Experimental and numerical investigation of mixed mode fatigue crack growth models in aluminum 6061-T6

International Journal of Fatigue 130 (2020) 105285

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Experimental and numerical investigation of mixed mode fatigue crack growth models in aluminum 6061-T6

T

S. Sajith, K.S.R.K. Murthy , P.S. Robi ⁎

Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

ARTICLE INFO

ABSTRACT

Keywords: Mixed mode Fatigue crack growth Numerical modeling Stress intensity factor Aluminum alloys Fractography

Damage tolerance philosophy and safe life design principles are widely used for critical structural components, and in such cases, fatigue life prediction using the numerical techniques is essential. In the present work, mixed mode fatigue crack growth experiments are performed using the compact tension shear specimens made of Al 6061-T6 alloy for mode mixity angles of 30°, 45°, and 60°. These experimental studies are supported by numerical and fractographic studies on the selected specimens. A three-parameter double exponential model for fitting the fatigue crack growth curves (crack length vs. the number of fatigue life cycles) is also proposed for both the mode I and mixed mode (I/II) loading conditions. Numerical prediction of mixed mode (I/II) fatigue life is carried out using the Paris’ law in combination with various K eq models. The results of the present investigation show that highly satisfactory best fits to the scattered experimental data can be obtained with the help of the proposed double exponential model. The predicted life is compared with the experimental fatigue life using various measures of error. Based on the overall performance of the models during the entire crack growth regime, the results of the present investigation clearly show that Irwin’s model and one of the Tanaka’s model are consistently found to predict the mixed mode fatigue life close to the experimental data. Whereas, Richard’s and Yan’s models, again based on the overall performance, are found to be conservative models consistently for the prediction of the mixed mode fatigue life. These conclusions are verified with the published results.

1. Introduction The presence of cracks and their exposure to the cyclic loads is very common in structural/engineering components. As a consequence, fatigue fractures turn into the most dominant mode of failure of engineering components. Using fracture mechanics principles, prediction of fatigue life is an essential tool in studies of structural integrity and damage tolerance analyses. Due to the complex nature of external forces, the geometry of components and boundary conditions, cracks in the components often subjected to mixed mode loading. A great deal of effort has been made to correlate mixed mode (I/II) fatigue crack growth (FCG) rate with various parameters such as stress intensity factors (SIFs) and J-integral etc. Among them, the modified form of Paris’ law proposed by Tanaka [1] is widely in use for correlating experimental FCG rates and the numerical prediction of residual life of engineering components. This law relates the mixed mode FCG rate with the equivalent stress intensity factor ( K eq) , which is a function of KI and KII (Fig. 1(a)). Subsequently, a good number of attempts have been made to improve the Tanaka’s approach by introducing various K eq models (based on various criteria) to correlate with the

⁎

experimental FCG data. These K eq models in Paris’ law can be broadly classified into three groups viz., models that are derived based on specific fatigue crack propagation theories [1], static mixed mode fracture theories [2–4], and empirical studies [5–7]. Tanaka [1] derived two K eq models based on the plastic yielding ahead of the crack tip under fatigue loading and tested the models using specimens made of aluminum. Biner [8] proposed a K eq model based on the sum of the individual strain energy release rates [3] in mixed mode loading (I/II) and employed the model to study the FCG rates in AISI 304 steel. Hussian et al. [2] presented a K eq model based on the computation of the strain energy release rate in mixed mode (I/II) loading conditions. To the knowledge of authors, this model has not yet been tested using fatigue experiments. Based on the maximum tangential stress criteria (MTS), Yan et al. [4] proposed a model which is a function of KI and KII . Based on the experimental and numerical studies empirical models have also been devised [5–7]. A summary of many such models is available in [7,9–16]. Apart from Tanaka’s modified form of Paris’ law, other mixed mode fatigue crack growth laws can be found in [17–21]. Although a large number of K eq models are available, it is apparent

Corresponding author. E-mail address: [email protected] (K.S.R.K. Murthy).

https://doi.org/10.1016/j.ijfatigue.2019.105285 Received 19 May 2019; Received in revised form 7 September 2019; Accepted 17 September 2019 Available online 18 September 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. (a) K eq as a function of KI and KII and (b) Fracture limit curve under mixed mode (I/II) loading. Table 1 Tensile properties of Al 6061-T6. Yield strength (MPa)

Ultimate tensile strength (MPa)

% Elongation

Young’s modulus (GPa)

Poisson’s ratio

270

307

14.8

68

0.33

Fig. 2. Schematic representation of the orientation of the crack with the rolling direction.

