Fatigue life prediction of cracked attachment lugs using XFEM

Accepted Manuscript Fatigue Life Prediction of Cracked Attachment Lugs Using XFEM M. Naderi, N. Iyyer PII: DOI: Reference:

S0142-1123(15)00060-2 http://dx.doi.org/10.1016/j.ijfatigue.2015.02.021 JIJF 3536

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

28 November 2014 25 February 2015 27 February 2015

Please cite this article as: Naderi, M., Iyyer, N., Fatigue Life Prediction of Cracked Attachment Lugs Using XFEM, International Journal of Fatigue (2015), doi: http://dx.doi.org/10.1016/j.ijfatigue.2015.02.021

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Fatigue Life Prediction of Cracked Attachment Lugs Using XFEM M. Naderi*, N. Iyyer Technical Data Analysis, Inc. (TDA), 3190 Fairview Park Drive, Suite 650, Falls Church, Virginia, 22042

Abstract: In the present paper, a three dimensional finite element method (FEM) is used to compute the stress intensity factor (SIF) in straight lugs of Aluminum 7075-T6. Extended finite element method (XFEM) capability available in ABAQUS is used to calculate the stress intensity factor. Crack growth and fatigue life of single through-thickness and single quarter elliptical corner cracks in attachment lug are estimated and then compared with the available experimental data for two different load ratios equal to 0.1 and 0.5. The SIF calculated from XFEM shows that the introduction of different loading boundary conditions significantly affect the estimated fatigue life.

Keywords: Attachment lugs; Stress intensity factor; Extended finite element method; Fatigue life

1. Introduction: Study and evaluation of the structural load carrying capability is of paramount importance to industry. Attachment lugs are one of the most critical components in airplane structures which transfer loads. Due to the geometry of the lugs, the level of the stress around the hole is high and does not promptly reduce as one proceeds from the hole into the material. Hence, attachment lugs are severely vulnerable to cracking following a catastrophic failure. In this prospect, an improved fracture mechanics based design may significantly prolong the fatigue life of attachment lugs while improving model accuracy and system safety.

The literature contains noteworthy studies of fatigue crack growth analysis for the fatigue life evaluation of structural components. Paris and Erdogan introduced a stress intensity factor (SIF) based empirical relationship for crack growth analysis [1] called Paris law equation. Walker [2] modified the Paris law

Corresponding author Tel.: +1 703 226 4073 E-mail address: [email protected]

Nomenclatures: a

crack depth

N

Number of cycle

aI , bI

nodal enriched degree of freedom vector

Q

Shape factor

af, cf

failure crack depth and length

R

load ratio

a0, c0

initial crack depth and length

rc

polar coordinate at crack tip

c

crack length

Ri

inner radius of hole

fw1

Finite width correction factor

Ro

outer radius of hole

f1

Finite width correction factor

t

lug thickness

G1

Pin-loaded lug correction factor

uI

nodal displacement vector

H(x)

Jump function

W

lug width

Kmax

Maximum SIF

x

a point on the crack

KI

mode I Stress intensity factor (SIF)

ϕ

angle of ellipse

σ0

remote stress

m, C, γ material parameters Me

Elastic magnification factor

ψα(x) asymptotic crack tip function

n

Outward normal of the contour

θ

NI(x)

Standard shape function

polar coordinate at crack tip

equation considering the mean stress effect. Elber [3], later modified the Paris law for crack growth rate considering crack closure concept. He experimentally found out that fatigue crack under cyclic tension is closed on itself about half of the maximum load. Schijve and Hoeymakers [4] and Friedrich and Schijve [5] studied lugs with a through-the-thickness and corner crack. They developed an empirical relationship based on experimental data for crack growth and stress intensity factor. Also a number of methods have been developed to relate fatigue crack growth to the maximum stress intensity factor and stress intensity factor range [6-9]. In the fatigue crack growth empirical equations, SIF is the necessary parameter related to the stress field ahead of the crack. Many analytical and numerical efforts are performed to obtain SIF for different crack configurations such as elliptical cracks, through-thickness cracks, elliptical corner cracks for infinite, finite plates as well as attachment lugs [10-22]. Raju and Newman [10-13] evaluated SIF for different corner cracks using some analytical solutions.

