Modeling of the anisotropic elastic properties of plasma-sprayed coatings in relation to their microstructure

Acta mater. 48 (2000) 1361±1370 www.elsevier.com/locate/actamat

MODELING OF THE ANISOTROPIC ELASTIC PROPERTIES OF PLASMA-SPRAYED COATINGS IN RELATION TO THEIR MICROSTRUCTURE I. SEVOSTIANOV{ and M. KACHANOV Department of Mechanical Engineering, Tufts University, 204 Anderson Hall, Medford, MA 02155, USA (Received 14 July 1999; accepted 4 October 1999) AbstractÐThe transversely isotropic elastic moduli of plasma-sprayed coatings are calculated in terms of microstructural parameters. The dominant features of the porous space are identi®ed as strongly oblate pores, that tend to be either parallel or normal to the substrate. ``Irregularities'' in the porous space geometryÐthe scatter in pore orientations and the di€erence between pore aspect ratios of the two pore systemsÐare shown to have a pronounced e€ect on the e€ective moduli. They may be responsible for the ``inverse'' anisotropy (Young's modulus in the direction normal to the substrate being higher than the one in the transverse direction) and for the relatively high values of Poisson's ratio in the plane of isotropy. The analysis utilizes results of Kachanov et al. (Appl. Mech. Rev., 1994, 47, 151) on materials with pores of diverse shapes and orientations. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Plasma spray; Microstructure; Elastic properties

1. INTRODUCTION

Plasma-sprayed ceramic coatings have a lamellar microstructure consisting of elongated, ¯at-like splats of diameters between 100 and 200 mm and thicknesses between 2 and 10 mm, formed by a rapid solidi®cation. The porous space comprises micropores and microcracks of diverse shapes and orientations (Fig. 1). Overall, it has a highly anisotropic structure that results in anisotropic e€ective moduli. In order to express the e€ective moduli in terms of microstructural parameters, the complexity of the porous space has to be reduced to several dominant elements. Following Bengtsson and Johannesson [2], Leigh et al. [4] and Leigh and Berndt [1], as well as a number of earlier works, we identify the dominant elements of the porous space as two families of oblate spheroidal pores, approximately parallel and approximately perpendicular to the substrate. We make two further observations on the porous space geometry. 1. While the pores tend to be parallel/perpendicular to the substrate, they actually have a substantial

{ To whom all correspondence should be addressed.

orientation scatter. 2. Average aspect ratios may be quite di€erent for the two families of pores. As seen in the analysis to follow, these two observations have important implications for the e€ective moduli. The e€ective anisotropic moduli of sprayed materials have been investigated both by experimentalists and from the point of view of theoretical modeling. As far as experimental data on anisotropic elasticity of plasma-sprayed coatings is concerned, the data of Parthasarathi et al. [5] appears to be the most informative one for the following two reasons. First, they used the ultrasonic method of measurements, which is, generally, more accurate than tests involving mechanical loading (that may produce, inadvertently, inelastic deformations). Second, the full set of the orthotropic constants was reported in this work. In the area of theoretical modeling, we mention the following contributions. Li et al. [6] proposed a model explaining relatively small Young's modulus in the deposition direction. The splats were assumed to be bonded along small areas and the low modulus was the result of bending of the unbonded parts. Their analysis, however, does not cover the

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 8 4 - 5

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SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

