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Fatigue crack growth of aluminium alloy 7075-T651 under proportional and non-proportional mixed mode I and II loads
Engineering Fracture Mechanics 174 (2017) 155–167
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Fatigue crack growth of aluminium alloy 7075-T651 under proportional and non-proportional mixed mode I and II loads Xiaobo Yu a,⇑, Ling Li b, Gwénaëlle Proust b a b
Aerospace Division, Defence Science and Technology Group, 506 Lorimer Street, Fishermans Bend, VIC 3207, Australia School of Civil Engineering, Faculty of Engineering & IT, The University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 24 October 2016 Received in revised form 6 January 2017 Accepted 7 January 2017 Available online 9 January 2017 Keywords: Fatigue crack growth Non-proportional Mixed mode Aluminium alloy 7075-T651
a b s t r a c t This study aims to experimentally investigate fatigue growth behaviour in AA7075-T651 under proportional and non-proportional mixed mode I and II loads. Fatigue tests were performed under cyclic tension and torsion using a thin-walled tubular specimen with a key-hole style crack starter. After the generation of a single-side mode I pre-crack, varied forms of mixed mode loads were applied, which in most cases led to coplanar growth for a short distance followed by a long and stable crack path deviation. It was found that under most of the non-proportional mixed mode load cases, the direction of the deviated crack path could not be reasonably predicted using the commonly accepted maximum tangential stress criterion. Meanwhile, in some cases, the crack path directions could be approximately predicted using the maximum shear stress criterion. It was also confirmed for the first time that a long, stable and non-coplanar shear mode fatigue crack growth could be produced in AA7075-T651 under non-proportional mixed mode I and II loads. The implications of transient co-planar as well as stable shear-mode fatigue crack growth were discussed, revealing the needs of future investigations. Ó2016 Commonwealth of Australia Crown Copyright Ó 2017 Published by Elsevier Ltd. All rights reserved.
1. Introduction Fatigue of aircraft structures is traditionally managed based on the assumption of uniaxial loads. This is a simplified and acceptable approach for most fatigue critical locations where the local stress field is predominantly uniaxial due to loading path restrictions. Nevertheless, as evidenced in a recent durability analysis of a new type of surveillance aircraft, severe multiaxial loads may exist at some of the primary structural locations. Multiaxial fatigue analysis, including fatigue crack growth (FCG) analysis under proportional and non-proportional mixed mode I and II loads, is needed for these locations. At present, experimental FCG investigations are predominantly undertaken for mode I or proportional mixed mode loads. Only a few studies [1–8] focus on non-proportional mixed mode loads. Limited results on A106-93 mild steel [3] reveal that a long and stable shear mode FCG – which is significantly different from the commonly understood open mode FCG – could be produced by non-proportional loads. To illustrate this difference, Fig. 1 compares the FCG behaviour under two load cases [3,9]: (i) proportional load case TD1, under which, the FCG deviated into a direction that can be approximately predicted by the maximum tangential stress (MTS) criterion [10], and striations were found along the deviated crack path; (ii) nonproportional load case TD2, under which the FCG deviated to a direction that is about 60° off the MTS prediction, and dimples ⇑ Corresponding author. E-mail addresses: [email protected] (X. Yu), [email protected] (L. Li), [email protected] (G. Proust). URL: http://www.dst.defence.gov.au (X. Yu). http://dx.doi.org/10.1016/j.engfracmech.2017.01.008 0013-7944/Crown Copyright Ó 2017 Published by Elsevier Ltd. All rights reserved.
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Nomenclature a half crack length dh diameter of the circular crack start notch hole D diameter of the tubular specimen Dmid diameter of the tubular specimen at mid wall KI and KII mode I and II stress intensity factor P axial load applied on the tubular specimen Q torsional load applied on the tubular specimen R radius of the tube r radius of a circular path around the crack tip t wall thickness of the tubular specimen YI and YII geometrical function used in calculating KI and KII YnI and YnII geometrical function, accounting for the effects of crack start notch on KI and KII YbI and YbII geometrical function, accounting for the effects of tubular curvature on KI and KII b dimensionless curvature parameter rmid tensile stress at the mid wall of an un-notched and un-cracked tubular specimen rhh tangential stress along a circular path around crack tip sðDÞ shear stress within an un-notched and un-cracked tubular specimen smid shear stress at the mid wall of an un-notched and un-cracked tubular specimen srh shear stress along a circular path around the crack tip l Poisson’s ratio DK IIth;shear threshold value of mode II stress intensity range, for shear mode crack growth from a mode I pre-crack under cyclic mode II and steady mode I load
were found along the deviated crack path. For this particular case, the deviation angle is approximately predictable by the maximum shear stress (MSS) criterion [11]. Here in this work, the term ‘‘proportional” refers to the case where the increments of two loading axes are proportional, which has the same meaning as synchronous. The shear mode FCG as shown in Fig. 1(e) is more than 8 mm long along the deviated path. Thus, it is not a short-distance propagation in shear mode occurring in an early stage of crack growth as discussed in Ref. [7]. Nor is it a transient behaviour occurring at a short-distance coplanar growth following the change of load mixture as reported in Ref. [12]. The persistence of the shear mode FCG as presented in Fig. 1(e) indicates the complexity of underlying mechanism under non-proportional loads. It means that a stage II fatigue crack can propagate in modes other than open mode, and in such a case, the crack path is absolutely unpredictable by the commonly accepted MTS criterion. Nevertheless, the above finding is so far only based on experimental results for A106-93 mild steel. It is not clear whether this finding also applies to other materials, such as aluminium alloy (AA) 7075-T651, which is typically used for aircraft structures, as no similar tests have been reported for the latter in the open literature. Hence, the purpose of this study is to investigate FCG behaviour in AA7075-T651 under non-proportional mixed mode I and II loads. In particular, it aims to clarify whether the long and stable shear mode FCG also occurs. For the sake of comparison, the specimen design and majority of the mixed-mode load cases used in the present study are the same as reported in Ref. [3].
2. Specimen and test procedures 2.1. Specimen The fatigue tests of the present study were performed on an Instron 8500 servo-hydraulic tension-torsion biaxial fatigue test machine with a load capacity of ±250 kN and displacement range of ±50 mm for the axial axis, and ±2000 Nm and ±45 degree for the torsional axis. The test machine is located in the Structural Laboratory, School of Civil Engineering at the University of Sydney. This is the same test machine as being used in Ref. [3], except that the electronic test controllers have been upgraded, and the load cells were recalibrated prior to the present study. The design of the specimen is the same as previously used in Ref. [3]. A schematic illustration of the specimen is given in Fig. 2, which includes a notched thin-walled tube and two solid plugs. Both the tube and plugs were machined from 2.500 AA7075-T651 solid bars, and the tight tolerance between the plugs and the tube was filled with Araldite epoxy adhesive. The plugs are reusable and their main function is to provide support to the tube under hydraulic grippers. The crack starter notch has a keyhole shape, and was introduced via drilling and electronic discharge machining.
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Fig. 1. FCG from a mode I pre-crack, under proportional (a–c), and non-proportional (d–f), mixed-mode loads [3,9].
Fig. 2. (a) Illustration of a tubular specimen for FCG test under cyclic tension and torsion. (b) The crack start notch with a single-sided mode I pre-crack. (Unit: mm; modified after [3]).
The tensile load P and torsional load Q were applied via hydraulic grippers, with the positive directions shown in Fig. 2a. The applied loads can also be described using reference stresses in an un-notched and un-cracked tube, at a transverse crosssection perpendicular to the tube axis. While away from the two ends of the specimen, the tensile stress r at the transverse cross-section is uniformly distributed across the wall thickness; meanwhile the shear stress, s, varies linearly from the inside to the outside wall, following the rule as depicted in Eq. (1):
sðDÞ ¼ smid D=Dmid
ð1Þ
where smid and Dmid (=56 mm) are the shear stress and diameter at the middle section of the tube wall, and sðDÞ and D are the shear stress and diameter at the point of interest. In the present study, the tube has a large mid-diameter-to-thickness ratio, Dmid =t ¼ 56=1:5 ¼ 37:3, which helps to suppress the variation of shear stress across the wall thickness. As calculated using
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Fig. 3. Load cases applied during the phase 2 of the fatigue tests. The small gaps between parallel loading and unloading strokes are added for the sake of illustration. (Unit: MPa).
