Three-dimensional fatigue crack growth simulation in ...

China Ocean Engineering

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Three-dimensional fatigue crack growth simulation in crack arrestor by using XFEM method

Journal: China Ocean Engineering

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Manuscript ID COE-2018-0230 Manuscript Type: Original Article

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Keywords:

extended finite element method, crack arrestor, fatigue crack growth, fatigue life prediction

Speciality: Dynamic response of floating structure

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Three-dimensional fatigue crack growth simulation in crack arrestor by using XFEM method Hamid Moaieri1, Hadi Tohidi2, Masoud Moghaddasi3 1. 2. 3.

Phd candidate, Marine Engineering, Amirkabir University of Technology, Iran Phd candidate, Mechanical Engineering, Islamic Azad University Science and Research Branch, , Iran MSc, Mechanical Engineering, Shahrekord University, Iran

Abstract: In this paper, fatigue crack growth in ship components that have crack arrestor, has been simulated using linear elastic fracture mechanics. In ship structures, a metal plate is used as arrestor which is placed on the damaged components in many ways, to prevent the crack growth. The extended finite element method has been used to simulate three-dimensional fatigue crack growth and obtain growth path and crack front. XFEM method reduces the calculation time significantly, because it does not require re-meshing at each step, and makes it possible to use larger meshing near the crack tip. For fatigue analysis in this study, appropriate mesh size was selected by using exact value of stress intensity factor, and semi-elliptical crack was placed at place of maximum Von Mises stress. Paris equation has been used to calculate the fatigue life of the desired component. The results of this simulation show that using crack arrestor can improve fatigue life of ship component. So far, no three-dimensional study is conducted by XFEM method on crack arrestors. Some validations have been carried out to ensure the accuracy of the simulation results. Fatigue crack growth path, fatigue life and the effect of arrestor dimensions was obtained. Keywords: ship, extended finite element method, crack arrestor, fatigue crack growth, fatigue life prediction.

the shape and of material properties. By using the SIF and Paris law, the fatigue crack growth at the plate is measured. In fact, the Paris law describes the crack growth rate in terms of material properties and stress intensity factor. Stress intensity factor can be achieved using numerical and theoretical methods. One of the numerical methods is the conventional finite element method, but it has high computational costs due to two reasons. Firstly, the use of very fine meshes around the crack tip to model the singular stress fields in this area, and secondly, after each step of crack growth, the mesh should be updated to match with the new shape of the crack front. These problems of using the finite element method have been recently fixed in developed finite element method. The basic idea of XFEM was presented by Belytschko and Black (1999). They added some degrees of freedom to the finite element approximations using partition of unity method to model the asymptotic singular field in the crack front. This reduced the dependence of mesh to crack geometry and made it possible for bigger meshing to be used in the crack tip. Belytschko and Dolbow (1999) added another group of degrees of freedom interpolated with Heaviside function to display the discontinuities of crack face and called it XFEM. By using this method, they modeled the crack independent of the mesh. In this model, the need to update the mesh at each stage after crack growth was fixed. Dolbow (1999) presented a solution to locally enrich displacement field in modeling any kind of discontinuities and improved the method. (Fares, 1989) (Xu et al., 1994) used integral methods to calculate the stress intensity factor. These methods are particularly suitable for the study of cracks in infinite objects; unlike the finite element where the whole model is meshed, here only the crack needs meshing. For example this method. If the crack grows in a single plane, only one-dimensional meshing is needed and computational costs are greatly reduced. This solution is called perturbation method that was developed by Bower and Oritz (1990, 1993) based on Rice’s works (1989). In this method, variation of the stress intensity factor is calculated after a small co-planar perturbation of crack front under constant loading with a one-dimensional integration. Later, Lazarus (2003) offered a different solution with improved numerical precision. Boundary element method is an alternative solution to model the fatigue crack growth (Cruse, 1988) (Gerstle et al., 1988) (Mi & Aliabadi, 1994), but it is not simply applicable to

