On the opening of a class of fatigue cracks due to thermo-mechanical fatigue testing of thermal barrier coatings

Computational Materials Science 50 (2011) 2561–2572

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On the opening of a class of fatigue cracks due to thermo-mechanical fatigue testing of thermal barrier coatings M.T. Hernandez a, D. Cojocaru a, M. Bartsch b, A.M. Karlsson a,⇑ a b

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716-3140, USA German Aerospace Center (DLR), D-51147 Cologne, Germany

a r t i c l e

i n f o

Article history: Received 19 February 2011 Received in revised form 22 March 2011 Accepted 30 March 2011 Available online 23 April 2011

a b s t r a c t The evolution of fatigue cracks observed in thermal barrier coatings (TBCs) subjected to an accelerated test scheme is investigated via numerical simulations. The TBC system consists of a NiCoCrAlY bond coat and partially yttria stabilized zirconia top coat with a thermally grown oxide (TGO) between these two coatings. The cracks of interest evolve in the bond coat parallel and near the interface with the TGO during thermo-mechanical fatigue testing. In their final stage, the cracks lead to partial spallation of the TBC. This study focuses on why the cracks open to their characteristic shape. To this end, finite element simulations are utilized. The crack surface separation is monitored for a range of material properties and oxidation rates. The simulations show that the inelastic response of the bond coat and the oxidation rate of the TGO govern the crack surface separation. Most interestingly, permanent separation of the crack surfaces is caused by a structural ratcheting in the vicinity of the crack. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Thermal barrier coatings (TBCs) are commonly used in aero-engines and land-based gas turbines to protect the load bearing superalloy from the high temperature gases. Even though TBCs have received significant attention during the last decade, they are still not considered a ‘‘prime-reliant’’ part of the structure. The coatings tend to fail prematurely and there are no comprehensive life prediction methodologies due to the complicated nature of the coating system. TBCs are heterogeneous, multilayered structures and their material properties, along with the morphology, evolve during use [1–10]. TBCs typically consist of (1) a metallic bond coat deposited on the superalloy; (2) an aluminum oxide (primarily alumina), commonly referred to as the thermally grown oxide (TGO), that forms and grows during thermal exposure; and (3) a ceramic top coat, typically yttria stabilized zirconia (YSZ). The YSZ top coat provides the thermal protection and sustains a temperature difference over its thickness, reducing the temperature of the metallic structures. Since YSZ is transparent to oxygen, oxygen reaches and reacts with the aluminum that is diffusing from the bond coat to form the alumina-based TGO. Thus, as the bond coat is depleted of aluminum, the chemical, physical and mechanical nature of the bond coat changes as part of normal operation. In addition, the top coat sinters during use, also changing its mechanical properties [11]. In all,

⇑ Corresponding author. Tel.: +1 302 831 6437; fax: +1 302 832 3619. E-mail address: [email protected] (A.M. Karlsson). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.03.041

several inter-related time- and cycle- dependent processes occur within the TBC system during service, which ultimately are associated with the failure of the system [3]. Laboratory tests are commonly used to understand and investigate TBC failures, since such tests are less expensive and time consuming compared to engine tests [12]. However, it is important to generate realistic damage mechanisms during testing; otherwise the results may be misleading. In order to generate realistic damage and failure as observed in coatings used on gas turbines blades, the thermal gradient mechanical fatigue (TGMF) test was developed [13]. This test imposes both a thermal gradient over the coating and a cyclic thermo-mechanical load. The specimen is a circular hollow cylinder, with the coating applied on the cylinder wall, designed to represent a coating applied to a non-flat surface. In this work, we examine the formation of a particular set of cracks (Fig. 1) that developed during an accelerated test scheme utilizing the TGMF test. The cracks propagate and the crack surfaces separate with cyclic thermo-mechanical loading, and eventually the coating spalls. The defects are investigated through numerical simulations, utilizing the finite element method (ABAQUS) [14], with the particular aim to understand why the cracks evolve to their characteristic shape with cyclic loading. The evolution of the bond coat crack is monitored and quantified via the separation of the two crack surfaces at the center of the crack, dc, defined in Fig. 2. The TGO thickness and growth kinetics are key parameters influencing the durability of TBC systems. The TGO grows thicker due to oxidation during thermal exposure, but some of the growth also occurs within the grain boundaries, lengthening the TGO.

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Top Coat

TGO Crack

Bond Coat Substrate

100 μm Axial Direction

Fig. 1. The bond coat crack developed during cyclic thermo-mechanical loading [7,13,18].

Since the overall structure prevents lengthening of the TGO, a compressive stress develops, commonly referred to as growth stress. In this work, the volume change of the oxide is induced into the models as eigenstrains [2,15]. Consequently, the total strain, etot , can be expressed as:

etot ¼ eE þ eP þ eT þ eG

ð1aÞ

where eE , eP , eT and eG are the elastic, inelastic, thermal and growth strain tensors, respectively [2,15]. The growth strain tensor can be represented by the components:

eG ¼ et e? þ el ek

ð1bÞ

where e\ and e|| indicate the directions perpendicular and parallel to the interface of the TGO and bond coat, respectively. Thus, et corresponds to the ‘‘thickening strain’’ and el corresponds to the ‘‘lengthening strain’’ (in-plane strain) achieved during high temperature exposure. In this work, the TGO is divided into two layers, where the lengthening strain is imposed uniformly through the TGO thickness, whereas the thickening strain is applied to the TGO elements next to the bond coat. This algorithm is used to closely simulate that TGO thickening occurs mostly at the interface to the bond coat [2,5]. The outline of this paper is as follow: based on the experimental observations, summarized in Section 2, we formulate in Section 3 a numerical model that captures key features of the crack opening. Sections 4 and 5 summarize the results from the numerical simulations and discuss the conditions for the crack to open to its characteristic shape. Our results show that the inelastic response in the bond coat, combined with the TGO growth, are the factors that drive the opening of the crack. We will show that this is associated with ratcheting of the plastic shear strain in the vicinity of the crack, where the plastic shear strain exhibits the same direction of plastic flow during loading and unloading.1 2. Experiments The test specimens are hollow, circular cylinders with inner and approximate outer diameters of 4 mm and 8.6 mm (after coating), respectively. They consist of a nickel-based directionally solidified superalloy, IN100DS coated via electron beam physical vapor deposition (EB-PVD) with nominally 110 lm thick NiCoCrAlY (in wt.%: 20Co, 21Cr, 12Al, 0.15Y + Ni) bond coat and nominally 220 lm thick top coat of YSZ (7–8 wt.% yttria) [16]. 1

We use the terminology ‘‘ratcheting’’ here to mean ‘‘step-wise, unidirectional change.’’ In the theory of plasticity, ‘‘ratcheting’’ refers to kinematic hardening during plastic yielding, which is not the meaning in this paper. In fact, linear-elastic, perfectly-plastic material properties are assumed, as discussed in later sections.

