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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Probabilistic modeling of fatigue related microstructural parameters in aluminum alloys Min Liao * Structures and Materials Performance Laboratory, Institute for Aerospace Research (IAR), National Research Council Canada (NRC), 1200 Montreal Road, M-14, Ottawa, Canada K1A 0R6

a r t i c l e

i n f o

Article history: Received 2 January 2007 Received in revised form 30 April 2008 Accepted 22 September 2008 Available online 7 October 2008 Keywords: Aluminum alloys Fractography Probabilistics Fatigue crack nucleation Micromechanics

a b s t r a c t This paper presents the results of probabilistic modeling of the fatigue related microstructural parameters in unclad 2024-T351 aluminum sheets. The statistical distributions of the constituent particle size, which were obtained from metallographic measurements from polished surfaces, were determined by graphical goodness-of-ﬁt tests. The distributions of the crack-nucleating particle sizes were determined using the data measured from various fatigue fracture surfaces. Initially, an extreme value theory based model was investigated to correlate the overall particle distribution with its fatigue subsets. Furthermore, a new Monte Carlo simulation was developed to determine the fatigue subsets using the microstructural parameters such as particle size, grain size, and grain orientation distributions, in association with qualitative criteria on fatigue crack nucleation and growth mechanisms. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction The National Research Council Canada (NRC) and other organizations are developing the holistic structural integrity process (HOLSIP) to augment and enhance traditional safe-life and damage tolerance paradigms in both design and sustainment stages. In HOLSIP, the life of a component is divided into four distinct phases: nucleation, short crack, long crack, and ﬁnal instability. Both cyclic and environmental effects (ex. fatigue and corrosion) are taken into account to assess structural life and residual strength capability. The HOLSIP-based models have demonstrated some success in the life estimation of various components, especially in the case of fatigue and corrosion interaction [1–4]. One important task in HOLSIP is material characterization, which uses the ‘initial discontinuity state’ (IDS) to describe the as-produced or as-manufactured state of the material in order to establish the initial analysis condition [5]. Examples of IDS include constituent particles, pores, and machining marks and scratches. In previous projects at NRC, considerable metallurgical studies and coupon fatigue tests have been carried out to physically measure the IDS, especially the constituent particles, for aluminum alloys used for aircraft structures [6,7]. A three stage approach was used to measure the IDS/particles, i.e., (1) microstructural analysis, (2) fatigue testing (ASTM E466 Standard [8]), and (3) post-fracture analysis. The material IDS/particle data were measured on polished material with the aid of scanning electron microscopy (SEM), and characterized by the area, height, and width. The fatigue subsets of the material IDS/particle distribution were obtained from the particles that nucleated the dominant fatigue cracks on the fracture surfaces. As an example, Fig. 1 presents the particles from a microstructural analysis, and the particle that nucleated the dominant fatigue crack, which was obtained from the post-fracture analysis. In this ﬁgure as well as this paper, LS, ST, and LT

* Tel.: +1 613 990 9812; fax: +1 613 952 7136. E-mail address: [email protected] 0013-7944/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.09.005

M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

669

Fig. 1. Material particle and crack-nucleating particle (unclad 2024-T351 thin sheet) [6].

stands for Longitudinal-Short, Short-Transverse, and Longitudinal-Transverse, respectively. The detailed test procedures can be found in [6]. The objectives of this work are to carry out a statistical analysis on the IDS/particle data and the fatigue subsets of the IDS/ particle data for unclad 2024-T351 sheets; and to investigate the correlation between the fatigue subsets and the overall IDS/ particle distribution, as well as, other microstructural parameters such as grain size and grain orientation distributions. 2. Statistical distributions of material IDS/particle This statistical analysis used the data obtained from unclad 2024-T351 sheets with different thickness, i.e., 0.06300 , 0.1600 , and 0.500 [6,7]. Polished samples were taken from the ST, LT, and LS planes with each sample covering an area of about 2 mm2, which included about 3000 to 15,000 particles depending on the plane. 2.1. Material IDS/particle distributions For aluminum alloys, the Lognormal distribution has been used for particles [9–11], pores [11], as well as other microstructural features, such as grain size [12]. Following a previous study [9], a three-parameter (3P) Lognormal distribution was found to best ﬁt most of the IDS/particle data, among the 2P Weibull, 3P Weibull, Gumbel, Frechet, and 2P Lognormal. For example, Fig. 2 shows the different distributions ﬁtted to the particle area, height, and width of the unclad 2024-T351 sheet (0.06300 ), on Normal probability paper. In these ﬁgures, the probability was calculated using symmetrical rank (or Hazen’s ranking method), i.e., pi = (i-0.5)/n. Overall the 3P Lognormal provided the best ﬁt to the data, especially in both tails of the distribution. The 3P Lognormal distribution can be expressed as,

lnðx sÞ l FðxÞ ¼ U ;

r

x > 0; r > 0

Fig. 2. Goodness-of-ﬁt plot of different distributions on normal probability paper (unclad 2024-T351 0.06300 sheet, ST plane).

