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Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure
International Journal of Fatigue 33 (2011) 1101–1110
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure R. Sunder a,⇑, B.V. Ilchenko b a b
BiSS Research, 41A, 1A Cross, AECS 2nd Stage, Bangalore 560 094, India Power Engineering Research Centre, Russian Academy of Sciences, P.B. 190, Kazan 420111, Russia
a r t i c l e
i n f o
Article history: Received 20 September 2010 Received in revised form 14 November 2010 Accepted 19 November 2010 Available online 30 November 2010 Keywords: Fatigue crack growth Biaxial spectrum loading
a b s t r a c t Pressurized transport aircraft fuselage panels see the combined action of gust loading and internal pressurization that causes a rather complex biaxial pseudo-random cyclic loading action. Stress biaxiality is also known to affect constant amplitude fatigue. Fatigue crack growth was studied using cruciform test coupons under constant amplitude and a modified TWIST load spectrum superposed with biaxial quasistatic load simulating internal cabin pressure. The experiments were performed on a specially designed four-actuator biaxial test system suited for high frequency in-phase and out-of-phase constant amplitude loading as well as biaxial pseudo-random spectrum loading. Observed crack growth rates were correlated with stress intensity values, corrected for biaxiality. Ó 2010 Published by Elsevier Ltd.
1. Introduction Fatigue damage accumulation is non-linear. This was obvious from early systematic experiments by Gassner [1], who established the inevitability of testing under loading conditions that closely simulate actual usage. The actual reasons for non-linear damage accumulation began to unravel many decades later, when tensile overloads were unexpectedly found to retard fatigue crack growth [2]. More research in the 1970s established that a number of load interaction mechanisms may be responsible for load sequence sensitivity of fatigue in metals [3]. These include crack tip residual stress, crack closure, crack-front incompatibility, crack-tip blunting/re-sharpening, etc. Over the years, a number of models have been proposed to estimate variable amplitude fatigue crack growth. Each model is invariably built around a single load interaction mechanism, usually, crack closure [4] or residual stress [5,6]. If a single-mechanism model does correctly estimate residual life, one would invariably have to attribute it to exaggerated dominance of one mechanism to approximate the actual result, which would not be much of an advancement over Gassner’s empirical approach of determining S–N curves under spectrum loading and computing associated damage sums. While research in uniaxial variable amplitude fatigue continues in an attempt to bridge this gap, problems of practical interest often relate to multi-axial loading conditions and call for the reexamination and extension of available understanding as stated above to the case of multi-axial loading. ⇑ Corresponding author. Tel.: +91 80 2341 4133; fax: +91 80 2341 0813. E-mail address: [email protected] (R. Sunder). 0142-1123/$ - see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2010.11.018
Available experimental data on fatigue crack growth under biaxial loading with cruciform specimens are limited and somewhat contradictory by comparison to that under uniaxial conditions. Liu and Dittmer observed that biaxiality does not affect crack growth rates under constant amplitude loading [7]. Yuuki et al. [8] also did not find any effect of biaxiality at low stresses, but found the effect noticeable at high stresses. Hopper and Miller found a consistent retarding effect of tensile transverse stress on growth rate [9]. Anderson and Garrett also observed a close relationship between crack growth rate and biaxial stress field [10]. Their measurements were macroscopic and cannot be used to judge whether stress field variations cause an instantaneous, or a gradual change in growth rate. Analytical studies also offer mixed conclusions. Ogura et al. point out that K is insensitive to biaxiality1 [11], and suggest that closure may be responsible for possible effects. However, McLung estimated that crack closure becomes sensitive to biaxiality only when stress exceeds 40% of yield stress [12], ruling out closure as a factor for most cases of practical interest. Adams found that tensile biaxiality affects plastic zone size [13]. This is supported by measurements by Liu and Dittmer [7], who as earlier noted did not however find any effect on constant amplitude crack growth rate. Liu and Dittmer however found a significant adverse effect of tensile stress biaxiality on crack growth under periodic overloads and under what appears to be a simple combat aircraft load spectrum. They attribute this to the reduced overload plastic zone size in the presence of tensile transverse load. Interestingly, this is the 1 One may note that standard expressions for K only represent the first term of series. The others are neglected as secondary, but they can be sensitive to biaxiality.
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Nomenclature
DK Bi Bi,max Bi,min COD DCPD FEM K Kmax Kmin
mode I stress intensity factor range, Kmax Kmin, p MPa m instantaneous ratio of transverse to axial force instantaneous ratio of transverse to axial force at maximum axial force instantaneous ratio of transverse to axial force at minimum axial force crack opening displacement d.c. potential drop technique finite-element method p mode I stress intensity factor, MPa m mode I stress intensity factor at Pax,max and Bi,max, p MPa m mode I stress intensity factor at Pax,min and Bi,min, p MPa m
opposite of what others have observed under constant amplitude loading [8–10]. Available experimental data underscore the need for much more experimental effort to cover cases of practical interest. Equally significantly, there has been a dearth of attention to the sensitivity of well-known load interaction mechanisms to biaxiality as well as the emergence of possible new mechanisms and factors associated with biaxial service loading that may not exist under uniaxial conditions. The goal of this study was to develop an easy to use digitally controlled biaxial test system for cruciform specimens and as a first exercise, investigate the growth of a circumferential crack in the upper panel of a pressurized fuselage cabin of a transport aircraft. The upper surface of the fuselage sees the same loads as the bottom surface of the wing. In case of pressurized cabins of transport aircraft, it will in addition, see the superposed action of internal pressure. Thus, the axial component of pressure-induced stress will add a tensile offset to the gust load spectrum. In addition, a hoop stress component that is twice the axial component will act in the transverse direction. This special case of biaxial loading involves pseudo-random flight spectrum loading with a tensile mean load offset, combined with a constant tensile transverse load. Under such loading, the fatigue crack tip will be exposed to tension–tension variable-amplitude loading, combined with a static transverse tensile load. The next section describes the experimental procedure including the test coupon used, the test system, gripping, loading and sense arrangements and load conditions. This is followed by a description and discussion of test results.
2. Experimental procedure The experiments were performed on cruciform test coupons using a digitally controlled four actuator 5 kN biaxial test system.
2.1. The test coupon The cruciform test coupon design is intended to serve ongoing research goals that include development of a standard coupon geometry to reproduce different levels of biaxiality, be suitable for materials in the 1–5 mm thickness interval, allow for useful center-crack growth data over at least 2a = 50 mm of crack growth, permit ease of reproducible gripping conditions and allow for future scale-up as test system force ratings are enhanced from the present 5 kN to 50 kN and over.
Km Kref Pax Pax,max Pax,min Pax,offs Ptv Pm PD R
p mode I stress intensity factor at Pm, MPa m mode I stress intensity factor at Pm + Pax,offs and Bi, p MPa m axial load, kN maximum axial load, kN minimum axial load, kN axial load, added to spectrum loading sequence as offset, kN transverse load, kN spectrum reference mean (axial) load, kN potential drop, V/A stress ratio, Kmin/Kmax
An important goal of coupon design is optimization of cruciform specimen corner radius for uniformity in stress distribution as well as maximum useful gage area for crack growth study. Many designs have been proposed in the literature for cruciform specimens subject to biaxial loading [14]. Some carry axial slits in the tab area to ensure uniformity of loading. Some carry reduced gage section thickness, intended to introduce edge displacement driven stress re-distribution with increasing crack size, while maintaining uniform gage edge area displacements. Irrespective of unique specimen features, an unavoidable result of finite specimen geometry is finite width effects that will induce crack growth driven changes in biaxial stress distribution across the gage area. At the same time, it is desirable to obtain stress intensity functions for the specific geometry in question. Based on these considerations, a simple and easy to machine uniform thickness cruciform specimen was designed as shown in Fig. 1 along with stress distribution under equai-biaxed loading. 2.2. Specimen gripping A demanding requirement of biaxial tension–tension testing is the need for coincidence of point of intersection of specimen axes and actuator axes. In the event they do not coincide and if the specimen mounting does not have the freedom to reorient itself due to rigid fastening and load transfer, part of the actuator force will be reacted as shear load by the actuators on the normal axis. This in turn can lead to sizeable error in loading along both axes, increased friction leading to potential damage to actuators and unintended damage to the specimen itself including shearing of loading tabs and non-uniform or angular crack growth. The magnitude of the above consequences of misalignment is sensitive to biaxial load train stiffness that translates any deviation between actuator and specimen axes intersection points into shear load component on the normal axis. This component may be relaxed by permitting sideways repositioning of specimen. This scheme was implemented through the loading bracket (see Fig. 1) to which the specimen tab is rigidly fastened. The bracket itself is provided with a slot for pin load transfer from the clevis mounted on the actuator rod end. The slot permits freedom of ±1.3 mm lateral movement of the specimen with associated shift between specimen and actuator axes. Given tab width of 30 mm and gage half-width of almost 40 mm, it was assumed that any distortion of load uniformity would be negligible. At the same time, real-time test control software ensures that even during fatigue cycling at up to 25 Hz, specimen center movement with respect to actuator axes intersection point does not exceed 0.5 mm in the
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Fig. 1. (Left) Cruciform specimen geometry used in tests. The geometry was optimized through FEM for uniformity of biaxial stress in the enclosed gauge circle. Dimensions in mm. 10 mm wire-cut notch at center serves as initial defect. 1 mm holes on either side are intended to fasten COD gauge tabs or PD signal leads. Steel specimen of 1 mm thickness and Al-alloy specimens of 2.7 mm thickness were used in tests. (Right) Backing plate for load transfer from specimen tabs to clevis grips. The slot shown provides a ±1.4 mm pin sliding tolerance to avoid shear loads due to minor misalignment. Any associated shift in neutral axis is assumed to cause negligible effect on uniformity of loading over gauge area.
