Influence of material models on the stress state in thermal barrier coating simulations

Surface & Coatings Technology 240 (2014) 301–310

Contents lists available at ScienceDirect

Surface & Coatings Technology journal homepage: www.elsevier.com/locate/surfcoat

Influence of material models on the stress state in thermal barrier coating simulations M. Bäker ⁎ Technische Universität Braunschweig, Langer Kamp 8, 38106 Braunschweig, Germany

a r t i c l e

i n f o

Article history: Received 20 September 2013 Accepted in revised form 17 December 2013 Available online 31 December 2013 Keywords: Thermal barrier coating Finite element modelling Plasticity Visco-plasticity

a b s t r a c t The stress state and failure mechanisms of thermal barrier coating systems are frequently studied using finite element simulations. One problem in evaluating the results of such simulations is that the material behaviour of the top coat and especially of the thermally grown oxide is not well-known. Either plasticity or viscoplasticity is frequently assumed to be the dominating stress relaxation mechanism with the implicit assumption that both mechanisms will lead to similar stress patterns. In this contribution, this assumption is critically evaluated. Different combinations of plasticity or viscoplasticity for the top coat and the TGO are studied. If viscoplasticity is not present, the stress evolution is dominated by plastic ratchetting, but this is counteracted by creep relaxation if the creep strength is sufficiently low. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Thermal barrier coatings are used to protect gas turbine blades from the extreme temperatures of the process gas that may exceed the melting temperature of the nickelbase superalloys used. Turbine blades are coated with a bond coat material (usually an MCrAlY) for oxidation protection and to improve the adhesion of the top coat, and by a top coat material (usually yttria-stabilised zirconia). In service, an alumina layer, called TGO or “thermally grown oxide”, forms at the interface between the oxygen-permeable top coat and the bond coat. Usually, the coating fails after the oxide layer reaches a critical thickness. Understanding the details of the failure mechanism is crucial to predict the life time of the coatings. In addition to lifetime experiments, finite element simulations have been used extensively in the past to calculate the stress evolution and the propagation of cracks in thermal barrier coatings. However, these simulations suffer from the problem that the material properties of the coating layers, especially the TGO, are not well-known. If all materials in a TBC system were purely elastic, extremely large stresses of more than 20 GPa would develop in the system, far beyond values that are actually measured [1]. Thus, stress-relaxation mechanisms must be present in the materials. The bond coat, being metallic and having a comparatively low melting temperature, can relax stresses by plastic and visco-plastic (creep) deformation. The TBC and TGO, being ceramics, can also relax by creeping, but stresses may also relax by micro-cracking and healing [2]. While the properties of the TBC can be measured using free-standing coatings [3,4], determining the behaviour of the TGO is extremely ⁎ Tel.: +49 531 391 3073. E-mail address: [email protected]. 0257-8972/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.surfcoat.2013.12.045

difficult. The melting temperature of Al2O3 is about 2053 °C [5], so that the creep rate at a typical testing temperature of 1000 °C should be low. Furthermore, during growth and subsequent holding at hot time, the microstructure of the TGO changes. The creep rate will reduce strongly with increasing grain size, which may also lead to a positiondependent creep rate in the TGO. Due to the uncertainty in the material behaviour, representing it correctly in a finite element simulation is difficult. Usually, rather simple material laws are used to gain at least qualitative insight into the behaviour of a TBC system. Furthermore, parameter variations can be performed to see the influence of the material on the stress state. Thus, there are widely differing choices for the material behaviour to be found in the literature, with some simulation models ignoring creep [6–8], others ignoring plasticity [9–12], and some including both effects [13–15]. This makes comparing different simulations and evaluating proposed failure mechanisms rather difficult. In this paper, the influence of the material behaviour on the stress state in a TBC system is studied in detail in a finite element model of an aAPS-sprayed thermal barrier coating. Simulations assuming plastic behaviour only or visco-plastic behaviour only or a combination of both are compared and differences in the resulting stress states are evaluated and explained. The paper is organised in the following way: Section 2 discusses the different causes of stresses in a TBC system in a generic way. The difference between plastic ratchetting and visco-plastic stress relaxation is explained in detail. In Section 3, the finite element model and the material laws used are presented. It is also explained how the material parameters were chosen so that a meaningful comparison between plastic and visco-plastic deformation is possible. Section 4 presents the results of the simulations for the different combinations of the material behaviour. It is shown that plastic and visco-plastic behaviour lead to a