Eq. (1), the shear stresses at the inside and outside walls are only 2.7% different from smid . In this paper, the stresses at the mid-wall, rmid , smid are taken as the reference stresses to describe the loads. The relationships between the applied loads and reference stresses are given in Eq. (2):
(
P ¼ prmid Dmid t
Q ¼ p2 smid ðD2mid þ t 2 Þt
ð2Þ
2.2. Test phases and load cases Ten specimens were tested. For each specimen, the fatigue test was performed in two phases. During phase 1 of the test, only tension was applied which cycled at 10 Hz between rmid = 25 MPa and 125 MPa, or equivalently between P = 6.6 kN and 33 kN. A single-sided circumferential crack, here noted as pre-crack and illustrated in Fig. 2b, was produced. The phase 1 procedure was completed when the total crack length, 2a, reached between 7.5 mm and 9 mm approximately. At this length, the effect of the crack start notch can be ignored, and the crack can be treated as a 2-D crack as it is long enough comparing to the wall thickness of the specimen. Meanwhile, the crack is not too long so that the effect of the tubular curvature can be ignored, and in combination with the applied loads, large scale yield is avoided. During phase 2 of the test, both tension and torsion were applied according to the smid versus rmid curves illustrated in Fig. 3. It is noted that the small gaps shown in the figure between parallel loading and unloading paths are added for the sake of clarity. For instance, the load case LC1 should be interpreted as smid cycles between 0 and 50 MPa while rmid remains
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(a) Command and feedback time histories for axial load
(b) Command and feedback time histories for torsional load Fig. 4. Illustration of phase and magnitude compensation onto command signals to achieve the designated tension and torsion time history.
constant at 125 MPa. For a fully opened circumferential crack, rmid leads to mode I stress intensity and smid leads to mode II stress intensity at the crack tip. The corresponding mode I and mode II stress intensity factors, noted as K I and K II respectively, can be calculated using Eq. (3),
(
pffiffiffiffiffiffi K I ¼ Y I rmid pa pffiffiffiffiffiffi K II ¼ Y II smid pa
ð3Þ
where Y I and Y II are geometry functions for mode I and mode II stress intensities, respectively. As derived in the Appendix, the geometry functions are evaluated by compounding two parameters, one for the notch effect, and the other for the curvature effect. It is shown that, both Y I and Y II can be approximated as 1.0 for the current notched tubular specimen while the pre-crack length 2a ranges between 7.5 mm and 10.5 mm, which is satisfied in all tests reported in this paper. Also as indicated in the Appendix, the possible errors induced by such approximation are less than 1%. Fig. 3 presents 12 load cases. At the pre-crack tip, load cases LC1 to LC3 nominally lead to constant K I plus cyclic K II conditions. Here the term ‘‘nominally” is added to indicate that only a two-dimensional planar crack is considered and also the resulting K I and K II do not include those induced by crack surface interference as reported in Refs. [13–16]. In the case of LC4, the variations of rmid and smid are in phase, thus it leads to a nominally proportional mixed mode I and II condition at the precrack tip. For all other load cases, the variations of rmid and smid lead to a non-proportional mode I and II condition at the precrack tip.
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Fig. 5. Illustration of achieved shear stress (smid ) versus tensile stress (rmid ) relationship for two typical load cases LC4 and LC5.
At phase 2 of the test, each specimen was subjected to a single load case, except for one specimen where three load cases (LC8, 9 and 10) were applied in a sequential order. It is noted that all the load cases, except LC13, have been previously tested for A106-93 mild steel [3], and that the load cases LC4 and LC5 depicted in Fig. 3 are identical to TD1 and TD2 in Fig. 1. 2.3. Quality control of applied load cases The time-histories of tension and torsion feedback were monitored, and when needed the load frequency was reduced to around 2 Hz, to ensure that the target relationships between P and Q, or equivalently between smid and, rmid , were achieved. As shown in Fig. 4, the command and feedback values of the loads are often different; hence compensations to the command values of the loads were introduced to obtain the desired feedback. The compensations include mean and amplitude of the cyclic load for each axis, and the relative phase between the axial and torsional axis. In this study, the compensations were manually tuned until the designed smid versus rmid relationship was reached with acceptable tolerance. Fig. 5 illustrates the achieved shear stress (smid ) versus tensile stress (rmid ) relationship for two typical load cases: LC4 and LC5, showing that the difference between the achieved and designed stresses is less than 5 MPa at key turning points. 2.4. Test condition and crack measurement All the fatigue tests were performed at room temperature in the laboratory environment. Fatigue crack growth was monitored using a portable optical microscope and still images were taken at given intervals of cycles. These still images were later used to derive the FCG rate. 3. Experimental results and analysis 3.1. Fatigue crack growth from circumferential pre-crack The fatigue cracks at the outer surfaces of all ten tubular specimens are shown in Fig. 6. These are the still images taken at the end of the phase 2 of the test and the cracks were kept open under applied loads in the test machine. The main interest here is the FCG behaviour during the phase 2 of the test, i.e. the FCG from the pre-crack tip. As summarized in Table 1, co-planar FCG prior to the change of FCG direction was observed under all load cases except LC3 where no FCG was observed from the pre-crack tip. In the cases of LC4 to LC13, where both rmid and smid were cyclically applied, the co-planar FCG was
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Fig. 6. Fatigue crack path at the outer surface of tubular specimens at the end of tests.
observed to be transient, typically less than 1.0 mm long. In the cases of steady rmid plus cyclic smid loads, the co-planar growth is relatively longer, especially in the case of LC2, where the coplanar growth extends for 4.5 mm and is followed by a +10.1° path which can also be approximated as co-planar growth. To further reveal the indications of the test results, a number of comparisons are made as follows. LC1, LC2 and LC3. These load cases tested here are similar to those achievable using a fixed-grip-shear specimens [17], where a static tensile (mode I) load was superimposed onto a cyclic shear (mode II) load. For the load case LC3 of the present study, no growth was observed from the pre-crack tip, which can be attributed to the severe crack closure and shear attenuation as the applied static tension is only one-fifth of the maximum rmid applied during the pre-cracking phase. Under the pffiffiffiffiffi pffiffiffiffiffi load cases LC1 and LC2, the nominal DK II at the pre-crack tip was 5.7 MPa m and 8.3 MPa m, respectively; and the FCG
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Table 1 Coplanar fatigue crack growth after the change from mode I to mixed mode I + II loads from a mode I pre-crack and a comparison between measured and predicted crack branching crack angles. Phase 2 load case
Pre-crack length, 2a (mm)
Coplanar FCG before branching
Crack branching direction angle (h)
Length (mm)
Cycle
Measured
MTS criterion*
MSS criterion*
70.5° 70.5° – 40.2° (37.9°)** 40.2° (40.2°)** 25.3° 61.3° 40.2° 40.2° 40.2°
0° 0° – +28.5° (+30.8°)** +28.5° (+28.5°)** +44.0° +8.1° +28.5° +28.5° +28.5°
LC1 LC2 LC3 LC4
8.3 7.8 8.3 8.1
1.3 4.5 no growth 0.54
3200 1850 n/a 650
66.1° +10.1° n/a 27.3°
LC5 LC6 LC8 LC11 LC12 LC13
8.0 8.4 7.5 7.7 9.1 8.3
0.4 0 0.6 0.4 0.8 0.3
750 411 4928 1832 568 1500
+41° +8° 52.5° +18.4° 43.2° 19.9°
* Unless otherwise stated, predictions were given based on the designed smid versus stresses used in the MTS and MSS criteria were evaluated every 0.1°. ** Based on the achieved smid versus rmid relationship as shown in Fig. 5.
rmid relationship as shown in Fig. 3. Also, the tangential and shear
Fig. 7. Fatigue crack growth observed at the outer surface of tubular specimens. Both the crack length and cycle number are zeroed at the crack path turning point.
