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substrate system
International Journal of Solids and Structures 44 (2007) 5272–5288 www.elsevier.com/locate/ijsolstr
Three-dimensional elastic deformation of a functionally graded coating/substrate system M. Kashtalyan *, M. Menshykova
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Centre for Micro- and Nanomechanics, School of Engineering and Physical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, UK Received 10 October 2006; received in revised form 12 December 2006; accepted 27 December 2006 Available online 31 December 2006
Abstract The concept of functionally graded material (FGM) is actively explored in coating design for the purpose of eliminating the mismatch of material properties at the coating/substrate interface, typical for conventional coatings, which can lead to cracking, debonding and eventual functional failure of the coating. In this paper, an FGM coating/substrate system of finite thickness subjected to transverse loading is analysed within the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. A comparative study of FGM versus homogeneous coating is conducted, and the dependence of stress and displacement fields in the coating substrate/system on the type of coating, geometry and loading is examined and discussed. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Functionally graded material; Coating; Three-dimensional elasticity theory
1. Introduction Coatings play an important role in a variety of applications protecting metallic or ceramic substrates against oxidation, heat penetration, wear and corrosion (Grainger and Blunt, 1998). Conventional coatings usually consist of one or two homogeneous layers deposited on a substrate. Due to the mismatch of thermomechanical properties between the coating and the substrate, conventional coatings are susceptible to cracking and debonding. To overcome this disadvantage and increase resistance of coatings to functional failure, the concept of functionally graded material (FGM) is being actively explored in coating design. FGM refers to heterogeneous composite material with gradient compositional variation of the constituents from one surface of the material to the other which results in continuously varying material properties (Koizumi, 1993; Suresh and Mortensen, 1998; Miyamoto et al., 1999). Currently, a number of approaches to generate FGM coatings for specific applications and/or loading conditions are being developed by materials scientists (Brovarone et al., 2001; Schulz et al., 2003; Yakovlev et al., 2004; Kim et al., 2005; Foppiano et al., 2006). *
1
Corresponding author. Tel./fax: +44 1224 272519. E-mail addresses: [email protected] (M. Kashtalyan), [email protected] (M. Menshykova). Tel.: +44 1224 273326; fax: +44 1224 272519.
0020-7683/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.12.035
M. Kashtalyan, M. Menshykova / International Journal of Solids and Structures 44 (2007) 5272–5288
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Theoretical modelling of FGM coatings has been focused predominantly on prediction of their fracture behaviour (Erdogan and Ozturk, 1995; Chen and Erdogan, 1996; Jin and Batra, 1996; Bao and Cai, 1997; Kadioglu et al., 1998; Huang et al., 2002; Chiu and Erdogan, 2003; Chi and Chung, 2003; Guo et al., 2004a; Choi, 2004; Huang et al., 2005). Several researches have addressed contact response of FGM coatings. Saizonou et al. (2002) carried out a numerical study (by boundary element method) of an FGM-coated elastic solid under normal and sliding contact loading. Guler and Erdogan (2004, 2006) investigated behaviour of FGM-coated solid under sliding contact loading provided by a sliding rigid stamp and contact of two FGM-coated elastic solids. Exponential variation of the Young’s modulus of the coating was adopted in both cases. Shodja and Ghahremaninejad (2006) examined an FGM-coated elastic solid under thermomechanical loading due to mixed normal and tangential Hertzian pressure. Thermomechanical properties were assumed to vary exponentially through the thickness. It should be noted that in all the above studies the thickness of the substrate was assumed to be infinite, and the investigations were carried out in the context of two-dimensional elasticity. For coating/substrate systems of finite thickness, Shbeeb and Binienda (1999) analysed interfacial fracture of a coating/substrate system with an FGM interphase; stress intensity factors and strain energy release rate for a crack at the interphase/substrate interface were calculated. Using the higher-order theory for FGMs, Pindera et al. (2002, 2005) examined fracture mechanisms in thermal barrier coatings with functionally graded bond coats under uniform cyclic thermal loading. Interfacial fracture of an FGM coating/substrate system subjected to concentrated transverse loading was studied by Guo et al. (2004b). Wang et al. (2005) carried out fracture analysis of an FGM-coated strip with a crack in the coating. To calculate stress intensity factors, the FGM coating was modelled as multi-layered medium, with elastic modulus varying linearly within each sub-layer, so that they are continuous on the sub-interfaces. Liew et al. (2006) investigated linear and non-linear vibrations of a coating/substrate system with an FGM interlayer. Again, all the above investigations were carried out in the context of two-dimensional elasticity theory. In the present paper, a functionally graded coating/substrate system of finite thickness subjected to transverse loading is analysed within the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. The analysis employs the three-dimensional elasticity solution for a functionally graded rectangular plate, recently developed by Kashtalyan (2004). Comparative study of FGM versus homogeneous coating is conducted. Dependence of stress and displacement fields in the coating/substrate system on the coating type, geometry and loading is examined and discussed. 2. Problem formulation A coating/substrate system of length a, width b and total thickness h is referred to a Cartesian co-ordinate system xyz (Fig. 1), so that 0 6 x 6 a, 0 6 y 6 b, 0 6 z 6 h. The substrate S has thickness h(1)(0 6 z 6 h(1)), Q ( x, y ) C
h (1) S
z
y x Fig. 1. Coating/substrate system under transverse loading.
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and the coating C has thickness h(2) = h h(1)(h(1) 6 z 6 h). Henceforth, all quantities associated with the substrate and the coating will be denoted with the superscripts (1) and (2), respectively. Let us assume that both the substrate and the coating are FGMs, with constant Poisson’s ratios m(1) = const and m(2) = const. The Young’s and shear moduli of the substrate and the coating are assumed to change continuously through the thickness, and the shear moduli are assumed to be exponential functions of the thickness in the form h z i ð1Þ GðkÞ ðzÞ ¼ gðkÞ exp cðkÞ 1 ; k ¼ 1; 2 h where c(k) are the inhomogeneity parameters, and g(k) = G(k)(h). The exponential variation of material properties with transverse co-ordinate was used by a number of researchers investigating functionally graded coatings, including Jin and Batra (1996); Chiu and Erdogan (2003); Guler and Erdogan (2004, 2006) and Shodja and Ghahremaninejad (2006). The coating and the substrate are assumed to be perfectly bonded, so that continuity of stresses and displacements exists at the coating/substrate interface, i.e. ð1Þ
ð2Þ
ð1Þ
ð2Þ
z ¼ hð1Þ : riz riz ¼ 0; ui ui ¼ 0; i ¼ x; y; z ðkÞ rij ðk
ð2Þ ðkÞ ui ðk
¼ 1; 2Þ are the components of the stress tensor, and ¼ 1; 2Þ are the components of the diswhere placement vector. The coating/substrate system is subjected to transverse loading Q on the top surface of the coating, so that ð2Þ ð2Þ z ¼ h : rzz ¼ Qðx; yÞ; rð2Þ xz ¼ ryz ¼ 0
ð3aÞ
pmx pny sin Qðx; yÞ ¼ qmn sin a b
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), is motivated by its practical significance, since any loading can be expressed as a Fourier series involving terms of this type. The bottom surface of the substrate is assumed to be load-free, i.e. ð1Þ ð1Þ ð1Þ ¼ rxz ¼ ryz ¼0 z ¼ 0 : rzz
ð4Þ
