Surface and Coatings Technology 149 (2002) 179–184
`
`On the delamination of thermal barrier coatings in a thermal gradient
`
`J.W. Hutchinson , A.G. Evans *
`a
`b,
`a
`Division of Engineering and Applied Science, Harvard University, Cambridge, MA 02138, USA
`b
`Princeton Materials Institute, Princeton University, Princeton, NJ 08540, USA
`
`Received 2 April 2001; accepted in revised form 30 July 2001
`
`Abstract
`
`Thermal barrier coating (TBC) systems are susceptible to delamination failures in the presence of a large thermal gradient.
`These failures, which occur within the TBC layer, are very different in character from those associated with the thermally grown
`oxide. Three possible causes of internal delamination are analyzed. In all cases, the thermomechanical properties of the TBC are
`allowed to vary because of sintering. (a) One mechanism relates to exfoliation of an internal separation in the TBC due to a
`through thickness heat flux. (b) Another is concerned with edge-related delamination within a thermal gradient. (c) The third is
`a consequence of sintering-induced stresses. The results of these analyses, when used in combination with available properties for
`the TBC, strongly suggest that the second mechanism (b) predominates in all reasonable scenarios. Consequences for the
`avoidance of this failure mode are discussed. 䊚 2002 Elsevier Science B.V. All rights reserved.
`
`Keywords: Thermal barrier coatings; Delamination; Thermal gradient; Sintering; Fracture toughness
`
`1. Introduction
`
`Multilayer thermal barrier systems are now commonly
`used in gas turbines. They comprise a single-crystal Ni
`alloy substrate, an intermediate Ni(Al) alloy layer (the
`bond coat) that acts as a barrier to oxidation, and an
`outer layer, typically yttria-stabilized zirconia, that pro-
`vides the thermal insulation w1–10x. A thermally grown
`oxide (TGO), generally a-Al O , forms between these
`2
`3
`two layers upon exposure to oxygen at high temperature.
`For high-performance systems, the TBC is manufactured
`by electron-beam physical vapor deposition (EB-PVD),
`imparting a columnar grain structure that provides strain
`tolerance w7–10x. Some failure modes originate in the
`vicinity of the interface, caused by the large residual
`compression that develops in the thin TGO layer upon
`thermal cycling w7–17x. Others occur internally, within
`the thermal barrier layer, especially in the presence of
`high heat flux (with an associated thermal gradient)
`
`* Corresponding author. Tel.: q1-609-258-4762; fax: q1-609-258-
`1177.
`E-mail address: anevans@princeton.edu (A.G. Evans).
`
`w18x. Mechanisms governing the former failure mode
`have been the subject of wide-ranging studies w6–
`8,10,13,19–22x. The latter mechanisms, which have not
`previously been examined in a systematic manner, will
`be addressed in the present article.
`In the presence of a sufficient thermal gradient, cracks
`form and propagate on delamination planes in the TBC
`parallel to the interface, resulting in regions that spall
`away, leaving a thin layer of zirconia still attached to
`the substrate w18x. This failure mode does not arise
`either when the system is thermally cycled within a
`furnace (furnace cycle tests) or when tested in a burner
`rig. It is only activated in a high heat-flux environment.
`The challenge is to identify the origins of the stress,
`and hence the delamination energy release-rate. Two
`distinct possibilities are envisaged: both are defined,
`analyzed and compared.
`
`1. An isolated crack parallel to the interface is envis-
`aged, subject to a thermal gradient, that experiences
`an energy release rate and exfoliates (Fig. 1a). A
`similar crack is connected to either a free edge (Fig.
`1b) or a crack through the thickness of the TBC.
`
`0257-8972/01/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved.
`PII: S0257-8972Ž 01. 01451-7
`
` 1
`
`UTC 2006
`General Electric v. United Technologies
`IPR2016-01289
`
`

`
`180
`
`J.W. Hutchinson, A.G. Evans / Surface and Coatings Technology 149 (2002) 179–184
`
`be addressed is shown in Fig. 2. Since the stresses
`caused by a thermal gradient (A) do not induce an
`energy release rate,
`the actual problem (B) can be
`solved by subtracting (A) from (B), resulting in the
`equivalent problem (C). In problem (C), the TBC away
`from the crack is at a uniform temperature (Ts0), while
`the crack faces experience a temperature differential,
`DTsT yT . Then, if the coating has thickness, h, and
`s
`i
`the crack is in the TBC close to the interface (afh in
`Fig. 2), the stresses s in the TBC at a distance y above
`0
`the crack are given by:
`Ž .
