NUMERICAL MODELLING OF CREEP DEFORMATION IN A CMSX-4 SINGLE

CRYSTAL SUPERALLOY TURBINE BLADE

David Dye1, Anxin Ma1 and Roger C Reed2

1 Department of Materials, Imperial College, London, SW7 2AZ, UK2 Department of Metallurgy and Materials, The University of Birmingham

Edgbaston, Birmingham, B15 2TT, UK

Keywords: superalloy, single crystal, dislocation density, slip system, damage, activation energy, creep, tertiarycreep, primary creep

Abstract

The creep deformation behaviour of a high pressure(cooled) turbine blade section is modelled using the fi-nite element method. Realistic estimates of the thermaland mechanical loading expected during service are em-ployed; properties for the single crystal superalloy CMSX-4 are assumed. The constitutive model accounts for botha

2<110> tertiary creep in the γ matrix and a <211>

shearing of the γ′ phase. The results indicate that theload in the web decreases during service due to creepdeformation, because of the primary creep effect. It isdemonstrated that a variation in the blade orientation of20◦ away from [001] causes a variation in the creep strainat 20 × 106 s (about 5500 h) of about a factor of two.Off-axis orientations experience higher stresses in the weband correspondingly greater primary creep strains in thecooler web locations. Uncertainties in the locations of thecooling holes of up to 0.2 mm are predicted to increasethe creep strain by a factor of around 15%.

Introduction

Nickel-based superalloy turbine blades are amongst themost important of the components required for the mod-ern aeroengine, and indeed the performance characteris-tics (e.g. specific fuel consumption, thrust) depend criti-cally upon the severity of the environment which can beendured. Work continues to develop new grades of thesealloys with improved high temperature capabilities, e.g.in creep, fatigue and oxidation. Usually, these compo-nents are fabricated from the nickel-based superalloys insingle crystal form, so that a strong degree of anisotropicbehaviour is inherited from the solidification processingused for their fabrication. As a class of engineering ma-terial developed specifically for high temperature appli-

cations, the superalloys should be regarded as a majorsuccess story.

For practical application under service conditions, anaccurate analysis of the mechanical response to the an-ticipated loading is obviously of great importance, notleast because the results of this will influence the servicelife which can be declared. Existing analysis capabilitiesfor the deformation behaviour of turbine blading remainunsatisfactory in a number of important respects. First,there is usually a significant degree of empiricism invokedso that the underlying micromechanics of deformation -in particular the interdependence of the different defor-mation modes - are not properly respected. Second, theanisotropy of deformation is difficult to account for, sothat major assumptions have needed to be introduced,with a resultant loss of physical faithfulness to the under-lying metal physics. Third, a coupling of all the relevanteffects - the heat transfer characteristics, the non-lineartemperature-dependent deformation, the geometrical un-certainties introduced from the manufacturing steps - re-mains a significant numerical challenge. It follows thatfurther work is required before one can say that existinganalysis capability is optimised. One can argue that moreaccurate predictive analyses will become more importantas conditions in the turbine become more aggressive, sinceit will not be possible to rely upon safety factors as largeas those used at present.

The work reported in this paper was carried out withthese factors in mind. Recently, the authors have devel-oped a model for the creep deformation of single crystalsuperalloys, which accounts for the important microstruc-tural degradation mechanisms occurring on the scale ofthe microstructure, and their inter-relationship: the glideof a/2 < 011> {111} dislocations in the matrix phaseof these materials which causes tertiary creep, and theshearing of the strengthening gamma prime precipitate

911

by a< 112> {111} ribbons which is responsible for pri-mary creep. Here, the model is applied to analyse theperformance of a cooled turbine blade which is subjectto a loading history which is likely to be representativeof that experienced during the initial stages of the flightcycle.

Modelling Approach

The numerical analysis reported in this paper requires theelastic and plastic portions of the total deformation to beseparated; for this purpose, a multiplicative decomposi-tion is used. The elastic part comprises the stretch androtation of the lattice, and the plastic part correspondsto the unrecoverable deformation caused by dislocationactivity. Because the material consists of two phases, itis necessary to decide how to treat the stress and straindistributions across the interfaces separating them; hereit has been assumed that the external stress can be av-eraged across the unit cell consisting of the γ − γ′ mi-crostructure, following Svoboda. Two distinct deforma-tion mechanisms are accounted for: (i) dislocation glid-ing in the channels of the matrix, so-called tertiary creep,and (ii) shearing of the precipitates by dislocation rib-bons, referred to as primary creep. The two mechanismsoccur with different overall glide directions and thereforehave different Schmid matrices and different deformationanisotropy.

