Aeta metall, mater. Vot. 38, No. 10, pp. 19771992, 1990 Printed in Great Britain
09567151/90 $3.00 + 0.00 Pergamon Press plc
CREEP DEFORMATION OF SINGLE CRYSTAL SUPERALLOYSMODELLING THE CRYSTALLOGRAPHIC ANISOTROPY R. N. G H O S H 1'2, R. V. C U R T I S I and M. M c L E A N I ' t IDivision of Materials Metrology, National Physical Laboratory, Teddington, Middx, England and 2National Metallurgical Laboratory, Jamshedpur831007, Bihar, India (Received 13 February 1990)
AkstractA generalised model of creep deformation in cubic single crystals is developed that considers the combined effects of viscous glide on two (or more) slip systems and accounts for tertiary creep by the accumulation of mobile dislocations with plastic strain. The model is applied to analyse a database of creep curves for the nickelbase single crystal superalloy SRR99 with the assumption that creep deformation occurs by glide on both the { 111} ( r 0 l ) and {001} (110) systems. A procedure for the calculation of creep curves and associated crystal rotations for arbitrary crystal orientations is described. The model predicts changes in the anisotropy of creep behaviour with stress and temperature that is in general agreement with the limited available experimental data. It also includes the contributions of all possible slip vectors to predict crystal rotations that are consistent with observations. RrsumrOn drveloppe un modrle grnrralis6 de drformation par fluage dans des cristaux cubiques, en considrrant les effets combinrs du glissement visqueux sur deux (ou plus) systrmes de glissement et en tenant compte du fluage tertiaire par accumulation de dislocations mobiles par suite de la drformation plastique. Le modrle est appliqu6 ~. l'analyse des donnres de courbes de fluage pour un superalliage SRR99 monocristallin ~i base de nickel en supposant que la drformation par fluage se produit par glissement sur les deux syst~mes {111}(]'01) et {001}(110). On d~crit une procrdure pour calculer les courbes de fluage et les rotations associres du cristal pour des orientations arbitraires du cristal. Le modrle prrdit des changements de ranisotropie du fluage en fonction de la contrainte et de la teml~rature, qui sont en accord avec les quelques donnres exp~rimentales disponibles. Le modrle tient compte aussi des contributions de tousles vecteurs de glissement possibles pour prrvoir des rotations du cristal qui sont en accord avec les observations.
ZusammenfassungEinveraUgemeinertes Modell der Kriechverformung in kubischen Einkristallen wird entwickelt. Es betrachtet den kombinierten Einflul3 der viskosen Gleitung auf zwei (order mehr) Gleitsystemen und berficksichtigt das terti/ire Kriechen durch Akkumulation beweglicher Versetzungen w~ihrend der plastischen Dehnung. Das Modell wird auf die Analyse von Kriechkurven von Einkristallen der NickelbasisSuperlegierung SRR99 unter der Annahme angewandt, dal3 die Kriechverformung durch Gleiten sowohl im {111}(1"01) wie auch im {001}(ll0)Gleitsystem erfolgt. Ein Verfahren fiir die Berechnung der Kriechkurven und der auftretenden Kristallrotationen wird fiir beliebige Einkristallorientierung vorgestellt. Das Modell sagt ~,nderungen in der Anisotropie des Kriechverhaltens mit der Spannung und der Temperatur voraus, die mit dem begrenzten zur Verffigung stehenden experimentellen Ergebnissen allgemein iibereinstimmen. Das Modell sehlieBt die Beitr/ige s/imtlicher mrglicher Gleitvektoren ein, um Kristallorientierungen voraussagen zu krnnen, die mit den Beobachtungen vertr/iglich sind.
I. INTRODUCTION The development of a new generation of single crystal superalloys and their introduction into commercial service as turbine blades in aeroengines have been among the major developments in gasturbine technology over the last decade [1]. Single crystal manufacturing technology has evolved so rapidly that it can be argued that these materials have been applied commercially before there has been a comprehensive scientific understanding of their behaviour. Consequently, it is unlikely that their full potential has been achieved. The increased temtPresent address: Department of Materials, Imperial College, London SW7 2BP, England.
