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Acta mater. Vol. 46, No. 10, pp. 3421±3431, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00043-3 1359-6454/98 $19.00 + 0.00

MODEL OF CREEP IN h001i-ORIENTED SUPERALLOY SINGLE CRYSTALS J. SVOBODA and P. LUKAÂSÏ Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Brno, Czech Republic (Received 6 November 1997; accepted 22 January 1998) AbstractÐA model of creep in h001i oriented nickel-base superalloy single crystals applicable both for low and high temperatures has been developed. This model extends the preceding model of the present authors [Svoboda, J., LukaÂsÏ , P., Acta mater., 1997, 45, 125] covering only the case of low temperature creep. The present model takes into account ®ve mechanisms, namely (i) dislocation slip in g channels and concurrent multiplication of dislocations, (ii) dislocation slip in 0 particles, (iii) dynamic recovery of the dislocation structure, (iv) morphological changes of 0 particles by migration of = 0 interfaces and (v) coarsening of the rafted structure. The calculated creep curves were compared with the creep curves measured on single crystals CMSX-4 at 7508C and 10008C; a good agreement was found. # 1998 Acta Metallurgica Inc.

1. INTRODUCTION

Superalloy single crystals are used for production of turbine blades because of their excellent resistance against high temperature creep. The typical microstructure of the nickel base superalloy single crystals consists of g matrix containing a high volume fraction (up to about 70%) of cuboidal g 0 particles which are coherent or semicoherent with the matrix. As the lattice parameters of the matrix and of the particles are slightly dierent, the g=g 0 interfaces aligned to {001} planes represent sources of mis®t stresses and possible sites for the deposition of mis®t dislocations. The reason for the high creep resistance of these crystals lies in their two phase structure with interfaces which are not easily penetrable for dislocations going from the g matrix to the g 0 particles. There is enough experimental evidence showing that the plastic deformation in both monocrystalline and polycrystalline superalloys strengthened by the g 0 particles takes place predominantly in the g matrix channels (e.g. [1±9]). The dislocations lying on the {111} planes bow out through the g channels and remain deposited on the g=g 0 interfaces. They cause a stress redistribution and thus worsen the conditions for further slip in the g channels. After the plastic strain of the superalloy caused by the dislocation slip in the g channels reaches about 0.1%, the resolved shear stress in the g channels is practically eliminated. The experimental fact that the creep strain of the superalloys typically exceeds 10% says that there must be mechanisms decreasing the dislocation density on the g=g 0 interfaces. Two such mechanisms are available in the superalloys: dynamic recovery and cutting of the g 0 particles. A combination of at least one of these mechanisms

with the dislocation slip in the g channels makes it possible to reach much higher creep strains. It is well known that the precipitation structure undergoes substantial changes during high temperature creep. The initially cuboidal precipitates change their form and may coalesce into rafts, then the rafted structure coarsens. The changes of the precipitation structure certainly aect the deformation process. Several semi-phenomenological models of creep deformation in the superalloy single crystals [3, 4, 10, 11] have been proposed. In all of them the creep is considered to be due to the visco-plastic ¯ow of the softer matrix around the hard particles. Another class of models aiming mainly to life prediction is represented by phenomenological models based on a damage parameter (e.g. [12]). None of these models is based directly on the basic mechanisms active during the creep process. The aim of this paper is to present a model based on the basic mechanisms occurring during creep of superalloy single crystals and to compare the theoretically calculated creep curves with the experimentally measured ones. The model presented in the following is a generalization and combination of our preceding partial models [13, 14] covering some of the involved mechanisms.

2. DESCRIPTION OF THE MODEL

The model takes into account the following mechanisms. * Dislocation slip in g channels and concurrent multiplication of dislocations. These dislocations remain deposited on the g=g 0 interfaces.

3421

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SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

* Dislocation slip in g 0 particles, in other words cutting of g 0 particles. The dislocations generated in the g matrix penetrate the g=g 0 interfaces and slip through the g 0 particles. * Dynamic recovery of the dislocation structure. The dislocation loops spanning around the g 0 particles move by the combination of slip and climb along the g=g 0 interfaces and shrink towards the apices of the g 0 particles. * Morphological changes of g 0 particles by migration of g=g 0 interfaces. The process of the rafting belongs to these morphological changes. * Coarsening of the rafted structure. The description of the dislocation slip in the g channels is based on the assumption that the dislocations move only when the energy released by their motion is higher than the energy of dislocations newly deposited on the g/g' interfaces. This assumption leads to the existence of a threshold stress, which is inversely proportional to the thickness of the g channel [13]. If the acting shear stress is higher than the threshold stress, the dislocations move with a velocity increasing with the dierence of the two stresses. The plastic strain rate is proportional to the density of mobile dislocations and to their velocity. The density of grown-in dislocations in virgin crystals is very low. After the external load is applied, the dislocations start to multiply. The time needed to reach a dislocation density high enough to cause a measurable plastic strain rate manifests itself as an incubation period on the creep curve. The dislocation slip in g channels changes the thickness of the g channels; the shape of the g' particles remains the same. The dislocation slip in the g' particles can be described analogously to the slip in the g channels. The threshold stress for cutting stems from the energy of anti-phase boundary which remains in the ordered g' particle after the cutting. The cutting leads to a change of the shape of the g' particles; the thickness of the g channels remains preserved. The mechanism of the dynamic recovery process, i.e. of the climb/slip motion of dislocation loops along the g/g' interfaces, is described in details in our preceding paper [13]. The jogs on dislocations deposited on the g/g' interfaces are considered to be the only sources and sinks of atoms in the whole system. The mixed dislocations can move along the g/g' interfaces only in a non-conservative way and their motion is thus accompanied by diusive ¯uxes. In the area swept by a dislocation a uniform layer of atoms is collected or deposited depending on the direction of the Burgers vector and the direction of the dislocation motion. In the model we assume that the diusion takes place exclusively in {This mechanism is often called ``directional coarsening'' which, however, does not adequately describe the substance of the process.