ð1Þ

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

where s is the threshold (location), l and r are the mean and standard deviation of the logarithmic particle size. The distribution parameters were estimated using maximum likelihood methods, and presented in Tables 1–3 for all investigated 2024-T351 sheets with different thickness. In general, the large particles, i.e., the right tail of the IDS/particle distribution are more critical to the fatigue life of the material [11,13,14]. Therefore, a weighted 3P Lognormal distribution was also examined to further improve the ﬁtting accuracy in the right tail of the IDS/particle distribution. The weighted distribution has the same formula as Eq. (1), but the parameters were determined by regression analysis using a weight function as follows,

wi ¼

1 1 pi

ð2Þ

where pi is again calculated using the symmetrical ranking method, i.e., pi = (i0.5)/n. Fig. 3 presents the goodness-of-ﬁt examples for the particle area data from the 0.06300 , 0.1600 , and 0.500 sheets. The Gumbel probability paper was used in these ﬁgures, which has higher resolution to show the probability difference on the right tail. The results show that the weighted 3P Lognormal distributions provided an even better ﬁt than the regular 3P Lognormal distributions. The distribution parameters of the weighted 3P Lognormal distributions are presented in Tables 4 to 6 for all investigated 2024-T351 sheets. 2.2. Comparison of material IDS/particle distributions Fig. 4 presents the non-parametric distributions of the IDS/particle area, width, and height on the ST plane for the 2024T351 sheets with different thickness. It is shown that the IDS/particle distributions vary with the thickness of the sheet. The Table 1 Distribution parameters for constituent particles of unclad 2024-T3 sheet (0.06300 ). 2024-T3, new, thin (0.06300 -CFSD) ST plane Area Three parameter lognormal

Threshold s Mean l Std. dev r Note

LS plane Width

Height

0.4813 0.4255 0.2107 0.1862 0.4352 0.2552 1.7264 0.8190 0.6936 Sample size: 11989, survey area: 2.44 mm2, lower limit: 0.5 lm2

Area

LT plane Width

Height

Area

Width

Height

0.4802 0.4231 0.1974 0.1880 0.4570 0.1992 1.7045 0.7873 0.7096 Sample size: 14194, survey area: 2.44 mm2, lower limit:0.5 lm2

0.4871 0.4116 0.2188 0.3255 0.4465 0.4869 1.8915 0.8284 0.8151 Sample size: 9676, survey area: 2.44 mm2, lower limit:0.5 lm2

LS plane

LT plane

Units: area in lm2, width and height in lm.

Table 2 Distribution parameters for constituent particles of unclad 2024-T3 sheet (0.1600 ). 2024-T3, new, thick (0.1600 -CFSD) ST plane Area Three parameter lognormal

Threshold s Mean l Std. dev. r Note

Width

Height

0.4926 0.4308 0.2143 0.3675 0.6258 0.3626 2.0889 0.9373 0.7962 Sample size: 6355, survey area: 2.44 mm2, lower limit: 0.5 lm2

Area

Width

Height

0.4943 0.4332 0.1764 0.3904 0.7051 0.3463 2.1525 0.9380 0.8120 Sample size: 7106, survey area: 2.44 mm2, lower limit:0.5 lm2

Area

Width

Height

0.4981 0.4301 0.4194 0.2117 0.4863 0.3137 2.3460 0.9739 1.0544 Sample size: 7704, survey area: 2.44 mm2, lower limit: 0.5 lm2

Units: area in lm2, width and height in lm.

Table 3 Distribution parameters for constituent particles of unclad 2024-T3 sheet (0.500 ). 2024-T3, new, thick (0.500 -DUST) ST plan Area* Three parameter lognormal

Threshold s Mean l Std. dev. r Note

LS plane Width

Height

0.5080 0.4379 0.2182 0.2667 0.2686 0.2437 1.9579 0.9506 0.8236 Sample size: 15256, survey area: 2.03 mm2, lower limit: 0.5 lm2

Units: area in lm2, width and height in lm. * Parameters determined by regression analysis.

