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Effects of recrystallization and Nb additions on texture and mechanical anisotropy of Zircaloy
Nuclear Engineering and Design 148 (1994) 1-15
ELSEVIER
Nuclear Engineering and Design
Effects of recrystallization and Nb additions on texture and mechanical anisotropy of Zircaloy * K. Linga Murty, Ravi Jallepalli, S.T. Mahmood North Carolina State University, Raleigh, NC 27695-7909, USA
(Received 1 June 1992;revised version 6 April 1993)
Abstract
The effect of recrystallization on the crystallographic textures and anisotropic mechanical properties of Zircaloy-4 sheets was investigated. In addition, the influence of niobium additions on these properties was studied using three different alloys. The mechanical anisotropy parameters were determined by mechanical testing of gridded tensile samples. The textures were characterized by X-ray pole figure measurements and crystallite orientation distribution functions (CODFs). The CODFs were combined with a crystal plasticity model to predict the anisotropy parameters based on the dominance individually of basal, prism and pyramidal slip systems. Good agreement was noted between the experimental results and model predictions based on prism slip for the recrystallized materials, while the results on cold-worked sheets differed from model predictions for all the three slip systems.
1. Introduction
Zirconium and its alloys are commonly used for many structural applications in water reactors. In particular, cladding, spacer grids and intermediate flow mixers for light water reactors (LWRs), channels for boiling water reactors (BWRs) and Calandria tubes for heavy water reactors (PHWRs) are fabricated using Zircaloy-4, an alloy of zirconium with tin (1.5 wt.%), iron (0.21 wt.%), and chromium (0.10 wt.%). Zirconium alloys have a hexagonal close-packed (h.c.p.) structure at the reactor operating temperature and below, and develop preferred orientations (or
* Extended version of the presentation given at the llth International Conference on Structural Mechanics in Reactor Technology, Division C, Tokyo,Japan, August 18-23, 1991.
textures) during fabrication processes involving various thermomechanical treatments. These textured materials exhibit anisotropic mechanical properties and the anisotropic nature of these characteristics influence their in-service behavior. Binary Z r - N b alloys such as Z r - l N b and Z r 2Nb have been in use in Russian and Canadian ( Z r - 2 N b ) reactors (Ells, 1974) and there exists a vast amount of literature on the physical metallurgy, deformation behaviors both in and out of reactor as well as the texture development of these alloys following various reduction schedules (Tenckhoff, 1988; Cheadle, 1967; Cheadle, 1972). Recent developments lead to niobium-added Zircaloys which are now being considered as replacements for Zircaloys in LWRs, and some nuclear reactor vendors (such as Westinghouse) recently announced the introduction of new generation alloys such as Zirlo T M alloys, which are
0029-5493/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0029-5493(94)00617-8
2
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1 15
Nb-modified Zircaloys (Sabol, 1989). These materials exhibit resistance to in-reactor corrosion and oxidation, particularly at long exposures. However, currently a very limited amount of information is available on the effect of Nb on crystallographic textures and accompanying mechanical anisotropy (Mahmood, 1991; Murty, 1991) of these Nb-modified Zircaloys, while the literature data refer to binary Z r - N b alloys (Ells, 1974; Cheadle, 1967; Cheadle, 1972). This information is very important for selecting optimum fabrication parameters (textures) as well as for accurate prediction of the dimensional stability of the final product (cladding, pressure tubes, spacer grids, channels, etc.). We report here the recent work on the crystallographic textures and mechanical anisotropy of Zircaloy-4 sheet in cold-worked stress-relieved (CWSR) and recrystallized (R x) conditions. Similar tests were also performed on three different Nb-added zirconium alloys to evaluate the effect of Nb on the crystallographic texture, mechanical anisotropy and formability of these materials. We do not consider the development of textures of these alloys but are concerned here with the characterization of the texture and the resulting mechanical anisotropy of the given alloys. Except for the alloy compositions of the test materials, no detailed information on the thermomechanical treatments (reduction schedules) is available. The development of the textures of Zircaloys (Tenckhoff, 1988; Britt, 1991) and binary Z r - N b alloys (Cheadle, 1967; Cheadle, 1972) have been considered in detail by various investigators and the reader is referred to the literature which includes both experiments and models (Tome, 1991) of texture development. The goal here is to experimentally determine the textures of given alloys, develop quantitative descriptions using crystallite
orientation distribution functions (CODFs), predict the mechanical anisotropy using CODFs in conjunction with plasticity (slip) models and finally correlate them with mechanical test data. The textures of the sheets were characterized by inverse pole figures obtained from the three orthogonal faces of the sheet samples and direct pole figures corresponding to the basal, prismatic and pyramidal planes from which the CODFs were derived. Upper-bound plasticity models were used in conjunction with CODFs to predict the mechanical anisotropy parameters (R and P) as well as the formability parameter B. The mechanical properties and the anisotropy parameters were determined from mechanical testing of gridded tensile samples. The experimental results correlated extremely well with the anisotropy parameters predicted by the prism slip dominance for all cases where the materials were recrystallized, whereas significant deviations were noted for the CWSR materials and plausible causes are described.
2. Experimental aspects Alloys with different compositions have been procured in the form of thin sheets of thickness varying from 0.75 to 1.14 mm. Table 1 gives the chemical compositions of these alloys. Zircaloy-4 sheets were received in the CWSR as well as the R x conditions. Alloy 1 is Zircaloy-2 with 1% Nb, alloy 2 is a ternary alloy with 0.3% Nb and 1.5% Sn and alloy 3 is Zircaloy-4 (with reduced Sn) with 1% Nb. Alloys 1 and 3 are in cold-worked condition, and alloy 2 is recrystallized. As pointed out earlier, no details of the thermomechanical treatments and reduction schedules of these materials are available.
Table 1 Chemical compositions (wt.%) of Zr-Nb
alloys
Nb
Sn
Fe
Cr
Mo
Ni
Zr
Zircaloy-4
-
1.5
0.21
0.1
-
-
Balance
Alloy 1
1.0
1.5
0.15
0.1
-
0.05
Balance
Alloy 2
0.3
1.5
-
-
0.3
-
Balance
Alloy 3
1.0
1.0
0.2
-
-
-
Balance
KL. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
2.1. Crystallographic texture The textures characterized by X-ray methods using both the inverse and direct pole figures using Schultz back-reflection technique; the details of the analyses and methodology may be found in Cheadle [1967, 1972]. The net integrated X-ray intensity under each allowed reflection in 1-20 scans was used to calculate the texture coefficients (TC) using the relation I(hkil)//Io(hkil)
TC(hki')
= [1/N
] E [ I
(1)
where I(hk, ) and Iochk,) are the measured and the random intensities respectively for the (hkil) reflection, N is the total number of allowed reflections, and TC(hkm gives the number of (hkil) planes, in the units of "times random" oriented along the specimen normal. The inverse pole figures were also constructed from these data obtained along the three orthogonal directions, roiling [RD], transverse [TD] and thickness or normal [ND] directions. The texture coefficients were then used to evaluate f-factors (Kearns, 1965) for each of the three orthogonal directions for each of the alloy sheets:
f=Jor~r/2 16 sin 4) COS2(~ d ~
obtained from X-ray diffraction were normalized to the units of "times-random" using the respective TCs, and the pole figures were plotted in the form of iso-intensity contours.
2.2. Mechanical testing A grid analysis technique was employed for experimental evaluation of the mechanical anisotropy parameters and mechanical properties along the rolling and transverse directions of the sheets. Tensile specimens with a gauge length of 31.75 mm and 6.35 mm width were machined from roiling and transverse directions of each of the sheets. Square grids (1.27 mm each side) were electrochemically etched onto one surface of the specimens to evaluate the axial and contractile width strain distributions. The changes in the grid dimensions along the tensile and transverse directions yielded information on the tensile and contractile strains, which made it possible to evaluate the transverse contractile strain ratios (CSRs) as a function of the tensile strain up to fracture. These CSRs are also the mechanical anisotropy parameters in the modified von-Mises-Hill yield criterion defining the generalized stress ~rg:
(2)
where 16 sin & is the volume fraction of the grains with their c axes oriented at a tilt angle & from the reference direction (sample thickness). An f-factor gives the effective fraction of basal poles oriented along a particular direction of the sheet and thus the sum of the f-factors determined along the three orthogonal directions is unity. A better representation of the crystallographic texture is obtained by a stereographic projection showing the variation in pole density with respect to the macroscopic coordinates (ND, RD, TD) for a particular set of crystallographic planes resulting in direct pole figures. A Norelco goniometer stage was employed, and basal (0002), prismatic (1010) and pyramidal (1012) pole figures were considered. The pole figure intensity data
3
R(OrND -- O'TD)2+ RP(O'TD -- O-RD)2-FP(O'RD -- trND) 2 P ( R + 1)
(3) w h e r e OND ~ OVTD a n d O'RD are the stresses along
the normal, transverse and rolling directions of the sheet, and R and P are the coefficients of anisotropy. R and P are given by R = ~ AeND
IffRD,
OrTD =
OrND = 0
(4a)
and A6RD / P =
, O'RD = O'N D = 0
(4b)
Z~END )eTD
Experimental and analytical procedures were outlined by Mahmood and Murty (1989).