full set of anisotropic moduli. Kroupa [7] modeled the porous space by two sets of spheroidal cavities. His model predicts relatively small changes in elastic moduli due to porosity, whereas in reality Young's moduli may be reduced up to twenty times, as compared to the bulk material (see, for instance [8]). Kroupa and Kachanov [9] modeled the porous space by two families of mutually perpendicular circular cracks and spherical pores. The limitation of their model is that it does not account for the contribution to the overall porosity due to cracks that are, in reality, not ideally thin but possess some initial opening. Leigh et al. [4] and Leigh and Berndt [1] modeled the porous space by two families of oblate spheroidal pores, parallel and perpendicular to the substrate. Their model identi®es the dominant features of the porous space geometry, but alongside with the other models mentioned above, it disregards the ``irregularity factors'' (1) and (2). As shown in the present work, these factors have a substantial impact on the e€ective moduli. In particular, they may be responsible for two interesting e€ects that do not seem to have been explained by previous models: the ``inverse'' character of anisotropy (Young's modulus in the direction normal to the substrate being higher than the one in the transverse direction) and relatively high values of Poisson's ratio in the plane of isotropy. The present work constitutes a further contribution to the ®eld. Similarly to Leigh et al. [4] and Leigh and Berndt [1], we identify the dominant elements of the porous space as consisting of two families of strongly oblate spheroidal pores (parallel/normal to the substrate), but we recognize that pore orientations have a signi®cant scatter and that (average) aspect ratios may be substantially di€erent for the two families. Our analysis utilizes results of Kachanov et al. [3] on materials with anisotropic mixtures of pores and

cracks of diverse shapes and orientations. The e€ect of elastic interactions between pores on the overall moduli is accounted for in the framework of Mori± Tanaka's scheme that appears to constitute a good approximation, at least, for strongly oblate pores (see numerical simulations on cracks of Kachanov [10]). 2. THEORETICAL BACKGROUND

We brie¯y outline results of Kachanov et al. [3] on materials with pores of diverse shapes and orientations that are utilized in the present analysis. We start with the observation that, for a volume V containing one cavity, strain per V under remotely applied stress skl can be represented as a sum: eij ˆ S 0ijkl skl ‡ Deij

…1†

where S 0ijkl are the matrix compliances and Deij is the contribution of the cavity. For the isotropic matrix,   1 ‡ n0 n0 S 0ijkl ˆ dij dkl dik djl ‡ dil djk ÿ E0 1 ‡ n0 where E0 and n0 are Young's modulus and Poisson's ratio of the matrix and dij is Kronecker's delta. Due to linear elasticity, Deij is a linear function of the applied stress: Deij ˆ Hijkl skl

…2†

thus giving rise to fourth rank cavity compliance tensor H. It was calculated for the ellipsoidal pores (as well as for a number of 2D shapes) by Kachanov et al. [3]. Components Hijkl for the strongly oblate spheroidal shapes (with semi-axes a1=a2 0 a>>a3), that are relevant for modeling of sprayed coatings, are as follows:

Fig. 1. Typical microstructure of a plasma-sprayed coating and its modeling by strongly oblate pores. (The photograph is taken from [2], with the permission of ASM International.)

SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

Hijkl

 Vcav 1 w2 …dik djl ‡ dil dkj † ˆ w1 dij dkl ‡ V 2E0 2 ‡

w3 …dij nk nl ‡ ni nj dkl † 2

w4 ‡ w6 =a …ni djk nl ‡ nj dik nl ‡ ni djl nk 4  ‡ nj dil nk † ‡ …w5 ‡ w7 =a†ni nj nk nl ‡

…3†

where n is the unit vector along the spheroid's axis of symmetry, a=a3/a and Vcav=4paa 3/3. Coecients w1±7 are functions of Poisson's ratio of the matrix: w1 ˆ ÿn0 ;

w2 ˆ 1 ‡ n0 ;

w3 ˆ ÿ…1 ‡ n0 †…1 ÿ 2n0 †; w5 ˆ 2 ‡ 5n0 ÿ 6n20 ; w7 ˆ

w6 ˆ

16…1 ÿ n20 † ; 3…1 ÿ n0 =2†

n20 †n0

8…1 ÿ : 3…1 ÿ n0 =2†

sSij ˆ …1 ÿ p†ÿ1 sij

…7†

and each pore is subjected to sSij , the increments of the e€ective compliancies due to pores are obtained from the ones given by the non-interaction approximation by an adjustment …8†

3. MODELING OF THE COATING MICROSTRUCTURE

…4†

For a solid with many cavities, Deij ˆ Sk De…k† ij , where De…k† are linear functions of applied stress skl. ij Determination of these functions (they re¯ect not only the pore shapes but interactions between pores as well) constitutes the most dicult part of the problem. Provided the mentioned functions are speci®ed, the e€ective compliances S eff ijkl follow from 0 eij ˆ S 0ijkl skl ‡ SDe…k† ij  …S ijkl ‡ DSijkl †skl

