Growth prediction of two interacting surface cracks of dissimilar sizes

Engineering Fracture Mechanics 77 (2010) 3120–3131

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Engineering Fracture Mechanics

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Growth prediction of two interacting surface cracks of dissimilar sizes

Masayuki Kamaya a,⇑, Eiichi Miyokawa b, Masanori Kikuchi b

a Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japanb Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 November 2009Received in revised form 27 April 2010Accepted 8 August 2010Available online 12 August 2010

Keywords:Stress intensity factorFinite element analysisCrack propagationMultiple cracksParallel cracks

0013-7944/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.engfracmech.2010.08.008

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

When multiple cracks approach one another, the stress intensity factor changes due to theinteraction of the stress field. This causes variation in the crack growth rate and shape ofcracks. In particular, when cracks are parallel to the loading direction, their shape becomesnon-planar due to the mixed mode stress intensity factor. In this study, the growth of inter-acting surface cracks was simulated by using the S-version finite element method, in whicha local detailed finite element mesh (local model) is superposed on a coarse finite elementmodel (global model) representing the global structure. First, simulations were performedfor fatigue crack growth experiments and the method validity was shown. Second, simula-tions were conducted for various relative sizes and spacings of twin cracks. It was shownthat the offset distance and the relative size were both important parameters to determinethe interaction between two surface cracks; the smaller crack stopped growing when thedifference in size was large. It was possible to judge whether the effect of interactionshould be considered based on the correlation between the relative spacing and relativesize.

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1. Introduction

Multiple cracks are likely initiated due to stress corrosion cracking and fatigue in nuclear power plant components. Sincemechanical interaction between multiple cracks accelerates the crack growth and reduces the time to failure, it is importantto incorporate the acceleration in predicting the crack growth for fitness-for-service (FFS) evaluation. In engineering assess-ments for FFS, influence of the interaction is taken into account conservatively by applying the combination rule [1–3]. In theASME boiler and pressure vessel code (e.g. Section XI, Articles IWA-3330) [4] and JSME fitness-for-service code [5], adjacentcracks are treated as a coalesced one when their relative spacing meets the criterion. However, it was pointed out that thecriterion for the combination rule is excessively conservative in some cases [6–10]. In particular, the effect of a difference incrack size (hereafter referred as ‘‘relative size effect”) is not taken into account in the aforementioned codes, although it hasrelatively large impact on the magnitude of the interaction [11,12]. The conservativeness in the current code increases as thedifference in crack size increases, because the magnitude of interaction is largest for the cracks of the same size [13]. Sincemost interacting cracks have dissimilar sizes, it is important to take the relative size effect into account in the criteria for areasonable assessment.

Although extensive experimental studies have been conducted in order to investigate the influence of interaction on thecrack growth [7,8,14,15], the results for dissimilar cracks are very limited, particularly for surface cracks in non-coplanar po-sition [16,17]. In order to clarify the influence of the difference in crack size in addition to the relative spacing of interacting

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Nomenclature

a1 crack depth of crack 1a2 crack depth of crack 2da/dN crack growth rateC material constant for crack growth ratec1 half crack length of crack 1c2 half crack length of crack 2{f} nodal force vectorH offset distance between cracksh height of the bodyDKI mode I stress intensity factor rangeDKII mode II stress intensity factor rangeDKIII mode III stress intensity factor rangeKIc1 KI at inner crack tip of crack 1KIc2 KI at outer crack tip of crack 2DKeq equivalent stress intensity factor range[K] stiffness matrixn material constant for crack growth rateND number of cycles until Rx becomes more than 40 mm for interacting cracksNS number of cycles until Rx becomes more than 40 mm for a single crackRx span length of two cracks (identical to 2c1 for a single crack)S horizontal distance between crack tipst thickness of the body{u} nodal displacement vectorW width of the bodyuo deflection angle in the crack growth simulation

M. Kamaya et al. / Engineering Fracture Mechanics 77 (2010) 3120–3131 3121

cracks, experiments under various geometrical cases have to be conducted. However, it is difficult to control the fatigue crackgrowth from machined twin notches of different size.