`Ž
`s y syE a DT 1yyyh
`¯
`(1)
`0
`tbc
`tbc
`¯Etbc
`is the thermal
`where
`is the modulus and a
`tbc
`expansion coefficient for the TBC. The effect of this
`stress on G is ascertained by adopting the Eshelby
`strategy w24x. The section of coating above the crack is
`detached, leaving a gap. A moment, M , as well as an
`o
`in-plane force, F , is imposed on the detached section
`o
`to assure that it fits exactly within the gap. The moment
`and in-plane force (per unit length) are given by:
`Ž
`.
`M sy 1y12 E a DT h
`¯
`Ž
`.
`F s 1y2 E a DT h
`¯
`tbc
`tbc
`The detached section is ‘welded in place’ and then
`the agent applying M and F lets go. The final moment
`o
`o
`and force, M and F, are obtained by coupling the ends
`the detached section to the remaining coatingy
`of
`
`.
`
`(2)
`
`tbc
`
`tbc
`
`2
`
`o
`
`o
`
`Fig. 2. The mechanics problem to be solved in order to assess the
`energy release rate for an isolated crack in the TBC subject to a heat
`flux.
`
`Fig. 1. Schematic of the three potential modes of delamination of a
`TBC in the presence of a thermal gradient.
`
`2. A shrinkage crack caused by sintering of the top
`layer of the TBC w18x, which may reorient into a
`delamination (Fig. 1c).
`In all cases, the TBC is assumed to be stress-free at
`a reference temperature, T , assumed to be equal to
`dep
`the deposition temperature (900–10008C) w23x. Devia-
`tions from this temperature induce stresses because of
`the constraint of the superalloy substrate. Additional
`stresses are created by the presence of a thermal gradient.
`Both contributions to the stress are considered.
`In the following, the energy release rate and the mode
`mixture are determined for each of the crack configu-
`rations and loadings. Comparing these with critical
`values for transverse cracking within the TBC predicts
`the most likely modes of delamination w18x.
`
`2. Exfoliation
`
`2.1. Isolated interface crack
`
`When a crack is isolated within the TBC, with no
`connection to the surface (Fig. 1a), the incidence of an
`energy release rate, G, is dictated by the conductivity of
`the crack. When the crack is conducting, such that the
`temperatures are the same on both crack faces, G is
`zero. The situation differs for insulated cracks that allow
`a temperature difference, DT, to develop between the
`faces, as shown in Fig. 1a. The mechanics problem to
`
` 2
`
`

`
`J.W. Hutchinson, A.G. Evans / Surface and Coatings Technology 149 (2002) 179–184
`
`181
`
`11
`
`tbc
`
`12
`
`tbc
`
`2
`
`3
`
`Fs
`
`22
`
`12
`
`o
`
`11
`
`o
`
`12 0
`
`substrate system by requiring continuity of moment,
`force, displacement and rotation. The displacement, u,
`and rotation, u, induced at the right-hand end in the
`remaining coatingysubstrate system are given by w26x:
`usa FyE qa MyE h
`¯
`¯
`(3)
`usa FyE hqa ME h
`¯
`¯
`tbc
`tbc
`12
`22
`The coefficients a depend on the ratio of the crack
`ij
`size to coating thickness, byh, and the two Dundurs’
`¯Ž
`a s E y
`elastic
`mismatch
`parameters,
`. Ž
`.
`D
`tbc
`y E qE
`¯
`¯
`¯
`and b . They have been tab-
`E
`D
`substrate
`tbc
`substrate
`ulated by Yu and Hutchinson w26x. The second Dundurs’
`parameter, b , plays a minor role and it will be taken
`D
`to be zero here. The corresponding displacement and
`rotation at the right end of the detached segment are
`given by:
`.
`Ž
`us F yF byE h
`¯
`tbc
`o
`.
