Growth evaluation of multiple interacting surface cracks. Part I: Experiments and simulation of coalesced crack

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Engineering Fracture Mechanics 75 (2008) 1336–1349

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Growth evaluation of multiple interacting surface cracks.Part I: Experiments and simulation of coalesced crack

Masayuki Kamaya *

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

Received 12 March 2007; received in revised form 5 July 2007; accepted 5 July 2007Available online 2 August 2007

Abstract

In order to develop a procedure for assessing the growth of interacting surface cracks, the relationship between the areaof the crack face and fatigue crack growth behavior was investigated. Fatigue crack growth tests were conducted usingstainless steel plate specimens with surface notches. Then, finite element analyses were performed to simulate the growthbehavior obtained by the experiment. It was shown that the change in area can be predicted by assuming the extension ofcrack front based on evaluated stress intensity factor at each position along the front. Based on experimental and analysisresults, it was revealed that the growth of interacting surface cracks as well as independent cracks can be represented wellby change in area and showed good correlation with the driving force based on area. It was also shown that, in the case ofparallel cracks, the area on the projected plane was dominant. It was concluded that, when the magnitude of the interac-tion is sufficiently large, by replacing the two cracks with a semi-elliptical crack of the same area on the projected plane, thegrowth in area can be predicted precisely.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Stress intensity factor; Finite element analysis; Crack propagation; Undulated crack front

1. Introduction

The coalescence and mechanical interaction of multiple cracks play an important role in the crackingbehavior due to degradation of plant components, especially in stress corrosion cracking (SCC) and fatigue[1–3]. The coalescence of multiple cracks causes an abrupt increase in crack size and accelerates the crackgrowth, and it was pointed out that, in small crack growth, the growth due to coalescence is predominantrather than growth of individual cracks [1,4]. The crack growth can be predicted by using the well-knownempirical relationship between crack growth rate and stress intensity factor (SIF) [5,6], and this approachhas been widely applied for assessing the structural integrity of engineering structures [7–10]. Therefore, bytaking into account the influence of the interaction on the SIF and the coalescence of multiple cracks, it ispossible to predict the cracking behavior of the interacting cracks.

0013-7944/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2007.07.015

* Tel.: +81 770 379114; fax: +81 770 372009.E-mail address: [email protected]

mailto:[email protected]

M. Kamaya / Engineering Fracture Mechanics 75 (2008) 1336–1349 1337

In the crack growth assessments, the surface cracks are usually modeled as semi-elliptical surface ones forsimplicity, and the SIF is calculated at both the deepest and surface points. Since extensive SIF solutions areavailable for semi-elliptical surface cracks under various loading and boundary conditions, the growth can bepredicted without difficulties by referring to existing solutions [11–13]. Since cracks tend to become semi-ellip-tical under simple loading conditions [14–16], the prediction based on the semi-elliptical shape gives reason-able solutions. However, the crack shape may largely deviate from the semi-ellipse due to coalescence ofadjacent cracks. If two semi-elliptical surface cracks coalesce, the resultant crack forms a re-entrant (dented)portion at the contact point [17–22]. The interaction between cracks also causes the crack shape to change.Due to the mechanical interaction, the stress at the crack tip is amplified or reduced, and this causes unequalchange in the SIF along the crack front [23]. The non-uniform SIF (that is, the crack growth rate) makes thecrack shape deviate from the semi-elliptical shape, causing in-plane and out-of-plane deformation of the crackface [22,24]. Since the magnitude of the interaction is dependent on various factors [21,25–41], such as relativelocation, relative size, crack shape, and number of cracks, it is difficult to quantify the change in SIF duringthe crack growth.