The Navier-type boundary conditions are assumed at the edges, so that ðkÞ ðkÞ x ¼ 0; a : rðkÞ xx ¼ 0; uy ¼ uz ¼ 0;
k ¼ 1; 2
ð5aÞ
ðkÞ ðkÞ y ¼ 0; b : rðkÞ yy ¼ 0; ux ¼ uz ¼ 0;
k ¼ 1; 2
ð5bÞ
The above boundary conditions, also known as Navier-type boundary conditions, have been widely used over the last three decades by researchers investigating thick plates in the context of three-dimensional elasticity theory, most recently by Gigliotti et al. (2007), but also by Liu (2000); Wu and Wardenier (1998); Fan and Ye (1990a,b); Wittrick (1987), and Srinivas and Rao (1970). The boundary conditions, Eq. (5) are representative of roller supports and analogous to simply supported edges in the plate theories. 3. Solution For an inhomogeneous isotropic material with the shear modulus G = G(z) and the Poisson’s ratio m = m(z) that depend on the transverse co-ordinate z only, Plevako (1971) derived three-dimensional equations of equilibrium in terms of displacements and developed general solutions to them. He showed that the displacements can always be expressed in terms of two functions which satisfy the second and the fourth order linear partial differential equations, respectively. If the Poisson’s ratios are constant, the displacements in the substrate and the coating can be expressed in terms of two functions, denoted L(k) = L(k)(x,y,z) and N(k) = N(k)(x,y,z)(k = 1,2), as follows 1 o2 oLðkÞ oN ðkÞ ðkÞ þ ð6aÞ uðkÞ ¼ m D x oz2 ox oy 2GðkÞ
M. Kashtalyan, M. Menshykova / International Journal of Solids and Structures 44 (2007) 5272–5288
uðkÞ y uðkÞ z
1 o2 oLðkÞ oN ðkÞ ðkÞ ¼ ðkÞ m D 2 oz oy ox 2G 1 o2 oLðkÞ o 1 o2 ðkÞ þ ¼ ðkÞ D 2 m D LðkÞ ; oz 2GðkÞ oz oz oz2 G
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ð6bÞ k ¼ 1; 2
ð6cÞ
From Eq. (6) and the constitutive equations for an isotropic inhomogeneous material (see Appendix A), the components of the stress tensor can be expressed in terms of functions L(k) and N(k) as 2 2 o o2 rðkÞ ¼ þ LðkÞ ð7aÞ zz ox2 oy 2 2 2 ðkÞ o o2 o2 LðkÞ ðkÞ o N ðkÞ þ G ð7bÞ þ rxz ¼ ox2 oy 2 oxoz oyoz 2 2 ðkÞ o o2 o2 LðkÞ ðkÞ o N G ð7cÞ rðkÞ ¼ þ yz ox2 oy 2 oyoz oxoz 2 o4 o2 N ðkÞ ðkÞ o ð7dÞ rðkÞ ¼ m D þ LðkÞ þ 2GðkÞ xx 2 2 2 oy ox oz oxoy 2 2 ðkÞ o4 ðkÞ o N ðkÞ ðkÞ o ðkÞ ð7eÞ ryy ¼ m D þ 2G L ox2 oy 2 oz2 oxoy 2 o2 o2 LðkÞ o o2 ðkÞ ðkÞ G rðkÞ ¼ m D ð7fÞ N ðkÞ ; k ¼ 1; 2 xy oz2 oxoy ox2 oy 2 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 2 1 1 o2 1 ðkÞ ðkÞ d D ðkÞ DL D 2 L ¼0 ð8aÞ 1 mðkÞ oz dz2 GðkÞ G DN ðkÞ þ
d oN ðkÞ ln GðkÞ ðzÞ ¼0 dz oz
ð8bÞ
It was shown by Plevako (1971) that if functions L(k) and N(k) allow separation of variables in the form ðkÞ
ðkÞ
LðkÞ ðx; y; zÞ ¼ w1 ðx; yÞ/1 ðzÞ;
ðkÞ
ðkÞ
N ðkÞ ðx; y; zÞ ¼ w2 ðx; yÞ/2 ðzÞ
ð9Þ ðkÞ
ðkÞ
then Eqs. (8a) and (8b) can be transformed into the Helmholz’s equation for functions w1 ðx; yÞ and w2 ðx; yÞ, ðkÞ ðkÞ and the forth- and the second-order ordinary differential equations for functions /1 ðzÞ and /2 ðzÞ. ðkÞ ðkÞ The form of functions w1 ðx; yÞ and w2 ðx; yÞ is influenced by the applied load and the boundary conditions on the edges of the plate. For sinusoidal loading, Eq. (3b), and Navier-type boundary on the edges, Eqs. (5a) and (5b), they can be chosen as (Kashtalyan, 2004) pmx pny ðkÞ pmx pny ðkÞ sin ; w2 ðx; yÞ ¼ cos cos ; k ¼ 1; 2 ð10Þ w1 ðx; yÞ ¼ sin a b a b Then boundary conditions on the edge, Eqs. (5a) and (5b) are satisfied exactly for any numbers n,m. The ðkÞ ðkÞ form of functions /1 ðzÞ and /2 ðzÞ depends on the functional variation of the shear moduli G(k) = G(k)(z). (k) When the shear moduli G = G(k)(z) of the substrate (k = 1) and the coating (k = 2) are exponential functions of the thickness co-ordinate in the form of Eq. (1), then Eqs. (8a) and (8b) are reduced to the following ordinary differential equations with constant coefficients (Kashtalyan, 2004) with respect to functions ðkÞ ðkÞ /1 ðzÞ and /2 ðzÞ
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h4
ðkÞ
ðkÞ
h2
ðkÞ
3 2 d4 /1 ðkÞ 3 d /1 ðkÞ 2 2 2 2 d /1 2c h þ ½c 2a h h dz4 dz3 dz2 ðkÞ ðkÞ d/ m ðkÞ ðkÞ 2 þ 2a2 cðkÞ h3 1 þ a2 h2 a2 h2 þ c /1 ¼ 0 dz 1 mðkÞ
ð11aÞ
ðkÞ
d2 /2 d/ ðkÞ þ cðkÞ h 2 a2 h2 /2 ¼ 0; dz2 dz
Here, a is a coefficient of the Helmholz’s equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r m2 n2 a¼p þ a b Following standard procedure, solutions to Eqs. (11a) and (11b) are found as ðkÞ " c z kðkÞ z lðkÞ z kðkÞ z lðkÞ z ðkÞ ðkÞ ðkÞ 4 cos þ A2 sinh cos /1 ðzÞ ¼ h exp A1 cosh 2h h h h h # kðkÞ z lðkÞ z kðkÞ z lðkÞ z ðkÞ ðkÞ sin þ A4 sinh sin þA3 cosh h h h h # ðkÞ " c z bðkÞ z bðkÞ z ðkÞ ðkÞ ðkÞ 2 þ A6 sinh /2 ðzÞ ¼ h exp A5 cosh 2h h h
ð11bÞ
ð12Þ
ð13aÞ
ð13bÞ
ðkÞ
Here, Aj ðj ¼ 1; . . . 6; k ¼ 1; 2Þ are 12 arbitrary constants that can be determined from the stress and displacement continuity conditions at the substrate/coating interface, Eq. (2), and the boundary conditions on the top and bottom surfaces of the coating/substrate system, Eqs. (3) and (4). Coefficients k(k) and l(k), and b(k), are the roots of characteristic equations corresponding to Eqs. (11a) and (11b), respectively. Expressions for them are given in Appendix B. ðkÞ ðkÞ ðkÞ ðkÞ Substitution of functions w1 ðx; yÞ and w2 ðx; yÞ, Eq. (10), and functions /1 ðzÞ and /2 ðzÞ, Eqs. (13a) and (13b), into Eq. (9) and then into Eqs. (6) and (7) gives the following expressions for displacements and stresses in a functionally graded coating/substrate system of finite thickness under transverse sinusoidal loading with exponential dependences of the shear moduli on the thickness co-ordinate uðkÞ x ¼
6 X
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
pmx pny sin a b
ð14aÞ
pmx pny cos a b
ð14bÞ
pmx pny sin a b
ð14cÞ
pmx pny sin a b
ð15aÞ
pmx pny sin a b
ð15bÞ
pmx pny cos a b
ð15cÞ
Aj U x;j ðzÞ cos
j¼1
uðkÞ y ¼
6 X
Aj U y;j ðzÞ sin
j¼1
uðkÞ z ¼
6 X
Aj U z;j ðzÞ sin
j¼1
rðkÞ zz ¼
6 X
Aj P zz;j ðzÞ sin
j¼1
rðkÞ xz ¼
6 X
Aj P xz;j ðzÞ cos
j¼1
rðkÞ yz ¼
6 X j¼1
Aj P yz;j ðzÞ sin
M. Kashtalyan, M. Menshykova / International Journal of Solids and Structures 44 (2007) 5272–5288
rðkÞ xx ¼
6 X
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
Aj P xx;j ðzÞ sin
j¼1
rðkÞ yy ¼
6 X
Aj P yy;j ðzÞ sin
j¼1
rðkÞ xy ¼
6 X
pmx pny sin a b
ð15dÞ
pmx pny sin a b
ð15eÞ
pmx pny cos a b
ð15fÞ
Aj P xy;j ðzÞ cos
j¼1 ðkÞ
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ðkÞ
Functions U i;j and P rt;j are specified in Appendix C. 4. Results and discussion In this section, a comparative study of two coating/substrate systems of finite thickness is presented. The first system, henceforth referred to as the HC system, has a homogeneous substrate with the shear modulus ð1Þ ð2Þ GHC ¼ const, and a homogeneous coating with the shear modulus GHC ¼ const. The second system, henceforth referred to as the FGMC system, has a homogeneous substrate, with the same shear modulus as in ð1Þ ð1Þ the HC system, i.e. GFGMC ¼ GHC . The coating in the FGMC system is functionally graded, with the shear ð2Þ ð1Þ ð2Þ modulus GFGMC that varies exponentially from GHC to GHC according to Eq. (1). Variation of the shear moduli through the thickness in the two systems is shown in Fig. 2. It simulates an important, from the engineering point of view, case of a soft substrate protected by a much stiffer coating. The homogeneous substrates in the HC and FGMC systems and the coating in the HC system are treated as functionally graded materials with the inhomogeneity parameters, c(k), sufficiently close to zero. It was shown by Kashtalyan (2004) that as the inhomogeneity parameter becomes smaller, the three-dimensional solution for an FGM rapidly converges to that for a homogeneous material. For example, for sufficiently small values of the inhomogeneity parameter (smaller than 105 by the absolute value), the difference between the computed out-of-plane displacements in functionally graded and homogeneous plates constituted less than 0.03% (Kashtalyan, 2004; Chernopiskii, 2006). ð2Þ The inhomogeneity parameter of the coating of the FGMC system, cFGMC , can be expressed in terms of the (2) stiffness gradient d , i.e. the ratio between the shear moduli values at the top and bottom surfaces of the coating, as ð2Þ
cFGMC ¼
h ln dð2Þ ; hð2Þ
ð2Þ
dð2Þ ¼
GHC ð1Þ
GHC
Fig. 2. Through-thickness variation of the shear moduli in two coating/substate systems: (a) HC system; (b) FGMC system.
