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Factors affecting the anisotropy of irradiation creep and growth of zirconium alloys
FACTORS AFFECTING THE ANISOTROPY OF IRRADIATION CREEP AND GROWTH OF ZIRCONIUM ALLOYS R. A. HOLT and E. F. IBRAHIM Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories. Chalk River, Ontario KOJ lJ0, Canada (Recewed
27 Norember 1978)
Abstract-The long term dimenslonal changes of zirconium alloys m a nuclear reactor core are attributed to generation of equal numbers of vacancies and interstitials by fast neutrons and their condensation at various sinks. The dimensional changes are anisotroptc because the microstructures are anisotropic and thus the sinks have anisotropic distributions. We derive relationships between anisotropy of m-reactor creep and growth of zirconium alloys with their crystallographic texture and grain shape. We conclude that growth occurs primarily by the partitioning of interstitials to dislocations or prismatic loops and vacancies to grain boundaries. Two strain producing mechanisms of creep are considered; climb and glide. Measured creep anisotropy agrees more closely with a glide model. Thus creep and growth can be treated additively to give predictions of the behaviour of zirconium alloy reactor components. R&sum&--On attribue les changements P long terme dans les dimensions des alliages de zirconium du coeur d’un reacteur nucleaire g la production par Ies neutrons rapides d’un nombre igal de lacunes et d’intersticiels et it leur condensation sur divers pieges. Les changements de dimension sont anisotropes. car les microstructures sont anisotropes et les pieges ont done une repartition anisotrope. Nous obtenons des relations entre l’anisotropie du fluage et de la croissance d’alliages de zirconium dans le rCacteur. et la texture cristallographique et la forme des grains. Nous en deduisons que la croissance rtsulte essentiellement de la dgregation des mtersticiels sur les dislocations ou les boucles prismatiques et de celle des lacunes sur les joints de grains. Nous considtrons deux m6canistnes de fluage produisant des d&formations: la montee et le glissement. L’anisotropie du fluage mesurte est en meilleur accord avec un mod6.le de ghssement. Le fluage et la croissance peuvent ainsi Btre trait& de manitre additive. et l’on obtient de bonnes privisions du comportement des constituants de riacteur en alliage de zirconium. Zusammenfnssung-Die langfristigen Abmessungtinderungen von Zirkonlegierungen im Kern eines Reaktors werden der Erzeugung von Leerstellen und Zwischengitteratomen in gleicher Anzahl und deren Kondensation an versctuedenen Senken zugeschrieben. Die Abmessungsiinderungen sind anisotrop. da die Mikrostruktur amsotrop ist und rolglich die Senken eine anisotrope Verteilung aufweisen. Es werden Zusammenhlnge aufgestellt zwlschen der Anisotropic des Kriechens und des Schwellens dieser Zirkonlegierungen im Reaktor und der kristallografischen Textur und der Kornform. Wir folgern, daB das Schwellen im wesentlichen durch das Ausheilen der Zwischengitteratome an Versetzungen und prismatischen Vertsetzungsringen und der Leerstellen an Korngrenzen entsteht. Zwei Mechanismen fir das Kriechen werden betrachtet: Gleiten und Klettern. Die gemessene Kriechanisotropie stimmt mlt dem Gleitmodell besser iiberein. Folglich kBnnen Kriechen und Schwellen additiv behandelt werden, urn zufriedenstellende Voraussagen iiber das Verhalten von Reaktorkomponenten aus Zirkonlegierungen zu erhalen.
1. INTRODUCTION The problem of predicting long term dimensional changes of zirconium alloy components in a fast neutron flux has been the subject of increasing study [l-q. Because the crystal structure of a-zirconium is close packed hexagonal, components fabricated from zirconium alloys have anisotropic properties. This leads to both anisotropic irradiation creep and to irradiation growth (dimensional change at constant volume in the absence of applied stress [8]). These dimensional changes affect the design of components in the cores of nuclear reactors. Thus it is important to predict the sign and magnitude of the dimensional changes and. if possible. to control them. \M
2’8.
R
Ross-Ross and Hunt [l] indicated the need for a mathematical treatment of anisotropy of in-reactor creep and Ross-Ross er al. [4] demonstrated anisotropit irradiation creep experimentally and proposed an analytical treatment to deal with this empirically, based on the approach of Hill [93. Ibrahim CZ], Schroeder and Holicky [3] and, more .recently, Franklin and Franz [6] recognized that anisotropic creep implied length change in an internally pressurized tube; Ross-Ross and Fidleris allowed for this in their design equations [YJ. Buckley [lo] first observed irradiation growth in zirconium single crystals and proposed a simple model relating growth to crystallographic texture.
1319
1320
HOLT
AND
IBRAHIM:
THE ANISOTROPY
Hesketh [l l] modified the model slightly but the relationship of growth strains to crystallographic texture remained the same. Wiedersich [12] suggested that growth could be related to grain shape and anisotropy of the distribution of orientations of dislocation Burgers vectors. Adamson [13] accumulated a large amount of data showing that irradiation growth was texture related and persisted to high fast neutron fluences. Much of the above information has been reviewed by Fidleris [S]. Alexander, Fidleris and Holt [7] treated long term dimensional changes of pressure tubes quantitatively as the sum of creep and growth components, both of which were related to crystallographic texture. In this paper we discuss the anisotropic microstructural factors affecting irradiation creep and growth of zirconium alloy components and derive quantitative relationships between microstructural parameters and the direction and magnitude of creep and growth strains. We compare the calculated anisotropy factors for creep and growth with experimental evidence. Finally, we illustrate the use of these factors in evaluating the behaviour of power reactor pressure tubes.
OF IRRADIATION
CREEP
+ (1120) for either loops [17] or straight dislocations [18]. Thus the strain produced by the climb mechanism in any grain depends on the orientation of the (1120) directions with respect to the stress and the direction of interest. The anisotropy factor for strain due to climb thus depends on the distribution of grain orientations, i.e. crystallographic texture. To calculate the anisotropy factor we assume [lS] Bbx a II - Cl.
The shape change of an a-zirconium single crystal due to the SIPA mechanism is shown schematically in Fig. 1. According to the climb-plus-glide model, enhanced climb in the presence of a fast neutron flux allows the dislocations to surmount obstacles and glide across obstacle free regions. Strain is produced by the glide process. Glide is always observed on : lOi0: and sometimes on ( IOil) or :0002) planes in z-zirconium. The Burgers vector is always 3 (1 120) except under conditions of extreme triaxial constraint. Thus the anisotropy of strain due to glide also depends on crystallographic texture. For glide we assume ;ib x rJ*.
2. GENERAL
APPROACH
Our approach is based on the assumption that the strain rate in a given direction of a material is the product of two components, a specific rate at which some strain producing mechanism occurs (slip. for example) and an anisotropy factor which resolves the strain produced by this hechanism into the direction of interest.t ’ = cd
AdK.
(1)
If we define a strain producing mechanism, we can define the anisotropy factor from the anisotropy of the microstructure. We can then predict the sign and relative magnitude of the strain rate with only limited knowledge of K.
(21
(3)
The shape changes due to the glide mechanisms are illustrated schematically in Fig. 1. For a given system of applied stress we calculate cl,, r~, and rb for each grain orientation. We then calculate & and fb and resolve them in the fabrication directions of the tube. The resolved strains, a, and fr,, are averaged according to the basal pole figure. Compatibility and constraint between grains are ignored. These calculations are described in detail in Appendix 1. 3.2 Experimental observations of creep anisotropy
Stress sensitivity of creep is close to one [6] in the range of operating conditions of interest for zirconium alloys in thermal power reactors (up to 2OOMPa, 500-6OOK. 10’6-10L8n m-* s-l). Crystal3. ANISOTROPY OF CREEP lographic textures of pressure tubes of two zirconium alloys are shown in Fig. 2, and anisotropy factors 3.1 Models for creep calculated for a stress sensitivity of one (see Appendix Currently favoured models of in-reactor creep fall 1) are given in Table 2. into two categories, those in which dislocation climb The anisotropy factors, lAAand err, represent the is the strain producing mechanism (loop alignment and SIPAS models [14, IS]) and those in which glide relative magnitudes of creep strains for specimens stressed uniaxially in the axial and transverse direcis the strain producing mechanism (climb-plus-glide tions respectively. We can compare the ratios of model [ 161). According to the climb models, interstitials pro- measured strains in different directions with calculated values to determine which mechanisms correduced by the fast neutron flux are attracted to straight dislocations or loops with their Burgers vectors. b, spond most closely to the observed behaviour. Causey [19] and Coleman et al. [ZO] obtained parallel to the applied stress. The excess vacancies go to dislocations or loops with their Burgers vectors creep strain-rate data from bent beam stress relaxaperpendicular to the applied stress (Fig. I). The most tion tests in two directions in pressure tube materials. commonly observed Burgers vector in a-zirconium is Changes of curvature in bent beam stress relaxation specimens are wholly from creep since growth does t The symbols used throughout are defined in Table 1. not affect curvature. The ratio of axial creep rate with : Stress-induced preferential absorption. an axial stress to transverse creep rate with a trans-
HOLT
AND
IBRAHIM:
THE ANISOTROPY OF IRRADIATION
0002
CREEP
1311
Plane
1
+ (1120) vector
(ISOO, Plane
(li01) Plane
/
(lioo),+
/
(11.20)
KlOO2),f ( 1120)
Prismatic
Basal
Slip
slip
cfioi),f (1150) Pyramidal shp
Fig. 1. Shape change of an a-Zr crystal for various creep mechanisms resulting from stress indicated. verse stress. R, corresponds to the ratio of anisotropy factors l~,&rr. These values are shown in Table 2. For both Zircaloy-2 and Zr-2.5wt.9, Nb the climb and prismatic slip models overpredict R. The contribution to strain of dislocations with b = 3 (IlzO) gliding on pyramidal or basal planes would reduce the value of R. An unrealistically large proportion of pyramidal slip would be required. However, the basal slip model greatly underpredicts R (Table 2). Good agreement with experimental results is obtained if basal slip accounts for 20”” of the strain and prismatic slip for 80”, of the strain. Based on published data for basal slip of zirconium[21] at high temperatures (i.e. > 850 K) and titanium alloys [22] at up to 500 K, this appears to be a reasonable assumption for the proportlon of basal slip at 500-600 K. The anisotropy predicted with the climb model is not modified by a change in glide plane. Thus a reduction in R can only be obtained if dislocations are present within which the Burgers vector has a component in the [0002] directlon. Since such dislocations are very rarely observed, the creep anisotropy can best be explained if glide is the strain producing mechanism. This conclusion is compatible with the results of t For vacancies. the sink may be a cluster of vacancies left by a displacement cascade-then no migration of the bacancles IS required.
