Biaxial creep of textured zircaloy II: Crystal mean plastic modelling

Materials Science and Engineering, 70 (1985) 181-190

181

Biaxial Creep of Textured Zircaloy II: Crystal Mean Plastic Modelling B. L. ADAMS Brigham Young University, Provo, UT84602 (U.S.A.) K. L. MURTY North Carolina State University, Raleigh, NC 27650 (U.S.A.)

(Received March 23, 1984; in revised form August 27, 1984)

ABSTRACT Crystal mean plastic bounding models are applied to highly textured cold-worked Zircaloy tubing and to highly textured recrystallized Zircaloy-4 tubing. Predictions o f creep behavior are compared with data obtained at 673 K under biaxial loading. Textural differences due to recrystallization are predicted to have a negligible effect on creep behavior whereas differences in the predominant slip mode (prismatic, basal and pyramidal slip) are shown to have significant impact in these materials. Prismatic slip is predicted to be dominant in the recrystallized tubes whereas basal and pyramidal slip are shown to be more important for the cold-worked tubes.

1. INTRODUCTION In Part I [1], we described the modelling of creep behavior of Zircaloy-4 tubes at 673 K under biaxial stress using a modified Hill approach. Evidently, by selecting the appropriate anisotropy parameters R and P, it is possible to correlate creep behavior for a single material over a range of stress states. A major drawback to this approach, however, is that R and P are not known functions of metallurgical conditions and hence there is no means for incorporating textural or microstructural input into the correlations other than to determine R and P empirically for a given material. Thus, extrapolating model predictions to new material conditions (e.g. partially recrystallized tubing) is n o t possible with the former approach. 0025-5416/85/$3.30

An alternative approach, which does incorporate the fundamental input of textural anisotropy, is the crystal mean plastic method. Here, single-crystal constitutive behavior is averaged over the range of crystalline orientations in the aggregate to predict bulk behavior. Several investigators have modelled deformation behavior by considering the orientation distribution of basal planes in orientation space [ 2 - 4 ] . Our approach differs from these efforts in that each orientation is weighted by the orientation distribution function in order to represent the complete effect of preferred orientation accurately. The orientation distribution function is determinable from X-ray intensity measurements of crystallographic plane pole distributions (pole figures). While other details of microstructure such as grain size, grain shape and precipitate distribution are omitted, it is quite clear that crystallographic texture has a significant influence on creep anisotropy. In addition, since the singlecrystal constitutive formulation considers specific slip systems (specific slip planes and slip directions), the averaging m e t h o d has the advantage of distinguishing between the contributions of these varied slip systems. Our modelling of Zircaloy suggests that the specific annealing and cold-working history of the tube can significantly alter the degree of operation of these slip systems. Hence an approach which considers anisotropy in terms of both texture and predominant slip system offers the materials engineer a means of selecting specific processing parameters which optimize the cladding's technological performance for a specified application. We have explored predictions of creep behavior based on bounding calculations similar © Elsevier Sequoia/Printed in The Netherlands

182 to those used to determine the lower bound [5] and the upper b o u n d [6] for the elastic stiffness in polycrystalline bodies. The upper bound approach for creep in polycrystals was first described by Hutchinson [7, 8]. We have adapted his approach, which he applied to isotropically distributed polycrystals, by incorporating explicit preferred crystallographic orientations in the averaging procedures. A complementary lower b o u n d calculation was also included. In this paper we briefly review the analysis of preferred orientation and the bounding model due to Hutchinson. The materials modelled were described in Part I [ 1 ] with the exception of the textural difference between the cold-worked stress-relieved tubing and the recrystallized (and ®) tubing. Orientation distribution functions are presented here for both material conditions. Testing and data analysis procedures were also presented in Part I [1] and will n o t be repeated here.