ð16Þ
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zz ð0:5a; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; Fig. 3. Through-thickness variation of the normalised out-of-plane normal stress r (b) n1 = 0.33; (c) n1 = 0.8.
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xz ð0; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 4. Through-thickness variation of the normalised transverse shear stress r n1 = 0.33; (c) n1 = 0.8.
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xx ð0:5a; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 5. Through-thickness variation of the normalised in-plane normal stress r n1 = 0.33; (c) n1 = 0.8.
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xy ð0; 0; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 6. Through-thickness variation of the normalised in-plane shear stress r n1 = 0.33; (c) n1 = 0.8.
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In this study, the stiffness gradient is taken as d(2) = 10 (stiff coating, soft substrate), and the thickness of the coating is h(2) = 0.1h. The chosen value of d(2) does not necessarily represent a certain material, but is rather used to establish the effect of the coating type (homogeneous or graded) on the stress and displacement fields in the coating/substrate system. The Poisson’s ratios for both systems are taken as m(1) = m(2) = 0.3.
Fig. 7. Through-thickness variation of the normalised in-plane displacement ux (0,0.5b,z) in HC and FGMC systems: (a) n1 = 0.1; (b) n1 = 0.33; (c) n1 = 0.8.
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One can note from Eqs. (12) and (15), and Appendices B and C, that for any given inhomogeneity ratios, c(k), stresses and displacements in a coating/substrate system depend on two dimensionless parameters, n1 ¼ mh a and n2 ¼ nhb, which characterise the geometry of the system and the applied load along x and y co-ordinate
Fig. 8. Through-thickness variation of the normalised transverse displacement uz(0.5a,0.5b,z) in HC and FGMC systems: (a) n1 = 0.1; (b) n1 = 0.33; (c) n1 = 0.8.
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axes, respectively. The same value of ni represents a thicker plate under loading of a larger wavelength and a thinner plate under short-wave loading. ij ¼ rij =qmn and normalised displaceFigs. 3–8 show through-thickness variation of the normalised stresses r ð2Þ
G
ui
ments ui ¼ qHCh in HC and FGMC systems for three different values of n1 assuming n1 = n2. mn zz , through-thickness variation is similar for all considered values of n1 For the out-of-plane normal stress r (Fig. 3a–c). The type of the coating (graded or homogeneous), and the loading and geometry of the system, has almost no effect on this stress component. xz in the FGMC system (Fig. 4a–c) is close to Through-thickness variation of the transverse shear stress r parabolic profile, typical for homogeneous plates, for all considered values of n1. It worth pointing out that in xz has a peak in the coating (Fig. 4c), which is elimthe HC system with n1 = 0.8 the transverse shear stress r inated when a homogeneous coating is replaced with a graded one. The type of the coating has a very strong influence on the through-thickness variation of the in-plane norxx (Fig. 5a–c). In contrast to the HC system, the in-plane normal stress in the FGMC systems is mal stress r always continuous across the coating/substrate interface. However, this is achieved at the expense of the increase of the stress magnitude at the top surface of the coating. For example, for n1 = 0.1 (Fig. 5a), the xx in the homogeincrease is almost twofold. In the same time, while for n1 = 0.1 the in-plane normal stress r neous coating is compressive, in the graded coating it has to change from compressive at the top surface to tensile at the interface in order to match the value of stress in the substrate (Fig. 5b). For n1 = 0.8, quite xx changes from compresthe opposite happens. While in the homogeneous coating the in-plane normal stress r sive at the top surface to tensile at the interface, in the FGM coating it becomes compressive throughout the FGM coating (Fig. 5c). Also, for higher values of the dimensionless parameter n1, variation of stress through the thickness becomes non-linear. xy (Fig. 6a–c). Through-thickness variSimilar observations can be made about the in-plane shear stress r xy is strongly influenced by the type of the coating. In the FGMC system, as ation of the in-plane shear stress r opposed to the HC one, the in-plane shear stress is always continuous across the coating/substrate interface, yet this is achieved at the expense of an increase in the stress magnitude at the top surface. For n1 = 0.1 xy in the FGMC sys(Fig. 6a), the increase is almost twofold. The absolute value of the in-plane shear stress r tem at the coating/substrate interface is always smaller than in the HC system. Through-thickness variation of the in-plane displacement ux in the FGMC and HC systems changes from linear or close to linear for small values of n1 (Fig. 7a and b) to highly non-linear as n1 increases (Fig. 7c). For small values of n1, the transverse displacement uz of the two systems is almost constant through the thickness (Fig. 8a) and similar to that of a homogeneous plate without a coating, which agrees well with the assumption of constant deflection made in various plate theories. However, the type of the coating significantly affects the magnitude of uz. In the FGMC system, the magnitude at the top surface of the coating, which is also the maximum absolute value of uz for the whole system, is by more than 30% higher than the corresponding value for the HC system, due to some loss of flexural stiffness provided by the coating (Fig. 8a). For higher values of n1 > 0.33 (Fig. 8b and c), variation of the transverse displacement through the thickness becomes highly non-linear. For n1 = 0.33, the maximum absolute value of the uz is reached at the coating/substrate interface (Fig. 8b), while for n1 = 0.8 it is reached at the top surface of the coating (Fig. 8c). This emphasizes once again the necessity of 3-D type analysis over those using plate theory assumptions. 5. Conclusions In the present paper, elastic deformation of the functionally graded coating/substrate system of finite thickness is investigated in the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. The approach makes use of the general solution of the equilibrium equations for inhomogeneous isotropic media obtained earlier by Plevako (1971) and its recent application to functionally graded plates with exponential variation of the Young’s modulus through the thickness by Kashtalyan (2004).
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Comparative study of two coating/substrate systems, one with a homogeneous coating and the other with a functionally graded one, revealed the effect of stiffness gradient in the coating on stresses and displacements in the system. Use of the functionally graded coating eliminates discontinuity of the in-plane normal stress across the coating/substrate. Also, for higher values of the dimensionless parameter that characterises geometry of the system and the applied loading, the in-plane normal stress can become compressive throughout the FGM coating, and the transverse shear stress can vary smoothly and have no peak in the coating.
Acknowledgements Financial support of this research by Engineering and Physical Sciences Research Council (UK) (EPSRC/ GR/A10031/2) is gratefully acknowledged.
Appendix A The constitutive equations for inhomogeneous isotropic material in terms of stresses and displacements
rðkÞ zz rðkÞ xx rðkÞ xy
! ðkÞ ðkÞ ouðkÞ ouz mðkÞ ouðkÞ ouðkÞ y ðkÞ oux ðkÞ ðkÞ ðkÞ ðkÞ z z þ þ þ ¼ 2G e ; rxz ¼ G ; ryz ¼ G oz 1 2mðkÞ oz ox oz oy ! ðkÞ ouðkÞ oux mðkÞ mðkÞ y ðkÞ ðkÞ ðkÞ þ þ ¼ 2GðkÞ e ¼ 2G eðkÞ ; r yy ðkÞ ox 1 2m oy 1 2mðkÞ ! ðkÞ ouðkÞ ouðkÞ ouðkÞ ouðkÞ y y ðkÞ oux þ þ z ¼G ; eðkÞ ¼ x þ oy ox ox oy oz ðkÞ
Appendix B Roots of characteristic equation corresponding to Eq. (11a) kðkÞ lðkÞ
!
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u u1 4 mðkÞ 2 ðkÞ 2 t b þ bðkÞ þ cð4Þ a2 h2 ¼ 2 1 mðkÞ
Roots of characteristic equation corresponding to Eq. (11b)
bðkÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cðkÞ þ a2 h2 ¼ 4
Appendix C ðkÞ
Functions U i;j involved in Eqs. (14a)–(14c).
