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Philosophical Magazine

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The growth of grain-boundary voids under stress

D. Hull & D. E. Rimmer

To cite this article: D. Hull & D. E. Rimmer (1959) The growth of grain-boundary voids understress, Philosophical Magazine, 4:42, 673-687, DOI: 10.1080/14786435908243264

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The Growth of Grain-boundary Voids Under Streset

By D. HULL and D. E. Rwad~~a

Atomic Energy Reeearch Establiehment, Harwell

[Received January 1, 19591

ABSTRACT The effect of combined hydrostatic preseure, P, and of b a r i d ternion,

u, on the rupture time of polycryetalline copper wire in the temperature range 400' to 600'0 hes been determined. Au the epeoime11.9 broke by intergranular freoture. due to the growth of voids along grain bounderies. Preliminary experiments indicated that the voids grow by the nddition of vacanckm under the action of the applied streee. A theory hes been developed Blpsuming that failure multa from the growth of void nuolei on the p i n boundary, whioh requires that the eotivation energy for fdm that of grain boundary *on and th8t the rupture t h e . t,. depends only on ( 0 - P ) . The experiments show that the eotivation energy in cloee to the expected value, and provided that u is conatant I, egreee with theory. Changes in u. when (0- P ) in oorntent, affect t, appreciably, end the reedta suggest that many of the void nuolei are strew induoed.

5 1. INTRODUCTION IN recent years attention has been drawn to the mechanism of failure of metala under certain creep conditions. Small voids have been observed at grain boundaries, particularly those transverse to the applied atreas when specimens are tested at high temperatures ; rupture reeulte from the growth and coalescence of these voids (Greenwood et al. 1964). This type of failure haa been observed in Magnox cans used for containing uranium fuel elements, and the present work w w undertaken to try to understand the mechanism of void nucleation and growth.

There appear to be several methods whereby voids might be nucleated on grain boundaries. Grain boundary aliding can open up a hole at the junction of three boundaries or a ledge on a single boundary. Impurity particles which lack cohesion with the lattice may act aa pre-existing voids. A dislocation pile-up breakmg through to the boundary might a h be a suitable nucleus. Of these four possibilities, all except the f h t must have vacancies supplied to them if they are to grow. Balluf3 and Seigle (1957) have shown that sufficient vacancies can be produced at transverse grain boundaries because of the tension acting across the boundary, and euggeet that this method of void growth is more likely to occur t h m that baaed on condensation of vacancies produced during strain (Mac& 1956). If the latter mechanism is operative voids would be produced by both

t Communicated by the Authors. P.M. 2x

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674 D. Hull and D. E. Rimmer on the

tensile and compressive atrains. However, no vacancies can be generated at a grain boundary by applying a compreeeive stress across it. It wm deoided, therefore; to carry out a eeriee of short time creep testa in which a uniaxial tensile s t r e ~ a and a hydrostatic pressure were applied simultaneously.

In the first past of this paper we outline a mechanism of rupture baaed on the growth of void nuolei on a grain boundary by the addition of vacancies €tom the boundary. The effeot of hydrostatic preaeure and uniaxial tensile stress on the time for failure of suoh 8 system ia odoulated. The reeulta of experiments deeigned to test this model are given and these are d i S C u s S d .

$2 . THEOBY OF RUPTIJRE BY VOID GROWTH The initial experimental results suggested that the mechanism propoeed

by BalluB and Seigle (1967) waa probably the operative one for vacancy formation. This process must certainly be occurring, r egdese of any other mode of void growth whioh may also operate, and i t permits an analysie in terms of parametere which may be estimated with maeonable precbion.

In this section we ahall derive an approximate expression for the rupture time, in order to show the sigdicant features of this model. A fairly rigorous solution for a system having a aimple geometry is given in the Appendix.

2.1. Estimate of Rupture Time The specimen is taken to be in the form of a wire and subjected to a

uniaxial streas, u, along ita length and a hydrostatio pmsure P. Void nuclei of an unspecified nature are aeaumed to lie on the grain boundaries. po will denote the radiua of a typioal nucleus and a their mean eeparation.