recent in-reactor creep tests suggesting that glide is the strain producing mechanism at conditions of interest for operating pressure tubes [23,24]. 4. ANISOTROPY
AND GROWTH
4.1 Models for growth Irradiation growth results from the redistribution of atoms displaced from the lattice by fast neutrons. The neutrons create equal numbers of vacancies and interstitials which migrate through the lattice until they recombine or reach a sink at which they can precipitate.? The volume change associated with the formation of the point defects is isotropic but that associated with their precipitation at sinks may be oriented. depending upon the type of sink. For example, for grain boundaries the strain is normal to the grain boundary. Vacancies cause a net contraction in this direction and interstitials cause a net expansion (12). The strain produced can be resolved into any direction of interest. d, and summed over all orientations of sinks. Assuming that the number of defects precipitating at a given sink is independent of the orientation of the sink E,J= (IV,, - N,$
z X,(w) cos2 0. 0,
= (R;‘,,- N,.,)RAd,.
(4) (51
1322
HOLT
AND
IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
CREEP
Table 1. Definition of symbols
XJlU) I s
Y :d, Pd Fd N,,
N,,
dQa G K,(S). K,(S) dJ 6 cd
subscrtpt Indicating general directions of interest subscripts indicating specific direction of interest, radial, transverse. axial stram in the ‘d’ direction strain rate in the ‘8 direction growth rate in the ‘d’ direction creep rate in the ‘d’ direction rate constant for a strain process anisotropy factor resolving strain in the direction of interest ‘d’ the Burgers vector subscript indicating the direction of the Burgers vector normal stress in the direction of the Burgers vector normal stress perpendicular to the Burgers vector and to [0002] shear strain rate on the slip plane in the direction of the Burgers vector shear stress on the slip plane in the direction of the Burgers vector stress sensitivity of glide anisotropy factor for the creep strain in the ‘e’ direction due to a uniaxial stress in the ‘8 direction ratio of anisotropy factors ~&err number of interstitials and vacancies respectively arriving at the type of sink denoted by subscript L3 Y subscript denoting dislocations atomic volume is the angle between the ‘d’ direction and the direction of strain for a given type of sink (e.g. ‘b direction for dislocations, direction of grain boundary normal for grain boundaries) is the fraction of sinks of type ‘y’ oriented as defined by o subscript denoting prismatic loops subscript denoting sub-grain boundaries subscript denoting grain boundaries subscript denoting clusters or voids anisotropy factor for type ‘y’ sinks with respect to the ‘d’ direction ‘the resolved prism pole component’ as defined in Appendix 1 resolved basal pole component in the ‘d’ direction number of excess interstitials or vacancies respectively arriving at type ‘y’ sinks displacements per atom (the number of times each atom has, on average, been displaced from its lattice site by fast neutrons anisotropy factor for growth in the ‘8 direction specific growth and cmp rates for material of structure ‘S displacement rate (dpa s- ‘) normalized stress creep anisotropy factor in the ‘d’ direction.
For growth to occur, different numbers of vacancies and interstitials must’precipitate at different sinks, i.e. N,, - No,, ,# 0
(6)
and the sinks must be characterized by anisotropic orientation distributions, X(w), which are different for each type of sink. If the point defect should precipitate at a void or point defect cluster there is no strain but a net density change occurs. To estimate the anisotropy of irradiation growth we must evaluate both the partitioning of point defects to the different sinks and the anisotropy of the sinks. 4.2 Anisotropy of sinks Precipitation of point defects at prismatic loops gives strain in the direction of the Burgers vectors of the loops. Only loops with a Burgers vector. b = 4 ( 1IlO), on { IlzO; planes have been observed in a-zirconium [17J In the absence of an applied stress, loops will form randomly on the ( 1lIOl planes and thus the anisotropy factor for prismatic loops is directly related
to crystallographic A dl=
pd -.
2
texture
Precipitation of point defects at dislocations causes them to climb giving strain in the b direction. Growth occurs if there is an anisotropic distribution of Burgers vectors resulting from a forming process [25]. Alternatively, in materials with anisotropic crystal structures, the anisotropy of the Burgers vector distribution could result strictly from crystallographic texture. In a-zirconium a Burgers vector, b = 4 (1120). is by far the most frequent. With the Burgers vectors randomly distributed about the basal plane normal A,iD= pd/2.
03)
The direction of strain when a point defect is absorbed at a grain boundary is the direction of the grain boundary normal. During forming processes the grains are elongated in the direction of working and compressed in directions normal to the direction of working. To derive the anisotropy factors for grain boundaries one must measure the angular distribution of grain boundary normals X&J) about the direction of interest. The anisotropy factor can be approximated by assuming a simple rectangular grain shape with axes in the three principal directions of
HOLT
IBRAHIM:
AND
THE ANISOTROPY
%Nb
Zr-2.5wt
OF IRRADIATION
CREEP
1323
Zirmloy - 2 Aual
Ask31
--“’ co5
Rodiai
Tmnsverse
Transverse
Fig. 2. Basal pole figures for typical cold-worked Zr-2.5 wt.‘, Nb and Zr-2 pressure tubes.
defects at smks is
strain during working and then (9)
Anisotropy of grain boundary orientations could explain the irradiation growth observed in materials with isotropic crystal structures [253 (see Appendix 2). Subgrain boundaries consist of networks of dislocations. Precipitation of point defects at subgrain boundaries will &use the component dislocations to climb giving strain in the directions of their Burgers vectors. In many cases the distribution of dislocation Burgers vectors in subgrain boundaries will be the same as for free dislocations and therefore A
dls
=
AdO
=
(10)
P&?.
Precipitation of point defects at free surfaces results in strain in the direction normal to the free surface. The proportion of point defects migrating to free surfaces would be negligibly small for most polycrystalline reactor parts. However, free surfaces should be considered in analysing irradiation growth in thin specimens in which electron or ion bombardment has been used to simulate neutron irradiation. 4.3 Parritioning
of point defects
The total strain due to the precipitation
of point
+ (Ni, - N,)A,
+ (N,, - N,,)&.
1 N,, - 1 N, = 0. Y
‘id = liTi,Adl + ICli,Adn R =
(fi,, + ii, -
fiT,A,, - i?,A,
a&J;- lQwAdg.
Ratio of fAA:crr
CA.4
Zircaloy-2
l
From stress-relaxation
tests C19.203
(13)
If a significant proportlon of vacancies go to clusters or voids swelling occurs. Since equal numbers of vacancies and interstitials are created by the
zirconium alloy pressure tubes
Measured* Prism slip. P Basal slip, B 80”, P + 20”” B Climb Measured* Prism slip. P Basal slip. B 80”” P + 20”, B Climb
(12)
Partitioning (as expressed by equation 6) occurs in the absence of applied stress due to the interaction between the large dilational strain field of interstitials and the long range strain heIds of some sinks (dislocations and loops). Interstitials are preferentially attracted to dislocations and to loops. Since vacancies have a very mu&h weaker strain field, excess vacancies precipitate at sinks without long range strain fields (clusters, voids, grain boundaries, subgrain boundaries and free surfaces). Thus
Table 2. Comparison of measured and calculated creep anisotropy For cold-worked
Zr-2.5 wt.?, Nb
(111
Also. since equal numbers of vacancies and interstitials are produced by the flux
2.10
-
0.54 1.79 2.10
0.83 1.36 0.93 0.83
2.18 0.5 1 1.85 2.18
0.93 1.41 1.03 0.93
-
%F 0:40
-
0.36 J&J 2.34
1324
HOLT AND IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
CREEP
Table 3. Possible point defect sinks Source of anrsotropy
Sink
Direction of strain
Clusters Voids Loops Dislocations
Isotropic (volume change) Isotropic (volume change1 In b direction In b direction
Subgrain boundaries Grain boundaries SUrfactS*
In b direction Normal to boundary
Texture Texture or anisotroptc strains durrng cold-work Texture or anisotropic strains during cold-work Grain shape
NormaI to surface
Macroscopic shape
* Can be ignored if grain size 4 minimum dimension.
neutron‘flux
_
^
Nil + NiD - ti,, - fi, Table 3 summarizes the possible and the source of their anisotropy.
