Calcium–magnesium–alumina–silicate (CMAS) delamination mechanisms in EB-PVD thermal barrier coatings

Surface & Coatings Technology 200 (2006) 3418 – 3427 www.elsevier.com/locate/surfcoat

Calcium–magnesium–alumina–silicate (CMAS) delamination mechanisms in EB-PVD thermal barrier coatings Xi Chen* Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, United States Received 3 August 2004; accepted in revised form 31 December 2004 Available online 2 March 2005

Abstract Thermal barrier coatings (TBCs) comprise thermally insulating materials having sufficient thickness and durability that they can sustain an appreciable temperature difference between the load bearing alloy and the coating surface. TBC exhibit multiple failure modes. Among them, in-service erosion caused by the deposition of significant amount of calcium–magnesium–alumina–silicate (CMAS) at high temperature was found to be one of the most prevalent failure modes. The large thermal expansion mismatch between the CMAS and TBC and the extra strain energy stored in the CMAS layer can lead to delamination cracks between the TBC and bond coat (BC). In this study, the energy release rate and mode mixity of a propagating delamination crack are calculated by using the finite element analysis. The columnar microstructure of EB-PVD TBC is factored into the approach. The effects of CMAS layer thickness, mechanical and thermal properties are examined, and the steady-state energy release rates are compared with a theoretical model. Two failure mechanisms associated with CMAS deposition are analyzed: cracking within individual columns and spallation of a large TBC layer. It is believed that both mechanisms have contributed to the CMAS delamination failure. Failure criterions are derived which provide useful insights on how to improve the resistance of CMAS delamination. D 2005 Elsevier B.V. All rights reserved. Keywords: Thermal barrier coating; CMAS; Delamination; Energy release rate; Fracture toughness

1. Introduction Thermal barrier coatings (TBCs) are widely used in turbine engines. They are regarded as one of the most successful innovations and applications of coatings in industry. Thermal barrier systems consist of a tri-layer (Fig. 1): the prevalent outer layer is yttria-stabilized-zirconia (YSZ), acting as the thermal barrier coating (TBC). A thermally grown oxide (TGO) exists between the TBC and a bond coat (BC) [1–4]. TBC exhibit multiple failure modes [1–3,5]. Prior assessments have focused primarily on modes governed by the energy density in the thermally grown oxide associated with thermal cycling, which causes failure by either large-scale buckling or edge delamination [1,5–9]; as well as modes related with foreign object damage (FOD)

* Tel.: +1 212 854 3787; fax: +1 212 854 6267. E-mail address: [email protected]. 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2004.12.029

[10,11] and erosion [12,13] at high temperature, which causes material removal due to the ingestion of foreign particles during turbine operation. Among other failure modes that have not been widely documented, spallation caused by environmental surface deposits at working temperature is particularly important [13–15]. The study emphasizes TBCs produced using electron beam physical vapor deposition (EB-PVD), which have a strain tolerant columnar microstructure and the gaps are filled with high-porosity oxides (Fig. 1) [1]. At temperature in excess of 1150 8C, sand particles and debris ingested during operation become molten and adhere to TBC surfaces. During this process, calcium, magnesium, alumina, and silicate (CMAS) are incorporated in the molten phase, and the excellent wetting characteristics of CMAS enable the deposition to infiltrate the TBC microstructure. The addition of the CMAS layer changes the near-surface mechanical properties. Thereafter, upon cooling, the CMAS layer solidifies into a fully dense stiff domain (Fig. 2a)

X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

3419

Thermal barrier system

TBC

Bond Coat

~100 µm

TGO

Superalloy

50 µm

Fig. 1. The thermal barrier system consisting of several layers: the Thermal Barrier Coating (TBC), Thermally Grown Oxide (TGO), Bond Coat (BC), and the superalloy substrate [1]. Note the columnar microstructure of EB-PVD TBC and the gaps are filled with high porous materials.

a

CMAS

TBC

b Undamaged TBC

Eroded TBC

Region exhibiting almost complete TBC removal

c

d

Delamination Coalescence Between Vertical Separations

50 µm

Fig. 2. Recent experimental observations [13]: a) CMAS deposits that penetrate between intact TBC columns; b) Center regions showing almost complete TBC removal caused by CMAS deposition (the right half of the specimen are eroded where the TBC columns are not completely removed); c) Side view of the spallation of TBC by CMAS deposition: in some regions the TBC columns are completely removed, whereas in other regions the eroded TBC columns show bterracingQ effect; d) Vertical separations that form in the TBC due to sintering, which serve as internal edges to initiate delamination.