from the literature that when a new model was developed, no verification of the existing models until that time was made. In structural integrity and damage tolerance assessments, the fatigue life under mixed mode (I/II) loading conditions is estimated using the numerical methods such as finite element method by employing the modified form of Paris’ law as

da = C ( K eq ) m dN

(1)

where C and m are material dependent constants obtained using standard mode I FCG experiments and K eq is the one of the above available equivalent stress intensity factor. Due to non-self-similar crack growth (Fig. 1(a)) in mixed mode loading, the accuracy of the predicted fatigue life depends on the accuracy of the SIF computation, the K eq model and the criterion employed for crack path prediction. As stated earlier, though many works are available providing a variety of K eq models for life prediction, the works on experimental and numerical verification of the effect of these models on the predicted life are scarce. It is clear that life predictions made without the knowledge about the capabilities of various K eq models proposed does not provide useful data. Therefore, it is obligatory that these models need to be examined (in a common experimental program) to understand whether they offer nearly accurate or conservative predictions compared to the

Fig. 3. (a) geometry of the CTS specimen, (b) geometry of Richard’s fixture and (c) Richard’s fixture with specimen in place.

experimental life data. Availability of such knowledge is of great importance in structural integrity and damage tolerance assessments. To the knowledge of authors, only a few works are carried out in this direction by Sajith et al. [15] using finite element simulations and Demir et al. [14] using experiments and numerical modeling. Nevertheless, 2

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where K eq = f ( KI ,

KI = KI ,max

KI ,min

KII ) and and

KII = KII ,max

KII ,min

(3)

Fig. 1(b) shows a typical fracture limit curve for mixed mode (I/II) loading where the threshold region and stable and unstable crack growth regions are shown. Fatigue crack growth occurs if the equivalent SIF exceeds the material-dependent threshold value KI , th [10,25],

K eq

(4)

KI , th

and unstable fracture occurs if

K eq max

(5)

KIC

where K eq max is maximum K eq and KIC is the plane strain fracture toughness of the material. For a finite, predefined crack increment a , the associated fatigue life can be estimated as a 0

da = C ( K eq )m

dN = N 0

(6)

If the increment ai (=ai + 1 ai ) is very small, Eq. (6) can be discretized and N in the above equation for an i th step can be estimated as

Fig. 4. Overall view of the experimental setup.

these limited studies are not sufficient to provide conclusive knowledge on the performance of various K eq models. Therefore, the purpose of the present investigation is to conduct mixed mode fatigue crack growth experiments using Al 6061-T6 supported by numerical simulations to provide a substantial understanding of the effect of different K eq models in prediction of the mixed mode fatigue life of engineering components. In the mixed mode (I/II) FCG experiments, the digital microscope is widely employed for capturing the images of the growing crack for the measurement of crack length, as the crack opening displacement (COD) gauge is not recommended for such experiments. Significant amount of scatter may arise in the measured data due to the dependence of measured crack length on the analysis of large number of captured images. Thus, another aim of the present work is to propose a three parameter double exponential curve-fitting model for the best fit of the experimental crack length (a) versus loading cycles (N ) data in mixed mode (I/II) loading conditions to minimize the scatter in the measured data. The proposed model is also best suited for the fitting of the mode I data. In the present work, mixed mode (I/II) fatigue crack growth experiments are performed using Richard’s apparatus and the corresponding compact tension specimen (CTS) [22]. The finite element simulation of mixed mode (I/II) fatigue crack growth is carried out, and the SIFs are estimated using the recently proposed finite element displacement-based technique [23]. Mode I FCG test is also conducted for obtaining material constants in Eq. (1). Then, the fatigue life is estimated using the different K eq models. The predicted life is compared with the experimental fatigue life along with the fractography of the growing crack at different stages of crack growth. The results of the present investigation concretely provide useful recommendations and capability of the selected K eq models for practical applications.

ai = Ni i m C ( K eq )

(7)

Eq. (7) is employed in the present investigation for the numerical estimation of the fatigue life for a predefined crack increment of a . i The K eq for an i th step is computed at a crack length of ai [26], and the mixed mode SIFs are calculated using a recently proposed crack flank displacement method [23]. As far as the Paris’ material constants C and m are concerned in Eq. (1), the current practice in numerical simulation of mixed mode fatigue crack propagation is to use the values from mode I experiments [4,15,25,27–29]. 2.2. Selected K eq models A brief discussion of various K eq models selected in the present investigation are described as follows: 2.2.1. Tanaka’s models Based on the Lardner’s [30] postulation (the fatigue crack growth is equal to the reverse component of displacement at the crack tip) and Weertman’s [31] dislocation based fatigue crack propagation theory Tanaka [1] proposed two forms of K eq models, respectively as

K eq = ( KI2 + 2 KII2 )1

2

(8)

K eq = ( KI4 + 8 KII4 )1

4

(9)

These forms of K eq have been widely employed for numerical fatigue crack growth studies [26,32,33]. 2.2.2. Irwin’s model Using the definition of Irwin’s [50] potential energy release rate G (=GI + GII ) in mixed mode (I/II) loading conditions and its relation to the SIFs KI and KII under plane stress conditions an equivalent SIF K eq can be derived as

2. Theoretical background This section briefly describes mixed mode Paris’ law, various K eq models, and the numerical estimation of the FCG rates and fatigue life.

