An ultrasonic method for dynamic monitoring of fatigue crack initiation and growth Bao Mi, Jennifer E. Michaels,a兲 and Thomas E. Michaels School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
共Received 13 June 2005; revised 24 October 2005; accepted 27 October 2005兲 Attached ultrasonic sensors can detect changes caused by crack initiation and growth if the wave path is directed through the area of critical crack formation. Dynamics of cracks opening and closing under load cause nonlinear modulation of received ultrasonic signals, enabling small cracks to be detected by stationary sensors. A methodology is presented based upon the behavior of ultrasonic signals versus applied load to detect and monitor formation and growth of cracks originating from fastener holes. Shear wave angle beam transducers operating in through transmission mode are mounted on either side of the hole such that the transmitted wave travels through the area of expected cracking. Time shift is linear with respect to load, and is well explained by path changes due to strain combined with wave speed changes due to acoustoelasticity. During subsequent in situ monitoring with unknown loads, the measured time of flight is used to estimate the load, and behavior of the received energy as a function of load is the basis for crack detection. Results are presented from low cycle fatigue tests of several aluminum specimens and illustrate the efficacy of the method in both determining the applied load and monitoring crack initiation and growth. © 2006 Acoustical Society of America. 关DOI: 10.1121/1.2139647兴 PACS number共s兲: 43.25.Zx, 43.35.Yb, 43.25.Dc, 43.35.Cg 关YHB兴
I. INTRODUCTION
The remaining life of many complex structures is often assumed to be correlated to usage, where usage is estimated by simple quantities such as flight hours for aircraft, years for bridges and roads, and number of starts for diesel engines.1 However, uneven and unknown loading conditions and unpredictable environmental disturbances cause some structural components to accumulate more damage than others with identical service lives. If a structural component is retired based simply on its service life, both safety and economics are compromised. Nondestructive evaluation 共NDE兲 has provided one way to obtain information necessary to determine the current health condition of a structure. NDE methods are implemented either with hand-held equipment following a well-defined inspection procedure or by using automated inspection systems. Either can be quite costly, particularly when intervals between inspections are short. Reducing inspection costs by integrating sensors with structural components has motivated development of in situ, real time methods for structural health monitoring 共SHM兲. In structures made of ductile alloys that are subjected to fatigue failure, a large part of the service life is spent in crack initiation and the presence of very small cracks. One of the major concerns with metallic aircraft in general is fatigue cracking, and developing methods for in situ monitoring of the onset and growth of cracks in such critical structures is of growing interest and importance for SHM.1,2 Ultrasonic methods have been extensively applied to crack detection and sizing for NDE during scheduled maintenance, and are thus a strong candidate for SHM.3
a兲
Electronic address:
[email protected]
74
J. Acoust. Soc. Am. 119 共1兲, January 2006
Pages: 74–85
Early research efforts on characterizing cracks ultrasonically have primarily been motivated by the need for NDE of critical components. The time-of-flight diffraction technique has been used for bulk crack detection and sizing.4,5 Surface wave scattering6 has been deployed for detection and sizing of surface-breaking cracks, and Lamb waves have been investigated for detecting rivet hole cracks in a plate.7,8 Physical and numerical models have been developed to predict the ultrasonic response to cracks in given materials or structures,9,10 and these models lay a foundation for interpretation of ultrasonic signals. Recent work on ultrasonic characterization of cracks and flaws has included SHM applications with the goal of in situ monitoring of structural integrity. Examples include using an array of surfacemounted PZT wafer elements for monitoring crack growth in a uniaxial specimen exposed to cyclic loading,11 modulation of surface acoustic waves by cracks opening and closing under load,12 vibration modal analysis to characterize fatigue cracks in a steel beam,13 and modulation of torsional waves in a cracked rod.14 Described here is an ultrasonic-based SHM method that has been developed for real time, in situ monitoring of fastener hole cracks in aluminum components. The fastener hole is monitored using an angle beam through transmission technique incorporating two transducers, one on each side of the hole. As the applied tensile stress is increased, the received signal shifts in time due to a combination of specimen elongation and change in ultrasonic velocity arising from the acoustoelastic effect. If a crack is present, the received signal also decreases in amplitude as the crack opens under stress. If this applied tensile stress is sufficiently large to open the crack, then the ratio of the received ultrasonic energy under stress to that with no stress is a reliable indicator of the
0001-4966/2006/119共1兲/74/12/$22.50
© 2006 Acoustical Society of America
FIG. 1. Specimen geometry including transducer locations and ultrasonic beam paths 共thickness h = 5.72 mm and hole diameter d = 4.83 mm兲.
