Creep deformation modelling of superalloy single crystals

CREEP DEFORMATION MODELLING OF SUPERALLOY

SINGLE CRYSTALS

J. SVOBODA{ and P. LUKAÂ SÏ

Institute of Physics of Materials, Academy of Sciences of the Czech Republic, ZÏ izÏ kova 22, CZ-61662Brno, Czech Republic

(Received 15 January 1999; accepted 22 February 2000)

AbstractÐA model of creep deformation in arbitrarily oriented nickel-base superalloy single crystals hasbeen developed. This model extends the preceding model of the present authors [Svoboda, J. and LukaÂ sÏ ,P., Acta mater., 1998, 46, 3421] covering the case of creep in h001i oriented crystals. The calculated creepcurves were compared with the creep curves of single crystals CMSX-4 of the orientations h001i, h011i andh111i measured at 7508C; a fair agreement was found. 7 2000 Acta Metallurgica Inc. Published by ElsevierScience Ltd. All rights reserved.

Keywords: Superalloys; Creep; Di�usion; Dislocations; Theory & modelling

1. INTRODUCTION

Superalloy single crystals have been used up to now

for production of critical parts of aero-engine gas

turbines. They are now considered also for use in

industrial gas turbines in order to achieve the

required increase in working temperature. The

behaviour of the superalloy single crystals is highly

anisotropic. Several experimental studies have

shown that the creep resistance depends strongly on

crystal orientation [1±7]. In the majority of studied

cases the orientation h001i has the highest creep re-

sistance. Nevertheless, this is not a generally valid

rule. For example, MacKay and Maier [1] found

for MAR-M247 and MAR-M200 single crystals

tested in the temperature range 760±7748C that the

rupture life was longest for h111i orientation, some-

what shorter for h001i orientation and shortest for

h011i orientation. On the other hand, Caron et al.

[3] found for CMSX-2 single crystals tested at

7608C considerably shorter creep lives for h111iorientation than for h001i orientation. Later they

showed [8] that the creep life of h111i oriented crys-

tals is highly dependent on the size of the g ' precipi-tates, while the creep life of h001i oriented crystals

depends on the size of g ' precipitates weakly. The

creep life of h111i crystals can be longer or shorter

than that of h001i crystals in dependence on the g 'particle size. Several papers show a drop in creep

rupture lives for crystals having orientation near to

h001i with increasing misorientation from the exact

h001i orientation. For example, Sass et al. [2] tested

CMSX-4 single crystals at 8008C under a constant

stress of 767 MPa. The longest lives were reached

by crystals with an orientation in close proximity of

[001]. With increasing misorientation from [001] the

creep strength dropped signi®cantly, with crystals

having an orientation close to the [001]±[011]

boundary of the standard stereographic triangle

exhibiting longer lifetimes than crystals having an

orientation closer to the [001]±[111] boundary.

Generally, it can be stated that the ranking of

orientations from the point of view of their creep

resistance depends on stress and temperature [7]

and on the size of the g ' precipitates [8].The proposed semi-phenomenological models of

creep in the superalloy single crystals are based

either on visco-plastic ¯ow of the softer matrix

around the hard particles or on crystal plasticity

assuming slip activity not only on octahedral slip

systems, but also on other types of systems. These

models do not consider the basic mechanisms active

during the creep process; their success in describing

the creep anisotropy is only partial [9±12].

In our preceding paper [13] we have proposed a

model based on the basic mechanisms for h001ioriented single crystals. In that paper it was shown

that the complicated creep curve is a result of inter-

action between the deformation processes and the

evolution of morphology of the g ' particles. The

aim of this paper is to present an extension of the

model for arbitrarily oriented crystals crept at low

Acta mater. 48 (2000) 2519±2528

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

PII: S1359 -6454 (00 )00078 -1

www.elsevier.com/locate/actamat

{ To whom all correspondence should be addressed.

temperature and to compare the theoretical creepcurves with the experimentally determined ones for

the temperature of 7508C on CMSX-4 single crys-tals.

