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Plastic Deformation and FractureBehavior of Zircaloy-2 Fuel CladdingTubes under Biaxial StressHideo MAKI a & Masatosi OOYAMA aa Hitachi Research Laboratory , Hitachi Ltd , Kuji-machi, Hitachi-shiPublished online: 15 Mar 2012.

To cite this article: Hideo MAKI & Masatosi OOYAMA (1975) Plastic Deformation and Fracture Behaviorof Zircaloy-2 Fuel Cladding Tubes under Biaxial Stress, Journal of Nuclear Science and Technology,12:7, 423-435

To link to this article: http://dx.doi.org/10.1080/18811248.1975.9733131

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journal of NUCLEAR SCIENCE and TECHNOLOGY, 12[7), pp. 423-435 (July 1975). 423

Plastic Deformation and Fracture Behavior of Zircaloy-2

Fuel Clzdding Tubes under Biaxial Stress

Hideo MAKI and Masatosi OOYAMA

Hitachi Research Laboratory, Hitachi Ltd.*

Received August 31, 1974 Revised December 3 , 1974

Various combinations of biaxial s t ress were applied on five batches of recrystallized zircaloy-2 fuel cladding tubes with different textures ; elongation in both axial and circumferential directions of the specimen was measured continuously up to 5% plastic deformation.

T h e anisotropic theory of plasticity proposed by Hill was applied to the resulting data , and anisotropy constants were obtained through the two media of plastic strain loci and plastic strain ratios. Comparison of t he results obtained with the two methods proved that t he plastic s t ra in loci provide da t a that a r e more effective in predicting quantitatively the plastic deformation behavior of the zircaloy-2 tubes. T h e anisotropy constants change their value with progress of plastic deformation, and judicious appli- cation of t he effective s t ress and effective s t ra in obtained on anisotropic materials will permit the relationship between stress and strain under various biaxialities of stresses t o be approximated by the work hardening law.

T h e tes t specimens used in the plastic deformation experiments were then stressed t o f racture under the same combination of biaxial s t ress as in the proceeding experi- ments, and the deformation in the fractured par t was measured. T h e result proved tha t t h e t i l t angle of the c-axis which serves a s the index of texture is related to f racture ductility under biaxial stress. Based on this relationship, i t was concluded that mate- rial with a t i l t angle ranging from 10" t o 15" is the most suitable for fuel cladding tubes, f rom the viewpoint of f racture ductility, at least in the case of unirradiated material.

KEYWORDS: zircaloy-2, fuel cans, texture, biaxial stress, plastic deforma- tion, fracture, anisotropy

I. INTRODUCTION

Zircaloy fuel cladding tubes of small diam- eter and thin wall, currently useful exten- sively in water reactors, are usually made from hot extruded raw tubing by alternative cold rolling and annealing repeated a couple of times. During this fabrication process, a peculiar close-packed hexagonal metal struc- ture is formed, which causes prominent an- isotropy in the mechanical properties.

During service in water reactors, the zir- caloy cladding undergoes plastic deformation, through its subjection not only to external pressure from coolant water and internal pressure from fission gas build-up, but also

to the contact load applied by the expansion of the sintered uranium oxide pellets, all of which combine to produce multiaxial stress.

In particular, the cladding tube is known to receive highly concentrated loading in the zones facing ridges formed along the string of pellets charged in the tube. Veeder") has analyzed the thermo-elastic deformation of individual sintered pellets of UOz, and proved that the pellets deform into hour-glass shape. Thus a string of such hour-glass forms would present ridged rings a t regular inter- vals, which would push against the cladding tube from inside to deform the tube with bamboo-like projecting nodes. ,Gittus") has * Kuji-machi , Hitachi-shi.

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424 J. Nucl. Sci. Technol.,

proposed a dynamical model to evaluate the stress and strain conditions in a restricted zone of the cladding tube facing the opening of radial pellet cracks during a power in- crease. Based on this model, it was analyzed that highly concentrated biaxial stress is produced in the cladding by the opening of radial cracks a t the pellet ridges. Later, Mogard et ~ 2 1 . ' ~ ) succeeded in experimentally simulating fuel failure with high reproduci- bility, which substantiated Gittus' prediction that the cladding tube was most liable to fail in the vicinity of the point facing a radial crack at the pellet ridge.

The cladding would be expected to fail a t the point where the local plastic deforma- tion exceeds its strain limit. In fuel design, therefore, the tube structure and service con- ditions should be determined such that the cladding tube would not be subjected to plastic deformation exceeding the strain limit at any point on its surface throughout its service life, and this necessitates clarification of the deformation behavior of the zircaloy cladding tube under multiaxial stress.

In the present study, we have developed an experimental apparatus that permits the application of any disired combination of bi- axial stress to zircaloy cladding tubes. This is done by superimposition of axial load and internal pressure, and the stress level can be gradually raised to specimen fracture while keeping the stress ratio unchanged. T h e principle of this apparatus is the same as reported by MehanC4), but in our apparatus the stress ratio can be controlled automati- cally and the amounts of elongation in both

axial and circumferential directions of a tube can be measured continuously to the extent of about 5%.