the g channels having a much higher diusivity than the ordered g' phase. Further we assume that the chemical composition of the transported mass is identical with the chemical composition of the g phase. This means that the recovery does not lead to any change of the shape of the g' particles but it does lead to the change of the thickness of the g channels. It is necessary to point out that the recovery can be active only if both horizontal and vertical interfaces are occupied by dislocations. The mechanism of the directional coalescence{ (rafting) of the g' particles by migration of the g/g' interfaces was described in our earlier paper [14]. During the migration the dierences in chemical composition of the phases must be compensated by diusional ¯uxes. In the model we assume that the g' particle preserves its volume and remains prismatic. The change of the shape of the g' particles is connected with a change of the thickness of the horizontal g channels (channels normal to the stress axis) and of the vertical g channels (channels parallel to the stress axis). After the rafted structure is developed by coalescence of the g' particles, the rafts can coarsen. In analogy to the normal growth of grains in polycrystals it is plausible to assume that the morphology remains self-similar. The kinetics of the coarsening can be described by the same rules as Ostwald ripening. 3. MATHEMATICAL TREATMENT

The mathematical treatment of the above listed mechanisms is basically of the same type as that presented in our previous papers [13, 14]. Therefore we shall present here only the main equations. The present model diers from the preceding less general models in the following points. (i) Large-strain treatment is used. (ii) The description of the migration of the g/g' interfaces can be used for any chemical composition of the g and g' phases. (iii) The rate of the dislocation slip in the g channels is related to both the local acting stress and the dislocation density. (iv) The jogs necessary for the dynamic recovery are assumed to be formed by thermally activated nucleation. (v) The mechanisms of cutting of the g' particles and coarsening of the rafted structure are incorporated into the model. The state of the system is geometrically described by the dimensions of the g' particle and by the thickness of the g channels in the unit cell, see Fig. 1. The uniaxial tensile stress is applied in the ydirection. The stress state is assumed to be homogeneous within the horizontal g channel, within the vertical g channel and within the g' particle; the normal stress components siH, siV and siP describe

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

the stress states in these particular volumes. Analogously, the states of the elastic and plastic strain are described by the normal components eiH, eiV, eiP and eiH,pl, eiV,pl, eiP,pl, respectively. Because of symmetry all the shear components are zero. The slip takes place on the slip systems of the type {111}h110i; there are altogether 12 of them, but four of them are inactive due to zero Schmid factor. The dislocations produced by the eight active slip systems (having non-zero y-component of the Burgers vector) are deposited on the g/g' interfaces during creep in superalloy. They are all of mixed character on the horizontal g/g' interfaces and of either screw or mixed character on the vertical g/g' interfaces. Their densities (de®ned as total length of dislocations per unit interface area) are rH, rV,screw and rV,mixed. These dislocation densities together with the geometrical parameters LH, LV, hH and hV determine the microstructure of the system. The evolution of the microstructure of the system is unambiguously described by the time derivatives of these microstructure parameters. These time derivatives depend on the actual values of the microstructure parameters, on the material properties, on the applied stress and on the temperature. The rates of change of the dislocation densities on the g/g' interfaces can be written in the form (all the densities concern strain producing mobile dislocations) _ÿ _ÿ r_ H r_ Hÿr H,r ÿ r H,P , r_ V,mixed

r_ V,mixed

ÿ

r_ ÿ V,mixed,r

ÿ

r_ ÿ V,mixed,P ,

_ÿ _ÿ r_ V,screw r_ V,screw ÿ r V,screw,r ÿ r V,screw,P :

1 2 3

Here the components r_ are due to the generation of dislocations during dislocation slip in g channels, the components r_ ÿ r describe the decrease of the dislocation density due to the dynamic recovery and the components r_ ÿ P characterize the decrease of the dislocation density on the g/g' interfaces due to the cutting of the g' particles. The rates of change of the g' particle dimensions and of the g channels thickness are given by _ H, L_ H ÿ2D_ LH e_ yP,pl ML

4

LV , L_ V ÿL_ H 2LH

5

_ hH e_ yH,pl Mh _ H, h_ H 2 u_ D

6

LV _ V, _ ÿ hV e_ yV,pl Mh h_ V ÿD_ ÿ u LH

7

The geometrical parameters are aected by all the considered mechanisms, namely by dislocation slip in g channels (_eyH,pl and e_ V,pl ), by dislocation slip

