precdicting corrosion crack rates growth

International Journal of Fracture 23 (1983) 213-228 © 1983 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands

Predicting stress corrosion crack growth rates when linear elastic fracture mechanics conditions are not operative

E. SMITH

Joint Manchester University/UMIST Metallurgy Department, Grosvenor Street, Manchester M1 7HS, UK

(Received September 28, 1982; in revised form March 16, 1983)

Abstract

The growth rate of a stress corrosion crack in situations where linear elastic fracture mechanics (LEFM) conditions are not operative is predicted on the basis that the crack tip opening angle (CTOA) is related to the growth rate d c/d t, the functional relation between the CTOA and d c/d t being obtained by coupling theoretical results for crack growth under small-scale yielding conditions in an inert environment, with the experimentally determined power law relation between the crack tip stress intensity K and dc/dt for environmentally-assisted crack growth under LEFM conditions. Then, by assuming that the same CTOA-dc/dt relation applies to non-LEFM conditions, and by determining the CTOA under these conditions, it is in principle possible to predict the stress corrosion crack growth rate under non-LEFM conditions. A specific model: the plane strain deformation of a solid with two symmetrically situated deep cracks, and with tension of the small remaining ligament, is analyzed in detail, and the effects of the extent of plastic deformation and loading pattern (i.e., displacement or load control), on the predicted stress corrosion crack growth rate, are examined in detail. The results are compared with those obtained via application of the K versus dc/dt relation, its conversion to a J versus dc/dt correlation and then the determination of J, and also with those obtained on the assumption that a K approach is directly applicable. The extent to which these latter approaches give conservative or non-conservative growth rate predictions when compared with the present paper's predictions, is discussed in relation to the extent of plastic deformation and loading pattern.

1. Introduction

If a stress corrosion crack is found dur ing the in-service inspect ion of an engineering component , it is impor tan t to be able to predict its subsequent growth rate under the operative loading condit ions, unt i l it reaches the critical length for uns table propagat ion.

The subsequent in-service inspect ion schedule can then be p lanned in a rat ional manner . At present, the generally accepted procedure for making such a lifetime predict ion is to use l inear elastic fracture mechanics (LEFM) laboratory experimental data that relate the crack tip stress intensi ty K with the crack growth rate v = d c / d t in an env i ronment that simulates, as closely as possible, the actual service environment . This procedure should be adequate if L E F M condi t ions are operative in service, but there are many practical

s i tuations where the stress levels are so high in relat ion to the material 's yield stress that L E F M condi t ions are unlikely to be operative. Examples are (a) the growth of cracks in the vicinity of key-ways in turb ine discs, and (b) the growth of cracks in AISI Type 304 stainless steel pipes in boi l ing water reactor coolant systems. For such cases, it is clearly desirable to have a soundly based methodology for predict ing in-service stress corrosion crack growth rates, so that a rat ional run-ret ire decision can be made if a crack is found dur ing in-service inspection.

One might, of course, disregard the fact that L E F M condi t ions are not operative and

213

214 E~ Smith

use the K approach as if they were operative. Alternatively one might use the K versus dc/dt relation, convert it to a J versus dc/dt relation, and then determine J for the situation under consideration. However, these approches do not satisfactorily address the growth problem, and against this background the author [1] has developed a methodology for predicting the growth rate of a stress corrosion crack when LEFM conditions are not operative. The basis of the methodology is that a steady-state environmental condition is assumed to exist in the vicinity of the crack tip and that the growth rate dc/dt depends on the crack tip opening angle (CTOA); this latter assumption is essentially equivalent to a correlation between growth rate and crack tip strain rate which has resulted from recenl mechanistic studies [2,3] of the stress corrosion cracking process. A functional relation between the CTOA and dc/dt is obtained by coupling theoretical results relating K with the CTOA for crack growth in an inert environment under small-scale yielding conditions, with the experimentally determined power law relation between the crack tip stress intensity K and dc/dt for environmentally assisted crack growth under LEFM conditions. Then, by assuming that the CTOA-dc/dt relation also applies to non-LEFM conditions and by determining the CTOA for the non-LEFM situation under consideration, it is in principle possible to predict the stress corrosion crack growth rate for that situation.

Earlier work [1], has focused on the model of an edge crack in a solid subjected to a sufficiently high sustained stress that plastic deformation is extensive, though contained: with this particular model, J increases during crack propagation. The growth rate predicted by the aforementioned procedure was shown to be greater than that obtained by application of the K versus dc/dt relation, its conversion to a J versus dc/dt correlation and then the determination of J for the particular case under consideration; the predictions are even more nonconservative if K values are determined at high stresses, assuming that a K approach is directly applicable. There are a limited number of experimental data suggesting that an LEFM approach underpredicts the growth rate at sustained high stress levels, and thereby support the general conclusions of this early work's analysis. For example, experiments [4] on iodine-induced stress corrosion cracks in internally pres- surized Zircaloy tubes show that the measured KEscc value (the limiting threshold value of K for growth) at close to yield stress levels is only one-third of the value as measured for long cracks at low stress levels in DCB tests (LEFM conditions). By implication, it is therefore expected that an LEFM approach underpredicts the growth rate at high stress levels, in agreement with the early work's theoretical predictions.