2. DESCRIPTION OF THE MODEL

First of all it is necessary to list the mechanismswhich are dominant during creep of CMSX-4 singlecrystals at 7508C. It is generally accepted that the

creep deformation starts by generation and glide ofdislocations of the type {111}h110i in g channels.Without co-operation with other mechanisms this

mechanism can cause plastic deformation in the gchannels of the same order as the mis®t d and thusthe plastic deformation of the whole crystal up to0.1%. Without dynamic recovery the process of

plastic deformation would stop at this level. Thereare two possible mechanisms of the dynamic recov-ery.

(i) The dislocations overcome the g ' particle bycombination of slip and climb; the dislocationloops annihilate in the vicinity of the apices of

the g ' particle.(ii) The dislocations or their reaction productscut through the g ' particle and then either an-

nihilate with the dislocations on the opposite g/g ' interface or they cut through the next g ' par-ticle.

In our preceding paper [13] the creep curves forthe orientation [001] could be well simulated by themechanism (i). This mechanism does not change the

shape of the g ' particles; during plastic deformationthe vertical g channels get narrower and thus thefurther plastic deformation becomes more di�cult.This results in the experimentally observed dramatic

decrease of the creep rate by two orders of magni-tude and the transition from quick creep to slowcreep. Modelling using the mechanism (i) success-

fully describes this transition [13]. Application ofthe mechanism (ii) did not lead to any hardening.Such a hardening could be explained only by a

change of cutting mechanism during transition fromquick to slow creep [14]. Sass et al. [5] studied themechanisms of creep in CMSX-4 single crystals at8508C. During creep of single crystals mainly with

the orientations near to h011i they identi®ed cuttingof the g ' particle by partial dislocation a=3�211�which left another partial dislocation a=6�211� onthe g/g ' interface and a superlattice stacking fault inthe g ' particle. After creeping h011i crystals to thestrain of 1.8% one to two cutting events were ident-

i®ed in each g ' particle; this corresponds to plasticdeformation of the g ' particle by less than 0.1%.Thus, there must be a dominant recovery mechan-

ism which does not leave traces; this is most prob-ably the mechanism (i) described above. Moreover,Sass et al. [5] observed only very infrequent cuttingfor the orientation h001i and practically no cutting

for the orientation h111i. This strongly supports theassertion that the mechanism (i) is dominant for all

the orientations. Cutting by the dislocation pairs,which was also observed in Ref. [5], is dominant athigh temperatures. It can, therefore, be expected

that it is not dominant at 7508C. In the modelling,the following mechanisms will be taken intoaccount.

1. Dislocation slip in g channels and concurrentmultiplication of dislocations. These dislocationsremain deposited on the g/g ' interfaces.

2. Dynamic recovery of the dislocation structure.The dislocation loops spanning around the g 'particles move by the combination of slip and

climb along the g/g ' interfaces and shrinktowards the apices of the g ' particles.

3. Morphological changes of g ' particles by mi-gration of g/g ' interfaces.

The preceding model assumed tensile stress in thedirection [001] and made use of considerable sym-metry of the case of the [001] oriented crystals. In

the general case the degree of symmetry is lower,the model will have to be more complicated andwill contain more parameters. To describe the

actual state of the structure in the [001] case it wasnecessary to use three parameters concerning dislo-cation density on the g/g ' interfaces and four par-

ameters describing the dimensions of the unit cell.

2.1. Speci®cation of the structure parameters

As well as in the model for h001i oriented crys-tals, also in the presented general model we assume

that only octahedral slip systems {111}h110i areactivated in the g channels. There are 12 slip sys-tems determined by the normalized vector of theslip direction aK � h110i=

��2p

and the normalized

vector of the normal to the slip plane cK �h111i= ��

3p: We shall distinguish between the g chan-

nels normal to x, y and z directions; that is why we

must consider 36 parameters characterizing densityof dislocations deposited on the g/g ' interfacesrs,K �s � x,y,z�:

Fig. 1. Scheme of the unit cell.