3423

Fig. 1. De®nition of the unit cell.

in g' particles (_eyP,pl ), by dynamic recovery (uÇ ), by _ and by coarsenmigration of the g/g' interfaces (D) ing of the precipitation structure after the com_ pletion of the rafting (M). The equations (1)±(7) can be integrated in time using the Euler's method to get the evolution of the system in time. Provided we know the values of the microstructure parameters, of the applied stress sy,app, of the mis®t parameter d and of the elastic constants in both the phases, we can determine all stress and elastic strain components in the g' particle and in the g channels [13]. We can thus in the following assume that the stresses and the elastic strains in the system are known. Consequently, the elastic energy stored in the unit cell is known too. The total (elastic + plastic) deformation of the single crystal in the direction of the applied load can be then expressed in the form ey,tot 0

1 1 e V 1 e fV P yP H yH B C [email protected] 2VV 1 eyV LH hH A: VP VH 2VV LH hH LV hV 2 1=3 8

For the superalloy with the volume fraction of the g' phase about 70% the volume of the corners, where the vertical and horizontal channels meet, represents about 10% of the total volume of the g phase. The volume of the corners within the unit cell is divided between the vertical and horizontal channels along the planes passing through the edges of the opposite precipitates. The volumes of the channels VH and VV in equation (8) are calculated in this way. The rate of the dislocation slip in the g channels, e_ pl , is given by the Orowan equation e_ pl brm v,

9

where rm is the density of mobile (capable of slipping) dislocations and v is their average velocity. The dislocations deposited on the g/g' interfaces cannot be identi®ed with the mobile dislocations. Only the dislocation segments running through the g channels are responsible for the slip. On the other hand, the dislocation segments running through the

3424

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

channels are parts of the large closed dislocation loops, the remaining parts of which are the dislocation segments lying on the g/g' interfaces. Thus it seems feasible to assume a proportionality between the density of mobile dislocations and the total density of dislocations deposited on the g/g' interfaces, i.e. rmArH+rV,mixed+rV,screw. Further we assume linear proportionality between the velocity v and the stress acting on the dislocations (resolved shear stress minus threshold stress). This assumption is in accord with the linear thermodynamics of irreversible processes asserting that the kinetics of the process (here the dislocation slip) is proportional to the driving force (stress acting on dislocation). The equation for the strain rate in the horizontal g channels can be then written in the form e_ yH,pl W rH rV,mixed rV,screw syH ÿ sxH ÿ soH ,

10

where the parameter W depends on temperature. The threshold stress is given by [13] soH 2Gb=hH :

11

The temperature dependence of the threshold stress is given by the temperature dependence of G. The contribution of the dislocation slip to the increment of the dislocation density on the g/g' interfaces is given by the equations presented in [13]; for the case of the horizontal g/g' interface it reads r_ H 2_eyH,pl =b:

12

The dislocation slip in the vertical g channels can be treated along the same lines as in the horizontal g channels. Generation of dislocations in the g' particles has never been observed. The cutting of the g' particle can be thus accomplished only by the segments of dislocations generated in the g channels that enter the g' particle and pass through its whole cross-section to the opposite g/g' interface. The sign of the Burgers vector of the dislocations deposited on the g/g' interfaces is opposite to the Burgers vector of the dislocation segments coming out of the g' particle and the segments can annihilate with their counterparts. If a dislocation segment coming out of the particle does not meet its counterpart to annihilate (in case of zero density of deposited dislocations on the g/g' interface), the segment goes through the g channel and stops at the next g/g' interface. In this case we equate the plastic strain rate in the g channel to that in the g' particle. The plastic strain rate in the g' particles can be described by the equation similar to equation (10) e_ yP,pl WP rH rV,mixed rV,screw syP ÿ sxP ÿ soP n ,

13

where WP, soP and n are temperature dependent parameters. The rate of decrease of the dislocation densities due to the annihilation on the g/g' interfaces is given by _ÿ _ÿ r_ ÿ HP 2r VP,mixed 2r VP,screw 2_eyP,pl =b:

14

If no annihilation is possible at a certain g/g' interface, the contributions from the slip in the corresponding g channel and in the g' particle are compensated (r_ ÿ r_ ÿ P 0, compare equations (12) and (14)). Recovery requires motion of dislocations along the g/g' interfaces. The screw dislocations whose Burgers vector lies in the plane of the g/g' interface can move by cross-slip. The mixed dislocations whose Burgers vector does not lie in the plane of the g/g' interface can move exclusively by combination of climb and slip. Therefore we can call the described recovery process ``dynamic recovery''. In our model we do not explicitly assume that the moving dislocations are part of the dislocation network on the g/g' interfaces. On the other hand, if they are the model remains intact. The rate of climb controls the rate of the dynamic recovery. The climb is due to the motion of jogs along the dislocation line. The motion of jogs is conditioned by absorption or emission of atoms. The jog can be considered as an ideal point source or sink of atoms and its motion in one direction requires a permanent diusional ¯ux of atoms to or from the jog. That is why the controlling mechanism for the climb of dislocations is diusion which transports the atoms from the jogs at dislocations on the vertical g/g' interfaces to the jogs at dislocations on the horizontal g/g' interfaces. The kinetics of the climb can be determined from the global principles of thermodynamics of irreversible processes [15]. The kinetic parameter uÇ (covering the changes of the geometrical parameters and the changes of the dislocation densities during the recovery) can be determined from [13] ÿ