Area*

LT plane Width

Height

0.5007 0.2195 0.2180 0.2430 0.1545 0.6571 2.2021 0.8557 0.9084 Sample size: 3220, survey area: 2.03 mm2, lower limit: 0.5 lm2

Area

Width

Height

0.5013 0.4226 0.4327 0.2556 0.2478 0.5114 2.0397 0.9895 1.0431 Sample size: 4924, survey area: 4.07 mm2, lower limit: 0.5 lm2

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680 99.999%

12

99.999%

12

99.966% 3P Lognormal (Weighed fit)

6

99.752%

4

98.185%

2

87.342%

0

36.788%

-2

0.062%

-4

1.942E-22%

0.1

1

10

100

1000

3P Lognormal 8

3P Lognormal (Weighed fit)

6

99.752%

4

98.185%

2

87.342%

0

36.788% 0.062%

-2 -4 0.1

2

1

10

100

99.995%

10 3P Lognormal

99.966%

1000

1.9E-22% 10000

Gumbel probability -ln(-lnP)

3P Lognormal

Area

99.995%

10

Gumbel probability -ln(-lnP)

Gumbel probability -ln(-(lnP)

99.995%

10 8

99.999%

12

Area

Area

8 6

99.752%

4

98.185%

2

87.342%

0

36.788% 0.062%

-2

1.9E-22%

-4 0.1

2

Area of particle (µm )

Area of particle (µm )

(a) area of particle on ST plane unclad 2024-T351 0.063” sheet

(b) area of particle on LT plane unclad 2024-T351 0.16” sheet

99.966%

3P Lognormal (Weighed fit)

1

10

100

1000

2

Area of particle (µm )

(c) area of particle on LS plane unclad 2024-T351 0.5” sheet

Fig. 3. Goodness-of-ﬁt plot of weighted and regular 3P Lognormal distributions on Gumbel probability paper.

Table 4 Weighted distribution parameters for particles of unclad 2024-T3 sheet (0.06300 ). 2024-T3, new, thin (0.06300 -CFSD) ST plane

Three parameter lognormal (weighted ﬁt)

Threshold s Mean l Std. dev. r

LS plane

LT plane

Area

Width

Height

Area

Width

Height

Area

Width

Height

0.4245 0.3196 1.5397

0.4478 0.3376 0.9360

0.2239 0.1767 0.7786

0.3781 0.3787 1.4667

0.4478 0.3883 0.8662

0.2239 0.1213 0.7866

0.3875 0.5217 1.6437

0.4149 0.4101 0.8827

0.2239 0.3972 0.9410

Units: area in lm2, width and height in lm.

Table 5 Weighted distribution parameters for particles of unclad 2024-T3 sheet (0.1600 ). 2024-T3, new, thin (0.1600 -CFSD) ST plane

Three parameter lognormal (weighted ﬁt)

Threshold s Mean l Std. dev. r

LS plane

LT plane

Area

Width

Height

Area

Width

Height

Area

Width

Height

0.0431 0.9150 1.4781

0.4478 0.5890 1.0036

0.2239 0.3591 0.8286

0.0044 0.9726 1.5037

0.0000 0.9782 0.8220

0.0089 0.4544 0.7624

0.3003 0.6096 1.8584

0.4478 0.4065 1.0286

0.4478 0.4629 1.0013

Units: area in lm2, width and height in lm.

Table 6 Weighted distribution parameters for particles of unclad 2024-T3 sheet (0.500 ). 2024-T3, new, thick (0.500 -DUST) ST plane

Three parameter lognormal (weighted ﬁt)

Threshold s Mean l Std. dev. r

LS plane

LT plane

Area

Width

Height

Area

Width

Height

Area

Width

Height

0.5087 0.1382 1.7187

0.4500 0.1293 1.1408

0.2256 0.3533 0.8299

0.5087 0.2224 1.8301

0.2256 0.2654 0.8850

0.2255 0.6904 0.9651

0.5088 0.2929 1.9658

0.4500 0.1941 1.0882

0.4511 0.7442 0.8902

Units: area in lm2, width and height in lm.

close-up plots on the right tails are also presented in this ﬁgure, which shows, for instance, the probabilities of having a particle over 200 lm2 on the ST plane are 0.02%, 0.2%, and 0.1% for 0.06300 , 0.1600 , and 0.500 thick sheets, respectively. 3. Statistical distributions of fatigue subsets of material IDS/particle The fatigue subsets of the material IDS/particle data were measured from the crack-nucleating particles on the fatigue fracture surface (ST plane), and characterized by area, width, and height of particle [6,7]. The fatigue specimens were hourglass-shaped smooth coupons, that were designed based on the ASTM E466 Standard [8].

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

1

1

0.9 0.8

0.998

0.6

0.063" CFSD (n:11989, total area: 2.44mm 2) 0.16" CFSD (n:6355, total area: 2.44mm 2)

0.5 0.4 0.3 0.2

0.5" DUST (n:15256, total area: 2.03mm 2)

0.1 0 0.1

Probability

Probability

0.7 0.996 0.063" CFSD (n:11989, total area: 2.44mm 2)

0.994 0.16" CFSD (n:6355, total area: 2.44mm 2)

0.992

0.5" DUST (n:15256, total area: 2.03mm 2)

0.99 1

10

100

1000

Area of particle (um2)

0

200

400

600

Area of particle (um2)

Fig. 4. Non-parametric distribution comparisons for area of particle (ST plane, 0.06300 , 0.1600 , 0.500 unclad 2024-T351 sheets).