 S eff ijkl skl

vides a reasonable approximation of the e€ective moduli of materials with pores (at least, for strongly oblate pores, see numerical simulations on cracks of Kachanov [10]). In the framework of MTS, the impact of interactions on the e€ective moduli is accounted for by rather simple means: since the average, over the solid phase, stresses sSij are expressed in terms of remotely applied sij and porosity p:

non-int DSijkl ˆ …1 ÿ p†ÿ1 DS ijkl :

w4 ˆ 2…1 ‡ n0 †;

1363

…5†

where DSijkl are changes in compliances due to cavities. The summation over cavities may be replaced by integration over orientations, if computationally convenient. In the non-interaction approximation, each cavity is placed in remotely applied stress skl and is not …k† in¯uenced by other cavities. Then Deij…k† ˆ H ijkl skl : For interacting cavities, we ®rst replace the problem by the equivalent one, with cavity surfaces loaded by tractions ti…k† ˆ nj…k† sji and stresses vanishing at in®nity. We further represent it as a superposition of N subproblems with one pore each; the traction on a pore in the kth subproblem is a sum of n…k† j sji and interaction tractions Dtj generated by pores in the remaining subproblems at the side of the considered pore in a continuous material. It appears physically reasonable to assume that, for cavities with uncorrelated mutual positions (thus excluding, for example, the spatially periodic arrangements, for which the e€ective response will depend on the speci®c arrangement), interaction tractions Dtj re¯ect simply the average stress sSij in the solid phase, so that Dti ˆ nj sSji : This assumption constitutes Mori±Tanaka's scheme (MTS) that pro-

It appears that a successful quantitative modeling should predict the basic features of the overall elastic anisotropy of the sprayed coatings. Among them: . Substantial anisotropy of Young's moduli. In some cases, this anisotropy may have an unexpected ``inverse'' characterÐthe sti€ness may be higher in the direction normal to the substrate [5]. . Relatively high Poisson's ratio in the plane of isotropy: n12 may reach 0.25±0.30, as reported by Parthasarathi et al. [5] and Rybicky et al. [11]. The existing models do not seem to provide an explanation of these features. These limitations appear to be rooted in ignoring ``irregularities'' of the porous space geometryÐfactors (1) and (2) mentioned in Section 1.

Fig. 2. Orientational distribution function Pl at l=1.

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SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

The present work demonstrates that much better agreement with the data can be achieved by realistic modeling of the irregular, statistical character of the coating microstructure. Namely, the ``inverse'' anisotropy can be directly related to the scatter in pore orientations and relatively high values of n12 are explained, primarily, by the di€erence between the average aspect ratios of pores of the two families. The analysis to follow is based on results brie¯y outlined in the previous section. For the computational convenience, we replace summation S H(i ) over pores by the integration over orientations. We express unit vector n(i ) along the ith spheroid's symmetry axis in terms of two angles 0 R j R p/2 and 0 R y R 2p (Fig. 2): n…j, y† ˆ cos y sin je1 ‡ sin y sin je2 ‡ cos je3 :

…9†

and introduce statistics P(j, y ) of cavity orientationsÐthe probability density function de®ned on the upper semi-sphere F of unit radius and subject to the normalization condition …… P…j, y†dj dy ˆ 1: …10† F

Following Sha®ro and Kachanov [13], we consider the orientational distribution that is intermediate between the random and the parallel ones, by specifying the following probability density Pl …j, y†  Pl …j† (its independence of y implies the transverse isotropy, with x3 being the symmetry axis) and contains lr0 as a parameter: Pl …j† ˆ