`Ž
`us12 M yM byE h
`¯
`tbc
`o
`Imposition of the continuity conditions for the two
`pieces using Eqs. (2)–(4) gives:
`.
`Ž
`.Ž
`.
`Ž
`byh a q12byh F y12 byh a M yh
`o
`/
`.
`Ž
`.Ž
`Ž
`.
`12 byh a qbyh My hy byh a F
`Myhs
`/
`.
`.Ž
`Ž
`/s a qbyh a q12byh ya
`2
`11
`22
`12
`The stress intensity factors induced by these displace-
`ments and rotations are expressed in terms of F and M.
`For b s0, they can be written as w26x:
`D
`K sc Fy hq2 3c Myh
`y
`y
`3y2
`K sc Fy hq2 3c Myh
`y
`y
`3y2
`DT
`II
`11
`12
`The coefficients c are again functions of byh and a ;
`D
`they have been tabulated by Yu and Hutchinson w26x.
`The energy release rate and mode mixture may be
`obtained from the stress intensity factors w25x as:
`1
`1
`1
`B
`E
`FŽ
`.
`C
`2 E
`E
`D
`G
`substrate
`EKII
`DT
`cstan
`F
`(7a)
`DT
`GKI
`In the limit, when byh41, G approaches the limiting
`steady-state value:
`Ž
`.
`G s 1y2 F yE hq6M yE h
`¯
`¯
`2
`o
`tbc
`¯ Ž
`.
`'E a DT hy6
`2
`(7b)
`tbc
`tbc
`as a function of byh is given in
`G yGss
`A plot of
`DT
`DT
`Fig. 3 for two values of a : one corresponding to no
`D
`s1y3
`¯
`¯
`E yE
`elastic mismatch and the other to
`.
`tbc
`substrate
`
`(4)
`
`(5)
`
`(6)
`
`Fig. 3. The energy release rate as a function of crack length, 2b,
`relative to coating thickness, h, for exfoliation of an isolated crack
`subject to a heat flux. The steady-state level is defined in the text.
`The steady-state limit wEq. (7b)x bounds the energy
`release rate from above and is only approached for fairly
`long cracks. The mode mixture associated with the
`entire range in Fig. 3 is predominately mode II with a
`small component of mode I w25x. The mode mixture has
`not been plotted.
`Recall that for an isolated crack, there are no other
`contributions to
`: that is, even when stresses arise
`DTGss
`in the TBC because of thermal expansion misfit and
`thermal gradients, they do not induce an energy release
`rate w25x. The situation changes when the delamination
`is connected to a free edge (Fig. 1c) w18x, as discussed
`next. Namely, the contribution in Eq. (6) still exists, but
`there is an extra contribution from the stresses in the
`TBC.
`
`2.2. Edge delamination
`
`When the TBC is at a higher temperature than the
`substrate and subject to a thermal gradient, two effects
`induce an energy release rate at an edge-connected
`delamination. Heating of the TBC above the stress-free
`temperature, T , causes a thermal expansion misfit that
`dep
`places the TBC in residual compression. A superposed
`thermal gradient in the TBC induces a bending moment.
`The net effect is a stress distribution in the TBC given
`by:
`.
`Ž
`s(y)ss 1yyyh qs yyh
`(8)
`o
`i
`where s is the stress at the TBC interface and s the
`o
`i
`stress at the surface, with y being the distance from the
`interface. These stresses create a force and a moment
`(per unit length) in the absence of the delamination,
`given by:
`.
`Ž
`.Ž
`Fs 1y2 s qs h
`o
`.
