Strength recovery by diffusive crack healing

STRENGTH

RECOVERY

BY DIFFUSIVE

CRACK HEALING

A. G. EVAKS and E. A. CHARLES Science Center. Rockwell International,

Thousand Oaks, CA 91360, U.S.A.

(Received 22 April 1976: in revisedform7 September 1976) Abstract-Studies of the morphology and the kinetics of diffusive crack recession in polycrystalline ceramics have suggested an approach for characterizing crack healing and strength recovery. The strength recovery model is shown to be consistent with the available strength recovery data. R&urn&-Des Ctudes de la morphologie et de la cinitique de la guirison par diffusion des fissures dans des cCramiques polycristallines ont sugg&r&une approche pour caracteriser la g&r&on des fissures et la restauration de la resistance. Ce modtle de restauration de la rCsistance wt compatible avec les don&es existantes. Zusammenfassung-Untersuchungen von Morphologie und Kinetik diffusiver RiDriickbildung in polykristallinen Keramiken haben einen Weg zur Charakterisierung des RiDausheilens und der Festigkeitserholung geGffnet. Es wird gezeigt, dal3 das Model1 der Festigkeitserholung mit den verftigbaren Daten vsrtrlglich ist.

1. INTRODUCIION

The strength of cracked materials can often be partially or wholly recovered at high temperatures by a diffusive crack healing process[I-31. The crack healing, which has been most extensively studied on single crystal systems [4, S], commences with a regression of the primary crack. This regression generates regular arrays of cylindrical voids in the immediate crack tip vicinity, perhaps as visualized (Fig. 1) by Nichols and Mullins[6], then, the cylinders, being unstable. quickly dissociate into spheres [6]. Crack healing studies in polycrystals have also indicated the existence of a distinct primary crack in the early stages of healing[7], and regular arrays of spherical voids after extensive healing [3]. However, irregular arrays of cracks have also been noted at an intermediate stage [2]. Studies of the relation between the recovered strength and the crack morphology [2,7] have used

J

(i)

01 (iii)

(ii)

u(yij (iv)

Fig. I. The dissociation of a plate into cylinders (after Nichols and Mullins) [6].

an assumed or theoretical relation for the change in the diameter of the spherical voids, and obtained a time dependence of the strength by equating the sphere diameter to the crack length. The strength functions derived therefrom afford, at best, only a partial description of the strength recovery[8]. A complete strength recovery model for polycrystals requires; (a) a characterization of the detailed crack healing morphology, (b) a study of the kinetics of the various crack regression processes, (c) the measurement andior analysis of crack propagation in partially healed structures. The present paper examines each of these problems, using alumina (lucalox) as a sample material, and then outlines a generalized strength recovery model. 2. CRACK HEALWG

STUDIES

2.1 Experimental

Precracks were formed in double torsion specimens [9] by loading in air at room temperature at a constant, small (2 x 10m6 ms-‘) displacement rate. to achieve crack propagation at a constant velocity (- 10W3ms-‘). The stress intensity factor for crack propagation at this velocity was determined to be 3.6 + 0.3 MNrne3”. The cracks were extended to a length of -2 cm. The precracked samples were annealed in air at 1400, 1600 or 18WC. The changes in the crack morphology caused by healing are depicted in Fig. 2. The primary crack was observed to recede continuously, leaving isolated irregularly spaced secondary cracks several grain diameters in length (Fig. 2a). The extent of secondary crack formation was greater on the surface than in 919

930

EVANS ,G;D CHARLES: STRENGTIf RECOVERY BY DIFFILSI’X-ECRACK HEALtNG -

UNHEALED ZONE

-

-PARTIALLY I

HEALED ZONE

Fig. Z(a). A transmission optical micrograph of a partially healed crack in alumina showing the r=eded primary crack and the secondary cracks.