In the earlier paper [1] it was suggested that the conclusion might have to be modified for situations where J does not increase during crack growth, and this particular point has been followed up in more recent work. Thus by inspecting the general relation between J and crack growth rate that is obtained by the procedure, the author has focused [5] on the extent to which experimental K-dc/dt data (supposedly obtained under LEFM conditions) are unique with regard to loading mode: increasing K, constant K or decreasing K. Uniqueness should be observed when the operative stress levels are low in relation to the yield stress, and this is often the case with high yield stress materials [6]. On the other hand, with low yield strength materials, it was shown that there ought to be differences in the K-dc/dt data for different loading modes because strict LEFM conditions are not really operative; the K-dc/dt curves should be higher for the increasing K loading mode than for the decreasing K loading mode. It was suggested that this might be the case at elevated temperatures when the' "effective" yield stress of a material is lowered by time-dependent deformation. In this context, experimental evidence [7] for 304 stainless steel tested in a corrosive environment at - 3 0 0 ° C shows that the measured crack tip stress intensity, for a given crack growth rate, is appreciably higher when experiments are conducted in a decreasing K rather than an increasing K loading mode. Other recent theoretical work [8] addressing the non-increasing J situation has considered the specific

Predicting stress corrosion crack growth rates 215

model: the plane strain deformation of a solid with two symmetrically situated deep cracks, and with tension of the small remaining ligament, for the case where there is general yield of the ligament. The loading points are situated on the central axis which bisects the ligament and the relative displacement of the loading points is maintained at a constant value during crack growth. With a plastic rigid model for the solid, J retains a constant value during crack growth, and using the predictive procedure described in this Introduction, it was shown that the LEFM procedure (i.e., direct usage of K values) in fact overpredicts the crack growth rate in this particular situation.

From the theoretical work just described, which is supported to a limited extent by the referenced experimental data, it is quite clear that as regards the growth rate of stress corrosion cracks at high stress levels, the extent of plastic deformation and loading pattern, i.e., displacement or load control, have a very significant effect on the degree of conservatism of LEFM procedures, and this should be appreciated when using lifetime prediction procedures. It is against this background that the present paper describes the results of a general study of the model of a solid with two symmetrically situated deep cracks, and with tension of the small remaining ligament. Using analytical results for the magnitude of the J integral and the load point displacement, it is possible to examine the effects of loading pattern and the extent of plastic deformation on the stress corrosion crack growth rate, all the way through from small-scale yielding, extensive though contained yield, to beyond general yield; the interplay between the degree of plastic deformation and loading pattern can therefore be examined. Again the predictions are compared with those obtained via the J - K - d c / d t correlation approach, and also with those obtained on the basis that a K approach is directly applicable. The extent to which these latter approches give conservative or nonconservative growth rate predictions are discussed in relation to the extent of plastic deformation and loading pattern.

2. Preliminary theoretical background

Following earlier work by Rice and Sorensen [9], Rice, Drugan and Sham [10] (hereafter referred to as RDS) have investigated the inert environment growth of a crack in a non-work-hardening plastic elastic material, under mode I plane strain small-scale yielding conditions. The results of an asymptotic analysis show that the crack tip stress state approximates to the classic Prandtl field, and they obtain an expression for the opening displacement at a distance r behind the growing crack tip. Coupling this expression with the criterion that a critical opening 8 c be maintained at a small characteristic distance r m

behind the tip (S J r m - O is essentially the crack tip opening angle CTOA defined with respect to the measurement position rm), RDS derive a crack growth equation

d J %0 flOo 2 [ rm ] = + l n , / (1)

de a a E ~ - ~

where c = crack length, a 0 = yield stress, E = Young's modulus, J denotes the far-field value of the J integral, and fl is a constant having the value 5.08 when Poisson's ratio ~, = 0.3. The asymptotic analysis does not give the value of the parameter a, though comparisons with finite element results suggest that a - 0.65 is approximately the same for static and growing cracks. Furthermore, R was shown to scale approximately with the plastic zone size, being about 15 to 30% larger, i.e., R = ~ E J / o o 2 with • - 0.23. RDS argued that the growth condition (1), as well as being valid for the small-scale yielding case, should also be valid for those highly constrained geometrical configurations where the Prandtl field is likely to be maintained, e.g., for larger-scale contained yielding, and also for some general yield states such as plane strain bending with a deep crack and for tension of a solid with two symmetrically situated deep edge cracks (this particular model

216 E. Smith

will be considered in detail later in this paper), though the values of R and ~ may be different for these various cases.

While the RDS growth criterion is based on a constant crack tip opening angle, RDS have noted that the criterion is also equivalent to the requirement that all points closer than a small characteristic distance r,,, above and below the crack tip should accumulate a plastic strain equal to or greater than a critical value as the crack approaches. Further- more, and most importantly from the present paper 's perspective, a growth equation similar to (1) is obtained if growth criteria analogous to those used in conjunction with the RDS model are coupled with a Dugdale-Bilby-Cottrell-Swinden (DBCS)-type modeL. [! l, 12] Thus, consider the general DBCS model, not necessarily restricted to the small-scale yielding case, where the tensile stress within the line plastic zone ahead of a crack has the value Y representative of the material 's tensile yield stress. Irrespective of the plastic zone size and any geometrical parameters inherent in the model, if as a criterion for continuing crack growth it is required that a constant CTOA O, measured with regard to a distance t;,, behind the tip, be maintained during growth, or it is required that a point within the line plastic zone at a small characteristic distance r m accumulates a displacement 8, as the crack approaches (this is Wnuk's final stretch criterion [13]), the ensuing differential equation for crack growth is [13,141

d J 4(1 - ~2) ln{ r,,_z, t dc OY + ~-~ s [ {9),

with 0---3,/r,,,, while s is a distance parameter which depends on both the extent of yielding and any geometrical parameters in the model, and enters into the expression for the crack opening 6 at a distance r behind the tip of a stationary crack length of c:

4 ( 1 - ~2)Yrln( ~ ) ~ = II) '[ ' lP(( ') + ; E f ~ ' (3)

where ~-Hp(C) is the crack tip displacement in the DBCS model. The similarities between Eqns. (1) and (2) are obvious, with s in the DBCS model being analogous to R in the RDS model; for the small-scale yielding situation s is linearly related to J and the plastic zone size, as also is R in the RDS model. The final stretch criterion applied to the DBCS model is analogous to the accumulated strain criterion for the RDS model, and has important physical significance, since if r,, is envisaged to be the size of the fracture process zone within which decohesion processes are operative, the criterion is equivalent to imposing the condition that a critical number of dislocations be emitted into the material during the fracture of a material element; such a criterion has obvious physical appeal. Though the DBCS model is highly idealized, because the resulting growth equations are similar to those obtained via the more realistic RDS model if analogous growth criteria are used. and also because the concept of growth being associated with the emission of a critical number of dislocations is physically reasonable, the idea of plastic growth being associated with a constant CTOA, which is equivalent to the accumulation of a critical strain or the emission of a critical number of dislocations per element of crack advance, has credibility. Furthermore, use of the DBCS type model for investigating plastic crack growth would appear to be justifiable, particularly for those geometrical configurations where high constraint is maintained.

In extending the preceding model to stress corrosion crack growth, the author [1] has proceeded from the basis that the magnitude of the crack tip opening angle (CTOA) 0, which is governed by the detailed fracture processes operative within the immediate vicinity of the crack tip~ is related to the growth rate v = d c / d t , with 0 decreasing as t:: decreases. RDS have also suggested that 0 decreases in the presence of an aggressive environment, while a correlation between growth rate and CTOA is also essentiall\'


equivalent to the correlation between growth rate and crack tip strain rate that has resulted from recent mechanistic studies [2,3] of the stress corrosion cracking process. This follows as a result of the RDS conclusion that a CTOA criterion is equivalent to an accumulated strain criterion. For then a CTOA-growth rate correlation implies an accumulated strain-growth rate correlation, and if "~ is the accumulated strain, the strain rate of the material near the crack tip is - yrm/v, whereupon it follows that there is a correlation between the growth rate and the crack tip strain rate. In physical terms the assumption of a correlation between 8 and v implies that crack advance is accompanied by the emission of dislocations from the crack tip region into the surrounding material; the more severe the environmental attack, and the more time that is allowed for this attack to proceed (e.g., the lower the crack speed), the smaller will be the number of dislocations emitted for each element of crack advance. Implicit in the use of a unique 8 - v relation is the assumption that a steady-state environmental condition is maintained in the vicinity of the crack tip during growth. If such a condition does not exist, the approach might have to be modified appreciably, and this could involve the use of bounding procedures based on specific environmental limits.

The governing relation between the CTOA and crack growth rate will now be obtained by coupling the theoretical results relating the crack tip stress intensity K with the CTOA for crack growth in an inert environment under small-scale yielding conditions, with the experimentally determined power law relation between K and dc/d t for environmentally assisted crack growth under LEFM conditions. In obtaining this relation the theoretical results from the DBCS model will be used, since this type of model will be used later for investigating the behavior of a solid with two symmetrically situated deep cracks. For small-scale yielding, the parameter s in relation (2) is equal to [13,14] ~reEJ/2(1 - i, 2)y2 _ ~reK2/2y 2 whereupon relation (2) becomes, after introduction of a/9 - v relation,

K2 2Y2rmexp{ ~rEO(v) -~r____K ~re , 4 ~ - ~ Y exp 2Y 2 -~c (4)

Relation (4) shows that K varies with crack speed; however, the K - v relation is unique only if the expression within the second exponential bracket is small. This will be the case at low applied stress levels as the following example clearly shows. [15] With a sustained stress test where a constant stress o is applied to a semi-infinite solid containing an edge crack of depth c, K - o v a - and the term within the second exponential bracket is equal to -'/r2o2//4Y 2, which is small if o << Y. In such cases, i.e., strict LEFM conditions, the relation between the crack tip stress intensity K and the crack growth rate v = dc /d t reduces to

2Y2rm eXPTre 4 0 _-- ;i-~ y K 2 -

and the relation between K and dc/d t is then unique in the sense that it is independent of the applied loadings and the crack size except, of course, through their coupled elects via the K parameter. This result emphasizes the importance and usefulness of laboratory K versus d c / d t data provided they are obtained under strict LEFM conditions. Such experimental data generally display a constant growth rate stage II regime which is often interpreted in terms of a controlling process involving the rate at which aggressive chemical species affect the material in the crack tip region. This is preceded by the stage I regime where, in contrast, there is a wide variation in crack growth rate for only a small change in K. Within this stage I regime, the K - v relation assumes the form

v =AKm (6)

where A and m are constants. Stress corrosion crack growth is governed by this relation

218 E. Smith

until K reaches Kp, when the stage II plateau regime commences, i.e., the law is valid for 0 < v < v e where v e is the plateau velocity. It follows from (5) and (6), by elimination of K. that the relation between 0 and ~ is

O ( v ) _ 4 ( l - v 2 ) y , [ ~e /v)2.. , , , rr-E- ,n 1 2y2r,~,t ~- } (7)

and this will be valid until v attains the plateau velocity cv; during the plateau regime. 0 will increase beyond the value 0 e, given by substituting v = % in (7), that it attains on reaching the plateau. (It should be noted that the power law description given by (6) is not strictly correct at very low velocities since it predicts a zero threshold stress intensity, and this conflicts with experimental evidence; nevertheless it is adequate for the purpose of the present study).