`Ž
`.Ž
`Msy 1y12 s ys h
`
`i
`
`2
`
`i
`
`o
`
`(9)
`
`DT
`I
`
`21
`
`22
`
`ij
`
`G s
`
`DT
`
`q
`
`tbc
`B
`y1C
`D
`
`K qK
`
`DT2
`I
`
`DT2
`II
`
`DT
`ss
`
`2
`o
`
`3
`
`tbc
`
` 3
`
`

`
`182
`
`J.W. Hutchinson, A.G. Evans / Surface and Coatings Technology 149 (2002) 179–184
`
`3. Sintering cracks
`
`When the top surface is at a sufficiently high temper-
`ature, the material begins to sinter, resulting in lateral
`shrinkage w18x. As the shrinkage occurs, in-plane tensile
`stresses are induced because of the constraint of the
`substrate. When this reaches the ‘sintering’ stress, s ,s
`the lateral shrinkage stops. This stress is the product of
`the surface energy, g , with curvature, k, of those
`s
`contacts between columns that experience neck growth
`and densification: s s2g k w28x. If it is large enough,
`s
`s
`the stress can cause ‘sintering’ cracks to form. If this
`stress develops over a depth, H, and a sintering crack
`forms through the thickness of the TBC w18x, the crack
`could reorient into a delamination at depth a. Subject to
`this scenario, the energy release available for the delam-
`ination can be derived as follows.
`The force and moment per unit length are:
`Fss Hs
`.
`Ž
`.
`Ž
`Ms 1y2 s H ayH
`s
`where the moment is again taken at approximately the
`midplane of the coating. The steady-state energy release
`rate expression in Eq. (10) continues to hold, so that:
`w
`x
`2B
`Es Hs
`F Ž
`∂
`2
`C
`¯D
`2Etbc
`G
`This result is plotted in Fig. 5.
`The corresponding result for a crack of length a
`extending from the surface through the thickness and
`subject to a stress s imposed over a segment of length
`s
`H from the free surface (a)H) is given by Tada et al.
`w29x for the case of no elastic mismatch:
`4 s H
`2B
`EB E
`a
`sC
`FC F
`¯D
`p E
`H
`GD G
`tbc
`w
`B
`= 1.30y0.18
`x
`C
`y
`D
`
`(13)
`
`G s
`
`sinter
`ss
`
`Ž
`.
`Hya 1q3 1yHya
`μ
`
`.
`
`(14)
`
`G
`
`through
`
`s
`
`H
`a
`
`E
`sin
`F
`G
`
`2
`
`z
`B E
`Hy1
`|
`C F
`a
`~
`D G
`
`(15)
`
`Fig. 4. A typical thermal profile within a TBC and substrate under
`operating conditions within a turbine w27x.
`
`ss
`
`'
`
`3
`
`(10)
`
`i
`
`i
`
`tbc
`
`i
`
`o
`
`(11)
`
`When a delamination crack emerges from an edge
`along the interface, the force and moment are released,
`giving rise to an energy release rate which approaches
`from below the steady-state energy release rate w25x:
`¯
`6M
`2
`2
`2
`2
`F
`s h
`Ds h
`q
`q
`edgeG s
`¯
`¯
`¯
`¯
`2E
`2E h E h
`24E
`tbc
`tbc
`tbc
`tbc
`¯ Ž
`.
`ss s qs y2
`is the average stress in the TBC
`where
`o
`Dsss yso
`is the stress difference between the
`and
`i
`top surface and the interface.
`A prototypical steady-state temperature distribution
`for a thermal barrier system (Fig. 4) w27x is used to
`relate
`being
`to the thermal environment, with T
`edgeGss
`dep
`the stress-free temperature, T the temperature at the
`o
`TBC surface, T the temperature at the interface with
`i
`the bond coat and T
`the alloy temperature at the
`cool
`cooling channels. For this scenario, the temperature in
`.
`¯ Ž
`Ts T qT
`y2
`the alloy is taken as
`, such that:
`cool
`Ž
`.
`DssE a T yT
`¯
`tbc
`¯Ž
`.
`x
`w
`∂
`μ
`.
`Ž
`T qT y2yT
`ss2E a TyT ya
`¯
`¯
`tbc
`s
`dep
`tbc
`o
`dep
`i
`Inspection of Eq. (10) and Eq. (11) indicates that for
`cases where T
`is in the range 900–10008C, and T is
`dep
`o
`above 12008C, the energy release is dominated by the
`¯s
`term in Eq. (10), which in turn is predominantly
`governed by the second term in Eq. (11), since the
`alloy temperature is near the deposition temperature. In
`other words, the dominant stress contribution is due to
`the elevation of the average temperature in the coating
`above the deposition temperature. Accordingly,
`the
`effective energy release rate becomes:
`¯E hatbc
`B
`E
`Fw
`T qT y2T
`G f
`C
`2
`D
`G
`This energy release is subject to strictly mode II (shear)
`is a compression force w25x.
