Crack growth-based fatigue life prediction using an equivalent initial ...

ARTICLE IN PRESS International Journal of Fatigue xxx (2009) xxx–xxx

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Crack growth-based fatigue life prediction using an equivalent initial flaw model. Part I: Uniaxial loading Yibing Xiang, Zizi Lu, Yongming Liu * Clarkson University, Potsdam, NY 13699, USA

a r t i c l e

i n f o

Article history: Received 24 February 2009 Received in revised form 18 May 2009 Accepted 8 July 2009 Available online xxxx Keywords: Fatigue crack growth EIFS Life prediction Notch Uniaxial loading

a b s t r a c t A general methodology is proposed in this paper for fatigue life prediction using crack growth analysis. Part I of the paper focuses on the fatigue life prediction for smooth and notched specimens under uniaxial loading. Part II of the paper focuses on the fatigue life prediction under proportional and non-proportional multiaxial loading. The proposed methodology is based on a previously developed equivalent initial flaw size (EIFS) concept. The EIFS is determined by the Kitagawa–Takahashi diagram and does not require back-extrapolation calculation. Fatigue lives of smooth specimens can be predicted using crack growth analysis with the initial crack length equaling to the EIFS. An asymptotic interpolation method is used to estimate the stress intensity factor (SIF) solution for short and long cracks at notches and is used for fatigue life prediction of notched specimens. The well-known fatigue notch effect is discussed using the proposed EIFS methodology. Various experimental data of different metallic materials are used to validate the proposed methodology and reasonable agreement is observed between model predictions and experimental data. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue life prediction and reliability evaluation are critical for the design and maintenance planning of many structural components, but are still challenging problems despite extensive research during the past several decades. Two major types of methodologies are available for fatigue life prediction. One approach is based on the material fatigue-life curves (e.g., S–N curves or –N curves) and a damage accumulation rule. The other approach is based on the fracture mechanics and crack growth analysis, which is the focus of the current study. A problem in the fracture mechanicsbased life prediction is to determine the initial crack size for crack growth analysis. One practice is to use an empirically assumed crack length, such as 0.25–1 mm for metals [1–3]. An alternative way is to use the results from nondestructive inspection (NDI) [4]. However, the initial flaw size can be below the current detection capability of the NDI technique. If the NDI detection limit is chosen as the initial flaw size, it will result in a very conservative design [5]. Other approaches have used experimentally measured defect size and shape for crack growth analysis. Merati and Eastaugh [3] used experimental studies to investigate the initial discontinuity state (IDS) for 7000 aluminum alloys, which can be used for fatigue modeling purpose.

* Corresponding author. Tel.: +1 315 268 2341; fax: +1 315 268 7985. E-mail address: [email protected] (Y. Liu).

The equivalent initial flaw size (EIFS) concept was developed nearly 30 years ago in an attempt to provide an initial damage state for fracture mechanics-based life prediction. The EIFS accounts for the initial quality, both from manufacturing and bulk material properties of structural details. The calculation of EIFS is usually performed using a back-extrapolation method. A brief review of the EIFS method can be found in [6]. A new EIFS calculation methodology was proposed by Liu and Mahadevan [6]. As mentioned in [6], the threshold stress intensity factor is determined using short crack growth data under ultrasonic frequency testing, which is originally used by Mayer [7]. The validation of the proposed EIFS with long crack growth data under commonly used load-shedding technique has not been justified. One of the objectives of this paper is to modify the developed EIFS methodology [6] to be applicable for long crack growth data. In the proposed approach, detailed mechanism modeling of microstructurally small crack growth is ignored. It is known that homogeneous linear elastic fracture mechanics (LEFM) breaks down for microstructurally small cracks (MSC). Multiple thresholds typically exist in terms of crack tip interactions with microstructure barriers such as grain boundaries [8]. Similitude assumption in the LEFM-based fatigue analysis is not true for microstructurally small crack, i.e., the crack growth rate is not the same for a microstructurally small crack and a large crack under the same applied stress intensity factor. The term ‘‘equivalent” in the EIFS indicates that the EIFS is not direct related to the measured physical size of the crack. As discussed in Liu and Mahadevan

0142-1123/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2009.07.011

Please cite this article in press as: Xiang Y et al. Crack growth-based fatigue life prediction using an equivalent initial flaw model. Part I: Uniaxial loading. Int J Fatigue (2009), doi:10.1016/j.ijfatigue.2009.07.011

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Y. Xiang et al. / International Journal of Fatigue xxx (2009) xxx–xxx