perature capabilities of single crystal superalloys such as P W A 1480, SRR99 and C M S X 2 which are about 50°C higher than those of earlier conventionally cast and directionally solidified alloys (e.g. M a r M200, M a r M002) can be exploited by established design procedures. However, the highly anisotropic behaviour of single crystals must be characterised before advanced computerbased design procedures can be fully implemented. There have been several experimental studies of the anisotropic creep behaviour of single crystal superalloys [2, 3] that have given apparently contradictory results. In particular, at relatively low creep temperatures ( ~ 760°C), where the level of anisotropy of stress rupture lives is often greatest, there has
1977
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GHOSH et al.: CREEP DEFORMATION OF SUPERALLOYS~MODELLING
been some controversy over whether (100) or (111) crystal orientations are strongest in uniaxial tensile creep. Recently, Caron et al. [4] have shown that the y' size and morphology, modified by various heat treatments, can have a profound effect on both the magnitude and order of this anisotropy. Nevertheless, there is a consensus that the degree of anisotropy of creep behaviour decreases with increasing temperature with (111) often having better creep performance than (001) orientations up to temperatures of ~ 1000°C. There are suggestions that the anisotropy may be reversed (i.e. (001) stronger than (111)) and increase again at even higher temperatures in experimental single crystal alloys such as PWA 1484, CMSX4 and MC2. Winstone [5] has reported the opposite trend for SRR99 with the cube orientation being strongest in creep at the lower temperatures. It is impractical to fully characterise the creep behaviour of single crystals of all possible orientations as functions of stress and temperature. Rather, there is a need to develop a rational basis for predicting the performance for arbitrary orientations and loading conditions from a limited database. The deformation mechanisms in y'strengthened superalloys can be very complex depending on the distribution of ~' and the conditions of deformation. Early work by Kear et al. [6] on conventional nickelbase superalloys in the single crystal form suggested that in tensile tests that octahedral slip, on {111 } (1 (}1), occurred at 750980°C and that cube slip was additionally activated at higher temperatures on {001}(110). More recent studies, for example by Sun and Hazzledine [7] on the modern single crystal alloy SRR99, have found no evidence of the activation of a primary cube slip system in compression tests but have confirmed the importance of crossslip on cube planes during macroscopic octahedral glide of dislocations. Studies of dislocation activity during the early stages of creep of CMSX2, by Caron et al. [4], also show the dominance of octahedral glide over a wide range of temperatures with dislocations having either ( I 0 1 ) or ( E l l ) Burgers vectors; there was no evidence of extensive cube slip although it is believed that some cube crossslip must affect the deformation rate on octahedral planes. It must also be borne in mind that these observations have been made on specimens deformed to low strains (~< 1%), to ensure visibility of the dislocations whereas tertiary creep occurs to strains of about 2530%. Moreover, Caron et al. [8] have previously observed cube slip at 760°C after tensile deformation of CMSX2. Following the empirical observation of Dyson and McLean [9] that the tertiary creep rate in conventional nickelbase superalloys, and indeed in other engineering materials, increases monotonically with accumulated plastic strain, physical models of creep in such materials have been proposed that depend on changes in the density and kinetics of
mobile dislocations [10, 11]. These physical models, which are generally consistent with microstructural observations [12, 13], have been expressed in an empirical formulation, using the formalism of continuum damage mechanics, and incorporated into a computerbased approach to the analysis, representation and simulation of the full shapes of creep curves. The system, designated CRISPEN, represents creep behaviour in isotropic engineering materials and has been successfully applied to a range of alloys [14]. The present paper describes an extension of this analysis to incorporate the anisotropic creep behaviour of single crystals, In common with the model used in the earlier version of CRISPEN, the present model is guided by current understanding of deformation mechanisms in modern single crystal superalloys. A preliminary account of the model has been given elsewhere [15]; the present paper gives a fuller development of the model and explores its implications in a range of calculations and data analysis. 2. PHYSICAL MODELS 2.1. lsotropic model
Strain rate acceleration during tertiary creep can be caused by a variety of damage mechanisms associated with (i) evolution of the microstructure of the materials, (ii) development of damage (e.g. porosity, cracks) that affects the internal material continuity or (iii) changes in the material geometry; these have been reviewed in some detail by Ashby and Dyson [16]. In many situations tertiary creep will have contributions from more than one of these mechanisms. However, the following discussion will be limited to modelling tertiary creep due to evolution of the intrinsic material microstructure in complex alloys [type (i) damage]. This is particularly relevant to (a) high ductility materials, such as directionally solidified and single crystal superalloys, where failure is usually determined by geometrical instabilities (e.g. necking) rather than by cavitation/cracking damage and (b) constant stress rather than constant load testing, although the difference between these two types of loading is relatively small in nickelbase superalloys. The models of dislocation creep in isotropic materials have been developed elsewhere [911] but will be summarised here to place the extension to anisotropic materials in context. Creep in simple solid solution metals, which are subject to plastic flow on loading by virtue of their low yield stresses, is characterised by a relatively stable dislocation substructure in which most dislocations are immobile; sophisticated models [17, 18] based on a balance between the slow coarsening of the dislocation substructure (recovery) and occasional rapid motion of isolated dislocation segments leading to dislocation generation (hardening) have been successful in accounting for the principal features of the creep behaviour
GHOSH et al.: CREEP DEFORMATION OF SUPERALLOYSMODELLING of simple metals. However, most engineering alloys have very high yield stresses because of additional strengthening mechanisms (e.g. particles, solutes) that essentially inhibit glide. It has been proposed that no stable dislocation structure is readily established in these materials; rather, most of the dislocations are mobile over all of the creep life, albeit with low velocities dependent on dislocation climb. In this case the creep rate ~ is given by the well known Orowan equation (1)
= pbv
where p is the (mobile) dislocation density, v is the dislocation velocity and b is the Burgers vector. The rate of generation of dislocations p is taken to be proportional to the dislocation density p and the dislocation velocity v as proposed by Taylor k
~_'~= kpv = ~ ~
(2)
where k is a constant. In the following discussion we take v to be a function of stress and temperature and to be independent of p; Dyson [11] has explored the implications of v being modified by internal stresses associated with p at very low applied stresses, but this will not be considered here. By defining a dimensionless damage parameter ~ (O=
P  Pi Pi
equations (1) and (2) can be rewritten as a coupled set of differential equations to represent the tertiary creep behaviour, p~ is the initial dislocation density (~ = ~i(1 + co)
(3)
6) = C'~ where k bp~
and ~ is the initial creep rate. Ion et al. [10] have shown that primary creep can be incorporated in this analysis by the introduction of a second state variable S and of an appropriate evolution law for S ~"  ~ i ( l   S ) ( ]
S = H~(I  S)
+ co)
R'S
(4)
~b = C'~ In view of the small levels of primary creep observed in most single crystal superalloys, the extension of the model to account for anisotropy will be restricted to a tertiary creep analysis in order to simplify the presentation. However, there is no difficulty in principle in introducing a primary creep contribution.
