Some considerations regarding the creep crack growth threshold

Scripta

MtiTALI,UIt(;ICA

V o l . 18, pp. 1 1 7 5 - 1 1 8 0 , 1984 P r i n t e d in t h e U . S . A .

Pergamon P r e s s Ltd. All ri~thts reserved

SOME CONSIDERATIONS REGARDING THE CREEP CRACK GROWTH THRESHOLD M.D. Thouless and A.G. Evans Materials and Molecular Research Division, Lawrence Berkeley Laboratory and Department of Materials Science and Mineral Engineering, University of California Berkeley, California 94720 (Received

August

I.

4,

1984)

Introduction

Observations of polycrystalline ceramics subject to creep have revealed that cracks are o n l y s u s c e p t i b l e to g r o w t h at stress i n t e n s i t i e s a b o v e a t h r e s h o l d , Kth , (fig. I) (I). At stress i n t e n s i t i e s b e l o w Kth , c r a c k s b l u n t and retract. F a i l u r e then occurs by crack coalescence across intervening shear bands. Crack propagation above Kth has been extensively modelled (2-6). Furthermore, adequate correlations with growth data have been achieved, using models based on the diffusive growth and coalescence of grain boundary cavities contained in a s m a l l d a m a g e zone ahead of the g r o w i n g crack (2). H o w e v e r , the t h r e s h o l d has yet to be adequately explained. The intent of this paper is to present some considerations pertinent to this issue. A threshold invariably emerges from models which consider the growth of cracks along the boundary between elastic bicrystals (3,6). This threshold arises because matter deposition on the grain boundary ahead of the growing crack induces a constraint on the damage zone which, at the crack g r o w t h t h r e s h o l d , r e d u c e s the damage zone stress to zero. However in a polycrystalline m a t e r i a l , v i s c o u s f l o w (e.g. by d i f f u s i v e creep) r e l a x e s the constraint. Consequently a zero stress condition at the crack tip cannot be sustained indefinitely. Hence, an analagous threshold does not emerge from models of crack propagation in viscoelastic solids. An alternative threshold mechanism must therefore be sought. A p o t e n t i a l a l t e r n a t i v e m e c h a n i s m e n t a i l s c o n s i d e r a t i o n of c a v i t y n u c l e a t i o n . When cavities nucleate, the nucleation rate decreases exponentially with stress, resulting in an apparent critical stress (7). This finite cavity nucleation stress provides a natural crack growth threshold in viscous solids. Specifically, the crack growth threshold occurs when the maximum crack tip tensile stress reduces below the cavity nucleation stress. The nature of this threshold and of the associated post threshold crack growth are examined in the following analysis.

2.

Creep Crack Propogation

A d e s c r i p t i o n of crack g r o w t h a b o v e the t h r e s h o l d , c o n s i s t e n t with the p r e s e n c e of a cavity nucleation stress, requires that the growth be evaluated by incorporating a sintering stress: the stress at which cavities in the damage zone exhibit a transition between growth and shrinkage. In this section, the effect of the sintering stress on the crack growth rate is investigated, w i t h e m p h a s i s on near t h r e s h o l d b e h a v i o r . For r e a s o n s of m a t h e m a t i c a l tractability, the damage zone is assumed to be confined to a single facet ahead of the crack, consistent with damage observations during crack growth in AI203 (I). Viscous constraint on the crack tip damage zone results the damage zone, given by (2); o = o a - (/~i 2) (Fv/F~/2)( ~%)nAi/2

in a stress,

i

1175 0 0 5 6 - 9 7 4 8 / 8 4 $ 3 . 0 0 + .00 C o p y r i g h t (c) 1984 P e r g a m o n P r e s s

~

, acting upon

(i)

Ltd.

CREEP CRACKGROWTHTHRESHOLD

1176

Vol.

18, No. i0

where Oa = K / ~ is the unconstrained stress on the center of the facet in the damage zone, K is the stress intensity factor, ~ is the grain size, n is the viscosity of the matrix, F v = (2~/3)(2-3cos ~ + cos3~), F b = sin2p, ~ is the dihedral angle, and A is the area fraction of cavitated grain boundary. The stress induces a void growth rate within the damage zone, ~4F312 1

oa

I

I - (/~i 2)(FvlF3/2 )(~I~)(~Ioa)AII2A_~s(I_A)/AI/2

(2)

AII212~n(1/A) - (I-A)(3-A)]

where ~s(I-A)/AI/2 is the normalized sintering stress and a s = 2 ~/~-~Fbl/2(~/%)/K/~. The grain size to cavity spacing ratio is (~/ ~, the surface tension is ~ and ~/~ is a ratio involving boundary (Db6b) and lattice (Dv) diffusivities, ~/~ = (i + 2Dv~/~Db6b)(l + Dv~/Db6b )-I Rearranging equation (2) gives,

/ A1/212gn(l/A)-(l-A)(3-A)]+(4/2-/9~ 2)(~/X)2 (¢/OA1/21 -

( _ ~ ) °a

s<,-A)iA Ii2.....