1 2 ‰…l ‡ 1†eÿlj ‡ l eÿlp=2 Š: 2p

…11†

This distribution ``bridges'' the random and the parallel orientation statistics: these extreme cases

correspond to l=0 and l=1. It covers two important asymptotics: (a) slightly perturbed parallel orientations (large l ); and (b) of weakly expressed orientational preference (small l ). Function (11) has the following features: it has maximum at j=p/2; and parameter l r 0 characterizes its ``sharpness''. Figure 3 provides an illustration and shows the patterns of orientational scatter of pores that correspond to several values of l. As discussed by Kachanov et al. [12], the e€ective moduli are relatively insensitive to the exact form of a function that has the above-mentioned features. The particular form (11) is chosen to keep the calculations, related to averaging over orientations, simple. We now calculate the anisotropic e€ective moduli of a coating. We denote by x3 the axis normal to the substrate, so that x1x2 is the plane of isotropy. As discussed above, the porous space is modeled by two families of strongly oblate pores. Each of the families has the orientational distribution (11): it has a preferential orientationÐ``horizontal'' or ``vertical'' (the distribution functions for the ``horizontal'' and ``vertical'' families di€er by shifting angle j on p/2) with the extent of scatter characterized by parameters l, generally di€erent for the two families. The aspect ratios a1 and a2 of pores may be di€erent for the ``horizontal'' and ``vertical'' families. The available microphotographs seem to indicate that the ``vertical'' pores tend to be narrower than the ``horizontal'' ones, so that a1 > a2. As seen in the text to follow, this di€erence may be responsible for the relatively high values of Poisson's ratio n12 in the plane parallel to the substrate. 3.1. Remark Characterization of each of the two pore families by a certain aspect ratio a1 or a2 does not imply the identical aspect ratios within a family: a is the average over the family aspect ratio. More precisely, this average is to be understood as follows (see [3] for details): hai ˆ …S…a3…k† †3 =S…a…k† †3 †1=3 : The calculated e€ective moduli are as follows:  ÿ1 E0 E1 ˆ E0 1 ‡ H 1111 1ÿp  ÿ1 E0 E3 ˆ E0 1 ‡ H 3333 1ÿp

Fig. 3. Dependence of the orientational distribution function Pl on angle j at several values of l and the corresponding orientational patterns.

n12 ˆ n0

E1 H 1122 ÿ E0 1ÿp

n13 ˆ n0

E1 H 1133 ÿ E0 1ÿp

…12†

SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

G13



 ÿ1 2G0  ˆ G0 1 ‡ H 1 ÿ p 1313

G12

H 1313 ˆ …13†

E1 ˆ 2…1 ‡ n12 †

H

 1 … p1 ‡ p2 †…w1 ‡ w2 † ‡ p1 f1 …l1 † ˆ 2E0

p1 f2 …l1 † …w4 ‡ w6 =a1 † 2 p2 ‡ p1 f6 …l1 †…w5 ‡ w7 =a1 † ‡ ‰ f5 …l2 † ‡ f6 …l2 †Š 2   …w5 ‡ w7 =a2 † H 1212 ˆ …H 1111 ÿ H 1122 †=2:

 …w3 ‡ w4 ‡ w6 =a2 † ‡ p1 f3 …l1 †…w5 ‡ w7 =a1 † 3p2 ‰ f3 …l2 † ‡ f4 …l2 † ‡ 2f6 …l2 †Š…w5 ‡ w7 =a2 † 8

f1 ˆ



18 ÿ l…l2 ‡ 3†eÿlp=2 6…l2 ‡ 9†

f2 ˆ

H 3333 ˆ

 1 … p1 ‡ p2 †…w1 ‡ w2 † ‡ … p1 f2 …l1 † 2E0 ‡ p2 f1 …l2 ††…w3 ‡ w4 † ‡ ‰ p1 f4 …l1 †   p1 p2 ‡ p2 f3 …l2 †Šw5 ‡ f2 …l1 † ‡ f1 …l2 † w6 a2 a2    p1 p2 ‡ f4 …l1 † ‡ f3 …l2 † w7 a2 a2

f3 ˆ

ÿl

f4 ˆ

 1 … p1 ‡ p2 †…w1 ‡ w2 † 2E0   p2 ‡ p1 f1 …l1 † ‡ … f1 …l2 † ‡ 3f2 …l2 †† w3 8 p2 ‰ f1 …l2 † ÿ f2 …l2 †Š…w4 ‡ w6 =a2 † 8 p2 ‡ p1 f5 …l1 †…w5 ‡ w7 =a1 † ‡ ‰ f3 …l2 † 8  ‡ f4 …l2 † ‡ 2f6 …l2 †Š…w5 ‡ w7 =a2 †