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1 Introduction 1

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Since ships are frequently subjected to complex loads during their service life; fluctuating stresses are always present at the structural members. This may cause fatigue failure in a member of the structure or the whole structure. This damage can cause catastrophic life, financial and environmental events. Therefore, fatigue strength assessment and prediction of the fatigue life are important in ship structural design. In materials science and material fatigue, the crack arrestor is a strong structure ring or a strip of material that prevents the occurrence of catastrophic fracture and damage by limiting the stresses surrounding crack. In ship structures, a metal plate is used as arrestor which is placed on the damaged components in different ways, to prevent the crack growth. In recent decades, fracture mechanics have been used to evaluate and develop the methods for fatigue crack growth. For example, (Hobbacher, 2009) (Sumi and Inoue, 2011) (Sumi, 2014) (Doshi and Vhanmane, 2013). (Paris et al., 1961) (Huang et al., 2008) (Cui et al., 2011) (Dong et al., 2015) employed fatigue life prediction method to study fracture mechanics. Fatigue life prediction is done using stress intensity factor (SIF). SIF is a complicated function of applied load, boundary conditions, crack growth, geometry of

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*Corresponding author Email: [email protected]

mc03.manuscriptcentral.com/coe


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nonlinear problems. For example, this method cannot be used in cases where the material has a non-linear behavior such as plasticity. Weight function is another method to calculate stress intensity factor. One limitation of this method is that the weight function is selected according to a specific configuration, so, crack growth direction should not be changed or new weight function should be derived (Fett and Munz, 1994) (Mazzu, 2013). Green's function method (Hills et al., 1984) (Nowell and Hills, 1987) can also be used for rapid calculation of stress intensity factor. In this method, Green’s function is the stress intensity factor, which is calculated for a point load. With this function, the stress intensity factor for any arbitrary loading is achieved with the integration, but finding the Green's function for complex geometries can be very difficult and time consuming. In this study, three-dimensional examination of fatigue crack growth in plates that have crack arrestor has been performed using XFEM. An initial crack has been placed where maximum stress is present. In this research, a program has been developed in Python language in ABAQUS environment, and the capacities of this program are used to model the crack using XFEM. Using this program, the crack growth path and its final form were obtained. Paris’ law was used to calculate the fatigue life. The procedure used was compared with previous experimental works for validation, which represents an acceptable approximation.

 1 H (x ) =  −1

if

( x − x ).n ≥ 0 *

(2)

otherwise

As shown in Fig. 1, x is the arbitrary point near crack face, x* is the nearest point to x, which is located on the crack face and n is the unit outward normal to the crack at x*. The asymptotic near tip function are as follows (Mohammadi, 2008): θ  θ  Fa ( x ) = [ r sin sin   , r coscos   , 2 2 θ  r sin (θ ) sin sin (θ ) sin   , 2 θ  r sin (θ ) cossin (θ ) cos  ] 2

(3)

Where θ and r are the local polar coordinates which are given in Fig. 1. These functions are extracted from classical solution to the θ

crack problem (Tanaka, 1974). The term √r sin � � takes 2 into account the discontinuity across the crack faces near the crack tip.

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2 Preliminaries

n 4  a = u ∑ N j ( x ) u j + H ( x ) a j + ∑ Fa ( x ) b j    j 1= a 1

(1)

Where Nj are conventional finite element shape functions, Uj are usual displacement of degrees of freedom, a j are added degrees of freedom to model the jump across crack face at the points far from the crack tip, Fa (x) are asymptotic near tip functions, which are shown in Eq. (3). a j are degrees of freedom of elements nodes that cut the crack faces, and baj are degrees of freedom are applied to nodes of an element that contains crack tip. The discontinuity jump function is given by the following equation (Simulia, 2014):

Fig. 1 Illustration of x and x* points.

