Modelling of TBC system failure: Stress distribution as a function of TGO thickness and thermal expansion mismatch

Engineering Failure Analysis 13 (2006) 409–426

www.elsevier.com/locate/engfailanal

Modelling of TBC system failure: Stress distribution asa function of TGO thickness and thermal expansion mismatch

M. Martena a,*, D. Botto b, P. Fino a, S. Sabbadini c, M.M. Gola b, C. Badini a

a Dipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italyb Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

c Ente Ricerca, Avio S.p.A., Via I Maggio 99, 10040 Rivalta di Torino, Italy

Received 12 July 2004; accepted 26 December 2004

Available online 31 March 2005

Abstract

Advances in gas turbine technology place an increasing demand on thermal protection systems of nickel-base super-

alloys in turbine blades. Current strategies for performance improvements are focused on thermal barrier coatings

(TBC). Typical current TBC system are composed of: top coat (TC), an yttria stabilised zirconia outer layer that pro-

vides thermal insulation; a bond coat (BC) layer, aluminium rich, supplying oxidation resistance and adhesion of TC to

the metal; a thermally grown oxide (TGO) scale, predominantly alumina, that is a reaction product formed between TC

and BC as a consequence of BC oxidation at high temperatures. At present, the capabilities of TBCs cannot be fully

exploited due to the lack of a reliable lifetime prediction model of the coating. Hence, continuous efforts are made

by materials scientists in this direction and this is the purpose of our work. To achieve this objective, a preliminary

activity is necessary to determine stress distribution in the system as a function of each factor affecting TBC behaviour.

First, we have developed a model of BC oxidation, based on Wagner�s theory, which predicts a parabolic law for the

growth of TGO scale. Then, using finite element method, we performed an analysis of stress distribution in the system

because of TGO thickening and thermal expansion mismatch. This is a prerequisite to understand failure mechanism

that are different depending on processing mode of TBC, either plasma spray (PS) or electron beam physical vapour

deposition (EB-PVD).

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Turbine blade; Surface coatings; Oxidation; Stress analysis; Failure

1350-6307/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfailanal.2004.12.027

* Corresponding author. Tel.: +39 011 5644672; fax: +39 011 5644699.

E-mail address: [email protected] (M. Martena).

mailto:[email protected]

410 M. Martena et al. / Engineering Failure Analysis 13 (2006) 409–426

1. Introduction

It is widely stated that the evolution of gas turbines can be mapped in terms of the turbine entry tem-

perature (TET) and for that reason, any new developments in aeroturbine engines require TETs to increase

even further than the present state-of-the art levels. This great demand is the driving force for the develop-ment of advanced turbine materials (superalloys), new cooling concepts, novel combustor designs and

above all, for the gradual introduction of TBCs in the engine hot sections.

A thermal barrier coating (TBC) system applied to turbine blades made up of nickel base superalloy can

lower the temperature of metallic substrate of about 100–150 �C [1–3], together with active cooling of back-

side metal, allowing an increase in the TET and as a consequence, in the engine efficiency.

A typical current TBC system is composed of a top coat (TC), which is the actual thermal barrier, with

the main function to reduce the heat transfer to the metal substrate; a bond coat (BC) deposited between

the metallic substrate and the TC to prevent underlying superalloy from oxidation and high temperaturecorrosion and to guarantee the coupling between the ceramic and the airfoil material; during service oper-

ations, the oxidation of BC at high temperatures produces a thermally grown oxide (TGO) scale located

between BC and TC. The thickness of this layer increases with increasing operation time and this element

seems to be almost the most important factor in determining lifetime of a coated component [4,5].

The TC is a thermally insulating and strain tolerant oxide; zirconia stabilised into its tetragonal form by

the addition of yttria in solid solution has emerged as the preferred material. It has low thermal conduc-

tivity, an order of magnitude below that of the Ni-base superalloy, and, as a ceramic, it also shows a rel-

atively high coefficient of thermal expansion, which comes close to that of the metal substrate. For thatreason, upon thermal shock, it allows to accommodate straining without spalling immediately for the ther-

mal expansion misfit. Two methods are used to deposit strain-tolerant TBCs. EB-PVD evaporates the oxide

from an ingot and directs the vapour onto the preheated component. The deposition condition are designed

to create a columnar grain structure. Plasma spray (PS) deposition is a lower cost alternative. The deposi-

tion is designed to incorporate intersplat porosity and a network of crack-like voids that again provides

strain tolerance, while lowering the thermal conductivity.