K eq = 2.1. Paris law for mixed mode (I/II) loading

KI2 + KII2

(10)

Biner [8] employed this model for mixed mode (I/II) fatigue crack growth studies.

Tanaka [1], proposed a modified form of the Paris’ law [24] and was first to correlate the fatigue crack growth rate da dN as a function of the mixed mode SIFs (Eq (1)),

da = C ( K eq ) m dN

N

2.2.3. Yan’s model Based on the equation of hoop stress under mixed mode loading conditions [34], Yan et al. [4] proposed another expression for equivalent SIF as

(2) 3

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Fig. 5. (a) Digital microscope setup for capturing the crack growth and (b) crack tip image captured using the microscope.

Fig. 6. Fatigue crack growth curves for mode I loading (a) a vs N curve and (b) da dN vs K curve.

Fig. 7. Fractured CTS specimens for loading angles (a) 30°, (b) 45°, and (c) 60°.

K eq =

1 cos c [ KI (1 + cos c ) 2 2

3 KII sin c ]

[36] an equivalent SIF K eq model is proposed by Hussain et al. [2] using variable mapping function as

(11)

where c is crack propagation direction (Fig. 1). This form of K eq is widely employed by many researchers [4,27–29,35] in their experimental and numerical studies.

K eq =

4 (3 + cos2

c

)2

1 1+

c c

c

(1 + 3cos2

c)

(9

KI2 + 4 sin 2 5cos2

c)

c

KI KII +

KII2

(12) The above expression is suggested to use in the modified Paris’ law by Miranda et al. [12] but it is not employed in the experimental studies

2.2.4. Hussain’s model Based on Griffith’s maximum fracture energy release rate principle 4

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Fig. 8. Crack initiation angle of mixed mode specimens (CTS) for loading angles (a) 30° (b) 45° and (c) 60°.

Fig. 9. Experimental data and the best fit line using three parameter double exponential model for (a) (d) = 60° .

p

30° (specimen 1)

3.95 × 10

6

30° (specimen 2) 45° 60°

q

k 1.08 × 10

1

2.59 × 10

6

5.05 × 10

6

3.21 × 10

6

= 30° (specimen 2), (c)

= 45° and

which is comparable to the equivalent stress in the classical stress hypothesis as

Table 2 Best fit parameters of the proposed double exponential model. Loading angle

= 30° (specimen 1), (b)

r2

1.36 × 10

4

2.48 × 10

1

1.22 × 10

4

4.24 × 10

4

3.19 × 10

4

1.79 × 10

2

1.33 × 10

4

K eq =

0.9975

KI 1 + 2 2

KI2 + 4(

KII )2

(13)

where = 1.155 is a constant [16]. This model is also widely employed in experiments and numerical simulations of fatigue crack growth [10,39].

0.9965 0.9983 0.9978

2.2.6. Demir’s model Recently, Demir et al. [14] proposed another model for K eq using the nonlinear regression analysis of the numerical and experimental results of the CTS specimen. This model is proposed for use at higher load mixity levels. The model is verified using a T-type mixed mode specimen. Their model is given by

to date. However, the above K eq is employed in numerical simulations by Sajith et al. [15]. 2.2.5. Richard’s model Richard and co-workers [37,38] have also proposed a form of K eq 5

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Table 3 List of K eq models used in the present study. Model

Tanaka 1

Tanaka 2

Irwin

Yan

Hussain

Richard

Demir

Eq. No.

Eq. (8)

Eq. (9)

Eq. (10)

Eq. (11)

Eq. (12)

Eq. (13)

Eq. (14)

