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The crack number density theory on air-plasma-sprayed thermal barrier coating
Accepted Manuscript The crack number density theory on air-plasma-sprayed thermal barrier coating
X.J. Wu PII: DOI: Reference:
S0257-8972(18)31268-4 https://doi.org/10.1016/j.surfcoat.2018.11.058 SCT 24024
To appear in:
Surface & Coatings Technology
Received date: Revised date: Accepted date:
13 February 2018 29 October 2018 19 November 2018
Please cite this article as: X.J. Wu , The crack number density theory on air-plasmasprayed thermal barrier coating. Sct (2018), https://doi.org/10.1016/j.surfcoat.2018.11.058
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ACCEPTED MANUSCRIPT The crack number density theory on air-plasma-sprayed thermal barrier coating
X.J. Wu Structures and Materials Performance Laboratory, Aerospace Research Center
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National Research Council of Canada
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Tel: 1-613-990-5051
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1200 Montreal Road, Ottawa, ON, Canada K1A 0R6
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Email: [email protected]
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ABSTRACT
A crack number density (CND) theory model is developed for air-plasma-sprayed
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thermal barrier coating (TBC), which describes the evolution of crack number and size
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distribution as function of exposure time. The model is compared in good agreement with experimental measurements from quasi-isothermal-cyclic oxidation tests. Both the CND
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model and experimental observations indicate that thermally-grown oxides (TGO) are responsible for crack nucleation and growth. The model can be used to define TBC failure (spallation) by coalescence of microcracks into a maximum allowable crack size with a given probability.
ACCEPTED MANUSCRIPT 1.
Introduction Thermal barrier coatings (TBC) are now widely used as protective coatings for
hot-section components in advanced gas turbine engines. A TBC system generally consists of a yittria-stabilized zirconia (YSZ) top coat and a metallic bond coat on a
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superalloy substrate component. When exposed to hot gas during engine operation, a
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layer of thermally grown oxides (TGO) will form at the interface between the ceramic
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topcoat and the metallic bond coat, which often causes cracking and eventually leads to TBC spallation.
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Many factors such as TBC microstructure, thermal expansion mismatch between
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different constituent layers and TGO formation have been found to contribute to TBC failure, which are reported in a vast amount of publications, but the author can only
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reference a few [1-15], due to the limited space. Regarding plasma-sprayed TBC, by the
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nature of its deposition process, there are always variations in chemical composition, grain structures and interface morphology in commercially produced TBC, which affect
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the coating life. Local depletion of aluminum could result in early formation of NiO,
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Cr2O3 and spinel (Ni(Cr,Al)2O4), instead of alumina (Al2O3), which would lead to early crack nucleation [10, 11], because oxidation would induce a large local volume strain, in
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addition to the thermal expansion mismatch strain. It has been generally understood that damage accumulation in air-plasma-spray (APS) TBC proceeds by collective nucleation and growth of numerous microcracks [5, 6, 10-14, 15]. It is the microstructural variations that result in a wide scatter of TBC life. Wu et al. once plotted the crack-size distributions in an air-plasma-sprayed TBC as lognormal distributions at different stages of thermal exposure [10]. In addition to metallurgical examination and characterization, many
ACCEPTED MANUSCRIPT mechanistic studies have also been devoted to describe the TBC failure phenomena. For example, stress and damage mechanics analyses have been performed using the finite element method (FEM) for TBC systems comprised of YSZ/TGO/bond-coat/substrate structure [2, 4, 7, 9], where the damage process is treated either as propagation of one
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dominant crack between the peaks of undulation of YSZ/bond coat interface [4], or as
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accumulation of the continuum damage with values from 0 to 1 at failure [7]. Under
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combined thermal-mechanical loading, surface cracks may also form vertical to the TBC/bond coat interface. Limited studies have been conducted to investigate the effects
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of segmented surface crack density on interfacial delamination in TBC under some
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extreme conditions such as tension at room temperature and thermal shock [16, 17]. The physical and statistical process of micro-cracking in APS-TBC has not been sufficiently
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described. For faithful assessment of TBC structural integrity, it is important to
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understand the physical crack evolution in TBC, because it is always the largest crack that poses the imminent threat to cause TBC spallation, and the probability of its
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occurrence determines the TBC life.
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In this paper, the crack number density (CND) theory originally proposed by Fang et al. [18] is used to describe the crack evolution process in an APS-TBC. The theory has
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been used to describe the population of short fatigue cracks in steels [19, 20]. As the CND theory generally describes the crack development process starting from a given crack nucleation distribution and growing at a certain rate, it should be applicable to TBC as well. In this regard, we assume that crack nucleation follows the Gamma-distribution and crack growth occurs by the Evans-He-Hutchinson sphere-wedging mechanism in relation to TGO growth [15]. The crack-size distribution data are used to calibrate the
ACCEPTED MANUSCRIPT crack number density function at the first observation moment and its evolutionary distributions are calculated from the CND model, which are then compared with the experimental measurements. Thus, the CND solution provides a full view of crack-size distribution in the time-axis. The Crack Number Density Theory
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2.
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The crack-number-density (CND) theory describes the evolution of crack number
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by a differential equation as [18]:
(1)
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n [An ] n N t c
rate, and nN is the crack nucleation rate.
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where n is the number of cracks at time t, c is the crack length, A = 𝑐̇ is the crack growth
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Eq. (1) states that the incremental crack number is the sum of contributions from
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crack nucleation and growth into the size c at any time t. The theory itself does not specify how cracks nucleate and how fast they will grow. Therefore, it has to be
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combined with relevant physical theories for description of a specific crack growth
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process.
The general solution of Eq. (1), with the initial value of n = 0, can be written in
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the following form [18]:
n N (c) t, n (c, t ) 1 c n (u )du, A(c) N
c c0 c c0
(2a)
where c0 is a threshold value below which cracks do not grow, and is an intermediate crack size variable that is evaluated from
t c
du A( u )
(2b)
ACCEPTED MANUSCRIPT where u is an intermediate integration variable. In TBC, the growth of TGO thickness (h) generally follows the parabolic oxidation law, which has been observed in many experimental studies [7, 9-14]: h 2kt
(3)
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where k is the oxidation constant, which depends on temperature in a typically Arrhenius
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form.
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A fracture mechanics model has been proposed by Evans et al. [15], which
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depicts crack growth as wedge-opened by a growing sphere corresponding to TGO formation along the undulating interface between the topcoat and bond coat. The stress
3
𝐾 = 2(1+𝑣)
ℎ
( ) √𝑅 3(1−𝑣)𝑚 𝑅
(4)
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√
𝑅 3/2 (𝑚−1)𝐸
( ) 𝜋 𝑐
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intensity factor K for such a crack configuration is expressed as
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where R is the radius of local asperity at crack initiation site, m is the ratio of volume change by oxidation, E is the Young’s modulus and is Poisson’s ratio of TBC.