4.4 Experimental
tiw=0.
(I41
sinks, their roles
observations of irradiation
growth
At low fluences, (c 1 dpa) the growth behaviour of a-zirconium alloys is complicated by transients due to stress relieving [26]. We will only describe the high fluence behaviour (> 1 dpa). Experimental observations of growth are [8,10,11,13,26-281: (1) Growth is always positive in the working diiection. (2) Growth can be positive or negative in directions normal to the working direction. {3) Growth in annealed materials tends to saturate or be reduced to a very low rate after 1 dpa.
(4) Growth in cold-worked materiab continues at a steady state rate at high dqses, > 1 dpa. (5) Growth of annealed material is recoverable on heating after irradiation. (6) Only a small proportion of the growth strain can be recovered on heating cold-worked materials after irradiation. (7) Density changes are smail compared with growth strains in cold-worked materials. (8) The growth strain in annealed mate&is cannot be accounted for by loops visible in the electron microscope. Based on observation 7 we can conclude that the proportion of vacancies migrating to clusters and voids is negligible and thus equation (13) becomes $d -=
R
Fig. 3. Shape change of a-Zr during growth. (1) Initial shape; (2) expansion in (ll~O> directions due to absorption of interstitials at dislocations: (31 isotropic contraction due to absorption of vacancies at boundartes of equiaxed grains; (41 anisotropic contraction due to absorption of vacancies at boundaries of grains flattened in the x-direction.
HOLT
AND
Table 4. Analysis of dimensional
IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
changes of Hanford ‘N’ reactor pressure tubes made by Chase Brass (Type 3) based on microstructural measurements on six tubes Radial
1.
2. 3. 4. 5. 6. 1. 8 9. 10.
Resolved basal pole component, F., Resolved prism pole component. P, Mean intercept grain size, pm Growth anisotropy factor (grain boundary anisotropy ignored) Growth anisotropy factor (grain boundary anisotropy included) Creep anisotropy factor (prism slip only) Creep anisotropy factor (prism slip + basal slip) Calculated rate (4 & 6) x low4 dpa-‘* Calculated rate (5 & 7) x 1O-4 dpa-’ Measured rate x 10m4dpa-’
* 1 x 1025nm-2
1325
CREEP
0.46 0.54 7.7 -0.38 -0.97 -0.44 -0.63 - 10.0 - 12.4 -
Transverse 0.46 0.54 10.3 - 0.38 -0.33 0.46 0.63 2.5 4.9 4.7
Axial 0.08 0.92 14.2 0.7 1.3 -0.0 0.00 7.5 1.5 79
> 1 MeV taken as equivalent to 1 dpa.
6. CONCLUSIONS
and we define the growth anisotropy factor
(16) For typical textures of zirconium alloy tubes and strip, this is always positive in the working direction and may be positive or negative in directions normal to the working direction. It is thus in good agreement with the experimental observations. Figure 3 illustrates schematically the shape changes for an a-zirconium crystal due to irradiation growth according to equation (15). 5. BEHAVIOUR OF ZIRCONIUM ALLOY PRESSURE TUBES Based on the foregoing assumptions, creep (glide) and growth (climb) are additive. The total rate of dimensional change of a zirconium alloy tube can thus be expressed by (7) id = K,(S)*+.Gd + K,(S)*4*5.C,.
(17)
Alexander er al. (7) interpreted the dimensional behaviour of the cold-worked Zr-2 pressure tubes in the Hanford ‘N’ reactor using this model. Using the analysis described here, they calculated anisotropy factors for creep assuming only prismatic slip and for growth assuming climb of f( 11~0). t lOi0; dislocations by absorption of interstitials but ignoring any contribution of grain shape. They underpredicted the diametral strain rates. Incorporating grain shape and basal slip into the model greatly improves the agreement with observed diametral strain rates, and allows the correlation of the behaviour of different types of pressure tubes in several reactors. For example, Table 4 shows the revised analysis of the behaviour of Hanford ‘N’ reactor pressure tube compared with that made by Alexander et al. [TJ showing improved agreement of the calculated transverse strain rate. The constants K,(S) and K,(S) were taken as 5.8 x 10e4 dpa-’ and 1.9 x 10-sdpa-’ respectively [29].
The anisotropy of irradiation induced long-term dimensional changes of zirconium alloys can be predicted from anisotropy of microstructure. once the strain producing mechanisms are identified. Anisotropy of irradiation growth is calculated from crystallographic texture and grain shape assuming that interstitials are attracted preferentially to 3 (1120) dislocations with excess vacancies arriving at grain boundaries. The anisotropy predicted by this model agrees with the experimental irradiation growth data. The anisotropy calculated from crystallographic texture assuming that creep occurs by climb or glide of j( 1120) dislocations on i lOi0; planes is greater than that measured experimentally. The anisotropy calculated for glide agrees with experimental measurements if about 20”,, as much slip of f (1120) dislocations occurs on (0002) as on ;loio; planes. Creep and growth can be treated additively to predict the anisotropy of zirconium alloy compcnents operating under stress in a fast neutron flux. Acknow/edgements-We thank A R. Causey for the use of unpublished information and C. E. Coleman and B. A. Cheadle for comments on the manuscript.
REFERENCES I. P. A. Ross-Ross and C. E. L. Hunt. J. nucl. Mater. 26, 2 (1968). 2. E. F. Ibrahim, Applications related phenomena in zirconium and its alloys, ASTM STP 458, 19 (1968). 3. J. Schroeder and M. Hohcky. J. nucl Mater. 33. 52 (1969). 4. P. A. Ross-Ross, V. Fidleris and D. E. Fraser. Can. Metall. Q. 11, 10 (1972). 5. P. A. Ross-Ross and V Fidleris. Inrernational Conference on Creep and Fatigue in Elevated Temperature Appliccltions. Philadelphia Sept. 1973. paper C216i73 6. D. G. Franklin and W. A Franz, Zirconium m the
nuclear industry. ASTM STP 633, 365 (1976). 7. W. Alexander. V. Fidleris and R. A. Holt. Zirconium in the nuclear Industry. ASTM STP 633, 344 (1976). 8. V. Fidleris. Atomic Energ! Rer. 13, 51 (1975). 9. R. Hill, The Mathematical Theory q/ Plastirrty Oxford University
Press, Oxford
(1950)
1326
HOLT
AND
IBRAHIM:
THE ANISOTROPY
10. S. N. Buckley, Properties of Reactor Materials and Eficts of Radiation Datnuge (edited by W. J. Littler),
p. 413. Butterworths, London (1962). 11. R. V. Hesketh, J. nucl. Muter. 30. 219 (1969). 12. H. Wiedersich, The Physical Merallurgy of Reactor Fuel Elements (edited bv J. E. Harris and E. C. Svkes). _ ” p. 144. Metals‘Society,-London (1973). 13. R. B. Adamson, Zirconium in the nuclear industry, ASTM STP 633, 326 (1976). 14. F. A. Nichols, J. nucl. Mater. 30, 249 (1969). 15. P. T. Hcald and M. V. Speight, Phi/. Mug. 29, 1075
OF IRRADIATION
CREEP
(O’ ‘p‘\
(1974).
16. S. R. MacEwen, J. nucl. Mater. 54, 85 (1974). 17. G. J. C. Carpenter and D. 0. Northwood, Proc. Int. Conf on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, Tennessee, 2, 783 (1975). 18. D. L. Douglass, The Metallurgy of Zirconium, Atomic Energy Rev. Supplement (1971). 19. A. R. Causey, J. nucl. Mater. 54, 64, (1974) and unpublished work. 20. C. E. Coleman. A. R. Causey and V. Fidleris, J. nucl. Mater. 60, 185 (1976). 21. A. Akhtar, Acta mera/L 21, 1 (1973). 22. T. Sakai and M. E. Fine, Scripra metoll. 8, 545 (1974). 23. S. R. MacEwen and V. Fidleris, Phil. Mug. 31, 1149
r
R
Fig. l-l. Pole figure grid.
( b)
(1975).
24. S. R. MacEwen and V. Fidleris. Atomic Energy of Canada Limited Report AECL-5552 (1976). 25. S. N. Buckley, UKAEA Conference, The Interactions Between Dislocations and Point Defects AERE R-5944, Vol. 2 Part 3, p. 547 (1968). 26. V. Fidleris. J. nucl. Mater. 46. 356 (1973). 27 J. E. Harbottle, Irradiation et&s dn structural alloys for nuclear reactor applications. ASTM STP 4&1, 287 (1970). 28. G. J. C. Carpenter, D. 0. Northwood. J. nucl. Mater. M, 260 (1975). 29. R. A. Halt. accepted for publication 1979.
APPENDIX
in J. nucl. Mater.