Using this apparatus, we measured t h e plastic deformation behavior of five different batches of zircaloy-2 fuel cladding tubes with different textures. The following items were examined :

(1) Plastic anisotropy of the cladding tubes. with different texture a t room tempera- ture

(2) Changes with temperature brought upon plastic anisotropy

(3) Applicability of the anisotropic theory of plasticity proposed by Hill") to the experimental results

(4) Desirable texture for the cladding tube from the viewpoint of fracture ductility

II. EXPERIMENTAL 1. Materials Five batches of recrystallized zircaloy-2

fuel cladding tubes with different textures were taken up in the experiments. Their manufacturing history, tensile properties and the tilt angle of the c-axis are as given in Table 1. All the tubes had an average inner diameter of 12.67 mm, while the average wall thickness was 0.90mm for Tube-A and 0.80 mm for Tubes-B, -C, -D and -E. By using thin films taken from these tubes, their (0002) and { l O i O } pole figures were determined b y Schulz's and Decker's methods. Since the preferred orientation of the (0002) pole pre- dominantly influences the anisotropy of the mechanical properties of a tube, we adopted

Table 1 Fabrication history, mechanical properties and te:bure of materials studied

Basalt pole

Fabrication history Mechanical properties

Symbols Method of Annealing Ultimate Yield Elongation tube reduction temperature tensile strength strength (G.L. =50 mm) Orientation

(") ("C) (kg/mm2) (kg/mm2) (%I A Pilger mill 650 60.7, 60.3 42.4, 42.9 22, 22 45 B Pilger mill 560 55.3, 58.0 42.4, 43.9 35, 33 20

D 3-1-011 mill 560 51.3, 50 .3 39.4, 39.7 42, 43 10 E 3-roll mill 560 51.3, 51. 6 41.5, 41.7 42, 39 5

C Pilger mill 560 54. 4, 57. 6 41. 3, 44. 0 36, 35 15

t The title angle of the (0002) basal pole with the radial toward the circumferential direction

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Vol. 12, No. 7 (July 1975) 425

as quantity indicative of specimen texture the tilt angle of this (0002) basal pole, as it is tilted from the radial direction toward the the circumferential direction of the tube spec- imen. Figure 1 shows the (0002) pole figure obtained with Tube-A.

R.D.

I /

Fig. 1 (0002) pole figure of Tube-A

Tubes-A, -B and -D were processed for final cold reduction on a Pilger mill, while Tubes-C and -E were processed on a 3-roll mill. The type of mill employed is mention- ed here only for reference, and are not con- sidered to affect the mechanical properties of the products to any significant extent. Also, the final annealing temperature of Tube -A is 650"C, while that of the other four tubes is 560°C, but again it is believed that this difference in heat treatment should pro- duce only negligible effect on their mechani- cal properties.

2. Experimental Apparatus Figure 2 illustrates the arrangement of

the experimental apparatus employed. The load is applied on the tubular specimen in the axial direction by a screw type tensile machine, and internal pressure is super- imposed thereon by the medium of water. The apparatus is so connected as to maintain constantly a predetermined ratio between internal pressure and tensile load, which latter is thus automatically increased when the internal pressure is raised. This facilit- ates adopting as parameter the ratio ga/ut between the axial stress ga in the specimen and the circumferential stress ut.

To measure axial deformation, an axial extensometer is fixed to the specimen, fitted

with two linear differential transformers. For observing dilatation in the radial direction, three linear differential transformer type extensometers are inserted through holes pierced through the wall of the electric fur- nace a t 120" angles between each other. The four measured values of axial load, internal pressure, axial elongation and radial dilata- tion are recorded continuously on two X - Y recorders. Experiments at high temperatures are conducted by heating up the electric furnace.

Tensile load

t

m l i f i e r Amplifier

I I Fig. 2 Schematic representation

of testing apparatus

3. Method of Experimentation and

Experiments were first conducted at room temperature for five different values of the stress ratio ua/at-O, 0.5, 1, 2 and 00, on all the five batches of cladding tubes listed in Table 1. The runs were repeated under each experimental condition to ascertain the re- producibility of the results. Experiments at 350°C were also undertaken on Tube-A for stress ratios of 0, 1 and 00, to examine the effect of heat on the anisotropy of plastic deformation.

Figure 3 shows the shapes and dimensions of the test specimens used. Specimen-I is for experiments with stress ratios oa/gt of 0.5, 1, 2 and 00. An end plug and a protec- tive ring were welded at each end of this 120 mm long specimen by electron beam weld- ing, and a small hole was bored through one end for applying the internal water pressure. A ridged ring was machined out on each end

Shapes of Test Specimens

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426 J . Nucl. Sci. Technol..

plug for attaching the axial extensometer. mental results The elastic constants used in the treatment were obtained from separate experiments in which the strain was meas- ured with higher accuracy.