In an engineering assessment for fitness-for-service (FFS), the influence of interaction on crack growth istaken into account conservatively by applying combination rules. For example, in the ASME Boiler and Pres-sure Vessel Code (e.g. Section XI, Articles IWA-3330) [7], adjacent cracks will join and assume a shape definedby a bounding semi-ellipse when their relative spacing satisfies the prescribed conditions. The depth of thisnewly formed crack is assumed to be equal to the depth of the deeper of the two pre-existing cracks andthe surface length is assumed to be equal to the sum of their length and ligament length of their neighboringcrack tips. It was pointed out that this recharacterization technique resulted in relatively large overestimationsof the actual crack growth [4,17,21–23,42,43]. In order to maintain the integrity of the plant components,somewhat redundant conservativeness is inevitable because the complex growth phenomena under the inter-action are summarized in the simple combination rule. Extensive studies have been conduced to understandthe crack growth behavior under multiple crack interaction and to reduce the conservativeness in the currentassessment procedures [4,17,18,21,22,43–47]. In these studies, through several experiments of fatigue crackgrowth from neighboring surface notches, change in crack growth rate due to the interaction and transitionalchange in crack shape during the coalescence were investigated. The results showed that little acceleration incrack growth was observed before the coalescence or overlapping (in non-coplanar case) [4,17,21,43–45,47].Then, after the contact (overlapping), the growth rate of the re-entrant portion increased and this made thecrack shape semi-elliptical [17,21,22,44,46,48,49]. The change in SIF during the change in crack shape fromcontact of two cracks to semi-elliptical shape was also evaluated [21,23,28,30,48–51]. The SIF at the re-entrantportion was relatively large and this was consistent with the fast growth in the experiments. However, it is notpractical to perform a quantitative growth evaluation of the transient growth for engineering purpose. There-fore, alternative simplified assessment procedures (combination rules) were provided for FFS assessment[4,18,22,43–45,47]. Recently, based on the SIF evaluation, the combination rule in ASME was modified inorder to reduce the conservativeness [37]. Although the proposed assessment procedure successfully reducedthe conservativeness contained in the existing code, the aim of these studies was to achieve a conservativecrack growth assessment for FFS. For an exact evaluation of the crack growth under the interaction, it isimportant to evaluate the transient growth behavior from the contact of two cracks to the saturation ofsemi-elliptical shape without using the recharacterization technique.

The author evaluated the SIF for various front shapes, which were determined using a cosine function basedon a semi-circular crack, by finite element analysis (FEA) [52], and pointed out that the average of the SIF alongthe crack front is almost the same if the area of the crack face is the same. This suggests that the size of areaplays an important role in the SIF of various shapes of crack. Murakami et al. also pointed out, based onthe SIF evaluation by the body force method, that the area is correlated with the maximum SIF of an arbi-trary-shaped surface crack and proposed an empirical equation for the maximum SIF calculation based onthe area of the crack face [28]. However, the relationship between the area and fatigue crack growth behaviorhas not been investigated. If the area is a representative parameter for the driving force of the crack growth, thetransient growth behavior during the coalescence can be evaluated by incorporating the change in area.

This study investigated the influence of change in crack shape caused by the coalescence on the growthbehavior. The correlation between the growth of interacting cracks and the area was also evaluated. At first,

1338 M. Kamaya / Engineering Fracture Mechanics 75 (2008) 1336–1349

fatigue crack growth tests were conducted. Stainless steel plate specimens were subjected to cyclic tensile load.One or two surface notches were formed on the surface of the specimen, then the fatigue crack growth behav-ior from the notch was investigated. Second, FEA was carried out to simulate the growth behavior during thecoalescence. The validity of the simulation procedure was confirmed by comparing with the experimentalresults. Finally, the role of area in growth behavior of cracks with the dented portion and interacting crackswere discussed. A procedure for evaluating the growth of interacting surface cracks will be shown.

2. Experimental procedure

The experiments were conducted with plate tensile specimens manufactured from a type 304 austeniticstainless steel. Composition in weight percent was 0.06C, 0.50Si, 1.05Mn, 8.56Ni, 18.8Cr, 0.17Mo, 0.23Cu,and the balance in Fe. The geometry of the plate specimen is shown in Fig. 1. Two different specimen thick-nesses, 10 mm and 15 mm, were employed. The width and length of the specimen were 50 mm and 580 mm,respectively. The length included the 200 mm of parallel portion and gripped portion. One or two surfacenotches were formed using an electrode discharge machine. The combinations of thickness of the specimen,number of notches, notch configuration and notch spacing are summarized in Table 1. The name of the spec-imen follows the convention: thickness of specimen (A: 10 mm, B: 15 mm), relative location (S: horizontal dis-tance, H: offset distance) and type of notch. The notch was semi-circular or semi-elliptical in shape and waslocated perpendicular to the load direction at the center of the specimen. In the case of two notches, the sizeand shape of the notches were the same and they were located at coplanar or non-coplanar (parallel) positions.