where positive c is defined as the angle measured in the anticlockwise direction with respect to the initial crack orientation direction. 2.4. Determination of mixed mode SIFs Finite element displacement-based SIF estimation technique proposed by Sajith et al. [23] is employed in the present work. It determines the crack opening displacement (COD) and crack sliding displacement (CSD) using a combination of singular and higher-order terms for the estimation of mixed mode (I/II) SIFs. This technique provides very accurate SIFs along with their signs, which is important in Eq. (16). The accuracy of the proposed technique is on the same order as that of the path independent integrals even when using the coarse meshes. Detailed implementation and other details of this technique can be found in [23]. 3. Experimental procedures 3.1. Materials, specimens’ geometry, and loading In the present investigation mode I and mixed mode (I/II) FCG experiments have been conducted on Al 6061-T6 specimens. In these experiments, the CTS specimens are loaded at angles 30°, 45° and 60° to the initial crack direction. BiSS-ITW make, Median 250 model, a 250 kN servohydraulic tensile testing machine is used in all experiments to load the specimens. Tensile test properties according to ASTM E8 of Al 6061T6 material are shown in Table 1. ASTM E647-15e1 standard guidelines are employed for conducting both the mode I and mixed mode experiments. All specimens of the present investigation (including tensile coupons) were cut from the plate in the rolling direction (L-T orientation), and the crack is in the transverse direction, as shown in Fig. 2. All specimens have been fabricated using wire EDM process. Mode I FCG tests have been conducted using compact tension (CT) specimen to obtain the Paris’ constants C and m in Eq. (1). The thickness of the CT specimen used in this study is 22 mm, width is 60 mm, and the other dimensions are as per ASTM E647-15e1 standard. On the other hand, the CTS specimen, as shown in Fig. 3(a) is employed for mixed mode FCG studies. The thickness of the CTS specimen is 15 mm and Fig. 3(a) shows all other dimensions of the specimen. Richard’s loading device [22], as shown in Fig. 3(b) is used for inducing the mixed mode loading condition in the CTS specimen. The loading device is made of alloy steel AISI 4340, as shown in Fig. 3(c). A fine notch of width 3 mm and the notch angle 30° is machined in CTS specimen using the wire EDM process (Fig. 3(a)). The initial length of the machined notch was 40 mm for the CTS specimen. A 5 mm fatigue pre-crack is introduced in the CTS. The specimen surfaces are mirrorpolished, and gridlines are drawn to facilitate the measurement of the growing crack length. Fig. 3(c) shows Richard’s fixture arrangement for mixed mode (I/II) fatigue testing along with the specimen in place. The overall view of the mixed mode (I/II) experimental setup is shown in Fig. 4. The mode I tests were performed at a maximum load Pmax = 14 kN , load ratio R = 0.1(R = Pmin Pmax ) , and loading frequency is set at 10 Hz from pre-cracking to the final rupture. The instantaneous crack lengths were recorded using the COD gauges attached to the CT specimen. The crack growth rates were calculated using the variable amplitude fatigue crack propagation (VAFCP) module of the universal testing machine. The VAFCP module is built in compliance with the ASTM E399-17 and ASTM E647-15e1 standards. The crack growth tests were continued

Fig. 10. Loading and boundary conditions used for the numerical simulation of the CTS specimens. is the angle between applied tensile load P and crack face normal (loading angle).

K eq = (1.0519 KI4

0.035 KII4 + 2.3056 KI2 KII2 )1

4

(14)

Apart from the above expressions, many others can be found in review papers [9,11] and articles [12,13]. 2.3. Fatigue crack path estimation Accurate prediction of the fatigue crack path is essential for the numerical prediction of fatigue life. It is worth noting that various criteria employed for numerical simulation of the crack path in mixed mode fatigue loading conditions are developed for elastostatic cases [9,11,40,41]. Among the established criterion, the maximum tangential stress (MTS) criterion proposed by Erdogan and Sih [34] is widely in use. According to this criterion, the crack propagates in a radial direction ( c ) from the crack tip in which the tangential stress ( ) becomes maximum ( ,max ) , and the unstable fracture takes place when the tangential stress reaches a critical value. By maximizing the tangential stress component with the polar coordinate of the crack tip = 0) , the crack extension direction local coordinate system ( ,max c can be found as

KI sin

c

+ KII (3 cos

c

(15)

1) = 0

The solution of the above equation can be expressed as c

= 2tan

1

1 KI 4 KII

c

= 2tan

1

1 KI 4 KII

+

1 4

( )

+8

for KII > 0

1 4

( )

+8

for KII < 0

KI 2 KII

KI 2 KII

(16) 6

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Fig. 11. Finite element meshes used for the numerical simulation of CTS specimen loaded at 30° (a) Initial mesh (b) View near the crack tip in initial mesh (c) Crack tip mesh at 10th step and (d) Crack tip mesh at 20th step.

cracking and FCG tests are carried out under load control at a constant load of Pmax = 16 kN , load ratio R = 0.1 and the frequency of 10 Hz. The images of the growing crack are captured using a digital microscope and the coordinates of the growing crack tip and length of the crack are measured using the captured images as explained in the next section. 3.2. Crack length measurement Due to the mode II loading, clip gauges are found not suitable while conducting the mixed mode fatigue experiments. The most common practice to measure the length of the growing crack (in a curvilinear fashion) in mixed mode loading (I/II) is to employ a digital microscope. Fig. 5(a) shows the microscope and the crack length capture setup. The digital microscope is mounted on the base of the traveling microscope to facilitate the hassle-free movement of the digital microscope along the propagating crack tip. Images of the growing crack tip were recorded at every 5-sec intervals using Dino-Lite Pro digital microscope. Corresponding to each image, the number of loading cycles were also recorded. Fig. 5(b) shows the typical image captured by the digital microscope. The crack length is measured using DinoCapture software from the images obtained during the crack propagation. The intersection of the crack tip with the gridlines is considered as a reference point during the crack length measurement. Thus, using the image analysis and recorded load cycles, the crack length versus fatigue life (a N ) curve is obtained.