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15
4
3. Results and discussion
preferentially oriented along the rolling and transverse directions. A smaller value for the prismatic texture coefficient along the RD of the recrystallized sheet when compared with the CWSR sample is indicative of the 30 ° rotation of the prismatic planes away from the RD upon recrystallization. Essentially similar results are noted for Nb-modified alloys, except for the fact that the TC values along TD are relatively low while alloy 1 exhibited a finite, albeit low, value along ND. These differences are more revealing in the direct pole figures, as noted later. A comparison of the f-factors shows that in each case the effective fraction of basal poles along the ND of the sheets is higher than those along the RD and the TD. The only exception is that alloys 1 and 2 exhibited relatively large values for the f-factor along TD. The sum of the f-factors along the three principal directions of each sheet is close to unity, as expected. Since the reduction schedules of these materials are not known, the reasons for these differences cannot be attributed to the Nb additions; further work is called for
3.1. Crystallographic textures As noted above, the first step in the characterization of crystallographic textures of the sheets was to determine the TCs for the different sets of planes, and f-factors for each of the three orthogonal directions of the sheets. The measured values of these parameters are given in Table 2. Both CWSR and R x Zircaloy-4 sheets exhibit high values of basal texture coefficients along the normal direction followed by TD with zero at RD indicating that the basal poles are confined to the N D - T D plane and concentrated close to ND. While the Nb-modified alloy 2 followed the same trend, different behaviors were noted for alloys 1 and 3. The TCs along ND and TD are essentially the same for alloy 1, while that for alloy 3 along TD is also very small, as for RD. The texture coefficients of prism poles are zero along N D for Zircaloys with non-zero values along the other two directions, indicating that the prism poles are Table 2 Texture coefficients and f-factors Texture coefficients
Zircaloy-4 CWSR
ND RD
TD Zircaloy-4 R x
ND RD
TD Alloy 1 (CWSR)
ND RD
TD Alloy 2 (R~)
ND RD
TD Alloy 3 (CWSR)
ND RD
TD
Basal
Prismatic
Pyramidal
f-factor
4.25 0.00 1.04
0.00 6.58 3.12
0.85 0.00 0.54
0.741 0.100 0.172 E f = 1.013
3.99 0.00 1.19
0.00 3.97 2.71
1.49 0.00 0.85
0.724 0.075 0.182 E f = 0.981
1.53 0.00 1.59
0.26 4.62 0.48
1.98 0.00 2.33
0.584 0.057 0.330 Y~f = 0.971
3.59 0.00 1.65
0.00 3.10 0.67
2.15 0.00 2.41
0.634 0.039 0.353 Y~f = 1.026
2.58 0.00 0.19
0.00 3.85 1.31
0.63 0.00 0.35
0.750 0.074 0.147 E f = 0.971
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
poles concentrated in the transverse plane normal to the roiling direction of the sheet. The intensity maxima in the basal pole figure of the CWSR Zircaloy-4 sheet have a value of 7.4 times random at +27 ° from the ND in the ND-TD plane, while recrystallization reduced the intensity maxima to about 4.3 and also tilted the basal poles closer to the ND, with the basal pole peak angle being _+20°. A more distinct difference in the textures of CWSR and R x Zircaloys is observed in the distribution of prismatic poles given in (1010) pole figures in Figs. l(a) and l(b). The majority of the prismatic poles in the CWSR are oriented along the RD while recrystallization rotates these poles around the c axes of the crystal-
using known thermomechanical treatments of these alloys. The direct pole figure technique is used in determining the basal, prismatic and pyramidal pole figures and either one or a combination of samples were used to obtain complete pole figures (Mahmood, 1989). The basal, prismatic, and pyramidal pole figures for all of the alloys are compiled in Fig. 1, with the center of each pole figure corresponding to the normal direction of the sheets. These pole distributions exhibit typical features of textures in deformed and recrystallized zirconium alloys (Murty, 1991; Mahmood, 1989; Murty, 1989). The basal pole figures, in each case, show bimodal distribution of basal
RD
RD
BO
!
"'"
.
TD
r_
BASAL (a)
RD
,.
TD
(b)
RD
q,
l
BASAL (ooo2)
PYRAMIDAL (1012)
0o'/o)
RD
TD
t TD
PRISMATIC
(0002)
5
PRISMATIC (lO10)
PYRAMIDAL (1012)
Fig. 1. Basal, prismatic and pyramidal pole figures of CWSR Zircaloy-4 (a); R x Zircaloy-4 (b); alloy-1 (c); alloy-2 (d); alloy-3 (e).
6
K.L. Murty et al./Nuclear Engineering and Destgn 148 (1994) 1-15
(as expected in low c/a ratio hcp metals such as Zr and Ti (Tenckhoff, 1988)) along with another pair orthogonal to it at _+ 15° from the ND in the N D - R D plane. This is distinctly evident in Fig. 2, where the intensities of the basal poles are plotted L~,s.the tilt angles from ND towards TD and RD respectively, which exhibit bimodal distributions along both directions. In contrast, Zircaloys are characterized by such a bimodal distribution only in the R D - T D plane with rapidly decreasing basal pole intensity towards RD from ND. Other texture modifications following Nb additions to zirconium and Zircaloys were also reported (Murty, 1991). In all cases, pyramidal pole distri-
lites, shifting the intensity maxima from the RD to 30 ° from the RD towards the TD. A second rather small peak in each case appears at about 60 ° from the main peak owing to the hexagonal symmetry. In general, the Nb additions seemed to preserve the recrystalliz.ation effect of the rotation of the prism planes. Alloys 1 and 3 have cold-worked textures while alloy 2 exhibits prism pole maxima at +_30° from RD towards TD as for the recrystallized Zircaloy. However, the basal pole distribution of alloy 3 is distinctly different in that the intensity maxima (approx. 4.5 times random) are at about _+27 ° from the ND in the N D - T D plane
RD
RD
RD
TD
•
"
Basal
Prism
Pyramidal
(00o2)
(lO~O)
0o~2)
RD
RD
RD
Basal
(0002)
Prism
Pyramidal
(1010)
(1012)
Fig. 1 (continued).
TD
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
7
RD
RD
RD
Basal (0002)
Prism (1010)
Pyramidal (1012)
Fig. ] (continued).
butions indicate relatively low peak intensities and are more spread out. We also note that alloy 2 exhibits relatively high peak intensities in both the basal pole (approx. 14 times random) and the pyramidal (approx. 5 times random) compared with others which exhibit about 4.5 and 2 or lower, respectively.
3.2. Crystallite orientation distribution functions The pole figure intensity data were used for a more quantitative texture analysis through the generation of CODFs for all the sheets. A C O D F is a distribution function which gives the probability that a crystallite lies within a certain range of orientations (0, qJ, ~b) with respect to the specified coordinates, which are generally the processing axes of the material (ND, RD, and T D for a sheet). C O D F is expressed as a series of generalized spherical harmonics (Murty, 1989; Roe, 1965; Bunge, 1982) in Roe's notation:
,o(0, 6, 6) I
I
E
E
I=0 m--1
n=-I
=E
Wlm Zlm,(cos O)e -im+ e-in6
(5) where WI~° are the appropriate series coefficients, Zim" are the augmented Jacobi polynomials and (0, ~O, ~b) are Euler angles (Roe, 1965)
connecting coordinate axes embedded in the crystallite with axes coincident with the specimen and the principal working directions of the material. In the present study, three pole figures (basal, prismatic and pyramidal) were used to evaluate the series coefficients W~m" up to I - - 1 6 . The error caused by truncating I at 16 is generally insignificant compared with the errors inherent in experimental texture measurements. The accuracy of the generated CODFs was checked by reconstructing the basal pole figure from the derived C O D F (Mahmood, 1989). CODFs are generally represented graphically in the form of Euler plots at fixed intervals of the angle & of rotation around the c axis up to & = 55°; beyond this, repetition occurs as a result of hexagonal symmetry. Fig. 3 is a compilation of the Euler plots where iso-intensity contours are plotted in "times-random" units at constant intervals of 1 with the base contour also at 1. The three orthogonal sheet directions ND, RD and T D correspond to (0, q~) angles of (90 °, 0°), (0 °, 0 °) and (0 °, 90 °) respectively. The constant 4, sections of the Euler plots can also be considered in terms of the ideal orientations (hkil)[uutw], where (hkil) refers to the crystallographic plane lying in the surface of the sheet and [ut, tw] is a crystallographic direction in this plane and parallel to RD (Mahmood, 1989).
8
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15 .
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
demonstrated previously (Murty, 1991; Mahmood, 1989) the simple slip models considered here yielded anisotropy parameters for Zircaloys, in good agreement with the experimental results both during low temperature deformation and high temperature creep.