[6], EIFS is to facilitate life prediction by using only long crack growth analysis and avoiding the difficulties of MSC regime modeling. The straightforward way for crack growth-based life prediction is to use the observed small crack size and a crack growth model including MSC regime. An alternative way is to use the EIFS and long crack growth only. If the areas below the inverse integration of the crack growth rate functions are identical, both approaches can give the same life prediction. The key issue is to properly choose the EIFS to match these two areas. The proposed method used fatigue limit data and Kitagawa diagram to calculate the EIFS and matches these two areas under infinite/very long life. The detailed MSC regime crack growth modeling is very complex and needs to include many different factors. In addition, the uncertainties associated with MSC growth are very huge. They are the main reasons that the EIFS approach is pursued for simplicity for the design and analysis. Detailed discussion and comparison between the proposed EIFS concept and the MSC modeling can be found [6]. Another difficulty in fatigue life prediction is the application to structural components with geometric effects (e.g., notches, holes, corners, etc.). Stress concentration significantly affects the fatigue life prediction. It is well documented that fatigue notch factor will be less than the theoretical stress concentration factor [9]. It is critical to include the stress concentration effect in a sound life prediction methodology, especially for the application to structural level fatigue analysis. Another objective in this paper is to propose a crack growth-based methodology to describe the fatigue notch effect and to apply the fatigue life prediction to notched specimens. This paper is organized as follows. A previously developed EIFS methodology is briefly reviewed and is modified to be applicable using long crack growth data. The developed EIFS methodology is combined with crack growth analysis for fatigue life prediction under constant amplitude uniaxial loading. Following this, an asymptotic interpolation method is introduced to calculate the stress intensity factor of notch cracks and is integrated with the EIFS methodology in order to predict the fatigue life of notched specimens. A detailed discussion of the fatigue notch factor is performed using the proposed crack growth-based methodology. Extensive experimental data of smooth and notched specimens are used to validate the proposed methodology under uniaxial loading.

2. EIFS methodology for fatigue life prediction 2.1. EIFS calculation using short crack measurement Liu and Mahadevan [6] proposed a new method to calculate the EIFS without back-extrapolation. Details can be found in the referred article. Only a brief review is given here. The basic idea is to match the experimentally observed infinite fatigue life and the life prediction from crack growth-based models. The concepts of fatigue limit and fatigue crack threshold stress intensity factor are used here. Fatigue limit is traditionally used within the safe-life design approach, which defines a loading criterion under which no failure occurs. Fatigue crack threshold is often used within the damage tolerance design approach, which defines a loading criterion under which the cracks will not grow significantly. A link between the traditional safe-life design approach and damage tolerance design approach has been proposed and is known as the Kitagawa–Takahashi (KT) diagram [10]. According to the KT diagram, the fatigue limit of the cracked specimen increases as the crack size decreases. The fatigue limit of the cracked specimen remains constant when the crack size is below a certain value, which is determined by the fatigue limit of the material (i.e., the value from smooth specimen testing) and the fatigue crack threshold intensity factor using linear elastic fracture mechanics (LEFM).

El Haddad et al. [11] proposed a model to express the fatigue limit Drf using the fatigue threshold stress intensity factor DK th and a fictional crack length a.

DK th ¼ Drf

pffiffiffiffiffiffi paY

ð1Þ

where Y is a geometry correction factor and depends on the crack configuration. For an infinite plate with a centered through crack of length 2a, Y is unity. For other crack configurations, Y can be obtained from stress intensity factor solutions available in the literature [12]. Eq. (1) is rewritten as