IN. EXPERIMENTAL RESULTS 1. Plastic Strain Loci Rewriting the yield criterion for isotropic

materials proposed by von Mises into cylin- drical coordinates, and disregarding the radial stress on account of the thin wall of the

Fig. 3 Forms of specimens cladding tube,

Specimen-II was used in the room tem- where C : Constant. oa2 + otz+ ( uu - uJ=C, (1)

perature experiment for zero stress ratio oa/ut, i.e. circumferential stress with no su- perimposed axial stress. The 150mm long tube specimen was fitted a t both ends with sleeves fixed to it with resin, and provided with ridged ring for the axial extensometer. A pressurizing mandrel made of stainless steel was passed through this specimen and the annular space between the two was sealed off at both ends with rubber O-ring. The use of these rubber rings rendered this form of specimen unsuitable for experiments a t high temperature, and for the 350°C run with oa /ot=O, a shortened version of Specimen- I was employed (effective length 70 mm). This modified form was adopted to prevent the

Equation ( 1 ) is a function representing a n ellipse on the plane coordinates of ou and ut- Since zircaloy is not subject to the yield phenomenon, Eq. ( 1) is adopted for conveni- ence as criterion representing the condition for determining the point of 0.2% plastic strain. If it is assumed that this criterion i s applicable also to plastic deformation exceed- ing 0.2%, elliptical functions representing the stress condition for producing arbitrary plas- tic strain levels of 1, 2, 3% or even more can be obtained by varying the values of the constant C. Equation (1 ) holds, independ- ently of the stress ratio, for any given value of the effective plastic strain defined by

( 2 ) z z specimen from buckling with the compressive Eeff=J$Eap + EapEtp+ ~ t p 2 ) ~ " ,

load that had to be applied to the specimen to neutralize the tensile component of the where Eeff: Effective Plastic strain stress generated by the internal pressure. But this expedient failed to completely obvi-

eap : True axial plastic strain etp: True circumferential plastic

strain, a te bending, which set in with increasing load and prevented the specimen from frac- turing under the load.

Generally, in all runs, the deformation be- havior was measured continuously until the specimen deformed up to about 576, then the axial load and internal pressure were both relieved to permit removal of the extenso- meters, and then the loading was resumed at the same stress ratio, and carried to spec- imen fracture. The values read from the curves registered on the X-Y recorders are

In analyzing the data, the numerically largest principal plastic strain cpmax was used for convenience. The two quantities E,E and cpmax give very nearly the same stress con- dition in the present instance.

In the biaxial stress experiment, the axial and circumferential strains of the tube can both be measured independently, and the plas- tic strain in the radial direction can be cal- culated from these two measured data by means of the formula -

used as input data for the computer code Eap+Etp+Erp=O, ( 3 ) "ANISOTROPY" for analysis of the experi- where crp: True radial plastic strain.

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Vol. 12, No. 7 (July 1975) 427

The results of the room temperature ex- periment are plotted in Fig. 4. Each plot represents the mean value between those ob- tained from two runs performed for each stress ratio. In the figure, the ellipse repre- senting von Mises’ criterion for causing 0.2% plastic strain is shown for comparison. If the material were ideally isotropic, the ex- perimental results for 0.2% plastic strain should coincide with the von Mises ellipse, and the corresponding data for 0.5, 1, 2, 3, 4 and 5% strain should depict ellipses around it of increasing size but of similar form. The deviation from ideal ellipse seen in the actu- ally obtained curves represents the anisotropy of the material. We shall call these curves “plastic strain loci“.

r+ CkQp6) Fig. 4 Plastic strain loci of tubes

obtained a t room temperature

In the case of Tube-Aof large tilt angle, the pattern of the plastic strain loci shows a tendency to bulge out in the direction of lower polar angles (ua/ut=0.5) particularly a t small values of deformation, but as the de- fxmation proceeds, with work hardening

acting to attenuate the anisotropy, the meas- ured plots gradually approach the true ellip- tical form. Thus, while with 0.2% off-set, the yield strength is distinctly smaller in the axial direction than in circumferential, they are almost balanced at 4% plastic deforma- tion. As the tilt angle of the specimen is reduced, the bulge of the plotted oval pro- trudes in the direction of cT,/a,=l. Also, the elongation of the oval is amplified with in- creasing plastic deformation indicating that the work hardening makes itself felt more prominently in the radial direction than in the axial and circumferential directions.

The plastic strain loci obtained with Tube -A at 350°C are shown in Fig.5. Compared with the results obtained on the same tube a t room temperature, the anisotropy is seen to be attenuated, although a t small values of plastic deformation the plotted ovals appear to retain some of protrusion in the direction of cTa/cTt=O or 0.5.

I

10 20 30 40 ct (kg/m$)

Fig. 5 Plastic strain loci of Tube-A obtained 350°C

2. Plastic Strain Ratio The value of E , ~ can be derived from the

values of E , ~ and Etp by means of Eq. ( 3 ) and consequently the deformation characteristics can be evaluated for all the three principal axial directions. In the case of an ideally isotropic material, the strain ratios in all three principal axial directions are determin- ed once the stress ratio is known.