Jo calculate anisotropy factors from texture we consider the grains in each orientation to behave as independent single crystals. For creep we calculate the shear and normal stresses on the crystallographic planes of interest. Using an assumed relation&up between stress and strain rate we calculate the shape change for each orientation. For growth the shape change is independent of stress. We resolve the shape change into dilational strains in the principal fabrication directions. The shape change of the component is then the average of these strains for all orientations in the pole figure. To do this. we divide a quarter pole figure up into a grid and represent each grid element by its centre (Fig. 1.1). The volume fraction represented by each element is TC(K=,
For basal slip,
and all other strains are zero. For prism slip.
1
Culculation of unisotropy factors from rexruret
V(K,, 4) =
Fig. l-2. Axes of HCP unit cell.
Q,: =
(1.3)
and all other strains are zero. For climb.
% = U&,X
(1.41
cr, = -E,,
(1.5,
and all other strains are zero. For climb due to interstitials (growth)
m the absence of stress
lyy = c,,.
(1.6)
For tubes the principle stresses are in the fabrication directions. i.e.
$)-sin rl
I: sin
Ll,(a,P
bRR
#
0 or = 0,
uTT
#
0 or = 0,
U.U
P 0 or = 0,
(1.7)
0~~
If we now examine the crystallographic structure .of the single crystal represented by any element (Fig. 1.2). Y represents the basal pole. and we define y as any prism plane normal in that gram. and 2 as the (1 120) direction for the prism plane (thus corresponding to the 6 direction). To represent a random distribution of (lIz0) about the basal pole we must consider all possible rotations of _I and 2 about x by an angle. d. t Additional symbols are defined in Table l-l. See the main text for reference list.
To calculate the stresses in each grain and the strains in the fabrication directions of the tube we need to know the direction cosines. These are calculated by rotation of co-ordinate axes from RTA to .yy~[20] as I XR
*
COs(K,),
lZr = sin(a,)cos(& LA r sin(aa)sin(&. I YR = - sin(aa)cos@),
1”~= cos(a&os(~)cos(#)
- sin(#sin(&
(1.8)
HOLT
AND
IBRAHIM:
THE
ANISOTROPY
OF IRRADIATION
- sm2 z”*cos’ zA,sm’ za - cos2 2”).
IrR = sin (ze) sin(h). sin(d) - sin@) cos(Cr). I:T = - cos(c+)cos(f#~)
eRA = x2*51* V*stn’ za.(sm2 zA*cos? z,isin2 zR - sm’ zrcos’
IrA = - cos(za) sin@) sin(d) + cos(& cos(d)
erR = x2*51.1
Then the stress m any grain are e*._= ea&~XR + ~rr~XTI:T + uAAlrAliA. =Yz=
ORRlyRbR uRR(1,R)2
=
+ +
uTTl,Tb urT(l:“)2
1327
eRr = X2.51 * V.sm’ zR*(sin’ zr.cosJ zT:sm2 za
IYA= cos(za) sin(&cos(rV - cos(4) sm(6).
o;,
CREEP
+ +
uAA~~A~;“~ U”“(LY~
0.9)
z,/(sm’ za - cos2 zr).
.sm2 zr*(s~n’ zR*cosZ zR,smz zr
- sm’ zA*cos2 z4 csm’ zr - cosr z”).
(1.10)
err = 12.51.1
(1.1 I)
erA = ~2*51*V*sm’zr*(sin’
The strains in the tube fabrtcatton directtons are
*stnLzr*smZzr.
- sm2 zacos’ eAR = x 2.5 I * I .stn’ -
EAT
=
sin2
zT.cosz
x2.51.)
zA~cosz z,ism? zr
xa smZ zr - co? z,). 2,*(sln’
yR*cosz
yT:smz
I,
-_
zR ;sin’
co?
yA
Y,).
*sm? z,*(stn: zr*cosz zrisin’ zl
- sm’ xR*cosZ za sm’ I, - cos’ zR). eAA= x2*5)* l’*sm’ z,*sm’ I, + fT..IXAI..”+ E,J,“I;“. The strains appropriate model are substituted
(1.14)
to any parttcular creep or growth from equations (1.1I to (1.8)and the
rest set to zero. The volume average strains are then calculated from ERR= z 1 C’(z,. 4) $eaR(za. 4.6). 1nd
(1.15)
For basal slip the strains are fRa = x 9.95. I,‘*cos2zR*sm2 za. ERr = y - 9.95. V*cos2 ZR’COSZ zr. ERA= r. - 9.95. I,‘+cos’ZR’COS2 Z”. Era = z - 9.95. Vcos’ Zr’COS2YR. lTT
For growth and for creep with a stress exponent of )I = 1 (i.e. low stress. m-reactor creep) the summations are greatly simplified. Growth phenomena which involve only normal strains and are isotropic about (0002‘ can be related to the resolved basal pole component Fd = 1 1 (I~. r$I cos’ zd Q@
(1.18)
or to the resolved prtsm pole component P, = 1 - Fd=
1 V(z,.#sm2zd. (1.19) %8 For creep with II = 1 the summatrons can be stmplified to give expresstons for relattve strain in uniaxial loadmg. For prism slip or chmb the strams are cRR
=
12*51*1’*sm
zR.smZ zR.
=
X9.95. C*cos2 zr*sin2 zr.
er” = 1 - 9~95*c’*cosz Zr’COS2ZA. f Aa = )3 - 9*95*l’cos2 z*cosz
CAT= l - 9.95. b ‘COSZz”‘cosr zr. cAA= X9.95. )‘*cos’ z,*sin2 zA. where 2.51 and 9.95 are normalization factors for lOO?;, prism and IO@“,,basal shp respectively. to give values of I for transverse creep with a random texture and hoop stress twice the axral stress (no radial stress) Thus for SO?, prism. 20”,, basal. e.g ERR= 0.8 ERR,,,,,.,,,) + 0.2.!&Rthr..,lt. From the grid in Fig. I.1 zr = arcc05, zA = arc cos,
(I - co? ~a)/( I + tan2 4). (I - co? z,).:(j + I itan2 I$).
Table 1-I. Definmon of symbols Y. I’.: zd 4 d 7C(z,. 4) V(z+ 9) uRR. OTT. u.,A eP.n. ETT. CA.4 0,; 0,; flrr 6.X: E,Z e,, er; L‘,. L’, c’, n,. nP
Ide
ZR .
axes of a h.c.p. umt cell (Fig. l-2) the angle between the ‘6 directton and Y (Fig. 1-I) the angle between the R-x plane and 7(F1g. 1-l) angle of rotation of r and r about Y texture coefficient for the grid element at zd, I#JFig. 1-l volume fraction represented by the grid element at ad. 4 are the normal stresses m the R. Tand A directions. respectively are the normal strains in the R. Tand A directtons IS the shear stress on the (ooO2) plane in the /ll?O\ directton IS the shear stress on the (ITOO)plane m the
HOLT
1328
AND
IBRAHIM:
APPENDIX
THE ANISOTROPY
2
Growth of isotropic materials?
Buckley [25] studied the growth of f.c.c. and b.c.c. materials during fission fragment irradiation. In the annealed condition these materials did not grow. Materials which had been cold-roiled IS”/, exhibited large positive growth strains in the rolling direction and small positive growth
n
Roller
/ Planes of maximum shear
CJ ROllW
Preferred dislocations
Fig. 2-1. Possible preferred 6 distribution rolling.
introduced
I-R.A. I I
-R.A
Fig. 2-2. Effect of cold-rolling on grain shape.
‘r
Rol liq
by
OF IRRADIATION
CREEP
strains in the transverse direction. Typically, the ratio (er/r,) was 0.01 to 0.07. He ascribed this behaviour to a preferred distribution of dislocation Burgers vectors due to the anisotropic nature of the rolling process (Fig. 2.1). Climb of these dislocations due to the absorption of interstttials created by the fission fragment bombardment would cause positive strains in the rolling direction and in the sheet normal direction. He assumed that the vacancies were stabilized as clusters (in whtch case no shrinkage would occur and a net density decrease would result) or loops (in which case there would be isotropic shrinkage). The former would give the observed dimensional changes with the hypothesized b distribution. The latter would give shrinkage in the transverse direction of twice the observed elongation in the rolling direction with the hypothesized b distribution (in conflict with the observed behaviour). An alternative explanation is that the grain shape resulting from cold-rolling is responsible for the observed behaviour. Assuming equiaxed grains in the annealed condition, cold-rolling produces grains which are elongated in the roiling direction and compressed in the short-transverse direction (Fig. 2.2). If the dislocations have an isotroptc b distribution and interstitials partition to grain boundaries, then the calculated growth strains would be as shown in Fig. 2.3 strictly due to changes in grain shape. The ratio er/e,, would be 0.09 at 15”; cold-work in reasonable agreement with the observed results. Changes in dislocation density (which have been ignored) might change the magnitudes of the strains, but not the relative behaviour in the three directions as a function of cold-work. Unfortunately, no information was reported on the density change accompanying growth or on whether the strains could be recovered by heating, both of which would help to distinguish between the models. In materials with anisotropic crystal structures, growth can occur with equiaxed grains (in the annealed condition for example). Figure 2.4 shows the effect of the change in grain shape due to cold-rolling on the calculated growth strains for a typical zirconium texture. The change in grain shape increases the axial growth strain independent of any effect of dislocation density.
dire&M
Fig. 2-3. Effect of rolling on grain shape anisotropy factors with random 6 distribution.