@ E 1 @ Rr , @ u 2 @ u_

15

where the total energy of the system E consists of the elastic energy, the energy of dislocations deposited on the g/g' interfaces and the energy of the loading system. The rate of heat production Rr in the system is in our case expressed by kTL2V p pLV Rr DO pO1=3 rH 2pLH O1=3 rV,mixed LV LH L2V 2 u_ : 16 6hV 6hH The rate of decrease of the dislocation density due to the recovery is given by _ÿ _ÿ _ r_ ÿ H: H,r 2r V,mixed,r 2r V,screw,r 4u=bL

17

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

The evolution of the shape of the g' particle due to the migration of the g/g' interfaces is modelled in detail in [14]. The dierences in the chemical composition of the two phases have to be compensated by diusional ¯uxes during the migration of the interfaces. The atoms of one component are replaced by the atoms of another component. In other words, the atoms exchange their positions on the distance comparable with the precipitate size. Such a diusion process does not require any sources or sinks of atoms and thus no jogs at dislocations are needed for the migration of the g/g' interfaces. Let the superalloy contain m elements of the concentration ci in the g phase and of the concentration ci' in the g' phase. Then during the migration of the horizontal g/g' interface by the distance D it is necessary to remove the layers (ci' ÿ ci)D of the ith element from the horizontal interface and deposit the layer (ci' ÿ ci)DLV/2LH at the vertical interface. In the ®nal form we can write for the rate of heat production in the system due to these exchange processes [14] m kT L3V LH L4V X ci 0 ÿ ci 2 _ 2 18 D : Rm 6O hV hH i1 ci Di For the sake of simplicity the atomic volume O is considered to be the same for all the atoms. The sum on the right-hand side of the last equation can be written in the form m X ci 0 ÿ ci 2 i1

ci Di

1 ; jD

19

the parameter j depends on temperature through the temperature dependence of ci and ci'. Similarly to the case of the dynamic recovery, the magnitude of the D_ can be determined from the linear equation ÿ

@ E 1 @ Rm : @ D 2 @ D_

20

The last mechanism treated in the model is the coarsening of the rafted structure. It was observed experimentally, e.g. [5], that after the directional coalescence of the g' particles is completed, the rafted structure coarsens. The small rafts are dissolving and the large rafts are getting larger. Coarsening is due to the migration of the g/g' interfaces controlled by diffusion in the g phase on the scale substantially exceeding the dimensions of our unit cell. Let S denote the actual scale of the rafted structure. The driving force for the coarsening having its origin in curvatures of interfaces is proportional to 1/S and the diusion distance is proportional to S. Thus the rate of change of the scale is proportional to 1/S2 and the rate of the relative increase of the dimensions in the unit cell is given by _ _ S aDj : M S kTS 3

21

3425

In the model the scaling volume S3 is replaced by the volume of the g' particle LHL2V; the value of the constant a can be determined by ®tting to the experiments. Further we suppose that the coarsening, i.e. the increase of all dimensions (LH, LV, hH and hV), starts after the rafted structure is reasonably developed. In the following numerical modelling we shall allow the coarsening to start after the value of hV or hH sinks to 5% of its original value ho. 4. EXPERIMENTAL DETAILS

The experimental measurements of the creep curves were performed on the superalloy single crystals CMSX-4. Single crystals were kindly provided by Howmet, England in the framework of COST 501 Round III collaborative program. The microstructure of the as-received rods consists of cuboidal g' precipitates embedded in a g matrix. The g' particle size lies between 0.4 and 0.5 mm and the volume fraction of the g' phase is about 70%. Superalloy CMSX-4 is a negative mis®t alloy with the mis®t d of about ÿ1 10ÿ3 [5]. The specimens having gauge length 50 mm and gauge diameter 3.5 mm were machined from the test bars. The deviation of the specimen axes from the h001i crystallographic direction did not exceed 88. The creep tests reported here were performed at temperatures 7508C, and 10008C in air. The testing temperature was measured by Pt/Pt±Rh thermocouples located in the nearest vicinity of the specimen gauge length; the temperature was kept constant within 18C. The temperature gradient along the gauge length did not exceed 0.58C. The elongation of the specimens was measured as a function of time. The data were recorded digitally by means of a PC and corrected for the ®nite rigidity of the creep machine loading system. The data were processed to get the dependence of the true (i.e. the logarithmic) total strain on time. Subsequently the strain rate vs strain curves were calculated. 5. RESULTS

In this part we shall compare the experimental creep curves with the theoretically calculated ones and we shall show the eect of the mis®t parameter on the theoretical creep curves. For the comparison we have chosen three creep curves measured at a ``low temperature'' (7508C) and two creep curves measured at a ``high temperature'' (10008C). All the ®ve curves are shown in total strain vs time co-ordinates in Fig. 2. The same curves in dierentiated form (total strain rate vs total strain) are presented in Figs 3 and 4 together with the theoretically calculated curves. The experimental curves for 7508C shown in Fig. 3 exhibit a relatively large scatter which seems to be typical for CMSX-4 single crystals. The stress exponent of

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SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

Fig. 2. Experimentally determined creep curves. (1) 7508C, 800 MPa; (2) 7508C, 775 MPa; (3) 7508C, 735 MPa; (4) 10008C, 200 MPa; (5) 10008C, 150 MPa.