302

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

distinct stress evolution in the TBC. This implies that the choice of material behaviour is rather crucial if the failure is to be modelled correctly. Finally, in Section 5, some general conclusions for the implementation of simulations are drawn. 2. Causes of stresses in TBC systems In simulations of a (plasma-sprayed) TBC system, the rough interface is usually approximated by a sine wave and the simulation volume is restricted to half of a wavelength due to symmetry. Usually, stresses at the two extreme positions of the geometry, called “peak” and “valley” are evaluated; occasionally, a position close to the peak (“off-peak”) is also used. It will be shown below that stresses here may be affected by the proximity to the TGO. Therefore, in addition to the valley position, a further point, called “mid-valley” is used for the evaluation: this point lies in the TBC directly above the valley, at a height that is the mean between the valley and the peak position (see Fig. 5 below). 2.1. Growth stresses The oxide layer between the bond coat and the TBC grows not only in perpendicular direction, but also laterally, see Fig. 1. Although the lateral growth rate is usually smaller than the perpendicular growth rate (a value of 1/10 is used here in accordance with [6]), this growth is important because it causes radial tensile stresses in the valley and radial compressive stresses in the peak region. Therefore, at the end of hot time, stresses in the valley region are expected to be tensile. The perpendicular growth also has an effect. Because of the increase in TGO thickness, both the valley and the peak region will be under compressive stresses in an axisymmetric specimen. The strength of this effect depends on the modelling assumptions. Physically, the TGO grows mainly on its lower side because oxygen diffusion is faster than aluminium diffusion (Fig. 1(b)). Therefore, the existing TGO will shield the TBC from these stresses. If the finite element model assumes growth only in the lower region of the TGO, this will be accounted for correctly. Occasionally, however, the simplifying assumption that the perpendicular growth is homogeneous throughout the TGO is made [16,17,12],

a) Lateral growth

see Fig. 1(c). In this case, the shielding effect will be smaller and the compressive stresses in the valley region will be overestimated. It will be shown below that this effect is less pronounced if creep in the TGO is taken into account because creep relaxes the growth stresses. The perpendicular growth stresses mainly affect the valley position directly at the interface between TGO and TBC. Since this is a local effect, it is helpful to study the stress not only directly at this position, but also further away from the interface. Therefore, stresses at the mid-valley position (with a radial coordinate halfway between the peak and the valley of the TBC) are also evaluated in Section 4.

2.2. Stresses on cooling Cooling causes stresses due to the mismatch of the coefficients of thermal expansion (CTE) of the materials. In addition, there is an effect due to the temperature dependence of Young's modulus which will be discussed first. Young's modulus decreases with temperature in all materials. Therefore, even if there were no thermal mismatch, stresses at the end of hot time would increase proportionally upon cooling because the problem can be considered as strain-controlled (Fig. 2(a)). There are two types of cooling stresses due to the CTE mismatch: locally, stresses are generated at the interfaces between bond coat, TGO, and TBC (Fig. 2(b)). For a thin or nonexistent TGO, the thermal mismatch between TBC and bond coat causes tensile stresses in the peak and compressive stresses in the valley region. If the TGO is sufficiently thick, these stresses are reversed, with the valley region being under tension and the peak region under compression. In addition, there is a global effect due to the thermal contraction of the substrate (Fig. 2(c)). Since the substrate is much thicker than the coating layers, it determines the axial extension of the system. Because the TGO is inclined and consists of a stiff material with a low CTE, it will change its orientation slightly during cooling because of this axial compression. This causes compressive stresses in the peak and tensile stresses in the valley region. Models that do not include the substrate do not show this effect [18].

b) Perpendicular growth

c) Homogeneous perpen-

at interface

dicular growth

Fig. 1. Stresses due to growth of TGO. a: lateral growth; b: perpendicular growth at interface to the bond coat; c: homogeneous perpendicular growth.

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

a) Young’s modulus in-crease

b) CTE mismatch at the

c) Axial compression from the

interface

substrate

303

Fig. 2. Different causes of stress due to cooling.

2.3. Plastic ratchetting vs. creep relaxation In this section, the difference between the stress states in plastically or visco-plastically deforming systems is discussed qualitatively. Plastically deforming systems can show ratchetting phenomena, whereas systems deforming by creep will exhibit stress relaxation and no pronounced ratchetting. To study the stress evolution qualitatively, we consider a simple system comprising two vertical strips of TGO and TBC materials that are constrained to have the same vertical strain. (This is a simplified version of the TGO/TBC interface, see also [19].) We start by assuming that the TBC deforms ideally plastically with yield strength σY at room temperature, and take only the CTE mismatch and the lateral growth into account, assuming that the TGO thickness and the TGO growth rate are constant. If the system starts stress free at room temperature, stresses in the TBC become compressive after the first heating, see Fig. 3(b). The TGO growth reduces these compressive stresses, so that the TBC is under tension after cooling. On the second cycle, the TBC is still compressed upon heating, but the overall stress level is now smaller. After TGO growth and cooling, the TBC deforms plastically and the stress after cooling corresponds to the yield stress. On subsequent cycling, the system always returns to this stress state at the end of each cooling cycle. In each cycle, the plastic ratchetting strain increases by the same amount. A more realistic description should take into account that the growth rate of the TGO decreases over time and that its thickness increases, so that the thermal mismatch stresses also increase, see Fig. 3(c). In this case, the stress becomes more compressive during each cycle after the first yield. It should be noted that this more realistic model is still strongly simplified because it assumes a uniaxial stress state. A more realistic simulation of plastic ratchetting and its effect on increasing the amplitude of the roughness profile can be found in [6,20]. For the case of pure creep, relaxation of stresses occurs at hot temperature, whereas the material is linear elastic elastic at low temperatures. The situation can be simplified in a similar way as before to