a¼

1

p

DK th Dr f Y

2 ð2Þ

Eq. (2) is the expression for the proposed EIFS. If the specimen has an initial crack length of EIFS and is under the stress rangeDrf ; the calculated fatigue life using a fracture mechanics-based approach is infinity, which is the experimental fatigue life under fatigue limit. In the proposed study, fatigue limit refers to the fatigue strength corresponding to 107–108 cycles [6]. Liu and Mahadevan [6] have showed that the proposed EIFS methodology works well for some experimental data, in which the threshold stress intensity factor is measured using short cracks under ultrasonic frequency testing. It is also mentioned that the proposed method needs proper modifications using the measurements under other experimental techniques. This is critical to make the proposed EIFS methodology applicable to general cases since majority of available crack growth database are obtained using long crack specimens under ASTM standard [13]. For example, the threshold stress intensity factor is usually obtained by the loading-shedding technique [13] and is different from the results for the short crack measurements. Details about the modification of the proposed EIFS methodology using long crack growth data is shown below. 2.2. EIFS calculation using long crack growth measurements In the proposed methodology, the fatigue threshold stress intensity factor and fatigue limit are two critical quantities in determining the EIFS and fatigue life. The accuracy of these two quantities has direct impact on the life prediction accuracy. It is known that different experimental procedures provide different threshold intensity factor measurements [14]. The ASTM E-647 uses the load-shedding method to determine the threshold stress intensity factor. An alternative procedure has been used by compression precracking followed by DK increasing. Schematic da/dN curves using these two experimental techniques are shown in Fig. 1 [14]. The load-shedding technique will produce a compressive residual stress ahead of the crack tip. As the load gradually decreases from a high DK value, the crack growth is retarded and stopped due to the combination of external applied load and internal residual stress. The obtained fatigue threshold stress intensity factor is only an extrinsic material property. The same material behavior can be also explained by the crack closure effect [15]. When the fatigue life is predicted for constant amplitude S–N testing, the load-shedding does not exist. Therefore, a life prediction directly based on the da/dN curve measured by load-shedding technique and the proposed EIFS methodology will result in a shorter life since the EIFS is larger due to the larger threshold stress intensity factor measurement (Eq. (2)). The compression precracking technique will have the opposite effect as it produces the tensile residual stress ahead of the crack tip. As the crack advance to be a long enough crack, the effect of different experimental techniques disappears as the crack tip is beyond the residual stress field produced by preloading. As a consequence, the life prediction


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Ideal FCG curve

da/dN

Compressive precracking followed by K increasing

Constant-R K decreasing

Kth

K

da/dN (m/cycle)

Fig. 1. Schematic representation of different experimental techniques for determining fatigue crack thresholds [14].

ASTM method Kmax method Fitting line

K (MPa*m1/2) Fig. 2. Linear trend of Kmax method in measuring fatigue crack threshold stress intensity factor for Ti-17.

based on experimental measurements using these two techniques will have a large error at high-cycle fatigue (affected by the near threshold crack growth) and a smaller error at the medium-cycle fatigue (affected by the Paris regime crack growth). A conclusion can be drawn that the proposed EIFS methodology requires an intrinsic fatigue threshold value (without residual stress effect from precracking). This requirement suggests that a more reliable experimental procedure for threshold measurement should be used. An interesting observation was reported recently in [16] for Ti-17. They compared the ASTM procedure and the constant Kmax method for the crack threshold stress intensity factor measurement. It is observed that the crack growth measured by the constant Kmax method continues the log-linear trend below the threshold stress intensity factor measured by the ASTM standard, which is stated to be closure free crack growth. A schematic plot is shown in Fig. 2. It is expected that the crack growth will eventually go to zero at a certain load level as it approaches the intrinsic threshold stress intensity factor. A theoretical explanation for the intrinsic threshold stress intensity factor is proposed in [17], which used the term ‘‘threshold corner” defined using the stress intensity factor at the crack growth rate of one Burger’s vector by extrapolating the crack growth curve from the Paris regime log-linearly. The threshold corner can be treated as an intrinsic threshold stress intensity factor and is related to the basic material properties, such as the interatomic spacing and Young’s modulus of materials. The threshold corner is usually less than the threshold measured by the ASTM standard. The above experimental and theoretical work suggests that an intrinsic threshold value can be defined using the long crack growth data from the Paris regime, which can be used in the proposed EIFS framework to predict fatigue life. An extension of the proposed EIFS framework to use the intrinsic threshold stress intensity factor is described below. The fatigue crack growth can be classified into two stages: stage I and stage II [18]. A schematic plot is shown in Fig. 3. The Paris regime crack growth is in the stage II and is caused by the dual slip system ahead of the crack tip. The crack growth direction is along the original crack plane. The stage I crack growth is caused by the

rapid crack growth

da/dN

near-threshold crack growth

Paris type crack growth Δ K th

ΔK

Dislocation Slip lines

Dislocation

β New crack front

Slip lines New crack front

Plane to measure the crack growth

Original crack gront Original crack gront

da

Stage I crack growth

da

Stage II crack growth

Fig. 3. Schematic illustration for the stage I and stage II crack growth.