On the other hand, the growth behavior of interacting cracks can be simulated by evaluating the stress intensity factor(SIF) and the growth rate obtained by experiments [18–21]. By performing the crack growth simulation under extensive con-ditions, it is possible to figure out the relative size effect of interacting cracks. Since SIF of the interacting cracks variesdepending on relative spacing and size of cracks, it is necessary to calculate SIF at each step during the simulation. Therefore,when the finite element method is applied for SIF evaluation, finite element mesh for the cracked part has to be re-con-structed according to the change in crack shape and position. However, mesh division for interacting cracks is not easy be-cause the mesh structure is intricate and it is difficult to get connectivity of nodes between interacting cracks [22]. Moreover,the crack shape is not always semi-elliptical due to an inhomogeneous interaction along the crack front. In particular, in thecase of non-coplanar cracks, the cracks deform three-dimensionally due to shear loading emanating from their non-symmet-ric relative position.

In this study, crack growth simulations were carried out to investigate the relative size effect on the growth of interactingcracks. The finite element analyses were conducted to evaluate SIF of interacting surface cracks in a non-coplanar position. Inorder to perform finite element analyses for cracks of complex shape, S-version finite element method (S-FEM) [23] was em-ployed by combining an auto-mesh generation technique, together with the fully automatic fatigue crack growth simulationsystem [24]. After a short review of the crack growth simulation system, crack growth simulations were made for crackgrowth experiments conducted in a previous study. Then, the effect of relative size and spacing of cracks on growth behaviorwas investigated based on results of simulations conducted under various conditions. Finally, the criterion for assessments inFFS was discussed.

2. Procedure for crack growth simulation

2.1. Model

Two surface cracks were assumed to be on a plate which had a thickness t, height h, and width W as shown in Fig. 1. Theplate was subjected to a cyclic axial loading of magnitude ro. At the beginning of the simulation, two cracks, crack 1 andcrack 2, were semi-elliptical in shape and located in parallel (non-coplanar position). The horizontal and offset distances be-tween the crack tips were denoted by S and H, respectively. The surface length and depth of cracks were expressed by 2c1, 2c2

and a1, a2 for crack 1 and crack 2, respectively. The span length of the two cracks on the projected plane perpendicular to theloading axis was denoted by Rx (Fig. 2). For comparison purpose, the growth of single crack was also simulated, in which Rx isthe same as 2c1.

W

h

t

2c1

a1

H

S

σo

Crack 1

Crack 2

a2

2c2

Fig. 1. Geometry of a plate containing two cracks.

Interacting cracks

Rx

Rx

Rx

Crack 2 is arrested

Single crack

Crack 2

Crack 2

Crack 1

Crack 1

Crack 1

Fig. 2. Definition of span length Rx.

3122 M. Kamaya et al. / Engineering Fracture Mechanics 77 (2010) 3120–3131

2.2. S-version finite element method (S-FEM)

SIF of the cracks were calculated by using S-FEM, which was originally proposed by Fish et al. [23]. Fig. 3a schematicallyshows a solid containing two cracks. In S-FEM, a coarse finite element mesh (global mesh) is assumed to represent the globalstructure and overlaid finite element meshes representing crack and its vicinity (local meshes) are superposed on the globalmesh as shown in Fig. 3b. Finite element formulation for this model is described by:

½KGG� ½KGL1� ½KGL2�½KL1L1� ½KL1L2�

sym: ½KL2L2�

264

375fuGgfuL1gfuL2g

8><>:

9>=>; ¼

ff Ggff L1gff L2g

8><>:

9>=>; ð1Þ

where {u} and {f} are vectors of unknown nodal displacement and nodal force, respectively. Superscript of G, L1 and L2 de-notes global mesh, local mesh 1 and 2, respectively. [KGG], [KL1L1] and [KL2L2] are stiffness matrixes of each mesh, and [KGL1]and [KL1L2] express relations between global and local meshes, which are obtained by:

½KGL1� ¼Z

XL1½BG�T½D�½BL1�dXL1 ð2Þ

½KL1L2� ¼Z

XL2½BL1�T½D�½BL2�dXL2 ð3Þ

itit

Local mesh 2

Local mesh 1

meshGlobalitit

(a) Solid with two cracks (b) Global and local mesh

Fig. 3. Schematic drawing representing basic concept of S-version finite element method.


where X denotes volume of global or local mesh and [B] is the displacement–strain matrix. By solving these equations, thedisplacement of each node can be obtained.

In S-FEM, it is not necessary to consider the geometrical connectivity between global and local meshes, and complexchanges in crack shape can be considered in finite element analyses by altering the local meshes without considering theirboundary shape. This enables us to perform crack growth simulation under various conditions of S, H and crack sizes withoutre-meshing the whole model. An auto-mesh generation technique and a fully automatic fatigue crack growth simulation sys-tem have been developed and details of the S-FEM for crack growth simulation are described elsewhere [24,25]. In this study,this system was applied to the interacting parallel surface cracks.

Fig. 4 shows an example of global and local meshes used for analyses. SIFs were calculated along the crack front at eachstep. As the crack tip was not in the pure mode I stress state, not only mode I SIF, KI, but also mode II SIF, KII, and mode III SIF,KIII, were evaluated from energy release rate, which was evaluated by the virtual crack closure method [26,27]. Poisson’s ra-tio of the material was taken to be 0.3.

2.3. Simulation procedure

The crack growth size per cycle (da/dN) was determined by:

dadN¼ CðDKeqÞn ð4Þ

where C and n are the material constants. DKeq denotes the amplitude of the equivalent SIF. da/dN and DKeq are given in m/cycle and MPam0.5, respectively. Several equations have been proposed for DKeq under the mixed mode loading condition[28,29], and that proposed by Richard et al. [29] was employed, which is defined by:

(a) Global mesh (b) Local mesh

Fig. 4. Global mesh and local mesh.

Crack −φo

Fig. 5. Crack growth direction.


DKeq ¼K I

2þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

I þ 4ð1:115K IIÞ2 þ 4ðK IIIÞ2q

ð5Þ

The change in crack shape (out-of-plane deformation) was determined by [29]:

uo ¼ � 140�jDK IIj

DK I þ jDK IIj þ jDK IIIj� 70�

jDK IIjDK I þ jDK IIj þ jDK IIIj

� �2" #

ð6Þ

where uo is the deflection angle defined in Fig. 5, and uo < 0� for DKII > 0 and uo > 0� for DKII < 0. In the simulations, the crackclosure was not considered and mean stress was assumed to be zero. Therefore, not the range but the amplitude of SIF wasconsidered in the crack growth analysis.

3. Simulations for experiments

3.1. Experimental conditions and analysis model

In the previous study [30], fatigue crack growth experiments were conducted using Type 304 stainless steel plate sub-jected to cyclic axial loading. The simulation was preformed for the following two experiments for twin interacting surfacecracks. The plate had dimensions of W = 50 mm, t = 15 mm and h = 200 mm [30]. In the experiments, by using an electrodedischarge machine, twin semi-circular surface notches of the same size were introduced perpendicular to the loading direc-tion at the center of the specimen. The positions of the notches were H = 5 mm and 10 mm under constant value of S = 0. Thespecimens were subjected to sinusoidal positive axial load at room temperature in laboratory air by an electro-hydraulic ser-vo-controlled fatigue machine. The frequency, maximum load and stress ratio (minimum load/maximum load) were keptconstant at 5 Hz, 100 kN, and 0.1, respectively, during the experiments except for special periods to introduce beachmarks.The minimum load was decreased to less than half of the maximum load in order to make beachmarks. Accordingly, the ap-plied cyclic stress range was 120 MPa, and in the simulations, SIF was calculated under applied load of 120 MPa for each step.

