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Stress ratio and the fatigue damage map—Part I: Modelling
International Journal of Fatigue 26 (2004) 739–746 www.elsevier.com/locate/ijfatigue
Stress ratio and the fatigue damage map—Part I: Modelling 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 work, the effect of the stress ratio R on the five areas that constitute the fatigue damage map is presented. The effect of R on crack arrest is modelled through the Kujawski’s empirical parameter a. The idea of matching crack growth rates was used to predict the effect of R on the transition from Stage I to Stage II growth. The transition from Stage II to Stage III was modelled through its direct relation to R. Finally, the areas of general yielding and toughness failure were modelled through the relationship between Dr and R. The accuracy of the modelling for each area is validated with experimental results. # 2003 Elsevier Ltd. All rights reserved. Keywords: Stress ratio; Fatigue damage map; Aluminium alloys
1. Introduction For many years, stress ratio (minimum to maximum stress ratio, R) effects on fatigue damage have been described through several crack closure mechanisms. Crack closure was first studied by Christensen [1] who examined the effect of fretting generated debris on the opening displacement range of cracks. In the early 1980s, Elber [2,3] highlighted the effect of the plastic deformation residue on the crack wake, commonly known as plasticity induced crack closure. The mechanism of the continuous breaking and reforming of oxide scale behind the crack tip, known as oxide induced crack closure, was also used by many researchers, particularly Paris et al. [4], Ritchie et al. [5] and Vasudevan and Suresh [6], to rationalise the anomalies in the near-threshold behaviour of fatigue damage in steels and aluminium alloys. In 1975, Purushothaman and Tien [7] observed that discrete points of contact between fracture surface asperities can also cause the premature closure of the crack. In addition to what is Corresponding author. Tel.: +44-114-225-4257; fax: +44-114-2255390. E-mail address: [email protected] (C.A. Rodopoulos).
0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.017
now designated as roughness induced crack closure, Suresh [8] introduced the phenomena of crack deflection (crack path tortuosity) and crack branching as the main controlling factors. In general, roughness induced crack closure is believed to be promoted by [9]: (a) near-threshold and plane strain conditions; (b) low stress intensity factor range, DK (DSIF), levels and R ratios; (c) coarse grains and shearable coherent precipitates and (d) crack deflections caused by grain boundaries. The mechanism of plasticity induced crack closure has been the target of extensive argumentation especially in explaining R ratio effects. Vasudevan et al. [10] suggested that there is no significant contribution to crack closure by the residual plastic stretch at the crack wake especially at the crack tip and under plane strain conditions. Similarly, Garret and Knot [11] argued that plasticity induced crack closure has little effect under plane strain conditions. More recently, Zhang et al. [12] found that near-threshold and stage I crack propagation in 2024-T351 tested in high vacuum is characterised by shear-band decohesion and thus it is unaffected by crack closure. With the possible dismissal of the effect of crack closure and its direct relationship
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with the stress ratio, an old chapter in fatigue research comes once again into the spotlight. The fatigue damage map (FDM) [13–15] was developed by Rodopoulos and de los Rios as a tool that allows the examination of the fatigue tendencies of engineering materials using readily available properties. In this first part, an attempt to incorporate the effects of stress ratio in the concept of the FDM is presented. Throughout the work, crack closure effects are not implicitly considered allowing the simplicity of the FDM to be maintained. However, for validation purposes, comparison to published data is also presented. 2. Threshold/crack arrest and stress ratio The long crack threshold stress intensity factor range, DKth, generally decreases with increasing R ratio and typically, but not always, tends to a constant value at high positive R ratios (R> 0:7) [16–20]. Discrepancies from the above ‘‘general rule’’ have been shown to depend on the yield stress, ry, of the material [21]. In [16,20], an extensive selection of DKth results on several steels, demonstrated that the tendency to a constant value behaviour is promoted by steels with low values of ry. In contrast, high strength steels showed an almost linear decay up to stress ratios of R¼ 0:9 [9]. In [22], Schmidt and Paris found that in the case of 2124-T3 the threshold saturation mechanism changes with the loading frequency. In their work, they reported saturation tendencies for R> 0:5 and R> 0:8 at frequencies of 300 and 580 Hz, respectively. Suresh et al. [23], demonstrated that in the case of 7075 aluminium alloy, DKth remains nearly constant along the whole range of positive R ratios when the material is tested in vacuum. In terms of a Kitagawa–Takahashi approach [9], the problem becomes more complicated since coarser grain materials showed higher long crack slope while at the same time exhibited lower fatigue limits compared to finer grain materials [9,16]. Therefore, it is clear that the ability of the material to arrest crack propagation is covered by complexity and contradiction. Thus, most threshold models contained experimental and semi-empirical parameters. A selection of threshold models can be found in [24]. In [25–28], it was proposed that the propagation/ arrest of a crack takes place when the dislocation pileup, representing the crack tip plastic zone, is unable to overcome the constraint provided by a dominant microstructural barrier, i.e. grain boundary, twin boundary, pearlite zone, etc. Based on the principles of the Navarro–Rios model [26–28], crack arrest was modelled by, mi rFL r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 rarrest ¼ ð1Þ m1 2a=D where rarrest is the crack arrest or threshold stress, a is