Fig. 5 presents the non-parametric distributions of the fatigue subsets for the area, width, and height of particle. These data were obtained from different fatigue tests under different conditions, such as thickness, stress level, and relative humidity. For the 0.500 sheet, multiple crack-nucleating particles were identiﬁed and measured on the fracture surface, especially in the case of the higher stress level of 48 ksi. Only the dominant particles, which resulted in the critical cracks that caused ﬁnal failure, are presented in Fig. 5. This ﬁgure indicates that most crack-nucleating particles are larger than what were measured on the polished surfaces (Figs. 2 and 3). In addition, the primary crack-nucleating particles appear to be larger in the thicker sheets, however, no obvious tendency can be found between the particle size and stress level. Since the fatigue subsets (crack-nucleating particles) vary with sheet thickness and stress level (maximum stress and stress ratio), it can be anticipated that signiﬁcant numbers of fatigue tests would be needed to generate the fatigue subsets under different thickness and stress levels. In contrast, the material IDS/particle distribution should be fairly consistent with the as-produced material for a speciﬁc thickness. A study was carried out to investigate the relationship between the material IDS/particle distribution and its fatigue subsets. 4. Correlation between material IDS/particle distribution and its fatigue subsets 4.1. Extreme value theory based model A probabilistic model was proposed in [11,13] to analytically determine fatigue subsets from a material IDS distribution for high Kt (notched), 7050-T7451 specimens. This model was modiﬁed in [9] to determine the fatigue subsets for low Kt (smooth), 2024-T351 specimens. The major difference between the high and low Kt cases is the volumetric effect on the stress distribution and stress gradient, which are more pronounced in the high Kt specimen. In brief, the model was based on the extreme value theory (or statistics of extreme) [15,16]. In the case where the material IDS/particle distribution follows a Lognormal distribution (Eq. (1)), the fatigue subset, i.e., the extreme value of the largest particles, is asymptotic to a Frechet distribution when the number of sampling particles is large, expressed as [9,15–17],

h a bi F S ðxÞ ¼ exp ð Þ ; xs

x P 0; a; b > 0

ð3Þ

where,

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S Þþlnð4 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pÞ þ l a ¼ exp r 2 lnðNS Þ r lnðlnðN 2 2 lnðN S Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 lnðN S Þ b¼ r

ð4Þ

and Ns is the number of particles in the fatigue ‘hot spot’, given by,

Ns ¼ Dp AHS

ð5Þ

In Eq. (5), Dp is the overall particle density which can be obtained from a basic metallographic examination on a polished surface, and AHS is the area of the highest stress region which is dependent on geometry, stress level and gradient and can be determined by a ﬁnite element analysis (FEA). For the low Kt (1.07) specimens that were used in this work, the areas of 95% and 99% of the highest stress region on the surface was determined from a 3D FEA, as shown in Fig. 6. It was empirically considered that the fatigue ‘hot spot’ can be covered by the 95% highest stress region [11,14]. Some of the modeling inputs and results are presented in Table 7. The material IDS distributions used in the model were the weighted 3P Lognormal distributions (Tables 4 to 6). A typical comparison of model and test results is presented in Fig. 7 for the unclad 2024-T351 0.06300 sheet. Overall, the modeling results did not change very much when the 95% or 99% highest

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

1 0.9 0.8

Probability

0.7 0.6 0.5 0.4 t0.063, LH, 44ksi, R0.1

0.3

t0.16, LH, 40ksi, R0.1

0.2

t0.5, 44ksi, R0.05 t0.5, 40ksi,R0.05

0.1

t0.5, 48ksi, R0.05

0 0

200

400

600

800

1000

Area of particle (µm2)

(a) area of particle 1 0.9

Probability

0.8 0.7 0.6 0.5 t0.063, LH, 44ksi, R0.1

0.4

t0.16, LH, 40ksi, R0.1

0.3

t0.5, 44ksi, R0.05

0.2

t0.5, 40ksi,R0.05

0.1

t0.5, 48ksi, R0.05

0 0

10

20

30

40

50

Width of particle (µm)

(b) width of particle 1 0.9

Probability

0.8 0.7 0.6 0.5 t0.063, LH, 44ksi, R0.1

0.4

t0.16, LH, 40ksi, R0.1

0.3

t0.5, 44ksi, R0.05

0.2

t0.5, 40ksi,R0.05

0.1

t0.5, 48ksi, R0.05

0 0

10

20

30

40

50

60

70

Height of particle (µm)

(c) height of particle Fig. 5. Non-parametric distributions of fatigue subsets, crack-nucleating particles on ST plane.