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X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

[13,14], which has a substantially lower coefficient of thermal expansion (CTE) than the TBC and superalloy substrate. Consequently, large compressive stresses develop as the system cools, which elevate the energy release rate of the system, lead to delamination of the underlying TBC and result in material removal (Fig. 2b,c) [13,14]. In Figs. 2b and c, it is evident that critical areas of TBC have been completely spalled off from the substrate due to CMAS delamination. Due to the complicated environment actually experienced by TBC, there are some other berodedQ regions with remnant TBC columns. For its practical importance, we focus on the most critical CMAS failure mode where the TBC columns are completely removed by delamination cracking along the TBC/BC interface. The intent of the present study is to decipher the role of CMAS deposition on the delamination of TBC. Recent experimental observations [13,14] suggest that upon cooling, the TBC has been completely removed in the regions deposited with predominant CMAS. Even with the presence of the large stresses caused by thermal expansion mismatches, an energy release rate for delamination crack only develops if binternal edgesQ exist, such as vertical separations caused by TBC sintering (Fig. 2d). These separations accommodate the in-plane displacements needed to extend the delamination [1,2,16]. By using the finite element method, the energy release rate of a propagating delamination crack extending from the end of a vertical sintering separation is calculated and compared the interface fracture toughness between TBC/BC. An analytical model is used to derive the steady-state energy release rate and critical CMAS thickness in closed-form, which are in good agreement with numerical results and experiments. Since the chemical composition and deposition rate of CMAS varies with operation environment [14], the CMAS thickness and physical properties (e.g. modulus and CTE) are varied in the numerical approach to verify their corresponding roles. The characterizations are combined to establish the most likely material removal scenario.

2. Model Inspired by the experimental observations (Fig. 2), the plane strain model of the TBC, thickness h TBC, is shown in Fig. 3a. The columns are assumed to be straight and parallel before deformation, all having width d, with gap, w, between columns. The number of columns that can be included in the model is restricted by computational time. In order to examine the edge effect and steady-state energy release rate, 200 columns are included in the numerical simulation. The thickness of CMAS is h CMAS and it penetrates the column gaps for a depth h pen. The remaining of the gaps is filled by high-porosity material, with a thickness, h TBC–h pen. The bond coat thickness is h BC and the superalloy substrate is taken to be infinitely deep. To obtain representative results, h TBC=h BC=100 Am [1],

h pen=10 Am, and h CMAS is varied between 0 and 40 Am to examine its role on energy release rate. The TBC column width is taken to be d=9 Am, and the gap width w=1 Am, such that the porosity of TBC is about 10%, a typical value observed in most thermal barrier systems [1,10–12]. The TGO is neglected in the present model. The material in the TBC columns is considered to be isotropic, with properties representative of porous zirconia. The TBC material is taken to be elastic–perfectly plastic governed by a Von Mises yield surface, with a Young’s Y modulus E TBC=100 GPa, and its yield stress r TBC is varying linearly with temperature: 1 GPa at room temperature and 100 MPa at 1150 8C [1,10]. The gaps are filled with a compliant, high porosity material with E gapc2 GPa, such that the transverse modulus of TBC is about 20 GPa [1]. The superalloy substrate is assumed to be elastic with E sub=100 Y GPa. For bond coat, E BC=100 GPa, r BC =500 MPa at ambient temperature and 500 MPa at working temperature. The CMAS is a ceramic which remains elastic at any temperature [13,14] with a variable stiffness. The Poisson’s ratio is taken to be 0.25 for all layers. The thermal expansion mismatch between the various layers is the source of residual stress, energy release rate, and delamination. The following typical values of CTE are adopted: a sub=14.10
6/8C, a BC=16.10
6/8C, a TBC=11.10
6/ 8C [1]. The CTE of CMAS is smaller than all other layers and is set to be a free variable. Finite element calculations were performed using the commercial code ABAQUS [17]. The option for finite deformation and strain was employed. As discussed above, the internal edges created by vertical sintering cracks are essential for developing the energy release rate. Therefore, only the thermal barrier system within the bpotential material removal zoneQ sandwiched between 2 sintering separations (A–A and B–B in Fig. 3a) is modeled. Within this domain, 200 TBC columns are incorporated in the simulation. The large number of TBC columns is necessary in order to study the edge effect of delamination. Fig. 3b shows the geometry and boundary condition used in the finite element analysis, where the internal vertical sintering edges in CMAS and TBC are traction-free. The rest of vertical boundaries are symmetric with respect to the x 1 axis. A typical mesh comprises about 400,000 4-node plane strain elements with reduced integration. The system is assumed to be stress-free at working temperature. Upon cooling to the ambient temperature (DT=1150 8C), both the normal and shear stress components developed at the TBC/ BC interface are recorded from the numerical analysis. By superposition, the solution of cracking in a residual stress field is equivalent to the stress state of a reduced problem, where the surface traction applied on the crack faces corresponds to the residual stress field [18]. In the second step of finite element analysis, the superposition principle is employed to determine the stress intensity for a delamination crack propagating in the residual stress field. As elaborated below, the detached CMAS/TBC strip