the crack length, rFL is the fatigue limit of the material (N> 107 cycles), D is the average transverse grain size, r1 is the crack closure stress and mi is the grain orientation factor. It should be noted that mi increases monotonically with crack length from a value of 1 until mi reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). The hypothesis m1 ¼ 1 is rationalised by the fact that, in many cases, crack nucleation takes place in grains that are most favourably oriented in relation to the applied stress that the resolved shear stress achieves maximum value. The threshold stress defined by Eq. (1), identifies two controlling parameters: (a) the strength of the grain boundary which is part of rFL (fatigue limit) and (b) the grain orientation, mi. Both parameters reflect the influence of microstructure on crack arrest. The first, by relating the strength of the boundary to the threshold stress for crack propagation and the latter, by incorporating the effect of the increasing number of grains traversed by the crack front as the crack grows. In 2001, Kujawski [29] proposed that the effect of the R-ratio on the threshold stress intensity factor range is given by, DKth ¼ ð1 RÞa DKth;0
for R 0
ð2aÞ
and DKth ¼ ð1 RÞ DKth;0
for R 0
ð2bÞ
where DKth,0 is the threshold DSIF corresponding to R ¼ 0 and a is a fitting parameter ranging between 0 and 1 (a value of a¼ 0:5 was suggested for aluminium alloys and martensitic steels). Navarro et al. [30] suggested that the plain fatigue limit is related to the threshold SIF through, rffiffiffiffiffiffiffiffi D ð3Þ DKth ¼ DrFL p 2 Using a similar argument for the fatigue limit the effect of R-ratio on the fatigue limit is given by, DrFL ¼ ð1 RÞa DrFLðR¼0Þ
for R 0
ð4aÞ
and DrFL ¼ ð1 RÞ DrFLðR¼0Þ
for R 0
ð4bÞ
Using Eqs. (1) and (4a,b), the effect of R-ratio on the crack arrest can be written as, Drarrest ¼
a mi ð1 RÞ DrFLðR¼0Þ r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 m1 2a=D
for R 0 ð5aÞ
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and Drarrest ¼
mi ð1 RÞDrFLðR¼0Þ r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 m1 2a=D
for R 0 ð5bÞ
Fig. 1 shows a plotting of Eq. (5a,b) for an aluminium alloy. It should be noted that the approach is accurate for opening mode growing cracks. For mode II or mode III growing cracks, the plain fatigue limit could be independent of R. To allow the use of experimental results to verify the accuracy of the method to predict the effect of R ratio on Drarrest, Eq. (5a,b) is transformed into the threshold value of the SIF, rffiffiffiffiffiffiffiffi mi D a ð1 RÞ DrFLðR¼0Þ p DKth ¼ Y for R 0 ð6aÞ 2 m1 rffiffiffiffiffiffiffiffi mi D DKth ¼ Y ð1 RÞDrFLðR¼0Þ p for R 0 ð6bÞ 2 m1 where Y is the crack correction factor. Assuming that the concept of similitude holds, Fig. 2 compares the threshold predictions by Eq. (6a,b) with experimental results published in the literature [32–37]. The results show the potential of Eq. (6) to predict experimental data.
3. Stage I/short crack growth Short fatigue cracks, also known as Stage I cracks, have been known to propagate at rates significantly higher than that of long cracks under the same nominal DK conditions [38–41]. This is because the size of
Fig. 2. Comparison between experimental threshold SIF values and Eq. (6a,b) as a function of R for 7075-T6. Two different K calibration (Y) values were used to account for two different crack shapes. The mechanical and physical properties used for the calculations are: DrFLðR¼0Þ ¼ 300 MPa [32], m1 ¼ 1 [14], D ¼ 25 108 lm [32,33], a ¼ 0:5 [29] and r1 ¼ 0. Scatter in the experimental data is attributed to different load shedding techniques and different quoted grain sizes. The independence of Y to crack length was only used for simplicity.
crack tip plastic zone size is significantly larger than that predicted by theories of continuum mechanics [42]. In [13,14,25,43,44], it was suggested that the Stage I crack growth terminates when the crack tip stress field is able to initiate plasticity on two successive grains without further growth of the crack. This assumption is rationalised considering that when the crack is small a single family of slip planes can accommodate the crack tip plasticity. At longer crack lengths, crack tip plasticity is more intense and can only be accommodated by multiple slip. The above was validated by Yoder et al. [43] and Curtis et al. [44], who examined the transition from Stage I to Stage II for a number of materials. According to the above rationale the transition co-ordinates (applied stress/crack length) from Stage I to Stage II (long) crack growth, is mathematically expressed as,
rI!II
" # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 c 4D r r1 þ r1 ¼ Y p y a þ 2D
ð7Þ
where rI!II is the applied stress level at the transition. In the case of a closure free cracks, Eq. (7) is written as, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 2 c 4D r rI!II ¼ ð8Þ Y p y a þ 2D Fig. 1. The effect of stress ratio on crack arrest stress range for 2024-T351 according to Eq. (5a,b). The mechanical and physical properties used for the calculations are: DrFLðR¼0Þ ¼ 200 MPa [25], mi ¼ 1 þ 0:35 lnð2a=DÞ [14], D¼ 52 lm [31], a¼ 0:5 [29] and r1 ¼ 0.
where rcy is the cyclic yield stress. A typical Stage I to Stage II transition for several aluminium alloys is shown in Fig. 3.