At the temperature at which the testa are being made (400O to 600"o) grain-boundary diffueion ia more efficient than lattice diffusion in trans- porting the surface atom of the void to the grain boundary. "his may be shown briefly by the following calculation :

The ratio of the number of atom traneferred from the void to the boundary by lattice -on to that by grain-boundary diffusion is given approximately by

where D, and D, are reepeotively lattice and grain-boundary diffusion coefficients, p ia the void radius and Sz the grain-boundary width.

D,. SZ - 3 x 10-l6 cm* eec-l for d v e r at 6 0 0 ' ~ (Hoffman and Turnbull 1961), and a a i d a r value is to be expected for copper. At the aame temperature D, (for copper) - 2 x 1O-l' cms sec-l. Even if p = 10-0 om, which must be an extreme upper limit to its possible size, the oontribution from lattice diffueion ia only 6% of the total, so that we are justified in

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ckowth of Qrain-bwndary Voids 676

neglecting it. If there ie considerable enhancement of lattice diffusion under stress, this argument may no longer be valid ; the resultti, however, seem to indioate that lattioe &ion does not occur.

In view of this i t is presumed that the atoms are released from within a ring round each void, whioh lie8 in, and has the same thickness, 62, as the grain boundmy.

The rate of growth of a void is determined by the gradient of chemical potential, Vf, in the plane of the grain boundary, since the diffusion flux, j , is given by (Herring 1950)

where k = Boltzmann'e constant, T = absolute temperature, volume, and Vf is to be evaluated at the void surfme.

= atomic

In the grain boundary, the potential is given by (Herring 1960)

f = -ao,51

where u,, is the normal tension acting acroo88 the boundary. At a point well away from any void it may be aaumed that this is roughly equal to the resultant of the externally applied stresees and ha8 a meximum value (a - P) on a grain boundary normal to the direction of u. On the surface of a void f = - 2yQ/p, where y is the surface tension of the

metal. Hence, aa a rough approximation we may take,

giving

This shows that only those nuclei for which

2.r Po' a-P

wil l grow. For voide smaller than this critical size the applied strese is not sufficient to overcome the eurfaoe tension and sintering oocurs. Bssuming that there are some nuclei which can grow, the larger ones will determine the rupture time and it then is reasonable to neglect 2y/p with respect to (a - P). The area of the void surface from which W o n tskee plaoe is 27rpSz,

so that the rate of increase in void volume is given by dV (Du8z)(o-P)~27rp" - N

dt kTa If the bubble retains spherical shape by surface diihion, we have

dp (Du6z)(u- P)Q - r v dt 2kTap

2x2

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676 D. Hull and D. E. Rimmer onthe

To obtain the rupture time, f,, this expression must be integrated between p=po and p - +.

h u m i n g that a B po this givea f lu '

It is shown in the Appendix that for an array of equally sized nuclei " ~(D,&) (o - P)O

lying on a square lattice the rupture time is given almost exactly by

t, =

. . . (11) The approximate result differs by a factor of 47T from the first term in the

bracket in eqn. (11). The effect of nucleus size, which waa neglected above, appeara in the third term and is usually negligible exoept for valuea of po only alightly greater than 2y/(u - P).

'It would appear that the square array is about the best simple periodio approximation that can be made to a random arrangement of nuclei, but in order to estimate the effect of the aaaumed spatial distribution on the result i t waa decided to oarry out an analyeis similar to that given in the Appendix, but for a single void f w h g atom into a concentric ring of grain-boundary having the same initial area, i.e. (aP-vpo2) . A close- packed array of such system is an alternative approximation that may be made to a real grain boundary. UBing a partioular set of parametem the time for failure on thie model waa about 70% of the breaking time ilsing the square lattice, The longer time taken by the square lattice i s due to the fact that the region of grain boundary aseociated with each void is of a more irregular shape than that of the other model (in which circular symmetry is ueed), and is therefore the more realistic.