presence of the crack, and growth of the crack can be monitored by tracking this energy ratio during the fatiguing process.15 When making these measurements in the laboratory, either the fatiguing process can be stopped and measurements made under controlled loading conditions, or a suitable data acquisition system can be interfaced to the fatigue machine to record applied loads and resulting strains. However, it is generally neither feasible nor possible to record actual applied stresses for such an SHM system that is deployed on a real structure. To overcome this problem, the method reported here utilizes the time shift of the ultrasonic signal to deduce the applied stress, and thus enables real time, in situ monitoring of crack growth by means of the through transmission energy ratio.16 Experimental procedures for both the fatigue tests and the ultrasonic measurements are described in Sec. II. Data analysis methods are explained in Sec. III, and a theoretical analysis of the time shift due to loading is given in Sec. IV. Results are shown in Sec. V and concluding remarks are made in Sec. VI. II. EXPERIMENTAL PROCEDURES
Coupons for fatigue testing were fabricated of 7075T7351 and 7075-T651 aluminum as shown in Fig. 1. Fastener holes are simulated by two holes of 4.83 mm diameter machined in the center of the 5.72-mm-thick coupons. A two-hole coupon was chosen to obtain a slightly asymmetric stress field typical of that experienced by a fastener hole at the end of a row of holes. Specimens were fatigued with a uniaxial load at a rate of 5 Hz using an aperiodic loading spectrum as illustrated in Fig. 2 to simulate the variable loads on a structure during service. The maximum load value in
FIG. 2. Typical 500-cycle segment from the fatigue loading spectrum where the 100% value corresponds to 75620 N 共17000 lb兲. J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
FIG. 3. Photograph of a two-hole fatigue coupon with attached angle beam transducers.
the spectrum; i.e., the 100% value, was 75620 N 共1700 lb兲, which corresponds to a maximum uniaxial stress of 279 MPa 共40447 psi兲 based upon the unreduced cross-sectional area. The average stress over the cross section going through the center of the holes was 350 MPa, which represents approximately 75% of the yield point for this material. The load cycles were grouped into blocks of 2640 cycles each, and typical time to failure was between 15 and 20 blocks. Fatigue testing was halted between blocks to take “static” measurements of ultrasonic response versus load. Ultrasonic measurements were also recorded during the fatiguing process, and these measurements are referred to as the “dynamic” measurements. The dynamic measurements were taken at random times so that the actual load associated with each waveform was not known, but a sufficient number of waveforms were recorded to ensure that waveforms were obtained throughout the entire load range. Two pairs of 70° shear wave angle beam transducers 共Panametrics model A5054, 10 MHz center frequency兲 were bonded to the coupons using a five minute, two-part epoxy as pictured in Fig. 3. Each transducer was positioned and aligned with the hole based upon the pulse-echo signal reflected from the hole. The distance from the transducer to the hole was set so that the shear wave half and full V reflections from the bottom and top of the hole were approximately balanced in amplitude. This configuration is equivalent to the transducer location where the center of the beam hits the midpoint of the hole, but due to beam spread, both corner reflections are present. This pulse-echo signal was used to monitor the transducer attaching process for all four transducers. In principle, the proper transducer location could be calculated from the beam angle and the thickness of the specimen. In practice, however, it was found necessary to monitor the pulse-echo signals during the transducer attachment process due to both beam skew and transducer-totransducer refracted angle variations. Both static and dynamic ultrasonic measurements were performed throughout the fatigue tests. For the static measurements, fatiguing was paused and static loads were applied from 0 to 44482 N 共0 to 10000 lb兲 in 4448 N 共1000 lb兲 increments, which corresponds to stresses ranging from 0 to a maximum of 164 MPa 共23793 psi兲 away from the holes. Dynamic measurements were taken at random times near the beginning of each block of fatigue cycles. All measurements were made using a Panametrics 5072PR Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
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FIG. 4. Typical angle beam through transmission signal showing the single and double V path arrivals.
pulser receiver, and were recorded with a Tektronix TDS5034 digital oscilloscope operating at a sampling frequency of 125 MHz. III. DATA ANALYSIS
When both angle beam transducers in the pair are aligned as described, the through-transmission ultrasonic signal has two dominant echoes as shown in Fig. 4. These echoes correspond to the single and double V paths illustrated in Fig. 1, and vary in two ways during fatigue loading. First, the amplitudes of the echoes vary as a function of both load and damage, and second, the echoes shift slightly in time with increasing load. Figure 5共a兲 shows the time shift prior to crack formation, and Fig. 5共b兲 shows the amplitude change that occurs after cracks have formed. Data analysis consists of characterizing the amplitude change by means of an energy ratio parameter, and measuring the time shift of both the single and double V echoes. A. Energy ratio from static measurements
Rnorm =
In previous work by the authors, a normalized energy ratio method was shown to be a robust and self-calibrating method for crack monitoring.15 To obtain this energy ratio, fatigue tests were interrupted to apply known static loads and make ultrasonic through transmission measurements as a function of load. This technique is reviewed here. The first step in determining the energy ratio is to calculate energy values for a windowed portion of the recorded through transmission ultrasonic waveform as a function of applied load expressed in terms of stress, , where x共t ; 兲 is the recorded waveform, E共兲 =
冕
t2
x2共t; 兲dt.
