Growth prediction of two interacting surface cracks of dissimilar sizes

Engineering Fracture Mechanics 77 (2010) 3120–3131

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Growth prediction of two interacting surface cracks of dissimilar sizes Masayuki Kamaya a,⇑, Eiichi Miyokawa b, Masanori Kikuchi b a b

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

a r t i c l e

i n f o

Article history: Received 9 November 2009 Received in revised form 27 April 2010 Accepted 8 August 2010 Available online 12 August 2010 Keywords: Stress intensity factor Finite element analysis Crack propagation Multiple cracks Parallel cracks

a b s t r a c t When multiple cracks approach one another, the stress intensity factor changes due to the interaction of the stress field. This causes variation in the crack growth rate and shape of cracks. In particular, when cracks are parallel to the loading direction, their shape becomes non-planar due to the mixed mode stress intensity factor. In this study, the growth of interacting surface cracks was simulated by using the S-version finite element method, in which a local detailed finite element mesh (local model) is superposed on a coarse finite element model (global model) representing the global structure. First, simulations were performed for fatigue crack growth experiments and the method validity was shown. Second, simulations were conducted for various relative sizes and spacings of twin cracks. It was shown that the offset distance and the relative size were both important parameters to determine the interaction between two surface cracks; the smaller crack stopped growing when the difference in size was large. It was possible to judge whether the effect of interaction should be considered based on the correlation between the relative spacing and relative size. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Multiple cracks are likely initiated due to stress corrosion cracking and fatigue in nuclear power plant components. Since mechanical interaction between multiple cracks accelerates the crack growth and reduces the time to failure, it is important to incorporate the acceleration in predicting the crack growth for fitness-for-service (FFS) evaluation. In engineering assessments for FFS, influence of the interaction is taken into account conservatively by applying the combination rule [1–3]. In the ASME boiler and pressure vessel code (e.g. Section XI, Articles IWA-3330) [4] and JSME fitness-for-service code [5], adjacent cracks are treated as a coalesced one when their relative spacing meets the criterion. However, it was pointed out that the criterion for the combination rule is excessively conservative in some cases [6–10]. In particular, the effect of a difference in crack size (hereafter referred as ‘‘relative size effect”) is not taken into account in the aforementioned codes, although it has relatively large impact on the magnitude of the interaction [11,12]. The conservativeness in the current code increases as the difference in crack size increases, because the magnitude of interaction is largest for the cracks of the same size [13]. Since most interacting cracks have dissimilar sizes, it is important to take the relative size effect into account in the criteria for a reasonable assessment. Although extensive experimental studies have been conducted in order to investigate the influence of interaction on the crack growth [7,8,14,15], the results for dissimilar cracks are very limited, particularly for surface cracks in non-coplanar position [16,17]. In order to clarify the influence of the difference in crack size in addition to the relative spacing of interacting

⇑ Corresponding author. Tel.: +81 770 379114; fax: +81 770 372009. E-mail address: [email protected] (M. Kamaya). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.08.008

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

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Nomenclature a1 a2 da/dN C c1 c2 {f} H h DKI DKII DKIII KIc1 KIc2 DKeq [K] n ND NS Rx S t {u} W

uo

crack depth of crack 1 crack depth of crack 2 crack growth rate material constant for crack growth rate half crack length of crack 1 half crack length of crack 2 nodal force vector offset distance between cracks height of the body mode I stress intensity factor range mode II stress intensity factor range mode III stress intensity factor range KI at inner crack tip of crack 1 KI at outer crack tip of crack 2 equivalent stress intensity factor range stiffness matrix material constant for crack growth rate number of cycles until Rx becomes more than 40 mm for interacting cracks number of cycles until Rx becomes more than 40 mm for a single crack span length of two cracks (identical to 2c1 for a single crack) horizontal distance between crack tips thickness of the body nodal displacement vector width of the body deflection angle in the crack growth simulation

cracks, experiments under various geometrical cases have to be conducted. However, it is difficult to control the fatigue crack growth 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 conditions, it is possible to figure out the relative size effect of interacting cracks. Since SIF of the interacting cracks varies depending 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-constructed according to the change in crack shape and position. However, mesh division for interacting cracks is not easy because 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 the case of non-coplanar cracks, the cracks deform three-dimensionally due to shear loading emanating from their non-symmetric relative position. In this study, crack growth simulations were carried out to investigate the relative size effect on the growth of interacting cracks. The finite element analyses were conducted to evaluate SIF of interacting surface cracks in a non-coplanar position. In order to perform finite element analyses for cracks of complex shape, S-version finite element method (S-FEM) [23] was employed by combining an auto-mesh generation technique, together with the fully automatic fatigue crack growth simulation system [24]. After a short review of the crack growth simulation system, crack growth simulations were made for crack growth experiments conducted in a previous study. Then, the effect of relative size and spacing of cracks on growth behavior was investigated based on results of simulations conducted under various conditions. Finally, the criterion for assessments in FFS 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. The plate was subjected to a cyclic axial loading of magnitude ro. At the beginning of the simulation, two cracks, crack 1 and crack 2, were semi-elliptical in shape and located in parallel (non-coplanar position). The horizontal and offset distances between 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 the loading axis was denoted by Rx (Fig. 2). For comparison purpose, the growth of single crack was also simulated, in which Rx is the same as 2c1.