2. DESCRIPTION OF P R E F E R R E D ORIENTATION IN ZIRCALOY CLADDING

2.1. The crystallographic pole figure The most c o m m o n representation of preferred orientation is the crystallographic pole figure. It is, simply, the distribution of the measured diffracted X-ray intensity in space of a particular crystallographic plane [9]. Specific planes are selected in an X-ray diffraction experiment by fixing the Bragg angle. With respect to an arbitrary set of processing axes, the diffraction direction (reciprocal lattice vector) may be rotated with t w o degrees of freedom. Most often, the crystallographic pole figure is plotted as a stereographic projection showing times-random iso-intensity contours of the intensity distribution referred to the processing axes. The pole figures shown in Fig. 1 were obtained by t w o methods. The basal distributions were measured using composite L o p a t a Kula specimens [10] prepared by laminating flattened sections of the tube wall (without a thickness reduction) and then slicing the laminate at the specified angle. In this case, one full quadrant of the stereographic projection is represented without serious geometric defocusing problems in the X-ray experiment. The prism pole figure distributions shown

were obtained from transverse slices of the tubing, without flattening, and they are restricted to the region + 75 ° from the axial direction to avoid defocusing aberrations. We have found, from other measurements of the prism pole distribution, that the region outside the 75 ° limit contains a negligibly small volume fraction of crystallites in the cladding. Also, all our Zircaloy pole figures show an orthotropic statistical distribution of crystallites (two mutually perpendicular mirror planes defined by the processing axes) even though this is not required a priori by the tube reduction process [11]. Figure 1 shows the ( 0 0 0 2 ) basal and ( 1 0 1 0 ) prism pole figures for cold-worked stress-relieved Zircaloy cladding and for recrystallized Zircaloy cladding. For the Zircaloy cladding we chose f to be the thickness direction, 0 the hoop or circumferential direction and ~ the axial or length direction. The intensity contours are interpreted as equally spaced, with peak values reading nominally ten times random. Both cladding types show a strong peak of the basal plane distribution located at approximately + 30 ° from the radial direction in the radial transverse plane. The prism pole distribution for cold-worked stress-relieved cladding is highly localized in the axial direction with peak intensities exceeding ten times random. As is particularly notable in Fig. 1, a dramatic redistribution of prism poles occurs with recrystallization. The poles are seen to shift from the axial direction to positions centered at approximately 30 ° from the axial direction, and the intensities are diffused considerably compared with the distribution for the cold-worked stress-relieved material. Peak values in the recrystallized case rarely exceed three to four times random in the prism pole distribution.

2. 2. The crystallite orientation distribution function There is a serious drawback in the pole figure representation of preferred orientations. While the pole figure specifies the distributed orientation of crystallographic plane normals, no information a b o u t the rotation of crystallites about the plane normal is available. Qualitatively, two or more pole figures can be used to spot certain strong preferred orientations, b u t this process is often misleading and it is not quantitative in nature. What is re-

183 /x. Z

/% r

I

1

Fig. 1. {0002} basal (left) and {1010~ prism pole distributions for cold-worked stress-relieved tubes (top) and for recrystallized tubes. Times-random intensity values are noted on each pole figure.

quired is a distribution function which gives the probability (or volume fraction) that a crystallite lies within a certain range of orientations with respect to specified macroscopic or processing axes. This function is called the orientation distribution function. Three angles (Euler angles) are required to specify the orientation of a crystallite with respect to macroscopic processing axes. Figure 2 illustrates one choice of definition for these angles, ~, 0 and ¢. Here we have adopted the notation of Roe [12]. The axes X, Y, Z are set coincident with an arbitrary orthogonal set Of crystallographic axes. In our case, [0001] basal and [ 1 0 i 0 ] and [1120] diagonal axes were selected. The coordinate system o x y z is fixed to the processing axes (x is the radial direction, y is the hoop or circumferential direction and z is the axial direction). ~ and qJ are the polar and azimuthal angles of the Z axis in the o x y z system. It should be noted that the polar rotation 0 is a b o u t the y axis following the initial rotation of ~ a b o u t z

~

.

.

A

r

Fig. 2. Orientation relationships between the crystallite coordinate system OXYZ, the specimen coordinate system oxyz and the diffraction vector r i.

