Influence of interaction between multiple cracks on stress corrosion crack propagation

Influence of interaction between multiplecracks on stress corrosion crack propagation

Masayuki Kamaya *, Nobuo Totsuka

Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japan

Received 4 September 2001; accepted 8 January 2002

Abstract

The interaction between multiple cracks has a major influence on crack growth behaviours.

This influence is particularly significant in stress corrosion cracking (SCC) because of the

relatively large number of cracks initiated due to environmental effects. In the SCC experi-

ments using alloy 600 in high-temperature environments, many cracks were observed on the

surface of fractured specimens. In this study, the interaction between multiple cracks was

evaluated using a crack growth simulation. Some improvements were made to the simulation

to reduce the calculation loads, which made it possible to conduct the simulation for many

cases. Through the simulation, the relationships between the interaction and relative position,

relative length, etc. of cracks were examined. The influence of the interaction between multiple

cracks on the experimental results is then discussed. � 2002 Elsevier Science Ltd. All rights

reserved.

Keywords: Multiple cracks; Interaction; Alloy 600; Crack growth simulation

1. Introduction

It is well known that the interaction between multiple cracks has a major influenceon crack growth behaviour. In stress corrosion cracking, due to the relatively largenumber of cracks that are initiated, the influence of this kind of interaction can be amajor problem, as several studies have reported [1–3]. Studies of the interaction

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Corrosion Science 44 (2002) 2333–2352

*Corresponding author. Tel.: +81-770-37-9111; fax: +81-770-37-2009.

E-mail address: [email protected] (M. Kamaya).

0010-938X/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0010-938X(02 )00039-2

between multiple cracks can be classified into two categories––studies by analyticalapproach [4–11] and by experimental approach [1–3,12–16].

The former approach involves numerical evaluations based on empirical rela-tionships between the crack growth rate and the stress intensity factor K. By takinginto account such a relationship, the effect of interaction on growth behaviour can beevaluated by analyzing the relationship between K and the relative position of in-teracting cracks. Since crack growth by fatigue is also depended on the K value, thiskind of approach is not special to stress corrosion cracking. The relative position ofinteracting cracks changes according to their growth, and this causes a change in theeffect of interaction between them. Therefore, it is necessary to examine not only therelationship between the K value and the relative position, but also the change in thisrelationship with time. The use of crack growth simulations is one possible means toexamine the influence of the interaction on the crack growth. The interacting crackgrowth behaviour can be imitated by analyzing the effect of the interaction on Kvalue. However, numerical analysis is required to calculate the K value of interactingcracks. And due to a limit in computing power and other factors, there are fewstudies using such a simulation [17,18].

In the experimental approach, the effect of interaction on the crack growth be-haviour can be evaluated directly without numerical calculation. Since it is difficult inthe experimental approach to consider the interaction effect using mechanical pa-rameters such as K value, the interaction effect is considered through the observationof crack growth behaviour. The interaction involves two typical features of crackgrowth behaviours––crack coalescence and stagnation. The coalescence of ap-proaching cracks, which is frequently observed in crack growth processes, becomes atrigger for crack growth acceleration. On the other hand, cracks may stop growingdue to the stress shielding effect caused by other relatively large cracks located in theneighborhood. Through the experimental approach, some characteristics aboutcrack growth behaviour have been found, such as a relative crack position factors forcrack coalescence and stagnation [2,15,16]. Also, Monte Carlo simulations weredeveloped to predict crack growth behaviours by taking into account the conditionsfor crack coalescence and stagnation [1,2,12]. This kind of simulation was aimed atpredicting crack growth behaviour by considering the interaction effect betweenmultiple cracks. However, in these simulations, the interaction effect is representedby only two behaviours––crack coalescence and stagnation––although the interac-tion changes continuously according to their relative position. This approximationmight cause a considerably large error in simulation results. In the analytical ap-proaches, the interaction effect was evaluated based on mechanical parameters suchas K values, which made it possible to discuss the intensity of the interactionquantitatively. In order to improve the accuracy and reliability of the simulation,numerical calculations must be used for evaluations based on the mechanical pa-rameters.