The BC is arguably the most crucial component of the TBC system [6]. Its chemistry and microstructure

influence durability through the structure and morphology of TGO created as it oxidised. Moreover, sys-tem performance is linked to its creep and yield characteristics. BCs are in two categories: one is based on

the NiCoCrAlY system and the second category consists of a Pt-modified diffusion aluminide.

The interface between BC and TGO, which morphology changes in service, is another critical element:

generally, high stresses are present in the interfacial zone, due to oxide thickening, to thermal expansion

misfit and to applied loads. As a result, crack can initiate and propagate, causing the ceramic layer to spall

off, leading to the system degradation. A physical description of these phenomena is not easy because it

requires in particular to know the fine structure and properties of the various constituents, which are dif-

ferent according to the type of BC and to the deposition method of ceramic barrier, as well as their evo-lution as a function of time and temperature.

At present, there is no adequate global description integrating all these data but several aspects are being

investigated in a detailed manner, i.e., the determination of the residual stresses in the alumina layer [7–9],

the coupling of the oxidation and mechanical aspects in finite element calculations of the stress field [10–13],

the delamination of compressed films in relations with imperfections and the interface geometry [14–17].

The knowledge derived from all these approaches, and its integration in a more physically based model

should serve designers and coaters to optimise actual and future thermal barrier systems with respect to

given service conditions. Currently, engine design still primarily relies on lifetimes extension benefits ofTBCs. However, it is obvious that prime reliant TBCs with predictable life-time performance are required

to implement TBCs as designed-in components and in this way to fully exploit their potential for significant

performance improvements.

M. Martena et al. / Engineering Failure Analysis 13 (2006) 409–426 411

From the above framework it is clear that, in recent years, many efforts have been devoted to understand

the behaviour and degration of TBCs under combined mechanical and thermal loadings. The great interest

in TBC shown by the scientific community and obviously by the industry is the reason for a very large num-

ber of scientific publications on this topic. Unfortunately, the informations about TBCs system behaviour

contained in these documents are often incomplete because they are referred to a characteristic system oronly to some specifics of a more general problem or simply because they are proprietary informations. So, it

is difficult more than dangerous for safety principles to estimate our own coatings lifetime only on the basis

of the existing literature background; whereas it is suitable to study our systems through an adequate exper-

imental activity and to describe its behaviour and predict its lifetime through an ad hoc model. The devel-

opment of a lifetime prediction model is the ultimate purpose of our work. In the next sections, our BC

oxidation model will be explained and it will be outlined the stress distribution in our system as a function

of TGO thickening, thermal expansion mismatch and BC/TGO interface morphology. In fact we think that

lifetime of the coatings is primarily influenced by these factors. The work is in particularly referred to TBCdeposited by PS method onto an MeCrAlY type BC protecting a nickel base superalloy substrate.

2. Modelling and experimental tests

2.1. TGO growth

The first step in the implementation of our computational model deals with the correct description of thegrowth rate of TGO because it plays a key role on durability of TBCs.

The BC alloy is designed as a local aluminium reservoir, enabling a-alumina to form as a protective scale

[18] in preference to other oxides, as oxygen ingresses as a gas through the porous TC. Alumina is the pre-

ferred oxide because in this case the scale is continuous, characterised by a slow growth rate and it has a

superior adherence to BC respect to different oxides.

Some authors suggest that alumina scale predominantly grows by outward aluminum diffusion, others

have reported that scale growth occurs by inward oxygen diffusion and some other results indicate that the

growth mechanism occurred by both cation and anion transport in the Al2O3 scale [12,19].