(both in the right-handed sense). The third angle ~ is in a right-handed sense about z ( = Z) following the rotations ~ and ~. The polar angles Oi and ×i fix the reciprocal lat-

184

tice direction for the ith diffraction condition (ith crystallographic plane) to the OXYZ and oxyz axes respectively. Also the colatitude angles ~i and 77i are shown in Fig. 2. (It should be noted that o and O coincide.) The orientation distribution function was defined in the space of Euler angles using the series m e t h o d of Bunge [13] and Roe [12]. Input for the series m e t h o d consists of data from a set of pole figures. If we define w ( ~ , 0, ¢) as the orientation distribution function, it can be represented as an infinite series of complete orthogonal polynomials defined over the range of Euler space: l

1

Y E w.,,,,z..,,(cosO)× /=0

m=-l

n=-I

× e x p ( - - i m ~ ) exp(--in¢)

2w 1

0-1

{x, )d{cosx)d

l

qi(x, rl) = ~

{2)

~ Qilmp~(cosx) exp(-im~?)

l=O rn=- I

(3) The functions P~(cos X) are the associated Legendre polynomials. When the orthogonality of these harmonics over the range of cos X and ~ are taken into consideration, the coefficients Qizm can be found by numerically integrating the expression

(1)

In Roe's formalism, Ztm,(COS 0) are the augmented Jacobi polynomials. The numerical coefficients Wire, are complex in general, but it can be proven that they are real for cubic or hexagonal polycrystals exhibiting an orthotropic statistical s y m m e t r y [10, 12]. Morris and Heckler [14] have further shown that, for hexagonal crystal symmetry, l and m are restricted to even values and n to integer multiples of six in the expansion of eqn. (1). Further, we have Wlmn = W z ~ = W~n = Wtmn. The restriction on l is a result of the c e n t r o s y m m e t r y inherent in the diffraction experiment and is not strictly demanded in the true orientation distribution function [12]. However, since the crystal constitutive law (described later) is inherently centrosymmetric, the odd portion of the true orientation distribution function has no physical meaning in the averaging process. Thus the even part of the orientation distribution function (l = 0, 2, 4, . . . ) is sufficient for the purposes of interest in this research, and this is the part which is readily determined from the X-ray pole figure data. With reference to Fig. 2, the distribution of pole intensities in the ith pole figure can be normalized using the integrated intensity to yield the function qi(×, ~?) {called the pole density function) defined in terms of the polar angle X and the colatitude angle ~7:

¢{x,,7) = I'{x,,7)/ f ,f

where Ii(×, ~?)is the measured intensity at X, 77 for the ith diffraction condition. The normalized intensity distribution is readily expanded in an infinite series of spherical harmonics weighted by the appropriate coefficients Qi:m. Thus,

1

2n

1

q'(x, l?)P~(cos X) × 0-1

× exp(im~) d(cos X) dT?

(4)

These series coefficients are then connected to the orientation distribution coefficients Wzmn by application of the Legendre addition theorem:

Qilm =

2~.(

2

~1'2

\2-~-1]

1

~ WlmnP~(cosOi) X n=-I

× exp(in~i)

(5)

In this equation the angles Oi and ¢i are the polar and colatitude angles relating the diffraction vector to the chosen lattice axes X, Y, Z, as shown in Fig. 2. These angles are specified by our choice of X, Y, Z for each pole figure. It is clear from eqn. (5) that, if sufficient sets of pole figure data are available, this system of linear equations can be solved for any order of the coefficients Wzm,. Because of physical and experimental constraints, usually only three to five pole figures are readily determinable, and as a result the Wtm~are typically determinable to the order of I ~< 21 in cubic and hexagonal polycrystals. For most purposes this order is sufficiently high to represent the orientation distribution function reliably. Experimental errors inherent in the measurements of pole figures mask the truncation error corresponding to this cut-off level. For the work that we have performed, three pole figures were determined, (0002),

185

( 1 0 i 0 ) and ( 1 0 i 2 } . With the aforementioned s y m m e t r y restrictions this fixed a truncation level of I at less than 18. The orientation distribution functions for cold-worked stress-relieved cladding and for recrystallized claddings are shown in Figs. 3 and 4 respectively, in the form of Euler plots at constant levels of the third angle ¢. Peak values in units of times-random intensity are given, and the intensity contours are evenly

I/,/~

v ~b=

v

10 °

spaced in times-random units. The randomization process that occurred during annealing, as manifest by the shift of the prism pole distribution shown in Fig. 1, is even more dramatically portrayed in the Euler plots. The peak times-random value of the orientation distribution function is about 720 for coldworked stress-relieved cladding compared with about 8 for the recrystallized material. Evidently the cold-worked stress-relieved

(~

%

~b = 2 0 °

@

,2o ( ~)

=50 o

~ = 30 °

(

~

©

720

Fig. 3. Euler plot representing the orientation distribution f u n c t i o n for c o l d - w o r k e d stress-relieved Zircaloy tubes. The peak c o n t o u r level is shown as 720 times r a n d o m ; o t h e r c o n t o u r s are evenly spaced. (It should be n o t e d that 0 and ~ vary from 0 to ~'/2. Slices at zr/18 are given for ¢.)

