Finite element analysis of stress distribution in thermal barrier coatings
Marcin Bia łas ⁎
当妈妈Institute of Fundamental Technological Rearch,Świ e ¸tokrzyska 21,00-049Warsaw,Poland
A B S T R A C T
A R T I C L E I N F O Article history:
快递单号自动识别Received 1April 2008
Accepted in revid form 27June 2008Available online 4July 2008Keywords:Creep
Burner rig test Plasma spraying Multilayer
Cohesive zone modelling
A numerical simulation of crack development within APS TBC systems is prented.The TGO thickening and
creep deformation of all system constituents is modelled.Two dimensional periodic unit cell is ud to examine the effect of interfacial asperity on stress distribution and subquent delamination of APS TBC.A study of cyclic loading and of creep of the ba material on the stress distribution clo to the asperity at the TGO/BC interface is made,revealing a small in fluence in fluence of both on the stress state in the thermal barrier coating system subjected to temperature loading.Cohesive zone elements at the oxide/ceramic interface model the development of the interfacial micro-crack.The finite element analysis shows that the development of the interfacial crack allows for a micro-crack formation within APS TBC.Subquent TGO growth results in a tensional zone within the oxide layer.Linking of the micro-cracks at the interface and within TBC through TGO could lead to a coating delamination in the unit cell.
©2008Elvier B.V.All rights rerved.
消防小报手抄报1.Introduction
During the last decade there has been an enormous effort to introduce single crystal and thermal barrier coating (TBC)technolo-gies for the manufacture of high temperature components for advanced power generation gas turbines.One of the driving forces for this development is the desire
to increa fuel gas temperatures,resulting in an improved thermal ef ficiency and thus making a primary contribution to the conrvation of energy resources and to the limitation of CO 2and other greenhou gas emissions.Mostly for this reason,the u of thermal barrier coated Ni-bad super-alloys for the thermally high loaded components will help to improve the gas turbine ef ficiency [1].
The potential of new generation single crystal super-alloys ud for blade production is basically given by their superior creep and fatigue resistance up to 1000°C [2,3].The increa of fuel gas temperature in recent years has led to temperatures at the material surface reaching the values up to 1250°C and a further temperature increa is envisaged.The thermal loading conditions can only be handled by a combination of modern cooling methods and protective coatings on top of the blades.In film cooling,the cooling air bled from the compressor is discharged through holes in the turbine blade wall or the end wall.The coolant injected from holes forms a thin thermal insulation layer on the blade surface to protect the blade from being overheated by the hot gas flow from the combustor.Typically,the
holes are in diameter not bigger than 0.5mm and are either normal to the surface or inclined at an angle of 15–30°[4–6].
Another technology allowing for an increa in turbines ef ficiency is high temperature protective coatings.Current protective coatings are two-layer systems,with a metallic,corrosion protective bond coat (MCrAlY or PtAl)on the super-alloy and a ceramic thermal barrier coating (mostly Yttria stabilid ZrO 2)on top and in contact with the hot gas [7].The thermal barrier coating provides a temperature drop of up to 200°C due to its low thermal conductivity,which is enhanced further by the intentionally porous micro-structure.Fig.1demonstrates the reduction of temperature achieved by thermal isolation through TBC and inner cooling.The potential for a temperature reduction by TBC application,however,has not been fully exploited so far becau in the ca of failure the internal and external cooling air must be suf ficient to keep the temperature in the structural material below the point at which failure occurs.To u the high potential of TBCs,the different aspects of exposure conditions and failure mechanisms must be understood and integrated into degrada-tion modelling and life prediction.