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h i qmn h pmh d2 ðkÞ ðkÞ z ðkÞ 2 2 ðkÞ ðkÞ exp c 1 m a h fj ðzÞ þ ðm 1Þ 2 fj ðzÞ ¼ ðkÞ 2g a h dz h i qmn h pnh d2 ðkÞ ðkÞ ðkÞ z ðkÞ 2 2 ðkÞ ðkÞ exp c 1 m a h fj ðzÞ þ ðm 1Þ 2 fj ðzÞ U y;j ðzÞ ¼ ðkÞ 2g b h dz 2 h i qmn h pmh d3 ðkÞ ðkÞ ðkÞ ðkÞ z ðkÞ ðkÞ d U z;j ðzÞ ¼ ðkÞ exp c 1 ðm 1Þ c f ðzÞ þ 3 fj ðzÞ 2g a h dz2 j dz d ðkÞ ðkÞ a2 h2 ðmðkÞ 2Þ fj ðzÞ mðkÞ cðkÞ fj ðzÞ ; j ¼ 1; . . . 4; dz
ðkÞ U x;j ðzÞ
qmn h pnh ðkÞ f ðzÞ 2gðkÞ b j q h pmh ðkÞ ðkÞ f ðzÞ U y;j ðzÞ ¼ mnðkÞ 2g a j ðkÞ
U x;j ðzÞ ¼
ðkÞ
U z;j ðzÞ ¼ 0;
j ¼ 5; 6:
ðkÞ
Functions P rt;j involved in Eqs. (15a)–(15f) ðkÞ
ðkÞ
P zz;j ðzÞ ¼ qmn a4 h4 fj ðzÞ 2 d ðkÞ ðkÞ 2 2 pmh f ðzÞ P xz;j ðzÞ ¼ qmn a h a dz j 2 d ðkÞ ðkÞ 2 2 pnh f ðzÞ P yz;j ðzÞ ¼ qmn a h b dz j " # 2 2 2 2 2 pnh pnh d pmh d ðkÞ ðkÞ ðkÞ ðkÞ P xx;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ mðkÞ f ðzÞ f ðzÞ b b a dz2 j dz2 j " # 2 2 2 2 2 pmh pmh d pnh d ðkÞ ðkÞ ðkÞ ðkÞ P yy;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ mðkÞ f ðzÞ f ðzÞ a a b dz2 j dz2 j pmh pnh d2 ðkÞ ðkÞ ðkÞ P xy;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ þ ð1 mðkÞ Þ 2 fj ðzÞ ; j ¼ 1; . . . 4; a b dz ðkÞ
P zz;j ðzÞ ¼ 0
h z i d pnh ðkÞ f ðzÞ ¼ qmn exp cðkÞ 1 b h dz j h z i d pmh ðkÞ ðkÞ f ðzÞ P yz;j ðzÞ ¼ qmn exp cðkÞ 1 a h dz j h z i pnh pmh ðkÞ ðkÞ P xx;j ðzÞ ¼ 2qmn exp cðkÞ 1 fj ðzÞ b a h h z i pnh pmh ðkÞ ðkÞ P yy;j ðzÞ ¼ 2qmn exp cðkÞ 1 fj ðzÞ b a h " 2 2 # h z i pmh pnh ðkÞ ðkÞ P xy;j ðzÞ ¼ qmn exp cðkÞ 1 fj ðzÞ; a b h
ðkÞ P xz;j ðzÞ
j ¼ 5; 6:
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ðkÞ
In the expressions above, functions fj ðzÞðj ¼ 1; . . . 6Þ are ðkÞ ðkÞ c z kðkÞ z lðkÞ z c z kðkÞ z lðkÞ z ðkÞ ðkÞ cos ; f2 ðzÞ ¼ exp cos f1 ðzÞ ¼ exp cosh sinh 2h h h 2h h h ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ c z k z l z c z k z l z ðkÞ ðkÞ sin ; f4 ðzÞ ¼ exp sin ; f3 ðzÞ ¼ exp cosh sinh 2h h h 2h h h ðkÞ ðkÞ c z bðkÞ z c z bðkÞ z ðkÞ ðkÞ ; f6 ðzÞ ¼ exp : f5 ðzÞ ¼ exp cosh sinh 2h h 2h h References Bao, G., Cai, H., 1997. Delamination cracking in functionally graded coating/metal substrate systems. Acta Materialia 45 (3), 1055–1066. Brovarone, C.V., Verne, E., Krajewski, A., Ravaglioli, A., 2001. Graded coatings on ceramic substrates for biomedical applications. Journal of the European Ceramic Society 21 (16), 2855–2862. Chen, Y.F., Erdogan, F., 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids 44 (5), 771–787. Chernopiskii, D.I., 2006. On the numerical investigation of the stress-strain state of non-thin elastic plate. International Applied Mechanics 42 (3), 119–126. Chi, S., Chung, Y.L., 2003. Cracking in coating-substrate system with multi-layered and FGM coatings. Engineering Fracture Mechanics 70 (10), 1227–1243. Chiu, T.C., Erdogan, F., 2003. Debonding of graded coatings under in-plane compression. International Journal of Solids and Structures 40 (25), 7155–7179. Choi, H.J., 2004. Impact response of a surface crack in a coating/substrate system with a functionally grade interlayer: antiplane deformation. International journal of Solids and Stuctures 41 (20), 5631–5645. Erdogan, F., Ozturk, M., 1995. Periodic cracking of functionally graded coatings. International Journal of Mechanical Sciences 33 (15), 2179–2195. Fan, J., Ye, J., 1990a. An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers. International Journal of Solids and Structures 26 (5-6), 655–662. Fan, J., Ye, J., 1990b. A series solution of the exact equation for thick orthotropic plates. International Journal of Solids and Structures 26 (17), 773–778. Foppiano, S., Marshall, S.J., Saiz, E., Tomsia, A.P., Marshall, G.W., 2006. Functionally graded bioactive coatings: reproducibility and stability of the coating under cell culture conditions. Acta Biomaterialia 2 (2), 133–142. Gigliotti, M., Jacquemin, F., Vautrin, A., 2007. Assessment of approximate models to evaluate transient and cyclical hygrothermoelastic stress in composite plates. International Journal of Solids and Structures 44 (3-4), 733–759. Grainger, S., Blunt, J., 1998. Engineering Coatings, Second ed. Woodhead Publishing Ltd, Abingdon. Guler, M.A., Erdogan, F., 2004. Contact mechanics of graded coatings. International Journal of Solids and Structures 41 (14), 3865– 3889. Guler, M.A., Erdogan, F., 2006. Contact mechanics of two deformable elastic solids with graded coatings. Mechanics of Materials 38 (7), 633–647. Guo, L.C., Wu, L.Z., Zeng, T., Ma, L., 2004a. The dynamics fracture behavior of a functionally graded coating-substrate system. Composite Structures 64 (3-4), 433–441. Guo, L.C., Wu, L.Z., Ma, L., 2004b. The interface crack problem under a concentrated load for a functionally graded coating-substrate composite system. Composite Structures 63 (3-4), 397–406. Huang, G.Y., Wang, Y.S., Gross, D., 2002. Fracture analysis of functionally graded coatings: antiplane deformation. European Journal of Mechanics—A/Solids 21 (3), 391–400. Huang, G.Y., Wang, YS., Yu, S.W., 2005. A new model for fracture analysis of functionally graded coatings under plane deformation. Mechanics of Materials 37 (4), 507–516. Jin, Z.H., Batra, R.C., 1996. Interface cracking between functionally graded coatings and a substrate under antiplane shear. International Journal of Engineering Science 34 (15), 1705–1716. Kadioglu, S., Dag, S., Yahsi, S., 1998. Crack problem for a functionally graded layer on an elastic foundation. International Journal of Fracture 94 (1), 63–77. Kashtalyan, M., 2004. Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics A/Solids 23 (5), 833–854. Kim, J.J., Kim, W.J., Choi, D.J., Park, J.Y., Ryu, W.S., 2005. Design of a C/SiC functionally graded coating for the oxidation protection of C/C composites. Carbon 43 (8), 1749–1757. Koizumi, M., 1993. Concept of FGM. Ceramic Transactions 34, 3–10. Liew, K.M., Yang, J., Wu, Y.F., 2006. Nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient. Computer Methods in Applied Mechanics and Engineering 195 (9–12), 1007–1026.
