Effect of a functionally graded interlayer on three-dimensional elastic deformation of coated plates...

Composite Structures 89 (2009) 167–176

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Composite Structures

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Effect of a functionally graded interlayer on three-dimensional elasticdeformation of coated plates subjected to transverse loading

M. Kashtalyan *, M. MenshykovaCentre for Micro- and Nanomechanics (CEMINACS), School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK

a r t i c l e i n f o

Article history:Available online 18 July 2008

Keywords:Functionally graded materialCoatingPlateThree-dimensional elasticity theory

0263-8223/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.compstruct.2008.07.007

* Corresponding author. Tel./fax: +44 1224 272519E-mail address: [email protected] (M. Kas

a b s t r a c t

Conventional coatings, which usually consist of one or two homogeneous layers deposited on a substrate,are susceptible to cracking and debonding due to the mismatch of thermomechanical properties betweenthe coating and the substrate. To overcome this disadvantage and increase resistance of coatings to func-tional failure, the concept of functionally graded material (FGM) is being actively explored in coatingdesign. Although an FGM coating certainly eliminates the abrupt change in material properties and stressdiscontinuity at the coating/substrate interface, it may be not an optimal solution to the problem.

This paper investigates the bending response of coated plates subjected to transverse loading withfunctionally graded interlayer between the substrate and the top coat. The Young’s modulus of the inter-layer is assumed to vary exponentially through the thickness. The analysis employs three-dimensionalelasticity solution for a functionally graded rectangular plate, recently developed by the first author,and is valid of plate of any thickness-to-length ratio. A comparative study of coated plates with and with-out an FGM interlayer is carried out, and the dependence of stress and displacement fields in the plate onthe stiffness gradient in the coating is examined and discussed.

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1. Introduction

Functionally graded material (FGM) refers to a heterogeneouscomposite material with gradient compositional variation of theconstituents from one surface of the material to the other whichresults in continuously varying material properties. The conceptof FGM, initially developed for super heat resistant materials tobe used in space planes or nuclear fusion reactors [1], is now ac-tively explored in a variety of engineering applications includingenergy conversion, dental and orthopaedic implants, sensors andthermogenerators, wear resistant coatings and joining of dissimilarmaterial. A number of reviews dealing with various aspects ofFGMs have been published over the years [2–5].

As far as coatings are concerned, incorporation of FGMs intocoating design can help eliminate the mismatch of thermomechan-ical properties between the coating and the substrate, whichmakes conventional one- or two-layer coatings susceptible tocracking and debonding, and thus increase the resistance of coat-ings to functional failure. Currently, a number of approaches togenerate FGM coatings for specific applications and/or loading con-ditions are being developed by materials scientists [6–10].

Fracture of FGM coatings has been a focus of numerous theoret-ical studies [11–22], with contact response also receiving someattention [23–26]. These investigations were carried out in the

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.htalyan).

context of two-dimensional elasticity, and in all of them, except[19,20], the thickness of the substrate was assumed to be infinite.

Although an FGM coating certainly eliminates the abruptchange in material properties and stress discontinuity at the coat-ing/substrate interface, it may be not an optimal solution to theproblem. While investigating interface cracking between ceramicand/or FGM coating and a substrate under antiplane shear loading,Jin and Batra [13] found that when the shear modulus of the coat-ing is higher than that of the substrate, the coating system with ahomogeneous top coat and a thin FGM interlayer reduces the stressintensity factor more significantly than the single layered FGMcoating.

Using an FGM interlayer (bond coat) between the substrate andthe top coat instead of a single layered FGM coating may be advan-tageous in many situations, however mechanical behaviour ofcoating/substrate systems with FGM interlayer has received con-siderably less attention in the literature. Choi [27–29] investigatedstatic and dynamic fracture behaviour of a coating/substrate sys-tem with graded interlayer. Stress intensity factors were calculatedfor a crack in the substrate located normally to the interface underfrictional Hertzian contact loading [27], a normal crack in the coat-ing under antiplane shear impact [28] and an oblique edge crack inthe coating under antiplane shear deformation [29]. Thermoelasticcontact problem for a coating/substrate system with graded inter-layer was investigated by Choi and Paulino [30]. A rigid flat punchwas assumed to slide over the surface of the coating involving fric-tional heat generation, and the exponential variation of the elastic

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x

zy

),( yxQ

S

CTC

IL

Fig. 1. Coated plate subjected to bending by transverse loading.