The specimen was subjected to sinusoidal positive axial load at room temperature in laboratory air by anelectro-hydraulic servo-controlled fatigue machine. The frequency, maximum load and stress ratio (minimumload/maximum load) were kept constant at 5 Hz, 100 kN, and 0.1, respectively, during the tests except specialperiods for introducing beachmarks. The minimum load was decreased to less than half of the maximum load

2c

View A

a

t

R55

160

50

580

200

A

2c

2c

H

EDMNotch

AS

S

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

Table 1Summary of experimental conditions and number of beachmarks obtained in the experiment

Specimen Number ofnotches

Relativeposition (mm)

Initial notchsize (mm)

Number ofbeachmarks

Plate thickness(mm)

Stress range(MPa)

H S c a

A-Single 1 2.5 2.5 4 10 180A-H0S5 2 0 5 2.5 2.5 5 10 180A-H2.5S2.5 2 2.5 2.5 2.5 2.5 3 10 180A-H2.5 2 2.5 0 2.5 2.5 3 10 180A-H2.5C2A4 2 2.5 0 2 4 4 10 180A-H2.5C4A2 2 2.5 0 4 2 3 10 180B-Single 1 5 5 5 15 120B-H0S5 2 0 5 5 5 5 15 120B-H2.5(1) 2 2.5 0 5 5 6 15 120B-H2.5(2) 2 2.5 0 5 5 5 15 120B-H5 2 5 0 5 5 8 15 120B-H10 2 10 0 5 5 8 15 120


in order to make beachmarks. It was confirmed, by using four strain gages, that no significant bending stresswas introduced.

The tests were continued until the specimen fractured, and crack profiles were obtained by tracing thebeachmarks on the fractured surface. Crack growth rates were calculated from change in step size betweenbeachmarks and number of cycles between them. The number of beachmarks obtained in the experimentsis shown in Table 1.

For comparison, a crack growth test using a compact tension (CT) specimen, which met the ASTM spec-ifications [53], was carried out. The size of the specimen was the standard 1TCT size of 12.5 mm thickness. Thestress intensity factor was changed from 17 MPa m0.5 to 28 MPa m0.5 under a constant stress ratio of 0.1. Thecrack length was monitored by the compliance method.

3. Experimental results and discussion

The surface of the specimen during and after the test of B-H2.5S2.5 (initial condition: S = 2.5 mm,H = 2.5 mm) is shown in Fig. 2. In the cases of initial notches being parallel (initial condition is H > 0),the inner crack tips of two facing cracks changed their progress direction so that they approached each other.On the other hand, the progress direction of the outer crack tips was almost straight, perpendicular to the loadaxis. In all cases of parallel initial notches, the cracks did not coalesce before the specimen fractured as shownin Fig. 2b; they propagated as independent cracks, and coalesced due to rupture of the specimen. The frac-tured surfaces of specimen B-H5 (initial condition: S = 0, H = 5 mm) and B-H10 (initial condition: S = 0,H = 10 mm) are shown in Fig. 3. Due to overlapping of the cracks, in the H > 0 case, only one of the twocracks exhibits a continuous profile. The shapes of crack profile distorted and deviated from semi-ellipticalshape due to out-of-plane deformation and change in crack growth rate caused by interaction between thecracks.

Fig. 4 shows the fractured surface of the coplanar notch cases. In the case of H = 0, the crack progressedstraight on the surface of the specimen and coalesced during the experiment. The interval of beachmarksbefore the crack tips of the two cracks touched was almost uniform over the crack front. This implies thatthe interaction had little influence on the growth rate before coalescence. Conversely, the re-entrant sectionformed by coalescence of cracks exhibited a relatively fast growth rate, which can be recognized from the largeinterval of the beachmarks at that portion. The uneven growth made the re-entrant section the smooth frontshape, and finally, made a uniform growth rate over the front.