2-4. EtTect of rolling on anisotropy factor with constant texture. t See main text for reference list.
27 Norember 1978)
Abstract-The long term dimenslonal changes of zirconium alloys m a nuclear reactor core are attributed to generation of equal numbers of vacancies and interstitials by fast neutrons and their condensation at various sinks. The dimensional changes are anisotroptc because the microstructures are anisotropic and thus the sinks have anisotropic distributions. We derive relationships between anisotropy of m-reactor creep and growth of zirconium alloys with their crystallographic texture and grain shape. We conclude that growth occurs primarily by the partitioning of interstitials to dislocations or prismatic loops and vacancies to grain boundaries. Two strain producing mechanisms of creep are considered; climb and glide. Measured creep anisotropy agrees more closely with a glide model. Thus creep and growth can be treated additively to give predictions of the behaviour of zirconium alloy reactor components. R&sum&--On attribue les changements P long terme dans les dimensions des alliages de zirconium du coeur d’un reacteur nucleaire g la production par Ies neutrons rapides d’un nombre igal de lacunes et d’intersticiels et it leur condensation sur divers pieges. Les changements de dimension sont anisotropes. car les microstructures sont anisotropes et les pieges ont done une repartition anisotrope. Nous obtenons des relations entre l’anisotropie du fluage et de la croissance d’alliages de zirconium dans le rCacteur. et la texture cristallographique et la forme des grains. Nous en deduisons que la croissance rtsulte essentiellement de la dgregation des mtersticiels sur les dislocations ou les boucles prismatiques et de celle des lacunes sur les joints de grains. Nous considtrons deux m6canistnes de fluage produisant des d&formations: la montee et le glissement. L’anisotropie du fluage mesurte est en meilleur accord avec un mod6.le de ghssement. Le fluage et la croissance peuvent ainsi Btre trait& de manitre additive. et l’on obtient de bonnes privisions du comportement des constituants de riacteur en alliage de zirconium. Zusammenfnssung-Die langfristigen Abmessungtinderungen von Zirkonlegierungen im Kern eines Reaktors werden der Erzeugung von Leerstellen und Zwischengitteratomen in gleicher Anzahl und deren Kondensation an versctuedenen Senken zugeschrieben. Die Abmessungsiinderungen sind anisotrop. da die Mikrostruktur amsotrop ist und rolglich die Senken eine anisotrope Verteilung aufweisen. Es werden Zusammenhlnge aufgestellt zwlschen der Anisotropic des Kriechens und des Schwellens dieser Zirkonlegierungen im Reaktor und der kristallografischen Textur und der Kornform. Wir folgern, daB das Schwellen im wesentlichen durch das Ausheilen der Zwischengitteratome an Versetzungen und prismatischen Vertsetzungsringen und der Leerstellen an Korngrenzen entsteht. Zwei Mechanismen fir das Kriechen werden betrachtet: Gleiten und Klettern. Die gemessene Kriechanisotropie stimmt mlt dem Gleitmodell besser iiberein. Folglich kBnnen Kriechen und Schwellen additiv behandelt werden, urn zufriedenstellende Voraussagen iiber das Verhalten von Reaktorkomponenten aus Zirkonlegierungen zu erhalen.
1. INTRODUCTION The problem of predicting long term dimensional changes of zirconium alloy components in a fast neutron flux has been the subject of increasing study [l-q. Because the crystal structure of a-zirconium is close packed hexagonal, components fabricated from zirconium alloys have anisotropic properties. This leads to both anisotropic irradiation creep and to irradiation growth (dimensional change at constant volume in the absence of applied stress [8]). These dimensional changes affect the design of components in the cores of nuclear reactors. Thus it is important to predict the sign and magnitude of the dimensional changes and. if possible. to control them. \M
2’8.
R
Ross-Ross and Hunt [l] indicated the need for a mathematical treatment of anisotropy of in-reactor creep and Ross-Ross er al. [4] demonstrated anisotropit irradiation creep experimentally and proposed an analytical treatment to deal with this empirically, based on the approach of Hill [93. Ibrahim CZ], Schroeder and Holicky [3] and, more .recently, Franklin and Franz [6] recognized that anisotropic creep implied length change in an internally pressurized tube; Ross-Ross and Fidleris allowed for this in their design equations [YJ. Buckley [lo] first observed irradiation growth in zirconium single crystals and proposed a simple model relating growth to crystallographic texture.
1319
1320
HOLT
AND
IBRAHIM:
THE ANISOTROPY
Hesketh [l l] modified the model slightly but the relationship of growth strains to crystallographic texture remained the same. Wiedersich [12] suggested that growth could be related to grain shape and anisotropy of the distribution of orientations of dislocation Burgers vectors. Adamson [13] accumulated a large amount of data showing that irradiation growth was texture related and persisted to high fast neutron fluences. Much of the above information has been reviewed by Fidleris [S]. Alexander, Fidleris and Holt [7] treated long term dimensional changes of pressure tubes quantitatively as the sum of creep and growth components, both of which were related to crystallographic texture. In this paper we discuss the anisotropic microstructural factors affecting irradiation creep and growth of zirconium alloy components and derive quantitative relationships between microstructural parameters and the direction and magnitude of creep and growth strains. We compare the calculated anisotropy factors for creep and growth with experimental evidence. Finally, we illustrate the use of these factors in evaluating the behaviour of power reactor pressure tubes.
OF IRRADIATION
CREEP
+ (1120) for either loops [17] or straight dislocations [18]. Thus the strain produced by the climb mechanism in any grain depends on the orientation of the (1120) directions with respect to the stress and the direction of interest. The anisotropy factor for strain due to climb thus depends on the distribution of grain orientations, i.e. crystallographic texture. To calculate the anisotropy factor we assume [lS] Bbx a II - Cl.
The shape change of an a-zirconium single crystal due to the SIPA mechanism is shown schematically in Fig. 1. According to the climb-plus-glide model, enhanced climb in the presence of a fast neutron flux allows the dislocations to surmount obstacles and glide across obstacle free regions. Strain is produced by the glide process. Glide is always observed on : lOi0: and sometimes on ( IOil) or :0002) planes in z-zirconium. The Burgers vector is always 3 (1 120) except under conditions of extreme triaxial constraint. Thus the anisotropy of strain due to glide also depends on crystallographic texture. For glide we assume ;ib x rJ*.
2. GENERAL
APPROACH
Our approach is based on the assumption that the strain rate in a given direction of a material is the product of two components, a specific rate at which some strain producing mechanism occurs (slip. for example) and an anisotropy factor which resolves the strain produced by this hechanism into the direction of interest.t ’ = cd
AdK.
(1)
If we define a strain producing mechanism, we can define the anisotropy factor from the anisotropy of the microstructure. We can then predict the sign and relative magnitude of the strain rate with only limited knowledge of K.
(21
(3)
The shape changes due to the glide mechanisms are illustrated schematically in Fig. 1. For a given system of applied stress we calculate cl,, r~, and rb for each grain orientation. We then calculate & and fb and resolve them in the fabrication directions of the tube. The resolved strains, a, and fr,, are averaged according to the basal pole figure. Compatibility and constraint between grains are ignored. These calculations are described in detail in Appendix 1. 3.2 Experimental observations of creep anisotropy
Stress sensitivity of creep is close to one [6] in the range of operating conditions of interest for zirconium alloys in thermal power reactors (up to 2OOMPa, 500-6OOK. 10’6-10L8n m-* s-l). Crystal3. ANISOTROPY OF CREEP lographic textures of pressure tubes of two zirconium alloys are shown in Fig. 2, and anisotropy factors 3.1 Models for creep calculated for a stress sensitivity of one (see Appendix Currently favoured models of in-reactor creep fall 1) are given in Table 2. into two categories, those in which dislocation climb The anisotropy factors, lAAand err, represent the is the strain producing mechanism (loop alignment and SIPAS models [14, IS]) and those in which glide relative magnitudes of creep strains for specimens stressed uniaxially in the axial and transverse direcis the strain producing mechanism (climb-plus-glide tions respectively. We can compare the ratios of model [ 161). According to the climb models, interstitials pro- measured strains in different directions with calculated values to determine which mechanisms correduced by the fast neutron flux are attracted to straight dislocations or loops with their Burgers vectors. b, spond most closely to the observed behaviour. Causey [19] and Coleman et al. [ZO] obtained parallel to the applied stress. The excess vacancies go to dislocations or loops with their Burgers vectors creep strain-rate data from bent beam stress relaxaperpendicular to the applied stress (Fig. I). The most tion tests in two directions in pressure tube materials. commonly observed Burgers vector in a-zirconium is Changes of curvature in bent beam stress relaxation specimens are wholly from creep since growth does t The symbols used throughout are defined in Table 1. not affect curvature. The ratio of axial creep rate with : Stress-induced preferential absorption. an axial stress to transverse creep rate with a trans-
HOLT
AND
IBRAHIM:
THE ANISOTROPY OF IRRADIATION
0002
CREEP
1311
Plane
1
+ (1120) vector
(ISOO, Plane
(li01) Plane
/
(lioo),+
/
(11.20)
KlOO2),f ( 1120)
Prismatic
Basal
Slip
slip
cfioi),f (1150) Pyramidal shp
Fig. 1. Shape change of an a-Zr crystal for various creep mechanisms resulting from stress indicated. verse stress. R, corresponds to the ratio of anisotropy factors l~,&rr. These values are shown in Table 2. For both Zircaloy-2 and Zr-2.5wt.9, Nb the climb and prismatic slip models overpredict R. The contribution to strain of dislocations with b = 3 (IlzO) gliding on pyramidal or basal planes would reduce the value of R. An unrealistically large proportion of pyramidal slip would be required. However, the basal slip model greatly underpredicts R (Table 2). Good agreement with experimental results is obtained if basal slip accounts for 20”” of the strain and prismatic slip for 80”, of the strain. Based on published data for basal slip of zirconium[21] at high temperatures (i.e. > 850 K) and titanium alloys [22] at up to 500 K, this appears to be a reasonable assumption for the proportlon of basal slip at 500-600 K. The anisotropy predicted with the climb model is not modified by a change in glide plane. Thus a reduction in R can only be obtained if dislocations are present within which the Burgers vector has a component in the [0002] directlon. Since such dislocations are very rarely observed, the creep anisotropy can best be explained if glide is the strain producing mechanism. This conclusion is compatible with the results of t For vacancies. the sink may be a cluster of vacancies left by a displacement cascade-then no migration of the bacancles IS required.