Fig. 12. Results of the mesh convergence study.

until the final rupture of the specimen. Since there are no standards for the mixed mode (I/II) fatigue crack growth test, ASTM E647-15e1 guidelines have been used in the present experiments. Two CTS specimens for the loading angle 30° (specimen 1 and specimen 2) and one specimen for each of the loading angles 45° and 60° have been tested here. For all these four specimens, pre-

7

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Fig. 13. (a) comparison of crack initiation angle with MTS and SED and (b) experimental and predicted crack paths using MTS of the three specimens.

the irregularity of the material microstructure [44]. Because of all these effects, smoothening the experimental data by best fitting techniques is essential for practical use of the measured data. In case of mode I loading, various curve fitting techniques are in practice viz., polynomial curve fitting [45], orthogonal polynomial approach [46], multi-segment three-parameter equations [47], cubic spline, smoothing spline techniques [48] and single-term exponential technique [49]. However, unsatisfactory results have been observed while using the above best-fit techniques for the present experimental data. For example, the polynomial fitting [45,46] utilized a single equation for the entire range of crack growth and resulted in many inflection points. The spline techniques [48] as well as the threeparameter model proposed by Smith [47], uses piecewise functions and results into multiple equations for the whole range of crack growth which is undesirable in case of the present data. Mohanty et al. [49], suggested the use of exponential functions (frequently used in the biological “law of growth”) may be suitable as the FCG rate increases considerably due to the growth of the crack uncontrollably at the final stages of the crack propagation. They suggested the use of a two-parameter single exponential function. However, no satisfactory best-fit trend has been noticed in the present case even after the use of the Mohanty et al. [49] model. But, this exercise showed promising results to work with the “law of growth” models. Thus, by adding an additional exponential term a three-parameter double exponential model for fitting the whole range of a N data obtained from both the mode I and mixed mode (I/II) fatigue experiments has been proposed here as

Fig. 14. Comparison of experimental and simulated fatigue life using various K eq models for the = 30° .

3.3. Proposed curve fitting As the mode I FCG test is standardized, the majority of the commercial testing systems contain software to provide smoothened experimental crack growth curve, i.e., a versus N graph by best fitting the experimental data. On the other hand, no such standardized test procedure is available for mixed mode FCG studies. Moreover, due to the presence of mode II displacement component, clip gauge is not recommended for the measurement of the crack growth in mixed mode experiments. In view of these issues and due to the growth of the crack in a curvilinear manner, the optical microscope is widely employed in mixed mode FCG tests. Even though at the microscopic level, the FCG process is smooth, the experimental a versus N data usually exhibit scatter [14,42] in the values. The crack length measurements using the digital microscope is subjected to human errors [43]. The scatter in fatigue crack growth data can also occur due to the variability in the testing procedure and

ai = (a 0

k ) e pNi + ke qNi

(17)

where ai is the ith step crack length and Ni is the corresponding number of cycles, a0 is the initial length of the crack, k is a constant, p and q are exponents to be determined. Theoretically, the proposed model passes through the point (0, a0) . Unlike the polynomial models, this model utilizes only three parameters (k, p and q) and can easily be fitted to the experimental data. In addition, the proposed function is continuous and differentiable over the range a0 af where af is the final or the required crack length. In the present work, all a versus N data has been best fitted using Eq. (17).

8

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Fig. 15. The percentage relative error in estimated fatigue life using various K eq models for the loading angle Table 4 RMS Error and error in L2 norm in estimated fatigue life using various K eq models for

= 30° (a) specimen 1 and (b) specimen 2.

= 30° .

Specimen

Error

Tanaka 1

Tanaka 2

Irwin

Yan

Hussain

Richard

Demir

Specimen 1

RMS L2

1318 5.44

876 3.67

1063 4.41

1496 6.17

1125 5.04

1442 5.95

1807 7.46

Specimen 2

RMS L2

1528 6.51

1214 5.15

1337 5.69

1671 7.13

1543 6.52

1627 6.94

1777 7.53

Fig. 16. Comparison of experimental and simulated fatigue life using various K eq models for = 45° .

Fig. 17. Comparison of experimental and simulated fatigue life using various K eq models for the = 60° .

4. Experimental results and discussion

4.2. Mixed mode (I/II) FCG experiments

4.1. Mode I fatigue crack growth experiment

Fig. 7 shows photographs of the fractured CTS specimen for 30°, 45°, and 60° loadings angles along with the crack paths. As shown in the subsequent sections, these paths are compared with the paths estimated using MTS criterion. The crack initiation angles measured using the optical microscope for all the three specimens are shown in Fig. 8. It should be noted that although not shown here, the fractured surface and crack initiation angles of the specimen 2 (for the loading angle 30°)

The crack length versus the number of cycles (a N ) and FCG rate K ) obtained from the mode I fatigue test are shown in curve (da dN Fig. 6 (a) and Fig. 6 (b), respectively. The determined Paris’ constants of a = 14 mm to 36 mm Al 6061-T6 using data from are C = 4.3378 × 10 7 (mm/cycle)/(MPa m1 2) and m = 2.6183.