Fig. 4. Dierentiated form of the experimental (points) and theoretical (full lines) creep curves at 10008C. 150 MPa, 200 MPa.

the minimum creep rate can hardly be reliably determined from the presented experimental curves measured at a narrow stress interval. The stress exponent corresponding to the theoretical curves is about 4. The calculation of the theoretical creep curves was performed along the lines shown in Section 3. This requires a large number of parameters. Some of the parameters can be measured by microscopic methods. It concerns the dimensions of the structure before loading (LV=LH=Lo=0.45 mm and hV=hH=ho=0.05 mm), atomic volume (O = 1.1 10ÿ29 m3) and Burgers vector (b = 2.5 10ÿ10 m). The initial densities of the grown-in dislocations are taken to be roH=1 105 mÿ1 and roV,screw=roV,mixed=5 104 mÿ1. The remaining parameters are listed in Table 1.

The elastic constants c11, c12, c11' and c12' in g and g' phases for the temperature of 7508C were taken from the paper by Pollock and Argon [16]. As the temperature dependence of the elastic constants presented by Pollock and Argon seems to be unrealistically weak, we have chosen the gradient of the temperature dependence found experimentally by Bayerlein and Sockel [17]. The value of the shear modulus G (needed for the calculation of the threshold stresses so) was chosen for the temperature of 7508C in such a way to get a reasonable agreement between the theoretical and the experimental creep curves. The temperature dependence of G was taken again from the paper by Bayerlein and Sockel. The lattice mismatch is more pronounced at higher temperatures [18]. For the diusion coecient we have chosen the expression ÿ290 kJ m2 sÿ1 22 D 5 10ÿ4 exp RT corresponding to the diusion coecient of nickel in an alloy of a composition similar to the g phase in superalloy CMSX-4 [19]. The exact numerical values of the remaining parameters do not follow from the direct independent measurements and will therefore be chosen to be ``reasonable'' and to lead to an agreement between the modelling and the experiments (see Section 6). The slip rate in the g channels [see equation (10)] is characterized by the parameter W and by the threshold stress so. The value of the constant W was taken to be directly proportional to D/kT W 8 10ÿ24

Fig. 3. Dierentiated form of the experimental (points) and theoretical (full lines) creep curves at 7508C. 735 MPa, 775 MPa, 800 MPa.

D m2 : kT

23

The threshold stresses soH and soV can be calculated for given G, b and the width of the g channels

ÿ 1 1 10 5 10 ÿ ÿ 150 ÿ 3 0 4 10ÿ39 400 320 4.4 10 2.9 10ÿ19 7508C 10008C

224 178

147 117

203 161

132 105

40 32

ÿ1.1 10 ÿ2 10ÿ3

ÿ3

7.8 10 6.3 10ÿ16

j p (m) soP (MPa) n

ÿ9 ÿ22

WP (msÿ1 Paÿn) soo (MPa) W (msÿ1 Paÿ1) D (m2 sÿ1) d G (GPa) c12' (GPa) c11' (GPa) c12 (GPa) c11 (GPa)

Table 1. Values of the constants used in the computation

ÿ19

a

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

3427

hH and hV [see equation (11)]. The initial value soH=soV is denoted by soo. The constants WP, n and soP characterize slip in the g' particles. This slip is considered to be active only at the high temperature. The remaining constants are: distance between the jogs p, parameter j determining the kinetics of migration of the g/g' interfaces (increasing with temperature) and parameter a characterizing coarsening of the rafted structure. This coarsening occurs only at high temperature. Figure 3 shows the calculated creep curves at 7508C for sy,app=735, 775 and 800 MPa and for the above listed parameters. The shape of these creep curves can be rationalized in the following way. The creep acceleration at the beginning of the test corresponds to the multiplication of dislocations in the g channels. After about 1% of creep strain both the creep rate and the density of dislocations saturate Ð the almost horizontal part of the curves in Fig. 3 (``quick creep'') corresponds to an equilibrium between the generation of dislocations in the g channels and the dynamic recovery. Both the processes, i.e. the dislocation creep in the g channels and the dynamic recovery, lead to a decrease of the thickness of the vertical g channels hV and, consequently, to an increase of the threshold stress soV. This results in a substantial decrease of the activity of the slip systems in vertical g channels corresponding to mixed dislocations on the vertical g/g' interfaces. Due to the dynamic recovery the dislocation density rV,mixed quickly decreases and recovery is thus also substantially reduced. For the further deformation it is necessary to have a mechanism leading to widening of the vertical g channels Ð the migration of the g/g' interfaces. The steady state by the end of the creep curves (``slow creep'') for e>8% represents the dynamic equilibrium between the slip in the g channels, the dynamic recovery and the migration of the g/g' interfaces. As the migration of the g/g' interfaces is the slowest process, it is the controlling mechanism for the ``slow creep'' in superalloys. The fact that the migration of the g/g' interfaces is the controlling mechanisms has been veri®ed in the following way: the parameter j was increased by a factor of 10; the creep rate increased also nearly by the factor of 10. It is useful to add that for the given parameters (corresponding to 7508C) no ``classical'' rafting can occur at any stress level for reasonable times. More speci®cally, the computation shows that for high applied stresses (about 700 MPa) the g/g' interface migrates in the opposite direction (the vertical g channels widen). For low applied stresses (about 200 MPa), when the vertical g channels are slowly closed by the migration, the development of the rafts requires about 10 years. At the temperature of 10008C the rafting is a common phenomenon. The parameters used in the