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crack is plotted to illustrate that the initial crack growth direction of an idealized crack just passing the threshold. The stage I crack will propagate in a zig-zag pattern and alternate its directions among different slip systems after passing the threshold [18]. In this study, we are only focusing on the limit state of threshold and not on the actual stage I crack growth. As shown in a previous study [8], two different threshold have been discussed: the material-based threshold and the mechanically-based threshold, characterized by microstructural fracture mechanics (MFM) and linear elastic fracture mechanics (LEFM), respectively. The former one on short cracks leads to the fatigue limit of materials and components. In order to calculate the smallest crack growth projection on the original crack plane, the magnitude of the Burgers vector is required for various crystal structures and atom properties [19]. A list of typical metals used in engineering is shown in Table 1. In Table 1, all the smallest crack growths are calculated using the angle of 70.5°. As we can see in Table 1, the smallest crack growth estimation of different metals is around 1010 m/cycle, especially for aluminum and steels. A value of da/dN less than one Burgers vector arises due to intermittent growth along a crack front. Statistical aspect is neglected in the current framework. In view of this, a modification of the proposed EIFS methodology is proposed using long crack growth measurement. First, the long crack growth measurement in the Paris regime is required. Then, the crack growth curve is extrapolated to the crack growth rate of 1010 m/cycle. The corresponding stress intensity factor coefficient is named ‘‘intrinsic threshold stress intensity factor” and is used to calculate the EIFS in Eq. (2). It is observed that this modification works very well for the materials collected in the current investigation. To show the effect of the proposed modification, an example for Al-2024 is shown here. The crack growth rate curve [20,21] with measured threshold stress intensity factor is shown in Fig. 5a. The experimental S–N curve data [15] and prediction results using the proposed EIFS methodology is shown Fig. 5b. As we discussed earlier, this leads to a shorter life prediction, especially for the high-cycle fatigue life. The modified crack growth curve using the ‘‘intrinsic threshold stress intensity factor” is shown in Fig. 6a for the same material and the life prediction results is shown in Fig. 6b. It is shown that the proposed EIFS modification works well for this material. As shown in the model validation section, the proposed EIFS methodology gives a satisfactory result for other materials collected in this study.

Atom b Burgers vector

β orentation of maximum shear stress plane

b

β

da = b *cos( β )

1: 70.5 degree for a mathmatical sharp crack using LEFM 2: 45 degree for smooth specimen 2: 45 ~ 70.5 degree for blunt crack (notch)

Fig. 4. Smallest crack growth rate along the original crack plane.

Table 1 Crystal parameters of different materials. Metal

Crystal structure

Atomic radius (1010 m)

Burgers vector, b

da/dN at the threshold (1010 m)

Aluminum Iron (a) Nickel Copper

fcc bcc fcc fcc

1.43 1.44 1.25 1.28

2.86 2.88 2.49 2.56

0.96 0.96 0.83 0.85

single slip system and the crack growth is along the slip plane ahead of the crack tip. The smallest crack growth is defined that the crack cannot propagate one interatomic spacing along the slip plane, i.e., one Burgers vector. When the intrinsic threshold stress intensity factor is defined using the data from the Paris regime, crack growth is still assumed to be on the original crack plane. The smallest crack growth along the original crack plane can be expressed as da ¼ b cosðbÞ and b is the angle between the original crack plane and the slip plane. Because the slip plane is caused by the shear stress, the angle b is along the maximum shear stress plane. For a mathematical sharp crack, it is 70.5° and for a complete blunt crack/flat surface, it is 45°. For an arbitrary shaped crack and crack tip radius (treated as a notch), the angle is between these two extreme values. In this study, b takes the value of 70.5° because it provides a slight conservative prediction and the crack tip radius in the range of interatomic spacing can be considered as a sharp crack in the classical fracture mechanics. A schematic illustration of the above discussions is shown in Fig. 4. In this paper, the proposed EIFS method does not adequately address the stage I crack. The schematic plot of the stage I crack, as shown in Fig. 3, is highly idealized and only presents the projection angle when the extrapolation is performed from long crack growth regime for an assumed coplanar initial crack. Only a kinked

Experimental data Fitting

da/dN (m/cycle)

1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 1

10

100 1/2

Δ K(MPa*m )

A life prediction methodology using the EIFS concept has been developed [6]. The procedures are summarized below. Detailed derivation and explanation can be found in [6].

b

500

Stress range (MPa)

a 1.E-04

2.3. Life prediction methodology

400

Experimental data Prediction

300 200 100 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

Fatigue life (Log (N))

Fig. 5. (a) Experimental da/dN data and regression curve for Al 2024-T3 and (b) experimental S–N curve data and life prediction.


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Experimental Data Fitting

a 1.E-04

b Stress range (MPa)

da/dN (m/cycle)

1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 1

10

100

Δ K(MPa*m1/2)

Experimental data Model prediction

500 400 300 200 100 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

Fatigue life (Log (N))

Fig. 6. (a) Experimental da/dN data and regression curve for Al 2024-T3 and (b) experimental S–N curve data and life prediction.