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Fig. 3. Stage I to Stage II transition curves for three aluminium alloys. In all three cases a closure free crack and Y ¼ 1 was considered. The mechanical properties are: (a) 2024-T351— rcy ¼ 450 MPa, D¼ 52 lm; (b) 7150-T651—rcy ¼ 490 MPa [15], D ¼ 58 lm and (c) 7075-T6—rcy ¼ 517 MPa, D¼ 108 lm [14].
Fig. 4. Determination of the dtip for the transition from Stage I to Stage II for a edge crack in a 2024-T351 sheet. The mechanical properties of the material are as in Fig. 3. The crack correction factor was taken from [48].
In several works [26,27,45,46], the propagation rate of a short fatigue crack was modelled through the elasto-plastic crack tip opening displacement, dtip,
In 1980, Allen [46] suggested that the effect of the ratio-R on the crack growth rate is given by,
da 1þR b ¼A ð12Þ d ; R0 dN 1 R tip
da ¼ Adbtip dN
ð9Þ
where the fitting parameters A and b depend on the material, stress-state, R-ratio, etc. The parameter dtip is given by Dugdale [47], " !# 8a c pr r ln sec dtip ¼ ð10Þ pG y 2rcy where G is the shear modulus. It is worth noting that the original Young’s modulus is replaced by the shear modulus to account for the shear mode loading of the short crack. By substituting Eq. (8) into Eq. (10), the dtip for the transition from Stage I to Stage II is given by, " !# 8a c prI!II r ln sec dtip ¼ ð11Þ pG y 2rcy In Fig. 4, the results from Eq. (11) are presented for 2024-T351 considering an edge crack in an infinite width sheet under uniaxial stress. The results indicate that for rI!II 5 rcy , dtip tends to a constant dctip value. The above comes as a verification to the fact that a constant size crack tip plastic zone corresponds to a constant SIF and consequently to a constant dtip. For rI!II ryc , dtip is found to tend to an infinite value.
where A, b are fitting parameters with b being independent of R and A for R¼ 0 (dtip ¼ Ddtip ). Hence, the expression 1 þ R=1 R represents a mathematical regulator of dtip that allows the use of the same set of fitting parameters autonomously to R. Considering that dctip depends implicitly on the material and the interaction between stress/crack/ component, it is rational to assume that the crack growth rate at the transition from Stage I to Stage II is also a constant given that the parameter A is a function of R. Modelling of the above rationale in terms of the R adjustment provided by Eq. (12) gives,
1R c R ð13Þ d Ddtip ¼ 1 þ R tip c where DdR tip is the adaptation of dtip in terms of R. Eq. (13), using Eqs. (8) and (10) can be also expressed in terms of the transition stress DrI!II , rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1 1R 2 c 4D R ry ð14Þ DrI!II ¼ Y 1þR p a þ 2D
In Fig. 5, the effect of R on the transition from Stage I to Stage II is illustrated. Extension of the Stage I growth area is provided by simultaneously plotting Eqs. (6a,b) and (14). The effect of R on the Stage I growth area is depicted in Fig. 6. In terms of the model discussed earlier, Fig. 6 shows that high R values will decrease Stage I growth. The tendency of the Stage I growth area to contract and
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Fig. 5. The effect of R on the transition from Stage I to Stage II for 2024-T351. The mechanical properties used for the calculations are as before. The crack correction factor was unity.
Fig. 7. Comparison between experimental and predicted transition from Stage I to Stage II growth rates for 2024-T351. The parameter Y was set to unity.
finally to disappear at R> 0:4 agrees with experimental results published by Pang and Song [49]. It is worth noting that Stage I growth is also promoted by Y < 1. In this section, the parameter Y was intentionally set to unity to avoid possible inaccuracies from the use of a continuum mechanics concepts to describe the stress field ahead of a short crack [25]. Eqs. (12) and (13) can be used to predict the crack growth rate at the transition from Stage I to Stage II growth. For the predictions, shown in Fig. 7, the Paris type fitting parameters for 2024-T351 at R¼ 0, C¼ 3:37 1011 and m¼ 3:32 pffiffiffiffi [49] (m/cycle growth, K in MPa m) were transformed into A¼ 5000 and b ¼ 1:88 assuming that the elastic fracture mechanics relationship between DK and dtip holds.
by the Paris law [25]. Termination of this steady-state growth will take place when the strain field ahead of the crack tip is sufficiently large as to achieve rupture strains which will degrade the flow resistance of the material [25]. As a result the crack will enter an unsteady state of growth where the dominance of Dr will diminish in favour of rmax (rupture stress control) [9]. In [25], the termination of the Stage II and the beginning of Stage III growth was modelled through its resemblance to static failure (cleavage, intergranular separation and fibrous failure),
4. Stage II—long cracks Stage II or long crack growth will commence when at a given applied stress, the crack attains a length beyond the limit curve of Stage I (Fig. 6). In Stage II, crack growth can be described with reasonable success
Fig. 6.
dtip ¼ lnð1 þ ef Þ q
ð15Þ
where ef is the monotonic elongation to failure and q is the length of crack tip plasticity. Considering that during Stage II growth, crack tip plasticity will continue to increase in multiples of D, the parameter q can be written as, q¼x
D and 2
x>4
The effect of R on Stage I growth for 2024-T351 for: (a) R¼ 0 and (b) R¼ 0:4. The parameter Y was unity.