168 M. Kashtalyan, M. Menshykova / Composite Structures 89 (2009) 167–176

and thermal constant through the thickness was considered. In allthe above studies the thickness of the substrate was assumed to beinfinite, and the investigations were carried out in the context oftwo-dimensional elasticity.

For coating/substrate systems of finite thickness with FGMinterlayer, Shbeeb and Binienda [31] calculated stress intensity fac-tors and strain energy release rate for a crack at the FGM inter-layer/substrate interface. Fracture mechanisms in thermal barriercoatings with functionally graded bond coats under uniform cyclicthermal loading were examined by Pindera et al. [32,33] using thehigher-order theory for FGMs. Linear and non-linear vibrationswere studied by Liew et al. [34]. Chung and Chen [35] investigatedbending behaviour of plates with FGM interlayer plates under uni-form loading. Through-thickness variation of the Young’s modulusaccording to sigmoid function was considered. The analysis wascarried out in the context of the classical plate theory, and thestudy was therefore restricted to thin plates with thickness-to-length ratio below 1/20.

Three-dimensional analysis of elastic deformation of a function-ally graded coating/substrate system was carried out Kashtalyanand Menshykova [36]. The coating was assumed to be single lay-ered, with the exponential variation of the Young’s modulusthrough the thickness. It was established that although use of thefunctionally graded coating eliminates stress discontinuity at thecoating/substrate interface, this is achieved at the expense of stres-ses at the top surface of the coating, which in some cases can in-crease almost twofold, and also the plate deflection, which canincreases due to some loss of flexural stiffness provided by thecoating. In contrast to [35], the solution proposed in [36] is validfor plates of with any thickness-to-length ratio.

The findings of [36] proved that using a single layered function-ally graded coating has certain adverse effects on the elastic defor-mation of coating/substrate systems subjected to transverseloading and were the motivation behind the present study, whichis aimed at investigating bending response of coated plates with afunctionally graded interlayer. The Young’s modulus of the inter-layer is assumed to vary exponentially through the thickness.The analysis employs the three-dimensional elasticity solutionfor a functionally graded rectangular plate, recently developed byKashtalyan [37]. A comparative study of coated plates with differ-ent structure of the coating is carried out, and dependence of stressand displacement fields in the plate on the stiffness gradient in thecoating is examined and discussed.

2. Problem formulation

A coated plate of length a, width b and total thickness h is re-ferred to a Cartesian co-ordinate system xyz (Fig. 1), so that0 6 x 6 a, 0 6 y 6 b, 0 6 z 6 h. The plate comprises a substrate ‘S’and a coating ‘C’, which consists of an interlayer ‘IL’ and a top coat‘TC’. The substrate/interlayer and the interlayer/top coat interfacesare denoted as z = h(1) and z = h(2), respectively. Henceforth, allquantities associated with the substrate, the interlayer and thetop coat will be denoted with the superscripts (1), (2) and (3),respectively.

Let us assume that all three layers (k = 1, 2, 3) are FGMs, withconstant Poisson’s ratios m(k) = const., and shear moduli that varycontinuously through the thickness according to exponentialfunctions

GðkÞðzÞ ¼ gðkÞ exp cðkÞzh� 1

� �h i; k ¼ 1;2;3; ð1Þ

where c(k) are the inhomogeneity parameters, and g(k) = G(k)(h). Anyhomogeneous layer (e.g. substrate) can be modelled as FGM withthe inhomogeneity parameter, c(k), sufficiently close to zero [37].The exponential variation of material properties with transverse

co-ordinate was used by a number of researchers investigatingfunctionally graded coatings, including Jin and Batra [13], Chiuand Erdogan [17], Guler and Erdogan [24,25] and Shodja and Ghahr-emaninejad [26].

Perfect bonding is assumed at all interfaces, so that continuityof stresses and displacements exists there

z ¼ hð1Þ : rð1Þiz � rð2Þiz ¼ 0; uð1Þi � uð2Þi ¼ 0; i ¼ x; y; z; ð2aÞz ¼ hð2Þ : rð2Þiz � rð3Þiz ¼ 0; uð2Þi � uð3Þi ¼ 0; i ¼ x; y; z; ð2bÞ

where rðkÞij ðk ¼ 1;2;3Þ are the components of the stress tensor, anduðkÞi ðk ¼ 1;2;3Þ are the components of the displacement vector.

