Some implications of recent developments in plastic fracture mechanics on stress corrosion cracking in engineering materials

Materials Science and Engineering, 44 (1980) 205 - 211

205

© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

Some Implications of Recent Developments in Plastic Fracture Mechanics on Stress Corrosion Cracking in Engineering Materials E. SMITH

Joint Manchester University-University of Manchester Institute of Science and Technology Metallurgy Department, Grosvenor Street, Manchester M1 7HS (Gt. Britain) (Received October 31, 1979)

SUMMARY

In the last few years there has been a remarkable increase in our understanding of the plastic fracture characteristics o f ductile engineering materials. The objective of the work reported in this paper was to transfer this understanding to the stress corrosion cracking (SCC) problem and particularly to see whether these recent developments have important implications to SCC. The important implications concern the safety margins with regard to the termination o f SCC by unstable plastic fracture and the use o f critical J values to predict stress corrosion crack growth behaviour at high stress levels. 1. INTRODUCTION

During the last few years there have been important developments concerning the behaviour of ductile engineering materials with regard to stable crack growth. Specifically it is now accepted that, although fracture will initiate at a crack tip when the critical values KI¢, Ji¢ and 81c of the crack tip characterizing parameters (the crack tip stress intensification, the J integral and the crackopening displacement respectively) are attained, failure of a specimen or structure need not necessarily ensue; the applied loads or displacements may have to be increased substantially before instability occurs. Furthermore, it is accepted that the margin between fracture initiation and instability, and therefore the extent of stable crack growth, depends both on the properties of the material under consideration and on the nature of the loading system and thereby the structure's geometrical configuration.

Clearly it would be very surprising if these considerations did not have implications for situations where a crack propagates by stress corrosion, fatigue, corrosion fatigue or creep to such a length that fracture then proceeds by the purely plastic mechanism that is more appropriate to a cracked structure subjected to monotonically increasing loads in the absence of environmental effects. It is against this background that in the work reported in this paper we focus on the interaction between stress corrosion cracking (SCC) and plastic fracture, the specific objective being to assess the implications of the recent plastic fracture developments to the SCC problem. It is our impression that the general approach adopted for the SCC problem is that a crack propagates by the stress corrosion mechanism until it attains a length at which the material's fracture toughness KIc is exceeded at the crack tip, whereupon the crack propagates unstably by a purely plastic mechanism. This need not necessarily be the case for two reasons. First, the specimen or structure may be subjected to stress levels which are high relative to the yield stress when linear elastic fracture mechanics (LEFM) procedures, on which the Kic concept is based, are not strictly valid. Secondly, KIc as measured in the laboratory can be associated with some crack growth or non-linearity of material behaviour, and therefore plastic fracture could initiate at KI values lower than KI~. These brief comments serve to indicate that there is an important interaction between the problems of SCC and plastic fracture, and it is therefore appropriate to consider the recent plastic fracture developments in relation to the SCC problem. Primarily in this paper we emphasize plastic fracture initiation and

206

plastic fracture instability in relation to stress corrosion fracture. To facilitate a readily comprehensible' discussion of the problem, we use the strip yield model to represent material non-linearity and we apply this model to the sustained loading situation, i.e. where a solid containing an edge crack is subjected to a constant applied stress; this model is appropriate to m a n y SCC experiments and also simulates a wide range of practical SCC situations.