A major weakness of TBC systems is the interface between the metallic bond coat and the ceramic TBC.At this interface an in-rvice degradation is obrved often leading to a macroscopic spallation of the ceramic layer [8].The interface regions undergo high stress due to the mismatch of thermal expansion between BC and TBC.Additionally,growth stress due to the development of thermally gr
own oxide (TGO)at the interface and stress due to interface rough-ness are superimpod.Stress relaxation leads generally to reduced stress levels at high temperature,but can give ri to enhanced stress accumulation after thermal cycling resulting in early crack initiation at the bond coat/alumina interface and spallation failure after-wards [1,9].One of the key issues of the prent paper is to investigate
Surface &Coatings Technology 202(2008)6002–6010
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0257-8972/$–e front matter ©2008Elvier B.V.All rights rerved.doi:
10.1016/j.surfcoat.2008.06.178
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j ou r n a l h o m e pa g e :ww w.e l s ev i e r.c o m/l o c a t e /s u r fc o a t
the in fluence of substrate creep upon stress distribution at the TBC/TGO interface and to examine whether only elastic substrate behaviour can be assumed during thermal loading.Moreover,the paper examines the effect of thermal cyclic loading and crack evo-lution within the TGO/BC interface.
Of particular interest in the prent contribution is the experi-mental work on TBCs in which CMSX-4super-alloy hollow cylindrical specimens were tested [7].The specimens were plasma sprayed with yttria stabilid zirconia (APS TBC)on NiCoCrAlY bond coat.Experimentally determined data for APS
TBC,TGO and BC creep [7,10,11]rve here as an input for material modelling of the constituents.The time-dependent model of CMSX-4prented in [2,3]has been ud.The oxidation process has been simulated by growth of thickness of TGO elements.The development of cracks at the TGO-BC interface has been simulated using cohesive zone elements.The results are next discusd in relation to the failure mechanism prented by Chang et al.and Freborg et al.[12,13].2.The simulation scheme
There have been many attempts to numerically investigate stress development in TBC systems [9,12–24].In most cas a 2D-unit cell reprenting a single asperity was ud.The stress field around the asperity was suppod to be reprentative for the entire interface area.The resulting micro-cracks could subquently link and thus
form a crack crossing a number of asperities at a macro level.Using a unit cell approach this paper follows the multi-scale modelling concept.
2.1.The numerical model
Fig.2illustrates the modelling concept.The test cylinder is suf ficiently long compared to its diameter for the problem to be approximated by a two-dimensional plane-strain ca.It has been assumed tha
t the strain in the axial direction is uniform.As propod in Fig.2the bond coat topography is idealid by circular gments within a unit cell.Symmetry of the problem allows meshing only one half of the undulation.The assumed boundary conditions put constraints on the displacement field in the cylindrical direction,only radial displacements are allowed to take place along lines OA and OB in Fig.2as schematically indicated on the finite element mesh in Fig.3.The cylinder was subjected to cyclic temperature loading with homogeneous temperature distribution.A single loading cycle is prented in Fig.4.商鞅的妻子是谁
The bond coat,the thermally grown oxide and the thermal barrier coating are treated as elastic and viscous materials.Their mechanical properties are functions of temperature as listed in Table 1(e [2]for the discussion of anisotropy of mechanical properties of CMSX-4).The value of temperature loading equal to 200°C was chon for the
野兽之灵
initial
Fig.1.Qualitative temperature distribution across the TBC
system.
Fig.2.The model of the cylindrical test specimen.The proportions of the layers in the figure do not remble their actual
理科分数线dimensions.
Fig.3.Local coordinate system for the oxidation modelling.Boundary conditions on the edges of the unit
cell.
Fig.4.Temperature pro file during a single loading cycle.
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M.Bia łas /Surface &Coatings Technology 202(2008)6002–6010
stress free state.It matches approximately with the coating stress free temperature for the air plasma spraying process[9,25].
The creep data for TBC and BC were experimentally determined by compression creep tests with stand-alone coatings(TBC,BC)and shear deformation experiments on TBC composites[7,10,11,25].The data for one dimensional ca were approximated using the following equation
:εTBC;BC creep ¼A′σn′e−ε
TBC;BC
creep
ε′þA″σn″e−
εTBC;BC
creep
ε″þAσnð1Þ
where A′,n′,ε′,A″,n″,ε″,A,n are temperature dependent material parameters.They are listed in Table2.The TGO time-dependent behaviour is reprented by the following formula
:
εTGO
creep
¼Aσn:ð2ÞThe TGO creep starts at750°C.The values of material parameters A and n in this ca are temperature independent and equal7.3e−4and 1,respectively[25].The one dimensional Eqs.(1)and(2)are generalid to a three dimensional ca by assuming incompressibility of the creep strain and the proportionality between the deviatoric stress and the deviator of creep strain rate.
