30)

where $δ_{0}$ denotes the maximum indentation depth and R is the radius of the spherical indenter.

The function $m (y)$ can be written as

m (y) = \frac{ay}{R}

(31)

The function $p (x)$ is bounded at both ends $x = \pm 1$ , and it is given by

p (x) = f (x) \sqrt{1 - x^{2}} (- 1 \leq x \leq 1)

(32)

The numerical method given in Appendix 2 is applied to solve equations (24) and (26) with the consideration of equations (31) and (32).

After the contact pressure $p (x)$ is obtained, the maximum of the electrical potential $ϕ_{0} = ϕ_{1} (0, h_{0})$ can be evaluated according to equation (17b).

Numerical results and discussion

For the indentation problem of the FGPM coating on a piezoelectric half-space (substrate), three kinds of commercially available piezoelectric materials are considered, whose properties are given in Table 1 (Giannakopoulos and Suresh, 1999). The thickness of the FGPM coating is chosen to be 20 mm. The radius of the cylindrical indenter a is 5 mm with a normal load of 100 N, and the radius of the spherical indenter R is taken as 200 mm under the applied mechanical force $P = 0.2 N$ . Throughout this article, the singular integral equation describing the corresponding contact problem is solved using the numerical method of Civelek (1972) with 30 discrete collocation points. In the following, we will investigate the effects of the gradient index $β h_{0}$ of the material properties on the contact pressure, the maximum indentation depth, and the electric potential.

Table 1.

Material constants.

Material	Elastic constants (GPa)					Piezoelectric constants (C/m²)			Dielectric constants (10⁻⁹ F/m)
Material	c ₁₁	c ₃₃	c ₄₄	c ₁₂	c ₁₃	e ₃₁	e ₃₃	e ₁₅	ε ₁₁	ε ₃₃
PZT-4	139.00	115.00	25.60	77.80	74.30	−5.200	15.10	12.70	6.461	5.620
PZT-5A	121.00	111.00	21.10	75.40	75.20	−5.400	15.80	12.30	8.107	7.346
BaTiO₃	166.00	162.00	42.90	76.60	77.50	−4.400	18.60	11.60	11.151	12.567

In order to check the correctness of the present results, we first calculate the maximum indentation for PZT-4 piezoelectric half-space under spherical indenter using the present methods for comparison with the approximate analytical results given by Giannakopoulos and Suresh (1999). Table 2 shows that the present results are in good agreement with the approximate analytical results.

Table 2.

Maximum indentation for PZT-4 piezoelectric half-space under spherical indenter (for $D = 2 R = 400 mm$ ).

a (mm)	$δ_{0}$ (mm)
a (mm)	$δ_{0} \approx 2 a^{2} / D$ (Giannakopoulos and Suresh, 1999)	Present results
0.01	5.00E−7	5.10314E−7
0.02	2.00E−6	2.04126E−6
0.03	4.50E−6	4.59282E−6
0.04	8.00E−6	8.16502E−6
0.05	1.25E−5	1.27578E−5
0.06	1.80E−5	1.83713E−5
0.07	2.45E−5	2.50054E−5
0.08	3.20E−5	3.26601E−5
0.09	4.05E−5	4.13354E−5
0.10	5.00E−5	5.10314E−5

Figure 3 demonstrates the effect of the gradient index $β h_{0}$ on the contact pressure for a rigid cylindrical indenter and a rigid spherical indenter with the PZT-4 substrate. The contact pressure $p (r)$ has a singularity at the boundary $r = a$ for the flat cylindrical indenter, while a smooth contact pressure is shown at the boundary $r = a$ for the spherical indenter. With the increase in the gradient index $β h_{0}$ , the contact pressure of the cylindrical indenter decreases in most part of the contact area while the contact pressure of the spherical indenter increases within the contact region. Figure 3(b) also shows that the smaller gradient index $β h_{0}$ , which means the softer surface, produces the larger radius of the contact region. The above results provide us a way to change the distribution of the contact pressure by adjusting the gradient index of the coating.

Figure 3.

Effect of the gradient index $β h_{0}$ on the contact pressure for (a) a rigid cylindrical indenter and (b) a rigid spherical indenter with the PZT-4 substrate.

The relations of P versus a, P versus $δ_{0}$ , and P versus $ϕ_{0}$ are shown in Figure 4 for different values of the material property gradient index $β h_{0}$ with the PZT-4 substrate and for the loading by a rigid spherical indenter. When the value of $β h_{0}$ increases, a larger applied normal load is needed to create the same contact region (Figure 4(a)) or the same maximum indentation depth $δ_{0}$ (Figure 4(b)). In contrast, a smaller normal load is needed to produce the same electrical potential with the increase in $β h_{0}$ . These behaviors imply that the indentation can offer a tool for the characterization of the mechanical and electrical properties of the FGPM.

Figure 4.