ð16Þ
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where x is the number of half grains constituting crack tip plasticity. Using the continuum mechanics solution at the transition from Stage II to Stage III growth, Eq. (15) can be written as, h i ð8a=pEÞrcy ln sec prII!III =2rcy xðD=2Þ ¼ lnð1 þ ef Þ; x > 4
ð17aÞ
where DrII!III ¼ rII!III ð1 RÞ
ð17bÞ
In Eq. (17a), the use of the elastic modulus E conforms with the opening mode characterising Stage II crack growth. However, stress range values can be obtained through Eq. (17b). A typical solution of Eq. (17a) using an iteration method is shown in Fig. 8. Solution of Eq. (17a,b) in terms of dtip can also be used to extract growth rate information (see Fig. 9).
Fig. 8. Transition from Stage II to Stage III growth for 2024-T351 at different positive R ratios. The mechanical properties E¼ 72:4 GPa and ef ¼ 18% were used in the calculation [25].
or considering the effect of the R-ratio as, Drtoughness ¼
5. Catastrophic failure The end of Stage III growth is followed by catastrophic failure. For conditions of cyclic loading catastrophic failure is defined either in terms of general yielding (also known as crack instability) or in terms of fracture toughness. In [13], it was suggested that general yielding takes place when the crack tip plasticity has reach ‘‘infinite’’ size. In the same work, it was suggested that, DrFL r1 rinstability ¼ mi pffiffiffiffiffiffiffiffiffiffiffi þ r1 þ rcy 2a=D
ð1 RÞKIc pffiffiffiffiffiffi Y pa
ð23Þ
6. Discussion This work represents an attempt to incorporate the effect of the stress ratio R into the FDM. The effect of R on crack arrest was modelled through Kujawski’s parameter a. The approach provides the necessary flexibility to allow for changes caused by different
ð18Þ
or, DrFL rinstability ¼ mi pffiffiffiffiffiffiffiffiffiffiffi þ rcy 2a=D
ð19Þ
for negligible crack closure. It should be noted that the crack closure stress will have a negligible effect at this stage. In [25], it was suggested that in engineering components where crack sizes are larger, Eqs. (19) and (20) can be written as, rinstability ¼ rcy
ð20Þ
or considering the effect of the R-ratio as, Drinstability ¼ ð1 RÞrcy
ð21Þ
Catastrophic failure in terms of fracture toughness and for plain strain conditions is, rtoughness ¼
KIc pffiffiffiffiffiffi Y pa
ð22Þ
Fig. 9. Comparison between experimental and predicted transition from Stage II to Stage III growth rates for 7075-T6. The experimental results represent the first point that the growth rate deviates from its linear behaviour. The mechanical properties used for the calculations, apart from those used before, were E¼ 70 GPa, ef ¼ 12% [33]. The growth rate was calculated using Eq. (9). The parameters A ¼ 3500 and b¼ 1:55 were extracted from the growth parameters C ¼ 3:18 108 and m¼ 1:24 taken from [33].
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Fig. 10. The FDM for 2024-T351 for: (a) R¼ 0 and (b) R¼ 0:5. In addition to the mechanical properties used before, a value of 38 MPa m1/2 was used for the plane strain fracture toughness [25].
materials, etc. Stress ratio effects on the transition from Stage I to Stage II growth was modelled by making use of Allen’s model. The analysis is possible due to the fact that transition conditions are characterised by a constant crack tip plastic zone and consequently by a constant value of CTOD, dtip. The above allows the use of the matching crack propagation concept for several values of R. The effect of R on the transition from Stage II to Stage III growth was found to be independent of R. Such result is reasonable considering that Stage III represents a stage of material degradation which solely depends on the strain conditions ahead of the crack tip. Similarly, catastrophic failure, both in terms of general yielding or fracture failure, was modelled directly through the relationship between rmax and R. The outcome of the above modelling is shown in Fig. 10.
7. Conclusions From the results presented in the previous sections and the FDMs shown in Fig. 10 the following conclusions can be drawn: (a) The effect of R on the three conditions that dominate fatigue crack growth, namely crack arrest, Stage I to Stage II growth and Stage II to Stage III growth can be accurately predicted by the FDM. (b) Positive R-ratios are expected to reduce Stage I growth. (c) The transition from Stage II to Stage III growth is related to the effect of R on maximum stress. Differences in the area denoted as Stage III growth are attribute to the effect of R-ratio on the conditions of fracture toughness and general yielding. (d) The size of the Stage I area can be used to extract information about the accuracy of the Paris Law to predict fatigue life.
Acknowledgements The authors are indebt to the Royal Academy of Engineering, The Engineering and Physical Science Research Council, Airbus UK, Hyundai Motor Company, INASCO Hellas and The British Council for the financial and otherwise support provided throughout this project.
