Creep of CMSX-4 superalloy single crystals: effects of misorientation and temperature

Creep of CMSX-4 superalloy single crystals: effects of misorientation and temperature

PII: Acta mater. Vol. 47, No. 5, pp. 1549±1563, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in ...

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PII:

Acta mater. Vol. 47, No. 5, pp. 1549±1563, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(99)00029-4 1359-6454/99 $20.00 + 0.00

CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS: EFFECTS OF MISORIENTATION AND TEMPERATURE N. MATAN, D.C. COX, P. CARTER, M.A. RIST, C.M.F. RAE and R.C. REED{ University of Cambridge/Rolls-Royce University Technology Centre, Department of Materials Science and Metallurgy, Pembroke Street, Cambridge CB2 3QZ, U.K. (Received 1 October 1998; accepted in revised form 4 January 1999; accepted 6 January 1999) AbstractÐIn order to better understand the creep performance of CMSX-4 superalloy single crystals, creep strain testing has been carried out on various specimens systematically misaligned by up to 208 from h001i. At 7508C, signi®cant primary creep is observed, the extent of which depends strongly upon small misorientations away from the h001i/h011i symmetry boundary. In this regime there is evidence of secondary creep  systems to which is associated with primary creep strain sucient for two or more further {111}h112i become active. At 9508C, tertiary creep is prevalent, there being little primary creep exhibited under the stress levels tested. Measurements of the shape change and the lattice rotation, as deduced from analysis of electron backscatter patterns in the scanning electron microscope (SEM), con®rms that the macroscopic  at 7508C but that this changes to {111}h110i  at 9508C. This conshape deformation is due to {111}h112i clusion is supported further by observations made using transmission electron microscopy (TEM). The extent of creep as a function of misorientation away from h001i has been rationalised with a numerical model which takes account of the underlying deformation mechanisms. In the case of primary creep, the model accounts for the initial orientation of the tensile axis, the lattice rotation, the hardening on the primary slip system, the extent of primary creep, the onset of secondary creep and the evolution of the macroscopic creep strain. There appears to be evidence to suggest that the maximum creep rate in the primary regime can be correlated to the secondary rate, which is invariant with increasing creep strain, i.e. a steady state is reached. At higher temperatures where tertiary creep is prevalent, the misorientation dependence of creep deformation is less strong; its extent is rationalised on the basis of Schmid's law. For the purposes of  and {111}h110i  deformation be associengineering design, it is suggested that the occurrence of {111}h112i ated with primary and tertiary creep; these should be modelled as two separate curves which are associated with hardening and softening, respectively. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Nickel alloys; Dislocation mobility; Mechanical properties; constitutive equations; creep; Microstucture

1. INTRODUCTION

Over the past 50 years, a class of material has been developed speci®cally for the hot zones of modern gas turbines: the nickel-base superalloys, e.g. Refs [1±3]. These alloys are now used widely for the fabrication of critical components such as turbine blades and nozzle guide vanes, e.g. Ref. [4]. Currently, no other material is able to o€er the turbine designer the combination of high-temperature strength and microstructural stability, resistance to fatigue and oxidation/corrosion, high sti€ness and acceptable density which these alloys provide, e.g. Refs [5±8]. It is well known that investment casting is used routinely nowadays for the fabrication of singlecrystal blading of very complex geometries, with intricate channels often incorporated so that cooler air can be forced to ¯ow within and along the blades during operation. Such developments have, {To whom all correspondence should be addressed.

over the past 20 years or so, contributed signi®cantly to a steady increase in the turbine entry temperatures of modern aero-engines [9]; the rate of approximately 58C per year averaged over the past two decades shows no sign of slowing. Less well known is that the single-crystal properties along h001i, which is chosen during the grain selection process since it is the preferred growth direction, are usually favourable compared to those along h011i or h111i. This is fortuitous. Had this not been the case, it is unlikely that single-crystal turbine blades would have found such widespread application. In practice, despite the preference for perfect alignment, a fraction of single-crystal turbine blades are used in a condition that has the centrifugal loading away from the exact h001i orientation, perhaps by up to 158 [10]. Practical considerations and cost implications necessitate that this is the case, and for this reason a trade-o€ between orientation and performance must always be made. This situation demands that attention should be paid to the

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quanti®cation of such misorientation on creep properties. Ideally, one should aim to rationalise and quantify the evolution of creep strain by considering the underlying modes of deformation and their temperature and stress dependence. This is the purpose of the present paper.