stress region was used. The model overestimated the fatigue subset distributions for all 2024-T351 sheets, even when the 99% highest stress region was used. In fact, a similar overestimation happened on the high Kt specimens fabricated from 7050 plates [11,13]. The detailed reasons for this overestimation are documented in [9]. The major reason was deemed that not all the largest particles in the highest stress region would nucleate cracks. Only those among the largest that are associated with other favourable microstructural features such as grain size, grain orientation, and grain boundary, would nucleate and propagate cracks to ﬁnal failure. Therefore, a critical density for crack-nucleating particles, DCP, was back calculated, which would result in the best ‘prediction’ to the measured fatigue subsets from the fracture surfaces, as shown in Fig. 7. The back calculation was based on the Kolmogorov–Smirnov (K–S) goodness-of-ﬁt criterion, i.e. to minimize the K-S statistic between the model and test distributions. Table 7 presents all the DCP results, which indicate that the critical density, DCP, is typically less than 0.5% of the overall particle density.

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

Fig. 6. One-quarter 3D FE model and results on low Kt smooth hourglass shape specimen. Table 7 Inputs and results for model for particle (width, height) of 2024-T3 sheets. Source

0.06300 sheet

Test

44 Ksi, R0.1

40 ksi, R0.1

40 ksi, R0.05

44 ksi, R0.05

48 ksi, R0.05

3D FEA

42.12

106.96

334.25

334.25

334.25

Metallurgical examination Ns = Dp*AHS

5818

2913

1583

1583

1583

245,054

311,574

529,118

529,118

529,118

Back calculation

20.1

23.7

1.3

3.9

3.8

3.5

1.9

3.5

4.8

8.1

0.35%

0.41%

0.04%

0.13%

0.24%

0.22%

0.12%

0.22%

0.30%

0.51%

Width Fatigue loading (peak stress, stress ratio) Highest stress region (95% rMAX), AHS (mm^2) Density for all particles, DP (LS plane) Number of particles in highest stress region Critical density for crack-nucleating particles, DCP Ratio of DCP / DP

Height

0.1600 sheet Width

Height

0.500 sheet Width

Height

0.500 sheet Width

Height

0.500 sheet Width

Height

4.2. Monte Carlo simulation based on material science and micromechanics Existing studies have shown that fatigue crack nucleation is a process that is fundamentally affected by the material microstructural features such as porosity, particle, and grain structure (size, orientation, and boundary) [5,14,18–21]. It is a combination of these microstructural features that dominate the crack nucleation process, as well as, the microstructural short crack growth [22,23]. The statistical distribution for each feature can be determined from the independent measurements of each feature. The joint statistical distribution of the multiple features (particle size, grain size, and grain orientation) needs the data from the dependent/concurrent measurements which contain the statistical correlation information among the multiple features. On a fatigue fracture surface, it is difﬁcult to carry out such measurements to determine the joint distribution affecting the fatigue process. On the other hand, current LEFM (linear elastic fracture mechanics) models, which are used in crack growth analysis tools like AFGROW and FASTRAN, can not use a joint (multi-dimensional) distribution of particle size, grain size, and grain orientation. Only a one-dimensional distribution, such as crack-nucleating particle size, can be used in these models/tools by assuming that the particle size is equal to a crack size. Statistically, this one-dimensional distribution (fatigue subset) can be described using the marginal or conditional distribution of the joint distribution of multiple features affecting the fatigue process. A new simulation method is proposed to determine the fatigue subsets of IDS/particle associated with grain size, orientation distributions, as well as stress levels. This simulation starts with distributions for the basic material parameters and uses the Monte Carlo technique to generate random, simpliﬁed microstructural samples for the fatigue critical area/volume. It then applies, qualitatively, crack nucleation criteria to determine the joint fatigue subset (joint distribution) of multiple microstructural features, and ﬁnally outputs the conditional fatigue subset (one-dimensional) of the crack-nucleating feature for life prediction purposes. A schematic ﬂow chart is given in Fig. 8 to describe the simulation procedures. Using existing data, simulations were carried out to predict the fatigue subsets of the crack-nucleating particles for the smooth (low Kt, hourglass) specimens. Previous fatigue tests have shown that the primary cracks were nucleated from the particles on the LS plane [6,7]. Other input parameters are summarized in Table 8. As mentioned above, due to the difﬁculty of concurrently measuring particle size, grain size, and grain orientation using the existing experimental techniques (these parameters were measured from separated tests, using different samples but same material), the real correlations among them were not known. In this preliminarily simulation, these parameters were assumed to be independent on each other. It should be noted that the correlation might affect the simulation results.