The accelerated testing scheme used for the experiments was developed in order to generate close-to-service conditions for the TBC systems. The test scheme is described in the work by Bartsch and co-workers [7,13,17,18], and consists of two steps: (1) pre-heat treatment, and (2) thermo-mechanical fatigue testing (TGMF). Since most failures of TBCs are associated with cracks developing in the vicinity of the TGO [19], the life time of a TBC can often be directly linked to the TGO thickness. As part of the accelerated test scheme, samples are therefore heat treated in a separate furnace prior to thermo-mechanical testing so they develop a ‘‘preexisting’’ TGO. The pre-heat treatment is cyclic, where each cycle is defined by 24 h at high temperature (1000 °C) and cooling to room temperature by removing the specimen from the furnace. The specimens of interest in this paper were subjected to pre-heat treatment accumulating 500 h at elevated temperature, resulting in TGO thicknesses of about 5 lm. After pre-heat treatment, the specimens are subjected to a synchronized, combined thermal and mechanical loading. Each thermo-mechanical load cycle takes 3 min and aims to represent the fatigue load of an average flight. The mechanical load corresponding to a nominal axial stress of 100 MPa is applied by a servohydraulic testing machine. The thermal load is imposed via a radiation furnace. For the test considered here, the maximum temperature set point at the outer surface of the specimen was 1000 °C and the minimum set-point value about 100 °C [18]. This temperature range corresponds to typical field conditions for thermal barrier systems [1,17,20,21]. High cooling rates are achieved with active air cooling from vents in a shutter, which is introduced into the furnace by a pneumatic device and encloses the specimen during the cooling cycle [17]. The defects of interest are characterized by a crack that develops in the bond coat parallel to the TGO [7,13,18], Fig. 1. After 1000 cycles, the cracks extended 50–200 lm in the axial direction and up to 5 mm in circumferential direction. The average spacing between the cracks measured in the axial direction is about 0.8 mm [13]. Based on experimental observations [7,13,17,18] and accompanying numerical simulations [9], it is concluded that the crack first develops across the TGO thickness. As the TBC is subjected to continued cyclic load, the crack propagates in two directions: into the top coat and into the bond coat. The top coat crack grows outwards, towards the surface. The crack that propagates in the bond coat kinks and grows parallel to the TGO-bond coat interface within the aluminum depleted zone. The crack initiation was investigated via numerical mechanics based simulations by Hernandez [9], where it was shown that the creep in the TGO is paramount in the initiation of the TGOthrough crack in the pre-oxidized samples. We believe that due to the aluminum depletion, the material properties in the bond coat close to the TGO are different from the properties away from this interface and that this property difference causes the crack to kink in the bond coat during the thermo-mechanical cyclic loading [22–29]. This study examines the mechanics and evolution of the bond coat crack shown in Fig. 1, once it has kinked. Thus, in the following we assume that the crack is formed and will focus on the separation of the crack surfaces at the center of the crack, dc, defined in Fig. 2. In particular, we are interested in understanding under what conditions and how the crack can open during thermo-mechanical cycling. 3. Numerical model The conditions for crack surface separation are investigated via numerical simulations by means of an axisymmetric finite element

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Applied Axial Stress

Substrate

Bond Coat

Top Coat

Crack TGO

22 (axial)

Axis of symmetry

11 (radial)

Fig. 2. Schematic representation of the axisymmetric model and the finite element mesh in the vicinity of the crack. The crack surface separation at the center of the crack, dc, is monitored as function of the number of cycles.

Table 1 Material properties at room temperature (RT) and 1000 °C (HT) [34]. Substrate

Bond coat

TGO

Top coat

RT

HT

RT

HT

RT

HT

RT

Elastic modulus, E (GPa)

218

153

140

70

360

318

13

14

Poisson ratio, m

0.3

0.22

0.28

Coefficient of thermal expansion, a (106 1/K)

11.5

9

11.5

0.3 18.8

(FE) model where the commercial software ABAQUS [30] is utilized. Bi-quadratic axisymmetric elements with reduced integration (CAX8R) are used throughout the model. The model consists of four regions, Fig. 2: (i) the substrate (2 mm thick), (ii) the bond coat (120 lm), (iii) the TGO (initially 5 lm) and (iv) the top coat (220 lm). The interfaces between the TGO and the bond coat, and the TGO and the top coat are modeled free of imperfections, which is a reasonable approximation based on the experimental observations [31].2 A crack of initial length a0 = 40 lm is modeled in the bond coat, 5 lm from the bond coat/TGO interface, Fig. 2. Normal contact (frictionless) was used on the crack faces. The length of the model in the axial direction is 1.0 mm and the inner radius of the modeled cylinder is Ri = 2.0 mm. A modeling frame, presented elsewhere [32,33], is used to generate the finite element model and to propagate the crack. The modeling frame uses ABAQUS Scripting Interface [14], where a FE-model is generated automatically. The crack propagation is achieved by releasing (previously constrained) nodes along a predefined propagation path. A refined mesh is modeled around the crack tip to capture the stress concentration associated with the crack tip. The refined region stretches significantly beyond the length of the crack after propagation. Each model has approximately 40,000 elements. Element sizes and time increments have been optimized by means of convergence analysis. To this end, we investigated finer meshes and smaller time increments, and obtained the same results as for the model used. For simplicity, rate-independent material properties are assumed and the thermal gradient over the coating is neglected. The assumption 2 In this study, we do not investigate buckling or other instabilities of the interface since these are highly unlikely, and never observed, with the TGO remaining attached to top coat [31].

8.6

0.24 16.6

6

8.7

HT

of time-independent properties is commonly made for TBCs subjected to cyclic loading when only accumulation of inelastic strain is important. Temperature-dependent material properties are used [34]. The elastic modulus, Poisson’s ratio, and the coefficient of thermal expansion are summarized in Table 1. These properties are temperature-dependent, but for simplicity we only include the values at ambient and operating conditions. The yield strengths and creep properties of the bond coat and the TGO at high temperature are hard to measure and not readily available [8]. The initiation of the cracks was investigated in a previous study [9], which included visco-plastic effects. That study showed that the bond coat stresses are more or less independent to a change of inelastic properties. This is due to that the onset of inelasticity in the bond coat occurs at significantly lower loads than the onset of inelasticity in the TGO (TGO remains elastic up to a substantially higher load). Thus, the stiffness of the TGO governs the system over the properties of the bond coat. As will be confirmed in the results section, it follows that the important parameter is the inelastic response in terms of accumulated plastic strain, and not the time-dependent effect per se. Thus, to keep the model with a tractable number of parameters, time independent plasticity is used in this study. Consequently, the high temperature yield strength (T = 1000 °C) of the bond coat and TGO are considered parameters. Two cases will be studied for the bond coat: rBC Y HT ¼ 10 MPa and 50 MPa. The onset of high temperature inelasticity of the TGO is captured as a time-independent yield strength at T = 1000 °C. Two values, rTGO Y HT ¼ 200 and rTGO ¼ 500 MPa, are investigated. At lower temperatures, the TGO Y HT is considered elastic. Numerically, this is implemented by letting the TGO have artificially high yield strength at temperatures below 900 °C (i.e., 10 GPa) so that the TGO remains elastic at lower temperatures. A linear variation of the yield strength between 900 °C

