MODELING OF THE ANISOTROPIC ELASTIC PROPERTIES OF PLASMA-SPRAYED COATINGS IN RELATION TO THEIR MICROSTRUCTURE

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1 Acta mater. 48 (2000) 1361± 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 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 /00/$ Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S (99)

2 1362 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 s kl can be represented as a sum: e ij ˆ S 0 ijkl s kl De ij 1 where S 0 ijkl are the matrix compliances and De ij is the contribution of the cavity. For the isotropic matrix, S 0 ijkl ˆ 1 n 0 d ik d jl d il d jk n 0 d ij d kl E 0 1 n 0 where E 0 and n 0 are Young's modulus and Poisson's ratio of the matrix and d ij is Kronecker's delta. Due to linear elasticity, De ij is a linear function of the applied stress: De ij ˆ H ijkl s kl 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 H ijkl for the strongly oblate spheroidal shapes (with semi-axes a 1 =a 2 0 a>>a 3 ), 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.)

3 H ijkl ˆ Vcav V the solid phase, so that Dt i ˆ n j s S ji : This assumption constitutes Mori±Tanaka's scheme (MTS) that pro- SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS w 1 d ij d kl w 2 2E 0 2 d ikd jl d il d kj vides a reasonable approximation of the e ective moduli of materials with pores (at least, for w 3 2 d strongly oblate pores, see numerical simulations on ijn k n l n i n j d kl 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 w 4 w 6 =a n i d jk n l n j d ik n l n i d jl n k 4 n j d il n k w 5 w 7 =a n i n j n k n l 3 solid phase, stresses s S ij are expressed in terms of remotely applied s ij and porosity p: where n is the unit vector along the spheroid's axis of symmetry, a=a 3 /a and V cav =4paa 3 /3. Coe cients w 1±7 are functions of Poisson's ratio of the matrix: w 1 ˆ n 0 ; w 2 ˆ 1 n 0 ; w 3 ˆ 1 n 0 1 2n 0 ; w 4 ˆ 2 1 n 0 ; w 5 ˆ 2 5n 0 6n 2 0 ; w 6 ˆ 16 1 n n 0 =2 ; w 7 ˆ 8 1 n2 0 n n 0 =2 : 4 For a solid with many cavities, De ij ˆS k De k ij, where De k ij are linear functions of applied stress s kl. 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 e ij ˆ S 0 ijkl s kl SDe k ij S eff ijkl s kl S 0 ijkl DS ijkl s kl where DS ijkl 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 s kl and is not in uenced by other cavities. Then De k ij ˆ H k ijkl s kl: For interacting cavities, we rst replace the problem by the equivalent one, with cavity surfaces loaded by tractions t k i ˆ n k j s ji 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 s ji and interaction tractions Dt j 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 Dt j re ect simply the average stress s S ij in 5 s S ij ˆ 1 p 1 s ij 7 and each pore is subjected to s S ij, the increments of the e ective compliancies due to pores are obtained from the ones given by the non-interaction approximation by an adjustment DS ijkl ˆ 1 p 1 DS non-int ijkl : 8 3. MODELING OF THE COATING MICROSTRUCTURE 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: n 12 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 P l at l=1.

4 1364 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 n 12 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 je 1 sin y sin je 2 cos je 3 : 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 P l j, y P l j (its independence of y implies the transverse isotropy, with x 3 being the symmetry axis) and contains lr0 as a parameter: P l j ˆ 1 2p l2 1 e lj l e lp=2 Š: 11 This distribution ``bridges'' the random and the parallel orientation statistics: these extreme cases 9 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 x 3 the axis normal to the substrate, so that x 1 x 2 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 a 1 and a 2 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 a 1 > a 2. As seen in the text to follow, this di erence may be responsible for the relatively high values of Poisson's ratio n 12 in the plane parallel to the substrate Remark Characterization of each of the two pore families by a certain aspect ratio a 1 or a 2 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 a k 3 3 =S a k 3 1=3 : 12 The calculated e ective moduli are as follows: E 1 ˆ E 0 1 E p H 1111 E 3 ˆ E 0 1 E p H 3333 Fig. 3. Dependence of the orientational distribution function P l on angle j at several values of l and the corresponding orientational patterns. E 1 n 12 ˆ n 0 H 1122 E 0 1 p E 1 n 13 ˆ n 0 H 1133 E 0 1 p