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1.0 0.9 0.8

Measured width (0.063) Cal. width (95% Smax area) Cal. width (99% Smax area) Back cal. width (95% Smax area)

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10

100

1000

Width of particle (μm) 1.0 0.9

Probability

0.8 0.7

Measured height (0.063) Cal. height (95% Smax area) Cal. height (99% Smax area) Back cal. height (95% Smax area)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 10

100

1000

Height of particle (μm) Fig. 7. Comparison of model and test results for crack-nucleating particle (width, height) for unclad 2024-T351 thin sheet (0.06300 ).

IDS/particle distribution

Grain size distribution Grain orientation, boundary …

Material IDS distributions

Conditional fatigue subset, particle size

Stress level (σ max, R)

Fatigue criteria (microstructure depended)

Critical fatigue area/volume

Joint distribution (fatigue subsets)

Conditional fatigue subset, grain size Conditional fatigue subset, grain orientation …

Fatigue subsets

Fig. 8. Correlating the material IDS distributions to its fatigue subsets.

The random samples of particle size, grain size, and grain orientation factor were independently generated using the Monte Carlo technique. The number of particles or grain samples was determined by dividing the fatigue critical area by the particle or grain density. The whole process was repeated N times in order to simulate N specimens/components, from which a fatigue subset with sample size N was generated. Fig. 9 presents a schematic particle and grain structure on the fatigue critical area. The fatigue criteria, which qualitatively account for the microstructural inﬂuences, were assumed as follows, Criterion 1: Cracks are nucleated at the largest particle on/near the free surface, i.e. Max (particle size). It is well known that most particles in aluminium sheets are cracked or debonded due to the rolling process [26,27], and thus large particles form large crack-like discontinuities. In theory, this criterion should result in the same fatigue subsets as the extreme value theory based model. Criterion 2: Cracks are nucleated at a large (not necessarily the largest) surface particle within/near a large (not necessarily the largest) surface grain, plus the particle size is less than the grain size, i.e., Max (particle size grain size)|(particle

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M. Liao / Engineering Fracture Mechanics 76 (2009) 668–680

Table 8 Input parameters for fatigue subset simulation. Material parameters

Characterization

Sources

Particle width distribution on LS plane Overall particle density on LS plane, Dp Grain size distribution along S direction

3P Lognormal distribution 5818(0.06300 ), 2913 (0.1600 ), 1583(0.500 ) 2P Lognormal distribution, mean: 14.22 lm (0.06300 ), 33.16 lm (0.1600 ), and 47.22 lm (0.500 ), coefﬁcient of variation 40% Unit area divided by grain area (L S), 1365 (0.06300 ), 129 (0.1600 ), 164 (0.500 )

Tables 1–3 Table 7 AGARD [24]

Grain density on LS plane Surface grain orientation distribution, reciprocal Schmidt factors, M

Exponential distribution, mean 2.21, coefﬁcient of variation 9.5%

Calculated from grain size distribution Assumption based on [25] analysis

External parameters

Characterization

Sources

Fatigue critical area (with 95% rmax on LS planes) Stress levels

42.12 mm2 (0.06300 ), 106.96 mm2 (0.1600 ), 334.25 mm2 (0.500 )

Table 7, determined from FEA [9] Table 7

Max. stress 40–48 ksi, R0.05 – 0.1

Loading Rolling ST

L LT

T S

LS

Number of grain = Area / grain density Number of particle = Area / particle density

Fig. 9. Schematic particle and grain structure on fatigue critical area (LS plane).

size < grain size). Therefore, larger grains would result in shorter nucleation life because more energy for dislocation development can be accumulated in order to overcome the speciﬁc fracture energy to form a new surface (crack) and to break through the blocking grain boundary [20]. Criterion 3: Cracks are nucleated at a large surface particle within/near a large surface grain with the favourable grain orientation to crack nucleation, plus the particle size is less than the grain size, i.e., Max (particle size grain size 1/ M (Schmidt factor))|(particle size < grain size). The grain orientation is quantiﬁed using the Schmidt factor for surface grain and the Taylor factor for the interior grain. In this criterion, only the Schmidt factor for the primary slipping plane is taken into account. The distribution of the reciprocal Schmidt factor M is assumed as an exponential distribution based on [25], Fig. 10. The assumed distribution has a minimum value of 2.0, mean value of 2.21, and coefﬁcient of variation of 9.5%, which are very close to the curve ﬁt in [25]. The minimum value of 2.0 means that both the normal of the slipping plane and the slipping direction are 45° to the loading axis, i.e., M = 1/[cos(45°) cos(45°)] = 2.0, which is the most favourable grain orientation for dislocation formation and thus crack nucleation [28]. Criterion 4: Criterion 3, plus non-arrest crack condition as a function of crack size, grain size, grain orientation, and stress level. This criterion is used to screen out the arrested cracks even if they meet Criterion 3 and nucleate, propagate within one or two grains. Also this criterion is used to consider the effect of stress level (max. stress and stress ratio). As a trial, a formula to calculate the stress to arrest a microcrack, which was originally developed in [29] and modiﬁed in [30] for different stress ratios, was used as follows,

