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Stress ratio and the fatigue damage map—Part II: The 2024-T351 aluminium alloy
International Journal of Fatigue 26 (2004) 747–752 www.elsevier.com/locate/ijfatigue
Stress ratio and the fatigue damage map—Part II: The 2024-T351 aluminium alloy C.A. Rodopoulos a,, J.-H. Choi b,c, E.R. de los Rios b, J.R. Yates b a
b
Materials Research Institute, Sheffield Hallam University, City Campus, Howard Street, Sheffield S1 1WB, UK Department of Mechanical Engineering, Structural Integrity Research Institute of the University of Sheffield (SIRIUS), The University of Sheffield, Sheffield S1 3JD, UK c Research & Development Division, Hyundai Motor Company, Whasung-City, Kyunggi-Do 445-850, South Korea Received 3 March 2003; received in revised form 11 September 2003; accepted 28 October 2003
Abstract In this second part an experimental matrix designed to verify the modelling in part I is presented. The obtained experimental data consist of crack growth rates, S–N curves, crack arrest curves and an extensive fractographic analysis to measure Stage I to Stage II and Stage II to Stage III transitions. Comparisons between the experimental data and the modelling confirmed that the concept of the fatigue damage map can be used to predict the effect of the stress ratio on the fatigue behaviour of the 2024-T351 aluminium alloy. Prediction error in the case of the crack arrest behaviour is also acknowledged. # 2003 Elsevier Ltd. All rights reserved. Keywords: Stress ratio; Short cracks; 2024; Fatigue damage map
1. Introduction The fatigue damage map (FDM) has been developed as a supplement to damage tolerance philosophy. Traditionally, damage tolerance requires knowledge of crack growth rates, stress/strain life curves and fracture mechanics [1]. Despite its successes, reflected by the much enjoyed safety in aviation, damage tolerant design remains a risky process. This is due to the following: (a) crack growth rates are usually taken from long cracks and therefore can only partially represent propagation in the short crack region. Hence, a large portion of the components life might be miscalculated [2]; (b) a procedure that seems to deliver accurate life predictions for a particular material might be completely inaccurate for another material (in [3], it was shown that the extent of short growth depends on the material) and (c) stress and strain life curves encapsulate the broad performance of materials without providing the necessary distinction between different
Corresponding author. Tel. +44-114-225-4257; fax: +44-114-2253501. E-mail address: [email protected] (C.A. Rodopoulos). 0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.018
stages, i.e. short crack growth, steady-state growth, etc. Consequently, material selection based on S–N and e N curves could be misleading, dangerous and costly. In the first part of this work, the modelling of the stress ratio on the FDM was presented. In the second part, the accuracy of the FDM in predicting the various fatigue stages under different conditions of loading (Rratio) is generally demonstrated with the exception in the prediction of crack arrest. The experimental work is based on the popular 2024-T351 aluminium alloy. 2. Experimental procedure Material, supplied by Airbus UK, was received in the form of 30.0 mm thick rolled plate. A full chemical composition is given in Table 1. Fatigue testing was performed in a four-point bending configuration. The stress gradient is given by the linear relationship Table 1 Chemical composition of 2024-T351 in wt% [4] Si
Fe
0.05 0.5
Cu
Mn
3.8–4.9 0.3–0.9
Mg
Cr
1.2–1.8 0.1
Zn
Ti
Al
0.25
0.15
Bal.
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C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Fig. 1. Test-piece dimensions. The dotted lines represent the gage area.