,
ND-RD L~
,o E m
3.3. CODF-plasticity model
o
E
-90
-60
-50 0 50 TillAngle Phi (degrees)
60
90
ND-TD to
L
Once the orientation distribution of crystallites in the polycrystalline materials is evaluated and the appropriate C O D F is known, the crystallite behavior in each orientation can be weighted by the volume fraction of crystallites in that orientation so that the behavior of the polycrystalline aggregate can be predicted. If m is the strain-rate sensitivity, the shear strain-rate on the sth slip system can be expressed as ~" = A r i ; [ , ) ' ] "
ID
I
-90
I
I
-60
I
I
I
-30
I
I
i
0
i
i
i
i
30
i
i
i
i
60
90
TIllAngle Phi (degrees)
Fig. 2. Basal pole intensity t,s. tilt angle for alloy 3 from ND towards TD and RD; note the bimodal distributions in both the N D - T D and N D - R D planes.
Physical and mechanical properties of single crystals are anisotropic and this extends to the bulk properties off the polycrystals whenever preferred orientations are present. To the first order, the bulk property is related to the microscopic property, p(O, 0, 05), by statistical averaging using the CODF, so that the average property of the polycrystalline aggregate (p(O, 05, ~b)) is given by
(p(O, qJ, da))= fo2~fo2~f]]p(O, dJ, O) > w ( 0 , ~, 05) d(cos 0) d~0 d~b
(6) In this model, the grain-to-grain interactions are not taken into account and an average through the texture is taken to represent the plastic behavior of the polycrystalline material. As has been
(7a)
where ~'i; is the reference shear stress on the sth slip system, m the strain-rate sensitivity and A the reference strain-rate. The reference shear stress ( r 0) is similar to the CRSS and defines the relative ease with which slip can occur on a given slip system. The total strain-rate in the crystal is expressed as the sum of contributions from each of the active slip systems: -c = x--, J,t. kz = Mijk/O'k/ Eij , 7 ]'£ij~ij kl
(7b)
]J~ij is the geometric tensor connecting slip coordinates with crystal coordinates. The shear stress is related to the crystallite stress by
where
where the superscript s specifies the slip system under consideration. In the upper-bound method, the strain (rate) state in each crystallite is regarded as being identical to the applied macroscopic strain tensor, and the stress state is calculated using Eqs. (7a)(7c). For m < 1 (i.e. the stress exponent > 1) an iterative numerical procedure was used in which successive approximations to the actual stress were obtained by means of a form of Newton's method (Adams, 1985). Once the stress state for a crystallite is determined, the bulk stress state is
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
>
~j._,
9
( )
0 8~ )
0
0
)
o
0 II 0
,E Z
C3~
©
0 0 0
~
0 0',
©
u© h2
r~ ~>~_j
C)
/,.%~ u ~
-n
c~
c
~D c ~
~ °
ii
A
i"~°~/
~ 0
~ ,
~®i °
~o
-x
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
then obtained by averaging the stress state of the individual grains weighted by CODF: ub
t'2rr , ' 2 7 r r 1
[%] =Jo Jo J-1~°(°'~°''t')°'j × d(cos 0) dO d4,
(8)
In this upper-bound method the strain compatibility is preserved while the stresses vary across crystallite boundaries. Thus the first-order distribution of internal stress is approximated. In the lower-bound method, the stress in each grain is taken to be the same as the applied macroscopic stress and the strains (or rates) vary from grain to grain depending on their orientations; these schemes are seen to be valid for high temperature creep (Murty, 1989). Self-consistent methods take into account both strain and stress compatibilites, and more detailed discussions of these various schemes may be found in the literature (Hutchinson, 1977).
11
Three different slip modes are considered in the present analyses: basal ({0001}(1120)), prism ({10]0}(1120)) and pyramidal ({1122}(1123)). Values of the reference stresses ( r 0) are chosen to emphasize the contribution of individual or particular combinations of slip modes to the polycrystalline behavior. It should be pointed out that in this model all of these slip systems are considered to be operative, thus preserving the condition of five independent slip systems for polycrystalline deformation, although the contribution of a given slip system or a combination of them is forced to become dominant by appropriate choice of r0 values. In addition to the power law (Eq. (7a)), a Bishop-Hill analysis corresponding to rigid plastic behavior was also made for comparison. The derived CODFs for the various alloys were used to calculate the bulk stresses for the three slip systems and these stresses were normalized to a constant rate of energy dissipation (I~ = rijS, ij) (Murty, 1991; Adams, 1985). The mechani-
Table 3 Anisotropy and formability parameters Material
Slip system
R
P
B
O.TD/O.RD a
Zircaloy-4 CWSR
Prism Basal Pyramidal Experimental
2.89 1.94 0.34 4.03 _+ 0.36
4.24 0.39 0.31 9.53 _+ 0.31
1.52 0.78 0.80 1.89
1.044 1.534 0.966 1.057
Zircaloy-4 Rx
Prism Basal Pyramidal Experimental
3.23 0.61 0.31 3.18 + 0.28
3.40 0.27 0.24 3.76 _+ 3.76
1.47 0.74 0.76 1.50
1.006 0.749 0.904 1.182
Alloy 1
Prism Basal Pyramidal Experimental
2.32 0.74 0.48 1.50 _+ 0.08
6.36 0.14 0.47 3.35 + 0.21
1.58 0.64 0.86 1.33
1.112 0.537 0.993 1.329
Alloy 2
Prism Basal Pyramidal Experimental
2.11 1.58 0.55 2.33 + 0.12
4.64 0.31 1.06 4.27 + 0.18
1.48 0.74 1.03 1.48
Prism Basal Pyramidal Experimental
4.84 2.15 0.24 4.50 + 0.23
4.56 0.38 0.23 4.22 + 0.21
1.68 0.78 0.78 1.63
Alloy 3
P / / P ~ + 1) a Model calculations based on V R ( ~ S (Eq. (9)).
1.101
0.622 1.204 1.196 0.995 0.635 0.983 1.017
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15
12
Strain (%) 0
12.1
3.1 4.4 5.7 6.9 7.9 9.2 10.1 11.0 11.9 Fig. 4. A typical sequence of grid photographs taken during a tensile test (alloy-3).
cal anisotropy parameters evaluated from these predicted bulk stresses are tabulated in Table 3. The results of the rigid-body analysis (BishopHill) were close to these values * and thus were not included here. 3.4. Anisotropic mechanical properties Fig. 4 depicts a typical sequence of grid photographs taken during a tensile test and the grid size measurements yield the true transverse and contractile strains. This technique can be used to study the spatial and temporal strain distributions. Fig. 5 depicts the engineering stress-strain curves for the R D and T D specimens derived from the total gage section of the specimens. Typical stress cs. strain curves derived from the data on the fractured grid elements are shown in Fig. 6 where both nominal and true stress-strain curves are included for the rolling and transverse directions. Similar results were obtained on all of the alloys. Tables 4 and 5 summarize the mechanical properties data for CWSR and R x Zircaloy sheets and Nb-modified zirconium alloys respectively. The yield strengths along the transverse directions o f the sheet for all the alloys are seen
to be higher than those along the rolling directions. In general, ductilities along the T D are lower than those along the RD, in particular for the R x materials (Zircaloy and alloy 2). These differences are more revealing in the work hardening exponents (n) and all alloys exhibit higher rates of strain hardening along R D than TD. Thus we note that both CWSR and R x materials exhibit anisotropy not only in the mechanical strengths and ductilities, but also in their strain hardening behaviors. Although recrystallization made the materials softer, it did not eliminate the texture and anisotropy, but rather it modified them. It is interesting to note that the tensile
1200
1000
R3 800
600
400 c 2OO
Alloy 3 0 000
* Predictions based on the Bishop-Hill analysis differed slightly more from the experimental results.
t 0 05
0.10
015
Engineering strain
Fig. 5. Engineering stress t,s. strain cubes along RD and TD.