2. B A C K G R O U N D FRACTURE

DISCUSSION O F PLASTIC

In the absence of an aggressive environment the failure sequence for a ductile specimen or structure containing a pre-existing crack is as follows: fracture initiation at the crack tip, stable crack growth under the prescribed loading conditions and finally instability. Fracture initiation is satisfactorily characterized in terms of a critical value Jic of the J integral or alternatively in terms of a critical value 5~c of the crack-tip-opening displacement, reducing to a characterization in terms of a critical value of the crack tip stress intensification in the LEFM regime. Not surprisingly, serious and indeed satisfactory attempts have been made to characterize plane strain crack growth of a ductile material by a suitable parameter, and in a recent comprehensive research programme [ 1 - 3 ] it has been concluded that the crack-tip-opening angle (CTOA) is such a parameter, in that this angle maintains a reasonably constant value during growth. Such a constancy is essentially equivalent to specifying that the curve describing the relation between the J integral and the crack growth increment Ac has a constant slope. In incorporating these results within a quantitative description of crack growth, the simplest procedure is undoubtedly that developed by Wnuk [4]. Based on the Dugdale-Bilby-Cottrell-Swinden [ 5, 6] (DBCS) strip yield crack model, in which yield is confined to an infinitesimally thin strip which sustains the yield stress Y (as in Fig. 1, for example), Wnuk's crack growth criterion (sometimes referred to as the "final stretch" criterion) is that the crack moves forwards a distance ~ if the displacement

accumulated while a material point is within a distance A from the tip attains a critical value 5. A and ~ are both envisaged as being characteristic of the material under consideration; is the fracture process zone size and it represents the spacing between the inhomogeneities that are responsible for the loss of cohesion by the material. By assuming that is small compared with the DBCS yield zone size, which in turn is assumed to be small compared with the crack size, Wnuk obtained a stable crack growth relation (i.e. a relation between J/K and the crack growth increment Ac) which is applicable to small-scale yielding LEFM conditions. Wnuk's approach is clearly highly idealized, particularly with regard to the representation of the plastic deformation accompanying crack growth. Nevertheless, his results are of the same mathematical form as those obtained from some recent theoretical and numerical calculations by Rice and Sorensen [ 7]. Their analysis of the deformation field, consistent with a Prandtl stress distribution at the tip of an advancing plane strain crack in a plastic-elastic solid, gives the functional form of the crack tip profile: the crack opening is of the form r In(constant/r) where r is the distance from the tip. This observation, combined with data generated from finite-element studies of cracks growing under small-scale yielding conditions, allows the derivation of a relation characterizing the deformation at an advancing crack tip. Assuming that a crack extends when a critical opening is attained at a small distance from the tip, which is t a n t a m o u n t to assuming that crack advancement is associated with a constant CTOA, Rice and Sorensen obtained the same stable crack growth relation for small-scale yielding conditions as that obtained by Wnuk. This similarity implies that the application of Wnuk's final stretch criterion to the DBCS model can be regarded as simulating the application of the CTOA criterion to more realistic models. Accordingly it is appropriate to use the Wnuk approach for a wider range of loading situations; thus Smith [8] has extended the small-scale yielding analysis to the large-scale yielding situation, which is more appropriate for m a n y ductile engineering materials. In Section 3 we describe an analysis for the case where crack growth proceeds under a sustained stress.

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fracture initiates when the crack length attains the critical value CIF given by

Cr

~E5ic CIF

8(1 --

1

v2)y In sec(~o/2Y)

(3)

To determine the length of crack which will

propagate by a purely plastic fracture mechaa

a

a

Fig. 1. T h e m o d e I p l a n e s t r a i n m o d e l o f a c r a c k o f d e p t h c in t h e s u r f a c e o f a s o l i d ; t h e y i e l d z o n e , which extends to a distance a from the surface, s u s t a i n s t h e y i e l d s t r e s s Y.

3. T H E O R E T I C A L A N A L Y S I S DESCRIBING CRACK GROWTH SUSTAINED STRESS

OF A MODEL UNDER A

sl°2 In

2e(a 2 - -

I]

(1)

where E and v are Young's modulus and Poisson's ratio respectively. The extent a of the spread of plasticity is given b y --

- sec

8Y(1 --

(2)

c

When a crack grows by a stress corrosion mechanism under the sustained stress o, plastic fracture will initiate at its tip when the crack tip displacement attains the critical value 5~ appropriate for plastic fracture. By setting ¢P(0) = 5,~, eqn. (1) shows that plastic

v2)c

=E

In sec

In sec

8Y(1--v2)c _A_ lnl _ _4 nE 2c 2ec

cosec 2

(~Y)I

=

(5) Thus, with ~/A simulating the CTOA 0, eqn. (5) gives the critical length CUF of crack that will grow unstably by a purely plastic fracture process under the sustairied stress o as CUF

c 2) c

(4)

