Simulation for intergranular stress corrosion cracking based on a three-dimensional polycrystalline model

Engineering Fracture Mechanics 76 (2009) 386–401

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Engineering Fracture Mechanics

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Simulation for intergranular stress corrosion cracking basedon a three-dimensional polycrystalline model

Masayuki Kamaya a,*, Mitsuhiro Itakura b

a Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japanb Center for Computational Science & e-System, Japan Atomic Energy Agency, Higashiueno 6-9-3, Taito-ku, Tokyo 110-0015, Japan

a r t i c l e i n f o

Article history:Received 8 February 2008Received in revised form 17 October 2008Accepted 14 November 2008Available online 25 November 2008

Keywords:Intergranular crackCrack initiationShort crackCrack growthDamage mechanicsPolycrystalline modelStress corrosion crackingFinite element analysis

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.11.004

* Corresponding author. Tel.: +81 770 379114; faE-mail address: [email protected] (M. Kamaya)

a b s t r a c t

In stress corrosion cracking of stainless steel, two different schemes of analysis of crackgrowth should be employed for the crack initiation phase and crack growth phase. How-ever, this distinction is not clear-cut in the crack initiation phase, since the vicinity ofthe pre-existing crack is a preferential area of crack initiation due to concentration ofstress. Therefore, initiation of crack tends to occur at the tip of a pre-existing crack andit can be regarded as crack growth. In this study, the contribution of this type of apparentcrack growth, referred to as initiation dominant growth (IDG), to crack growth was evalu-ated by a Monte Carlo simulation. A three-dimensional polycrystalline body was generatedby Voronoi tessellation. The cracks were assumed to grow along grain boundaries. Theeffect of stress-concentration around pre-existing cracks was taken into account by apply-ing the finite element method. Initiation and propagation of the cracks were modeledbased on concepts of damage mechanics. The simulation could reproduce the changes innumber of cracks and the sum of crack length obtained experimentally as well as prefer-ential crack initiation at the stress-concentration zones and suppression of crack initiationin stress-shielding zones. It was shown that the contribution of IDG to crack growth waslarge for small cracks, and that damage by crack initiation accounted for more than 50%of total damage even when the length of a crack was 0.6 mm at the surface.

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1. Introduction

In predicting the duration of plant components, it is important to know the period of incubation prior to crack initiationsand the growth of the initiated small cracks, because these periods comprise most of the duration. Since multiple smallcracks tend to be initiated in a confined area, the influence of coalescence and mechanical interaction between cracks shouldbe taken into account. In particular, in stress corrosion cracking (SCC), coalescence of small cracks is believed to trigger stea-dy growth to macroscopic cracks [1,2]. The incubation period has random nature and the growth rate of small cracks is notsteady depending on microstructure. Therefore, to assess the initiation and growth of small cracks, Monte Carlo simulationshave been developed for creep [3] and fatigue [4,5] as well as SCC [6,7]. In these simulations, the mechanical interactionsbetween multiple cracks and the coalescence of approaching cracks were considered in determining growth behavior. How-ever, the cracks were assumed to be initiated randomly in space.

In the previous study by the current authors, the crack initiation and growth of multiple cracks due to SCC were inves-tigated using sensitized Type 304 stainless steel under uniform tensile load [8]. The surface of the specimens was ob-

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M. Kamaya, M. Itakura / Engineering Fracture Mechanics 76 (2009) 386–401 387

served through a transparent acrylic cell. It was revealed that the period of incubation prior to crack initiation dependson the applied stress [8]. Namely, a high stress reduced the incubation period. Since stress tends to be large locally in thevicinity of pre-existing cracks due to stress-concentration around the tip, in SCC of stainless steel under uniform tensilestress, the vicinity of pre-existing cracks was found to be a preferred area for crack initiation. On the other hand,crack initiation was suppressed along crack faces on the surface due to the stress-shielding effect. If a crack is initiatedat the tip of a pre-existing crack, this initiation can be regarded as growth of a pre-existing crack from the view pointof change in crack shape. The apparent growth caused by crack initiation was predominant, and such growth was referredto as initiation dominant growth (IDG) [9]. In order to incorporate the effect of the IDG on the crack growth, it isinevitable to consider the influence of the local stress on crack initiation, although it has not been done in the previousstudy.