2.1.1. BC oxidation model: isothermal aging

We have developed a theoretical model of BC oxidation based on Wagner�s theory [20] which takes into

account both cation and anion diffusion.

When BC is oxidised at high temperature, a coherent layer of aluminium oxide is formed; hereby the two

reactants, aluminium and oxygen, become separated from each other by a reaction product. In spite of their

separation, the reaction advances, so one of the reactants or both diffuse across the oxide layer. Assuming

that diffusion is the only rate determining process, the concentrations or the activities of the reactants at theboundaries of the oxide layer are independent of time, thermodynamic equilibrium is established at inter-

faces and mass flux is normal to the interfaces, a parabolic law can describe the growth kinetic of the scale,

so growth rate decreases with increasing oxidation time [21]:

Dx ¼ ðkptÞ1=2: ð1Þ

In the above equation, kp is the parabolic rate constant given by:

kp ¼Z po

O2

piO2

DO þ zAl

zOj jDAl

� �d ln pO2

; ð2Þ


where poO2is the oxygen partial pressure at the gaseous atmosphere/TGO interface; piO2

is the oxygen partial

pressure at the BC/TGO interface or better the equilibrium pressure of alumina decomposition reaction

into its constituents, evaluated by thermodynamic equilibrium condition; zAl and zO are aluminium and

oxygen ions valence; DO and DAl are oxygen and aluminium self-diffusion coefficient within the TGO.

In our model, we used the self-diffusion coefficient of aluminium in polycrystalline aluminium oxidedetermined by Paladino and Kingery [22] and represented by the equation:

DAl ¼ D0Al exp � QRT

� �; ð3Þ

where D0Al = 28 (cm2/s) is aluminium diffusivity in alumina and Q = (477,318 ± 62,805) (J/mol) is the acti-

vation energy for aluminium diffusion in alumina.

It is believed that the results obtained are representative of bulk diffusion in Al2O3. Cation diffusion in

ionic solid has not been found to be strongly dependent upon the presence of grain boundaries [23,24].

The presence of grain boundaries does, however, enhance anion diffusion in aluminium oxide [25]. Tak-

ing into account grain boundary and bulk diffusion of oxygen (respectively, Dgb and Db), an effective dif-

fusion coefficient can be defined as:

Deff ¼ ð1� f ÞDb þ fDgb; ð4Þ

where f is the fraction of sites associates with grain boundaries diffusion given by:

f ¼ 3dU

: ð5Þ

d is the grain boundary width (1 nm) and U is the average grain size (1 lm).

From diffusion data reported in [25], a law for effective self diffusion coefficient of oxygen in alumina as a

function of temperature has been extrapolated:

DO ¼ D0O exp � QRT

� �; ð6Þ

where D0O = 2.188 (cm2/s) is oxygen diffusivity in alumina and Q = 426,000 (J/mol) is the activation energy

for oxygen diffusion in alumina.

The values of oxygen and aluminium self-diffusion coefficient, estimated by Eqs. (6) and (3) at the appro-

priate temperature and introduced in Eq. (2), allow to obtain the thickness of TGO grown after a time t of

isothermal exposure through the Eq. (1).

2.1.2. BC oxidation model: cyclic exposure

For isothermal exposures, degradation from oxidation is defined on the basis of scale growth model

alone. However, during cyclic exposures, the protective scale efficiency may be compromised because some

portion of the oxide already formed is interested by damaging phenomena. These are due to extremely high

stresses experienced by the TGO for thermal expansion misfit [26–28]. The primary effect is that a certain

amount of the scale is cracked and growth of the oxide occurs at rates higher than those for the undamaged

scale. Actually, the growth process is not completely diffusion controlled, but some portion of the metallicsubstrate is directly exposed to oxidant atmosphere by the cracks network. The damaged amount of TGO is

therefore just as important as growth rate in determining cyclic oxidation kinetic. Indeed, taking into ac-

count only diffusion phenomena in describing growth of the scale during thermal cycling induce to under-

estimate the scale thickness; on the other hand, assuming that after each cycle the scale is entirely damaged

and completely inefficient as protective barrier induce to overestimate the TGO thickness. So, it is necessary

to find an acceptable compromise between these two extreme conditions.