It therefore seem meeonable to uae eqn. (1 1) for the interpretation of experimental reeults, with the following proviso. A randomly disperaed array of mean aeparation a should bring about fracture in a shorter time than the uniform set considered in obtaining eqn. (1 l), aa some will join up quickly and increaae the looal stress concentration. A value of u determined from an experimental result wil l therefore be somewhat less than the mean distance between nuclei.

2.2. Theoretical Predictions The eignificant features of eqn. (11) are: (i) The activation energy of

the procesa is that for grain-boundary diffusion. (ii) The quantitiea u and P only occur in the form (a - P) so that for identical epecimena at the mme temperature the breaking-time is a function only of (a-2'). If (0 - P) 2y/po the three terms on the right-hand side of eqn. (1 1) are in the approximate ratio

Takmg y = l@ dyne cm-I, a-2 x 1 0 4 cm, the first term dominates

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Qrozdh of Qrain-boundary Void.9 677

if ( u - P ) ~ 6 x 10' dyne cm-L= 750 p.s.i. In the experiments to be described this is true, although significant contributiona come from the other two terms. Thus the shape of the t, versus ( u - P ) curve is approximately hyperbolic down to (a- P) = 2y/p0, at which point the third term becomes infinite. Physically this is the point at which the applied stress is balanced by the surface tension of the void nucleus, 80 that the voids never grow. Figure 1 shows the expected variation oft, with (u- 2').

Fig. 1

I I I I

I I

$lpo (Z} Theoretical Variation oft, with (u- P ) .

$ 3 . EXPERIMENTAL DETAILS The tests were made on a relatively impure copper (99.8%) which

mntained small copper oxide inclusions in short stringers along the direction of drawing. The specimens were in the form of wire 0.020 in. diameter, which waa etched in dilute nitric acid to reduce the diameter to between 0.013 and 0.010 in. along a 1 in. gauge length. Copper beede were welded onto the ends of the wire specimens to hold them in the apparatus. The specimens were annealed at 600'0 for one hour before testing. Combined hydrostatic and uniaxial tensile streases were applied using the apparatus illustrated in fig. 2. T h e tensile streaaee were obtained using a spring D attached to one end of the epecimen C and held in a silica tube B, approxi- mately 12 in. long. This aeeembly wae iliserted inside a pressure bomb A (Fennell et d. 1958) with an internal diameter 6/16 in., capable of holdmg pressures of 6000 p.s.i. for a few h o w . All the testa were made in argon. The time to h t u r e waa meaaured by meana of an electric clock circuit

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678 D. Hull and D. E. R i m e r on the

through the specimen. The preeeure bomb waa inserted in a furnace tube and temperatures between 400' and 600'0 maintained to +3Oc. The creep strain was not measured during the test, but measurements on fractured specimens showed that the creep strain waa lees than 3 to 5%.

'she specimens were examined metallographica,lly after fracture by sectioning along their length followed by polishing on fine alumina.

Fig. 2

Diegrem of epperetue d for applying simultaneous hydrostetio preeeure 8nd b 8 i d tension.

9 4. EXPERIMENTAL OBSEBVATIONS

4.1. E'ed of Arm-hydrostatic Cmpre&ve Streaeee

An experiment waa carried out to determine whether voids were formed in specimene tested with o = P . In this cam there is no resolved strees along the tensile axis, but there are compressive streaaes normal to this axis. The apecimena were teeted a t 480"c with a tensile stress sufficient to break the specimens in less than 20min (without the hydrostatio preaaure). The apecimena did not fracture after 8 hours and subsequent micro-examination revealed that no voida had formed.