0.01

0

1  103

1  102

Thickness (lm)

5

5.13

7.43

100 600

400 Applied Stress

0

0 N

Cycle

and the assumed high temperature yield strength is imposed. This approach is used for simulating the TGO high temperature behavior [2,5,10,15,31,35–37] as a computationally time efficient approach. The well-known von Mises yield criteria (e.g., Ref. [38]) has been used throughout the numerical analysis. As discussed previously, the TGO growth can be divided into ‘‘lengthening’’ and ‘‘thickening’’ components, Eq (1). The growth of the TGO due to oxidation is simulated by imposing stress free strains at T = 1000 °C via the subroutine UEXPAN [30], following an approach employed in the literature, e.g., Refs. [2,5,10,15,31, 35,36]. For all cases considered, the lengthening strain is set to 10% of the thickening strain [2,37]. Throughout the simulations, the thickening strain is a parameter and is set to three values: et = 0, et = 1  103 or et = 1  102 per cycle. The final TGO thickness achieved after 100 simulated cycles for each applied growth strain is shown in Table 2. The initial condition is the assumption of stress free state at uniform temperature of 1000 °C, to simulate the processing and deposition conditions. The structure is then uniformly cooled to 0 °C, which is the starting point for the cyclic load. Subsequently, the simulated load cycle, Fig. 3, is designed so that the effects due to thermal cycling, mechanical loading, and TGO growth can be identified. The applied temperature is assumed uniform within the model (i.e. no spatial gradients) and it is cycled between 1000 °C and 0 °C. The mechanical load (axial force) is applied at high temperature (1000 °C). The TGO growth strain (et, el) is only applied during the high temperature and high axial force sequence of the thermo-mechanical load cycle. The simulations consist of 100 thermo-mechanical load cycles. This is sufficient to capture the overall, qualitative, behavior of the system. Both propagating and stationary crack tips are considered. For the case of a propagating crack, the two crack tips are advanced with the constant rate of da/dN = 0.5 lm/cycle (i.e., 1 element/cycle) at the end of each cycle, using a node release technique. Thus, the final crack length after 100 cycles is nominally 140 lm. The stationary crack will remain at 40 lm length after 100 cycles. The constant crack growth rate prescribed in the model was imposed for simplicity, aiming to keep the number of parameters

δc(μm)

Axial Loading

0

1

2

ε =0

N (cycles)

3

4

t

-4

1

N+1

Fig. 3. Simulated load cycle showing the applied nominal stress (axial stresses) and temperature used in the numerical model.

t

t

0

δc

ε =1·10 -3

ε =1·10 -2

x 10 B Plastic Shear Strain (cycle by cycle)

0.8 ε =1·10 -2 t

0.6

0.4

ε =1·10 -3 t

ε =0

0.2

0

t

0

1

2

3

4

N (cycles) C Radial Stresses (cycle by cycle)

ε =1·10 -2 t

0.2

0.1

ε =1·10 -3 t

0 Axial Loading

200

0.002

-0.1

0

TGO Growth

TGO Growth Strain

0.004

Plastic Shear Strain

800

0.006

Radial Stresses (MPa)

Temperature (°C)

Temperature

0.008

Nominal Mechanical Stress (MPa)

1000

Cooling

Thickening strain per cycle, eT

A Crack Opening Displacement (cycle by cycle) Heating

Table 2 Final TGO thickness achieved after 100 cycles for each applied growth strain.

High TGO Growth Temperature Axial Unloading

M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

Axial Unloading

2564

ε =0 t

1

2

3

4

N (cycles) TGO Fig. 4. For the cases of stationary crack, rBC Y HT ¼ 10 MPa, rY HT ¼ 200 MPa and selected TGO growth strain, et: (A) Crack surface separation, dc (including the definition of the per cycle rate of crack surface separation, d_ c ¼ ddc =dN) (B) Plastic shear strain; and (C) Radial stresses in the bond coat (only the high temperature portion of the cycles is shown); during the first four cycles.

in the model low, allowing for a tractable numerical scheme. This simplification can easily be removed in future studies, where the crack propagation rate could be made solution dependent.

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4. Results

0.12

and the following observations are made. Extending the observations for 100 cycles it is clear that the TGO growth strain is an important factor driving the separation of the crack surfaces, Fig. 5A. Without growth strain, et = 0, the crack opens during the cooling-heating sequence of the cycle, but closes back to vanishing dc after each completed cycle (Fig. 4A and Fig. 5A), thus d_ c ¼ 0 and consequently dc = 0 for et = 0. However, when a growth strain is imposed (et = 1  103 and 1  102), the crack surfaces separate, d_ c > 0 (Fig. 5A). Moreover, the high temperature bond coat yield strength, rBC Y HT , governs the evolution of dc, where vanishing dc is observed for a high value _ (rBC Y HT ¼ 50 MPa), but a distinct increasing trend, dc > 0, is observed for lower values (rBC ¼ 10 MPa), Fig. 6A. Furthermore, a higher vaY HT lue of the high temperature TGO yield strength, rTGO Y HT ¼ 500 MPa, generates a larger dc, compared to rTGO Y HT ¼ 200 MPa, Fig. 6A. Interestingly, for the combination of rBC Y HT ¼ 10 MPa, 2 rTGO , the crack first opens (d_ c > 0) Y HT ¼ 200 MPa and et = 1  10 with each cycle, but this trend is reversed after about 50 cycles and the crack surface separation becomes smaller for each cycle (d_ c < 0). This case will be analyzed later. We note that the TGO growth strain has been shown to be critical for various types of failure evolution of TBCs [2,9,15,37,39,40]. So far, we have only considered cases where the crack did not propagate (e.g., assuming constant crack length). When allowing the crack to extend with each thermo-mechanical cycle (via a node release technique developed previously [32,33]) at a propagation rate of da/dN = 0.5 lm/cycle, the results show that the cyclic rate of crack face separation, d_ c , does not significantly depend on if the crack is propagating or not, Fig. 6. However, the crack surface separation is larger within each cycle (during the cooling and heating sequence) as the crack gets longer, Fig. 6A and B. Thus, crack propagation affects the response during cooling and heating but does not change the overall trend of cyclic response of the crack surface separation.3

3 However, it is possible the opening of the crack may affect the crack propagation rate, which is beyond the current scope of study.

t

δc(μm)