At a rough estimate, we assume that a fraction x of the TGO actual thickness increase in a specimen

subjected to a number n of thermal cycles is due to damaging phenomena and the remaining fraction

1 � x is due to diffusion phenomena:

Ds ¼ Dsdð1� xÞ þ xDsc; ð7Þ
where Dsd is the thickness increase resulting from diffusion phenomena and Dsc is that due only to cracking
phenomena, computed multiplying the TGO thickness grown for diffusion phenomena in the first thermal

cycle by the number of thermal cycles. We believe that the amount x of damaged protective barrier is not

constant during the life of the component, but it linearly increases from a minimum value 0 at the beginning

of thermal treatment up to a maximum value xmax when specimen failure occurs.

As it will be shown in next section, this simple approximation leads to a good agreement between exper-

imental and numerical results.

2.1.3. Experimental tests for checking theoretical oxidation models

To check the adequacy of the theoretical models illustrated above, we performed some experiments of

isothermal ageing and thermal cycling of specimens shaped as a plate and consisting of a superalloy sub-

strate, an MeCrAlY type BC and a PS deposited ceramic barrier, both covering only one face of the spec-

imens. All samples were sectioned at different phase of thermal treatment, both isothermal and cyclic. The

polished cross-sections were examined using scanning electron microscopy to measure TGO thickness as a

function of exposure time.The isothermal test was done holding the sample in air at a temperature of 1070 �C until its failure

occurs. During the test, the TGO thickens up to a value around 6 lm at final failure, when the ceramic

Fig. 1. Micrography of a section of the specimen after 95 h of isothermal exposure at 1070 �C.

Fig. 2. Micrography of a section of the specimen at failure, after 585 h of isothermal exposure at 1070 �C.


have detached from the substrate. In Figs. 1 and 2 are reported two micrography representing, at the

same magnification, two specimen sections, the first taken at the beginning of thermal treatment and

the second at failure, after 585 h of thermal exposure. The increase of TGO thickness as a function of

exposure time is quite evident.

At failure, TC detached from the specimen primarily at TC/TGO interface, although the fracture surface

have crossed in some zone the TGO scale. As will be shown by stress distribution presented in a next section,

the maximum stress in TBC system is in the oxide layer, so the site of failure is located near the TGO.

The values of measured and calculated TGO thickness are reported in the Fig. 3: match between the twocurves is quite good.

The cyclic test was performed in air using an apparatus specifically designed and assembled. Each cycle,

between 20 and 1070 �C, lasted 40 min and the sample remained at maximum temperature 12 min. The

growth of TGO in thermal cycling is more rapid than in the isothermal treatment at the same maximum

temperature because thermal shock due to frequent heating and cooling damage the protective barrier,

as already explained. In Fig. 4 is shown a section of the specimen cycled: the oxide scale is not continuous

as in the case of isothermal exposure, but its integrity seems compromised by some cracked and spalled

zone.The TGO thickness have been evaluated by numerical methods assuming that the fraction of damaged

protective barrier at system failure was xmax = 0.12. This value is consistent with SEM analysis. Even in the

case of thermal cycling there is a good match between numerical and experimental results, as represented in

Fig. 5.

Fig. 4. Micrography of a section of the specimen subjected to thermal cycling: damage of TGO is evident.

Fig. 3. Comparison between calculated and experimental value of TGO thickness for isothermal treatment at 1070 �C.


2.2. Stress computation

2.2.1. Analytical model

For first roughly results a simple analytical model [29] can be used to compute the stress from the ther-

mal expansion misfit upon cooling. In this simple model, depicted in Fig. 6, the TC effect is neglected,

Fig. 5. Comparison between calculated and experimental value of TGO thickness for thermal cycling.