Fig. 2(b). Polarized Iight reflected micrographs showing the intragtanular

the sub-surface, as detected by through focusing in transmitted illumination (lucalox has the advantage of optical translucency). In regions between the secondary cracks, pardai~y healed grains, such as those depicted in Fig. t(b) were frequently observed. These grains consisted of central unheated ‘islands’, with continuous cylinders at their periphery that protruded at irregular intervals into the surronn~mg grain. The protruding cylinders terminated in spheres, and arrays of truncated cylinders and spheres occurred throughout the remainder of the grain. The quantitative dete~ination of crack recession rates by direct crack observation was not a straightforward procedure because secondary crack iormation resufted in a gradual transition from the completely unhealed region, through a large pa~~al~y healed zone of secondary cracks, to a ‘compfeteIy* healed region containing void arrays, as shown schematicaliy in Fig. 3. The difficulties were compounded by the dependence of the healing on the proximity to the specimen surface. However. for present purposes, we have arbitrarily selected a reference condition to represent the primary crack Front; that is the fine along the primary crack plane where SOYiof the crack surface is in its original. unhealed condition (see Fig. 3). Applying this condition, an average value for the crack recession along the total crack front could

healing morphokq.

be found. This choice allowed the crack recession to be traced quite accurately, except during the initial healing stage (when the partially healed zone was being deveio~ed~ The primary crack recession @,-a) obtained in this way did not occur at a constant rate but di~nished with time; a logarithmic plot of the data indicating this effect is shown in Fig. 4. The time dependence of the recession rate appeared to be essentially independent of the initial length of the crack, a*. The healing rate generally increased as the temperature increased (Fig” 4), but strong influences of the environment and of crack contaminants were also noted, which can _geatIy exceed the temperature effect. The secondary crack healing could not be traced with the same consistency as the primary crack healing. Crack recession mostly appeared to occur in a continuous fashion, at a steadily di~is~g rate; but some dissociation of the larger cracks into several smaller secondary cracks was also noted 2.2

Andy&

The int~grant~lar healing morpholoa- (summarized in Fig. 21 and available diffusion rheory[6, NJ suggest the following intragranular healing sequence. Initially, an annulus develops at the _-in periphery

EV.XNS .KXDCH.‘.RLES:

STRESGTH

RECOVERY BY DIFFCSKE

CRACK HE.ALING

-NO PHYS!CZL SEPARAT:ON

-VOIDS

PARTIALLY HEALED~+-'COMPLETE'CRACK

HEALI!dG

33x 1 REFERENCE PRIMARY CRACK FRONT (80: PHYSICAL SE?ARATION)

Fig. 3. A schematic showing the crack healing morphology in polycrystals: the reference primary crack front used for recession rate measurements is also indicated.

postulated by Nichols and Mullins)[6]; the annulus forms protruding cylinders at perturbations (i.e. grain comers. grain boundary ledges) along the grain circumference; each protruding cylinder develops a spherical tip [lo]: the protruding cylinders being unstable [6] to longitudinal perturbations of wavelength > 2;ir, (where r, is the cylinder radius) cause, in some instances, the formation of isolated cylinders; while the spherical tips [2] also being unstable form isolated spheres, at a time (to) and a size (r,,), given under steady state conditions+ of surface diffusion control by ClOl; (as

ri = B2ro,

where rr, is the sphere radius, D is the surface diffusion coefficient, ‘/ is the surface tension, R is the atomic volume. v is the number of diffusing atoms per unit surface area, k is Boltzmann’s constant, T is the temperature and p, and bz are dimensionless constants. The crack healing sequence in polycrystals GUI be derived from the intragmnular healing behavior, if the characteristic separation of the crack faces is known. When a crack forms in a polycrystal some lateral motion of the opposite faces occurs. as the local internal strains at the grain boundaries are relieved (this is especially prevalent in non-cubic materials). The

lateral motion causes a local disregister, which prevents complete crack closure when the applied forces are removed, and results in a non-uniform separation of the crack faces. The average separation (ii) of the faces of the crack (length ao) while under an applied stress, increases with distance x from the tip. and for x <
ix

,-,

where K is the stress intensity factor and E is Young’s modulus. A possible relation for the residual opening IO“ : I

/

I

_I

,()‘EL

10-a

tThe more complex problem of spheroidation Ender transient conditions is not considered herein. because we anticipate that the steady state condition will appll to most crack healing situations (i.e. with spheroidation occurring over reiat&l~ long periods of time).