Growing of the alumina scale at high temperature at the BC/TBC interface is simulated using the swelling option in ABAQUS,e for example[9,17].The oxidation process is considered in the temperature range950°C–1050°C.It is modelled as an orthotropic swelling strain of the TGO layer.Its initial thickness was assumed to be equal to 0.5µm.The relation between the thickness of oxide layer and time was experimentally determined for three different temperatures[26] and approximated by the following equation
d tðÞ¼A0e−
E a
RT t
n
ð3Þ
with T being temperature and t time.The values of coefficients A0,n, E a and R are listed in Table3.The one dimensional oxidation strain is defined as
εox¼ln d
d0
ð4Þ
where d0is the initial thickness of TGO layer,d0=0.5µm.Making u of Eq.(3)the one dimensional oxidation strain rate takes now the form
:
εox¼nA
0e−E a RT
ffiffiffiffiffi
d0
n
p
eεox n
;εox0ðÞ¼0:ð5Þ
The oxidation strain rate in tensor notation is provided by the formula
:εoxidation
TGO
¼1
3
:εoxð6Þwhereεox is given by Eq.(5)and R takes the form
¼x e x e xþy e y e yþz e z e z:ð7ÞVectors e i,i=x,y,z define a local coordinate system with axes x and y respectively perpendicular and parallel to the initial TGO layer,e Fig.3.Thus,Eq.(6)makes a generalisation of the one dimensional oxidation law(5)to a three dimensional ca.By tting R x=3,R y= R z=0we describe solely the thickness growth of the TGO layer (direction x in Fig.3).
The preliminary calculations have been made to check the effect of lateral oxidation(R y≠0).They showed that with the TGO layer relaxing at high temperature the deformation arising due to lateral oxidation has only a minor influence on stress distribution.The TGO creep(Eq.(2))is implemented in the prent FEM model and the assumption R x=3,R y=R z=0is ud.
2.2.The time-dependent behaviour of CMSX-4
The ba material CMSX-4was modelled as an elastic body with plastic and creep effects taking place in the temperature higher than 800°C.Micro-structural dependent constitutive equations[2,3]have been adopted.They aim at the evolution ofγ/γ′particles in high temperatures using the orthotropic Hill's potential,who anisotropy coefficients are related to the edge length ofγ′particles.The model is described in detail in[2].Here only its main features are shortly summarid.
The inelastic strain rate can be expresd as
:εin
ij
¼3
2
K h001iσeff V
N−1
A ijkl s eff
kl
ð8Þ
with K〈001〉and N being material parameters.The anisotropy tensor A ijkl depends upon the evolution of dimensions ofγ′particles.The s kl eff is a deviator of the effective stress.The effective stressσkl eff has been introduced in order to estimate the micro-structural impact on the
Table1
The temperature dependance of Young modulus E〈100〉in direction〈100〉,Poisson ratio νand thermal expansion coefficientαfor CMSX-4,TGO,BC and TBC(after[25])
T[°C]E〈100〉[MPa]να[1/°C]E[MPa]να[1/°C] CMSX-4TGO
201232860.359632 1.20023e−053803650.27 5.07939e−06 2201155460.365476 1.2105e−053690600.27 5.90395e−06 4201078060.370447 1.25212e−053612250.27 6.72851e−06 6201000660.374547 1.34314e−053518760.277.55307e−06 820923260.377774 1.46843e−053360320.278.37763e−06 1020845860.380129 1.57968e−053087080.279.2022e−06
BC TBC
201518570.31886 1.23579e−05175000.29.68e−06 2201507460.32701 1.30407e−0516340.90.29.67479e−06 4201452530.33434 1.39115e−0515181.80.29.70578e−06 6201323370.34086 1.49703e−0514022.70.29.80975e−06 8201089210.34656 1.62171e−0512863.60.2 1.00235e−05 1020718900.35145 1.76519e−0511704.50.2 1.03837e−05
Table2
Creep data for BC and TBC(after[25])
T[°C]A n A′n′ε′A″n″ε″BC
750 1.25e−14 4.5 1.25e−9 4.50.09 1.25e−12 4.50.24 850 1.4e−11 3.8 1.4e−6 3.80.08 1.4e−9 3.80.23 950 2.3e−9 3.1 2.3e−4 3.10.07 2.3e−7 3.10.22 10509.5e−8 2.559.5e−3 2.550.069.5e−6 2.550.21
TBC
750 2.e−22 4.5 1.e−10 4.50.01 2.2e−18 4.50.05 850 2.e−20 4.32 1.e−10 4.320.02 2.e−16 4.