Crack growth rates obtained using the specimens containing a single notch are shown in Fig. 5. The crackgrowth rate was the average of two rates obtained using three beachmarks, which are the previous and follow-ing beachmarks of the center one. The SIF was calculated using the equation shown in the reference [11]

Fig. 2. Cracks on the specimen surface (specimen B-H2.5S2.5, initial condition: S = 2.5 mm, H = 2.5 mm).

Fig. 3. Fractured surface of specimen (initial notch is parallel).


assuming semi-elliptical shape based on the depth and surface length at the center beachmark. Crack growthrates obtained using the CT specimen are also shown in Fig. 5. As shown in the figure, the relationship

Fig. 4. Fractured surface of specimen (initial notch is coplanar).

10

100

1000

10 100

Stress intensity factor range, MPam0.5

Cra

ck g

row

th r

ate,

10-9

m/c

ycle

A-Single

B-Single

A-Single

B-Single

CT Specimen

11 2.523.5 10CGR K−= × Δ

Fig. 5. Crack growth rate obtained using single notched specimens and compact tension specimen (open symbol: growth rate at surface,close symbol: growth rate at the deepest point).


between the crack growth rate, CGR, and the SIF range, DK, can be approximated by using the power law ofthe SIF, and the following equation was obtained from the data from the CT specimen:

CGR ¼ 3:5� 10�11DK2:52: ð1Þ

The growth rate is given in m/cycle and DK in MPa m0.5. The growth rates at the deepest points almost matchthose obtained using the CT specimen, although those at the surface point were relatively slow. As shown inFigs. 3 and 4, at the surface point, the crack profiles do not cross the specimen surface perpendicularly. Thisimplies that the crack growth at the surface was relatively slow. It was pointed out that the growth rate at thesurface is slower than that at the deepest point under the same SIF [11], although the magnitude of the dif-ference is difficult to quantify [22].

The crack growth rates at the deepest point are shown in Fig. 6. In the case of two cracks, the SIF wascalculated assuming the semi-elliptical shape based on the maximum depth and span length at the surfaceof the two cracks on the projected plane, and the growth rates were measured at the deepest point. Therefore,

1

10

100

1000

10 100

Stress intensity factor range, MPam0.5Stress intensity factor range, MPam0.5

B-H2.5(1)

B-H2.5(2)

B-H5

B-H10

B-H0S5

B-Single

CT1

10

100

1000

10 100

Cra

ck g

row

th r

ate,

10-9

m/c

ycle

Cra

ck g

row

th r

ate,

10-9

m/c

ycle

A-H2.5

A-H2.5C2A4

A-H2.5C4A2

A-H2.5S2.5

A-H0S5

A-Single

CT

(a) Thickness of specimen: 10 mm (b) Thickness of specimen: 15 mm

Fig. 6. Crack growth rate at the deepest point.


the magnitude of SIF is near to that of a coalesced crack rather than an individual crack. The growth ratesshowed good correlation with the SIF range, although they differed from the data obtained using the CTspecimen.

As pointed out by the author previously, based on SIF evaluations, the interacting cracks can be regardedas a coalesced crack from a mechanical point of view when they are close enough [33,54]. Therefore, the crackgrowth rates in the cases of H = 2.5 mm and H = 0 (after coalescence) were almost identical to that of the CTspecimen, when the SIF is calculated assuming the cracks have coalesced. On the other hand, as shown in thecases of B-H10 and B-H5, a large offset distance caused a relatively slow growth rate due to small interactionbetween cracks. The magnitude of the interaction is dependent on the shape of the crack as well as the relativelocation, and varies with crack tip position along the crack front. Therefore, the influence of the interaction onthe growth rate varies during the growth along the front, making it difficult to assess the crack growth.