In this study, the effects of IDG on crack growth were evaluated using a Monte Carlo simulation that accounts for theinfluence of local stress on crack initiation. The target of the simulation was SCC of stainless steel, which was tested inthe previous study [8]. In the tests, multiple cracks were initiated and grew along grain boundaries. The changes in crackdistribution, number of cracks, and crack length were recorded using an in-situ observation system. In the simulation, a poly-crystalline body was generated by Voronoi tessellation, and finite element analysis (FEA) was performed for the polycrystal-line body to evaluate the local stress distribution induced by pre-existing cracks [10]. The initiation and propagation ofcracks were modeled based on concepts of damage mechanics. We first present the procedure and results of the SCC test.We then present details of the finite element analysis of the polycrystalline body in addition to the damage model of crackinitiation and propagation. Then, simulation results obtained with several different model parameters are shown. Finally, thecontribution of IDG to the observed SCC behavior is discussed.

(a) 33h

1mm

(b) 45h

(c) 75h

Fig. 1. Surface images observed by CCD camera system in the experiment [9].

388 M. Kamaya, M. Itakura / Engineering Fracture Mechanics 76 (2009) 386–401

2. Cracking behavior observed experimentally [8]

In the experiment, SCC initiation behavior of sensitized stainless steel was investigated through observation of specimensurface using an in-situ crack observation system [11,12]. The material was Type 304 stainless steel (C:0.07, Si:0.70, Mn:1.56,P:0.023, S:0.003, Ni:8.72, Cr:18.52 mass%, Fe: bal.), which was annealed at 1323 K for 1.8 ks as solution treatment, followedby 1023 K for 6 ks and 773 K for 86.4 ks as double sensitization treatment. The 0.2% proof strength and tensile strength of thematerial were 200 MPa and 711 MPa, respectively. A plate tensile specimen (gauge size: 20 mm long, 4 mm wide, and 2 mmthick) was used and the surface was polished using #600 emery paper.

The specimen was subjected to a constant stress of 160 MPa in a solution of 10�5 kmol/m3 Na2SO4 exposed to air in orderto dissolve oxygen as an oxidant. The test temperature was 368 K. The specimen surface was observed with a CCD camerasystem every 3 h from the start of the SCC tests. Details of this apparatus are described elsewhere [8].

The first cracks were observed at 15 h from the start of the test, and the specimen was fractured at 78 h. Fig. 1 shows theobserved surface of the specimen at 33 h, 45 h, and 75 h [8]. Many cracks were initiated, some of which coalesced during thetests. The cracks increased in size due to coalescence with adjacent cracks rather than growth of individual cracks. Thechange in number of cracks during the test is shown in Fig. 2a [8]. The increase in number of cracks saturated at about60 h, and was followed by decrease in number of cracks. Fig. 2b shows the change in sum of crack length on the surface[8]. The changes in observed images confirmed that new crack initiation tended to occur near pre-existing cracks, and thatit contributed to crack growth by coalescence.

0

20

40

60

80

100

120

0 20 40 60 80

Exposure time, h

Tot

al c

rack

leng

th, m

m

0

50

100

150

200

250

300

350

400

450

500

0 20 40 60 80

Exposure time, h

Num

ber

of c

rack

s

(a) Change in the number of cracks with exposure time.

(b) Change in sum of crack length with exposure time.

Fig. 2. Experimental results [9].