Constitutive Laws for the Matrix Phase, γ

The matrix material has an fcc crystal structure. If mo-bile and immobile dislocations are not distinguished, theOrowan equation can be used to calculate the shear rateγα for each slip system α according to

γfcc = ρfccbλα

fccFattack exp

(−

Q110slip

kBT

)

× exp

(|τ + τmis| − τpass − τoro

kBTVC

)(1)

where b is the magnitude of the Burgers vector, Fattack

the dislocation vibration frequency which must be lessthan the Debye frequency, Q is an activation energy whichis expected to be close to the activation energy for self-diffusion, kB is Boltzmann’s constant, T is the temper-ature and τ is the resolved shear stress on the system.τpass and τoro are resistance stresses coming from disloca-tion interactions and configuration curving, respectively,while τmis is the average misfit stress in the matrix phase.

Vc is an activation volume and λ the average jump dis-tance. A detailed description and derivation is given else-where [1]. The dislocation density evolves according to

ρfcc =cmult1bλα

fcc

γfcc − cannh1ρfccγfcc (2)

where cmult1 and cannh1 are fitting parameters for the dis-location multiplication and annihilation rates.

Constitutive Laws for the Precipitate, γ′

The precipitate material has an L12 ordered crystal struc-ture. It is assumed that a/2< 011> {111} dislocationscombine to form the a< 112> {111} ribbons observed toform during primary creep [2] and that these shear theγ′. Therefore the dislocation ribbons are not requiredto bend around inside the γ channel and therefore theOrowan stresses acting on them are small. Consequently,the following flow rule for the γ′ is employed:

γL12 = ρL12bλα

L12Fattack exp

(−

Q112slip

kBT

)

exp

(|τL12| − τpass

kBTVC

)(3)

Since the ribbon dislocations extend across both the chan-nel and precipitate, one part of each ribbon dislocationexperiences a forward misfit stress and the other a back-ward misfit stress. Therefore it is rather difficult to con-sider the effect of misfit for the ribbons. For this reason,the effect of misfit is omitted for the L12 superdisloca-tions.

The dislocation multiplication/annihilation term hasan additional component to describe the combination ofγ dislocations. Each possible set which might possiblycombine to form an a < 112> ribbon is considered inturn, hence

ρL12 = +cmult21 min(ρIfcc, ρIIfcc)Γ

+cmult22

bλγL12 − cannh1ρL12γL12 (4)

where Γ is the dislocation combination frequency and ρIFCCand ρIIFCC are the densities of each of two dislocation can-didates for combination. The term Γ is described as athermally activated process, according to

Γ = Fattack exp

(|τ | − τAPB − G |δmis| − τpass

kBTVC

)(5)

where G is the shear modulus and the resistance stressτAPB is defined via the anti-phase boundary energy γAPB

912

by τAPB = γAPB/b and the misfit parameter δ is defined asδmis = 2 (aγ′ − aγ) / (aγ′ + aγ).

For each possible dislocation combination, one has toaccount for the formation of the a<112> ribbon disloca-tions from distinct a/2<110> matrix dislocations. Forthis to occur, it is assumed that two pairs of dislocationspresent on a common slip plane combine according toequations such as

2 ×(a

2[110] +

a

2[101]

)= a[211] (6)

i.e.

2 × (bIFCC + bII

FCC) = bL12. (7)

Damage Caused by Vacancy Accumulation

Damage causing eventual creep fracture is attributed tothe condensation of voids from vacancies, leading to theformation of porosity and microcracks [3]. Here, it isassumed that the phenomenon of void condensation iscaused by dislocation evolution; each unit volume is con-sidered to represent of order (ρ/b)b3 = ρb2 vacancies,which are released by dislocation annihilation. Voids thenform which both (i) reduce the effective internal cross-section and (ii) increase the climb rate of γ/γ′ phaseboundary dislocations.