Figure 6 illustrates the plastic strain char- acteristics of five different specimens meas-

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428 J. Nucl. Sci . Technol.,

ured under various combinations of biaxial stress. For each series of diagrams a pair of variables was chosen from the three ele- ments erp, eaP and etp, and the coordinates are scaled in such manner that in each case the ideally isotropic relation is represented by straight line with 45" inclination (the double-dotted chain line drawn in each diagram). For each row of diagrams representing one of the specimen batches A to E, -erp was chosen for ordinate for oa/ot=m, 2, 0.5 and 0. Mutual comparison of these curves reveals that the specimen shows increasing resistance to deformation in the radial direction as compared with the axial and circumferential directions, as the tilt angle becomes smaller. It can be understood that radial deformation would be inhibited as an increasing propor- tion of the texture is oriented with their c-axis close to the radial direction, since a close-packed hexagonal crystal is not liable to deform in the c-axis direction.

To /G 6 5 4

- 2 I a5 0

3 Tube-A 2 I 0

5 4 3 Tu be-B - - $ 2

E l 2 0 5

u 4 2 3 Tu be-C - LI

I ' * O

5 4

3 Tube-D 2 I 0

5 4

3 Tube-E 2 I

o o 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 3 True plastic slrain (%)

Fig. 6 Plastic strain ratios of tubes obtained at room temperature

For the stress ratio Ga/ot=l, etp and Eap

were chosen for the coordinates. This series

of diagrams indicates that, with increasing tilt angle, the specimen tends to deform more easily in the axial than in the circumferential direction. In the case of oa/ot=O, the curves are seen to present a peculiar behavior. The specimen does not show any change in the radial direction until that in the circumferen- tial direction reaches about 1.2% ; in fact, the tube wall even appears to thicken slightly with elongation. This particular phenomenon was not observed in separate experiments conducted with stress-relieved materials. With further straining to 2 ~ 3 % in the circumfer- ential direction, the curves decline toward horizontal direction, and in the cases of Tube -A--C, thereafter steepen their slope again in parabolic form. The Apparent contraction in the axial direction may be imputed to local necking.

Figure 7 shows the results at 350°C with Tube-A under the conditions of oa/ot=m and 1. Compared with the room-temperature data of Fig. 6, it is observed that in the case of ua/ut=m, the curve is somewhat closer to isotropic behavior, while for ualut=l the curve falls away from the ideal line, i.e. a t elevated temperatures the deformability is enhanced more prominently in the axial than in the circumferential direction, at least in the range of several percent plastic defor- mation.

Ture plastic strain (%)

Fig. 7 Plastic strain ratios of Tube-A obtained at 350°C

As mentioned earlier, a slight compressive load had to be applied to realize the condition of ua/ot=O to compensate the axial load gen- erated by the internal pressure at elevated

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Vol. 12, No. 7 (July 1975) 429

temperature. In this run the strain ratio curve could not be drawn with high precision on account of difficulties in obtaining steady measurement of axial elongation, but the results were nevertheless indicative of strong- er increase of deformability in the radial than in the axial direction, which tended to ap- proach the deformation behavior to isotropic character.

3. Fracture Behavior The fracture data are shown in Table 2.

In this table, the fracture stress is given in terms of engineering stress, and the circum- ferential fracture stress is calculated with the expression

PfD,O $-- 200t0, u -- (4)

where uf : Fracture stress (kg/mm2) PJ : Maximum pressure (kg/cm2)

Dia: Initial inner diameter of tube

to : Initial wall thickness of tube (mm)

(mm).

The fracture strain is given in terms of true strain. For stress ratios of 00 and 2, the circumferential and radial fracture strains Etf and c r f were obtained by measuring re- spectively the diameter and wall thickness of the fractured part, and the axial fracture strain Eaf was derived from these two values, using Eq. ( 3 ) . For u,/u,=l, 0.5 and 0, the corresponding strains were determined from the circumference and the wall thickness at the part of the specimen where its diameter swelled out to its largest value. The remain- ing Ear was again calculated with Eq. ( 3 ) .

We take as fracture elongation the largest among e a f , Etf and E,f in absolute value. Fig- ure 8 shows the dependence of the fracture elongation on the biaxial stress ratio. It is seen that stress biaxiality has the effect of reducing the fracture elongation to values as low as )$ or J.s of the corresponding value for ua/ut=O. Also, as the tilt angle is dimin- ished, the stress ratio for minimum fracture elongation would appear to shift from 0.5 toward higher ua/ut.