f5 ˆ

3…l2 ‡ 1†…l2 ‡ 29† ‡ 360 ÿlp=2 e 8…l2 ‡ 9†…l2 ‡ 25†

24 ‡ …l2 ‡ 1†…l2 ‡ 21† …l2 ‡ 9†…l2 ‡ 25†

‡

H 1133 ˆ

  1 p2 … p1 ‡ p2 †w1 ‡ p1 f2 …l1 † ‡ … f1 …l2 † 2E0 2  ‡ f2 …l2 †† w3 ‡ p1 f6 …l1 †…w5 ‡ w7 =a1 † ‡

p2 ‰ f5 …l2 † ‡ f6 …l2 †Š…w5 ‡ w7 =a2 † 2



…l2 ‡ 9†…l2 ‡ 25† ÿ 120 ÿlp=2 e 5…l2 ‡ 9†…l2 ‡ 25†

15 …l2 ‡ 9†…l2 ‡ 25† ÿl

f6 ˆ

…l2 ‡ 3†…3 ‡ l eÿlp=2 † 3…l2 ‡ 9†

45 …l2 ‡ 9†…l2 ‡ 25†

‡l

H 1122 ˆ

…14†

Coecients w1±7 are functions of Poisson's ratio n0 of the matrix and are given by (4), and f1±6 are functions of scatter parameters l1 and l2, given by:

p2  …w3 ‡ w4 ‡ w6 =a1 † ‡ ‰5f2 …l2 † ‡ 3f1 …l2 †Š 8

‡

1 1 p2 … p1 ‡ p2 †w2 ‡ ‰ f1 …l2 † ‡ f2 …l2 †Š 2E0 2 4  …w4 ‡ w6 =a2 † ‡

where H ijkl are obtained by integrations of Hijkl [given by (3)] over orientations, with distribution function (11):

 1111

1365

…l2 ‡ 1†…l2 ‡ 29† ‡ 120 ÿlp=2 e 8…l2 ‡ 9†…l2 ‡ 25†

3…l2 ‡ 25† ÿ 60 …l2 ‡ 9†…l2 ‡ 25† ‡l

…l2 ‡ 1†…l2 ‡ 30† ‡ 156 ÿlp=2 e : 3…l2 ‡ 9†…l2 ‡ 25†

…15†

In two important asymptoticsÐsmall orientational scatter (large l ) and weakly expressed orientational preference (small l )Ðthe expressions for f1±6 simplify as follows. . In the case of large l (g=1/l is small): f1 13g2 …1 ÿ 9g2 †; f3 145g4 ;

f2 1…1 ‡ 3g2 †…1 ÿ 9g2 †;

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SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

f4 11 ÿ 12g2 ÿ 478g4 ; 2

f5 115g4 ;

2

f6 13g …1 ÿ 29g †:

…16†

. In the case of small l: 1 l f1 1 ÿ ; 3 18

1 l f1 1 ‡ ; 3 9

1 149 f3 1 ÿ l ; 5 600

1 7 f4 1 ‡ l ; 5 75 f5 1

1 14 ÿl ; 15 180

f6 1

1 62 ‡l : 15 225

…17†

3.2. Remark An interesting observation concerning these two asymptotic cases is that, in the case of small l, functions f1±6 contain terms linear in l, whereas in the case of large l, terms linear in g=1/l are absent. Hence, in the ®rst case, the e€ective moduli have higher sensitivity to the orientational perturbations. The calculated e€ective moduli (13) depend on the following parameters: . l1 and l2 that characterize the orientational scatter of the ``horizontal'' and ``vertical'' families of pores; . a1 and a2 (average aspect ratios for pores of the ®rst and the second families); . partial porosities p1 and p2 for each of the two families. Table 1 compares predictions of our model with the ultrasonically measured anisotropic elastic constants for sprayed Al2O3 [5]; elastic moduli of bulk Al2O3 are taken in our calculations as E0=380 GPa, n0=0.25. In these experiments, the anisotropy was assumed to be of the orthotropic type and nine elastic sti€nesses Cij were measured. These sti€nesses are related to the ``engineering constants'' Ei, Gij, nij as given in Appendix A. The anisotropy was actually very close to the transversely isotropic one; the di€erences in moduli corresponding to di€erent