1979
The four parameters (~, H, R', C') completely define the shape of the creep curve that can be generated by numerical integration of equation set 4. A software package, designated CRISPEN that operates on IBM compatible personal computers, has been described previously [14] that utilises equation set 4 to develop a database of these model parameters that can represent the detailed straintime characteristics of creep curves for a range of test conditions and allows calculation of strain as a function of time for arbitrary uniaxial loading conditions. This tensile formulation of creep deformation has been applied to a database of [001] oriented single crystals of the nickelbase superalloy SRR99 by Curtis et al. [19]. 2.2. Anisotropic model
The development of equation set 3 is phenomenological. The equations associate the progressive increase in creep rate with the generation of additional mobile dislocations without specifying how they move or multiply (e.g. by glide, climb, or a combination of both). The resulting set of equations describe an isotropic creep behaviour. However, we have reviewed in Section 1 the evidence relating to dislocation motion in single crystal superalloys showing that it appears to occur by viscous glide on a few well defined slip systems. Ghosh and McLean [15] have given a preliminary account of a model in which the tensile creep strain E is produced by combining the time dependent shear strains ),k on all operative slip systems. In particular, the model envisages the changing anisotropy of creep behaviour in single crystal superalloys to result from differing contributions of octahedral and cubic slip. However, the model will be first described here in general form before considering the specific case of superalloys. Plastic deformation in a crystal by glide results in simple shear displacements which can be regarded as the combination of strain and rotation components. Therefore, when a single crystal of arbitrary orientation is subjected to uniaxial deformation, as in a creep test, its orientation is likely to change. However, the indices of the slip system with respect to the crystallographic axes are invariant. Hence, it is convenient to represent the components of deformation using the crystallographic axes as the main frame of reference. If yk is the magnitude of shear strain on the kth slip system represented by ( n ~ n k n k ) [ b k b k b k] where n k and b k represent the components of the u n i t vectors defining the slip plane and slip direction respectively (Fig. 1), the resulting displacement gradient component e~ is given by ek=
? k bik nj.k
(5)
The suffixes i, j represent components along the three cubic crystal axes and have integer values of 1, 2 or 3 only. Equation (5) represents the contribution from a single slip system k.
1980
GHOSH et al.: CREEP DEFORMATION OF SUPERALLOYSMODELLING respect to the cubic crystal axes (i.e. zero rotation), but does lead to a change in magnitude of the orientation vector. This condition occurs when the tensile axis coincides with a crystal axis of high symmetry such as [001], [011] or [111]. However for crystals stressed in an arbitrary orientation, defined within the stereographic triangle by the parameters (0, ~,) defined in Fig. 2, the deformation gradient matrix will be asymmetrical leading to both rotation and elongation. The appropriate matrix can be constructed from equations (5) and (6) when the magnitude of ?k for specific slip systems are known. Following the same reasoning as was used to derive the tensile creep rate equations, but restricting dislocation activity to (n k n~ n k) [b k b k b k] slip vectors, the shear creep rate ~k can be expressed by a modified Orowan equation
O,E
,O,K.O2K.O3 ~0
/.~,~
'I:K'"YK
~ = pkbvk.
(9)
The change in dislocation density on this plane can be written
1
Pk=~kPkVk

(lO)
~k~)k b
Fig. 1. Schematic illustration of the relationship of shear stress C and shear strain ?k in the crystallographic direction (n~, n~, n~)[b~, b~, b~] in relation to the tensile stress a and tensile strain e.
where ctk is a constant. Again, defining a dimensionless damage parameter
If we allow contributions to the creep deformation from simultaneous slip on N slip systems, the total displacement gradient eo is given by the summation
where p~ is the initial dislocation density pertaining to the kth slip system, it follows that a simplified set of equations can describe the shear creep rate ~k
N
eq= ~ e~.
(6)
~ = ~(1 + ~ )
(ll)
k=l
As a result of this small deformation, the original tensile axis represented by the vector [tl t2t3] rotates to a new position [T~ 7"2T3] relative to the crystallographic frame of reference (Fig. 2). Using matrix notation, the deformation can be represented by the following transformation [T~] [ l + e H T2 = e21 T3 e31
e~2 e l 3 .J i l l ] 1[e22 e23 / h . e32 1 Jr e33J t 3
(7)
tl=sin ¢psin O
The transformation matrix in the above equation is known as the deformation gradient matrix and it completely defines both the change in length and orientation of the crystal. For example, the axial strain E can be determined from the difference in magnitude of the vectors representing the displacement of the tensile axis before and after the deformation E=
Tt 7
where [3k = ~k/bp~. By combining equation set 10 with equations (5) and (6), individual tertiary tensile creep curves can be specified by the two modelparameters (~ik, ilk) in combination with the appropriate crystallo
^
~
(tl,t2,t3)
(8)
A symmetric deformation gradient matrix gives no change in the orientation of the tensile axis with
Fig. 2. Schematic illustration defining the orientation parameters (0, ~) and the rotation of the tensile axis from (t. t2, 13) to (T I, T2, T3).