understand the stress evolution: With constant TGO growth rate and TGO thickness, the system will settle to a constant state where the strain introduced by the TGO growth and the stress relaxation are balanced, see Fig. 4(a) [10]. In a more realistic case, where the TGO growth rate decreases, the creep strain rate will exceed the growth strain rate at some point so that stresses relax at hot time. It can thus be expected that for sufficiently thick TGO (and sufficiently creep-soft materials), the system becomes stress-free at hot time. If we assume that creep only occurs during hot time, there is no effect of the cycling and the stress state would be the same if the specimen were held constantly at hot time. When creep occurs during cooling, cooling has an additional effect which does not cancel on heating: consider a two-layer system where one layer becomes tensile on cooling and where creep in this layer is slow. We then have creep relaxation of the tensile stresses during cooling and re-heating because the stresses are tensile always, resulting in creep ratchetting. If all materials show creep (as in the case of a TBC system), there is an additional effect due to the different temperature dependence of the creep relaxation. In a TBC system, the bond coat usually is still creepsoft at temperatures where the TGO and TBC can be expected not to creep significantly [5]. This will cause some stress relaxation at lower temperatures and thus a small ratchetting effect. These creep ratchetting effects can be observed in the results shown below — there is a small difference in the stress state between the beginning and the end of the cool/heat-part of the cycle. Compared to creep relaxation at hot time, this effect is small. 3. Model The considerations in the previous section were based on a strongly simplified geometry. To quantitatively study the influence of the material behaviour, a typical finite element model with a sinusoidal interface is used in the following.

304

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

a) Simple model

b) Idealised case

c) More realistic case

Fig. 3. Schematic explanation of plastic ratchetting in the TBC. In the idealised case, the TGO growth rate is constant and the increasing thickness of the TGO is not taken into account. In the more realistic case, the TGO growth rate decreases and the thermal mismatch stresses increase over time. The numbers denote the end of the cooling step in the respective cycle. Red arrows correspond to the heating step, green arrows to the hold time, and blue arrows to the cooling step.

3.1. Model geometry The geometry of the model is taken from [21]: the coating is applied to an infinite cylinder of a superalloy substrate with radius 7 mm, see Fig. 5. The superalloy substrate has a radius of 7 mm. The average thickness of bond coat and top coat is 150 μm. To simulate an infinitely extended system, the bottom nodes of the model are constrained in axial direction. The top nodes are glued to a rigid surface so that they are free to move in radial direction, but can only move collectively in the axial direction i.e., the axial displacement of all top nodes has to be identical. The interface region between bond coat and TBC is modelled with a sinusoidal geometry with a half-wavelength of 12 μm and a total height of 10 μm as shown in the figure. Initially, the thermally grown oxide between bond and top coat has a thickness of 1 μm. The model was meshed using approximately 15,000 four-node elements with selectively reduced integration. The mesh was refined in the interface region close to the TGO.

Each system was run for 150 cycles (except for two cases which did not converge). Each cycle starts at a temperature of 20 °C. Within 60 s, it is heated isothermally to 1000 °C, held at this temperature for 10,000 s, and cooled back in another 60 s to 20 °C. The system is assumed to be stress-free initially. Stresses are evaluated at two positions: previous simulations [10,22] have shown that except for small TGO thicknesses, stresses in the peaks of the asperities are compressive and those in the valley are tensile. Therefore, the stress valley region is crucial for the formation of delaminations. In evaluating stress contour plots, it was found that the maximum tensile stress frequently does not occur directly at the valley position, but about above the valley position at a distance that is close to the amplitude (half of the total height of the asperities). Therefore, this mid-valley position (see Fig. 5) was also used to analyse stresses. Since the vertical distance between peak and valley may change during the simulation due to a change in the slope of the interface, the midvalley position is re-calculated after each cycle. This and the remeshing procedure used may lead to slight oscillations in the stresses due to a

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

a) Idealised case 3

305

b) More realistic case 4

5

3

2

4

5

2

1

1

Fig. 4. Schematic explanation of stress evolution in the TBC with creep. In the idealised case, the TGO growth rate is constant and the increasing thickness of the TGO is not taken into account. In the more realistic case, the TGO growth rate decreases and the thermal mismatch stresses increase over time.

change in the element chosen for evaluation or to a change in the element's shape on remeshing. 3.2. Choosing material parameters Elastic and thermal material parameters of the involved materials are given in Table 1. The substrate is assumed to deform elastically only; since it is much thicker than all coating layers, creep in the substrate only has a negligible effect on the stresses at the TGO–TBC interface. At high temperature, the oxide layer is growing due to oxidation of the bond coat. This process causes a volume increase of the material which in turn causes growth stresses. The growth is modelled as an anisotropic swelling of the oxide layer with a parabolic growth constant of 1.5 · 10−17 m2/s and an anisotropy ratio of 10 (i.e., growth in thickness direction is ten times larger than growth in lateral direction [6]). During each cycle, a constant growth rate of the TGO is assumed. After cooling to room temperature, the TGO thickness is measured and the growth rate is adapted to the current thickness. Note that, since the thickness of the TGO changes not only by growth, but also by creep relaxation, this implies that the TGO thickness is not exactly the same at the end of different simulations. A more detailed simulation of the oxide growth process that calculates the movement of the phase