Based on this premise and assuming that a single va-cancy has a volume b3, the ratio of void volume to thetotal volume Rvoid is then taken to be proportional to theannihilation terms for the channel dislocation and ribbondislocations. Hence one defines

Rvoid = cvoid (fγ˙ρfcc + fγ′ ˙ρL12) (8)

where each f denotes the fraction of one of the two phasespresent and cvoid is a constant. The damage parameterD is then defined by

D = (Rvoid)2

3 (9)

Note that D acts to increase the effective stress by de-creasing the available section area in the normal way;hence the resolved shear stress τ is then replaced by aneffective shear stress τeff according to τeff = τ/(1 − D).Because the three matrix channels (two horizontal andone vertical) will have different stress levels, the vacancyconcentration will be different in the horizontal and ver-tical channels. Therefore the entire model in fact consid-ers each channel separately, Figure 1. In particular, theimbalance of vacancy concentration around any given γ′

precipitate will cause atoms to migrate around it.

channel

thickness h

precipitate size lX

precipitate

volume

fraction fg'

channel

volume

fraction fg/3

Z

Figure 1: Schematic illustration of the microstructuralunit of a single crystal superalloy.

Obtaining the Stress and Strain States for the Material

In principle, it is possible to identify a unit cell of materialcomposed of the precipitate and matrix channels, to meshthese in the finite element analysis as a submodel andthen obtain an exact solution for the stress and straineverywhere within the unit cell; however, this approachis computationally expensive and excessively complex toapply to the analysis of components. Therefore, it hasnot been used here. Instead, Svoboda’s scheme [4] hasbeen used to derive the overall flow behaviour from thatof each microstructural component, as follows. In Figure1 we identify a precipitate of size lx, ly, lz and channelsof thickness hx, hy, hz. The change in dimensions of theprecipitate (occurring by diffusional processes) must obeythe rule hx + lx = hy + ly = hz + lz = L0 and lxlylz = l30 .So the volume fractions f can be defined as a function of(lx, ly, lz) as follows

f pct = lxlylz/L3

0= (l0/L0)

3

f chnx = (L0 − lx)lylz/L3

0

f chny = lx(L0 − ly)lz/L3

0

f chnz = 1 − f chn

x − f chny − f pct (10)

For all single crystal superalloys, a non-zero misfit δwill cause a significant discontinuity in the elastic strainsacross the matrix/precipitate interface. However, plasticdeformation results in the establishment of misfit disloca-tions at the interface that can completely relax the misfitstrains / stresses. For the γ channel with normal along the1 direction (x), f chn

x , then the following tensor equationcan be used for the in-plane components ij = 22, 33, 23,

σchnx = Cchn(C

−1

pctTpct + δmisI−

∑

α

ρα

FCCb2Mα) (11)

where Cchn and Cpct are the single crystal stiffness tensors

for the channel and precipitate and Mα is the Schmid

913

P13

P10

P1

P2P3

P4

P5

P6

P7

P8

P9P11

P12

1101110010501000 950 900 850 800 750 700

Temp, Co

Figure 2: The input temperature field (top) and materialpoints identified in the turbine blade section.

matrix for slip system α in the starting configuration.For the other two channels, the i, j are permuted. For thestresses normal to the interface, i.e. the ij = 11, 12, 13components for the f chn

x channel, it is assumed that

σchnx = Tpct. (12)

Finally, a force balance can be used to find the local stresswithin the microstructural component, in terms of theapplied stress tensor

σext = f pctσpct + f chnx σchn

x + f chny σchn

y + f chnz σchn

z . (13)

Application to turbine blade creep

The model has been applied to analyse a cooled high pres-sure turbine blade, one of the most critical components inthe gas turbine. The primary loads on the turbine bladesection are centrifugal (along the long axis of the blade)but these are augmented by considerable thermal stressescaused by the high temperature gradients.