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As one can see from Eq. (4), the stress intensity factor increases with the TGO
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thickness, but inversely proportional to crack length, c, to the 3/2 power. As TGO thickens, the stress intensity factor may reach a critical value, i.e. the fracture toughness
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KIC, to cause crack growth. But, crack extension immediately lowers the stress intensity factor such that the crack growth remains stable in the TBC. By re-arrangement of Eq. (4), we have:
(𝑚−1)𝐸𝑅ℎ
𝑐 = [2𝑚(1−𝑣2 )
√𝜋𝐾𝐼𝐶
2 3
] = 𝐶ℎ2/3
(5a)
where (𝑚−1)𝐸𝑅
𝐶 = [2𝑚(1−𝑣2 )
√𝜋𝐾𝐼𝐶
]
2 3
(5b)
ACCEPTED MANUSCRIPT Experimental studies have shown that a power-law relationship does exist between the maximum crack size and the equivalent TGO thickness, with the power exponent closely matching the theoretical value of the above model [12-14]. From Eq. (5), we can derive the crack growth rate as function of crack length, c, as 2 2 2 𝐴 = 𝑐̇ = 3 𝐶ℎ−1/3 ℎ̇ = 3 𝑘𝐶ℎ−4/3 = 3 𝑘𝐶 3 𝑐 −2
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(6)
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We assume that c0 =0 and crack nucleation in the APS-TBC follows a Gamma
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distribution, (, ) with = 2 and = 1/(ϕh), where ϕ is the shape factor, such that 𝛽𝛼
(7)
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𝑛𝑁 = 𝐵 (𝛼) 𝑐 𝛼−1 𝑒 −𝛽𝑐
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where B is a scaling constant.
Substituting Eq. (6) and (7) into Eq. (2) through integration, we obtain the crack
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number density function as 𝐵
where, from Eq. (2b), 3
(8b)
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𝜂 = √𝑐 3 − 2𝑘𝐶 3 𝑡
(8a)
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𝑛(𝑐, 𝑡) = 𝐴(𝛼) [𝛽(𝜂𝑒 −𝛽𝜂 − 𝑐𝑒 −𝛽𝑐 ) + (𝑒 −𝛽𝜂 − 𝑒 −𝛽𝑐 )]
Experimental Validation
A)
The Experiment
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3.
The TBC samples were manufactured by air-plasma spray, which consisted of yttria partially stabilized zirconia (ZrO2-8%Y2O3) top coat, Ni-22Cr-10Al-1Y (wt%) bond coat and Inconel 625 substrate. The sample size was 12.5 mm in diameter. The thickness of the ceramic coat was 250-310 m, and that of the bond coat was 160-180 m. A total of six samples were exposed to stagnant air at 1200oC in a furnace. The
ACCEPTED MANUSCRIPT thermal exposure cycle consisted of 15-minutes of ramping-up, 23-hours of isothermal soak at 1200°C, and 45-minutes of cooling-down to the ambient temperature (25°C). At each time interval of 24 hrs, 48 hrs, 72 hrs, 120 hrs, and 144 hrs, one sample was retrieved. The last sample failed after 161 hrs with more than 30% of the topcoat spalled.
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All samples were then sectioned along the diameter line and examined under a Philips
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XL30S field emission gun scanning electron microscope (SEM). The SEM was operated
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at an accelerating voltage between 5 kV and 20 kV. A statistically-significant number of micrographs were taken along the cut-section at locations where cracks were found
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randomly. Crack size measurements were taken from micrographs for a particular
B)
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exposure-time using the same magnification scale. Results
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The evolution of the TBC microstructures have been examined in the previous
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study, using SEM and EDX techniques [11]. The micrographs are organized in time sequence from the as-sprayed condition to the subsequently exposed conditions, as shown
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in Figure 1, to reflect the physical cracking evolution process in relation to TGO
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formation and growth, as it will be described using the CND theory. The as-sprayed microstructure is a typical laminar microstructure consisting of
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thermal spray splats, Figure 1a. The topcoat contains discontinuities such as pores and splat boundaries; the bond coat is relatively dense but has been partially oxidized during the plasma spraying process. Upon thermal exposure in air at 1200°C, mixed oxides containing (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO grew rapidly along with the continuous formation of an Al2O3 layer in the TBC/bond coat interface region, as shown in Figure 1b to 1d. These mixed oxides and spinel have been reported in a previous paper [11].
ACCEPTED MANUSCRIPT Similar TGO products were also observed in other studies, e.g. [6]. Cracking was mostly seen to be associated with the formation of (Cr,Al)2O3, spinel (Ni(Cr,Al)2O4) and nickel oxide (NiO) [11]. Formation of the mixed oxides was quite heterogeneous, which introduced local volume change. The crystalline structures and volumes of the basic
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phases in the bond coat and TGO are given in Table 1. By comparison, it is clear that
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oxides have larger volumes per unit cell than the β-NiAl phase in the bond coat,
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especially the spinel (Ni(Cr,Al)2O4). The metallurgical evidence shown in Figure 1 and Table 1 supports the wedge-cracking mechanism, as represented by Eq. (4). Cracks
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usually nucleated within or near the mixed oxides (Figures 1b and 1c) and grew into the
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ceramic topcoat. At a later stage, some microcracks would coalesce but the large cracks appeared along the pre-existing discontinuities (splat boundaries) to form a long
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dominant crack in the ceramic topcoat near the interface region (Figure 1d), which would
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lead to TBC spallation.
(a)
(b)
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top coat
crack
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bond coat
(d)
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(c)
Figure 1. Microstructures of air-plasma sprayed TBC: a) pristine, and after b) 24 hrs, c)
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48 hrs, and d) 144 hrs. of exposure.
To evaluate TGO growth, an equivalent TGO thickness, eq, is defined as [12]: ∑(𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑇𝐺𝑂 𝑎𝑟𝑒𝑎) ∑(𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑇𝐶/𝐵𝐶 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒)
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𝛿𝑒𝑞 =
(9)
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This takes into account the entire TGO, including the mixed oxides and spinel, which are quite
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heterogeneous. Then, TGO growth is assumed to follow the parabolic law, Eq. (3), with an
oxidation constant k = 0.9 μm2/hr at the test temperature. The oxidation curve is shown in
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Figure 2.
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Figure 2. TGO thickness as function of exposure time at 1200oC.