worst case. This was confirmed by equal force readouts at both ends of the axis indicating lack of any load transfer to the normal axis actuators. 2.3. Test system configuration A number of solutions have been described in the literature for testing cruciform specimens [7,15–18]. This study was aimed at developing one to suit sustainable long-term experimental research involving a testing process that to the operator does not appear any more complex than that associated with conventional uniaxial servohydraulic test systems, even though it is practically the equivalent of four independent test systems: 1. Independent digital control of the four servo-actuators with any desired servo-feedback including actuator stroke, force readout, local strain or displacement readout at the desired point on the gage area. 2. Ring-type self-reacting load frame permitting scale-up in terms of size and force rating of future systems. 3. Single digital controller with capability of highly synchronized multi-channel servo-control, to perform as a single system with the same flexibility, performance and hardware components as conventional uniaxial test systems for ease of support and maintenance. 4. Unified, accessible and expandable software platform to permit ease of developing new test applications without the need to comprehend the complexity of a multi-channel control scheme. Thus, a new application to impose thermo-mechanical loading features can be added without changing software that controls other system functions. The test system instrumentation:
was
integrated
with
the
Application software developed for this study included: 1. Biaxial fatigue cycling under in-phase and out-of-phase tensile load cycling at up to 50 Hz with precision control of both load magnitude as well as phase. Logging of PD versus cycle count and provision for future experiments under K-control. 2. Pseudo-random spectrum loading with precision adaptive control along one axis and quasi-static transverse tensile load. Such testing simulates the superposed action of aircraft gust loading and internal pressurization. Periodic logging during the test of low load rate Load versus COD response in addition to logging of PD. Load – COD response can provide an idea of how transverse load affects both compliance and crack closure. 2.4. Biaxial load frame The load frame (Fig. 2) is formed by a pair of parallel steel rings, mounted on rigid spacers including four radial mounted servocontrolled 5 kN, 50 mm stroke actuators. As the frame is totally
following
1. Direct current potential drop (DCPD) instrumentation with reference and specimen potential drop measurement under pulsed current to eliminate thermal emf component. 2. Crack opening displacement (COD) gage based measurements of displacement along the loading axis between points on either side of the crack. 3. Optional transverse strain measurement through extensometer.
Fig. 2. Load frame developed for biaxial testing. Four identical and independently controlled 5 kN servo-hydraulic actuator assemblies are mounted between pair of precision machined steel rings. In actual testing, two of the actuators applied required static, cyclic or pseudo-random loads, while the other two maintain specimen center position.
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self-reacting, only a stand is provided to retain the frame in the vertical position at a convenient height for the operator. The stiffness of the frame ensures diametral deflections less than 0.05 mm at maximum applied load. The four actuators are wired and controlled in much the same way as four independent uniaxial test systems. The pressure, return and drain lines from the actuators are hard piped to a common service manifold on the base stand at the foot of the load frame, which is equipped with components required to ensure safe and reliable operation of the actuators without noticeable cross-talk even under dynamic conditions. Each actuator is equipped with a load cell and clevis grip, with the latter electrically insulated from the actuator in order to comply with the requirements of DC potential drop (PD) instrumentation. Lugs carrying the current for the PD measurement are fastened on opposing backing plates of the specimen along the loading axis perpendicular to the crack plane. The PD signal leads are spot welded about 2.5 mm above and below the specimen center. 2.5. Biaxial control implementation Numerous test control mode combinations are possible. However, the following method appeared to be the most suitable. One X and one Y actuator are in Stroke Control. Their task is to maintain the specimen center as close to the intersection point of the two loading axes as possible. The other two actuators are in Load Control. However, the load feedbacks of two actuators on the same axis are interchanged, i.e., the load cell mounted on the stationary actuator that is in Stroke Control provides the feedback for the actuator that is in Load Control and vice versa. As the actuator rod applying the load will see considerably greater movement than the one maintaining constant stroke, its own load cell readout may be distorted by inertial forces from its own mass as well as that of the gripping and the specimen assembly. By sensing force readout from the opposing actuator as feedback, the fidelity as well as accuracy of force response is maintained without the need for expensive acceleration compensated load cells at frequencies in excess of 10–25 Hz. The opposing actuator sees negligible reactive movement during dynamic cycling. While mounting the specimen, all four actuators are kept in Stroke Control. Switch to Load Control on two of the actuators is performed after ensuring some tensile load is imposed in order to avoid backlash-induced damage to the specimen. At this point, the actuators in Stroke Control are moved so as to ensure sufficient clearance due to tensile motion on the backing plate slots. This is a manual adjustment that can be corrected at any time without test stoppage. In future, a local position feedback transducer may be introduced for automatic correction of specimen center location. An important problem of biaxial loading is the inevitable crosstalk between axial and transverse servo-control loops, associated with Poisson’s Ratio. This is manifest at frequencies in excess of a few Hz and can lead to a cyclic loading around existing mean on the transverse axis of up to 30% of axial force, and vice versa. This makes digital adaptive control an inevitable requirement for required quality and performance of biaxial cyclic testing. The problem can be somewhat reduced under quasi-static transverse loading by adding compliant links on the transverse axis to make it less sensitive to cross-talk. 2.6. Summary of experiments Two sets of tests were performed. The first set of experiments was performed under constant amplitude loading on 1 mm thick mild steel sheet specimens. The primary goal of these tests was to explore the viability of the test system and process: to validate assumptions in calculating stress intensity range for cyclic loading
under varying biaxiality and also, to validate the FEM-based analytical K-calibration functions and confirm reproducibility of PD-calibration function for crack size estimates. The tests were performed with biaxiality ratio of unity and stress ratio R = 0.1, with Pax,max = 4.5 kN. One test was performed under strictly inphase loading, while the other was with 180-degree out-of-phase loading. This combination represents extreme variation in biaxiality ratio and serves to support the validations listed above. The second set of experiments was performed under MarkerTWIST [19], a modified version of the TWIST load spectrum, designed to facilitate quantitative fractography. Two types of cracks are possible when a pressurized fuselage cabin reacts TWIST-type of loading on the aircraft wing. If the dominant effect is that of cabin pressurization and depressurization over each flight, fuselage cracking will be along the airframe axis. This is because the hoop stress is twice the axial stress. However, the upper surface of the fuselage particularly near the wing mounting area will see substantial axial loads to react the wing gust, maneuver and landing loads. If the effect of these loads is substantial, cracking will be in the transverse direction in the form of circumferential cracks as the pressurized fuselage experiences cyclic bending. The experiments performed simulate such a scenario. The Marker-TWIST spectrum was applied in the axial direction with a static tensile axial offset simulating cabin pressure and a static tensile transverse load of twice the magnitude simulating the hoop stress component. 2.7 mm thick Al-alloy 2024-T3 was chosen for these experiments in view of its relevance to airframe applications and also the potential of fractography. All the specimens were tested with rolling direction oriented along the axial axis of loading. Table 1 lists the properties of the two materials used in tests. The Marker-TWIST spectrum is truncated three levels down. As a consequence, just 10 blocks (1000 flights) are adequate to completely reproduce the spectrum. Further, two more derivatives of Marker-TWIST were used (see Fig. 3). These are Marker-MiniTWIST that reduces cycle duration by about 90% and Marker-MicroTWIST that just over one cycle on an average for every flight (or 125 per block). These two derivatives were employed to advance crack growth at smaller crack size that is associated with much lower growth rates. In terms of crack extension per flight or per block, Marker-MicroTWIST causes about 50% retardation, but reduces test duration by almost two orders of magnitude when compared to the full spectrum. All the tests were performed with a spectrum mean load of 1.5 kN and individual load amplitudes suitably scaled as per the TWIST loading standard. The superposed axial and transverse loading details and other related information appears in Table 2. The first three tests were performed to investigate the effect on spectrum load fatigue of different levels of fuselage cabin pressure given unchanged spectrum load levels. Note that the axial load offset shifts the spectrum loads upwards. From considerations of crack closure, increased shift should cause increased growth rate. At the same time, such a shift is accompanied by a doubling of transverse tensile load that may be expected to retard fatigue. The fourth test was focused on the effect of transverse tensile load. In this test, the axial offset was kept constant at 1.5 kN, while the transverse load was toggled between 0 and 3 kN every 1 mm of half-crack extension in an effort to characterize associated change in average crack growth rate. A fifth test was performed with transverse load changed every few blocks as described in Table 2. The objective of this test was to generate a fatigue fracture that could carry microscopic evidence of how instantaneous level of transverse load affects spectrum load fatigue crack extension. The tests were performed uninterrupted at constant load rate of about 60 kN/s. This results in test frequency of about 25 Hz for the smallest amplitude and of the order of 5 Hz for the highest load levels. Iterative adaptive control ensured reproduction of required
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R. Sunder, B.V. Ilchenko / International Journal of Fatigue 33 (2011) 1101–1110 Table 1 Material properties. No.
Material
Thickness (mm)
Modulus (GPa)
0.2% yield stress (MPa)
Ultimate stress (MPa)
1 4
Mild steel 2024-T3 Al-alloy
1 2.7
203 73
260 350
410 380
Fig. 3. Marker-TWIST – modified TWIST load spectrum in the form of programmed block loading used in biaxial spectrum load tests on 2024-T3 alloy. Marker-TWIST permits quantitative fractography to determine relative contribution to crack growth of individual load levels in the spectrum. Fatigue crack growth tests were performed under repeated application of this block.
Table 2 Loading details for individual tests. Test no.