The material fatigue crack growth curve can be expressed as

da=dN ¼ C½DK DK th

m

ð3Þ

where C and m are fitting parameters. DK th is the ‘‘intrinsic threshold stress intensity factor” discussed in the last section. It is noted that C also depends on the stress ratio and can be expressed as

C ¼ f ðRÞ

ð4Þ

where R is the stress ratio, f is a generic function and can be calibrated using experimental data under different stress ratios. It is noted that Eq. (3) cannot be applied to microstructural small crack growth since it will propagate in an irregular behavior. The objective of Eq. (3) is to include long crack growth data in the EIFS calculation and life prediction. Detailed discussion the small crack growth model and the proposed EIFS concept has been discussed in [6]. Eq. (3) can be rewritten as

dN ¼

1 da C½DK DK th m

ð5Þ

Integrating both sides, fatigue life N can be obtained as

N¼

Z 0

N

dN ¼

Z

ac

ai

1 da C½DK DK th m

ð6Þ

where ac is the critical length at failure and can be calculated using fracture toughness and applied stress levels. ac also depends on the specimen geometry and loading types. For high-cycle fatigue problem, the time spent at fast crack growth regime is negligible. Change of ac will not affect life prediction much. ai is the EIFS which is determined using Eq. (2). In the proposed study, a semi-circular surface crack is assumed. The geometry correction factor for a surface crack can be expressed as [3,4,12]:

2 14 3 a2 2 2 sin / þ cos / c 6 7 Y ¼4 5M f EðkÞ

ð7Þ

where a is the depth and 2c is the surface length of a semi-elliptical flaw. The term in square brackets represents the solution to the equivalent embedded elliptical flaw. The parameter / is the angle in the parametric equation of ellipse and E(k) is the complete integral of the second kind. The correction factor Mf for finite width W and finite thickness t is calculated as

a2 a4 gfw Mf ¼ M1 þ M2 þ M3 t t

ð8Þ

where

a c 0:89 M 2 ¼ 0:54 0:2 þ ac

M 1 ¼ 1:13 0:09

1 a24 M 3 ¼ 0:5 a þ 14 1 0:6 þ c c a2 ð1 sin /Þ2 g ¼ 1 þ 0:1 þ 0:35 t rffiffiffi! pc a fw ¼ sec 2W t The stress intensity factor solution at the surface crack tip (/ = 0) is used for EIFS calculation. It should be noted that surface crack growth is actually a 3D crack problem, where the crack front experiences different crack growth behavior. In the current study, only crack growth at the surface crack tip is considered and more complex situations need further study. In the proposed EIFS methodology, the assumption of crack configuration will not influence the results obtained as long as the EIFS calculation and the fatigue life prediction use the same assumption. Liu and Mahadevan [6] have demonstrated that the life prediction results are independent of the crack configuration assumption. The above discussion is for elastic analysis and is appropriate for high-cycle fatigue analysis, in which materials are usually remaining elastic during the entire fatigue life. For medium and low-cycle fatigue problems, materials will experience some plastic deformation. Elastic analysis is not sufficient for fatigue life prediction. To include the effect of plastic deformation, a correction factor considering the reversed plastic zone is considered. Wilkinson [22] derived cyclic reversed plastic zone size as

q ¼ a sec

prmax ð1 RÞ 1 4ry

ð9Þ

where q is the plastic zone size using the dislocation theory. It should be noted that this result is similar to Dugdale’s model using continuum mechanics with elastic–perfect plasticity model. A modification is proposed as

q ¼ a sec

prmax ð1 RÞ 1 4r0

ð10Þ

where r0 is the cyclic ultimate strength, which represents the fatigue strength when the failure cycle is one. In the current simulation, it takes the value of the material ultimate strength. Due to the different material cyclic hardening or softening behavior, this parameter needs to be calibrated using the material cyclic constitutive relationship. However, the experimental data for the collected


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materials are not reported and we choose the material ultimate strength instead. Stress intensity factor range can be expressed considering plastic correction as