turned into 66.1° (close to the MTS prediction given in the next section) and +10.1° (close to the MSS prediction given in the next section) directions, respectively. It means that, under the cyclic smid plus static rmid load conditions shear mode FCG can be produced under a higher DK II for AA7075-T651. Similar observations were reported in Ref. [17] for AA7075-T6. However, under the same load case as LC2, no shear mode FCG was observed for A106-93 mild steel [3]. This result reconfirms an earlier review finding [3] that under cyclic mode II loads, a shear mode FCG is more likely in aluminium alloys than in steels. LC4, LC5, LC11, LC12 and LC13. Across these five load cases, the maximum and minimum values of rmid and smid are all identical. Nevertheless, the FCG directions as shown in Fig. 6 and tabulated in Table 1 are all very different, varying from 43.2° for LC12 to +41° for LC5. Fig. 7 compares FCG rates along different paths. Here the crack length versus cycle relationships were derived from the still images taken at given intervals of cycles at the outer surface of the specimen. Both the cycle counting and crack length were zeroed when the direction of FCG started to deviate from the circumferential direction. The crack length was measured along the deviated crack path. At the 1 mm crack length (after the change of FCG direction), the averaged FCG rate differs by 5 times between the slowest and fastest cases. For longer cracks, the averaged FCG rates are much closer, except for LC13 which was significantly slowed after the crack length (along the deviated path) reached 1.5 mm. LC4, LC5 The comparison between LC4 and LC5 is here further elaborated. Under LC4 (proportional rmid versus smid variation), the crack path turns into a 27.3° direction. Meanwhile, under LC5 (non-proportional rmid versus smid variation), the crack path turns into a +41° direction. These two directions are more than 60° apart, showing the dependency of crack path,
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Fig. 8. Fatigue crack surface images captured using deep focusing optical microscope examination technology. (Courtesy: Drew Donnelly of DST group for capturing the images).
Fig. 9. Crack angle predicted by the MTS, MSED and MSS criteria.
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and hence the FCG mechanism, on the relative phases between the rmid and smid variations. As will be shown in the next section, the +41° direction is close to the prediction given by the MSS criterion. A similar dependency was previously observed for A106-93 mild steel [3] as shown in Fig. 1. It means that for both the mild steel [3] and AA7075-T651 (this study), a long and stable shear mode FCG can be produced by applying non-proportional mixed mode loads. Fig. 8 compares the fatigue crack surfaces between LC4 and LC5. The images were taken using deep focusing optical microscopic examination technology. Prior to the change of FCG direction, the fatigue crack surfaces appear similar between LC4 and LC5, and also similar between the pre-cracking and co-planar growth. After the change of FCG direction, LC5 produced a much rougher zig-zagged fracture surface than LC4, indicting a possible different mechanism of FCG at this magnification level. LC11 and LC12. For these two load cases, the smid - rmid curves are the same except that one follows a clockwise direction, and the other follows an anti-clockwise direction. The crack paths as measured on the specimens are +18.4° and 43.2°, respectively, again more than 60° apart. It is noted that under most loading cases, a second crack also emanated from the far edge of the circular hole. However, the second crack only emerged after more than 4000 cycles of the applied mixed mode load. Therefore, their effect on crack path deviation could be ignored in the above analysis.
Fig. 10. Derivation of maximum tangential and shear stress range for LC4 and LC5.
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3.2. Prediction of fatigue crack growth direction under mixed mode I and II loads A wide range of criteria has been proposed to predict crack growth direction under mixed mode I and II load conditions. These include: maximum tangential stress (MTS) criterion [10], maximum principal stress criterion [18], maximum tangential strain criterion [19,20], maximum principal strain criterion [21], minimum strain energy density (MSED) criterion [22,23], minimum dilatational strain energy density criterion [24], maximum energy release rate criterion [25,26], local symmetry criterion [27], maximum crack growth rate criterion [28], and maximum shear stress (MSS) criterion [11]. Most of the criteria were initially proposed for brittle fracture and latter extended to fatigue crack growth. The predictions given by these criteria are not exactly the same, but differences arising from the varied criteria are not significant except for the MSS criterion. A comparison between the MTS, MSED and MSS criteria is given in Fig. 9. In this section, the MTS and MSS criteria are further compared with the experimental results. The mathematical expressions of the two criteria are given in Eqs. (4) and (5),
MTS criterion :
@ Drhh ¼0 @h
and
@ 2 Drhh
MSS criterion :
@ Dsrh ¼0 @h
and
@ 2 Dsrh
@h2
@h2
< 0; along C
< 0; along C
ð4Þ
ð5Þ
where C is a circular path around the crack tip, rhh and srh are tangential and shear stress on C calculated using Eq. (6) which is simplified from Ref. [29] assuming K-dominance.
8 9 > < rhh > =
KI h ffi cos rrh ¼ pffiffiffiffiffiffiffiffi > > 2 2pr : ;
(
cos2
h 2
sin 2h cos 2h
)
8 9 h 2 h K II < 3sin 2 cos 2 = þ pffiffiffiffiffiffiffiffiffi 2 2pr : cos 2h 1 3 sin 2h ;
ð6Þ
where K I and K II are calculated using Eq. (3), and r is the radius of the circular path C. It is noted that the predicted FCG direction is not affected by the absolute value of r. Hence normalised stresses are used in Eq. (4) and (5). Fig. 10 plots nomalised tangential and shear stress curves, showing that Drhh and Dsrh are calculated from enveloping maximum and minimum curves. The three curves labelled A, B and C correspond to the turning points indicated on the smid rmid curves. While calculating Drhh and Dsrh , the effects of cracking closure, shear attenuation and mixed mode coupling [13–16] were ignored. Nevertheless, Drhh only includes the above-zero range. Table 1 also compares the measured FCG directions with those predicted by MTS and MSS criteria. Unless otherwise stated, the predicted crack branching angles were calculated using the as-designed rmid versus smid relationship illustrated in Fig. 3. For load cases LC4 and LC5, the as-achieved rmid versus smid curves (as plotted in Fig. 5) were also used to predict crack branching angles, showing little differences to the results. Hence in the paper, the predictions based on the as-designed rmid versus smid curves are used for further comparison. The comparison of crack branching angles given in Table 1 show that only four of the measured crack directions, or two of the six non-proportional load cases, can be approximately (with less than ±15° error) predicted using the MTS criterion. Meanwhile, three of the directions can be approximately predicted by the MSS criterion. It is interesting to note that in the case of LC4 (proportional mixed mode loads), the experimentally observed crack path is close to the prediction given by the MTS prediction. Meanwhile, in the case of LC5 (non-proportional mixed mode loads), the experimentally observed crack path is close to the prediction by the MSS criterion. A similar correlation was previously observed for A106-93 mild steel [3]. 4. Discussion and future study 4.1. Co-planar FCG In previous studies [11,12,30], co-planar FCG was used as an indicator of shear-mode crack growth when cyclic mode II load (with superimposed steady mode I load) was applied onto a mode I pre-crack. Nevertheless, it is worthwhile to differentiate the co-planar FCG into two types: (i) stable and (ii) transient. The stable co-planar FCG can only be produced when cyclic mode II and steady mode I loads are applied onto a mode I pre-crack and when DK II;eff is larger than a threshold value for shear mode crack growth, DK IIth;shear . Ref. [30] tested five aluminium alloys and demonstrated that DK IIth;shear is inversely-proportional to the square root of average grain size, and that pffiffiffiffiffi DK IIth;shear for AA7075 ranges between 8 and 9 MPa m. In the present study, stable co-planar growth was observed under pffiffiffiffiffi LC2, where DK II = 8.3 MPa m for a pre-crack length of 2a = 8 mm. In comparison, mode I deviation was observed under pffiffiffiffiffi LC1, where DK II = 5.7 MPa m for a pre-crack length of 2a = 8.3 mm. Hence, the present study re-confirmed that, for AA7075-T651, stable co-planar shear mode FCG can be produced when DK II is larger than a threshold value. This threshold pffiffiffiffiffi value is estimated between approximately 5.7 and 8.3 MPa m.