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
13
Table 4 Mechanical properties of CWSR and R x Zircaloy-4 Proper~
CWSR
0.2% yield strength (MPa) ( ± 15%) Tensile strength (MPa) ( ± 5%) Total elongation ( ± 0.01) Strain hardening exponent (n) a
Rx
RD
TD
RD
TD
436 692 0.15 0.05 ± 0.03
461 682 0.12 0.04 ± 0.02
380 480 0.35 0.22 ± 0.04
449 511 0.29 0.12 ± 0.03
ao-=gen"
s t r e n g t h s of alloy 3 a r e essentially i d e n t i c a l indicating p l a n a r isotropy, a n d the r a t e s o f strain h a r d e n i n g a r e also similar a l o n g R D a n d T D . T h e ratios o f t h e yield s t r e n g t h s a l o n g R D a n d T D a r e r e l a t e d to t h e m e c h a n i c a l a n i s o t r o p y p a r a m e t e r s . R e a r r a n g e m e n t o f t h e m o d i f i e d Hill's e q u a t i o n (Eqn. (3)) gives
~rnD
VR(P+I
)
m e a s u r i n g t h e axial tensile a n d t r a n s v e r s e contractile strains in t h e grid e l e m e n t s which ultim a t e l y f r a c t u r e d in t h e R D a n d T D s p e c i m e n s . Fig. 7 shows the w i d t h c o n t r a c t i o n as a f u n c t i o n o f l o n g i t u d i n a l e l o n g a t i o n for b o t h t h e R D a n d T D s p e c i m e n s for alloy 3; similar d a t a w e r e also o b t a i n e d on t h e o t h e r alloys. A l i n e a r v a r i a t i o n is n o t e d b e t w e e n the c o n t r a c t i l e a n d tensile strains a n d the s l o p e s o f t h e s e lines w e r e u s e d to d e t e r m i n e the R a n d P p a r a m e t e r s (Eqns. (4a) a n d (4b)). T h e s e results a r e t a b u l a t e d in T a b l e 3 along with t h e m o d e l p r e d i c t i o n s (Eqn. (8)). T h e s e grid e x p e r i m e n t s also yield i n f o r m a t i o n on t h e l o c a l i z e d n e c k i n g b e h a v i o r s as d e t a i l e d by M a h m o o d a n d M u r t y (1989). T a b l e 3 also i n c l u d e s t h e f o r m a b i l i t y p a r a m e ter B d e f i n e d as ( B a c k o f e n , 1972)
(9)
T h e ratios o f t h e e x p e r i m e n t a l yield stresses a r e c o m p a r e d in T a b l e 3 with t h e m o d e l p r e d i c t i o n s b a s e d on t h e r i g h t - h a n d side f a c t o r o f the a b o v e e q u a t i o n for t h e v a r i o u s slip systems for all of t h e alloys. A l t h o u g h we n o t e a g o o d a g r e e m e n t with p r i s m slip p r e d i c t i o n s in all cases, t h e c o r r e l a t i o n is m i s l e a d i n g since m a n y c o m b i n a t i o n s o f R a n d P v a l u e s give rise to similar yield stress ratios, and a direct correlation between the measured R a n d P v a l u e s with the m o d e l p r e d i c t i o n s is m o r e appropriate. T h e grid analysis t e c h n i q u e e n a b l e s an evaluation of t h e m e c h a n i c a l a n i s o t r o p y p a r a m e t e r s R a n d P. T h e s e p a r a m e t e r s w e r e d e t e r m i n e d by
OvI
B -
(lOa) 2O'iv
w h e r e o-l a n d ¢r~v a r e t h e p l a n e strain stress in t h e first q u a d r a n t a n d t h e equibiaxial stress in the f o u r t h q u a d r a n t of the yield locus respectively. T h e significance of the p a r a m e t e r B lies in the fact t h a t t h e h i g h e r the B value, t h e b e t t e r t h e
Table 5 Mechanical properties of Nb-modified alloys Property
Alloy 1
0.2% yield strength (MPa) Tensile strength (MPa) Total elongation Strain hardening exponent (n)
489
_+ 17
650
+ 23
565
+ 20
676
+ 24
892
_+31
907
± 32
736
+ 26
672
± 23
712
± 25
660
+ 23
1116
+ 39
1088
+ 38
RD
Alloy 2 TD
0.24 + 0.02 0.21 ± 0.02
RD
0.29 ± 0.03 0.08 + 0.01
Alloy 3 TD
0.30 + 0.02 0.17 + 0.02
RD
0.22 + 0.03 0.05 ± 0.02
TD
0.12_+ 0.01 0.06 + 0.01
0.11 ± 0.01 0.05 + 0.01
14
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
cup formability of the material. The formability parameter is calculated using the anisotropy parameters R and P (Murty, 1989): B= ( R + l)(R+4R+P) ( 1Ob)
4R(R+P+ 1)
d
Table 3 lists the predictions based on the upperbound model for the three slip systems with the strain-rate sensitivity parameter m of 0.01 applicable for zirconium alloys at ambient. For recrystallized alloys and also for alloy 3 we note a good correlation between the experimental results and the model predictions based on the dominance of prism slip. Such a correlation is expected since low c/a ratio materials such as Zr and Ti are
0.6
Slope=0.818 r
._ z ;;
Rz4.50
A l l o y 3 RD
0.6
0.4 . 6 .= F 02.
0.2
0.0
0.4
0.6
0.6
Axial strain
0.8
Slope=0.809
800 -
._c m i r E ?
Engmeering
600 -
Pz4.22
Alloy 3 TD
0.6
0.4
0.2
400. a 200
0.0 00
f
n “0.0
1600
0.5 Slralll
,
1.0
I
1400 1200
2
1000
% 600 : ' 6
600 400
0' U.0
I 0.5
Strain
I 1.0
Fig. 6. Nominal and true stress-strain curves for RD and TD specimens obtained from fractured grid elements.
i3
I
0.2
0.4
0.6
0.6
Axial strain Fig. 7. Transverse contractile strain (cw) cs. longitudinal tensile strain (E,) for RD (top) and TD (bottom) specimens; the slopes of the lines are related to R and P.
known to deform by prism slip, and the critical resolved shear stress for prismatic slip is much lower than that for basal and pyramidal slip (Mahmood, 1989; Murty, 1989). Distinct deviations are noted for CWSR Zircaloy-4 and alloy 1, which is also cold-worked. Particularly noteworthy is the differences in the P parameter, and these deviations are believed to arise from possible contributions to the deformation from twinning in addition to slip. Moreover, it has been shown that basal slip might make a non-negligible contribution to slip in cold-worked Zircaloys even at low temperatures (Tome, 1991). Further work is needed to unravel the reasons for these findings. The anisotropy parameters R and P for
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
alloy 3 are close to each other, just as in recrystallized Zircaloy-4, revealing that these materials exhibit planar isotropy as noted from the yield stresses (Table 5). However, the values of R and P are far removed from unity, indicating the existence of mechanical anisotropy when considered in three dimensions.
4. Conclusions Mechanical properties, anisotropy and formability parameters were studied on Zircaloy and Nb-added zirconium alloy sheets using gridded tensile specimens along the roiling and transverse directions. Crystallographic textures of these materials were characterized using X-ray diffraction. The texture coefficients and f-factors were evaluated from 1-20 scans along the three orthogonal directions. Basal, prismatic and pyramidal pole figures were determined, from which CODFs were evaluated. Upper-bound crystal slip models were used in conjunction with CODFs to calculate the mechanical anisotropy parameters of the textured bulk samples. The model predictions b a s e d o n p r i s m slip a g r e e d w i t h t h e e x p e r i m e n t a l results on the recrystallized materials while cold w o r k i n g r e s u l t e d in l a r g e d e v i a t i o n s .
Acknowledgments Financial support through grants from the National Science Foundation and the Electric Power Research Center at North Carolina State University are gratefully acknowledged. We wish to express our sincere appreciation to Mr. Jeff C. Britt for assistance and for various discussions. Acknowledgments a r e d u e t o T h e U n i v e r s i t y o f Western Australia for the award of a Gledden Senior Visiting Fellowship to KLM, during which time the manuscript was prepared.
15
References B.L. Adams and K.L. Murty, Mater. Sci. Eng., 70 (1985) 181. W.A. Backofen, Deformation Processing, Addison Wesley, Reading, MA, 1972, p. 47. J.C. Britt and K.L. Murty, Texture development in zirconium alloys, in Proc. Symp. on Zirconium Alloys for Reactor Components (ZARC-91), Bhabha Atomic Research Center, Bombay, December 1991, pp. 1-45. H.J. Bunge, Texture Analysis in Materials Science, Butterworths, London, 1982. B.A. Cheadle, C.E. Ells and W. Evans, J. Nucl. Mater., 23 (1967) 199. B.A. Cheadle, S.A. Aldridge and C.E. Ells, Can. Met. Quart., 11 (1972) 121. C.E. Ells, Deformation of irradiated zirconium-niobium alloys, in Zirconium in Nuclear Applications, ASTM STP 551, ASTM, Philadelphia, PA, 1974, pp. 311-327, and refs. 1-7 in the paper. J.W. Hutchinson, Metall. Trans., A8 (1977) 1465. J.J. Kearns, Thermal Expansion and Preferred Orientation in Zircaloy, WAPD-TM-472, Westinghouse Electric Corporation, Pittsburgh, PA, 1965. S.T. Mahmood and K.L. Murty, Localized plastic flow, anisotropic mechanical properties and crystallographic texture in Zircaloy sheet, J. Mater. Eng., 11 (1989) 315. S.T. Mahmood, J. Ravi and K.L. Murty, Effects of niobium additions on textures and mechanical properties of zirconium alloys, in Proc. l l t h Int. Conf. on SMiRT, Tokyo, August 18-23, 1991, paper C03/2. K.L. Murty, Applications of crystallographic textures of zirconium alloys in the nuclear industry, in Zirconium in the Nuclear Industry: Eighth International Symposium, ASTM STP 1023, American Society for Testing and Materials, Philadelphia, PA, 1989, pp. 570-595. K.L. Murty, Mater. Forum, 15 (1991) 217. R.J. Roe, J. Appl. Phys., 36 (1965) 2024. G.P. Sabol, G.R. Kilp, M.G. Balfour and E. Roberts, Development of a cladding alloy for high burnup, in Zirconium in the Nuclear Industry: Eighth International Symposium, ASTM STP 1023, ASTM, Philadelphia, PA, 1989, pp. 227-244. E. Tenckhoff, Deformation Mechanisms, Texture and Anisotropy in Zirconium and Zircaloy, ASTM STP 966, ASTM, Philadelphia, PA, 1988. C. Tome, R.A. Lebensohn and U.F. Kocks, Acta Metall. Mater., 39 (1991) 2667, and refs. 1-8 therein.