5 G 13 ˆ G 0 1 2G p H 1313 G 12 ˆ SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS 1365 E n where H ijkl are obtained by integrations of H ijkl [given by (3)] over orientations, with distribution function (11): H 1313 ˆ 1 1 2E 0 2 p 1 p 2 w 2 p 2 4 f 1 l 2 f 2 l 2 Š w 4 w 6 =a 2 p 1f 2 l 1 w 4 w 6 =a 1 2 p 1 f 6 l 1 w 5 w 7 =a 1 p 2 2 f 5 l 2 f 6 l 2 Š w 5 w 7 =a 2 H 1111 ˆ 1 p 1 p 2 w 1 w 2 p 1 f 1 l 1 2E 0 w 3 w 4 w 6 =a 1 p 2 8 5f 2 l 2 3f 1 l 2 Š w 3 w 4 w 6 =a 2 p 1 f 3 l 1 w 5 w 7 =a 1 3p 2 8 f 3 l 2 f 4 l 2 2f 6 l 2 Š w 5 w 7 =a 2 H 3333 ˆ 1 2E 0 p 1 p 2 w 1 w 2 p 1 f 2 l 1 H 1212 ˆ H 1111 H 1122 =2: 14 Coe cients w 1±7 are functions of Poisson's ratio n 0 of the matrix and are given by (4), and f 1±6 are functions of scatter parameters l 1 and l 2, given by: f 1 ˆ 18 l l2 3 e lp=2 6 l 2 9 f 2 ˆ l2 3 3 l e lp=2 3 l f 3 ˆ l 2 9 l 2 25 H 1122 ˆ 1 2E 0 p 2 f 1 l 2 w 3 w 4 p 1 f 4 l 1 p1 p 2 f 3 l 2 Šw 5 f 2 l 1 p 2 f 1 l 2 a 2 a 2 p1 f 4 l 1 p 2 f 3 l 2 w 7 a 2 a 2 p 1 p 2 w 1 w 2 p 1 f 1 l 1 p 2 8 f 1 l 2 3f 2 l 2 w 6 w 3 p 2 8 f 1 l 2 f 2 l 2 Š w 4 w 6 =a 2 p 1 f 5 l 1 w 5 w 7 =a 1 p 2 8 f 3 l 2 f 4 l 2 2f 6 l 2 Š w 5 w 7 =a 2 l 3 l2 1 l l 2 9 l 2 e lp=2 25 f 4 ˆ 24 l2 1 l 2 21 l 2 9 l 2 25 l l2 9 l l 2 9 l 2 e lp= f 5 ˆ l 2 9 l 2 25 l l2 1 l l 2 9 l 2 e lp=2 25 f 6 ˆ 3 l l 2 9 l 2 25 l l2 1 l l 2 9 l 2 e lp=2 : H 1133 ˆ 1 p 1 p 2 w 1 p 1 f 2 l 1 p 2 2E 0 2 f 1 l 2 f 2 l 2 w 3 p 1 f 6 l 1 w 5 w 7 =a 1 p 2 2 f 5 l 2 f 6 l 2 Š w 5 w 7 =a 2 In two important asymptoticsðsmall orientational scatter (large l ) and weakly expressed orientational preference (small l )Ðthe expressions for f 1±6 simplify as follows.. In the case of large l (g=1/l is small): f 1 13g 2 1 9g 2 ; f g 2 1 9g 2 ; f 3 145g 4 ;

6 1366 SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS f g 2 478g 4 ; f 5 115g 4 ; f 6 13g g 2 :. In the case of small l: 3.2. Remark f l 18 ; f l 9 ; f f l 7 75 ; l ; 16 f l ; f l : 17 An interesting observation concerning these two asymptotic cases is that, in the case of small l, functions f 1±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:. l 1 and l 2 that characterize the orientational scatter of the ``horizontal'' and ``vertical'' families of pores;. a 1 and a 2 (average aspect ratios for pores of the rst and the second families);. partial porosities p 1 and p 2 for each of the two families. Table 1 compares predictions of our model with the ultrasonically measured anisotropic elastic constants for sprayed Al 2 O 3 [5]; elastic moduli of bulk Al 2 O 3 are taken in our calculations as E 0 =380 GPa, n 0 =0.25. In these experiments, the anisotropy was assumed to be of the orthotropic type and nine elastic sti nesses C ij were measured. These sti nesses are related to the ``engineering constants'' E i, G ij, n ij 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 p 1 =0.075 (interlamellar pores) and p 2 =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 a 1 =0.05 and a 2 =0.08 for the inter- and intralamellar pores, respectively. The orientational scatter parameters for the two pore systems were chosen as l 1 =1.0, l 2 =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 l 1 and l 2, 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 In uence of the overall porosity on the overall moduli We examine the dependence of the moduli on the overall porosity p=p 1 +p 2 (the focus of attention of many previous works). We keep the other parameters xed by assuming that p 1 =p 2, the orientational scatter parameters for the two families, are l 1 =5.0, l 2 =10.0 and the average aspect ratios a 1 =a 2 =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 C ij with the measured ones a Elastic sti nesses Results of ultrasonic measurements [5] Predicted sti nesses Disagreement (%) C C C C C C C C C a Numeration of coordinate axes x 1, x 2 and x 3 used in [5] is changed to conform to ours: and an apparent misprint in [5] related to the identi cation of C 44, C 55 and C 66 is corrected.