0.08

ε =1·10 -2

0.06

t

0.04 ε =0 t

0.02

0

0

20

40

-3

6

x 10 B Plastic Shear Strain

60

80

N (cycles)

100

High Temperature

3

TGO Growth

4

2

Axial Unloading

5

Axial Loading

ð2Þ

ε =1·10 -3

Plastic Shear Strain

d_ c ¼ ddc =dN

0.1

ε =1·10 -2 t

1 ε =0 t

ε =1·10 -3 t

0

0

0.3

20

40

60

80

N (cycles)

100

C Radial Streses

0.2

Radial Stress (MPa)

The separation of the crack surfaces, dc, Fig. 2, is investigated for selected parameters: (i) the high temperature yield strengths of TGO the bond coat, rBC Y HT , and TGO, rY HT ; (ii) the applied growth strain per cycle, et; and (iii) the crack propagation rate, da/dN. For clarity, we first discuss the evolution during the first four cycles, Fig. 4A, before discussing the long term response. The simulation starts with assuming stress free conditions at 1000 °C, from where the structure is cooled to 0 °C (minimum cycle temperature) during the initial step of the calculation. The load cycle described in Fig. 3 starts thereafter. The results show that the crack surfaces separate (e.g., the crack ‘‘opens’’) significantly during cooling and the separation is reduced during heating, Fig. 4A. However, the system remains elastic throughout the cooling-heating sequence, thus no permanent crack surface separation is obtained in this part of the cycle. When the structure is back at high temperature, the axial force is applied (with constant load rate), resulting in an increase in the crack separation. While both the temperature and the axial force are held constant, the TGO is allowed to grow, resulting in insignificant change of dc. Finally, the axial force is removed (with constant load rate), resulting in an increase of dc for et > 0. Based on these observations, it is useful to introduce the ‘‘percycle crack surfaces separation rate,’’ defined in Fig. 4A

A Crack Opening Displacement

ε =1·10-3 t

0.1

ε =0 t

0

-0.1 ε =1·10 -2 t

-0.2

-0.3

0

20

40

60

80

100

N (cycles) TGO Fig. 5. For the cases of stationary crack, rBC Y HT ¼ 10 MPa, rY HT ¼ 200 MPa and selected TGO growth strains, et: Evolution of the (A) Crack surface separation, dc; (B) Plastic shear strain; and (C) Radial stresses (stress values are sampled after the nominal mechanical load is imposed); as a function of the number of cycles.

5. Discussion The results presented in the previous section suggest three possible evolutions of the crack surface separation (Fig. 5A and Fig. 6 and Table 3) that are governed by the high temperature bond coat TGO yield strength, rBC Y HT , the high temperature TGO yield strength, rY HT , and the TGO growth strain. We classify the three observed evolutions based on the crack surface separation, dc, and the per cycle

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(2) Cyclic crack surface separation, d_ c > 0 leading to permanent crack surface separation, dc > 0. This is observed for moderate TGO growth strains (et = 1  103) and for a low bond coat yield strength (rBC Y HT ¼ 10 MPa); (3) Cyclic crack surface separation d_ c > 0 leading to dc > 0, followed by a reduction in cyclic crack surface separation, d_ c < 0. That is, in this case, the crack surfaces first separate incrementally, but the trend is reversed and the opening is reduced with each cycle. However, dc remains positive during the simulations. This is observed for large TGO growth strains (et = 1  102) and low bond coat yield strength (rBC Y HT ¼ 10 MPa).

ε =1·10 -3

A Stationary Crack

t

0.3

δc (μm)

0.25 σTGO=500MPa

0.2

y HT

σTGO=200MPa

0.15

y HT

0.1

σBC=10MPa y HT

σBC=50MPa

0.05 0

y HT

0

20

40

60

80

100

N (cycles ) ε =1·10 -3

B Propagating Crack

t

0.3

σTGO=500MPa y HT

δc(μm)

0.25 σTGO=200MPa

0.2

y HT

0.15 σBC=10MPa

σBC=50MPa

y HT

0.1

y HT

0.05 0

0

20

40

60

80

100

N (cycles ) Fig. 6. Crack surface separation, dc, as a function of the number of cycles for selected high temperature bond coat and TGO yield strengths: (A) Stationary, and (B) Propagating crack. (et = 1  103).

rate of crack separation, defined in equation (2) as d_ c ¼ ddc =dN according to the following: (1) No cyclic crack surface separation, d_ c ¼ 0, resulting in that the crack surfaces remains closed dc = 0. This is observed for vanishing TGO growth strain (et = 0) or for a high bond coat yield strength (rBC Y HT ¼ 50 MPa);

Even though the crack surface separation remains almost unchanged during the actual TGO growth (Fig. 4A), the TGO growth directly influences the state of stress in the bond coat, thus becoming a governing factor for the crack opening displacement. In Fig. 6B it was shown that crack propagation by itself (i.e., the length of the crack) does not contribute significantly to permanent crack surface separation (even though the length of the crack appears to be related to the magnitude of the instantaneous crack surface separation during the heating–cooling part of the cycle). In fact, in the following, we will show that a permanent change in the crack surface separation after each cycle, d_ c > 0, Fig. 4A, is associated with forward and reverse yielding in the bond coat, along with ratcheting of the plastic shear strain in the vicinity of the crack. In other words, the direction of the plastic shear flow is the same during loading and unloading. Ratcheting is unusual in engineering structures, but it has been documented for example in composites [41], electronic devices (multilayers) [42–45] and TBCs [2,37,46,47]. It is important to note that linear-elastic, perfectly-plastic material properties are assumed. Thus, the ratcheting is NOT caused by a kinematic hardening of the material, but truly a structural phenomenon, linked to the combination of morphology, material properties, and loading.

5.1. Crack surface separation 5.1.1. State of stress during cyclic loading In Section 4 it was shown that cyclic crack surface separation, d_ c ¼ ddc =dN, occurs only for low bond coat yield strengths (Fig. 6A). In order to elucidate the effect of bond coat yield strength on crack surface separation, we will compare the near and far field stress components. The ‘‘near field’’ represents the state of stress and strain obtained due to the perturbation of the crack. The ‘‘far field’’ state of stress and strain corresponds to the state the system

Table 3 Selected cases considered in the analysis and their general response.

rBC Y HT (MPa)

rTGO Y HT (MPa)

et

da/dN (lm/cycle)

General response of the crack surfaces during 100 cycles

10 10 10 50 50 50 10 10 10 10 10 50 10 50 10

200 200 200 200 500 500 500 500 200 200 200 500 500 500 500

0 1  103 1  102 1  103 1  103 1  102 1  103 1  102 0 1  103 1  102 1  102 1  102 1  103 1  103

0 0 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5

No separation Increasing separation Initial separation followed No separation No separation No separation Increasing separation Initial separation followed No separation Increasing separation Initial separation followed No separation Initial separation followed No separation Increasing separation

by decreased separation

by decreased separation

by decreased separation by decreased separation

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M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

0.16

A Far Field

Stress (MPa) σ22, σ33

5 µm

550 µm

40

σ

11

Forward Yielding

Reverse Yielding

20

22

0.04

0 σ33 σ

60

3 µm

Axial Loading

12

Axial Unloading

TGO Growth

0 0.16

B Near Field

20 µm

Fig. 7. Schematic showing the location where the stresses are sampled from the numerical simulations to represent the ‘‘far field’’ and ‘‘near field’’ with respect to the crack (not drawn to scale).