The plate is subjected to transverse loading Q on the top surfaceof the coated plate, so that

z ¼ h : rð3Þzz ¼ Qðx; yÞ; rð3Þxz ¼ rð3Þyz ¼ 0; ð3aÞ

Qðx; yÞ ¼ �qmn sinpmx

asin

pnyb

; ð3bÞ

where qmn is the intensity of the loading at the centre of the plate,and m and n are wave numbers. The choice of loading, Eq. (3b), ismotivated by its practical significance, since any loading can be ex-pressed as a Fourier series involving terms of this type. The bottomsurface of the plate is assumed to be load-free, i.e.

z ¼ 0 : rð1Þzz ¼ rð1Þxz ¼ rð1Þyz ¼ 0: ð4Þ

The Navier-type boundary conditions are assumed at the edges ofthe plate, so that

x ¼ 0; a : rðkÞxx ¼ 0; uðkÞy ¼ uðkÞz ¼ 0; k ¼ 1;2;3; ð5aÞy ¼ 0; b : rðkÞyy ¼ 0; uðkÞx ¼ uðkÞz ¼ 0; k ¼ 1;2;3: ð5bÞ

The boundary conditions, Eqs. (5), are representative of roller sup-ports and analogous to simply supported edges in the platetheories.

3. Solution

If the Poisson’s ratios are constant, the displacements in anylayer of the plate can be expressed in terms of two functions[36–38], denoted L(k) = L(k)(x, y, z) and N(k) = N(k)(x, y, z) (k = 1,2), asfollows

uðkÞx ¼ �1

2GðkÞmðkÞD� o2

oz2

!oLðkÞ

oxþ oNðkÞ

oy; ð6aÞ

uðkÞy ¼ �1


oz2

!oLðkÞ

oy� oNðkÞ

ox; ð6bÞ

uðkÞz ¼ �1

GðkÞD� o2

oz2

!oLðkÞ

ozþ o

oz1


oz2

!LðkÞ

" #;

k ¼ 1;2;3: ð6cÞ

M. Kashtalyan, M. Menshykova / Composite Structures 89 (2009) 167–176 169

From Eqs. (6) and the constitutive equations for an isotropic inho-mogeneous material (see Appendix A), the components of the stresstensor can be expressed in terms of functions L(k) and N(k) as

rðkÞzz ¼o2

ox2 þo2

oy2

!2

LðkÞ; ð7aÞ

rðkÞxz ¼ �o2

ox2 þo2

oy2

!o2LðkÞ

oxozþ GðkÞ

o2NðkÞ

oyoz; ð7bÞ

rðkÞyz ¼ �o2

ox2 þo2

oy2

!o2LðkÞ

oyoz� GðkÞ

o2NðkÞ

oxoz; ð7cÞ

rðkÞxx ¼ mðkÞo2

oy2 Dþ o4

ox2oz2

!LðkÞ þ 2GðkÞ

o2NðkÞ

oxoy; ð7dÞ

rðkÞyy ¼ mðkÞo2

ox2 Dþ o4

oy2oz2

!LðkÞ � 2GðkÞ

o2NðkÞ

oxoy; ð7eÞ

rðkÞxy ¼ � mðkÞD� o2

oz2

!o2LðkÞ

oxoy� GðkÞ

o2

ox2 �o2

oy2

!NðkÞ;

k ¼ 1;2;3: ð7fÞ

Functions L(k) = L(k)(x, y, z) and N(k) = N(k)(x, y, z) in Eqs. (6) and (7)must satisfy the following partial differential equations

D1

GðkÞDLðkÞ

� �� 1

1� mðkÞD� o2

oz2

!LðkÞ

d2

dz2

1

GðkÞ

� �¼ 0; ð8aÞ

DNðkÞ þ ddz

ln GðkÞðzÞ oNðkÞ

oz¼ 0; k ¼ 1;2;3: ð8bÞ

It was shown by Plevako [38] that if functions L(k) and N(k) allowseparation of variables in the form

LðkÞðx; y; zÞ ¼ wðkÞ1 ðx; yÞ/ðkÞ1 ðzÞ; NðkÞðx; y; zÞ ¼ wðkÞ2 ðx; yÞ/

ðkÞ2 ðzÞ; ð9Þ

then Eqs. (8a,b) can be transformed into the Helmholz’s equationfor functions wðkÞ1 ðx; yÞ and wðkÞ2 ðx; yÞ, and the forth- and the sec-ond-order ordinary differential equations for functions /ðkÞ1 ðzÞ and/ðkÞ2 ðzÞ.