As mentioned above, neither of these approaches has been entirely adequate forevaluating the influence of interaction on crack growth behaviour. In the presentstudy, to evaluate this influence, a crack growth simulation was developed. In orderto take into account the effect of interaction, in this simulation the K values of in-

2334 M. Kamaya, N. Totsuka / Corrosion Science 44 (2002) 2333–2352

teracting crack tips were calculated using the body force method, and some im-provements and approximations were applied to the simulation in order to reducecalculation loads. This made it possible to simulate crack growth behaviour to studythe interaction between multiple cracks for many cases. Using this simulation, theinfluence of the interaction between multiple cracks on crack growth behaviour wasevaluated based on the many calculation results.

2. Experiments

In order to examine the crack interaction effect, experiments using alloy 600 wereconducted. The subject of the experiments was stress corrosion cracking in the pri-mary water environment of a pressurized water reactor, which is called primarywater stress corrosion cracking (PWSCC).

2.1. Experimental method

The studied material was alloy 600, mill annealed treated at 980 �C. The chemicalcomposition and mechanical properties are presented in Tables 1 and 2 respectively.The test specimens were processed into the shape shown in Fig. 1 and then polishedwith #1200 emery finish. The specimens were placed in a simulated pressurized waterreactor primary environment (500 ppm B, 2 ppm Li, 4 ppm H2, <5 ppb O2) at 360 �Cand subjected to constant tensile load of 529 MPa. All the tests were carried out in anautoclave that was directly connected to a recirculation test loop. The test conditionsare presented in Table 3. In order to investigate the relationship of crack progressand exposure time, specimens were taken out at specific exposure times and a parallelportion was cut out from each specimen. After observing the surface of the specimenusing a scanning electron microscope (SEM), the cross-section was polished to ex-amine the crack depth. Using an optical microscope with a magnification of 1000, allcracks deeper than 3 lm were measured.

Table 1

Chemical composition of test material (wt.%)

C Si Mn P S Ni Cr Fe Ti Al B

0.03 0.3 0.35 0.007 0.001 73.85 15.9 9.27 0.25 0.038 0.0003

Table 2

Mechanical properties of test material

Test temperature

(K)

0.2% yield strength

(MPa)

Tensile strength

(MPa)

Elongation

(%)

Reduction in

area (%)

Room 234 627 49 33.5

623 186 591 47.4 31.1

M. Kamaya, N. Totsuka / Corrosion Science 44 (2002) 2333–2352 2335

2.2. Experimental results and discussion

2.2.1. Crack growth behaviourThe surfaces of specimens were observed using SEM as shown in Fig. 2. Only a

few cracks were observed on the surfaces of the specimens in tests B, C and D, asshown in Fig. 2(a), while many cracks were observed in the specimen of test A,shown in Fig. 2(b). Intergranular cracking was observed in the fractured surface ofthe specimen of test A, with 35.1% of the total fractured surface for the specimenA(2). The intergranular part is separated into multiple sites. This indicates thatgrown cracks were initiated from multiple sites and that the specimen fractured aftercoalescence of multiple cracks.

The maximum crack depth of each specimen is listed in Table 3. The relationshipwith exposure time is plotted in Fig. 3. Although cracks grow at a relatively slow rateuntil 1500 h, crack growth rate increases drastically after this point. The change incrack growth rate correlates with the change in the number of cracks. Therefore, theinteraction between multiple cracks might be a key factor in crack growth acceler-ation after 1500 h.

Fig. 1. Geometry of test specimen (mm).

Table 3

Summary of test conditions and results

Specimen Exposure time (h) Number of cracks

(/30 mm)

Max. crack depth (lm)

A(l) 1724a 255 383.1

A(2) 1807a 313 260.2

A(3) 1687a 371 214.7

A(4) 1769a 449 286.6

B(l) 1500 107 96.4

B(2) 1500 89 50.2

B(3) 1500 92 55.7

C(l) 1000 31 30.5

C(2) 1000 54 63.8

D(l) 500 24 29.6

D(2) 500 15 31.2

a Specimen was fractured at this time.


Fig. 2. SEM microphotograph of the surface of the specimen (exposure time: 1807 h).

Fig. 3. Relationship between the maximum crack depth and exposure time.


2.2.2. Evaluation of crack growth rate by maximum crack depthThe growth rate of the maximum crack (damax=dt) can be estimated by the gra-

dient of the line that connects the maximum crack depths at each exposure time, asshown in Fig. 3. Since the exposure times of test A are not constant, the mean valueof fracture time was used for the calculation of crack growth rate. These growth ratesare plotted in Fig. 4, which shows the relationship between estimated crack growthrate and stress intensity factor K. In this figure, the K value of each data point wascalculated by assuming that cracks took the shape of a semi-circle located in aninfinite plane [19]. By approximating these data points, the crack growth rate as afunction of K was formulated as

damaxdt

¼ 5:24� 10�14K3:46 ð1Þ

The growth rate is given in m/s and K value in MPaffiffiffiffim

p.