2. BACKGROUND

The creep deformation behaviour of superalloy single crystals is now known to be strongly temperature, stress, orientation and alloy dependent [1, 6, 11]. At temperatures in excess of 8508C, all known superalloy single crystals exhibit a logarithmic strain softening behaviour consistent with tertiary creep; there is little evidence of a steady-state regime in which the creep strain rate remains constant [12]. For crystals orientated near h001i the dominant deformation mode is glide/ climb by dislocations of the form {111}h1 10i, i.e. so-called octahedral slip [13]. Material loaded uniaxially along a direction away from h001i, particularly near h111i, exhibits macroscopic deformation consistent with slip on {001}h1 10i, i.e. cube slip [14]; this has been interpreted [15] as a consequence of the repeated cross-slip of interfacial screw dislocations. The propensity for cube slip at any given orientation increases with increasing temperature [12, 16]. At lower temperatures, particularly in the vicinity of 7508C, superalloy single crystals are capable of exhibiting considerable amounts of primary creep, the extent of which has been found to vary with temperature and small misorientations away from h001i [17±19]. There is a common consensus that primary creep is due to deformation on {111}h11 2i [17±21]; however, after some deformation a ``secondary'' mode of deformation has been reported which is associated with lattice rotation sucient for two {111}h11 2i-type systems to become active [17]. Experience with the ®rst, second and third generation crystals has revealed that as the alloys have become more creep resistant, the propensity for primary creep has increased. How does one go about rationalising these many e€ects, in particular the temperature dependence of the creep anisotropy? To what extent can the creep behaviour be extrapolated across the regimes of temperature, load and orientation? How does one deal with the possibility of creep driven by a triaxial state of stress? Furthermore, is our understanding of the underlying deformation modes sucient for a uni®ed creep deformation formalism to be proposed, which is suitable for the purposes of engineering design? These are questions for which no categorical answers exist at present, although a clearer picture is emerging.

3. EXPERIMENTAL DETAILS

3.1. Material and specimen preparation The CMSX-4 single-crystal material used here was provided by Rolls-Royce plc in the fully heattreated condition. For the determination of the e€ect of misorientation on creep deformation, two sets of four blanks were machined from a singlecrystal slab of dimensions 20 cm by 6 cm by 1 cm. One of the long axes of the slab was aligned nominally along h001i. By using material from a single casting it was hoped to avoid spurious e€ects which might otherwise have arisen from di€erences in microstructure and composition. It was arranged for the test pieces in each set to be misaligned systematically from each other by a total of approximately 208. For the study of the e€ect of stress on creep deformation, further 1 cm diameter h001i cast rods were used. 3.2. Creep testing and orientation determination Creep strain test pieces of diameter 5.6 mm and gauge length 28 mm were prepared. Considerable care was taken to ensure that the surface ®nish was better than 1.6 mm along the gauge length, so that premature failure of the specimens would be avoided. Creep strain testing was carried out using 20 kN constant load creep testing machines; such testing was compliant with the British Standard UDC 629.7 [22], to which the interested reader is referred. Four of the test pieces taken from the single-crystal slab were tested at 9508C and 185 MPa; the other four were tested at 7508C and 750 MPa. Test pieces taken from the cast rods were tested at 7508C and 9508C at various levels of stress.

Fig. 1. The location of the creep specimens within the standard stereographic triangle, as determined by the indexing of back Laue patterns via the SCORPIO method [23].

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS Table 1. Initial orientations of the single crystal specimens G 4 J and K 4 N. The angle a0 refers to the initial angle from the h001i/ h011i symmetry axis measured along the great circle containing the h112i pole associated with the slip system with the largest Schmid factor. The angle y is the angle made with [001] Specimens G H I J K L M N

Initial orientations

y (degrees)

a0 (degrees)

[ÿ72 146 986] [ÿ51 177 983] [ÿ16 268 963] [ÿ6 410 912] [ÿ50 77 996] [ÿ36 134 990] [ÿ11 300 954] [ÿ4 339 940]

9.4 10.6 15.6 24.2 5.27 7.98 17.47 19.83

± ± ± ± 4.2 3.2 1.2 0.4

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used to locate, trace and record the geometry of the surface of the crept test pieces. In practice, 30 data points were taken for each circumferential scan; 20 of these were made for each sample. A further 8 axial scans were made along the length of the gauge with data points taken at 0.5 mm intervals. The data were collected under computer numeric control and recorded digitally to facilitate further analysis. 3.4. Analysis by transmission electron microscopy Several specimens from material crept at 7508C and 9508C were examined in order to deduce the