Equation (1) therefore gives the condition for a crack of length c to grow under the sustained stress o as

rE

dP(s)- 8Y(1--v2)C []n(a )

2c

¢(0,c + 4)--~(A,c) = 5

8Y(1 -- v2)(c + 4)

Let us consider the mode I plane strain model of a crack of length 2c situated within an infinite solid subjected to an externally applied tensile stress o; this model also applies to the behaviour of a crack of depth c in the surface of a semi-infinite solid (Fig. 1). Bflby et al. [6] have analysed this model and their results give the relative displacement ¢(s) across the yield zone, which sustains the yield stress Y and extends to x = +a, as

+-

nism, Wnuk's final stretch criterion, described in Section 2, is applied. As indicated in Section 2, this criterion is that the crack tip moves forwards a distance 4 if the displacement accumulated while a material point is within a distance 4 from the tip attains a critical value 5, i.e.

2e c°t2

exp 4 ( 1 - - v 2 ) y

(6)

Relations (3) and (6) provide the basis for assessing whether unstable plastic fracture occurs as soon as plastic fracture is initiated. Instability and initiation occur simultaneously if CIF :> CUF, i.e. if

2e(lrE~ic/4(1 -- v2)yA ) ln(sec2(Tro/2Y)) > (7) exp(EO/4(1 -- v2)Y} tan2(no/2Y) Since the right-hand side decreases as the stress increases, this relation shows that if instability and initiation occur simultaneously at low stresses they also occur simultaneously at high stress levels. Inspection of relation (7) shows that instability and initiation are

208 coincident at low stress levels (i.e. small-scale yielding conditions) provided that

IreEaic exp 1 4(1 2(1 -- p2)YA >

~----'V'2y} )l

(8)

a relation that has been obtained in earlier analyses [4, 7] in which researchers have specifically focused on the small-scale yielding situation. With aic ~ 4, relation (8) shows that plastic fracture initiation and instability are coincident if

but since the right~hand side is approximately 10, at least for most steels, relation (9) simplifies to

EO --

Y

:~ 10

(10)

This simple relation provides an approximate indication as to whether plastic fracture initiation and instability coincide. Thus a high yield stress, i.e. a high value of Y/E, and a low crack growth resistance, i.e. a low 0, both favour instability; for the purposes of the present discussion such a material will hereafter be referred to as being unstable. A low yield strength, i.e. a low value of Y/E, and a high crack growth resistance, i.e. a high 0, both favour stability and such a material will hereafter be referred to as being stable. The 0 and Y/E terms have been conveniently combined by Paris et al. [9] who have represented the material's resistance to stable plastic fracture by a parameter Treat, referred to as the material's tearing modulus, defined by the relation EO Tmat

-'~ ~

Y

(11)

Thus stable materials are characterized by a tearing modulus in excess of 10, whereas unstable materials are characterized by a tearing modulus lower than 10. However, these definitions refer specifically to the sustained stress situation and, as emphasized particularly by Paris and coworkers, the stability condition is strongly influenced by the nature of the loading system.