Spectrum
Axial load offset (kN)
Static transverse load, Ptv (kN)
Remarks
1 2 3 4
Marker-MiniTWIST Marker-MiniTWIST Marker-MiniTWIST #1 Marker-MicroTWIST #2 Marker-MiniTWIST #3 Marker-TWIST Marker-TWIST
1.5 1.0 0.5 1.5
3 2 1 Toggled between 0 and 3 kN after every 1 mm crack increment to isolate effect of transverse load Two blocks at 3 kN transverse load followed by 1 block at 1.5 kN and one at 0.15 kN
Transverse load is twice axial load offset to simulate hoop stress from cabin pressure
5
1.5
load history with peak error under +5% and average error under 1%, irrespective of transverse load magnitude. The transverse load, as well as the axial load offset are maintained static even though in reality, the cabin pressure component will diminish to zero at the conclusion of each flight. It is assumed that this aspect would have a negligible effect on test results because most cyclic loads in flight are experienced once the fuselage is pressurized. Initial half crack size was 10 mm for the steel specimens and 15 mm in the Al-alloy specimens. Testing was stopped at a half crack size of 45 mm, or, when difference between the two half crack sizes exceeded 5 mm. Perhaps because of specimen alignment features, and also because average growth rates were low enough to keep the crack-front flat, crack direction never deviated noticeably from original plane, even beyond the enclosed gauge circle.
Spectrum #1 to half crack 33 mm, #2 to 37 mm and #3 beyond 37 mm For fractographic study of effects at microscopic level
crack always normal to the axial loading direction was assumed. Given relatively thin material thickness of 1–5 mm over gage area of about 75 mm diameter, the analyses assumed plane stress conditions to prevail. Calculated K-values are summarized in Fig. 4. Polynomial fits of these curves were used to determine stress intensity whilst processing experimental data. Constant amplitude fatigue crack growth rates are typically correlated against cyclic stress intensity range, DK = Kmax Kmin. In the case of biaxial loading, particularly under out-of-phase conditions estimation of Kmax and Kmin demands strict definition. In this study, it is assumed that:
2.7. Stress intensity calculation
1. Peak and valley points in the applied axial load cycle determine Kmax and Kmin values in estimating DK. 2. Kmax and Kmin values are calculated as Mode I values pertaining to the instantaneous level of biaxiality at the two points in question.
Finite Element Method (FEM) analyses of the cracked geometry under different ratios of stress biaxiality provided Mode I stress intensity estimates for discrete crack sizes and load biaxiality ratios assuming load transfer at the specimen fasteners. A straight
Thus, it is assumed that crack driving force under cyclic loading involving biaxial loading will be determined exclusively by instantaneous level of biaxiality at the peak and valley of the axial load cycle, i.e., by instantaneous computed Kmax and Kmin values. In
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Fig. 4. Computed stress intensity factor from FEM analysis assuming plane stress conditions. Results obtained for unit thickness (mm) and unit load (kN) for different magnitude of transverse stress ranging from zero (uniaxial loading) to 3 kN. Intermediate conditions may be interpolated from these curves approximated as polynomial functions.
the event of synchronous in-phase loading, both peak and valley in the axial load cycle will see the same extent of biaxiality. However, in the event of 180-degree out-of-phase loading, peak axial load will see minimum biaxiality, while the valley will experience extreme biaxiality. The above scheme to calculate Kmax and Kmin values also applies to the case when transverse load remains constant (static load). Depending on actual combination of axial and transverse loading and actual K-functions for different biaxiality, cases may not be ruled out, where the position in time of Kmax and Kmin may not coincide with that of axial peak and valley points. However, this possibility does not arise with any of the cases considered in this study. The above considerations governed calculation of DK values from the K-calibration functions for different levels of biaxility while plotting the test results for constant amplitude loading. The question of characteristic K-estimates for spectrum loading is a little more complex. Under uniaxial loading, spectrum load crack growth rates are typically plotted against Kmean (associated with spectrum mean stress), Kmax (from spectrum max stress), or, against Krms. This raises the question regarding what level of biaxiality is to be assumed because in the present experiments, transverse load was always constant, implying varying biaxiality depending on instantaneous axial load. Considering that the spectrum in use is a transport spectrum, where most load cycles are symmetric around the mean load, K was calculated for spectrum mean stress of 1.5 kN plus the actual axial load offset imposed.
3. Test results and discussion Constant amplitude test results obtained under in-phase and out-of-phase biaxial loading are summarized in Fig. 5. The data on the left are plotted against DK computed ignoring biaxiality, i.e., assuming uniaxial loading. When viewed in this format, crack growth rates under out-of-phase loading appear as much as five times greater than those under in-phase loading at a given DK. Given that DK is estimated as the difference between Kmax and Kmin and given the low stress ratio, it would follow that Kmax in the absence of transverse tensile load as in out-of-phase loading is more
damaging than in the presence of a high tensile transverse load as in the case of in-phase loading. The ability of numerical estimates of Mode I stress intensity to account for biaxiality appears to be demonstrated successfully by the data plotted on the right. Crack growth rates for the two vastly different conditions of cyclic biaxiality appear to merge into a single scatter band when plotted against Mode I DK computed after accounting for biaxiality. These data appear to confirm that Mode I DK may serve as an adequate similarity criterion for comparable stress ratios, independent of variations in biaxiality ratio, provided of course, crack path remains normal to axial loading. The data in Fig. 5 suggest that just as in the case of uniaxial loading, the primary variables are computed values of Mode I Kmax and Kmin, which in turn are determined by instantaneous biaxiality at the peak and valley in the axial loading cycle. This conjecture was put to the test of the biaxial spectrum loading, whose results appear in Figs. 6 and 7. Fig. 6 shows test results obtained under Marker-TWIST spectrum loading with and without the transverse load component. When plotted against crack size, spectrum load crack growth rates show substantial and varying sensitivity to transverse tensile load (Fig. 6, Left). Tensile transverse load tends to retard crack growth. However when plotted against Kref, that accounts for both the axial load offset as well as applied transverse load, the data seem to merge into a single scatter band. Considering crack extension occurs during the upper half of the rising portion of the load cycle, one may assume that static transverse load and cyclic in-phase transverse load with the same maximum will cause similar crack growth behaviour. One may conclude that provided most fatigue damage accrues around a given Kref as is in the case of aircraft gust load spectra, even when offset by a constant load, or, when controlled by a given Kmax, as in the case of constant amplitude loading at low stress ratio, crack growth rates under biaxial loading may be satisfactorily correlated with such a Kref or Kmax, corrected for transverse load at the same point in the load cycle. An important follow-up conclusion would be that whether the transverse component of biaxial loading is static or cyclic in-phase or out-of-phase is of little relevance. The above conclusion appears to apply even with regard to Marker-TWIST spectrum loading at different levels of axial offset load
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Fig. 5. Constant amplitude crack growth rates under in-phase and out-of-phase cycling with Pax,max = 4.5 kN and stress ratio R = 0.1 at 20 Hz and biaxiality ratio of unity. Material: 1 mm thick mild steel sheet, with axial loading along rolling direction. Accounting for instantaneous biaxiality at Pax,max and at Pax,min while calculating DK appears to merge in-phase and out-of-phase data into a narrow scatter band.
Fig. 6. (Left) Crack growth rates under Marker-TWIST spectrum loading with and without transverse load of 3 kN. Axial load offset by 1.5 kN in both cases. (Right) Same data plotted against Kref computed for sum of axial offset load and spectrum mean load (3 kN) and accounting for biaxiality. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
and transverse load equal to twice the axial offset load to simulate axial and hoop stress due to fuselage cabin pressure. Fig. 7 shows
crack growth rates obtained under the same spectrum mean load of 1.5 kN but three different axial load offsets of 0.5, 1 and 1.5 kN
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Fig. 7. (Left) Fatigue crack growth rates under Marker-TWIST load spectrum with spectrum mean load of 1.5 kN under different levels of axial load offset and corresponding transverse load. (Right) Same data plotted against Kref computed for sum of spectrum mean and offset loads with associated biaxiality. Dotted trend line appears to suggest that growth rates are likely to merge into a single scatter band. Characteristic stress intensity appears to account for biaxiality even though axial load offset to the load spectrum may vary considerably. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
and corresponding transverse tensile load of 1, 2 and 3 kN. Under uniaxial loading, any axial tensile offset in loading would accelerate crack growth rate. However, as a consequence of the simultaneous doubling of transverse load, crack growth rate actually retards with increasing axial load offset. This is evident from the data on the left. However, when plotted against Kref corrected for biaxiality, data from al three tests appear to fall into a single scatter band. The above data support the possibility that characteristic stress intensity is indeed determined by instantaneous axial load at the
point corresponding to spectrum mean load, Pm + Pax,offs, combined with biaxiality, Bi. The engineering significance of this finding lies in the identification of a single, unifying and computable fracture mechanics parameter that may account for the noticeable effect of load biaxiality under both constant amplitude as well as spectrum load fatigue crack growth. However, secondary effects may accrue from other factors sensitive to biaxiality. In this context both compliance and closure under constant transverse load become relevant parameters for examination because of their potential sensitivity to biaxiality.