Effects of the gradient index $β h_{0}$ on the relations of (a) P versus a, (b) P versus $δ_{0}$ , and (c) P versus $ϕ_{0}$ for a rigid spherical indenter with the PZT-4 substrate.

Figures 5 to 8 give the distribution of the contact pressure, contact region, indentation depth, and electrical potential for different piezoelectric substrates and for the loading by a spherical indenter with $β h_{0} = 0.3 and - 0.3$ . The numerical calculations are carried out for an FGPM coating bonded to the BaTiO₃, PZT-4, and PZT-5A substrates, respectively. Figure 5 shows that the maximum of the contact pressure for the BaTiO₃ substrate is larger than that of the PZT-4 substrate, and the latter is larger than that of the PZT-5A substrate. The relations of P versus a (Figure 6) and P versus $δ_{0}$ (Figure 7) for BaTiO₃, PZT-4, and PZT-5A substrates are illustrated in Figures 6 to 8, respectively. To create the same contact region (Figure 6), the same maximum indentation depth $δ_{0}$ (Figure 7), or the same maximum electrical potential $ϕ_{0}$ (Figure 8), the largest applied force is needed for the BaTiO₃ substrate and the smallest one is required for the PZT-5A substrate. The above results are due to the fact that the elastic moduli of the BaTiO₃ substrate is larger than that of the PZT-4 substrate, and the latter is larger than that of the PZT-5A substrate while the piezoelectric constants $e_{31}$ of the BaTiO₃ substrate are smaller than that of the PZT-4 substrate, and the latter is smaller than that of the PZT-5A substrate.

Figure 5.

Distribution of the contact pressure for different piezoelectric substrates and for the loading by a spherical indenter with $β h_{0}$ equals (a) 0.3 and (b) −0.3.

Figure 6.

Relation of P versus a for different piezoelectric substrates and for the loading by a spherical indenter with $β h_{0}$ equals (a) 0.3 and (b) −0.3.

Figure 7.

Relation of P versus $δ_{0}$ for different piezoelectric substrates and for the loading by a spherical indenter with $β h_{0}$ equals (a) 0.3 and (b) −0.3.

Figure 8.

Relation of P versus $ϕ_{0}$ for different piezoelectric substrates and for the loading by a spherical indenter with $β h_{0}$ equals (a) 0.3 and (b) −0.3.

Conclusion

Axisymmetric indentation problem of a FGPM layer coated on a half-space substrate indented by a perfect electrical insulator indenter is studied in this article using the method of Hankel transform. The elastic, piezoelectric, and dielectric constants of the FGPM coating are assumed to vary exponentially along the thickness direction. The general solution is obtained for the displacement component perpendicular to and on the surface from the singular integral equation. The numerical calculations for a cylindrical indenter and a spherical indenter are conducted for three different piezoelectric substrates. The effects of the gradient index on the electrical and mechanical responses are investigated. The numerical results show that (1) the gradient index of FGPM coating have a great effect on the relations of force versus indentation and force versus electrical potential; (2) the distribution of the contact pressure can be altered by adjusting the gradient index of the FGPM coating; and (3) the relations of P versus a, P versus $δ_{0}$ , and P versus $ϕ_{0}$ can potentially offer a tool for the characterization of the mechanical and electrical properties of the FGPM coating.

Footnotes

Appendix 1

The functions $γ_{ij}$ satisfy the following characteristics equation

(33)

\begin{matrix} (m_{3} n_{3}' + n_{2} m_{4}') γ_{ij}^{6} + (m_{3} n_{2}' + n_{2} m_{3}' + m_{2} n_{3}' + n_{1} m_{4}') γ_{ij}^{5} \\ + (m_{3} n_{1}' + n_{2} m_{2}' + m_{2} n_{2}' + n_{1} m_{3}' + m_{1} n_{3}' + n_{0} m_{4}') γ_{ij}^{4} \\ + (m_{3} n_{0}' + n_{2} m_{1}' + m_{2} n_{1}' + n_{1} m_{2}' + m_{1} n_{2}' + n_{0} m_{3}' + m_{0} n_{3}') γ_{ij}^{3} \\ + (m_{2} n_{0}' + n_{2} m_{0}' + m_{1} n_{1}' + n_{1} m_{1}' + m_{0} n_{2}' + n_{0} m_{2}') γ_{ij}^{2} \\ + (m_{1} n_{0}' + n_{1} m_{0}' + m_{0} n_{1}' + n_{0} m_{1}') γ_{ij} + (m_{0} n_{0}' + n_{0} m_{0}') = 0 \end{matrix}

where $i = 1, 2, \dots, 6, j = 1, 2$ ;