`¯s
`loading, since
`
`edge
`ss
`
`2
`tbc
`
`o
`
`i
`
`x
`dep
`
`2
`
`(12)
`
`Fig. 5. The energy release rate for sintering-induced delamination as
`a function of crack depth.
`
` 4
`
`

`
`J.W. Hutchinson, A.G. Evans / Surface and Coatings Technology 149 (2002) 179–184
`
`183
`
`Table 1
`Parameter range for analysis of energy release rates and delamination
`
`TBC modulus, E (GPa)
`tbc
`TBC thermal expansion coefficient, a (ppm 8C )y1
`tbc
`Stress-free temperature, T
`(8C)
`dep
`Surface temperature of TBC, T (8C)
`o
`Interface temperature, T (8C)
`2
`TBC thickness, h (mm)
`TBC toughness (J m )y2
`Mode I
`Mode II
`
`20–100
`13
`900–1000
`1200–1300
`1050
`100
`
`5–20
`60–80
`
`This result is also plotted in Fig. 5. The energy release
`rate of the mode I through-crack is always greater that
`that of the delamination crack at the same depth a, but
`the difference becomes small when the depth is four- or
`five-fold the sintering layer thickness. Since the delam-
`ination crack would normally extend along the trajectory
`with K s0, the preferred crack plane would be at ay
`II
`Hs3.86, such that the steady-state energy release rate
`becomes w25x:
`G s0.343 s HyE
`¯
`sinter
`2
`ss
`s
`tbc
`Such a crack would be strictly mode I (opening).
`
`(16)
`
`4. Predominant mechanisms
`
`4.1. Material properties
`
`likely to cause
`Insights into the phenomena most
`delamination in the presence of a thermal gradient can
`be gained by comparing the energy release rates ascer-
`tained from Eqs. (7a), (7b), (12) and (16) for several
`prototypical scenarios and relating the absolute levels to
`the fracture toughness of the TBC. The parameter ranges
`indicated on Table 1 are used to conduct the estimates.
`¯Etbc
`The range in
`is used to explore the effect of
`sintering, which can elevate the high-temperature mod-
`ulus from approximately 20 up to approximately 100
`GPa w30,31x. The stress-free temperatures reflect the
`range used in commercial practice for EB-PVD coatings
`w23x. The surface temperature is taken to range from
`that used at present to temperatures expected for more
`aggressive designs, based on high-performance TBCs.
`The sintering stress is typical of that found for micron-
`sized necks in powder compacts w28x. The fracture
`toughness values have the following origins. The mode
`II toughness has been measured at ambient temperature
`by various impression tests: it is of order of G f60 J
`II
`w21,32,33a,33bx. The mode I toughness has not
`y2
`m
`been measured for EB-PVD materials. For the present
`purposes, it is estimated, based on (i) similarity with
`plasma spray coatings w34x and (ii) typical ratios of
`mode Iymode II toughness for oxides w35x. This assess-
`ment infers a toughness of G s5–20 J m , encom-
`y2
`I
`passing the mode I toughness for polycrystalline alumina
`
`(the TGO), as well as that for the TGOybond coat
`interface w8,21x. The through-thickness toughness is
`unknown, but should be much smaller than the trans-
`verse toughness (probably lower than 1 J m ).y2
`The sintering stress, s s2g k, with g f1 J m , isy2
`s
`s
`s
`entirely dependent on the curvature of the necks at the
`contacts between adjacent columns. Images of these
`necks w21,23x
`indicate typical values of kf2=106
`m ('1 mm neck diameters), such that s f4 MPa.
`y1
`s
`Slightly larger values (up to 10 MPa) are conceivable
`in some cases.
`4.2. Through-cracks w18x
`
`The only source of an energy release rate for cracks
`that might extend through the TBC is that related to the
`sintering stress wEq. (15)x. The largest realistic values
`arise when there is a pre-existing crack, afH, where-
`f0.1 J m . While this is quite small,
`y2
`upon G
`through
`even for the low estimate of the toughness cited above,
`it must be large enough to cause through-cracks at
`temperatures where diffusional (creep) processes facili-
`tate crack extension (similar to sintering cracks in other
`applications).