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The transient co-planar FCG, on the other hand, occurs in a wide range of mixed mode load conditions but only extends for a short distance typically less than 1 mm. The transient co-planar FCG can also be viewed as delayed crack path deviation. In the case of a one-off transition from mode I to mixed mode loads, the delayed crack path deviation may not be a significant phenomenon. However, in the case of a random mixed mode spectrum, the FCG behaviour might be affected due to continued delayed response to the change of mixed mode load conditions. More investigations are needed to further understand this transient behaviour and its effects, including testing using in-service multiaxial loading spectra. 4.2. Stable shear-mode FCG Prior to the present study, stable shear mode FCG in aluminium alloys was only reported in the form of co-planar growth from a mode I pre-crack. The present results indicate that shear mode FCG can also be produced and is stable along a deviated crack path. Here, the shear mode FCG is only a loosely defined term, referring to a FCG of which the overall trend of the crack path is approximately (with less than ±15° error) predictable using the MSS criterion. As revealed in the present study for AA7075-T651, the non-proportionality between the mode I and II cyclic loads is a key pre-requisition to produce stable shear mode FCG. This finding is in line with the previous study on A106-93 mild steel [3]. It explains why only mode I crack deviation was observed in many of the previous laboratory tests where the applied mixed mode I + II cycles were proportional. As also revealed in the present study, not all FCG paths can be predicted by either the MTS or MSS criterion. Hence, it is not sufficient to categorize the FCG into only two types: open-mode and shear-mode. A more general description and criterion are needed. The criterion should be able to take into account not only the shape but also the clockwise/anti-clockwise direction of the loading curve which describes the mode I versus mode II relationship in the form of an x versus y plot. More research is needed along this line. 5. Conclusion This study clarifies that for AA7075-T651, the FCG under non-proportional mixed mode I and II loads is significantly different from the open mode FCG commonly expected under mode I or proportional mixed mode loads. Similarly to previous results for mild steel, a long and stable shear mode FCG can be produced in AA7075-T651 under non-proportional loads. The commonly accepted MTS criterion does not apply under most non-proportional load cases. The implications of transient co-planar FCG as well as stable shear-mode FCG are discussed, indicating the needs of further investigations over two areas: (i) effects of delayed response to change of load mixture, in particular under in-service multiaxial loading spectra; and (ii) enhanced description of FCG, other than simply referred as open or shear mode FCG, and associated criterion for prediction of crack path. Acknowledgement This paper is extended from an ICMFF11 conference paper, which was also published on Frattura ed Integrità Strutturale, 38 (2016) 148-154. The facture surface examination of the specimens tested under LC4 and LC5 was performed by Drew Donnelly of DST group. The authors would like to thank also Kenneth Tam for the data analysis of the crack growth with the number of cycles. Appendix A. Mode I and II stress intensity factors of a circumferential crack emanating from a hole in a thin-walled circular tube For a through-thickness circumferential crack in a tube as shown in Fig. 2, the mode I stress intensity factor, K I , can be expressed in the form of Eq. (A-1):
K I ¼ Y I rmid
pffiffiffiffiffiffi pa
ðA-1Þ
where a is the half crack length as illustrated in Fig. 2, rmid is the reference tensile stress as defined in Eq. (2), and Y I is a geometrical function accounting for the effects of crack starter notch and the tubular curvature. For the present thinwalled tube (Dmid =t ¼ 37:3) and a crack length within the range of 2a ¼ 7:5 mm to 10.5 mm, i.e. 2a=t = 5–7, the through thickness variation of K I is ignored. Hence, the function Y I is calculated by compounding two parameters, YnI for notch effect, and Y bI for curvature effect. Bowie [31] and Newman [32] showed that the effects of notch is less than 1% when the crack length is over two times longer than the notch size. In the present study, the notch hole diameter dh = 2.0 mm, hence the ratio between the crack length and the notch size, 2a=dh , is larger than 3.25 meaning that Yn = 1.0 can be assumed. For a circumference crack which is small in relative to the tube diameter, Y bI can be calculated using Eq. (A-2), which was simplified from Eqs. (15b and 15d) of Ref. [33] by assigning a = 90°:
Y bI ¼ 1 þ
p 8
b2
ðA-2Þ
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where b is a dimensionless curvature parameter defined as,
b2 ¼
a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8Rt 12ð1 l2 Þ
ðA-3Þ
where l is Poisson’s ratio, a is half crack length, and R and t are the radius and the wall thickness of the tube, respectively. With R ¼ 0:5Dmid ¼ 28 mm, t ¼ 1:5 mm, 2a ¼ 10:5 mm for longest crack length under consideration, and l = 0.33 for AA7075-T751, the curvature effect is very small with Y bI ¼ 1:0003. Hence in this study, Y I ¼ Y nI Y bI is taken as 1.0 while calculating K I at the pre-crack tip for 2a ranged between 7.5 mm and 10.5 mm. Similarly, the mode II stress intensity factor, K II , can be expressed in the form of Eq. (A-4):
K II ¼ Y II smid
pffiffiffiffiffiffi pa
ðA-4Þ
where a is the half crack length and smid is the reference shear stress. The geometrical function YII is compounded from two parameters, YnII for notch effect, and Y bII for curvature effect. For a certain crack length, YnII can be transformed from the data given in Table 5 of Ref. [34]. The longest crack listed in the table has 2a ¼ 1:5dh when re-expressed using the variables defined in the present paper. For this crack, the notch factor YnII is found to be 0.98 transformed from the tabulated FII value of 1.200. For a longer crack (2a P 3:25dh ), which is of interest to the present study, YnII is assumed to reach its asymptotic value of 1.0. In analogy to Y bI , the curvature factor Y bII can be calculated using Eq. (A-5), which is simplified from Eq. (15c & e) of Ref. [33] by assigning a ¼ 90°:
Y bII ¼ 1 þ
p 8
b2
ðA-5Þ
A comparison of Eqs. (A-2) and (A-5) indicates that Y bII = Y bI therefore the curvature effort to K II is also negligible. Hence in this study, Y II is also taken as 1.0 while calculating K II at the pre-crack tip for 2a ranged between 7.5 mm and 10.5 mm. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
Nayebhashemi H, Taslim ME. Effects of the transient mode-II on the steady-state crack-growth in mode-I. Eng Fract Mech 1987;26(6):789–807. Wong SL et al. A branch criterion for shallow angled rolling contact fatigue cracks in rails. Wear 1996;191(1–2):45–53. Jan. Yu X. Fatigue crack propagation under mixed mode loads, PhD thesis. Australia: The University of Sydney; 1999. Doquet V, Pommier S. Fatigue crack growth under non-proportional mixed-mode loading in ferritic-pearlitic steel. Fatigue Fract Eng Mater Struct 2004;27(11):1051–60. Highsmith Jr S. Crack path determination for non-proportional mixed-mode fatigue PhD thesis. USA: Georgia Institute of Technology; 2009. Doquet V et al. Influence of the loading path on fatigue crack growth under mixed-mode loading. Int J Fract 2009;159(2):219–32. Oct. Shamsaei N, Fatemi A. Small fatigue crack growth under multiaxial stresses. Int J Fatigue 2014;58:126–35. Zerres P, Vormwald M. Review of fatigue crack growth under non-proportional mixed-mode loading. Int J Fatigue 2014;58:75–83. Yu X. On the fatigue crack growth analysis of spliced aircraft wing panels under sequential axial and shear loads. Eng Fract Mech 2014;123:116–25. Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. J Basic Eng 1963;85(4):519–25. Otsuka A, Mori K, Miyata T. The condition of fatigue crack growth in mixed mode condition. Eng Fract Mech 1975;7(3):429–32. IN15-IN18,433-439. Smith MC. Some aspects of mode II fatigue crack growth PhD thesis. UK: University of Cambridge; 1984. Yu X, Abel A. Crack surface interference under cyclic mode I and steady mode II loading: Part I: experimental study. Eng Fract Mech 2000;66 (6):503–18. 8/1/. Yu X, Abel A. Crack surface interference under cyclic mode I and steady mode II loading: Part II: Modelling analysis. Eng Fract Mech 2000;66 (6):519–35. 8/1/. Yu X, Abel A. Modelling of crack surface interference under cyclic shear loads. Fatigue Fract Eng Mater Struct 1999;22(3):205–13. Yu X, Abel A. Mixed-mode crack surface interference under cyclic shear loads. Fatigue Fract Eng Mater Struct 2000;23(2):151–8. Otsuka A et al. ICF5, Cannes (France) 1981; 2013. Maiti SK, Smith RA. Theoretical and experimental studies on the extension of cracks subjected to concentrated loading near their faces to compare the criteria for mixed-mode brittle-fracture. J Mech Phys Solids 1983;31(5). 