4. Procedure of analysis

In order to evaluate the influence of shape of crack profile on the growth behavior, the crack growth ofspecimen A-H0S5 (initial condition: S = 5 mm, H = 0) and B-H0S5 (initial condition: S = 5 mm, H = 0) weresimulated by FEA. In these tests, initial notches were semi-circular and in coplanar positions. These crackscoalesced during the tests and the coalesced crack had a re-entrant portion around the contact point, whilethe shape of the crack became smooth at the end of the tests. The initial crack shape of the simulation wasobtained from the first beachmark after the two cracks coalesced. Then, the growth behavior of the crack withthe re-entrant portion was evaluated.

In the simulation, an automatic mesh generating program was used, which can generate a finite elementmesh for an arbitrary-shaped crack based on crack tip coordinates. This program was developed by usingthe command language provided in the commercial software PATRAN [52]. The geometry of the FEA modelis shown in Fig. 7. The thickness and half-width of the plate were the same as those of the experiments, whilethe half-length was set to B = 100 mm. The initial crack shape was obtained from the crack profile of the frac-tured specimen. As shown in Fig. 4, the shapes of the coalesced crack were almost symmetrical. Therefore, inthe simulations, the initial cracks were assumed to have symmetrical shape, which was the same as the largercrack at the coalesced position. Due to the symmetries of the problem, only one quarter of the plate was mod-eled by finite elements. The mesh for the initial position of specimen A-H0S5 is shown in Fig. 8. The meshconsists of 26500 8-noded solid elements for the case shown in Fig. 8. The general-purpose finite element pro-gram ABAQUS, Version 6.5 [55] was used for FEA. In ABAQUS, the SIF is derived from the energy releaserate, which is obtained by the virtual crack extension integral method [56]. The finite element models have 73

Fig. 7. Geometry of cracked plate and boundary conditions.

Fig. 8. An example finite element mesh (specimen A-H0S5).


nodes along the crack front and the SIF was evaluated at each node. Poisson’s ratio of the material was takento be 0.3. The accuracy of this type of analysis was confirmed in the previous study [52].

Based on the evaluated SIF at each node position except the surface point, the extension length Dd wascalculated by the following equation:

Dd ¼ 3:5� 10�11DK2:52 dN ð2Þ

where dN is the interval of the step and was controlled so that the maximum Dd did not exceed 0.2 mm at anyportion along the crack front. The crack front extended perpendicularly to the crack front by length of Dd ateach node position, except the surface point and symmetry point. At these points, the crack tip position wasset to the same as the neighboring node in order to make the crack front perpendicular to the edge of the finiteelement model.

5. Analysis results

The evaluated SIF of mode I, KI, for the initial crack shape is shown in Fig. 9, where SIF is normalized as

F 1 ¼KI

r0

ffiffiffiffiffiffi

pap ð3Þ

where r0 denotes the applied tensile stress and a is the maximum depth and it was 4.97 mm and 8.36 mm in thecase of A-H0S5 and B-H0S5, respectively. The SIF takes various values along the crack front, and becomes

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 11 21 31 41 51 61 71

Crack tip position

Nor

mal

ized

str

ess

inte

nsity

fac

tor

F1

A-H0S5

B-H0S5

12

3

73 72

I1

o

KF

aσ π=

Fig. 9. Normalized stress intensity factor along the crack front obtained from the initial crack profile.

Fig. 10. Loci of crack profile obtained by the simulations.


large at the re-entrant portion and the surface. The small fluctuation in the SIF along the front resulted fromfaint un-smoothness of the crack shape.

Fig. 10 shows loci of crack profile during the simulations. At an early stage of the simulation, the dentedportions grew at a relatively fast rate due to large SIF. The fast growth made the crack shape semi-elliptical,and this was consistent with the experimental result. In order to evaluate the influence of change in the frontshape, simulations in which the initial shape was semi-elliptical were performed. In the simulations, the area ofthe initial semi-elliptical crack was the same as that of each test (simulation). The result corresponding to B-H0S5 is shown in Fig. 10c. Regardless of the initial condition, the final crack shape is similar.


The aspect ratio, which is defined as (depth at symmetrical line: D)/(half of surface length: R), during thesimulation is shown in Fig. 11, together with the experimental results. The simulations showed slightly smalleraspect ratio than the experiments. In the calculation of the SIF, the plane strain condition was assumed includ-ing at near the surface point. Furthermore, the exceptional treatment in the crack extension procedure at thesurface point may cause the difference. As shown in Fig. 4, growth rate at the surface was relatively slow in theexperiments and this was not observed in the simulations. In any case, the difference is small and the aspectratio converges to a constant value regardless of the initial crack shape.