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Liu, F.L., 2000. Static analysis of thick rectangular laminated plates: three-dimensional elasticity solutions via differential quadrature element method. International Journal of Solids and Structures 37 (51), 7671–7688. Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A., Ford, R.G., 1999. Functionally Graded Materials: Design. Processing and Applications. Kluwer Academic Publishers, Boston, MA. Pindera, M.J., Aboudi, J., Arnold, S.M., 2002. Analysis of spallation mechanism in thermal barrier coating with graded bond coats using the higher-order theory for FGMs. Engineering Fracture Mechanics 69 (14-16), 1587–1606. Pindera, M.J., Aboudi, J., Arnold, S.M., 2005. Analysis of the spallation mechanism suppression in plasma-sprayed TBCs through the use of heterogeneous bond coat architectures. International Journal of Plasticity 21 (6), 1061–1096. Plevako, V.P., 1971. On the theory of elasticity of inhomogeneous media. Journal of Applied Mathematics and Mechanics 35 (5), 806–813. Saizonou, C., Kouitat-Njiwa, R., von Stebut, J., 2002. Surface engineering with functionally graded coatings: a numerical study based on the boundary element method. Surface and Coatings and Technology 153 (2-3), 290–297. Schulz, U., Peters, M., Bach, F.W., Tegeder, G., 2003. Graded coatings for thermal, wear and corrosion barriers. Materials Science and Engineering A 362 (1-2), 61–80. Shbeeb, N.I., Binienda, W.K., 1999. Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness. Engineering Fracture Mechanics 64 (6), 693–720. Shodja, H.M., Ghahremaninejad, A., 2006. An FGM coated elastic solid under thermomechanical loading: a two dimensional linear elastic approach. Surface and Coatings Technology 200 (12-13), 4050–4064. Srinivas, S., Rao, A.K., 1970. Bending, vibration and buckling of simply-supported thick orthotropic rectangular plates and laminates. International Journal of Solids and Structures 6 (6), 1463–1481. Suresh, S., Mortensen, A., 1998. Fundamentals of Functionally Graded Materials. Maney Publishing, London. Wang, Y.S., Huang, G.Y., Gross, D., 2005. The fracture analysis of a graded coating/substrate system of finite thickness with arbitrary spatial variations of coating properties: plane deformation. Engineering Fracture Mechanics 72 (12), 1942–1953. Wittrick, W.H., 1987. Analytical three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin’s plate theory. International Journal of Solids and Structures 23 (4), 441–464. Wu, Z.J., Wardenier, J., 1998. Further investigation on the exact elasticity solution for anisotropic thick rectangular plates. International Journal of Solids and Structures 35 (7-8), 747–758. Yakovlev, A., Bertrand, Ph., Smurov, I., 2004. Laser cladding of wear resistant metal matrix composite coatings. Thin Solid Films 453–454, 133–138.
Three-dimensional elastic deformation of a functionally graded coating/substrate system M. Kashtalyan *, M. Menshykova
1
Centre for Micro- and Nanomechanics, School of Engineering and Physical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, UK Received 10 October 2006; received in revised form 12 December 2006; accepted 27 December 2006 Available online 31 December 2006
Abstract The concept of functionally graded material (FGM) is actively explored in coating design for the purpose of eliminating the mismatch of material properties at the coating/substrate interface, typical for conventional coatings, which can lead to cracking, debonding and eventual functional failure of the coating. In this paper, an FGM coating/substrate system of finite thickness subjected to transverse loading is analysed within the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. A comparative study of FGM versus homogeneous coating is conducted, and the dependence of stress and displacement fields in the coating substrate/system on the type of coating, geometry and loading is examined and discussed. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Functionally graded material; Coating; Three-dimensional elasticity theory
1. Introduction Coatings play an important role in a variety of applications protecting metallic or ceramic substrates against oxidation, heat penetration, wear and corrosion (Grainger and Blunt, 1998). Conventional coatings usually consist of one or two homogeneous layers deposited on a substrate. Due to the mismatch of thermomechanical properties between the coating and the substrate, conventional coatings are susceptible to cracking and debonding. To overcome this disadvantage and increase resistance of coatings to functional failure, the concept of functionally graded material (FGM) is being actively explored in coating design. FGM refers to heterogeneous composite material with gradient compositional variation of the constituents from one surface of the material to the other which results in continuously varying material properties (Koizumi, 1993; Suresh and Mortensen, 1998; Miyamoto et al., 1999). Currently, a number of approaches to generate FGM coatings for specific applications and/or loading conditions are being developed by materials scientists (Brovarone et al., 2001; Schulz et al., 2003; Yakovlev et al., 2004; Kim et al., 2005; Foppiano et al., 2006). *
1
Corresponding author. Tel./fax: +44 1224 272519. E-mail addresses: [email protected] (M. Kashtalyan), [email protected] (M. Menshykova). Tel.: +44 1224 273326; fax: +44 1224 272519.
0020-7683/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.12.035
M. Kashtalyan, M. Menshykova / International Journal of Solids and Structures 44 (2007) 5272–5288
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Theoretical modelling of FGM coatings has been focused predominantly on prediction of their fracture behaviour (Erdogan and Ozturk, 1995; Chen and Erdogan, 1996; Jin and Batra, 1996; Bao and Cai, 1997; Kadioglu et al., 1998; Huang et al., 2002; Chiu and Erdogan, 2003; Chi and Chung, 2003; Guo et al., 2004a; Choi, 2004; Huang et al., 2005). Several researches have addressed contact response of FGM coatings. Saizonou et al. (2002) carried out a numerical study (by boundary element method) of an FGM-coated elastic solid under normal and sliding contact loading. Guler and Erdogan (2004, 2006) investigated behaviour of FGM-coated solid under sliding contact loading provided by a sliding rigid stamp and contact of two FGM-coated elastic solids. Exponential variation of the Young’s modulus of the coating was adopted in both cases. Shodja and Ghahremaninejad (2006) examined an FGM-coated elastic solid under thermomechanical loading due to mixed normal and tangential Hertzian pressure. Thermomechanical properties were assumed to vary exponentially through the thickness. It should be noted that in all the above studies the thickness of the substrate was assumed to be infinite, and the investigations were carried out in the context of two-dimensional elasticity. For coating/substrate systems of finite thickness, Shbeeb and Binienda (1999) analysed interfacial fracture of a coating/substrate system with an FGM interphase; stress intensity factors and strain energy release rate for a crack at the interphase/substrate interface were calculated. Using the higher-order theory for FGMs, Pindera et al. (2002, 2005) examined fracture mechanisms in thermal barrier coatings with functionally graded bond coats under uniform cyclic thermal loading. Interfacial fracture of an FGM coating/substrate system subjected to concentrated transverse loading was studied by Guo et al. (2004b). Wang et al. (2005) carried out fracture analysis of an FGM-coated strip with a crack in the coating. To calculate stress intensity factors, the FGM coating was modelled as multi-layered medium, with elastic modulus varying linearly within each sub-layer, so that they are continuous on the sub-interfaces. Liew et al. (2006) investigated linear and non-linear vibrations of a coating/substrate system with an FGM interlayer. Again, all the above investigations were carried out in the context of two-dimensional elasticity theory. In the present paper, a functionally graded coating/substrate system of finite thickness subjected to transverse loading is analysed within the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. The analysis employs the three-dimensional elasticity solution for a functionally graded rectangular plate, recently developed by Kashtalyan (2004). Comparative study of FGM versus homogeneous coating is conducted. Dependence of stress and displacement fields in the coating/substrate system on the coating type, geometry and loading is examined and discussed. 2. Problem formulation A coating/substrate system of length a, width b and total thickness h is referred to a Cartesian co-ordinate system xyz (Fig. 1), so that 0 6 x 6 a, 0 6 y 6 b, 0 6 z 6 h. The substrate S has thickness h(1)(0 6 z 6 h(1)), Q ( x, y ) C
h (1) S
z
y x Fig. 1. Coating/substrate system under transverse loading.