2520 SVOBODA and LUKAÂ SÏ : CREEP DEFORMATION IN SINGLE CRYSTALS

The morphology of the unit cell can be describedby the dimensions of the g ' particle Lx, Ly, Lz and

by the thickness of g channels hx, hy, hz (Fig. 1).The g channels are deformed plastically and thechannels and the precipitates are mutually shifted

along the interfaces due to recovery. For the gchannel normal to the x axis the plastic strains ofthe channel ex,12, ex,13 and the shifts along the g/g 'interfaces wx,2, wx,3 must be introduced for the fulldescription of the morphology. Analogous par-ameters must be introduced for the g channels nor-

mal to the y and z axes. Thus altogether 54independent parameters are needed to describe thestructure of the unit cell. The stress and elasticstrain components in the g channels and in the g 'particles can be considered as dependent par-ameters, which can be evaluated from the appliedstress, mis®t parameter, elastic constants and struc-

ture parameters.

2.2. Stress±strain analysis

Altogether there are 24 (6 in the g ' particle and 3� 6 in the g channels) stress and 24 elastic strain

components in the unit cell. Thus, 48 equations areneeded for their evaluation. Hook law represents 24equations (similar to equations (20)±(26) in Ref.

[15]); the elastic constants c11, c12, c44, c '11, c '12 andc '44 have to be employed. Let us consider equili-brium conditions between the applied stress and theinternal stresses for cross-sections marked in Fig. 1.

For each cross-section we get one equation for thenormal components (similar to equation (16) inRef. [15]) and two equations for the shear com-

ponents. A further four cross-sections are normal tothe x and z axes. For all the cross-sections weobtain 18 equations altogether. However, three

couples of the equations for shear components inthe cross-sections of type No. 2 in Fig. 1 are identi-cal for the initial state with the cubic symmetry and

very similar in the deformed state. That is why eachcouple of these equations was summed and con-sidered as one equation. As a result, the equilibriumcondition o�ers 15 equations. The remaining nine

equations are the contact conditions on the g/g 'interfaces analogous to equations (13)±(15) in Ref.[15]. For the g/g ' interface normal to the x axis it

holds that

ex,ij ÿ eP,ij � d dij ÿX12K�1

rx,KbK,ij �1�

where ij � 22, 33, 23 and

bK,ij � 1=2�aK,icK,j � aK,jcK,i �: �2�

Two further triplets of equations for the g/g ' inter-faces normal to the y and z axes can be obtained ina similar way. The set of 48 linear equations for 48unknown stress and elastic strain components can

be solved. Fortunately, the set can be analyticallypre-solved; the remaining set, which has to be

solved numerically, consists of six equations.

2.3. Calculation of the total energy of the system

To be able to calculate the kinetics of the recov-ery and of the migration of the g/g ' interfaces, it isnecessary to calculate the total energy of the system(of the unit cell) as a function of the state par-ameters and the applied stress. The total energy

consists of the elastic energy, the energy of depos-ited dislocations and the potential energy of theloading system. The elastic energy can be calculated

from the stress and elastic strain componentsobtained. The energy of dislocations can be calcu-lated from the dislocation densities rs,K (see

Appendix A.2). The energy of the loading system isgiven by the tensorial product of the applied stresssapp,ij and the total strain of the unit cell �eij multi-plied by the volume of the unit cell. The total strain

component �e11 is given by

�e11 � ln

� �Lx � hx ��L0 � h0 � �1� �e11 �

��3�

The components �e22 and �e33 can be calculated in asimilar way. The component �e12 is given by

�e12 � ln

�wx,2 � wy,1 � hxex,12 � hyey,12

2�L0 � h0 � �1� �e12 ��4�

The components �e13 and �e23 can again be calculatedin a similar way. The components of the mean elas-tic deformation �eij are given by

�eij �ÿVPeP,ij � Vxex,ij � Vyey,ij � Vzez,ij

�ÿVP � Vx � Vy � Vz

� : �5�

2.4. Dislocation slip in the g channels

Now we can assume that the stress tensor is

known in each g channel. For the Kth slip systemthe resolved shear stress in the g channel normal tothe x axis is given by:

tx,K �X3i,j�1

sx,ijbK,ij �6�

Part of the energy released by slip is deposited inthe form of the energy of dislocations on the g/g 'interfaces. This results in the threshold stress tx,K0(see Appendix A.2). Then, following the lines of lin-ear thermodynamics of irreversible processes, the

rate of dislocation slip is proportional to �tx,K ÿtx,K0� and to the dislocation density at the g/g ' inter-faces corresponding to the slip system. The slip ratecan be expressed by:

SVOBODA and LUKAÂ SÏ : CREEP DEFORMATION IN SINGLE CRYSTALS 2521

_gx,K �Wÿrx,K � ry,K � rz,K

��tx,K ÿ tx,K0 � �7�

The rate of deposition of dislocations is pro-portional to the slip rate and can be expressed byequation (8) (analogous to equation (6) in Ref.

[15]):

_r�1�x,K � _gx,K

��2=3pb

: �8�

The total rate of the plastic deformation in the gchannel is given by

_ex,ij �X12K�1

_gx,KbK,ij �9�

and the change of the width of the g channel isgiven by:

_h�1�x � hx_ex,11: �10�

The upper index in parentheses denotes the serialnumber of the mechanism contributing to thechange of the parameter. This index is omitted

when only one mechanism operates. The slip in theg channels normal to the y and z axes can bedescribed analogously.

2.5. Dynamic recovery

Recovery requires motion of dislocations alongthe g/g ' interfaces. Let us consider dislocations onthe g/g ' interfaces corresponding to one slip system.

Interfaces of one orientation are occupied by screwdislocations, while interfaces of the remaining orien-tations are occupied by mixed dislocations. Thescrew dislocations can move along the interface by

combination of cross-slips. The motion of themixed dislocations is possible only if layers ofatoms are deposited and collected on interfaces

with di�erent orientations. Jogs on dislocations pro-duced by thermal activation act as the only sourcesand sinks of vacancies. The jogs can be considered

to be ideal point sources or sinks of vacancies.During the recovery the vacancies must be emittedby the jogs on interfaces of one orientation, trans-ported by di�usion and absorbed by jogs on inter-

faces of the other orientation. The areal density ofjogs on the g/g ' interfaces is decisive for the rate ofrecovery. It is not important whether the dislo-

cations mutually interact or form the network.The kinetics of recovery can be calculated in the

same way as in Ref. [13]. Di�usion connected with

the recovery dissipates the total energy of the sys-tem and the amount of the dissipated energy mustbe equal to the amount of the energy released in

the system due to recovery. In Ref. [13] the rate ofenergy dissipation is expressed by equation (16).The ®rst two terms in the equation correspond toradial ¯uxes around jogs; the third and the fourth

terms correspond to the transport of vacancies inthe g channels. For typical values of p120b and

r110ÿ7 m the ®rst two terms dominate. Thismeans that the recovery is controlled by the di�u-sive transport of vacancies in the area surrounding

jogs. This enables the energy dissipation connectedwith the transport of vacancies in g channels to beneglected and assumes that the recovery processes

in individual slip systems are mutually independent.In the preceding papers [13, 15] it was proved

that the recovery decreases the dislocation densities

rx,K, ry,K, and rz,K by the same amount DrK. Thenit is possible to calculate the value of ÿDE=DrK nu-merically. This value represents the driving forcefor the recovery. It is necessary to stress that DrKcauses the changes in thickness of g channels, causesshifts along the g/g ' interfaces and in¯uences thecontact conditions at the g/g ' interfaces. All these

changes must be involved in the calculation of DE.In the ®nal form the rate of change of dislocationdensities owing to recovery is given by:

_r�2�x,K � _r�2�y,K � _r�2�z,K �DEDrK

2pDOL30kTpb

� bK,11Lx

rx,K� bK,22Ly

ry,K� bK,33Lz

rz,K

!ÿ1: �11�

The change in the thickness of the g channel isgiven by:

_h�2�x �

X12K�1

bK,11Lxb _r�2�x,K: �12�

Shifts along the g/g ' interfaces caused by the recov-

ery can be expressed by:

_wx2 �X12K�1

bK,12Lxb _r�2�x,K �13�

_wx3 �X12K�1

bK,13Lxb _r�2�x,K: �14�

For the y and z directions the correspondingequations are analogous.