The relationship between the fracture elongation and the tilt angle is shown in

Table 2 Fracture data

Fracture Stress Speci- Fracture elongation (%) ratio men stress oa /o t No. (kg/mm2)

Eaf Etf Erf

A -1 60.7 62.3 -28.9 A -2 60.3 72.7 -31.8 B -1 56.7 84.3 -47.3 B -2 56.9 88.3 -48.0

oo c-1 52.1 104.2 -61.4 c -2 56.5 94.9 -50.7 D -1 52.7 92.0 -60.1 D -2 52.6 93.3 -59.2 E -1 52.9 79.6 -56.2 E -2 54.4 90.0 -63.1

-33.4 - 40.9 - 37.0 -40.2 -42.8 - 44.2 - 31.9 - 34.1 -23.4 -26.9

- - - 2 A-3 74.0 A -4 74.3 28.2 - 6.5 -21.7

A -5 71.6 A -6 71.1 B -5 72.3 B -6 72.0

1 c-5 70.5 C -6 71.0 D -5 73.2 D -6 72.7

E -6 73.2 E-5 76.0

8.3 7.6 -15.9 8.6 11.5 -20.1 7.8 8.1 -15.9 7.3 8.0 -15.3 7.9 10.1 -18.0 8.1 10.1 -18.2 5.6 8.0 -13.6 6.4 7.3 -13.4 7.1 6.9 -14.0 5.1 8.3 -13.4

A - 7 A -8 B -7 B -8

0.5

D -7 D -8 E -7 E -8

68.9 69.4 67.0 67.4 66.3 66.5 65.9 66.7 67.2 66.1

59.7 58.5 50.3 49.5 52.4 52.5 51.7 51.2 52.4 49.7

- 1.6 - 1.7 - 4.5 - 3.8 - 8.1 - 9.3 - 14.0 - 15.0 -

- 9.2

-22.6 -26.6 -28.0 - 28.2 -30.5 -32.4 -31.7 -32.5 -39.1 -36.8

19.1 14.8 17.6 18.2 25.9 27.1 27.5 27.8

20.5 -

- 17.4 - 13.1 - 13.1 - 14.4 - 17.8 - 17.8 - 13.5 -12.8

- 11.3 -

35.8 38.4 48.7 48.4 53.8 55.5 49.1 49.1 54.7 49.5

- 13.2 - 11.8 -20.7 -20.2 -23.3 - 23.1 - 17.4 - 16.6 - 15.6 - 12.7

~

Fig. 9, with the stress ratio as parameter. There is seen a general tendency for the fracture elongation to increase with decreas- ing tilt angle for the cases of u a / u t = ~ , 0.5 and 0, while the trend appears to be reversed in the case of ua/ut=l. In the particular case of ua/ut=0.5, the elongation has a maxi- mum point around 10"-15" angle.

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430 J . Nucl. Sci. Techno[.,

0 0 5 I 2 m 0 0.5 I 2 0 0 5 1 2 m 0 0 5 1 2

100

80

60

40

2 0

0 0 0 5 1 2 m

Stress ratio O‘a/dt

Fig. 8 Relationship between gation and biaxiality

absolute value of fracture elon- oJut at room temperature

I I I I I 0 10 2 0 30 40 50

T t l t angle [ a )

Fig. 9 Absolute value of fracture elonga- tion in relation to tilt angle of c-axis under various biaxial stress ratios u U / u t a t room temperature

IV. DISCUSSION

1. Hill’s Theory Hillc5’ proposed a theory of yielding and

plastic flow of anisotropic metals, by devel- oping the yield criterion for isotropic metals established by von Mises. From Hill’s theory, disregarding the radial stress-since the wall thickness of the tube is sufficiently small in comparison with the diameter, we obtain the yield criterion

F ( u ~ ~ ) = F u , ~ + G ~ , ~ + H ( u , - u ~ ) ~ , ( 5 ) where F(o i j ) : Plastic potential

F, G, H : Anisotropy constants (con- stants with which ratios among F, G and H are meaningful).

We now assume that this yield criterion can be extended to plastic strain exceeding the yield, and that differential strains are proportional to the differentiation of the plas- tic potential in reference to stress :

where d l : Positive proportional constant. By substituting Eq. ( 5 ) into Eq. (6 ) , we derive equations expressing the principal strains :

dEap=dR[H(a,- at) +Gaul dEt,=dR[Fa,+ H(cT-u,)] } ( 7 ) ~E, ,=~R[-GG~, -FF~J

by eliminating dR from Eq. (7) ,

2. Determination of Anisotropy

We will now arrange the results of the present experiment by applying Hill’s theory for anisotropic metals. There are two methods of determining the anisotropy constants of zircaloy-2 cladding tubes :

(1) To substitute the stress conditions ob- tained in the experiment into Eq. ( 5 ) .

(2) To substitute the plastic strain ratios obtained in the experiment into Eq. (8).

We first redrew the plastic strain loci by using anisotropy constants obtained by the

Constants

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second method. The result revealed that the calculated plastic strain loci do not agree satisfactorily with experimental values. We then adopted the first method, as described below.