K

2E \i 2n

d

I

IO"

soi

t(s)

IO'

Fig. 4. The time dependence of the isothermal crack recession in,& obtained in alumina (lucalox) at zero applied stress.

g-__

EVANS ~1’1)CHARLES: STRESGTH RECOVERY BY DIFFI;SNE

CRACK HEALWC

of rhs crack. E,. in the unload position m@hi thus be;

where ~a is a ‘residual’ stress intensification factor, n is a dimensionless constant that lies bet%-een 0.5 (the on-load position dependence) and 0 (the complete closure condition) and s0 is a no~~i~t~on constant. Further progress now requires two assumptions. Firstiy, we consider that there is an equilibrium cyfinder radius rf. for a particular grain (see Fig. 2b) that i.s reiated to the initial separation of the crack faces, /tR. such that; r, = (14R/ttO)mr*

(4)

where u0 and r* are constants with the Dimensions of length and m is a dimensionless constant. Next, it is required to assume a rate controlling step in the intragranular crack recession process (i.e. the formation of the peripheral cylinder, or the dissociation of the protruding cylinders into spheres and short cylinders). We assume that dissociation is rate limiting, but note that cylinder formation should exhibit similar dimensional characteristics fl2]. Havmg adopted these assumptions, it is now possible to rationalize the crack healing morphology observed in the polycrystals. Combining equations (if and (4) and noting that the sphere separation is proportional to the cylinder radius[lOJ, we obtain the time. rr, for complete healing of a grain (radius G) of uniform initial separation (t(a) as;

where i., is a dimensionless constant. Equation (5) predicts that the regions of closest crack surface proximity (~a) would heal first, followed by regions of larger separation, to leave isolated unhealed segments of the original crack, This would account for the secondary cracks observed in the partiali? healed zone (Figs. 2a and 3). Additionally. comparing equations (3) and (5) shows that the initial crack dissociations should occur in the vicinity of the original crack tip (small x). This accounts for the observed gradual recession of the primary crack. .An approximate functional form for the average recession rate (-m) can be obtained from equations (3) and (5). and is given by:

where 2; has been replaced by a,-n. ~te~ra~ion of equation (6) gives the instantaneous crack length;

lO+L 4.5

1

I

1

5.0

5.5

5.0

iIT(Kx104)

Fig. 5. The crack healing data at constant time plotted

as a function of reciprocal temperature. where 31is a constant with the dimensions of length, and @is a dimensionless time. This result is expected to afford an approximate description of crack healing when the spheroidization process is controlled by steady state surface diffusion. The crack healing data. plotted as the time dependence of the isothermal crack recession (Fig. 4), and the temperature dependence of the crack recession at constant time (Fig. S), are found to be consistent with this formalism such that the crack recession is characterized by;

where Q is the apparent activation energy, 65 kcai,’ mol, and El and w are constants equal to 54 and 0.6, respectively. This activation energy is, in fact, similar to the value for surface diffusion [IQ. 3. CRACK PROPACATIOS

Precracked and partially healed double-torsion samples were tested in dry nitrogen gas (which limits slow crack growth) at room temperature at a consrant displacement rate of 3 x 10m6ms-‘. Acoustic emission was monitored continuously during the tests and the crack motion was followed optically. The results of a typical test on a partially heated sample was summarized in Fig. 6(a). Acoustic emission si~ifying some crack motion was first detected at low load ievels, -0.05 P, {where PCis the maximum propagation load obtained on that sample) and the average acoustic emission rate increased as the load increased. Crack motion within the partially healed zone was first detected optically at -0.5 P,. The optically observed change in crack length, Ae, was correlated with the critical stress intensity factor, Kf by computing i(; from the load (using K = tP,

EVANS AVD CHARLES:

STRENGTH

RECOVERY BY DIFFUSIVE

CRACK HEALING

923

3.2 Analysis

I5

G x

1

to i

”

V

It

t

,

t

1 Q

Time tsl

’

8,

I

bl

rA$-PRECRACKED

mary crack should be negligibk. The observation that ItJlc, is approximately equal to unity in the partially

ZK, PARTIPLLY

.U

Y

2

ZONE

OF CRACK

HEALED

HEALING-

1

I,

I

PARTIAL HEALING 0

1 ‘COMPLETE * , /HEALING ; 03

bo(cm)

Fig. 6(a). The time dependence of the load and the acoustic emission rate obtained in the testing of a partially healed crack in a double torsion specimen at a constant displacement rate. (b) The effect of a change in crack length, Aa, on the critical stress intensity factor, K;, for a partially healed and an as-precracked sample tested at a constant displacement rate.