2.6. Migration of g/g ' interfaces

The physical substance of the migration of g/g 'interfaces leading to the morphological changes insuperalloys is described in detail in Ref. [13]. In the

present model we assume that the g ' particle canchange its shape with two degrees of freedominstead of only one as shown in Ref. [13].

Migration of the g/g ' interfaces is a di�usion-con-trolled process dependent on temperature andapplied stress. At temperatures above 9008C therate of migration is very high and the migration


leads to rafting. At 7508C the rate of migration issubstantially slower, but nevertheless it plays a role

in the creep process. For example, in h001i orientedcrystals a deformation of 20% would completelyclose the vertical g channels if the g ' particles were

perfectly rigid. It is just the migration of the g/g 'interfaces which keeps the vertical g channels openduring the slow creep. The change in width of the

vertical g channels is at most several tens of percent. This cannot be experimentally measuredowing to the inherent irregularity of the distribution

of the g ' particles in the CMSX-4 single crystals.The rates of migration _Dx, _Dy, and _Dz normal to

the g/g ' interfaces are bound by the condition ofvolume conservation:

_Dx

Lx�

_Dy

Ly�

_Dz

Lz� 0 �15�

The rate of energy dissipation due to di�usion ofatoms connected with the migration of g/g ' inter-faces is given by:

RM � kTL60

12DjO

24� Ly

hxLz� Lx

hyLz

� _Dx

Lxÿ

_Dy

Ly

! 2

��

Lz

hxLy� Lx

hzLy

� _Dx

Lxÿ

_Dz

Lz

! 2

��

Lz

hyLx� Ly

hzLx

� _Dy

Lyÿ

_Dz

Lz

! 235 �16�

The substitutions

U � Dx

Lxÿ Dy

Lyand V � Dx

Lxÿ Dz

Lz�17�

enable the number of kinetic variables to bereduced to two and the rate of energy dissipation

gains the form:

RM � kTL6

12DjO

"�Ly

hxLz� Lx

hyLz

�_U2

��

Lz

hxLy� Lx

hzLy

�_V2

��

Lz

hyLx� Ly

hzLx

�ÿ_Uÿ _V

� 2# �18�

The rates U.and V

.can be calculated from the set of

two linear equations of motion for the system:

ÿ @E@U� 1

2

@RM

@ _Uand ÿ @E

@V� 1

2

@RM

@ _V�19�

The partial derivative @E=@U was calculated nu-merically as DE=DU keeping DV � 0: DU causeschanges in the dimensions of Lx, Ly, Lz and hx, hy,

hz. This is the reason that the total energy E ischanged. @E=@V is calculated analogously. After U

.

and V.are evaluated, the values of _Dx, _Dy, and _Dz

can be expressed. The evolution of the morphologyof the system is then given by:

_Lx � _Dx, _h�3�x � ÿ _Dx, _Ly � _Dy,

_h�3�y �ÿ _Dy, _Lz � _Dz,; _h

�3�z � ÿ _Dz �20�

2.7. Calculation of creep curves

The coordinates in the preceding calculations are

identical with the crystal directions of the typeh001i. The applied stress has to be expressed in thiscoordinate system. Then the evolution of the state

parameters (the relevant equations for individualcontributions are given in previous paragraphs) canbe calculated by integration in time. From the ten-

sor of the total strain given by equations (3) and (4)it is then possible to calculate the component of thetotal strain in the direction of the uniaxial appliedstress. This is the sought value.

3. RESULTS AND DISCUSSION

In this section the experimental creep curves arecompared with the theoretically calculated ones.

For the comparison we have chosen creep curvesfor three multiple-slip oriented crystals (h001i, h011iand h111i) measured at a ``low temperature'' of

7508C under constant load conditions. These exper-imental curves (points) are presented in Figs 2±4 indi�erentiated form (total strain rate vs total strain)

together with the theoretically calculated curves(full lines).The calculation of the theoretical creep curves

was performed along the lines shown in Section 2.