In the simulations, the initial crack size was set to 2c1 = 2c2 = 10 mm and a1 = a2 = 5 mm. The constants for crack growthrate were C = 3.5 � 10�11 and n = 2.52, which were obtained using the same material [30].

3.2. Results of the simulations

Fig. 6 shows the changes of mode I SIF and shapes of cracks during the simulation for the case of H = 5 mm (hereafter, thiscase is denoted as H5, and H10 is for the case of H = 10 mm). By growing the cracks in the surface direction, the inner side of

0

5

10

15

20

25

30

35

-20 -10 0 10 20

Position x, mm

Stre

ss in

tens

ity f

acto

r K

I, M

Pam

0.5

x

N = 0

N = 1.0 × 105

N = 1.8 × 105

Fig. 6. Changes of KI and crack shape obtained by simulation for test H5 (initial condition: 2c1 = 2c2 = 10 mm, S = 0, H = 5 mm).


the cracks overlapped, and then, the growth direction changed so that they approached each other. At the beginning of thesimulation, SIF was nearly constant along a crack, and then, became large at the surface outside of two cracks (hereafter,outer crack tips) and small at the surface inside (hereafter, inner crack tips). In general, the variation of SIF along the crackfront greatly depends on the front shape. It tends to be large at the dent portion, whereas it is small at the bulged portion[31]. This characteristic made the front shape smooth and induced uniform SIF along the crack front. However, when thecracks were overlapped on the projected plane, SIF at the overlapped position decreased largely due to the shielding effect.The reduction in SIF caused arrest of crack growth at the overlapped position and deformed the front shape. On the otherhand, at the outer crack tips, SIF became large not only by interaction effect but by the edge effect. The normalized spanlength Rx/W was more than 0.85 at the end of the experiment.

It should be noted that KII and KIII were small compared with KI, although they affected the growth direction of the innercrack tip when the cracks were overlapped. This meant that the growth rate of the interacting cracks was dominated by themode I SIF even under the mixed mode loading condition.

Fig. 7 shows crack configurations on the surface of the specimens obtained by the simulations and corresponding exper-imental results of H5 and H10. The direction of progression of the inner crack tips of two facing cracks changed so that theyapproached each other. On the other hand, the direction of progression of the outer crack tips was almost straight and per-pendicular to the loading direction. The simulations reproduced these characteristics.

Fig. 8 shows the loci of the crack profiles on the projected plane perpendicular to the axial direction obtained by the sim-ulations together with the corresponding fractured surfaces of the specimens. The loci are drawn every five steps. The crackprofiles obtained by the experiments were not semi-elliptical, particularly at the overlapped portion; the spacing betweenthe loci was relatively narrow at the inner side. This was due to the small SIF caused by the stress shielding effect as shown inFig. 6. The direction of progression of the inner crack tips changed so that they approached each other as shown in Fig. 7. Thisalso contributed to the change in crack profiles at the overlapped portion. Regardless of the complex change in the crackshape, the loci obtained by the simulations agreed well with the experimental results.

The changes in crack depth and span length Rx, which is defined in Fig. 2, are shown in Fig. 9a. The assumed incubationperiods before the initiation of the fatigue crack growth from the machined notch were 150,000 and 300,000 cycles for H5and H10, respectively, which were determined by trial and error. The depth in the figure corresponds to the maximumdepth of the crack. Although the change in crack depth was almost the same as the experimental results, some deviationswere observed in the surface length. The width and thickness of the plate were 50 mm and 15 mm, respectively, whereasthe size of the cracks was more than Rx = 40 mm and a = 10 mm at the end of the experiments. Such a small ligament en-hanced SIF as shown in Fig. 6, and might cause further acceleration of the crack growth due to plastic deformation near thecrack tips.