In a comprehensive experimental and numerical work, Kathiresan and Hsu[14] reported the damage tolerance design of straight and tapered lugs with through-thickness and corner crack. They used conventional finite element method with spring elements connected the pin and lug. Also, Rigby and 1|P a ge

Aliabadi [15] used boundary element method (BEM) to evaluate SIF for cracks at straight attachment lugs. They studied both uniform and cosine pressure distribution on the calculated SIF. Pian et al. [16] applied both uniform and cosine bearing pressure distribution to determine SIF using stress hybrid approach with special crack tip element. Aberson and Anderson [17] used a special crack-tip singularity element to compute the stress intensity factors for a crack in a nonsymmetrical aircraft lug of an engine pylon. Using two dimensional FEM, Gencoz et al. [18] reproduced the stresses observed in photoelastic models of the lugs with a special stress distribution. Pin-loaded lugs were analyzed in the presence of cracks emanating from circular holes of the lugs by Narayana et al.[19].They used finite element model with special singular six-node quadrilateral elements at the crack tip. The non-linear load contact behavior at the pin-hole interface was considered with by an inverse technique. Saouma and Zatz [20] proposed an innovative two-dimensional finite element method to determine the crack initiation, crack path and fatigue life. Boljanovic et al. [22] studied the strength of the attachment lugs with through-thickness and corner cracks. They used analytical and quarter-point singular finite element to evaluate SIF and estimated the fatigue life of the cracked attachment lugs by introducing stress ratio effect through the maximum stress intensity factor and stress intensity factor range. In addition to the numerical techniques used in the above cited articles to evaluate SIF and fatigue crack growth, a new finite element technique called Extended Finite Element Method (XFEM) was recently developed to handle arbitrary cracking in the material without re-meshing the crack tip [23-25].

The present paper aims to utilize XFEM to model fatigue crack growth of attachment lugs. ABAQUS commercial software [26] with XFEM capability is chosen for modeling and simulation software. Single through-thickness and single quarter elliptical corner cracks are considered as crack configurations in the present work. SIF is calculated using analytical equations and XFEM. Then, the fatigue life is estimated and compared with experimental data.

2. Theoretical Background:

In this section the analytical solution for SIF of through-thickness crack as well as elliptical corner cracks in finite plate and extended finite element (XFEM) model is reviewed. More details of the LEFM and XFEM are referred to the literature [10-13, 23-25].

2.1 Stress Intensity Factor of through-thickness and corner crack:

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Newman and Raju [10-12] investigated the stress intensity factors for single through-thickness and corner cracks in finite plates. The SIF equation for single through thickness crack in the lug can be expressed as [3, 10-12, 22] follows.

 πR  K I = σ 0 πc f w1 f1G1 sec i  W 

(1)

where c, Ri, W, r, ϕ and σ0 are crack length, hole radius, lug width and remote stress, respectively. The KI is stress intensity factor. The interaction between pin and lug is taken into account as follows [12]

G1 =

1 W + 2 π (2 Ri + c )

Ri Ri + c

(2)

The correction factor for single crack is expressed as follows [12]

f1 = 0.707 − 0.18λ + 6.55λ2 − 10.548λ3 + 6.85λ4

(3)

where

λ=

1 c 1 + cos(0.85ϕ ) Ri

(4)

Parameter ϕ is the elliptical angle in the case of quarter elliptical corner crack and ϕ=0 for throughthickness crack. The finite width correction factor is expressed as

 π 2 Ri + c  f w1 = sec   2 W −c 

(5)

For quarter elliptical corner crack in lug, the Equation (1) is re-write as follows

KI = σ0 π

a  πR  M e f1G1 sec i  gϕ Q W 

(6)