Fig. 8. Effect of constant transverse load on load line unloading compliance with Pmax = 3 kN. Compliance ratio appears to show a small but reproducible non-monotonous trend with respect to crack size. However, its sensitivity to biaxiality appears negligible by comparison to that of stress intensity. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
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3.1. Compliance measurements A constant amplitude test was performed without transverse load at Pmax = 3.5 kN and Pmin = 0.5 kN. Every 2 mm of crack extension, the test was interrupted for unloading compliance measurements with and without transverse load over a data window 50%–100% of Pmax. The results appear in Fig. 8. They show that transverse load has a discernible effect on unloading compliance. This is highlighted by the plot of compliance ratio versus crack size. Unloading compliance appears to be attenuated by tensile transverse load. Liu attributes it to possible finite width effect [7] because the effect was not observed at smaller crack size. Compliance ratio increases towards axial-only loading with increasing crack size, then diminishes again. This may be due to variation in biaxiality caused by non-uniform stress distribution across the gage section. Irrespective of the actual underlying reasons for compliance ratio variation, ignoring it can cause errors in crack size estimate of 3–5 mm. As compliance is a mechanics, rather than
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mechanism driven parameter, the observed effects may be studied through analytical simulation and perhaps as a variable to be accounted for in biaxial variable amplitude fatigue. As pointed by Adams [13] unloading compliance will determine crack opening displacement and may therefore be one way in which transverse load affects fatigue crack growth. However, one may note that at the point where compliance readouts are almost the same with and without transverse load (at about 30 mm crack size), spectrum load growth rates still remain very sensitive to transverse load. 3.2. Crack closure McClung analytically determined that closure is insensitive to biaxiality until applied stress levels exceed 40% of yield stress. To experimentally investigate whether closure may be sensitive to transverse load, load–COD data were logged under slow unloading with three different transverse loads at a crack size of 30 mm. Compliance compensated COD data for the three cases appear in Fig. 9. The leftward shift between curves was caused by reduction in COD readout at maximum load under the influence of transverse load. From these three curves, one may get the impression that tensile transverse load in fact increases closure level – even though applied stress levels are less than 25% of yield stress. One may note however, that the measurement being remote from the crack tip area, may actually not reflect actual crack tip response, but rather, a far field wake response that has no bearing on crack growth behaviour. This aspect demands further study using near-tip measurement techniques such as fractography or laser indentation interferometry [18]. 4. Fractography of spectrum load biaxial fracture
Fig. 9. Load versus linear response compensated COD obtained during interrupted spectrum load test. Measurements made during slow unloading at three different levels of constant transverse load. Circles mark visually apparent points of deviation from linear response, suggesting decrease in crack closure load with decreasing tensile transverse load. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
The reasonable correlation of both constant amplitude as well as spectrum load crack growth rates with a characteristic stress intensity computed accounting for load biaxiality suggests that the effect of biaxiality may be adequately handled by computational mechanics. As shown in Fig. 8, compliance is also sensitive to biaxiality and numerical simulation of its response needs to be verified. Crack closure also appears to be sensitive to biaxiality as indicated in Fig. 9, though COD based measurements of closure are often questionable. Parameters such as Kref and compliance are mechanics driven and sensitive to instantaneous biaxial load at a given crack size. Variable amplitude fatigue is known to be load sequence sensitive. So-called crack growth life prediction models relate this sen-
Fig. 10. Composite fractograph built from multiple micrographs covering about 0.35 mm of crack extension at crack size of 38 mm in Test #5. Identified intervals are presumed to be over complete blocks of Marker-TWIST. Reduced crack extension in the first two segments suggest it must have been under maximum transverse load because this block was repeated twice, followed by one at 1.5 kN and another one at 1.0 kN after which the sequence repeats. This appears to confirm the sensitivity of crack growth at microscopic level to instantaneous levels of biaxiality. Monotonic plastic zone size is of the order of 0.3 mm. Detailed description of identification of fractographic features from this and other experiments under Marker-TWIST can be found in Ref. [21]. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
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sitivity to plastic zone size and crack closure, both of which are known to be affected by biaxiality. Fig. 9 reflects the effect of biaxiality on closure. Plastic zone size under uniaxial loading will be considerably greater than at the same load and crack size subject to transverse tensile load. Once a plastic zone forms under uniaxial loading, its size cannot diminish instantaneously with the application of a tensile transverse load. In the present experiments, calculated spectrum load plastic zone size was of the order of 0.2– 0.4 mm at crack size close to 40 mm. Thus, in Test No. 5, this size would be of the order of crack growth interval covered by the entire fractograph shown in Fig. 10. Crack increment in a single block of the load spectrum is but a small fraction of monotonic plastic zone size. Yet, crack growth was extremely sensitive to change in transverse tensile load as shown by noticeable block-to-block changes in growth rate. It would follow that for the test conditions in question, crack growth rate is much more sensitive to change in instantaneous tensile transverse load, rather than to monotonic plastic zone size. Crack extension in individual spectrum load blocks is clearly discernible on the fractograph. From the sequence combination of repeat blocks applied at different transverse load, one can readily relate crack extension to individual blocks. It may be noted that only the transverse load component was changed from block-toblock. This underscores once again, the significance of tensile transverse load in controlling the mechanics of fatigue crack growth. More significantly, it would appear that computational procedures to merely account for the effect of instantaneous biaxiality on characteristic stress intensity may be adequate to extrapolate uniaxial fatigue crack growth rate behaviour to monotonic or cyclic biaxial loading conditions. It may be noted that the above conclusions are based on exploratory tests on sheet specimens under predominantly plane stress conditions of constant amplitude and TWIST-type of spectrum loading on steel and Al-alloy specimens. Their applicability needs to be verified on a broader base of test conditions and materials. 5. Conclusions 1. A four actuator biaxial test system was developed to test cruciform type of test coupons under constant amplitude and pseudo-random flight spectrum loading. Synchronous quadchannel digital control software was implemented to perform a wide variety of constant amplitude and spectrum load tests at test frequencies up to 25 Hz. Control scheme and specimen gripping features preclude the absence of side loads due to potential specimen misalignment. 2. Mode I K-calibration functions were established through FEM analysis for a straight central crack subject to plane stress conditions under varying degrees of biaxiality. 3. Constant amplitude fatigue crack growth tests were performed on steel sheet specimens under in-phase and out-ofphase loading. Tests were performed on Al-alloy specimens under a modified TWIST spectrum with different levels of offset in axial load and corresponding transverse tensile load to simulate load conditions on the top panel of a pressurized cabin over the wing. 4. Fatigue crack growth rates are noticeably sensitive to load biaxiality under both constant amplitude as well as spectrum loading. However, when plotted against characteristic Mode I stress intensity computed after accounting for instantaneous biaxiality, crack growth rates appear to fall into a single scatter band.
5. Characteristic stress intensity for constant amplitude in-phase and out-of-phase loading is computed accounting for instantaneous biaxiality at the peak and valley of axial load component. The effect of biaxiality at intermediate axial load levels may be negligible. 6. Under transport aircraft spectrum loading, characteristic stress intensity is computed from K at the spectrum mean load level after accounting for instantaneous biaxiality at this load.
Acknowledgements Prof. Valery Shlyannikov’s constant encouragement, support and useful suggestions made this collaborative effort possible. Sitdikov performed the FEM calculations of stress intensity. Virupaksha, Ramesh and Raghu carefully set up the experiments and processed test results. Vivek headed the design team that built the biaxial load frame. Graduating engineering students Elena Zemlyanaya, K.N. Raghavan, G.V. Sarthak, K.S. Sridhar and D. Syed participated in the testing effort. The fractograph in Fig. 10 was provided by John Porter at the University of Dayton research Institute. Extremely useful critique and suggestions from Sebastian Henkel of Freiberg Technical University are deeply appreciated. References [1] Gassner E. Strength experiments under cyclic loading in aircraft structures. Luftwissen 1939;6:61–4 [in German]. [2] Schijve J. Fatigue crack propagation in light alloy sheet material and structures, vol. 3. Pergamon Press; 1961. [3] Schijve J. Observations on the prediction of fatigue crack propagation under variable-amplitude loading. Fatigue crack growth under spectrum loads, ASTM STP 1976;595:3–23. [4] Newman JC. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. In: Chang JB, Hudson CM, editors. Methods and models for predicting fatigue crack growth under random loading, ASTM STP 1981;748: 53–84. [5] Wheeler OE. J Basic Eng Trans ASME 1972:181–6. [6] Willenborg JD, Engle RM, Wood HA. A crack growth retardation model using an effective stress concept AFFDL-TM-FBR-71-1. Air Force Flight Dynamics Laboratory; 1971. [7] Liu AF, Dittmer DF. Effect of multi-axial loading on crack growth, Vol 1 – technical summary. Air Force Flight Dynamics Laboratory Report AFFDL-TR78-175, vol. 1–3; 1978. [8] Yuuki R, Akita K, Kishi N. The effect of biaxial stress state and changes of state on fatigue crack growth behaviour. Fatigue Fract Eng Mater Struct 1989;12(2):93–103. [9] Hopper CD, Miller KJ. Fatigue crack propagation in biaxial stress fields. J Strain Anal 1977;12(1). [10] Anderson PRG, Garrett GG. Fatigue crack growth rate variations in biaxial stress fields. Int J Fract 1980;16:R111–R1116. [11] Ogura K, Ohji K, Ohkubo Y. Fatigue crack growth under biaxial loading. Int J Fract 1974;10:609. [12] McClung RC. Closure and growth of Mode I cracks in biaxial fatigue. Fatigue Fract Eng Mater Struct 1989;12(5):447–60. [13] Adams NJI. Some comments on the effect of biaxial stress on fatigue crack growth and fracture. Eng Fract Mech 1973;5:983–91. [14] Lebedev AA, Muzyka NR. Design of cruciform specimens for fracture toughness tests in biaxial tension, Review paper, Strength Mater 1998; 30(3): 243–54. [15] Muzyka NR. Equipment for testing sheet structural materials under biaxial loading. Part 2. Testing by biaxial loading in the plane of the sheet. Strength Mater 2002;23(4). [16] Radon JC, Leevers PS, Culver LE. J Exp Mech 1977:228–32. [17] Wu XD, Wan M, Zhou XB. Biaxial tensile testing of cruciform specimen under complex loading. J. Mater Process Technol 2005;168(1):181–3. [18] Makinde A, Thibodeau L, Neale KW. Development of an apparatus for biaxial testing using cruciform specimens. Exp Mech 1992;32(2). [19] Sunder R. Development of the Marker-TWIST Load Sequence for Quantitative Fractographic Studies. Int J Fatigue 2010. [21] Porter JW, Rosenberger AH, Sunder R. Contribution of individual cycles to crack extension under a transport aircraft wing load spectrum – A fractographic analysis. In: International conference on fatigue damage of structural materials VIII, Hyannis; Sep 19–24, 2010. [To be submitted to Int J Fatigue 2010].