~b = 0 °

~=

10 °

¢ = 20 °

¢ =

50 °

8

Fig. 4. Euler plot representing the o r i e n t a t i o n distribution f u n c t i o n for recrystallized Zircaloy tubes. The peak c o n t o u r level is shown as eight times r a n d o m ; o t h e r c o n t o u r s are evenly spaced. (It should be n o t e d that 0 and vary f r o m 0 to ~/2. Slices at ?z/18 are given for ¢.)

186 texture is very nearly single crystal in nature, whereas the recrystallized cladding shows a more normal degree of preferred orientation. The severity of the cold-work stress-relieved texture posed a major problem in this research. Inherent in the series m e t h o d of representation is the occurrence of "negative" regions in the Euler plots. These are typically of the order of 5%-10% of the peak values for textures of ordinary severity. We found negative regions of the order of 30% of the peak for the cold-worked stress-relieved texture in our work, which simply reflects the difficulty of fitting the orthogonal functions to the measured distribution with the prescribed truncation. In Figs. 3 and 4 the negative regions were simply omitted for the distributions shown. 3. CRYSTAL MEAN PLASTIC BOUNDING FORMULATION Hutchinson [7, 8] first adapted the slip models of crystal plasticity to power law steady state creep behavior of polycrystalline aggregates, including isotropically distributed hexagonal polycrystals. His uniform strain rate upper b o u n d estimate is readily adapted to a full textural description of the metal. Three prominent slip modes have been considered: ( 0 0 0 1 ) ( 1 1 2 0 ) basal, (1010)(119.0) prismatic and (119.9.)(1123) pyramidal. The creep rate in the crystallite is given as

~'{ij(s)~ (s)

ei.i : Z

(6)

(s)

where ~¢~) is the slip rate on the sth slip system and p~j<') is the geometric tensor connecting slip coordinates with crystallite coordinates. For power law steady state creep the slip rate can be expressed in the form

~
A =

T(s) n-1 - -

r ~')

(7)

r0¢~lro¢')[ where A is the reference strain rate, r0 ~') is the reference shear stress, which is always a positive quantity in this context, and n is the stress exponent. The shear stress is also related to the crystallite stress oii ~ using the fundamental tensor:

r <') = o ~ j ~ <') (8) For isothermal modelling it is n o t possible to separate the reference strain rate A and the

reference shear stress r0 ('). For this reason it is convenient to refer to a reference creep compliance M0(') which is defined as A M0 - - (r0<'>) n

(9)

If the stress is presumed known in eqn. (8) (i.e. oij c = oij is the macroscopic applied

stress), a lower b o u n d estimate ~ii lb of the creep rate can be calculated from the orientation distribution function (here represented and w(~) where $ denotes orientation) using the averaging procedure: ~ijlb = {~ijc}

= .I w(~)~ii ° d$

(10)

In the lower b o u n d estimate, stress equilibrium is artificially reserved at the expense of violating strain rate compatibility. The upper bound estimate is computed by iteratively solving eqn. (6) for the stress oij ~ at a fixed (imposed) strain rate ~ij ~ = O~j (the macroscopic strain rate). Upper b o u n d stresses oij ub are estimated from the textural average of crystallite stresses: Oi.i ub = (Oij c }

= f w ( ~ ) o i j c cl~

(11)

Thus strain compatibility is preserved but stresses vary across crystallite boundaries; in this manner the first-order distribution of internal stress is approximated. It should be noted that the upper b o u n d method substantially overestimates the magnitude of the internal stress. A more detailed discussion of upper and lower bounding principles has been given elsewhere [15, 16]. Equations {6)-(11) allow calculation of the creep dissipation rate W = oiieij. For biaxial stressing a locus of combined stresses can be plotted (oo versus a~) by fixing W at a constant value. However, this requires that numerical predictions be scaled to equivalent values of W and/~/0 C'). 3.1. Scaling principles for b o u n d i n g e s t i m a t e s