Drarrest ¼

mi DrFLðR0Þ r1 pﬃﬃﬃﬃﬃﬃ ð1 RÞa þ r1 m1 2a=D

mi ¼ ½1 þ Alnð2a=DÞ m1 where mi – Taylor factor for ith grain, DrFLðR0Þ – fatigue limit at stress ratio of zero, DrFLðR0Þ ¼ 29 ksi for 2024-T351,

ð6Þ

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4

function

probability density

5

Exponential

3

(0.21, min 2.00) 2 1 0 0

2

4

6

8

10

Reciprocal Schmidt factor (M) Fig. 10. Exponential distribution of reciprocal Schmidt factor M.

r1 – crack closure stress, and r1 = 0 for short crack growth within 1–2 grains, a – crack size, which is assumed as the size of cracked particles on LS plane, D – grain size, which is a lognormal distribution along S direction, a – coefﬁcient of stress ratio R, which is 0.5 for aluminium alloys [29]. Using Eq. (6), a correlation between the maximum arrest stress and crack/grain size ratio was determined and is presented in Fig. 11. This ﬁgure indicates that a crack could nucleate and propagate from a smaller particle under a higher stress given that the ﬁrst grain size is the same. The Monte Carlo simulation was carried out on 2024-T351 0.06300 , 0.1600 , and 0.500 sheets under different stress levels. The simulated fatigue subsets of the material IDS/particle size distribution are presented in Figs. 12–14, along with the measured crack-nucleating particles. For each criterion, 100 specimens were ‘virtually tested’ (simulated) and a large number of particles were sampled for each specimen. When more specimens (>100) were used, the simulated distributions were smoother, however their shape and location were not changed signiﬁcantly. The results from these ﬁgures indicate that, The new Monte Carlo simulation based on the Criterion 1 produced almost the same distributions (Fretch) as those determined by the extreme value theory based model in section 4.1. This is a mutual veriﬁcation for both methods. However, both methods overestimated the fatigue subset of the material IDS/particle width distributions. For the 0.06300 sheet, Criterion 2 produced a very good prediction for the fatigue subset of the material IDS/particle width distributions while for the 0.1600 and 0.500 sheets, Criteria 3 and 4 produced better estimations than Criterion 2. In all cases Criterion 3 gave distinct results from Criterion 2, which means that the grain orientation factor used in Criterion 3 had a signiﬁcant effect on the predictions. In all cases, Criterion 3 and 4 gave very similar results to each other, which means that the non-arrest crack condition used in Criterion 4 had very little effect on the predictions. It is known that the stress levels do have an effect on the measured

Max. arrest stress (ksi)

60 55 50 45

R0.0, first grain

40 35 30 25 20 15 10 5 0 0

5

10

15

20

2a/D (crack size/grain size) Fig. 11. Effect of crack and grain sizes on arrest stress.

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1.0 0.9 0.8

Probability

0.7

Width (crack-nucleating particle) Width (Frechet, extreme value model) Width (Monte Carlo, Criterion1) Width (Monte Carlo, Criterion2) Width (Monte Carlo, Criterion3) Width (Monte Carlo, Criterion4)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

20

40

60

80

100

Particle width (μm) Fig. 12. Conditional particle size distribution, fatigue subset of 2024-T351 0.06300 sheet, maximum stress 44 ksi, stress ratio R0.1.

1.0 0.9

Probability

0.8 0.7

Width (crack-nucleating particle) Width (Frechet, extreme value model) Width (Monte Carlo, Criterion1) Width (Monte Carlo, Criterion2) Width (Monte Carlo, Criterion3) Width (Monte Carlo, Criterion4)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

50

100

150

200

250

300

Particle width (μm) Fig. 13. Conditional particle size distribution, fatigue subset of 2024-T351 0.1600 sheet, maximum stress 40 ksi, stress ratio R0.1.

