Creep deformation modelling of superalloy single crystals

Creep deformation modelling of superalloy single crystals

Acta mater. 48 (2000) 2519±2528 www.elsevier.com/locate/actamat CREEP DEFORMATION MODELLING OF SUPERALLOY SINGLE CRYSTALS J. SVOBODA{ and P. LUKAÂSÏ ...

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Acta mater. 48 (2000) 2519±2528 www.elsevier.com/locate/actamat

CREEP DEFORMATION MODELLING OF SUPERALLOY SINGLE CRYSTALS J. SVOBODA{ and P. LUKAÂSÏ

Institute of Physics of Materials, Academy of Sciences of the Czech Republic, ZÏizÏkova 22, CZ-61662 Brno, Czech Republic (Received 15 January 1999; accepted 22 February 2000) AbstractÐA model of creep deformation in arbitrarily oriented nickel-base superalloy single crystals has been developed. This model extends the preceding model of the present authors [Svoboda, J. and LukaÂsÏ , P., Acta mater., 1998, 46, 3421] covering the case of creep in h001i oriented crystals. The calculated creep curves were compared with the creep curves of single crystals CMSX-4 of the orientations h001i, h011i and h111i measured at 7508C; a fair agreement was found. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science 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 resistance. 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, somewhat 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 h111i orientation than for h001i orientation. Later they showed [8] that the creep life of h111i oriented crystals is highly dependent on the size of the g ' precipitates, 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

{ To whom all correspondence should be addressed.

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 h001i oriented single crystals. In that paper it was shown that the complicated creep curve is a result of interaction 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

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 0 7 8 - 1

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SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

temperature and to compare the theoretical creep curves with the experimentally determined ones for the temperature of 7508C on CMSX-4 single crystals. 2. DESCRIPTION OF THE MODEL

First of all it is necessary to list the mechanisms which are dominant during creep of CMSX-4 single crystals at 7508C. It is generally accepted that the creep deformation starts by generation and glide of dislocations of the type {111}h110i in g channels. Without co-operation with other mechanisms this mechanism can cause plastic deformation in the g channels of the same order as the mis®t d and thus the plastic deformation of the whole crystal up to 0.1%. Without dynamic recovery the process of plastic deformation would stop at this level. There are two possible mechanisms of the dynamic recovery. (i) The dislocations overcome the g ' particle by combination of slip and climb; the dislocation loops annihilate in the vicinity of the apices of the g ' particle. (ii) The dislocations or their reaction products cut through the g ' particle and then either annihilate with the dislocations on the opposite g/ g ' interface or they cut through the next g ' particle. In our preceding paper [13] the creep curves for the orientation [001] could be well simulated by the mechanism (i). This mechanism does not change the shape of the g ' particles; during plastic deformation the vertical g channels get narrower and thus the further plastic deformation becomes more dicult. This results in the experimentally observed dramatic decrease of the creep rate by two orders of magnitude and the transition from quick creep to slow creep. Modelling using the mechanism (i) successfully describes this transition [13]. Application of the mechanism (ii) did not lead to any hardening. Such a hardening could be explained only by a change of cutting mechanism during transition from quick to slow creep [14]. Sass et al. [5] studied the mechanisms of creep in CMSX-4 single crystals at 8508C. During creep of single crystals mainly with the orientations near to h011i they identi®ed cutting of the g ' particle by partial dislocation a=3‰211Š which left another partial dislocation a=6‰211Š on the g/g ' interface and a superlattice stacking fault in the g ' particle. After creeping h011i crystals to the strain of 1.8% one to two cutting events were identi®ed in each g ' particle; this corresponds to plastic deformation of the g ' particle by less than 0.1%. Thus, there must be a dominant recovery mechanism which does not leave traces; this is most probably the mechanism (i) described above. Moreover, Sass et al. [5] observed only very infrequent cutting for the orientation h001i and practically no cutting