DK ¼ rmax

pffiffiffiffiffiffiffi 0 pa0 Y

ð11Þ

where Y0 is the geometry correction factor using the equivalent crack length a0 considering plastic correction. a0 can be expressed as

a0 ¼ a þ q

ð12Þ

The units used in the proposed study are SI units, in which da/dN is m/cycle and DK th;i is MPa/m0.5. Once the crack growth curve and the stress intensity factor solution are determined, Eqs. (10)–(12) are used for life prediction followed the same procedure. One implicit assumption in the fatigue crack growth analysis is the so called ‘‘similitude”, which indicates that the crack growth rate only depends on the applied stress intensity factor range. However, this is only applicable to linear elastic fracture mechanics (LEFM). When the crack tip plastic zone is comparable to the crack size (e.g., short cracks), the LEFM cannot be used. It will produce different crack growth due to the plasticity effect. The proposed the plasticity correction factor is to consider the short crack and high loading case (e.g., low-to-medium-cycle fatigue). For large scale yielding case, the proposed method cannot be used and further theoretical and experimental work is required. 3. Life prediction for notched specimens 3.1. Asymptotic solution for stress intensity factor considering notch effect Stress intensity factor solution is required for fatigue crack growth and life prediction using the fracture mechanics-based approach. An asymptotic solution has been proposed by Liu and Mahadevan [23]. Detailed derivation and verification of the asymptotic interpolation can be found in [23]. The general formula for the SIF solution is expressed as

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n h a io K ¼ 1:122r p a þ d 1 exp K 2t 1 d

ð13Þ

where a is the crack length, d is notch depth, and Kt is the stress concentration factor. 1.122 is the surface correction factor. The SIF solution (Eq. (13)) under several extreme is discussed here. conditions For a short crack ða=d ! 0Þ, a=d K 2t 1 is approaching zero for a finite stress concentration factor. Using the first order Taylor series expansion of the exponential function

h a i a 2 exp K 2t 1 ¼ 1 Kt 1 d d

ð14Þ

Eq. (13) can be expressed as

K ¼ 1:122K t r

pffiffiffiffiffiffi pa

ð15Þ

For a long crack ða=d ! 1Þ, a=d K 2t 1 is approaching infinity when the stress concentration factor does not equal to unity. The h i exponential function exp a=d K 2t 1 is approaching zero. Eq. (13) can be expressed as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ¼ 1:122r pða þ dÞ

ð16Þ

Other solutions and formula for different notch configurations have been developed and can be used for estimate the stress intensity factors of cracks at the notch root [23]. Due to the space limit, it is not repeated here. Interested readers can found details in the referred articles [23].

3.2. Fatigue notch factor effect It is well known that the notched specimen has the fatigue notch effect, i.e., the strength reduction factor of notched specimen is usually between unity and the elastic stress concentration factor. Fatigue notch factor Kf is defined as [24]

Kf ¼

unnotched fatigue strength notched fatigue strength

ð17Þ

To include the fatigue notch effect, the smooth specimen is assumed to have a semi-circular micro-notch whose length is on the order of the EIFS. Detailed discussion on the fatigue notch factor using the asymptotic stress intensity factor solution and the EIFS concept is shown below. Considering the asymptotic solution of a notch crack (Eq. (13)), a smooth specimen achieves its fatigue limit ðrsmooth Þ when the applied SIF equals to the threshold stress intensity factor.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " !#)! u u a K 2t K th ¼ Y arsmooth tp a þ dm 1 exp 1 dm a2

ð18Þ

where a is the crack length equaling to the EIFS and dm is the depth of the micro-notch. The fatigue limit for a notched specimen ðrnotch Þ is achieved when

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " !#)! u u a K 2t K th ¼ Y arnotch tp a þ d 1 exp 1 d a2 where d is the realistic notch depth. Combing Eqs. (18) and (19), a relationship between rnotch can be expressed as:

rsmooth rnotch

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n h 2 ioffi K a þ d 1 exp da a2t 1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n h 2 io K a þ dm 1 exp dam a2t 1

ð19Þ

rsmooth and

ð20Þ

EIFS is usually several magnitudes smaller compared to the specimen dimensions (i.e., microns compared to millimeters and higher). Eq. (20) can be simplified as

rsmooth 1 ¼ K t rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n h io rnotch 2 dm a 1þ

a

ð21Þ

1 exp dm K t 1

Eq. (21) can be rewritten as

Kf 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ r n h io Kt 1 þ dam 1 exp dam K 2t 1 1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n h ioffi 2 1 1 þ Rns 1 exp Rns K t 1

ð22Þ

It can be seen that Kf/Kt is a value between 1 and 1/Kt. It is shown in Eq. (22) that the fatigue notch factor is related to the ratio of dm/a. Kf equals to unity when dm/a is approaching infinity and Kf equals to Kt when dm/a is approaching zero. For other cases, Kf is between the above two extremes. dm/a is different for different materials. In practical applications, one fatigue limit data of notched specimen is used calculate this ratio and other cases can be predicted. In the current investigation, the values of this ratio for the collected materials are shown in Table 2.


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Y. Xiang et al. / International Journal of Fatigue xxx (2009) xxx–xxx Table 2 dm/a Values of four different materials.