Finite Element Implementation

The creep model has been implemented as a user ma-terial subroutine within Abaqus [5]. A 2-element highmesh of 2434 eight-noded hexahedra (C3D8H) was gen-erated for the blade section studied by MacLachlan [6].During creep deformation, planes normal to the bladeaxis (denoted z) were kept coplanar, such that the topand bottom surfaces of the mesh remain symmetrical in-plane. Rigid body motion was excluded by constrainingtwo nodes on the bottom plane. The centrifugal loadswere added by applying a surface pressure to the top sur-face. Three time steps were included in the finite element(FE) analysis. First, heating from ambient to the operat-ing temperature field was applied, Figure 2, enabling thethermal stresses to be calculated. Second, the centrifu-gal load was added. Finally, in the third step creep de-formation was simulated under the influence of both thethermal and centrifugal loads. Figure 2 indicates thatthe temperatures range from ∼ 750 to 1100◦C and soboth primary creep by glide of L12 superdislocations andtertiary creep by fcc dislocation glide may both occur si-multaneously in different parts of the blade section. Notethat the aerodynamic loads have been ignored in the mod-elling.

In the present model a < 112 > {111} ribbon shearof the γ′ is only a significant contributor to creep atlow temperatures, because the threshold stress requiredis high and therefore at temperatures in the region of700 − 1100◦C a/2 < 110 > {111} creep of the γ matrixdominates. This is appealing because the temperaturecut-off of primary creep arises naturally from the model.

From e.g. Ref [7] the thermal expansion coefficients ofthe γ (denoted αchn) and γ′ (denoted αpct) are knownto be considerably different and also temperature depen-dent. Here, an average thermal expansion for the mi-crostructure has been employed, such that the thermalexpansion amounts to the application of a load, as fol-lows

α = (f chnx + f chn

y + f chnz )αchn + f chnαpct (14)

where (αchn, αpct) take values of 1.63 × 10−5 ◦C−1 and1.26×10−5 ◦C−1 respectively, which are the values foundin the literature at 850◦C.

Following [6], a centrifugal load of 350 MPa has beenapplied, which approximates to the worst-case operatingconditions experienced during takeoff. The sensitivity toblade orientation has been examined, Table 1 and Fig 3.

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[011]

[111]

[001]

A

DB

G

L

J

M

F

O P

Q

Figure 3: Initial orientations tested in the basic triangle;here only [001] and Q are examined.

Evolution of the Stress Distribution

Although the thermal expansion is isotropic, elasticanisotropy gives rise to an orientation dependence of thethermal stress distribution, Figure 4(a-b). The stressesin the perfect [001] orientation vary by around 400 MPa,with the highest stresses found in the blade core, which isthe coolest part of the blade, and the lowest stresses foundin the blade tip. In the Q orientation the stress variationis much more significant, ∼ 700 MPa, but is broadly sim-

Orientation φ1 Φ φ2

001 0 0 0A −148.5 1.5 180B −178.6 2.4 180D −167.8 3.7 180F −162.8 5.0 180G −147.0 5.3 180J −165.0 8.0 180L −142.3 9.6 180M −155.6 12.7 180O −177.9 17.5 180P −179.3 19.8 180Q −166.4 23.0 180

Table 1: Input initial orientations for used in the turbineblade model, following those used for monotonic creeptesting by Drew[8]. The Euler angles φ1, Φ and φ2 repre-sent rotations around Z, X and Z respectively.

ilar in pattern. The thermal loads are not large enoughto cause primary creep in the core, but are high enoughto cause tertiary creep and rafting throughout the bladeskin.

When the simulated service load is applied, Figure 4(c-d), the stress distribution pattern is very similar to thethermal case but the loads are of course more tensile. No-tably, this load is large enough, > 550 MPa, to causeprimary creep in the Q orientation in the blade web.Therefore primary creep of the web would be expectedto dominate the response during the initial stages of life.

As shown by MacLachlan [6], because the blade sectionnormal to the blade axis must remain coplanar there musttherefore be a load shift from the web (primary creep) tothe blade skin (tertiary creep) during the early stages ofdeformation. After around 5000 h of creep, Figure 4(e,f),the stress distribution therefore equilibrates to give muchless variation across the blade section. It is important tonote that this response would not be predicted by a modelof only tertiary creep, which would predict that an ever-increasing fraction of the load would be borne by the coldweb. Notably, the eventual stress distribution suggeststhat these equilibrated stresses are universally lower thanthe threshold stress of 550 MPa for precipitate cutting.