FIG. 5. Effects of tensile loading on the received through transmission angle beam signals. 共a兲 Before the onset of cracking, the ultrasonic signal is shifted in time as load is applied but the amplitude is unaffected. 共b兲 After crack initiation, increasing load causes the ultrasonic signal to both shift in time and decrease in amplitude.
共1兲
R Rinitial
.
共3兲
For the work reported here, a 6.0 s time window encompassing both the single and double V echoes was used for calculation of energy as shown by the larger box in Fig. 4. Static ultrasonic measurements were made and energy ratios were calculated after each block of fatigue cycles. These energy ratios are the ones used as benchmarks for comparison with those obtained from the dynamic measurements. It was experimentally determined by observing the through transmission response that a load of 22241 N 共5000 lbs兲 was sufficient to open a crack, and the corresponding stress of 82 MPa 共11896 psi兲 is the value used for ref.
t1
Next, a ratio is formed of the energy at two different loads: a reference load and no load, R=
E共 = ref兲 . E共 = 0兲
共2兲
Finally, the energy ratio is normalized by the energy ratio value obtained before the fatigue loading was initiated, i.e., the no-damage condition, 76
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B. Determination of load from time shift
The axial loading exerted on the specimen shifts the ultrasonic signals in time, as can be seen in Figs. 5共a兲 and 5共b兲. Following previous work by Mi et al.,16 the time shift between the signals under the two loading conditions, referred to as the “delta time-of-flight” 共⌬TOF兲, was calculated from the time of the peak of the cross-correlation function of the two signals. A 1.0 s time window was used to individuMi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
FIG. 7. Loads calculated from the change in ultrasonic time of flight as determined from 50 consecutive dynamic measurements.
FIG. 6. Change in time of flight of the ultrasonic signal versus load referenced to the zero-load time at the start of each fatigue block. 共a兲 Single V arrival. 共b兲 Double V arrival.
ally gate the single and double V echoes as shown by the two smaller boxes in Fig. 4 since these two peaks are shifted by different amounts. The resolution of the peak location is within one sampling interval of the digitized signal, i.e., 8 ns at the sampling frequency of 125 MHz, and it was improved to 0.8 ns by using piecewise cubic spline interpolation. Experimentally determined values of ⌬TOF versus applied load are shown in Fig. 6 for a specimen that failed in the 17th block of fatigue cycles. For this specimen, static measurements were taken for the initial no-damage condition and after each block of fatigue cycles, and the ⌬TOF values are referenced to the zero-load state at each block. Figure 6共a兲 shows the single V path results, and Fig. 6共b兲 shows the double V path results. The curves are very consistent until near the end of life when sizable cracks are present. If the curves are referenced to the initial no-damage state, there is a progressively increasing offset at zero load of up to approximately 20 ns as fatiguing progresses. It is believed that most if not all of this offset is due to a combination of permanent specimen deformation increasing the ultrasonic path and an increase in temperature during fatiguing caused by internal friction. Note that the double V path curves are less affected by the presence of a large crack than the single V path curves. This result is not unexpected because, for this specimen, the amplitude of the double V path echo is much larger than that of the single V path echo. Thus its shape would tend to be less affected by the crack because small signals scattered from the crack would contribute less to the overall echo, and the signal could sustain more of a loss in amplitude before it is reduced to the same level as the background coherent and J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
random noise. If two echoes have similar shapes, then the cross-correlation method will provide a good estimate of the time shift between them. Therefore, ⌬TOF curves from the shear wave double V path are better suited than those from the single V path for deducing actual loads from the recorded dynamic waveforms. As shown in Fig. 6, axial loading exerted on the specimen shifts the ultrasonic signals linearly about 50 ns per 150 MPa for the single V path signal, and about 75 ns per 150 MPa for the double V signal. The statically determined ⌬TOF versus applied load data from the initial no-damage state are referred to as calibration curves for both single and double V arrivals. These piecewise linear curves were subsequently used to determine the effective applied load from measured single and double V echo times as referenced to zero-load values. Typical results are shown in Fig. 7 from 50 dynamically measured ultrasonic waveforms using both single and double V calibration curves; they are in good agreement with each other. The estimated loads, although not measured directly and thus unverified, fall within the expected range based upon the load spectrum. However, as mentioned previously, the double V ⌬TOF curve is least affected by the presence of cracks emerging from the fastener holes and thus is more accurate near the end of fatigue life. C. Energy ratio from dynamic measurements