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and the coating C has thickness h(2) = h h(1)(h(1) 6 z 6 h). Henceforth, all quantities associated with the substrate and the coating will be denoted with the superscripts (1) and (2), respectively. Let us assume that both the substrate and the coating are FGMs, with constant Poisson’s ratios m(1) = const and m(2) = const. The Young’s and shear moduli of the substrate and the coating are assumed to change continuously through the thickness, and the shear moduli are assumed to be exponential functions of the thickness in the form h z i ð1Þ GðkÞ ðzÞ ¼ gðkÞ exp cðkÞ 1 ; k ¼ 1; 2 h where c(k) are the inhomogeneity parameters, and g(k) = G(k)(h). The exponential variation of material properties with transverse co-ordinate was used by a number of researchers investigating functionally graded coatings, including Jin and Batra (1996); Chiu and Erdogan (2003); Guler and Erdogan (2004, 2006) and Shodja and Ghahremaninejad (2006). The coating and the substrate are assumed to be perfectly bonded, so that continuity of stresses and displacements exists at the coating/substrate interface, i.e. ð1Þ
ð2Þ
ð1Þ
ð2Þ
z ¼ hð1Þ : riz riz ¼ 0; ui ui ¼ 0; i ¼ x; y; z ðkÞ rij ðk
ð2Þ ðkÞ ui ðk
¼ 1; 2Þ are the components of the stress tensor, and ¼ 1; 2Þ are the components of the diswhere placement vector. The coating/substrate system is subjected to transverse loading Q on the top surface of the coating, so that ð2Þ ð2Þ z ¼ h : rzz ¼ Qðx; yÞ; rð2Þ xz ¼ ryz ¼ 0
ð3aÞ
pmx pny sin Qðx; yÞ ¼ qmn sin a b
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), is motivated by its practical significance, since any loading can be expressed as a Fourier series involving terms of this type. The bottom surface of the substrate is assumed to be load-free, i.e. ð1Þ ð1Þ ð1Þ ¼ rxz ¼ ryz ¼0 z ¼ 0 : rzz
ð4Þ
The Navier-type boundary conditions are assumed at the edges, so that ðkÞ ðkÞ x ¼ 0; a : rðkÞ xx ¼ 0; uy ¼ uz ¼ 0;
k ¼ 1; 2
ð5aÞ
ðkÞ ðkÞ y ¼ 0; b : rðkÞ yy ¼ 0; ux ¼ uz ¼ 0;
k ¼ 1; 2
ð5bÞ
The above boundary conditions, also known as Navier-type boundary conditions, have been widely used over the last three decades by researchers investigating thick plates in the context of three-dimensional elasticity theory, most recently by Gigliotti et al. (2007), but also by Liu (2000); Wu and Wardenier (1998); Fan and Ye (1990a,b); Wittrick (1987), and Srinivas and Rao (1970). The boundary conditions, Eq. (5) are representative of roller supports and analogous to simply supported edges in the plate theories. 3. Solution For an inhomogeneous isotropic material with the shear modulus G = G(z) and the Poisson’s ratio m = m(z) that depend on the transverse co-ordinate z only, Plevako (1971) derived three-dimensional equations of equilibrium in terms of displacements and developed general solutions to them. He showed that the displacements can always be expressed in terms of two functions which satisfy the second and the fourth order linear partial differential equations, respectively. If the Poisson’s ratios are constant, the displacements in the substrate and the coating can be expressed in terms of two functions, denoted L(k) = L(k)(x,y,z) and N(k) = N(k)(x,y,z)(k = 1,2), as follows 1 o2 oLðkÞ oN ðkÞ ðkÞ þ ð6aÞ uðkÞ ¼ m D x oz2 ox oy 2GðkÞ
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uðkÞ y uðkÞ z
1 o2 oLðkÞ oN ðkÞ ðkÞ ¼ ðkÞ m D 2 oz oy ox 2G 1 o2 oLðkÞ o 1 o2 ðkÞ þ ¼ ðkÞ D 2 m D LðkÞ ; oz 2GðkÞ oz oz oz2 G
5275
ð6bÞ k ¼ 1; 2
ð6cÞ
From Eq. (6) and the constitutive equations for an isotropic inhomogeneous material (see Appendix A), the components of the stress tensor can be expressed in terms of functions L(k) and N(k) as 2 2 o o2 rðkÞ ¼ þ LðkÞ ð7aÞ zz ox2 oy 2 2 2 ðkÞ o o2 o2 LðkÞ ðkÞ o N ðkÞ þ G ð7bÞ þ rxz ¼ ox2 oy 2 oxoz oyoz 2 2 ðkÞ o o2 o2 LðkÞ ðkÞ o N G ð7cÞ rðkÞ ¼ þ yz ox2 oy 2 oyoz oxoz 2 o4 o2 N ðkÞ ðkÞ o ð7dÞ rðkÞ ¼ m D þ LðkÞ þ 2GðkÞ xx 2 2 2 oy ox oz oxoy 2 2 ðkÞ o4 ðkÞ o N ðkÞ ðkÞ o ðkÞ ð7eÞ ryy ¼ m D þ 2G L ox2 oy 2 oz2 oxoy 2 o2 o2 LðkÞ o o2 ðkÞ ðkÞ G rðkÞ ¼ m D ð7fÞ N ðkÞ ; k ¼ 1; 2 xy oz2 oxoy ox2 oy 2 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 2 1 1 o2 1 ðkÞ ðkÞ d D ðkÞ DL D 2 L ¼0 ð8aÞ 1 mðkÞ oz dz2 GðkÞ G DN ðkÞ þ
d oN ðkÞ ln GðkÞ ðzÞ ¼0 dz oz
ð8bÞ
It was shown by Plevako (1971) that if functions L(k) and N(k) allow separation of variables in the form ðkÞ
ðkÞ
LðkÞ ðx; y; zÞ ¼ w1 ðx; yÞ/1 ðzÞ;
ðkÞ
ðkÞ
N ðkÞ ðx; y; zÞ ¼ w2 ðx; yÞ/2 ðzÞ
ð9Þ ðkÞ
ðkÞ
then Eqs. (8a) and (8b) can be transformed into the Helmholz’s equation for functions w1 ðx; yÞ and w2 ðx; yÞ, ðkÞ ðkÞ and the forth- and the second-order ordinary differential equations for functions /1 ðzÞ and /2 ðzÞ. ðkÞ ðkÞ The form of functions w1 ðx; yÞ and w2 ðx; yÞ is influenced by the applied load and the boundary conditions on the edges of the plate. For sinusoidal loading, Eq. (3b), and Navier-type boundary on the edges, Eqs. (5a) and (5b), they can be chosen as (Kashtalyan, 2004) pmx pny ðkÞ pmx pny ðkÞ sin ; w2 ðx; yÞ ¼ cos cos ; k ¼ 1; 2 ð10Þ w1 ðx; yÞ ¼ sin a b a b Then boundary conditions on the edge, Eqs. (5a) and (5b) are satisfied exactly for any numbers n,m. The ðkÞ ðkÞ form of functions /1 ðzÞ and /2 ðzÞ depends on the functional variation of the shear moduli G(k) = G(k)(z). (k) When the shear moduli G = G(k)(z) of the substrate (k = 1) and the coating (k = 2) are exponential functions of the thickness co-ordinate in the form of Eq. (1), then Eqs. (8a) and (8b) are reduced to the following ordinary differential equations with constant coefficients (Kashtalyan, 2004) with respect to functions ðkÞ ðkÞ /1 ðzÞ and /2 ðzÞ
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h4
ðkÞ
ðkÞ
h2
ðkÞ
3 2 d4 /1 ðkÞ 3 d /1 ðkÞ 2 2 2 2 d /1 2c h þ ½c 2a h h dz4 dz3 dz2 ðkÞ ðkÞ d/ m ðkÞ ðkÞ 2 þ 2a2 cðkÞ h3 1 þ a2 h2 a2 h2 þ c /1 ¼ 0 dz 1 mðkÞ
ð11aÞ
ðkÞ
d2 /2 d/ ðkÞ þ cðkÞ h 2 a2 h2 /2 ¼ 0; dz2 dz
Here, a is a coefficient of the Helmholz’s equation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r m2 n2 a¼p þ a b Following standard procedure, solutions to Eqs. (11a) and (11b) are found as ðkÞ " c z kðkÞ z lðkÞ z kðkÞ z lðkÞ z ðkÞ ðkÞ ðkÞ 4 cos þ A2 sinh cos /1 ðzÞ ¼ h exp A1 cosh 2h h h h h # kðkÞ z lðkÞ z kðkÞ z lðkÞ z ðkÞ ðkÞ sin þ A4 sinh sin þA3 cosh h h h h # ðkÞ " c z bðkÞ z bðkÞ z ðkÞ ðkÞ ðkÞ 2 þ A6 sinh /2 ðzÞ ¼ h exp A5 cosh 2h h h
ð11bÞ
ð12Þ
ð13aÞ
ð13bÞ
ðkÞ
Here, Aj ðj ¼ 1; . . . 6; k ¼ 1; 2Þ are 12 arbitrary constants that can be determined from the stress and displacement continuity conditions at the substrate/coating interface, Eq. (2), and the boundary conditions on the top and bottom surfaces of the coating/substrate system, Eqs. (3) and (4). Coefficients k(k) and l(k), and b(k), are the roots of characteristic equations corresponding to Eqs. (11a) and (11b), respectively. Expressions for them are given in Appendix B. ðkÞ ðkÞ ðkÞ ðkÞ Substitution of functions w1 ðx; yÞ and w2 ðx; yÞ, Eq. (10), and functions /1 ðzÞ and /2 ðzÞ, Eqs. (13a) and (13b), into Eq. (9) and then into Eqs. (6) and (7) gives the following expressions for displacements and stresses in a functionally graded coating/substrate system of finite thickness under transverse sinusoidal loading with exponential dependences of the shear moduli on the thickness co-ordinate uðkÞ x ¼
6 X
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
pmx pny sin a b
ð14aÞ
pmx pny cos a b
ð14bÞ
pmx pny sin a b
ð14cÞ
pmx pny sin a b
ð15aÞ
pmx pny sin a b
ð15bÞ
pmx pny cos a b
ð15cÞ
Aj U x;j ðzÞ cos
j¼1
uðkÞ y ¼
6 X
Aj U y;j ðzÞ sin
j¼1
uðkÞ z ¼
6 X
Aj U z;j ðzÞ sin
j¼1
rðkÞ zz ¼
6 X
Aj P zz;j ðzÞ sin
j¼1
rðkÞ xz ¼
6 X
Aj P xz;j ðzÞ cos
j¼1
rðkÞ yz ¼
6 X j¼1
Aj P yz;j ðzÞ sin
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rðkÞ xx ¼
6 X
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ
Aj P xx;j ðzÞ sin
j¼1
rðkÞ yy ¼
6 X
Aj P yy;j ðzÞ sin
j¼1
rðkÞ xy ¼
6 X
pmx pny sin a b
ð15dÞ
pmx pny sin a b
ð15eÞ
pmx pny cos a b
ð15fÞ
Aj P xy;j ðzÞ cos
j¼1 ðkÞ
5277
ðkÞ
Functions U i;j and P rt;j are specified in Appendix C. 4. Results and discussion In this section, a comparative study of two coating/substrate systems of finite thickness is presented. The first system, henceforth referred to as the HC system, has a homogeneous substrate with the shear modulus ð1Þ ð2Þ GHC ¼ const, and a homogeneous coating with the shear modulus GHC ¼ const. The second system, henceforth referred to as the FGMC system, has a homogeneous substrate, with the same shear modulus as in ð1Þ ð1Þ the HC system, i.e. GFGMC ¼ GHC . The coating in the FGMC system is functionally graded, with the shear ð2Þ ð1Þ ð2Þ modulus GFGMC that varies exponentially from GHC to GHC according to Eq. (1). Variation of the shear moduli through the thickness in the two systems is shown in Fig. 2. It simulates an important, from the engineering point of view, case of a soft substrate protected by a much stiffer coating. The homogeneous substrates in the HC and FGMC systems and the coating in the HC system are treated as functionally graded materials with the inhomogeneity parameters, c(k), sufficiently close to zero. It was shown by Kashtalyan (2004) that as the inhomogeneity parameter becomes smaller, the three-dimensional solution for an FGM rapidly converges to that for a homogeneous material. For example, for sufficiently small values of the inhomogeneity parameter (smaller than 105 by the absolute value), the difference between the computed out-of-plane displacements in functionally graded and homogeneous plates constituted less than 0.03% (Kashtalyan, 2004; Chernopiskii, 2006). ð2Þ The inhomogeneity parameter of the coating of the FGMC system, cFGMC , can be expressed in terms of the (2) stiffness gradient d , i.e. the ratio between the shear moduli values at the top and bottom surfaces of the coating, as ð2Þ
cFGMC ¼
h ln dð2Þ ; hð2Þ
ð2Þ
dð2Þ ¼
GHC ð1Þ
GHC
Fig. 2. Through-thickness variation of the shear moduli in two coating/substate systems: (a) HC system; (b) FGMC system.