where 5’is a function of the specimen dimensions) (PI. The results for a partially healed sample and an asprecracked sample are plotted in Fig. 6(b). This plot shows that K: increases while the crack propagates through the secondary crack zone, but reaches a value SO.9 K, we11 before entering the ‘completely’ healed void array zone. Finally, in the latter zone, K: is approximately equal to I(,, and is thus independent of the crack increment. A functional relation for K; (a) that provides a good fit to the observed K; variation (Fig. 6b) is; K; = (Z/rc)K, tan-’ &/a*)

f%

where a* is Q constant with the dimensions of length,? for the data in Fig. 6(b) a* is 0.05 cm. This function applies when the partially healed zone has fully developed, i.e. at times in excess of a critical normalized time, 8*. For times <6*, while the partially healed zone is being developed, a pertinent expression for K: might be; K; = K* (0) + (2/n) AK (f?)tan-’ (Au/u*),

The crack propagation resistance of a partially healed structure (compared to the resistance of the virgin material) is related to the interaction between the primary crack, the secondary cracks and the void arrays (see Fig. 3). The voids are separated by a distance largerthan their diameter and thus, as shown by solutions for the interaction of two collinear cracks (see Appendix), the interaction of the voids with the primary crack should be negligible. Similarly, in the outer regions of the partially healed zone, where the average secondary crack separation exceeds approximately one-third the average diameter (see Appendix)+ the interaction of the secondary cracks with the pri-

(10)

where the sum of EC* and AK is equal to rC;, t Physically, a* is a measure of that zone of healing wherein the crack repropagation resistance is less than half that in the fully healed or virgin material.

healed and ‘completely’ healed regions (which consist of void arrays and a small density of secondary cracks) is thus explained. In the inner zone, where the average secondary crack separation is less than one-third the average crack diameter, the crack propagation resistance should decrease, and approach zero at the primary crack front. Although quantitative analysis of crack propagation in this regime is not presently possible (see Appendix), these tendencies are qualitatively consistent with the observed crack growth resistance (Fig. 6b). 4. STRENGTH The crack pro~gation healed structure is deters ship [ 141

RECOVERY stress, bg, in a partially by the usual relation-

(11) where Y is a constant that depends on the shape of the primary crack and its relative size. Substituting the functionai relation for K; derived in the preceding section and forming dimensionless q~tities gives;

,/‘I + Au/a (e < e*t

(12)

The dimensionless crack propagation stress is plotted, for @> 8*, in Fig. 7, as 5 function of the change in the normalized crack length. Also shown are approximate # values for 6 c f?* at a*/a = 0.01. It is apparent from Fig. 7 that the crack propagation stress initially increases with increase in crack length, reaches a maximum and then decreases. The fracture stress,

EVANS M(D CHARLES:

924

STRENGTH

RECOVERY BY DIFFUSIVE

CRACK HEALING

For 0 > P’, we obtain;

c; l

Further,

we

note

from

Fig.

7,

that

when

a, > a S 0.01 a*,

@* z 1 for 0 < 8*, and equation (IS) applies for all e. Otherwise, a more complex func-

0.1

t?

6

0.01 0.001

0.1

0.0 I

100

I

Aa/a

Fig. 7. The dimensionless crack propagation stress plotted as a function of the normalized crack length increment; the numbers adjacent to each curve indicate the associated values of a*/a.

0.6

oL--+-J 0.01

IO

0.1

a ‘/a

Fig. 8. The effect of a*/~ on the peak value of the dimensionless crack

propagation stress, Qt*.

of, of a partially healed structure is determined by the maximum value of di, @*:

&=

rc,

cp*.

tion pertains, which can only be evaluated by supplying pertinent information about K*(t). Applying the functional relation for 6 determined in the present study (eqn 8) and obtaining @* from Fig. 8 gives the time dependence of the strength recovery depicted schematically in Fig. 9. The principal features of the strength recovery are; (a) the kinetics should be expressible in terms of the nomalized time, 8; (b) for large a0 (small ae) a linear relation between log [l - (ao/ar)2] and log t should apply during initial recovery (with a slope z 0.6) and (c) the late stage recovery should occur much more sfowIy than the initial recovery. This latter stage should commence when the partially healed zone is approximately one-tenth the size of the remaining primary crack. Complete strength recovery is achieved when the propagation stress for the most deleterious inherent flaw in the materialt becomes less than the propagation stress for the most serious partially healed crack. This could occur during the initial recovery phase or during late stage recovery, depending on the relative and absolute magnitudes of the inherent flaws and the induced cracks. It is unlikely that the inherent flaws will be so small that fracture controlled by individual secondary cracks will be realized,: and we thus regard the above as a complete strength recovery description for most ~lyc~stalline ceramics. Strength recovery data for Al,O,[lSj, MgO[lSJ and UOZ [2,7J obtained from the literature are plotted according to the scheme suggested above in Fig. 10. The data are widely scattered, but appear to exhibit the essential features predicted by Fig. 9. The strength recovery model thus appears reasonable.