Various empirical data have been reported on the crack growth rate of alloy 600in PWSCC, as plotted in Fig. 4 [20–25]. The crack growth rates shown in Fig. 4 wereobtained by experiments preformed under almost the same conditions as those ofthis study. There is large dispersion between the experimental data. However, thedependency of crack growth rate on K value clearly can be seen in the small K valueregion. The K values corresponding to this study are relatively low compared withthe other data plotted in Fig. 4. Generally, as shown in Fig. 4, cracks remain in short

Fig. 4. Relationship between stress intensity factor and crack growth rates obtained by experiments and

reference papers.


crack regions for a majority of the time to fracture. Therefore, the crack growth ratescorresponding to small K value regions are important for knowing crack growthbehaviour and predicting the time to fracture. In this study, crack growth rates wereobtained using smooth planar specimens instead of the more generally used compacttension specimens. By using this kind of specimen, relatively small crack depths canbe measured by observing the cross-section for obtaining the crack growth ratecorresponding to small K value regions. However, on the surface of planar speci-mens, as previously mentioned, the interaction between initiated multiple cracksinfluences crack growth behaviour. Therefore, it is important to understand theinfluence of the interaction in order to estimate crack growth rate precisely.

3. Crack growth simulation

As observed in experiments, many cracks were initiated on the specimen and theyseem to affect each other. It was also pointed out that the interaction might cause asharp change in crack growth rate just before fracture. In addition, there is a cor-relation between crack growth rate and stress intensity factor K. By taking intoaccount this fact that crack growth rate relates to the K value, crack growth be-haviour can be predicted by evaluating the K value with consideration to the in-teraction effect. In this study, a crack growth simulation was developed in order toevaluate the crack growth behaviour. In this simulation, the growth behaviours ofinteracting cracks are simulated by calculating growth rate according to Eq. (1), andestimating K values by numerical calculation. In this way, the influence of the in-teraction between multiple cracks on their growth behaviour are discussed.

Usually, cracks take complicated three-dimensional shapes as shown in Fig. 2.However, in this study, cracks were taken as simplified two-dimensional throughcracks. Some analytical and experimental investigations have been reported com-paring the mechanical interference effects of through cracks and semi-elliptical sur-face cracks. These have shown that semi-elliptical surface cracks have an equal orlesser interference effect than through cracks. The effects are equal when the depth ofsurface crack approaches an infinite value in relation to the surface length [26,27].Also, the interference effects of surface cracks mainly act at surface points. On theother hand, few effects act at the deepest point [28]. Therefore, the simplified throughcrack model can be regarded as a kind of the semi-elliptical surface crack from thepoint of view of interaction effects.

3.1. Simulation procedures

The target of this simulation is the growth behaviour of two cracks located asshown in Fig. 5. The outline of the simulation procedure is shown in Fig. 6. Thebody force method was used to calculate K values. This method can derive accurateK values that take into account the interaction between multiple cracks [29]. Crackgrowth rates were calculated according to estimated K values and Eq. (1).


Cracks were assumed to change their growth direction according to the rhmax

criterion [30]. By this criterion, crack propagation direction hmax which is defined inFig. 7, is determined by substituting KI and KII values into the following equation:

h ¼ � cos�13K2

II þ KI

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8K2

II þ K2I

p9K2

II þ K2II

!ð2Þ

where the sign in Eq. (2) is positive in the case of KII=KI < 0 and negative in the othercases. The equivalent stress intensity factor Keq which was used to calculate the crackgrowth rate in Eq. (1) is described by the following equation:

Keq ¼ coshmax2

KI cos2 hmax

2

�� 3

2KII sin hmax

�ð3Þ

Fig. 5. Two through cracks in infinite plane subjected to tensile stress.

Fig. 6. Crack growth calculation process.


In the simulation, K values were calculated for each specific exposure time, called‘‘step’’ in this study. Curved crack shapes are expressed by a series of short straightlines. And, as shown in Fig. 7, at each step, a straight line, whose length is deter-mined by crack growth length, is added to the end of the curved crack. The intervalof the step was controlled so that the crack growth length, Dc, does not exceed 1% ofcrack length. A tensile stress of 529 MPa was applied perpendicular to the initialcrack direction as shown in Fig. 5.