4. IMPLICATIONSOF OUR ANALYSIS

4.1. The transition between stress corrosion crack propagation and plastic fracture In the analysis performed in Section 3 we clearly demonstrated that for unstable materials, i.e. those with a low tearing modulus (indicating a high yield strength and/or a low CTOA) for plastic fracture propagation, when a stress corrosion crack grows under a sustained stress and becomes sufficiently long for plastic fracture to be initiated at its tip, then unstable plastic fracture should ensue immediately. This is possibly one reason why high strength materials are, in general, particularly susceptible to failure by SCC since the critical size to which the crack must grow by stress corrosion before unstable plastic fracture occurs is governed by the initiation of plastic fracture rather than by the onset of instability. If cracks are found in these materials in sustained stress situations and if their lengths exceed the length predicted on the basis that the crack tip stress intensification K~ exceeds Kic (see relation (3) for low o/Y and set K1 = o(~c) 1/2 and gxc = {EY81c/(1 p2)}1/2), these lengths must have been sustained presumably because of crack branching; this would be the case particularly at high stress levels (a -~ Y) since the LEFM approach then overpredicts the critical crack length at which plastic fracture is initiated. Consequently any safety margins must arise from branching, although it should be noted that if fluctuating stresses are present as well as the sustained stress (i.e. a corrosion fatigue situation) there is less tendency for crack branching; this point of course emphasizes merely one reason why fluctuating stresses have an adverse effect on the SCC process. However, with stable materials, i.e. those with a high tearing modulus (indicating a low yield strength and/or a high CTOA) for plastic fracture propagation, our analysis shows that, although plastic fracture is initiated when a growing stress corrosion crack attains a critical length given by relation (3), plastic fracture instability does not automatically ensue. In this case the crack is expected to grow by an environmentally assisted plastic fracture process, probably at a faster rate than if there were no plastic fracture. Indeed, there is general experimental support for this view, -

-

209

~ [I

II

Plastic fracture initiation could cause increased growth rate

KI

Fig. 2. Schematic representation of the relation between the crack tip stress intensification KI and the crack growth rate dc/dt.

since it is accepted that the crack growth rate increases as we move from the essentially constant-speed stage II of the K versus dc/dt curve (Fig. 2) to stage III which precedes instability. With such materials, if cracks are found whose lengths exceed the length predicted on the basis that unstable fracture occurs when critical values KIc, Jx~ and ¢ ic are attained at the tip, their existence could to some e x t e n t be due to the fact that plastic fracture cannot become unstable even though it can be initiated. Arising from the preceding discussion is the clear indication that when considering a material's SCC susceptibility in relation to eventual plastic fracture instability it is equally important to focus both on the material's resistance to plastic fracture propagation as measured, for example, by its CTOA or tearing modulus and on its resistance to plastic fracture initiation as measured by ~i¢ or J~c. This is a feature that should be taken into account in any SCC discussion; it has stemmed directly from recent developments in elastic-plastic fracture mechanics. It is a feature that is especially important in regard to the effects of metallurgical variables; thus, for example, it is equally important to consider both the influence of embrittling elements on plastic crack growth resistance and their effects on plastic fracture initiation. Of course, the importance of a material's resistance to growth depends on the nature of the loading system. In this paper we focus on the sustained stress situation which is particularly easy to analyse but which is not conducive to stability. With other loading systems, where there is a measure of displace-

m e n t control, our considerations will be of even greater significance in that there is a wider safety margin with regard to unstable failure, and this should be considered in any overall failure assessment procedure.

4.2. Stage I stress corrosion cracking So far we have focused on the interaction between plastic fracture processes, as would occur in the absence of an aggressive environment, and SCC; it is expected that these considerations will be particularly relevant to the later stages of stage II and also stage III crack growth (see Fig. 2). It is also worth speculating whether the recent developments in plastic fracture mechanics have implications for the K-sensitive stage I region and the early part of stage II where the crack tip displacem e n t is probably insufficient for the value 5ic appropriate to plastic fracture initiation to be attained. There appears to be no question that plastic deformation accompanies stress corrosion crack growth even though purely plastic fracture might not be initiated at a crack tip; for a purely elastic situation the KI value for propagation is about 1 klbf in -3/2 (1 MN m-a/u) and threshold stress intensities for stress corrosion crack growth exceed this value. In developing a model incorporating plastic deformation, together with a fracture process zone at the crack tip, it is logical, at least for an initial discussion, to proceed as in our earlier analyses and to assume that crack propagation is associated with a CTOA which increases with crack velocity. Thus, again focusing consideration on the model in Fig. 1 where a solid subjected to a sustained tensile stress o contains an edge crack of depth c, because J = YdPtip relations (1) and (5) give dJ

dc

-YO+

4 y 2 ( 1 -- v2) In I a2A ~rE 2e(a 2 --c2)c I (12)