The question frequently arises as to how numerical bounding solutions can be scaled to reflect changes in the rate W of creep dissipa-

187 tion or the reference creep compliance Mo (~). First, let us consider the lower b o u n d formulation when the imposed stress state is altered in scale. Let the scale factor be ~ such that

oil* = ~¢rij

(12)

The rate of creep dissipation in the lower b o u n d formulation can then be written, for the scaled stress state, as (s)

×

sgn(ppq(s)(~))~Opq}

= h Ih l" sgn(X) Ou ~, PU(s)(g)M0 (~) X (s)

× I ppq (')(p,)opq I" sgn(ppq(S)(~)Opq }

--IXl"+lw

(13)

Equations (12) and (13) can be combined to give the scaling law

The two equations, eqns. (16) and (19), can be readily used to adjust the scale for creep locus determinations or, alternatively, to find values for the reference creep compliance consistent with measured experimental values.

4. RESULTS AND DISCUSSION The results of crystal mean plastic modelling are illustrated in Figs. 5-7. In these plots the hoop and axial stresses were normalized to the uniaxial axial stress E. The rate W of creep dissipation was fixed at 3 × 10 -2 MPa h -1 using the scaling laws presented in eqns. (16) and (19). Stress normalization in this representation does not permit the results to be interpreted together as " u p p e r " and

[W*~I/(n+l) sgn(X) oil

oij*= (~-)

(14)

for changes in the rate of creep dissipation. If the reference creep compliance is altered by the numerical factor (~ (>~ 0) such that Mo (s) * = ~M0 (~)

(15)

%

11 ~

o

%

then it follows from eqn. (13) that eqn. (14) can be rewritten as /~O(S)

/

eu* = X [~, I"eu

I

(16)

where both changes are considered. Equation (16) is equally applicable to the upper bound case where the imposed strain rate is altered by the scalar factor X such that

//

Because o u = ou(~) it can readily be shown from eqn. (13) that

/*

// I

I 2

0

[3

&

---'--- I| / /

AO ff

½

(W*]1/(n+l) sgn(X) ou(~)

mo(~) (W*t~/("+~)

/

(18)

In other words, increasing the imposed strain rate by the factor X I)~ In does not alter the orientation-dependent distribution of strain rates; they are simply scaled according to eqn. (18). It follows that the averaged value of oij* (~) is just

(ou* (~)} - Morn, \ - ~ /

/

//

(a)

(17)

ou*($ ) -/~/0(,), \ ~ - /

/

/

/

ij

sgn(X) o u

J~/o (s)

/

//

o

(~)1/(n+1)

oii* - 3;/0(,),

/

0

/

/

/

/

/

/

//

/// // i

(b)

i 1

A,"

I

I 2

ff0/x

Fig. 5. Upper bound predictions ( ) and lower bound predictions (- - - ) of creep dissipative loci for easy (1010} (1120) prismatic slip (Toprism < 10T0 basal, TO pyramidal) : ( a ) c o l d - w o r k e d s t r e s s - r e l i e v e d t e x t u r e

sgn(h) ou($) (19)

data (©); (b) reerystallized texture data (o) and ® data from ref. 17 (&).

188

o

,l

oo O

i (a)

%

(a)

/

0

0

O A

[]

O

/x

AO

c9

%

I I

I I I

0

(b)

I

%

1

A

I

Fig. 6. Upper b o u n d predictions ( ) and lower b o u n d predictions (- - - ) of creep dissipative loci for easy ( 0 0 0 1 } ( 1 1 2 0 ) basal slip (To basal • 10T0 prism, T0PYramidal): (a) cold-worked stress-relieved t e x t u r e data (o); (b) recrystallized t e x t u r e data (D) and ® data from ref. 17 (A).