This requires a large number of parameters. Someof the parameters can be measured by microscopicmethods. These concern the dimensions of the

structure before loading �L0 � 0:45 mm andh0 � 0:05 mm), atomic volume �O � 1:1� 10ÿ29 m3�and Burgers vector �b � 2:5� 10ÿ10 m). The initial

density of the grown-in dislocations in each slip sys-tem is taken to be 1 � 104/m. The elastic constantsc11, c12, c44, c '11, c '12 and c '44 in g and g ' phases forthe temperature of 7508C were taken from the

paper by Pollock and Argon [16]. The remainingparameters are listed in Table 1.For the di�usion coe�cient we have chosen the

value corresponding to the di�usion coe�cient of

Table 1. Values of the constants used in the computation�T � 7508C)

G (GPa) d D (m2/s) W (m/s/Pa) p (m) j

40 ÿ1.1� 10ÿ3 7.8� 10ÿ19 2� 10ÿ21 5� 10ÿ9 10


nickel in an alloy of a composition similar to the gphase in the superalloy CMSX-4 [17].

There are three ®t parameters, the values ofwhich cannot be taken from independent sources,namely W (which determines slip in the g channels;

p (which determines the kinetics of recovery); and j(which determines the kinetics of migration of theg/g ' interfaces. These parameters, their choice and

their physical acceptability were discussed in detailin Ref. [13].Results of the modelling are shown in Figs 2±7.

The parameters W, p and j were chosen in such away to obtain the best agreement between the ex-perimental creep curve for h001i orientation and thecalculated creep curve for this orientation. Both the

curves in di�erentiated form are shown in Fig. 2.Using then the same values of all the parametersthe creep curves for the orientations h011i and

h111i were computed. The agreement for these twoorientations (Figs 3 and 4 ) is somewhat looser, butstill it is possible to speak about a fair agreement

especially when taking into consideration the factthat the inherent scatter of the creep rates in nomin-ally the same creep tests can reach factor 3.

The modelling also makes it possible to identifythe mechanisms involved in the particular stages ofthe creep deformation. The case of h001i orientationwas discussed in Ref. [13]. The mechanisms for

h011i and h111i orientations will be described in thefollowing. In the case of h011i orientation the gchannels within the unit cell can be divided into

two roof channels and one vertical channel. At theapplied stress of 650 MPa all the channels are plas-tically deformed and the recovery process takes

place. The creep acceleration at the beginning ofthe test is caused by the multiplication of the dislo-

cations. The slip and recovery processes lead to thewidening of the roof channels and to a narrowing

of the vertical channel. The sharp decrease of thecreep rate (at about 4% strain) is due to the nar-rowing of the vertical channel to such an extent

that slip becomes di�cult. Slow creep, controlled bythe widening of the vertical channel due to the mi-gration of the g/g ' interfaces, starts at a strain of

about 5%. In the case of the h111i orientation allthe channels are equivalent and the simultaneousactivity of the slip systems in the g channels leads

to shear in directions h011i. The recovery of thedeposited dislocations on the g/g ' interfaces doesnot cause any change of channel thickness, butcauses shifts in the g/g ' interfaces in the directions

h011i. Owing to the symmetry there is no migrationof the g/g ' interfaces. The model does not predictany hardening of the crystal for the discussed h111iorientation. The experimental creep curve (Fig. 4)exhibits a hardening after about 0.5% deformation.This might be due to the decrease of the distance

between the tips of the g ' particles. This processmay cause narrowing at various points in the chan-nels and might result in the hardening. For all the

orientations it holds that the theoretically predictedonset of creep (increase of creep rate) at the begin-ning of the tests is more revealing than the exper-imentally measured prediction. This is probably due

to the fact that the model assumes the multipli-cation of dislocations within every unit cell. In rea-lity the dislocations have to spread out from the

nests of the grown-in dislocations. This process isslower [16].The model makes it possible to compute the

times needed to reach a given strain. This is shownin Figs 5±7 for the case of a 2% creep strain.