Fig. 6. Schematic illustration of the main steps that lead to the field stress due to the oxide formation: (a) heating of the BC specimen

with no oxide formed; (b) oxide growing; (c) cooling to ambient temperature with the BC and TGO layers free to shrink; (d) stress

arising when compatibility requirement is imposed.



because of its lower stiffness compared with BC and TGO. The planar dimension are considered to be much

larger than the thickness, so that the edge effects are negligible. We assume that there is no stress in the

normal direction, and that, because of geometrical symmetry, stress are equal and uniform in the lateral

direction and in plane strain condition. Young�s modulus and Poisson ratio are temperature independent;

the BC mechanical behaviour is elastic–perfectly plastic, with no creep features, and temperature dependentyield stress.

At the beginning, step (a) in Fig. 6, BC, the only constituent present, is heated to high temperature,

namely the peak cycle temperature. After exposure to high temperature oxidation starts, step (b) in

Fig. 6, and a thin oxide film, the TGO, begins to growth. In the subsequent cooling to the minimum tem-

perature, namely the room temperature, step (c) in Fig. 6, each layer, if left free to shrink, undergoes a dif-

ferent thermal strain, due to the dissimilar thermal expansion coefficient

ethTGO ¼ aTGO � DT ;ethBC ¼ aBC � DT :

ð8Þ

Here and below, the subscript ‘‘TGO’’ and ‘‘BC’’ refer to oxide and metal layer, respectively, a being the

thermal expansion coefficient and DT = T � Tref being the temperature differential between current and ref-

erence temperature. Since the oxide and the metal mutually constrain to each other elastic strains must ariseto match the strain compatibility requirement, step (d) in Fig. 6. Hence, the following strain compatibility

equation hold

eelBC � eelTGO ¼ �ðethBC � ethTGOÞ: ð9Þ

The elastic strains are

eelTGO ¼ 1� mTGO

ETGO

rTGO;

eelBC ¼ 1� mBCEBC

rBC;

ð10Þ

where r is the biaxial stress, E the Young�s modulus and m the Poisson ratio. Substituting the material

constitutive behaviour, Eq. (10), and the thermal strain, Eq. (8), the compatibility Eq. (9) is expressed

as

1� mBCEBC

rBC � 1� mTGO

ETGO

rTGO ¼ �ðaBC � aTGOÞ � DT : ð11Þ

A simple force balance equation gives the following relation between stress in oxide and metal

rTGO � hTGO þ rBC � hBC ¼ 0: ð12Þ

Substituting Eq. (12) in Eq. (11) and rearranging the terms the misfit stress rTGO, in a planar thin film, is

given as

rTGO ¼ Da � DT1�mTGO

ETGOþ 1�mBC

EBC

hTGO

hBC

h i ð13Þ

with Da = (aBC � aTGO). For very thin oxide film, namely for hTGO/hBC � 1, Eq. (13) may be written in the

short form

rTGO ¼ ETGO

1� mTGO

Da � DT : ð14Þ


The thermal misfit stresses, both in BC and TGO, after cooling to ambient temperature T = 20 �C and its

variation with the thickness ratio hTGO/hBC are plotted in Fig. 7. A reference temperature Tref = 1070 �C has

been considered. The TGO undergoes high compression stress, up to 4 GPa, especially for thin thickness,

when oxidation starts. A compressed thin layer motivates instability and cracking of TGO at the beginning

of the oxidation process [30–32]. Yielding of BC, rYBC ¼ 400 MPa, mitigates the TGO stress, but only for

high thickness ratio. For the current material properties and with a difference in temperature DT = 1020 �C,the thickness ratio that allows BC to yield is about 0.12. The residual stress on re-heating is depicted in Fig.

8. The TGO layer undergoes a tension residual stress heavily dependent on BC yield stress: the worst case is

for high BC yield stress. Acceptable results should be those related the yield stress rYBC ¼ 100 MPa, as at

high temperature this value seems be more realistic for BC.

Fig. 8. Stress in BC and TGO for different thickness ratio hTGO/hBC: oxidation temperature.

Fig. 7. Stress in BC and TGO for different thickness ratio hTGO/hBC: room temperature.


2.2.2. Finite element model

In order to relax restrictions due to simplified geometry and to simulate more complex material proper-

ties a finite element software has been used. The finite element model of a TBC strip, of size denoted w, is

depicted in Fig. 9: BC, TGO and TC layers are modelled. The upper TC face is left stress and constrain free.