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Stress ratio and the fatigue damage map—Part I: Modelling 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 work, the effect of the stress ratio R on the five areas that constitute the fatigue damage map is presented. The effect of R on crack arrest is modelled through the Kujawski’s empirical parameter a. The idea of matching crack growth rates was used to predict the effect of R on the transition from Stage I to Stage II growth. The transition from Stage II to Stage III was modelled through its direct relation to R. Finally, the areas of general yielding and toughness failure were modelled through the relationship between Dr and R. The accuracy of the modelling for each area is validated with experimental results. # 2003 Elsevier Ltd. All rights reserved. Keywords: Stress ratio; Fatigue damage map; Aluminium alloys
1. Introduction For many years, stress ratio (minimum to maximum stress ratio, R) effects on fatigue damage have been described through several crack closure mechanisms. Crack closure was first studied by Christensen [1] who examined the effect of fretting generated debris on the opening displacement range of cracks. In the early 1980s, Elber [2,3] highlighted the effect of the plastic deformation residue on the crack wake, commonly known as plasticity induced crack closure. The mechanism of the continuous breaking and reforming of oxide scale behind the crack tip, known as oxide induced crack closure, was also used by many researchers, particularly Paris et al. [4], Ritchie et al. [5] and Vasudevan and Suresh [6], to rationalise the anomalies in the near-threshold behaviour of fatigue damage in steels and aluminium alloys. In 1975, Purushothaman and Tien [7] observed that discrete points of contact between fracture surface asperities can also cause the premature closure of the crack. In addition to what is Corresponding author. Tel.: +44-114-225-4257; fax: +44-114-2255390. E-mail address: [email protected] (C.A. Rodopoulos).
0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.017
now designated as roughness induced crack closure, Suresh [8] introduced the phenomena of crack deflection (crack path tortuosity) and crack branching as the main controlling factors. In general, roughness induced crack closure is believed to be promoted by [9]: (a) near-threshold and plane strain conditions; (b) low stress intensity factor range, DK (DSIF), levels and R ratios; (c) coarse grains and shearable coherent precipitates and (d) crack deflections caused by grain boundaries. The mechanism of plasticity induced crack closure has been the target of extensive argumentation especially in explaining R ratio effects. Vasudevan et al. [10] suggested that there is no significant contribution to crack closure by the residual plastic stretch at the crack wake especially at the crack tip and under plane strain conditions. Similarly, Garret and Knot [11] argued that plasticity induced crack closure has little effect under plane strain conditions. More recently, Zhang et al. [12] found that near-threshold and stage I crack propagation in 2024-T351 tested in high vacuum is characterised by shear-band decohesion and thus it is unaffected by crack closure. With the possible dismissal of the effect of crack closure and its direct relationship
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with the stress ratio, an old chapter in fatigue research comes once again into the spotlight. The fatigue damage map (FDM) [13–15] was developed by Rodopoulos and de los Rios as a tool that allows the examination of the fatigue tendencies of engineering materials using readily available properties. In this first part, an attempt to incorporate the effects of stress ratio in the concept of the FDM is presented. Throughout the work, crack closure effects are not implicitly considered allowing the simplicity of the FDM to be maintained. However, for validation purposes, comparison to published data is also presented. 2. Threshold/crack arrest and stress ratio The long crack threshold stress intensity factor range, DKth, generally decreases with increasing R ratio and typically, but not always, tends to a constant value at high positive R ratios (R> 0:7) [16–20]. Discrepancies from the above ‘‘general rule’’ have been shown to depend on the yield stress, ry, of the material [21]. In [16,20], an extensive selection of DKth results on several steels, demonstrated that the tendency to a constant value behaviour is promoted by steels with low values of ry. In contrast, high strength steels showed an almost linear decay up to stress ratios of R¼ 0:9 [9]. In [22], Schmidt and Paris found that in the case of 2124-T3 the threshold saturation mechanism changes with the loading frequency. In their work, they reported saturation tendencies for R> 0:5 and R> 0:8 at frequencies of 300 and 580 Hz, respectively. Suresh et al. [23], demonstrated that in the case of 7075 aluminium alloy, DKth remains nearly constant along the whole range of positive R ratios when the material is tested in vacuum. In terms of a Kitagawa–Takahashi approach [9], the problem becomes more complicated since coarser grain materials showed higher long crack slope while at the same time exhibited lower fatigue limits compared to finer grain materials [9,16]. Therefore, it is clear that the ability of the material to arrest crack propagation is covered by complexity and contradiction. Thus, most threshold models contained experimental and semi-empirical parameters. A selection of threshold models can be found in [24]. In [25–28], it was proposed that the propagation/ arrest of a crack takes place when the dislocation pileup, representing the crack tip plastic zone, is unable to overcome the constraint provided by a dominant microstructural barrier, i.e. grain boundary, twin boundary, pearlite zone, etc. Based on the principles of the Navarro–Rios model [26–28], crack arrest was modelled by, mi rFL r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 rarrest ¼ ð1Þ m1 2a=D where rarrest is the crack arrest or threshold stress, a is