Equation (7) gives the variation in the CTOA with stress corrosion crack growth rate c, and this equation will now be used to predict the growth rate of a stress corrosion crack in situations when LEFM conditions are no longer applicable. In other words, it is presumed that the 0 - v relation is independent of the extent of yielding, an assumption whose analogue in the inert environment crack growth case is that the CTOA is independent of the extent of yielding. Basically, the approach is to determine the parameters d J / d c and s in relation (2) and thereby obtain 0, whereupon use of relation (7) gives the crack tip velocity for the particular situation under consideration. Indeed, elimination of 0 between relations (2) and (7) gives v in terms of the parameters d J / d c and s. i.e.,

By way of contrast, if the stress corrosion crack growth rate is predicted on the basis of the power law relation (6), and the K - J conversion formula K 2 = E J~(1 --tQ), the resulting growth rate v~j is

EJ /2

vKj = A { -, 1 - v ' f (L))

while the crack growth rate VLF obtained on the assumption that the K approach is directly applicable, even though LEFM conditions are inoperative, is given by relation (6). Inspection of relation (8) immediately shows that there are significant differences in the crack growth rate predicted via the present paper's procedure and the growth rates obtained via LEFM procedures. This paper's procedure highlights the role of the parameters s and dJ/dc, particularly the latter since it is involved in the exponential bracket, i.e.. the gradient of J, as distinct to J itself, plays a crucial role when crack growth proceeds under non-LEFM conditions. Of course, in the LEFM case, where the applied stress levels are low, the procedures lead to the same results, as the simple analysis for the sustained stress example has already shown.

3. Analysis of the model of a solid with two symmetrically situated deep cracks with tension of the remaining ligament: contained yield case

In predicting stress corrosion crack growth rates so that accurate account may be taken of the loading pattern, i.e., displacement or load control, and also the extent of plastic deformation, the present section investigates the plane strain deformation of a solid (Fig. 1) with two symmetrically situated deep cracks, and with tension of the small remaining ligament, for the case where yield is contained. A load P is applied at a point a great distance from the ligament along the central axis which bisects the ligament, and A is the


P' AT

< 2h ~-

Figure 1. The plane strain model of a solid containing two symmetrically situated deep cracks. A load P is applied to the solid at a great distance from the ligament, A being the load point displacement. The solid is loaded in series with a linear spring of compliance C M such that the total load point displacement is Ar=CMP + A.

load point displacement• The solid is loaded in series with a linear spring of compliance CM, which can be identified with the testing machine compliance; the total load point displacement is therefore A r = CMP + A. As indicated in the preceding section, this is a highly constrained geometrical configuration where the Prandtl field is maintained, and for which the RDS model, and thereby the simulation DBCS model, are especially appropriate. The objective of this section's analysis is to use analytical expressions for d J / d c and s, substitute them into (8), and consequently predict the stress corrosion crack growth rate.

In analyzing the model, the results [16] for a periodic system of coplanar cracks in an infinite solid will be used, the solid being cut along vertical surfaces so as to give the model of a solid of total width 2h containing two symmetrically situated cracks of depth c, the solid deforming under plane strain conditions due to the application of an applied tensile stress o (this cutting procedure will be exact for the analogous mode III model)• Using the DBCS representation of yield, the results [16] for the contained yield situation show that s and J are given respectively by the expressions

[sin2(~-~ ) -sln'z{~re(~ )J ' ] tan(~-~ ) s = 2ec (10)

• 2 { ~ra ~rc

and

~ ~ 2 ~ h Y s i n ~ sin(x + xI,) } j = 8 ( l f f / 2 cosx __ ln -- dx (10 rr2E "* ~/1 - sin2a sin2x sin(x ~t')

where (a - c) = Rp is the size of the plastic zone at each crack tip and a = ~ra/2h is given by the expression

sin(° ) sin a - cos = sin q' (12)

220 E. Smith

For the special case where the cracks are deep in comparison with the solid width, relations (10) and (11) simplify to

2eL s (13)

( l + ' y 2 )

and

J = 8 ( 1 - v 2 ) Y 2 L In + In ~ (14) ~rE ¢ ] + y2 + 72 , Y

where 2 L is the ligament width and

( 1 - t ) (15)

V'l - ( 1 - t ) -~

with t being the ratio of the plastic zone size ( R , ) at each tip to half the ligament width (L) ; thus, for small-scale yielding, t ~ 0 while "~ -~ ~ , and t -~ 1 and "~ ~ 0 as the general yield state is approached. Applying these results to the case where a load P is applied at a point a great distance away from the ligament, and along the line of symmetry which bisects the ligament (thickness B), relations (12)-(15) give

with

while

and

(17)

J = 4(1 - v2)Y 2 u(X) : 4(1 - v2)y2L erE wE [(1 + )k) In(1 + X) + (1 - X) ln(l - )k ) ] (19)

It should be noted that )t is small for small-scale yielding conditions while X = 1 at general yield. Though relation (t9) gives J, it does not, however, give d J / d c , since this depends on the loading pattern; d J / d ¢ will now be determined.