GHOSH et al.:
CREEP DEFORMATION OF SUPERALLOYSMODELLING
graphic transformations. Clearly (~ik, ilk) are closely related to the analogous parameters (~i, C') that are used in the CRISPEN tensile formulation through geometrical factors that depend on the active slip systems and which can be derived from equation (6). In order to interpolate and extrapolate from an extensive database it is necessary to describe the model parameters (;~, ilk) as functions of stress and temperature. There are several ways of representing the dependence of creep rate on stress and temperature (e.g. as power law, exponential or hyperbolic functions). Curtis and McLean [19] have shown that in the tensile formulation of creep deformation, the initial characteristic creep rates ~i for the single crystal superalloy SRR99 were well represented by an exponential function of stress. Adopting a parallel formulation for shear creep rates here, we take Yik= a~ e x p ( a ~ z k  Q~T)
(12)
where a~, a~ and Q~ are constants, R is the gas constant and z k is the resolved shear stress on the kth slip system. The strain softening coefficient fig is also described by a parallel procedure as that used by Curtis and McLean [19] for the tensile formulation. At low stresses flk has a constant value that is independent of stress and temperature; at higher stresses flk is described by the following exponential formulation
flk = a~ exp
1
where a k, a~ and Q~ are also constants. At a given temperature and stress the minimum value of flk given by the two algorithms is applicable. 3. APPROACH TO DATA ANALYSIS Association of the model parameters in the shearcreep formulation of the linearstrainsoftening model of tertiary creep, described in 2.2, with characteristics of creep curves obtained from conventional creep tests requires a knowledge (or assumption) of the specific slip systems that are operating during a
1981
particular test. For nickelbase superalloys in general, and f.c.c, single crystal materials in particular, the experimental evidence restricts the choice to three possible systems {lll}(TO1) { 111 } (TI2> [001](110). The Schmid factors for various vectors of these types associated with tensile stresses along the principal crystal directions[001], [11 i] and [01 l]are listed in Table 1. In the present analysis only slip in the (T01) direction will be considered for both octahedral and cube plane glide. Under creep conditions, particularly in conditions of low stress and long lives, dislocation activity is largely restricted to the f.c.c. 3, matrix. The smallest perfect dislocation for octahedral slip in the f.c.c, lattice has a Burger's vector of l(T01). Although this can dissociate into two partials with Burger's vectors of magnitude ~(TI2) separated by a stacking fault, macroscopic deformation will require translation of the complete dissociated complex with total Burger's vector ½(i01). Movement of isolated ~ ( I I 2 ) partials would lead to profuse generation of stacking faults which are not observed. When cutting of the coherent y' particles occurs, movement of a dislocation of ½(T01) Burger's vector creates an antiphase boundary in the ordered y' lattice which is annhilated by the movement of a second identical dislocation. There have been isolated observations of perfect dislocations with very large Burger's vectors, such as (TT2), that can be considered to be formed by combining several normal ½(101) dislocations. However, these should be energetically unstable since the energy of six isolated ½(I01) dislocations needed to form a (TT2) Burger's vector would be half of that of the product. Thus macroscopic slip in a (TT2) direction is unlikely to be a dominant factor in creep deformation of superalloys, although microscopic evidence of its occurrence is quite possible. Much of the support for (1"1"2) slip has been based on observations of lattice rotations that have been inconsistent with (1"01) slip. However, we shall demonstrate below that there is an alternative explanation for this
Table 1. Schmidfactorsand the number of possibleslipsystem{111}(r01), {111}(i]2) and {001}(110) slip associatedwith (001), (011) and (111) orientedcrystals {lll}(T01) {11I} (T'i'2) {001}(110) No. of slip No. of slip No. of slip Tensile axis systems SF systems SF systems SF (001) 8 0.4082 4 0.4714 6 0.0000 4 0.0000 8 0.2357 (011) 4 0.4082 2 0.4714 4 0.3536 8 0.0000 4 0.2357 2 0.0000 6 0.0000 (111) 6 0.2722 3 0.3143 3 0.4714 6 0.0000 6 0.1571 3 0.0000 3 0.0000
1982
GHOSH et al.: CREEP DEFORMATION OF SUPERALLOY~MODELLING
type of behaviour based on competition between octahedral and cube slip. For tensile stresses along [001] there is zero resolved shear stress on the [010] and [100] cube planes, so {001 } (110) slip should not operate during creep of [001] oriented single crystals. Consequently, analysis of creep data for [001] oriented crystals should provide a measure of the appropriate model parameters associated with {111} (T01) and similar types of slip vectors. For creep curves produced by a tensile stress along the symmetrical [001] axis there are simple crystallographic transformations between the parameters of the tensile formulation of tertiary creep by linear strain softening (ii, C') and the parameters of the shear formulation (~k, ilk). These are listed in Table 2 where it is assumed that all eight { 111 } (T01) type vectors with finite Schmid factors of 0.408 operate. It is therefore convenient, in practice, to adopt the data analysis techniques described by Ion et al. [10] and Barbosa et al. [14] to establish (~i, C') and to compute databases for the shear model (~k, ilk) by means of these transformations. Tables 1 and 2 include the Schmid and geometrical factors associated with {111} ( i i 2 ) glide although they a r e not specifically considered in the following detailed analysis. Figure 3 shows a comparison of the creep curves for a [001] oriented specimen of SRR99 produced by both the tensile and shear formulations using values of the model parameters derived in this manner; the original experimental data are also shown to demonstrate the level of agreement between the model and the data. The curves have been calculated using the measured values of the model parameters (~i,C') or ($k, flk) by numerical integration of
Table 2. Geometricalfactors for convertingtensilestrain into shear strain along differentslip directions Shear strain (?) Tensile strain (E) {III} {111}(TT2> {001}<110> <001 ) 8/.,/6 8/3x/2 0 (011 ) 4/x/6 4/3x/2 ,/2 ( 111) 4/x/6 4/3v/2 x/2 For
8
example: E(001> = ~ ; 7 {111}( l'01>. Vo
equation sets (3) or (11). This was accomplished on an IBM AT compatible personal computer using a modified RungeKutta method in which the time steps were adapted to the current gradient of the straintime plot. By examining creep curves for [001] oriented crystals produced over a range of test conditions it is possible to identify the functional dependence of ~* and fl* for { l l l } (TOl) glide on stress and temperature. With this information the contribution of octahedral glide to the deformation of arbitrary orientation can be calculated. This simplistic approach will always predict that creep performance scales inversely with the Schmid factor associated with the appropriate octahedral shear slip mode but that the magnitude of the anisotropy depends on the stress sensitivity of ~ k. This procedure predicts a much higher level of anisotropy than is generally observed and that [001] always has the lowest possible creep strength, contrary to observation. For [111] oriented crystals all of the likely types of slip systems have finite Schmid factors although that for the {001 } (110) type deformation is considerably greater than those for any of the octahedral glide
SRR99 4 5 0 M P o 850°C <001> 0.17 
I
I
[3
Sheer
 
l
I
I
I
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i
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0 0
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I 3
,
I 4
i
I 5
,
I 6
i
I 7
,
I 8
i 90
T i m e lOOh
Fig. 3. Comparison of the fits of the tensile and shear formulations of the model to a typical experimental creep curve for (001) tensile orientation at 450 MPa/850°C.