The evolution of the normal stress for both the [001]and Q orientations is shown in Figure 5. Notably, thestress decreases in the first ∼ 1000 h in the two largersections of the web (P2 and P7), which is balanced by acorresponding increase in the rest of the blade, includingthe remaining web location, P4. In the perfect [001] orien-tation, the stress at the blade tip remains nearly constant,but in the Q orientation it increases. In the blade skinthe stresses are nearly constant over life. After around6×106 s or ∼ 1650 h, the stresses remain nearly constantthroughout the blade.

Creep Strain Variation Within the Blade

The significant temperature and normal stress distribu-tion throughout the blade can produce variations in themechanisms for creep deformation, Figure 6, which de-composes the creep strain into the shear caused by eachof the fcc and L12 ribbon dislocations for the [001] and Qorientations. Clearly, the points in the web (P2 and P7)creep mostly by primary ribbon creep, whilst the pointsin the blade skin and trailing edge creep mostly by ter-tiary creep. In the Q orientation, the amount of primarycreep in the web is greater than the [001], which reflectsthe fact that they are both more highly stressed and arein a poorer orientation for primary creep resistance. Inaddition, the threshold effect for primary creep is clearly

915

Normal Stress, MPa

400342283225167108

50 -8 -67-125-183-241-300

800733667600533467400

333 267

200133

670

600550500450400350300

250 200

150100

500

a-b c-d e-f

(a)[001]

(c)[001]

(d)Q

(e)[001]

(b)Q

(f )Q

~5500h Creep

Thermal + Load

Thermal only

Figure 4: Normal stress distribution in the turbine bladesection. (a) and (b) are the stress due to the thermal loadonly, (c) and (d) are the stresses after application of thethermal and centrifugal loads and (e) and (f) are the stressafter 2× 107 s =∼ 5500 h exposure to creep. (a), (c) and(e) correspond to a blade with a perfect [001] orientationand (b), (d) and (f) to orientation Q, respectively.

No

rmal

str

ess

(MPa

)

800

700

600

500

400

300

200

100

0

Time (s x10 )60 5 10 15 20

P7

P13

P5

P6

P2

P10

P4 P9

P3

P1

P11P8

P12

[001]

No

rmal

str

ess

(MPa

)

800

700

600

500

400

300

200

100

0

Time (s x10 )60 5 10 15 20

P7

P13

P2

P5

P6

P10

P4 P9

P3

P1

P11

P8

P12

Q

Figure 5: Evolution of the normal stress at selected pointsin the blade for the [001] orientation (top) and Q (bot-tom).

observed.

Orientation Dependence of the Blade Response

The 12 orientations shown in Table 1 have been studied.The maximum distances of the orientations listed in table1 from the [001] orientation are about 20◦ inside the basictriangle [8]. Figure 7 shows the total and creep deforma-tion along the blade axis for each orientation studied. Thevariation observed between the samples is dominated bythe elastic response, but the creep strains also vary by upto 0.03%. As expected, the [001] is the best orientation,but the creep response does not vary directly accordingto distance from [001]. For example, the M and Q ori-entations have almost the same overall response but arenearly 10◦ different in primary orientation and the P andO orientations are the worst. Also note that the creep

916

Time (s x10 )60 5 10 15 20

Sh

ea

sr s

trin

a d

ue

to

fc

c d

islo

cati

on

(x1

0-3)

[001]

P13

P5P1

P3

P4

P6

P8

P9

P10P12 P11

25

20

15

10

5

0

Time (s x10 )60 5 10 15 20

Sh

ea

sr s

tra

in d

ue

to

L1

2 d

islo

cati

on

(x1

0-3)

14

12

10

8

6

4

2

0

P2

P4

P7

P9

Q

Time (s x10 )60 5 10 15 20

Sh

ea

sr s

tra

in d

ue

to

fc

c d

islo

cati

on

(x1

0-3) 25

20

15

10

5

0

P13

P5 P1

P4 P9

Q

P6 P3 P10

P12

Time (s x10 )60 5 10 15 20

Sh

ea

sr s

tra

in d

ue

to

L1

2 d

islo

cati

on

s (x

10-3)

P2

P4

P7

P9

11

10

9

8

7

6

5

4

3

2

1

0

[001]

Figure 6: Shear strain evolution at different locations inthe blade section due to the fcc γ and <112> γ′ disloca-tions, for blades in both the [001] and Q orientations.