The load point displacement A can be separated into elastic components AEL and Ap/ with AEL being that displacement which is produced by the load P when there is no plastic deformation; AeL is equal to A - AeL. Similarly, the J integral can be separated into elastic and plastic components JEL and Jm: Using the same dimensional arguments as Paris, Ernst and Turner [17], ApL can be expressed in the equivalent functional forms

a~ ,~ : LgIX ) t X= f ( ApL/L ) t (20)

where ~ = P / 2 B L Y . It is possible to represent ApL in this simple functional form since L and B are the only length parameters associated with the solid's geometry, if the load is applied at a great distance from the ligament, and the ligament size is small. Now Jm can be expressed in the form [17]

JPL- - f a ' I 3P dApc 1 fa,,, OP A ....

s = 2eL)t 2 (18)


where A c is the crack area and 8A,. = - 2BSL, and consequently the second of relations (20) gives

2 ] (22)

whereupon the first of relations (20) then gives

Jet. ag(a) [Xg(e)d e (23) 2LY 2 Jo

Thus, if

foXg(e)de = I (X) '

or (24) d / ( X )

g ( X ) = dX

Eqn. (23) reduces to

d l 21 JpL (X) dX X LYX (25)

which integrates to give

X 2 fXJpL(e)de I = ~ -1o ~5 (26)

Now expression (19) gives the value of the J integral, and because the elastic component JeL is

4(1 -- v 2) y2LX2 (27) JEL = rrE

it follows that the plastic component JVL is given by the expression

4(1 - vZ)y2L [(1 + 2~) ln(1 + )~) + (1 - )~) ln(1 - ),) - X 2 ] (28) JP L = "IrE

whereupon substitution in relation (26) gives

I 2(1 ~ _ ; 2 ) Y [ 3 x 2 _ ( 1 +X)21n(1 + ) l ) _ ( l _ X ) 2 1 n ( l _ X ) ] (29)

and relations (20) and (24) give

dI 4 ( 1 - v 2 ) y L [ 2 ~ - ( I + X ) I n ( I + X ) ] a P L = L g ( ~ ) = L d x = ~r2 + (1 + ) ~ ) I n ( 1 - 2 l ) (30)

The elastic component AeL of the load point displacement of the solid is given by the expression [ 18]

< _ - ( ' -" '" - 7 %- (31)

whereupon the total load point displacement is given by relations (30) and (31) as

A r= CMP + AeL + AEL

4(1 - v2)yL [ 2 X - (1 + X) ln(1 + X) + (1 - X) ln(l - X)] = CMP+ wE

[ ~ h 4 I n ( T r L ) I ( 1 E - ~ ) P + - -- (32)

222 E. Smith

Now if stress corrosion crack growth proceeds under constant total load point displacement conditions, i.e., displacement control, then d A z / d L = O, or

dP PdC L. dAez d ==0 t33)

with C c being given by relation (31), while

d~L~ = g(X) + L . + ~ , , (34)

using relation (30) and noting that X = P / 2 B L Y . Elimination of d A e z / d L between (33) and (34) gives

[ ~ L d g ( X ) l d P Qvt + (-l:: q P d X -d-£ +

and since relation (19) gives

- +d57 [1 = 0 (35)

~rE d J - u( )t ) + L + ~ ~-£ 4(1 - v2)Y 2 dL

(36)

elimination of d P / d L between relations (35) and (36) gives

~rE d J 4(1 - v2)Y 2 dc

~rE d J 4(1 - v2)Y 2 dL

, dg( ~ ) PdC v ] LX du(X) . (2t) +X d u ( x ) P dX [ XL d g ( X ) ] dX

137)

whereupon substitution for the functions u ( )t ) and g ( )t ) via relations (19) and (30) gives

~rE d J 4(1 - v2)Y 2 dc

- l n ( 1 - X2)

. { 1 +X • ] i l +X ~rE PdC, I ~ ' ln[ l - - i -L~)[2X-ln[ l - - -Z~)~ 4 ( l _ v 2 ) y ~ T f ]

+ (381

- X l n ( l - X a ) - } 4 (1-~Ez)Y-~( M + C ~ ) j

With this expression for d J / d c and with expression (18) for the parameter s, it is in principle possible to determine the stress corrosion crack growth rate by making the appropriate substitutions in the general growth equation (8).

Rather than determining the actual growth rate, however, it is more instructive to compare the rate v with that velocity VKj (see Eqn. (9)) predicted on the basis of the power law relation (6) and the K - J conversion formula, i.e.,

~reEJ exp 4(1 5f fZ)y2 dc (39)

this result being obtained from relations (8) and (9). It then follows by substitution using


(18). (19), (31), and (38) that

vKJJ (1 - X2)[(1 + X) ln(1 + ~,) + (1 - ~,) ln(1 - )~ ) ]

1+xi\2 -(In(l-~JJ (40) (c . + ca) × exp - l n ( 1 - X 2)-~ 2 ( 1 - p 2 )

with C E being given by (31). This relation allows a comparison to be made between v and v/<s for different loading patterns and for differing amounts of plastic deformation within the contained yield regime. For small-scale yielding when ~ is small, it is immediately seen that V/VKj tends to unity, irrespective of the specimen and machine compliances C a and C M. This is entirely in accord with the viewpoint expressed in the preceding section, where it was shown that the stress corrosion crack growth rate can be characterized by J(K) if small-scale yielding conditions are operative.