389-+. Chang KJ. On the maximum strain criterion - a new approach to the angled crack problem. Eng Fract Mech 1981;14(1):107–24. Chambers AC, Hyde TH, Webster JJ. Mixed mode fatigue crack growth at 550-degree-C under plane stress conditions in jethete M152. Eng Fract Mech 1991;39(3):603–19. Mageed AMA, Pandey RK. Mixed-mode crack-growth under static and cyclic loading in A1-alloy sheets. Eng Fract Mech 1991;40(2). 371-&. Sih GC. Strain energy density factor applied to mixed mode crack problems. Int J Fract 1974;10(3):305–21. Sih GC, Barthelemy BM. Mixed-mode fatigue crack-growth predictions. Eng Fract Mech 1980;13(3):439–51. Theocaris P, Andrianopoulos N. A modified strain-energy density criterion applied to crack propagation. J Appl Mech 1982;49(1):81–6. Palaniswamy K, Knauss WG. Propagation of a crack under general, in-plane tension. Int J Fract Mech 1972;8(1):114–7. Hussain M, Pu S, Underwood J. Fracture analysis: proceedings of the 1973 national symposium on fracture mechanics, Part II, ASTM International; 1974. Gol’dstein RV, Salganik RL. Brittle-fracture of solids with arbitrary cracks. Int J Fract 1974;10(4):507–23. Hourlier F, Pineau A. Advances in fracture research, fracture 81 (ICF5), in: D. Francois, e. a. (ed.); 1981. Anderson TL, Anderson T. Fracture mechanics: fundamentals and applications. CRC Press; 2005. Plank R, Kuhn G. Fatigue crack propagation under non-proportional mixed mode loading. Eng Fract Mech 1999;62(2–3):203–29. Bowie O. Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular hole. J Math Phys 1956;35(1):60–71. Newman Jr, J. An improved method of collocation for the stress analysis of cracked plates with various shaped boundaries; 1971. Lakshminarayana H, Murthy M. On stresses around an arbitrarily oriented crack in shell. Int J Fract 1976;12(4). Isida M, Chen DH, Nisitani H. Plane problems of an arbitrary array of cracks emanating from the edge of an elliptical hole. Eng Fract Mech 1985;21 (5):983–95.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Fatigue crack growth of aluminium alloy 7075-T651 under proportional and non-proportional mixed mode I and II loads Xiaobo Yu a,⇑, Ling Li b, Gwénaëlle Proust b a b
Aerospace Division, Defence Science and Technology Group, 506 Lorimer Street, Fishermans Bend, VIC 3207, Australia School of Civil Engineering, Faculty of Engineering & IT, The University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 24 October 2016 Received in revised form 6 January 2017 Accepted 7 January 2017 Available online 9 January 2017 Keywords: Fatigue crack growth Non-proportional Mixed mode Aluminium alloy 7075-T651
a b s t r a c t This study aims to experimentally investigate fatigue growth behaviour in AA7075-T651 under proportional and non-proportional mixed mode I and II loads. Fatigue tests were performed under cyclic tension and torsion using a thin-walled tubular specimen with a key-hole style crack starter. After the generation of a single-side mode I pre-crack, varied forms of mixed mode loads were applied, which in most cases led to coplanar growth for a short distance followed by a long and stable crack path deviation. It was found that under most of the non-proportional mixed mode load cases, the direction of the deviated crack path could not be reasonably predicted using the commonly accepted maximum tangential stress criterion. Meanwhile, in some cases, the crack path directions could be approximately predicted using the maximum shear stress criterion. It was also confirmed for the first time that a long, stable and non-coplanar shear mode fatigue crack growth could be produced in AA7075-T651 under non-proportional mixed mode I and II loads. The implications of transient co-planar as well as stable shear-mode fatigue crack growth were discussed, revealing the needs of future investigations. Ó2016 Commonwealth of Australia Crown Copyright Ó 2017 Published by Elsevier Ltd. All rights reserved.
1. Introduction Fatigue of aircraft structures is traditionally managed based on the assumption of uniaxial loads. This is a simplified and acceptable approach for most fatigue critical locations where the local stress field is predominantly uniaxial due to loading path restrictions. Nevertheless, as evidenced in a recent durability analysis of a new type of surveillance aircraft, severe multiaxial loads may exist at some of the primary structural locations. Multiaxial fatigue analysis, including fatigue crack growth (FCG) analysis under proportional and non-proportional mixed mode I and II loads, is needed for these locations. At present, experimental FCG investigations are predominantly undertaken for mode I or proportional mixed mode loads. Only a few studies [1–8] focus on non-proportional mixed mode loads. Limited results on A106-93 mild steel [3] reveal that a long and stable shear mode FCG – which is significantly different from the commonly understood open mode FCG – could be produced by non-proportional loads. To illustrate this difference, Fig. 1 compares the FCG behaviour under two load cases [3,9]: (i) proportional load case TD1, under which, the FCG deviated into a direction that can be approximately predicted by the maximum tangential stress (MTS) criterion [10], and striations were found along the deviated crack path; (ii) nonproportional load case TD2, under which the FCG deviated to a direction that is about 60° off the MTS prediction, and dimples ⇑ Corresponding author. E-mail addresses: [email protected] (X. Yu), [email protected] (L. Li), [email protected] (G. Proust). URL: http://www.dst.defence.gov.au (X. Yu). http://dx.doi.org/10.1016/j.engfracmech.2017.01.008 0013-7944/Crown Copyright Ó 2017 Published by Elsevier Ltd. All rights reserved.
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Nomenclature a half crack length dh diameter of the circular crack start notch hole D diameter of the tubular specimen Dmid diameter of the tubular specimen at mid wall KI and KII mode I and II stress intensity factor P axial load applied on the tubular specimen Q torsional load applied on the tubular specimen R radius of the tube r radius of a circular path around the crack tip t wall thickness of the tubular specimen YI and YII geometrical function used in calculating KI and KII YnI and YnII geometrical function, accounting for the effects of crack start notch on KI and KII YbI and YbII geometrical function, accounting for the effects of tubular curvature on KI and KII b dimensionless curvature parameter rmid tensile stress at the mid wall of an un-notched and un-cracked tubular specimen rhh tangential stress along a circular path around crack tip sðDÞ shear stress within an un-notched and un-cracked tubular specimen smid shear stress at the mid wall of an un-notched and un-cracked tubular specimen srh shear stress along a circular path around the crack tip l Poisson’s ratio DK IIth;shear threshold value of mode II stress intensity range, for shear mode crack growth from a mode I pre-crack under cyclic mode II and steady mode I load
were found along the deviated crack path. For this particular case, the deviation angle is approximately predictable by the maximum shear stress (MSS) criterion [11]. Here in this work, the term ‘‘proportional” refers to the case where the increments of two loading axes are proportional, which has the same meaning as synchronous. The shear mode FCG as shown in Fig. 1(e) is more than 8 mm long along the deviated path. Thus, it is not a short-distance propagation in shear mode occurring in an early stage of crack growth as discussed in Ref. [7]. Nor is it a transient behaviour occurring at a short-distance coplanar growth following the change of load mixture as reported in Ref. [12]. The persistence of the shear mode FCG as presented in Fig. 1(e) indicates the complexity of underlying mechanism under non-proportional loads. It means that a stage II fatigue crack can propagate in modes other than open mode, and in such a case, the crack path is absolutely unpredictable by the commonly accepted MTS criterion. Nevertheless, the above finding is so far only based on experimental results for A106-93 mild steel. It is not clear whether this finding also applies to other materials, such as aluminium alloy (AA) 7075-T651, which is typically used for aircraft structures, as no similar tests have been reported for the latter in the open literature. Hence, the purpose of this study is to investigate FCG behaviour in AA7075-T651 under non-proportional mixed mode I and II loads. In particular, it aims to clarify whether the long and stable shear mode FCG also occurs. For the sake of comparison, the specimen design and majority of the mixed-mode load cases used in the present study are the same as reported in Ref. [3].