22 33

σBC=10MPa yHT

Stress (MPa) σ ,σ

11 (radial)

Near Field

0.08

σ

-20

22 (axial)

0.12

11 12

σBC=10MPa yHT

Stress (MPa) σ ,σ

Far Field

60

TGO

Reverse Yielding

Forward Yielding

40 σ

20

σ

σ 22

0

experiences absent a crack. The locations of the points4 where the fields are extracted are defined in Fig. 7. A representative cycle (20th cycle) was selected for comparison of the near and far field stress components for the two cases rBC Y HT ¼ 10 MPa and 50 MPa (assuming et = 1  103 and rTGO ¼ 200 MPa), Figs. 8 and 9, respectively. The Y HT subscripts 11, 12, 22, 33 will be used to describe the radial, shear, axial, and circumferential (stress and strain) components, respectively. Due to the symmetry of the problem, no shear stresses, r12, arise in the far field (a crack free bond coat). Moreover, an analytical model (Appendix A) suggest that the radial stresses in the bond coat are tensile for non-vanishing growth strain and for the TGO thicknesses of interest (P5 lm – TGO thickness at the start of the test, Fig. A1. Tensile radial stresses are beneficial for crack opening (crack surface separation) of the investigated crack. The numerical results confirm that there are no shear stresses in the far field, Fig. 8A and Fig. 9A. Moreover, the radial stresses, r11, in the numerical simulations are positive for rBC Y HT ¼ 10 MPa and 50 MPa (et = 1  103), Figs. 8 and 9. The difference in the stresses between the analytical (Fig. A1) and numerical model are due to inelastic properties (in the bond coat and TGO) considered in the numerical model. Thus, the TGO growth in combination with the yielding of the bond coat and TGO induces small, but biased positive radial stresses (r11 > 0) for the two yield strengths considered. This consequently provides a ‘‘Mode 1’’ loading, encouraging crack opening when a crack of the considered geometry is present. Due to the stress field in the vicinity of a crack tip, shear stresses arise in the near field, whereas the radial stresses are augmented and remain positive throughout the cycle for the cases considered (Fig. 8B and Fig. 9B). Based on classic fracture mechanics theories, the shear stress developed due to the crack tip stresses are directly proportional to the far field stress, with its magnitude depending on location relative to the crack tip [48]. Thus, if the far field radial stress changes sign, the shear stress changes sign.

4 Obviously, the stress in the near field strongly depends on where the point representing the near field is located. However, in the discussion that follows, we will investigate how the stresses qualitatively change as a function of the load cycle, with the understanding that the magnitude will vary depending on location. All other locations in the vicinity of the crack show similar behavior.

12

0.08

0.04 σ

-20 20.4

0.12

11

Stress (MPa) σ11,σ12

Bond Coat

33

20.6

20.8

0 21

Cycle Fig. 8. Bond coat stress components for (A) far field and (B) near field, during the 20th cycle (high temperature portion) for the case where rBC Y HT ¼ 10 MPa, 3 rTGO : The subscripts 11, 12, 22, 33 correspond to the Y HT ¼ 200 MPa and et = 1  10 radial, shear, axial, and circumferential stress components, respectively. The gray regions indicate bond coat yielding.

The axial and circumferential components (r22, r33) depend on the bond coat yield strengths considered but are not perturbed by the presence of the crack (within the resolution of the figures). For low rBC Y HT , yielding occurs both during loading (‘‘forward yielding’’) and unloading (‘‘reverse yielding’’), Fig. 8. However, for rBC Y HT ¼ 50 MPa, bond coat yielding occurs only during the first cycles (not shown) when the maximum axial load is reached, but not during unloading. Even though the stress almost reaches the yield strength, yielding does not ensue, and plastic strain does not accumulate during continued cycling (Fig. 9). For intermediate rBC Y HT , the magnitude of the forward and reverse yielding as well as the plastic strain accumulation, is proportional to rBC Y HT . (Interestingly, the simulations show that if the bond coat exhibits cyclic yielding, the behavior is always associated with both forward and reverse yielding (not shown for brevity). The yield strength of rBC Y HT ¼ 50 MPa appears to be the limit, where no permanent crack surface separation is obtained and no plastic strain accumulation occurs.) 5.1.2. Plastic strain and ratcheting We next compare the near field stress and the plastic strain components for the three growth strains considered: et = 0, et = 1  103 TGO and et = 1  102, for rBC Y HT ¼ 10 MPa and rY HT ¼ 200 MPa (Figs. 10 and 11). The results are sampled during the 20th cycle, where incremental crack surface separation is observed for non-vanishing growth strain rates Fig. 5A. Both the radial, r11, and shear stresses, r12, increase with increasing TGO growth strain, Fig. 10. Plastic deformation occurs

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M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

0.16

A Far Field

20

0.28

A ε =0 t

σ

12

22

20

0.08 σ

33

0

0.04

-20 60

Axial Loading

TGO Growth

Axial Unloading

0

0.08

20

0

0.04 σ

33

20.6

20.8

11

0.04

TGO Growth

33

Axial Unloading

20.8

-0.04 21

Cycle 0.28 22

10

0.2 σ

33

0

0.12 σ

σ 11

12

-10

0.04

0 21

-20 20.4

Fig. 9. Bond coat stress components for (A) far field and (B) near field, during the 20th cycle (high temperature portion) for the case where rBC Y HT ¼ 50 MPa, 3 rTGO . The subscripts 11, 12, 22, 33 correspond to the Y HT ¼ 200 MPa and et = 1  10 radial, shear, axial, and circumferential stress components, respectively.