As plastic deformation becomes more extensive though still contained, i.e., as X increases, the effects of the various parameters on the predicted growth rate can be assessed by expanding the right-hand side of (40) in powers of )~ whereupon it follows that

1 - - = I + X 2 - )

t vKj j EB(CM + Ca)

+ (higher powers of X) (41)

Since vKj does not depend on d J/dc, inspection of relation (41) immediately shows that the crack growth rate v for load control conditions (C g = o0), is greater than for displacement control conditions (C g :x 0), even at the same J value (i.e., the same value of

), and in both cases is greater than the growth rate v~s predicted by the J approch. This result accords with the comments in the Introduction, where it was indicated that the author's earlier work [5] had shown that the J ( K ) - dc/dt curve should be higher for a situation where J(K) increases with crack length than when J(K) decreases with crack length. The effects become more marked as plastic deformation becomes progressively more extensive. Thus, focusing on the constant load condition (C g = ~) , relation (40) reduces to

(42) V~gj (1 - X2)[(1 + 2,) ln(1 + X) + (1 - X) ln(1 - X)]

and Table 1 shows the ratio v/v~g for various values of X - P/2LBY and the plastic zone size Rp at each crack tip. Since m can be large, for example a value m = 8 has been suggested [19] for intergranular stress corrosion cracking in type 304 stainless steel used in boiling water reactor coolant pipes, the predicted crack growth rate v can be far in excess of the growth rate VKS predicted by the K - J correlation procedure and will be even greater if the K approach is used directly. Of course, for these load control conditions, in the limiting case where X = 1 there is general yield across the intervening ligament, and unstable non-environmentally-assisted failure will occur since the critical CTOA for non-environmentally-assisted crack growth is attainable. These results for load control conditions are in general accord with those obtained from the author's earlier analysis [1] of the model of an edge crack in a semi-infinite solid subject to a sufficiently high sustained stress that plastic deformation is extensive.

Relation (40) allows the effect of departing from load control conditions to be assessed for different amounts of plastic deformation. Thus giving rrEBCe/2(1 -p2) the typical

224 E. Smith

T A B L E 1

T h e p red ic t ed c rack g r o w t h ra te for d i f fe ren t a m o u n t s of p las t ic d e f o r m a t i o n u n d e r load con t ro l cond i t i ons

( C M = ~ ) , a n d also for the d i s p l a c e m e n t con t ro l cond i t ions : C M = 0 a n d qrEBC E/2(1 -- v 2 ) = 5

p Rp

2LBY L L O A D C O N T R O L DISP. C O N T R O l .

(_v)2 ..... ( f )2 ,,, ~)KJ t']ffj

0.2 0.02 1.02 0.99

0.4 0.08 1.15 1.00

0.6 0.20 1.46 1.03

0.8 0.40 2.41 1.08

0.9 0.57 4.31 1.18

0.95 0.70 7.98 1.28

value of 5, this corresponding to D < h and h - IOL, and assuming a rigid test machine (CM = 0), when crack growth proceeds as a result of a constant displacement applied to the solid, relation (40) gives the predicted growth rate, which is shown in the last column of Table 1. For this situation, although the predicted growth rate is still greater than the growth rate VKj predicted by the K - J correlation procedure, with the effect becoming greater as plastic deformation becomes more extensive, the effects are significantly less than for the load control case.

4. Analysis of the model of a solid with two symmetrically situated deep cracks with tension of the remaining ligament: post-general yield case

As indicted in the preceding section, with the non-work-hardening material examined in the present study, stress corrosion crack growth under load control conditions gives way to unstable non-environmentally-assisted failure when plastic deformation traverses the ligament. Under displacement control conditions, however, it is possible for stress corrosion crack growth to occur after general yield; this facet of the problem will now be considered by extending the preceding section's analysis. As before, the solid is assumed to be loaded in series with a linear spring, and the total load point displacement A r is fixed during growth. For the general yield state, this displacement is given by the expression

A T = C M P + ApL q- AEL q- AGy

8(1 =2LBYC~+ - ~ ) ~ L 1[ - ln2]+ - - I n

IrE rr , ~ E

+ A c t (43)

where AeL is given by relation (30) with X = 1, i.e., it is the value of ApL just at general yield, A EL is given by relation (31) with P having the general yield value 2 LB Y and A~; ~ is the additional plastic contribution to the solid's load point displacement, due to the post-general yield plastic deformation. The J integral now has the value

J = 8(1 - v 2 ) y 2 L In 2 • rE + Y&(;r (44)

where the first term on the right-hand side is the value of J just at general yield and is given by relation (19) with X = 1, while the second term is due to the additional plastic displacement associated with the post-general yield deformation. Since d A T / d L = O, Eqn.


(43) becomes, upon differentiation,

4 ~rL - 2BYCM + 8(1- v2)Y [1-1n 2] + I D-~ln(-2-h )I EVZ)2Y

8(1 - vZ)Y+ dAor erE dL = 0 (45)

while differentiation of (44) gives

d J 8(1 - v2)Y 2 In 2 YdAor ~- - - ( 4 6 )

d L = erE dL

Proceeding along similar lines to those of the preceding section, elimination of the dAoy/dL term between relations (45) and (46) gives

~rE dJ ¢rEB(CM + CE) = - 4 1 n 2 + (47)

4(1 - / , 2 ) y 2 dc 2(1 - v 2)

where C E is again given by expression (31). With expression (47) for dJ/dc and with s = 2eL for the general yield situation (see (18) with k = 1), the ratio of the growth rate v to the velocity Vxg, predicted on the basis of the power law relation (6) and the K - J conversion formula, is given by relations (39), (44) and (47) as

(~EB(CM+CE) } (D._.._~_ ]2/rn 1 exp 2--~---- ~ ; (48)

VKj] 32 1-~ 8 ( 1 - v 2 ) L Y l n 2

In this case, it is also instructive to make the comparison with the growth rate VLF, predicted on the basis that the K approach is directly applicable, regardless of the fact that LEFM conditions are not operative; noting that K = 2YI/-L/~r, relations (6), (8), (47) and the knowledge that s = 2eL, gives