Tota

l No

rmal

Str

ain

, %

1.80

1.78

1.76

1.74

1.72

1.70

1.68

1.66

1.64

1.62

1.60

Time (s x106)0 5 10 15 20

Cre

ep N

orm

al S

trai

n, %

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Time (s x106)0 5 10 15 20

Figure 7: Orientation dependence of the total and creepnormal strain in the blade section.

responses, even at the whole blade level, do not map ontoa single master curve, with some orientations showing alower initial primary creep rates but larger rates at latertimes.

Figure 7(top) surprisingly indicates that orientation Qgives the best deformation resistance among the test ori-entations, as suggested previously [8]. In contrast to com-mon perception, this study shows that the larger the dis-tance from the [001] orientation, the better the deforma-tion resistance. However, it is important to recognise thatthis is a consequence of the variation in elastic responserather than of the creep response, Figure 7(bottom).

Effect of Casting Shape Variations on the Creep Response

During the investment casting of superalloy blades, thelocation of the ceramic cores used to form the coolingpassages can be difficult to control. This can be thecase both due to difficulties in casting preparation or dueto motion of the cores during charging with the molten

917

0.728

0.700

0.642

0.583

0.525

0.467

0.408

0.350

0.292

0.233

0.175

0.117

0.058

0.000

Shear strain due to ribbon dislocations, %

x

y

Figure 8: Shear strain due to L12 superdislocation rib-bons obtained for modification of the cooling hole positionby 0.2 mm in the +y (top) and −y (bottom) directions.

alloy[9]. Therefore it is inevitable to have certain varia-tion in the core locations compared to the design ideal.Here we examine the effect of displacements of the coresof 0.2 mm from their ideal positions in each direction.

Interestingly, Figure 8 shows that changing the coolinghole position changes the amount of primary creep experi-enced in the web significantly. The total strain, Figure 9,varies by about 1× 10−4 between the four displacements,with a displacement of +y = 0.2 mm found to minimizethe strain response. A similar pattern is observed in theoverall creep strain in the blade, but again elastic effectsdominate the response. The overall effect on the amountof creep observed found to vary by around 10% due tovariations in cooling hole position alone. Interestingly,deflections towards the vacuum (concave, −y) face ap-pear to be most critical. Similarly, deflections in the −xdirection appear to have very little effect while deflectionsin the +x direction do affect the response.

These considerations suggest that the optimal positionfor the cooling holes may be a deflection in the +y di-rection, but this model has ignored the effect of such de-flections on the temperature distribution (which has been

Centre

-y 0.2mm

-x 0.2mm

+x 0.2mm +y 0.2mm

Time (s x10 )60 1 2 3 4 5 6 7 8

Cre

ep n

orm

al s

trai

n (x

10-4

)

3.5

3.0

2.5

2.0

1.5

4.0

4.5

1.0

0.5

0

Time (s x 10 ) 6 0 1 2 3 4 5 6 7 8

Tota

l no

rmal

str

ain

, %

1.75

1.74

1.73

1.72

1.71

1.70

1.76

Centre

-y 0.2mm

-x 0.2mm

+x 0.2mm

+y 0.2mm

Figure 9: Total strain along blade axis direction for dif-ferent modification of cooling hole positions.

scaled to keep the surfaces at the same temperature). Ofcourse, there are additional design considerations, such asthe vibrational modes, that affect cooling hole location.

Comparison to the Blade Section Model of MacLachlan

The evolution of the normal stress at different locationsin the blade, Figure 6, is different to the predictions madepreviously by MacLachlan [10, 6]. In that model a soften-ing factor was introduced for primary creep to allow oneslip system to dominate primary creep until the orienta-tion reaches the duplex slip boundary. Therefore, ratherlarge rotations and creep strains are predicted and so aload shift is predicted twice: at first load shifts from thecold web to the hot skin and subsequently load shifts backto the web. In the present model slip system activationdepends only on the resolved shear stress and resistancefrom dislocation interactions. Therefore little softening iscaused by crystal rotation and only one load shift, from

918

the web to the skin, is predicted. Currently, there is noexperimental data available to clarify which behaviour ac-tually occurs in practice, although additional experimen-tation on the orientation evolution during a creep testwould allow elucidation of which approach is correct.