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Cracks in the one-day (A1D), two-day (A2D), three-day (A3D) and six-day (A6D) exposed samples were counted and measured from multiple micrographs (taken at
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different locations) in each sample cross-section. In some micrographs, only one crack
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was found within the numbered (1, 2, 3, …) frame; in others, multiple cracks were found within one frame, which were then labelled, e.g., 10L for the crack on the left, and 10R for the crack on the right side of the frame, as shown in Figure 3. The crack size is defined as the linear length from tip to tip. The crack length measurement results are shown in Table 2. The crack-size distribution as represented by these data are shown in Figure 4.
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Figure 3. Crack labelling in Frame 10 and Frame 30 for the 24 hr. exposed sample.
cryst.
units/cell
NiAl
cubic
Ni3Al
cubic
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Al2O3
hcp
NiAl2O4
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24 hrs.
1 2 3 4 5 6 7
crack length (m) 45.6 11.2 8.8 17.6 18.4 18.4 21.6
1 2 3L 3R 4L 4R 5
1
45.38
cubic
8
522.00
cubic
8
575.00
hcp
6
288.47
cubic
4
72.90
Table 2.
Crack Length Measurements
48 hrs.
image #
24.06
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NiO
1
254.00
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Cr2O3
vol. (Å3)
6
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NiCr2O4
image #
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Table 1. Phase Volumes
crack length (m) 4.8 26.4 9.6 15.2 15.2 50.4 50.4
72 hrs. image #
1 2 3 4 5 6 7
crack length (m) 13.6 12.8 24.8 101.6 17.6 22.4 37.6
144 hrs. image #
1L 1R 2L 2R 3L 3R 4L
crack length (m) 54 76 40 112 94 182 42
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66 74 70 44 44 56 36 0 28 72 116 96 92 0 0 150 86 244 0 72 60 44 60 74 98 0 60 36 38 284 132 256 86 144 26 46 38 56 98 58 90 26 50 56 68 78 52 78 178 84 102 86 58 46 0 0
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4R 5L 5M 5R 6L 6M 6R 7 8L 8R 9L1 9L2 9R 10L 10R 11L 11R 12 13 14L 14R 15L 15R 16L 16R 17 18 19L 19R 20L 20R 21L 21R 22 23L 23R 24L 24R 25L 25R 26L 26R 27L 27M 27R 28L 28M 28R 29L 29R 30L 30R 31L 31R 32 33
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14.4 70.4 96 20 32.8 28.8 25.6 47.2 42.4 30.4 25.6 24.8 13.6 26.4 42.4 36.8 84 24 52 29.6 65.6 30.4 82.4 45.6 53.6 25.6 21.6 48.8 34.4 51.2 29.6 29.6 35.2 51.2 36.8 59.2 44.8 46.4 40 33.6 23.2 40 24.8 28.8 19.2 94.4 32.8 40 50.4 24 51.2 27.2 16.8 18.4 55.2 49.6
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8L 8R 9 10 11L 11R 12 13 14L 14R 15L 15R 16L 16R 17 18 19 20L 20R 21 22 23 24L 24R 25 26 27 28 30L 30M 30R 31L 31R 33 36 37 38 39 40L 40R 41 42L 42R 43 44 45 46 47 48 50 51L 51R 52L 52R 53L 53R
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56 20.8 26.4 6.4 32.8 52 42.4 44 14.4 20.8 14.4 14.4 20 0 17.6 17.6 42.4 88 65.6 32 22.4 40.8 69.6 24 12.8 33.6 19.2 17.6 13.6 36 32.8 28.8 41.6 30.4 21.6 12 15.2 0 15.2 38.4 40 49.6 20 52.8 18.4 44 20 26.4 37.6 27.2 36 25.6 16 41.6 28 20.8
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6L 6R 7 8L 8R 9L 9R 10L 10R 11 12L 12R 13 14 15L 15R 16 17 18L 18R 19 20 21 22 23 24L 24R 25 26L 26R 27 28L 28R 29L 29R 30 31 32 33 34L 34R 35 36 37 38 39 40 41 42 43 44L 44R 45 46L 46R 47L
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114.4 20.8 20 20 37.6 14.4 65.6 12 12 17.6 14.4 30.4 34.4 25.6 32 27.2 37.6 15.2 28.8 14.4 56 18.4 16 35.2 36 11.2 24.8 12.8 12.8 25.6 12 13.6 26.4 13.6 24.8 30.4 28 36 24.8 16 23.2 14.4 19.2 14.4 10.4 30.4 27.2 42.4 24 10.4 12.8 36 30.4 20.8 20 28
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8 9 10L 10R 11 12 13 14L 14R 15 17 18L 18R 19 20 21L 21R 23L1 23L2 23R 24 25 26 27 28L 28R 29 30L1 30L2 30R 31L 31R 32L 32R 33 34 35L 35R 36L 36R 37 38 41 44 45 47L 47R 48 49L 49R 50 51 52 53 55 56L
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4.
54L 54R 55L 55R 56 57L 57R 58L 58R 59 60 61 62 63 64 65L 65R
19.2 32 28 43.2 18.4 25.6 16 47.2 35.2 74.4 20.8 22.4 30.4 41.6 28 20 20.8
34 35 36L 36M 36R 37L 37R 38L 38R 39 40 41 42L 42R
166 192 38 74 72 72 104 106 92 60 0 0 94 42
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66.4 28.8 46.4 49.6 36.8 51.2 12 19.2 21.6 20 12.8 31.2 37.6 41.6
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47R 48L 48R 49 50 51 52L 52R 53L 53R 54L 54R 55 56
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64.8 39.2 39.2 18.4 18.4 12 26.4 56 20 16 12
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56R 58L 58R 59 60L1 60L2 60R 63 64 65L 65R
Discussion
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The experimental crack size distributions are shown in Figure 4 (a-d), for
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different exposure times. When normalized with the total number of cracks for the
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particular exposure time, the data represent the probability density of the crack size distribution in the TBC. By the heterogeneous nature of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO
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stochastic in nature.