After testing, the initial orientations of the test pieces taken from the single-crystal slab were determined by the automatic indexing of back Laue patterns using the SCORPIO method, which is reviewed in Ref. [23]. For each specimen, three independent SCORPIO measurements were made on the uncrept grip regions and the results averaged. The positions of the specimens determined in this way are plotted in the standard stereographic triangle in Fig. 1, and are labelled accordingly; they are also listed in Table 1. Since the tensile axes of all specimens can be assumed to lie within a single plane de®ned by the surface of the original single-crystal slab, the deviations in the orientation data away from this plane are most likely to represent experimental scatter associated with specimen preparation and the SCORPIO measurements. Similarly, SCORPIO measurements were used to determine the orientation of the single-crystal rods from which all other test pieces were machined; only rods within 108 of h001i were employed, in order to minimise the e€ects of misorientation. 3.3. Determination of lattice rotation and shape deformation For the determination of the lattice rotations which occurred upon deformation, test pieces were surface ground to dimensions 4 mm  4 mm  60 mm using a milling machine. The surfaces were then polished according to standard metallurgical practice to a 1 mm ®nish, and then for a further twenty minutes with colloidal silica. Using TSL OIM software [24] installed on a JEOL 5800LV scanning electron microscope (SEM), electron-backscattered patterns (EBSPs) were taken within 40 mm  40 mm squares at 3 mm intervals along the entire gauge length. For each sample, typically 500 patterns were taken; these were subsequently indexed and the lattice rotations determined using the methods reported in Refs [25±27]. For the determination of the macroscopic shape deformation, a Leitz PMM12106 threedimensional co-ordinate measurement device was

Fig. 2. Experimental creep curves from CMSX-4 tested at 7508C: (a) at 750 MPa at varying amounts of misorientation away from h001i and (b) at various stress levels for samples orientated within 108 of h001i.

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deformation mechanism during the early stages of creep. Discs were cut from the specimens at low-index orientations. After thinning by mechanical abrasion, these were electropolished at 50 V in a solution of 10% perchloric acid in acetic acid. The resulting foils were examined in a JEOL 2000CX transmission electron microscope operating at 200 kV.

4. RESULTS

4.1. The e€ect of temperature, misorientation and stress on creep strain evolution 4.1.1. Creep deformation at 7508C. The evolution of creep strain at 7508C and 750 MPa for samples K, L, M and N is illustrated in Fig. 2(a). In this regime, one can see that the e€ect of misorientation

Fig. 3. Experimental creep curves from CMSX-4 tested at 9508C: (a) at 185 MPa at varying amounts of misorientation away from h001i, with creep strain plotted against time and (b) the same data, but with creep strain rate plotted against creep strain; (c) at various stress levels for samples orientated within 108 of h001i.

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

away from h001i is very strong. A signi®cant amount of primary creep is exhibited, the extent of which seems to depend upon the misorientation away from the h001i/h011i symmetry boundary. This is broadly consistent with observations made elsewhere [17, 18, 28]. The e€ect of the magnitude of the applied stress on creep strain evolution at 7508C is illustrated in Fig. 2(b). One can see that, when the stress is large enough to promote primary creep, the creep strain evolution is relatively insensitive to its exact magnitude. The ``threshold'' stress for primary creep at this temperature appears to lie between 600 and 750 MPa. 4.1.2. Creep deformation at 9508C. The evolution of creep strain at 9508C and 185 MPa for samples G, H, I and J is illustrated in Fig. 3(a). Consistent with observations reported elsewhere [12, 16] in this regime the creep strain rate increases monotonically with creep strain in a linear fashion [see Fig. 3(b)]; the e€ect of misorientation away from h001i is relatively weak. On the other hand, the results from the creep testing of samples machined from the cast bars [see Fig. 3(c)], illustrate that there is a relatively strong dependence upon the magnitude of the applied stress. 4.2. Observations made using transmission electron microscopy 4.2.1. 7508C/750 MPa. Figure 4 is a micrograph from the sample tested to 1.4% strain and sectioned on the (111) plane. The initial orientation of this sample was very close to that of sample L which had a primary creep strain of 5%; hence at 1.4%