for the orientation h111i. This strongly supports the assertion that the mechanism (i) is dominant for all the orientations. Cutting by the dislocation pairs, which was also observed in Ref. [5], is dominant at high temperatures. It can, therefore, be expected that it is not dominant at 7508C. In the modelling, the following mechanisms will be taken into account. 1. Dislocation slip in g channels and concurrent multiplication of dislocations. These dislocations remain 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 shrink towards the apices of the g ' particles. 3. Morphological changes of g ' particles by migration of g/g ' interfaces. The preceding model assumed tensile stress in the direction [001] and made use of considerable symmetry 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 and will contain more parameters. To describe the actual state of the structure in the [001] case it was necessary to use three parameters concerning dislocation density on the g/g ' interfaces and four parameters describing the dimensions of the unit cell. 2.1. Speci®cation of the structure parameters As well as in the model for h001i oriented crystals, also in the presented general model we assume that only octahedral slip systems {111}h110i are activated in the g channels. There are 12 slip systems determined by the normalized vector of the p slip direction aK ˆ h110i= 2 and the normalized vector pof  the normal to the slip plane cK ˆ h111i= 3: We shall distinguish between the g channels normal to x, y and z directions; that is why we must consider 36 parameters characterizing density of dislocations deposited on the g/g ' interfaces rs,K …s ˆ x,y,z†:

Fig. 1. Scheme of the unit cell.

SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

The morphology of the unit cell can be described by 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 the channels and the precipitates are mutually shifted along the interfaces due to recovery. For the g channel normal to the x axis the plastic strains of the channel ex,12, ex,13 and the shifts along the g/g ' interfaces wx,2, wx,3 must be introduced for the full description of the morphology. Analogous parameters must be introduced for the g channels normal to the y and z axes. Thus altogether 54 independent parameters are needed to describe the structure of the unit cell. The stress and elastic strain components in the g channels and in the g ' particles can be considered as dependent parameters, which can be evaluated from the applied stress, mis®t parameter, elastic constants and structure 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 are needed for their evaluation. Hook law represents 24 equations (similar to equations (20)±(26) in Ref. [15]); the elastic constants c11, c12, c44, c '11, c '12 and c '44 have to be employed. Let us consider equilibrium conditions between the applied stress and the internal stresses for cross-sections marked in Fig. 1. For each cross-section we get one equation for the normal components (similar to equation (16) in Ref. [15]) and two equations for the shear components. A further four cross-sections are normal to the x and z axes. For all the cross-sections we obtain 18 equations altogether. However, three couples of the equations for shear components in the cross-sections of type No. 2 in Fig. 1 are identical for the initial state with the cubic symmetry and very similar in the deformed state. That is why each couple of these equations was summed and considered as one equation. As a result, the equilibrium condition 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

12 X ˆ d dij ÿ rx,K bK,ij

…1†

Kˆ1

where ij ˆ 22, 33, 23 and bK,ij ˆ 1=2…aK,i cK,j ‡ aK,j cK,i †:

…2†

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

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be solved. Fortunately, the set can be analytically pre-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 recovery and of the migration of the g/g ' interfaces, it is necessary to calculate the total energy of the system (of the unit cell) as a function of the state parameters and the applied stress. The total energy consists of the elastic energy, the energy of deposited dislocations and the potential energy of the loading system. The elastic energy can be calculated from the stress and elastic strain components obtained. The energy of dislocations can be calculated from the dislocation densities rs,K (see Appendix A.2). The energy of the loading system is given by the tensorial product of the applied stress sapp,ij and the total strain of the unit cell e ij multiplied by the volume of the unit cell. The total strain component e 11 is given by  e 11 ˆ ln

…Lx ‡ hx † …1 ‡ e11 † …L0 ‡ h0 †

 …3†

The components e 22 and e 33 can be calculated in a similar way. The component e 12 is given by  e 12 ˆ ln

wx,2 ‡ wy,1 ‡ hx ex,12 ‡ hy ey,12 …1 ‡ e12 † 2…L0 ‡ h0 †

 …4†

The components e 13 and e 23 can again be calculated in a similar way. The components of the mean elastic deformation eij are given by eij ˆ

ÿ  VP eP,ij ‡ Vx ex,ij ‡ Vy ey,ij ‡ Vz ez,ij  ÿ : VP ‡ 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 system the resolved shear stress in the g channel normal to the x axis is given by: tx,K ˆ

3 X

sx,ij bK,ij

…6†

i,jˆ1

Part of the energy released by slip is deposited in the 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 linear thermodynamics of irreversible processes, the rate of dislocation slip is proportional to …tx,K ÿ tx,K0 † and to the dislocation density at the g/g ' interfaces corresponding to the slip system. The slip rate can be expressed by:

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g_ x,K

ÿ  ˆ W rx,K ‡ ry,K ‡ rz,K …tx,K ÿ tx,K0 †

…7†

The rate of deposition of dislocations is proportional to the slip rate and can be expressed by equation (8) (analogous to equation (6) in Ref. [15]): _ x,K r_ …1† x,K ˆ g

p 2=3 : b

…8†

The total rate of the plastic deformation in the g channel is given by e_ x,ij ˆ

12 X g_ x,K bK,ij

…9†

Kˆ1

and the change of the width of the g channel is given by: …1† h_x ˆ hx e_ x,11 :

…10†

The upper index in parentheses denotes the serial number of the mechanism contributing to the change of the parameter. This index is omitted when only one mechanism operates. The slip in the g channels normal to the y and z axes can be described analogously. 2.5. Dynamic recovery Recovery requires motion of dislocations along the g/g ' interfaces. Let us consider dislocations on the g/g ' interfaces corresponding to one slip system. Interfaces of one orientation are occupied by screw dislocations, while interfaces of the remaining orientations are occupied by mixed dislocations. The screw dislocations can move along the interface by combination of cross-slips. The motion of the mixed dislocations is possible only if layers of atoms are deposited and collected on interfaces with di€erent orientations. Jogs on dislocations produced by thermal activation act as the only sources and sinks of vacancies. The jogs can be considered to be ideal point sources or sinks of vacancies. During the recovery the vacancies must be emitted by the jogs on interfaces of one orientation, transported by di€usion and absorbed by jogs on interfaces of the other orientation. The areal density of jogs on the g/g ' interfaces is decisive for the rate of recovery. It is not important whether the dislocations 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 system and the amount of the dissipated energy must be equal to the amount of the energy released in the system due to recovery. In Ref. [13] the rate of energy dissipation is expressed by equation (16). The ®rst two terms in the equation correspond to radial ¯uxes around jogs; the third and the fourth

terms correspond to the transport of vacancies in the g channels. For typical values of p120b and r110 ÿ7 m the ®rst two terms dominate. This means that the recovery is controlled by the di€usive transport of vacancies in the area surrounding jogs. This enables the energy dissipation connected with the transport of vacancies in g channels to be neglected 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. Then it is possible to calculate the value of ÿDE=DrK numerically. This value represents the driving force for the recovery. It is necessary to stress that DrK causes the changes in thickness of g channels, causes shifts along the g/g' interfaces and in¯uences the contact 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 dislocation densities owing to recovery is given by: …2† _ …2† _ z,K r_ …2† ˆ x,K ˆ r y,K ˆ r



DE 2pDO DrK L30 kTpb

bK,11 Lx bK,22 Ly bK,33 Lz ‡ ‡ rx,K ry,K rz,K

! ÿ1 :

…11†

The change in the thickness of the g channel is given by: …2† h_x ˆ

12 X Kˆ1

bK,11 Lx br_ …2† x,K :

…12†

Shifts along the g/g ' interfaces caused by the recovery can be expressed by: w_ x2 ˆ

12 X bK,12 Lx br_ …2† x,K

…13†

12 X bK,13 Lx br_ …2† x,K :

…14†

Kˆ1

w_ x3 ˆ

Kˆ1

For the y and z directions the corresponding equations are analogous. 2.6. Migration of g/g ' interfaces The physical substance of the migration of g/g ' interfaces leading to the morphological changes in superalloys is described in detail in Ref. [13]. In the present model we assume that the g ' particle can change its shape with two degrees of freedom instead of only one as shown in Ref. [13]. Migration of the g/g ' interfaces is a di€usion-controlled process dependent on temperature and applied stress. At temperatures above 9008C the rate of migration is very high and the migration

SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

leads to rafting. At 7508C the rate of migration is substantially slower, but nevertheless it plays a role in the creep process. For example, in h001i oriented crystals a deformation of 20% would completely close 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 open during the slow creep. The change in width of the vertical g channels is at most several tens of per cent. This cannot be experimentally measured owing to the inherent irregularity of the distribution of the g ' particles in the CMSX-4 single crystals. The rates of migration D_ x , D_ y , and D_ z normal to the g/g ' interfaces are bound by the condition of volume conservation: D_ x D_ y D_ z ‡ ‡ ˆ0 Lx Ly Lz