Table 3 Summary of experimental S–N curve data collection for smooth specimens.

Materials

dm/a

Material

Stress ratio

References

2024-T3 7075-T6 4340 Ti–6Al–4V

1.2 2.5 0 0.8

Al 2024-T3 Al 2024-T3 Al 7075-T6 Al 7075-T6 Steel 4340 Ti–6Al–4V

R = 1, R = 0 R = 0.1 R=0 R = 1 R = 1 R = 1, R = 0.1, R = 0.5

[15] [27] [15] [28] [29] [30]

4. Validation for smooth and notched specimens 4.1. Experimental data

Table 4 Summary of experimental S–N data for notched specimens.

Experimental data on fatigue crack growth testing are collected for several materials under different stress ratios. They are: (1) Al 2024-T3 material under three different stress ratios [25] (R = 1, R = 0, R = 0.1). Al 7075-T6 material under two different stress ratios [25] (R = 1, R = 0). Steel 4340 material under stress ratios [26] (R = 1). Ti–6Al–4V under three different stress ratios [26] (R = 1, R = 0.1, R = 0.5). The experimental da/dN–DK curves for different materials are shown in Fig. 7. The fatigue SN testing results for both smooth and notched specimens of these materials under the different stress ratios are also collected to validate the model predictions. Details of the collected data are listed in Tables 3 and 4. The basic mechanical properties and fatigue properties (fatigue limit and fatigue threshold intensity factors) are listed in Table 5. The fatigue limit is defined as the fatigue strength coefficient at 107–108 fatigue cycles [6]. The fatigue threshold stress intensity factor is estimated using the method described in Section 2.2. The ai used in this study is in the order of microns and is different for different materials. The typical measured initial defect size (e.g., pores or inclusions) is typically an order of magnitude larger than

Stress ratio

References

Notch type

Al 2024-T3 Al 7075-T6 Steel 4340 Ti–6Al–4V

R = 1, R = 0 R = 1, R = 0 R = 1 R = 1, R = 0.1, R = 0.5

[15] [15] [15,29] [30]

Circular hole Circular hole Single edge semi-circular notch Double notches

Table 5 Summary of experimental data of different materials.

ry

ru

(MPa)

(MPa)

2024-T3

360 360 360

7075-T6

520 501 1456 1003

Materials

4340 Ti–6Al–4V

dadn R = 0 dadn R = 0.1 dadn R = -1

1.E-05

C

R

m

DK th (MPa m0.5)

Drf (MPa)

490 490 490

4.9423E11 2.238E10 2.6029E10

1 0 0.1

2.6526

1.9999 1.146 1.0839

281.3 200 171

575 569 1548 1014

7.2965E10 1.6170E10 2.0470E11 7.3617E11 1.6580E10 2.2273E10

0 1 1 1 0.1 0.5

2.3398

0.5202 1.0034 2.6509 1.4687 1.0434 0.9215

227.2 402.5 800 804 502 360

1.E-05

R=0

1.E-06

1.E-07

1.E-08

1.E-09

1.E-07

1.E-08

1.E-09

1

10

100

Δ K (MPa*M

1/2

1

10 1/2 Δ K (MPa*M )

)

(a) Al-2024 1.E-05

1.E-05

dadn R = 0 dadn R = 0.5 dadn R = 0.7 dadn R = -1

1.E-06

100

(b) Al-7075 R = 0.7 R = 0.5 R = 0.1 R=0 R = -1 R = -3 R = -5

1.E-06

da/dN (m/cycle)

da/dN (m/cycle)

2.1671 2.3779

R = -1

1.E-06

da/dN (m/cycle)

da/dN (m/cycle)

Material

1.E-07

1.E-08

1.E-07

1.E-08

1.E-09

1.E-09 1

10 100 1/2 Δ K (MPa*M )

1000

1

(c) Steel 4340

10 Δ K (MPa*M

100 1/2

1000

)

(d)Ti-6Al-4V

Fig. 7. Experimental da/dN–DK curves for different materials.