GHOSH et al.:
CREEP DEFORMATION OF SUPERALLOYSMODELLING
systems. Where both cube and octahedral slip have been activated and the kinetics of dislocation motion are similar, there will be little error in assuming that the tensile deformation of [ l l l ] oriented crystals is restricted to shear on { 1 0 0 } ( l l 0 ) systems. (As discussed above the octahedral shear contributions are negligible). Consequently a similar procedure, as was used for [100] creep data, can be used to transform the tensile model parameters (~, C') for [ll 1] oriented cr).stals to those of the shear formulation for cube slip. The relevant transformations are also given in Table 2. Figure 4 compares creep curves calculated using the tensile and shear formulations of the model, for optimised values of the model parameters, with the experimentally measured creep data for [111] oriented crystals of SRR99. Having established the characteristics for octahedral and cube shearcreep by analysis of creep data for orientations where only one type of deformation dominates, it is possible to calculate the creep strains and crystal rotations associated with arbitrary orientations where both types of slip systems are active. This assumes that the strains from the two deformation modes are additive. 4. ANALYSIS OF CREEP CURVES 4. I. [(901] O r i e n t a t i o n s
A database of 27 tensile creep curves obtained by constantstress testing of [100] oriented specimens of single crystal SRR99 in the temperature range 750950°C has been analysed by the procedure described in Section 3. The data were supplied by Dr M. Winstone of the Royal Aerospace Establishment who carried out tests on specimens prepared
SRR99 0.18 
'
400MPo '
'
850°C '
'
from 12mm diameter cylindrical rods of SRR99 that were supplied in the fully heattreated form by RoilsRoyce pic. Specimens for which the tensile axes were within 5° of [001] were considered to be of exact [001] orientation for the purposes of the following analysis. The precision of the measured crystallographic orientations determined by Laue Xray analysis is probably no better than + 2 °. Data for creep at 1050°C were available but have not been included in this analysis because of the additional complication of 7' rafting which appears to radically alter the deformation mechanism. The analysis of these data by the tensile strain softening model, incorporating a description of primary creep, will be described in detail elsewhere [19]. In the context of the present paper they have been reanalysed in terms of the dominant tertiarycreep contribution alone and the shear modelparameters (~k, flk) have been calculated. These are shown as functions of stress and temperature in Fig. 5. The lines consistent with the optimum analytical descriptions of these parameters for { 111 } (]'01) slip using equations (12) and (13) are shown. The constants of equations (12) and (13) are given in Table 3. The analytical representations of ~k and ilk, using the optimum values of the constants (Table 3), have been used to calculate the creep curves for each condition of stress and temperature for which measurements were made on [001] crystals. A comparison of the measured and predicted times to 1 and 5% strain, and to fracture is given in Fig. 6. Generally the calculation and measurement are in agreement within _+30% although prediction of times to low strains at 750°C are considerably in error because of
<111> '
'
~
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Fig. 4. As Fig. 3 but for (111) tensile orientation tested at 400 MPa/850°C. AM 38/10~M
1983
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GHOSH
CREEP
et al.:
105
DEFORMATION
OF
SUPERALLOYSMODELLING
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1000
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Shear stress Fig. 5. V a r i a t i o n o f m o d e l p a r a m e t e r s f o r {I 11} ( T 0 1 ) glide, d e r i v e d f r o m < 0 0 1 ) tensile c r e e p d a t a , as f u n c t i o n s o f stress a n d t e m p e r a t u r e . (a) ~, a n d (b) 8 k.
the relatively large primary creep strains that occur under these conditions and that are not included in the analysis. It is noteworthy that an uncertainty
in temperature measurement of ___2°C, which is within the range of good creep testing practice, would account for a scatter in lives of ~ _+10%.
Table 3. Material constants for creep of SRR99 in tensile and shear formulations Tensile formulation for Parameters a~ a~ a~ a*4 Qk Qg
Shear formulation for
(001)
(111)
{111}(101)
{001}(110)
Unit
9.3 x l0 is 1,77x 10 2 1.36 x 10 2 9 . 4 4 x l 0 3 552 131
2.53 x 1 0 5 5 . 5 7 x 10 3 7.56 x 10s 4.55x10 3 85.6 82
2.85 x 10 ]3 4 . 3 4 x 10 2 4.44 x 10 2 2 . 3 x 10 2 552 131
1.79 x 10 5 1.18× 10 2 1.07 × 106 9.65×103 85.6 82
s ~ MPa i
RT/ C = , , e×p~ ÷ ~ 
, ) or 680 (sma,er or the t~o value~).
MPa i kJ m o l i kJ mol I
GHOSH et al.:
103
CREEP DEFORMATION OF SUPERALLOYSMODELLING
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103
Calculated time to I°I.strain (h)
104
(b ) + 750°C ~ 800oc [] 850°C 0 900oc 0 950°C A 1000oc
¢C
Shear formulat ion
103
O L,.
102
~
J _~" ..Af~ +
+
O
10
/
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/
""
+
L
101
i
i lllliil
10I
1
b , ,~,,,,I 10
, I III111[ I I I IIIII] I I 111111 102 103 104
Calculated time to 5°1ostrain (h)
104
(C)
Shear formulation + 750°C 800°C
103
/
D 850°C O 900oC
S
J 0 / + 0
¢ 950oC
~
102
I[
10
1
IVl
J
Ilill,I 10
I
I Illltl
I
lO2
I
I IllllJ
+
+
I
103
I
I IIIII
104
Calculated life (h) Fig. 6. Comparison of measured times to various creep strains with the values calculated using the analytical representation of the model parameters shown in Fig. 5. For <001 > tensile specimens at various stresses and temperatures. (a) Time to I% strain. (b) Time to 5% strain. (c) Time to rupture.