3.2. Simulation results and discussion

3.2.1. Crack growth behaviourSimulations were conducted for two cracks of length 2c1 ¼ 2c2 ¼ 0:1 mm under

the initial position of ðS=c;H=cÞ ¼ ð8; 2Þ, ð8; 8Þ, ð2; 2Þ and ð2; 8Þ, where c ¼ c1 ¼ c2The crack configurations obtained by the simulation are shown in Fig. 8(a)–(d) re-spectively. At the early stage of the simulation, the two cracks progress perpendic-ularly to the stress direction. However, if the cracks cross, the inner crack tips of twofacing cracks begin to change their progress direction so that they approach eachother. On the other hand, the progress directions of the outer crack tips are almoststraight. These crack configurations can be seen in Fig. 2(b) and other studies [15].

Fig. 8(e) shows the result of a simulation for different crack lengths, 2c1 ¼ 0:1 mmand 2c2 ¼ 0:06 mm, with relative positions being set to ðS=c1;H=c1Þ ¼ ð8; 8Þ. In thiscase, the growth behaviour of crack 2 becomes dormant due to the stress shieldingeffect caused by crack 1. This suggests that in addition to relative position, the rel-ative crack length is also an important factor determining the interaction betweenmultiple cracks.

3.2.2. Influence of interaction on crack growth behaviourSimulations were conducted to obtain the relationship between time and crack

length under the initial conditions of 2c1 ¼ 2c2 ¼ 0:1 mm. Fig. 9(a) shows the resultsfor the case of initial relative position conditions ðS=c;H=cÞ ¼ ð8; 2Þ, ð8; 8Þ and ð8; 80Þ.And Fig. 9(b) shows the case of ðS=c;H=cÞ ¼ ð2; 2Þ, ð2; 8Þ and ð2; 80Þ. The H=c ¼ 80

Fig. 7. Schematic representation of curved crack used in the simulation.


case can be regarded as a case of no interaction, since the H=c ¼ 80 is equivalent toH=c ¼ 1 because the interaction is sufficiently small compared to the case S=c ¼ 8and S=c ¼ 2. The parameter Rx represents the distance of the outer crack tip, which isshown in Fig. 5. Fig. 9(a) shows that the crack growths are accelerated by the inter-action effect. And this effect becomes large as the distance of cracks becomes small.However, as shown in the case H=c ¼ 8 of Fig. 9(b), the crack growth under smallS condition is not always accelerated by the interaction. In the case ofðS=c;H=cÞ ¼ ð2; 8Þ, the crack growth is restricted by the stress shielding effect. Thefinal crack configuration of this case is shown in Fig. 8(d). From the appearance of thiscrack configuration, the acceleration effect seems to exceed the shielding effect. So, inthis case, the interaction effect changes from negative to positive for the crack growth.This suggests that, since the interaction effect changes according to the crack growth,

Fig. 8. Results of the crack growth simulation. Initial conditions (a) S=c1 ¼ 8, H=c1 ¼ 2, 2c1 ¼ 2c2 ¼ 0:1

mm, (b) S=c1 ¼ 8, H=c1 ¼ 8, 2c1 ¼ 2c2 ¼ 0:1 mm, (c) S=c1 ¼ 2, H=c1 ¼ 2, 2c1 ¼ 2c2 ¼ 0:1 mm, (d)

S=c1 ¼ 2, H=c1 ¼ 8, 2c1 ¼ 2c2 ¼ 0:1 mm, (e) S=c1 ¼ 8, H=c1 ¼ 8, 2c1 ¼ 0:1 mm, 2c2 ¼ 0:06 mm.


the influence of the interaction on growth behaviour should be evaluated by takinginto account the entire crack growth process by using the crack growth simulation.

Two approaching cracks may coalesce and become one large crack. Since co-alescence causes crack extension, this also causes crack growth acceleration. In the

Fig. 9. Relationship between crack length and exposure time. Initial conditions (a) S=c1 ¼ 8,

2c1 ¼ 2c2 ¼ 0:1 mm, (b) Initial conditions S=c1 ¼ 2, 2c1 ¼ 2c2 ¼ 0:1 mm.


simulation, crack coalescence can be imitated by replacing two cracks with a singlecrack of length Rx when the distance of the inner crack tip S becomes zero. Theresults of the coalescence case are shown in Fig. 10. These results correspond to thecase H=c ¼ 2 and H=c ¼ 8 in Fig. 9(a). This figure means that if the offset distance Hof two cracks is short enough, the cracks would grow by almost the same rate as ifcombined. This fact suggests that the interaction between multiple cracks is deter-mined by the relative crack positions and that crack coalescence is not intrinsic to theinteraction on crack growth behaviour.