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strain this sample is still in the early stages of primary creep. Dislocation ribbons of net Burgers vector ah112i are the most striking feature of the structure crept under these conditions (see A in Fig. 4). Similar ribbons have been reported in MarM200 [18, 19] and SRR99 [29]; it is these which govern the macroscopic deformation. The con®guration consists of four dislocations and may be represented by a a a h112i ‡ SISF ‡ h112i ‡ APB ‡ h112i ‡ SESF 3 6 6 a ‡ h112i 3 where SISF and SESF refer to superlattice intrinsic and extrinsic stacking faults, respectively, and APB to an anti-phase boundary. Where the ribbon passes through the g', the two a/6h112i dislocations are tightly constricted by the high energy of the APB lying between them. As it passes through g, the two a/6h112i dislocations separate because there is no APB in this phase. The stacking faults on (111) do not show the familiar striped contrast because, in this orientation, they are parallel to the foil surface and hence appear as dark regions. Interestingly, ribbons are also present on two other {111} planes in small numbers (B and C). The contrast from the dislocations bounding the ribbons was consistent with the Burgers vector a/3h112i. The sections of superlattice stacking faults on the inclined {111} planes show striped contrast. Although they appear as isolated stacking faults due to the sectioning of the specimen, they are aligned over long distances, indicating that they are part of the same dislocation

Fig. 4. Transmission electron micrograph of CMSX-4 tested in primary creep to 1.4% strain at 7508C and 750 MPa. Foil normal is {111}.

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ribbon weaving in and out of the same specimen section. Thus, there is clear evidence that at least three {111}h112i systems are operating even though the strain is less than a quarter of the primary strain expected from a specimen of this orientation. The implications of this last observation are discussed further in Section 5. Also present in large numbers are a/2h110i dislocations (see D, Fig. 4), which expand in the g channels as creep deformation proceeds. Co-planar pairs of loops of di€erent Burgers vectors have combined to form perfect a/2h112i dislocations which can then enter the g' trailing a superlattice stacking fault, leaving a Shockley partial dislocation at the g/g' interface; the mechanism is that given in Refs [30, 31]. The reaction produces half of the ah112i dislocation ribbon, a suitable representation being a a a a ‰011Š ‡ ‰101Šÿ ÿ4 ÿ ‰112Š ‡ ‰112Š: 2 2 3 6 A second pair of the same dislocations gliding on the same plane can combine with the Shockley partial remaining at the g/g' interface and enter g' as four dislocations separated by SISF, APB and SESF, as at the dislocation ribbon indicated at A in Fig. 4. Hence four dislocations of the type a/2h110i are needed to form the ah112i ribbon, for example a a a a a ‰011Š ‡ ‰101Š ‡ ‰011Š ‡ ‰101Šÿ ÿ4 ÿ ‰112Š 2 2 2 2 3 a a a ‡ ‰112Š ‡ ‰112Š ‡ ‰112Š: 6 6 3 Although this is not likely to be a frequent occurrence, once established, this combination can move

freely through g and g'. Moreover, as the total Burgers vector is ah112i, a large amount of strain is accumulated as it does so. As the dislocation density increases, in large part due to the a/2h110i dislocations wrapping themselves around the g', the dislocation ribbons become less mobile. It is suggested that this causes the reduction in the strain rate and thus the cessation of primary creep. Hence the operation of {111}h1 10i slip in the g plays an important role both in initiating primary creep and bringing it to an end. 4.2.2. 9508C/185 MPa. A TEM image of a sample crept at 9508C/185 MPa to 0.07% strain (280 h) is shown in Fig. 5. The foil has as its normal the [001] tensile axis. There is no evidence of a breakdown in the g/g' coherency in areas untouched by the creep deformation. A number of dislocations were analysed and all were found to have Burgers vectors consistent with the type a/2h110i. The dislocations running from top right to bottom left have Burgers vectors of two types: a/2[101] (c, e and f) and a/2[01 1] (b, d, g and h). Cutting across these at right angles are much rarer examples of a/2[10 1] (a). In all material crept under these conditions, no dislocations were seen in the g' either as superlattice stacking faults (SSFs) or as pairs of similar a/2h110i dislocations separated by anti-phase boundaries (APBs). In Fig. 5, for example, activity appears to be predominantly on two systems. The dislocations loop between the g' precipitates, sometimes extending slightly into the vertical channels as at a. As they wrap around the [001] face of the g' precipitates, the line direction is [110], hence both dislocations have mixed character at the g/g' interface in

Fig. 5. Transmission electron micrograph of CMSX-4 tested in tertiary creep to 0.07% strain (280 h) at 9508C and 185 MPa. Foil normal is {100}.