3. Procedure of analysis

3.1. Polycrystalline body

Fig. 3 illustrates the geometry of the polycrystalline body subjected to analysis. The body was 3 mm long, 1 mm wide, and0.5 mm thick, and almost a quarter of the size of the specimen used in the experiment. A uniform tensile stress, So, was ap-plied on the top and bottom planes along the y-axis. The body was divided into 6000 grains by random Voronoi tessellation.Random numbers were used for determining coordinates of nucleus for each crystal grain. The cumulative probability ofgrain boundary length at the surface is shown in Fig. 4 together with that of the test material. The length of twin boundaries

So

3 mm

1 mm 0.5 mmzx

y

Fig. 3. Geometry and boundary condition of analyzed body.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.05 0.10 0.15 0.20

Length of grain boundary c, mm

Cum

ulat

ive

prob

abili

ty F

Experiment

Simulation

c

Fig. 4. Distribution of grain boundary length.


was excluded from the data for the test material. The agreement of these data verifies the statistically successful reproduc-tion of grain arrangement. Cracks were assumed to initiate and grow through the grain boundaries.

3.2. Local stress calculation

The driving force of crack initiation and propagation was evaluated by FEA. Finite element nodes were placed at the ver-tices of grains, centers of grain surfaces, and centers of grains. Each grain was decomposed into tetrahedral elements, con-taining two vertex nodes, a surface center node, and a grain center node (Fig. 5). To glue the grains together, a thin elementwas placed between the elements at each grain boundary. Gluing was achieved by linear elastic spring of which stiffness was200 TPa. When a crack was initiated or passed the grain boundary, the elastic constants of the corresponding gluing were setto zero. Elastic stress was calculated for the cracked body. The local stress field around pre-existing cracks was thus consid-ered in the simulation. Isotropic elasticity was assumed and Young’s modulus and Poisson’s ratio were 200 GPa and 0.3,respectively. In-house solver was used for FEA.

In this simulation, the normal stress acting on each grain boundary (hereafter, GB normal stress, denoted as Sn) (Fig. 6)was used for crack initiation and growth as described later. GB normal stress was calculated from displacement of the facecenter node (Fig. 5) of each grain boundary using elasticity of glue element. Fig. 7a shows the relationship between GB nor-

Fig. 5. Construction of finite elements. (a) Nodes are placed at vertex, face center, and body center of each grain. (b) Each grain is glued together by thintriangular element.

Fig. 6. Definition of GB normal stress and inclination of grain boundary p.


mal stress of grain boundaries on the surface (existing at z = 0, y < 0.3 mm and y > 2.7 mm) and its inclination against to theloading direction (angle p in Fig. 6) calculated for uncracked polycrystalline body. The normalized GB normal stress, Sn/So,should be identical to cos2p, although some deviations are observed. The relationship between the deviation of Sn/So fromcos2p and the length of grain boundary on the surface is shown in Fig. 7b. The error in the calculation of Sn was causedby coarse mesh division and was significant for relatively short grain boundaries. It should be noted that number of grainboundaries with large error of more than 0.2 in (Sn/So�cos2p) was 26, and this was less than 1.7% of total number of grainboundaries.

3.3. Model of crack initiation and growth

The initiation of cracks was modeled based on damage mechanics as applied in the previous study [9]. This model de-scribed well the cracking behavior of SCC of Type 304 stainless steel in high temperature water. In this model, damage is

=1582

0.0

0.2

0.4

0.6

0.8

1.0

0. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Normalized GB normal stress S n/S o

Incl

inat

ion

of g

rain

bou

ndar

y co

s2p

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 0.05 0.10 0.15 0.20

Length of grain bundary,mm

Err

or i

n G

B n

orm

al s

tres

s S n

/So

-cos

2 p

N =1582

(a) Change in GB normal stress with inclination of grain boundaries

(b) Relationship between the magnitude of error and length of grain boundaries

N

Fig. 7. Error in finite element analysis for GB normal stress.


assumed to accumulate at grain boundaries before cracking. When the accumulated damage exceeds the resistance to crack-ing, a crack is initiated at the grain boundary. The rate of accumulation of damage depends on the stress acting on the grainboundary. At a grain boundary near the tip of a pre-existing crack, stress is exceedingly high due to the stress singularity atthe crack tip, and cracking easily occurs. Initiation of cracking near the tip of a pre-existing crack causes IDG, or induces coa-lescence to the pre-existing crack. Not only crack initiation but also crack propagation was considered in this model. Cracksgrew in size by breakdown of crack tips in the process zone; this type of crack growth is referred to as ‘‘propagation” in thisstudy.