the crack length, rFL is the fatigue limit of the material (N> 107 cycles), D is the average transverse grain size, r1 is the crack closure stress and mi is the grain orientation factor. It should be noted that mi increases monotonically with crack length from a value of 1 until mi reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). The hypothesis m1 ¼ 1 is rationalised by the fact that, in many cases, crack nucleation takes place in grains that are most favourably oriented in relation to the applied stress that the resolved shear stress achieves maximum value. The threshold stress defined by Eq. (1), identifies two controlling parameters: (a) the strength of the grain boundary which is part of rFL (fatigue limit) and (b) the grain orientation, mi. Both parameters reflect the influence of microstructure on crack arrest. The first, by relating the strength of the boundary to the threshold stress for crack propagation and the latter, by incorporating the effect of the increasing number of grains traversed by the crack front as the crack grows. In 2001, Kujawski [29] proposed that the effect of the R-ratio on the threshold stress intensity factor range is given by, DKth ¼ ð1 RÞa DKth;0
for R 0
ð2aÞ
and DKth ¼ ð1 RÞ DKth;0
for R 0
ð2bÞ
where DKth,0 is the threshold DSIF corresponding to R ¼ 0 and a is a fitting parameter ranging between 0 and 1 (a value of a¼ 0:5 was suggested for aluminium alloys and martensitic steels). Navarro et al. [30] suggested that the plain fatigue limit is related to the threshold SIF through, rffiffiffiffiffiffiffiffi D ð3Þ DKth ¼ DrFL p 2 Using a similar argument for the fatigue limit the effect of R-ratio on the fatigue limit is given by, DrFL ¼ ð1 RÞa DrFLðR¼0Þ
for R 0
ð4aÞ
and DrFL ¼ ð1 RÞ DrFLðR¼0Þ
for R 0
ð4bÞ
Using Eqs. (1) and (4a,b), the effect of R-ratio on the crack arrest can be written as, Drarrest ¼
a mi ð1 RÞ DrFLðR¼0Þ r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 m1 2a=D
for R 0 ð5aÞ
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and Drarrest ¼
mi ð1 RÞDrFLðR¼0Þ r1 pffiffiffiffiffiffiffiffiffiffiffi þ r1 m1 2a=D
for R 0 ð5bÞ
Fig. 1 shows a plotting of Eq. (5a,b) for an aluminium alloy. It should be noted that the approach is accurate for opening mode growing cracks. For mode II or mode III growing cracks, the plain fatigue limit could be independent of R. To allow the use of experimental results to verify the accuracy of the method to predict the effect of R ratio on Drarrest, Eq. (5a,b) is transformed into the threshold value of the SIF, rffiffiffiffiffiffiffiffi mi D a ð1 RÞ DrFLðR¼0Þ p DKth ¼ Y for R 0 ð6aÞ 2 m1 rffiffiffiffiffiffiffiffi mi D DKth ¼ Y ð1 RÞDrFLðR¼0Þ p for R 0 ð6bÞ 2 m1 where Y is the crack correction factor. Assuming that the concept of similitude holds, Fig. 2 compares the threshold predictions by Eq. (6a,b) with experimental results published in the literature [32–37]. The results show the potential of Eq. (6) to predict experimental data.
3. Stage I/short crack growth Short fatigue cracks, also known as Stage I cracks, have been known to propagate at rates significantly higher than that of long cracks under the same nominal DK conditions [38–41]. This is because the size of
Fig. 2. Comparison between experimental threshold SIF values and Eq. (6a,b) as a function of R for 7075-T6. Two different K calibration (Y) values were used to account for two different crack shapes. The mechanical and physical properties used for the calculations are: DrFLðR¼0Þ ¼ 300 MPa [32], m1 ¼ 1 [14], D ¼ 25 108 lm [32,33], a ¼ 0:5 [29] and r1 ¼ 0. Scatter in the experimental data is attributed to different load shedding techniques and different quoted grain sizes. The independence of Y to crack length was only used for simplicity.
crack tip plastic zone size is significantly larger than that predicted by theories of continuum mechanics [42]. In [13,14,25,43,44], it was suggested that the Stage I crack growth terminates when the crack tip stress field is able to initiate plasticity on two successive grains without further growth of the crack. This assumption is rationalised considering that when the crack is small a single family of slip planes can accommodate the crack tip plasticity. At longer crack lengths, crack tip plasticity is more intense and can only be accommodated by multiple slip. The above was validated by Yoder et al. [43] and Curtis et al. [44], who examined the transition from Stage I to Stage II for a number of materials. According to the above rationale the transition co-ordinates (applied stress/crack length) from Stage I to Stage II (long) crack growth, is mathematically expressed as,
rI!II
" # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 c 4D r r1 þ r1 ¼ Y p y a þ 2D
ð7Þ
where rI!II is the applied stress level at the transition. In the case of a closure free cracks, Eq. (7) is written as, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 2 c 4D r rI!II ¼ ð8Þ Y p y a þ 2D Fig. 1. The effect of stress ratio on crack arrest stress range for 2024-T351 according to Eq. (5a,b). The mechanical and physical properties used for the calculations are: DrFLðR¼0Þ ¼ 200 MPa [25], mi ¼ 1 þ 0:35 lnð2a=DÞ [14], D¼ 52 lm [31], a¼ 0:5 [29] and r1 ¼ 0.
where rcy is the cyclic yield stress. A typical Stage I to Stage II transition for several aluminium alloys is shown in Fig. 3.