r=rmax ¼ 0:36z, where z is the position of the bending fibre from the neutral/central fibre. Test-piece dimensions are shown in Fig. 1. The surface of the specimens was polished using a succession of finer grade emery papers and diamond pastes to a 1/4 lm finish. The mirror finish was selected in order to minimise secondary crack population. All tests were carried out using specimens oriented with the tension axis parallel to the rolling direction and the fatigue surface perpendicular to the rolling plane. The mechanical and some physical properties of the material can be found in Table 2. Fatigue tests were performed in load control with a sinusoidal wave form at a frequency of 15 Hz. Three different stress ratios R (minimum load to maximum load) values were used: 0.1, 0.3 and 0.5. The experimental data in the form of S–N curves are shown in Fig. 2. The results indicate that the fatigue resistance of the material drops as the stress ratio increases. This behaviour becomes clear when the life at R¼ 0:3 is compared to that at R¼ 0:5. However, in terms of the fatigue limit ðN> 107 Þ the effect of the stress ratio becomes minimum. The above indicates that high cycle fatigue is not sensitive to the maximum stress. Crack monitoring was performed using a video camera mounted on an optical microscope and a dedicated image capturing card supported by a computer. Crack measurements were performed using image analysis software (SigmaScan by SPSS). The software is based on the tracking of the crack using a mouse. Several crack growth data, da/dN, are presented in Fig. 3.
Fig. 2. S–N curves for the 2024-T351 at R¼ 0:1, 0.3 and 0.5. The arrows indicate run-outs.
From the experimental results, the following remarks can be made: (a) the effect of the microstructure, i.e. grain boundaries, on crack growth is more dominant at
Table 2 Mechanical and physical properties of 2024-T351 at room temperature [4,5] (fatigue properties for growth perpendicular to rolling direction) Elastic modulus (GPa) Shear modulus (GPa) Poisson’s ratio Mon. yield stress (MPa) Cyclic yield stress (MPa) Density (g/cm3) Hardness (HB) Fracture toughness (MPa m1/2) Elongation (%) Ult. tensile strength (MPa) Max. stress fatigue limit ðR¼ 1Þ
72–76 27–31 0.33 325–340 420–450 2.77 115–120 34–38 18–21 475–520 120–140
Fig. 3. (a) Crack growth rate vs crack length for Dr¼ 295 MPa at R¼ 0:1 and R¼ 0:3. (b) Crack growth rate vs crack length for Dr ¼ 257 MPa at R¼ 0:1 and R¼ 0:3.
C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
R¼ 0:1 than at R¼ 0:3. The above is manifested by the number and magnitude of the acceleration–deceleration peaks; (b) the growth rate at R¼ 0:3 is higher than that at R¼ 0:1, especially at high Dr levels; (c) rmax dominates short crack growth rate, considering that comparison was made at similar Dr values; and (d) at R¼ 0:3, the tendency of the crack to achieve a consistently increasing growth rate comes sooner than at R¼ 0:1. 3. Verification of the FDM 3.1. Crack arrest The method used for the determination the crack arrest behaviour was repeated Dr-shedding. The test procedure follows the ASTM recommendation [6]. Deviation from the above was that the crack was allowed to initiate naturally at a stress level of 10% above the fatigue limit. In the case where secondary cracks were believed to interfere with the propagation of the main crack, the test was repeated. The crack arrest behaviour, at the three selected stress ratios, is shown in Fig. 4. Predictions shown in Fig. 4 were determined using (see part I) 0:5 2a ð1 RÞ DrFLðR¼0Þ pffiffiffiffiffiffiffiffiffiffiffi Drarrest ¼ 1 þ 0:35 ln D 2a=D for R 0
ð1Þ
where the parameter½1 þ 0:35 lnð2a=DÞ is an experimentally determined grain orientation distribution for aluminium alloys [7]. From the results, it is clear that none of the three curves follows the 0.5 gradient (elastic stress singularity) established by the work published by Kitagawa and Takahashi [8]. In addition, all curves exhibited an
Fig. 4. Crack arrest curves at R¼ 0:1, 0.3 and 0.5. The discrepancy from the 0.5 stress gradient is obvious. The curves have been constructed from numerous tests.