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

the most highly stressed horizontal channels [6, 13]. This is in agreement with the work of Gabb et al. [32], who noted that dislocations formed at the interface to relieve mis®t during annealing without stress were edge dislocations, in contrast to the mixed dislocations formed during a/2h110i{111} octahedral slip.

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4.3. Shape deformation measurements Following Ref. [16], the shape change due to creep deformation is presented as a polar plot of r ÿ rref where r is the local radius and rref is a reference radius, which is chosen to be near to the minimum radius measured in any given specimen. Figure 6(a) shows such a plot for specimen L,

Fig. 6. Cross-sectional specimen shape change of (a) sample L crept at 7508C and 750 MPa and (b) sample I crept at 9508C and 185 MPa. The shape changes at the two temperatures indicate that the modes of lattice deformation are di€erent, i.e. {111}h11 2i at 7508C and {111}h1 10i at 9508C.

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which was deformed to a nominal strain of 14% at 7508C and 750 MPa. Note that the plots have been drawn such that h001i directions are orientated

approximately vertically and horizontally. The data quoted are from various positions along the length of the specimen, and correspond therefore to var-

Fig. 7. Stereogram illustrating the positions of the systems with the highest Schmid factors for samples L and I: (a) systems of the form {111}h11 2i, and (b) systems of the form {111}h1 10i. Also given are the Schmid factors in the initial orientation.

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

ious degrees of creep strain; this has been estimated by determining the fractional reduction of area assuming that any given transverse section is elliptical. Figure 6(b) is a similar plot for sample I which was deformed to 5.5% strain at 9508C and 185 MPa. In the following Section 5, quantitative analysis of the experimental data is attempted using anisotropic deformation models which account for the underlying mechanisms which govern slip. As will be seen, the mode of lattice deformation changes from {111}h11 2i at 7508C to {111}h1 10i at 9508C. Although this is most readily deduced from comparisons of the experimental and computed data, the following argument is sucient to con®rm this assertion. Consider the stereograms in Fig. 7, on which the slip systems with the highest Schmid factors for the various systems are plotted. During deformation, the planes slip over one another in the direction of the Burgers vector; if slip is accommodated by a single slip system, a specimen of initially circular cross-section becomes elliptical, such that the diameter shrinks along the axis onto which the active Burgers vector projects. Moreover, on the stereogram the tensile axis can be considered in such circumstances to move towards the pole of the Burgers vector corresponding to the slip system with the largest Schmid factor. At 7508C this is ( 111)[1 12] and at 9508C (111)[ 101]. The shape changes are distinct, and this fact allows the modes of lattice deformation to be deduced unambiguously.

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edged. An internal state-variable approach based on the governing equations of Gilman [33] will be developed. It is assumed that the mobile dislocation density rm varies with the macroscopic creep strain epsilon according to [33] ÿ  rm ˆ r0 ‡ ME expfÿfEg …1† where r0 is the initial dislocation density, M is a dislocation multiplication rate and f is a dislocation attrition coecient. The macroscopic creep strain

5. QUANTITATIVE INTERPRETATION AND ANALYSIS

5.1. 7508C/750 MPa In this regime the challenge is to quantify the creep associated with macroscopic deformation on {111}h11 2i. The data in Fig. 2(a) suggest that for realistic simulations the initial crystallographic orientation and lattice rotation need to be acknowl-

Fig. 8. Schematic diagram illustrating the de®nition of the angle a0, which is the angle between the initial orientation of the specimen and the h001i/h011i symmetry boundary, measured along the great circle containing the h112i pole associated with the {111}h11 2i slip system with the largest Schmid factor.

Fig. 9. (a) Plot of a0 vs E at maximum E_ for specimens K 4 N; (b) plot of ln Okp vs t0, the resolved shear stress in the initial orientation. Solid lines represent the best ®ts to the experimental data.