2.2. Fatigue crack growth Accumulation of damages from fatigue causes crack growth. Any alternative loading which alters the stress intensity factor may trigger failure mechanism. Paris law is used to calculate the fatigue life cycles from the point when the initial crack begins to grow until it reaches the critical length where the failure occurs. The Paris law formula is as follows (Fares, 1989):

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2.1 Extended finite element method XFEM method uses extra degrees of freedom and special functions to represent the crack geometry in the finite element model. Two types of function are used for this purpose. These functions include near-tip Asymptotic field function and discontinuous function to model singularity of crack tip and the jump in the displacement field across the crack face, respectively. By enriching the conventional finite element approximations by partition of unity method using these functions and degrees of freedom, displacement field will be as follows (Mohammadi, 2008):

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da n = c ( ∆K ) dN

(4)

In the above relation, c and n are matter constants which are obtained by three-point bending test. 𝑎𝑎 is crack length, 𝑁𝑁 is number of load cycles, and ∆K is stress intensity factor range which is equal to the value below (Fares, 1989):

∆K= K max − K min

(5)

"𝑐𝑐" 𝑎𝑎𝑎𝑎𝑎𝑎 "𝑛𝑛" are Paris law constants which could be obtained by experiments on specimens with through-thickness crack; however, by assumption of plane stress or strain condition on any points along crack front, this law can be applied to any point on curvilinear crack front. In this paper, out-of-plane displacement is small, and the


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calculated (K ΙΙΙ ) terms are close to zero, so the effect of crack stress concentration is in the plane perpendicular to the crack front, as a result, in-plane stresses and strains are much larger than out-of-plane terms. That is why the plane stress and plane strain assumption have been used. With regard to assumption of plane strain condition, Keq can be used instead of K in Eq. (5). This relation is defined as follows (Tanaka, 1974):

= K eq

4

K 4Ι + 8K 4ΙΙ

(6)

3. Modeling approach 3.1. Three-dimensional crack growth modeling The total fatigue life of the piece has been divided into ∆N=1,000. For reducing the computation time, it is better to choose the size of the crack growth in each step as large as possible. However, if growth in each step is too large, crack growth path will not be calculated correctly. To select the size of the step, a convergence analysis was carried at the initial steps of crack growth. At each step of crack growth, stress intensity factors of different modes are calculated in the crack front points (Fig. 2a). These points are the mesh node points, so their number depends on the size of the mesh. The shape of crack front at each step is obtained by fitting the curve passing through these points. Equivalent stress intensity factor (𝐾𝐾𝑒𝑒𝑒𝑒 ) are calculated at these points, the size of the crack growth is obtained by Paris equation at any point (Eq. 4). The crack propagation angle is obtained at maximum circumferential stress criterion (Fares, 1989):

Fig. 2 Building new crack surface: (a) some points are chosen along the initial crack front; (b) new points are found; and (c) new surfaces are fitted.

intensity factor reaches a critical value. This process of 3D fatigue crack propagation is executed in ABAQUS by using its scripting environment, which takes advantage of Python language. Fatigue analysis flowchart is given in Fig. 4.

   

(7)

Fig. 3 Local coordinate system in crack front.

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The angle is measured from the vector ‘𝒒𝒒’ to vector ‘𝒏𝒏’ in local coordinate system shown in Fig. 3. Each point extends based on the magnitude and direction to achieve the new coordinate of each point (Fig. 2b). A surface will be fitted on these new points and the previous points. This surface is considered as level surface 𝝓𝝓 = 𝟎𝟎 of function 𝝓𝝓. Another surface will be fitted on the new points perpendicular to 𝝓𝝓= 0 surface. This surface is considered as level surface 𝝍𝝍 = 𝟎𝟎 of function 𝝍𝝍. The values of these two functions will be used to define the location of crack in ABAQUS. The values of these functions at the nodes of the elements cut by the crack will be fed to ABAQUS input file (*.inp). The crack growth steps will continue until the stress

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 3K 2 + K 2 + 8K 2 K 2 ΙΙ Ι Ι ΙΙ  K Ι2 + 9K ΙΙ2 

θ = cos −1 

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3.2. Geometry and properties of material In order to study fatigue crack growth on metal plates that having crack arrestors, a metal plate with dimensions that is shown in Fig. 5, is intended. The initial location of the crack and its orientation is specified in this figure. Crack angle is considered 0o, 30o, 45o, 60o and 90o. Arrestor is considered with different sizes and thickness. Thicknesses of one, two and three millimetres are considered for the arrestor; the arrestor’s dimensions are shown in Table 1. Different modes of arrestor installation are shown in Fig. 6.



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Fig. 4 Fatigue analysis flowchart

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Fig. 5 Dimensions of base plate, initial location of the crack and its orientation.