The bottom BC face is constrained only in the y direction. To simulate generalised plane strain the nodes onthe right face of the strip belong to a rigid surface, free to move along x direction. For the same reason,

generalised plane strain on z direction, axi-symmetric elements, eight nodes quadratic, are used. The axis

of symmetry is on the left face. The TGO undulation has been assigned a representative S-shaped and a

typical thickness: the model is parametric and a wide range of undulation amplitude, to be denoted A,

and TGO size, hTGO, can be performed. Young�s modulus E, Poisson ratio m and thermal expansion coef-

ficient a for all the three material are temperature independent. Behaviour of TGO and TC is considered

elastic while BC is elastic-perfectly plastic with temperature dependent yield strength rYBC. Material prop-

erties, taken from [33] are listed in Table 1. The appropriate dimensions for amplitude undulation of oxi-dised TGO can be estimated from SEM micrographies such as those given in Figs. 1, 2 and 5. Here, the

amplitude undulation A varies from 10 up to 30 lm, the thickness hTGO up to 6 lm, and the semi period,

coincident with the strip size w, is about 26 lm.

Fig. 9. Finite element model of the TBC system.

Table 1

Summary of material properties

Young�s modulus (MPa) Poisson ratio Thermal expansion coefficient (1/K) Yield stress (MPa)

20 �C 1070 �C

BC 200,000 0.3 15E � 6 400 100

TGO 400,000 0.3 8E � 6 – –

TC 20,000 0.3 11E � 6 – –


2.2.3. Analysis

FE analysis are carried out simulating the system at different step of TGO growth without taking into

account a continuous TGO growing. Simulation starts with the TBC system temperature at the peak cycle

temperature, TMAX = 1070 �C. At this temperature, when oxidation began, the TBC is considered stress

free: specifically the reference temperature TREF is considered equal to the maximum temperature. The

TBC system is then cooled to minimum temperature, Tmin = 20 �C, and then again re-heated to peak cycle

temperature. TBC temperature has been assumed to be homogeneous and varying linearly during cooling

and subsequent re-heating. Stress distribution for small undulation, A = 10 lm, and thin TGO layer,hTGO = 0.5 lm, is depicted in Fig. 10. Upon cooling the BC do not yield, Fig. 10(a), but nevertheless the

TGO undergoes tension stress of value close to its tensile strength, Fig. 10(b), especially near to the upper

undulation peak. As the undulation amplitude increase, A = 30 lm, a minor BC zone starts to yield, Fig.

11(a). Yielding starts near to the lower undulation peak. The TGO undergoes tensile stress greater than its

tensile strength, Fig. 11(b). Residual stress after re-heating of TBC system to the peak cycle temperature is

negligible, Fig. 12. Figs. 13 and 14 show stress distribution for small undulation, A = 10 lm, and thick

TGO layer, hTGO = 4.0 lm. Upon cooling almost all of the BC straight under the undulation is yielded,

Fig. 13(a). The TGO tensile stress increase reaching a maximum value, about 900 MPa, on the upper peakof the undulation facing the BC, Fig. 13(b). Subsequent re-heating to the peak cycle temperature leaves

Fig. 10. Undulation amplitude A = 10 lm – TGO thickness 0.5 lm – room temperature; (a) equivalent stress req (Von Mises) in BC

(MPa) and (b) maximum principal stress r1 in TGO (MPa).



Fig. 12. Undulation amplitude A = 30 lm – TGO thickness 0.5 lm – oxidation temperature; (a) equivalent stress req (Von Mises) in

BC (MPa) and (b) maximum principal stress r1 in TGO (MPa).


residual stress such to produce reverse yielding in BC, Fig. 14(a). All the TGO undergoes a tensile stress

but with maximum value, about 400 MPa, lower than the previous value obtained upon cooling. A simi-

lar stress distribution is obtained increasing the undulation amplitude, A = 30 lm, with the same TGO

thickness, Figs. 15 and 16. In this case the TGO maximum stress value after cooling is about 1580 MPa

(Fig. 15(b)).