However, compared to MacLachlan’s model [10, 6], thecurrent work predicts a similar dependence of creep resis-tance on orientation, that is that the closer to the [001]orientation, the better the creep resistance. In addition,MacLachlan’s model predicts that the overall blade strainrate accelerates with strain, whereas here slowdown is ob-served. This is because in the present model the drop inload in the web results in a cessation of primary shearand therefore in a drop in the overall blade strain rate.

Furthermore, besides creep deformation anisotropy, inthis paper we have considered consequent change in theelastic response. This implies that the overall strain is notminimised in the [001] orientation. Although thermal gra-dient and centrifugal loading cause larger normal stressfor the Q orientation, the elastic anisotropy, the differentSchmid factors for different slip systems and different dis-location interactions produce better overall deformationresistance for orientations away from [001].

Summary and Conclusions

The creep deformation behaviour of a high pressure(cooled) turbine blade in CMSX-4 has been developed,which properly respects the loading conditions antici-pated during operation. The underlying modes of micro-mechanical behaviour in primary and tertiary creep (andtheir inter-relationship) are accounted for.

The following conclusions can be drawn from this work.

1. The analysis indicates that load shakedown occursduring the initial stages of life, with primary creep inthe webs of the blade resulting in load shedding to theperiphery (skin) of the blade and a consequent cessationof primary creep in the web.

2. Variations in orientation about [001] can cause sig-nificant variations in the normal stress in the blade web,which produces different primary creep responses in thatlocation. The total strain is minimized, due to elasticanisotropy, in the Q orientation, but the [001] orientationminimizes the creep response.

3. The effect of variation in the location of the coolingchannels has been analysed and found to cause up to a10% variation in the creep response. However, this ef-fect appears to be smaller than that due to uncertaintiesin the orientation of the blade with respect to the [001]crystallographic axis.

Acknowledgements

Funding from EPSRC is gratefully acknowledgedunder grants GR/T27877/01, EP/D04619X/1 andEP/C536312/1. The blade section data used as inputto the model was supplied by Dr. D.W. MacLachlan atRolls-Royce plc, Derby, UK. Useful discussions with Dr.N Jones at Rolls-Royce and Dr. M Winstone at DSTLare also gratefully acknowledged.

References

[1] A. Ma, D. Dye, and R.C. Reed. A model for thecreep deformation behaviour of single crystal super-alloy CMSX-4. Acta Materialia, page in press, 2008.

[2] C.M.F. Rae and R.C. Reed. Primary creep in singlecrystal superalloys: origins, mechanisms and effects.Acta Materialia, 55:1067–1081, 2007.

[3] R.C. Reed, D.C. Cox, and C.M.F. Rae. Damageaccumulation during creep deformation of a singlecrystal superalloy at 1150◦C. Materials Science andEngineering, A448:88–96, 2007.

[4] J. Svoboda and P. Lukas. Creep deformation mod-elling of superalloy single crystals. Acta Materialia,48:2519–2528, 2000.

[5] Abaqus/Standard User’s Manual. Hibbit, Karlssonand Sorensen, Inc., 1080 Main Street, Pawtucket, RI02860 4847, USA, 2007.

[6] D.W. Maclachlan and D.M. Knowles. The effect ofmateiral behaviour on the analysis of single crystalturbine blade: Part ii - component analysis. FatigueFract Engng Mater Struct, 25:399–409, 2002.

[7] T.M. Pollock A.S. Argon. Creep resistance of CMSX-3 nickel based superalloy single crystals. Acta Mate-rialia, 40:1–30, 1992.

[8] G.L. Drew. Thermal-mechanical fatigue of the singlecrystal nickel based superalloy CMSX-4. PhD thesis,Cambridge University, 2003.

[9] R.C. Reed. The SUPERALLOYS: fundamentals andapplications. Cambridge Press, 2006.

[10] D.W. Maclachlan and D.M. Knowles. The effect ofmateiral behaviour on the analysis of single crystalturbine blade: Part i - material model. Fatigue FractEngng Mater Struct, 25:385–398, 2002.

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