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is nearly similar to that exhibited by the specimen 1 in Figs. 7(a) and 8(a). Fig. 9 shows a N plots of the raw experimental data (of all the four specimens) obtained by analyzing crack growth images captured using the digital microscope (Section 3.2) and best-fit curves to the raw data using the proposed double exponential model in Eq. (17). Coefficients in Eq. (17) corresponding to the four specimens are shown in Table 2. The corresponding goodness of fit (r 2 ) is also given in Table 2. It can be noticed from Fig. 9 and Table 2 that the proposed model exhibits a highly satisfactory representation of the original raw data of mixed mode experiments. 5. Numerical modeling, results, and discussions 5.1. Details of numerical modeling In this section, the finite element simulation of FCG in the three CTS specimens corresponding to the three loading angles has been carried out. Then the mixed mode fatigue life is computed using seven selected K eq models and is listed in Table 3. Using Eq. (7) and the procedure explained in Section 2.1, fatigue life is computed corresponding to each of the selected K eq models. Simulations of all the three specimens have been carried out in ANSYS®. Eight noded quadratic quadrilateral elements are used and plane stress conditions are assumed in all the simulations. Quarter point elements (QPEs) have been deployed at all the tips of a growing crack for modeling the inverse square root singularly. Sixteen number of QPEs elements were employed around the crack tip, and the length of these elements is set at 0.08 mm in all the three simulations. Fig. 10 shows boundary conditions and loads applied for finite element analysis of a typical specimen. Bottom holes of the CTS specimen are constrained. The pin loads P1 to P3 developed by the loading device at the top holes (Fig. 10) are given by [39,50]

Fig. 18. The percentage relative error in estimated fatigue life using various K eq models for the = 45° .

P1 = P (0.5 cos P2 = P sin P3 = P (0.5 cos

K eq

5.2. Convergence of computed mixed mode SIFs

Error type

Tanaka 1

Tanaka 2

Irwin

Yan

Hussain

Richard

Demir

RMS L2

1633 7.58

925 5.03

1006 5.27

1953 8.98

5332 28.02

1854 8.54

1571 7.55

Table 6 RMS Error and error in L2 norm in estimated fatigue life using various models for the = 60° .

(18)

(c b) sin )

where is the loading angle (Fig. 10), c and b are defined in Fig. 10. The loads (P1 to P3) were calculated for different loading angles (30°, 45°, and 60°), and the two-dimensional finite element analysis was carried out for each loading angle. After each crack growth step complete re-meshing approach has been employed in the present work. Fig. 11 shows meshing features for loading angle 30° to represent typical features of all the three specimens. Fig. 11(a) shows the initial finite element mesh employed for FCG simulation CTS specimen loaded at 30°. Fig. 11(b) shows mesh near the crack tip in the initial mesh. Fig. 11(c) and (d) show mesh near the crack tip at the 10th and 20th step of the crack growth, respectively. The following sections describe the results of finite element analyses of all the three experimental specimens.

Fig. 19. The percentage relative error in estimated fatigue life using various K eq models for the = 60° . Table 5 RMS Error and error in L2 norm in estimated fatigue life using various models for the = 45° .

+ (c b) sin )

In order to demonstrate the convergence of the mixed mode SIFs, KI and KII calculated using the finite element based displacement technique [23], the CTS specimen as shown in Fig. 10 with = 45o is considered here. A solution of the mixed mode SIFs of this configuration is given by Richard [5].

K eq

Error type

Tanaka 1

Tanaka 2

Irwin

Yan

Hussain

Richard

Demir

RMS L2

5012 13.04

4180 10.87

2975 7.84

5570 14.50

5481 15.77

5363 13.96

1401 4.11

KI = KII =

P Wt P Wt

a (1 a

cos a W)

sin (1 a W )

0.26 + 2.65a (W 1 + 0.55a (W

a)

0.23 + 1.40a (W 1

0.67a (W

a)

0.08(a (W

a))2

a)

a) + 2.08(a (W

a))2

(19)

where P is the uniaxial force (Eq. (18)) applied to Richard’s loading device (Fig. 3(c)) a is the crack length, W is the specimen width and t is

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Fig. 20. Fractographs of the mode I CT specimen showing the fractured surfaces at different crack lengths (a) 12.6 mm, (b) 15.8 mm, (c) 24.5 mm, and (d) 34.8 mm. [arrows indicate the direction of crack growth, (1) secondary cracks (2) striations (3) transgranular facets].