…15†

The rate of energy dissipation due to di€usion of atoms connected with the migration of g/g ' interfaces is given by: 2 !2  _  6 kTL0 4 Ly Lx Dx D_ y RM ˆ ‡ ÿ 12DjO hx Lz hy Lz Lx Ly  ‡  ‡

 _ Dx D_ z Lz Lx ‡ ÿ hx Ly hz Ly Lx Lz

!2

!23  _ _ Ly Dy Dz 5 Lz ‡ ÿ hy Lx hz Lx Ly Lz

…16†

The substitutions Uˆ

Dx Dy ÿ Lx Ly

and



Dx Dz ÿ Lx Lz

…17†

enable the number of kinetic variables to be reduced to two and the rate of energy dissipation gains the form: "  Ly kTL6 Lx 2 RM ˆ U_ ‡ 12DjO hx Lz hy Lz  ‡  ‡

 Lz Lx 2 V_ ‡ hx Ly hz Ly

#  2 Ly ÿ _ Lz U ÿ V_ ‡ hy Lx hz Lx

…18†

. . The rates U and V can be calculated from the set of two linear equations of motion for the system: ÿ

@E 1 @ RM ˆ @U 2 @ U_

and

ÿ

@E 1 @ RM ˆ @V 2 @ V_

…19†

The partial derivative @ [email protected] U was calculated numerically as DE=DU keeping DV ˆ 0: DU causes changes in the dimensions of Lx, Ly, Lz and hx, hy,

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hz. This is the reason that the total energy E is. changed. . @ [email protected] V is calculated analogously. After U and V are evaluated, the values of D_ x, D_ y, and D_ z can be expressed. The evolution of the morphology of the system is then given by: …3† L_ x ˆD_ x , h_x ˆ ÿD_ x , L_ y ˆ D_ y , …3† …3† h_y ˆ ÿ D_ y , L_ z ˆ D_ z ,; h_z ˆ ÿD_ z

…20†

2.7. Calculation of creep curves The coordinates in the preceding calculations are identical with the crystal directions of the type h001i. The applied stress has to be expressed in this coordinate system. Then the evolution of the state parameters (the relevant equations for individual contributions are given in previous paragraphs) can be calculated by integration in time. From the tensor of the total strain given by equations (3) and (4) it is then possible to calculate the component of the total strain in the direction of the uniaxial applied stress. This is the sought value. 3. RESULTS AND DISCUSSION

In this section the experimental creep curves are compared with the theoretically calculated ones. For the comparison we have chosen creep curves for three multiple-slip oriented crystals (h001i, h011i and h111i) measured at a ``low temperature'' of 7508C under constant load conditions. These experimental curves (points) are presented in Figs 2±4 in di€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. Some of the parameters can be measured by microscopic methods. These concern the dimensions of the structure before loading …L0 ˆ 0:45 mm and h0 ˆ 0:05 mm), atomic volume …O ˆ 1:1  10 ÿ29 m3 † and Burgers vector …b ˆ 2:5  10 ÿ10 m). The initial density of the grown-in dislocations in each slip system is taken to be 1  104/m. The elastic constants c11, c12, c44, c '11, c '12 and c '44 in g and g ' phases for the temperature of 7508C were taken from the paper by Pollock and Argon [16]. The remaining parameters 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 ˆ 7508C) G (GPa) 40

d

D (m2/s)

W (m/s/Pa)

p (m)

j

ÿ1.1  10ÿ3

7.8  10ÿ19

2  10ÿ21

5  10ÿ9

10

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SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