Fig. 3. Calculated (full line) and experimentally (points)determined di�erentiated creep curves of the superalloysingle crystal CMSX-4 (orientation h011i; 7508C;

650 MPa).


735 MPa).


Figure 5 shows how this this time depends on

applied stress for all three crystal orientations. Thebasic structural parameters were chosen to modelthe superalloy single crystals of CMSX-4

�L0 � 0:45 mm, h0 � 0:05 mm, volume fraction of theg ' phase of 72.9%). It is clear that the ranking ofthe orientations is h111i, h001i, and h011i. For a

lower volume fraction of the g ' phase, i.e. 54.8%�L0 � 0:45 mm, h0 � 0:1 mm), the ranking remains(Fig. 6), but the strength decreases by almost a fac-tor of 2. If a structure coarsened by a factor of 1.6

�L0 � 0:72 mm, h0 � 0:08 mm� at the same volume

fraction of the g ' phase as in the ®rst case (72.9%)is assumed, a softer structure (Fig. 7) with respectto the original one (Fig. 5) is obtained. The pre-

sented diagrams show that the model makes it poss-ible to simulate the creep behaviour as dependenton structural parameters.

The model is an idealization of reality. It assumesperfectly periodic structure of the g ' particles. Thisis the reason a decrease of the applied stress

leadsÐaccording to the modelÐin some types of gchannels to deactivation of some slip systems and


800 MPa).

Fig. 5. Calculated time to 2% creep strain while dependenton the applied stress for three crystal orientations. Theinput parameters �L0 � 0:45 mm, h0 � 0:05 mm, volumefraction of the g ' phase being 72.9%) correspond to single

crystals of CMSX-4.

Fig. 6. Calculated time to 2% creep strain while dependenton applied stress for three crystal orientations. The inputparameters are L0 � 0:45 mm, h0 � 0:10 mm, and volume

fraction of the g ' phase is at 54.8%.

Fig. 7. Calculated time to 2% creep strain while dependenton applied stress for three crystal orientations. The inputparameters are L0 � 0:72 mm, h0 � 0:08 mm, and volume

fraction of the g ' phase is at 72.9%.


consequently to a halting of the recovery processand thus, to a zero creep rate. In reality the

decrease of the applied stress leads to the deactiva-tion only in narrower channels and the slip/recoveryprocess can continue in the wider channels. In con-

clusion it can be stated that the model is well appli-cable for high enough stresses at which all of the gchannels are activated. To model properly the case

of considerably lower stresses it would probably benecessary to take into account the distribution ofthe width of the g channels.

4. CONCLUSIONS

The modelling and its confrontation with the ex-perimental results make it possible to state the fol-

lowing conclusions.

1. A unit cell model of creep of arbitrarily orientedsuperalloy single crystals taking into account slip

in g channels, dynamic recovery of dislocationstructure and migration of g/g ' interfaces hasbeen developed. The model is based exclusivelyon the concept of linear thermodynamics.

2. The model is applicable for low temperaturesand medium applied stresses. The calculatedcreep curves were compared with the experimen-

tal creep curves of single crystals CMSX-4 of theorientations h001i, h011i and h111i measured at7508C; a fair agreement was found.

3. By varying the input parameters such as dimen-sions and volume fraction of the g ' particles it ispossible to evaluate the e�ect of these parameterson the resulting creep behaviour. As an example,

the e�ect of the dimensions and volume fractionof the g ' particles on the time needed to reach astrain of 2% for the orientations h001i, h011iand h111i has been calculated. The resultsdemonstrate a strong e�ect.

AcknowledgementsÐThis research was supported by theGrant Agency of the Academy of Sciences of the CzechRepublic under contracts A2041608 and A2041002 and bythe Ministry of Education of the Czech Republic undercontracts OC P3.30 and OC 522.80. This support is grate-fully acknowledged.