As described in the introduction, many studies have been done to investigate thecrack coalescence phenomenon, and some geometric conditions for crack coales-cence were proposed [2,15,16]. These conditions, however, differ depending on ma-terials and test conditions. On the other hand, Fig. 10 suggests that if the simulationis conducted to consider the influence of interaction on the crack growth behav-iour––that is, if the crack configuration is not significant––crack coalescence doesnot need to be considered in the simulation.

3.2.3. Influence of interaction on K value of crack tipsThe relationship between exposure time and the K value of outer crack tip KðBFMÞ

are shown in Fig. 11(a)–(c). In these figures, the distances of inner crack tip S andKðcÞ and KðRxÞ are also plotted, where KðcÞ represents the K value calculated usingcrack length c1 according to the following equation:

KðcÞ ¼ rffiffiffiffiffiffiffipc1

p ð4Þ

Fig. 10. Effect of interaction of crack tips on relationship between crack length and exposure time (initial

conditions S=c1 ¼ 8, 2c1 ¼ 2c2 ¼ 0:1 mm).


Fig. 11. Relationship between stress intensity factor and exposure time. Initial conditions (a) S=c1 ¼ 8,

H=c1 ¼ 2, 2c1 ¼ 2c2 ¼ 0:1 mm, (b) S=c1 ¼ 2, H=c1 ¼ 8, 2c1 ¼ 2c2 ¼ 0:1 mm, (c) S=c1 ¼ 8, H=c1 ¼ 8,

2c1 ¼ 0:1, 2c2 ¼ 0:06 mm.


This KðcÞ is quoted to express the K value without interaction. The other K value,KðRxÞ is calculated by replacing c1 with Rx in Eq. (4). The KðRxÞ represents the K valuecorresponding to the coalesced crack. Fig. 11(a) and (b) show the relationship be-tween K value and the exposure time, which corresponds to the case of Fig. 8(a) and(d), respectively. Parameter S in Fig. 11 represents the change in the relative positionof the two cracks. As two cracks approach, S becomes small. And after they crosseach other, S would have a negative value. On the other hand, the intensity of theinteraction between the two cracks can be evaluated by comparing three kinds of Kvalues. In the initial stage of the simulation, cracks are located far enough apart thatKðBFMÞ takes almost the same value as KðcÞ corresponding to a case of no interaction.In other words, at this stage the crack growth of the outer crack tip is not acceleratedby the interaction. However, as the cracks grow, KðBFMÞ shifts to the KðRxÞ curve, whichrepresents the K value of the combined crack. And finally, KðBFMÞ takes almost thesame value as the K value of the coalesced crack KðRxÞ This causes the crack growthacceleration shown in Figs. 9 and 10. The crack configuration of the last stage ofFig. 11(a) and (b) correspond to Fig. 8(a) and (b), respectively. Although, no co-alescence can be seen in these figures, cracks are accelerated by the interaction. Therelationship between time and Rx in Fig. 11(b) corresponds to the case H=c ¼ 8 in Fig.9(b). In this case, from the point of view of total crack growth behaviour, the crackgrowth is restricted by the stress shielding effect, while the crack growth is acceleratedby the interaction effect at the final stage of the simulation, as shown in Fig. 11(b).

Fig. 11(c) shows the result for the case of different crack lengths, with a crackconfiguration as shown in Fig. 8(e). In this case, the KðBFMÞ curve is almost identicalto KðcÞ at all times and KðcÞ approaches KðBFMÞ This means that crack growth is neveraccelerated by the interaction effect, so that crack 1 grows as an independent crack.