recent in-reactor creep tests suggesting that glide is the strain producing mechanism at conditions of interest for operating pressure tubes [23,24]. 4. ANISOTROPY
AND GROWTH
4.1 Models for growth Irradiation growth results from the redistribution of atoms displaced from the lattice by fast neutrons. The neutrons create equal numbers of vacancies and interstitials which migrate through the lattice until they recombine or reach a sink at which they can precipitate.? The volume change associated with the formation of the point defects is isotropic but that associated with their precipitation at sinks may be oriented. depending upon the type of sink. For example, for grain boundaries the strain is normal to the grain boundary. Vacancies cause a net contraction in this direction and interstitials cause a net expansion (12). The strain produced can be resolved into any direction of interest. d, and summed over all orientations of sinks. Assuming that the number of defects precipitating at a given sink is independent of the orientation of the sink E,J= (IV,, - N,$
z X,(w) cos2 0. 0,
= (R;‘,,- N,.,)RAd,.
(4) (51
1322
HOLT
AND
IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
CREEP
Table 1. Definition of symbols
XJlU) I s
Y :d, Pd Fd N,,
N,,
dQa G K,(S). K,(S) dJ 6 cd
subscrtpt Indicating general directions of interest subscripts indicating specific direction of interest, radial, transverse. axial stram in the ‘d’ direction strain rate in the ‘8 direction growth rate in the ‘d’ direction creep rate in the ‘d’ direction rate constant for a strain process anisotropy factor resolving strain in the direction of interest ‘d’ the Burgers vector subscript indicating the direction of the Burgers vector normal stress in the direction of the Burgers vector normal stress perpendicular to the Burgers vector and to [0002] shear strain rate on the slip plane in the direction of the Burgers vector shear stress on the slip plane in the direction of the Burgers vector stress sensitivity of glide anisotropy factor for the creep strain in the ‘e’ direction due to a uniaxial stress in the ‘8 direction ratio of anisotropy factors ~&err number of interstitials and vacancies respectively arriving at the type of sink denoted by subscript L3 Y subscript denoting dislocations atomic volume is the angle between the ‘d’ direction and the direction of strain for a given type of sink (e.g. ‘b direction for dislocations, direction of grain boundary normal for grain boundaries) is the fraction of sinks of type ‘y’ oriented as defined by o subscript denoting prismatic loops subscript denoting sub-grain boundaries subscript denoting grain boundaries subscript denoting clusters or voids anisotropy factor for type ‘y’ sinks with respect to the ‘d’ direction ‘the resolved prism pole component’ as defined in Appendix 1 resolved basal pole component in the ‘d’ direction number of excess interstitials or vacancies respectively arriving at type ‘y’ sinks displacements per atom (the number of times each atom has, on average, been displaced from its lattice site by fast neutrons anisotropy factor for growth in the ‘8 direction specific growth and cmp rates for material of structure ‘S displacement rate (dpa s- ‘) normalized stress creep anisotropy factor in the ‘d’ direction.
For growth to occur, different numbers of vacancies and interstitials must’precipitate at different sinks, i.e. N,, - No,, ,# 0
(6)
and the sinks must be characterized by anisotropic orientation distributions, X(w), which are different for each type of sink. If the point defect should precipitate at a void or point defect cluster there is no strain but a net density change occurs. To estimate the anisotropy of irradiation growth we must evaluate both the partitioning of point defects to the different sinks and the anisotropy of the sinks. 4.2 Anisotropy of sinks Precipitation of point defects at prismatic loops gives strain in the direction of the Burgers vectors of the loops. Only loops with a Burgers vector. b = 4 ( 1IlO), on { IlzO; planes have been observed in a-zirconium [17J In the absence of an applied stress, loops will form randomly on the ( 1lIOl planes and thus the anisotropy factor for prismatic loops is directly related
to crystallographic A dl=
pd -.
2
texture
Precipitation of point defects at dislocations causes them to climb giving strain in the b direction. Growth occurs if there is an anisotropic distribution of Burgers vectors resulting from a forming process [25]. Alternatively, in materials with anisotropic crystal structures, the anisotropy of the Burgers vector distribution could result strictly from crystallographic texture. In a-zirconium a Burgers vector, b = 4 (1120). is by far the most frequent. With the Burgers vectors randomly distributed about the basal plane normal A,iD= pd/2.
03)
The direction of strain when a point defect is absorbed at a grain boundary is the direction of the grain boundary normal. During forming processes the grains are elongated in the direction of working and compressed in directions normal to the direction of working. To derive the anisotropy factors for grain boundaries one must measure the angular distribution of grain boundary normals X&J) about the direction of interest. The anisotropy factor can be approximated by assuming a simple rectangular grain shape with axes in the three principal directions of
HOLT
IBRAHIM:
AND
THE ANISOTROPY
%Nb
Zr-2.5wt
OF IRRADIATION
CREEP
1323
Zirmloy - 2 Aual
Ask31
--“’ co5
Rodiai
Tmnsverse
Transverse
Fig. 2. Basal pole figures for typical cold-worked Zr-2.5 wt.‘, Nb and Zr-2 pressure tubes.
defects at smks is
strain during working and then (9)
Anisotropy of grain boundary orientations could explain the irradiation growth observed in materials with isotropic crystal structures [253 (see Appendix 2). Subgrain boundaries consist of networks of dislocations. Precipitation of point defects at subgrain boundaries will &use the component dislocations to climb giving strain in the directions of their Burgers vectors. In many cases the distribution of dislocation Burgers vectors in subgrain boundaries will be the same as for free dislocations and therefore A
dls
=
AdO
=
(10)
P&?.
Precipitation of point defects at free surfaces results in strain in the direction normal to the free surface. The proportion of point defects migrating to free surfaces would be negligibly small for most polycrystalline reactor parts. However, free surfaces should be considered in analysing irradiation growth in thin specimens in which electron or ion bombardment has been used to simulate neutron irradiation. 4.3 Parritioning
of point defects
The total strain due to the precipitation
of point
+ (Ni, - N,)A,
+ (N,, - N,,)&.
1 N,, - 1 N, = 0. Y
‘id = liTi,Adl + ICli,Adn R =
(fi,, + ii, -
fiT,A,, - i?,A,
a&J;- lQwAdg.
Ratio of fAA:crr
CA.4
Zircaloy-2
l
From stress-relaxation
tests C19.203
(13)
If a significant proportlon of vacancies go to clusters or voids swelling occurs. Since equal numbers of vacancies and interstitials are created by the
zirconium alloy pressure tubes
Measured* Prism slip. P Basal slip, B 80”, P + 20”” B Climb Measured* Prism slip. P Basal slip. B 80”” P + 20”, B Climb
(12)
Partitioning (as expressed by equation 6) occurs in the absence of applied stress due to the interaction between the large dilational strain field of interstitials and the long range strain heIds of some sinks (dislocations and loops). Interstitials are preferentially attracted to dislocations and to loops. Since vacancies have a very mu&h weaker strain field, excess vacancies precipitate at sinks without long range strain fields (clusters, voids, grain boundaries, subgrain boundaries and free surfaces). Thus
Table 2. Comparison of measured and calculated creep anisotropy For cold-worked
Zr-2.5 wt.?, Nb
(111
Also. since equal numbers of vacancies and interstitials are produced by the flux
2.10
-
0.54 1.79 2.10
0.83 1.36 0.93 0.83
2.18 0.5 1 1.85 2.18
0.93 1.41 1.03 0.93
-
%F 0:40
-
0.36 J&J 2.34
1324
HOLT AND IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
CREEP
Table 3. Possible point defect sinks Source of anrsotropy
Sink
Direction of strain
Clusters Voids Loops Dislocations
Isotropic (volume change) Isotropic (volume change1 In b direction In b direction
Subgrain boundaries Grain boundaries SUrfactS*
In b direction Normal to boundary
Texture Texture or anisotroptc strains durrng cold-work Texture or anisotropic strains during cold-work Grain shape
NormaI to surface
Macroscopic shape
* Can be ignored if grain size 4 minimum dimension.
neutron‘flux
_
^
Nil + NiD - ti,, - fi, Table 3 summarizes the possible and the source of their anisotropy.