the specimen thickness. A value of P = 16 kN , a = 45 mm , W = 90 mm and t = 15 mm are considered for finite element analysis. Eight finite element meshes (similar to Fig. 11(a)) with increasing mesh density of the selected specimen have been employed to compute the mixed mode SIFs using the displacement technique [23]. Fig. 12 shows the plot of the variation of present computed mixed mode SIFs and the reference values obtained from Eq. (19) as a function of the length of the QPEs at the crack tip. It can be noticed from Fig. 12 that as the meshes are refined the computed SIFs KI and KII are converging to the reference solutions. It can also be observed from the results in Fig. 12 that the QPE’s length of 0.08 mm is adequate for the accurate estimation of mixed mode SIFs. Therefore, as stated in Section 5.1, QPE length is set at 0.08 mm in all the simulations of the present investigation.

xi = xi

1

+ a cos(

r i)

and

yi = yi

1

+ a sin(

r i)

(20)

where r i is the angle between the crack growth direction and the x -axis for the ith step determined using Eq. (16). This procedure is continued until the crack reaches a length of 60 mm. Then the fatigue lives are calculated using Eq. (7) and the K eq models given in Table 3 at each step of the crack growth. All necessary codes for computation of the fatigue life are developed in MATLAB®. 5.4. Fatigue life prediction of CTS specimen for the loading angle of 30° Following the finite element simulation procedure described in the previous section, the predicted mixed mode fatigue life using various K eq models (Table 3) for CTS specimen loaded at an angle of 30° is shown in Fig. 14. As stated earlier, two CTS specimens (specimen 1 and specimen 2) have been experimentally tested with a loading angle of 30°. The actual crack path and predicted crack path using MTS criterion are shown in Fig. 13(b). It can be seen from Fig. 13(b) that the estimation of the fatigue crack path by the MTS criterion is accurate. In order to assess the capability of these models considered, predicted fatigue lives are compared with the experimental fatigue life for the loading angle 30° (obtained in Section 3.2) as shown in Fig. 14. This figure also shows experimental data of both the specimens (specimen 1 and specimen 2). Fig. 15 shows the plots of percentage relative error in predicted life (with respect to the experimental life) by each of the seven models. Table 4 presents gross error (considering the whole range of crack growth of 15 mm) computed in terms of RMS and L2 norm values.

5.3. Prediction of the crack path for all loading angles using MTS criterion The crack growth direction ( c ) at all stages of crack growth is calculated using the MTS criteria [34], as given in Eq. (16). Fig. 13(a) shows the comparison of the experimental crack kinking angles (the first step of the crack growth) with the both MTS and strain energy density (SED) criterion for all the three loading angles. It can be noticed from Fig. 13 that the MTS criterion accurately predicts the measured angles and the crack path for all the loading angles. In all the simulations of the present study, a crack increment of a = 0.5 mm is set in all crack growth increments. After i 1th step of crack growth, the crack tip is advanced to the next position (xi , yi ) such that

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Fig. 21. Fractographs of the mixed mode 30° CTS specimen showing the fractured surfaces at different crack lengths (a) 45.5 mm, (b) 52.8 mm, (c) 56.2 mm, and (d) 60.6 mm. [arrows indicate the direction of crack growth, (1) secondary cracks (2) striations (3) transgranular facets].

Results in Fig. 15 also show that some models have small errors during the initial stages and acquired large errors at later stages of crack propagation and vice-versa. Considering the overall performance of the models during the entire range of crack propagation, it is evident from Figs. 14 and 15 that, predictions using Irwin’s and Tanaka 2 K eq models are close to the experimental data as compared to the other models. Results presented in Table 4 also shows that Irwin and Tanaka 2 models have made more accurate predictions as compared with the other models as both the RMS and L2 norm error values are minimum for these models. Again considering the overall performance during the entire crack growth range, Figs. 14 and 15 and Table 4 also show that Richard’s, Yan’s and Demir’s models provide conservative estimates of the mixed mode fatigue life. It is worth noting here that similar results have also been observed in the published experimental data of different mixed mode (I/II) experiments [15]. It can be noticed that the same conclusions can be observed using the experimental data of specimen 1 and specimen 2.

in terms of RMS and L2 norm values corresponding to 45° and 60°, respectively. The actual crack paths and predicted crack paths using MTS criterion for the loading angles 45° and 60° are shown in Fig. 13(b). It can be seen from Fig. 13(b) that the estimation of the fatigue crack paths by the MTS criterion is accurate in all the loading angles. Referring to Figs. 16–19, as in the case of 30°, some models exhibited relatively small errors during the initial stages of crack propagation and acquired large errors at later stages. Nonetheless, considering the overall performance of the models during the entire range of crack propagation, an interesting observation can be noticed from the results presented in Figs. 16–19 that for the loading angles 45° and 60°, again the Irwin’s and Tanaka 2 models consistently predicted the mixed mode fatigue life more closely to the experimental data. Furthermore, similar to the case of the loading angle 30°, Yan’s and Richard’s models provided a conservative life estimate as compared to the other models. It is even more interesting to note here that similar results have also been observed in the published experimental data of different mixed mode (I/II) experiments [15]. Results in Tables 5 and 6 also clearly indicate that on overall, Irwin’s and Tanaka 2 model predictions are close to the experimental data while Yan’s and Richard’s predictions are conservative as compared to other loads. However, for 60° loading angle, Demir’s [14] model also provided very accurate predictions as compared to other loading angles. As claimed by the authors [14], this model is devised to provide accurate predictions at higher mode mixities.