Where a is the crack depth. The relationship for fw1, f1, G1 are discussed above. Parameters Q, and Me are the elliptical shape factor, elastic magnification factor, and the front-face correction factor, respectively. The relationship for Q, Me, and gφ are discussed in [10-12, 22]. The above equations are used for analytical solution of SIF in through-thickness and corner cracked lugs. Crack growth process under cyclic loading can be theoretically investigated through the relationship proposed by Walker [28] equation. Walker modified Paris law [1] by introducing the stress ratio and maximum stress intensity factor as given by:

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[

da dc γ or = C (1 − R ) K max dN dN

]

m

(7)

where N is the number of cycles to failure, C, γ and m are material parameter experimentally obtained. The number of cycle to failure can be calculated by integrating Equation (7) at the depth or length direction. Crack direction in thickness and surface is called depth and length direction, respectively.

N =∫

a0

N =∫

da

af

[

C (1 − R ) K max,a γ

m

dc

cf

c0

(8)

]

[

γ

C (1 − R ) K max,c

(9)

]

m

where a0, and c0 are the initial crack depth and crack length, while af, and cf are the failure crack depth and crack length. Kmax,a and Kmax,c are the maximum stress intensity factor for depth and length direction respectively.

2.2 Extended Finite Element Method: The XFEM is a partition of unity based method [13] in which, the classical finite element approximation is enhanced by means of enrichment functions. These enrichment functions consist of the asymptotic crack tip functions that capture the singularity at the crack tip.

The features of XFEM which was first developed by Belytschko and Black [23] are adding a priori knowledge about the solution (using enrichment functions) in the finite element space and modeling discontinuities and singularities independent of the mesh. The enrichment functions typically consist of discontinuous asymptotic crack tip functions that capture the singularity around the crack tip and a discontinuous function representing the jump in displacement across the crack surfaces. The displacement approximation function with the partition of unity enrichment is as follows [24]

uXFEM ( x) = ∑ N I ( x)uI + I ∈S

4

∑ N I ( x)H ( x)aI +

∑ N I ( x)∑ψ α ( x)bIα

I ∈S h

I ∈S c

(14)

α =1

where NI (x) are standard shape functions, uI is the nodal displacement vector, ψ α (x) represents asymptotic crack tip solution function, aI is the nodal enriched degree of freedom vector, H(x) is the jump function across the crack surface and bαI represents the nodal enriched degree of freedom vector. The S are the nodal sets in the domain, Sh is the nodal set cut by the crack, and Sc is the nodal set surrounding the crack tip. For a general crack, the jump function is as follows [24]

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1 if ( x − x * ).n > 0 H ( x) =  − 1 otherwise

(15)

where n is the outward normal vector, x* is a point on the crack close to x. For an isotropic material, the asymptotic crack tip function is

θ θ θ θ   ψ α ( x ) =  rc sin , rc cos , rc sin sin θ , rc cos sin θ  

2

2

2

2



(16)

where r and θ are local polar coordinates defined at the crack tip. 3. Problem Statement and Model Description: This paper describes the finite element analysis of stress intensity factor calculation and fatigue life estimation of the attachment lugs using extended–finite element method (X-FEM). Figure 1a shows the specimen geometry and dimensions. Typical through-thickness and quarter-elliptical corner cracks are presented in Figure 1b-c. In the case of corner crack, it is assumed that the crack depth to length ratio is kept constant during computation. The specimen is cyclically loaded at the pin location and a crack is located perpendicular to the load at the bore of the hole. The material considered for the lug in this paper is Al-7075 T6 with elastic modulus of 70 Gpa. Table 1 summarizes the dimensions of the lug and material properties for fatigue life estimation [14, 28]. Two different load ratios (ratio of minimum stress to maximum stress) of 0.1 and 0.5 are considered in the simulation of fatigue life. The material constants m and C in Equations 8) and (9) are assumed the same for both depth and surface direction.