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Fig. 3. Stage I to Stage II transition curves for three aluminium alloys. In all three cases a closure free crack and Y ¼ 1 was considered. The mechanical properties are: (a) 2024-T351— rcy ¼ 450 MPa, D¼ 52 lm; (b) 7150-T651—rcy ¼ 490 MPa [15], D ¼ 58 lm and (c) 7075-T6—rcy ¼ 517 MPa, D¼ 108 lm [14].
Fig. 4. Determination of the dtip for the transition from Stage I to Stage II for a edge crack in a 2024-T351 sheet. The mechanical properties of the material are as in Fig. 3. The crack correction factor was taken from [48].
In several works [26,27,45,46], the propagation rate of a short fatigue crack was modelled through the elasto-plastic crack tip opening displacement, dtip,
In 1980, Allen [46] suggested that the effect of the ratio-R on the crack growth rate is given by,
da 1þR b ¼A ð12Þ d ; R0 dN 1 R tip
da ¼ Adbtip dN
ð9Þ
where the fitting parameters A and b depend on the material, stress-state, R-ratio, etc. The parameter dtip is given by Dugdale [47], " !# 8a c pr r ln sec dtip ¼ ð10Þ pG y 2rcy where G is the shear modulus. It is worth noting that the original Young’s modulus is replaced by the shear modulus to account for the shear mode loading of the short crack. By substituting Eq. (8) into Eq. (10), the dtip for the transition from Stage I to Stage II is given by, " !# 8a c prI!II r ln sec dtip ¼ ð11Þ pG y 2rcy In Fig. 4, the results from Eq. (11) are presented for 2024-T351 considering an edge crack in an infinite width sheet under uniaxial stress. The results indicate that for rI!II 5 rcy , dtip tends to a constant dctip value. The above comes as a verification to the fact that a constant size crack tip plastic zone corresponds to a constant SIF and consequently to a constant dtip. For rI!II ryc , dtip is found to tend to an infinite value.
where A, b are fitting parameters with b being independent of R and A for R¼ 0 (dtip ¼ Ddtip ). Hence, the expression 1 þ R=1 R represents a mathematical regulator of dtip that allows the use of the same set of fitting parameters autonomously to R. Considering that dctip depends implicitly on the material and the interaction between stress/crack/ component, it is rational to assume that the crack growth rate at the transition from Stage I to Stage II is also a constant given that the parameter A is a function of R. Modelling of the above rationale in terms of the R adjustment provided by Eq. (12) gives,
1R c R ð13Þ d Ddtip ¼ 1 þ R tip c where DdR tip is the adaptation of dtip in terms of R. Eq. (13), using Eqs. (8) and (10) can be also expressed in terms of the transition stress DrI!II , rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1 1R 2 c 4D R ry ð14Þ DrI!II ¼ Y 1þR p a þ 2D
In Fig. 5, the effect of R on the transition from Stage I to Stage II is illustrated. Extension of the Stage I growth area is provided by simultaneously plotting Eqs. (6a,b) and (14). The effect of R on the Stage I growth area is depicted in Fig. 6. In terms of the model discussed earlier, Fig. 6 shows that high R values will decrease Stage I growth. The tendency of the Stage I growth area to contract and
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Fig. 5. The effect of R on the transition from Stage I to Stage II for 2024-T351. The mechanical properties used for the calculations are as before. The crack correction factor was unity.
Fig. 7. Comparison between experimental and predicted transition from Stage I to Stage II growth rates for 2024-T351. The parameter Y was set to unity.
finally to disappear at R> 0:4 agrees with experimental results published by Pang and Song [49]. It is worth noting that Stage I growth is also promoted by Y < 1. In this section, the parameter Y was intentionally set to unity to avoid possible inaccuracies from the use of a continuum mechanics concepts to describe the stress field ahead of a short crack [25]. Eqs. (12) and (13) can be used to predict the crack growth rate at the transition from Stage I to Stage II growth. For the predictions, shown in Fig. 7, the Paris type fitting parameters for 2024-T351 at R¼ 0, C¼ 3:37 1011 and m¼ 3:32 pffiffiffiffi [49] (m/cycle growth, K in MPa m) were transformed into A¼ 5000 and b ¼ 1:88 assuming that the elastic fracture mechanics relationship between DK and dtip holds.
by the Paris law [25]. Termination of this steady-state growth will take place when the strain field ahead of the crack tip is sufficiently large as to achieve rupture strains which will degrade the flow resistance of the material [25]. As a result the crack will enter an unsteady state of growth where the dominance of Dr will diminish in favour of rmax (rupture stress control) [9]. In [25], the termination of the Stage II and the beginning of Stage III growth was modelled through its resemblance to static failure (cleavage, intergranular separation and fibrous failure),
4. Stage II—long cracks Stage II or long crack growth will commence when at a given applied stress, the crack attains a length beyond the limit curve of Stage I (Fig. 6). In Stage II, crack growth can be described with reasonable success
Fig. 6.
dtip ¼ lnð1 þ ef Þ q
ð15Þ
where ef is the monotonic elongation to failure and q is the length of crack tip plasticity. Considering that during Stage II growth, crack tip plasticity will continue to increase in multiples of D, the parameter q can be written as, q¼x
D and 2
x>4
The effect of R on Stage I growth for 2024-T351 for: (a) R¼ 0 and (b) R¼ 0:4. The parameter Y was unity.