5.5. Fatigue life prediction using CTS specimen for the loading angles of 45° and 60° A similar analysis was carried out for CTS specimens with the loading angles of 45° and 60°. Following the simulation procedure as is carried out in the previous section, the predicted fatigue life using all the seven K eq models for the loading angles 45° and 60° are shown in Figs. 16 and 17, respectively. Figs. 18 and 19 show the plot of percentage relative error in predicted life by each model for the loading angles 45° and 60°, respectively. Similarly, Tables 5 and 6 present error

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Fig. 22. Fractographs of the mixed mode 45° CTS specimen showing the fractured surfaces at different crack lengths (a) 45.4 mm, (b) 51.6 mm, (c) 55.2 mm, and (d) 57.7 mm.

6. Fatigue fractography

[53]. In mixed mode loading, as mode II SIF diminishes and mode I SIF increases continuously, the crack surfaces open up more, reducing the interference between surfaces, and thereby producing relatively smooth surfaces. A notable distinction between mode I (Fig. 20) and mixed mode (Figs. 21–23) loading lies in the roughness of the fracture surface. Due to the continual decrease of KII , the fracture morphology changes from dominantly fatigue fracture to a mixture of fatigue fracture and transgranular quasi-cleavage failure [13,53].

Fractography studies of both mode I and mixed mode specimens are carried out, and results are presented in Fig. 20 for mode I loading, Figs. 21–23 for mixed mode loading angles of 30°, 45°, and 60° respectively. These images have been obtained using a Zeiss Sigma field emission scanning electron microscope (FESEM). Fatigue crack propagates by the formation of the ductile striations owing to the ductility of the material [51]. The presence of ductile striations can be observed in fractographs of both the mode I (Fig. 20(c) and (d)) and mixed mode loading. The spacing between the striations can be attributed to the FCG rate at that instant. It can be observed from fractographs of both the mode I and mixed mode loading that the width of the striations is increasing as the crack grows further, indicating the acceleration of the crack growth. In general, striations align perpendicular to the macroscopic crack growth direction (Figs. 20(c), 21(c), 22(d) and 23(d)). However, due to the variations in local stresses and microstructure, a change in crack growth plane and striation alignment direction [52] can be observed in Fig. 20(d). The fractured surface of the mode I specimen (Fig. 20) shows the presence of transgranular crack growth [42]. The presence of secondary cracks is also evident from the fractography of both the modes I and mixed mode (I/II) samples (Figs. 20–23). At the initial stages of the mixed mode fatigue crack growth, the fractured surfaces of all the specimens (Fig. 21(a), 22(a) and 23(a)) have higher roughness due to the presence of mode II component (which causes severe rubbing of the crack surfaces). Subsequently, the surfaces are relatively smooth at the later stages of the crack growth (Fig. 21(c–d), 22(c–d), and 23(c–d))

7. Conclusions In the present investigation, mixed mode (I/II) fatigue crack growth studies in CTS specimen made of Al 6061-T6 alloy are carried out experimentally and numerically for various mode mixities. Using experimental and numerical results, an attempt has been made in this work to study the capability of various K eq models available for use in the mixed mode Paris’ law for numerical prediction of the fatigue life. A new best-fit technique for representation of the experimental a N data based on the double exponential function is also proposed and demonstrated. Results of the present investigation show that very satisfactory best-fits can be obtained for both the mode I and mixed mode (I/II) experimental data with the help of the proposed best-fit model. Based on the overall performance during the entire range of the fatigue crack propagation, the results of the present study clearly demonstrate that Irwin’s K eq model is consistently predicting the mixed mode fatigue life closer to the experimental data. Moreover, one of the two Tanaka’s model is also consistently found to be a promising model for the prediction of life close to the experimental data. Again based on the

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Fig. 23. Fractographs of the mixed mode 60° CTS specimen showing the fractured surfaces at different crack lengths (a) 45.5 mm, (b) 49.2 mm, (c) 53.5 mm, and (d) 57.8 mm.

overall performance, Richard’s and Yan’s model are found to be conservative as compared to the other models. Interestingly, the K eq model proposed by Demir et al. [14] is found highly accurate in prediction of life for 60° as specified by Demir et al. [14] and is conservative for other loading angles.

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