20

Axial Loading

TGO Growth

20.6

Stress (MPa) σ22 ,σ33

only during loading and unloading of the axial load, Fig. 11. For the radial, axial, and circumferential components, the sign of the plastic strain increment depends directly on if the axial load is applied or removed. For example, the axial plastic strain increment, DeP22, is positive during loading, but negative during unloading. Plastic shear strain, eP12, develops in the near field, and increases with increasing TGO growth strain. Without TGO growth, the plastic shear strain remains close to zero (Fig. 11A), with a vanishing increment during axial loading and unloading. However, when a growth strain is applied, the plastic shear strain increment has the same sign during both loading and unloading (Fig. 11B and C). Consequently, as the structure is cycled, the plastic shear strain increases monotonically, reproducing the ratcheting that characterizes the crack surface separation (Fig. 4A and B). The plastic strain increases during both loading and unloading due to the biased radial stress that develops during the TGO growth. The radial stress is always positive (or always negative as will be seen later) during the cyclic loading, even though the magnitude is cycled. The particular details of the ratcheting are as follows. According to the J2-flow theory [38], the plastic strain increment tensor and the deviatoric stress tensor must satisfy the expression:

where dePij is the plastic strain increment, and Sij is the deviatoric stress defined by

-0.04 21

Cycle 0.28

-2 C ε t=1·10 22

10

0.2 σ

σ

11

0

σ

12

0.12 33

-10

-20 20.4

0.04

Axial Loading

Axial Unloading

TGO Growth

20.6

20.8

-0.04 21

Cycle Fig. 10. Bond coat stress components in the near field, during the 20th cycle (high temperature portion) for (A) et = 0; (B) et = 1  103; and (C) et = 1  102 with TGO rBC Y HT ¼ 10 MPa, rY HT ¼ 200 MPa: The gray regions indicate bond coat yielding. (The subscripts 11, 12, 22, 33 correspond to the radial, shear, axial, and circumferential stress components, respectively.).

Sij ¼ rij 

1 rkk dij 3

ð3bÞ

with

 dij ¼

ð3aÞ

Axial Unloading

20.8

σ

fi; j ¼ 1; 2; 3g

σ

-3 B ε t=1·10

Cycle

dePij ¼ Sij dk;

12

20.6

,σ

11

Axial Loading

22

σ

0.12 33

σ

σ

Stress (MPa) σ

0.12

22

11 12

12

Stress (MPa) σ ,σ

33

,σ

σ

20 σ

22

Stress (MPa) σ

0

-20 20.4

0.16

yHT

-20 20.4

0.2

-10

σ BC=50MPa

40

10

12

B Near Field

σ

22

Stress (MPa) σ11,σ12

σ

Stress (MPa) σ22 ,σ33

Stress (MPa) σ

σ

0.12

11

Stress (MPa) σ ,σ 11

σ

40

22

,σ

33

yHT

Stress (MPa) σ11,σ12

σBC=50MPa

Stress (MPa) σ11,σ12

60

1 i¼j 0

i–j

 and

rkk ¼ r11 þ r22 þ r33

ð3cÞ

dk is a scalar factor of proportionality. Eq. (3) holds by definition during the cycle for all the components. Consequently, the following must hold:

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M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

0

P12

ε

0 P33

-4

-5.5 ε Axial Loading

-8 20.4

Axial Unloading

Cycle

20.8

-4

-4

Plastic Strain ε P11,ε P22 ,ε P33

x 10

x 10

B ε t=1·10

-5 21

5.5

-3

ε

4

P22

4.5

ε

0

P12

3.5

ε

P33

-4

Axial Loading

-8 20.4

Axial Unloading

TGO Growth

20.8

0.05

S33

-5

0 S11

S12

3 x 10

Axial Loading

TGO Growth

B Plastic Strain Increment

Δε

Far Field

P22

Δε

Δε

-1.5

Δε

P33

Δε

P11

P12

4

P11

Δε

P33

0

-4 Δε

-3 20.4

-6

8 x 10

1.5

0

-0.05

Axial Unloading

20.6

20.8

P22

-8 21

Fig. 12. Bond coat (A) deviatoric stress and (B) plastic strain increment in the far field during the 20th cycle (high temperature portion), where rBC Y HT ¼ 10 MPa, 3 rTGO . The gray regions indicate bond coat yielding. Y HT ¼ 200 MPa, et = 1  10

P11

20.6

0

Cycle

2.5 ε

0.1

S22

-4

TGO Growth

20.6

5

-10

P11

0.15

Far Field

A Deviatoric Stress

Deviatoric Stress (MPa) S12

ε

Plastic Strain ε P12

5.5

Deviatoric Stress (MPa) S11 , S22 , S33

P22

Plastic Strain Increment Δε P11 Δε P22,Δε P33

ε

Plastic Strain ε P12

Plastic Strain ε P11,ε P22,ε P33

A ε t=0

4

8

10

x 10 1.5

x 10

Plastic Strain Increment Δε P12

-4

-4

8

1.5 21

Cycle -4

-4

x 10

x 10

9.5

-2 C ε t=1·10

4

0

-4

-8 20.4

Axial Loading

ε

P22

ε

P12

8.5

7.5

ε

P33

ε

P11

6.5

20.8

5.5 21

      deP11 deP22 deP33 ¼ ¼ ¼ dkloading > 0 S11 loading S22 loading S33 loading

ð5Þ

where dk > 0 is obtained by visual inspection of Fig. 12. Similarly, during unloading, it holds

ð6Þ

Cycle Fig. 11. Bond coat plastic strain components for the bond coat in the near field during the 20th cycle (high temperature portion) for (A) et = 0; (B) et = 1  103; and TGO (C) et = 1  102 with rBC Y HT ¼ 10 MPa, rY HT ¼ 200 MPa. (The subscripts 11, 12, 22, 33 correspond to the radial, shear, axial, and circumferential stress components, respectively.).

deP11 deP12 deP22 deP33 ¼ ¼ ¼ ¼ dk S11 S12 S22 S33

Figs. 12 and 13 show the deviatoric stresses and plastic strain increments in the far field and near field, respectively, assuming 3 TGO rBC . The results are Y HT ¼ 10 MPa, rY HT ¼ 200 MPa and et = 1  10 sampled during the 20th cycle. As mentioned previously, the far field represents the state of stress of the unperturbed system assuming that no crack is present, and due to the symmetry of the problem, no shear stresses (i.e., S12 = 0), and consequently no plastic shear strain, develop (Fig. 12). During loading, Eq. (4) holds:

      deP11 deP22 deP33 ¼ ¼ ¼ dkunloading > 0 S11 unloading S22 unloading S33 unloading

Axial Unloading

TGO Growth

20.6

Plastic Strain ε P12

Plastic Strain ε P11,ε P22,ε P33

8

ð4Þ

In other words, the components of the plastic strain increment and the corresponding deviatoric stress are all related via the same constant for each considered increment.