In the simulation, in accordance with the model, damage accumulated at each grain boundary. It has been pointedout that the normal force acting on the grain boundary is predominant in crack initiation in SCC. Cracks tended to beinitiated at the grain boundaries perpendicular to the loading direction [13]. Damage then accumulated based on GBnormal stress. The rates of accumulation of damage due to initiation (Di) and propagation (Dp) were assumed to be asfollows:

dDi

dt¼ Sn

So

� �m

; ð1Þ

dDp

dt¼ Fp

Sn

So

� �m

; ð2Þ

where Fp is a constant controlling the ratio of the contribution of propagation. The constant m represents the stress depen-dency of accumulation of damage. GB normal stress depends on the inclination of the grain boundary against the direction ofloading as shown in Fig. 7a. Cracks thus tend to be initiated at grain boundaries perpendicular to the direction of loading. Thedamage accumulation rate for Di successfully simulated crack initiation behavior of SCC. The effectiveness of GB normalstress on crack propagation can be deduced from well known relationship between crack growth rate and stress intensityfactors of Mode I. The total damage at each grain boundary (D) was calculated with the following equation:

Generate polycrystalline body

Divide into finite elements

Random numbers

Finite element method

Calculate rate of damage accumulation (dD/dt)

Break the g.b. when:

(only one g.b. breaks at a time)

Calculate GB normal stress (Sn)at each g.b.

Start

End

End of simulation ?

calculate time interval (dt)

t t + dt

( )k kN

k

dDR

dt>Σ

←

Fig. 8. Procedure of the simulation.

Fig.


dDdt¼ dDi

dtþ dDp

dtðfor grain boundaries at crack tipsÞ; ð3Þ

dDdt¼ dDi

dtðfor grain boundaries at surfaceÞ: ð4Þ

Damage accumulated at grain boundaries existing on the surface and ahead of the crack tip. If the grain boundary satisfiedthe following equation, a crack was initiated at the grain boundary:

XNk

k

dDðkÞ

dt> R; ð5Þ

where Nk is the number of steps and dD(k)/dt is the dD/dt of kth step. The parameter R corresponds to the resistance to crack-ing, and is thought to be a function of constant value and a time-dependent variable, which is related to, for example, envi-ronmental factors such as grain boundary corrosion and passivation, which plays an important role in the resistance to crackinitiation in stainless steel. With the assumption that the contribution of time-dependent factors to the change in R is neg-

A CB

Cracks are initiated from this surface

AB C

(b) 0.71 to(number of cracked g.b.: 700)

(c) 1.0 to(number of cracked g.b.: 1500)

(a) 0.52 to(number of cracked g.b.: 350)

A CBA CB

9. Crack distribution obtained by the simulation (m = 4, Fp = 1, VCSL = 0). Circled marks in (b) are the cracks newly initiated between the times.


ligible, R remained a constant during the simulation. The same magnitude of R = 1 was assigned to each grain boundary ex-cept some special grain boundaries. Several studies pointed out that there are special grain boundaries which have highresistance to SCC [14–16]. Such grain boundaries were modeled by an infinite resistance to cracking. The ratio of resistantgrain boundary was represented by VCSL. For example, VCSL = 0.3 means 30% grain boundaries, which were selected randomly,possess high resistance, whereas VCSL = 0 means there is no resistant grain boundary and all the grain boundaries have thesame resistance R. In order to exclude edge effects, grain boundaries at positions with y < 0.3 mm and y > 2.7 mm were as-sumed to be the resistant grain boundaries.