" l o w e r " bounds because different values of the reference creep compliance Mo (s) were necessarily chosen. Selecting a consistent value for M0 (') is possible through a leastsquares fit of both the upper and the lower bound predictions to the database. In practice, however, we found the computational effort to be prohibitive. In the light of this restriction, the predictions of Figs. 5 - 7 should be treated as individual rather than coupled predictions, without the usual bounding interpretation. The data shown in these figures were described in Part I [1]. Figure 5 illustrates the comparisons of experimental data with upper and lower bound predictions for easy prism slip (T0prism 10T0bas~, roPY~mid~). Although minor differences may be apparent between the cold-

(b)

%

II

A

J

Fig. 7. Upper b o u n d predictions ( -) and lower b o u n d predictions (- - - ) of creep dissipative loci for easy (1122} ( 1123 ) pyramidal slip (TOpyramidal < 10T0Prism , TObasal) : (a) cold-worked stress-relieved texture data (o); (b) recrystallized texture data (D) and ® data from ref. 17 (A).

worked stress-relieved texture and the recrystallized texture, the shapes are very similar. (Upper b o u n d predictions were not available for oz/o0 ~ 1, but the general shape is expected to be similar to the lower bound results.) Both models predict substantial creep resistance for internal pressurization loading (uz/ue = 0.5). For easy basal slip (Tobas~ ~ 10r0 prism, V0PYramida), as illustrated in Fig. 6, substantial creep resistance in the axial direction is predicted compared with that in the hoop direction. The effect is most pronounced for the lower b o u n d model, but the upper b o u n d model closely fits the experimental data for the creep of cold-worked stress-relieved material. For easy pyramidal slip (T0 pyramidal ~ 107"0 prism, TobaSal), we predict a nearly isotropic behavior for both upper and lower b o u n d models as shown in

189

Fig. 7. This is not surprising considering that pyramidal slip alone provides the five degrees of freedom required for general deformation. One of the important results represented in these figures is the negligible effect of crystallite rotations a b o u t the c axis on creep constitutive behavior. Even though the coldworked stress-relieved texture contains predominantly (1010} prism poles lying at ~/6 from those in the recrystallized texture, the creep predictions are nearly indistinguishable. The severity of the cold-worked stressrelieved texture compared with that of the recrystallized material also apparently has little effect on these predictions. Thus we are led to the conclusion that it is primarily the mean position of the (0002} basal poles which dominates creep behavior. This evidence tends to validate the assumptions made in earlier models [2, 3]. It is also abundantly clear that the slip mode is predicted to have a major effect on constitutive behavior when differences in slip mode operation are possible. Adams and Hirth [18], for example, previously suggested that the excellent fit of the easy basal slip predictions of Fig. 6 to the cold-worked stress-relieved data might imply a relaxation of basal slip with respect to prismatic slip at creep temperatures of 673 K. This could be due to prismatic slip hardening during prior cold working at ambient temperatures. We must be careful not to presume that relaxation of r0 bas~' with respect to ro "~sm implies that more basal slip occurs than prismatic slip; some upper b o u n d simulations using the Taylor model, in fact, suggest that this is an incorrect conclusion over a substantial range of strain rate ratios [18]. It is clear that some mechanism other than prismatic slip is required to account for creep behavior in the cold-worked stressrelieved material. The fact that pyramidal slip is much closer to the cold-worked stressrelieved data (as shown in Fig. 7) than is prismatic slip is strongly corroborated by recent transmission electron microscopy observations which show a substantial d c o m p o n e n t to creep in cold-worked stress-relieved materials [19]. This is not surprising considering the fact that neither basal nor prismatic slip nor a combination of the t w o can supply the five degrees of freedom required for constrained deformation. Thus some degree of pyramidal slip (or twinning) is expected in

any compatible deformation model even though reference shear stresses for this mode may be high. Also, we note that, if a different scaling point were chosen for the pyramidal predictions of Fig. 7(b) such a s o~ = 0 o / 0 2 = 1 rather than o = 0, the data could be well correlated with the predicted shape for most of the range 0 ~ ~ ~ 2. We have found, in some limited simulations using combinations of the three slip modes, that it is possible to fit the wide range of behavior seen in the partially recrystallized Zircaloy materials, as well as in the coldworked stress-relieved and recrystallized cases, with judiciously chosen sets of reference stresses. Without further corroborative data from transmission electron microscopy observations, however, such predictions would be of limited usefulness.