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SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

computations of the curves presented in Fig. 4 are listed in Table 1. There are four mechanisms contributing to the creep deformation at high temperatures and at stresses of practical interest. They are (i) dislocation slip in only one type of the g channels (horizontal for negative mis®t alloys or vertical for positive mis®t alloys), (ii) widening of the active type of the g channels as a consequence of the g/g' interface migration leading to rafting, (iii) cutting of the g' precipitates and (iv) coarsening of the rafted structure. No dynamic recovery is possible due to the fact that only one type of the g channels undergoes plastic deformation. According to our calculations the whole process of rafting is completed during the ®rst 0.1% of plastic strain. The rafted structure then coarsens. Slip in the active g channels, the rafting and the consequent coarsening of the rafted structure lead to a substantial increase of the shear stresses in the g' rafts (see Section 6). Thus the rate of cutting increases with increasing creep strain. The contributions of the above listed mechanisms are schematically depicted in Fig. 5. The ®rst minimum corresponds to the end of the plastization of the active channels (horizontal ones for negative mis®t alloys) without any substantial change of their width. In the course of the rafting the width of the active channels increases. This leads to a decrease of the threshold stress in the active g channels and consequently to further deformation of the superalloy. The position of the second minimum is given by the interplay of the diminishing eect of the widening of the active channels and of the increasing eect of the cutting. The process of rafting is accomplished at or slightly beyond the second minimum. The computed creep curves for the applied stresses sy,app=150 and 200 MPa are presented in Fig. 4. The shape of the curves corresponds to that depicted schematically in Fig. 5 and agrees reasonably well with the experimental curves.

Fig. 5. Schematics of the mechanisms determining the shape of the creep curve at 10008C.

Fig. 6. Computed dierentiated creep curves for dierent values of the mis®t parameter at 7508C and 735 MPa.

Figures 6 and 7 show the dependence of the theoretical creep curves on the value of the mis®t parameter d. With the exception of the mis®t parameter the computation was carried out for the constants listed in Table 1. Figure 6 corresponds to 7508C and sy,app=735 MPa, Fig. 7 to 10008C and sy,app=150 MPa. It can be seen that the eect of the mis®t parameter is very strong. At 7508C it determines the extent of the ``quick creep'' region, which practically vanishes for d < ÿ13.5 10ÿ4. For d = ÿ 14 10ÿ4 the shape of the creep curve changes, namely the curve displays a deep minimum at small strains. At 10008C the mis®t in¯uences predominantly the initial part of the creep curve. Both the values of the strain rate corresponding to the ®rst minimum and to the maximum decrease with the decreasing absolute value of the mis®t parameter. For d $ (ÿ13.8 10ÿ4, 0) no plastic deformation occurs in g channels and there is no creep; more speci®cally, the ®rst minimum goes to zero and the creep ceases.

Fig. 7. Computed dierentiated creep curves for dierent values of the mis®t parameter at 10008C and 150 MPa.

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS 6. DISCUSSION

The above presented model is based on ®ve interacting mechanisms. The kinetics of each mechanism must be described by kinetic parameters. In the interaction of the mechanisms further material parameters (e.g. mis®t and elastic constants) play a role. All these parameters may depend on temperature. This might lead to the conclusion that every shape of the creep curve can be obtained by using a suitable combination and temperature dependence of these parameters; then the presented model would be of no value. The aim of the following discussion is to disprove this possibility. There are three categories of the parameters, namely (i) the parameters whose values and temperature dependence can be directly measured, (ii) the parameters whose values can be only roughly estimated and (iii) the parameters whose values can be determined only by the comparison of the results of the modelling with the experiments. To the ®rst category belong the initial values of the parameters L, h, b, O, r, elastic constants c11, c12, c11', c12', G and diusion coecient D. To the second category belong the mis®t parameter d and the parameter j, whose value can be estimated very roughly using equation (19). To the third category belong the parameter W characterizing the kinetics of the slip in g channels, WP, n and soP determining the kinetics of the cutting of the g' precipitates, further p strongly aecting the kinetics of the dynamic recovery and a determining the rate of the coarsening of the rafted structure. The eect of all the parameters of the category (ii) and (iii) will be discussed in the following. The eect of the mis®t parameter d on the creep curves is documented in the preceding section. The results of the modelling indicate that the value of d for the alloy CMSX-4 is not suitable; namely vdv is small at low temperatures and high at high temperatures. If it were just the opposite, the creep properties would be Ð according to the theoretical prediction Ð much better. The parameter W together with the value of ro determine the length of the incubation period and the rate of the onset of the ``quick creep'' at low temperatures. If the value of W is not extremely small, the rest of the creep curve is independent of W. In fact, there is no possibility to determine the parameter W (see equation (10)) from an independent measurement and must be taken to ®t the experimental creep curves. The shape of the creep curves is rather insensitive to the choice of W; a change of W by a factor of 10 does not appreciably change the shape of the modelled creep curve. In simulations at low temperatures presented above we have assumed that there is no cutting of the g' precipitates. Then the parameter j determines the rate of the ``slow creep''. The value of j estimated from the rate of the ``slow creep'' reasonably