\begin{matrix} m_{3} = \in_{330} (c_{130} + c_{440}) + e_{330} (e_{150} + e_{310}); \\ m_{2} = [\in_{330} (2 c_{130} + c_{440}) + e_{330} (e_{150} + 2 e_{310})] (β / s); \\ m_{1} = (c_{130} \in_{330} + e_{310} e_{330}) (β / s)^{2} \\ - e_{150} (e_{150} + e_{310}) - \in_{110} (c_{130} + c_{440}); \\ m_{0} = - (e_{150} e_{310} + c_{130} \in_{110}) (β / s); \\ m_{4}' = c_{330} \in_{330} + e_{330}^{2}; \\ m_{3}' = 2 (c_{330} \in_{330} + e_{330}^{2}) (β / s); \\ m_{2}' = (c_{330} \in_{330} + e_{330}^{2}) (β / s)^{2} \\ - c_{440} \in_{330} - 2 e_{150} e_{330} - c_{330} \in_{110}; \\ m_{1}' = - [c_{440} \in_{330} + 2 e_{150} e_{330} + c_{330} \in_{110}) (β / s); \\ m_{0}' = c_{440} \in_{110} + e_{150}^{2} \\ n_{2} = (e_{150} + e_{310}) [\in_{330} (c_{130} + c_{440}) + e_{330} (e_{150} + e_{310})] \\ - c_{440} (e_{150} \in_{330} - e_{330} \in_{110}); \\ n_{1} = {e_{150} [\in_{330} (c_{130} + c_{440}) + e_{330} (e_{150} + e_{310})] \\ + (e_{150} + e_{310}) (c_{130} \in_{330} + e_{310} e_{330}) \\ - c_{440} (e_{150} \in_{330} - e_{330} \in_{110})} (β / s); \end{matrix}

\begin{array}{l} n_{0} = e_{150} (c_{130} \in_{330} + e_{310} e_{330}) {(β / s)}^{2} + c_{110} (e_{150} \in_{330} - e_{330} \in_{110}); \\ n_{3}^{'} = - (e_{150} + e_{310}) (\in_{330} c_{330} + e_{330}^{2}); \\ n_{2}^{'} = - (e_{310} + 2 e_{150}) (\in_{330} c_{330} + e_{330}^{2}) (β / s); \\ n_{1}^{'} = - e_{150} (c_{330} \in_{330} + e_{330}^{2}) {(β / s)}^{2} + (e_{150} + e_{310}) [c_{440} \in_{330} + e_{150} e_{330}] \\ - (e_{150} \in_{330} - e_{330} \in_{110}) (c_{130} + c_{440}) \end{array}

and $n_{0}' = e_{330} [e_{150}^{2} + c_{440} \in_{110}] (β / s)$ .

The functions $γ_{i 1}$ ( $i = 1, 2, \dots, 6$ ) are the characteristic roots of equation (33) for the FGPM coating, while the functions $γ_{i 2}$ ( $i = 4, \dots, 6$ ) are the characteristic roots of equation (33) for the piezoelectric half-space ( $j = 2$ with $β = 0$ ). The following order of $γ_{ij}$ is assumed

(34)

ℜ [γ_{i 1}] < ℜ [γ_{i + 1, 1}], ℜ [γ_{i 2}] < ℜ [γ_{i + 1, 2}] (i = 1, 2, \dots, 5)

The functions $p_{kij} (s)$ $(k, i = 1, 2, \dots, 6, j = 1, 2)$ are given by

(35a)

p_{1 ij} (s) = c_{130} a_{ij} (s) + γ_{ij} (s) {c_{330} - e_{330} b_{ij} (s)}

(35b)

p_{2 ij} (s) = c_{440} {γ_{ij} (s) a_{ij} (s) - 1} + e_{150} b_{ij} (s)

(35c)

p_{3 ij} (s) = e_{310} a_{ij} (s) + γ_{ij} (s) {e_{130} + \in_{330} b_{ij} (s)}

(35d)

p_{4 ij} (s) = \frac{1}{s}

(35e)

p_{5 ij} (s) = \frac{a_{ij} (s)}{s}

(35f)

p_{6 ij} (s) = \frac{b_{ij} (s)}{s}

where $a_{ij} (s)$ and $b_{ij} (s)$ are given by

\begin{matrix} a_{ij} (s) = \frac{n_{3}' γ_{ij}^{3} + n_{2}' γ_{ij}^{2} + n_{1}' γ_{ij} + n_{0}'}{n_{2} γ_{ij}^{2} + n_{1} γ_{ij} + n_{0}} \\ b_{ij} (s) = \\ \frac{{(c_{130} + c_{440}) γ_{ij} + c_{130} (β / s)} a_{ij} (s) + c_{330} γ_{ij}^{2} + c_{330} (β / s) γ_{ij} - c_{440}}{e_{330} γ_{ij}^{2} + e_{330} (β / s) γ_{ij} - e_{150}} \end{matrix}

Appendix 2 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (project no. 11262013; T.-J.L.), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (project no. NJYT-13-B09; T.-J.L.), and the Natural Science Foundation of Inner Mongolia (project no. 2012MS0119; T.-J.L.) and by the China Scholarship Council (CSC).

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