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An investigation of non-linear elastic behavior of CFRP laminates and strain measurement using Lamb waves
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 64 (2004) 2509–2516 www.elsevier.com/locate/compscitech
An investigation of non-linear elastic behavior of CFRP laminates and strain measurement using Lamb waves N. Toyama *, J. Takatsubo Research Institute of Instrumentation Frontier, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Received 10 September 2003; received in revised form 10 May 2004; accepted 12 May 2004 Available online 6 July 2004
Abstract This paper investigates the non-linear elastic behavior of unidirectional and cross-ply CFRP laminates and proposes a new method to measure tensile strain using Lamb waves. YoungÕs modulus was measured as a function of strain in situ using Lamb wave velocity during a tensile test. The stiffening effect of the carbon fibers on [0]8 specimens and the softening effect of the epoxy matrix on [90]8 specimens were accurately evaluated. YoungÕs modulus of the 0 ply was obtained as a quadratic function of strain. Using the function and the rule of mixture, the dependence of YoungÕs modulus on strain was accurately predicted for cross-ply laminates. Based on the results, the tensile strain was quantitatively correlated with the corresponding arrival time of the Lamb waves. The strains obtained from the proposed method agreed well with those from the strain gauge. Finally, the effect of transverse cracks on the in situ YoungÕs modulus of the cross-ply laminate under a tensile load was investigated. This method clearly detected even a small decrease in the YoungÕs modulus due to the transverse cracks in stiffening cross-ply laminate. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Carbon fibres; A. Polymer matrix-composites (PMCs); B. Non-linear behaviour; C. Elastic properties; D. Non-destructive testing
1. Introduction The tensile modulus of carbon fiber increases with increasing strain [1–3]. Curtis et al. [1] first demonstrated that the dynamic YoungÕs modulus of a single carbon fiber measured using an ultrasonic method significantly increased as the tensile load increased. They found that this phenomenon was completely reversible and was due to the improvement in orientation of imperfectly aligned crystallites with an increasing tensile load. This non-linear stress–strain response is also observed for carbon fiber products such as unidirectional and multidirectional CFRP laminates. The stiffening behavior of unidirectional CFRP laminates has been extensively investigated. *
Corresponding author. Tel.: +81-29-861-3025; fax: +81-29-8613126. E-mail address: [email protected] (N. Toyama). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.05.007
van Dreumel et al. [4], Ishikawa et al. [5], and Stecenko et al. [6] obtained more than a 20% increase in the YoungÕs modulus of unidirectional CFRP laminates between zero and failure strain. Ishikawa et al. [5] also proposed an empirical equation relating YoungÕs modulus and strain. Furthermore, Lagace [7] investigated the effects of the non-linearity caused by the stiffening fibers and softening resin matrix on the overall behavior of the numerous angle-ply CFRP laminates. These studies measured YoungÕs modulus by the stress–strain curves, and a relatively large scatter was observed. Ultrasonic Lamb waves have significant potential for large-area non-destructive evaluation for composite laminates because they can propagate across a long distance. However, the Lamb wave evaluation is more complicated than the conventional ultrasonic method due to its dispersive nature, i.e., the wave velocity depends on the frequency and plate thickness. To
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overcome this difficulty, a fundamental symmetric (S0) mode with a low frequency-thickness product below 1 MHz mm has often been used because it is almost non-dispersive in that range. The stiffness of composite laminates decreases with the initiation of damage such as transverse cracks, and the Lamb wave velocity is directly related to in-plane stiffness. Consequently, the cracks can be detected through a decrease in the Lamb wave velocity [8–10]. Furthermore, the authors [10,11] detected delamination and quantitatively evaluated its length in cross-ply CFRP laminates through an increase in the wave velocity. These studies utilized the high sensitivity of the Lamb wave velocity for the relatively small change in the in-plane stiffness caused by damage. Besides the detection of damage, Lamb waves have also been used for stiffness measurement in plates [12–14]. However, these studies have been conducted under strain-free conditions, and, as far as we know, there have been few reports measuring YoungÕs modulus of CFRP laminates in situ using Lamb waves under loads. In this study, we measured the velocity of the S0 mode for unidirectional and cross-ply CFRP laminates in situ under tensile loads, and investigated their non-linear elastic behavior in detail. Following this experimental investigation, a change in YoungÕs modulus with strain for cross-ply laminates is predicted using the in situ YoungÕs modulus of the 0 ply and the rule of mixture, and a new method to measure the tensile strain using Lamb waves is proposed. Furthermore, the effect of the transverse cracks on the in situ YoungÕs modulus is investigated for a cross-ply laminate. This research demonstrates that the Lamb wave velocity is a powerful tool for measuring tensile strain for long range, sensitive detection of transverse cracks in loaded CFRP laminates.