fatigue subsets [31]; the non-arrest crack condition used in Criterion 4 and the related inputs may be questionable and need further investigation. Overall, when more microstructural features, such as grain size and orientation, are considered, the simulation resulted in better predictions when compared to the test measurements. This is an encouraging result especially considering it can be produced from a simple Monte Carlo simulation, within a few minutes on a regular personal computer, and the simulated one-dimensional fatigue subsets can be used by the existing LEFM model/tool. It should be pointed out that the current simulation system is fundamentally not suitable for those cases where the crack nucleation mechanism is different from those in Criterion 1 to 4, for example, high cycle fatigue. 5. Discussions From the statistical analysis, it was seen that the material IDS/particle distribution usually had a very large sample size and a long right tail. Provided that a proper microstructural analysis is done, the material IDS/particle distribution would serve as a parent distribution, which is consistent with the as-produced material. For fatigue life estimation, the right tail portion of the material IDS/particle distribution is more important since fatigue cracks usually nucleate from large particles (but not necessarily the largest). The 3P weighted Lognormal distribution should be used since it provided a very good ﬁt to the right tail. More often, the fatigue subsets of the material IDS/particle distributions are used, in a probabilistic model, for predicting the fatigue life distributions [11,13,32,33], as well as fatigue risk assessment [34]. As the particle sizes are very small, a small/ short crack model is needed, similar to the one developed in [33], to correlate the crack-nucleating particle with the fatigue life. However, the issue to use this fatigue subset is that it is affected by many intrinsic and extrinsic factors, such as material state, manufacturing, stress level, geometry, and environment. It is impractical to perform a signiﬁcant number of tests to generate the required fatigue subsets for all these conditions. Therefore, the developed correlation between the material IDS distribution and its fatigue subset can be very helpful to provide an appropriate fatigue subset for the fatigue life estimation, under different conditions. Further investigations are needed to improve this simulation system, such as,

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1.0 0.9

Probability

0.8 0.7 Width (crack-nucleating particle) Width (Frechet, extreme value model) Width (Monte Carlo, Criterion1) Width (Monte Carlo, Criterion2) Width (Monte Carlo, Criterion3) Width (Monte Carlo, Criterion4)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

20

40

60

80

100

120

Particle width (μm)

(a) maximum stress 40 ksi, stress ratio R0.05. 1.0 0.9

Probability

0.8 0.7 Width (crack-nucleating particle) Width (Frechet, extreme value model) Width (Monte Carlo, Criterion1) Width (Monte Carlo, Criterion2) Width (Monte Carlo, Criterion3) Width (Monte Carlo, Criterion4)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

20

40

60

80

100

120

Particle width (μm)

(b) maximum stress 44 ksi, stress ratio R0.05. 1.0 0.9

Probability

0.8 0.7

Width (crack-nucleating particle) Width (Frechet, extreme value model) Width (Monte Carlo, Criterion1) Width (Monte Carlo, Criterion2) Width (Monte Carlo, Criterion3) Width (Monte Carlo, Criterion4)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

20

40

60

80

100

120

Particle width (μm)

(c) maximum stress 48 ksi, stress ratio R0.05. Fig. 14. Conditional particle size distributions, fatigue subset of 2024-T351 0.500 sheet.

More trials on high Kt specimens. Continuing theoretical and experimental studies on the physics of crack nucleation and short crack growth in order to improve/strengthen the fatigue criteria. More studies on the material characterization method (mathematical/statistical) to better describe the correlation among the different features. More studies on the microstructure geometry model in order to increase the ﬁdelity in this material simulation system. For example, the microstructure orientation mapping, and Voronoi tessellation methods may be used to build a better geometrical model for microstructure [35,36]. 6. Conclusion remarks Overall the 3P Lognormal distribution was found to be the best-ﬁt distributions for the material IDS/particle size (area, width, and height), and the weighted 3P Lognormal distribution was found to provide the best ﬁt to the right

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tail of the distributions. The distribution parameters were obtained for the unclad 2024-T351 0.06300 , 0.1600 , and 0.500 sheets. The model based on the extreme value theory determined that the critical density of crack-nucleating particles (Dcp), for different thickness of 2024 sheets, was typically less than 0.5% of the density of all particles. This effort also indicated that in addition to particle size, other microstructural features need to be considered for predicting the fatigue subsets of the material IDS/particle size distributions. A new Monte Carlo simulation was developed to quickly compute the fatigue subsets of the material IDS/particle size distributions, which results can be used by existing LEFM models. Qualitative fatigue criteria were established based on the physical understanding of crack nucleation and short crack growth, including the effects of particle size, grain size, orientation, as well as stress levels. The preliminary predictions were promising, which were better than those from the extreme value theory based model, and agreed fairly well with experimental measurements. Further investigations are proposed to improve this new simulation method. Acknowledgements This work was carried out with the ﬁnancial support of Defence Research and Development Canada (DRDC) and National Research Council Canada (NRC), Project ‘‘POD and Probabilistic Risk Assessment” (46_QJ0_24). The original metallurgical data were obtained from the previous CFSD (Corrosion and Fatigue Structure Demonstration) Project, funded by USAF through a contract from Lockheed Martins Aeronautics, and the DUST (Dual Use Science and Technology) Project, partially funded by the USAF through a collaboration with APES Inc. References [1] Komorowski JP. New tools for aircraft maintenance. Aircraft Engng Aerospace Technol 2003;75(5):453–60. [2] Brooks C, Honeycutt K, Prost-Domasky S, Peeler D. 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