X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

a

3421

b Potential material removal zone: 200 TBC columns B A

CMAS penetration zone

hCMAS

hCMAS

CMAS

Free A sintering internal edge in TBC

hpen Vertical Sintering Crack

200 TBC columns

B

CMAS

Free sintering internal edge in TBC

TBC hTBC

TBC

TBC d

w

a

x2

d

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10

BC

x1

A

Short column crack

hBC

Bond coat

Bond coat

B

Superalloy B

A Symmetry in bond coat and superalloy

c A

200 TBC columns

B

CMAS Free edge in

TBC

TBC

Free edge in TBC

x2 Crack length a

x1

Long delamination crack

Bond coat

B

A Symmetry in bond coat and superalloy

Fig. 3. Schematic drawings of the CMAS delamination problem: a CMAS layer is deposited on top of the TBC, and it penetrates the TBC columns once it melts. As the system is cooled, cracking may be induced near the base of the TBC if the energy release rate induced by CMAS deposition is high. a) Schematic of several components in the TBC system and their relevant dimensions. Only the section within the potential material removal zone is modeled in the numerical analysis. b) Short column cracks near the sintering separation. These cracks are located at the TBC/BC interface. c) A long delamination crack propagating along the TBC/BC interface.

experiences a residual bending moment with a sign that causes the delaminated surfaces to remain in contact. In order to gain helpful insights, friction contact is assumed between the delaminated surfaces. This assumption allows us to focus only on the role of CMAS deposition vs. energy release rate (instead of exploring the effect of crack surface roughness). Two types of delamination crack are investigated: short cracks within individual columns (Fig. 3b), and long delamination/spallation crack which breaks many columns behind the crack front (Fig. 3c). Both types of cracks are located on the TBC/BC interface, in consistent with

experimental observations of the critical situation (for complete TBC column removal, Fig. 2b–d). The crack length is denoted as a in both cases. Their energy release rates and mode mixity are calculated as the cracks proceed.

3. Numerical results 3.1. Residual stress A representative contour plot of residual stress normal to the interface (r 22) is shown in Fig. 4a, with a CMAS=8.10
6/

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X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

a

x2

σ 22 / σ

Y TBC

CMAS

0.1

0.05

TBC 0

Interior edge

x1 Bond coat

b

10

20

30

40

Column Number from Interior Edge

50

10

Normalized Shear Stress at TBC/BC Interface

0.6

h CMAS = 20 µ m ECMAS = 100 GPa α CMAS = 8·10-6 /º C

0.4 0.2 0.0

Y σ22 / σ TBC

Y σ22 / σ TBC

Normalized Normal Stress at TBC/BC Interface

c

Column Number from Interior Edge

-0.2

CMAS -0.4

Residual tension by TBC column bending

-0.6

TBC

-0.8

x2 =0

BC

-1.0 0

1

2

3

20

30

40

50

0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25

h CMAS = 20 µ m ECMAS = 100 GPa α CMAS = 8·10-6 /º C

-0.30

x2=0

-0.35 4

5

Distance from Interior Edge x1 /h TBC

6

0

1

2

3

4

5

6

Distance from Interior Edge x1 /h TBC

Fig. 4. The residual stress field near the interior edge, with a CMAS=8.10
6/8C, E CMAS=100 GPa, and h CMAS=20 Am: a) Contour plot of the residual stress field normal to the interface; b) The variation of normalized normal stress at the TBC/BC interface, with a schematic showing the TBC column bending effect; c) The variation of normalized shear stress at the TBC/BC interface.