ð16Þ
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zz ð0:5a; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; Fig. 3. Through-thickness variation of the normalised out-of-plane normal stress r (b) n1 = 0.33; (c) n1 = 0.8.
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xz ð0; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 4. Through-thickness variation of the normalised transverse shear stress r n1 = 0.33; (c) n1 = 0.8.
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xx ð0:5a; 0:5b; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 5. Through-thickness variation of the normalised in-plane normal stress r n1 = 0.33; (c) n1 = 0.8.
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xy ð0; 0; zÞ in HC and FGMC systems: (a) n1 = 0.1; (b) Fig. 6. Through-thickness variation of the normalised in-plane shear stress r n1 = 0.33; (c) n1 = 0.8.
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In this study, the stiffness gradient is taken as d(2) = 10 (stiff coating, soft substrate), and the thickness of the coating is h(2) = 0.1h. The chosen value of d(2) does not necessarily represent a certain material, but is rather used to establish the effect of the coating type (homogeneous or graded) on the stress and displacement fields in the coating/substrate system. The Poisson’s ratios for both systems are taken as m(1) = m(2) = 0.3.
Fig. 7. Through-thickness variation of the normalised in-plane displacement ux (0,0.5b,z) in HC and FGMC systems: (a) n1 = 0.1; (b) n1 = 0.33; (c) n1 = 0.8.
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One can note from Eqs. (12) and (15), and Appendices B and C, that for any given inhomogeneity ratios, c(k), stresses and displacements in a coating/substrate system depend on two dimensionless parameters, n1 ¼ mh a and n2 ¼ nhb, which characterise the geometry of the system and the applied load along x and y co-ordinate
Fig. 8. Through-thickness variation of the normalised transverse displacement uz(0.5a,0.5b,z) in HC and FGMC systems: (a) n1 = 0.1; (b) n1 = 0.33; (c) n1 = 0.8.
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axes, respectively. The same value of ni represents a thicker plate under loading of a larger wavelength and a thinner plate under short-wave loading. ij ¼ rij =qmn and normalised displaceFigs. 3–8 show through-thickness variation of the normalised stresses r ð2Þ
G
ui
ments ui ¼ qHCh in HC and FGMC systems for three different values of n1 assuming n1 = n2. mn zz , through-thickness variation is similar for all considered values of n1 For the out-of-plane normal stress r (Fig. 3a–c). The type of the coating (graded or homogeneous), and the loading and geometry of the system, has almost no effect on this stress component. xz in the FGMC system (Fig. 4a–c) is close to Through-thickness variation of the transverse shear stress r parabolic profile, typical for homogeneous plates, for all considered values of n1. It worth pointing out that in xz has a peak in the coating (Fig. 4c), which is elimthe HC system with n1 = 0.8 the transverse shear stress r inated when a homogeneous coating is replaced with a graded one. The type of the coating has a very strong influence on the through-thickness variation of the in-plane norxx (Fig. 5a–c). In contrast to the HC system, the in-plane normal stress in the FGMC systems is mal stress r always continuous across the coating/substrate interface. However, this is achieved at the expense of the increase of the stress magnitude at the top surface of the coating. For example, for n1 = 0.1 (Fig. 5a), the xx in the homogeincrease is almost twofold. In the same time, while for n1 = 0.1 the in-plane normal stress r neous coating is compressive, in the graded coating it has to change from compressive at the top surface to tensile at the interface in order to match the value of stress in the substrate (Fig. 5b). For n1 = 0.8, quite xx changes from compresthe opposite happens. While in the homogeneous coating the in-plane normal stress r sive at the top surface to tensile at the interface, in the FGM coating it becomes compressive throughout the FGM coating (Fig. 5c). Also, for higher values of the dimensionless parameter n1, variation of stress through the thickness becomes non-linear. xy (Fig. 6a–c). Through-thickness variSimilar observations can be made about the in-plane shear stress r xy is strongly influenced by the type of the coating. In the FGMC system, as ation of the in-plane shear stress r opposed to the HC one, the in-plane shear stress is always continuous across the coating/substrate interface, yet this is achieved at the expense of an increase in the stress magnitude at the top surface. For n1 = 0.1 xy in the FGMC sys(Fig. 6a), the increase is almost twofold. The absolute value of the in-plane shear stress r tem at the coating/substrate interface is always smaller than in the HC system. Through-thickness variation of the in-plane displacement ux in the FGMC and HC systems changes from linear or close to linear for small values of n1 (Fig. 7a and b) to highly non-linear as n1 increases (Fig. 7c). For small values of n1, the transverse displacement uz of the two systems is almost constant through the thickness (Fig. 8a) and similar to that of a homogeneous plate without a coating, which agrees well with the assumption of constant deflection made in various plate theories. However, the type of the coating significantly affects the magnitude of uz. In the FGMC system, the magnitude at the top surface of the coating, which is also the maximum absolute value of uz for the whole system, is by more than 30% higher than the corresponding value for the HC system, due to some loss of flexural stiffness provided by the coating (Fig. 8a). For higher values of n1 > 0.33 (Fig. 8b and c), variation of the transverse displacement through the thickness becomes highly non-linear. For n1 = 0.33, the maximum absolute value of the uz is reached at the coating/substrate interface (Fig. 8b), while for n1 = 0.8 it is reached at the top surface of the coating (Fig. 8c). This emphasizes once again the necessity of 3-D type analysis over those using plate theory assumptions. 5. Conclusions In the present paper, elastic deformation of the functionally graded coating/substrate system of finite thickness is investigated in the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. The approach makes use of the general solution of the equilibrium equations for inhomogeneous isotropic media obtained earlier by Plevako (1971) and its recent application to functionally graded plates with exponential variation of the Young’s modulus through the thickness by Kashtalyan (2004).
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Comparative study of two coating/substrate systems, one with a homogeneous coating and the other with a functionally graded one, revealed the effect of stiffness gradient in the coating on stresses and displacements in the system. Use of the functionally graded coating eliminates discontinuity of the in-plane normal stress across the coating/substrate. Also, for higher values of the dimensionless parameter that characterises geometry of the system and the applied loading, the in-plane normal stress can become compressive throughout the FGM coating, and the transverse shear stress can vary smoothly and have no peak in the coating.
Acknowledgements Financial support of this research by Engineering and Physical Sciences Research Council (UK) (EPSRC/ GR/A10031/2) is gratefully acknowledged.
Appendix A The constitutive equations for inhomogeneous isotropic material in terms of stresses and displacements
rðkÞ zz rðkÞ xx rðkÞ xy
! ðkÞ ðkÞ ouðkÞ ouz mðkÞ ouðkÞ ouðkÞ y ðkÞ oux ðkÞ ðkÞ ðkÞ ðkÞ z z þ þ þ ¼ 2G e ; rxz ¼ G ; ryz ¼ G oz 1 2mðkÞ oz ox oz oy ! ðkÞ ouðkÞ oux mðkÞ mðkÞ y ðkÞ ðkÞ ðkÞ þ þ ¼ 2GðkÞ e ¼ 2G eðkÞ ; r yy ðkÞ ox 1 2m oy 1 2mðkÞ ! ðkÞ ouðkÞ ouðkÞ ouðkÞ ouðkÞ y y ðkÞ oux þ þ z ¼G ; eðkÞ ¼ x þ oy ox ox oy oz ðkÞ
Appendix B Roots of characteristic equation corresponding to Eq. (11a) kðkÞ lðkÞ
!