ELSEVIER
Nuclear Engineering and Design
Effects of recrystallization and Nb additions on texture and mechanical anisotropy of Zircaloy * K. Linga Murty, Ravi Jallepalli, S.T. Mahmood North Carolina State University, Raleigh, NC 27695-7909, USA
(Received 1 June 1992;revised version 6 April 1993)
Abstract
The effect of recrystallization on the crystallographic textures and anisotropic mechanical properties of Zircaloy-4 sheets was investigated. In addition, the influence of niobium additions on these properties was studied using three different alloys. The mechanical anisotropy parameters were determined by mechanical testing of gridded tensile samples. The textures were characterized by X-ray pole figure measurements and crystallite orientation distribution functions (CODFs). The CODFs were combined with a crystal plasticity model to predict the anisotropy parameters based on the dominance individually of basal, prism and pyramidal slip systems. Good agreement was noted between the experimental results and model predictions based on prism slip for the recrystallized materials, while the results on cold-worked sheets differed from model predictions for all the three slip systems.
1. Introduction
Zirconium and its alloys are commonly used for many structural applications in water reactors. In particular, cladding, spacer grids and intermediate flow mixers for light water reactors (LWRs), channels for boiling water reactors (BWRs) and Calandria tubes for heavy water reactors (PHWRs) are fabricated using Zircaloy-4, an alloy of zirconium with tin (1.5 wt.%), iron (0.21 wt.%), and chromium (0.10 wt.%). Zirconium alloys have a hexagonal close-packed (h.c.p.) structure at the reactor operating temperature and below, and develop preferred orientations (or
* Extended version of the presentation given at the llth International Conference on Structural Mechanics in Reactor Technology, Division C, Tokyo,Japan, August 18-23, 1991.
textures) during fabrication processes involving various thermomechanical treatments. These textured materials exhibit anisotropic mechanical properties and the anisotropic nature of these characteristics influence their in-service behavior. Binary Z r - N b alloys such as Z r - l N b and Z r 2Nb have been in use in Russian and Canadian ( Z r - 2 N b ) reactors (Ells, 1974) and there exists a vast amount of literature on the physical metallurgy, deformation behaviors both in and out of reactor as well as the texture development of these alloys following various reduction schedules (Tenckhoff, 1988; Cheadle, 1967; Cheadle, 1972). Recent developments lead to niobium-added Zircaloys which are now being considered as replacements for Zircaloys in LWRs, and some nuclear reactor vendors (such as Westinghouse) recently announced the introduction of new generation alloys such as Zirlo T M alloys, which are
0029-5493/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0029-5493(94)00617-8
2
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1 15
Nb-modified Zircaloys (Sabol, 1989). These materials exhibit resistance to in-reactor corrosion and oxidation, particularly at long exposures. However, currently a very limited amount of information is available on the effect of Nb on crystallographic textures and accompanying mechanical anisotropy (Mahmood, 1991; Murty, 1991) of these Nb-modified Zircaloys, while the literature data refer to binary Z r - N b alloys (Ells, 1974; Cheadle, 1967; Cheadle, 1972). This information is very important for selecting optimum fabrication parameters (textures) as well as for accurate prediction of the dimensional stability of the final product (cladding, pressure tubes, spacer grids, channels, etc.). We report here the recent work on the crystallographic textures and mechanical anisotropy of Zircaloy-4 sheet in cold-worked stress-relieved (CWSR) and recrystallized (R x) conditions. Similar tests were also performed on three different Nb-added zirconium alloys to evaluate the effect of Nb on the crystallographic texture, mechanical anisotropy and formability of these materials. We do not consider the development of textures of these alloys but are concerned here with the characterization of the texture and the resulting mechanical anisotropy of the given alloys. Except for the alloy compositions of the test materials, no detailed information on the thermomechanical treatments (reduction schedules) is available. The development of the textures of Zircaloys (Tenckhoff, 1988; Britt, 1991) and binary Z r - N b alloys (Cheadle, 1967; Cheadle, 1972) have been considered in detail by various investigators and the reader is referred to the literature which includes both experiments and models (Tome, 1991) of texture development. The goal here is to experimentally determine the textures of given alloys, develop quantitative descriptions using crystallite
orientation distribution functions (CODFs), predict the mechanical anisotropy using CODFs in conjunction with plasticity (slip) models and finally correlate them with mechanical test data. The textures of the sheets were characterized by inverse pole figures obtained from the three orthogonal faces of the sheet samples and direct pole figures corresponding to the basal, prismatic and pyramidal planes from which the CODFs were derived. Upper-bound plasticity models were used in conjunction with CODFs to predict the mechanical anisotropy parameters (R and P) as well as the formability parameter B. The mechanical properties and the anisotropy parameters were determined from mechanical testing of gridded tensile samples. The experimental results correlated extremely well with the anisotropy parameters predicted by the prism slip dominance for all cases where the materials were recrystallized, whereas significant deviations were noted for the CWSR materials and plausible causes are described.
2. Experimental aspects Alloys with different compositions have been procured in the form of thin sheets of thickness varying from 0.75 to 1.14 mm. Table 1 gives the chemical compositions of these alloys. Zircaloy-4 sheets were received in the CWSR as well as the R x conditions. Alloy 1 is Zircaloy-2 with 1% Nb, alloy 2 is a ternary alloy with 0.3% Nb and 1.5% Sn and alloy 3 is Zircaloy-4 (with reduced Sn) with 1% Nb. Alloys 1 and 3 are in cold-worked condition, and alloy 2 is recrystallized. As pointed out earlier, no details of the thermomechanical treatments and reduction schedules of these materials are available.
Table 1 Chemical compositions (wt.%) of Zr-Nb
alloys
Nb
Sn
Fe
Cr
Mo
Ni
Zr
Zircaloy-4
-
1.5
0.21
0.1
-
-
Balance
Alloy 1
1.0
1.5
0.15
0.1
-
0.05
Balance
Alloy 2
0.3
1.5
-
-
0.3
-
Balance
Alloy 3
1.0
1.0
0.2
-
-
-
Balance
KL. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
2.1. Crystallographic texture The textures characterized by X-ray methods using both the inverse and direct pole figures using Schultz back-reflection technique; the details of the analyses and methodology may be found in Cheadle [1967, 1972]. The net integrated X-ray intensity under each allowed reflection in 1-20 scans was used to calculate the texture coefficients (TC) using the relation I(hkil)//Io(hkil)
TC(hki')
= [1/N
] E [ I
(1)
where I(hk, ) and Iochk,) are the measured and the random intensities respectively for the (hkil) reflection, N is the total number of allowed reflections, and TC(hkm gives the number of (hkil) planes, in the units of "times random" oriented along the specimen normal. The inverse pole figures were also constructed from these data obtained along the three orthogonal directions, roiling [RD], transverse [TD] and thickness or normal [ND] directions. The texture coefficients were then used to evaluate f-factors (Kearns, 1965) for each of the three orthogonal directions for each of the alloy sheets:
f=Jor~r/2 16 sin 4) COS2(~ d ~
obtained from X-ray diffraction were normalized to the units of "times-random" using the respective TCs, and the pole figures were plotted in the form of iso-intensity contours.
2.2. Mechanical testing A grid analysis technique was employed for experimental evaluation of the mechanical anisotropy parameters and mechanical properties along the rolling and transverse directions of the sheets. Tensile specimens with a gauge length of 31.75 mm and 6.35 mm width were machined from roiling and transverse directions of each of the sheets. Square grids (1.27 mm each side) were electrochemically etched onto one surface of the specimens to evaluate the axial and contractile width strain distributions. The changes in the grid dimensions along the tensile and transverse directions yielded information on the tensile and contractile strains, which made it possible to evaluate the transverse contractile strain ratios (CSRs) as a function of the tensile strain up to fracture. These CSRs are also the mechanical anisotropy parameters in the modified von-Mises-Hill yield criterion defining the generalized stress ~rg:
(2)
where 16 sin & is the volume fraction of the grains with their c axes oriented at a tilt angle & from the reference direction (sample thickness). An f-factor gives the effective fraction of basal poles oriented along a particular direction of the sheet and thus the sum of the f-factors determined along the three orthogonal directions is unity. A better representation of the crystallographic texture is obtained by a stereographic projection showing the variation in pole density with respect to the macroscopic coordinates (ND, RD, TD) for a particular set of crystallographic planes resulting in direct pole figures. A Norelco goniometer stage was employed, and basal (0002), prismatic (1010) and pyramidal (1012) pole figures were considered. The pole figure intensity data
3
R(OrND -- O'TD)2+ RP(O'TD -- O-RD)2-FP(O'RD -- trND) 2 P ( R + 1)
(3) w h e r e OND ~ OVTD a n d O'RD are the stresses along
the normal, transverse and rolling directions of the sheet, and R and P are the coefficients of anisotropy. R and P are given by R = ~ AeND
IffRD,
OrTD =
OrND = 0
(4a)
and A6RD / P =
, O'RD = O'N D = 0
(4b)
Z~END )eTD
Experimental and analytical procedures were outlined by Mahmood and Murty (1989).