Fig. 6 Different modes of arrestor installation


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formulas (Fares, 1989): Table 1 Case

a[mm] 50

2

3

4

Arrestor’s dimensions Arrestor b[mm] area [mm2] 1.15×104 115

40

115

30

115

25

115

40

35

30

45

25

45

25

35

40

80

30

80

20

80

30

70

30

60

9.2×103

K Ι = λs σ

Sample

6.9×103

F24

5.6×103

F31

5.4×103

F32

4.5×103

F34

6.4×103

f (ϕ )



a





(8)

Where a and c are crack dimensions and ϕ is the angle indicating the position on the crack front as shown in Fig. 8. 00

F33

3.5×103

Q

a Q = 1 + 1.464( )1.65 c a f (ϕ ) (sin 2 ϕ + ( ) 2 cos 2 ϕ )0.25 = c

F23

5.75×103

πa

λs = 1.13 − 0.09  * (1 + 0.1(1 − sin ϕ ) 2 ) c

F21 F22

F41

4.8×103

F42

3.2×103

F43

4.2×103

Fo 3.6×103

F44 F45

The steel used is ASTM-A516 which is broadly used in marine works. Its mechanical properties are shown in Table 2. Material constants have been taken from BS standard, one of the most reliable sources for studying fatigue. Table 2

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Mechanical properties of ASTM-A516

Tensile stress (MPa)

Fracture stress (MPa)

Fracture strain (%)

365

550

31.4

n

2.88

C

iew 6.77×1013

A trade-off between the error and computation time was used to select the mesh size. Mesh study was based on the values of Stress Intensity Factors. Stress intensity factors are obtained at two points (Fig. 8) for different mesh sizes. a=3 mm and c=4 mm are considered as the initial size of the Crack. The results are shown in Table 3. Mesh 3 is selected among the different mesh sizes, because of its acceptable computational cost and error. At the same time, the theoretical value of stress intensity factor for a semi-elliptical surface crack in a semi-infinite solid subjected to stress σ can be calculated using the following

Fig. 7 Modeling base metal in ABAQUS

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3.3. Finite element model Modeling was performed as shown in Fig. 7 and was placed under static loading. The fatigue has fluctuating nature. Stress ratio (R) was used in order to consider the fluctuating nature. Since the stress intensity factor is linear and has a direct relationship with stress ratio, in order to obtain the lowest stress intensity factor, we should multiply its maximum value by R, and then use relations 5 and 6 to calculate ∆K and Keq. Boundary conditions are shown in Fig. 7. Two ends of the model in x direction are restricted in Y direction. In addition Base metals’ elements are constrained in Z direction. These conditions are taken into consideration so that the loading and boundary conditions have the highest correlation with exploitation of this piece. 3.3.1. Validation via theoretical model

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Fig. 8 Semi-elliptical crack in a semi-infinite body

4. Results Applying the above steps for modeling, fatigue crack growth was simulated to final fracture. In Table 4, the fatigue life for base plate without arrestor and in Table 5, the fatigue life of F2 arrestor is given. As seen in the Table 5, fatigue life increases by increasing the thickness of the arrestor. By installment of arrestors 60° cracks do not continue to grow. This is due to the fact that stress intensity factors do



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not reach their threshold value.

Table 3

Validation of Stress intensity factors and their associated errors for different mesh sizes at two different points on crack front Number of elements along crack front

Theoretical values (Eq. 8)

𝝋𝝋 = 𝟎𝟎

Error (%)

16.846 22.846 19.503 20.012

17.36 12.07 4.33 1.83

𝛑𝛑 𝟐𝟐 21.382 𝝋𝝋 =

20.836

Mesh 1 Mesh 2 Mesh 3 Mesh 4

10 20 30 40

18.846 23.039 21.754 20.705

Fo Table 4

Case

Angle

Fatigue life

0

43205

30

49685

45

60919

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F1

Fatigue life for base plate without arrestor

rR 60

86410

90

Crack does not grow

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Error (%)

11.86 7.75 1.74 3.16

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Table 5 Sample

Angle

0

30

45 F21

Thickness (mm)