ARTICLE IN PRESS Y. Xiang et al. / International Journal of Fatigue xxx (2009) xxx–xxx

that arrived at in this EIFS method. There are possible explanations for the large difference between observed initial defects and the proposed EIFS. The observed initial defect usually has irregular morphologies and cannot satisfy the mathematical sharp requirement for an ideal crack in fracture mechanics. If the produced plastic zone is matched between the observed initial defect and the ‘‘equivalent” ideal crack, the resulted equivalent crack size will be smaller than the observed defect.

a

R= 0 R = 0.1 R = -1 Model Prediction

900

Stress Range (MPa)

800 700 600 500 400 300

4.2. Life prediction for smooth specimens Fatigue life predictions using the proposed EIFS methodology are shown in Fig. 8. Solid lines are model predictions and solid marks are experimental data. Different R-ratio experimental data are included. Overall, the proposed method gives a very good agreement between model predictions and experimental observations. Some discrepancies are observed for medium-cycle to

700

b

500 400 300 200

200 100 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

R=0 R=-1 Model Prediction

600 Stress Range (MPa)

8

100 1.E+03

1.E+08

1.E+04

1.E+06

1.E+07

1.E+08

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

R=-1 2000 R=0.5 R=0.1 1800 Model Prediction 1600 1400 1200 1000 800 600 400 200 0 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

Fatigue Life(log(N))

Fatigue Life(Log N)

c 2000

Model Prediction

1800

R=-1

d Stress Range (MPa)

Stress Range(MPa)

1.E+05

Fatigue Life(Log N)

Fatigue Life(Log N)

1600 1400 1200 1000 800

Stress Range (MPa)

a

400

R=0

350

R = -1 Model Prediction

300 250 200 150 100 50 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

b

500

R=0

450

R=-1

Stress Range (MPa)

Fig. 8. Experimental data and life prediction for: (a) Al-2024; (b) Al-7075; (c) steel 4340; and (d) Ti–6Al–4V.

400

Model Prediction

350 300 250 200 150 100 1.E+04

1.E+09

1.E+05

R=0

800

R=0

Stress Range (MPa)

c 900

Model Prediction

700 600 500 400 300 200 1.E+03

1.E+04

1.E+05 1.E+06 Fatigue Life(Log N)

1.E+07

1.E+08

d

600

Stress Range (MPa)

Fatigue life (Log(N))

500

1.E+06 1.E+07 Fatigue Life(Log N)

1.E+08

R=-1 R=0.1 R=0.5 Model Prediction

400 300 200 100 1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

Fatigue Life(Log N)

Fig. 9. Experimental data and life prediction for: (a) Al-2024; (b) Al-7075; (c) steel 4340; and (d) Ti–6Al–4V.


ARTICLE IN PRESS Y. Xiang et al. / International Journal of Fatigue xxx (2009) xxx–xxx

low-cycle fatigue regime. It might be caused by the plastic correction factor and needs further study. 4.3. Life predictions for notched specimens Fatigue life predictions using the proposed EIFS methodology are shown in Fig. 9. Solid lines are model predictions and solid marks are experimental data. Again, the proposed method gives a very good agreement between model predictions and experimental observations. 5. Conclusions In this paper, a general methodology for life prediction of smooth and notched specimens using crack growth analysis is proposed. This methodology is based on a previously developed EIFS concept. Proper modification is proposed to use long crack growth data only for fatigue life prediction. An asymptotic SIF solution for notch cracks is combined with the proposed EIFS methodology for life prediction of notched specimens. The prediction results are compared with a wide range of experimental data available in the literature. Some conclusions can be drawn as below: 1. Overall good agreement between the model predictions and experimental data are observed for smooth specimens and notched specimens collected in this work. 2. Previously developed EIFS methodology cannot be directly used for life prediction using threshold stress intensity factor measured using load-shedding technique. The proposed modification only needs long crack growth data in the Paris regime. 3. The well known notch factor can be explained using the proposed methodology, which relates to the parameter dm/a. It has relationship with material properties and is different for different materials. 4. The proposed method is only validated using the collected materials under constant amplitude loading. Further model development and model validation are required. The current work focuses on the uniaxial constant amplitude loading. Life prediction under random variable amplitude loading is different than that under constant amplitude loading. For example, the overload can dramatically change the crack growth curve compared to that under the constant amplitude loading. Several methodology for crack growth under variable amplitude loading have been proposed in the literature, such as crack closure concept and reverse plastic zone concept. General variable loading case needs further investigation. If the uncertainties are considered, i.e., both fatigue limit and crack growth laws are random, the proposed EIFS becomes a random variable based on the El Haddad’s model. Other uncertainties from material properties, such as the crack growth curves, need to be included for probabilistic analysis. Future work is required to include various uncertainties in the life prediction for accurate fatigue reliability analysis. Acknowledgements The research reported in this paper was supported by funds from NASA Ames research center (Contract No. NNX09AY54A, Project Manager: Dr. Kai Goebel) and by funds from the Federal Aviation Administration William J. Hughes Technical Center (Contract No. DTFACT-06-C-00017, Project Manager: Dr. John Bakuckas). The support is gratefully acknowledged.

9

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