The change in area on the projected plane is shown in Fig. 9b. It was pointed out that the change in area, rather than theboundary size such as Rx and a, is appropriate for representing the crack growth behavior because the boundary size is sen-sitive to the crack shape. In the experiments, due to the overlapping of the cracks, only one of the two cracks exhibited acontinuous profile (see Fig. 8). Therefore, the area was measured at one of the two crack profiles from the symmetry lineon the projected plane, and denoted as the half area, Ah. Since the interaction was larger in the case of H5, the growth rate

10 mm

(a) H5 (initial condition: S = 0, H = 5 mm)

(b) H10 (initial condition: S = 0, H = 10 mm)

Simulation Experiment

Simulation Experiment

Fig. 7. Results of crack growth simulation and corresponding experimental results (initial crack length 2c1 = 2c2 = 10 mm).

(a) H5 (initial condition: S = 0mm, H = 5mm)

(b) H10 (initial condition: S = 0mm, H = 10mm)

10mm

10mm

Simulation

Experiment

Simulation

Experiment

Fig. 8. Loci of projected crack profile obtained by the simulation and fractured surface of specimen.


became faster than that of H10. Even for H10, the growth rate was faster than that of the specimen containing only one crack[30]. In all cases, the simulations agreed well with the experimental results.

4. Simulations for various cases

4.1. Analysis model

The simulation was conducted under various geometrical cases in order to investigate the relative size effect on crackgrowth behavior. Here, fatigue crack growth of A533B steel for reactor pressure vessel was assumed. The growth rate of thismaterial was obtained as C = 1.67 � 10�12 and n = 3.23 [17]. The plate was assumed to have the dimensions W = 500 mm,t = 300 mm and h = 500 mm and was subjected to a cyclic axial loading of amplitude of 45 MPa without mean stress. Theinitial size of crack 1 was 2c1 = 10 mm and a1 = 4 mm, and the size of crack 2 was changed, although the ratio of crack depthto surface length (a1/c1 and a2/c2) at the beginning of the simulations was 0.8 for all cases.

4.2. Results and discussion

4.2.1. Growth of cracks of dissimilar sizesThe simulation was conducted for dissimilar cracks: the initial size of crack 2 was set to 2c2 = 5 mm. The initial distance

between the cracks was S = 10 mm and H = 10 mm. The change in the crack shape and SIF are shown in Fig. 10. Since SIFdepended on the size of the crack, the growth of crack 1 was faster than that of crack 2 according to the power law expressedby Eq. (4). The difference in growth rate enhanced the difference in the crack size. Finally, crack 2 was included in crack 1 onthe projected plane and the SIF of crack 2 decreased due to the shielding effect. Thus, the growth of interacting dissimilarcracks may result in the arrest of the smaller cracks when the difference in the size is large. On the other hand, when thesize of cracks is the same, two cracks continue to grow keeping the interacting position as shown in Fig. 6. For predictingthe crack growth in FFS assessment, it is important to know how such growth behaviors affect the fatigue life.

0

50

100

150

200

250

300

Number of cycles (N), 105

Hal

f Are

a, m

m2

FEM (B-H5) FEM (B-H10 )

Experiment (B-H5)Experiment (B-H10)

0

10

20

30

40

50

0 1 2 3 4 5 6

0 1 2 3 4 5 6

Number of cycles (N), 105

Span

leng

th a

nd d

epth

, mm

Span length (B-H5)Depth (B-H5)

Span length (B-H10)Depth (B-H10)

(a) Change in crack size

(b) Change in area

Open: FEMSolid: Experiment

Fig. 9. Crack growth curves obtained by the simulation and experiments.

0

2

4

6

8

10

12

14

16

18

20

-40 -30 -20 -10 0 10 20

Position x, mm

Stre

ss in

tens

ity f

acto

r K

I, M

Pam

0.5

x

N = 0

N = 3.9 × 106

N = 5.1 × 106

Fig. 10. Changes of KI and crack shape obtained by simulation (initial condition: 2c1 = 10 mm, 2c2 = 5 mm, S = 10 mm, H = 10 mm).