749
almost identical pattern for a 200 lm. The above indicates the independency of the initial crack arrest capacity of the material to stress ratio while initial cracking took place in a grain larger than the average value of 52 lm. Herein, it should be noted that the above questions the accuracy of Eq. (1) to predict crack arrest. A possible explanation is the tendency of the 2024-T351 to promote shear-band decohesion at ‘‘short’’ crack lengths. The above has been observed by Zhang et al. [9]. The crack arrest curve at R¼ 0:5 and for a> 200 lm shows a faster decay compared to R¼ 0:1 and 0.3. Similar behaviour has been frequently observed by many others [10], where the threshold stress intensity factor was reported to decrease with R. 3.2. Stage I to Stage II growth In part I, it was suggested that the effect of the stress ratio on the transition from Stage I to Stage II crack growth is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 1R 2 c 4D R ry DrI!II ¼ ; R0 ð2Þ Y 1þR p a þ 2D for negligible crack closure. To verify the accuracy of Eq. (2), scanning electron microscopy (SEM) was used to determine the transition from Stage I to Stage II crack growth. Micrographs, as those shown in Fig. 5, were used to determine changes in the amount of crystallographic facets present. Transition from Stage I to Stage II crack growth was assumed to take place when the faceted area falls below 10%. This was achieved by using a digital threshold filter on the original image. Measurements were performed using SigmaScan by SPSS. SEM measurements versus the predictions by
Fig. 5. SEM image showing the transition (indicated by the sketched dashed border) from Stage I to Stage II growth at R¼ 0:1 and Dr¼ 270 MPa. The area encapsulated by the border represents faceted Stage I growth. Photo taken from [11].
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C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Eq. (2), for the selected stress ratios, are shown in Fig. 6. From Fig. 6, the accuracy of Eq. (2) in predicting the transition from Stage I to Stage II growth is apparent. The prediction error in the case of R¼ 0:5 is attributed to the minimum Stage I growth, SEM indicates values of approximately half the grain, while the FDM predicts zero Stage I growth at the selected stress level. However, such error could be associated to numerous sensitivities covering microstructural short cracks (less than a grain in size), i.e. grain size; crack correction factor, etc. 3.3. Stage II to Stage III growth As discussed in part I, the transition from Stage II to Stage III crack growth is dominated by the degradation of the material at a microscopic scale. The theoretical analysis revealed that such condition can be approximate by h i ð8a=pEÞrcy ln secðprII!III Þ=2rcy ¼ lnð1 þ ef Þ; x > 4 xðD=2Þ ð3Þ In Eq. (3), the effect of the stress ratio can be introduced through the relationship between DrðDrII!III Þ and R. It should be noted that the correction factor Y is missing from Eq. (3). This is rationalised by the fact that the effect of the crack shape on the stress field of very long cracks is minimum. In a way similar to that used in the previous section, SEM was used to identify the transition from Stage II to Stage III. A typical image is shown in Fig. 7. In Fig. 8, SEM measurements are compared to FDM predictions (Eq. (3)).
Fig. 7. Transition from Stage II to Stage III crack growth at R¼ 0:1 and Dr¼ 270 MPa. The transition was identified by the presence of striation markings and microvoids. Photo taken from [10].
Fig. 8. Comparison between SEM measurements and FDM predictions (Eq. (3)) for the transition from Stage II to Stage III crack growth.
4. Implementing the FDM to damage tolerant design
Fig. 6. Comparison between SEM measurements and FDM predictions (Eq. (2)) for the transition from Stage I to Stage II crack growth. A crack correction factor Y ¼ 0:68 was used.
At the beginning of this work it was pointed out that the S–N curve provides a broad characterisation of the fatigue life of a material. In addition, correlation between the S–N curve and the Paris–Erdogan crack propagation power law is unworkable, as far as the limits of Stage II crack growth are not known. This much sought after correlation is now possible with the introduction of the FDM. In the previous section, it was proposed that the limits of Stage II crack growth are given by Eqs. (2) and (3). Assuming the accuracy of the power law da=dN ¼ CDKm to characterise Stage II crack growth, the fati-
C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Fig. 9. Life allocation for Stage I crack growth at R¼ 0:1. The parameters C and m (da/dN in m/cycle), determined from Fig. 3, are 1:5 1011 and 3.4, respectively (for growth units in m/cycle and Ds in MPa).