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15
4
3. Results and discussion
preferentially oriented along the rolling and transverse directions. A smaller value for the prismatic texture coefficient along the RD of the recrystallized sheet when compared with the CWSR sample is indicative of the 30 ° rotation of the prismatic planes away from the RD upon recrystallization. Essentially similar results are noted for Nb-modified alloys, except for the fact that the TC values along TD are relatively low while alloy 1 exhibited a finite, albeit low, value along ND. These differences are more revealing in the direct pole figures, as noted later. A comparison of the f-factors shows that in each case the effective fraction of basal poles along the ND of the sheets is higher than those along the RD and the TD. The only exception is that alloys 1 and 2 exhibited relatively large values for the f-factor along TD. The sum of the f-factors along the three principal directions of each sheet is close to unity, as expected. Since the reduction schedules of these materials are not known, the reasons for these differences cannot be attributed to the Nb additions; further work is called for
3.1. Crystallographic textures As noted above, the first step in the characterization of crystallographic textures of the sheets was to determine the TCs for the different sets of planes, and f-factors for each of the three orthogonal directions of the sheets. The measured values of these parameters are given in Table 2. Both CWSR and R x Zircaloy-4 sheets exhibit high values of basal texture coefficients along the normal direction followed by TD with zero at RD indicating that the basal poles are confined to the N D - T D plane and concentrated close to ND. While the Nb-modified alloy 2 followed the same trend, different behaviors were noted for alloys 1 and 3. The TCs along ND and TD are essentially the same for alloy 1, while that for alloy 3 along TD is also very small, as for RD. The texture coefficients of prism poles are zero along N D for Zircaloys with non-zero values along the other two directions, indicating that the prism poles are Table 2 Texture coefficients and f-factors Texture coefficients
Zircaloy-4 CWSR
ND RD
TD Zircaloy-4 R x
ND RD
TD Alloy 1 (CWSR)
ND RD
TD Alloy 2 (R~)
ND RD
TD Alloy 3 (CWSR)
ND RD
TD
Basal
Prismatic
Pyramidal
f-factor
4.25 0.00 1.04
0.00 6.58 3.12
0.85 0.00 0.54
0.741 0.100 0.172 E f = 1.013
3.99 0.00 1.19
0.00 3.97 2.71
1.49 0.00 0.85
0.724 0.075 0.182 E f = 0.981
1.53 0.00 1.59
0.26 4.62 0.48
1.98 0.00 2.33
0.584 0.057 0.330 Y~f = 0.971
3.59 0.00 1.65
0.00 3.10 0.67
2.15 0.00 2.41
0.634 0.039 0.353 Y~f = 1.026
2.58 0.00 0.19
0.00 3.85 1.31
0.63 0.00 0.35
0.750 0.074 0.147 E f = 0.971
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
poles concentrated in the transverse plane normal to the roiling direction of the sheet. The intensity maxima in the basal pole figure of the CWSR Zircaloy-4 sheet have a value of 7.4 times random at +27 ° from the ND in the ND-TD plane, while recrystallization reduced the intensity maxima to about 4.3 and also tilted the basal poles closer to the ND, with the basal pole peak angle being _+20°. A more distinct difference in the textures of CWSR and R x Zircaloys is observed in the distribution of prismatic poles given in (1010) pole figures in Figs. l(a) and l(b). The majority of the prismatic poles in the CWSR are oriented along the RD while recrystallization rotates these poles around the c axes of the crystal-
using known thermomechanical treatments of these alloys. The direct pole figure technique is used in determining the basal, prismatic and pyramidal pole figures and either one or a combination of samples were used to obtain complete pole figures (Mahmood, 1989). The basal, prismatic, and pyramidal pole figures for all of the alloys are compiled in Fig. 1, with the center of each pole figure corresponding to the normal direction of the sheets. These pole distributions exhibit typical features of textures in deformed and recrystallized zirconium alloys (Murty, 1991; Mahmood, 1989; Murty, 1989). The basal pole figures, in each case, show bimodal distribution of basal
RD
RD
BO
!
"'"
.
TD
r_
BASAL (a)
RD
,.
TD
(b)
RD
q,
l
BASAL (ooo2)
PYRAMIDAL (1012)
0o'/o)
RD
TD
t TD
PRISMATIC
(0002)
5
PRISMATIC (lO10)
PYRAMIDAL (1012)
Fig. 1. Basal, prismatic and pyramidal pole figures of CWSR Zircaloy-4 (a); R x Zircaloy-4 (b); alloy-1 (c); alloy-2 (d); alloy-3 (e).
6
K.L. Murty et al./Nuclear Engineering and Destgn 148 (1994) 1-15
(as expected in low c/a ratio hcp metals such as Zr and Ti (Tenckhoff, 1988)) along with another pair orthogonal to it at _+ 15° from the ND in the N D - R D plane. This is distinctly evident in Fig. 2, where the intensities of the basal poles are plotted L~,s.the tilt angles from ND towards TD and RD respectively, which exhibit bimodal distributions along both directions. In contrast, Zircaloys are characterized by such a bimodal distribution only in the R D - T D plane with rapidly decreasing basal pole intensity towards RD from ND. Other texture modifications following Nb additions to zirconium and Zircaloys were also reported (Murty, 1991). In all cases, pyramidal pole distri-
lites, shifting the intensity maxima from the RD to 30 ° from the RD towards the TD. A second rather small peak in each case appears at about 60 ° from the main peak owing to the hexagonal symmetry. In general, the Nb additions seemed to preserve the recrystalliz.ation effect of the rotation of the prism planes. Alloys 1 and 3 have cold-worked textures while alloy 2 exhibits prism pole maxima at +_30° from RD towards TD as for the recrystallized Zircaloy. However, the basal pole distribution of alloy 3 is distinctly different in that the intensity maxima (approx. 4.5 times random) are at about _+27 ° from the ND in the N D - T D plane
RD
RD
RD
TD
•
"
Basal
Prism
Pyramidal
(00o2)
(lO~O)
0o~2)
RD
RD
RD
Basal
(0002)
Prism
Pyramidal
(1010)
(1012)
Fig. 1 (continued).
TD
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
7
RD
RD
RD
Basal (0002)
Prism (1010)
Pyramidal (1012)
Fig. ] (continued).
butions indicate relatively low peak intensities and are more spread out. We also note that alloy 2 exhibits relatively high peak intensities in both the basal pole (approx. 14 times random) and the pyramidal (approx. 5 times random) compared with others which exhibit about 4.5 and 2 or lower, respectively.
3.2. Crystallite orientation distribution functions The pole figure intensity data were used for a more quantitative texture analysis through the generation of CODFs for all the sheets. A C O D F is a distribution function which gives the probability that a crystallite lies within a certain range of orientations (0, qJ, ~b) with respect to the specified coordinates, which are generally the processing axes of the material (ND, RD, and T D for a sheet). C O D F is expressed as a series of generalized spherical harmonics (Murty, 1989; Roe, 1965; Bunge, 1982) in Roe's notation:
,o(0, 6, 6) I
I
E
E
I=0 m--1
n=-I
=E
Wlm Zlm,(cos O)e -im+ e-in6
(5) where WI~° are the appropriate series coefficients, Zim" are the augmented Jacobi polynomials and (0, ~O, ~b) are Euler angles (Roe, 1965)
connecting coordinate axes embedded in the crystallite with axes coincident with the specimen and the principal working directions of the material. In the present study, three pole figures (basal, prismatic and pyramidal) were used to evaluate the series coefficients W~m" up to I - - 1 6 . The error caused by truncating I at 16 is generally insignificant compared with the errors inherent in experimental texture measurements. The accuracy of the generated CODFs was checked by reconstructing the basal pole figure from the derived C O D F (Mahmood, 1989). CODFs are generally represented graphically in the form of Euler plots at fixed intervals of the angle & of rotation around the c axis up to & = 55°; beyond this, repetition occurs as a result of hexagonal symmetry. Fig. 3 is a compilation of the Euler plots where iso-intensity contours are plotted in "times-random" units at constant intervals of 1 with the base contour also at 1. The three orthogonal sheet directions ND, RD and T D correspond to (0, q~) angles of (90 °, 0°), (0 °, 0 °) and (0 °, 90 °) respectively. The constant 4, sections of the Euler plots can also be considered in terms of the ideal orientations (hkil)[uutw], where (hkil) refers to the crystallographic plane lying in the surface of the sheet and [ut, tw] is a crystallographic direction in this plane and parallel to RD (Mahmood, 1989).
8
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15 .
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
demonstrated previously (Murty, 1991; Mahmood, 1989) the simple slip models considered here yielded anisotropy parameters for Zircaloys, in good agreement with the experimental results both during low temperature deformation and high temperature creep.
,
ND-RD L~
,o E m
3.3. CODF-plasticity model
o
E
-90
-60
-50 0 50 TillAngle Phi (degrees)
60
90
ND-TD to
L
Once the orientation distribution of crystallites in the polycrystalline materials is evaluated and the appropriate C O D F is known, the crystallite behavior in each orientation can be weighted by the volume fraction of crystallites in that orientation so that the behavior of the polycrystalline aggregate can be predicted. If m is the strain-rate sensitivity, the shear strain-rate on the sth slip system can be expressed as ~" = A r i ; [ , ) ' ] "
ID
I
-90
I
I
-60
I
I
I
-30
I
I
i
0
i
i
i
i
30
i
i
i
i
60
90
TIllAngle Phi (degrees)
Fig. 2. Basal pole intensity t,s. tilt angle for alloy 3 from ND towards TD and RD; note the bimodal distributions in both the N D - T D and N D - R D planes.