`4.3. Delaminations w18x
`
`The energy release rates for delamination due to the
`sintering stress wEq. (16)x is approximately a factor of
`10 less than that for the through-crack, and the toughness
`is much higher. These two factors exclude delamination
`as a result of the sintering stress. This would still be
`true if the sintering stress were appreciably larger (by
`an order of magnitude) than that cited in Table 1.
`To address delamination of isolated cracks, the max-
`imum achievable values in the thermal environment of
`the TBC are found by equating T to T , with the
`o
`i
`assumption that the TBC is transparent and radiation
`heats the top surface of the crack. Then
`ranges
`DTGss
`y2
`from 1 to 17 J m , with the largest value referring to
`a combination of the largest modulus (upon sintering of
`the TBC) and the greatest surface temperature. Even
`the extreme value is lower than the mode II toughness
`relevant to this type of loading.
`Delamination from edges or through-cracks is much
`more probable, with the likelihood dependent on the
`incidence of TBC sintering w18x. The highest energy
`release rates arise whenever some sintering has occurred
`), and when the system is¯E ™100 GPa
`
`(such that
`tbc
`subject to a combination of the highest surface temper-
`ature with the lower deposition temperature. Then,
`y2
`is over 200 J m , well above the mode II
`edge
`Gss
`toughness (again the operative mode of loading), where-
`upon delamination appears to be inevitable. It is still
`just below the mode II toughness (approx. 50 J m )y2
`for a surface temperature of 12008C and a deposition
`
` 5
`
`

`
`184
`
`J.W. Hutchinson, A.G. Evans / Surface and Coatings Technology 149 (2002) 179–184
`
`temperature of 10008C, conditions often encountered in
`advanced turbines. Delaminations can thus be envisaged
`as the TBC sinters and when surface temperatures reach
`extreme levels, either in the presence of a free edge or
`when a through-crack exists in the TBC (caused by
`sintering) w18x.
`In the absence of sintering (¯E f20 GPa
`
`) these
`tbc
`energy release rates decrease to approximately 40 and
`y2
`10 J m , respectively, values either slightly or appre-
`ciably smaller than the mode II toughness, but similar
`to the mode I toughness. The implication is that delam-
`ination is unlikely, even at extremes of temperature,
`unless transverse loads are imposed that decrease the
`mode mixture (larger component of mode I).
`In some instances, edge delamination may be exac-
`erbated when the crack is insulating, such that the stress
`intensities from Eq. (6) superpose on those associated
`with Eq. (10). Conditions wherein this might occur
`remain to be addressed.
`
`5. Conclusion
`
`While three possible mechanisms of delamination in
`a thermal gradient have been analyzed, only one appears
`to be effective: namely, the edge delamination result
`expressed by Eq. (12). This result has several implica-
`tions for conditions likely to activate this failure mode
`in preference to others (governed by the TGO). The
`,¯Etbc
`most important parameters are the TBC modulus,
`as well as the difference between the deposition and
`surface temperatures, T yT . The delamination like-
`dep
`o
`lihood increases as either of these quantities increase.
`The modulus is primarily affected by sintering, governed
`in turn by both the material and the surface temperature.
`The temperature difference is associated with manufac-
`turing conditions, T , as well as the design of the
`dep
`turbine and the thermal conductivity of the TBC, which
`affect T .o
`
`References
`w1x R.A. Miller, J. Am. Ceram. Soc. 67 (1984) 517.
`w2x J.T. DeMasi-Marcin, D.K. Gupta, Surf. Coat. Technol. 68y69
`(1994).
`w3x R. Hillery (Ed.), Coatings For High-Temperature Structural
`Application, NRC Report, National Academy Press, 1996.
`w4x T.E. Strangman, Thin Solid Films 127 (1985) 93–105.
`w5x P.K. Wright, A.G. Evans, Curr. Opin. Solid-State Mater. Sci. 4
`(1999) 255–265.
`
`w6x P.K. Wright, Mater. Sci. Eng. A 245 (1998) 191–200.
`w7x A.G. Evans, D.R. Mumm, J.W. Hutchinson, G.H. Meier, F.S.