Dynamic measurements are made by randomly acquiring waveforms during fatigue loading as previously explained, and the ⌬TOF calibration curves are used to assign an applied load to each dynamically measured signal.16 The specific steps used for calculating energy ratios from these dynamic measurements are as follows: 1. A calibration curve relating applied load to ⌬TOF is obtained from static measurements prior to fatiguing. 2. Ultrasonic waveforms are randomly recorded during fatigue loading. 3. The ⌬TOF is obtained by cross-correlating each signal with the corresponding static no-load signal. 4. The ⌬TOF curves obtained from static calibrations are used to assign an effective axial load value to each waveform. Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
77
5. The energy is calculated for each waveform as per Eq. 共1兲. 6. Energy is plotted versus load, and data are locally averaged to reduce noise. 7. The energy at both zero load and the reference load is obtained by a local linear fit. 8. The energy ratio is computed from these energy values as per Eq. 共2兲. 9. The energy ratio is normalized by that obtained at the beginning of the fatigue test as per Eq. 共3兲. In this manner the normalized ultrasonic energy ratio can be monitored using dynamic data acquired during the fatigue life of a specimen. IV. THEORETICAL ANALYSIS OF TIME SHIFT CAUSED BY LOADING
Critical to determining applied load from dynamic measurements is consistent time shifting of the ultrasonic signal resulting from the load. Many factors can affect the ultrasonic time of flight during loading; these include changes in geometry, acoustic velocity, crack size, ultrasound beam bending, plastic deformation, and measurement conditions. Among these, changes in geometry and acoustic velocity are hypothesized to be the two dominating factors. The effects of the crack itself and plastic deformation around the hole on the time of flight are very localized, affecting only a small portion of the ultrasonic beam path, and can be safely ignored, particularly during early stages of fatiguing. Near the end of life the crack size becomes significant and its effect on the ultrasonic time of flight becomes obvious, as can be seen in Figs. 6共a兲 and 6共b兲. Modeling the effect of the crack is extremely difficult, if not impossible, due to uncertainties in initiation, orientation, and location. In the following analysis, change of geometry and acoustoelasticity will be considered in order to determine if they are indeed the dominant factors. For a generic ultrasonic wave arrival, the time of flight is given by TOF =
P , v
relative time of flight caused by acoustoelasticity 共change in ultrasonic velocity兲. A. Time shift due to geometry
Axial loading results in an elongation of the specimen in the loading direction and a contraction in the transverse directions due to the Poisson effect. These strains are assumed to be uniform between the two transducers, and local plastic deformation around the hole is ignored. The ultrasonic beam path is approximated by straight lines from the center of the transmitter to the edge of the hole and ending at the receiver. The ultrasonic velocity for the entire path is assumed to be the shear velocity; the effect of the creeping wave around the edge of the hole is ignored. Referring to Fig. 1, the single and double V paths in the unstrained specimen are P1 = 2
P2 = 2
1 TOF TOF P ⌬P + ⌬v = ⌬P − 2 ⌬v . P v v v
D2 + h2 +
冉冊 冉冊 d 2
D2 + 共2h兲2 +
2
共8兲
,
d 2
2
,
共9兲
where P1 is the single V path, P2 is the double V path, D is the distance from the transducer to the center of the hole, h is the thickness of the specimen, and d is the diameter of the fastener hole. Changes in the paths P1 and P2 due to small changes in D and h are expressed as ⌬P1 =
P1 P1 ⌬D + ⌬h, D h
共10兲
⌬P2 =
P2 P2 ⌬D + ⌬h. D h
共11兲
If ⑀ is the strain in the direction of loading, then ⌬D = ⑀D and ⌬h = −⑀h, where is Poisson’s ratio. Then ⌬P1 and ⌬P2 can be expressed in terms of ⑀, ⌬P1 =
4共D2 − h2兲 ⑀, P1
共12兲
⌬P2 =
4共D2 − 4h2兲 ⑀. P2
共13兲
共4兲
where P is the total propagation path and v is the ultrasonic velocity. Small changes in time of flight are due to small changes in both the path and the ultrasonic velocity,
冑 冑
Thus, changes in time of flight due to geometry for single and double V paths are
共5兲
⌬TOFG1 =
⌬P1 4共D2 − h2兲 = ⑀, P 1v v
共14兲
The two components of the total change in time of flight are defined as
⌬TOFG2 =
⌬P2 4共D2 − 4h2兲 = ⑀. P 2v v
共15兲
⌬TOF =
1 ⌬TOFG = ⌬P, v ⌬TOFA = −
P ⌬v , v2
共6兲
共7兲
where ⌬TOFG is the relative time of flight caused by geometry 共change in propagation path兲, and ⌬TOFA is the 78
J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
Figure 8 shows the calculated ⌬TOF versus load curves for the single V and double V arrivals based on geometry change only. The distance D from the center of the hole to the nominal exit point of the shear wave from the transducer was measured to be 17.7 mm. The nominal thickness of the specimen, h, was 5.72 mm and the hole diameter, d, was 4.83 mm. Compared to the experimental data of Fig. 6, the Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
FIG. 9. Coordinate system definition for acoustoelasticity analysis of angle beam ultrasonic wave propagation.