The cracks were assumed to be initiated at the z = 0 face, and to grow into the body. The time interval separating stepswas controlled so that only one grain boundary split at each step. FEA was performed at each step considering the crack dis-tribution in order to obtain GB normal stress for damage calculation by Eqs. (3) and (4). The simulation was continued untilthe number of cracked grain boundaries reached 1500. The flow chart of the simulation is shown in Fig. 8.

4. Results of analysis and discussion

4.1. Results of simulation for experiment

Fig. 9 shows the changes in distribution of cracks, which are shown in thick lines on the three surfaces A, B, and C. Theconstants were set to m = 4 and Fp = 1, which were determined by trial and error, so that the crack distribution and thechange in number of cracks becomes similar to that of the experiments. Ratio of resistant grain boundary was VCSL = 0.The cracks grow in the depth direction as well as along the surface. The circles shown in Fig. 9b indicate newly initiated

0

5

10

15

20

25

0.0 0.2 0.4 0.8 1.0

Time, t /t o

Sum

of

crac

k le

ngth

,mm

Fig.9(a) Fig.9(b)

Fig.9(c)

0

50

100

150

200

250

0.0 0.2 0.4 0.6 0.8 1.0

Time, t /t o

Num

ber

of c

rack

s

Fig.9(a)Fig.9(b) Fig.9(c)

(a) number of cracks

(b) sum of crack length at the surface

0.6

Fig. 10. Change in the number of cracks and sum of crack length at the surface with exposure time (simulation: m = 4, Fp = 1 , VCSL = 0).


cracks between 0.52to and 0.71to, where to is the total time of simulation. A cluster of cracked grain boundaries in contactwith each other were regarded as one crack. Cracks were preferentially initiated near pre-existing cracks or in uncrackedligaments between cracks. Stress between the cracks was relatively large due to mechanical interaction between them,and the cracks were preferentially initiated in such regions. These cracks induced coalescence of cracks and contributedto the decrease in the number of cracks as shown in Fig. 10a, which shows the change in the number of cracks. The changein the number of cracks exhibits a peak at 0.8to, as noted experimentally. As schematically shown in Fig. 11, when crackswere initiated in the stress-concentration zone, they contributed to the growth of main cracks by coalescence, and the num-ber of cracks thereby decreased. At the same time, crack initiation was suppressed in the stress-shielding zones, while crackinitiation at this region makes the number of cracks increase, since it is difficult for coalescence to occur there as shown inFig. 11b. The simulation properly reflected initiation behavior, that is to say, IDG as well as the coalescence of the cracks asobserved in the experiment. It should be noted that cracks were counted regardless of size, although the cracks over 0.05 mmcould be clearly identified in the experiment. As shown in Fig. 4, more than half of grain boundaries are less than 0.05 mm.Therefore, the number of small cracks tends to be larger in the simulation than that in the experiment. However, the differ-

(b) Randomly initiated cracks hardly coalesce with other cracks and the number of cracks increases

(a) Initiations at vicinity of pre-exiting crack tip contribute to the growth of main crack and the number of cracks is reduced.

Stress concentration zone

Stress shielding zone Newly initiated crack

Pre-existing crack

Fig. 11. A schematic drawing of representing the influence of crack initiation on the crack growth and change in number of cracks.


ence in the small cracks is deduced to show little influence on the conclusions of this study, because the small cracks haveminor effect on local stress, which is the main concern of this study. Fig. 10b shows the change in the sum of crack length atthe surface. It increased monotonically after the first crack initiation, as observed experimentally.

As shown in Fig. 7, some errors were included in calculation, especially for grain boundaries of large Sn. This reduced incu-bation period of several grain boundaries and caused discontinuous change in number of cracks at t = 0.35to as shown inFig. 10a. However, since the length of such grain boundaries is relatively short, it is expected that the influence of the errorin Sn calculation on the results and discussion is limited.