8C, E CMAS=100 GPa, and h CMAS=20 Am. Here, only the several columns that are closest to the interior edge are shown. In order to emphasize the column bending effect (see below), the gap elements are removed from the contour plot. In Fig. 4b, the stress at the TBC/BC interface is normalized by the yield stress of TBC at room temperature, and plotted as a function of normalized distance away from the interior edge, x 1/h TBC. It is noticeable that due to the columnar microstructure and thermal expansion misfit among CMAS/TBC/BC layers, the TBC columns undergo extensive bending near the edge. A schematic drawing of TBC column bending is shown in the insert of Fig. 4b. The most significant bending stress arises at the upper-left and lower-right corners in each column (Fig. 4a and b). The maximum tensile normal stress peaks at about the 6th column from the interior edge, and then decays away from the sintering separation (Fig. 4b). It is expected that due to these high local bending stresses, individual columns

adjacent to the sintering edge will crack. The cracks are likely to initiate on the right side of each column and then propagating left, breaking the whole column (the highly porous gaps are assumed to have no fracture toughness). Once these short column cracks coalesce, a delamination crack will form near the interior edge (Figs. 2d and 3c). The distribution of shear stress (r 12) along the interface is given by Fig. 4c. Once again, the stress oscillation within each column is due to the column bending effect. The inplane stress component, r 11, is relaxed at the free edge and transits to a constant value r i=E i Da i DT in the interior—the transition distance is set by the shear lag in the substrate, typically several film thickness [19]. Here, i refers to the various layers (e.g. CMAS, TBC, BC), Da i is the mismatch of CTE between each layer and the substrate, and E i is the Young’s modulus of corresponding layers. Since both r 22 and r 12 are very small in the interior of the film (when x 1/ h TBC is large), the steady-state energy release rate of a

X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

3.2. Short cracks in individual columns

a

50

2

hCMAS = 20 µ m

Delamination Crack

45

Energy Release Rate G (J/m )

delamination crack is dictated by r 11, which can be derived in a closed form (see below). Near the edge, however, both large r 22 and r 12 caused by column bending dictate the stress intensity of a short delamination crack, and finite element analysis must be used to explore the stress intensities. The residual normal and shear traction at the TBC/BC interface (Fig. 4b and c) are then imposed on the surfaces of a crack. The fracture characteristics of both short column cracks and long delamination cracks are discussed below.

3423

ECMAS = 100 GPa α CMAS = 8 • 10 -6 / o C

CMAS

40 35

TBC

30 25

a

20 15 10

Bahavior when crack tip is in the middle of each column

5

Energy release rates and mode mixity of short cracks within selected columns near the interior edge are given in Fig. 5a and b, respectively. The column numbers are counted from the free edge (see Fig. 3b). The numerical

0 0

b 80 Column #10 2

Energy Release Rate G (J/m )

Column #6

a 50

BC

40

Column #3

h CMAS = 20 µ m

30

E CMAS = 100GPa Mode I Toughness

20 10

1.67

50

Gsstheory = 59 J / m 2

45

from eq. (7)

4

2.5

3.33

40 35 30 25 20 15

hCMAS = 20 µ m

10

ECMAS = 100 GPa α CMAS = 8 • 10 -6 / o C

5 0

Crack tip is in the middle of each column

Delamination Crack

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized Crack Length a / d

Reference problem

h CMAS = 20 µ m

40

E CMAS = 100GPa

σ0 30

Column #3

a d

20

Column #10

Short Column Cracks 0 0.2

5

10

15

20

25

30

35

40

45

50

Fig. 6. a) The oscillation of energy release rate of a delamination crack within the first several columns from the interior edge. The solid curve denotes the behavior when the crack tip is located in the middle of each column. b) The energy release rate of a long delamination crack when the crack tip is in the middle of each column, with a CMAS=8.10
6/8C, E CMAS=100 GPa, and h CMAS=20 Am. The steady-state energy release rate G sstheory predicted from the analytical model (7) is in good agreement with the numerical trend.