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APPENDIX A

A.1. Nomenclature

Lx, Ly, Lz dimensions of the g ' par-ticle

L0 initial dimension of theg ' particle

hx, hy, hz thickness of the g chan-nels

h0 initial thickness of the gchannels

Vx, Vy, Vz, VP volumes of g channelsand the g ' particle

d mis®t parametersx,ij, sy,ij, sz,ij, sP,ij stress components in the

g channels and the g 'particle

K serial number of the slipsystem �K � 1±12)

tx,K, ty,K, tz,K resolved shear stress forthe Kth slip system in gchannels

tx,K0, ty,K0, tz,K0 threshold resolved shearstress for slip in g chan-nels

aK normalized vector in theslip direction

cK normalized vector normalto the slip plane

ex,ij,ey,ij, ez,ij, eP,ij elastic strain componentsin g channels and the g 'particle

ex,ij, ey,ij, ez,ij plastic strain componentsin the g channels

dij Kronecker symbolG, c11, c12, c44, c '11, c '12, c '44 elastic constantsrx,K, ry,K, rz,K density of dislocations

deposited on the g/g 'interfaces


wx,2, wx,3, wy,1, wy,3, wz,1, wz,2 sliding along the g/g 'interfaces due to recoveryof dislocations

W parameter characterizingthe rate of slip in g chan-nels

p mean distance betweenjogs measured along thedislocation line

k Boltzmann constantT absolute temperatureb magnitude of the Burgers

vectorD average coe�cient of

self-di�usion inthe gphase [13]

O atomic volumej parameter characterizing

the di�erence in chemicalcomposition of the g andg ' phases [13]

A.2. Energy of dislocations deposited on the g/g 'interfaces

Let us consider slip system a�� 2p =2,��2p=2, 0�, c

��ÿ ��3p=3,

��3p=3,

��3p=3� and the g/g ' interfaces lying

in the planes (001) and (010). The direction of thedislocation line in the plane (001) is g1 � �

��2p=2,��

2p=2, 0� and thus the dislocation is of the screw

type. The direction of the dislocation line in theplane (010) is g2��

��2p=2, 0,

��2p=2� and thus the dis-

location is of a mixed type. These two cases rep-

resent all the possible con®gurations of dislocationsof the systems {111}h110i deposited on the g/g 'interfaces. The mutual interaction of dislocations of

the di�erent Burgers vectors and/or di�erent direc-tions of dislocation lines will be neglected. FigureA1 shows the dislocations deposited on two neigh-

bouring g/g ' interfaces. The direction g is perpen-dicular to the paper, the dislocations in the upperinterface have opposite Burgers vector with respect

to the dislocations in the lower interface. We shallcalculate how the energy of the dislocation is chan-ged due to the interaction with other dislocations.For the isolated pair of dislocations produced by

slip we assume that the line energy is Ed � 1=2Gb 2:The procedure worked out by Hirth and Lothe [18]can be used for the evaluation of the change of theenergy when the distance of two parallel dislo-

cations is changed from Ra to R. The value Ra canbe understood as the distance at which the ®eld ofthe dislocation is screened by other dislocations. In

our case we take Ra � L0=2: The change of theenergy Ed is half the interaction energy with all theother dislocations lying close to the distance Ra.

The interaction energy for two screw dislocationscan be expressed as

W12,screw � ÿGb1b2

2pln2R

L�A1�

and for mixed dislocations as

W12,mixed � ÿGb1b2

2p

�1

2� 3

4�1ÿ n��

ln2R

L

� sin 2Y4�1ÿ n�

!�A2�

Fig. A1. Schematic drawing for the calculation of the energy of dislocations deposited on the g/g ' inter-faces.

Fig. A2. Dislocation line energy while dependent on thedislocation density for the con®guration shown in Fig. A1.


Then it is possible to calculate the function Ed (r )for screw and mixed dislocations. This is shown in

Fig. A2. For typical dislocation density r= 1 2±4� 10

7/m both dependencies can be approximated

by a straight line:

Ed � 1=2Gb 2ÿ1� 1:1� 10ÿ8r

��A3�

(r in mÿ1) The threshold stress for slip can be cal-

culated from the condition that the energy releasedby the slip is equal to the increase of the energy of

dislocations deposited on the g/g ' interface. For thechannel of unit area it holds that:

t0h dg � 2 d�Edr� � Gb 2ÿ1� 2:2� 10ÿ8r

�dr �A4�

Using relation (8) we get

t0 ��2p

Gb��3p

h

ÿ1� 2:2� 10ÿ8r

��A5�


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