: t

)

(3)

v

Crack growth can occur only if ~s(I-A)/AI/2 < I. Hence, the left-hand side of equation (3) can be expanded by the binomial theorem. Making the substitution A I/2 = x, the expansion gives, /2~ x n~--~----

n ~

~! r\ (-l)r r!(n-r~ x2

-4gnx + 4x2- x 4 - 3 + 49~/~ g

/4F3/2 \ =

b

~,

~

~

x

(4)

°a

Equation (4) can now be integrated in the form,

n=0

2~n r~__O r! (n-r) !

x 2+2r-n I-4 Znx-3+ ~4~-

(~I ~ I + 4x4+2r-n

6+2r-n 1

4F3 /2 (-~)3(-~)

(5)

X° tf -x

dt °

CREEP CRACK GROWTHTHRESHOLD

Vo]. 18, No. 10

i177

to give

Z 2~n r~=~0

(6) r,(n-r)!

\ ~ /

n

where F(x) -

x 3+2r-n

4~'nx

+

4 3+2r-n

4 5+2r-n

4F2+ 3 - 9~ 2

xS+2r-n

I 7+2r-n

x7+2r-n

The lower limit on the c a v i t y size X o is dictated by the crack g r o w t h t h r e s h o l d criterion. For initial purposes, the threshold Kth is arbitrarily selected (e.g. to coincide with experimental observations) and presumed to be abrupt. More detailed examination of the threshold, involving considerations of cavity nucleation rates is presented in the following section. The specific relation between X o and the threshold stress intensity factor Kth , is d e t e r m i n e d by e q u a t i n g the stress on the center of the grain Facet w i t h i n the damage zone to the sintering stress,

° a -- Kth/¢~-~ = ~s(1-Xo2)/X°

(7)

The upper limit Xf r e p r e s e n t s the c a v i t y area fraction at which facet f a i l u r e occurs. Herein Xf is regarded as a variable. However, Xf may, perhaps, be specified by the transition from equilibrium to crack-like cavity morphology (2). Upon facet failure, the crack advances by one facet dimension, ~ , resulting in a crack velocity a = ~/tf. Hence, substituting tf = I/& and recalling that c a = K/¢~--~, the crack velocity can be derived from eqn (6) as

•

La

_

(4Fb3/21 _ _

{ n~

(~)(~£) ~ 3 ~

2~ s

~

(-l)r

n'

r!(n-'r)!

IF (Xf)-

F(Xo)

]}-l

(8)

\%7~ v /

Kc~-

The resultant

relative

stress,

S , acting upon the damage zone is given by

S ~ o/[(27sF~/2/X) (1-A)/A 1/2] A I /2 -

4~

(9(9

~s I-A)] -

j

4~'2

The crack velocity predicted by equation (8) has been plotted in figure (2) to establish the e f f e c t s of i/ ) and K ~ on the crack growth. C o m p a r i s o n with the r e s u l t s p r e d i c t e d in the absence of a s i n t e r i n g stress (Fig. 2) r e v e a l s that this stress exerts a n e g l i g a b l e influence on the crack growth rate. The corresponding stresses acting on the damage zone are shown in Figure (3), as a Function of (~/~) and c a v i t y size. =If r : (n-3)/2 the first term is r e p l a c e d by +inx 2inx-3 + (4v2/9~2)(~/~)2(4/0, if r = (n-5)/2 the second term becomes ÷4enx, and if r(n-7)/2 the last term is -~nx.

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The Threshold Stress Intensity Factor

In the preceding analysis, it was assumed that a sharp threshold stress exists, below which it is impossible to nucleate cavities and hence, to propagate a crack. A simple calculation is included in this section to demonstrate the basis for this assumption. The nucleation rate (p) for grain boundary cavities subject to a normal stress, aa, is given by (7)

=

4~ 7 a ~4/3

DD6 b

(Pmax-P)

{ exp -

4¥3Fv~< OaPT +

~a~l k~--I

(I0)

where p is the number of cavities per unit area, Pmax is the number of possible nucleation sites per unit area , ~ is the atomic volume, k is Boltzmann's constant and T is the absolute temperature. This equation can be re-expressed in terms of the viscosity (2), ~,

7kT%3

{

4y3Fv

tag }

=

4~7/3 an

(Pmax-P)

exp

- Oa2kT

+

(11)

k-~

Near the threshold, a damage zone confined to a single grain is most plausible, a ~ K/~as before. With this premise, equation (11) can be integrated to give

~n

0max

=

ykT £3/~-£ 4~27/3Kn

exp

-

v

+

K2kT

t ~-kT

so that

(12) n

where t n is the time required to nucleate cavities with a density n n. Experimental observations indicate that a certain cavity density always accompanies crack propagation, suggesting that a minimum density Pn is needed to allow continuous cavity growth in the damage zone. Adopting this hypothesis, t n becomes the minimum residence time needed to observe crack propagation. Consequently, comparison of the residence time t n with the steady state crack velocity, a , eqn (8) reveals a regime of nucleation limited crack growth, which occurs when tn > /~. Nucleation limited crack growth occurs with the velocity,

a n = Z/t n or,

(13) ~a n

4fj 13K/~_ n[~maxl Pmax_P ~ ]

exp

(K~) 2kT

+

~/~-~ kT

The resulting velocity plots are shown in figure (4). The following parameters have been assumed, ~ = I0-29m3, 7 = I Jm -2, Z= I0-6m, T = 1600k, ~ = 75 °. The development of a very sharp threshold is noted, due to the inclusion of a nucleation limited regime. However, the magnitude of the threshold is several times greater than that observed experimentally (I). This discrepancy probably arises because additional stresses caused by grain boundary sliding transients aid in the cavity nucleation process (8). 4.