1985
GHOSH et al.: CREEP DEFORMATION OF SUPERALLOYSMODELLING
1986
4.2. [I11] Orientations
(a) 10  7
Fewer creep data are available for [111], than for [001], oriented crystals, consequently a slightly different strategy had to be adopted to estimate the parameters associated with cube shear deformation. It was, first of all, established that a model based on octahedral glide alone, using the parameters evaluated in Section 4.1, predicted strains that were very much less than those observed; this is illustrated in Fig. 7. The experimental creep curves for [111] oriented crystals were, therefore, analysed as described in Section 3 to give the parameters associated with {001}<110> cube slip. Only five creep curves for [111] oriented SRR99 were suitable for analysis and these had a small range of lives and creep rates. A best fit of the parameters ~k and flk to equations (12) and 0 3 ) was determined numerically. These quantities are displayed in Fig. 8 as functions of the compound parameters (a*2rk  Q~/RT) and ( Q ~ / R T  a ~ r k) respectively for the optimised values of the constants a2*, Q~, a~ and Q~. The values of the various constants for the creep deformation of l! 11] oriented crystals in terms of the tensile and shear formulations of the model are listed in Table 3. The latter assumes that shear deformation in [ l l l ] oriented tensile creep specimens only occurs by {001} slip. Figure 9 compares the creep lives and times to 5 and 10% strain calculated using the values of constants listed in Table 3 with available creep data. The agreement to about _+30% approaches the intrinsic scatter in creep results. 5. S I M U L A T I O N
OF CREEP
FOR ARBITRARY
CRYSTAL ORIENTATIONS The parametric representations of timedependent glide on {lll} and {001}<110> have been 0.30
5RR99 450MPo , , J i D <>
,
850°C , ,
,
,
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,
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,
,
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r
i
,
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/ <111>
2
3
4
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o
7
~
<001>
8
9
Temperature x 1050°C + 850°C 950°C D 750°C D
loe
Fig. 7. Comparison of measured creep curves for <00l > and <111> oriented specimens of SRR99 tested at 450MPa/ 850°C with the calculated curves assuming that only {111} <1"01> glide operates.
I
I
I
I
I
I
I
68
66
64
62
6
58
56
(o2'mshear
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54
)
(b) 1000 Temperature x 750°C
+ 850°C ~ 950oC
100
13 I 0 5 0 ° C
pk 10
1 16
I
I
I
I
I
I
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15
14
13
12
11
10
9
(OlRTa4~'sheor
stress
)
Fig. 8. Variation in model parameters for {001} < 110> glide, derived from <111> tensile creep data, as functions of a parametric representation of stress and temperature. (a) ~k and (b) ilk. used to calculate the creep strains resulting from the combined actions of octahedral and cube slip. As discussed above, the creep deformation of [001] and [111] oriented crystals should each be dominated by a single type of slip system: octahedral for [001] and cube for [111]. However, the differing stress and temperature dependencies of the various model parameters will influence the relative creep performance of these two orientations. Figure 10 shows a selection of creep curves calculated for a range of temperatures and stresses together with the available experimental curves when available; the predictions generally agree with the trends in the anisotropy in creep behaviour. At both 850 and 950°C, [001] oriented crystals have the longer creep lives at low stresses but [111] oriented crystals show better creep performance in high stress/short life tests. In considering the effect of temperature at a constant stress, the calculations for 450 MPa show a trend to increasing anisotropy with increasing temperature [111] crystals having progressively better creep performance than [001]. By contrast [001] is superior at low temperatures. Data for [011] orientations are often presented in the literature as an indicator of the overall anisotropy. Like [001] and [111], this is a symmetrical orientation which does not change during plastic deformation. Simulation of creep for exact [011]
~oo
T i m e 100 h
x 7
GHOSH et al.:
CREEP DEFORMATION OF SUPERALLOYSMODELLING
(a) 03
P r e d i c t e d t i m e in h
10 2 Temperot ure
x
950
+
1050 850
13 t
i
i
i
i
i
~ii
i
i
,
750 ,
,
,
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103
Ex~zerimentol t i m e 1o 5"1, s t r o i n , h
(b) Predicted t i r r e in h 10 3
tO 2
/
10
L
10
I
I
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I
I I
102
L
[
x
950
+
1050
I
I
I
l
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Fig. 9. As Fig. 6 but for < 111 > creep data. (a) Time to 5% strain. (b) Time to 10% strain.