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Table 2. Values of the creep constants chosen for the analyses presented in Section 5 Parameters k

A (/s) Bk (/MPa) Ck Dk (/MPa) Ek (degree) Fk n

7508C

9508C ÿ7

1.00  10 0.00 1.40  10ÿ15 7.00  10ÿ2 14.67 2.02  10ÿ5 0.49

1.60  10ÿ12 5.79  10ÿ2 3.75  10ÿ8 3.73  10ÿ2 ± ± ±

rate E_ is related to rm via Orowan's equation E_ ˆ rm bv

…2†

where b is the Burgers' vector and v is the dislocation velocity, which is assumed constant. One then has E_ ˆ …G_ ‡ OE† expfÿfEg

a macroscopic strain E of 1/f. Figure 9(a) shows that a plot of a0, the initial value of alpha, vs E at maximum E_ , is consistent with the proposed dependence. Similarly, Fig. 9(b) demonstrates that ln{O} is proportional to t0, the resolved shear stress in the initial orientation. By appealing to equation (3) and incorporating these empirical observations, the following evolution law for the shear strain rate g_ kp in primary creep on the kth system is suggested:   9 ÿ k g_ kp ˆ G_ p ‡ Okp gkp exp ÿ fkp gkp > > > > > >  > k = _G ˆ Ak exp B k tk p p p k ˆ 1ÿ ÿ412 ÿ …4†  > > Okp ˆ C kp exp Dkp tk > > > > > ; fk ˆ E k =ak p

p

…3†

_ where G=r 0bv is the initial creep rate from the initial mobile dislocation density, and O = Mbv is a softening coecient which is proportional to the rate of dislocation multiplication. Our creep experiments indicate that the attrition term f is orientation dependent and that O increases exponentially with applied stress. Speci®cally, f is inversely proportional to a, the angle between the orientation of the crystal and the h001i/h011i symmetry boundary, measured along the great circle containing the h112i pole towards which the primary slip system (i.e. that with the highest Schmid factor) causes rotation. This relationship is demonstrated if we assume that both a and t, the shear stress on the primary slip system, remain constant during our creep tests. The initial value of a is denoted a0 (see Fig. 8). Under these circumstances, one can show that use of equation (3) yields a macroscopic strain rate E_ which reaches a maximum value of O/(fE) at

Fig. 10. Plot of secondary creep rate vs maximum strain rate; experimental data for samples K 4 N and a number of other sources [17, 18, 28, 30, 34].

Fig. 11. Comparison of the experimental (symbols) and modelled creep strain data (solid lines) for samples K 4 N: (a) creep strain vs time and (b) creep strain rate vs creep strain.

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where the subscript p refers to primary creep and therefore deformation on {111}h11 2i. For both the k G_ p and Okp terms a dependence upon the shear stress tk is allowed for via the constants Akp , B kp and C kp , Dkp , respectively. Possibly a threshold stress needs to be considered additionally but at this stage the data to con®rm this do not seem to be available. The term E kp is the proportionality constant for the orientation dependence of dislocation attrition. Acknowledging the qualitative assumptions made elsewhere [17, 18, 28] we associate primary creep and rotation towards the h001i/h011i symmetry boundary with deformation on a single slip system, and assume this to be the one with the highest Schmid factor (see Fig. 8). For each of crystals K 4 N this is ( 111)[1 12]; see Fig. 7(a). Optimum values for the creep parameters, derived from a global ®t to all our creep data, are given in Table 2. Note that although a and t were assumed constant in order to establish the nature of the constitutive equations, ak and tk are updated during model simulations, thus allowing crystal lattice rotation to evolve during deformation. In the regime before a steady-state creep rate E_ ss is attained [see Fig. 2(a)] the reduction in the observed creep rate is associated with lattice rotation sucient for deformation on a secondary system to become favourable [17±19]. It is necessary to estimate an appropriate magnitude of the steadystate rate E_ ss , but at this stage it is dicult to see how an a priori prediction can be made. However, the evidence from the tests reported here and also from a number of sources [17, 18, 28, 30, 34] suggests that E_ ss correlates with the maximum in the macroscopic primary creep rate E_ max (see Fig. 10). A suitable empirical relationship appears to be n ÿ E_ ss A E_ max …5† where n is a constant which appears to be approximately 0.5. This suggests that the contributions to the secondary creep rate which arise from the activation of two or possibly more further slip systems can be considered to arise from an expression of the form   ÿ kˆ1 n mk g_ kss ˆ F kp g_ p,max  X k …6† m where g_ kss is the shear strain rate on the kth system, each of which has a Schmid factor mk. The sum is taken over all the slip systems which are assumed to be active during secondary creep. The proportionality constant chosen is related to the best-®t line in Fig. 10; the constants F kp and n are given in Table 2. Equation (6) amounts to a partitioning of the macroscopic secondary creep rate between the operative secondary systems. Equations (4)±(6) and the data given in Table 2 are sucient for a prediction of the evolution of

Fig. 12. Calculations of the shape deformation for sample L assuming various secondary systems of the form {111}h11 2i for comparison with experimental data of Fig. 6(a): (a) (111)[ 1 12]; (b) ( 1 11)[112]; (c) (1 11)[ 112]. In each case the primary slip system is taken to be ( 111)[1 12].