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formation, due to local compositional inhomogeneity, crack nucleation is certainly
To describe the crack evolution process using Eq. (8), the crack growth constant
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C and the distribution shape factor ϕ need to be calibrated, based on the crack size measurement data. Despite that the crack growth constant C depends on the Young’s modulus E, the fracture toughness KIC, the local oxidation-induced volume change m and the asperity radius R, as given by Eq. (5), which may vary per local cracking condition, in the present statistical analysis using the CND theory for the first time on TBC, both C and ϕ are assumed to be constants for the exposure condition: C = 0.394 μm3/2 and ϕ =
ACCEPTED MANUSCRIPT 0.24 are determined by calibrating to the 24 hr. distribution at 1200oC. The subsequent distribution curves are calculated using Eq. (8) as follows. Using Eq. (8), the total number of cracks at any given exposure time can be evaluated as ∞
𝑁(𝑡) = ∫0 𝑛(𝑐, 𝑡)𝑑𝑐 Hence, the CND probability is given by 𝑛(𝑐,𝑡) 𝑁(𝑡)
=
1 [𝛽(𝜂𝑒 −𝛽𝜂 −𝑐𝑒 −𝛽𝑐 )+(𝑒 −𝛽𝜂 −𝑒 −𝛽𝑐 )] 𝐴 ∞1 ∫0 𝐴[𝛽(𝜂𝑒 −𝛽𝜂 −𝑐𝑒 −𝛽𝑐 )+(𝑒 −𝛽𝜂 −𝑒 −𝛽𝑐 )]𝑑𝑐
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𝜌(𝑐, 𝑡) =
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(10)
(11)
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Note that the scaling constant B is cancelled out from the numerator and
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denominator in Eq. (11). Thus, the CND probability distributions are computationally obtained from Eq. (11), assuming crack nucleation in a Gamma distribution with index
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= 2, and β = 1/(ϕh), as shown in Figure 4 (a-d) for each exposure stage. This is a
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simplified engineering analysis, and its validity is proven in the statistical sense. Figure 4 demonstrates that the CND theory is capable of describing the entire crack-size evolution
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history at a given temperature with constant values of C and ϕ, in good agreement with
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the experimental data. In future studies, the dependence of C and ϕ on processing and
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temperature should be characterized.
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(b)
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(a)
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(d)
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(c)
Figure 4. Crack number distributions at a) 24 hrs, b) 48 hrs, c) 72 hrs and d) 144 hrs.
A general trend can be observed from experimental data and CND calculations, Figure 4, that both the mean crack size and distribution width increases with the exposure time. The CND theory combined with the Evans-He-Hutchinson model provides a physics-based description of crack nucleation and growth phenomena in relation to TGO
ACCEPTED MANUSCRIPT growth. Particularly, crack nucleation is related to the heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO with local volume change (strain), and crack growth is driven by further TGO growth. Under the test conditions of the present study, (Cr,Al)2O3·Ni(Cr,Al)2O4·NiO
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oxides formed at the early stage of thermal exposure. This occurred at sites where the
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local aluminum concentration was low and nickel had segregated as a result of
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compositional inhomogeneity in the air-plasma-sprayed TBC system. The X-ray mapping of elements was shown in a previous paper [11]. These zones were situated along the
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original ceramic/bond coat interface region, and it is believed that the mixed oxides were
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produced because, locally, there was insufficient aluminum reacting to form an initial alumina layer to serve effectively as a diffusion barrier to Ni or Cr. Lower concentration
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of aluminum in the nickel segregated areas, particularly in air-plasma sprayed NiCrAlY,
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may change the thermodynamics of oxide formation such that it favors Ni(Cr,Al)2O4/NiO rather than Al2O3, as opposed to the opposite in the ideal case [21, 22]. These
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observations have provided the physical basis of relating crack nucleation to TGO
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formation in a stochastic manner. Assuming crack nucleation follows the Gamma distribution, it is shown that the
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theoretical CND description of crack evolution agree rather well with the actual cracksize measurement data, as a function of exposure time. Apparently, the widening of the distribution can be attributed to TGO thickening, as it is controlled by the distribution shape parameter β, which is inversely proportional to the TGO thickness, h. The mechanism of crack growth is believed to be due to TGO growth as a wedge to drive the crack open. From the fracture mechanics point of view, crack nucleation and growth in
ACCEPTED MANUSCRIPT TBC occurs by the same wedge-opening mechanism, the former is more related to heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO clusters, while the latter is driven by the entire TGO growth. It is envisaged that when the local stored elastic energy arising from volume change due to oxidation becomes equal to the decohesion energy,
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crack nucleation occurs, and the crack advances in association with the thickening of the
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TGO layer to keep the energy in balance, as implied by Eqs (4) and (5). The hypothesis
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of this mechanism operation is supported by the metallurgical evidence shown in this study. Particularly, because of uneven undulation of the TBC/bond coat interface, the
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largest crack can extend over several peaks, into a length of hundreds microns. While in a
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typical FEM model of uniform undulation with uniform TGO, crack length across two adjacent peaks (typical ~ 50μm) would cause TBC failure. The practical implication of
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CND theory vs. FEM idealization is very important. As non-destructive inspection (NDI)
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techniques are being developed to assess the integrity of TBC coated parts in service, realistic relationships of crack length vs. NDI signals must be established. Then, the CND
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model can be used for probabilistic TBC life prediction with an allowable maximum
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crack size, i.e., in terms of the probability to reach the maximum crack size at any exposure time. In practice, the distribution parameters C and ϕ can be determined
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(calibrated) with experimental observation of crack-size distribution only once at a given exposure time. The CND model provides a full picture of damage evolution in TBC in the time axis, whereas the deterministic models are mostly fitted to the experimental life without probability information. When TBC is operated at high temperatures, sintering may occur. Sintering can either cause changes in material properties or even may heal some cracks. In this case, in
ACCEPTED MANUSCRIPT addition to changing the crack growth constant C, a threshold value c0 may be set in Eq. (2), below which cracks do not grow. Then, a new CND solution needs to be obtained by integration, Eq. (2). However, there is no evidence that has happened in the present
Conclusion
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5.
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investigation. The crack-size evolution follows well with c0 = 0, as shown in Figure 4.
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A crack number density theory model has been developed for air-plasma sprayed TBC. The CND model in combination with the Evans-He-Hutchinson sphere-wedging
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crack model recognizes the nature of crack nucleation in association with the volumetric
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misfit as induced by heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO and considers the crack growth mechanism as driven by TGO growth. The model describes
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the evolution of crack size distribution in the time-axis under a quasi-isothermal-cyclic
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oxidation condition. Good agreements have been found with the crack-size measurements from the furnace-exposed air-plasma sprayed TBC samples tested at 1200oC. In principle,
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the method can be extended to other isothermal and cyclic oxidation conditions, as long
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as the Arrhenius relation of the oxidation constant is determined. This may lend a
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powerful tool for probabilistic TBC life prediction.
Acknowledgement
The TBC samples were provided by Industrial Materials Institute of National Research Council of Canada.
ACCEPTED MANUSCRIPT References
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ACCEPTED MANUSCRIPT Highlights:
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1. For the first time, crack evolution in air-plasma sprayed TBC is described by the crack number density theory. 2. The solution is derived from the crack number density evolutionary equation, assuming the wedge-opening mechanism of crack nucleation and growth in TBC, as suggested by EvansHe-Hutchinson, and crack nucleation following the Gamma distribution. 3. The integrated crack size distribution matches well with the experimental observations on an air-plasma sprayed TBC samples tested at 1200oC. 4. The model then offers a method for probabilistic TBC life prediction.