4.2. Hetallography It was confirmed that all the specimens festad had ruptured by the

formation of voids on grain boundaries. Figure 3 (Pl. 77) shows the general appearance of a specimen, the majority of voids having formed on boundariee normal to the applied strew. A few stringers of oxide particles are also visible. Figure 4 (Pl. 77) is a high magnification photograph which shows clearly the voids along grain boundaries. Voids do not form on twin boundariea, even when the boundariea are transverse to the applied stress. The photographs show that the voids in fractured specimens are cloae together and often coalesce. The eke and number of voids varied with tbe applied stress. This is illustrated in the photographs (figs. 5 and 6 (Pl. 78)) of two specimens teeted at the tame temperature, the fh t streaaed at 4730 p.8.i. and the second at 3000 p.6.i. At the high stress a large number of small voids have formed compared with a smaller number of larger voids at the low streaa.

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G-rowth of Gain-boundary Vo'ids

4.3. Activation Energy of the Proceaa Leuding lo Rupture A series of specimem was tested at a constant uniaxial tension and the

time to fracture meaeured for a range of testing temperatures between 370" and 490Oc. The results, which are plotted in fig. 7, give an activation

87 9

Fig. 7

Variation of log 1, with 1/T for a=4730p.e.i., P=O.

energy of 26 000 k 3000 calslmole. This figure comparea well with the activation energy for grain-boundary difFwion determined for eilver (20200 csls/mole) (Hoffman and Turnbull 1961).

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680 D. Hull and D. E. Rimmer on the

Fig. 8

PIKSSURL P, p.nJ.

Variation of 1, with P. Experimental points with theoretical C U T V ~ B . T=41Ooo. Fig. 0

\

I I

4000 5 0 0 b o o 0 TENSION 9 p.s.1.

Variation of t , with (I for constant values of ((I - P). (Taken from smoothed experimental resultti at 410% )

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Growth of Gain-baundury Voids

4.4. Effect of Strem (u) and Pressure ( P ) at Conslanl Temperature

'rhe variation of the time to rupture (t,) with hydrostatic pressure for a number of values of u waa obtained at 410"c and the results are plotted in fig. 8. The variation of rupture time with stress for a number of values of ( u - P ) has been obtained from these results and is plotted in fig. 9. Equation (11) predicts that the rupture time &odd be dependent only on (u - P), i.e. a variation of u should not affect the rupture time providing ( 0 - P ) remains constant. As shown in fig. 9, the rupture time alters appreciably with change in u and this is discussed below in 9 6.

68 1

5 5. DISCUSSION

I n Q 4.1 an experiment designed to test the theory of void growth by vacancy condensation under strain waa described. It waa shown that although a certain stress applied in tension would easily break the specimen, when applied in compression it could not do EO. Further, in the latter cme no visible voids were produced. This suggests that the vacancies for void growth are produced by the tension across the grain boundaries rather than deformation inside the grain. It seemed reasonable therefore to try and correlate the experimental results with a theory baaed on the grain-boundary diffusion model from which eqn. (1 1) was derived.

The activation energy of the processes leading to rupture in the tem- perature range used corresponds with that for grain-boundary diffusion rather than lattice diffusion. This supporte one of the basic wumptions of the the0 ry that the voids grow by the dihsion of atoms from the void surfrtce along the grain boundary.

Examination of the reaulta in fig. 8 compared with eqn. (1 1) shows that the theory cannot explain all the experimental results, since if (u - P) were the only stress factor affecting t, then for a constant value of (a - P), e.g. 3000 p.s.i., the breaking time should be constant, whereas at u= 4730p.s.i. t, = 2.2 x 10' MC, and at u = 6060 p.s.i., t, = 1-25 x 10' E ~ C . However, we shall consider first how eqn. (11) agrees with the effect of ( 0 - P ) on t, providing u is constant, and then examine the variation of t , with u. Figure 8 shows some observed values oft, a8 a function of P for a number of values of u. Equation ( l l ) , with which it is to be compared, contains two unknown^, a and p o . The appropriate value to use for po is difficult to determine. It is however possible to find a lower limit for it because it is known that l ,+a as (a- P)+2y/p0 (EM Q 2.2). Figure 8 shows that f , is still finite when (u - P) = 2000 p.6.i. It is unlikely that this, the smallest experimental value, is by coincidence very close to the critical one. If we take 1600 p.6.i. = 108 dyne cm-2 aa the critical value

which is reasonable. Using this figure, any one of the experimental results

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682 D. Hull and D. E. Rimmer on the

I =;;*) 6400 6ooo

giving t, for specified values of u and P wil l enable a value for a to be calcula- ted by eqn. (1 1). Thie may then be used to predict valuee of 1, for compari- son with the remaining experimental values. If results for which P = 0 are used to determine a, the term containing po is small and the assumptions made in ita eetimation have little effect on the caloulated value of a.