3.2.4. Influence of crack shape on interaction effectAs shown in Fig. 8, the inner tips of two approaching cracks change their di-

rection according to Eq. (3), because KII becomes significant by the interaction effect.This inner crack tip behaviour is very important for evaluating the crack coalescencephenomenon. However, as discussed previously, the interaction between multiplecracks is determined by relative position and crack coalescence is not intrinsic to theinteraction on crack growth behaviour. So, this suggests that interaction dose notdepend on inner crack tip behaviour. Next, simulations were conducted under theassumptions that the cracks propagate without changing their progress direction.This assumption can be realized by keeping hmax ¼ 0 in the simulation. In this case,cracks maintain a linear shape and the relative offset distance remains constant at theinitial value, H, for the whole simulation. Simulations were conducted under thesame initial conditions as the case of Fig. 8(a) and (b), by maintaining a linear shape.The results on the relationship between time and distance Rx are shown in Fig. 12,and the relationship between time and the K value of the outer crack tip KðBFMÞ isshown in Fig. 13. These two figures suggest that the interaction between multiplecracks does not depend on the inner crack tip shape. The intensity of the interactionincreases as the cracks get closer to each other. Therefore, if the objective of the


simulation is defined as evaluating the relationship between time and parameter Rx

the assumptions relating to inner crack tip behaviour, such as those for Eqs. (2) and(3), as well as treatment of mode II, become insignificant to the simulation. Thecrack shape can be approximated by a line in the simulation.

As described in Fig. 7, in the simulation a curved crack shape is expressed by aseries of short straight lines. The number of these short lines increases as the numberof steps in the simulation increases, thereby increasing calculation time. However, by

Fig. 12. Comparison of results of crack growth simulation with curved and linear shape cracks

(2c1 ¼ 2c2 ¼ 0:1 mm) (open: curved shape cracks, solid: linear shape cracks).

Fig. 13. Comparison of results of crack growth simulation with curved and linear shape cracks

(2c1 ¼ 2c2 ¼ 0:1 mm) (open: curved shape cracks, solid: linear shape cracks).


approximating the curved crack shape by a line, the calculation time can be reduceddrastically. This approximation makes it possible to perform the simulation fornumerous cases.

As for the simulation that would be done in the latter section of this paper, thecracks are assumed to propagate with a direction of hmax ¼ 0 and to maintain alinear shape and initial value of distance H.

3.2.5. Relationship between interaction and initial positionAs shown in Fig. 9, the influence of the interaction on the crack growth behaviour

depends on the initial relative position of cracks. Also, the interaction does not al-ways accelerate crack growth. In order to evaluate the relationship between therelative position of two cracks and the influence of the interaction on crack growthbehaviour, simulations for 2c1 ¼ 2c2 ¼ 0:1 mm with various position conditionswere conducted. The time for the distance Rx to reach Rx ¼ 4 mm, which is defined asts were investigated for various initial S and H values. Fig. 14 shows the contour plotof ts normalized by to which corresponds to the case ts for H ¼ 1. Fig. 14 shows thatif the initial position conditions S, H were included in the region ts=to > 1, the crackgrowth would be accelerated by the interaction. On the other hand, crack growthwould be restricted by the interaction if the initial conditions were included in theregion of ts=to < 1. Since the ts=to < 1 area occupies the most part in Fig. 14, cracksare accelerated in the case of most combinations of S and H. Therefore, it is deducedthat crack growth was accelerated by the interaction in experiments, due to the manycracks observed on the surface, as shown in Fig. 2. This acceleration may havecaused the drastic change in crack growth rate after 1500 h.

Fig. 14. Contour plot of ts=to.


3.2.6. Influence of relative crack lengthAs shown in Figs. 8(e) and 11, the interaction between multiple cracks is influ-

enced not only by the relative position but also the relative length of cracks. Takinginto account the fact that cracks usually have different lengths, it is essential toevaluate the influence of the interaction between cracks of different length. For ex-ample, simulations were conducted to estimate the time ts with initial conditions ofðS=c1;H=c1Þ ¼ ð8; 2Þ and ð8; 8Þ for various crack length conditions. The ts is the timethat the distance Rx takes to reach Rx ¼ 4 mm, as quoted in Fig. 14. And crack lengthwas changed from c2 ¼ 0 to 0.1 mm under a constant c1 ¼ 0:1 mm condition. Theresults are shown in Fig. 15. In this figure ts is normalized by t2o which corresponds

Fig. 15. Relationship between normalized exposure time to Rx ¼ 4 mm and crack length ratio c2=c1. Initialconditions (a) S=c1 ¼ 8, H=c1 ¼ 2, 2c1 ¼ 0:1 mm, (b) S=c1 ¼ 8, H=c1 ¼ 8, 2c1 ¼ 0:1 mm.