Fig. 17 shows the normal stress on TGO–TC interface for constant TGO thickness, hTGO = 4.0 lm, and

different amplitude undulation. For undulation amplitude less than 10.0 lm the normal stress is almost

constant along the strip width, while for amplitude greater than 10.0 lm the normal stress reaches a peak.

The value of the peak do not vary significantly as the wave amplitude increase, while the peak position

moves from the centre of the strip to the undulation lower peak. In an analogous manner Fig. 18 shows

the normal stress on TGO–TC interface with constant undulation amplitude, A = 22 lm, and variable

TGO thickness. For TGO thickness lower than 0.7 lm no normal stress is present. For thickness greater

than 0.7 lm the normal stress shows a peak whose value increase with increasing TGO thickness and whoseposition moves from the lower wave peak to the centre of the strips.

Fig. 15. Undulation amplitude A = 30 lm – TGO thickness 4.0 lm – Room temperature; (a) equivalent stress req (Von Mises) in BC





Fig. 17. Normal stress on TBC–TC interface (MPa).Undulation amplitude variable from 10 up to 30 lm – TGO thickness 4.0 lm.

Fig. 18. Normal stress on TBC–TC interface (MPa). TGO thickness variable from 0.7 up to 5.4 lm – Undulation amplitude 22.0 lm.


3. Conclusion

A model to describe TBC system behaviour and a first comparison to experimental results has been pre-

sented. The model contains the most important elements considered as important for TBC failure, in fact

stress distribution in the system as a result of TGO growth, thermal misfit and BC roughness has been

determined.

First, a theoretical oxidation model has been developed. It is based on Wagner�s theory and it predicts a

parabolic law for the growth of TGO. The model fits well experimental data in the case of isothermal expo-sure. A simple correction is necessary to explain the faster growth of TGO during thermal cycling. It takes

into account that BC oxidation process is accelerated by the formation of a certain number of cracks in

TGO, as a consequence of the high level of stress determined by subsequent heating and cooling.

A finite element model, with BC elastic–plastic behaviour, was used to compute the thermal misfit stress

in TBC system with TGO undulation. The undulation geometry and its characteristic dimension was


obtained from SEM analysis on a few TBC specimen. A single temperature cycle was simulated: the TBC

strip was cooled to room temperature and subsequently re-heated to the maximum peak temperature. The

combined effect of wave amplitude and TGO thickness was explored, and the stress distributions for a

meaningful range of these values evaluated. The primary finding was that even for smooth undulation

and thin TGO thickness the maximum principal stress in TGO is tensile and close or greater than theTGO tensile strength. Since the TGO is a brittle material such stress can break the TGO layer at the begin-

ning of the oxidation, when the TGO layer is very thin, as seen from experimental observation. Other find-

ings were:

1. Thicker TGO layer increase the BC stress, upon cooling to room temperature, until yielding. The sub-

sequent re-heating to maximum peak temperature results in BC residual stress; this residual stress can be

so high to produce reverse yielding in the BC. Cyclic plasticity of BC accumulates plastic strain and

cause fatigue crack to open.2. The normal stress on TGO–TC interface shown a maximum value near the lower peak of the TGO

undulation. The position of the maximum normal stress is consistent with experimental observation that

shown crack starting just near the lower peak of the TGO wave. Stresses normal to TC/TGO interface,

which are responsible for the TBC delamination, increase with increasing undulation amplitude, at least

in a wide range, and above all with increasing TGO thickness.

Even if containing some simplifying hypothesis, the model here presented is able to explain most of the

experimental results at least in a qualitative way.

Acknowledgements

This investigation was carried trough within the frame of the 5th EC Framework Programme: Compet-

itive and Sustainable Growth, project G4RD-CT-2001-00504 ‘‘New Increased Temperature Capability

Thermal Barrier Coatings’’. The authors are solely responsible for the content of this publication.

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Modelling of TBC system failure: Stress distribution as a function of TGO thickness and thermal expansion mismatch

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