Column #6

10

0.1

0

Normalized Crack Length a / (d+w) (Column Number)

b

Mode Mixity (degree)

3

55

TBC

2

Energy Release Rate G (J/m )

70

d

0.83

60

Short Column Cracks

2

Normalized Crack Length a / (hCMAS+hTBC)

a

60

1

Normalized Crack Length a / (d+w) (Column Number)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Crack Length a / d

Fig. 5. a) The energy release rate, and b) mode mixity of short cracks in the 3rd, 6th, and 10th column from the sintering edge. aC MAS=8.10
6/8C, E CMAS=100 GPa, and h CMAS=20 Am.

results are calculated with the same representative CMAS physical properties, a CMAS=8.10
6/8C, E CMAS=100 GPa, and h CMAS=20 Am. The first important feature of column cracking is that the energy release rate G increases rapidly with the crack length a/d. In fact, G goes to infinity when the crack is going to break the entire column (a/dY1). This observation is in consistent with the classic crack mechanics reference solution, sketched in the insert of Fig. 5b: For a single edge crack in a finite width plate subjected to a linearly distributed residual bending stress, as a/dY1, the

X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

pffiffiffiffiffiffi mode I stress intensity KI Y0:374r0 pa=ð1
a=d Þ3=2 and approaches infinity [20]. Although the geometry and residual stress distribution are more complicated in the present study, our numerical solution agrees with such trend. The second primary feature is that with the high bending normal stress at the TBC/BC interface, column cracking is essentially mode I (Fig. 5b). Comparing with the typical mode I TBC fracture toughness (C IC=10 J/m2) shown in Fig. 5a, it is conceivable that with the assistance of some initial flaws, several columns adjacent to the interior edge are susceptible to break. The delamination crack then emerges with the coalescence of short column cracks, discussed next.

Normalized Crack Length a / hTBC 1

2

3 theory ss

G

60

Energy Release Rate G (J/m2)

3424

4

= 59 J / m 2

} EE

CMAS

=1

TBC

50

Gsstheory = 37 J / m 2

40

ECMAS

}E

= 0.5

TBC

30

20

h CMAS = 20 µm

α CMAS = 8 • 10 -6 / o C

10

Delamination Crack 0 0

10

20

30

40

50

Normalized Crack Length a / (d+w) (Column Number)

a

Normalized Crack Length a / hTBC 1

90

2

3

Gsstheory = 81J / m 2

Delamination Crack

hCMAS

Energy Release Rate G (J/m2)

80

}h

ECMAS = 100GPa α CMAS = 8 10 6 / o C

70

Fig. 8. The effect of CMAS stiffness on the energy release rate: E CMAS/ E TBC=1 and 0.5. The energy release rate increases with increasing CMAS stiffness.

4

= 0.3

TBC

Gsstheory = 59 J / m 2

60

}

50 theory ss

G

40

= 38 J / m

3.3. Long delamination cracks

hCMAS = 0.2 hTBC

2

} hh

CMAS

= 0.1

TBC

30 20

Gsstheory = 11J / m 2

hCMAS

}h

10

=0

TBC

0 0

10

20

30

40

50

(no CMAS)

Normalized Crack Length a / (d+w) (Column Number) Normalized Crack Length a / hTBC

b

1

90

Mode Mixity ψ (degree)

89

2

3

4

h CMAS / h TBC = 0

88 87 86 h CMAS / h TBC = 0. 1

85 84

h CMAS / h TBC = 0.2

83

h CMAS / h TBC = 0.3

82

ECMAS = 100 GPa

81

σ CMAS = 8 • 10 -6 / o C

Delamination Crack

80 0

10

20

30

40

50

Normalized Crack Length a / (d+w) (Column Number) Fig. 7. The effect of CMAS thickness: a) The energy release rate, and b) mode mixity of delamination cracks when h CMAS/h TBC=0 (no CMAS), 0.1, 0.2, and 0.3. The energy release rate increases with increasing CMAS thickness. Note that the delamination is pure mode II, and the steady-state energy release rates obtained from finite element analysis agree with the analytical model (7).