gue life of Stage II can be predicted by NII ¼
Eq: ð3Þ1m=2 Eq: ð2Þ1m=2 pffiffiffi CðY Dr pÞð1 m=2Þ
ð4Þ
where Eqs. (3) and (2) represent solutions of the above equation in terms of crack length and C, m are experimentally determined parameters. It should be noted that the crack correction factor, Y, should be set to unity since the effect of the crack geometry is already included into Eq. (2). Using the above rationale, the fatigue life of Stage I crack growth is then given by NI ¼ Nf NII
ð5Þ
where Nf is the total fatigue life at a particular Dr value taken from the S–N curve. It is worth noting that Eq. (5) considers that Stage III crack growth will have a negligible effect on life allocation due to the high propagation rate. Fig. 9 shows the life allocation for the case of R¼ 0:1.
5. Discussion and conclusions The FDM was developed in order to provide a tool able to predict the fatigue tendencies of polycrystalline materials using readily available and cost-effective data. Such a task is achieved by the modelling of the five distinct fatigue mechanisms that operate throughout the fatigue life of a component, namely: crack arrest, Stage I (short) crack growth, Stage II (long crack) steadystate growth, Stage III unsteady growth and failure by either general yielding or fracture. In the first part of this work, the effect of stress ratio, R, was introduced into the concept of the FDM. In the
751
second part, the modelling was verified for the 2024T351. Arrest curves of naturally initiated cracks were obtained following a typical load shedding technique. The results showed a linear behaviour for crack lengths up to 200 lm at all three stress ratios. The above can be explained by the tendency of the material to promote shear-band decohesion. For crack lengths a> 200 lm, the data indicate an almost linear decay which does not follow the gradient of the elastic stress singularity. This indicates the sensitivity of parameters like DKth to the testing procedure and also possible inaccuracies from the application of constant life models (Gerber, Goodman, etc.). The faster decay exhibited at R¼ 0:5 compared to the almost identical pattern shown at R¼ 0:1 and 0.3 can be rationalised by the fact that at R¼ 0:5 the tendency of the material to a stageitype of growth is negligible. Thus, at R¼ 0:5, the stress field is characterised by a strong mode I stress field in contrast to the mixed mode that dominates lower stress ratios. It should be noted that the use of Eq. (1) revealed certain prediction errors compared to experimental data and further research is required. Fractographic analysis performed on several specimens revealed the accuracy of the modelling. In detail, examination of the faceted fracture surface revealed that: (a) the initial assumption of two grain crack tip plasticity represents an accurate representation of the stress field that characterises a newly developed Stage II crack and (b) Stage I crack growth diminishes with stress ratio. The latter can be useful in cases where the accuracy of the Paris–Erdogan model is in question. The accuracy of the FDM was also verified in the case of the transition from Stage II to Stage III growth. Herein, Stage III was also found to diminish with stress ratio. Knowledge of such condition is critical in cases where the material is characterised by low ductility and consequently the use of fracture toughness as a boundary condition for growth is unacceptable. Knowledge of the different stages that constitute fatigue damage can provide useful information for the designing of damage tolerant components. Herein, knowledge of the transition from Stage I to Stage II growth can provide an understanding of the portion of total life attributed to Stage I crack growth and hence quantify the potential error caused by the universal application of a Paris–Erdogan crack propagation law type solution.
References [1] Fatigue design handbook—AE10. 2nd ed. Warrendale: SAE; 1988. [2] Miller KJ. The short crack problem. Fatigue Fract Eng Mater Struct 1982;5(3):223–32. [3] Rodopoulos CA, de los Rios ER. Theoretical analysis on the behaviour of short fatigue cracks. Int J Fatigue 2002;24:719–24.
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[4] Boyer HE, Gall TL, editors. Metals handbook. 9th ed. Metals Park (OH): American Society of Metals; 1997. [5] Callister Jr WD. Materials science and engineering—an introduction. 2nd ed. USA: Wiley; 1991. [6] McClung RC. Analysis of fatigue crack closure during simulated threshold testing. In: Newman JC, Piasik RS, editors. Fatigue crack growth thresholds, endurance limit and design. ASTM STP 1372. PA: ASTM; 2000. [7] Curtis SA, Solis Romero J, de los Rios ER, Rodopoulos CA, Levers A. Predicting the interfaces between fatigue crack growth regimes in 7150-T651 aluminium alloy using the fatigue damage map. Mater Sci Eng A 2003;A344:79–85.