substrate

boundary and includes a model of diffusion can be found in [15] for the case of EB-PVD coatings. The main goal of this study is to compare the stress development using different assumptions for the plastic or visco-plastic material behaviour of the coating. For a meaningful comparison to be possible, stresses in the materials should be of the same magnitude. To ensure this, the following approach was chosen: initially, we assumed no plastic deformation in all materials and used two different values for the creep strengths of the TGO and the TBC, called creep hard and creep soft (see Table 2), taking standard values for the creep behaviour of the bond coat (Table 3). These creep parameters were based on previous studies [10] and were chosen so that stresses at the end of the hold time were small for the creep-soft case and considerably larger for the creephard case. Although the parameters are varied in a reasonable range for the materials investigated, they should not be used uncritically in other simulations, since the main intention of the parametric study is not to provide creep data, but rather to study the general effects and the interplay of plasticity and visco-plasticity. To determine plausible parameters for the yield strength, finite element simulations of the two systems were performed. The maximum von Mises stress in the three materials was determined at the end of the first holding time (in the first cycle) while still at hot temperature

bond coat 10µm TBC 12µm

TGO 7mm

150µm

150µm s

Fig. 5. Model geometry. The model is cylindrically symmetric, with the axial direction of the cylinder being vertical. A detail of the finite element mesh in the TGO region is also shown.

306

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

Table 1 Material properties used in the model, taken from [10].

Young's modulus (20 °C) [GPa] Young's modulus (1100 °C) [GPa] Poisson number (20 °C) Poisson number (1000 °C) CTE (20 °C) [K−1] CTE (1000 °C) [K−1]

Substrate

Bond coat

Oxide

TBC

184

200

400

48

Table 3 Plastic and viscoplastic properties of the bond coat. In the simulations with creep only and plasticity only, the bond coat also deforms only by creep and plasticity, respectively. In all other cases, creep and plasticity are both active. A at 1000 °C −7

145

110

325

22

0.3 0.3 12 · 10−6 16 · 10−6

0.3 0.33 13.6 · 10−6 17.6 · 10−6

0.23 0.25 8.0 · 10−6 9.3 · 10−6

0.1 0.12 9 · 10−6 12.2 · 10−6

1.39 · 10

−3

MPa

−1

s

n

Act. energy

Rp at 0 °C

Rp at 1000 °C

3

284 kJ/mol

930 MPa

2 MPa

10-5

Cs-n1

Cs-n2

and after cooling to room temperature. These von Mises stress values (slightly rounded) were chosen as yield stress for an ideally plastic material law, defining plastically soft and plastically hard materials. Note that these parameters were not chosen with the intention of being realistic descriptions of the material, but rather to see how the material assumptions influence the stress development in general. Since the simulation showed that the material behaviour was dominated by creep relaxation when creep and plasticity were combined, the creep strength in the creep-hard case was further increased by a factor of 10, defining an “ultra-hard” material. Determining creep properties for the TGO is rather difficult as explained above. Although parameter variations allow to understand the influence of creep relaxation, it may be objected that the assumed creep exponent of 1 is rather unrealistic. Therefore, reference [23] was used to determine more realistic creep parameters for the TGO. The following creep law was used   n σ −Q ϵ˙ ¼ A0 m exp ; RT δ

ð1Þ

where A0 is a constant, σ is the stress, n is the Norton creep exponent, δ is the grain size, m is the grain size exponent, Q is the activation energy and T is the temperature. The creep rate at 1200 °C was read off from Fig. 1 in [23] and was extrapolated to different temperatures using an activation energy of 483 kJ/mol and Arrhenius law. To define a creep-soft and a creep-hard case, two different values of the grain size (0.75 μm and 2.4 μm) were used, assuming a grain size exponent in the creep rate of 3. The Norton creep exponent was taken to be 2. Note that the data were used for undoped Al2O3; if the material contains even small amounts of dopant elements, the activation energy for creep becomes larger and the creep rate reduces considerably. Fig. 6 gives an overview of the different creep laws used for the TGO, showing that a wide range of creep rates is covered by the model.

creep rate / 1/s

10-6

Ch-n1

10-7

Ch-n2

Cuh-n1

10-8 10-9

Cs-n1: n=1, creep soft Ch-n1: n=1, creep hard Cuh-n1: n=1, creep ultra-hard Cs-n2: n=2, creep soft Cs-n2: n=2, creep hard

10-10 10-11

0

100

200

300

400

500

600

700

800

900 1000

Stress / MPa Fig. 6. Creep rates of the different TGO material models at 1000 °C.