In the near field, the perturbation of the crack, in combination with the oxide growth, imposes a biased and positive shear stress, r12 > 0, (Fig. 8B and Fig. 9B) throughout the load cycle, whereas the other stress components remain mostly unchanged compared to the far field. Thus, according to Eqs. (5) and (6), it must hold that during loading and unloading deP12/S12 > 0, which is confirmed in Fig. 13. Consequently, it follows that the plastic shear strain increment must be positive both during loading and unloading, since S12 = r12 > 0 (Fig. 13). In summary, a biased radial stress develops (always positive in this case), due to the TGO growth (also seen in the analytical model, Appendix A). The sign of the shear stresses relates directly

M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

A Deviatoric Stress

5

0.1

S22 S12

0

0.05

S33

-5

0 S11

-10

Plastic Strain Increment Δε P11 Δε P22,Δε P33

0.15

Near Field

3 x 10

Axial Loading

-4

TGO Growth

Δε

Δε

Δε

P12

Δε

P22

P12

4 P11

Δε

0

-1.5

Δε

Δε

P33

P11

P33

0

-4 Δε

-3 20.4

-6

8 x 10

Near Field

B Plastic Strain Increment

1.5

-0.05

Axial Unloading

20.6

20.8

P22

Plastic Strain Increment Δε P12

Deviatoric Stress (MPa) S11 , S22 ,S33

10

Deviatoric Stress (MPa) S12

2570

-8 21

Cycle Fig. 13. Bond coat (A) deviatoric stress and (B) plastic strain increment in the near field during the 20th cycle (high temperature portion), where rBC Y HT ¼ 10 MPa, 3 rTGO . The gray regions indicate bond coat yielding. Y HT ¼ 200 MPa, et = 1  10

to the radial stress, and consequently does not change its sign during loading and unloading of the axial stress. Thus, the plastic shear strain increment is always positive according to Eqs. (3-6), and the plastic deformation occurs in the same direction during loading and unloading. We refer to this as ratcheting, where the bond coat accumulates plastic deformation in each cycle. The accumulation of plastic shear strain causes permanent deformation and crack surface separation, Fig. 5A and B. 5.1.3. Reduction in the crack surface separation The case of cyclic crack surface separation, d_ c ¼ ddc =dN > 0 leading to dc > 0, followed by a reduction in crack surface separation, d_ c < 0, will now be investigated. This is observed for large TGO growth strains (et = 1  102) and low bond coat yield strength (rBC Y HT ¼ 10 MPa), Fig. 5A. The distinguishing feature for the higher TGO growth strain is the change in sign for the radial bond coat stress during cycling (Fig. 5C). For low TGO growth strain, et = 1  103, the radial bond coat stresses remain positive and reach a cyclic steady state, Fig. 5C. In this case, the cyclic crack surface separation increases for each cycle d_ c ¼ ddc =dN > 0. However, for a higher TGO growth strain, et = 1  102, the radial bond coat stresses start out positive, but the magnitude decreases with cycling, to finally become negative after about 50 cycles (Fig. 5C), at which time the shear stress (not shown) and the plastic shear strain increment (Fig. 5B) in the near field becomes negative. Consequently, the cyclic crack surface separation rate becomes negative after about 50 cycles, d_ c ¼ ddc =dN < 0. The plastic shear strain now ratchets with a reverse sign to what was discussed above.

A high growth strain results in a significantly thicker TGO with high compressive stresses. In fact, the changes in the radial bond coat stress occur when overall yielding of the TGO starts due to the incremental increase in TGO compressive stress (not shown for brevity) after about 4 cycles. Thus, for the higher growth strain, this appears to dominate the response of the bond coat, and consequently the radial stresses become compressive in the bond coat, resulting in a cyclic reduction of the crack surface separation. In the real system, the TGO growth can be described with a power law of the form: thickness = k(time)1/n, where time is the time at the temperature causing oxidation, k is a growth constant and n is an exponent that lies typically between 2 and 3.3 [22,49,50]. Thus, the TGO grows fast when initially subjected to high temperature followed by a decreasing growth rate as time progresses. The TBC system under consideration had been subjected to significant time-at-temperature before the TGMF test, and the TGO growth rate can be assumed relatively low during the test. The numerical results showed that relatively low growth rate corresponds to positive cyclic crack surface separation, d_ c > 0, resulting in a permanent crack surface separation, dc > 0 govern the system. In all, the numerical simulations support the experimental observations. 6. Concluding remarks In this work we investigated via finite element analysis a class of fatigue cracks observed in thermal barrier coatings (TBCs) after an accelerated test scheme. The test scheme has two parts: The specimens are first pre-heat treated and then subjected to combined thermo-mechanical testing. Previous work showed that the fatigue cracks initiate in the TGO and propagate with thermomechanical cycling through the top coat and bond coat. Once the crack reaches the bond coat it kinks due to property differences in this layer and it opens up in the radial direction. This investigation focused on the crack opening in the bond coat, and the crack surface separation, dc, is monitored. The influence the inelastic high temperature strength of the bond coat, TGO rBC Y HT , and TGO, rY HT , the TGO growth strain, et, and the crack propagation rate, da/dN, have on dc was investigated. The analyses showed that rBC Y HT and et govern the crack surface separation. Three distinctive crack opening evolutions were observed during the 100 cycles investigated: (i) No permanent crack surface separation, observed for vanishing TGO growth strain or for a high bond coat yield strength; (ii) Crack surface separation, observed for moderate TGO growth strains and for a low bond coat yield strength; (iii) Initial crack surface separation followed by a reduction in crack surface separation, observed for large TGO growth strains and low bond coat yield strength. It was also shown that the crack tip propagation rate (or the length of the crack) does not change the qualitative behavior. Crack surface separation occurs during both axial mechanical loading and unloading. The simulations show that the TGO growth strain changes the stress state in the bond coat and introduces biased radial stresses. In addition, plastic deformation occurs only during loading and unloading of the mechanical load. By comparing the bond coat stress field with and without a crack, the numerical simulations show that shear stresses arise in the near field of the crack tip. Interestingly, the plastic shear strain increases in the same direction during loading and unloading of the mechanical axial stress, causing permanent crack surface separation. This result can be qualitatively verified by observing the SEM image of the defect, where the bond coat grains appear to be deformed or distorted due to shear stresses (Fig. 1).

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M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

The numerical simulations also revealed the hypothetical scenario of cyclic and increasing crack surface separation followed by cyclic decreasing crack surface separation, which is associated with low high temperature bond coat yield strength and large TGO growth strains rates. For this case, the radial stresses in the bond coat become compressive, resulting in the reversal of the crack surface separation. In the experimental investigation, a low TGO growth strain rate is obtained due to the pre-oxidation of the specimen. Thus, the scenario suggested by the numerical simulations that crack surface separation occurs for low but non-vanishing TGO growth is consistent with the experiments.