The anisotropy constants have their sig- nificance, not in their absolute values but in the relative values. Hence in the present instance we set a t unity the value of H. With the five batches of test materials, we obtained the plastic strain loci shown in Fig. 4 for room temperature. Then, substituting into Eq. (4) the three sets of biaxial stress conditions obtained under oa/ot=m, 1 and 0, and solving simultaneous equations, we ob- tained the values of F and G. Table 3 shows the results of this analysis. The anisotropy constants F, G and H are indexes indicating resistivity against deformation of the cladding

Table 3 Anisotropy constants for plasticity of tubes a t room temperature

M ~ ~ ~ ~ m Anisotropy constants s t ra in (%) F G H 0.2 0.51 1.07 1.00 1. 0 0.58 0.84 1.00

A 2.0 0. 74 0. 78 1. 00 3. 0 0. 70 0.65 1.00 4.0 0. 72 0. 62 1.00

0.2 0. 63 0.81 1.00 1. 0 0. 42 0.51 1.00

B 2. 0 0.48 0.42 1. 00 3. 0 0. 51 0. 41 1. 00 4.0 0.49 0.36 1.00

0.2 0.50 0. 78 1.00 1.0 0. 34 0.58 1. 00

C 2.0 0. 33 0. 46 1.00 3.0 0.30 0.43 1.00 4. 0 0.35 0.43 1. 00

0.2 0.57 0.81 1.00 1. 0 0.32 0.44 1.00

D 2.0 0. 38 0. 31 1. 00 3.0 0.37 0.32 1.00 4.0 0.42 0.32 1.00

0.2 0.27 0. 49 1. 00 1. 0 0.25 0. 36 1. 00

3.0 0.31 0.26 1.00 4. 0 0.33 0.21 1. 00

E 2.0 0.29 0.29 1.00

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tube in the axial, circumferential and radial directions, respectively. That is, an increase in one of these relative values indicates a corresponding enhancement of the resistivity of the material to deformation in the relevant direction. It is revealed from the table that as the tilt angle becomes smaller, the de- formability increases in the axial and circum- ferential directions while diminishing in the radial direction. The data also indicate quantitatively that the anisotropy of the materials changes with the progress of plas- tic deformation. And once this deformation is started, deformation in the radial direction becomes rapidly inhibited, except in the case of Tube-A.

Figure 10 is a graphic representation of this trend. The plastic strain loci in the figure were drawn by substituting the an- isotropy constants obtained with Tube-B into Eq. ( 5 ) . By nature, the curves pass through the experimental points for oa/ut=w, 1 and 0, but it is noteworthy that they also repre- sent fairly well the remaining experimental points

- E E , D x I

d

for ua/ot=2 and 0.5.

0 20 40 60

6'1 ( k g / r n r n 2 1

Fig. 10 Comparison between experimen- ta l results and plastic strain loci calculated with anisotropy constants on Tube-B

Table 4 gives the anisotropy constants ob- tained with Tube-A at 350°C. This experi- ment was conducted only for ualut=m, 1 and 0, and the anisotropy constants were calcu- lated therefrom. The curves shown in Fig. 5 were drawn by substituting the anisotropy

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constants thus obtained into Eq. ( 5 ). Com- parison of the values of Table 4 with those of Tube-A in Table 3 provides quantitative evidence that the deformation behavior ap- proaches isotropy a t high temperature.

characteristics differ appreciably according to the direction. This means that, with pro- gress of plastic deformation, the deformabil- ity would be affected differently in the three principal directions of the specimen, which can be considered to be the cause of the changes observed in the anisotropy constants with increasing Dlastic deformation.

Table 4 Anisotropy constants for plas- ticity of Tube-A a t 350°C

Maximum Anisotropy constants

strain (%) F G H plastic

- - Generally considered, effective stress and

effective plastic strain are correlated by the . ~. 0.2 0.82 1. 77 1.00 equation

2. 0 0. 76 0.93 1. 00 where eeff: Effective stress 3. 0 0. 80 0. 80 1. 00 u,g : Effective plastic strain 4. 0 0.81 0. 74 1. 00 A : Constant

1.0 0.70 1.08 1.00 ~ e f f = A ~ e f f n , ( 9 )

n : Work hardening coefficient.

Substituting the anisotropy constants given in Tables 3 and 4 into Eq. ( 8 ) permits the plastic strain ratios under various biaxial stress conditions to be derived. Since the anisotropy constants vary with plastic defor- mation, we separately calculated the plastic strain ratios corresponding to the four ranges of strain, i.e., 0-1.5%, 1.5-2.5%, 2.5-3.5% and 3.5-4.5%, by applying the anisotropy constants at 1, 2, 3 and 4%, respectively, and the resulting values were joined together in succession to obtain the calculated curves shown in Figs. 6 and 7 by dotted lines. Ex- cept for the cases of ua/ut=l and 0, the cal- culated values agree well with the experi- mental results.

No explanation can at present be given to the peculiar behavior of the samples when oa/al=O, in which the cladding tube does not lose its thickness in the initial stage of de- formation. We shall here limit ourselves to noting that when a cladding tube is subjected to deformation under conditions such that aa/ot=O-which could be brought about in practice through mechanical interaction with fuel pellets, the tube thickness could some- times remain unchanged even after deforma- tion has proceeded in the axial and circum- ferential directions to the extent of several percent.

3. Work Hardening Characteristics According to the plastic strain loci shown

in Fig. 4, it is seen that the work hardening

Effective plastic strain is defined by Eq. ( 2 ) , and effective stress by

in so far as concerns isotropic materials. In the case of anisotropic materials, Eqs.