Fatigue life

1 2 3 1 2 3 1 2 3 1

50198 60455 67831 59576 77133 92414 75892 94003 117615 121030 Crack does not grow Crack does not grow Crack does not grow

2 60 3 90

F22

2 60

--

--

Crack does not grow

45

60

90

---

0 30 45

F24

-1 2 3 1 2 3 1 2 3 1 2

60 3

90

--

2 3

--

1.16 1.40 1.57 1.17 1.45 1.71 1.19 1.43 1.81 1.35

1 2 3 1 2 3 1 2 3 1

--

Fatigue life 47525 55302 61786 54159 65839 79992 67373 78509 100998 108811 Crack does not grow Crack does not grow Crack does not grow 45797 54006 59620 53043 63066 76018 64952 73433 93009 102473 Crack does not grow Crack does not grow

Crack does not grow

Fatigue life increased ratio 1.10 1.28 1.43 1.09 1.32 1.61 1.11 1.28 1.65 1.26 ---1.06 1.25 1.38 1.07 1.27 1.53 1.07 1.22 1.54 1.16 ---

--

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90

F23

--

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3

Thickness (mm)

30

iew

45

50198 60445 67831 58131 72098 84961 72493 87479 110263 117254 Crack does not grow Crack does not grow

Angle

0

ev

30

1 2 3 1 2 3 1 2 3 1

Sample

rR

0

--

Fatigue life of F2 arrestor

Fatigue life increased ratio 1.16 1.40 1.57 1.20 1.550 1.86 1.25 1.54 1.93 1.40

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Fig. 9 shows the crack length diagram against fatigue life for F2 arrestor with a crack angle of zero and its comparison with the sample without arrestor.



8 80

80

60

F21-0 deg-1 mm

50

F22-0 deg1mm

40

F23-0 deg1mm

Crack length (mm)

Crack length (mm)

F31-0 d 1 mm

70

70

F24-0 deg-1 mm

30

60

F32-0 d 1 mm

50 Example of jump in diagram

F33-0 d 1 mm

40

F34-0 d 1 mm

30 20

F1

F1

10

20

0

10

0

10000

20000

0 0

10000

20000

30000 40000 Load cycle

50000

60000

30000 Load cycle

40000

50000

Fig. 10 Crack length versus load cycles for F3 arrestor

Fig. 9 Crack length versus load cycles for F2 arrestor

Fo

90

70

40 30

Figure 11 shows the crack length diagram against fatigue life for F2 arrestor with a crack angle of thirty. In this case, it is observed that the fatigue life and crack length are increased compared to the case without arrestor. The point that should be noted is that in this case, the effect of a size is greater than crack angle is zero, because when the crack grows at a 30 degree angle, not only the length, but also the width of the arrestor is important. Figure 12 shows the crack length against fatigue life for all the samples modeled in a thickness of 1 mm for an angle of zero degrees.

F1-30 deg

10 0 0

20000

40000

60000

80000

Load cycle

Fig. 11 Crack length versus load cycles for F2 arrestor with a crack angle of 30 80

70

F1

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According to the results of these calculations, we can understand that the fatigue life is not only related to the arrestor area, but more importantly the dimensions in which the crack grow. In other words, in this case, crack grows in direction with “𝑏𝑏”, and “𝑎𝑎”, “𝑏𝑏” has a greater impact than "𝑎𝑎", and samples that had larger “𝑏𝑏”, had longer fatigue life.

F24-30 deg-1 mm

20

F211mm F221mm F231mm

60

Crack length (mm)

dN

altered and the fatigue life was calculated for each of these cases.

F23-30 deg-1 mm

50

On

da

rate ( ) is reduced. In this case, the variables a and b were

F22-30 deg-1 mm

60

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Figure 10 shows the crack length diagram against fatigue life for F2 arrestor with a crack angle of zero. In this mode, the fatigue life and crack length are increased compared to base plate. There is a jump in the diagram of each sample; the jumps are due to crack arrival in the arrestor, where the crack growth

F21-30 deg-1 mm

80

Crack length (mm)

As seen in Fig. 9, by installing the crack arrestor, fatigue life and the maximum length of crack increase. Considering this diagram, fatigue life changes and the results of F21 and F22 samples are very close together and are almost overlapping. It could be interpereted that increasing the length of dimension “a”,the side which is not in initial crack direction, would not have considerable effect on life cycle after a certain value; In this case 𝑎𝑎 = 40𝑚𝑚𝑚𝑚. Maximum Crack length in the base plate without arrestor was 61.6 mm, and in the case that F21 arrestor was used, its length reached 70 mm.