X.J. Wu PII: DOI: Reference:
S0257-8972(18)31268-4 https://doi.org/10.1016/j.surfcoat.2018.11.058 SCT 24024
To appear in:
Surface & Coatings Technology
Received date: Revised date: Accepted date:
13 February 2018 29 October 2018 19 November 2018
Please cite this article as: X.J. Wu , The crack number density theory on air-plasmasprayed thermal barrier coating. Sct (2018), https://doi.org/10.1016/j.surfcoat.2018.11.058
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ACCEPTED MANUSCRIPT The crack number density theory on air-plasma-sprayed thermal barrier coating
X.J. Wu Structures and Materials Performance Laboratory, Aerospace Research Center
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National Research Council of Canada
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Tel: 1-613-990-5051
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1200 Montreal Road, Ottawa, ON, Canada K1A 0R6
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Email: [email protected]
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ABSTRACT
A crack number density (CND) theory model is developed for air-plasma-sprayed
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thermal barrier coating (TBC), which describes the evolution of crack number and size
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distribution as function of exposure time. The model is compared in good agreement with experimental measurements from quasi-isothermal-cyclic oxidation tests. Both the CND
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model and experimental observations indicate that thermally-grown oxides (TGO) are responsible for crack nucleation and growth. The model can be used to define TBC failure (spallation) by coalescence of microcracks into a maximum allowable crack size with a given probability.
ACCEPTED MANUSCRIPT 1.
Introduction Thermal barrier coatings (TBC) are now widely used as protective coatings for
hot-section components in advanced gas turbine engines. A TBC system generally consists of a yittria-stabilized zirconia (YSZ) top coat and a metallic bond coat on a
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superalloy substrate component. When exposed to hot gas during engine operation, a
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layer of thermally grown oxides (TGO) will form at the interface between the ceramic
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topcoat and the metallic bond coat, which often causes cracking and eventually leads to TBC spallation.
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Many factors such as TBC microstructure, thermal expansion mismatch between
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different constituent layers and TGO formation have been found to contribute to TBC failure, which are reported in a vast amount of publications, but the author can only
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reference a few [1-15], due to the limited space. Regarding plasma-sprayed TBC, by the
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nature of its deposition process, there are always variations in chemical composition, grain structures and interface morphology in commercially produced TBC, which affect
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the coating life. Local depletion of aluminum could result in early formation of NiO,
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Cr2O3 and spinel (Ni(Cr,Al)2O4), instead of alumina (Al2O3), which would lead to early crack nucleation [10, 11], because oxidation would induce a large local volume strain, in
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addition to the thermal expansion mismatch strain. It has been generally understood that damage accumulation in air-plasma-spray (APS) TBC proceeds by collective nucleation and growth of numerous microcracks [5, 6, 10-14, 15]. It is the microstructural variations that result in a wide scatter of TBC life. Wu et al. once plotted the crack-size distributions in an air-plasma-sprayed TBC as lognormal distributions at different stages of thermal exposure [10]. In addition to metallurgical examination and characterization, many
ACCEPTED MANUSCRIPT mechanistic studies have also been devoted to describe the TBC failure phenomena. For example, stress and damage mechanics analyses have been performed using the finite element method (FEM) for TBC systems comprised of YSZ/TGO/bond-coat/substrate structure [2, 4, 7, 9], where the damage process is treated either as propagation of one
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dominant crack between the peaks of undulation of YSZ/bond coat interface [4], or as
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accumulation of the continuum damage with values from 0 to 1 at failure [7]. Under
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combined thermal-mechanical loading, surface cracks may also form vertical to the TBC/bond coat interface. Limited studies have been conducted to investigate the effects
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of segmented surface crack density on interfacial delamination in TBC under some
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extreme conditions such as tension at room temperature and thermal shock [16, 17]. The physical and statistical process of micro-cracking in APS-TBC has not been sufficiently
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described. For faithful assessment of TBC structural integrity, it is important to
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understand the physical crack evolution in TBC, because it is always the largest crack that poses the imminent threat to cause TBC spallation, and the probability of its
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occurrence determines the TBC life.
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In this paper, the crack number density (CND) theory originally proposed by Fang et al. [18] is used to describe the crack evolution process in an APS-TBC. The theory has
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been used to describe the population of short fatigue cracks in steels [19, 20]. As the CND theory generally describes the crack development process starting from a given crack nucleation distribution and growing at a certain rate, it should be applicable to TBC as well. In this regard, we assume that crack nucleation follows the Gamma-distribution and crack growth occurs by the Evans-He-Hutchinson sphere-wedging mechanism in relation to TGO growth [15]. The crack-size distribution data are used to calibrate the
ACCEPTED MANUSCRIPT crack number density function at the first observation moment and its evolutionary distributions are calculated from the CND model, which are then compared with the experimental measurements. Thus, the CND solution provides a full view of crack-size distribution in the time-axis. The Crack Number Density Theory
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2.
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The crack-number-density (CND) theory describes the evolution of crack number
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by a differential equation as [18]:
(1)
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n [An ] n N t c
rate, and nN is the crack nucleation rate.
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where n is the number of cracks at time t, c is the crack length, A = 𝑐̇ is the crack growth
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Eq. (1) states that the incremental crack number is the sum of contributions from
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crack nucleation and growth into the size c at any time t. The theory itself does not specify how cracks nucleate and how fast they will grow. Therefore, it has to be
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combined with relevant physical theories for description of a specific crack growth
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process.
The general solution of Eq. (1), with the initial value of n = 0, can be written in
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the following form [18]:
n N (c) t, n (c, t ) 1 c n (u )du, A(c) N
c c0 c c0
(2a)
where c0 is a threshold value below which cracks do not grow, and is an intermediate crack size variable that is evaluated from
t c
du A( u )
(2b)
ACCEPTED MANUSCRIPT where u is an intermediate integration variable. In TBC, the growth of TGO thickness (h) generally follows the parabolic oxidation law, which has been observed in many experimental studies [7, 9-14]: h 2kt
(3)
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where k is the oxidation constant, which depends on temperature in a typically Arrhenius
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form.
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A fracture mechanics model has been proposed by Evans et al. [15], which
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depicts crack growth as wedge-opened by a growing sphere corresponding to TGO formation along the undulating interface between the topcoat and bond coat. The stress
3
𝐾 = 2(1+𝑣)
ℎ
( ) √𝑅 3(1−𝑣)𝑚 𝑅
(4)
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√
𝑅 3/2 (𝑚−1)𝐸
( ) 𝜋 𝑐
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intensity factor K for such a crack configuration is expressed as
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where R is the radius of local asperity at crack initiation site, m is the ratio of volume change by oxidation, E is the Young’s modulus and is Poisson’s ratio of TBC.