In the case of the bond coat, the creep properties were not varied because these can be determined experimentally with satisfactory accuracy. The plastic properties were determined in the same way as for the other materials, but were not varied during the simulations. In the simulations with creep only and plasticity only, the bond coat also deforms only by creep and plasticity, respectively. In all cases where creep and plasticity were combined, both creep and plasticity from Table 3 were used in the bond coat. The mechanism of stress relaxation in the TGO may be dominated by microcracks that form and heal, especially at high temperature [2]. To see how this would affect the stress state, a simplified description of the material was used where plastic yielding occurs at a strain of 0.3% at hot temperature and 1% at room temperature.

4. Results In this section, results for simulation using creep only, plasticity only, and creep and plasticity combined are shown. Table 4 gives an overview of all simulated combinations, denoting the symbols used in the stress

Table 2 Materials properties table for the TGO and TBC. The activation energy is used to calculate the creep strength at the different temperatures occurring during heating and cooling. n denotes the creep exponent used, A is the pre-factor at 1000 °C for the Norton creep law ϵ˙ ¼ Aσ n . The yield stress is denoted by Rp. Creep data for the TGO at n = are taken from [23]. Creep TGO

TBC

A at 1000 °C Creep soft Creep hard Creep ultra-hard n = 2 creep soft n = 2 creep hard

−8

−1

n −1

10 s Pa 10−9 s−1 Pa−1 10−10 s−1 Pa−1 8.147 4 · 10−25 s−1 Pa−2 4.888 44 · 10−26 s−1 Pa−2

1 1 1 2 1

Act. energy 4.087 42 4.087 42 4.087 42 4.83 4.83

· · · · ·

A at 1000 °C 5

10 105 105 105 105

J/mol J/mol J/mol J/mol J/mol

5 5 5 5 5

· · · · ·

−8

−1

−1

10 s Pa 10−9 s−1 Pa−1 10−10 s−1 Pa−1 10−9 s−1 Pa−1 10−9 s−1 Pa−1

n

Act. energy

1 1 1 1 1

2.44 2.44 2.44 2.44 2.44

· · · · ·

105 105 105 105 105

J/mol J/mol J/mol J/mol J/mol

Plasticity TGO

Plast. hard Plast. soft

TBC

Rp at 0 °C

Rp at 1000 °C

Rp at 0 °C

Rp at 1000 °C

6200 MPa 4600 MPa

1600 MPa 240 MPa

500 MPa 380 MPa

270 MPa 33 MPa

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

plast infinite

Table 4 Overview of the simulations and corresponding symbols discussed in Sections 4.1–4.3. Creep hard

Creep ultra-hard

Creep inf.

● ○ +

■ □ ×

▼ ▽ −

▲ △ −

history plots. In all plots, dotted lines mark the stress at the end of the heating phase, dashed lines the stress at the end of the holding phase, and solid lines the stress at the end of the cooling phase. Symbols are only drawn at every tenth cycle to improve visibility. Finally, in Subsection 4.4, the effect of possibly more realistic assumptions on the behaviour of the TGO is studied.

200

100 50 0 -50 -100

4.1. Creep only

creep soft creep hard

150

Stress / MPa

Plast soft Plast hard Plast inf.

Creep soft

307

0

50

100

150

200

250

300

350

400

Time / h

First, we consider the case of creep only in all materials. Fig. 7 shows a contour plot of the radial stresses near the interface after 75 cycles for thee creep-hard case. Compressive stresses in the TGO are large and the TGO growth has put the peak region under compression, whereas stresses in the valley region are tensile. It can be clearly seen that the stresses directly at the valley position are smaller than those in the mid-valley position as explained above. Fig. 8 shows the stress evolution for creep soft and creep hard materials at the mid-valley position of the TBC. Initially, the stress at the end of the first heating phase is compressive in the system, as expected from Fig. 4. Due to creep of the materials, these stresses rapidly relax after a few cycles. After the initial stage, the stresses at the end of the heating and the end of the holding time are relaxing due to creep. Stresses caused by TGO growth are relaxed more efficiently in later cycles because the TGO growth rate decreases with increasing thickness. The stresses at the end of the cooling phase also become smaller during the first 75 h due to relaxation of the growth stresses. In later stages, the stresses at the end of cooling are dominated by the CTE mismatch between TBC and TGO and by the axial compression effect and are thus increasing again because of the increasing TGO thickness. Directly at the valley position, the picture is similar (Fig. 9), with stresses relaxing during hot time. For the case of a creep-hard material, compressive stresses due to the perpendicular growth of the TGO do not relax completely (the stress level is about 25 MPa), so the tensile stresses after cooling are also reduced compared to the creep-soft case. Comparing Figs. 8 and 9, it can be seen that the mid-valley stresses are larger than the valley stresses. This is due to the axial compression effect: the valley position is under larger hydrostatic pressure when the substrate material is shrinking in axial direction, whereas the midvalley position is fully exposed to the tensile stresses caused by the shift in geometry due to compression and by the CTE mismatch. This suggests that if a crack is initiated somewhere in the valley region, it

Fig. 8. Radial stress evolution at the mid-valley position for creep soft and creep hard materials with no plasticity. Here and in all subsequent figures of stress evolution, three curves are drawn for each simulation: the solid curve marks the stresses at the end of cooling, the dotted curve at the end of heating and the dashed curve at the end of hold time. For better visibility, in this and the following figures every tenth cycle is marked with a symbol.

will not be directly initiated at the interface, but above it. (A contour plot shows that the stress maximum is usually close to the mid-valley position.) Experimentally, it is observed that cracks may indeed not start directly at the interface, but above it [24]; however, this is not always the case because the fracture toughness of the interface may be lower than that of the TBC itself.