`Pettit, Prog. Mater. Sci., in press.
`w8x V. Sergo, D.R. Clarke, J. Am. Ceram. Soc. 81 (1998) 3237–
`3242.
`w9x J.R. Nicholls, Mater. Sci. Forum 935 (1997) 251–254.
`w10x S.R. Choi, J.W. Hutchinson, A.G. Evans, Mech. Mater. 31
`(1999) 431–447.
`w11x D.M. Lipkin, D.R. Clarke, Oxid. Met. 45 (1996) 267–280.
`w12x V.K. Tolpygo, D.R. Clarke, Oxid. Met. 49 (1998) 187–211.
`w13x M.J. Stiger, N.M. Yanar, M.G. Topping, F.S. Pettit, G.H. Meier,
`Z. Metall. 90 (1999) 1069–1078.
`w14x R.J. Christensen, V.K. Tolpygo, D.R. Clarke, Acta Mater. 45
`(1997) 1761–1766.
`w15x V.K. Tolpygo, D.R. Clarke, Acta Mater. 46 (1998) 5153–5166.
`w16x X-Y. Gong, D.R. Clarke, Oxid. Met. 50 (3y4) (1999) 355–376.
`w17x W.J. Quadakkers, A.K. Tyagi, D. Clemens, R. Anton, L. Singh-
`eiser, in: J.M. Hampikian, N.B. Dahotre (Eds.), Elevated Tem-
`perature Coatings: Science Technology, TMS, Warrendale, PA,
`1999, p. 119.
`w18x D. Zhu, R.A. Miller, B.A. Nagaraj, R.W. Bruce, Surf. Coat.
`Technol. 138 (2001) 1–8.
`w19x M. Gell, K. Vaidyanathan, B. Barber, J. Cheng, E. Jordan, Met.
`Mater.Trans. 30A (1999) 427.
`w20x A.G. Evans, J.W. Hutchinson, M.Y. He, Acta Mater. 47 (1999)
`1513–1522.
`w21x D.R. Mumm, A.G. Evans, Acta Mater. 48 (2000) 1815–1827.
`w22x C. Mennicke, D.R. Mumm, D.R. Clarke, Z. Metall. 90 (1999)
`1079–1085.
`w23x A. Mariochocchi, A. Bartz, D. Wortman, Thermal Barrier
`Coating Workshop, NASA CP 3312, 1995, p. 79.
`w24x J.D. Eshelby, Proc. R. Soc. A 244 (1952) 87–112.
`w25x J.W. Hutchinson, Z. Suo, Adv. Appl. Mech. 29 (1992) 62–191.
`w26x H.-H. Yu, J.W. Hutchinson, Int. J. Solids Struct., submitted.
`w27x R. Siegel, C.M. Spruckler, Mater. Sci. Eng. A 245 (1998) 150–
`159.
`w28x W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to
`Ceramics, Wiley and Sons, New York, 1976.
`w29x H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks
`Handbook, 3rd, ASME Press, New York, 2000.
`w30x C.A. Johnson, J.A. Ruud, R. Bruce, D. Wortman, Surf. Coat.
`Technol. 108y109 (1998) 80.
`w31x S.R. Choi, D. Zhu, R.A. Miller, Ceram. Eng. Sci. 19 (1998)
`293–301.
`w32x Y.C. Tsui, T.W. Clyne, in: C.C. Berndt (Ed.), Thermal Spray
`Practical Solutions for Engineering Problems, ASM Internation-
`al, 1996, p. 275.
`w33ax A. Vasinonta, J.L. Beuth, Eng. Fracture Mech., in press.
`w33bx R. Handoko, J.L. Beuth, G.H. Meier, F.S. Pettit, M.J. Stiger,
`Proceedings of the Materials Division Symposium on Durable
`Surfaces, ASME Orlando, November 2000, Trans Tech
`Publications, Switzerland, 2000.
`w34x G. Quian, T. Nakamura, C.C. Berndt, Mech. Mater. 27 (1998)
`91–110.
`w35x A.G. Evans, J.W. Hutchinson, Y. Wei, Acta Mater. 47 (1999)
`4093–4113.
`
` 6