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure R. Sunder a,⇑, B.V. Ilchenko b a b
BiSS Research, 41A, 1A Cross, AECS 2nd Stage, Bangalore 560 094, India Power Engineering Research Centre, Russian Academy of Sciences, P.B. 190, Kazan 420111, Russia
a r t i c l e
i n f o
Article history: Received 20 September 2010 Received in revised form 14 November 2010 Accepted 19 November 2010 Available online 30 November 2010 Keywords: Fatigue crack growth Biaxial spectrum loading
a b s t r a c t Pressurized transport aircraft fuselage panels see the combined action of gust loading and internal pressurization that causes a rather complex biaxial pseudo-random cyclic loading action. Stress biaxiality is also known to affect constant amplitude fatigue. Fatigue crack growth was studied using cruciform test coupons under constant amplitude and a modified TWIST load spectrum superposed with biaxial quasistatic load simulating internal cabin pressure. The experiments were performed on a specially designed four-actuator biaxial test system suited for high frequency in-phase and out-of-phase constant amplitude loading as well as biaxial pseudo-random spectrum loading. Observed crack growth rates were correlated with stress intensity values, corrected for biaxiality. Ó 2010 Published by Elsevier Ltd.
1. Introduction Fatigue damage accumulation is non-linear. This was obvious from early systematic experiments by Gassner [1], who established the inevitability of testing under loading conditions that closely simulate actual usage. The actual reasons for non-linear damage accumulation began to unravel many decades later, when tensile overloads were unexpectedly found to retard fatigue crack growth [2]. More research in the 1970s established that a number of load interaction mechanisms may be responsible for load sequence sensitivity of fatigue in metals [3]. These include crack tip residual stress, crack closure, crack-front incompatibility, crack-tip blunting/re-sharpening, etc. Over the years, a number of models have been proposed to estimate variable amplitude fatigue crack growth. Each model is invariably built around a single load interaction mechanism, usually, crack closure [4] or residual stress [5,6]. If a single-mechanism model does correctly estimate residual life, one would invariably have to attribute it to exaggerated dominance of one mechanism to approximate the actual result, which would not be much of an advancement over Gassner’s empirical approach of determining S–N curves under spectrum loading and computing associated damage sums. While research in uniaxial variable amplitude fatigue continues in an attempt to bridge this gap, problems of practical interest often relate to multi-axial loading conditions and call for the reexamination and extension of available understanding as stated above to the case of multi-axial loading. ⇑ Corresponding author. Tel.: +91 80 2341 4133; fax: +91 80 2341 0813. E-mail address: [email protected] (R. Sunder). 0142-1123/$ - see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2010.11.018
Available experimental data on fatigue crack growth under biaxial loading with cruciform specimens are limited and somewhat contradictory by comparison to that under uniaxial conditions. Liu and Dittmer observed that biaxiality does not affect crack growth rates under constant amplitude loading [7]. Yuuki et al. [8] also did not find any effect of biaxiality at low stresses, but found the effect noticeable at high stresses. Hopper and Miller found a consistent retarding effect of tensile transverse stress on growth rate [9]. Anderson and Garrett also observed a close relationship between crack growth rate and biaxial stress field [10]. Their measurements were macroscopic and cannot be used to judge whether stress field variations cause an instantaneous, or a gradual change in growth rate. Analytical studies also offer mixed conclusions. Ogura et al. point out that K is insensitive to biaxiality1 [11], and suggest that closure may be responsible for possible effects. However, McLung estimated that crack closure becomes sensitive to biaxiality only when stress exceeds 40% of yield stress [12], ruling out closure as a factor for most cases of practical interest. Adams found that tensile biaxiality affects plastic zone size [13]. This is supported by measurements by Liu and Dittmer [7], who as earlier noted did not however find any effect on constant amplitude crack growth rate. Liu and Dittmer however found a significant adverse effect of tensile stress biaxiality on crack growth under periodic overloads and under what appears to be a simple combat aircraft load spectrum. They attribute this to the reduced overload plastic zone size in the presence of tensile transverse load. Interestingly, this is the 1 One may note that standard expressions for K only represent the first term of series. The others are neglected as secondary, but they can be sensitive to biaxiality.
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Nomenclature
DK Bi Bi,max Bi,min COD DCPD FEM K Kmax Kmin
mode I stress intensity factor range, Kmax Kmin, p MPa m instantaneous ratio of transverse to axial force instantaneous ratio of transverse to axial force at maximum axial force instantaneous ratio of transverse to axial force at minimum axial force crack opening displacement d.c. potential drop technique finite-element method p mode I stress intensity factor, MPa m mode I stress intensity factor at Pax,max and Bi,max, p MPa m mode I stress intensity factor at Pax,min and Bi,min, p MPa m
opposite of what others have observed under constant amplitude loading [8–10]. Available experimental data underscore the need for much more experimental effort to cover cases of practical interest. Equally significantly, there has been a dearth of attention to the sensitivity of well-known load interaction mechanisms to biaxiality as well as the emergence of possible new mechanisms and factors associated with biaxial service loading that may not exist under uniaxial conditions. The goal of this study was to develop an easy to use digitally controlled biaxial test system for cruciform specimens and as a first exercise, investigate the growth of a circumferential crack in the upper panel of a pressurized fuselage cabin of a transport aircraft. The upper surface of the fuselage sees the same loads as the bottom surface of the wing. In case of pressurized cabins of transport aircraft, it will in addition, see the superposed action of internal pressure. Thus, the axial component of pressure-induced stress will add a tensile offset to the gust load spectrum. In addition, a hoop stress component that is twice the axial component will act in the transverse direction. This special case of biaxial loading involves pseudo-random flight spectrum loading with a tensile mean load offset, combined with a constant tensile transverse load. Under such loading, the fatigue crack tip will be exposed to tension–tension variable-amplitude loading, combined with a static transverse tensile load. The next section describes the experimental procedure including the test coupon used, the test system, gripping, loading and sense arrangements and load conditions. This is followed by a description and discussion of test results.
2. Experimental procedure The experiments were performed on cruciform test coupons using a digitally controlled four actuator 5 kN biaxial test system.
2.1. The test coupon The cruciform test coupon design is intended to serve ongoing research goals that include development of a standard coupon geometry to reproduce different levels of biaxiality, be suitable for materials in the 1–5 mm thickness interval, allow for useful center-crack growth data over at least 2a = 50 mm of crack growth, permit ease of reproducible gripping conditions and allow for future scale-up as test system force ratings are enhanced from the present 5 kN to 50 kN and over.
Km Kref Pax Pax,max Pax,min Pax,offs Ptv Pm PD R
p mode I stress intensity factor at Pm, MPa m mode I stress intensity factor at Pm + Pax,offs and Bi, p MPa m axial load, kN maximum axial load, kN minimum axial load, kN axial load, added to spectrum loading sequence as offset, kN transverse load, kN spectrum reference mean (axial) load, kN potential drop, V/A stress ratio, Kmin/Kmax
An important goal of coupon design is optimization of cruciform specimen corner radius for uniformity in stress distribution as well as maximum useful gage area for crack growth study. Many designs have been proposed in the literature for cruciform specimens subject to biaxial loading [14]. Some carry axial slits in the tab area to ensure uniformity of loading. Some carry reduced gage section thickness, intended to introduce edge displacement driven stress re-distribution with increasing crack size, while maintaining uniform gage edge area displacements. Irrespective of unique specimen features, an unavoidable result of finite specimen geometry is finite width effects that will induce crack growth driven changes in biaxial stress distribution across the gage area. At the same time, it is desirable to obtain stress intensity functions for the specific geometry in question. Based on these considerations, a simple and easy to machine uniform thickness cruciform specimen was designed as shown in Fig. 1 along with stress distribution under equai-biaxed loading. 2.2. Specimen gripping A demanding requirement of biaxial tension–tension testing is the need for coincidence of point of intersection of specimen axes and actuator axes. In the event they do not coincide and if the specimen mounting does not have the freedom to reorient itself due to rigid fastening and load transfer, part of the actuator force will be reacted as shear load by the actuators on the normal axis. This in turn can lead to sizeable error in loading along both axes, increased friction leading to potential damage to actuators and unintended damage to the specimen itself including shearing of loading tabs and non-uniform or angular crack growth. The magnitude of the above consequences of misalignment is sensitive to biaxial load train stiffness that translates any deviation between actuator and specimen axes intersection points into shear load component on the normal axis. This component may be relaxed by permitting sideways repositioning of specimen. This scheme was implemented through the loading bracket (see Fig. 1) to which the specimen tab is rigidly fastened. The bracket itself is provided with a slot for pin load transfer from the clevis mounted on the actuator rod end. The slot permits freedom of ±1.3 mm lateral movement of the specimen with associated shift between specimen and actuator axes. Given tab width of 30 mm and gage half-width of almost 40 mm, it was assumed that any distortion of load uniformity would be negligible. At the same time, real-time test control software ensures that even during fatigue cycling at up to 25 Hz, specimen center movement with respect to actuator axes intersection point does not exceed 0.5 mm in the
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Fig. 1. (Left) Cruciform specimen geometry used in tests. The geometry was optimized through FEM for uniformity of biaxial stress in the enclosed gauge circle. Dimensions in mm. 10 mm wire-cut notch at center serves as initial defect. 1 mm holes on either side are intended to fasten COD gauge tabs or PD signal leads. Steel specimen of 1 mm thickness and Al-alloy specimens of 2.7 mm thickness were used in tests. (Right) Backing plate for load transfer from specimen tabs to clevis grips. The slot shown provides a ±1.4 mm pin sliding tolerance to avoid shear loads due to minor misalignment. Any associated shift in neutral axis is assumed to cause negligible effect on uniformity of loading over gauge area.