Physical and mechanical properties of single crystals are anisotropic and this extends to the bulk properties off the polycrystals whenever preferred orientations are present. To the first order, the bulk property is related to the microscopic property, p(O, 0, 05), by statistical averaging using the CODF, so that the average property of the polycrystalline aggregate (p(O, 05, ~b)) is given by
(p(O, qJ, da))= fo2~fo2~f]]p(O, dJ, O) > w ( 0 , ~, 05) d(cos 0) d~0 d~b
(6) In this model, the grain-to-grain interactions are not taken into account and an average through the texture is taken to represent the plastic behavior of the polycrystalline material. As has been
(7a)
where ~'i; is the reference shear stress on the sth slip system, m the strain-rate sensitivity and A the reference strain-rate. The reference shear stress ( r 0) is similar to the CRSS and defines the relative ease with which slip can occur on a given slip system. The total strain-rate in the crystal is expressed as the sum of contributions from each of the active slip systems: -c = x--, J,t. kz = Mijk/O'k/ Eij , 7 ]'£ij~ij kl
(7b)
]J~ij is the geometric tensor connecting slip coordinates with crystal coordinates. The shear stress is related to the crystallite stress by
where
where the superscript s specifies the slip system under consideration. In the upper-bound method, the strain (rate) state in each crystallite is regarded as being identical to the applied macroscopic strain tensor, and the stress state is calculated using Eqs. (7a)(7c). For m < 1 (i.e. the stress exponent > 1) an iterative numerical procedure was used in which successive approximations to the actual stress were obtained by means of a form of Newton's method (Adams, 1985). Once the stress state for a crystallite is determined, the bulk stress state is
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
>
~j._,
9
( )
0 8~ )
0
0
)
o
0 II 0
,E Z
C3~
©
0 0 0
~
0 0',
©
u© h2
r~ ~>~_j
C)
/,.%~ u ~
-n
c~
c
~D c ~
~ °
ii
A
i"~°~/
~ 0
~ ,
~®i °
~o
-x
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
then obtained by averaging the stress state of the individual grains weighted by CODF: ub
t'2rr , ' 2 7 r r 1
[%] =Jo Jo J-1~°(°'~°''t')°'j × d(cos 0) dO d4,
(8)
In this upper-bound method the strain compatibility is preserved while the stresses vary across crystallite boundaries. Thus the first-order distribution of internal stress is approximated. In the lower-bound method, the stress in each grain is taken to be the same as the applied macroscopic stress and the strains (or rates) vary from grain to grain depending on their orientations; these schemes are seen to be valid for high temperature creep (Murty, 1989). Self-consistent methods take into account both strain and stress compatibilites, and more detailed discussions of these various schemes may be found in the literature (Hutchinson, 1977).
11
Three different slip modes are considered in the present analyses: basal ({0001}(1120)), prism ({10]0}(1120)) and pyramidal ({1122}(1123)). Values of the reference stresses ( r 0) are chosen to emphasize the contribution of individual or particular combinations of slip modes to the polycrystalline behavior. It should be pointed out that in this model all of these slip systems are considered to be operative, thus preserving the condition of five independent slip systems for polycrystalline deformation, although the contribution of a given slip system or a combination of them is forced to become dominant by appropriate choice of r0 values. In addition to the power law (Eq. (7a)), a Bishop-Hill analysis corresponding to rigid plastic behavior was also made for comparison. The derived CODFs for the various alloys were used to calculate the bulk stresses for the three slip systems and these stresses were normalized to a constant rate of energy dissipation (I~ = rijS, ij) (Murty, 1991; Adams, 1985). The mechani-
Table 3 Anisotropy and formability parameters Material
Slip system
R
P
B
O.TD/O.RD a
Zircaloy-4 CWSR
Prism Basal Pyramidal Experimental
2.89 1.94 0.34 4.03 _+ 0.36
4.24 0.39 0.31 9.53 _+ 0.31
1.52 0.78 0.80 1.89
1.044 1.534 0.966 1.057
Zircaloy-4 Rx
Prism Basal Pyramidal Experimental
3.23 0.61 0.31 3.18 + 0.28
3.40 0.27 0.24 3.76 _+ 3.76
1.47 0.74 0.76 1.50
1.006 0.749 0.904 1.182
Alloy 1
Prism Basal Pyramidal Experimental
2.32 0.74 0.48 1.50 _+ 0.08
6.36 0.14 0.47 3.35 + 0.21
1.58 0.64 0.86 1.33
1.112 0.537 0.993 1.329
Alloy 2
Prism Basal Pyramidal Experimental
2.11 1.58 0.55 2.33 + 0.12
4.64 0.31 1.06 4.27 + 0.18
1.48 0.74 1.03 1.48
Prism Basal Pyramidal Experimental
4.84 2.15 0.24 4.50 + 0.23
4.56 0.38 0.23 4.22 + 0.21
1.68 0.78 0.78 1.63
Alloy 3
P / / P ~ + 1) a Model calculations based on V R ( ~ S (Eq. (9)).
1.101
0.622 1.204 1.196 0.995 0.635 0.983 1.017
K.L. Murty et al./Nuclear Engineering and Design 148 (1994) 1-15
12
Strain (%) 0
12.1
3.1 4.4 5.7 6.9 7.9 9.2 10.1 11.0 11.9 Fig. 4. A typical sequence of grid photographs taken during a tensile test (alloy-3).
cal anisotropy parameters evaluated from these predicted bulk stresses are tabulated in Table 3. The results of the rigid-body analysis (BishopHill) were close to these values * and thus were not included here. 3.4. Anisotropic mechanical properties Fig. 4 depicts a typical sequence of grid photographs taken during a tensile test and the grid size measurements yield the true transverse and contractile strains. This technique can be used to study the spatial and temporal strain distributions. Fig. 5 depicts the engineering stress-strain curves for the R D and T D specimens derived from the total gage section of the specimens. Typical stress cs. strain curves derived from the data on the fractured grid elements are shown in Fig. 6 where both nominal and true stress-strain curves are included for the rolling and transverse directions. Similar results were obtained on all of the alloys. Tables 4 and 5 summarize the mechanical properties data for CWSR and R x Zircaloy sheets and Nb-modified zirconium alloys respectively. The yield strengths along the transverse directions o f the sheet for all the alloys are seen
to be higher than those along the rolling directions. In general, ductilities along the T D are lower than those along the RD, in particular for the R x materials (Zircaloy and alloy 2). These differences are more revealing in the work hardening exponents (n) and all alloys exhibit higher rates of strain hardening along R D than TD. Thus we note that both CWSR and R x materials exhibit anisotropy not only in the mechanical strengths and ductilities, but also in their strain hardening behaviors. Although recrystallization made the materials softer, it did not eliminate the texture and anisotropy, but rather it modified them. It is interesting to note that the tensile
1200
1000
R3 800
600
400 c 2OO
Alloy 3 0 000
* Predictions based on the Bishop-Hill analysis differed slightly more from the experimental results.
t 0 05
0.10
015
Engineering strain
Fig. 5. Engineering stress t,s. strain cubes along RD and TD.