nickel in an alloy of a composition similar to the g phase in the superalloy CMSX-4 [17]. There are three ®t parameters, the values of which 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 the g/g ' interfaces. These parameters, their choice and their physical acceptability were discussed in detail in Ref. [13]. Results of the modelling are shown in Figs 2±7. The parameters W, p and j were chosen in such a way to obtain the best agreement between the experimental creep curve for h001i orientation and the calculated 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 parameters the creep curves for the orientations h011i and h111i were computed. The agreement for these two orientations (Figs 3 and 4 ) is somewhat looser, but still it is possible to speak about a fair agreement especially when taking into consideration the fact that the inherent scatter of the creep rates in nominally the same creep tests can reach factor 3. The modelling also makes it possible to identify the mechanisms involved in the particular stages of the creep deformation. The case of h001i orientation was discussed in Ref. [13]. The mechanisms for h011i and h111i orientations will be described in the following. In the case of h011i orientation the g channels within the unit cell can be divided into two roof channels and one vertical channel. At the applied stress of 650 MPa all the channels are plastically deformed and the recovery process takes place. The creep acceleration at the beginning of the test is caused by the multiplication of the dislo-

cations. The slip and recovery processes lead to the widening of the roof channels and to a narrowing of the vertical channel. The sharp decrease of the creep rate (at about 4% strain) is due to the narrowing of the vertical channel to such an extent that slip becomes dicult. Slow creep, controlled by the widening of the vertical channel due to the migration of the g/g ' interfaces, starts at a strain of about 5%. In the case of the h111i orientation all the channels are equivalent and the simultaneous activity of the slip systems in the g channels leads to shear in directions h011i. The recovery of the deposited dislocations on the g/g ' interfaces does not cause any change of channel thickness, but causes shifts in the g/g ' interfaces in the directions h011i. Owing to the symmetry there is no migration of the g/g ' interfaces. The model does not predict any hardening of the crystal for the discussed h111i orientation. 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 process may cause narrowing at various points in the channels and might result in the hardening. For all the orientations it holds that the theoretically predicted onset of creep (increase of creep rate) at the beginning of the tests is more revealing than the experimentally measured prediction. This is probably due to the fact that the model assumes the multiplication of dislocations within every unit cell. In reality the dislocations have to spread out from the nests of the grown-in dislocations. This process is slower [16]. The model makes it possible to compute the times needed to reach a given strain. This is shown in Figs 5±7 for the case of a 2% creep strain.

Fig. 2. Calculated (full line) and experimentally (points) determined di€erentiated creep curves of the superalloy single crystal CMSX-4 (orientation h001i; 7508C; 735 MPa).

Fig. 3. Calculated (full line) and experimentally (points) determined di€erentiated creep curves of the superalloy single crystal CMSX-4 (orientation h011i; 7508C; 650 MPa).

SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

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Fig. 4. Calculated (full line) and experimentally (points) determined di€erentiated creep curves of the superalloy single crystal CMSX-4 (orientation h111i; 7508C; 800 MPa).

Fig. 6. Calculated time to 2% creep strain while dependent on applied stress for three crystal orientations. The input parameters are L0 ˆ 0:45 mm, h0 ˆ 0:10 mm, and volume fraction of the g' phase is at 54.8%.

Figure 5 shows how this this time depends on applied stress for all three crystal orientations. The basic structural parameters were chosen to model the superalloy single crystals of CMSX-4 …L0 ˆ 0:45 mm, h0 ˆ 0:05 mm, volume fraction of the g ' phase of 72.9%). It is clear that the ranking of the orientations is h111i, h001i, and h011i. For a lower volume fraction of the g ' phase, i.e. 54.8% …L0 ˆ 0:45 mm, h0 ˆ 0:1 mm), the ranking remains (Fig. 6), but the strength decreases by almost a factor of 2. If a structure coarsened by a factor of 1.6

…L0 ˆ 0:72 mm, h0 ˆ 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 respect to the original one (Fig. 5) is obtained. The presented diagrams show that the model makes it possible to simulate the creep behaviour as dependent on structural parameters. The model is an idealization of reality. It assumes perfectly periodic structure of the g ' particles. This is the reason a decrease of the applied stress leadsÐaccording to the modelÐin some types of g channels to deactivation of some slip systems and

Fig. 5. Calculated time to 2% creep strain while dependent on the applied stress for three crystal orientations. The input parameters …L0 ˆ 0:45 mm, h0 ˆ 0:05 mm, volume fraction of the g' phase being 72.9%) correspond to single crystals of CMSX-4.

Fig. 7. Calculated time to 2% creep strain while dependent on applied stress for three crystal orientations. The input parameters are L0 ˆ 0:72 mm, h0 ˆ 0:08 mm, and volume fraction of the g' phase is at 72.9%.