[8] Kitagawa H, Takahashi S. Applicability of fracture mechanics to very small cracks or cracks in the early stage. Presented in the Second International Conference on Mechanical Behaviour of Materials, ICM2. Metals Park (OH): ASM; 1976, p. 627–31. [9] Zhang XP, Wang CH, Ye L, Mai YW. In situ investigation of small fatigue crack growth in poly-crystal and single-crystal aluminium alloys. Fatigue Fract Eng Mater Struct 2001;25:141–50. [10] Lawson L, Chen EY, Meshii M. Near-threshold fatigue: a review. Int J Fatigue 1999;21:S15–34. [11] Choi JH. The effects of stress ration on the fatigue behaviour of 2024-T351 aluminium alloy. MPhil thesis, University of Sheffield, 2002.
Stress ratio and the fatigue damage map—Part II: The 2024-T351 aluminium alloy C.A. Rodopoulos a,, J.-H. Choi b,c, E.R. de los Rios b, J.R. Yates b a
b
Materials Research Institute, Sheffield Hallam University, City Campus, Howard Street, Sheffield S1 1WB, UK Department of Mechanical Engineering, Structural Integrity Research Institute of the University of Sheffield (SIRIUS), The University of Sheffield, Sheffield S1 3JD, UK c Research & Development Division, Hyundai Motor Company, Whasung-City, Kyunggi-Do 445-850, South Korea Received 3 March 2003; received in revised form 11 September 2003; accepted 28 October 2003
Abstract In this second part an experimental matrix designed to verify the modelling in part I is presented. The obtained experimental data consist of crack growth rates, S–N curves, crack arrest curves and an extensive fractographic analysis to measure Stage I to Stage II and Stage II to Stage III transitions. Comparisons between the experimental data and the modelling confirmed that the concept of the fatigue damage map can be used to predict the effect of the stress ratio on the fatigue behaviour of the 2024-T351 aluminium alloy. Prediction error in the case of the crack arrest behaviour is also acknowledged. # 2003 Elsevier Ltd. All rights reserved. Keywords: Stress ratio; Short cracks; 2024; Fatigue damage map
1. Introduction The fatigue damage map (FDM) has been developed as a supplement to damage tolerance philosophy. Traditionally, damage tolerance requires knowledge of crack growth rates, stress/strain life curves and fracture mechanics [1]. Despite its successes, reflected by the much enjoyed safety in aviation, damage tolerant design remains a risky process. This is due to the following: (a) crack growth rates are usually taken from long cracks and therefore can only partially represent propagation in the short crack region. Hence, a large portion of the components life might be miscalculated [2]; (b) a procedure that seems to deliver accurate life predictions for a particular material might be completely inaccurate for another material (in [3], it was shown that the extent of short growth depends on the material) and (c) stress and strain life curves encapsulate the broad performance of materials without providing the necessary distinction between different
Corresponding author. Tel. +44-114-225-4257; fax: +44-114-2253501. E-mail address: [email protected] (C.A. Rodopoulos). 0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.018
stages, i.e. short crack growth, steady-state growth, etc. Consequently, material selection based on S–N and e N curves could be misleading, dangerous and costly. In the first part of this work, the modelling of the stress ratio on the FDM was presented. In the second part, the accuracy of the FDM in predicting the various fatigue stages under different conditions of loading (Rratio) is generally demonstrated with the exception in the prediction of crack arrest. The experimental work is based on the popular 2024-T351 aluminium alloy. 2. Experimental procedure Material, supplied by Airbus UK, was received in the form of 30.0 mm thick rolled plate. A full chemical composition is given in Table 1. Fatigue testing was performed in a four-point bending configuration. The stress gradient is given by the linear relationship Table 1 Chemical composition of 2024-T351 in wt% [4] Si
Fe
0.05 0.5
Cu
Mn
3.8–4.9 0.3–0.9
Mg
Cr
1.2–1.8 0.1
Zn
Ti
Al
0.25
0.15
Bal.