(13)

Values of @* obtained from Fig. 7 for 8 > 6* are plotted in Fig. 8 as a function of a*/~. The fracture stress can be deduced directly from Fig. 8 when a* and a are known. The kinetics of strength recovery can now be obtained by substituting a from equation (7) into (13) and by noting that a, is related to the initial strength of the precracked structure, a, by:

f STRENGTH RECOVERY

(14)

OJl t Inherent Baws such as inclusions and pores are not susceptible to significant healing and do not change sign& cantly with time. $ Fracture from void arrays is, of course, even fess Iikely.

+

l_OG

Fig. 9. A schematic of the time, r, dependence of the strength recovery predicted by the present model.

EVANS k&D CHARLES:

STRENGTH

RECOVERY BY DIFFUSIVE

(Gupta)15 20

1

,’

30

50

I loo

II 2w 3w

I 5

I 500

I 10

I 20

925

CRACK HEALING

II 30

(Gupta)" 50

100

t (mln)

1.0

,q

0.7

2

‘0

3

I

0.5

c

0.3


""2

11111 I

2

3

5

7

10

jlf3J+J-: 0.3-0

(Roberts and Wrc& I

I

20

30

"02

(Gandyopadhyry et al,'

0.2

1

I

II

2

3

I 5

10

t (hr) Fig. 10. Strength recovery data plotted according to the scheme suggested by Fig. 9. 5. CONCLUSION

AND IMPLICATIONS

Studies of crack healing and crack repropagation in alumina have provided an approach for characterizing crack recession and strength recovery in polycrystalline ceramics. The strength recovery model is reasonably consistent with the available strength recovery data. Attempts at extending this approach to include the effects of crack density [ 161 and of the stress intensity factor [17] thus seem merited. Also, closer studies of the effects of crack contaminants and of microstructure on the healing rate would provide invaluable information about the details of the healing process. Acknowledgements-The authors acknowledge the financial support obtained under the Rockwell International Independent Research and Development program, and are indebted to Dr. M. Isida for supplying the data used in Fig. 1l(a).

REFERENCES 1. F. F. Lange and K. C. Radford, J. Am. ceram Sot. 53, 420 (1970). 2. J. T. A. Roberts and B. J. Wrona, J. Am. cemm Sot. 56, 297 (1973). 3. T. K. Gupta, J. Am. ceram Sot. 58, 143 (1975).

4. C. F. Yen and R. L. Coble, J. Am. ceram Sot. 55, 507 (1972). 5. B. R. Lawn and B. J. Hockey, J. muter. Sci. 10, 1275 (1975). 6. F. A. Nichols and W. W. Mullins, Tmns. &.ME 233, 1840 (1965). 7. G. Bandyopadhyay, J. T. A. Roberts and G. C. T. Wie, J. Am. ceram Sot. In press. 8. R. Dutton, J. Am ceram Sot. 56, 660 (1973). 9. D. P. Williams and A. G. Evans, J. Testing Ecaluation 1, 264 (1973). 10. F. A. Nichols and W. W. Mullins. J. aool. Phvs. 36. 1826 (1965). 11. B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids. Cambridge University Press, London (1975). 12. F. A. Nichols, Private communication. 13. W. M. Robertson and R. Chang, Materials Science Research (edited by W. W. Krigel and H. Palmow). Vol. 3, p. 49. Plenum Press, New York (1966). 14. A. G. Evans and G. TaDDin Proc. Br. cerum SOC. 20, .. 275 (1972). 15. T. K. Gupta, J. Am ceram Sot. To be published. 16. G. Bandvonadhvav and C. R. Kennedy, J. Am ceram Sot. To be- published. 17. A. G. Evans, Ceramics for High Performance Applications (edited by J. J. Burke, A. E. Gorum and R. N. Katz). Brook Hill, MA (1974). 18. M. Isida, Methods of Analysis and Solutions of Crack Problems (edited by G. C. Sih), p. 56. Soordhoff, Leyden (1973). 19. G. C. Sih, Handbook ofStress Intensity Factors. Lehigh University (1973). I.