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Fig. 13. Comparison of experimental (points) and simulated (solid lines) lattice rotation data for sample L. Calculations made assuming various secondary systems of the form {111}h11 2i: (a) (111)[ 1 12]; (b) ( 1 11)[112]; (c) (1 11)[ 112]. In each case the primary slip system is taken to be ( 111)[1 12].

macroscopic creep strain, using a standard forward integration of equation (4) with time. The contribution from each of the active slip systems is computed from the deformation gradient matrix as in Refs [12, 16]; the total strain is determined from the primary and secondary creep strains by simple addition. In the ®rst instance it is assumed that two systems operate during secondary creep, and that these are the two with the largest initial Schmid factors: ( 111)[1 12] (the primary system) and (111)[ 1 12]. The results of the simulations are given in Fig. 11, from which it can be seen that the major features of the creep curves are reproduced reasonably well. In fact, we have found that the predicted creep curves are relatively insensitive to the choice of secondary system; various combinations of the systems operating in this regime [i.e. (1 and 2), (1 and 3), (1 and 4); see the notation in Fig. 7(a)] do not give rise to noticeably di€erent results. This situation arises because of the partitioning implied by the form of equation (6). On the other hand, the predicted shape deformation is sensitive to the choice of secondary slip system (see Fig. 12), which should be compared with Fig. 6(a), in which the experimental data are plotted. The secondary system is taken to be (111)[ 1 12] in (a), ( 1 11)[112] in (b) and (1 11)[ 112] in (c) [see the stereogram in Fig. 7]; in each case the

primary system is assumed to be ( 111)[1 12], as before. The lattice rotation simulations also yield distinct results (see Fig. 13). When this work began, it was hoped that a comparison of the experimental results and theoretical predictions would allow the active secondary system to be identi®ed unambiguously. However, it appears that the situation is somewhat more complicated than was originally envisaged. The shape deformation data favour (1 11)[ 112] (that with the fourth largest Schmid factor) whilst suggesting that a contribution from (111)[ 1 12] and/or ( 1 11)[112] is occurring. The lattice rotation data favour (111)[ 1 12] (that with the second largest Schmid factor), but suggest that there is not a sharp transition between primary and secondary creep. The TEM observations also suggest that a number of systems is active from a very early stage of deformation. Thus there exists the possibility of factors other than Schmid's law playing a role in shaping the macroscopic deformation, and this warrants further study. 5.2. 9508C/185 MPa For the quantitative interpretation of the data in Fig. 3, an anisotropic formalism for tertiary creep [12] will be used; the e€ect of misorientation near the h001i pole is accounted for by considering octahedral slip {111}h1 10i. The appropriate evol-

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

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ution law for the shear strain rate g_ kt on the kth system is 9 k > g_ kt ˆ G_ t ‡ Okt gkt > > =  _Gk ˆ Ak exp B k tk k ˆ 1ÿ ÿ412 ÿ …7† t t t > >  k k > ; k k Ot ˆ C t exp Dt t where the subscript t refers to tertiary creep and therefore deformation on {111}h1 10i. As before, the k term G_ t is a constant which is a measure of the initial creep rate, and Okt is a softening coecient which is related to rate of dislocation multiplication; for both of these terms a dependence upon the shear stress tk is allowed for via the constants Akt , B kt , C kt and Dkt . Using a technique similar to that reported in Ref. [12] and the creep data from the cast bars given in Fig. 3(b), optimum values of these constants were determined, and these are listed in Table 2. Using these data, the creep curves, lattice rotation and shape deformation were determined following Refs [12, 16]. The simulated creep curves are given in Fig. 14, and these should be compared with the experimental data in Fig. 3(a). The simulations suggest that G, H, I and J have crept more rapidly than a sample machined with its axis exactly along the h001i pole would have done; this is because the Schmid factor varies from 0.41 on the pole, reaching a maximum of 0.50 at about 258 away from it, e.g. Ref. [6]. It appears that each of the samples G, H, I and J crept more slowly than equation (7) and the data in Table 2 predict. We believe that this is because the data have been derived from crept material taken from cast rods rather than the larger cast slabs, so that di€erences

in the initial microstructure give rise to di€erent rek sponses and in particular, di€erent values of G_ t . This suggestion warrants further investigation [35]. Moreover, it appears that the ranking of samples G 4 J with regard to creep strain accumulation cannot be rationalised perfectly with the slip-system model, although one can argue that samples I and J have crept more rapidly than samples G and H, as predicted. It is possible that comparison between experiment and theory is hampered by experimental scatter associated with the creep testing and because of uncertainties in the initial orientation of the test

Fig. 14. Creep simulations at 9508C and 185 MPa, which should be compared with the experimental data in Fig. 3(a).