5. S U M M A R Y A N D C O N C L U S I O N S

Upper and lower b o u n d approximations to creep constitutive behavior have been correlated with biaxial creep data at 673 K for cold-worked stress-relief-annealed Zircaloy tubes and for recrystallized Zircaloy tubes. These bounding models account for preferred crystallographic orientations in the Zircaloy by averaging single-crystal behavior over the range of preferred orientations using the orientation distribution function. The following conclusions can be drawn from this work. (i) Cold-worked Zircaloy tubes exhibit severe preferred orientations approximately single crystal in nature. Recrystallized tubes have a more normal degree of texture severity. (ii) The major textural differences between the cold-worked Zircaloy tubes and the recrystallized Zircaloy tubes, i.e. texture severity and prism pole rotation, do not noticeably influence creep constitutive behavior. Only the mean orientation of the d axis is predicted to be important in explaining creep anisotropy. (iii) A dominant influence on creep behavior is predicted for changes in slip m o d e ( e . g . prismatic, basal or pyramidal slip). Prismatic slip is predicted to exhibit exaggerated creep strengths in the equal biaxial stress state whereas basal slip is predicted to be weak in this state but stronger in the uniaxial axial state than uniaxial hoop stressing for the tex-

190 t u r e s studied. P y r a m i d a l slip is p r e d i c t e d t o be n o m i n a l l y i s o t r o p i c f o r all t e x t u r e s . (iv) C o l d - w o r k e d Z i r c a l o y t u b i n g is pred i c t e d t o e x h i b i t relatively e a s y basal a n d p y r a m i d a l slip in c o n t r a s t w i t h relatively easy p r i s m a t i c slip f o r t h e r e c r y s t a l l i z e d tubes.

ACKNOWLEDGMENTS T h e a u t h o r s wish t o a c k n o w l e d g e the various c o n t r i b u t i o n s f r o m c o l l a b o r a t i v e w o r k w i t h S. G. M c D o n a l d a n d G. P. S a b o l o f the Westinghouse Research and Development C e n t e r a n d w i t h G. S. Clevinger a n d R. J. B e a u r e g a r d o f the L y n c h b u r g R e s e a r c h Center, B a b c o c k a n d Wilcox Co. B. L. A d a m s a c k n o w l e d g e s s u p p o r t f o r the r e s e a r c h p r o v i d e d b y t h e N a t i o n a l Science Foundation under Grants DMR-8112970 and D M R - 8 4 0 6 6 0 6 . K. L. M u r t y a c k n o w l e d g e s s u p p o r t u n d e r N a t i o n a l Science F o u n d a t i o n Grant DMR-8313157.

4 D. G. Franklin and W. A. Franz, Proc. 3rd C o n f on Zirconium in the Nuclear Industry, in A S T M Spec. Tech. Publ. 633, 1977, p. 315. 5 A. Reuss, Z. Angew. Math. Mech., 9 (1929) 49. 6 W. Voight, Lerbuch der Krystallphysic, Tauber, Leipzig, 1928, p. 962. 7 J. W. Hutchinson, Proc. R. Soc. London, Set. A, 348 (1976) 101. 8 J. W. Hutchinson, Metall. Trans. A, 8 (1977) 1465. 9 H. J. Bunge, in H. J. Bunge and C. Esling (eds.), Quantitative Texture Analysis, Deutsche Gesellschaft fi]r Metallkunde, Oberursel, 1982, p. 85. 10 S. L. Lopata and E. B. Kula, Trans. AIME, 224 (1962) 865. 11 H.J. Bunge, Texture Analysis in Materials Science, Butterworths, London, 1982, pp. 100, 244. 12 R. J. Roe, J. Appl. Phys., 36 (1965) 2024. 13 H. J. Bunge, Z. Metallkd., 56 (1965) 872. 14 P. R. Morris and A. J. Heckler, Trans. AIME, 245 (1969) 1877. 15 F. A. McClintock, in F. A. McClintock and A. S. Argon (eds.), Mechanical Behavior o f Materials, Addison-Wesley, London, 1966, p. 359. 16 B. L. Adams, Proc. A S M Conf. on Texture, Microstructure and Mechanical Properties Relationships o f Materiais, Palm Beach Gardens, ~'L, November 1981, American Society for Metals,

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