3429

well agrees with the value of j estimated using equation (19) and published values of the concentrations ci and ci'. At high temperatures the parameter j determines the time to rafting. From the comparison of the modelling with the experiments it follows that the value of j increases with increasing temperature. It means that the dierence between the values ci and ci' decreases (see equation (19)). We have performed also activation analysis of ``slow creep'' [21]. The results show that Ð due to the increase of j with increasing temperature Ð the activation energy of the steady-state creep is much higher than the activation energy of self-diusion in g phase. The cutting of the g' precipitates is generally assumed to play a role at high stresses for low temperatures or at high temperatures for all stresses. After the rafting is accomplished at high temperatures, the cutting is the necessary prerequisite for further creep process. Then the values of the parameters WP, n and soP can be determined by the comparison of the simulated and experimentally determined creep curves. It is not easy to estimate the contribution of the cutting to the total creep strain at low temperatures. The threshold stress soP for the cutting is given by the relation p g soP 6 APB : 24 b At room temperature the energy of the anti-phase boundary is estimated to be 0.2 J mÿ2 [20]. Assuming that at room temperature the energy is equal to the free energy, this leads to the value soP12000 MPa. We estimate that at 7508C the value of the threshold stress is lower by a factor of 2. The resulting value soP11000 MPa is comparable with the value syPÿsxP coming from the simulations. In our foregoing paper [13] we have shown that the stress syPÿsxP signi®cantly increases during creep. Then the stress sensitivity of the cutting [see equation (13)] must be very high. If the cutting represented the major contribution to the rate of the ``slow creep'', there should be a threshold stress. This was not observed. We can thus believe that the cutting is not the dominant mechanism of ``slow creep'' at low temperatures. The parameter p (mean distance between the jogs on dislocations moving in a non-conservative way along the g/g' interfaces) controls the rate of the ``quick creep'' at low temperatures. The value chosen for simulations (p = 20b) does not follow from any direct measurement, but seems to be reasonable. At high temperatures the non-conservative motion of dislocations along the g/g' interfaces does not take place and the value of p does not play any role. The parameter a controls the kinetics of the coarsening of the rafted structure and can be determined by comparison of the simulation with the

3430

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS

microscopic observations. At low temperatures there is no rafting and the parameter a has no sense. At high temperatures the rafted structure typically coarsens by a factor of 2 or 3 during creep. The chosen value of a corresponds to this degree of coarsening. The coarsening of the rafted structure leads to the increase of the value syPÿsxP and consequently to the acceleration of the cutting and of the creep rate. After the rafting is completed, the stresses syH and syP are identical with the applied stress sy,app. Plastic deformation of the horizontal g channels is very easy at high temperatures and it can be assumed that syHÿsxH will be practically identical with soH. Using equation (11) and the condition of equilibrium of forces for the rafted structure in the x-direction sxH hH sxP LH 0

25

hH 2Gb ÿ syP ÿ sxP sy,app 1 : LH LH

26

we get

For the alloy with volume fraction of the g' particles equal to 70% the bracket has a constant value of 1.43 and the stress syPÿsxP increases during coarsening of the rafted structure (LH increases). The eect of coarsening can contribute up to 30 MPa to the value of syPÿsxP, which can signi®cantly accelerate the process of cutting. Figure 8 shows at which regions of the creep curves the particular parameters discussed above play the dominant role. These diagrams simultaneously oer basic features of the sensitivity analyses for the parameters used in the calculations. The parts of the creep curves, to which the particular parameters are assigned, are strongly sensitive to the values of these parameters. The sensitivity to the remaining parameters is much lower. The last point we would like to discuss is the predictive capability of the model. The model has both the interpolative and extrapolative predictive capability. For example, The parameters for the calculation of the creep curves shown in Fig. 3 were partly taken from independent measurements, partly were determined by ®tting the calculated curves to the experimental points (see Section 5). Having done that it is easy to predict the creep curves for other applied stresses. The only prerequisite for this procedure is the assumption that the mechanisms for other stresses are the same.

Fig. 8. Schematic drawing showing regions of the dominance of the particular kinetic parameters.

experimental results obtained on CMSX-4 single crystals. (i) A good agreement between the calculated and experimental creep curves can be stated both for low and high temperatures. (ii) To each part of the creep curve the controlling mechanisms can be assigned. (iii) The morphological change of the g' particles leading to rafting is possible under low applied tensile stresses only. Under high applied tensile stresses the g' particles elongate in the direction of the applied stress and rafting does not occur. (iv) The value of the mis®t parameter of the CMSX-4 alloy is not the optimum one with respect to the creep properties at both low and high temperatures. AcknowledgementsÐThis research was supported by the Grant Agency of the Academy of Sciences of the Czech Republic under the contracts Nos. A2041608 and A241407. This support is gratefully acknowledged.