Fig. 16. Simulated shape deformation for sample I, which was crept at 9508C; this should be compared with the experimental data in Fig. 6(b).

Fig. 15. Creep strain evolution at 9508C. Experimental data: solid lines. Modelled data: dashed lines denoting that the limiting cases arise from the e€ect of misorientation of 208 from h001i. Creep is slowest on the h001i axis.

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MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

Fig. 17. Comparison of experimental (points) and simulated (solid lines) lattice rotation data for sample I. Dashed line shows simulation extrapolated to a strain of 50%.

pieces. However, we have found it instructive to consider the predicted variation in creep evolution to be expected from single crystals misorientated by up to 208 from h001i, and which therefore exhibit primary Schmid factors which vary from 0.41 to 0.49. The results are shown in Fig. 15. The variation in behaviour, particularly at the lower stress levels, is considerable. Figure 15 demonstrates that a considerable part of the scatter associated with the behaviour of samples G 4 J in this testing regime is likely to arise from di€erences in the Schmid factor. The simulated shape deformation is presented in Fig. 16, and this should be compared with the experimental data in Fig. 6(b). It is clear that the anisotropic thinning of the sample is predicted reasonably well. The simulated lattice rotation, Fig. 17, is also in good agreement with the experimental result although the magnitude of the lattice rotation in this regime is not large. 6. SUMMARY AND CONCLUSIONS

The following conclusions can be drawn from this work. 1. Direct measurement of the shape changes arising from the creep deformation of CMSX-4 single crystals has shown that the mode of lattice deformation changes from {111}h112i at 7508C to {111}h110i at 9508C. Observations made using TEM microscopy support this conclusion.

2. The extent of primary creep deformation at 7508C depends strongly upon the angle between the tensile axis and the h001i/h011i symmetry boundary, as measured along the great circle containing the h112i pole, towards which the tensile axis should ideally rotate. This dependence has been interpreted in a quantitative sense. 3. As primary creep deformation proceeds, the creep rate drops as secondary slip systems become activated. The lattice rotation which occurs is a superposition of the deformation associated with at least two {111}h112i slip systems. Evidence from TEM microscopy supports this. Thus, in this regime it appears that secondary creep arises as a consequence of primary creep, and therefore that the primary and secondary modes are inextricably related. 4. In the regime investigated, secondary deformation is associated with a steady-state creep rate. Examination of the creep data reported here and elsewhere suggests that the magnitude of this steady-state creep rate correlates with the maximum rate in primary creep. 5. An anisotropic creep deformation model for the behaviour associated with {111}/h112i deformation has been proposed. The model accounts for the initial orientation of the tensile axis, the lattice rotation, the hardening on the primary slip system, the extent of primary creep, the onset of secondary creep and the evolution of the macroscopic creep strain.

MATAN et al.: CREEP OF CMSX-4 SUPERALLOY SINGLE CRYSTALS

6. At 9508C the tertiary creep evolution depends more strongly upon the applied stress than on the misorientation from h001i. It appears that the misorientation dependence can be rationalised on the basis of Schmid's law. The anisotropic creep model accounts satisfactorily for the shape deformation and the lattice rotation. 7. For the purposes of engineering design, it is suggested that the occurrence of {111}h11 2i and {111}h1 10i deformation should be modelled as two separate curves which are associated with hardening and softening, respectively. It is likely that the creep strains from the two modes can be treated as additive; however, more work is required to con®rm this. It is suggested that the phenomena occurring in the vicinity of 8508C should be investigated thoroughly. AcknowledgementsÐThe authors would like to acknowledge the Government of Thailand (The Development and Promotion of Science and Technology Talent Project), the Engineering and Physical Sciences Research Council (EPSRC), Rolls-Royce plc and the Defence and Evaluation Research Agency (DERA) for sponsoring this work. Helpful discussions with Bob Broom®eld and Steve Williams of Rolls-Royce plc and Mike Winstone and George Harrison of DERA are acknowledged.

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