FIG. 8. Calculated change in time of flight versus load for both single and double V arrivals due to geometrical effects only.
geometry change accounts for less than half of the change in time of flight due to loading. It is interesting to note that in Fig. 8 the double V curve shows a smaller time shift than the single V curve because more of the double V propagation path is in the thickness direction, which is reduced due to the Poisson effect. Since the actual behavior is reversed with the double V curve exhibiting the larger time shift, it is clear that geometry changes alone cannot explain the observed time shifts. B. Time shift due to acoustoelasticity
The acoustoelastic effect refers to the change in acoustic wave velocities in a material as a function of applied stress. The theory of acoustoelasticity is based on a continuum theory of small disturbances superimposed on an elastically deformed body. There is a long history of theoretical and experimental work on acoustoelasticity with an early version of the modern theory developed by Hughes and Kelly17 in 1953. Pao et al. reviewed the theory of acoustoelasticity in 1984 with an emphasis on measurement of residual stresses.18 Pao and Gamer considered acoustoelasticity in orthotropic media due to an arbitrary homogeneous elastic or plastic strain field,19 and Qu and Liu investigated acoustoelasticity for guided wave propagation in layered media.20 Acoustoelasticity has also been experimentally investigated for damage detection.21 For a specific experiment, an acoustoelastic constant K can be defined that linearly relates change in velocity v to applied stress , ⌬v = K . v
共16兲
The constant K is dependent upon the specific wave mode, propagation direction, and stress orientation, and is a function of second- and third-order elastic constants. For the work presented here, consider a coordinate system with the x1 axis aligned along the direction of propagation of the shear wave, as shown in Fig. 9. We are interested in the vertically polarized shear wave 共SV wave兲 with particle motion in the x2 direction and assume a uniaxial stress field, neglecting the stress concentration in the vicinity of the hole. The effective acoustoelastic constant for this SV wave due to the applied uniaxial stress field is of the form J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
⌬v = 关K1 sin2 + K2 cos2 兴 = K共兲 , v
共17兲
where is the refracted angle, and K1 and K2 are defined as K1 =
K2 =
4共 + 兲 + m + 41 n
,
共18兲
共 + 2兲 + m + 41 n . 22共3 + 2兲
共19兲
22共3 + 2兲
In these equations and are the Lamé constants, and l, m, and n are the Murnaghan third-order elastic constants. The derivation of these equations is given in the Appendix. It can be seen that K1 is the acoustoelastic constant for a shear wave propagating in the direction of applied stress 共 = 90° 兲, and that K2 is the acoustoelastic constant for a shear wave propagating perpendicular to the direction of applied stress and with parallel polarization 共 = 0 ° 兲. The change in wave velocity is ⌬v = vK共兲 .
共20兲
The resulting change in time of flight due to this change in wave velocity for the single and double V paths is determined by substituting Eq. 共20兲 into Eq. 共7兲, ⌬TOFA1 = −
冋
=−
⌬TOFA2 = −
册
K共1兲P1 P1 2 ⌬v = − v v
冋
冋
册
2K共1兲冑D2 + h2 + 共d/2兲2 , v
册
K共2兲P2 P2 ⌬v = − v2 v =−
冋
共21兲
册
2K共2兲冑D2 + 共2h兲2 + 共d/2兲2 . v 共22兲
Note that 1 and 2 refer to the refracted angles for the two V paths, which are different. If i and the corresponding acoustoelastic constant K共i兲 are known, then the time shift can be readily calculated for any value of stress. Calculation of K共兲 can be done either directly from Eq. 共17兲 if K1 and K2 are known from experimental measurements, or by first calculating K1 and K2 from the elastic constants as per Eqs. 共18兲 and 共19兲. Numerical values for acoustoelastic constants for most materials are not readily available due to the difficulty in making the measurements. Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
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TABLE I. Lamé second-order elastic constants 共 and 兲 and Murnaghan third-order elastic constants 共l, m and n兲 for 7075-T651 aluminum.22 Constant
Value 共GPa兲
l m n
54.9 26.5 −252.2 −325.0 −351.2
Similarly, since values for third-order elastic constants are usually determined from measurements of acoustoelasticity, they are also not readily available. Here we use values of , , l, m, and n determined by Stobbe22 for 7075-T651 aluminum and summarized in Table I to calculate K1 and K2; values are −1.571⫻ 10−11 and −3.458⫻ 10−11 Pa−1, respectively. The velocity of the shear wave speed in the undeformed material is determined from c = 冑 / and is calculated to be 3.076 mm/ s using a value of 2800 kg/ m3 for the density . The refracted angles of the single and double V paths were calculated to be 72° and 57°, respectively, based upon a measured value of 17.7 mm for D. Figure 10共a兲 shows the comparison between the experimentally measured ⌬TOF and the theoretical predictions using the above parameters. The theoretical calculation includes the ⌬TOF based on the elastic deformation 关Eqs. 共14兲 and 共15兲兴 and the ⌬TOF based on the acoustoelastic effect 关Eqs. 共21兲 and 共22兲兴. Note that the two theoretical curves fall between the two experimental curves, suggesting that there is
a geometrical term not accounted for in the model rather than errors in the elastic constants. One assumption that is clearly an approximation is that the single and double V propagation paths originate from the same point on the surface of the specimen. In reality, due to beam spreading in the transducer wedge, the single V path originates slightly closer to the hole and the double V path originates slightly further away from the hole; these distances were estimated to be 1 mm based upon visual inspection of the wedge geometry, resulting in a value of D equal to 16.7 mm for the single V path and 18.7 mm for the double V path. Using these values, theoretical curves were recalculated and are shown in Fig. 10共b兲, now exhibiting excellent agreement with the experimental results. The fact that the theoretical calculations agree well with the experimental measurements supports the assumption that the ⌬TOF is caused primarily by elastic deformation and acoustoelasticity resulting from a uniaxial stress field, and that other effects can be neglected.