4.2.2. Fatigue life under interactionThe number of cycles necessary for the cracks to grow more than Rx = 40 mm was defined as ND. In order to evaluate the

influence of the interaction on the fatigue life, ND was compared with that of the single crack, which was denoted as NS. In thecase of a single crack, the crack size was set to 2c1 = 10 mm under the same a1/c1 as the case of the twin crack. The ratio ND/NS


was 1.00 for the case of 2c2 = 5 mm (S = H = 10 mm), in which the smaller crack (crack 2) was arrested during the simulationas shown in Fig. 10. Rx was equivalent to 2c1 (see Fig. 2), and the change in SIF of crack 1 due to the interaction was negligiblysmall. Therefore, in this case, the interaction had no influence on the fatigue life. On the other hand, ND/NS was 0.88 for thecase of 2c2 = 10 mm (S = H = 10 mm), in which the cracks grew maintaining the interacting position as shown in Fig. 6. Theenhanced SIF by interaction reduced the fatigue life. For a conservative integrity assessment for FFS, the effect of the inter-action should be taken into account in predicting the crack growth.

From above results, it is deduced that the effect of the interaction on the fatigue life (ND/NS) correlates with the growth ofthe cracks (arrest or not). In order to confirm this correlation, ND was investigated for various sizes of 2c2 (c2 < c1) under con-ditions of 2c1 = 10 mm and S = H = 10 mm. Fig. 11 shows the change in ND/NS with c2 normalized by c1. When c2 < 0.6c1, ND/NS

was almost unity, and in such cases, the growth of crack 2 was arrested. On the other hand, in the cases of ND/NS < 1, thecracks maintained the interacting position. Thus, when the difference in the crack size is large enough, it is not necessaryto take the interaction effect into account in the growth prediction for FFS assessment; crack 1 can be treated as a singlecrack for predicting the crack growth. The judgment whether the interaction should be considered can be made based onthe growth of the smaller crack; when the growth of the smaller crack (crack 2) is arrested, the fatigue life can be regardedas being equivalent to that of the single crack for which length is 2c1.

4.2.3. Effect of relative positionThe intensity of interaction between multiple cracks depends not only on the relative size but also on the relative spacing

(S and H). Then, the simulation was performed for various relative spacings and the effect of relative spacing on the fatiguelife was investigated. In other words, whether the crack growth was arrested or not was investigated for various conditionsof relative spacing. The occurrence of the arrest could be judged by comparing SIF of interacting crack tips. When c2 < c1, SIFof the inner crack tip of crack 1, KIc1, was larger than that of the outer crack tip of crack 2, KIc2 (see Fig. 12). And then, KIc1

decreased due to the shielding effect when the inner crack tips were overlapped as shown in Fig. 6. However, if the magni-tude of interaction was small and KIc1 was larger than KIc2 even if the cracks were overlapping, the inner crack tip of crack 1kept growing and the smaller crack was shielded completely by crack 1 and stopped growing. Thus, the crack arrest occurredwhen KIc1 > KIc2 after the two cracks overlapped.

Simulations were performed for the possible combinations of H/c1 = 1, 2, 3 and S/c1 = 1, 2, 3 under the fixed initial cracksizes of 2c1 = 10 mm and 2c2 = 5 mm. Fig. 13 summarizes the simulation results. Solid symbols show that crack 2 stoppedgrowing, whereas open symbols denote the conditions at which crack 2 does not stop growing. When the initial offsetdistance was small, the crack 2 continued growing under a large interaction. The horizontal distance S also influenced the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

ND

/Ns

c2/c1

Crack 2 continues growing

Crack 2 stops growing

Fig. 11. Effect of relative size on fatigue life (initial condition: 2c1 = 10 mm, S = 10 mm, H = 10 mm).

Crack 2 continues growing when

KIc1 < KIc2

Crack 2

Crack 2

Crack 1

Crack 1

KIc1

KIc2

KIc1

KIc2

Crack 2 is arrested when

KIc1 > KIc2

Fig. 12. Judgment criterion for arrest of crack 2.