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σo t

Crack 2 a2

H

h

2c2

a1 2c1

S

Crack 1

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

Crack 2 Crack 1

Interacting cracks Rx Crack 2

Crack 1

Crack 2 is arrested Rx

Crack 1 Single crack Rx Fig. 2. Definition of span length Rx.

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 schematically shows a solid containing two cracks. In S-FEM, a coarse finite element mesh (global mesh) is assumed to represent the global structure and overlaid finite element meshes representing crack and its vicinity (local meshes) are superposed on the global mesh as shown in Fig. 3b. Finite element formulation for this model is described by:

38 G 9 8 G 9 ½K GL2  > < fu g > = > < ff g > = 6 7 4 ½K L1L1  ½K L1L2  5 fuL1 g ¼ ff L1 g > : L2 > ; > : L2 > ; fu g ff g sym: ½K L2L2  2

½K GG 

½K GL1 

ð1Þ

where {u} and {f} are vectors of unknown nodal displacement and nodal force, respectively. Superscript of G, L1 and L2 denotes 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:

½K GL1  ¼ ½K L1L2  ¼

Z XL1

½BG T ½D½BL1 dXL1

ð2Þ

½BL1 T ½D½BL2 dXL2

ð3Þ

Z

XL2

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

ti

Global mesh

3123

ti

Local mesh 1

Local mesh 2

(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, the displacement of each node can be obtained. In S-FEM, it is not necessary to consider the geometrical connectivity between global and local meshes, and complex changes in crack shape can be considered in finite element analyses by altering the local meshes without considering their boundary shape. This enables us to perform crack growth simulation under various conditions of S, H and crack sizes without re-meshing the whole model. An auto-mesh generation technique and a fully automatic fatigue crack growth simulation system 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 each step. 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 ratio of the material was taken to be 0.3. 2.3. Simulation procedure The crack growth size per cycle (da/dN) was determined by:

da ¼ CðDK eq Þn dN

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.

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−φo

Crack

Fig. 5. Crack growth direction.

DK eq ¼

KI 1 þ 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2I þ 4ð1:115K II Þ2 þ 4ðK III Þ2

ð5Þ

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

"

uo ¼  140

 2 # jDK II j jDK II j  70 DK I þ jDK II j þ jDK III j DK I þ jDK II j þ jDK III j

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 crack closure was not considered and mean stress was assumed to be zero. Therefore, not the range but the amplitude of SIF was considered 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 subjected to cyclic axial loading. The simulation was preformed for the following two experiments for twin interacting surface cracks. The plate had dimensions of W = 50 mm, t = 15 mm and h = 200 mm [30]. In the experiments, by using an electrode discharge machine, twin semi-circular surface notches of the same size were introduced perpendicular to the loading direction at the center of the specimen. The positions of the notches were H = 5 mm and 10 mm under constant value of S = 0. The specimens were subjected to sinusoidal positive axial load at room temperature in laboratory air by an electro-hydraulic servo-controlled fatigue machine. The frequency, maximum load and stress ratio (minimum load/maximum load) were kept constant 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 applied 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 growth rate were C = 3.5  1011 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, this case 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

Stress intensity factor KI, MPam0.5

35

N=0 N = 1.0 × 10 5

30

N = 1.8 × 10 5

25 20 15 10 5

x 0 -20

-10

0

10

20

Position x, mm 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).