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F241mm

50

F311mm F321mm

40

F331mm F341mm

30

F411mm

20

F421mm F431mm

10

F441mm

0

0

10000

20000

30000

40000

50000

60000

F451mm

Load cycle

Fig. 12 Crack length versus load cycles for all the samples with thickness of 1 mm and crack angle of zero Figure 13 gives the comparison between cases 1, 2 and 3 F2


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arrestor and the sample without arrestor. As it is seen, fatigue life and crack length has a direct relationship with the thickness of the arrestor. Increased arrestor thickness results in increased fatigue life. 100

F21-0 deg-1 mm F22-0 deg1mm F23-0 deg1mm F24-0 deg-1 mm F1

90 80

Crack length (mm)

70 60 50 40 30 20 10 0 0

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F21-0 deg-2 mm F22-0 deg-2 mm F23-0 deg-2 mm F24- 0 deg-2 mm F21-0 deg-3 mm F22-0 deg-3 mm F23-0 deg-3 mm F24-0 deg-3 mm

10000 20000 30000 40000 50000 60000 70000 Load cycle

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Fig. 13 Crack length versus load cycles for F2 arrestor in 1, 2 and 3 mm thickness

In Fig. 14, the fatigue crack growth path for base plate (without arrestor) at a crack angle of zero is shown. In Fig. 15, crack growth path is given for F21 arrestor which was the strongest arrestor. As can be seen in this figure, the crack growth rate in the side in which the arrestor is used is lower than the other side; for this reason, it will have shorter length in the same number of cycles.

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Fig. 14 Crack growth path in base plate with crack angle of 0 degree

(b) Fig. 15 Crack growth path in F21 arrestor: (a) plate side that is in contact with arrestor; (b) plate flat side



10

In Fig. 16, the crack growth path is drawn for the base plate (without arrestor) at an angle of 45 degrees; in this case crack growth rate is lower than the angle of zero. The crack initially grows diagonally and increased length results in more tendency to the vertical direction, and finally, the crack cuts the piece vertically.

• •

•

•

Increased crack arrestor thickness results in fatigue life and crack length increase. By increasing the arrestor’s length, in perpendicular to crack direction, its effect on life cycle would decrease after a certain value. It is also worth noting that the length of the dimension, in direction with the crack growth has a greater role to increase the fatigue life of the piece. Arrestor installation causes crack growth in an asymmetric semi- elliptical growth.

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In Fig. 17, the crack growth path is given for F21 arrestor for crack angle of 45 degrees. In each row of figures, the number of steps passed for crack growth is shown. As seen in this figure, crack growth in the side without arrestor is faster and more unstable. Moreover, in the side without arrestor, the crack growth path had greater curvature.

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5. Conclusion In this study, linear elastic fracture mechanics were used along with XFEM for simulating crack growth in planes having crack arrestors and predicting the fatigue life. XFEM method reduces the calculation time significantly, because it does not require for remeshing at each step, and makes it possible to use larger meshing near the crack tip. For fatigue analysis in this study, appropriate mesh size was selected by using exact value of stress intensity factor, and semi-elliptical crack was placed at place of maximum Von Mises stress. Fatigue life of the piece was predicted for different arrestor dimensions and crack angles. The crack growth path was also achieved in each step. The following results can be inferred from this study:


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(a)

(b)

Fig. 17 Crack growth path in F21 arrestor with crack angle of 45 degrees: (a) plate side that is in contact with arrestor; (b) plate flat side

References 1 Hobbacher, A. F. (2009). The new IIW recommendations for fatigue assessment of welded joints and components–A comprehensive code recently updated. International Journal of Fatigue, 31(1), 50-58. DOI: 10.1016/j.ijfatigue.2008.04.002 (Journals) 2 Sumi, Y., & Inoue, T. (2011). Multi-scale modeling of

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