(a)

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(b)

(c)

Figure 1: a) Geomtry of the lugfour models, b) Throu-thickness crack, c) Corner crack Table 1: Dimenssions of the lugs (All dimenssions are in mm) [14, 28] C m Ri Ro/Ri W L t a/c γ -8 19.05 1.5, 2.25, 3 2Ro 200 12.7 1 2.71 x 10 3.7 0.645

4. Finite Element Model: ABAQUS commercial software with the capability of XFEM is used in the current work for SIF evaluation and fatigue crack growth analysis. Three dimensional brick or hexahedron elements (C3D8) are the considered mesh types in the present simulations. Different mesh sizes are considered for sensitivity analysis. The average aspect ratio of structured C3D8 brick elements around the cracked region area is set around 1 to 2 in the models. Figure 2 shows typical FE mesh with structured mesh around the crack region. In the attachment lugs different loading conditions are analyzed in this paper. Load is transferred through pin to the lug with the assumption that distributed pressure along the thickness is unchangeable. As shown in Figure 3, different techniques are examined to model applied pressure between the interface of pin and hole. In Figure 3a, the contact elements are used to simulate the pin load. A frictionless contact is considered between the pin and hole interface. A high stiffness material is assumed for the pin. It is extremely difficult to determine the exact distribution of contact pressure inside the hole. To overcome this difficulty, as shown in Figure 3b, a cosine pressure distribution is assumed. The uniform pressure distribution along the contact interface of pin/hole is considered as another technique of modeling load distribution (Figure 3c).

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Figure 2: Typical mesh of the current FE model

(a)

(b)

(c) Figure 3: Three methods of modeling pin/hole interface: a) full contact problem, b) Cosine pressure distribution, c) Uniform pressure distribution

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5. Numerical Procedure: For lug problems, stress intensity factor is calculated using Python script which has interface with the commercial software ABAQUS. It is a cumbersome task to use ABAQUS CAE module to model different possibilities of lug with different crack length and location and post-process the SIF calculation. The flowchart of the SIF calculation procedure is presented in Figure 4. The Python script starts with input data of the lug geometry, initial crack geometry and location. Three-dimensional model is then built in ABAQUS along with for example boundary conditions, X-FEM crack definition, mesh generation and etc. ABAQUS solves stress field and calculates SIF for each crack length and location by saving the *.odb files. Next, Python script reads all *.odb files and extract SIF into a *.txt file for all possible crack length and location. It is noted that only first mode stress intensity factor (KI) is brought in this paper. SIF values computed from XFEM are used for calculation of a polynomial expression. Matlab curve fit toolbox is employed to fit a best polynomial for the calculated SIF obtained from XFEM in the form of n

K max = ∑ bi a i

(17)

i =0

where a represents the crack length, bi is the coefficient of polynomial and n is the degree of the polynomial which is determined by Matlab curve fit toolbox. A Matlab program was written to read polynomial curve fits and then calculate fatigue life using Equations (8) and (9).

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Figure 4: Flowchart of Python script to extract SIF

6. Results and Discussion: In what follows, different numerical examples of lug with either through-thickness crack or quarter elliptical crack are presented. SIF and fatigue life are evaluated for the lug subjected to constant amplitude fatigue loading. A mesh sensitivity analysis is performed for validation of the numerical results using XFEM. Two different load ratios (R) of 0.1and 0.5 as well as three different Ro/Ri of 1.5, 2.25, and 3 are considered. The maximum remote stress is 41.38 Mpa. Experimental data are reported in Ref. [14]. In XFEM simulation, maximum five contour integrals are requested in which the first two are set aside in average SIF calculation.

Table 2 summarizes the calculated SIF based on analytical solution and XFEM for different mesh sizes including Mesh 1, Mesh 2, and Mesh 3. The number of elements through-thickness in Mesh 1, Mesh 2, and Mesh 3 are 10, 15, 20, respectively. Minimum number of elements 10 through thickness is chosen such that enough number of elements in front of the crack tip exists to guarantee the last element ring falls within the model’s boundary. The results of Table 2 corresponds to through-thickness cracked lug with 9|P a ge

Ro/Ri =3 and R=0.5. Constant pin pressure is assumed for the simulation results of Table 2. In all simulation, the elements’ aspect ratio near the crack region is around 1 to 2. It can be seen that XFEM results are almost insensitive to the mesh refinement. The obtained difference between SIF of two different meshes is small and acceptable.