4.2. Plasticity only Next we study the case where all materials deform only plastically. In the mid-valley position (Fig. 10), radial stresses at the end of cooling are tensile, but they become smaller and shift to the compressive region. This is expected from Fig. 3(c) due to the CTE mismatch and also because of the axial strain imposed by the substrate. During hot time, the stresses become more tensile due to the growth of the TGO. A qualitatively similar stress evolution (evaluated at a position between midvalley and peak) was shown in [25]. For plastically soft materials, these stresses are larger than the yield stress at room temperature so that the stress at the beginning of hot time becomes compressive. Furthermore, the slope of the curve is smaller because part of the growth stresses is already relaxed by plastic deformation during hot time (both TBC and TGO are yielding). Stresses in the valley become compressive at large cycle numbers, see Fig. 11. This is due to the perpendicular growth of the TGO which causes compressive stresses directly at the interface, so that the stress shifts towards the compressive during holding time. Furthermore, the

Fig. 7. Contour plot of the radial stress evolution for creep hard materials after 75 cycles.

308

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

creep infinite

plast infinite 100

400

creep soft creep hard

80

300

60

250

Stress / MPa

Stress / MPa

40 20 0 -20 -40

200 150 100 50 0

-60

-50

-80

-100

-100

plast soft plast hard

350

0

50

100

150

200

250

300

350

-150

400

0

50

100

150

Time / h

200

250

300

350

400

Time / h

Fig. 9. Radial stress evolution at the valley position for creep soft and creep hard materials with no plasticity.

Fig. 11. Radial stress evolution at the valley position for plastically soft and hard materials with no creep.

valley position is more tightly constrained by the surrounding TGO and tensile stresses induced by axial compression are smaller. The assumption of a perpendicular TGO growth everywhere in the TGO is a simplification, as explained in Section 2.1. If growth was restricted to the lower region of the TGO, these stresses would be shielded by the stiff upper part of the TGO and the compression of the valley region would be strongly reduced. Especially with a plastically hard material, where no yielding occurs during hot time, the implementation of the TGO growth thus strongly influences the stress state. This is different if creep is active because then stresses always relax partially during hot time, especially at larger TGO thickness when the growth rate is small.

state becomes slightly compressive as explained in the previous section. However, since these stresses relax quickly, ratchetting strains cannot accumulate. If the materials have a higher yield strength (case “plast hard”; this simulation was stopped after 200 h due to convergence problems), the difference to the case of no plasticity is small. Cooling stresses are slightly larger than without plastic deformation. This is caused by plastic yield in the bond coat which bears less of the tensile stresses exerted by the TGO cooling and this increases the load of the TBC slightly. If the creep strength of the TGO and TBC is increased by an order of magnitude (Fig. 13), the picture is similar: the stress state is still dominated by creep relaxation at high temperature. Again, in the case of plastically soft materials, there is some plastic ratchetting, leading to compression at the beginning of the hold time. As in the previous case, the stresses are slightly larger with plasticity than without; this is again due to plasticity in the bond coat. In both cases, the end state is dominated more by creep relaxation than by plastic ratchetting. Therefore, two additional simulations were performed with creep rates of TGO/TBC being reduced further by a factor of 10, see Fig. 14. In this case, stress states are very similar to the case of no creep, Fig. 10 (note that “no creep” also means “no creep” in the bc). The evolution of the valley stresses is very similar — they correspond closely to the case of no plasticity unless the creep strength is very large. Table 5 gives an overview of the dominating stress evolution mechanism at the mid-valley position for all cases considered. It can be seen

4.3. Plasticity and creep combined In the previous two sections, it was shown that the stress state of the TBC strongly depends on the chosen stress-relaxation mechanism. A realistic description of the material behaviour should take into account both plasticity and creep. In this section, the effects of combined plasticity and creep are considered. Fig. 12 shows the mid-valley stresses for creep-soft materials with different values of the yield strength. In general, there is no strong influence due to plastic yield and the stress evolution is similar to the case of no plasticity, marked “plast. infinite”, which is the same as in Fig. 8. Only if the yield stress is small (case “plast soft”), there is a small amount of plastic ratchetting: the stress at the end of cooling is slightly lower due to plastic yield. After heating from room temperature, the stress

creep infinite 400

creep soft

plast soft plast hard

350

200 150

250 200

Stress / MPa

Stress / MPa

300

plast soft, creep soft plast hard, creep soft plast inf., creep soft

150 100 50

100 50 0

0 -50

-50 -100

0

50

100

150

200

250

300

350

400

Time / h

-100

0

50

100

150

200

250

300

350

400

Time / h Fig. 10. Radial stress evolution at the mid-valley position for plastically soft and hard materials with no creep.