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As one can see from Eq. (4), the stress intensity factor increases with the TGO
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thickness, but inversely proportional to crack length, c, to the 3/2 power. As TGO thickens, the stress intensity factor may reach a critical value, i.e. the fracture toughness
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KIC, to cause crack growth. But, crack extension immediately lowers the stress intensity factor such that the crack growth remains stable in the TBC. By re-arrangement of Eq. (4), we have:
(𝑚−1)𝐸𝑅ℎ
𝑐 = [2𝑚(1−𝑣2 )
√𝜋𝐾𝐼𝐶
2 3
] = 𝐶ℎ2/3
(5a)
where (𝑚−1)𝐸𝑅
𝐶 = [2𝑚(1−𝑣2 )
√𝜋𝐾𝐼𝐶
]
2 3
(5b)
ACCEPTED MANUSCRIPT Experimental studies have shown that a power-law relationship does exist between the maximum crack size and the equivalent TGO thickness, with the power exponent closely matching the theoretical value of the above model [12-14]. From Eq. (5), we can derive the crack growth rate as function of crack length, c, as 2 2 2 𝐴 = 𝑐̇ = 3 𝐶ℎ−1/3 ℎ̇ = 3 𝑘𝐶ℎ−4/3 = 3 𝑘𝐶 3 𝑐 −2
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(6)
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We assume that c0 =0 and crack nucleation in the APS-TBC follows a Gamma
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distribution, (, ) with = 2 and = 1/(ϕh), where ϕ is the shape factor, such that 𝛽𝛼
(7)
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𝑛𝑁 = 𝐵 (𝛼) 𝑐 𝛼−1 𝑒 −𝛽𝑐
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where B is a scaling constant.
Substituting Eq. (6) and (7) into Eq. (2) through integration, we obtain the crack
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number density function as 𝐵
where, from Eq. (2b), 3
(8b)
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𝜂 = √𝑐 3 − 2𝑘𝐶 3 𝑡
(8a)
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𝑛(𝑐, 𝑡) = 𝐴(𝛼) [𝛽(𝜂𝑒 −𝛽𝜂 − 𝑐𝑒 −𝛽𝑐 ) + (𝑒 −𝛽𝜂 − 𝑒 −𝛽𝑐 )]
Experimental Validation
A)
The Experiment
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3.
The TBC samples were manufactured by air-plasma spray, which consisted of yttria partially stabilized zirconia (ZrO2-8%Y2O3) top coat, Ni-22Cr-10Al-1Y (wt%) bond coat and Inconel 625 substrate. The sample size was 12.5 mm in diameter. The thickness of the ceramic coat was 250-310 m, and that of the bond coat was 160-180 m. A total of six samples were exposed to stagnant air at 1200oC in a furnace. The
ACCEPTED MANUSCRIPT thermal exposure cycle consisted of 15-minutes of ramping-up, 23-hours of isothermal soak at 1200°C, and 45-minutes of cooling-down to the ambient temperature (25°C). At each time interval of 24 hrs, 48 hrs, 72 hrs, 120 hrs, and 144 hrs, one sample was retrieved. The last sample failed after 161 hrs with more than 30% of the topcoat spalled.
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All samples were then sectioned along the diameter line and examined under a Philips
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XL30S field emission gun scanning electron microscope (SEM). The SEM was operated
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at an accelerating voltage between 5 kV and 20 kV. A statistically-significant number of micrographs were taken along the cut-section at locations where cracks were found
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randomly. Crack size measurements were taken from micrographs for a particular
B)
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exposure-time using the same magnification scale. Results
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The evolution of the TBC microstructures have been examined in the previous
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study, using SEM and EDX techniques [11]. The micrographs are organized in time sequence from the as-sprayed condition to the subsequently exposed conditions, as shown
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in Figure 1, to reflect the physical cracking evolution process in relation to TGO
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formation and growth, as it will be described using the CND theory. The as-sprayed microstructure is a typical laminar microstructure consisting of
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thermal spray splats, Figure 1a. The topcoat contains discontinuities such as pores and splat boundaries; the bond coat is relatively dense but has been partially oxidized during the plasma spraying process. Upon thermal exposure in air at 1200°C, mixed oxides containing (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO grew rapidly along with the continuous formation of an Al2O3 layer in the TBC/bond coat interface region, as shown in Figure 1b to 1d. These mixed oxides and spinel have been reported in a previous paper [11].
ACCEPTED MANUSCRIPT Similar TGO products were also observed in other studies, e.g. [6]. Cracking was mostly seen to be associated with the formation of (Cr,Al)2O3, spinel (Ni(Cr,Al)2O4) and nickel oxide (NiO) [11]. Formation of the mixed oxides was quite heterogeneous, which introduced local volume change. The crystalline structures and volumes of the basic
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phases in the bond coat and TGO are given in Table 1. By comparison, it is clear that
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oxides have larger volumes per unit cell than the β-NiAl phase in the bond coat,
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especially the spinel (Ni(Cr,Al)2O4). The metallurgical evidence shown in Figure 1 and Table 1 supports the wedge-cracking mechanism, as represented by Eq. (4). Cracks
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usually nucleated within or near the mixed oxides (Figures 1b and 1c) and grew into the
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ceramic topcoat. At a later stage, some microcracks would coalesce but the large cracks appeared along the pre-existing discontinuities (splat boundaries) to form a long
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dominant crack in the ceramic topcoat near the interface region (Figure 1d), which would
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lead to TBC spallation.
(a)
(b)
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top coat
crack
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bond coat
(d)
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(c)
Figure 1. Microstructures of air-plasma sprayed TBC: a) pristine, and after b) 24 hrs, c)
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48 hrs, and d) 144 hrs. of exposure.
To evaluate TGO growth, an equivalent TGO thickness, eq, is defined as [12]: ∑(𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑇𝐺𝑂 𝑎𝑟𝑒𝑎) ∑(𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑇𝐶/𝐵𝐶 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒)
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𝛿𝑒𝑞 =
(9)
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This takes into account the entire TGO, including the mixed oxides and spinel, which are quite
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heterogeneous. Then, TGO growth is assumed to follow the parabolic law, Eq. (3), with an
oxidation constant k = 0.9 μm2/hr at the test temperature. The oxidation curve is shown in
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Figure 2.
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Figure 2. TGO thickness as function of exposure time at 1200oC.