7 SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS Scatter of pore orientations and its impact on the moduli: ``inverse'' anisotropy Scatter of pore orientations, characterized by parameters l 1 and l 2 (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, E 3 > E 1 ). 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 l 2 > l 1. Then we obtain curves of E 1 /E 3 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 E 1 /E 3, G 12 /G 13 and n 12 /n 13 (that characterize the anisotropy) on the scatter parameter l 2 for several xed values of parameter l 1. Our results establish the condition of ``inverse'' anisotropy E 3 > E 1 in terms of microstructural parameters: H 3333 < H with H 3333 and H 1111 expressed in terms of partial porosities p 1 and p 2 of the two families of pores, their orientation scatter, l 1 and l 2, and the average pore aspect ratios a 1 and a 2 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 l 2 > l 1, such a reduction is stronger in the direction normal to the substrate. anisotropy depends not only on the ratio l 1 /l 2 of the orientational scatters for the two pore families, but on the average aspect ratios a 1 and a 2 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 n 12 /n 13 depends on scatter parameters l 1 and l 2, as well as on a 1 and a 2, through their ratios a 1 /a 2 and l 1 /l 2 only High values of n 12 in relation to pore aspect ratios and to the orientational scatter Relatively high values of Poisson's ratio n 12 in the plane parallel to the substrate (n 12 may approach Poisson's ratio n 0 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 n 12. Figure 7 illustrates the dependence of n 12 on the average aspect ratio a 2 of the ``vertical'' pores at several xed values of a 1. An interesting observation is that n 12 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 p 1 =p 2 (=p/2), aspect ratios a 1 =a 2 =0.1 and scatter parameters l 1 =5, l 2 =10. Fig. 5. Ratios of the overall moduli, that characterize the extent of anisotropy, as functions of scatter parameter l 2, at aspect ratios a 1 =a 2 =0.1 and partial porosities p 1 =p 2 =0.05, for several values of l 1 (1, l 1 =0; 2, l 1 =2.5; 3, l 1 =5; 4, l 1 =7.5; 5, l 1 =10).

8 1368 SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS tend to correspond to higher values of l 1, 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 a 2, for the following sets of parameters: 1, a 1 =0.05, l 1 =2.5, l 2 =5.0; 2, a 1 =0.05, l 1 =5.0, l 2 =10.0; 3, a 1 =0.1, l 1 =2.5, l 2 =5.0; 4, a 1 =0.1, l 1 =5.0, l 2 =10.0; 5, a 1 =0.2, l 1 =2.5, l 2 =5.0; 6, a 1 =0.2, l 1 =5.0, l 2 =10.0. decreasing function of a 2, depending on the value of a 1. Besides being in uenced by the diversity of pore aspect ratios, the value of n 12 also depends on the orientational scatter. Namely, higher values of n 12 Fig. 7. Ratio n 12 /n 0 as a function of aspect ratio a 2 at several values of a 1 for two combinations of l 1 and l 2. 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 n 12 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. AcknowledgementsÐThe authors thank F. Kroupa for

9 SEVOSTIANOV and KACHANOV: PLASMA-SPRAYED COATINGS 1369 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 l 1 and l 2 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, Bengtsson, P. and Johannesson, T., J. Thermal Spray Technol., 1995, 4, Kachanov, M., Tsukrov, I. and Sha ro, B., Appl. Mech. Rev., 1994, 47, Leigh, S.-H., Lee, G.-C. and Berndt, C. C., in Proc. 15th Int. Thermal Spray Conf., Nice, France, Parthasarathi, S., Tittmann, B. R., Sampath, K. and Onesto, E. J., J. Thermal Spray Technol., 1995, 4, Li, C. J., Ohmori, A. and McPherson, R., in Proc. of Int. Conf. AUSTCERAM-92, Melbourne, Australia, 1992, p Kroupa, F., Kovove Materialy, 1995, 33, 418 [in Czech]. 8. Tsui, Y. C., Doyle, C. and Clyne, T. W., Bimaterials, 1998, 19, 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 Kachanov, M., Appl. Mech. Rev., 1992, 45, Rybicky, E. F., Shadley, J. R., Xiong, Y. and Greving, D. J., J. Thermal Spray Technol., 1995, 4, 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 Sha ro, B. and Kachanov, M., J. appl. Phys., in press.

10 1370 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 E i, shear moduli G ij and Poisson's ratios n ij ); and elastic sti nesses C ij. These two systems are interrelated as follows: C 11 ˆ 1 n 23 n 32 DE 1 C 12 ˆ n 12 n 13 n 32 DE 1 ˆ n 21 n 23 n 31 DE 2 C 22 ˆ 1 n 31 n 13 DE 2 C 23 ˆ n 23 n 21 n 13 DE 2 ˆ n 32 n 31 n 12 DE 3 C 33 ˆ 1 n 12 n 21 DE 3 C 13 ˆ n 13 n 12 n 23 DE 1 ˆ n 31 n 32 n 21 DE 3 C 44 ˆ G 23 C 55 ˆ G 31 C 66 ˆ G 12 where D=1/(1 n 12 n 21 n 23 n 32 n 31 n 13 2n 12 n 23 n 31 ).