2. Experimental procedures 2.1. Materials The materials studied were unidirectional and crossply CFRP (T800H/3631, Toray Industries, Inc.) laminates with lay-ups of [0]8, [90]8, [0/90]2S and ½0=903 = 0S . The composite plate (300 mm · 300 mm) for each lay-up was fabricated by hand lay-up of unidirectional prepregs. The laminates were cured by a hotpress machine in accordance with the manufacturerÕs recommended processes. The average thickness of each ply after the curing was 0.135 mm. A sample from each plate was weighed and measured, and the average density obtained was 1530 kg/m3. A tensile coupon 210 mm long and 15 mm wide for each laminate was cut from the plate using a diamond-wheel saw, and 30 mm GFRP end tabs were bonded at both ends of the specimens.
2.2. Test procedures Fig. 1 illustrates the experimental setup for Lamb wave generation and detection during a tensile test. A broadband longitudinal transducer (M5W, Fuji Ceramics Corporation) with a diameter of 5 mm was used as a transmitting transducer. Another longitudinal transducer (M304A, Fuji Ceramics Corporation, 300 kHz) with a diameter of 4 mm was used as a receiving transducer. These transducers were glued normally to the surface of the specimen and their positions were on the center line. The distance between the transducers was 100 mm. A pulse generator (Model 5077PR, Panametrics, Inc.) generated pulses of 500 kHz to excite the transmitting transducer. The pulse generator also transmitted a trigger signal to a digital oscilloscope (TDS7054, Tektronix, Inc.) to set the initial time. The detected signals were amplified (A1002, Fuji Ceramics Corporation) and acquired at a sampling rate of 2.5 GS/s by the digital oscilloscope, which averaged 100 samples to improve the signal-to-noise ratio. The Lamb wave velocity under strain-free conditions was measured before the tensile test. The thin specimen combined with the low frequency used to generate the Lamb wave yield a frequency-thickness product of 0.5 MHz mm. Only the S0 and the fundamental anti-symmetric (A0) modes propagate in this region, and the velocity of the S0 mode is much higher than that of the A0 mode (e.g., about seven times higher for the [0]8 laminate). Therefore, the leading part of the detected wave was easily identified as the S0 mode, which is almost non-dispersive and more sensitive to the change in the in-plane stiffness of the laminate than the A0 mode. First, in order to measure the differences in arrival times of the S0 mode for different propagating distances, only the transmitter was glued and the receiver was placed on the specimen surface via coupling water 20–120 mm from the transmitter in steps of 20 mm. A least-squares fit from a plot of arrival time and distance was performed to obtain the velocity of the S0 mode (termed wave velocity) and the arrival time at zero distance. The arrival time of the S0 mode at the receiver was
Fig. 1. Schematic illustration of the experimental setup.
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516 1600
0.3
0.1
Voltage (V)
[0/90]2S 800
0.0 -0.1
[0/903/0]S
400
0 % 0.4% 0.8%
0.2
[0]8
1200
Stress (MPa)
2511
-0.2 0 0.0
0.2
0.4
0.6
0.8
-0.3
1.0
8
10
12
Strain (%) Fig. 2. Typical stress–strain curves for [0]8, [0/90]2S, and ½0=903 =0S laminates. The broken line indicates the initial linear slope.
determined to be the zero-cross point after the first positive peak, as illustrated below. The tensile test was performed at room temperature with a cross-head speed of 0.5 mm/min using a universal testing machine (AG-I, Shimadzu Corporation). The tensile strain was measured with a strain gauge with a gauge length of 5 mm attached to the center surface of the specimen, and the tensile load was measured with a 50 kN load cell. Fig. 2 presents the typical stress–strain responses of [0]8, [0/90]2S and ½0=903 = 0S laminates. The broken line indicates the initial linear slope. The stiffening effects of the carbon fibers on these laminates were clearly observed. Since the central focus was on the non-linear elastic behavior of the undamaged CFRP laminates, the strain range from zero up to 1.0% was investigated.
14
16
18
Time (µs) Fig. 3. Leading parts of the detected waveforms at different strains during the tensile test for the [0]8 specimen, #1. The dot indicates the arrival time used in this study.
0.05% step of strain exceeded 30 ns. Thus, it was accurately and easily measured using this method. The wave velocity at strain e can be calculated as V ðeÞ ¼
Lð1 þ eÞ tR ðeÞ t0
ð1Þ
where L is the initial distance between the transducers at zero strain, and tR(e) and t0 are the arrival times at the receiver for strain e and at zero distance. Fig. 4 also depicts the calculated wave velocity as a function of strain. The wave velocity increased to 10,953 m/s at 1.0% strain, which was 1.12 times as high as that under strain-free conditions. The wave velocity in the low frequency-thickness product regime for the principal axis of the orthotropic laminate is expressed as [8] sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11 E1 V ¼ ¼ ð2Þ qh qð1 m12 m21 Þ
3. Experimental results 3.1. Non-linear elastic behavior of [0]8 laminate 12.8
11200
10800
12.4 12.2
10400 12.0 11.8
10000
Wave velocity (m/s)
Arrival time Wave velocity
12.6
Arrival time (µs)
Fig. 3 illustrates the leading parts of the typical waveforms detected by the receiver at different strains during the tensile test for the [0]8 specimen, #1. Although the distance between the transducers increased with increasing strain, the entire waveform apparently shifted left and its amplitude decreased. In order to investigate this phenomenon in detail, the arrival times were measured in 0.05% increments of strain, from zero to 1.0%. The dot in Fig. 3 indicates the arrival time defined in this study. Fig. 4 depicts the experimental results of the variations in arrival time with strain. The arrival time decreased non-linearly with increased strain. The signal-to-noise ratio of the detected signal was sufficiently high, and the change in the arrival time for each
11.6 11.4 0.0
0.2
0.4
0.6
0.8
9600 1.0
Strain (%) Fig. 4. Experimental results of arrival time and wave velocity as functions of strain for specimen #1.