The change in area of the crack face during the simulations is shown in Fig. 12, together with the exper-imental results. The simulation results were almost identical to the experimental results. Furthermore, the sim-ulations for a semi-elliptical crack showed almost the same results as the experiment and its simulation. Theseconsistencies imply the validity of the current simulation for predicting the change in area. The change in areacan be predicted by assuming the extension of crack front based on the evaluated SIF at each position alongthe front. It should be noted that the cycles for the beachmarks are not included in the number of cycles, andarea of beachmarks is not excluded from the total area, because the area of beachmarks is small.

Fig. 13 shows the change in the growth rate in area (hereafter, area growth rate and denoted as dA/dN). Aparameter representing the crack driving force, which is denoted as Df and defined by the following equation,is taken as the horizontal axis:

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400 500

Area size, mm2

Asp

ect r

atio

D/

R

Simulation (A-H0S5)Simulation (B-H0S5)Experiment (A-H0S5)Experiment (B-H0S5)Simulation (Semi-ellipse A)Simulation (Semi-ellipse B)

Fig. 11. Change in aspect ratio during the simulations and experiments.

0

100

200

300

400

500

600

0 10000 20000 30000 40000Number of cycles

Are

a of

cra

ck f

ace,

mm

2


Fig. 12. Change in area during the simulations and experiments.

1

10

100

1 10

Crack driving force σ o(A )0.5

, MPa m

Are

a gr

owth

rat

edA

/dN

, 10-9

m2 /c

ycle


Fig. 13. Change in area growth rate against the crack driving force.


Df ¼ Dr0

ffiffiffi

Ap

ð4Þ

where Dr0 is the stress range and A is the area of the crack face. Also, good agreement in the area growth ratewas observed between the experiment, the simulation for the re-entrant crack and the simulation for the semi-elliptical crack.

6. Discussion

The results shown in Figs. 12 and 13 reveal that the area growth rate is the same for the same area andapplied stress range. Since the cracks tended to be semi-elliptical regardless of their initial shape, the areagrowth rate can be taken as the crack growth rate in length, which is represented by Eq. (1). In this simulation,crack growth was modeled as growth of each point along the crack front, and growth rate was determined bythe evaluated SIF and Eq. (1). Therefore, the resultant area growth rate does not necessarily correlate witharea. Also, the area growth rate might be affected by material constants in the crack growth equation, lengthof crack front and distribution of the SIF along the crack front. Especially, the exponential constant, mp, inthe growth rate equation, which is mp = 2.52 in Eq. (1), could have an influence on the area growth rate. Then,simulations were conducted for the cases of mp = 1.5 and mp = 3.3, respectively. Fig. 14 shows the change inarea of the crack face obtained by the simulations. The initial crack shape and other conditions were the sameas the previous simulations. Even under different value of material constant mp, the area growth rate is almostthe same for cracks of the same area and applied stress range regardless of the crack shape. As mentionedpreviously, the author pointed out that the average of the SIF along the crack front is almost the same forvarious shapes of crack front when the area of the crack face is the same [52]. The dependency of area onthe SIF was also pointed out by Murakami and Nemat-Nasser [28]. From the results of the current study,it can be concluded that the area of the crack face is the predominant parameter for the growth of interactingcracks under the test conditions.