The form of functions wðkÞ1 ðx; yÞ and wðkÞ2 ðx; yÞis influenced by theapplied load and the boundary conditions on the edges of the plate.For sinusoidal loading, Eq. (3b), and Navier-type boundary on theedges, Eqs. (5a,b), they can be chosen as [36,37]

wðkÞ1 ðx; yÞ ¼ sinpmx

asin

pnyb

; wðkÞ2 ðx; yÞ

¼ cospmx

acos

pnyb

; k ¼ 1;2;3: ð10Þ

Then boundary conditions on the edge, Eqs. (5a,b) are satisfied ex-actly for any numbers n, m. The form of functions /ðkÞ1 ðzÞ and /ðkÞ2 ðzÞdepends on the functional variation of the shear moduliG(k) = G(k)(z). When the shear moduli G(k) = G(k)(z) of the substrate(k = 1), the interlayer (k = 2), and the top coating (k = 3) are expo-nential functions of the thickness co-ordinate in the form of Eq.(1), then Eqs. (8a,b) are reduced to the following ordinary differen-tial equations with constant coefficients [36,37] with respect tofunctions /ðkÞ1 ðzÞ and /ðkÞ2 ðzÞ

h4 d4/ðkÞ1

dz4 � 2cðkÞh3 d3/ðkÞ1

dz3 þ ½cðkÞ2 � 2a2h2�h2 d2/ðkÞ1

dz2

þ 2a2cðkÞh3 d/ðkÞ1

dzþ a2h2 a2h2 þ mðkÞ

1� mðkÞcðkÞ

2� �

/ðkÞ1 ¼ 0; ð11aÞ

h2 d2/ðkÞ2

dz2 þ cðkÞhd/ðkÞ2

dz� a2h2/ðkÞ2 ¼ 0; k ¼ 1;2;3: ð11bÞ

Here a is a coefficient of the Helmholz’s equation

a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ma

� �2þ n

b

� �2r

: ð12Þ

Following standard procedure, solutions to Eqs. (11a,b) are found as

/ðkÞ1 ðzÞ ¼ h4 expcðkÞz2h

� �

� AðkÞ1 coshkðkÞz

hcos

lðkÞzhþ AðkÞ2 sinh

kðkÞzh

coslðkÞz

h

"

þAðkÞ3 coshkðkÞz

hsin

lðkÞzhþ AðkÞ4 sinh

kðkÞzh

sinlðkÞz

h

#; ð13aÞ

/ðkÞ2 ðzÞ ¼ h2 exp � cðkÞz2h

� �AðkÞ5 cosh

bðkÞzhþ AðkÞ6 sinh

bðkÞzh

" #: ð13bÞ

Here AðkÞj (j = 1, . . ., 6; k = 1, 3) are eighteen arbitrary constants thatcan be determined from the stress and displacement continuityconditions at the interfaces, Eq. (2), and the boundary conditionson the top and bottom surfaces of the plate, Eqs. (3) and (4). Coef-ficients k(k) and l(k), and b(k), are the roots of characteristic equa-tions corresponding to Eqs. (11a) and (11b), respectively.Expressions for them are given in Appendix B.

Substitution of functions wðkÞ1 ðx; yÞ and wðkÞ2 ðx; yÞ, Eq. (10), andfunctions /ðkÞ1 ðzÞ and /ðkÞ2 ðzÞ, Eqs. (13a,b), into Eq. (9) and then intoEqs. (6), (7) gives the following expressions for displacements andstresses in a functionally graded coating/substrate system of finitethickness under transverse sinusoidal loading with exponentialdependences of the shear moduli on the thickness co-ordinate

uðkÞx ¼X6

j¼1

AðkÞj UðkÞx;j ðzÞ cospmx

asin

pnyb

; ð14aÞ

uðkÞy ¼X6

j¼1

AðkÞj UðkÞy;j ðzÞ sinpmx

acos

pnyb

; ð14bÞ

uðkÞz ¼X6

j¼1

AðkÞj UðkÞz;j ðzÞ cospmx

asin

pnyb

: ð14cÞ

rðkÞzz ¼X6

j¼1

AðkÞj PðkÞzz;jðzÞ sinpmx

asin

pnyb

; ð15aÞ

rðkÞxz ¼X6

j¼1

AðkÞj PðkÞxz;jðzÞ cospmx

asin

pnyb

; ð15bÞ

rðkÞyz ¼X6

j¼1

AðkÞj PðkÞyz;jðzÞ sinpmx

acos

pnyb

; ð15cÞ

rðkÞxx ¼X6

j¼1

AðkÞj PðkÞxx;jðzÞ sinpmx

asin

pnyb

; ð15dÞ

rðkÞyy ¼X6

j¼1

AðkÞj PðkÞyy;jðzÞ sinpmx

asin

pnyb

; ð15eÞ

rðkÞxy ¼X6

j¼1

AðkÞj PðkÞxy;jðzÞ cospmx

acos

pnyb

; k ¼ 1;2;3: ð15fÞ

Functions UðkÞi;j and PðkÞrt;j are specified in Appendix C.