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
13
Table 4 Mechanical properties of CWSR and R x Zircaloy-4 Proper~
CWSR
0.2% yield strength (MPa) ( ± 15%) Tensile strength (MPa) ( ± 5%) Total elongation ( ± 0.01) Strain hardening exponent (n) a
Rx
RD
TD
RD
TD
436 692 0.15 0.05 ± 0.03
461 682 0.12 0.04 ± 0.02
380 480 0.35 0.22 ± 0.04
449 511 0.29 0.12 ± 0.03
ao-=gen"
s t r e n g t h s of alloy 3 a r e essentially i d e n t i c a l indicating p l a n a r isotropy, a n d the r a t e s o f strain h a r d e n i n g a r e also similar a l o n g R D a n d T D . T h e ratios o f t h e yield s t r e n g t h s a l o n g R D a n d T D a r e r e l a t e d to t h e m e c h a n i c a l a n i s o t r o p y p a r a m e t e r s . R e a r r a n g e m e n t o f t h e m o d i f i e d Hill's e q u a t i o n (Eqn. (3)) gives
~rnD
VR(P+I
)
m e a s u r i n g t h e axial tensile a n d t r a n s v e r s e contractile strains in t h e grid e l e m e n t s which ultim a t e l y f r a c t u r e d in t h e R D a n d T D s p e c i m e n s . Fig. 7 shows the w i d t h c o n t r a c t i o n as a f u n c t i o n o f l o n g i t u d i n a l e l o n g a t i o n for b o t h t h e R D a n d T D s p e c i m e n s for alloy 3; similar d a t a w e r e also o b t a i n e d on t h e o t h e r alloys. A l i n e a r v a r i a t i o n is n o t e d b e t w e e n the c o n t r a c t i l e a n d tensile strains a n d the s l o p e s o f t h e s e lines w e r e u s e d to d e t e r m i n e the R a n d P p a r a m e t e r s (Eqns. (4a) a n d (4b)). T h e s e results a r e t a b u l a t e d in T a b l e 3 along with t h e m o d e l p r e d i c t i o n s (Eqn. (8)). T h e s e grid e x p e r i m e n t s also yield i n f o r m a t i o n on t h e l o c a l i z e d n e c k i n g b e h a v i o r s as d e t a i l e d by M a h m o o d a n d M u r t y (1989). T a b l e 3 also i n c l u d e s t h e f o r m a b i l i t y p a r a m e ter B d e f i n e d as ( B a c k o f e n , 1972)
(9)
T h e ratios o f t h e e x p e r i m e n t a l yield stresses a r e c o m p a r e d in T a b l e 3 with t h e m o d e l p r e d i c t i o n s b a s e d on t h e r i g h t - h a n d side f a c t o r o f the a b o v e e q u a t i o n for t h e v a r i o u s slip systems for all of t h e alloys. A l t h o u g h we n o t e a g o o d a g r e e m e n t with p r i s m slip p r e d i c t i o n s in all cases, t h e c o r r e l a t i o n is m i s l e a d i n g since m a n y c o m b i n a t i o n s o f R a n d P v a l u e s give rise to similar yield stress ratios, and a direct correlation between the measured R a n d P v a l u e s with the m o d e l p r e d i c t i o n s is m o r e appropriate. T h e grid analysis t e c h n i q u e e n a b l e s an evaluation of t h e m e c h a n i c a l a n i s o t r o p y p a r a m e t e r s R a n d P. T h e s e p a r a m e t e r s w e r e d e t e r m i n e d by
OvI
B -
(lOa) 2O'iv
w h e r e o-l a n d ¢r~v a r e t h e p l a n e strain stress in t h e first q u a d r a n t a n d t h e equibiaxial stress in the f o u r t h q u a d r a n t of the yield locus respectively. T h e significance of the p a r a m e t e r B lies in the fact t h a t t h e h i g h e r the B value, t h e b e t t e r t h e
Table 5 Mechanical properties of Nb-modified alloys Property
Alloy 1
0.2% yield strength (MPa) Tensile strength (MPa) Total elongation Strain hardening exponent (n)
489
_+ 17
650
+ 23
565
+ 20
676
+ 24
892
_+31
907
± 32
736
+ 26
672
± 23
712
± 25
660
+ 23
1116
+ 39
1088
+ 38
RD
Alloy 2 TD
0.24 + 0.02 0.21 ± 0.02
RD
0.29 ± 0.03 0.08 + 0.01
Alloy 3 TD
0.30 + 0.02 0.17 + 0.02
RD
0.22 + 0.03 0.05 ± 0.02
TD
0.12_+ 0.01 0.06 + 0.01
0.11 ± 0.01 0.05 + 0.01
14
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
cup formability of the material. The formability parameter is calculated using the anisotropy parameters R and P (Murty, 1989): B= ( R + l)(R+4R+P) ( 1Ob)
4R(R+P+ 1)
d
Table 3 lists the predictions based on the upperbound model for the three slip systems with the strain-rate sensitivity parameter m of 0.01 applicable for zirconium alloys at ambient. For recrystallized alloys and also for alloy 3 we note a good correlation between the experimental results and the model predictions based on the dominance of prism slip. Such a correlation is expected since low c/a ratio materials such as Zr and Ti are
0.6
Slope=0.818 r
._ z ;;
Rz4.50
A l l o y 3 RD
0.6
0.4 . 6 .= F 02.
0.2
0.0
0.4
0.6
0.6
Axial strain
0.8
Slope=0.809
800 -
._c m i r E ?
Engmeering
600 -
Pz4.22
Alloy 3 TD
0.6
0.4
0.2
400. a 200
0.0 00
f
n “0.0
1600
0.5 Slralll
,
1.0
I
1400 1200
2
1000
% 600 : ' 6
600 400
0' U.0
I 0.5
Strain
I 1.0
Fig. 6. Nominal and true stress-strain curves for RD and TD specimens obtained from fractured grid elements.
i3
I
0.2
0.4
0.6
0.6
Axial strain Fig. 7. Transverse contractile strain (cw) cs. longitudinal tensile strain (E,) for RD (top) and TD (bottom) specimens; the slopes of the lines are related to R and P.
known to deform by prism slip, and the critical resolved shear stress for prismatic slip is much lower than that for basal and pyramidal slip (Mahmood, 1989; Murty, 1989). Distinct deviations are noted for CWSR Zircaloy-4 and alloy 1, which is also cold-worked. Particularly noteworthy is the differences in the P parameter, and these deviations are believed to arise from possible contributions to the deformation from twinning in addition to slip. Moreover, it has been shown that basal slip might make a non-negligible contribution to slip in cold-worked Zircaloys even at low temperatures (Tome, 1991). Further work is needed to unravel the reasons for these findings. The anisotropy parameters R and P for
K.L. Murty et al. / Nuclear Engineering and Design 148 (1994) 1-15
alloy 3 are close to each other, just as in recrystallized Zircaloy-4, revealing that these materials exhibit planar isotropy as noted from the yield stresses (Table 5). However, the values of R and P are far removed from unity, indicating the existence of mechanical anisotropy when considered in three dimensions.
4. Conclusions Mechanical properties, anisotropy and formability parameters were studied on Zircaloy and Nb-added zirconium alloy sheets using gridded tensile specimens along the roiling and transverse directions. Crystallographic textures of these materials were characterized using X-ray diffraction. The texture coefficients and f-factors were evaluated from 1-20 scans along the three orthogonal directions. Basal, prismatic and pyramidal pole figures were determined, from which CODFs were evaluated. Upper-bound crystal slip models were used in conjunction with CODFs to calculate the mechanical anisotropy parameters of the textured bulk samples. The model predictions b a s e d o n p r i s m slip a g r e e d w i t h t h e e x p e r i m e n t a l results on the recrystallized materials while cold w o r k i n g r e s u l t e d in l a r g e d e v i a t i o n s .
Acknowledgments Financial support through grants from the National Science Foundation and the Electric Power Research Center at North Carolina State University are gratefully acknowledged. We wish to express our sincere appreciation to Mr. Jeff C. Britt for assistance and for various discussions. Acknowledgments a r e d u e t o T h e U n i v e r s i t y o f Western Australia for the award of a Gledden Senior Visiting Fellowship to KLM, during which time the manuscript was prepared.
15
References B.L. Adams and K.L. Murty, Mater. Sci. Eng., 70 (1985) 181. W.A. Backofen, Deformation Processing, Addison Wesley, Reading, MA, 1972, p. 47. J.C. Britt and K.L. Murty, Texture development in zirconium alloys, in Proc. Symp. on Zirconium Alloys for Reactor Components (ZARC-91), Bhabha Atomic Research Center, Bombay, December 1991, pp. 1-45. H.J. Bunge, Texture Analysis in Materials Science, Butterworths, London, 1982. B.A. Cheadle, C.E. Ells and W. Evans, J. Nucl. Mater., 23 (1967) 199. B.A. Cheadle, S.A. Aldridge and C.E. Ells, Can. Met. Quart., 11 (1972) 121. C.E. Ells, Deformation of irradiated zirconium-niobium alloys, in Zirconium in Nuclear Applications, ASTM STP 551, ASTM, Philadelphia, PA, 1974, pp. 311-327, and refs. 1-7 in the paper. J.W. Hutchinson, Metall. Trans., A8 (1977) 1465. J.J. Kearns, Thermal Expansion and Preferred Orientation in Zircaloy, WAPD-TM-472, Westinghouse Electric Corporation, Pittsburgh, PA, 1965. S.T. Mahmood and K.L. Murty, Localized plastic flow, anisotropic mechanical properties and crystallographic texture in Zircaloy sheet, J. Mater. Eng., 11 (1989) 315. S.T. Mahmood, J. Ravi and K.L. Murty, Effects of niobium additions on textures and mechanical properties of zirconium alloys, in Proc. l l t h Int. Conf. on SMiRT, Tokyo, August 18-23, 1991, paper C03/2. K.L. Murty, Applications of crystallographic textures of zirconium alloys in the nuclear industry, in Zirconium in the Nuclear Industry: Eighth International Symposium, ASTM STP 1023, American Society for Testing and Materials, Philadelphia, PA, 1989, pp. 570-595. K.L. Murty, Mater. Forum, 15 (1991) 217. R.J. Roe, J. Appl. Phys., 36 (1965) 2024. G.P. Sabol, G.R. Kilp, M.G. Balfour and E. Roberts, Development of a cladding alloy for high burnup, in Zirconium in the Nuclear Industry: Eighth International Symposium, ASTM STP 1023, ASTM, Philadelphia, PA, 1989, pp. 227-244. E. Tenckhoff, Deformation Mechanisms, Texture and Anisotropy in Zirconium and Zircaloy, ASTM STP 966, ASTM, Philadelphia, PA, 1988. C. Tome, R.A. Lebensohn and U.F. Kocks, Acta Metall. Mater., 39 (1991) 2667, and refs. 1-8 therein.