worst case. This was confirmed by equal force readouts at both ends of the axis indicating lack of any load transfer to the normal axis actuators. 2.3. Test system configuration A number of solutions have been described in the literature for testing cruciform specimens [7,15–18]. This study was aimed at developing one to suit sustainable long-term experimental research involving a testing process that to the operator does not appear any more complex than that associated with conventional uniaxial servohydraulic test systems, even though it is practically the equivalent of four independent test systems: 1. Independent digital control of the four servo-actuators with any desired servo-feedback including actuator stroke, force readout, local strain or displacement readout at the desired point on the gage area. 2. Ring-type self-reacting load frame permitting scale-up in terms of size and force rating of future systems. 3. Single digital controller with capability of highly synchronized multi-channel servo-control, to perform as a single system with the same flexibility, performance and hardware components as conventional uniaxial test systems for ease of support and maintenance. 4. Unified, accessible and expandable software platform to permit ease of developing new test applications without the need to comprehend the complexity of a multi-channel control scheme. Thus, a new application to impose thermo-mechanical loading features can be added without changing software that controls other system functions. The test system instrumentation:
was
integrated
with
the
Application software developed for this study included: 1. Biaxial fatigue cycling under in-phase and out-of-phase tensile load cycling at up to 50 Hz with precision control of both load magnitude as well as phase. Logging of PD versus cycle count and provision for future experiments under K-control. 2. Pseudo-random spectrum loading with precision adaptive control along one axis and quasi-static transverse tensile load. Such testing simulates the superposed action of aircraft gust loading and internal pressurization. Periodic logging during the test of low load rate Load versus COD response in addition to logging of PD. Load – COD response can provide an idea of how transverse load affects both compliance and crack closure. 2.4. Biaxial load frame The load frame (Fig. 2) is formed by a pair of parallel steel rings, mounted on rigid spacers including four radial mounted servocontrolled 5 kN, 50 mm stroke actuators. As the frame is totally
following
1. Direct current potential drop (DCPD) instrumentation with reference and specimen potential drop measurement under pulsed current to eliminate thermal emf component. 2. Crack opening displacement (COD) gage based measurements of displacement along the loading axis between points on either side of the crack. 3. Optional transverse strain measurement through extensometer.
Fig. 2. Load frame developed for biaxial testing. Four identical and independently controlled 5 kN servo-hydraulic actuator assemblies are mounted between pair of precision machined steel rings. In actual testing, two of the actuators applied required static, cyclic or pseudo-random loads, while the other two maintain specimen center position.
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self-reacting, only a stand is provided to retain the frame in the vertical position at a convenient height for the operator. The stiffness of the frame ensures diametral deflections less than 0.05 mm at maximum applied load. The four actuators are wired and controlled in much the same way as four independent uniaxial test systems. The pressure, return and drain lines from the actuators are hard piped to a common service manifold on the base stand at the foot of the load frame, which is equipped with components required to ensure safe and reliable operation of the actuators without noticeable cross-talk even under dynamic conditions. Each actuator is equipped with a load cell and clevis grip, with the latter electrically insulated from the actuator in order to comply with the requirements of DC potential drop (PD) instrumentation. Lugs carrying the current for the PD measurement are fastened on opposing backing plates of the specimen along the loading axis perpendicular to the crack plane. The PD signal leads are spot welded about 2.5 mm above and below the specimen center. 2.5. Biaxial control implementation Numerous test control mode combinations are possible. However, the following method appeared to be the most suitable. One X and one Y actuator are in Stroke Control. Their task is to maintain the specimen center as close to the intersection point of the two loading axes as possible. The other two actuators are in Load Control. However, the load feedbacks of two actuators on the same axis are interchanged, i.e., the load cell mounted on the stationary actuator that is in Stroke Control provides the feedback for the actuator that is in Load Control and vice versa. As the actuator rod applying the load will see considerably greater movement than the one maintaining constant stroke, its own load cell readout may be distorted by inertial forces from its own mass as well as that of the gripping and the specimen assembly. By sensing force readout from the opposing actuator as feedback, the fidelity as well as accuracy of force response is maintained without the need for expensive acceleration compensated load cells at frequencies in excess of 10–25 Hz. The opposing actuator sees negligible reactive movement during dynamic cycling. While mounting the specimen, all four actuators are kept in Stroke Control. Switch to Load Control on two of the actuators is performed after ensuring some tensile load is imposed in order to avoid backlash-induced damage to the specimen. At this point, the actuators in Stroke Control are moved so as to ensure sufficient clearance due to tensile motion on the backing plate slots. This is a manual adjustment that can be corrected at any time without test stoppage. In future, a local position feedback transducer may be introduced for automatic correction of specimen center location. An important problem of biaxial loading is the inevitable crosstalk between axial and transverse servo-control loops, associated with Poisson’s Ratio. This is manifest at frequencies in excess of a few Hz and can lead to a cyclic loading around existing mean on the transverse axis of up to 30% of axial force, and vice versa. This makes digital adaptive control an inevitable requirement for required quality and performance of biaxial cyclic testing. The problem can be somewhat reduced under quasi-static transverse loading by adding compliant links on the transverse axis to make it less sensitive to cross-talk. 2.6. Summary of experiments Two sets of tests were performed. The first set of experiments was performed under constant amplitude loading on 1 mm thick mild steel sheet specimens. The primary goal of these tests was to explore the viability of the test system and process: to validate assumptions in calculating stress intensity range for cyclic loading
under varying biaxiality and also, to validate the FEM-based analytical K-calibration functions and confirm reproducibility of PD-calibration function for crack size estimates. The tests were performed with biaxiality ratio of unity and stress ratio R = 0.1, with Pax,max = 4.5 kN. One test was performed under strictly inphase loading, while the other was with 180-degree out-of-phase loading. This combination represents extreme variation in biaxiality ratio and serves to support the validations listed above. The second set of experiments was performed under MarkerTWIST [19], a modified version of the TWIST load spectrum, designed to facilitate quantitative fractography. Two types of cracks are possible when a pressurized fuselage cabin reacts TWIST-type of loading on the aircraft wing. If the dominant effect is that of cabin pressurization and depressurization over each flight, fuselage cracking will be along the airframe axis. This is because the hoop stress is twice the axial stress. However, the upper surface of the fuselage particularly near the wing mounting area will see substantial axial loads to react the wing gust, maneuver and landing loads. If the effect of these loads is substantial, cracking will be in the transverse direction in the form of circumferential cracks as the pressurized fuselage experiences cyclic bending. The experiments performed simulate such a scenario. The Marker-TWIST spectrum was applied in the axial direction with a static tensile axial offset simulating cabin pressure and a static tensile transverse load of twice the magnitude simulating the hoop stress component. 2.7 mm thick Al-alloy 2024-T3 was chosen for these experiments in view of its relevance to airframe applications and also the potential of fractography. All the specimens were tested with rolling direction oriented along the axial axis of loading. Table 1 lists the properties of the two materials used in tests. The Marker-TWIST spectrum is truncated three levels down. As a consequence, just 10 blocks (1000 flights) are adequate to completely reproduce the spectrum. Further, two more derivatives of Marker-TWIST were used (see Fig. 3). These are Marker-MiniTWIST that reduces cycle duration by about 90% and Marker-MicroTWIST that just over one cycle on an average for every flight (or 125 per block). These two derivatives were employed to advance crack growth at smaller crack size that is associated with much lower growth rates. In terms of crack extension per flight or per block, Marker-MicroTWIST causes about 50% retardation, but reduces test duration by almost two orders of magnitude when compared to the full spectrum. All the tests were performed with a spectrum mean load of 1.5 kN and individual load amplitudes suitably scaled as per the TWIST loading standard. The superposed axial and transverse loading details and other related information appears in Table 2. The first three tests were performed to investigate the effect on spectrum load fatigue of different levels of fuselage cabin pressure given unchanged spectrum load levels. Note that the axial load offset shifts the spectrum loads upwards. From considerations of crack closure, increased shift should cause increased growth rate. At the same time, such a shift is accompanied by a doubling of transverse tensile load that may be expected to retard fatigue. The fourth test was focused on the effect of transverse tensile load. In this test, the axial offset was kept constant at 1.5 kN, while the transverse load was toggled between 0 and 3 kN every 1 mm of half-crack extension in an effort to characterize associated change in average crack growth rate. A fifth test was performed with transverse load changed every few blocks as described in Table 2. The objective of this test was to generate a fatigue fracture that could carry microscopic evidence of how instantaneous level of transverse load affects spectrum load fatigue crack extension. The tests were performed uninterrupted at constant load rate of about 60 kN/s. This results in test frequency of about 25 Hz for the smallest amplitude and of the order of 5 Hz for the highest load levels. Iterative adaptive control ensured reproduction of required
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Material
Thickness (mm)
Modulus (GPa)
0.2% yield stress (MPa)
Ultimate stress (MPa)
1 4
Mild steel 2024-T3 Al-alloy
1 2.7
203 73
260 350
410 380
Fig. 3. Marker-TWIST – modified TWIST load spectrum in the form of programmed block loading used in biaxial spectrum load tests on 2024-T3 alloy. Marker-TWIST permits quantitative fractography to determine relative contribution to crack growth of individual load levels in the spectrum. Fatigue crack growth tests were performed under repeated application of this block.
Table 2 Loading details for individual tests. Test no.