(10) and ( 2 ) would be replaced, respectively, by

Oeff =(O~-cJa(Tt+Ut2)1/2 (10)

Granted that the foregoing relationship between effective stress and effective plastic strain are applicable to zircaloy-2 cladding tube, it should be possible to express the ex- perimental results on logarithmic coordinates in the form of straight lines. Using as an example the experimental results obtained with Tube-B in the range of deformation above 176, the values of E,E and ueff were calculated by Eqs. ( 2 ) and (lo), and the values of cr& and E:E by Eqs. (11) and (12). The resulting relationships are shown in Fig. 11.

This arrangement made with the applica- tion of geff and E,R makes it necessary to substitute different values for the constant A and for the work hardening coefficient n in Eq. ( 9 ) depending on the stress ratio. Actually, however, there holds an almost linear relationship between OLE and &, so that the values of A and n do not vary very

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Vol. 12, No. 7 (July 1975) 433

much with the stress conditions.

I , , I , , , , I 1 2 5 1 2 5 1 0

Eefi ?/.I G f f ( % I

Fig. 11 Logarithmic representation of equivalent stress-strain relation- ship in case of Tube-B

4. Desirable Texture for Cladding Tube

The cladding tube is usually given a final cold reduction of 50 to 70%. Such cold work- ing produces a texture with the [lOiO], Cll303 and COO011 directions respectively parallel to the axis, the circumference and the radius of the formed cladding tube, or else with [OOOl] inclined at some angle with the radial toward the circumferential direction. Recrys- tallization annealing has the effect of rotating the texture by approximately 30" about the basal pole, which mutually exchanges the directions of [llSO] and ClOiO], and makes them parallel to the axis and the circumfer- ence, respectively. In the cladding tubes studied, a distinct directionality was observed for [OOOl], as revealed in Fig. 1, but the [llSO] direction was not found to be orientated so sharply to the specimen axis, indicating that some grains or segments had, instead, their ClOiO] direction oriented toward the axis.

There are basically two modes of plastic deformation : slip and twinning. The critical resolved shear stress is lower for slip than for twinning. The systems of deformation are: {lOiO) Ci2iOl slip, (lOi2) ClOil] twins, and-less frequently-{ 1121) [Ti261 twins un- der tension along the basal pole, and {1132} [llB3] twins under compression in the same direction'6'.

From present knowledge, it is difficult to

explain quantitatively the experimental re- sults shown in Fig. 9. We will nevertheless attempt to derive therefrom an understanding of the relation between tilt angle and fracture elongation. To simplify the discussion, we suppose a single crystal of which the [1120] and [lOiO] directions are parallel to the axis and the circumference of a tubular specimen, and with the [OOOl] direction tilted at an angle with the radial toward the circumfer- ential direction of the tube.

If the fracture elongation were determin- ed by the deformability in the stressed di- rection alone, the fracture elongation should be, in the case of oa /o t=a , independent of the amount of tilt angle, since the axial tube deformation would, in this case, be brought about essentially by means of the {lOiO} Ci2iOl slip. The actually observed data pre- sented in Fig. 9 are indicative of a more complex interrelationship between the frac- ture elongation and the texture existing in the specimen. If the material deforms in the direction of stress, it must deform also in the other unstressed directions in accordance with the constant volume law expressed Eq. (3). Hence, the deformability property in the axial direction must be affected by the corresponding properties in both circumfer- ential and radial directions. When the tilt angle is zero, the tube would shrink by slip with the least resistance in the circumferen- tial direction while the greatest resistance would be presented in the radial direction. Reduction in wall thickness must occur by the same system of deformation system as that which operates in a tube compressed in the radial direction-k. { 1122} [llS3] twin, and for this twin a tilt angle of about 22" should be the most favorable, since the CllH31 direction coincides with the maximum shear- ing direction, which is at 45" with the direc- tion of radial compression. While the de- formability in the circumferential direction diminishes with increasing tilt angle, the various effects such as described above must combine to produce the actually observed result of maximum fracture elongation at about 15" tilt angle.

This tilt angle for maximum fracture

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elongation should shift toward smaller values when the stress ratio ua/ut=O, since in this case the deformability in the circumferential direction,-which is that of the axis of stress -brings to bear a more important effect on the fracture elongation than under ua/ut=co. When ga/~.t=O.5r it would be expected that the tilt angle for maximum fracture elonga- tion would be situated at some point inter- mediate between the cases of uajut=O and 00.

The experimental results are somewhat indi- cative of such a relation between maximum fracture elongation and tilt angle, but not in a decisively evident manner.

If we suppose a hydraulic pressure super- imposed on the tube in such manner as to eliminate the stresses both in the axial and in the circumferential directions under ua/ut =1, the deformation characteristics of the tube under such condition should be equiva- lent to those of a tube compressed in the radial direction alone. Hence, as the tilt angle is increased up to go", the fracture elongation would be expected to increase. It would also be expected that the fracture elongation should show a high value at a tilt angle of about 22", if the {1132} [1123] twin contri- butes significantly to the deformability in the radial direction. Further studies are called for in order to understand the experimental results against the surmise set forth above.