It waa found that for each value of (I, a different value of a is needed ta fit the experimental point at P = 0. However, if for any given u, the appropriate value of a is ued to derive the t,(P) relation, this agrees well with the experimental pointe. The full lines in fig. 8 are the theoretioal ourves, using the following valuee of a:

.-I 1-1

Table Giving the Distance Between Voida, a, for Values of Applied Teneion, u

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Growth of Qrain-bvundary Voids 6 83

In concluaion, the experiments show that the grain-boundary voids which are frequently observed to be the wu88 of oreep failure, grow by the diffusion of surface atom to the surrounding grain boundary under the action of the applied stress. It appears that many of the void nuclei are indnoed by the streas, but their exact nature is uncertain.

ACKNOWLEDQMENTS It b a pleaeure to thank Rofeesor A. H. Cottrell, Dr. W. M. Lomer and

Mr. R. S. Barnea for many useful discussions, and Measrs. D. C. Wynne and B. Hudson for help with some of the experimental work.

REFEEENOES B A ~ . , L ~ , R. W., and SEIOLE, L. L., 1967, Ada Met., 5,449. CHEN, C. W., end &cm..m, E. S., 1966, Acta Met., 4, 666. Fmmm,, J., €boom, N. H., and BARNES, R. S., 1968, Meld Tram., 25,

G m s , R. D., 1956, Act0 Md., 4,98. GREENWOOD, J. N., Mn.r.RR, D. R., and S m , J. W., 1964, Ada Md. , 2,260. -0, C., 1950, J . appl. Phya., 21,437. H O ~ M A N , R. E., and TOBNBm, D., 1961, J . appl. Phy8., 22, 634. Ma-, E. S., 1966, Tram. Amer. I&. min. (d.) Zngr.9, 206, 106.

332.

A P P E N D I X The Rupture Time for a 8qwzre Array of E q d y Sized NotcIei

Figure 10 shows a square array of spherical voids with repeat distance a and having grown to a radius p, lying on a grain-boundary normal to the direction of (I. The centre of a void is taken aa the origin of reotangular and polar axes.

In order to calculate Vf at the void surface it is necessary to define the problem more exactly than in 5 2, and we shall make the following additional aesumptions :

(i) The ring of void surface lying in the grain boundary, which acts aa a source of dii€uaing atoms has a radial width w, about equal to that of one atomio layer (see fig. 10). The atoms are released uniformly from within tbis volume.

(ii) The atoms are deposited uniformly over the grain boundary, so aa to minimize lattice strains.

(iii) The void retains its apherioal shape throughout its growth. It must be pointed out that whilst there is mechanism available (i.e. surface difhion) for doing this, a sphere is not the equilibrium shape for a void under the conditions that have been postulated. Owing to the presence of the other voids the natural shape of ita grain-boundary cross section is a deformed circle with bulges towards the oornem of the unit square.

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684 D. Hull and D. E. Rimmer on the

These become more predominant &B the void grows. In order to retain a circular cross section it is necessary to impose small restraining forces on the void. In view of the other approximations necessary in this problem i t has been thought permissible to aasume that the bubbles remain spherical, yet ignore the forces holdmg them to this shape. Although the effect of this on the h a 1 result is unimportant, it will be seen later that it produces some slight inconsistencies in the mathematics which have been accepted in view of the great simplification in the working that haa resulted.

Fig. 10

I 0

-UNIT CELL

Cross section through voids lying in the plane of the grain boundary.