When growth proceeds under small-scale yielding conditions (a -- c ~ c), because J = 8(1 -- u2)y2(a --c)/E relation (12) reduces to

dJ_ 4Y2(1- v2) in( J~) de

.E

-7

with J~s being given by

(13)

210 j~

2 ( 1 - v2)Y2A

=

~eE

{~EO

1

exp 4 ( 1 _ v2)y

(14)

Relations (13) and (14) provide an alternative, but equivalent, basis for developing an initiation-instability criterion (i.e. relation (8)). For a constants/situation (in fact a constant-K situation since the emphasis, for the moment, is on the small-scale yielding situation) the introduction of the assumption that the CTOA 0 increases with crack velocity during stress corrosion crack propagation enables relations (13) and (14) to give the following relation between the applied J value and the crack velocity v:

j(v) = 2(1--v2) Y2A { ~EO(v) I ~eE exp 4(1-- v2)Y

(15)

For there to be very little change in J for wide variations in crack velocity, as in stage I, the exponential term must be very small, i.e. the CTOA must be very small. If this is so, propagation proceeds at an essentially constant J value over a wide range of crack velocities; this value is approximately Jiscc = K~scc(1 v2)/E, whereupon -

-

Kiscc = Y

~ YA 112

where small-scale yielding conditions are inoperative, relation (6) gives the size of crack that will grow by stress corrosion, with a threshold CTOA 00, as Ciscc = ~e

co 2/ °)ex,l \ 2Y

~4(1 -- v2)y

(17)

reducing to ClSCC = 2e cOt2

(18)

since it has already been argued that the exponential term is approximately unity near threshold conditions. However, if it is assumed that a crack grows by stress corrosion when J exceeds the Jiscc value given by relation (15) but with the exponential term equal to unity, then because

J= 8 y 2 ( 1 - - v 2 ) c In secl ~ a l rE ~2Y]

(19)

for the situation under consideration the predicted critical length for stress corrosion crack growth obtained by this procedure is C# which is given by

(16) c~ -

Since the general experimental observation is that K~scc decreases with increasing yield strength, at least for high strength steels, the effective fracture process zone size A must decrease with increasing yield strength. In more detailed studies we shall focus on the possible reasons for this decrease, and these will of course involve the detailed microstructural mechanisms by which the stress corrosion crack grows. This particular point will not be developed further in this paper, the main purpose of this part of the discussion being to use the experience gained from recent plastic fracture mechanics developments to emphasize the point that any model describing SCC threshold conditions should incorporate plastic deformation effects in a logical manner.

4.3. Stress corrosion crack growth at high stress levels In extending the preceding discussion on low stress thresholds to higher stress levels

2e In sec2(iro/2Y)

(20)

Relations (18) and (20) give

Cj

tan2(~o/2Y)

Ciscc

In sec2(Iro/2Y)

(21)

Thus, if o = 0.9Y, then C#/Ciscc ~ 10 and there is obviously an appreciable difference between the critical crack sizes Cj and CIscc. These results therefore suggest that large, and more significantly non-conse~ative, errors could arise in describing stress corrosion crack growth threshold conditions at high stress levels by the use of the same critical J value as that which is appropriate for growth at low stress levels. There will probably be significant errors even when the applied stress is not particularly high; with a equal to 0.5Y the ratio of the critical crack sizes obtained using the two approaches is about 1.4. Thus it is again seen that the transference of experience from the plastic fracture mechanics field leads to an important implication, in this case