748
C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Fig. 1. Test-piece dimensions. The dotted lines represent the gage area.
r=rmax ¼ 0:36z, where z is the position of the bending fibre from the neutral/central fibre. Test-piece dimensions are shown in Fig. 1. The surface of the specimens was polished using a succession of finer grade emery papers and diamond pastes to a 1/4 lm finish. The mirror finish was selected in order to minimise secondary crack population. All tests were carried out using specimens oriented with the tension axis parallel to the rolling direction and the fatigue surface perpendicular to the rolling plane. The mechanical and some physical properties of the material can be found in Table 2. Fatigue tests were performed in load control with a sinusoidal wave form at a frequency of 15 Hz. Three different stress ratios R (minimum load to maximum load) values were used: 0.1, 0.3 and 0.5. The experimental data in the form of S–N curves are shown in Fig. 2. The results indicate that the fatigue resistance of the material drops as the stress ratio increases. This behaviour becomes clear when the life at R¼ 0:3 is compared to that at R¼ 0:5. However, in terms of the fatigue limit ðN> 107 Þ the effect of the stress ratio becomes minimum. The above indicates that high cycle fatigue is not sensitive to the maximum stress. Crack monitoring was performed using a video camera mounted on an optical microscope and a dedicated image capturing card supported by a computer. Crack measurements were performed using image analysis software (SigmaScan by SPSS). The software is based on the tracking of the crack using a mouse. Several crack growth data, da/dN, are presented in Fig. 3.
Fig. 2. S–N curves for the 2024-T351 at R¼ 0:1, 0.3 and 0.5. The arrows indicate run-outs.
From the experimental results, the following remarks can be made: (a) the effect of the microstructure, i.e. grain boundaries, on crack growth is more dominant at
Table 2 Mechanical and physical properties of 2024-T351 at room temperature [4,5] (fatigue properties for growth perpendicular to rolling direction) Elastic modulus (GPa) Shear modulus (GPa) Poisson’s ratio Mon. yield stress (MPa) Cyclic yield stress (MPa) Density (g/cm3) Hardness (HB) Fracture toughness (MPa m1/2) Elongation (%) Ult. tensile strength (MPa) Max. stress fatigue limit ðR¼ 1Þ
72–76 27–31 0.33 325–340 420–450 2.77 115–120 34–38 18–21 475–520 120–140
Fig. 3. (a) Crack growth rate vs crack length for Dr¼ 295 MPa at R¼ 0:1 and R¼ 0:3. (b) Crack growth rate vs crack length for Dr ¼ 257 MPa at R¼ 0:1 and R¼ 0:3.
C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
R¼ 0:1 than at R¼ 0:3. The above is manifested by the number and magnitude of the acceleration–deceleration peaks; (b) the growth rate at R¼ 0:3 is higher than that at R¼ 0:1, especially at high Dr levels; (c) rmax dominates short crack growth rate, considering that comparison was made at similar Dr values; and (d) at R¼ 0:3, the tendency of the crack to achieve a consistently increasing growth rate comes sooner than at R¼ 0:1. 3. Verification of the FDM 3.1. Crack arrest The method used for the determination the crack arrest behaviour was repeated Dr-shedding. The test procedure follows the ASTM recommendation [6]. Deviation from the above was that the crack was allowed to initiate naturally at a stress level of 10% above the fatigue limit. In the case where secondary cracks were believed to interfere with the propagation of the main crack, the test was repeated. The crack arrest behaviour, at the three selected stress ratios, is shown in Fig. 4. Predictions shown in Fig. 4 were determined using (see part I) 0:5 2a ð1 RÞ DrFLðR¼0Þ pffiffiffiffiffiffiffiffiffiffiffi Drarrest ¼ 1 þ 0:35 ln D 2a=D for R 0
ð1Þ
where the parameter½1 þ 0:35 lnð2a=DÞ is an experimentally determined grain orientation distribution for aluminium alloys [7]. From the results, it is clear that none of the three curves follows the 0.5 gradient (elastic stress singularity) established by the work published by Kitagawa and Takahashi [8]. In addition, all curves exhibited an
Fig. 4. Crack arrest curves at R¼ 0:1, 0.3 and 0.5. The discrepancy from the 0.5 stress gradient is obvious. The curves have been constructed from numerous tests.