2. Specimen and test procedures 2.1. Specimen The fatigue tests of the present study were performed on an Instron 8500 servo-hydraulic tension-torsion biaxial fatigue test machine with a load capacity of ±250 kN and displacement range of ±50 mm for the axial axis, and ±2000 Nm and ±45 degree for the torsional axis. The test machine is located in the Structural Laboratory, School of Civil Engineering at the University of Sydney. This is the same test machine as being used in Ref. [3], except that the electronic test controllers have been upgraded, and the load cells were recalibrated prior to the present study. The design of the specimen is the same as previously used in Ref. [3]. A schematic illustration of the specimen is given in Fig. 2, which includes a notched thin-walled tube and two solid plugs. Both the tube and plugs were machined from 2.500 AA7075-T651 solid bars, and the tight tolerance between the plugs and the tube was filled with Araldite epoxy adhesive. The plugs are reusable and their main function is to provide support to the tube under hydraulic grippers. The crack starter notch has a keyhole shape, and was introduced via drilling and electronic discharge machining.
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157
Fig. 1. FCG from a mode I pre-crack, under proportional (a–c), and non-proportional (d–f), mixed-mode loads [3,9].
Fig. 2. (a) Illustration of a tubular specimen for FCG test under cyclic tension and torsion. (b) The crack start notch with a single-sided mode I pre-crack. (Unit: mm; modified after [3]).
The tensile load P and torsional load Q were applied via hydraulic grippers, with the positive directions shown in Fig. 2a. The applied loads can also be described using reference stresses in an un-notched and un-cracked tube, at a transverse crosssection perpendicular to the tube axis. While away from the two ends of the specimen, the tensile stress r at the transverse cross-section is uniformly distributed across the wall thickness; meanwhile the shear stress, s, varies linearly from the inside to the outside wall, following the rule as depicted in Eq. (1):
sðDÞ ¼ smid D=Dmid
ð1Þ
where smid and Dmid (=56 mm) are the shear stress and diameter at the middle section of the tube wall, and sðDÞ and D are the shear stress and diameter at the point of interest. In the present study, the tube has a large mid-diameter-to-thickness ratio, Dmid =t ¼ 56=1:5 ¼ 37:3, which helps to suppress the variation of shear stress across the wall thickness. As calculated using
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Fig. 3. Load cases applied during the phase 2 of the fatigue tests. The small gaps between parallel loading and unloading strokes are added for the sake of illustration. (Unit: MPa).
Eq. (1), the shear stresses at the inside and outside walls are only 2.7% different from smid . In this paper, the stresses at the mid-wall, rmid , smid are taken as the reference stresses to describe the loads. The relationships between the applied loads and reference stresses are given in Eq. (2):
(
P ¼ prmid Dmid t
Q ¼ p2 smid ðD2mid þ t 2 Þt
ð2Þ
2.2. Test phases and load cases Ten specimens were tested. For each specimen, the fatigue test was performed in two phases. During phase 1 of the test, only tension was applied which cycled at 10 Hz between rmid = 25 MPa and 125 MPa, or equivalently between P = 6.6 kN and 33 kN. A single-sided circumferential crack, here noted as pre-crack and illustrated in Fig. 2b, was produced. The phase 1 procedure was completed when the total crack length, 2a, reached between 7.5 mm and 9 mm approximately. At this length, the effect of the crack start notch can be ignored, and the crack can be treated as a 2-D crack as it is long enough comparing to the wall thickness of the specimen. Meanwhile, the crack is not too long so that the effect of the tubular curvature can be ignored, and in combination with the applied loads, large scale yield is avoided. During phase 2 of the test, both tension and torsion were applied according to the smid versus rmid curves illustrated in Fig. 3. It is noted that the small gaps shown in the figure between parallel loading and unloading paths are added for the sake of clarity. For instance, the load case LC1 should be interpreted as smid cycles between 0 and 50 MPa while rmid remains
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(a) Command and feedback time histories for axial load
(b) Command and feedback time histories for torsional load Fig. 4. Illustration of phase and magnitude compensation onto command signals to achieve the designated tension and torsion time history.
constant at 125 MPa. For a fully opened circumferential crack, rmid leads to mode I stress intensity and smid leads to mode II stress intensity at the crack tip. The corresponding mode I and mode II stress intensity factors, noted as K I and K II respectively, can be calculated using Eq. (3),
(
pffiffiffiffiffiffi K I ¼ Y I rmid pa pffiffiffiffiffiffi K II ¼ Y II smid pa
ð3Þ
where Y I and Y II are geometry functions for mode I and mode II stress intensities, respectively. As derived in the Appendix, the geometry functions are evaluated by compounding two parameters, one for the notch effect, and the other for the curvature effect. It is shown that, both Y I and Y II can be approximated as 1.0 for the current notched tubular specimen while the pre-crack length 2a ranges between 7.5 mm and 10.5 mm, which is satisfied in all tests reported in this paper. Also as indicated in the Appendix, the possible errors induced by such approximation are less than 1%. Fig. 3 presents 12 load cases. At the pre-crack tip, load cases LC1 to LC3 nominally lead to constant K I plus cyclic K II conditions. Here the term ‘‘nominally” is added to indicate that only a two-dimensional planar crack is considered and also the resulting K I and K II do not include those induced by crack surface interference as reported in Refs. [13–16]. In the case of LC4, the variations of rmid and smid are in phase, thus it leads to a nominally proportional mixed mode I and II condition at the precrack tip. For all other load cases, the variations of rmid and smid lead to a non-proportional mode I and II condition at the precrack tip.
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Fig. 5. Illustration of achieved shear stress (smid ) versus tensile stress (rmid ) relationship for two typical load cases LC4 and LC5.
At phase 2 of the test, each specimen was subjected to a single load case, except for one specimen where three load cases (LC8, 9 and 10) were applied in a sequential order. It is noted that all the load cases, except LC13, have been previously tested for A106-93 mild steel [3], and that the load cases LC4 and LC5 depicted in Fig. 3 are identical to TD1 and TD2 in Fig. 1. 2.3. Quality control of applied load cases The time-histories of tension and torsion feedback were monitored, and when needed the load frequency was reduced to around 2 Hz, to ensure that the target relationships between P and Q, or equivalently between smid and, rmid , were achieved. As shown in Fig. 4, the command and feedback values of the loads are often different; hence compensations to the command values of the loads were introduced to obtain the desired feedback. The compensations include mean and amplitude of the cyclic load for each axis, and the relative phase between the axial and torsional axis. In this study, the compensations were manually tuned until the designed smid versus rmid relationship was reached with acceptable tolerance. Fig. 5 illustrates the achieved shear stress (smid ) versus tensile stress (rmid ) relationship for two typical load cases: LC4 and LC5, showing that the difference between the achieved and designed stresses is less than 5 MPa at key turning points. 2.4. Test condition and crack measurement All the fatigue tests were performed at room temperature in the laboratory environment. Fatigue crack growth was monitored using a portable optical microscope and still images were taken at given intervals of cycles. These still images were later used to derive the FCG rate. 3. Experimental results and analysis 3.1. Fatigue crack growth from circumferential pre-crack The fatigue cracks at the outer surfaces of all ten tubular specimens are shown in Fig. 6. These are the still images taken at the end of the phase 2 of the test and the cracks were kept open under applied loads in the test machine. The main interest here is the FCG behaviour during the phase 2 of the test, i.e. the FCG from the pre-crack tip. As summarized in Table 1, co-planar FCG prior to the change of FCG direction was observed under all load cases except LC3 where no FCG was observed from the pre-crack tip. In the cases of LC4 to LC13, where both rmid and smid were cyclically applied, the co-planar FCG was
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Fig. 6. Fatigue crack path at the outer surface of tubular specimens at the end of tests.