4. Results and discussion

To validate the developed three-dimensional solution for acoated plate with a functionally graded interlayer, it is comparedto the three-dimensional solutions available in the literature, forexample to the solution for an isotropic homogeneous plate[39,40]. Stresses and displacements in the homogeneous platecan be obtained from the proposed solution for a coated plate witha graded interlayer by numerically setting c(k) ? 0 in Eqs. (14) and(15). Table 1 shows the normalised out-of-plane displacement G0uz

qh

in the coated plated with a graded interlayer and the homogeneous

Table 1Normalised transverse displacements in a coated plate with a functionally gradedinterlayer and a homogeneous isotropic plate

Gðhð2Þ ÞG0

Normalised transverse displacement G0 uzqh

a2 ;

b2 ; 0:5h

� �Coated plate with a functionally graded interlayer0.9 �1.3597560.99 �1.3442010.999 �1.3427190.9999 �1.3425720.99999 �1.3425570.999999 �1.342555

Homogeneous isotropic plate [39,40]1.0 �1.343

(i)

(ii)

(iii)

G

SG

CG

Sh h

z

G

SG

CG

Sh h

z

G

SG

CG

)(5.0 CSIL GGG +=

Sh h

z

Fig. 2. Through-thickness variation of the shear moduli in coated plates withdifferent structure of the coating: (i) H–H coating (homogeneous interlayer,homogeneous top coat); (ii) FGM–H coating (FGM interlayer, homogeneous topcoat); (iii) FGM coating.


plate for a range of shear modulus values G(h(2))/G0 at the coating/interlayer interface, where G0 is the shear modulus of the homoge-neous plate; the Poisson’s ratios of the plate are taken as 0.3. Theshear modulus in the coated plate varies according to Eq. (1) withG(0) = 0.999999G0, G(h(1)) = G(h) = G0. The displacements areshown in the centre of the plate (x = 0.5a, y = 0.5b), and the platedimensions are taken as a = b and a/h = 3 (thick square plate).The plate is simply supported on its edges and loaded on its topsurface by the sinusoidal loading, Eq. (3), with m = n = 1. The re-sults presented in Table 1 indicate that as G(h(2)) tends to G0, solu-tion for a coated plate with a functionally graded interlayer rapidlyconverges to that for the homogeneous isotropic plate [39,40].When the difference between G(h(2)) and G0 becomes smaller than10�5, the discrepancy between the computed out-of-plane dis-placements of the coated plate with a functionally graded inter-layer and the homogeneous isotropic plate constitutes less than0.03%.

To establish the effect of functionally graded interlayer onbending response of coated plates, comparative study of coatedplates with the same substrate, but different structure of the coat-ing was carried out; the results are presented and discussed in thissection. Three cases are considered:

(i) H–H coating, consisting of a homogeneous interlayer withthe shear modulus 0.5(GS + GC) and a homogenous top coatwith the shear modulus GC, of equal thicknesses;

(ii) FGM–H coating, consisting of a functionally graded inter-layer and a homogeneous top coat of equal thicknesses;the shear modulus of the top coat is GC, and the shear ofthe interlayer varies exponentially from GS to GC accordingto Eq. (1), where GS is the shear modulus of the substrate;

(iii) FGM coating, i.e. functionally graded coating with the shearmodulus that varies exponentially from GS to GC according toEq. (1).

In all three cases, the total thickness of the coating hC is thesame, the substrate is homogeneous, with the shear modulus GS.Variation of the shear moduli through the thickness in cases (i)–(iii) is shown in Fig. 2.

The homogeneous substrates in cases (i)–(iii), the homogeneousinterlayer and top coat in case (i) and the homogeneous top coat incase (ii) are modelled as functionally graded materials with theinhomogeneity parameters, c(k), sufficiently close to zero. As wasshown by Kashtalyan [37], when the inhomogeneity parameter be-comes smaller, the three-dimensional solution for an FGM rapidlyconverges to that for a homogeneous material. For example, forsufficiently small values of the inhomogeneity parameter (smallerthan 10�5 by the absolute value), the difference between the com-puted out-of-plane displacements in functionally graded andhomogeneous plates constituted less that 0.03% [37].