Spectrum
Axial load offset (kN)
Static transverse load, Ptv (kN)
Remarks
1 2 3 4
Marker-MiniTWIST Marker-MiniTWIST Marker-MiniTWIST #1 Marker-MicroTWIST #2 Marker-MiniTWIST #3 Marker-TWIST Marker-TWIST
1.5 1.0 0.5 1.5
3 2 1 Toggled between 0 and 3 kN after every 1 mm crack increment to isolate effect of transverse load Two blocks at 3 kN transverse load followed by 1 block at 1.5 kN and one at 0.15 kN
Transverse load is twice axial load offset to simulate hoop stress from cabin pressure
5
1.5
load history with peak error under +5% and average error under 1%, irrespective of transverse load magnitude. The transverse load, as well as the axial load offset are maintained static even though in reality, the cabin pressure component will diminish to zero at the conclusion of each flight. It is assumed that this aspect would have a negligible effect on test results because most cyclic loads in flight are experienced once the fuselage is pressurized. Initial half crack size was 10 mm for the steel specimens and 15 mm in the Al-alloy specimens. Testing was stopped at a half crack size of 45 mm, or, when difference between the two half crack sizes exceeded 5 mm. Perhaps because of specimen alignment features, and also because average growth rates were low enough to keep the crack-front flat, crack direction never deviated noticeably from original plane, even beyond the enclosed gauge circle.
Spectrum #1 to half crack 33 mm, #2 to 37 mm and #3 beyond 37 mm For fractographic study of effects at microscopic level
crack always normal to the axial loading direction was assumed. Given relatively thin material thickness of 1–5 mm over gage area of about 75 mm diameter, the analyses assumed plane stress conditions to prevail. Calculated K-values are summarized in Fig. 4. Polynomial fits of these curves were used to determine stress intensity whilst processing experimental data. Constant amplitude fatigue crack growth rates are typically correlated against cyclic stress intensity range, DK = Kmax Kmin. In the case of biaxial loading, particularly under out-of-phase conditions estimation of Kmax and Kmin demands strict definition. In this study, it is assumed that:
2.7. Stress intensity calculation
1. Peak and valley points in the applied axial load cycle determine Kmax and Kmin values in estimating DK. 2. Kmax and Kmin values are calculated as Mode I values pertaining to the instantaneous level of biaxiality at the two points in question.
Finite Element Method (FEM) analyses of the cracked geometry under different ratios of stress biaxiality provided Mode I stress intensity estimates for discrete crack sizes and load biaxiality ratios assuming load transfer at the specimen fasteners. A straight
Thus, it is assumed that crack driving force under cyclic loading involving biaxial loading will be determined exclusively by instantaneous level of biaxiality at the peak and valley of the axial load cycle, i.e., by instantaneous computed Kmax and Kmin values. In
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Fig. 4. Computed stress intensity factor from FEM analysis assuming plane stress conditions. Results obtained for unit thickness (mm) and unit load (kN) for different magnitude of transverse stress ranging from zero (uniaxial loading) to 3 kN. Intermediate conditions may be interpolated from these curves approximated as polynomial functions.
the event of synchronous in-phase loading, both peak and valley in the axial load cycle will see the same extent of biaxiality. However, in the event of 180-degree out-of-phase loading, peak axial load will see minimum biaxiality, while the valley will experience extreme biaxiality. The above scheme to calculate Kmax and Kmin values also applies to the case when transverse load remains constant (static load). Depending on actual combination of axial and transverse loading and actual K-functions for different biaxiality, cases may not be ruled out, where the position in time of Kmax and Kmin may not coincide with that of axial peak and valley points. However, this possibility does not arise with any of the cases considered in this study. The above considerations governed calculation of DK values from the K-calibration functions for different levels of biaxility while plotting the test results for constant amplitude loading. The question of characteristic K-estimates for spectrum loading is a little more complex. Under uniaxial loading, spectrum load crack growth rates are typically plotted against Kmean (associated with spectrum mean stress), Kmax (from spectrum max stress), or, against Krms. This raises the question regarding what level of biaxiality is to be assumed because in the present experiments, transverse load was always constant, implying varying biaxiality depending on instantaneous axial load. Considering that the spectrum in use is a transport spectrum, where most load cycles are symmetric around the mean load, K was calculated for spectrum mean stress of 1.5 kN plus the actual axial load offset imposed.
3. Test results and discussion Constant amplitude test results obtained under in-phase and out-of-phase biaxial loading are summarized in Fig. 5. The data on the left are plotted against DK computed ignoring biaxiality, i.e., assuming uniaxial loading. When viewed in this format, crack growth rates under out-of-phase loading appear as much as five times greater than those under in-phase loading at a given DK. Given that DK is estimated as the difference between Kmax and Kmin and given the low stress ratio, it would follow that Kmax in the absence of transverse tensile load as in out-of-phase loading is more
damaging than in the presence of a high tensile transverse load as in the case of in-phase loading. The ability of numerical estimates of Mode I stress intensity to account for biaxiality appears to be demonstrated successfully by the data plotted on the right. Crack growth rates for the two vastly different conditions of cyclic biaxiality appear to merge into a single scatter band when plotted against Mode I DK computed after accounting for biaxiality. These data appear to confirm that Mode I DK may serve as an adequate similarity criterion for comparable stress ratios, independent of variations in biaxiality ratio, provided of course, crack path remains normal to axial loading. The data in Fig. 5 suggest that just as in the case of uniaxial loading, the primary variables are computed values of Mode I Kmax and Kmin, which in turn are determined by instantaneous biaxiality at the peak and valley in the axial loading cycle. This conjecture was put to the test of the biaxial spectrum loading, whose results appear in Figs. 6 and 7. Fig. 6 shows test results obtained under Marker-TWIST spectrum loading with and without the transverse load component. When plotted against crack size, spectrum load crack growth rates show substantial and varying sensitivity to transverse tensile load (Fig. 6, Left). Tensile transverse load tends to retard crack growth. However when plotted against Kref, that accounts for both the axial load offset as well as applied transverse load, the data seem to merge into a single scatter band. Considering crack extension occurs during the upper half of the rising portion of the load cycle, one may assume that static transverse load and cyclic in-phase transverse load with the same maximum will cause similar crack growth behaviour. One may conclude that provided most fatigue damage accrues around a given Kref as is in the case of aircraft gust load spectra, even when offset by a constant load, or, when controlled by a given Kmax, as in the case of constant amplitude loading at low stress ratio, crack growth rates under biaxial loading may be satisfactorily correlated with such a Kref or Kmax, corrected for transverse load at the same point in the load cycle. An important follow-up conclusion would be that whether the transverse component of biaxial loading is static or cyclic in-phase or out-of-phase is of little relevance. The above conclusion appears to apply even with regard to Marker-TWIST spectrum loading at different levels of axial offset load
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Fig. 5. Constant amplitude crack growth rates under in-phase and out-of-phase cycling with Pax,max = 4.5 kN and stress ratio R = 0.1 at 20 Hz and biaxiality ratio of unity. Material: 1 mm thick mild steel sheet, with axial loading along rolling direction. Accounting for instantaneous biaxiality at Pax,max and at Pax,min while calculating DK appears to merge in-phase and out-of-phase data into a narrow scatter band.
Fig. 6. (Left) Crack growth rates under Marker-TWIST spectrum loading with and without transverse load of 3 kN. Axial load offset by 1.5 kN in both cases. (Right) Same data plotted against Kref computed for sum of axial offset load and spectrum mean load (3 kN) and accounting for biaxiality. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
and transverse load equal to twice the axial offset load to simulate axial and hoop stress due to fuselage cabin pressure. Fig. 7 shows
crack growth rates obtained under the same spectrum mean load of 1.5 kN but three different axial load offsets of 0.5, 1 and 1.5 kN
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Fig. 7. (Left) Fatigue crack growth rates under Marker-TWIST load spectrum with spectrum mean load of 1.5 kN under different levels of axial load offset and corresponding transverse load. (Right) Same data plotted against Kref computed for sum of spectrum mean and offset loads with associated biaxiality. Dotted trend line appears to suggest that growth rates are likely to merge into a single scatter band. Characteristic stress intensity appears to account for biaxiality even though axial load offset to the load spectrum may vary considerably. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
and corresponding transverse tensile load of 1, 2 and 3 kN. Under uniaxial loading, any axial tensile offset in loading would accelerate crack growth rate. However, as a consequence of the simultaneous doubling of transverse load, crack growth rate actually retards with increasing axial load offset. This is evident from the data on the left. However, when plotted against Kref corrected for biaxiality, data from al three tests appear to fall into a single scatter band. The above data support the possibility that characteristic stress intensity is indeed determined by instantaneous axial load at the
point corresponding to spectrum mean load, Pm + Pax,offs, combined with biaxiality, Bi. The engineering significance of this finding lies in the identification of a single, unifying and computable fracture mechanics parameter that may account for the noticeable effect of load biaxiality under both constant amplitude as well as spectrum load fatigue crack growth. However, secondary effects may accrue from other factors sensitive to biaxiality. In this context both compliance and closure under constant transverse load become relevant parameters for examination because of their potential sensitivity to biaxiality.