The effect of different loading boundary conditions on the calculated SIF is summarized in Table 3 for the model with through-thickness crack and Mesh 1. It is seen that SIF obtained from the cosine pin pressure loading condition is close to the analytical solution. However, for actual pin and constant pin pressure loading condition, the SIF is slightly overestimated which in turn results in more conservative fatigue life. Table 4 compares the results of stress intensity factor computed analytically and numerically with different loading boundary conditions. The results of Table 4 corresponds to corner cracked lug with Ro/Ri =2.25. The Von Misses stress distribution is shown in Figure 4 for both through-thickness and corner crack configuration with two different crack length including 5 and 10 mm. The results of Figure 5 resembles to the lug with Ro/Ri =2.25 and the maximum applied load is 41.38 Mpa. It is noted that the linear elasticity assumption is considered in the numerical simulation.

Table 2: SIF applying analytical and numerical XFEM method with various mesh sizes KI (Mpa m1/2) c (mm) Analytical XFEM, Mesh 1 XFEM, Mesh 2 XFEM, Mesh 3 5 16.47 16.19 16.12 16.04 10 17.22 17.54 17.41 17.39 15 17.72 18.15 18.1 18.11 20 18.62 19.09 18.97 18.89 25 20.39 20.69 20.68 20.53 30 24.26 23.97 23.83 23.5

Table 3: SIF for through-thickness crack applying analytical and numerical XFEM method KI (Mpa m1/2) c Model with Cosine Pin Model with Constant Pin Model with Actual (mm) Analytical Pressure Pressure Pin 5 16.47 16.19 16.74 18.6 10 17.22 17.54 19.22 19.94 15 17.72 18.15 20.85 21.05 20 18.62 19.09 22.68 23.7 25 20.39 20.69 25.43 25.65 30 24.26 23.97 29.49 30.5 10 | P a g e

Table 4: SIF of corner crack applying analytical and numerical XFEM method KI (Mpa m1/2) Model with Cosine Pin Model with Constant Pin φ (o) Analytical Pressure Pressure 0 11.25 11.24 12.24 15 12.9 10.5 11.4 30 12.78 10.2 11.1 45 12.9 10.6 11.22 60 13.4 11.45 11.92 75 14.3 12.8 13.2 90 15.8 14.5 15.1

Model with Actual Pin 13.6 10.4 9.6 10.42 10.55 12.2 15.5

In the framework of linear elastic fracture mechanics (LEFM), fatigue life estimation of lugs with either through-thickness or corner crack starts with SIF calculation by applying Equations (1) and (6) for adequate increment of crack length. Then, number of loading cycles up to failure can be estimated by applying Equations (8) and (9). The calculated crack length versus number of loading cycles is plotted in Figure 6a-c for the lug with through-thickness crack. The maximum applied load is 41.38 Mpa with two different load ratios of 0.1 and 0.5 as well as Ro/Ri=2.25, and 3. The results are compared with experimental data reported in [14]. It can be seen that simulated SIF considering cosine pin pressure distribution are close to the analytical solution. However, calculated SIF with assumption of actual pin and constant pin pressure distribution are relatively close to each other with significant difference calculated SIF of analytical solution and cosine pin pressure assumption. For the load ratio of 0.5 (Figure 6a and 6c), the estimation of number of loading cycles to failure for the model with actual pin and with constant pin pressure boundary condition is in fair correlation with experimental data. In contrast, a significant difference is obvious in the fatigue life estimated from analytical solution and the model with cosine pin pressure boundary condition with experimental data.

For the load ratio of 0.1 (Figures 6b and 6d), in contrast to Figure 6a and 6c, the fatigue life estimated from the model with actual pin and with constant pin pressure boundary condition is underestimated. However, the estimation of number of loading cycles to failure computed from analytical method and the model with cosine pin pressure is in fair correlation with experimental data.

The fatigue life evaluations are presented in Figure 6a-b for the lug with single quarter elliptical crack. The maximum applied load is 41.38 Mpa and Ro/Ri=1.5 with two the load ratios of 0.5 (Figure 7a) and 0.1

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(Figure 7b). The crack length c is presented in upper half of the figure and crack length a is plotted in lower half of the figure.