Gittus"' has inferred from dynamic analy- sis that small areas of the cladding tube facing ridges and radial cracks of UO, pellets would be subjected to biaxial stresses of uu/ut=l, but this has not yet been experi- mentally verified.

Lengthwise cracks are invariably observed to constitute the failure in overpower-to-fail- ure experiments conducted by many investi- gator~'~"~'""~'. Some of these cracks have V- or X-shaped edges. The mechanism of this phenomenon may be explained as follows. The ridge and radial cracks of a UO, pellet pressing against the inner cladding tube gen- erates intensive stress concentration in a restricted area of the tube, but the stress would weaken in proportion to the distance from the center of stress concentration. Con- strained by the surrounding material, the

stress condition approaches ualut=l, which leads to generation of the V- or X-shaped crack.

The stress ratio ua/ut at the center of the crack where it initiated must in every case range between 0 and 1. We will here assume it to equal 0.5, which should be a reasonable supposition even if it may not be precisely correct. If the value of ua/ut at the center of the crack is zero, the most suitable tilt angle for the cladding tube should be smaller than Z O O , judging from the relationship between tilt angle and fracture elongation given in Fig. 9. This optimal tilt angle would be 10" to 15" when ua/ut=0.5. While there has been no conclusive study made so far concerning the stress prevailing in fuel cladding under actual conditions, it may still be suggested that a cladding tube should be less liable to fail if its tilt angle is made to range between 10" to 15" even when the stress ratio is closed to zero.

I t must be recognized however that the present results, as exemplified in Fig. 9, have been obtained on non-irradiated tubes, and it remains to be ascertained whether the same tendency can be observed in irradiated spec- imens. This is an important problem for further study.

V. CONCLUSIONS

Various combinations of biaxial stress were applied on five batches of recrystallized zircaloy-2 fuel cladding tubes with different textures, and their plastic deformation and fracture behavior was studied.

(1) The anisotropic theory of plasticity proposed by Hill was applied to the ex- perimental results, and anisotropy con- stants were obtained through the two media of plastic strain loci and plastic strain ratios. Comparison between the results obtained with the two methods proved that the plastic strain loci provide data that are more effective in predict- ing quantitatively the plastic deformation behavior.

(2) These anisotropy constants change their value with progress of plastic de-

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formation. Judicious utilization of these constants will permit the general trend of change in plastic strain ratios under various combinations of biaxial stress to be predicted by calculation.

(3) By using the effective stress and effec- tive strain defined for anisotropic mate- rials, the relationship between plastic strain and various biaxial stress condi- tions can be explained by means of the work hardening law.

(4) The experimental results obtained with Tube-A indicated that the plastic defor- mation behavior a t 350°C is more iso- tropic than that a t room temperature.

(5) The effect of differences in the com- bination of biaxial stress on the fracture elongation was evaluated quantitatively, which proved that the tilt angle of the c-axis is related to fracture elongation under biaxial stress. Based on this re- lationship, it was concluded that the material with a tilt angle ranging from 10" to 15" is the most suitable for fuel cladding tubes. Further studies are nec- essary to confirm that this can be said also of irradiated materials.

ACKNOWLEDGMENTS We express our thanks to Dr. T. Murata,

manager of the 6th Department, and Mr. T.

Iizuka, manager of the 5th Department, Hi- tachi Research Laboratory, who encouraged us with their valuable advice throughout the present study. Thanks are due to Mr. T. Moriyama for his assistance in carrying out the difficult experiments. We are also indebt- ed to Dr. T. Okubo, Assistant Professor, Fac- ulty of Engineering, University of Tokyo, and Dr. Y. Mishima, Professor, Faculty of Engi- neering, University of Tokyo, for their valuable advice in analyzing the data.

-REFERENCES-

VEEDER, J. : AECL-2660, (1967). GITTUS, J.H. : TRG-Rep.-1547(S), (1967). DJURLE, S., LYSELL, G., MOGARD, H.: Proc. 4 t h Geneva Conf., AICONF., 49/P/315, (1971). MEHAN, R.L.: Trans. ASME. D, 83, 499 (1961). HILL, R. : "The Mathematical Theory of Plas- ticity", (1956), Oxford, Clarendon. TENCKHOFF, E. : 2. Metallk., 6 3 , 192 (1972). MOGARD, H. : Presented a t CREST Specialist Meeting Water Reactor Fuel Elements, Saclay, France, 22-24 Oct. 1973. AAS, S., OLSHAUSEN, K.D., VIDEM, K . : Pre- print Int . Conf. Nucl. Fuel Performance, Lon- don, England, 15-1 9 Oct. 1973, P. 55. LYSELL, G., VALLI, G.: ibid., P. 73. ROLSTAD, E., SVANHOLM, K.: ibid., P. 77. BAIN, AS., WOOD, J.C., COLEMAN, C.E.: ibid., P. 56. LYONS, M.F., ROWLAND, T.C., WEISS, D.T.: ibid.. P. 68.

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