Fig. 12. Radial stress evolution at the mid-valley position for creep-soft materials.

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

creep hard 250

Table 5 Dominating mechanism of stress evolution for the different cases.

plast soft, creep hard plast hard, creep hard plast inf., creep hard

200

Plast. soft Plast. hard Plast. infinite

Stress / MPa

150

Creep soft

Creep hard

Creep ultra-hard

Creep infinite

Relaxation Relaxation Relaxation

Relaxation Relaxation Relaxation

Ratchetting Ratchetting –

Ratchetting Ratchetting –

100 50 0 -50 -100

0

50

100

150

200

250

300

350

400

Time / h Fig. 13. Radial stress evolution at the mid-valley position for creep-hard materials.

that ratchetting effects dominate only if the creep strength of the materials is very high. 4.4. More realistic scenarios If we use a creep exponent of 2 instead of 1 in the TGO, the stresses in the TBC are qualitatively similar, see Fig. 15. Two cases were compared where only the properties of the TGO are changed. For the creep-soft TGO with n = 2, the stress curve is rather similar to the case of a creep-hard TGO with n = 1. Stress relaxation in the TGO is still sufficient to allow the TBC stresses to relax to small values during hot time. The similarity of the two cases is not surprising because creep rates are similar, see Fig. 6. For the creep-hard case with n = 2, stresses in the TBC do not relax during hold time even at large cycle numbers and stay compressive. This is similar to the stress in the valley region in Fig. 9. Since the creep rate in the TGO is very low in this case (see Fig. 6), the interface between TGO and TBC becomes very steep and the compression effect that is only seen in the valley for the case of a creep-hard TGO at n = 1 extends further and reaches the mid-valley region. This is confirmed by a contour plot which shows very large compressive stresses at the valley position that gradually fall off. The largest tensile stresses are thus reached in a position that lies higher than the mid-valley position used in the evaluation. Despite these differences, the overall evolution of the stress is not too different from the generic case studied before when n = 2 is assumed.

If the TGO material contains even a small amount of dopant elements, the creep rate reduces drastically [23]. In this case, it can be assumed that other stress-relaxation mechanisms may occur in the TGO, for example microcracking and healing [2]. Fig. 16 shows the result for a simulation assuming microcracks in the TGO that limit the strain to 0.3% at hot temperature and 1% at room temperature, in comparison with the case of creep hard TGO and TBC. The curves are similar to each other, but stresses do not relax fully during hot time. This is similar to the previous case of a creep-hard TGO with n = 2 (Fig. 15) because in both cases the stresses in the TGO do not relax during hot time (either because creep is too slow or because stresses below the yield limit do not relax). 5. Discussion In this work, the influence of the choice of the stress-relaxing mechanism in simulations of thermal barrier coatings was analysed. It was found that due to ratchetting, plastic materials tend to have a completely different stress evolution from visco-plastic materials that relax stresses at hot time. It was also found that, in all cases, stresses in the mid-valley position are more tensile than directly in the valley position. This is partially due to the axial compression effect, but also to the chosen implementation of TGO growth. If the perpendicular TGO growth is assumed to be homogeneous throughout the TGO, compressive stresses in the valley region are overestimated. This is also corroborated by a comparison of the simulations done in this paper with those in [10,22], where the TGO growth occurred only in the lowest element layer of the TGO and where larger tensile stresses are observed in the valley position. The following conclusions for the implementation of finite element models of TBC systems can be drawn: • Visco-plastic deformation causes stress relaxation at high temperature and thus counteracts the effects of plastic ratchetting. Models without visco-plasticity should be avoided unless the creep strength of all materials (and especially the TBC) is known to be high; but even in this case creep will limit the ratchetting stress at some point. Without creep, ratchetting effects will be overestimated and the simulated stress evolution will not be realistic.

creep hard

creep ultra-hard 400

250

plast soft, creep ultra plast hard, creep ultra

350

Creep soft TGO n=2, creep hard TBC Creep hard TGO n=2, creep hard TBC creep hard TGO/TBC

200

300

Stress / MPa

150

250

Stress / MPa

309

200 150 100

100 50 0

50 -50

0 -50 -100

-100 0

50

100

150

200

250

300

350

400

0

50

100

150

200

250

300

350

400

Time / h

Time / h Fig. 14. Radial stress evolution at the mid-valley position for very creep-hard materials.

Fig. 15. Radial stress evolution at the mid-valley position for creeping TBC/BC and creep exponent n = 2 in the TGO, compared to the case of creep hard materials from Fig. 13.

310

M. Bäker / Surface & Coatings Technology 240 (2014) 301–310

creep hard 200

stress relaxation mechanism of the TGO may be of less importance; nevertheless the overall stress level in the TGO still has to be calculated correctly. Although parameter variations may be helpful to understand how stresses may evolve, it has to be made sure that the underlying assumptions on the deformation mechanisms do not unduly influence the results.

crack TGO, creep hard TBC creep hard TGO/TBC

Stress / MPa

150 100

Acknowledgements

50

Thanks to Prof. Michael Schütze for helpful discussions concerning stress relaxation mechanisms in the TGO and to Philipp Seiler for reading and improving the manuscript. Part of this research was founded by the Deutsche Forschungsgemeinschaft, projects Ba-1795/2-2 and Ba1795/9-3.