4.4 Experimental
tiw=0.
(I41
sinks, their roles
observations of irradiation
growth
At low fluences, (c 1 dpa) the growth behaviour of a-zirconium alloys is complicated by transients due to stress relieving [26]. We will only describe the high fluence behaviour (> 1 dpa). Experimental observations of growth are [8,10,11,13,26-281: (1) Growth is always positive in the working diiection. (2) Growth can be positive or negative in directions normal to the working direction. {3) Growth in annealed materials tends to saturate or be reduced to a very low rate after 1 dpa.
(4) Growth in cold-worked materiab continues at a steady state rate at high dqses, > 1 dpa. (5) Growth of annealed material is recoverable on heating after irradiation. (6) Only a small proportion of the growth strain can be recovered on heating cold-worked materials after irradiation. (7) Density changes are smail compared with growth strains in cold-worked materials. (8) The growth strain in annealed mate&is cannot be accounted for by loops visible in the electron microscope. Based on observation 7 we can conclude that the proportion of vacancies migrating to clusters and voids is negligible and thus equation (13) becomes $d -=
R
Fig. 3. Shape change of a-Zr during growth. (1) Initial shape; (2) expansion in (ll~O> directions due to absorption of interstitials at dislocations: (31 isotropic contraction due to absorption of vacancies at boundartes of equiaxed grains; (41 anisotropic contraction due to absorption of vacancies at boundaries of grains flattened in the x-direction.
HOLT
AND
Table 4. Analysis of dimensional
IBRAHIM:
THE ANISOTROPY
OF IRRADIATION
changes of Hanford ‘N’ reactor pressure tubes made by Chase Brass (Type 3) based on microstructural measurements on six tubes Radial
1.
2. 3. 4. 5. 6. 1. 8 9. 10.
Resolved basal pole component, F., Resolved prism pole component. P, Mean intercept grain size, pm Growth anisotropy factor (grain boundary anisotropy ignored) Growth anisotropy factor (grain boundary anisotropy included) Creep anisotropy factor (prism slip only) Creep anisotropy factor (prism slip + basal slip) Calculated rate (4 & 6) x low4 dpa-‘* Calculated rate (5 & 7) x 1O-4 dpa-’ Measured rate x 10m4dpa-’
* 1 x 1025nm-2
1325
CREEP
0.46 0.54 7.7 -0.38 -0.97 -0.44 -0.63 - 10.0 - 12.4 -
Transverse 0.46 0.54 10.3 - 0.38 -0.33 0.46 0.63 2.5 4.9 4.7
Axial 0.08 0.92 14.2 0.7 1.3 -0.0 0.00 7.5 1.5 79
> 1 MeV taken as equivalent to 1 dpa.
6. CONCLUSIONS
and we define the growth anisotropy factor
(16) For typical textures of zirconium alloy tubes and strip, this is always positive in the working direction and may be positive or negative in directions normal to the working direction. It is thus in good agreement with the experimental observations. Figure 3 illustrates schematically the shape changes for an a-zirconium crystal due to irradiation growth according to equation (15). 5. BEHAVIOUR OF ZIRCONIUM ALLOY PRESSURE TUBES Based on the foregoing assumptions, creep (glide) and growth (climb) are additive. The total rate of dimensional change of a zirconium alloy tube can thus be expressed by (7) id = K,(S)*+.Gd + K,(S)*4*5.C,.
(17)
Alexander er al. (7) interpreted the dimensional behaviour of the cold-worked Zr-2 pressure tubes in the Hanford ‘N’ reactor using this model. Using the analysis described here, they calculated anisotropy factors for creep assuming only prismatic slip and for growth assuming climb of f( 11~0). t lOi0; dislocations by absorption of interstitials but ignoring any contribution of grain shape. They underpredicted the diametral strain rates. Incorporating grain shape and basal slip into the model greatly improves the agreement with observed diametral strain rates, and allows the correlation of the behaviour of different types of pressure tubes in several reactors. For example, Table 4 shows the revised analysis of the behaviour of Hanford ‘N’ reactor pressure tube compared with that made by Alexander et al. [TJ showing improved agreement of the calculated transverse strain rate. The constants K,(S) and K,(S) were taken as 5.8 x 10e4 dpa-’ and 1.9 x 10-sdpa-’ respectively [29].
The anisotropy of irradiation induced long-term dimensional changes of zirconium alloys can be predicted from anisotropy of microstructure. once the strain producing mechanisms are identified. Anisotropy of irradiation growth is calculated from crystallographic texture and grain shape assuming that interstitials are attracted preferentially to 3 (1120) dislocations with excess vacancies arriving at grain boundaries. The anisotropy predicted by this model agrees with the experimental irradiation growth data. The anisotropy calculated from crystallographic texture assuming that creep occurs by climb or glide of j( 1120) dislocations on i lOi0; planes is greater than that measured experimentally. The anisotropy calculated for glide agrees with experimental measurements if about 20”,, as much slip of f (1120) dislocations occurs on (0002) as on ;loio; planes. Creep and growth can be treated additively to predict the anisotropy of zirconium alloy compcnents operating under stress in a fast neutron flux. Acknow/edgements-We thank A R. Causey for the use of unpublished information and C. E. Coleman and B. A. Cheadle for comments on the manuscript.
REFERENCES I. P. A. Ross-Ross and C. E. L. Hunt. J. nucl. Mater. 26, 2 (1968). 2. E. F. Ibrahim, Applications related phenomena in zirconium and its alloys, ASTM STP 458, 19 (1968). 3. J. Schroeder and M. Hohcky. J. nucl Mater. 33. 52 (1969). 4. P. A. Ross-Ross, V. Fidleris and D. E. Fraser. Can. Metall. Q. 11, 10 (1972). 5. P. A. Ross-Ross and V Fidleris. Inrernational Conference on Creep and Fatigue in Elevated Temperature Appliccltions. Philadelphia Sept. 1973. paper C216i73 6. D. G. Franklin and W. A Franz, Zirconium m the
nuclear industry. ASTM STP 633, 365 (1976). 7. W. Alexander. V. Fidleris and R. A. Holt. Zirconium in the nuclear Industry. ASTM STP 633, 344 (1976). 8. V. Fidleris. Atomic Energ! Rer. 13, 51 (1975). 9. R. Hill, The Mathematical Theory q/ Plastirrty Oxford University
Press, Oxford
(1950)
1326
HOLT
AND
IBRAHIM:
THE ANISOTROPY
10. S. N. Buckley, Properties of Reactor Materials and Eficts of Radiation Datnuge (edited by W. J. Littler),
p. 413. Butterworths, London (1962). 11. R. V. Hesketh, J. nucl. Muter. 30. 219 (1969). 12. H. Wiedersich, The Physical Merallurgy of Reactor Fuel Elements (edited bv J. E. Harris and E. C. Svkes). _ ” p. 144. Metals‘Society,-London (1973). 13. R. B. Adamson, Zirconium in the nuclear industry, ASTM STP 633, 326 (1976). 14. F. A. Nichols, J. nucl. Mater. 30, 249 (1969). 15. P. T. Hcald and M. V. Speight, Phi/. Mug. 29, 1075
OF IRRADIATION
CREEP
(O’ ‘p‘\
(1974).
16. S. R. MacEwen, J. nucl. Mater. 54, 85 (1974). 17. G. J. C. Carpenter and D. 0. Northwood, Proc. Int. Conf on Fundamental Aspects of Radiation Damage in Metals, Gatlinburg, Tennessee, 2, 783 (1975). 18. D. L. Douglass, The Metallurgy of Zirconium, Atomic Energy Rev. Supplement (1971). 19. A. R. Causey, J. nucl. Mater. 54, 64, (1974) and unpublished work. 20. C. E. Coleman. A. R. Causey and V. Fidleris, J. nucl. Mater. 60, 185 (1976). 21. A. Akhtar, Acta mera/L 21, 1 (1973). 22. T. Sakai and M. E. Fine, Scripra metoll. 8, 545 (1974). 23. S. R. MacEwen and V. Fidleris, Phil. Mug. 31, 1149
r
R
Fig. l-l. Pole figure grid.
( b)
(1975).
24. S. R. MacEwen and V. Fidleris. Atomic Energy of Canada Limited Report AECL-5552 (1976). 25. S. N. Buckley, UKAEA Conference, The Interactions Between Dislocations and Point Defects AERE R-5944, Vol. 2 Part 3, p. 547 (1968). 26. V. Fidleris. J. nucl. Mater. 46. 356 (1973). 27 J. E. Harbottle, Irradiation et&s dn structural alloys for nuclear reactor applications. ASTM STP 4&1, 287 (1970). 28. G. J. C. Carpenter, D. 0. Northwood. J. nucl. Mater. M, 260 (1975). 29. R. A. Halt. accepted for publication 1979.
APPENDIX
in J. nucl. Mater.