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SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

consequently to a halting of the recovery process and thus, to a zero creep rate. In reality the decrease of the applied stress leads to the deactivation only in narrower channels and the slip/recovery process can continue in the wider channels. In conclusion it can be stated that the model is well applicable for high enough stresses at which all of the g channels are activated. To model properly the case of considerably lower stresses it would probably be necessary to take into account the distribution of the width of the g channels. 4. CONCLUSIONS

The modelling and its confrontation with the experimental results make it possible to state the following conclusions. 1. A unit cell model of creep of arbitrarily oriented superalloy single crystals taking into account slip in g channels, dynamic recovery of dislocation structure and migration of g/g ' interfaces has been developed. The model is based exclusively on the concept of linear thermodynamics. 2. The model is applicable for low temperatures and medium applied stresses. The calculated creep curves were compared with the experimental creep curves of single crystals CMSX-4 of the orientations h001i, h011i and h111i measured at 7508C; a fair agreement was found. 3. By varying the input parameters such as dimensions and volume fraction of the g ' particles it is possible to evaluate the e€ect of these parameters on the resulting creep behaviour. As an example, the e€ect of the dimensions and volume fraction of the g ' particles on the time needed to reach a strain of 2% for the orientations h001i, h011i and h111i has been calculated. The results demonstrate a strong e€ect.

6. LukaÂsÏ , P., CÏadek, J., SÏustek, V. and Kunz, L., Mater. Sci. Engng A, 1996, 208, 149. 7. Bullough, C. K., Toulios, M., Oehl, M. and LukaÂsÏ , P., in Materials for Power Engineering 1998, ed. J. Lecomte-Beckers, F. Schubert and P. J. Ennis. Forschungszentrum, JuÈlich, 1998, p. 861, Part II. 8. Caron, P. and Khan, T., in Strength of metals and alloys in Proc. ICSMA-8, Vol. II, ed. P.O. Kettunen et al., 1988, p. 893. 9. MuÈller, L. and Feller-Kniepmeier, M., Scripta metall. mater., 1993, 29, 81. 10. MuÈller, L., Glatzel, U. and Feller-Kniepmeier, M., Acta metall. mater., 1993, 41, 3401. 11. Li, S. X. and Smith, D. J., Scripta metall. mater., 1995, 33, 711. 12. Toulios, M., Mohrmann, R. and Fleury, G., in Materials for Power Engineering 1998, ed. J. LecomteBeckers, F. Schubert and P. J. Ennis. Forschungszentrum, JuÈlich, 1998, p. 879, Part II. 13. Svoboda, J. and LukaÂsÏ , P., Acta mater., 1998, 46, 3421. 14. Mughrabi, H., Schneider, W., Sass, V. and Lang, C., in Strength of materials in Proc. ICSMA-10, ed. H. Oikawa et al., 1994, p. 705. 15. Svoboda, J. and LukaÂsÏ , P., Acta mater., 1997, 45, 125. 16. Pollock, T. M. and Argon, A. S., Acta metall. mater., 1994, 42, 1859. 17. RuÊzÏicÏkovaÂ, J. and Million, B., Mater. Sci. Engng, 1981, 50, 59. 18. Hirth, J. P. and Lothe, J., Theory of Dislocations. McGraw-Hill, New York, 1968. APPENDIX A

A.1. Nomenclature Lx, Ly, Lz L0 hx, hy, hz h0 Vx, Vy, Vz, VP d sx,ij, sy,ij, sz,ij, sP,ij

AcknowledgementsÐThis research was supported by the Grant Agency of the Academy of Sciences of the Czech Republic under contracts A2041608 and A2041002 and by the Ministry of Education of the Czech Republic under contracts OC P3.30 and OC 522.80. This support is gratefully acknowledged.