C. Temperature effects
Although not specifically investigated here, a discussion of time shifts would not be complete without considering temperature effects. Temperature changes are unavoidable outside of the laboratory and are well known to cause time shifts in received ultrasonic signals due to both thermal expansion and change in acoustic velocity with temperature. Changes in temperature can contribute to both offset and change in slope of the ⌬TOF versus load curves. In considering temperature changes, we assume that they are quasistatic, occurring over a longer time scale than load changes. Thus load effects are considered to be superimposed on the temperature effects. Changes in propagation path P and acoustic velocity v due to a change in temperature of ⌬T are ⌬P = P␣⌬T,
共23兲
⌬v = ⌬T,
共24兲
where ␣ is the coefficient of thermal expansion and is the change in acoustic velocity for a unit change in temperature. Substituting these expressions into Eq. 共5兲 gives the change in time of flight due to a temperature change of ⌬T assuming that there is no applied load, ⌬TOF =
FIG. 10. Theoretical change in time of flight compared to measurements for data recorded after six fatigue blocks. 共a兲 Calculations using the same transducer-hole distance D for both single V and double V paths. 共b兲 Transducer-hole distances adjusted to compensate for differences in single V and double V paths. 80
J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
冋 册
P ␣− ⌬T. v v
共25兲
Typical values for ␣ and for aluminum are 2.43 ⫻ 10−5 / ° C and −0.752 m / s / ° C, respectively.23 Using these values, we calculate the quantity in square brackets in Eq. 共25兲 to be 2.64⫻ 10−4 / ° C. For the double V path, a 10 ° C change in temperature results in a ⌬TOF of 36 ns, which would cause the overall ⌬TOF versus load curve to be offset by this amount. Although this time shift is quite Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
FIG. 11. Energy versus load at various fatigue blocks where energy is normalized to the zero load value for each block.
significant, compensation can be made by referencing the ⌬TOF versus load curves to zero load values, as was done in Fig. 6. To understand how a temperature change affects the slope of the ⌬TOF versus load curve, the initial linear dimensions and velocity are perturbed as per Eqs. 共23兲 and 共24兲 to reflect the quasistatic temperature change of ⌬T. The various changes in time of flight are then calculated as per Eqs. 共14兲, 共15兲, 共21兲, and 共22兲, dropping terms higher than quadratic in the perturbations ⑀, ⌬P / P, and ⌬v / v. The result is a general expression for how the change in time of flight due to applied load is affected by a quasistatic change in temperature,
冋 冉 冊 册
⌬TOF共 ;⌬T兲 = ⌬TOF共 ;⌬T = 0兲 1 + ␣ −
v
⌬T . 共26兲
The quantity in square brackets changes the slope of the ⌬TOF versus load curve. For an extreme temperature change of 50 ° C, this quantity is only 1.0132, causing an insignificant error of approximately 1 ns for a typical ⌬TOF of 80 ns. Thus, as long as changes in time of flight are referenced to no-load values at the same temperature, and temperature changes are truly quasistatic relative to changes in load, temperature effects can be safely neglected.
V. RESULTS
Since practical implementation of this in situ ultrasonic approach depends upon dynamic data acquisition without interruption of service, the key is to accurately estimate the load under which the dynamic measurements are taken because the ultrasonic features are dependent upon load. This is especially true when cracks are present since the opening and closing of the cracks by the load modulate the ultrasonic signals. As an example, Fig. 11 shows the ultrasonic energy as a function of applied load for various fatigue blocks. This figure demonstrates that the energy changes with applied load, particularly for the later fatigue blocks as the crack size increases. J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
FIG. 12. Comparison of energy versus fatigue cycles as determined from both static and dynamic measurements. 共a兲 No load. 共b兲 Load of 82 MPa.