The financial support of the National Science Foundation (DMR0710210) and the German Science Foundation (BA2304/2-1) are acknowledged. Appendix A. Analytical model An analytical, linear-elastic model was developed to elucidate the fundamental state of stress in the bond coat as a function of TGO growth, TGO thickness and the applied mechanical (axial) load. The tensile radial stress that develop under the influence of these parameters and boundary conditions can be interpreted as a Mode I (‘‘crack opening’’) stress for the cracks investigated. Consider a hollow, circular cylinder consisting of five layers, numbered 1, 2, 3, 4 and 5. Let ai be the inner diameter and bi the outer diameter of layer i, where i = 1 (the substrate), 2 (the bond coat), 3 (the TGO next to the bond coat, 4 (TGO next to the top coat) or 5 (top coat). We note that b1  a2, b2  a3, b3  a4 and b4  a5, and that both thickening and lengthening strain is imposed in layer 3 but only lengthening strain is imposed in layer 4. A uniform specimen with no cracks in the bond coat is assumed. The evolution at the maximum temperature of thermo-mechanical cycle is considered, which coincides with the reference temperature. Thus, there are no thermal effects in the problem. The analytical, elastic solution (assuming steady state and generalized plane strain conditions in the length direction) is derived ðiÞ ðiÞ to predict the radial and tangential stresses (rr ðrÞ; rh ðrÞ), as a ðiÞ function of the radius, r, and the axial stress rZ ðrÞ, for a long circular hollow cylinder with sub layers i [51,52]:

rð3Þ r ðrÞ ¼ rðiÞ h ðrÞ ¼ rðiÞ h ¼





Ei ð1 þ mi Þ



 C 1i C 2i  2 ; ð1  2mi Þ r



E3 ð1 þ m3 Þ



Ei ð1 þ mi Þ

Ei ð1 þ mi Þ

i ¼ f1; 2; 4; 5g

 C 13 C 23 E3 et t3  2  ; ð1  2m3 Þ r ð1  m3 Þr

ðA1aÞ



 C 1i C 2i þ 2 ; ð1  2mi Þ r

ðA1bÞ





4 X

2p

Z

bi

ai

i¼1

rriZ ðrÞdr

ðA2Þ

where

A ¼ 2p

Z

b4

rdr

ðA3Þ

a1

The boundary conditions are defined by the vanishing pressure on both inner and outer surfaces:

rrð1Þ ða1 Þ ¼ 0; rð5Þ r ðb5 Þ ¼ 0

ðA4Þ

Equilibrium of the interfacial radial stress is obtained by

Acknowledgements

rðiÞ r ðrÞ ¼

r Z A ¼

C 1i C 2i þ ð1  2mi Þ r 2

 

i ¼ f1; 2; 5g

Ei el ; ð1  mi Þ

i ¼ f3; 4g

ðA1cÞ

ðA1dÞ

where Ei corresponds to Young’s modulus and mi to Poisson’s ratio in layer i, respectively. C1i and C2i (i = 1, 2, 3, 4 or 5) are constants, which are determined based on the boundary and continuity conditions. For the TGO, i = 3 and 4, eigenstrains are included in the radial and tangential stress expression to simulate the TGO growth, Eqs. (A1b) and (A1d). These terms account for the thickening, et, and lengthening, el, TGO growth, respectively, as defined in Eq. (1). The relationship (A1b) assumes that t 3  r, where t3 is the thickðiÞ ness of the TGO. The axial stress in each layer, rZ ðrÞ, is related to  Z , and is determined by the force the applied axial stress r equilibrium

ð3Þ rrð1Þ ðb1 Þ ¼ rð2Þ rð2Þ r ða2 Þ; r ðb2 Þ ¼ rr ða3 Þ; ð3Þ ð4Þ ð4Þ rr ðb3 Þ ¼ rr ða4 Þ; rr ðb4 Þ ¼ rð5Þ r ða5 Þ

ðA5Þ

Continuity of the radial displacement and axial strain is obtained by ð2Þ uð1Þ r ðb1 Þ ¼ ur ða2 Þ;

ð3Þ uð2Þ r ðb2 Þ ¼ ur ða3 Þ;

ð4Þ uð3Þ r ðb3 Þ ¼ ur ða4 Þ;

ð5Þ uð4Þ r ðb4 Þ ¼ ur ða5 Þ

ð2Þ ð3Þ eð1Þ eð2Þ Z ðb1 Þ ¼ eZ ða2 Þ; Z ðb2 Þ ¼ eZ ða3 Þ; ð4Þ ð5Þ eð3Þ eð4Þ Z ðb3 Þ ¼ eZ ða4 Þ; Z ðb4 Þ ¼ eZ ða5 Þ

ðA6Þ

ðA7Þ

where

urðiÞ ¼ C 1i r þ

C 2i ðiÞ  mi eZ r; r

ðA8Þ

eðiÞ Z ¼

 1  ðiÞ ðiÞ rz  mi rðiÞ ; r  rh Ei

eðiÞ Z ¼

 1  ðiÞ ðiÞ rz  mi rðiÞ þ el ; r  rh Ei

i ¼ f1; 2; 5g

i ¼ f3; 4g

ðA9Þ

ðA10Þ

Thus, the conditions in Eqs. (A2-A10) will solve the ten unknowns constants, Cij, {i = 1, 2; j = 1..5} and the five axial stress ðjÞ components over the cross section rZ ðrÞ {j = 1..5} in Eqs. (A1). The material properties used in the analytical model are the high temperature material properties as listed in Table 1. The dimensions used are: the substrate, t1 = 2 mm thick, the bond coat, t2 = 120 lm, the none-thickening TGO, t4 = 4 lm, and the top coat t5 = 220 lm. t3 is the thickness of the TGO that is allowed to grow in its thickness direction, and is a parameter in the study, so to investigate the influence of the thickness on the radial bond coat stresses. Thus, the bond coat radial stress as a function of TGO growth strain, et and el, thickness, t3, and applied axial stress,  z Þ can be investigated. Since this expression is rð2Þ ¼ f ðel ; et ; t3 ; r; r r rather cumbersome, we omit it for brevity, and note that the solution can be easily obtained using a suitable computer code such as MapleTM from Maplesoft, assuming constant axial stress within each layer to address the complication the integral Eq. (A2) imposes. Fig. A1 shows the radial stress at a point in the bond coat, close to interface with the TGO, as a function of total TGO thickness (t3 + t4) for selected thickening growth strains of et = 0, et = 1  Z ¼ 0 103, et = 1  102 with el = 0.1et, and applied axial stress r and 100 MPa. For vanishing growth strain and axial load, the radial bond coat stress remains zero, independent of TGO thicknesses. The radial bond coat stress increases both with increasing TGO thickness and growth strain. An axial stress results in increasing radial bond coat stress. Thus, for non-vanishing growth strain, the radial bond coat stresses are tensile and increase with increasing TGO thickness.

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M.T. Hernandez et al. / Computational Materials Science 50 (2011) 2561–2572

0.4 0 MPa 100 MPa

Radial Stresses (MPa)

0.35 0.3

ε =1·10-2

0.25

t

0.2 0.15

-3

ε =1·10 t

ε =0

0.1

t

0.05 0 5

5.5

6

6.5

7

7.5

8

TGO Thickness (μm) Fig. A1. According to the analytical model, radial stresses in the bond coat as a function of total TGO thickness for selected growth strains, using material  z of 0 and 100 MPa. properties from Table 1 and imposing an axial stress r

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