If the area correlates with the crack growth driving force, it is not necessary to consider the exact shape offront or the fluctuation of the SIF along the front in the growth prediction of a crack with complex frontshape. The prediction based on a semi-elliptical crack of the same area provides a good approximation inthe change of crack size. So, cracks of various shapes can be recharacterized as a semi-elliptical crack ofthe same area. The same is true for interacting cracks. Fig. 15 shows the relationship between the area growthrate and the crack driving force. In the cases of parallel initial notches, the area was double the measured sizewhich was taken from one of the two cracks on the projected plane. Therefore, the area corresponds to thatmeasured on the projected plane that is perpendicular to the load axis. The data corresponds to various con-

0

100

200

300

400

500

600

0 50 100 150 200 250Number of cycles

Are

a of

cra

ck f

ace,

mm

2

Are

a of

cra

ck f

ace,

mm

2

Simulation (A-H0S5)Simulation (B-H0S5)Simulation (Semi-ellipse A)Simulation (Semi-ellipse B)

0

100

200

300

400

500

600

0 10 20 30 40Number of cycles 106

Simulation (A-H0S5)Simulation (B-H0S5)Simulation (Semi-ellipse A)Simulation (Semi-ellipse B)

(a) Exponential constant mp = 3.3 (a) Exponential constant mp = 1.5

11 3.33.5 10CGR KΔ−= × 11 1.53.5 10CGR KΔ−= ×

Fig. 14. Change in area during the simulations under different crack growth equation.

0.1

1.0

10.0

100.0

1 10

Are

a gr

owth

rat

edA

/dN

, 10-9

m2 /c

ycle

Are

a gr

owth

rat

edA

/dN

, 10-9

m2 /c

ycle

B-H2.5(1)

B-H2.5(2)

B-H5

B-H10

B-H0S5

B-Single0.1

1.0

10.0

100.0

1 10Crack Driving Force σ o(A )0.5, MPa m Crack Driving Force σ o(A )0.5, MPa m

A-H2.5

A-H2.5C2A4

A-H2.5C4A2

A-H2.5S2.5

A-H0S5

A-Single

(a) Thickness of specimen: 10 mm (b) Thickness of specimen: 15 mm

Fig. 15. Change in area against the crack driving force.


ditions such as single crack, parallel cracks and co-planar re-entrant crack. There is little scatter in the growthrate in area. This implies that the area is also a representative parameter in the growth of interacting surfacecracks as well as independent cracks. The agreement in the area growth rate between interacting cracks andsingle cracks suggests that, for the prediction of crack growth, interacting cracks can be replaced with a singlecrack of the same area on the projected plane. This replacement (recharacterizing) can be applied for assessingthe crack growth of interacting cracks. It should be noted that this technique is not for FFS but for assessmentof realistic growth behavior of interacting cracks, because no deliberate conservatism is included in thisrecharacterization.

In the previous study, the author evaluated the SIF for interacting parallel surface cracks under variousrelative positions, and pointed out that the interaction is significant when the offset distance H is smaller than0.8a when the size of interacting cracks is the same [38]. The offset distance at the first beachmark was 1.38a inthe case of B-H10 and it was the largest among all cases of parallel cracks, whereas the 0.63a for B-H5 was thesecond largest. Therefore, after the first beachmark, the magnitude of the interaction was large enough so thatthe two cracks could be regarded as a coalesced crack, although the area growth rate slightly depended onoffset distance (e.g. B-H10 and B-H5). The quantitative evaluation of the growth of parallel cracks will be dis-cussed in Part II of the current work [57].


7. Summary and conclusions

In this study, in order to investigate the relationship between the area of crack face and fatigue crackgrowth behavior, fatigue crack growth tests were conducted using stainless steel plate specimens with surfacenotches. Then, FEA was performed to simulate the growth behavior obtained by the experiment. Based onthese results, the role of area in the growth behavior of cracks with a dented portion and interacting crackswas discussed. The conclusions obtained are as follows:

(1) A procedure for simulating crack growth considering the change in crack shape was shown. The changein area of crack face can be predicted by assuming the extension of crack front based on evaluated SIF ateach position along the front.

(2) The area is the representative parameter in the growth of interacting surface cracks as well as indepen-dent cracks. The growth in area showed good correlation with the driving force based on area.

(3) The growth of the crack with a re-entrant portion can be evaluated by recharacterizing the crack as asemi-elliptical crack of the same area.

(4) The recharacterization can be applied for interacting parallel cracks. In this case, the area on the pro-jected plane is dominant.

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Mech, in press, doi:10.1016/j.engfracmech.2007.07.014.

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Documents

Growth evaluation of multiple interacting surface cracks. Part I: Experiments and simulation of coalesced crack