320.08 950 3.e−18 4.15 3.e−10 4.150.039.e−15 4.150.12 1050 3.77e−16 3.98 3.77e−11 3.980.04 3.02e−13 3.980.18 1150 4.8e−14 3.8 1.44e−10 3.80.05 4.8e−12 3.80.25Table3
Parameters for oxidation kinetics(after[25])
作业的重要性
T[°C]A0n E a R 950 1.57e−150.331800008.314 1000 1.57e−150.3251800008.314 1050 1.57e−150.32251800008.314 R
R
R R
R
6004M.Białas/Surface&Coatings Technology202(2008)6002–6010
global deformation.It is the difference between the applied stress and an internal stress
σeff kl ¼σij −σi
ij :
ð9Þ
The internal stress σij i describes the resistance of the micro-structure against the deformation due to the applied stress.It is compod as follows
σi ij ¼σb ij þσp
大漠飞沙
ij
ð10Þ
where the “back stress ”σij b reprents the dislocation movement
within the γ′channels and the “friction stress ”σij p
is caud by the dislocation –particle interaction.The formation of both is given in detail in [2].
2.3.Modelling of crack development
To simulate the damage initiation and propagation cohesive zone elements are located at the TGO/BC interface.The mixed-mode decohesion model of Camanho and Dávila [27]implemented in A
baqus [28]is ud.The following quadratic nominal stress criterion to model damage initiation is applied h σn i n 2þτs
s
2¼1ð11Þ
where σn 0and τs 0
are interface critical stress respectively in opening or sliding modes.The linear damage evolution as prented in Fig.5is ud to model the descending branch of the traction –paration relation.The dependence of the fracture energy on the mode mix has been de fined by the following criterion
G n n
þG s
s ¼1:ð12Þ
In the expression above G n c and G s c
are the critical energy relea rates respectively in mode one and two.It has been assumed that cohesive elements do not undergo damage under pure compression.As reported in the literature [10,20,21]the following material data were ud for BC/
TGO interface:G c n =20J/m 2,σ0n =200MPa,G c s =60J/m 2,τ0
s =100MPa.
In reality the cracking in TBC systems is not con fined only to the TGO/BC interface.It spreads into TGO and TBC and finally leads to coating spallation.With the prence of a temperature gradient throughout the protective layer also TBC chipping can be obrved,e [29].Therefore,the prent analysis should be understood only as a preliminary step of the proper modelling of TBC cracking.Though,it agrees well with the general scenario of TBC delamination process as will be demonstrated in the following ctions.
3.Results and discussion
In the prent ction the results of finite element calculations will be prented.Subctions 3.1and 3.2consider stress calculations
without system cracking,whereas Subction 3.3describes crack propagation at the TGO/BC interface.