Figure 12 compares results from static and dynamic measurements in the form of energy versus fatigue cycle curves for the unloaded and loaded specimen. For the dynamic measurements, the load was estimated from the time of flight relative to the minimum recorded for that fatigue block 共assumed to be the no-load reference兲 using the ⌬TOF calibration curve established from the undamaged specimen. The energy was calculated by averaging the energy from signals in the neighborhood of the target load. Overall, the static and dynamic curves agree reasonably well for both no load as shown in Fig. 12共a兲 and under a load of 82 MPa as shown in Fig. 12共b兲 except for the first few blocks. This inconsistency, which is not always observed to be present, is suspected to be due to coupling stabilization and initial strain hardening. The through-transmission energy curve measured at 82 GPa begins to decrease when the cracks inside the fastener hole approach a surface length of approximately 0.25 mm 共0.01 in.兲, which is when they are just visible using a 10⫻ magnifier. For the last 40% of the fatigue life, the drop is approximately linear and approaches a relative energy of zero at 100% of expended life. The no-load curve also exhibits a drop, but not until approximately 80% of life expended. After 80% of life, the no-load and 82 GPa load curves maintain a clear separation for the remainder of the specimen life. This difference in response between no-load and loaded crack conditions is consistent with the assumption that the opening of the crack under load blocks the ultrasonic energy from propagating across the crack. Figure 13 shows a comparison of the normalized energy ratio for two specimens as calculated from Eq. 共3兲. The first specimen was fatigued until failure with the crack fully deMi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
81
FIG. 13. Comparison of normalized energy ratio versus fatigue cycles as determined from both static and dynamic measurements for two specimens. 共a兲 S3-0001 共7075-T7351兲. 共b兲 S4-0030 共7075-T651兲.
veloped, and the second specimen was fatigued until shortly after the crack was detectable. The cross-sectional images of the cracks after the specimens were fractured are shown in Fig. 14. In both cases, the dynamic measurements agree very well with the static measurements. Not considered here is estimation of crack sizes, which is the topic of another paper,24 but results here confirm that crack size estimates based upon dynamic data should be in good agreement with those determined from static measurements.
VI. SUMMARY AND CONCLUSIONS
An ultrasonic method has been developed to monitor initiation and growth of fatigue cracks emerging from fastener holes during the fatiguing process, and has been verified in aluminum two-hole coupons. This method uses permanently mounted miniature angle beam transducers, which transmit ultrasonic shear waves through a region near a fastener hole that is targeted for monitoring. The energy of the ultrasonic waves transmitted through the affected region is reduced due to the presence of cracks and, further, this energy is modulated by the applied load during fatiguing due to opening and closing of the crack. A normalized energy ratio is calculated as a robust metric for monitoring progression of crack growth, which requires measurement of ultrasonic waveforms as a function of applied load. The work reported here shows results from development of a new method based upon making “dynamic” measurements during fatiguing without either interrupting the test or measuring loads. Waveforms are acquired at random times unsynchronized with the 82
J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
FIG. 14. Cross-sectional images of two holes after coupons were fractured. 共a兲 S3-0001 共7075-T7351兲. 共b兲 S4-0030 共7075-T651兲.
load spectrum, and small shifts in the arrival times of various shear wave components are used to estimate the uniaxial load associated with each waveform, thereby enabling calculation of energy ratios as a function of load. Results agree well with those obtained from static measurements for which fatiguing was interrupted and ultrasonic measurements were made as a function of known applied load. A theoretical analysis of time shift versus applied load shows that change in path due to elastic deformation combined with the acoustoelastic effect adequately explains the measurements. Both temperature changes and plastic deformation can be ignored as long as the times are referenced to those obtained under no load at the same temperature and state of plastic deformation. Alternatively, measured temperatures could be used to compensate the ultrasonically measured times. The eventual application of this method is expected to be in situ ultrasonic monitoring of crack initiation and growth near “hot spots” in service critical structures. The opening and closing of cracks under applied load modulate the ultrasonic energy that can be transmitted through a crack. In order to properly interpret results under unmeasured and variable loading, it will be necessary to know the actual stress conditions near the crack, and the time-of-flight method demonstrated in this paper is expected to play a key role. Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
ACKNOWLEDGMENTS
⌫␣␥␦ = C␣␥␦ + C␣␦
The work of D. M. Stobbe and A. C. Cobb in assisting with the measurements and analysis is gratefully acknowledged. The authors acknowledge and appreciate the support of the Defense Advanced Research Projects Agency 共DARPA兲, Defense Sciences Office, Structural Integrity Prognosis System Program, Contract No. HR0011-04-0003 to Northrop Grumman Corporation, Integrated Systems. This paper is approved for public release with unlimited distribution.
共A3兲 Here the C␣␥␦⑀ are the third-order elastic constants. These equations are derived by Pao,19 Dorfi,25 and others for an arbitrary homogeneous strain field. Here we consider propagation of a plane wave in the x1 direction, which is rotated from the direction of applied uniaxial stress as shown in Fig. 9 where is the refracted angle. The Christoffel equation obtained from Eq. 共A1兲 is 共A␣1␥1 − oV2␦␣␥兲U␣ = 0.