749
almost identical pattern for a 200 lm. The above indicates the independency of the initial crack arrest capacity of the material to stress ratio while initial cracking took place in a grain larger than the average value of 52 lm. Herein, it should be noted that the above questions the accuracy of Eq. (1) to predict crack arrest. A possible explanation is the tendency of the 2024-T351 to promote shear-band decohesion at ‘‘short’’ crack lengths. The above has been observed by Zhang et al. [9]. The crack arrest curve at R¼ 0:5 and for a> 200 lm shows a faster decay compared to R¼ 0:1 and 0.3. Similar behaviour has been frequently observed by many others [10], where the threshold stress intensity factor was reported to decrease with R. 3.2. Stage I to Stage II growth In part I, it was suggested that the effect of the stress ratio on the transition from Stage I to Stage II crack growth is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 1R 2 c 4D R ry DrI!II ¼ ; R0 ð2Þ Y 1þR p a þ 2D for negligible crack closure. To verify the accuracy of Eq. (2), scanning electron microscopy (SEM) was used to determine the transition from Stage I to Stage II crack growth. Micrographs, as those shown in Fig. 5, were used to determine changes in the amount of crystallographic facets present. Transition from Stage I to Stage II crack growth was assumed to take place when the faceted area falls below 10%. This was achieved by using a digital threshold filter on the original image. Measurements were performed using SigmaScan by SPSS. SEM measurements versus the predictions by
Fig. 5. SEM image showing the transition (indicated by the sketched dashed border) from Stage I to Stage II growth at R¼ 0:1 and Dr¼ 270 MPa. The area encapsulated by the border represents faceted Stage I growth. Photo taken from [11].
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C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Eq. (2), for the selected stress ratios, are shown in Fig. 6. From Fig. 6, the accuracy of Eq. (2) in predicting the transition from Stage I to Stage II growth is apparent. The prediction error in the case of R¼ 0:5 is attributed to the minimum Stage I growth, SEM indicates values of approximately half the grain, while the FDM predicts zero Stage I growth at the selected stress level. However, such error could be associated to numerous sensitivities covering microstructural short cracks (less than a grain in size), i.e. grain size; crack correction factor, etc. 3.3. Stage II to Stage III growth As discussed in part I, the transition from Stage II to Stage III crack growth is dominated by the degradation of the material at a microscopic scale. The theoretical analysis revealed that such condition can be approximate by h i ð8a=pEÞrcy ln secðprII!III Þ=2rcy ¼ lnð1 þ ef Þ; x > 4 xðD=2Þ ð3Þ In Eq. (3), the effect of the stress ratio can be introduced through the relationship between DrðDrII!III Þ and R. It should be noted that the correction factor Y is missing from Eq. (3). This is rationalised by the fact that the effect of the crack shape on the stress field of very long cracks is minimum. In a way similar to that used in the previous section, SEM was used to identify the transition from Stage II to Stage III. A typical image is shown in Fig. 7. In Fig. 8, SEM measurements are compared to FDM predictions (Eq. (3)).
Fig. 7. Transition from Stage II to Stage III crack growth at R¼ 0:1 and Dr¼ 270 MPa. The transition was identified by the presence of striation markings and microvoids. Photo taken from [10].
Fig. 8. Comparison between SEM measurements and FDM predictions (Eq. (3)) for the transition from Stage II to Stage III crack growth.
4. Implementing the FDM to damage tolerant design
Fig. 6. Comparison between SEM measurements and FDM predictions (Eq. (2)) for the transition from Stage I to Stage II crack growth. A crack correction factor Y ¼ 0:68 was used.