orientations using the model parameters derived above often predicts a creep performance that is significantly better than that of [001] which is contrary to experimental measurements. However, [001] and [ I l l ] are stable orientations where small deviations from the exact orientation do not lead to large crystal rotations; by contrast slightly misoriented [011] crystals rotate away from the desired orientation with accumulated plastic strain. Figure 11 shows the calculated rotations and associated creep curves for a series of crystal orientations that deviated by about 5° from [001], [111] and [011]. The misorientation has least effect on the creep curves for crystals close to [111] since the predicted rotations bring each crystal closer to the nominal I l l 1] orientation. This does not occur for either [001] or [011] where rotations retain or increase the original misorientation and this leads to inferior creep performance than that of the exact low index orientation. The database established above has been used to calculate creep curves for [001], [011] and [111] oriented crystals of SRR99 tested at 850°C with a stress of 400 MPa. Data for these test conditions are available from a previous programme and only the [111] creep curves have been used in the establish
1987
ment of the parameter database. The calculated and measured creep curves are shown in Fig. 12. These have the same order and approximately the same magnitude of anisotropy in creep performance. The calculation for [011] and [001] have assumed exact low index orientation; closer agreement would be obtained by assuming a small deviation from this orientation which is probably closer to the experimental reality. The calculations, however, cannot account for the lower ductility of the [011] crystal. For arbitrary orientations, particularly close to the central zone of the unit stereographic triangle, both octahedral and cube slip can, in principle, make significant contributions which will be very sensitive to stress and temperature. Figure 13 shows specimen calculations for creep curves for a series of initial orientations and the crystal rotations associated with the creep strains are also shown. It should be noted that these rotations result from the activation of many slip vectors of two general families of slip systems and are quite different from those that are generally quoted as arising from the operation of a single slip vector. Experimental data are available for the orientation A, and the measured creep curve is included for comparison. Creep curves for a given stress and temperature have been calculated for initial starting orientations distributed uniformly throughout the unit stereographic triangle. Starting points were taken at orientations (0, ~), defined in Fig. 2, where 0 and ~b are progressively increased in steps of 5°. The anisotropy in creep performance can be displayed as contours of either the times to achieve specific strains or the strains at specific times as shown in Fig. 14. The calculated rotations can be extremely sensitive to the stress and temperature. Figure 15(a) shows the changes in orientations associated with combined octahedral and cube glide at different stresses for the same initial orientation [123] at 850°C. The associated creep curves are shown in Fig. 15(b). The change in rotation is a consequence of octahedral slip becoming less dominant, and of cube slip becoming more important as the stress decreases. 6, DISCUSSION The most controversial assumptions of the present model are the neglect of any contribution to m a c r o scopic deformation from {lll} slip vectors and the assumption that the {001} <110> in addition to the widely reported {111 } system can contribute to creep deformation over a range of stresses and temperatures. As has been detailed in Section 3 there is certainly microscopic evidence of dislocations with Burgers vectors of various magnitudes in the direction . However, considerations of dislocation energetics, particularly in conditions of low stress where dislocation activity is largely restricted to the f.c.c. 7 phase, would suggest that these occur
1988
GHOSH et al.:
(0)
SRR99 4 0 0 M P o 850°C . . . . . . .
0 30
CREEP DEFORMATION OF SUPERALLOYSMODELLING
.
.
.
SRR99 z,5OMPa , , , ,
030
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Fig. 10. Calculated creep curves, assuming both {111} (T01) and {001 } < 110> glide, for <001 > and < 111 ) tensile creep at various stresses and temperatures. Experimental creep curves, when available are shown for comparison. (a) 400 MPa, 850°C. (b) 450 MPa, 850°C. (c) 500 MPa, 850°C. (d) 450 MPa, 900°C. (e) 200 MPa, 950°C. (f) 300 MPa,
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950°C. (g) 4 5 0 M P a , 850°C.
GHOSH et al.:
CREEP D E F O R M A T I O N OF S U P E R A L L O Y S   M O D E L L I N G
1989
(a) SRR99 450MPQ 850°C 0 3O
E3
(:2 <111> <> < 47 62 63> x
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Time 100h
(c)
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. . . .
850°C ,
,
,
,
(d)
<0CI >
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x
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+
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3
2
4
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9
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+
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+
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S t r e s s 450MPo
[D
<>x
0 .x
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D
+
01e
SRR99
[3
+
0
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~
o D
D
+
orTo
I
~
5 Time lOOh
I 6
Xx \ ,
I 7
,
[
8
,
I 9
100
II111<~o~>+1ooq <11o>
Fig. 11. Calculated creep curves and crystal rotations for SRR99 at 450 MPa/850°C in exact and for deviations of ~ 5 ~' from low index orientations. (a, b) Near . (c, d) Near <001>. (e, f) Near <011 >.
1990
GHOSH
CREEP DEFORMATION OF SUPERALLOYSMODELLING
et al.:
SRR99 400MPa 850°C
can be accounted for by differing proportions of cube and octahedral slip. In an associated study [21], c~ < 111> x we are monitoring crystal rotations to a high pre<> <001> [] cision using the electronbackscatterpattern techx <011> T xX < 111> Experimental []  nique and this clearly indicates complex rotations at x all temperatures considered, particularly under conditions where {111} slip has not been reported. For example, during creep at 850°C and 300 MPa a crystal misoriented by about 15 ° from [111] progressively rotates towards the [111] pole whereas ~' oo {II1} slip would cause quite different ro0.10 o°°° tations. There is certainly direct evidence of the occurrence of extensive cube slip during tensile testing S.××~7 oo o o ° ° r ~ Z ~ × / .,,oo<> ° [8] and of significant crossslip on cube planes during ~x x <> 0l 0 o0,O.O000 00'0'0000<>v vv creep of single crystal superalloys [7]. It is therefore 1 2.0 likely that macroscopic deformation will reflect this Time 1000h cube slip component. Fig. 12. Comparison of calculated and measured creep The present analysis provides a description of curves for <001>, <111) and <011> oriented SRR99 rested the anisotropy of creep behaviour that is generally at 400 MPa and 850°C. consistent with the available data. Sufficient information on [001] oriented specimens is available to either as part of a dissociated complex of partial give a good representation of viscous glide on dislocations and stacking faults that combine to have { 111 } . However, few suitable data are available the ideal ½ Burger's vector, or are a consequence for < 111 > oriented specimens and, consequently, the of the local combination of ideal lattice dislocations present representation of timedependent glide on to produce an unstable superdislocation. Indeed re {001}<110> is subject to considerable uncertainty. cent TEM evidence suggests that the latter only The calculations presented in this paper for crystal occurs at high stresses and low temperatures where orientations where combined octahedral and cube extensive 7' cutting occurs [4]. Much of the argument slip can occur are, therefore, liable to considerable for { l l l } < i i 2 > deformation has been based on the uncertainty and should be taken as indications of interpretation of macroscopic crystal rotations that the type of information that can be obtained rather are observed to occur during plastic deformation and than an optimum quantitative description. In spite of that have been shown to be incompatible with simple this, the calculated and measured anisotropies in {111} slip [20]. Lattice rotation measurements creep curves are in agreement within the normal are rarely sufficiently precise to identify the operative scatter in creep behaviour. The measured and calcuslip system unambiguously, however they have been lated crystal rotations, however, can differ considerused as evidence that simple {111} slip does ably. It is likely that analysis of crystal rotations will not occur. The present calculations show quite provide greater discrimination of the relative contriclearly that a wide range of orientation trajectories butions of octahedral and cube slip than can be 0 30
'
'
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(a) SRR99 0.30
300MPo
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,
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,
,
.