Due to the local bending stresses, the stress intensity oscillates within each column as a delamination crack proceeds. Fig. 6a shows the energy release rate of a short delamination crack as the crack propagates through the first 4 columns from the interior edge. The variations of G within each column are self-similar, and they follow the basic trend of bending stresses (Fig. 4b). The self-similarity of delamination G preserves for all TBC columns examined in this study, and the average of G within a column occurs when the crack tip is roughly in the middle of each column. Note that the average of G across a column when the column number gets large is the steady-state energy release rate G ss. Therefore, in order to reveal the steady-state trend, the evolution of G when the crack tip is located in the middle of each column is plotted in Fig. 6b, with h CMAS=20 Am, a CMAS=8.10
6/8C and E CMAS=100 GPa. It can be seen that G gradually increases with crack growth, and G ss is attained after the delamination crack has broken through about 30–40 columns (or 3–4 delaminated CMAS/TBC strip thickness).1 The G ss obtained from finite element analysis is aout 60 J/m2, agrees well with that predicted from the analytical model ( G theory =59 J/m2 from Eq. (7), ss see below). The effects of CMAS physical properties are now examined. For different CMS thickness, the energy release rate and mode mixity of delamination cracks are plotted in Fig. 7a and b, respectively. The steady state energy release rates predicted from the analytical model 1 It should be noted that if the film is homogeneous, a recent study by Yu and Hutchinson [21] has shown that the edge effect is smaller: G ss is achieved when the delamination length is about 1–2 film thickness. In the present study, however, the columnar microstructure of TBC has induced more compliance in the system, causing larger edge effect.

X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

Normalized Crack Length a / hTBC 1

2

3

4

Gsstheory = 152 J / m 2

Delamination Crack

160

Energy Release Rate G (J/m2)

G sstheory are also given in Fig. 7a and they agree well with the numerical analysis. It is found that G increases with increasing CMAS thickness, owing to the extra strain energy stored in the thicker CMAS layer. In all cases, G ss is reached when the delamination length is about 3–4 CMAS/TBC strip thickness. Both TBC and CMAS have smaller CTE than that of the substrate, and they are both under residual compression as the system is cooled. The corresponding residual bending moment causes the delaminated CMAS/TBC composite to bend downwards and remain in contact with the BC/substrate [16]. Thus, the resulting delamination cracking is essentially mode II when friction is neglected between the crack surfaces. This can be verified from Fig. 7b. Comparing with a typical mode II TBC toughness, C IC=60–80 J/m2, a critical CMAS thickness for spallation can be deduced, h critical CMASc20~30 Am. This critical thickness is in consistent with experimental observation [13], where the CMAS thickness on the remaining TBC sections was found to be smaller than this value. The effects of CMAS stiffness and CTE mismatch are shown in Figs. 8 and 9, respectively. Since the residual stress scales with E CMAS and Da CMAS=a sub
a CMAS, the

3425

140

}

120

∆α CMAS = 12 • 10 -6 / o C

100

∆α CMAS = 6 • 10 -6 / o C

80

Gsstheory = 59 J / m 2

60

}

40 20

∆α CMAS =α

sub

−α CMAS

h CMAS = 20 µm ECMAS = 100GPa

0 0

10

20

30

40

50

Normalized Crack Length a / (d+w) (Column Number) Fig. 9. The effect of CMAS thermal expansion misfit on the energy release rate: Da CMAS=12.10
6/8C and 6.10
6/8C. The energy release rate increases with increasing expansion misfit between CMAS and substrate.

energy release rate increases with increasing E CMAS and Da CMAS. In both cases, the edge effect is about the same and cracking is still essentially mode II.

4. An analytical model for steady-state cracking After the melting and penetration of ingested debris, a large in-plane residual compression r CMAS is developed in the CMAS layer upon cooling to ambient (Fig. 3). For plane strain problem, only the stress component in x 1 direction contributes to the strain energy far away from the edge. The delamination is located at the TBC/BC interface with a depth, h CMAS+h TBC (Fig. 3a). The steady-state energy release rate, G theory , can be obtained by taking the difference between the ss strain energy density (SED) far ahead of the delamination crack tip from that far behind the tip, i.e., Gtheory ¼ ðSEDÞdown
ðSEDÞup ¼ ss

 res 2 3 3 ri hi 1 X r2i hi 1 X
; 2 i¼1 Ei 2 i¼1 Ei

ð1Þ

where i=1,2,3 refers to the three layers of the delaminated strip: CMAS, the CMAS penetration layer (a mixed CMAS/TBC composite), and TBC, respectively. r i =E i Da i DT=E i (a sub
a i )DT is the magnitude of in-plane residual stress induced in each layer (far away from the free edge) by thermal expansion misfit. Since the detached CMAS/TBC strip remains in contact with the BC/substrate, the thermal misfit between the CMAS and TBC indicates that there are some residual stresses within the delaminated layers, denoted by rres for the three layers discussed above. It is straightforward that i h1 ¼ hCMAS ; h2 ¼ hpen ; and h3 ¼ hTBC
hpen ; E1 ¼ ECMAS ; E2 ¼ Epen ¼ ð f =ECMAS þ ð1
f Þ=ETBC Þ
1 ;