0 -50 -100 0

50

100

150

200

250

300

350

400

Time / h Fig. 16. Radial stress evolution at the mid-valley position for creeping TBC/BC and plastically deforming TGO in comparison to creep in all materials.

• Although stress relaxation of the TGO strongly affects the results, the stresses in the TBC are not affected too much by the details of the TGO stress relaxation mechanism: from Figs. 15 and 16, it can be seen that relaxation due to pseudo-plasticity in the TGO or creep with a creep exponent of n = 1 or n = 2 all leads to similar results for the stress. • Stresses should be evaluated not only at the valley position, but also further away from the interface (as done in this paper using the mid-valley position) because they might be larger in this region. • If a detailed simulation of the stress evolution is required, TGO growth should be modelled as realistically as possible with perpendicular growth only at the TGO–bond coat interface. This is especially important in cases where the creep strength of the material is large. If the TGO or the TBC is creep-soft, this effect is less important. • As already shown in [18], the axial compression induced by the substrate may have an important effect on the stress state. Therefore, the substrate must always be included in models of TBC systems and cannot be neglected. In conclusion, a realistic simulation of the stress evolution in a TBC system requires detailed knowledge of the material behaviour and of the relative strength of plastic yield and creep relaxation. Frequently, stresses in the TBC are of highest interest because failure is known to be due to the formation of TBC cracks. In this case, the details of the

References [1] V.K. Tolpygo, J.R. Dryden, D.R. Clarke, Acta Mater. 46 (3) (1998) 927–937. [2] Michael Schütze, Derek Roy Holmes, Robert Barry Waterhouse, Protective Oxide Scales and Their Breakdown, Wiley, Chichester, 1997. [3] H. Echsler, D. Renusch, M. Schütze, Mater. Sci. Technol. 20 (7) (2004) 869–876. [4] X. Wang, S. Tint, M. Chiu, A. Atkinson, Acta Mater. 60 (8) (2012) 3247–3258. [5] R. Bürgel, H.J. Maier, T. Niendorf, Handbuch Hochtemperatur-Werkstofftechnik, 4th edition Vieweg + Teubner, Braunschweig/Wiesbaden, 2011. [6] A.M. Karlsson, A.G. Evans, Acta Mater. 49 (10) (2001) 1793–1804(Cited By (since 1996): 86). [7] M.T. Hernandez, A.M. Karlsson, M. Bartsch, Surf. Coat. Technol. 203 (23) (2009) 3549–3558. [8] W. Xie, E. Jordan, M. Gell, Mater. Sci. Eng. A 419 (1-2) (2006) 50. [9] A.M. Freborg, B.L. Ferguson, G.J. Petrus, W.J. Brindley, Mater. Sci. Eng. A245 (1998) 182–190. [10] J. Rösler, M. Bäker, K. Aufzug, Acta Mater. 52 (16) (2004) 4809–4817. [11] E.P. Busso, J. Lin, S. Sakurai, M. Nakayama, Acta Mater. 49 (9) (2001) 1515–1528. [12] M. Białas, Surf. Coat. Technol. 202 (24) (2008) 6002–6010. [13] M. Ranjbar-Far, J. Absi, G. Mariaux, F. Dubois, Mater. Des. 31 (2) (2010) 772–781. [14] M. Ranjbar-Far, J. Absi, G. Mariaux, D.S. Smith, Mater. Des. 32 (10) (2011) 4961–4969. [15] Audric Saillard, Mohammed Cherkaoui, Laurent Capolungo, Esteban P. Busso, Philos. Mag. 90 (19) (2010) 2651–2676. [16] J. Aktaa, K. Sfar, D. Munz, Acta Mater. 53 (16) (2005) 4399. [17] M. Bäker, Comput. Mater. Sci. 64 (2012) 79–83. [18] M. Bäker, J. Rösler, E. Affeldt, Comput. Mater. Sci. 47 (2) (2009) 466–470. [19] J. Rösler, M. Bäker, M. Volgmann, Acta Mater. 49 (18) (2001) 3659–3670. [20] M.Y. He, J.W. Hutchinson, A.G. Evans, Acta Mater. 50 (5) (2002) 1063–1073. [21] M. Bäker, J. Rösler, Comput. Mater. Sci. 52 (1) (2012) 236–239. [22] M. Bäker, J. Rösler, G. Heinze, Acta Mater. 53 (2) (2005) 469–476. [23] J. Cho, C.M. Wang, H.M. Chan, J.M. Rickman, M.P. Harmer, Acta Mater. 47 (15) (1999) 4197–4207. [24] A. Rabiei, A.G. Evans, Acta Mater. 48 (15) (2000) 3963–3976. [25] M.Y. He, J.W. Hutchinson, A.G. Evans, Mater. Sci. Eng. A 345 (1-2) (2003) 172–178(Cited By (since 1996): 28).