observed to be transient, typically less than 1.0 mm long. In the cases of steady rmid plus cyclic smid loads, the co-planar growth is relatively longer, especially in the case of LC2, where the coplanar growth extends for 4.5 mm and is followed by a +10.1° path which can also be approximated as co-planar growth. To further reveal the indications of the test results, a number of comparisons are made as follows. LC1, LC2 and LC3. These load cases tested here are similar to those achievable using a fixed-grip-shear specimens [17], where a static tensile (mode I) load was superimposed onto a cyclic shear (mode II) load. For the load case LC3 of the present study, no growth was observed from the pre-crack tip, which can be attributed to the severe crack closure and shear attenuation as the applied static tension is only one-fifth of the maximum rmid applied during the pre-cracking phase. Under the pffiffiffiffiffi pffiffiffiffiffi load cases LC1 and LC2, the nominal DK II at the pre-crack tip was 5.7 MPa m and 8.3 MPa m, respectively; and the FCG
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Table 1 Coplanar fatigue crack growth after the change from mode I to mixed mode I + II loads from a mode I pre-crack and a comparison between measured and predicted crack branching crack angles. Phase 2 load case
Pre-crack length, 2a (mm)
Coplanar FCG before branching
Crack branching direction angle (h)
Length (mm)
Cycle
Measured
MTS criterion*
MSS criterion*
70.5° 70.5° – 40.2° (37.9°)** 40.2° (40.2°)** 25.3° 61.3° 40.2° 40.2° 40.2°
0° 0° – +28.5° (+30.8°)** +28.5° (+28.5°)** +44.0° +8.1° +28.5° +28.5° +28.5°
LC1 LC2 LC3 LC4
8.3 7.8 8.3 8.1
1.3 4.5 no growth 0.54
3200 1850 n/a 650
66.1° +10.1° n/a 27.3°
LC5 LC6 LC8 LC11 LC12 LC13
8.0 8.4 7.5 7.7 9.1 8.3
0.4 0 0.6 0.4 0.8 0.3
750 411 4928 1832 568 1500
+41° +8° 52.5° +18.4° 43.2° 19.9°
* Unless otherwise stated, predictions were given based on the designed smid versus stresses used in the MTS and MSS criteria were evaluated every 0.1°. ** Based on the achieved smid versus rmid relationship as shown in Fig. 5.
rmid relationship as shown in Fig. 3. Also, the tangential and shear
Fig. 7. Fatigue crack growth observed at the outer surface of tubular specimens. Both the crack length and cycle number are zeroed at the crack path turning point.
turned into 66.1° (close to the MTS prediction given in the next section) and +10.1° (close to the MSS prediction given in the next section) directions, respectively. It means that, under the cyclic smid plus static rmid load conditions shear mode FCG can be produced under a higher DK II for AA7075-T651. Similar observations were reported in Ref. [17] for AA7075-T6. However, under the same load case as LC2, no shear mode FCG was observed for A106-93 mild steel [3]. This result reconfirms an earlier review finding [3] that under cyclic mode II loads, a shear mode FCG is more likely in aluminium alloys than in steels. LC4, LC5, LC11, LC12 and LC13. Across these five load cases, the maximum and minimum values of rmid and smid are all identical. Nevertheless, the FCG directions as shown in Fig. 6 and tabulated in Table 1 are all very different, varying from 43.2° for LC12 to +41° for LC5. Fig. 7 compares FCG rates along different paths. Here the crack length versus cycle relationships were derived from the still images taken at given intervals of cycles at the outer surface of the specimen. Both the cycle counting and crack length were zeroed when the direction of FCG started to deviate from the circumferential direction. The crack length was measured along the deviated crack path. At the 1 mm crack length (after the change of FCG direction), the averaged FCG rate differs by 5 times between the slowest and fastest cases. For longer cracks, the averaged FCG rates are much closer, except for LC13 which was significantly slowed after the crack length (along the deviated path) reached 1.5 mm. LC4, LC5 The comparison between LC4 and LC5 is here further elaborated. Under LC4 (proportional rmid versus smid variation), the crack path turns into a 27.3° direction. Meanwhile, under LC5 (non-proportional rmid versus smid variation), the crack path turns into a +41° direction. These two directions are more than 60° apart, showing the dependency of crack path,
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Fig. 8. Fatigue crack surface images captured using deep focusing optical microscope examination technology. (Courtesy: Drew Donnelly of DST group for capturing the images).
Fig. 9. Crack angle predicted by the MTS, MSED and MSS criteria.
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and hence the FCG mechanism, on the relative phases between the rmid and smid variations. As will be shown in the next section, the +41° direction is close to the prediction given by the MSS criterion. A similar dependency was previously observed for A106-93 mild steel [3] as shown in Fig. 1. It means that for both the mild steel [3] and AA7075-T651 (this study), a long and stable shear mode FCG can be produced by applying non-proportional mixed mode loads. Fig. 8 compares the fatigue crack surfaces between LC4 and LC5. The images were taken using deep focusing optical microscopic examination technology. Prior to the change of FCG direction, the fatigue crack surfaces appear similar between LC4 and LC5, and also similar between the pre-cracking and co-planar growth. After the change of FCG direction, LC5 produced a much rougher zig-zagged fracture surface than LC4, indicting a possible different mechanism of FCG at this magnification level. LC11 and LC12. For these two load cases, the smid - rmid curves are the same except that one follows a clockwise direction, and the other follows an anti-clockwise direction. The crack paths as measured on the specimens are +18.4° and 43.2°, respectively, again more than 60° apart. It is noted that under most loading cases, a second crack also emanated from the far edge of the circular hole. However, the second crack only emerged after more than 4000 cycles of the applied mixed mode load. Therefore, their effect on crack path deviation could be ignored in the above analysis.
Fig. 10. Derivation of maximum tangential and shear stress range for LC4 and LC5.
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3.2. Prediction of fatigue crack growth direction under mixed mode I and II loads A wide range of criteria has been proposed to predict crack growth direction under mixed mode I and II load conditions. These include: maximum tangential stress (MTS) criterion [10], maximum principal stress criterion [18], maximum tangential strain criterion [19,20], maximum principal strain criterion [21], minimum strain energy density (MSED) criterion [22,23], minimum dilatational strain energy density criterion [24], maximum energy release rate criterion [25,26], local symmetry criterion [27], maximum crack growth rate criterion [28], and maximum shear stress (MSS) criterion [11]. Most of the criteria were initially proposed for brittle fracture and latter extended to fatigue crack growth. The predictions given by these criteria are not exactly the same, but differences arising from the varied criteria are not significant except for the MSS criterion. A comparison between the MTS, MSED and MSS criteria is given in Fig. 9. In this section, the MTS and MSS criteria are further compared with the experimental results. The mathematical expressions of the two criteria are given in Eqs. (4) and (5),
MTS criterion :
@ Drhh ¼0 @h
and
@ 2 Drhh
MSS criterion :
@ Dsrh ¼0 @h
and
@ 2 Dsrh
@h2
@h2
< 0; along C
< 0; along C
ð4Þ
ð5Þ
where C is a circular path around the crack tip, rhh and srh are tangential and shear stress on C calculated using Eq. (6) which is simplified from Ref. [29] assuming K-dominance.