The inhomogeneity parameter cIL of the FGM interlayer (case(ii)) can be expressed in terms of the stiffness gradient GC/GS, i.e.the shear moduli ratio between its upper and lower interfaces

cIL ¼h

0:5hCln

GC

GS: ð16Þ

Likewise, the inhomogeneity parameter cC of the FGM coating (case(iii)) and can be expressed in terms of the shear moduli ratio GC/GS

between the top surface and the coating/substrate interface

cC ¼hhC

lnGC

GS: ð17Þ

In this study, we consider the stiffness gradient GC/GS > 1, whichrepresents an important from the engineering point of view


scenario of a soft substrate protected by a much stiffer coating. Thechosen values of GC/GS do not necessarily represent certain materi-als, but are rather used to establish the effect of the coating struc-ture and the stiffness gradient on stress and displacement fieldsin the coated plates. The Poisson’s ratios of the substrate and thecoating in all there cases are taken as 0.3, and the total thicknessof the coating is hC = 0.2h.

Figs. 3–6 show through-thickness variation of the normalisedstresses �rij ¼ rij=qmn and normalised displacements �ui ¼GCuiðqmnhÞ�1 in coated square plates with different structure ofthe coatings (a = b, m = n = 1). Results are shown for thick plateswith a/h = 3 (Figs. 3 and 4) and moderately thick plates witha/h = 10 (Figs. 5 and 6), with different values of the stiffness gradi-ent: GC/GS = 2 (Figs. 3 and 5) and GC/GS = 10 (Figs. 4 and 6).

It can be seen that for the out-of-plane normal stress �rzz (Figs.3a, 4a, 5a and 6a), through-thickness variation is similar for all con-sidered cases, stiffness gradient values and length-to-thickness

Fig. 3. Through-thickness variation of the normalised stresses and displacements�rzzð0:5a;0:5b; zÞ; (b) transverse shear stress �rxzð0; 0:5b; zÞ; (c) in-plane normal stress�uxð0; 0:5b; zÞ; (f) transverse displacement �uzð0:5a; 0:5b; zÞ.

ratios (Fig. 3a–c). The structure of the coating (H–H-, FGM–H- orFGM-), has almost no effect on this stress component.

Through-thickness variation of the transverse shear stress �rxz

(Figs. 3b, 4b, 5b and 6b) for both FGM–H and FGM-coatings is closeto parabolic profile, typical for homogeneous plates, for all consid-ered values of GC/GS and a/h. For GC/GS = 10 (Figs. 4b and 6b), themaximum value of the transverse shear stress decreases slightlyin the plate with FGM-coating (case (iii)) compared to the othertwo cases.

The structure of the coating has a very strong influence on thethrough-thickness variation of the in-plane normal stress �rxx

(Figs. 3c, 4c, 5c and 6c). In contrast to the H–H-coated plate, thein-plane normal stresses in the plates with FGM–H- and FGM-coating are continuous across the coating/substrate interface.However, in the FGM-coated plates this is achieved at the expenseof the increase of the stress magnitude at the top surface of thecoating. Meanwhile, in the plate with FGM–H-coating, stresses

in thick (a/h = 3) coated plates with GC/GS = 2: (a) out-of-plane normal stress�rxxð0:5a; 0:5b; zÞ; (d) in-plane shear stress �rxyð0; 0; zÞ; (e) in-plane displacement


in the top coat are almost identical to those in the H–H-coatedplate.

Similar observations can be made for the in-plane shear stress�rxy (Figs. 3d, 4d, 5d and 6d). Through-thickness variation of thein-plane shear stress �rxy is strongly influenced by the structureof the coating. In the plates with FGM–H- and FGM-coating, thein-plane shear stress are continuous across all interfaces, howeverin the FGM-coated plates this is achieved at the expense of an in-crease in the stress magnitude at the top surface.

Through-thickness variation of the in-plane displacement �ux

(Figs. 3e, 4e, 5e and 6e) is similar in all considered cases, changingfrom linear or close to linear for moderately thick plates (Figs. 5eand 6e) to highly non-linear for thick plates (Figs. 3e and 4e).