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u u1 4 mðkÞ 2 ðkÞ 2 t b þ bðkÞ þ cð4Þ a2 h2 ¼ 2 1 mðkÞ
Roots of characteristic equation corresponding to Eq. (11b)
bðkÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cðkÞ þ a2 h2 ¼ 4
Appendix C ðkÞ
Functions U i;j involved in Eqs. (14a)–(14c).
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h i qmn h pmh d2 ðkÞ ðkÞ z ðkÞ 2 2 ðkÞ ðkÞ exp c 1 m a h fj ðzÞ þ ðm 1Þ 2 fj ðzÞ ¼ ðkÞ 2g a h dz h i qmn h pnh d2 ðkÞ ðkÞ ðkÞ z ðkÞ 2 2 ðkÞ ðkÞ exp c 1 m a h fj ðzÞ þ ðm 1Þ 2 fj ðzÞ U y;j ðzÞ ¼ ðkÞ 2g b h dz 2 h i qmn h pmh d3 ðkÞ ðkÞ ðkÞ ðkÞ z ðkÞ ðkÞ d U z;j ðzÞ ¼ ðkÞ exp c 1 ðm 1Þ c f ðzÞ þ 3 fj ðzÞ 2g a h dz2 j dz d ðkÞ ðkÞ a2 h2 ðmðkÞ 2Þ fj ðzÞ mðkÞ cðkÞ fj ðzÞ ; j ¼ 1; . . . 4; dz
ðkÞ U x;j ðzÞ
qmn h pnh ðkÞ f ðzÞ 2gðkÞ b j q h pmh ðkÞ ðkÞ f ðzÞ U y;j ðzÞ ¼ mnðkÞ 2g a j ðkÞ
U x;j ðzÞ ¼
ðkÞ
U z;j ðzÞ ¼ 0;
j ¼ 5; 6:
ðkÞ
Functions P rt;j involved in Eqs. (15a)–(15f) ðkÞ
ðkÞ
P zz;j ðzÞ ¼ qmn a4 h4 fj ðzÞ 2 d ðkÞ ðkÞ 2 2 pmh f ðzÞ P xz;j ðzÞ ¼ qmn a h a dz j 2 d ðkÞ ðkÞ 2 2 pnh f ðzÞ P yz;j ðzÞ ¼ qmn a h b dz j " # 2 2 2 2 2 pnh pnh d pmh d ðkÞ ðkÞ ðkÞ ðkÞ P xx;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ mðkÞ f ðzÞ f ðzÞ b b a dz2 j dz2 j " # 2 2 2 2 2 pmh pmh d pnh d ðkÞ ðkÞ ðkÞ ðkÞ P yy;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ mðkÞ f ðzÞ f ðzÞ a a b dz2 j dz2 j pmh pnh d2 ðkÞ ðkÞ ðkÞ P xy;j ðzÞ ¼ qmn mðkÞ a2 h2 fj ðzÞ þ ð1 mðkÞ Þ 2 fj ðzÞ ; j ¼ 1; . . . 4; a b dz ðkÞ
P zz;j ðzÞ ¼ 0
h z i d pnh ðkÞ f ðzÞ ¼ qmn exp cðkÞ 1 b h dz j h z i d pmh ðkÞ ðkÞ f ðzÞ P yz;j ðzÞ ¼ qmn exp cðkÞ 1 a h dz j h z i pnh pmh ðkÞ ðkÞ P xx;j ðzÞ ¼ 2qmn exp cðkÞ 1 fj ðzÞ b a h h z i pnh pmh ðkÞ ðkÞ P yy;j ðzÞ ¼ 2qmn exp cðkÞ 1 fj ðzÞ b a h " 2 2 # h z i pmh pnh ðkÞ ðkÞ P xy;j ðzÞ ¼ qmn exp cðkÞ 1 fj ðzÞ; a b h
ðkÞ P xz;j ðzÞ
j ¼ 5; 6:
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ðkÞ
In the expressions above, functions fj ðzÞðj ¼ 1; . . . 6Þ are ðkÞ ðkÞ c z kðkÞ z lðkÞ z c z kðkÞ z lðkÞ z ðkÞ ðkÞ cos ; f2 ðzÞ ¼ exp cos f1 ðzÞ ¼ exp cosh sinh 2h h h 2h h h ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ c z k z l z c z k z l z ðkÞ ðkÞ sin ; f4 ðzÞ ¼ exp sin ; f3 ðzÞ ¼ exp cosh sinh 2h h h 2h h h ðkÞ ðkÞ c z bðkÞ z c z bðkÞ z ðkÞ ðkÞ ; f6 ðzÞ ¼ exp : f5 ðzÞ ¼ exp cosh sinh 2h h 2h h References Bao, G., Cai, H., 1997. Delamination cracking in functionally graded coating/metal substrate systems. Acta Materialia 45 (3), 1055–1066. Brovarone, C.V., Verne, E., Krajewski, A., Ravaglioli, A., 2001. Graded coatings on ceramic substrates for biomedical applications. Journal of the European Ceramic Society 21 (16), 2855–2862. Chen, Y.F., Erdogan, F., 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids 44 (5), 771–787. Chernopiskii, D.I., 2006. On the numerical investigation of the stress-strain state of non-thin elastic plate. International Applied Mechanics 42 (3), 119–126. Chi, S., Chung, Y.L., 2003. Cracking in coating-substrate system with multi-layered and FGM coatings. Engineering Fracture Mechanics 70 (10), 1227–1243. Chiu, T.C., Erdogan, F., 2003. Debonding of graded coatings under in-plane compression. International Journal of Solids and Structures 40 (25), 7155–7179. Choi, H.J., 2004. Impact response of a surface crack in a coating/substrate system with a functionally grade interlayer: antiplane deformation. International journal of Solids and Stuctures 41 (20), 5631–5645. Erdogan, F., Ozturk, M., 1995. Periodic cracking of functionally graded coatings. International Journal of Mechanical Sciences 33 (15), 2179–2195. Fan, J., Ye, J., 1990a. An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers. International Journal of Solids and Structures 26 (5-6), 655–662. Fan, J., Ye, J., 1990b. A series solution of the exact equation for thick orthotropic plates. International Journal of Solids and Structures 26 (17), 773–778. Foppiano, S., Marshall, S.J., Saiz, E., Tomsia, A.P., Marshall, G.W., 2006. Functionally graded bioactive coatings: reproducibility and stability of the coating under cell culture conditions. Acta Biomaterialia 2 (2), 133–142. Gigliotti, M., Jacquemin, F., Vautrin, A., 2007. Assessment of approximate models to evaluate transient and cyclical hygrothermoelastic stress in composite plates. International Journal of Solids and Structures 44 (3-4), 733–759. Grainger, S., Blunt, J., 1998. Engineering Coatings, Second ed. Woodhead Publishing Ltd, Abingdon. Guler, M.A., Erdogan, F., 2004. Contact mechanics of graded coatings. International Journal of Solids and Structures 41 (14), 3865– 3889. Guler, M.A., Erdogan, F., 2006. Contact mechanics of two deformable elastic solids with graded coatings. Mechanics of Materials 38 (7), 633–647. Guo, L.C., Wu, L.Z., Zeng, T., Ma, L., 2004a. The dynamics fracture behavior of a functionally graded coating-substrate system. Composite Structures 64 (3-4), 433–441. Guo, L.C., Wu, L.Z., Ma, L., 2004b. The interface crack problem under a concentrated load for a functionally graded coating-substrate composite system. Composite Structures 63 (3-4), 397–406. Huang, G.Y., Wang, Y.S., Gross, D., 2002. Fracture analysis of functionally graded coatings: antiplane deformation. European Journal of Mechanics—A/Solids 21 (3), 391–400. Huang, G.Y., Wang, YS., Yu, S.W., 2005. A new model for fracture analysis of functionally graded coatings under plane deformation. Mechanics of Materials 37 (4), 507–516. Jin, Z.H., Batra, R.C., 1996. Interface cracking between functionally graded coatings and a substrate under antiplane shear. International Journal of Engineering Science 34 (15), 1705–1716. Kadioglu, S., Dag, S., Yahsi, S., 1998. Crack problem for a functionally graded layer on an elastic foundation. International Journal of Fracture 94 (1), 63–77. Kashtalyan, M., 2004. Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics A/Solids 23 (5), 833–854. Kim, J.J., Kim, W.J., Choi, D.J., Park, J.Y., Ryu, W.S., 2005. Design of a C/SiC functionally graded coating for the oxidation protection of C/C composites. Carbon 43 (8), 1749–1757. Koizumi, M., 1993. Concept of FGM. Ceramic Transactions 34, 3–10. Liew, K.M., Yang, J., Wu, Y.F., 2006. Nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient. Computer Methods in Applied Mechanics and Engineering 195 (9–12), 1007–1026.
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