0

1

2

3

4

0 1 2 3 4

H/c

1

S/c1



Fig. 13. Result of parametric studies fro relative position (initial condition: 2c1 = 10 mm, 2c2 = 5 mm).


interaction; crack 2 was arrested even under small H when S was small, although the magnitude of the influence of S seemedto be relatively small. Thus, both S and H should be taken into account in order to incorporate the effect of relative spacing onthe crack growth prediction.

In the JSME FFS code [5], whether the interacting cracks are combined in growth prediction is judged based on the relativespacing (S and H) at the initial condition. If the relative spacing at the beginning of the growth prediction meets the criterion,two cracks are combined when the distance S becomes zero during their growth. Although the current criterion is reasonablefrom the results shown in Fig. 13, the three-dimensional figure for S, H and c2/c1 has to be prepared in order to incorporatethe crack size effect into the current criterion. The effect of distance S can be excluded from the criterion by making the judg-ment when the cracks are overlapped. Fig. 14 shows the relationship between the H/2c1 and c2/c1 when the distance S be-comes zero, where H is the offset distance at the beginning of the simulations. This figure was obtained from the simulationsunder various cases, summarized in Table 1. The solid symbol denotes that the growth of crack 2 was arrested, and the opensymbol shows crack 2 continued to grow keeping the interacting position. Whether the crack was arrested correlates wellwith the combination of H/c1 and c2/c1. When H/c1 was small and c2/c1 was large, the interaction of the two cracks reducedthe fatigue life, whereas it had no effect in the opposite case. Accordingly, by considering the situation only for S = 0, the ef-fect of distance S can be excluded from the criterion. Fig. 14 can provide an alternative criterion to Fig. 13 for taking intoaccount the relative size effect.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

H/2

c 1

2c2/2c1



2c1 2c2

H

Fig. 14. Relationship between H/2c1 and c2/c1 when distance S becomes zero.

Table 1Summary of initial conditions of the simulations used for Fig. 13.

Case S [mm] H [mm] c2/c1 Remark

1 2.5 10 0.82 2.5 10 0.73 2.5 10 0.64 2.5 10 0.55 5 5 0.5 Fig. 126 5 15 0.57 5 10 0.88 5 10 0.69 5 10 0.5 Fig. 12

10 10 2.5 0.511 10 2.5 0.412 10 2.5 0.313 10 2.5 0.214 10 5 0.615 10 5 0.5 Fig. 1216 10 5 0.417 10 5 0.318 10 10 1.0 Fig. 1019 10 10 0.9 Fig. 1020 10 10 0.8 Fig. 1021 10 10 0.7 Fig. 1022 10 10 0.6 Fig. 1023 10 10 0.5 Figs. 10 and 1224 10 15 0.925 10 15 0.826 10 15 0.727 10 15 0.628 10 15 0.5 Fig. 1229 10 20 1.030 10 20 0.931 10 20 0.732 15 5 0.5 Fig. 1233 15 10 0.834 15 10 0.735 15 10 0.636 15 10 0.5 Fig. 1237 15 15 0.5 Fig. 1238 20 10 0.839 20 10 0.740 20 10 0.5


5. Summary and conclusions

In this study, the interaction effect of two surface cracks was evaluated using a fully automatic fatigue crack growth sim-ulation system. At first, simulations were performed to simulate experiments conducted in a previous study. Then, the effectof the interaction was investigated for various cases. The following conclusions were obtained:

1. The simulations successfully simulated the fatigue crack growth of two interacting surface cracks. In particular, the three-dimensional change in the crack shape agreed well with the experimental results.

2. The smaller crack stopped growing when the difference in size of interacting cracks was large.3. When smaller cracks stopped growing, the interaction effect on the fatigue life of the larger cracks was negligibly small.4. The offset distance and the relative size were important parameters to determine the interaction between two surface

cracks, and it was possible to judge whether the effect of interaction should be considered from the correlation betweenH/c1 and c2/c1 when the distance S became zero.

References

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Documents

Growth prediction of two interacting surface cracks of dissimilar sizes