Jo calculate anisotropy factors from texture we consider the grains in each orientation to behave as independent single crystals. For creep we calculate the shear and normal stresses on the crystallographic planes of interest. Using an assumed relation&up between stress and strain rate we calculate the shape change for each orientation. For growth the shape change is independent of stress. We resolve the shape change into dilational strains in the principal fabrication directions. The shape change of the component is then the average of these strains for all orientations in the pole figure. To do this. we divide a quarter pole figure up into a grid and represent each grid element by its centre (Fig. 1.1). The volume fraction represented by each element is TC(K=,
For basal slip,
and all other strains are zero. For prism slip.
1
Culculation of unisotropy factors from rexruret
V(K,, 4) =
Fig. l-2. Axes of HCP unit cell.
Q,: =
(1.3)
and all other strains are zero. For climb.
% = U&,X
(1.41
cr, = -E,,
(1.5,
and all other strains are zero. For climb due to interstitials (growth)
m the absence of stress
lyy = c,,.
(1.6)
For tubes the principle stresses are in the fabrication directions. i.e.
$)-sin rl
I: sin
Ll,(a,P
bRR
#
0 or = 0,
uTT
#
0 or = 0,
U.U
P 0 or = 0,
(1.7)
0~~
If we now examine the crystallographic structure .of the single crystal represented by any element (Fig. 1.2). Y represents the basal pole. and we define y as any prism plane normal in that gram. and 2 as the (1 120) direction for the prism plane (thus corresponding to the 6 direction). To represent a random distribution of (lIz0) about the basal pole we must consider all possible rotations of _I and 2 about x by an angle. d. t Additional symbols are defined in Table l-l. See the main text for reference list.
To calculate the stresses in each grain and the strains in the fabrication directions of the tube we need to know the direction cosines. These are calculated by rotation of co-ordinate axes from RTA to .yy~[20] as I XR
*
COs(K,),
lZr = sin(a,)cos(& LA r sin(aa)sin(&. I YR = - sin(aa)cos@),
1”~= cos(a&os(~)cos(#)
- sin(#sin(&
(1.8)
HOLT
AND
IBRAHIM:
THE
ANISOTROPY
OF IRRADIATION
- sm2 z”*cos’ zA,sm’ za - cos2 2”).
IrR = sin (ze) sin(h). sin(d) - sin@) cos(Cr). I:T = - cos(c+)cos(f#~)
eRA = x2*51* V*stn’ za.(sm2 zA*cos? z,isin2 zR - sm’ zrcos’
IrA = - cos(za) sin@) sin(d) + cos(& cos(d)
erR = x2*51.1
Then the stress m any grain are e*._= ea&~XR + ~rr~XTI:T + uAAlrAliA. =Yz=
ORRlyRbR uRR(1,R)2
=
+ +
uTTl,Tb urT(l:“)2
1327
eRr = X2.51 * V.sm’ zR*(sin’ zr.cosJ zT:sm2 za
IYA= cos(za) sin(&cos(rV - cos(4) sm(6).
o;,
CREEP
+ +
uAA~~A~;“~ U”“(LY~
0.9)
z,/(sm’ za - cos2 zr).
.sm2 zr*(s~n’ zR*cosZ zR,smz zr
- sm’ zA*cos2 z4 csm’ zr - cosr z”).
(1.10)
err = 12.51.1
(1.1 I)
erA = ~2*51*V*sm’zr*(sin’
The strains in the tube fabrtcatton directtons are
*stnLzr*smZzr.
- sm2 zacos’ eAR = x 2.5 I * I .stn’ -
EAT
=
sin2
zT.cosz
x2.51.)
zA~cosz z,ism? zr
xa smZ zr - co? z,). 2,*(sln’
yR*cosz
yT:smz
I,
-_
zR ;sin’
co?
yA
Y,).
*sm? z,*(stn: zr*cosz zrisin’ zl
- sm’ xR*cosZ za sm’ I, - cos’ zR). eAA= x2*5)* l’*sm’ z,*sm’ I, + fT..IXAI..”+ E,J,“I;“. The strains appropriate model are substituted
(1.14)
to any parttcular creep or growth from equations (1.1I to (1.8)and the
rest set to zero. The volume average strains are then calculated from ERR= z 1 C’(z,. 4) $eaR(za. 4.6). 1nd
(1.15)
For basal slip the strains are fRa = x 9.95. I,‘*cos2zR*sm2 za. ERr = y - 9.95. V*cos2 ZR’COSZ zr. ERA= r. - 9.95. I,‘+cos’ZR’COS2 Z”. Era = z - 9.95. Vcos’ Zr’COS2YR. lTT
For growth and for creep with a stress exponent of )I = 1 (i.e. low stress. m-reactor creep) the summations are greatly simplified. Growth phenomena which involve only normal strains and are isotropic about (0002‘ can be related to the resolved basal pole component Fd = 1 1 (I~. r$I cos’ zd Q@
(1.18)
or to the resolved prtsm pole component P, = 1 - Fd=
1 V(z,.#sm2zd. (1.19) %8 For creep with II = 1 the summatrons can be stmplified to give expresstons for relattve strain in uniaxial loadmg. For prism slip or chmb the strams are cRR
=
12*51*1’*sm
zR.smZ zR.
=
X9.95. C*cos2 zr*sin2 zr.
er” = 1 - 9~95*c’*cosz Zr’COS2ZA. f Aa = )3 - 9*95*l’cos2 z*cosz
CAT= l - 9.95. b ‘COSZz”‘cosr zr. cAA= X9.95. )‘*cos’ z,*sin2 zA. where 2.51 and 9.95 are normalization factors for lOO?;, prism and IO@“,,basal shp respectively. to give values of I for transverse creep with a random texture and hoop stress twice the axral stress (no radial stress) Thus for SO?, prism. 20”,, basal. e.g ERR= 0.8 ERR,,,,,.,,,) + 0.2.!&Rthr..,lt. From the grid in Fig. I.1 zr = arcc05, zA = arc cos,
(I - co? ~a)/( I + tan2 4). (I - co? z,).:(j + I itan2 I$).
Table 1-I. Definmon of symbols Y. I’.: zd 4 d 7C(z,. 4) V(z+ 9) uRR. OTT. u.,A eP.n. ETT. CA.4 0,; 0,; flrr 6.X: E,Z e,, er; L‘,. L’, c’, n,. nP
Ide
ZR .
axes of a h.c.p. umt cell (Fig. l-2) the angle between the ‘6 directton and Y (Fig. 1-I) the angle between the R-x plane and 7(F1g. 1-l) angle of rotation of r and r about Y texture coefficient for the grid element at zd, I#JFig. 1-l volume fraction represented by the grid element at ad. 4 are the normal stresses m the R. Tand A directions. respectively are the normal strains in the R. Tand A directtons IS the shear stress on the (ooO2) plane in the /ll?O\ directton IS the shear stress on the (ITOO)plane m the
HOLT
1328
AND
IBRAHIM:
APPENDIX
THE ANISOTROPY
2
Growth of isotropic materials?
Buckley [25] studied the growth of f.c.c. and b.c.c. materials during fission fragment irradiation. In the annealed condition these materials did not grow. Materials which had been cold-roiled IS”/, exhibited large positive growth strains in the rolling direction and small positive growth
n
Roller
/ Planes of maximum shear
CJ ROllW
Preferred dislocations
Fig. 2-1. Possible preferred 6 distribution rolling.
introduced
I-R.A. I I
-R.A
Fig. 2-2. Effect of cold-rolling on grain shape.
‘r
Rol liq
by
OF IRRADIATION
CREEP
strains in the transverse direction. Typically, the ratio (er/r,) was 0.01 to 0.07. He ascribed this behaviour to a preferred distribution of dislocation Burgers vectors due to the anisotropic nature of the rolling process (Fig. 2.1). Climb of these dislocations due to the absorption of interstttials created by the fission fragment bombardment would cause positive strains in the rolling direction and in the sheet normal direction. He assumed that the vacancies were stabilized as clusters (in whtch case no shrinkage would occur and a net density decrease would result) or loops (in which case there would be isotropic shrinkage). The former would give the observed dimensional changes with the hypothesized b distribution. The latter would give shrinkage in the transverse direction of twice the observed elongation in the rolling direction with the hypothesized b distribution (in conflict with the observed behaviour). An alternative explanation is that the grain shape resulting from cold-rolling is responsible for the observed behaviour. Assuming equiaxed grains in the annealed condition, cold-rolling produces grains which are elongated in the roiling direction and compressed in the short-transverse direction (Fig. 2.2). If the dislocations have an isotroptc b distribution and interstitials partition to grain boundaries, then the calculated growth strains would be as shown in Fig. 2.3 strictly due to changes in grain shape. The ratio er/e,, would be 0.09 at 15”; cold-work in reasonable agreement with the observed results. Changes in dislocation density (which have been ignored) might change the magnitudes of the strains, but not the relative behaviour in the three directions as a function of cold-work. Unfortunately, no information was reported on the density change accompanying growth or on whether the strains could be recovered by heating, both of which would help to distinguish between the models. In materials with anisotropic crystal structures, growth can occur with equiaxed grains (in the annealed condition for example). Figure 2.4 shows the effect of the change in grain shape due to cold-rolling on the calculated growth strains for a typical zirconium texture. The change in grain shape increases the axial growth strain independent of any effect of dislocation density.
dire&M
Fig. 2-3. Effect of rolling on grain shape anisotropy factors with random 6 distribution.
2-4. EtTect of rolling on anisotropy factor with constant texture. t See main text for reference list.