Through-thickness variation of the transverse displacement �uz

(Figs. 3f, 4f, 5f and 6f) strongly depends on the thickness of theplate and the structure of the coating. Transverse displacementof FGM–H-coated plates (i.e. with functionally graded interlayer)

Fig. 4. Through-thickness variation of the normalised stresses and displacements i�rzzð0:5a;0:5b; zÞ; (b) transverse shear stress �rxzð0; 0:5b; zÞ; (c) in-plane normal stress�uxð0; 0:5b; zÞ; (f) transverse displacement �uzð0:5a;0:5b; zÞ.

is significantly smaller than that of plates with FGM-coating. Inthick plates (Figs. 3e and 4e), it becomes highly non-linear, whichemphasizes once again the necessity of 3-D type analysis overthose using plate theory assumptions.

5. Conclusions

In the present paper, elastic deformation of coated plated sub-jected to transverse loading is investigated in the context ofthree-dimensional elasticity theory. The Young’s modulus of thecoating is assumed to vary exponentially through the thickness,and the Poisson’s ratio is assumed to be constant. The approachmakes use of the general solution of the equilibrium equationsfor inhomogeneous isotropic media obtained earlier by Plevako[38] and its recent application to functionally graded plates withexponential variation of the Young’s modulus through the thick-ness by Kashtalyan [37].

n thick (a/h = 3) coated plates with GC/GS = 10: (a) out-of-plane normal stress�rxxð0:5a; 0:5b; zÞ; (d) in-plane shear stress �rxyð0; 0; zÞ; (e) in-plane displacement

Fig. 5. Through-thickness variation of the normalised stresses and displacements in moderately thick (a/h = 10) coated plates with GC/GS = 2: (a) out-of-plane normal stress�rzzð0:5a;0:5b; zÞ; (b) transverse shear stress �rxzð0; 0:5b; zÞ; (c) in-plane normal stress �rxxð0:5a; 0:5b; zÞ; (d) in-plane shear stress �rxyð0; 0; zÞ; (e) in-plane displacement�uxð0; 0:5b; zÞ; (f) transverse displacement �uzð0:5a; 0:5b; zÞ.


Based on the comparative study of coated plates with differentstructure of the coating, the effect of functionally graded interlayeron stresses and displacements in plate subjected to transverseloading was established. The benefits of using a coating comprisingfunctionally graded interlayer and homogeneous top coat com-pared to a single functionally graded coating of the same totalthickness are twofold. Firstly, discontinuity of the in-plane normalstress across interfaces is eliminated without increasing the stressmagnitude at the top surface of the coating; secondly, increase intransverse displacement of the coated plate is significantly smallerwhen a functionally graded interlayer is used.

Acknowledgements

Financial support of this research by Engineering and PhysicalSciences Research Council (UK) (EPSRC EP/E020976/1 and EP/E030351/1) is gratefully acknowledged.

Appendix A

The constitutive equations for inhomogeneous isotropic mate-rial in terms of stresses and displacements

rðkÞzz ¼2GðkÞouðkÞz

ozþ mðkÞ

1�2mðkÞeðkÞ

!; rðkÞxz ¼GðkÞ

ouðkÞx

ozþouðkÞz

ox

!;

rðkÞyz ¼GðkÞouðkÞy

ozþouðkÞz

oy

!;

rðkÞxx ¼2GðkÞouðkÞx

oxþ mðkÞ

1�2mðkÞeðkÞ

!; rðkÞyy ¼2GðkÞ

ouðkÞy

oyþ mðkÞ

1�2mðkÞeðkÞ

!;

rðkÞxy ¼GðkÞouðkÞx

oyþouðkÞy

ox

!; eðkÞ ¼ouðkÞx

oxþouðkÞy

oyþouðkÞz

oz:

Fig. 6. Through-thickness variation of the normalised stresses and displacements in moderately thick (a/h = 10) coated plates with GC/GS = 10: (a) out-of-plane normal stress�rzzð0:5a;0:5b; zÞ; (b) transverse shear stress �rxzð0; 0:5b; zÞ; (c) in-plane normal stress �rxxð0:5a; 0:5b; zÞ; (d) in-plane shear stress �rxyð0; 0; zÞ; (e) in-plane displacement�uxð0; 0:5b; zÞ; (f) transverse displacement �uzð0:5a;0:5b; zÞ.