At the beginning of this work it was pointed out that the S–N curve provides a broad characterisation of the fatigue life of a material. In addition, correlation between the S–N curve and the Paris–Erdogan crack propagation power law is unworkable, as far as the limits of Stage II crack growth are not known. This much sought after correlation is now possible with the introduction of the FDM. In the previous section, it was proposed that the limits of Stage II crack growth are given by Eqs. (2) and (3). Assuming the accuracy of the power law da=dN ¼ CDKm to characterise Stage II crack growth, the fati-
C.A. Rodopoulos et al. / International Journal of Fatigue 26 (2004) 747–752
Fig. 9. Life allocation for Stage I crack growth at R¼ 0:1. The parameters C and m (da/dN in m/cycle), determined from Fig. 3, are 1:5 1011 and 3.4, respectively (for growth units in m/cycle and Ds in MPa).
gue life of Stage II can be predicted by NII ¼
Eq: ð3Þ1m=2 Eq: ð2Þ1m=2 pffiffiffi CðY Dr pÞð1 m=2Þ
ð4Þ
where Eqs. (3) and (2) represent solutions of the above equation in terms of crack length and C, m are experimentally determined parameters. It should be noted that the crack correction factor, Y, should be set to unity since the effect of the crack geometry is already included into Eq. (2). Using the above rationale, the fatigue life of Stage I crack growth is then given by NI ¼ Nf NII
ð5Þ
where Nf is the total fatigue life at a particular Dr value taken from the S–N curve. It is worth noting that Eq. (5) considers that Stage III crack growth will have a negligible effect on life allocation due to the high propagation rate. Fig. 9 shows the life allocation for the case of R¼ 0:1.
5. Discussion and conclusions The FDM was developed in order to provide a tool able to predict the fatigue tendencies of polycrystalline materials using readily available and cost-effective data. Such a task is achieved by the modelling of the five distinct fatigue mechanisms that operate throughout the fatigue life of a component, namely: crack arrest, Stage I (short) crack growth, Stage II (long crack) steadystate growth, Stage III unsteady growth and failure by either general yielding or fracture. In the first part of this work, the effect of stress ratio, R, was introduced into the concept of the FDM. In the
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second part, the modelling was verified for the 2024T351. Arrest curves of naturally initiated cracks were obtained following a typical load shedding technique. The results showed a linear behaviour for crack lengths up to 200 lm at all three stress ratios. The above can be explained by the tendency of the material to promote shear-band decohesion. For crack lengths a> 200 lm, the data indicate an almost linear decay which does not follow the gradient of the elastic stress singularity. This indicates the sensitivity of parameters like DKth to the testing procedure and also possible inaccuracies from the application of constant life models (Gerber, Goodman, etc.). The faster decay exhibited at R¼ 0:5 compared to the almost identical pattern shown at R¼ 0:1 and 0.3 can be rationalised by the fact that at R¼ 0:5 the tendency of the material to a stageitype of growth is negligible. Thus, at R¼ 0:5, the stress field is characterised by a strong mode I stress field in contrast to the mixed mode that dominates lower stress ratios. It should be noted that the use of Eq. (1) revealed certain prediction errors compared to experimental data and further research is required. Fractographic analysis performed on several specimens revealed the accuracy of the modelling. In detail, examination of the faceted fracture surface revealed that: (a) the initial assumption of two grain crack tip plasticity represents an accurate representation of the stress field that characterises a newly developed Stage II crack and (b) Stage I crack growth diminishes with stress ratio. The latter can be useful in cases where the accuracy of the Paris–Erdogan model is in question. The accuracy of the FDM was also verified in the case of the transition from Stage II to Stage III growth. Herein, Stage III was also found to diminish with stress ratio. Knowledge of such condition is critical in cases where the material is characterised by low ductility and consequently the use of fracture toughness as a boundary condition for growth is unacceptable. Knowledge of the different stages that constitute fatigue damage can provide useful information for the designing of damage tolerant components. Herein, knowledge of the transition from Stage I to Stage II growth can provide an understanding of the portion of total life attributed to Stage I crack growth and hence quantify the potential error caused by the universal application of a Paris–Erdogan crack propagation law type solution.
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