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the cracks overlapped, and then, the growth direction changed so that they approached each other. At the beginning of the simulation, 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 crack front 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 the cracks 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 other hand, at the outer crack tips, SIF became large not only by interaction effect but by the edge effect. The normalized span length 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 inner crack tip when the cracks were overlapped. This meant that the growth rate of the interacting cracks was dominated by the mode 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 experimental results of H5 and H10. The direction of progression of the inner crack tips of two facing cracks changed so that they approached each other. On the other hand, the direction of progression of the outer crack tips was almost straight and perpendicular 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 simulations together with the corresponding fractured surfaces of the specimens. The loci are drawn every five steps. The crack profiles obtained by the experiments were not semi-elliptical, particularly at the overlapped portion; the spacing between the loci was relatively narrow at the inner side. This was due to the small SIF caused by the stress shielding effect as shown in Fig. 6. The direction of progression of the inner crack tips changed so that they approached each other as shown in Fig. 7. This also contributed to the change in crack profiles at the overlapped portion. Regardless of the complex change in the crack shape, 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 incubation periods before the initiation of the fatigue crack growth from the machined notch were 150,000 and 300,000 cycles for H5 and H10, respectively, which were determined by trial and error. The depth in the figure corresponds to the maximum depth of the crack. Although the change in crack depth was almost the same as the experimental results, some deviations were observed in the surface length. The width and thickness of the plate were 50 mm and 15 mm, respectively, whereas the size of the cracks was more than Rx = 40 mm and a = 10 mm at the end of the experiments. Such a small ligament enhanced SIF as shown in Fig. 6, and might cause further acceleration of the crack growth due to plastic deformation near the crack 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 the boundary size such as Rx and a, is appropriate for representing the crack growth behavior because the boundary size is sensitive to the crack shape. In the experiments, due to the overlapping of the cracks, only one of the two cracks exhibited a continuous profile (see Fig. 8). Therefore, the area was measured at one of the two crack profiles from the symmetry line on the projected plane, and denoted as the half area, Ah. Since the interaction was larger in the case of H5, the growth rate

Simulation

Experiment

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

Experiment

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

10 mm

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

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Simulation

Experiment

10mm

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

Experiment

10mm

(b) H10 (initial condition: S = 0mm, H = 10mm) 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 crack growth behavior. Here, fatigue crack growth of A533B steel for reactor pressure vessel was assumed. The growth rate of this material was obtained as C = 1.67  1012 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. The initial 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 depth to 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 sizes The 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 SIF depended on the size of the crack, the growth of crack 1 was faster than that of crack 2 according to the power law expressed by Eq. (4). The difference in growth rate enhanced the difference in the crack size. Finally, crack 2 was included in crack 1 on the projected plane and the SIF of crack 2 decreased due to the shielding effect. Thus, the growth of interacting dissimilar cracks may result in the arrest of the smaller cracks when the difference in the size is large. On the other hand, when the size of cracks is the same, two cracks continue to grow keeping the interacting position as shown in Fig. 6. For predicting the crack growth in FFS assessment, it is important to know how such growth behaviors affect the fatigue life.

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Span length and depth, mm

50

Span length (B-H5) Depth (B-H5) Span length (B-H10) Depth (B-H10)

40

Open: FEM Solid: Experiment

30

20

10

0

0

1

2

3

4

5

6

5

6

Number of cycles (N), 105

(a) Change in crack size 300 FEM (B-H5) FEM (B-H10 ) Experiment (B-H5) Experiment (B-H10)

Half Area, mm2

250 200 150 100 50 0

0

1

2

3

4

Number of cycles (N), 105

(b) Change in area Fig. 9. Crack growth curves obtained by the simulation and experiments.

Stress intensity factor KI, MPam0.5

20 18 16 14 12 10 8 6 4

N=0 N = 3.9 × 106

2

N = 5.1 × 106

0 -40

-30

-20

x -10

0

10

20

Position x, mm 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 interaction The 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 the case 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

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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 simulation as 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 negligibly small. Therefore, in this case, the interaction had no influence on the fatigue life. On the other hand, ND/NS was 0.88 for the case of 2c2 = 10 mm (S = H = 10 mm), in which the cracks grew maintaining the interacting position as shown in Fig. 6. The enhanced SIF by interaction reduced the fatigue life. For a conservative integrity assessment for FFS, the effect of the interaction 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 of the cracks (arrest or not). In order to confirm this correlation, ND was investigated for various sizes of 2c2 (c2 < c1) under conditions 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, the cracks maintained the interacting position. Thus, when the difference in the crack size is large enough, it is not necessary to take the interaction effect into account in the growth prediction for FFS assessment; crack 1 can be treated as a single crack for predicting the crack growth. The judgment whether the interaction should be considered can be made based on the growth of the smaller crack; when the growth of the smaller crack (crack 2) is arrested, the fatigue life can be regarded as being equivalent to that of the single crack for which length is 2c1. 4.2.3. Effect of relative position The 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 fatigue life was investigated. In other words, whether the crack growth was arrested or not was investigated for various conditions of relative spacing. The occurrence of the arrest could be judged by comparing SIF of interacting crack tips. When c2 < c1, SIF of 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 magnitude of interaction was small and KIc1 was larger than KIc2 even if the cracks were overlapping, the inner crack tip of crack 1 kept growing and the smaller crack was shielded completely by crack 1 and stopped growing. Thus, the crack arrest occurred when 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 crack sizes of 2c1 = 10 mm and 2c2 = 5 mm. Fig. 13 summarizes the simulation results. Solid symbols show that crack 2 stopped growing, whereas open symbols denote the conditions at which crack 2 does not stop growing. When the initial offset distance was small, the crack 2 continued growing under a large interaction. The horizontal distance S also influenced the