Fig. 8. Effect of constant transverse load on load line unloading compliance with Pmax = 3 kN. Compliance ratio appears to show a small but reproducible non-monotonous trend with respect to crack size. However, its sensitivity to biaxiality appears negligible by comparison to that of stress intensity. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
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3.1. Compliance measurements A constant amplitude test was performed without transverse load at Pmax = 3.5 kN and Pmin = 0.5 kN. Every 2 mm of crack extension, the test was interrupted for unloading compliance measurements with and without transverse load over a data window 50%–100% of Pmax. The results appear in Fig. 8. They show that transverse load has a discernible effect on unloading compliance. This is highlighted by the plot of compliance ratio versus crack size. Unloading compliance appears to be attenuated by tensile transverse load. Liu attributes it to possible finite width effect [7] because the effect was not observed at smaller crack size. Compliance ratio increases towards axial-only loading with increasing crack size, then diminishes again. This may be due to variation in biaxiality caused by non-uniform stress distribution across the gage section. Irrespective of the actual underlying reasons for compliance ratio variation, ignoring it can cause errors in crack size estimate of 3–5 mm. As compliance is a mechanics, rather than
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mechanism driven parameter, the observed effects may be studied through analytical simulation and perhaps as a variable to be accounted for in biaxial variable amplitude fatigue. As pointed by Adams [13] unloading compliance will determine crack opening displacement and may therefore be one way in which transverse load affects fatigue crack growth. However, one may note that at the point where compliance readouts are almost the same with and without transverse load (at about 30 mm crack size), spectrum load growth rates still remain very sensitive to transverse load. 3.2. Crack closure McClung analytically determined that closure is insensitive to biaxiality until applied stress levels exceed 40% of yield stress. To experimentally investigate whether closure may be sensitive to transverse load, load–COD data were logged under slow unloading with three different transverse loads at a crack size of 30 mm. Compliance compensated COD data for the three cases appear in Fig. 9. The leftward shift between curves was caused by reduction in COD readout at maximum load under the influence of transverse load. From these three curves, one may get the impression that tensile transverse load in fact increases closure level – even though applied stress levels are less than 25% of yield stress. One may note however, that the measurement being remote from the crack tip area, may actually not reflect actual crack tip response, but rather, a far field wake response that has no bearing on crack growth behaviour. This aspect demands further study using near-tip measurement techniques such as fractography or laser indentation interferometry [18]. 4. Fractography of spectrum load biaxial fracture
Fig. 9. Load versus linear response compensated COD obtained during interrupted spectrum load test. Measurements made during slow unloading at three different levels of constant transverse load. Circles mark visually apparent points of deviation from linear response, suggesting decrease in crack closure load with decreasing tensile transverse load. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
The reasonable correlation of both constant amplitude as well as spectrum load crack growth rates with a characteristic stress intensity computed accounting for load biaxiality suggests that the effect of biaxiality may be adequately handled by computational mechanics. As shown in Fig. 8, compliance is also sensitive to biaxiality and numerical simulation of its response needs to be verified. Crack closure also appears to be sensitive to biaxiality as indicated in Fig. 9, though COD based measurements of closure are often questionable. Parameters such as Kref and compliance are mechanics driven and sensitive to instantaneous biaxial load at a given crack size. Variable amplitude fatigue is known to be load sequence sensitive. So-called crack growth life prediction models relate this sen-
Fig. 10. Composite fractograph built from multiple micrographs covering about 0.35 mm of crack extension at crack size of 38 mm in Test #5. Identified intervals are presumed to be over complete blocks of Marker-TWIST. Reduced crack extension in the first two segments suggest it must have been under maximum transverse load because this block was repeated twice, followed by one at 1.5 kN and another one at 1.0 kN after which the sequence repeats. This appears to confirm the sensitivity of crack growth at microscopic level to instantaneous levels of biaxiality. Monotonic plastic zone size is of the order of 0.3 mm. Detailed description of identification of fractographic features from this and other experiments under Marker-TWIST can be found in Ref. [21]. Material: 2.7 mm thick 2024-T3 sheet with axial loading along rolling direction.
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sitivity to plastic zone size and crack closure, both of which are known to be affected by biaxiality. Fig. 9 reflects the effect of biaxiality on closure. Plastic zone size under uniaxial loading will be considerably greater than at the same load and crack size subject to transverse tensile load. Once a plastic zone forms under uniaxial loading, its size cannot diminish instantaneously with the application of a tensile transverse load. In the present experiments, calculated spectrum load plastic zone size was of the order of 0.2– 0.4 mm at crack size close to 40 mm. Thus, in Test No. 5, this size would be of the order of crack growth interval covered by the entire fractograph shown in Fig. 10. Crack increment in a single block of the load spectrum is but a small fraction of monotonic plastic zone size. Yet, crack growth was extremely sensitive to change in transverse tensile load as shown by noticeable block-to-block changes in growth rate. It would follow that for the test conditions in question, crack growth rate is much more sensitive to change in instantaneous tensile transverse load, rather than to monotonic plastic zone size. Crack extension in individual spectrum load blocks is clearly discernible on the fractograph. From the sequence combination of repeat blocks applied at different transverse load, one can readily relate crack extension to individual blocks. It may be noted that only the transverse load component was changed from block-toblock. This underscores once again, the significance of tensile transverse load in controlling the mechanics of fatigue crack growth. More significantly, it would appear that computational procedures to merely account for the effect of instantaneous biaxiality on characteristic stress intensity may be adequate to extrapolate uniaxial fatigue crack growth rate behaviour to monotonic or cyclic biaxial loading conditions. It may be noted that the above conclusions are based on exploratory tests on sheet specimens under predominantly plane stress conditions of constant amplitude and TWIST-type of spectrum loading on steel and Al-alloy specimens. Their applicability needs to be verified on a broader base of test conditions and materials. 5. Conclusions 1. A four actuator biaxial test system was developed to test cruciform type of test coupons under constant amplitude and pseudo-random flight spectrum loading. Synchronous quadchannel digital control software was implemented to perform a wide variety of constant amplitude and spectrum load tests at test frequencies up to 25 Hz. Control scheme and specimen gripping features preclude the absence of side loads due to potential specimen misalignment. 2. Mode I K-calibration functions were established through FEM analysis for a straight central crack subject to plane stress conditions under varying degrees of biaxiality. 3. Constant amplitude fatigue crack growth tests were performed on steel sheet specimens under in-phase and out-ofphase loading. Tests were performed on Al-alloy specimens under a modified TWIST spectrum with different levels of offset in axial load and corresponding transverse tensile load to simulate load conditions on the top panel of a pressurized cabin over the wing. 4. Fatigue crack growth rates are noticeably sensitive to load biaxiality under both constant amplitude as well as spectrum loading. However, when plotted against characteristic Mode I stress intensity computed after accounting for instantaneous biaxiality, crack growth rates appear to fall into a single scatter band.
5. Characteristic stress intensity for constant amplitude in-phase and out-of-phase loading is computed accounting for instantaneous biaxiality at the peak and valley of axial load component. The effect of biaxiality at intermediate axial load levels may be negligible. 6. Under transport aircraft spectrum loading, characteristic stress intensity is computed from K at the spectrum mean load level after accounting for instantaneous biaxiality at this load.
Acknowledgements Prof. Valery Shlyannikov’s constant encouragement, support and useful suggestions made this collaborative effort possible. Sitdikov performed the FEM calculations of stress intensity. Virupaksha, Ramesh and Raghu carefully set up the experiments and processed test results. Vivek headed the design team that built the biaxial load frame. Graduating engineering students Elena Zemlyanaya, K.N. Raghavan, G.V. Sarthak, K.S. Sridhar and D. Syed participated in the testing effort. The fractograph in Fig. 10 was provided by John Porter at the University of Dayton research Institute. Extremely useful critique and suggestions from Sebastian Henkel of Freiberg Technical University are deeply appreciated. References [1] Gassner E. Strength experiments under cyclic loading in aircraft structures. Luftwissen 1939;6:61–4 [in German]. [2] Schijve J. Fatigue crack propagation in light alloy sheet material and structures, vol. 3. Pergamon Press; 1961. [3] Schijve J. Observations on the prediction of fatigue crack propagation under variable-amplitude loading. Fatigue crack growth under spectrum loads, ASTM STP 1976;595:3–23. [4] Newman JC. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. In: Chang JB, Hudson CM, editors. Methods and models for predicting fatigue crack growth under random loading, ASTM STP 1981;748: 53–84. [5] Wheeler OE. J Basic Eng Trans ASME 1972:181–6. [6] Willenborg JD, Engle RM, Wood HA. A crack growth retardation model using an effective stress concept AFFDL-TM-FBR-71-1. Air Force Flight Dynamics Laboratory; 1971. [7] Liu AF, Dittmer DF. Effect of multi-axial loading on crack growth, Vol 1 – technical summary. Air Force Flight Dynamics Laboratory Report AFFDL-TR78-175, vol. 1–3; 1978. [8] Yuuki R, Akita K, Kishi N. The effect of biaxial stress state and changes of state on fatigue crack growth behaviour. Fatigue Fract Eng Mater Struct 1989;12(2):93–103. [9] Hopper CD, Miller KJ. Fatigue crack propagation in biaxial stress fields. J Strain Anal 1977;12(1). [10] Anderson PRG, Garrett GG. Fatigue crack growth rate variations in biaxial stress fields. Int J Fract 1980;16:R111–R1116. [11] Ogura K, Ohji K, Ohkubo Y. Fatigue crack growth under biaxial loading. Int J Fract 1974;10:609. [12] McClung RC. Closure and growth of Mode I cracks in biaxial fatigue. Fatigue Fract Eng Mater Struct 1989;12(5):447–60. [13] Adams NJI. Some comments on the effect of biaxial stress on fatigue crack growth and fracture. Eng Fract Mech 1973;5:983–91. [14] Lebedev AA, Muzyka NR. Design of cruciform specimens for fracture toughness tests in biaxial tension, Review paper, Strength Mater 1998; 30(3): 243–54. [15] Muzyka NR. Equipment for testing sheet structural materials under biaxial loading. Part 2. Testing by biaxial loading in the plane of the sheet. Strength Mater 2002;23(4). [16] Radon JC, Leevers PS, Culver LE. J Exp Mech 1977:228–32. [17] Wu XD, Wan M, Zhou XB. Biaxial tensile testing of cruciform specimen under complex loading. J. Mater Process Technol 2005;168(1):181–3. [18] Makinde A, Thibodeau L, Neale KW. Development of an apparatus for biaxial testing using cruciform specimens. Exp Mech 1992;32(2). [19] Sunder R. Development of the Marker-TWIST Load Sequence for Quantitative Fractographic Studies. Int J Fatigue 2010. [21] Porter JW, Rosenberger AH, Sunder R. Contribution of individual cycles to crack extension under a transport aircraft wing load spectrum – A fractographic analysis. In: International conference on fatigue damage of structural materials VIII, Hyannis; Sep 19–24, 2010. [To be submitted to Int J Fatigue 2010].