Let us consider a unit cell centred on the origin and defined by 1.1 < 4 4 IY I < !la-

Vy is a meaaure of the strength of a given point aa a source of or sink for atoms. In view of assumption (i) and (ii) above it is determined (apart from a scaling factor) at all points inside the unit cell since we have,

Vy=A when p c r c p + w , . . . . . (1) Vzf=O when O c r < p ,

V f = B when r > p + w , , * (2) B(u"7+)+Aw*2?rp=O . . . . .

1 where A and B me related by the equation,

as the total number of atoms is constant.

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Growth of Gain-boundary Voids 685

But Vy has periodicity a in x , y, and may be written in the form 27r + O D + w

m-- w m-- Q

Vy= 2 2 C,,,,,oosa(mz+ny) . . . (3)

where

from eqm. (1). b = Q . w 9 p ,

(4)

27rp(ma + ns)u* C,,,,=(Bor 0 ) - a(mz + nz)ur

a8 a 27rApw J, ( 27rp(mL + nn)ua) . . . .

where the h t term is B if m = n = 0 and zero otherwise. Hence

C

from eqn. (2). Otherwiee,

- = - Cmn 2np(mr + n2)l'*) - ( - 5') Jo ( 27p(mS + na)uz

a . . . (6)

The most general integral of eqn. (3) which has the required symmetry is

B

27r cmn cos-(mx+ny)+K . . (6) f=-- a'

~ r r e (ma+nz) a where K is a conatant, obtained from the condition,

- * (7) f= -u,a. . . . . . . a,, will vary over the unit oell owing to the streaa concentration set up round the void. However, the total force over the unit cell due to the applied Btreea and pressure must be (u-P)aa.

Hence integrating eqn. (7) over a unit cell and allowing for surfaoe tension,

a[%py- (u-P)az] = I ~ : l + * f d z d y - I ~ I C f i d B d r . . . (8) -to 0 0

On physical grounds we may expect f to have the value - 2yQ/p at all pointa ineide the void. If it were calculated from eqn. (6) thia result would not be obtained becauee the restoring f o r m round the void have been neglected. - (2yQ/p) will iu fact be the mean value inside the region and for the purposes of carrying out the second integration in eqn. (8) we shall

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086

take it aa constant. value, a fact which will be used later.

D. Hull and D. E. Rimmer on the

The origin is one of the positions which haa the mean Hence

2yQ C?[2npy - (U - P)ua] = Kd + - . mpa P

giving K = - O(U-P).

Beeuming a radial flow of atom at the void eurface which is reaaonable if p a the diffusion flux is given by,

j=-A( 'f) kTQ 5 r-p+w*

If we let 1-0 and evaluate (a!/&-) at r = p it must be remembered that (af/&-) is now diecontinuous at this point (see fig. 11) and the value required is the maximum (numerically) which is equal to twice that obtained by computing the Fourier s u m at r =p.

Fig. 11

1 _c r

Approximate radial variation of grsdient of chemical potential.

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Qrowth of Qrain-boundary Voids 687

The number of atom, per second, coming out of an elementary area pddSz of void is,

2 ~ , p d e s ~ af kTQ (5); -

Therefore, the r a b at which atom leave the void ia given by,

The coefficiente Cmn are not known, but C,,,,/B is given in eqn. (5). However, (LB strrted earlier the value off is known at the oentre of the void.

Substituting

f- - when z==y=O P

into eqn. (9) givee

-= dt kT a

where

The rate of change of void radius is, d p = - R .-= dN 27rn(D,Sz)(u - P - 2 y / p ) S dt h p ' dt kTap

The time to h t u r e ie obtained by integrating this expression between p = po and p = &z. S ie a function of p/a only, and haa been daulated for a set of values in the expeoted range of integration. These suggest that the approximation S = 1 may be used BB it only deviates sigdhantly from this value for a short time just before the voide oodeeoe.

Thie givee,

where t, = time to failure. On integrating and wuming po 4 a, we obtain

. . . (11)

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