directions parallel to the substrate were small (consistently with the lamellar character of the microstructure). In our calculations, the total porosity p = 0.15, according to the data of [5]. Other, ``®ner'' microstructural parameters (partial porosities for the inter- and intralamellar pores, orientational scatter and average shapes of pores) were not reported in [5]. Therefore, we estimate them, in our calculations, from the data reported in other works [1, 2, 6, 8]. More speci®cally, the partial porosities were taken as p1=0.075 (interlamellar pores) and p2=0.055 (intralamellar pores). The average aspect ratios of pores (that, in view of approximate constancy of the average openings, characterize the average pore lengths) were taken as a1=0.05 and a2=0.08 for the inter- and intralamellar pores, respectively. The orientational scatter parameters for the two pore systems were chosen as l1=1.0, l2=5.0, correspondingly. As seen from Table 1, the di€erences between our predictions and the experimental data do not exceed 10%. This agreement appears to validate our theoretical model. Moreover, the fact that agreement is good for the chosen values of l1 and l2, of the orientational scatter parameters, provides an insight into the actual geometry of the porous space. 4. DISCUSSION

We discuss here the dependence of the calculated moduli on microstructural parameters. 4.1. In¯uence of the overall porosity on the overall moduli We examine the dependence of the moduli on the overall porosity p=p1+p2 (the focus of attention of many previous works). We keep the other parameters ®xed by assuming that p1=p2, the orientational scatter parameters for the two families, are l1=5.0, l2=10.0 and the average aspect ratios a1=a2=0.1. The corresponding dependencies are shown in Fig. 4. As expected, the moduli decrease monotonically as p increases.

Table 1. Comparison of the predicted sti€nesses Cij with the measured onesa Elastic sti€nesses C11 C22 C33 C44 C55 C66 C12 C23 C13

Results of ultrasonic measurements [5]

Predicted sti€nesses

Disagreement (%)

100.85 98.57 134.90 38.59 38.21 46.15 21.31 33.03 33.96

107.54 107.54 123.31 35.01 35.01 44.06 19.43 35.36 35.36

6.2 8.3 8.6 9.3 8.4 4.5 8.8 6.6 4.0

a Numeration of coordinate axes x1, x2 and x3 used in [5] is changed to conform to ours: 123 4 231 and an apparent misprint in [5] related to the identi®cation of C44, C55 and C66 is corrected.

SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

4.2. Scatter of pore orientations and its impact on the moduli: ``inverse'' anisotropy Scatter of pore orientations, characterized by parameters l1 and l2 (for the ``horizontal'' and ``vertical'' families, respectively), has pronounced e€ects on the e€ective elastic moduli. In particular, the scatter appears to be responsible for the ``inverse'' anisotropy (higher sti€ness in the direction normal to the substrate, E3 > E1). Consistently with the microphotographs (Fig. 1), indicating that the orientational scatter for the horizontal pores may be larger than the one for the vertical pores, we assume l2 > l1. Then we obtain curves of E1/E3 that may lie, partly or fully, below the level of 1.0 (see Fig. 5). The latter ®gure also illustrates the dependence of moduli ratios E1/E3, G12/G13 and n12/n13 (that characterize the anisotropy) on the scatter parameter l2 for several ®xed values of parameter l1. Our results establish the condition of ``inverse'' anisotropy E3 > E1 in terms of microstructural parameters: H 3333 < H 1111

…18†

with H 3333 and H 1111 expressed in terms of partial porosities p1 and p2 of the two families of pores, their orientation scatter, l1 and l2, and the average pore aspect ratios a1 and a2 by formulas (14). Physically, the possibility of the ``inverse'' anisotropy due to the orientational scatter is explained by the fact that orientational perturbations reduce the impact of pores on Young's modulus in the direction normal to the pores. Since l2 > l1, such a reduction is stronger in the direction normal to the substrate.