Concludin~ Remarks

The preceding analysis reveals that the existence of a threshold determined by the sintering stress does not influence the post threshold crack velocity (fig. 2). Considerations

Vol.

[8,

No.

l()'

CREEP {£R.X~£K (;ROWTIt

[IIRE!SItOLD

11,~!}

of the sintering stress can thus be conveniently excluded from analysis of the post threshold crack v e l o c i t y . The p r e s e n c e of a crack g r o w t h t h r e s h o l d has been predicted, based on the e x i s t e n c e of c a v i t y n u c l e a t i o n c o n t r o l l e d crack growth. A p r e l i m i n a r y a n a l y s i s of c a v i t y nucleation rates within the damage zone reveals that this threshold is relatively abrupt, in accord with e x p e r i m e n t a l o b s e r v a t i o n s (I). C o n s e q u e n t l y , at stress i n t e n s i t i e s b e l o w Kth growth becomes nucleation limited and crack blunting occurs in preference to crack growth. Acknowledgments This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, M a t e r i a l s Sciences D i v i s i o n of the U.S. D e p a r t m e n t of E n e r g y under C o n t r a c t No. DE-ACO3-76SFO0098. References I. 2. 34. 5. 6. 7. 8.

W. M. Blumenthal, "Mechanical response of ceramics to creep loading," Ph.D. thesis, Univ. of California, Berkeley, CA, 1983. M.D. Thouless, C. H. Hsueh and A. G. Evans, "A damage m o d e l of creep crack growth in polycrystals," Acta Metall. 31, pp. 1675-1687 (1983). T. J. Chuang, "A d i f f u s i v e c r a c k - g r o w t h m o d e l for creep fracture," J. Am. Ceram. Soc., 65, pp. 93-103 (1982). T~J. Chuang and J. R. Rice, "The shape of i n t e r g r a n u l a r creep cracks g r o w i n g by surface diffusion," Acta Metall. 21, pp. 1625-1628 (1973). D. S. W i l k i n s o n and V. Vitek, "The p r o p a g a t i o n of cracks by c a v i t a t i o n : a general theory," Acta Metall. 30, pp. 1723-1732 (1982). R. Raj and S. Balk, "Creep crack propagation by cavitation near crack tips," Met. Sci. 14, pp. 385-394 (1980). R. Raj and M. F. Ashby, "Intergranular fracture at elevated temperature," Acta Metall. 23, pp. 653-666 (1975). A. G. Evans, J. R. Rice, J. P. Hirth, " S u p p r e s s i o n of C a v i t y F o r m a t i o n in Ceramics: Prospects for Superplasticity," J. Am. Ceram. Soc., 63, pp. 368-375, 1980.

I - 2 # m AI~O~

E ).I--

0 ,.J W L} tJ

012

0.4

0.6

0.8

1.0

K/Kr FIG.

1

A schematic plot of crack velocity against stress intensity factor illustrating the existence of a threshold (courtesy W. B]umenthal).

1180

CREEP CRACK GROWTH THRESHOLD

Vol.

IOO

i0(

..- Y

.J

18,

No.

10

I I I I I UNCONSTRAINED

~/X :5

'E z ¢-

CONSTR AINED

/

o

/ / /

- -

WITH SINTERING STRESS

- - - - - - WITHO~IT SINTERING S T R ~ ,

Z_

\

Threshold, Kth 1£~03-1

] 2 • 103

I

I

STRESS INTENSITY

FACTOR,

I

~

K~F/Nm

~

I

I

IO

104 RELATIVE

I

FIG. 2

i

I

I

A II~

A plot of relative stress against the relative cavity size for unconstrained and constrained cavity growth for a fixed value of K/~ = 4 x 103Nm -I (Kth/~ = 2 x 103Nm-l).

28. i

SIZE,

FIG. 3

A plot of na against K/~ when A = 0.2, K h = 2 x 103Nm -I, for (~/%) = 5, i0 and

iO 6

CAVITY

I ~

I

I

I

/ CAVITY / GROWTH LIMITE[ /~

T

/

E Z

CRACKGROWTH (~d)

// // IO5

/

3 bJ

/

/

>

/

FIG. 4

/

A plot of the crack velocity against stress, showing the regions where the velocity is controlled either by cavity nucleation or by cavity growth.

CAVITY NUCLEATION LIMITED CRACK GROWTH (~6m)

/ /

5

~'Threshold', Kth 10 4

,

I

I

I

I

I I II

I

J.

IO4 STRESS INTENSITY FACTOR, K ~ ' / N m "1

I

5x 104