ð16Þ
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where x is the number of half grains constituting crack tip plasticity. Using the continuum mechanics solution at the transition from Stage II to Stage III growth, Eq. (15) can be written as, h i ð8a=pEÞrcy ln sec prII!III =2rcy xðD=2Þ ¼ lnð1 þ ef Þ; x > 4
ð17aÞ
where DrII!III ¼ rII!III ð1 RÞ
ð17bÞ
In Eq. (17a), the use of the elastic modulus E conforms with the opening mode characterising Stage II crack growth. However, stress range values can be obtained through Eq. (17b). A typical solution of Eq. (17a) using an iteration method is shown in Fig. 8. Solution of Eq. (17a,b) in terms of dtip can also be used to extract growth rate information (see Fig. 9).
Fig. 8. Transition from Stage II to Stage III growth for 2024-T351 at different positive R ratios. The mechanical properties E¼ 72:4 GPa and ef ¼ 18% were used in the calculation [25].
or considering the effect of the R-ratio as, Drtoughness ¼
5. Catastrophic failure The end of Stage III growth is followed by catastrophic failure. For conditions of cyclic loading catastrophic failure is defined either in terms of general yielding (also known as crack instability) or in terms of fracture toughness. In [13], it was suggested that general yielding takes place when the crack tip plasticity has reach ‘‘infinite’’ size. In the same work, it was suggested that, DrFL r1 rinstability ¼ mi pffiffiffiffiffiffiffiffiffiffiffi þ r1 þ rcy 2a=D
ð1 RÞKIc pffiffiffiffiffiffi Y pa
ð23Þ
6. Discussion This work represents an attempt to incorporate the effect of the stress ratio R into the FDM. The effect of R on crack arrest was modelled through Kujawski’s parameter a. The approach provides the necessary flexibility to allow for changes caused by different
ð18Þ
or, DrFL rinstability ¼ mi pffiffiffiffiffiffiffiffiffiffiffi þ rcy 2a=D
ð19Þ
for negligible crack closure. It should be noted that the crack closure stress will have a negligible effect at this stage. In [25], it was suggested that in engineering components where crack sizes are larger, Eqs. (19) and (20) can be written as, rinstability ¼ rcy
ð20Þ
or considering the effect of the R-ratio as, Drinstability ¼ ð1 RÞrcy
ð21Þ
Catastrophic failure in terms of fracture toughness and for plain strain conditions is, rtoughness ¼
KIc pffiffiffiffiffiffi Y pa
ð22Þ
Fig. 9. Comparison between experimental and predicted transition from Stage II to Stage III growth rates for 7075-T6. The experimental results represent the first point that the growth rate deviates from its linear behaviour. The mechanical properties used for the calculations, apart from those used before, were E¼ 70 GPa, ef ¼ 12% [33]. The growth rate was calculated using Eq. (9). The parameters A ¼ 3500 and b¼ 1:55 were extracted from the growth parameters C ¼ 3:18 108 and m¼ 1:24 taken from [33].
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Fig. 10. The FDM for 2024-T351 for: (a) R¼ 0 and (b) R¼ 0:5. In addition to the mechanical properties used before, a value of 38 MPa m1/2 was used for the plane strain fracture toughness [25].
materials, etc. Stress ratio effects on the transition from Stage I to Stage II growth was modelled by making use of Allen’s model. The analysis is possible due to the fact that transition conditions are characterised by a constant crack tip plastic zone and consequently by a constant value of CTOD, dtip. The above allows the use of the matching crack propagation concept for several values of R. The effect of R on the transition from Stage II to Stage III growth was found to be independent of R. Such result is reasonable considering that Stage III represents a stage of material degradation which solely depends on the strain conditions ahead of the crack tip. Similarly, catastrophic failure, both in terms of general yielding or fracture failure, was modelled directly through the relationship between rmax and R. The outcome of the above modelling is shown in Fig. 10.
7. Conclusions From the results presented in the previous sections and the FDMs shown in Fig. 10 the following conclusions can be drawn: (a) The effect of R on the three conditions that dominate fatigue crack growth, namely crack arrest, Stage I to Stage II growth and Stage II to Stage III growth can be accurately predicted by the FDM. (b) Positive R-ratios are expected to reduce Stage I growth. (c) The transition from Stage II to Stage III growth is related to the effect of R on maximum stress. Differences in the area denoted as Stage III growth are attribute to the effect of R-ratio on the conditions of fracture toughness and general yielding. (d) The size of the Stage I area can be used to extract information about the accuracy of the Paris Law to predict fatigue life.
Acknowledgements The authors are indebt to the Royal Academy of Engineering, The Engineering and Physical Science Research Council, Airbus UK, Hyundai Motor Company, INASCO Hellas and The British Council for the financial and otherwise support provided throughout this project.
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