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anisotropy depends not only on the ratio l1/l2 of the orientational scatters for the two pore families, but on the average aspect ratios a1 and a2 of pores as well. This is illustrated by Fig. 6. Physically, this is explained by the fact that orientational perturbations of pores that have ``rounder'' shapes have a milder in¯uence on the e€ective moduli (in the limit of the spherical shape, this in¯uence vanishes). Another interesting observation is that it may be assumed, with good accuracy, that ratio n12/n13 depends on scatter parameters l1 and l2, as well as on a1 and a2, through their ratios a1/a2 and l1/l2 only. 4.4. High values of n12 in relation to pore aspect ratios and to the orientational scatter Relatively high values of Poisson's ratio n12 in the plane parallel to the substrate (n12 may approach Poisson's ratio n0 of the bulk material) were reported in the literature [1, 5, 11]. This may be unexpected, in view of the fact that porosity, generally, reduces Poisson's ratios. Our results show that the di€erence in the average aspect ratios of pores between the two pore families may be responsible for the high values of n12. Figure 7 illustrates the dependence of n12 on the average aspect ratio a2 of the ``vertical'' pores at several ®xed values of a1. An interesting observation is that n12 may be either an increasing or

4.3. ``Inverse'' anisotropy in relation to the diversity of pore aspect ratios Condition (18) for the appearance of ``inverse''

Fig. 4. E€ective elastic moduli (normalized to the matrix moduli) as functions of the overall porosity p at partial porosities p1=p2(=p/2), aspect ratios a1=a2=0.1 and scatter parameters l1=5, l2=10.

Fig. 5. Ratios of the overall moduli, that characterize the extent of anisotropy, as functions of scatter parameter l2, at aspect ratios a1=a2=0.1 and partial porosities p1=p2=0.05, for several values of l1 (1, l1=0; 2, l1=2.5; 3, l1=5; 4, l1=7.5; 5, l1=10).

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SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS

tend to correspond to higher values of l1, i.e. to smaller orientational scatter in the ``horizontal'' family of pores. 5. CONCLUSIONS

Fig. 6. Ratios of the overall moduli, that characterize the extent of anisotropy, as functions of ratio a2, for the following sets of parameters: 1, a1=0.05, l1=2.5, l2=5.0; 2, a1=0.05, l1=5.0, l2=10.0; 3, a1=0.1, l1=2.5, l2=5.0; 4, a1=0.1, l1=5.0, l2=10.0; 5, a1=0.2, l1=2.5, l2=5.0; 6, a1=0.2, l1=5.0, l2=10.0.

decreasing function of a2, depending on the value of a1. Besides being in¯uenced by the diversity of pore aspect ratios, the value of n12 also depends on the orientational scatter. Namely, higher values of n12

A quantitative model that re¯ects the realistic, ``irregular'' features of plasma sprayed coatings and expresses their anisotropic e€ective moduli in terms of these features is constructed. Two basic families of strongly oblate pores, that constitute the porous space, are assumed to have the orientational scatter (that are usually di€erent for the two families) and di€erent average aspect ratios. Our analysis shows that these ``irregularities'' in the porous space geometry have a pronounced e€ect on the overall moduli. These e€ects, as calculated from our model, are in good agreement with ultrasonic measurements of the anisotropic elastic constants reported by Parthasarathi et al. [5]. In particular, the mentioned imperfections provide an explanation of two important features of the e€ective anisotropy, that have not been explained by the previous models: . The possibility of ``inverse anisotropy'' (sti€ness may be higher in the direction normal to the substrate than in the transverse direction, as in the data of Parthasarathi et al. [5]). We directly relate it to the scatter of pore orientations. More precisely, the e€ect is due to the di€erence between the orientational scatters of pores normal and parallel to the substrate. Generally, the larger scatter for the pores parallel to the substrate enhances the ``inverse'' anisotropy. The ``inverse'' anisotropy also depends on the relation between the (average) aspect ratios of the two families of pores. The condition for ``inverse'' anisotropy in terms of the microstructural parameters is given by inequality (18). . Relatively high values of Poisson's ratio n12 in the plane parallel to the substrate. They are explained, mainly, by the diversity of pore aspect ratios. These results further advance modeling of coating microstructures, as compared to the studies where the pore space was modeled as consisting of perfectly regular arrangements of pores. The analysis of the e€ects of ``irregularities'' of the microstructure given here has important implications for the optimization of technological parameters (such as substrate temperature). Indeed, such microstructural features as the extent of scatter of pore orientations are related to technological regimes, as seen from Fig. 8. Therefore, the quantitative relations between these features and the overall elastic moduli derived in the present work can be applied to the optimization of the regimes.