Appendix B

Roots of characteristic equation corresponding to Eq. (11a)

kðkÞ

lðkÞ

!¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12�bðkÞ

2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðkÞ

4 þ cð4Þ2a2h2 mðkÞ1� mðkÞ

r !vuut :

Roots of characteristic equation corresponding to Eq. (11b)

bðkÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficðkÞ2

4þ a2h2

s:

Appendix C

Functions UðkÞi;j involved in Eqs. (14a–c).

UðkÞx;j ðzÞ ¼ �qmnh2gðkÞ

pmha

exp �cðkÞzh� 1

� �h i

� �mðkÞa2h2f ðkÞj ðzÞ þ ðmðkÞ � 1Þ d2

dz2 f ðkÞj ðzÞ" #

;

UðkÞy;j ðzÞ ¼ �qmnh2gðkÞ

pnhb


� �h i

� �mðkÞa2h2f ðkÞj ðzÞ þ ðmðkÞ � 1Þ d2


;

UðkÞz;j ðzÞ ¼ �qmnh2gðkÞ

pmha


� �h i

� ðmðkÞ � 1Þ �cðkÞd2

dz2 f ðkÞj ðzÞ þd3

dz3 f ðkÞj ðzÞ !"

�a2h2 ðmðkÞ � 2Þ ddz

f ðkÞj ðzÞ � mðkÞcðkÞf ðkÞj ðzÞ� �#

; j ¼ 1; . . . ;4;


UðkÞx;j ðzÞ ¼ �qmnh2gðkÞ

pnhb

f ðkÞj ðzÞ;

UðkÞy;j ðzÞ ¼qmnh2gðkÞ

pmha

f ðkÞj ðzÞ;

UðkÞz;j ðzÞ ¼ 0; j ¼ 5;6:

Functions PðkÞrt;j involved in Eqs. (15a–f)

PðkÞzz;jðzÞ ¼ qmna4h4f ðkÞj ðzÞ;

PðkÞxz;jðzÞ ¼ qmna2h2 pmha

� �2 ddz

f ðkÞj ðzÞ;

PðkÞyz;jðzÞ ¼ qmna2h2 pnhb

� �2 ddz

f ðkÞj ðzÞ;

PðkÞxx;jðzÞ ¼ qmn mðkÞa2h2 pnhb

� �2

f ðkÞj ðzÞ� mðkÞpnh

b

� �2 d2

dz2 f ðkÞj ðzÞ"

� pmha

� �2 d2

dz2 f ðkÞj ðzÞ#;

PðkÞyy;jðzÞ ¼ qmn mðkÞa2h2 pmha

� �2

f ðkÞj ðzÞ� mðkÞpmh

a

� �2 d2

dz2 f ðkÞj ðzÞ"

� pnhb

� �2 d2

dz2 f ðkÞj ðzÞ#;

PðkÞxy;jðzÞ ¼ qmnpmh

a

� �pnh

b

� �mðkÞa2h2f ðkÞj ðzÞþ ð1� mðkÞÞ d2


;

j¼ 1; . . . ;4;

PðkÞzz;jðzÞ ¼ 0;

PðkÞxz;jðzÞ ¼�qmnpnh

b

� �exp cðkÞ

zh�1

� �h i ddz

f ðkÞj ðzÞ;

PðkÞyz;jðzÞ ¼ qmnpmh

a

� �exp cðkÞ

zh�1

� �h i ddz

f ðkÞj ðzÞ;

PðkÞxx;jðzÞ ¼ 2qmnpnh

b

� �pmh

a

� �exp cðkÞ

zh�1

� �h if ðkÞj ðzÞ;

PðkÞyy;jðzÞ ¼ �2qmnpnh

b

� �pmh

a

� �exp cðkÞ

zh�1

� �h if ðkÞj ðzÞ;

PðkÞxy;jðzÞ ¼ qmnpmh

a

� �2

� pnhb

� �2" #

exp cðkÞzh�1

� �h if ðkÞj ðzÞ j¼ 5;6:

In the expressions above, functions f ðkÞj ðzÞ (j = 1, . . ., 6) are

f ðkÞ1 ðzÞ ¼ expcðkÞz2h

� �cosh

kðkÞzh

coslðkÞz

h;


� �sinh

kðkÞzh

coslðkÞz

h;


� �cosh

kðkÞzh

sinlðkÞz

h;


� �sinh

kðkÞzh

sinlðkÞz

h;

f ðkÞ5 ðzÞ ¼ exp � cðkÞz2h

� �cosh

bðkÞzh

;

f ðkÞ6 ðzÞ ¼ exp � cðkÞz2h

� �sinh

bðkÞzh

:

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