7. CONCLUSIONS

The presented model makes it possible to calculate the creep curves of h001i oriented superalloy single crystals containing a high volume fraction of g' particles. The following conclusions can be drawn from the model and from its comparison with the

REFERENCES 1. Henderson, P. J. and McLean, M., Acta metall., 1983, 31, 1203. 2. Feller-Kniepmeier, M. and Link, T., Mater. Sci. Eng. A, 1989, 113, 191.

SVOBODA and LUKAÂSÏ: CREEP IN SUPERALLOY SINGLE CRYSTALS 3. Pollock, T. M. and Argon, A. S., Acta metall. mater., 1992, 40, 1. 4. Feller-Kniepmeier, M. and Kuttner, T., Acta metall. mater., 1994, 42, 3167. 5. Mughrabi, H., Schneider, W., Sass, V. and Lang, C., In Strength of Materials, ed. H. Oikawa et al., ICSMA-10. The Japan Institute of Metals, 1994, p. 705. 6. Rouault-Rogez, H., Dupeux, M. and Ignat, M., Acta metall. mater., 1994, 42, 3137. 7. Kuhn, H. A., Biermann, H., UngaÂr, T. and Mughrabi, H., Acta metall. mater., 1991, 39, 2783. 8. Feller-Kniepmeier, M. and Link, T., Metall. Trans., 1989, 20A, 1233. 9. Li, J. and Wahi, R. P., Acta metall. mater., 1995, 43, 507. 10. MuÈller, L., Glatzel, U. and Feller-Kniepmeier, M., Acta metall. mater., 1993, 41, 3401. 11. MuÈller, L. and Feller-Kniepmeier, M., Scr. metall. mater., 1993, 29, 81. 12. Li, S. X. and Smith, D. J., Scripta metall. mater., 1995, 33, 711. 13. Svoboda, J. and LukaÂsÏ , P., Acta mater., 1997, 45, 125. 14. Svoboda, J. and LukaÂsÏ , P., Acta mater., 1996, 44, 2557. 15. Svoboda, J. and Turek, I., Philos. Mag. B, 1991, 64, 749. 16. Pollock, T. M. and Argon, A. S., Acta metall. mater., 1994, 42, 1859. 17. Bayerlein, U. and Sockel, H. G., In Superalloys 1992, ed. S. D. Antolovich et al. TMS, Warrendale, 1992, p. 695. 18. Royer, A., Bastie, P. and Bellet, D., Philos. Mag. A, 1995, 72, 669. 19. RuzickovaÂ, J. and Million, B., Mater. Sci. Eng., 1981, 50, 59. 20. Hemker, K. J. and Mills, M. J., Philos. Mag. A, 1993, 68, 305. 21. Svoboda, J. and LukaÂsÏ , P., Mater. Sci. Eng. A, 1997, 234±236, 173.

siH (i = x, y, z) siV (i = x, y, z) siP (i = x, y, z) soH soP eiH (i = x, y, z) eiV (i = x, y, z) eiP (i = x, y, z) eiH,pl (i = x, y, z) eiV,pl (i = x, y, z) eiP,pl (i = x, y, z) G, c11, c12, c11', c12' rH rV,screw rV,mixed D u MÇ W WP n p k T b ci c i'

APPENDIX

Di

Nomenclature LH and LV hH hV VP, VH and VV d

0

dimensions of the g particle thickness of the horizontal g channel thickness of the vertical g channel volumes of the particle, horizontal and vertical channel mis®t parameter

D O j a gAPB

3431

normal stress components in the horizontal g channel normal stress components in the vertical g channel normal stress components in the g 0 particle threshold stress in the horizontal channel threshold stress for cutting of the g 0 particles normal elastic strain components in the horizontal g channel normal elastic strain components in the vertical g channel normal elastic strain components in the g 0 particle normal plastic strain components in the horizontal g channel normal plastic strain components in the vertical g channel normal plastic strain components in the g 0 particle elastic constants density of dislocations deposited on the horizontal g=g 0 interfaces density of screw dislocations deposited on the vertical g=g 0 interfaces density of mixed dislocations deposited on the vertical g=g 0 interfaces shift of the horizontal g=g 0 interface to the center of the g 0 particle due to the migration of the g=g 0 interfaces thickness of the layer of atoms deposited on the horizontal g=g 0 interfaces due to the dynamic recovery the rate of the relative increase of the dimensions in the unit cell due to coarsening of the rafted structure parameter characterizing the rate of slip in the g=g 0 channels parameter characterizing the rate of slip in the g 0 particles stress exponent mean distance between the jogs measured along the dislocation line Boltzmann constant absolute temperature magnitude of the Burgers vector concentration of the ith element in the g phase concentration of the ith element in the g 0 phase diusion coecient of the ith element in the g phase average coecient of self-diusion in g phase atomic volume parameter characterizing dierence in the chemical composition of the g and g 0 phases parameter characterizing the rate of coarsening of the rafted structure free energy of the anti-phase boundary with h111i orientation