( v ]2/m =~-~2 ex p 7rEB(CM+CE) (49) ~LF] 2(1 - - V 2 )

Where the general yield state is just attained (A6r = 0), relation (48) shows that for the case where the test machine is rigid (CM=0) and for ~rEBCE/2(1- v2)=5, then (V/VKj) 2/m = 6.70, which accords with the increasing values of this ratio as general yield is approached (see the last column in Table 1). If the post-general yield deformation becomes extensive (i.e., AGy increases), the ratio decreases and can become less than unity, particularly if CE has a smaller value. In this case, the K - J correlation procedure is conservative, in relation to the present paper's procedure, as regards its growth rate predictions. Direct application of the K approach can also give conservative growth rate predictions if C E becomes smaller (see relation 49). This conclusion is in accord with the results from an earlier analysis [8] of the general yield state under displacement control conditions where it was shown that (V/VLF) could be less than unity if a plastic rigid solution was used for J.

5 . D i s c u s s i o n

The present paper has used a crack tip opening angle procedure for predicting the growth rate of a stress corrosion crack in situations where LEFM conditions are unlikely to be operative. The procedure follows in a logical manner from established research on inert environment crack growth, and is based on the assumption that the CTOA associated with growth is uniquely related to the growth rate dc/dt, an assumption that is essentially

226 E. Smith

equivalent to the correlation between growth rate and crack tip strain rate that has arisen [2,3] as a result of mechanistic studies of the stress corrosion cracking process. The functional relation between the CTOA and dc/dt has been obtained in this paper by coupling the theoretical results for plane strain crack growth under small-scale yielding LEFM conditions (for an inert environment), with an experimentally determined power law relation between the crack tip stress intensity K and the growth rate dc/dt in an aggressive environment; however, the theoretical results could equally well be coupled with any experimental correlation between K and dc/dt. Then, presuming that the CTOA-dc/dt relation is independent of the extent of yielding, an assumption whose analogue in the inert environment growth case is that the CTOA is independent of the extent of yielding, it is in principle possible to predict the growth rate in situations where LEFM conditions are no longer applicable.

A specific model: the plane strain deformation of a solid with two symmetrically situated deep cracks, and with tension of the small remaining ligament, has been investigated in detail, this being possible because simple analytical results are available for the J integral and the plastic displacement, irrespective of the extent of yielding at the crack tip. The effects of the extent of spread of plastic deformation and the loading pattern (i.e., displacement or load controlled deformation), together with their interactive effects, have therefore been explored in detail, the predicted growth rate being compared with that obtained via application of the K-dc/dt experimental relation, its conversion to a J-dc/dt relation, and then the determination of J; they have also been compared with the growth rate determined on the assumption that a K approach is directly applicable. The extent to which these latter approaches lead to nonconservative growth rate predictions in relation to the extent of spread of plastic deformation and the loading pattern have been discussed; the theoretical predictions have also been related to earlier theoretical work, and correlated in a general manner with experimental results on the crack growth rate at high stress levels (e.g., iodine stress corrosion cracking in Zircaloy tubes) [4] and with results on the effect of the loading pattern on the crack growth rate (e.g., 304 stainless steel in a corrosive environment). [7] There is, therefore, a direct, albeit very general, link between this paper's theory and experimental results; on this basis, the theoretical work could form the springboard for experimental programs designed to actually measure growth rates and correlate them with the extent of spread of plastic deformation and loading mode. when there would then be a quantitative, and more specific, check on the theory.

In view of the importance of the conclusions that have been drawn from this investigation, it is worth highlighting some qualifying comments. Firstly, and most importantly, the methodology is based on the premise that a steady-state environmental condition exists in the vicinity of the crack tip. As indicated in Section 2. if such a condition does not exist, and whether or not this is the case could emerge from the detailed experimental programs proposed in the preceding paragraph, the methodology might have to be modified appreciably, and this could involve the use of bounding procedures based on specific environmental limits. Secondly, the model investigated in this paper has been analyzed via the highly idealized DBCS-type representation of plastic deformation. Though there are good grounds, as discussed in Section 2, for believing that its predictions have value, nevertheless this representation of plastic deformation does have limitations, which have been emphasized by Rice. [20] It is therefore desirable that more realistic theoretical approaches be used to support the simple approach developed in this paper. Thirdly, the effects of work-hardening should be incorporated within the methodology. The material behavior in the present study is non-work-hardening and. though this leads to a simple analysis, there are obvious limitations. For example, it is not immediately possible to draw conclusions regarding the growth of a stress corrosion crack

Predict ing stress corrosion crack growth rates 227

unde r load cont ro l condi t ions af ter general yield, though this diff icul ty can be overcome in an a p p r o x i m a t e m a n n e r by replac ing the yield stress Y by an effective yield stress YEFF which allows for work-hardening . In this way, the au thor has s tudied [21] load cont ro l g rowth af ter general yield; not surpris ingly, it has been shown that the growth rate can be high in this s i tuat ion. Four th ly , the effects of t ime-dependen t de fo rma t ion should also be accoun ted for in a realist ic manner . In the ear l ier work [5], this has been done in a very s imple way by assuming that the yield stress is reduced by t ime-dependen t de fo rma t ion and an "effect ive" yield stress pa r ame te r employed. N o t surprisingly, such de fo rma t ion can cause p rob lems when searching for pa ramete r s to character ize creep crack growth, a p r o b l e m that is in some respects ana logous to stress cor ros ion crack growth, and which has received a p re l iminary s tudy [22] a long s imilar l ines to those used in this paper .