3.1.The in fluence of the time-dependent behaviour of CMSX-4
In the prent ction a comparison of the results obtained with and without the creep of the ba material CMSX-4will be prented.No crack development at the TGO/BC interface is considered at this stage.Fig.6(bond coat peak)and Fig.7(BC/CMSX-4interface)prent the development of stress σ22in points A and B at the end of the cooling pha (20°C)throughout 100cycles.It can be en that at point A the stress obtained for the ca of CMSX-4creep are slightly bigger than tho obtained without CMSX-4creep and the difference increas with the number of cycles.However,for practical interest it does not em to be relevant.Only in the region clo to the substrate the in fluence of CMSX-4creep can be obrved —the rheological behaviour of CMSX-4leads to 30%increa in the σ22stress after 100cycles.The stress remain below 10MPa and therefore have no relevance for deformation and degradation of the TBC system.It all leads to the conclusion that during thermal loading the creep of CMSX-4can be easily neglected,since it does not play almost any role on the stress redistribution around the asperity.Probably combined mechanical and thermal loading will lead to a more pronounced effect of substrate creep than it is obrved in the prent ca.3.2.The effect
of cyclic loading
Karlsson et al.[22]mention five conditions which together are prerequisites for the delamination:(a)the temperature cyclic loading,(b)thermal expansion mis fit,(c)bond coat and TGO yielding,(d)
bond
Fig.5.Pure mode constitutive
equations.
Fig.6.Stress σ22at bond coat/TGO interface (point A)after cooling down to 20°C
versus number of
cycles.
Fig.7.Stress σ22at the bond coat/substrate interface (point B)after cooling down to 20°C versus number of cycles.
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M.Bia łas /Surface &Coatings Technology 202(2008)6002–6010
coat oxidation,and (e)geometrical imperfections at the interface.They examined the effect of cyclic loading upon stress in the TGO for EB-PVD TBCs with and without plastic yielding of the bond coat and TGO and with other constituents being elastic.They showed that the thermal cycling caus substantially larger radial enlargement of the TBC-TGO-BC system when the TGO is allowed to yield at the peak temperature.
The effect of cyclic loading on the stress distribution will be now addresd for the prent ca of APS TBC with all constituents modelled as prented in Section 2.Only CMSX-4will be treated as an elastic body due to the negligible impact of its creep upon stress distribution around the asperity,the other constituents are considered as elastic with creep.No crack development at the TGO/BC interface is considered at this stage.
Two thermal loading histories will be ud to conduct the calculations:(i)a cycle scenario with a single sub-cycle prented in Fig.4—50multicycles will be considered;(ii)a single thermal cycle with the dwell time at peak temperature being the total for 50cycles of ca (i)(100h).Cooling and heating
rates are the same for both cas:the temperature difference between ambient (20°C)and peak (1050°C)is achieved during 103s.The two loading histories are contrasted in Fig.8.
Fig.9prents σ22stress after cooling down to room tempera-ture for the two loading histories.It can be en that for the cas considered monotonic and cyclic loadings yield almost the same result:the distributions of σ22stress are almost identical,the differences of minimal and maximal values of σ22stress are not higher than 2.5%.Thus,it is en that for the prent analysis cyclic loading has no real effect on stress distribution.This is easily understood for the ca without crack development where only viscous materials were assumed not giving ri to any cyclic loading related effects.
Fig.10prents the tensile σ22stress within TBC at the beginning and at the end of 100h hold up at peak temperature.This result is equivalent to 50cycles of the shape prented in Fig.4.Compressive zones are indicated in grey,since they are irrelevant for crack evo-lution.In fact,we e that the oxidation process results in the creation of a tensile zone in the middle of the asperity —point B in Fig.10(B).The stress σ22in TBC increa as the TGO thickens during the dwell time and after 100h reach about 35MPa.The location of the maximal σ22stress at the beginning of the oxidation process is point A in Fig.10(A).However,the value of the σ22at this point d
ecreas as the TGO thickens and the value of the σ22stress in point B in Fig.10(B)increas.The change of the maximal σ22stress with time is prented in Fig.11.After 2hours dwell time (one typical loading cycle)it reaches 14MPa.Values of the critical normal stress for TBC reported in
the
Fig.8.A single loading cycle with the dwell time of m multiple loading
cycles.
Fig.9.The stress σ22at ambient temperature for a single cycle and a multi-cycle ca after 100h dwell time at peak temperature.
6006M.Bia łas /Surface &Coatings Technology 202(2008)6002–6010
literature vary from10to100MPa[7,30].It is assumed here that there is no crack formation within TBC during thefirst hold-up pha.