ð2Þ 

lateral and E3 ¼ ETBC ¼ f =Egap þ ð1
f Þ=ETBC


1

;

a1 ¼ aCMAS ; a2 ¼ apen ¼ f aCMAS þ ð1
f ÞaTBC ; and a3 ¼ aTBC ;

ð3Þ ð4Þ

where f=w/(d+w). Since the delaminated strip remains flat due to the frictionless contact between the crack surfaces, the residual stress within each of the three layers can be determined by rres i ¼ Ei ðai DT
eres Þ;

ð5Þ

where e res is the residual strain of the detached strip. Since the detached strip is traction free, force equilibrium requires 3 X i¼1

rres i hi ¼ 0:

ð6Þ

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X. Chen / Surface & Coatings Technology 200 (2006) 3418–3427

By solving e res from Eqs. (5) and (6), finally, the steady-state energy release rate G theory can be obtained in a closed form: ss  Gtheory ¼ ss

DT

P3

i¼1

2

P3

Ei hi Dai

i¼1

Ei hi

2 ð7Þ

:

increases with h CMAS, E CMAS, and Da CMAS, due to more strain energy density contributed by CMAS. Note that G theory ss These trends agree with numerical results (Figs. 7–9). For the same reasoning, the delamination is preferred at the bottom of TBC where the energy density is maximum, which is in consistent with our basic assumption (Fig. 3). As revealed in Figs. 6–9, the steady-state energy release rates predicted by this analytical model agree well with finite element simulation. Finally, by equating G theory to the mode II TBC toughness, C IIC, a critical CMAS thickness can be derived: ss  ffi pffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P CIIC þ CIIC CIIC þ 2Da1 DT 2 j¼2;3 hj Ej Da1
Daj
Da1 DT 2 j¼2;3 hj Ej Daj critical : ð8Þ hCMAS ¼ E1 ðDa1 DT Þ2 At this critical thickness of CMAS deposition, the spallation of the CMAS/TBC strip is likely to occur. From (8), h critical CMAS decreases with increasing E CMAS and Da CMAS. Since both E CMAS and Da CMAS depend on the composition, in order to resist CMAS-induced delamination, it is therefore more desirable to have a composition that leads to a compliant CMAS layer with a large coefficient of thermal expansion.

5. Conclusion The addition of the CMAS layer to the thermal barrier system changes the near-surface mechanical properties and elevates the energy release rate. Moreover, the large thermal expansion mismatch between the CMAS layer and TBC leads to high thermal stress upon cooling to ambient, and the TBC system is susceptible to delamination. In this study, by incorporating the columnar microstructure of TBC, the finite element analysis is used to analyze the energy release rate and mode mixity of both short column cracks and long delamination (spallation) cracks. The steady-state energy release rates are compared with an analytical model. The effects of CMAS physical properties are examined and their implication for spallation failure are deduced. The analysis of the residual stress field and strain energy density induced by CMAS deposition indicates that cracking is likely to occur at the bottom of TBC, provided vertical sintering edges exist. The short column cracks near the interior separation are dictated by the high bending stresses caused by extensive TBC column bending (Fig. 4). The energy release rates of column cracks increase quickly with crack growth, and the coalescence of these short cracks may lead to delamination. The delamination is pure mode II, and it approaches steady state when the crack length is about three to four times of the detached CMAS/TBC strip thickness. The edge effect of delamination arises from the compliant columnar microstructure of the TBC. The delamination energy release rate increases with CMAS thickness, stiffness, and thermal expansion misfit with substrate. The steadystate energy release rate obtained from numerical analysis agrees well with that predicted from the analytical model. The analysis infers that a critical CMAS thickness for critical TBC spallation is roughly, h CMAS c20~30 Am. This is in

qualitative agreement with the observation from parallel experiments [13]. Finally, in order to improve the critical can be increased resistance of CMAS delamination, h CMAS if CMAS is more compliant with a higher coefficient of thermal expansion.

Acknowledgement The author is grateful for helpful discussions with Professor Anthony G. Evans at the University of California at Santa Barbara, and Professor John W. Hutchinson at Harvard University. This work was supported in part by ONR 04-123219, and in part by NSF CMS-0407743.

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