4.2. Effects of factors inducing damage on crack distribution

The crack distributions at the end of the simulation are shown in Fig. 12 for the cases of m = 2 and 8 (Fp = 1, VCSL = 0). Anincrease in the constant m corresponds to a higher dependence of stress on accumulation of damage. Crack initiation wasthus intensified in the stress-concentration zones and suppressed in the stress-shielding zones. The density of the crackedgrain boundaries increased as m increased. Since the number of cracked grain boundaries at the end of the simulation wasthe same, fewer cracks on the surface imply active growth in the depth direction. In the case of m = 8, a large crack appeared

A CB(a) m = 2 (b) m = 8

A CB

Fig. 12. Crack distribution obtained by the simulation (Fp = 1, VCSL = 0).

0

50

100

150

200

250

300

350

0.0 0.4 0.8 1.2 1.6 2.0

Normalized GB normalstress S n/S o

Num

ber

of g

rain

bou

ndar

ies

t = 0 (N=1582)m = 2 (N=906)m = 4 (N=1079)m = 8 (N=1251)

Fig. 13. Distribution of GB normal stress at the surface. (simulation: Fp = 1, VCSL = 0).


at the bottom of the body and passed through the thickness of the plate. The area of the stress-shielding zone is large fordeeper cracks [9]. Accordingly, only a limited number of cracks were initiated near the main crack.

Fig. 13 shows the distribution of GB normal stress acting on uncracked grain boundaries on the surface. In the initial con-dition, GB normal stress depends only on the inclination of the grain boundary against the loading direction as shown inFig. 7a. At the grain boundaries of large GB normal stress, damage accumulated rapidly and cracks were initiated after a shortperiod of incubation. Therefore, the ratio of uncracked grain boundaries with large GB normal stress was small at the end ofthe simulation. Meanwhile, the shielding effect increased the ratio of grain boundaries with small GB normal stress. As noted,if a crack becomes long and deep, the range of the stress-shielding zone is extended. Hence, the ratio of uncracked grainboundaries with small GB normal stress was relatively high in the case of m = 8.

The changes in number of cracks are shown in Fig. 14a. In the case of m = 2, the number of cracks exhibited a peak at 0.8to,as seen in the case of m = 4, although it decreased rapidly after the peak. The existence of too many cracks on the surfaceresulted in coalescence due to limitations in the path of cracks. The change in the sum of crack length on the surface is shownin Fig. 14b. There was no significant difference between the cases, although the values obtained depended on the condition.

Fig. 15 shows the results of the simulation for the conditions of Fp = 0 and 4 under m = 4 (VCSL = 0). The large Fp increasesthe contribution of propagation to crack growth. The crack distributions were similar to those for m = 2 and 8, which areshown in Figs. 12a and 11b, respectively. The change in number of cracks and sum of crack length on the surface are shownin Fig. 16. The effect of Fp was similar to that of parameter m as regards the distribution of cracks. A large Fp resulted in accel-eration of crack growth on the surface and in the depth direction as well as suppression of the number of cracked grainboundaries on the surface.

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0

Time, t /t o

Sum

of

crac

k le

ngth

, mm

m = 2

m = 4

m = 8



0

50

100

150

200

250

0.0 0.2 0.4 0.6 0.8 1.0

Time, t /t o

Num

ber

of c

rack

s

m = 2

m = 4

m = 8

Fig. 14. Change in the number of cracks and sum of crack length at the surface with exposure time (simulation: Fp = 1 , VCSL = 0).

A CB

(a) Fp = 0 (b) Fp = 4

A CB

Fig. 15. Crack distribution obtained by the simulation (m = 4, VCSL = 0).