The location of maximalσ22stress within TGO has also been indicated in Fig.10(B).After100h of dwell time it reaches about 110MPa being much below the critical stress for TGO equal1200MPa [31].
Fig.6prents the values ofσ22stress at the peak of the TGO/BC interface at room temperature after different number of cycles.It can be obrved,that already after2hours dwell time(one loading cycle) big tensional stress appear at the interface reaching about300MPa. With the interfacial strength being50–80MPa[32]we conclude that thefirst micro-crack would appear at the peak of the TGO/BC interface already during thefirst cooling down to ambient.The development of this micro-crack and its influence on the stress redistribution within the system will be analyd in the next ction.
3.3.Crack development at the TGO/BC interface
Fig.12shows the stressσ22at the end of thefirst cycle(2h of dwell time)with and without modelling of the TGO/BC interface crack. We can e that the crack development has a considerable influence on the stress distribution around the asperity.The development of the interfacial crack allows for a creation of a tensile zone within the TBC (region A in Fig.12)with the maximalσ22stress reaching87MPa already after2h of dwell time.Due to the relaxation taking place at high temperature,heating and hold-up phas produce stress much smaller than tho obrved during the cooling down.Thus,when considering crack development at the interface it is the cooling pha being the most crucial for a further development of damage within the system.
Fig.13shows the influence of cyclic loading upon the stress redistribution around the asperity when the development of the interfacial crack is modelled.It can be en that even now both monotonic and cyclic loadings(100hours dwell time obtained for two loading histories prented in Fig.8)result in a similar stress redistribution after cooling down to ambient temperature.Fig.14 shows the length of the interfacial crack versus time.It is en that the crack forms already during thefirst cooling down to ambient and its length stabilis already after3cycles.For the single cycle ca the duration of the high temperature exposure has no effect upon length of the crack after cooling down:it reaches the s
ame length after4,20, 40and100h of exposure at peak temperature.The cyclic loading manifests itlf only by a longer crack during the stabilisation pha when compared with that after a single cycle loading.The differences in length are in fact small and one could say that cyclic loading has a negligible effect upon the length of the interfacial crack.Mróz and Białas[33]and Białas and Mróz[34]have showed that a cohesive interface subjected to a cyclic shear loading can accumulate damage regardless of loading direction(incremental failure).The cohesive model of Mróz and Białas[33]however does not assume unloading to the initial state as it is the ca in the model by Camanho and Dávila [27,28]ud in the prent analysis.The minor influence of cyclic thermal loading upon crack development at the TGO/BC interface should therefore be explained by a small amplitude of cyclic loading at the interface itlf,where upon unloading the normal and shear stress decrea without additional damage accumulation.In this context,it is only the dwell time alone that influences the stress distribution around the asperity and not the way it is realid in practice(cyclic or monotonic loading).
The stabilisation of crack length after a few cycles contradicts the experience that failure occurs after a sufficient number of cycles.This result just shows that all the crucial factors for coating delamination are not included yet in the prent FE model.One of them could be the cracking within TBC,that will be considered in the upcoming rearch.
There are two crucial aspects related to the stress redistribution within the system due to crack formation at the TGO/BC interface: (i)development of a tensile zone within TBC(region A indicated in Fig.12and in Fig.13)and(ii)development of a tensile zone within TGO (region B in Fig.13).The zones can be obrved only during the cooling pha and theσ22stress within them reach their biggest values at ambient temperature.The two aspects will be addresd next.
As already indicated in Fig.12the tensional zone around point A forms in the ca of interfacial cracking already during thefirst
cooling Fig.11.The maximal value of stressσ22within TBC during100hours dwell time at peak temperature.Points A and B are depicted in Fig.10
.
Fig.10.The stressσ22during100hours dwell time at peak temperature(equivalent to
50sub-cycles prented in Fig.4):(A)at the beginning of the dwell time;(B)after100h
at the end of the dwell time.Compressive zones in grey.
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M.Białas/Surface&Coatings Technology202(2008)6002–6010