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Cracks in the one-day (A1D), two-day (A2D), three-day (A3D) and six-day (A6D) exposed samples were counted and measured from multiple micrographs (taken at
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different locations) in each sample cross-section. In some micrographs, only one crack
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was found within the numbered (1, 2, 3, …) frame; in others, multiple cracks were found within one frame, which were then labelled, e.g., 10L for the crack on the left, and 10R for the crack on the right side of the frame, as shown in Figure 3. The crack size is defined as the linear length from tip to tip. The crack length measurement results are shown in Table 2. The crack-size distribution as represented by these data are shown in Figure 4.
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Figure 3. Crack labelling in Frame 10 and Frame 30 for the 24 hr. exposed sample.
cryst.
units/cell
NiAl
cubic
Ni3Al
cubic
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Al2O3
hcp
NiAl2O4
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24 hrs.
1 2 3 4 5 6 7
crack length (m) 45.6 11.2 8.8 17.6 18.4 18.4 21.6
1 2 3L 3R 4L 4R 5
1
45.38
cubic
8
522.00
cubic
8
575.00
hcp
6
288.47
cubic
4
72.90
Table 2.
Crack Length Measurements
48 hrs.
image #
24.06
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NiO
1
254.00
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Cr2O3
vol. (Å3)
6
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NiCr2O4
image #
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Table 1. Phase Volumes
crack length (m) 4.8 26.4 9.6 15.2 15.2 50.4 50.4
72 hrs. image #
1 2 3 4 5 6 7
crack length (m) 13.6 12.8 24.8 101.6 17.6 22.4 37.6
144 hrs. image #
1L 1R 2L 2R 3L 3R 4L
crack length (m) 54 76 40 112 94 182 42
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66 74 70 44 44 56 36 0 28 72 116 96 92 0 0 150 86 244 0 72 60 44 60 74 98 0 60 36 38 284 132 256 86 144 26 46 38 56 98 58 90 26 50 56 68 78 52 78 178 84 102 86 58 46 0 0
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4R 5L 5M 5R 6L 6M 6R 7 8L 8R 9L1 9L2 9R 10L 10R 11L 11R 12 13 14L 14R 15L 15R 16L 16R 17 18 19L 19R 20L 20R 21L 21R 22 23L 23R 24L 24R 25L 25R 26L 26R 27L 27M 27R 28L 28M 28R 29L 29R 30L 30R 31L 31R 32 33
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14.4 70.4 96 20 32.8 28.8 25.6 47.2 42.4 30.4 25.6 24.8 13.6 26.4 42.4 36.8 84 24 52 29.6 65.6 30.4 82.4 45.6 53.6 25.6 21.6 48.8 34.4 51.2 29.6 29.6 35.2 51.2 36.8 59.2 44.8 46.4 40 33.6 23.2 40 24.8 28.8 19.2 94.4 32.8 40 50.4 24 51.2 27.2 16.8 18.4 55.2 49.6
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8L 8R 9 10 11L 11R 12 13 14L 14R 15L 15R 16L 16R 17 18 19 20L 20R 21 22 23 24L 24R 25 26 27 28 30L 30M 30R 31L 31R 33 36 37 38 39 40L 40R 41 42L 42R 43 44 45 46 47 48 50 51L 51R 52L 52R 53L 53R
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56 20.8 26.4 6.4 32.8 52 42.4 44 14.4 20.8 14.4 14.4 20 0 17.6 17.6 42.4 88 65.6 32 22.4 40.8 69.6 24 12.8 33.6 19.2 17.6 13.6 36 32.8 28.8 41.6 30.4 21.6 12 15.2 0 15.2 38.4 40 49.6 20 52.8 18.4 44 20 26.4 37.6 27.2 36 25.6 16 41.6 28 20.8
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6L 6R 7 8L 8R 9L 9R 10L 10R 11 12L 12R 13 14 15L 15R 16 17 18L 18R 19 20 21 22 23 24L 24R 25 26L 26R 27 28L 28R 29L 29R 30 31 32 33 34L 34R 35 36 37 38 39 40 41 42 43 44L 44R 45 46L 46R 47L
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114.4 20.8 20 20 37.6 14.4 65.6 12 12 17.6 14.4 30.4 34.4 25.6 32 27.2 37.6 15.2 28.8 14.4 56 18.4 16 35.2 36 11.2 24.8 12.8 12.8 25.6 12 13.6 26.4 13.6 24.8 30.4 28 36 24.8 16 23.2 14.4 19.2 14.4 10.4 30.4 27.2 42.4 24 10.4 12.8 36 30.4 20.8 20 28
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8 9 10L 10R 11 12 13 14L 14R 15 17 18L 18R 19 20 21L 21R 23L1 23L2 23R 24 25 26 27 28L 28R 29 30L1 30L2 30R 31L 31R 32L 32R 33 34 35L 35R 36L 36R 37 38 41 44 45 47L 47R 48 49L 49R 50 51 52 53 55 56L
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4.
54L 54R 55L 55R 56 57L 57R 58L 58R 59 60 61 62 63 64 65L 65R
19.2 32 28 43.2 18.4 25.6 16 47.2 35.2 74.4 20.8 22.4 30.4 41.6 28 20 20.8
34 35 36L 36M 36R 37L 37R 38L 38R 39 40 41 42L 42R
166 192 38 74 72 72 104 106 92 60 0 0 94 42
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66.4 28.8 46.4 49.6 36.8 51.2 12 19.2 21.6 20 12.8 31.2 37.6 41.6
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47R 48L 48R 49 50 51 52L 52R 53L 53R 54L 54R 55 56
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64.8 39.2 39.2 18.4 18.4 12 26.4 56 20 16 12
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56R 58L 58R 59 60L1 60L2 60R 63 64 65L 65R
Discussion
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The experimental crack size distributions are shown in Figure 4 (a-d), for
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different exposure times. When normalized with the total number of cracks for the
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particular exposure time, the data represent the probability density of the crack size distribution in the TBC. By the heterogeneous nature of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO
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stochastic in nature.