Acknowledgments

This work has been conduc ted as par t of the Electr ic Power Research Ins t i tu te p rog ram on env i ronmenta l ly -ass i s t ed f racture in nuclear mater ia ls , and the au thor thanks several colleagues, especial ly Dr. R.L. Jones and Mr. J .D. Gi lman , for va luable discussions in this general p rob l em area, and also for their encouragement .

References

[ 1 ] E. Smith, Materials Science and Engineering 55 (1982) 97. [2] F.P. Ford, in Aspects of Fracture Mechanics in Pressure Vessels and Piping, Eds. S.S. Palusamy and S.G.

Sampath, ASME Pressure Vessels and Piping Conference, Orlando, Florida, U.S.A. (1982) 229. [3] P.M. Scott and A.E. Truswell, in Aspects of Fracture Mechanics in Pressure Vessels and Piping, Eds. S.S.

Palusamy and S.G. Sampath, ASME Pressure Vessels and Piping Conference, Orlando, Florida, U.S.A. (1982) 271.

[4] R.L. Jones, F.L. Yagee, R.A. Stoehr and D. Cubicciotti, Journal of Nuclear Materials 82 (1979) 26. [5] E. Smith, paper accepted for publication in RES MECHANICA. [6] H.R. Smith, D.E. Piper and F.K. Downey, Journal of Engineering Fracture Mechanics 1 (1968) 123. [7] D.M. Norris, T.U. Marston, R.L. Jones and S.W. Tagart, EPRI Pressure Boundary Technology Programme,

Special Report NP-1540-SR (1980) 2. [8] E. Smith, Materials Science and Engineering, 60 (1983) 185. [9] J.R. Rice and E.P. Sorensen, Journal of Mechanics and Physics of Solids 26 (1978) 163.

[10] J.R. Rice, W.J. Drugan and T.L. Sham, in Proceedings of Twelfth National Symposium on Fracture Mechanics, ASTM STP 700 (1980) 189.

[l 1] D.S. Dugdale, Journal of Mechanics and Physics 8 (1960) 100. [12] B.A. Bilby, A.H. Cottrell and K.H. Swinden, Proceedings of the Royal Society A 262 (1963) 304. [ 13] M.P. Wnuk, International Journal of Fracture 15 (1979) 553- 58 I. [14] E. Smith, Journal of Engineering Materials and Technology 103 (1981) 148. [15] E. Smith, in Advances in Fracture Research, Proc. 5th Int. Conf. on Fracture, Cannes, March 1981, D.

Francois (ed.), Pergamon, Oxford (1980) 1019. [16] B.A. Bilby, A.H. Cottrell, E. Smith and K.H. Swinden, Proceedings of the Royal Society A279 (1964) 1. [17] P.C. Paris, H. Ernst and C.E. Turner, in Fracture Mechanics: Twelfth Conference, ASTM STP 700, American

Society for Testing and Materials (1980) 338. [18] J.R. Rice, in Mechanics and Mechanisms of Crack Growth, Proceedings of Conference at Cambridge, U.K.,

M.J. May, Ed., British Steel Corporation Physical Metalurgy Centre Publication (1974) 14. [19] G.R. Egan and R.C. Cipolla, ASME Paper 78-MAT-23, American Society for Mechanical Engineers (1978). [20] J.R. Rice, Discussion to Reference [3]. [21] E. Smith, unpublished work. [22] R. Pilkington and E. Smith, Journal of Engineering Materials and Technology 102 (1980) 347.

R6sum6

On peut prb.dire la vitesse de croissance dc /d t d'une fissure de corrosion sous tension dans les cas o6 la th6orie lin6aire et 61astique (LEFM) n'est pas applicable en se basant sur la relation que pr6sente dc /d t avec rangle

228 E. Smith

d'ouverture de l'extr6mit6 de la fissure (CTOA). Cette relation est obtenue en couplant des r6sultats theoriques de croissance de fissure sous des conditions d'6coulement plastique a petite 6chelle dans un environnemenl inerte, avec la relation parabolique exp6rimentale liant l'intensit6 de contrainte K '~ I'extr6mit6 de la fissure el d e/dt obtenu sous des conditions de m~canique de rupture lin/~aire dans un environnement actif.

En supposant que la m6me relation CTOA-dc/dt est applicable aux conditions o/l la LEFM n'est pas valide, et en d~terminant le CTOA sous ces conditions, il est en principe possible de pr6dire la vitesse de croissance d'une fissure en corrosion sous tension. On analyse en d&ail un modele specifique: celui de l'etat plan de d6formation d'un solide poss6dant deux fissures profondes symbtriques, soumis ~ tension sur le ligament restant. On examine ~galement dans le d6tail les effets de l'~tendue de la d~formation plastique et du mode de mise en charge (p.ex contr61e des d6placements ou des charges) sur les pr6visions de vitesse de croissance de la fissure de corrosion sous tension.

Les r+sultats sont compares avec ceux que fournit l'application de la relation K-dc/dt, de sa conversion en une correlation J-dc/dt conduisant/i la d6termination de J, ainsi qu'avec les r6sultats obtenus en supposant qu'une approche par K est directement applicable.

La mesure dans laquelle ces derni~res approches produisent des pr6dictions de vitesse de fissuration conservatives ou non est discut6e en relation avec 1'6tendue de la d6formation plastique et le mode de raise en charge.

precdicting corrosion crack rates growth

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