to the ts case of c2 ¼ 0. In both cases––ðS=c1;H=c1Þ ¼ ð8; 2Þ and ð8; 8Þ––ts takes itsminimum value at c2=c1 ¼ 1 and increases as c2 becomes small compared to c1. And,finally, ts is saturated to the normalized ts ¼ 1. This tendency of change in the pa-rameter ts against c2=c1 was seen under the other initial position conditions, exceptfor the case where initial positions are defined by ts=to > 1, as shown in Fig. 14. Inthat case, the normalized ts values exceeded 1 for all c2=c1 conditions. If the initiallength c2 is relatively small, the growth rate of crack 2 becomes relatively smallcompared to that of crack 1, due to the difference in crack length. This difference ingrowth rate tends to increase the difference in crack length. Finally, crack 2 iscovered by crack 1, as shown in Fig. 8(e), and the growth of crack 2 stagnates due tothe stress shielding effect of crack 1. Therefore, maximum crack growth accelerationoccurs for interactions where cracks are of the same length. On the other hand, if thelength of crack 2 is significantly shorter than that of crack 1, the interaction becomessufficiently small to allow the effect of crack 2 on the growth behaviour of crack 1 tobe ignored. Therefore, the normalized ts is saturated at 1.

As shown in Fig. 14, for most combinations of relative position S and H, crackstend to be accelerated by interaction, thereby making ts smaller. In order to considerthe influence of the interaction on crack growth behaviour, the influence of thedifference in crack lengths must be considered, as in Fig. 14. The interaction can beignored when the length of crack 2 decreases to a certain level, which is c2=c1 ¼ 0:36in Fig. 15(a) and c2=c1 ¼ 0:73 in Fig. 15(b). This threshold was evaluated for all S, Hcases and the results are plotted in Fig. 16. If the relative length of crack 2 is smallerthan that of the corresponding data plotted in Fig. 16, the interaction can be ignoredfor the growth evaluation of crack 1. The c2=c1 value becomes small as the offset

Fig. 16. Contour plot for threshold crack length ratio c2=c1.


distance H becomes small. This means that cracks are easily accelerated by theinteraction when they are arranged in a line perpendicular to the direction ofstress, even if they are of different length. On the other hand, when their offsetdistanceH is large, the interaction tends to become small, depending on their relativelength.

In the experiment, as shown in Fig. 2(b), many short cracks were observed on thesurface. However, according to the results shown in Fig. 16, most of these shortcracks have little influence on the growth behaviour of the long, main crack. This isbecause these crack lengths can be regarded as sufficiently short relative to thethreshold given in Fig. 16. This means that the interaction between multiple crackshas little influence on the main crack growth behaviour if the main crack becomessignificant and sufficiently long compared with other short sub-cracks. On the con-trary, if all cracks remain almost the same length, cracks would be easily acceleratedby the interaction. Thus, the influence of the interaction of multiple cracks on crackgrowth depends largely on the stage of crack growth.

4. Summary and conclusions

Through the PWSCC experiments and crack growth simulations, the influence ofmultiple cracks on crack growth behaviour was evaluated. The following conclusionscould be made:

1. The influence of the interaction between multiple cracks on crack growth behav-iour greatly depends on the relative position of cracks. Also, the influencecan be classified into two categories, acceleration and suppression of crackgrowth.

2. In most cases, approaching cracks tend to be accelerated by interaction.3. This acceleration is brought by the enhanced K value due to the interaction of

outer crack tips. If the offset distance of two cracks is sufficiently small, the growthrate is very similar to that of a combined crack.

4. Therefore, coalescence did not need to be considered in the simulation if the ob-jective of the simulation was defined as evaluating the relationship between timeand crack length.

5. Even if crack shape is approximated by a line in the simulation, there is little in-fluence on the crack growth behaviour because the interaction does not depend onthe change in progress direction of the inner crack tips.

6. Interaction is influenced not only by the relative position but also by the relativelength of cracks. This influence is strongest when crack lengths are equal and de-creases as the difference in crack length increases. If the crack length difference isgreater than a certain level, the interaction is sufficiently small to allow the influ-ence of the interaction on crack growth behaviour to be ignored.

7. According to these results, it can be deduced from these experiments that cracksgrowth is accelerated by the interaction between multiple initiated cracks, whichcan cause drastic changes in crack growth rates at the final stages of experiments.


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Influence of interaction between multiple cracks on stress corrosion crack propagation