Fig. 7. Ratio n12/n0 as a function of aspect ratio a2 at several values of a1 for two combinations of l1 and l2.

AcknowledgementsÐThe authors thank F. Kroupa for

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Fig. 8. Two coating microstructures corresponding to temperatures 800 and 4008C of the substrate. Their modeling by two families of strongly oblate pores and choice of scatter parameters l1 and l2 to match the observed patterns. (The photographs are taken from [2], with the permission of ASM International.)

helpful discussions and valuable comments. They also thank S.-H. Leigh for pointing out misprints in papers [1, 4]. This work was supported by NASA through a contract to Tufts University. The second author (M.K.) acknowledges partial support of the von Humboldt Research Award for senior scientists. REFERENCES 1. Leigh, S.-H. and Berndt, C. C., Acta mater., 1999, 47, 1575. 2. Bengtsson, P. and Johannesson, T., J. Thermal Spray Technol., 1995, 4, 245. 3. Kachanov, M., Tsukrov, I. and Sha®ro, B., Appl. Mech. Rev., 1994, 47, 151. 4. Leigh, S.-H., Lee, G.-C. and Berndt, C. C., in Proc. 15th Int. Thermal Spray Conf., Nice, France, 1998. 5. Parthasarathi, S., Tittmann, B. R., Sampath, K. and Onesto, E. J., J. Thermal Spray Technol., 1995, 4, 367.

6. Li, C. J., Ohmori, A. and McPherson, R., in Proc. of Int. Conf. AUSTCERAM-92, Melbourne, Australia, 1992, p. 816. 7. Kroupa, F., Kovove Materialy, 1995, 33, 418 [in Czech]. 8. Tsui, Y. C., Doyle, C. and Clyne, T. W., Bimaterials, 1998, 19, 2015. 9. Kroupa, F. and Kachanov, M., in Proc. 19th Int. Symp. Modeling of Structure and Mechanics of Materials from Microscale to Product, Roskilde, Denmark, 1998, p. 325. 10. Kachanov, M., Appl. Mech. Rev., 1992, 45, 305. 11. Rybicky, E. F., Shadley, J. R., Xiong, Y. and Greving, D. J., J. Thermal Spray Technol., 1995, 4, 377. 12. Kachanov, M., Tsukrov, I. and Sha®ro, B., in Fracture and Damage in Quasibrittle Structures, ed. Z. Bazant, et al. Chapman & Hall, London, 1994, p. 19. 13. Sha®ro, B. and Kachanov, M., J. appl. Phys., in press.

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SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS APPENDIX A

Two systems of elastic constants are commonly used for the orthotropic materials: ``engineering constants'' (comprising Young's moduli Ei, shear moduli Gij and Poisson's ratios nij); and elastic sti€nesses Cij. These two systems are interrelated as follows: C11 ˆ …1 ÿ n23 n32 †DE1 C12 ˆ …n12 ÿ n13 n32 †DE1 ˆ …n21 ÿ n23 n31 †DE2

C22 ˆ …1 ÿ n31 n13 †DE2 C23 ˆ …n23 ÿ n21 n13 †DE2 ˆ …n32 ÿ n31 n12 †DE3 C33 ˆ …1 ÿ n12 n21 †DE3 C13 ˆ …n13 ÿ n12 n23 †DE1 ˆ …n31 ÿ n32 n21 †DE3 C44 ˆ G23

C55 ˆ G31

C66 ˆ G12

where D=1/(1 ÿ n12n21 ÿ n23n32 ÿ n31n13 ÿ2n12n23n31).