For the crack length along the bore, a, the model with cosine pin pressure tends to adequately predict the fatigue life when compared with experiments from engineering point of view. It can be seen that conservative estimations are resulted from analytical methods and XFEM with actual pin ad constant pin pressure boundary condition.

For the crack length along the surface, c, the results obtained from analytical method and the XFEM model with actual pin tends to the conservative life. However, from engineering point of view, the estimated life resulted from the model with actual cosine and constant pin pressure boundary condition has an adequate trend when compared to experimental data.

(a)

(c)

(b)

(d) 12 | P a g e

Figure 5: Von Misses stress distribution of attachment lug . 5a) Through-thickness crack, a=5 mm; 5b) Through-thickness crack, a=10 mm; 5c) Corner crack, a=c=5 mm; 5d) Corner crack, a=c=10 mm

(a)

(b)

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(c)

(d) Figure 6: Crack length versus number of loading cycles to failure for the single through-thickness cracked lug. The results are compared with experiment. a) Experiment 1: ABPLC84, Experiment 2: ABPLC91; b) Experiment 1: ABPLC85, Experiment 2: ABPLC89; c) Experiment 1: ABPLC46, Experiment 2: ABPLC93.

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(a)

(b) Figure 7: Crack length versus number of loading cycles to failure for the single corner cracked lug. The results are compared with experiment . a) Load ratio = 0.5, Experiment 1: ABPLC18, Experiment 2: ABPLC22; b) Load ratio = 0.1, Experiment 1: ABPLC17, Experiment 2: ABPLC21.

7. Conclusions:

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In the present work, a XFEM-based computational procedure has been developed for fatigue crack growth and SIF calculation. Crack growth and fatigue life of single through-thickness and single quarter elliptical corner cracks in attachment lug are estimated and then compared with the available

experimental data. In the FE model, the load is applied through three different methods including the model with cosine pin pressure, constant pin pressure and actual pin. Mesh sensitivity analysis is performed for three different mesh sizes. The model with pin usually shows higher SIF and lower fatigue life when compared with experimental data and other models. Correlations of analytical- and XFEM-experimental data of through-thickness and corner cracks are presented. The estimated fatigue life obtained from different methods shows fair correlations in some cases and poor correlations in other case. Therefore, some important points can be concluded from this study. 1- For finite element models, it is seen that the loading boundary condition including mode with actual pin, constant pin pressure and cosine pin pressure effects the fatigue life by a factor of about 2 to 3. 2- Comparing with the experimental data, fatigue life with the actual pin in the model is generally conservative by the factor of about 2 to 3. 3- Analytical solutions together with the Walker’s crack growth law usually tend to result conservative fatigue life in the case of corner crack. However, for through-thickness crack, both conservative and un-conservative answer fatigue life are expected. 4- For through-thickness cracks, the Walker’s crack growth law and XFEM model with cosine boundary condition predicts the fatigue life similar to analytical method. In contrast, XFEM model with constant pin pressure and actual pin predict the life in the similar manner. 5- For through-thickness cracks, the Walker’s crack growth law and XFEM model with constant or cosine pin pressure assumption in finite element model results in adequate fatigue life trend in engineering point of view. 6- It is important to mention that one may use the Walker’s crack growth law and XFEM model with actual pin or constant pin pressure to usually make conservative fatigue life for all lug cases studied in the present work.

8. References: [1] Paris PC, Erdogan FA. A critical analysis of crack propagation. J Basic Eng Trans SME, Series D 1963; 55: 528-534.