K

REFERENCES

aK

1. MacKay, R. A. and Maier, R. D., Metall. Trans. A, 1982, 13, 1747. 2. Sass, V., Schneider, W. and Mughrabi, H., Scripta metall. mater., 1994, 31, 885. 3. Caron, P., Khan, T. and Nakagawa, Y. G., Scripta metall., 1986, 20, 499. 4. Kakehi, K., Sakaki, T., Gui, J. M. and Misaki, Y., in Creep and Fracture of Engineering Materials and Structures, ed. B. Wilshire and R. W. Evans. The Institute of Materials, London, 1993, p. 221. 5. Sass, V., Glatzel, U. and Feller-Kniepmeier, M., Acta mater., 1996, 44, 1967.

cK

tx,K, ty,K, tz,K tx,K0, ty,K0, tz,K0

ex,ij,ey,ij, ez,ij, eP,ij ex,ij, ey,ij, ez,ij dij G, c11, c12, c44, c '11, c'12, c '44 rx,K, ry,K, rz,K

dimensions of the g ' particle initial dimension of the g' particle thickness of the g channels initial thickness of the g channels volumes of g channels and the g' particle mis®t parameter stress components in the g channels and the g' particle serial number of the slip system …K ˆ 1±12) resolved shear stress for the Kth slip system in g channels threshold resolved shear stress for slip in g channels normalized vector in the slip direction normalized vector normal to the slip plane elastic strain components in g channels and the g' particle plastic strain components in the g channels Kronecker symbol elastic constants density of dislocations deposited on the g/g' interfaces

SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

2527

Fig. A1. Schematic drawing for the calculation of the energy of dislocations deposited on the g/g ' interfaces. wx,2, wx,3, wy,1, wy,3, wz,1, wz,2 W p k T b D O j

sliding along the g/g' interfaces due to recovery of dislocations parameter characterizing the rate of slip in g channels mean distance between jogs measured along the dislocation line Boltzmann constant absolute temperature magnitude of the Burgers vector average coecient of self-di€usion inthe g phase [13] atomic volume parameter characterizing the di€erence in chemical composition of the g and g' phases [13]

slip we assume that the line energy is Ed ˆ 1=2Gb 2 : The procedure worked out by Hirth and Lothe [18] can be used for the evaluation of the change of the energy when the distance of two parallel dislocations is changed from Ra to R. The value Ra can be understood as the distance at which the ®eld of the dislocation is screened by other dislocations. In our case we take Ra ˆ L0 =2: The change of the energy Ed is half the interaction energy with all the other dislocations lying close to the distance Ra. The interaction energy for two screw dislocations can be expressed as W12,screw ˆ ÿ

…A1†

and for mixed dislocations as W12,mixed

A.2. Energy of dislocations deposited on the g/g ' interfaces p p Let slip pus  consider p p  system a ˆ ‰ 2=2, 2=2, 0Š, c ˆ ‰ÿ 3=3, 3=3, 3=3Š and the g/g ' interfaces lying in the planes (001) and (010). The direction of pthe dislocation line in the plane (001) is g ˆ ‰ 2=2, 1 p 2=2, 0Š and thus the dislocation is of the screw type. The directionpof dislocation line in the  the p  plane (010) is g2 ˆ ‰ 2=2, 0, 2=2Š and thus the dislocation is of a mixed type. These two cases represent all the possible con®gurations of dislocations of 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 directions of dislocation lines will be neglected. Figure A1 shows the dislocations deposited on two neighbouring g/g ' interfaces. The direction g is perpendicular to the paper, the dislocations in the upper interface have opposite Burgers vector with respect to the dislocations in the lower interface. We shall calculate how the energy of the dislocation is changed due to the interaction with other dislocations. For the isolated pair of dislocations produced by

Gb1 b2 2R ln L 2p

Gb1 b2 ˆÿ 2p



sin 2 Y ‡ 4…1 ÿ n †

 1 3 2R ‡ ln 2 4…1 ÿ n † L

! …A2†

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

SVOBODA and LUKAÂSÏ: CREEP DEFORMATION IN SINGLE CRYSTALS

2528

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 7  10 /m both dependencies can be approximated by a straight line: ÿ  Ed ˆ 1=2Gb 2 1 ‡ 1:1  10 ÿ8 r ÿ1

…A3†

(r in m ) The threshold stress for slip can be calculated from the condition that the energy released by the slip is equal to the increase of the energy of

dislocations deposited on the g/g ' interface. For the channel of unit area it holds that: ÿ  t0 h dg ˆ 2 d…Ed r† ˆ Gb 2 1 ‡ 2:2  10 ÿ8 r dr …A4† Using relation (8) we get p  2Gb ÿ t0 ˆ p 1 ‡ 2:2  10 ÿ8 r 3h

…A5†