.
.
.
.
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X
< 2 3 10>

950°C ,
(b)
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X
experimenta{
SRR99 Temp 950°C
O20 <>
~
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1
×
0.10
0 x 0//
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/
/
A / (measured) Q D n
f .~ ~ ? ? ? [], []) , 1
'rime lOOh
I
I
A (calc) in n
i
I
16
Hill +loo~l
Fig. 13. Calculated (a) creep curves and (b) crystal rotations for complex crystal orientations of SRR99 tested at 300 MPa, 950%. An experimental creep curve is shown for reference.
GHOSH et al.:
CREEP DEFORMATION OF SUPERALLOYSMODELLING
(ol
/,At,
Str(ain (at 208h o
o2o
~
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1991
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illle('~
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/ ~
Time to reach (a str(ain of 10%
~
o 5oo 750h
~ 20
(b )
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El
/2~
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/
/ (2¢ //~ O
/) x/E<> x/
0
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\
/
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Fig. 14. Calculated anisotropy of creep performance for SRR99 at 400 MPa/850°C displayed as contours in the unit stereographic triangle. (a) Creep strain after 208 h. (b) Times to achieve 10% strain.
Fig. 15. Calculated creep performance for SRR99 with the complex orientation (123) at 850°C and various stresses (a) creep curves, (b) crystal rotations.
obtained by analysis of creep curve shapes. Full validation of the model will require additional creep data for [111] oriented specimens to establish the parameters for cube glide with more precision, creep data for selected complex orientations to compare with calculations and measurement of the associated crystal rotations.
(d) The predicted crystal rotations are sensitive to the relative contributions of cube and octahedral slip, and are consistent with complex crystal rotations that have been observed.
7. CONCLUSIONS (a) A general model of creep deformation by viscous glide on specific crystallographic slip systems has been described that accounts for tertiary creep and crystal rotations. (b) Creep data for SRR99 are represented by the model by assuming that deformation occurs by {111}(I01) and {001}(110) dislocations. (c) The model predicts changing anisotropies in creep behaviour with stress and temperature that are in general agreement with experimental observations.
was supported by a British Council/Overseas Technical Cooperation Training award and the research programme was partially funded by the Ministry of Defence. The authors thank Dr M. Winstone (RAE, Pyestock) for provision of creep data and Drs L. N. McCartney and T. B. Gibbons for helpful comments on the original text. AcknowledgementsRNG
REFERENCES 1. D. N. Duhl, in Superalloys II (edited by C. T. Sims et al.). Wiley, New York (1987). 2. R. A. MacKay and R. D. Mater, MetalL Trans. 13A, 1747 (1982). 3. M. R. Winstone and J. E. Northwood, Proc. Int. Conf. on Metals Engineering. p. 1, Univ. of Aston (1981). 4. P. Caron, Y. Ohta, Y. G. Nakagawa and T. Khan, in Superalloys 1988 (edited by D. N. Duhl et al.), p. 215. Metall. Sac., Warrendale, Pa (1988).
1992
GHOSH et al.: CREEP DEFORMATION OF SUPERALLOYSMODELLING
5. M. Winstone, Proc. M T U Semin. on Single Crystals, Munich (1989). 6. B. H. Kear and B. J. Piearcey, Trans. metall. Soc. A.I.M.E. 239, 1209 (1967). 7. Y. Q. Sun and P. M. Hazzledine, Phil. Mag. A 58, 603 (1988). 8. P. Caron and T. Khan, Proc. 1st A S M Europe Tech. Conf. Advanced Materials and Processing Techniques for Structural Applications (edited by T. Khan and A. Lasalmonie), p. 59, ONERA, Chatillon (1987), 9. B. F. Dyson and M. McLean, Acta metall. 31, 17 (1983). 10. J. C. Ion, A. Barbosa, M. F. Ashby, B. F. Dyson and M. McLean, NPL Report DMA A115 (1986). 11. B. F. Dyson, Rev. Phys. Appl. 23, 605 (1988). 12. P. J. Henderson and M. McLean, Acta metall. 31, 1203 (1983).
13. P. N. Quested, P. J. Henderson and M. McLean, Acta metall. 36, 2743 (1988). 14. A. Barbosa, N. G. Taylor, M. F. Ashby, B. F. Dyson and M. McLean, in Superalloys 1988 (edited by D. N. Duhl et al.), p. 683. Metall. Soc., Warrendale, Pa (1988). 15. R. N. Ghosh and M. McLean, Scripta metall. 23, 1301 (1989). 16. M. F. Ashby and B. F. Dyson, Proc. 6th Int. Conf. on Fracture (ICF6), New Delhi, p. 3, Pergamon Press, Oxford (1984). 17. D. McLean, Rep. Prog. Phys. 29, 1 (1966). 18. R. Lagenborg, J. Mater. Sci. 3, 596 (1968). 19. R. V. Curtis and M. McLean. To be published. 20. A. A. Hapgood and J. W. Martin, Mater. Sci. Engng 82, 27 (1986). 21. R. V. Curtis, P. N. Quested and M. McLean. To be published.