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[2] Walker EK. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7076-T6 aluminum. In: Effect of environment and complex load history on fatigue life. ASTM STR 462, Philadelphia: American Society for Testing and Materials; 1970. p. 1–4. [3] Elber W. The significance of fatigue crack closure. ASTM STR 486, 1971; 230-242. [4] Schijve J., Hoeymakers A.H.W. Fatigue crack growth in lugs, Fatigue of Engineering Materials and Structures 1979; 1:185-201. [5] Friedrich S., Schijve J. Fatigue crack growth of corner cracks in lug specimens. Report LR-375, Delft Un. of Tech., Fac. of Aerospace Eng., 1983. [6] Erdogan F, Roberts R. A comparative study of crack propagation in plates under extension and bending. In: Proc int conf on fracture, Sendai, Japan; 1965. [7] Kujawski D. A new driving force parameter for crack growth in aluminum alloy. Int. J. Fatigue 2001;23:733–40. [8] Glinka G, Robin C, Pluvinage G, Chehimi CA. Cumulative model of fatigue crack growth and the crack closure effect. Int. J. Fatigue 1984;6(1):37–47. [9] Wu SX, Mai YW, Cotterell B. A model of fatigue crack growth based on Dugdale model and damage accumulation. Int J Fract 1992;57:253–67. [10] Newman JR JC. Fracture analysis of surface- and through-cracked sheets and plates. Engng Fracture Mech 1973; 5: 667-689. [11] Raju IS, Newman Jr JC. Stress intensity factor for two symmetric corner cracks. ASTM STP 677 1979; 411-430. [12] Newman Jr JC. Predicting failure of specimens with either Surface cracks or corner at holes. NASA TN D-8244; 1976. [13] Newman JR JC, Raju IS. Stress intensity factor for cracks in three dimensional finite bodies subjected to tension and bending loading. NASA technical Memorandum 85793, 1984. [14] Kathiresan, K., Hsu, T.M., Advanced life analysis methods-crack growth, analysis methods for attachment lugs, AFWAL-TR-84-3080 Volume II, 1984. [15] ] Rigby R., Aliabadi M.H. Study on expression of SIF for tapered lug subjected to oblique pin-load, Engineering Failure Analysis 1997; 4: 133-146. [16] Pian T.H.H., Mar J.W., Orringer o., Stalk G. Numerical computation of stress intensity factors for aircraft structural details by finite element methods. AFFDL-TR-76-12, 1976. [17] Aberson J. A., Anderson J. M. Cracked Finite-Elements Proposed for NASTRAN. Third NASTRAN Users’ Colloquium, NASA TMX-2893, 1973: 531-550. [18] Gencoz O., Goranson U.G., Merrill R.R. Application of finite element analysis techniques for predicting crack propagation in lugs, Int J Fatigue 1980; 2: 121-129. 17 | P a g e

[19] Narayana K.B , Dayananda T.S., Dattaguru B., Ramamurthy T.S., Vijayakumar K. Cracks emanating from pin-loaded lugs. Engn Fracture Mech 1994; 47: 29-38. [20] Saouma V.E., Zatz I.J. An automated finite element procedure for fatigue crack propagation analyses. Engng Fracture Mech 1984; 20: 321-333. [21] Shin CS. Some aspects of corner fatigue crack growth from holes. Int J Fatigue 1991; 13: 233-240. [22] Boljanovic S., Maksimovic S. Fatigue crack growth modeling of attachment lugs. Int J Fatigue 2014; 58: 66-74. [23] Belytschko, T., Black, T. (1999): Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, Vol. 45, No. 5, 1999, pp. 601-620. [24] Moës N., Dolbow J., Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46: 132-150. [25] Sukumar N., Moës N., Moran B., Belytschko T. Extended finite element method for threedimensional crack modelling. International Journal for Numerical Methods in Engineering 2000; 48: 1549-1570. [26] Dassault Systèmes : Abaqus 6.13 Online Documentation, Dassault Systèmes, Providence, Rhode Island 2013. [27] Babuska I., Melenk J.M. The partition of unity method. International Journal of Numerical Methods in Engineering 1996; 40: 727–758. [28] Dowling N.E. Mechanical behavior of materials. Pearson Education, Inc., Upper Saddle River, NJ, 2007.

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Highlights: 1. Stress intensity factor estimation in attachment lugs 2. Extended finite element method to calculate SIF in attachment lugs 3. Fatigue life estimation and crack growth in straight lugs

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