APPENDIX: EQUATIONS OF ACOUSTOELASTICITY FOR ANGLE BEAM SV WAVES
2u ␥ = ou¨␣ , a  a ␦
共A1兲
冤
冤
共A2兲
2共 + 兲sin2 − cos2 2共3 + 2兲
− sin cos 2
0
− sin cos 2
− sin2 + 2共 + 兲cos2 2共3 + 2兲
0
0
0
− 2共3 + 2兲
Both second- and third-order elastic constants are needed to calculate the A␣1␥1. Calculation of all nine of the A␣1␥1 coefficients is not necessary to determine the velocity of the SV wave. Consider the eigenvalue problem of Eq. 共A4兲 expressed in matrix form, J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
共A5兲
For a homogeneous and isotropic material, which is assumed to be the case here, there are two independent second-order elastic constants and three independent third-order elastic constants. We use the Lamé constants and and the Murnaghan constants l, m, and n. Relationships between various third-order elastic constants are given by Norris.26 The resulting strain tensor can then be calculated as
The C␦⑀ are the second-order elastic constants, e⑀i is the initial finite strain tensor, ␦␣␥ is the Kronecker delta function, and ⌫␣␥␦ is given by
˜⑀ =
冥
sin2 − sin cos 0 ˜ = − sin cos cos2 0 . 0 0 0
where ai are the natural coordinates, ui are the components of displacement, and o is the density in the natural state. Summation over repeated indices is implied in this and subsequent equations. The coefficients are given by A␣␥␦ = C␦⑀e⑀i ␦␣␥ + ⌫␣␥␦ .
共A4兲
In this equation, V is the acoustic velocity and U␣, the eigenvector, is the polarization vector. We are interested in the SV wave, which corresponds to the eigenvector that is primarily in the x2 direction. Calculation of the nine A␣1␥1 coefficients requires knowledge of the stress and strain tensors. The stress tensor in this rotated coordinate system resulting from a uniaxial stress of is given by
Acoustoelasticity theory is based upon the propagation of a small-amplitude elastic wave in a material subjected to a finite strain. Here we follow the derivation given by Pao19 for propagation of plane elastic waves in a homogeneous orthotropic medium with initial strains. The equations of motion in the natural 共unstrained兲 coordinates for the elastic wave are
A␣␥␦
u␥i u␣i + C␥␦ + C␣␥␦⑀e⑀i . a a
冤
冥
共A6兲
.
A1111 − oV2
A1121
A2111
A2121 − oV
A3111
A3121
A1131 2
A2131 A3131 − oV2
冥冤 冥 冤 冥
0 U1 U2 = 0 . 0 U3
Mi, Michaels, and Michaels: Dynamic monitoring of fatigue cracks
共A7兲
83
If there is no initial strain, then the off-diagonal terms are zero and the polarization vectors are aligned with the coordinate axes. The finite strain field both perturbs the diagonal terms and introduces small off-diagonal components, but the diagonal terms dominate. Thus, only the A2121 term is needed to provide a good estimate of the SV wave velocity.19 After much algebra and dropping all rotation terms, this component is A2121 = +
冋
2共 + 兲 + m + 41 n
共3 + 2兲
册
共A8兲
and the corresponding velocity of the SV wave is V=
冑 冋
册
1 2共 + 兲 + m + 4 n + . o o 共3 + 2兲
共A9兲
Recognizing that the second term under the radical is much smaller than the first, the velocity can be approximated by V=
冑
再 冋
2共 + 兲 + m + 41 n 1+ o 22共3 + 2兲
册冎 冉 冊 = Vo 1 +
⌬V , Vo
共A10兲
where Vo = 冑 / o. This equation clearly reduces to the usual expression for the shear wave speed when the applied stress is zero. It is interesting to note that this equation for V has no dependence on , the refracted angle. Recall that the above analysis is in reference to the natural, or unstrained, coordinates. A correction term must be applied to obtain the velocity v in the initial, or strained, coordinate system to compensate for the change in path length due to the initial strain,
冉
v = V共1 + ⑀11兲 = Vo 1 +
冉
冊
冊
⌬V 共1 + ⑀11兲 Vo
⌬V + ⑀11 . Vo
⯝ Vo 1 +
共A11兲
Here ⑀11 is the strain in the direction of propagation and is given in Eq. 共A6兲. Upon substitution and after some algebra, we obtain v=
冑
再
冋
4共 + 兲 + m + 41 n 2 1 + sin o 22共3 + 2兲
+ cos2
冋
共 + 2兲 + m + 41 n 22共3 + 2兲
册冎
Relative changes in velocity are thus
再 冋
4共 + 兲 + m + 41 n ⌬v = sin2 22共3 + 2兲 vo
冋
.
册 册冎
共 + 2兲 + m + 41 n + cos 22共3 + 2兲 2
册
,
共A12兲
共A13兲
where vo = 冑 / o = Vo. This equation is in agreement with Eqs. 共17兲–共19兲 of Sec. IV B, and is also consistent with equations derived by Hughes and Kelly17 for = 0° and = 90°. In particular, = 90° corresponds to shear wave propagation along the direction of applied stress, and 84
J. Acoust. Soc. Am., Vol. 119, No. 1, January 2006
= 0° corresponds to shear wave propagation perpendicular to the applied stress and with parallel polarization. 1
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