4.3. Ratio of IDG on crack growth

By experiment and in current simulation, it has been shown that IDG plays an important role in crack growth of SCC. Toquantify the contribution of crack initiation (IDG) to growth, the ratio of Di to total damage was examined. Fig. 17 shows therelationship between maximum surface crack length and the ratio of Di to total damage D of cracked grain boundariesbelonging to the crack. Since the largest crack was not the same throughout the simulation, the result shows discontinuouschange at several points. Damage was fully induced by Di when a crack was initiated. Then the damage due to crack prop-agation Dp began to accumulate. The surface length at the end of the simulation was approximately 0.6 mm. At that length,the ratio of Di was more than 50% of the total damage. This implies that the effect of IDG on crack growth is large and shouldbe considered in predicting the growth of small cracks. Use of the current three-dimensional polycrystalline model togetherwith modeling of initiation and growth by damage mechanics is one way of performing such prediction.

4.4. Influence of resistant grain boundaries

The simulations were performed under different ratio of resistant grain boundaries (VCSL = 0.1, 0.3 and 0.5) under the con-ditions of m = 4 and Fp = 1. Fig. 18 shows the crack distribution for VCSL = 0.5. Although a half of the grain boundary could notsplit, the crack distribution did not show significant difference from that of VCSL = 0, which is shown in Fig. 8. The change innumber of cracks is shown in Fig. 19. Increase in the ratio of resistant grain boundary made the time of simulation longer aspointed out in Ref. [16]. However, the characteristics of change in number of cracks were almost the same. Namely, the num-ber of cracks exhibited a peak and followed by gradual decrease. As discussed in this study, the local stress plays am impor-tant role in the crack initiation and growth. Cracks can grow by IDG even if the resistant grain boundaries exist ahead of cracktip.

5. Summary and conclusions

In this study, to simulate the cracking behavior of SCC and investigate the influence of crack initiation on growth behavior,a three-dimensional polycrystalline body was used for a Monte Carlo simulation. In the simulation, both the local stress dis-tribution caused by pre-existing cracks and inclination of the grain boundaries were taken into account in determining crackinitiation and propagation, based on damage mechanics. The simulation can imitate the changes in number of cracks andsum of crack lengths obtained experimentally. The conclusions obtained are as follows:

(1) Preferential crack initiation in the stress-concentration zones can be simulated by considering influence of local stresson crack initiation.

(2) Suppression of new crack initiation in the stress-shielding zones together with coalescence of cracks resulted indecrease in the number of cracks.

0

5

10

15

20

25

Time, t /t o

Sum

of

crac

k le

ngth

,mm

Fp = 0

Fp = 1

Fp = 4

0

50

100

150

200

250

Time, t /t o

Num

ber

of c

rack

s

Fp = 0

Fp = 1

Fp = 4



0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 16. Change in the number of cracks and sum of crack length at the surface with exposure time (simulation: m = 1 , VCSL = 0).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8

Surface length, mm

Dam

age

by in

itiat

ion/

Tot

al d

amag

e

Fig. 17. Relationship between surface crack length and ratio of damage by initiation to total damage of cracked grain boundaries. (simulation: m = 4, Fp = 1,VCSL = 0).


Fig. 18. Crack distribution obtained by the simulation (m = 4, Fp = 1, VCSL = 0.5).

0

50

100

150

200

250

0.0 1.0 2.0 3.0 4.0 5.0

Time, t /t o(VCSL = 0)

Num

ber

of c

rack

s

V = 0.0

V = 0.1

V = 0.3

V = 0.5

VCSL

VCSL

VCSL

VCSL

Fig. 19. Change in the number of cracks with exposure time normalized by total time for VCSL = 0 (simulation: m = 4, Fp = 1).


(3) The dependence of stress on accumulation of damage and the relative contribution of crack propagation affect the dis-tribution of cracks.

(4) The contribution of IDG to crack growth was greater than 50% when the length of the crack was 0.6 mm at the surface.

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Simulation for intergranular stress corrosion cracking based on a three-dimensional polycrystalline model