8 9 > < rhh > =
KI h ffi cos rrh ¼ pffiffiffiffiffiffiffiffi > > 2 2pr : ;
(
cos2
h 2
sin 2h cos 2h
)
8 9 h 2 h K II < 3sin 2 cos 2 = þ pffiffiffiffiffiffiffiffiffi 2 2pr : cos 2h 1 3 sin 2h ;
ð6Þ
where K I and K II are calculated using Eq. (3), and r is the radius of the circular path C. It is noted that the predicted FCG direction is not affected by the absolute value of r. Hence normalised stresses are used in Eq. (4) and (5). Fig. 10 plots nomalised tangential and shear stress curves, showing that Drhh and Dsrh are calculated from enveloping maximum and minimum curves. The three curves labelled A, B and C correspond to the turning points indicated on the smid rmid curves. While calculating Drhh and Dsrh , the effects of cracking closure, shear attenuation and mixed mode coupling [13–16] were ignored. Nevertheless, Drhh only includes the above-zero range. Table 1 also compares the measured FCG directions with those predicted by MTS and MSS criteria. Unless otherwise stated, the predicted crack branching angles were calculated using the as-designed rmid versus smid relationship illustrated in Fig. 3. For load cases LC4 and LC5, the as-achieved rmid versus smid curves (as plotted in Fig. 5) were also used to predict crack branching angles, showing little differences to the results. Hence in the paper, the predictions based on the as-designed rmid versus smid curves are used for further comparison. The comparison of crack branching angles given in Table 1 show that only four of the measured crack directions, or two of the six non-proportional load cases, can be approximately (with less than ±15° error) predicted using the MTS criterion. Meanwhile, three of the directions can be approximately predicted by the MSS criterion. It is interesting to note that in the case of LC4 (proportional mixed mode loads), the experimentally observed crack path is close to the prediction given by the MTS prediction. Meanwhile, in the case of LC5 (non-proportional mixed mode loads), the experimentally observed crack path is close to the prediction by the MSS criterion. A similar correlation was previously observed for A106-93 mild steel [3]. 4. Discussion and future study 4.1. Co-planar FCG In previous studies [11,12,30], co-planar FCG was used as an indicator of shear-mode crack growth when cyclic mode II load (with superimposed steady mode I load) was applied onto a mode I pre-crack. Nevertheless, it is worthwhile to differentiate the co-planar FCG into two types: (i) stable and (ii) transient. The stable co-planar FCG can only be produced when cyclic mode II and steady mode I loads are applied onto a mode I pre-crack and when DK II;eff is larger than a threshold value for shear mode crack growth, DK IIth;shear . Ref. [30] tested five aluminium alloys and demonstrated that DK IIth;shear is inversely-proportional to the square root of average grain size, and that pffiffiffiffiffi DK IIth;shear for AA7075 ranges between 8 and 9 MPa m. In the present study, stable co-planar growth was observed under pffiffiffiffiffi LC2, where DK II = 8.3 MPa m for a pre-crack length of 2a = 8 mm. In comparison, mode I deviation was observed under pffiffiffiffiffi LC1, where DK II = 5.7 MPa m for a pre-crack length of 2a = 8.3 mm. Hence, the present study re-confirmed that, for AA7075-T651, stable co-planar shear mode FCG can be produced when DK II is larger than a threshold value. This threshold pffiffiffiffiffi value is estimated between approximately 5.7 and 8.3 MPa m.
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The transient co-planar FCG, on the other hand, occurs in a wide range of mixed mode load conditions but only extends for a short distance typically less than 1 mm. The transient co-planar FCG can also be viewed as delayed crack path deviation. In the case of a one-off transition from mode I to mixed mode loads, the delayed crack path deviation may not be a significant phenomenon. However, in the case of a random mixed mode spectrum, the FCG behaviour might be affected due to continued delayed response to the change of mixed mode load conditions. More investigations are needed to further understand this transient behaviour and its effects, including testing using in-service multiaxial loading spectra. 4.2. Stable shear-mode FCG Prior to the present study, stable shear mode FCG in aluminium alloys was only reported in the form of co-planar growth from a mode I pre-crack. The present results indicate that shear mode FCG can also be produced and is stable along a deviated crack path. Here, the shear mode FCG is only a loosely defined term, referring to a FCG of which the overall trend of the crack path is approximately (with less than ±15° error) predictable using the MSS criterion. As revealed in the present study for AA7075-T651, the non-proportionality between the mode I and II cyclic loads is a key pre-requisition to produce stable shear mode FCG. This finding is in line with the previous study on A106-93 mild steel [3]. It explains why only mode I crack deviation was observed in many of the previous laboratory tests where the applied mixed mode I + II cycles were proportional. As also revealed in the present study, not all FCG paths can be predicted by either the MTS or MSS criterion. Hence, it is not sufficient to categorize the FCG into only two types: open-mode and shear-mode. A more general description and criterion are needed. The criterion should be able to take into account not only the shape but also the clockwise/anti-clockwise direction of the loading curve which describes the mode I versus mode II relationship in the form of an x versus y plot. More research is needed along this line. 5. Conclusion This study clarifies that for AA7075-T651, the FCG under non-proportional mixed mode I and II loads is significantly different from the open mode FCG commonly expected under mode I or proportional mixed mode loads. Similarly to previous results for mild steel, a long and stable shear mode FCG can be produced in AA7075-T651 under non-proportional loads. The commonly accepted MTS criterion does not apply under most non-proportional load cases. The implications of transient co-planar FCG as well as stable shear-mode FCG are discussed, indicating the needs of further investigations over two areas: (i) effects of delayed response to change of load mixture, in particular under in-service multiaxial loading spectra; and (ii) enhanced description of FCG, other than simply referred as open or shear mode FCG, and associated criterion for prediction of crack path. Acknowledgement This paper is extended from an ICMFF11 conference paper, which was also published on Frattura ed Integrità Strutturale, 38 (2016) 148-154. The facture surface examination of the specimens tested under LC4 and LC5 was performed by Drew Donnelly of DST group. The authors would like to thank also Kenneth Tam for the data analysis of the crack growth with the number of cycles. Appendix A. Mode I and II stress intensity factors of a circumferential crack emanating from a hole in a thin-walled circular tube For a through-thickness circumferential crack in a tube as shown in Fig. 2, the mode I stress intensity factor, K I , can be expressed in the form of Eq. (A-1):
K I ¼ Y I rmid
pffiffiffiffiffiffi pa
ðA-1Þ
where a is the half crack length as illustrated in Fig. 2, rmid is the reference tensile stress as defined in Eq. (2), and Y I is a geometrical function accounting for the effects of crack starter notch and the tubular curvature. For the present thinwalled tube (Dmid =t ¼ 37:3) and a crack length within the range of 2a ¼ 7:5 mm to 10.5 mm, i.e. 2a=t = 5–7, the through thickness variation of K I is ignored. Hence, the function Y I is calculated by compounding two parameters, YnI for notch effect, and Y bI for curvature effect. Bowie [31] and Newman [32] showed that the effects of notch is less than 1% when the crack length is over two times longer than the notch size. In the present study, the notch hole diameter dh = 2.0 mm, hence the ratio between the crack length and the notch size, 2a=dh , is larger than 3.25 meaning that Yn = 1.0 can be assumed. For a circumference crack which is small in relative to the tube diameter, Y bI can be calculated using Eq. (A-2), which was simplified from Eqs. (15b and 15d) of Ref. [33] by assigning a = 90°:
Y bI ¼ 1 þ
p 8
b2
ðA-2Þ
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where b is a dimensionless curvature parameter defined as,
b2 ¼
a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8Rt 12ð1 l2 Þ
ðA-3Þ
where l is Poisson’s ratio, a is half crack length, and R and t are the radius and the wall thickness of the tube, respectively. With R ¼ 0:5Dmid ¼ 28 mm, t ¼ 1:5 mm, 2a ¼ 10:5 mm for longest crack length under consideration, and l = 0.33 for AA7075-T751, the curvature effect is very small with Y bI ¼ 1:0003. Hence in this study, Y I ¼ Y nI Y bI is taken as 1.0 while calculating K I at the pre-crack tip for 2a ranged between 7.5 mm and 10.5 mm. Similarly, the mode II stress intensity factor, K II , can be expressed in the form of Eq. (A-4):
K II ¼ Y II smid
pffiffiffiffiffiffi pa
ðA-4Þ
where a is the half crack length and smid is the reference shear stress. The geometrical function YII is compounded from two parameters, YnII for notch effect, and Y bII for curvature effect. For a certain crack length, YnII can be transformed from the data given in Table 5 of Ref. [34]. The longest crack listed in the table has 2a ¼ 1:5dh when re-expressed using the variables defined in the present paper. For this crack, the notch factor YnII is found to be 0.98 transformed from the tabulated FII value of 1.200. For a longer crack (2a P 3:25dh ), which is of interest to the present study, YnII is assumed to reach its asymptotic value of 1.0. In analogy to Y bI , the curvature factor Y bII can be calculated using Eq. (A-5), which is simplified from Eq. (15c & e) of Ref. [33] by assigning a ¼ 90°:
Y bII ¼ 1 þ
p 8
b2
ðA-5Þ
A comparison of Eqs. (A-2) and (A-5) indicates that Y bII = Y bI therefore the curvature effort to K II is also negligible. Hence in this study, Y II is also taken as 1.0 while calculating K II at the pre-crack tip for 2a ranged between 7.5 mm and 10.5 mm. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
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