_

EVANS

926

CHARLES:

AVD

STRENGTH

RECOVERY BY DIFFUSIVE

determined by Isida [18]. This solution can be expressed in the form;

APPENDIX Crack Interaction

Analysis

*I

(MAJOR

OR MINOR)

APPROXIMATE

2 0.6

0.2 -

-o,-

*

j

<(ICzMINOR

0.1

t 0.2

0

I 0.3

20

K,

K = I

I 0.s

CRACK I I

I

2

I

3

$6,

__=F

where K, is the stress intensity factor at the major crack in the absence of the minor crack, of, is the half length of the smaller crack, a, is the half length of the major crack, d is the crack separation and F is tbe function plotted in Fig. ll(a).t The stress intensity factor is always larger at the major crack showing that crack extension will occur by the propagation of this crack into the minor crack (at least in the absence of blunting). This solution does not apply however to the interaction of the major crack with individual secondary cracks or spherical voids, which have an approximately circular profile on the fracture plane (see insert in Fig. tlb). The solution to this problem is much too complex to attempt herein; but some important insights can be gained from the following approximate solution. The stress intensity factor at one end of a through crack in a variable stress field is [19];

0.8 y’

c1

K

The stress intensity factors K at two colIinear through cracks in a tensile field (see insert in Fig. lla) has been

~,/a,

CRACK HEALIXG

IO

5

cl /a,

Fig. 11(a). The stress intensity factors for two collinear through cracks, K, relative to the stress intensity factor for the major crack in the absence of the minor crack, K,, plotted as a function of the minor crack size relative to the crack separation. Also shown is the approximate solution for nJui zz 0 obtained by neglecting interaction effects

ku j-;*u(y)J2~r3y.642)

where y is the distance measured from the opposite end of the crack Hence, if self interaction effects are neglected, the stress intensity factor at the minor crack can be obtained by inserting the stress field of the major crack into equation (A2). The stress a(y) is determined by the distance, r, from the major crack tip, and for r < ai [ll]; +Z&, where r is related to y by;

1.6

r = d + Za, - y.

14

Eliminating r from equations (A3) and (A4), substituting into equation (A2), and expressing the results in terms of dimensionless variables, gives for a, =Za( ; c

I.2

t-$&l

0

1.0

y’ ;

(A31

\

\

0.8

01

0.1

I

0.20.305

i

I

PENNY MINOR l&/a,=0

\

2

(5 - z)(z + If

(A4)

d3,

(AS)

where t: = 2aJa and z = (2a, - y)/d. Integrating eqn (AS) numerically gives the result plotted in Fig. 11(a). Comparison with the exact solution (Fig. ila) shows that the ap proximate result is reasonable for d/a, f 0.3. With this background, an equivalent calculation can be performed for the penny shaped secondary crack, which should provide a reasonable measure of K in the same range of d/a,. The stress intensity factor for a penny crack in a variable stress field is [19];

SHAPEO CRACK,

I

I

I

I

I

t

2

3

s

IO

x [l + 2~~(~~cornacosnB]d*d.,

(A6)

d/a, Fig. 11(b). Approximate stress intensity factors for a through crack and a collinear any-shard minor crack, compared with the stress intensity factors for two collinear through cracks from Fig. 1i(a). f We are indebted to Dr. Isida for supplying the results in the range d/a, c: 1, which do not appear in Ref. 18.

where b is the distance from the center of the crack, B defines a line of stress symmetry and c( is the ~cIination with respect to the line of symmetry. To determine K at the symmetry points (0 = 0), equation (A6) can be reduced by means of the relation; f_ Fcosna n=0

=

1 - rcosa l-2rcosufr”

647)

EVANS

AND

CHARLES:

STRENGTH

RECOVERY BY DIFFUSIVE

to:

Uv;“l_) Cl - 2(b/afcm u - (b,k@]

b db dz. (AS)

The stress, (r, is given by equation (A3) with r=d+a-bcosa.

iA91

Substituting c into equation (AS) and expressing the result in terms of dimensionless variables gives K at the minor crack as;

CRACK HEALING

927

where 0.3.