211

the highlighting of the possibility that stress corrosion crack growth at high stress levels under a sustained stress cannot be satisfactorily represented using the J integral, i.e. by assuming growth to occur at the same critical J level as is appropriate for the growth of cracks at low stresses. 4.4. Stress corrosion cracking o f turbine materials In conclusion it might be appropriate to relate the implications discussed in this paper to a specific SCC problem, namely that of SCC in turbine materials. Experimental information [10] for at least some of these materials suggests that they are sufficiently tough under normal circumstances that plastic fracture instability and initiation are not coincident, i.e. the tearing modulus or equivalently the CTOA is adequate. It is obviously desirable, however, to ensure that this is always the case, since the preceding discussion has clearly indicated that the safety margins with regard to unstable fracture are reduced when plastic fracture instability and initiation are coincident. Thus, as well as focusing on the effects of metallurgical variables on fracture toughness (Kit, Jic, 5ic) values via, for example, the influence of temper-embrittling elements, attention should also be given to their influence on the material's tearing modulus or CTOA. Furthermore, because some of the service failures have been associated with high applied stress levels [11], e.g. cracks forming in the vicinity of key-ways, there is a good case for basing lifetime predictions in these situations on data obtained by monitoring the growth of cracks under sustained stress conditions. Such data should be more reliable for the problem in question than data obtained from K versus dc/dt experiments, although t h e K versus dc/dt data will of course be of considerable value when comparing the behaviour of various materials and also the effects of different environments. 5. CONCLUSIONS

It is quite evident that recent developments in the plastic fracture mechanics field, particularly with regard to the discrimination between plastic fracture initiation and plastic fracture instability, have important implica-

tions for the SCC problem. The more important implications concern the following: (1) safety margins with regard to the termination of SCC by unstable plastic fracture; (2) the focusing of attention on the importance of a material's plastic crack growth resistance in addition to its plastic fracture initiation resistance, particularly with regard to the effects of metallurgical variables on these parameters; (3) the possible undesirability of using critical J values obtained from low stress LEFM tests to predict stress corrosion crack growth behaviour at high stress levels (this procedure could be nonconservative, i.e. it could overpredict the critical length of crack which will grow by stress corrosion). ACKNOWLEDGMENTS

This research was conducted as part of the Electric Power Research Institute research programme on SCC in turbine disc materials. The author would like to thank his colleagues at the Electric Power Research Institute, especially Drs. R. L. Jones and M. Kolar, for discussions on SCC and plastic fracture during the last few years. REFERENCES 1 T. U. Marston (ed.),Res. Rev. Doc. E P R I NP-701SR, February 1978, Electric Power Research Institute. 2 C. F. Shih, Proc. 5th S M I R T Conf., Berlin, August 1979, North-Holland, Amsterdam, Paper G 6/5, to be published. 3 M. F. Kanninen, D. Brock, G. T. Hahn, C. W. Marschall, E. F. Rybicki and G. M. Wilkowski, Nucl. Eng. Des., 48 (1978) 117. 4 M. P. Wnuk, in G. C. Sih (ed.), Proc. Conf. on Dynamic Crack Propagation, Lehigh University, Pennsylvania, 1973, Noordhoff, Leyden, 1973, p. 273. 5 D. S. Dugdale, J. Mech. Phys. Solids, 8 (1960) 100. 6 B. A. Bilby, A. H. Cottrell and K. H. Swinden, Proc. R. Soc. London, Ser. A, 272 (1963) 304. 7 J. R. Rice and E. P. Sorensen, J. Mech. Phys. Solids, 26 (1978) 163. 8 E. Smith, Int. J. Fract., to be published. 9 P. C. Paris, H. Tada, A. Zahoor and H. Ernst, A treatment of the subject of tearing instability, U.S. NRC Rep. NUREG-0311, 1977, U.S. Nuclear Regulations Commission, Washington, D.C. 10 W. A. Logsdon, Mechanics of crack growth, Am. Soc. Test. Mater., Spec. Tech. Publ. 590, 1976, p. 43. 11 J. M. Hodge and I. L. Mogford, Proc. Inst. Mech. Eng., 193 (1979) 93.