1.2 1.0

ND /Ns

0.8 0.6 0.4

Crack 2 continues growing

0.2

Crack 2 stops growing 0.0 0.0

0.2

0.4

0.6

0.8

1.0

c2/c1 Fig. 11. Effect of relative size on fatigue life (initial condition: 2c1 = 10 mm, S = 10 mm, H = 10 mm).

Crack 2 KIc2 Crack 2 continues growing when

Crack 1

KIc1 < KIc2

KIc1 Crack 2 KIc2

Crack 2 is arrested when

Crack 1 KIc1

KIc1 > KIc2

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

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4

Crack 2 continues growing Crack 2 stops growing

H/c1

3

2

1

0

0

1

2

3

4

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 seemed to be relatively small. Thus, both S and H should be taken into account in order to incorporate the effect of relative spacing on the crack growth prediction. In the JSME FFS code [5], whether the interacting cracks are combined in growth prediction is judged based on the relative spacing (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 reasonable from the results shown in Fig. 13, the three-dimensional figure for S, H and c2/c1 has to be prepared in order to incorporate the crack size effect into the current criterion. The effect of distance S can be excluded from the criterion by making the judgment when the cracks are overlapped. Fig. 14 shows the relationship between the H/2c1 and c2/c1 when the distance S becomes zero, where H is the offset distance at the beginning of the simulations. This figure was obtained from the simulations under various cases, summarized in Table 1. The solid symbol denotes that the growth of crack 2 was arrested, and the open symbol shows crack 2 continued to grow keeping the interacting position. Whether the crack was arrested correlates well with the combination of H/c1 and c2/c1. When H/c1 was small and c2/c1 was large, the interaction of the two cracks reduced the fatigue life, whereas it had no effect in the opposite case. Accordingly, by considering the situation only for S = 0, the effect of distance S can be excluded from the criterion. Fig. 14 can provide an alternative criterion to Fig. 13 for taking into account the relative size effect.

2c1

2c2 H

1.0

H/2c1

0.8

0.6

0.4

0.2

0.0 0.0

Crack 2 continues growing Crack 2 stops growing 0.2

0.4

0.6

0.8

1.0

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

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M. Kamaya et al. / Engineering Fracture Mechanics 77 (2010) 3120–3131 Table 1 Summary of initial conditions of the simulations used for Fig. 13. Case

S [mm]

H [mm]

c2/c1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

2.5 2.5 2.5 2.5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 20 20 20

10 10 10 10 5 15 10 10 10 2.5 2.5 2.5 2.5 5 5 5 5 10 10 10 10 10 10 15 15 15 15 15 20 20 20 5 10 10 10 10 15 10 10 10

0.8 0.7 0.6 0.5 0.5 0.5 0.8 0.6 0.5 0.5 0.4 0.3 0.2 0.6 0.5 0.4 0.3 1.0 0.9 0.8 0.7 0.6 0.5 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.7 0.5 0.8 0.7 0.6 0.5 0.5 0.8 0.7 0.5

Remark

Fig. 12

Fig. 12

Fig. 12

Fig. 10 Fig. 10 Fig. 10 Fig. 10 Fig. 10 Figs. 10 and 12

Fig. 12

Fig. 12

Fig. 12 Fig. 12

5. Summary and conclusions In this study, the interaction effect of two surface cracks was evaluated using a fully automatic fatigue crack growth simulation system. At first, simulations were performed to simulate experiments conducted in a previous study. Then, the effect of 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 threedimensional 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 between H/c1 and c2/c1 when the distance S became zero.

References [1] Leek TH, Howard IC. An examination of methods of assessing interacting surface cracks by comparison with experimental data. Int J Pressure Vessels Pip 1996;68:181–201. [2] Guide to methods for assessing the acceptability of flaws in metallic structures BS 7910. London: British Standards Institution; 2005. [3] Fitness-for-service API 579. Washington, DC: American Petroleum Institute; 2000. [4] ASME, Rule for In-service inspection of nuclear power plant components: boiler and pressure vessel code section XI 2007. New York: ASME; 2003. [5] JSME fitness-for-service code S NA1-2008. Tokyo: JSME; 2008. [6] Frise PR, Bell R. Modelling fatigue crack growth and coalescence in notches. Int J Pressure Vessels Pip 1992;51:107–26.

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