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formation, due to local compositional inhomogeneity, crack nucleation is certainly
To describe the crack evolution process using Eq. (8), the crack growth constant
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C and the distribution shape factor ϕ need to be calibrated, based on the crack size measurement data. Despite that the crack growth constant C depends on the Young’s modulus E, the fracture toughness KIC, the local oxidation-induced volume change m and the asperity radius R, as given by Eq. (5), which may vary per local cracking condition, in the present statistical analysis using the CND theory for the first time on TBC, both C and ϕ are assumed to be constants for the exposure condition: C = 0.394 μm3/2 and ϕ =
ACCEPTED MANUSCRIPT 0.24 are determined by calibrating to the 24 hr. distribution at 1200oC. The subsequent distribution curves are calculated using Eq. (8) as follows. Using Eq. (8), the total number of cracks at any given exposure time can be evaluated as ∞
𝑁(𝑡) = ∫0 𝑛(𝑐, 𝑡)𝑑𝑐 Hence, the CND probability is given by 𝑛(𝑐,𝑡) 𝑁(𝑡)
=
1 [𝛽(𝜂𝑒 −𝛽𝜂 −𝑐𝑒 −𝛽𝑐 )+(𝑒 −𝛽𝜂 −𝑒 −𝛽𝑐 )] 𝐴 ∞1 ∫0 𝐴[𝛽(𝜂𝑒 −𝛽𝜂 −𝑐𝑒 −𝛽𝑐 )+(𝑒 −𝛽𝜂 −𝑒 −𝛽𝑐 )]𝑑𝑐
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𝜌(𝑐, 𝑡) =
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(11)
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Note that the scaling constant B is cancelled out from the numerator and
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denominator in Eq. (11). Thus, the CND probability distributions are computationally obtained from Eq. (11), assuming crack nucleation in a Gamma distribution with index
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= 2, and β = 1/(ϕh), as shown in Figure 4 (a-d) for each exposure stage. This is a
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simplified engineering analysis, and its validity is proven in the statistical sense. Figure 4 demonstrates that the CND theory is capable of describing the entire crack-size evolution
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history at a given temperature with constant values of C and ϕ, in good agreement with
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the experimental data. In future studies, the dependence of C and ϕ on processing and
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temperature should be characterized.
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(a)
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(c)
Figure 4. Crack number distributions at a) 24 hrs, b) 48 hrs, c) 72 hrs and d) 144 hrs.
A general trend can be observed from experimental data and CND calculations, Figure 4, that both the mean crack size and distribution width increases with the exposure time. The CND theory combined with the Evans-He-Hutchinson model provides a physics-based description of crack nucleation and growth phenomena in relation to TGO
ACCEPTED MANUSCRIPT growth. Particularly, crack nucleation is related to the heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO with local volume change (strain), and crack growth is driven by further TGO growth. Under the test conditions of the present study, (Cr,Al)2O3·Ni(Cr,Al)2O4·NiO
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oxides formed at the early stage of thermal exposure. This occurred at sites where the
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local aluminum concentration was low and nickel had segregated as a result of
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compositional inhomogeneity in the air-plasma-sprayed TBC system. The X-ray mapping of elements was shown in a previous paper [11]. These zones were situated along the
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original ceramic/bond coat interface region, and it is believed that the mixed oxides were
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produced because, locally, there was insufficient aluminum reacting to form an initial alumina layer to serve effectively as a diffusion barrier to Ni or Cr. Lower concentration
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of aluminum in the nickel segregated areas, particularly in air-plasma sprayed NiCrAlY,
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may change the thermodynamics of oxide formation such that it favors Ni(Cr,Al)2O4/NiO rather than Al2O3, as opposed to the opposite in the ideal case [21, 22]. These
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observations have provided the physical basis of relating crack nucleation to TGO
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formation in a stochastic manner. Assuming crack nucleation follows the Gamma distribution, it is shown that the
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theoretical CND description of crack evolution agree rather well with the actual cracksize measurement data, as a function of exposure time. Apparently, the widening of the distribution can be attributed to TGO thickening, as it is controlled by the distribution shape parameter β, which is inversely proportional to the TGO thickness, h. The mechanism of crack growth is believed to be due to TGO growth as a wedge to drive the crack open. From the fracture mechanics point of view, crack nucleation and growth in
ACCEPTED MANUSCRIPT TBC occurs by the same wedge-opening mechanism, the former is more related to heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO clusters, while the latter is driven by the entire TGO growth. It is envisaged that when the local stored elastic energy arising from volume change due to oxidation becomes equal to the decohesion energy,
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crack nucleation occurs, and the crack advances in association with the thickening of the
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TGO layer to keep the energy in balance, as implied by Eqs (4) and (5). The hypothesis
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of this mechanism operation is supported by the metallurgical evidence shown in this study. Particularly, because of uneven undulation of the TBC/bond coat interface, the
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largest crack can extend over several peaks, into a length of hundreds microns. While in a
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typical FEM model of uniform undulation with uniform TGO, crack length across two adjacent peaks (typical ~ 50μm) would cause TBC failure. The practical implication of
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CND theory vs. FEM idealization is very important. As non-destructive inspection (NDI)
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techniques are being developed to assess the integrity of TBC coated parts in service, realistic relationships of crack length vs. NDI signals must be established. Then, the CND
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model can be used for probabilistic TBC life prediction with an allowable maximum
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crack size, i.e., in terms of the probability to reach the maximum crack size at any exposure time. In practice, the distribution parameters C and ϕ can be determined
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(calibrated) with experimental observation of crack-size distribution only once at a given exposure time. The CND model provides a full picture of damage evolution in TBC in the time axis, whereas the deterministic models are mostly fitted to the experimental life without probability information. When TBC is operated at high temperatures, sintering may occur. Sintering can either cause changes in material properties or even may heal some cracks. In this case, in
ACCEPTED MANUSCRIPT addition to changing the crack growth constant C, a threshold value c0 may be set in Eq. (2), below which cracks do not grow. Then, a new CND solution needs to be obtained by integration, Eq. (2). However, there is no evidence that has happened in the present
Conclusion
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investigation. The crack-size evolution follows well with c0 = 0, as shown in Figure 4.
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A crack number density theory model has been developed for air-plasma sprayed TBC. The CND model in combination with the Evans-He-Hutchinson sphere-wedging
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crack model recognizes the nature of crack nucleation in association with the volumetric
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misfit as induced by heterogeneous formation of (Cr,Al)2O3· Ni(Cr,Al)2O4·NiO and considers the crack growth mechanism as driven by TGO growth. The model describes
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the evolution of crack size distribution in the time-axis under a quasi-isothermal-cyclic
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oxidation condition. Good agreements have been found with the crack-size measurements from the furnace-exposed air-plasma sprayed TBC samples tested at 1200oC. In principle,
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the method can be extended to other isothermal and cyclic oxidation conditions, as long
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as the Arrhenius relation of the oxidation constant is determined. This may lend a
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powerful tool for probabilistic TBC life prediction.
Acknowledgement
The TBC samples were provided by Industrial Materials Institute of National Research Council of Canada.
ACCEPTED MANUSCRIPT References
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1. For the first time, crack evolution in air-plasma sprayed TBC is described by the crack number density theory. 2. The solution is derived from the crack number density evolutionary equation, assuming the wedge-opening mechanism of crack nucleation and growth in TBC, as suggested by EvansHe-Hutchinson, and crack nucleation following the Gamma distribution. 3. The integrated crack size distribution matches well with the experimental observations on an air-plasma sprayed TBC samples tested at 1200oC. 4. The model then offers a method for probabilistic TBC life prediction.