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where A11 is the longitudinal in-plane stiffness for the entire plate, q is the density, and h is the thickness of the plate. E1 is the longitudinal YoungÕs modulus, and m12 and m21 are the major and minor PoissonÕs ratios. For the CFRP laminates investigated in this study, 1 m12m211 and hence Eq. (2) reduces to
mens was similar. The plotted data were averaged and found to represent the relationship between YoungÕs modulus and strain for the 0 ply. Ishikawa et al. [5] proposed its relation as the quadratic expression
E1 ¼ qV 2
where EL0 is the YoungÕs modulus at zero strain, and EL1 and EL2 are two empirical coefficients that may be referred to as higher-order YoungÕs moduli. The quadratic relation of Eq. (4) was fitted with the averaged data using the least-squares method, and the following equation for the YoungÕs modulus was obtained as a function of strain
ð3Þ
Fig. 5 presents the calculated YoungÕs modulus using Eq. (3) deduced from the results of the wave velocity measurement and the tangent modulus derived from the stress–strain curve for specimen #1. The leastsquares fit from a plot of strain and stress between e ± 0.05% was performed to obtain the tangent modulus at strain e. The YoungÕs moduli obtained by the different methods exhibited similar values, but the scatter in the results from the stress–strain curve was larger, especially in the lower strain level. This was primarily due to the poor sensitivity of the load cell under the lower loads. In contrast, the YoungÕs moduli obtained from the wave velocity increased progressively with increasing strain. Based on these results, YoungÕs modulus could be measured more accurately using the Lamb wave velocity. Lagace [7] reported that the unloading path followed the original loading path in the stress–strain curve for unidirectional CFRP laminates. Thus, YoungÕs modulus during unloading was also measured. Values identical to those of the loading case were obtained, and it was confirmed that the dependence of YoungÕs modulus on strain was completely reversible within the strain range. Variations of YoungÕs modulus with strain for four additional specimens were measured and the results are also summarized in Fig. 5. Even though the YoungÕs modulus of each specimen varied to some extent, which may have been due to misalignments of the fibers and/or differences in the fiber volume fractions, the non-linear dependence of YoungÕs modulus on strain for all speci-
EL ðeÞ ¼ EL0 þ EL1 e þ EL2 e2
ð4Þ
EL ðeÞ ¼ 144:07 þ 4133:6e 70331e2 ðGPaÞ
ð5Þ
Fig. 6 depicts the normalized YoungÕs modulus as a function of strain for five [0]8 specimens. The YoungÕs modulus was normalized by the measurement at zero strain. Fig. 6 also shows a fitted curve deduced from Eq. (5). The YoungÕs modulus exhibited an almost similar dependence on strain, especially at the lower strain levels, consistent with the fitted curve even for the different specimens. This fact is the key to accurate measurement of the strain for the cross-ply laminates, as will be described later. An increase of 24% of the YoungÕs modulus from zero to 1.0% strain was obtained from Eq. (5), and this is similar to the magnitude reported by van Dreumel and Kamp [4], Ishikawa et al. [5], and Stecenko and Stevanoic [6] in various unidirectional CFRP laminates. It should be noted that this remarkable increase in YoungÕs modulus with strain may cause the disparity between the measured and calculated Lamb wave velocity. In fact, some authors [14–16] have reported that the calculated Lamb wave velocity of CFRP laminates using YoungÕs modulus derived from the stress–strain curve was higher than that measured under strain-free conditions. They explained that the discrepancy was due to
180
#1 #2 #3 #4 #5 #1 Stress-strain
1.30
Normalized Young's modulus
Young's modulus (GPa)
190
170
160
150
140 0.0
0.2
0.4
0.6
0.8
1.0
Strain (%) Fig. 5. Strain dependence of YoungÕs modulus derived from the wave velocity for five [0]8 specimens. The data from the stress–strain curve is also shown for specimen #1.
1.25
#1 #2 #3 #4 #5 Least-squares fit
1.20 1.15 1.10 1.05 1.00 0.0
0.2
0.4
0.6
0.8
1.0
Strain (%) Fig. 6. Experimental and fitted normalized YoungÕs modulus as a function of strain for five [0]8 specimens.
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
3.2. Non-linear elastic behavior of [90]8 laminate The same experiments were performed on the three [90]8 specimens. The failure strains for this laminate were between 0.4% and 0.5%. Ranges exceeding this strain could not be investigated. Fig. 7 depicts the variations of YoungÕs modulus with strain for three specimens. An average decrease of about 2.5% of the YoungÕs modulus between zero and 0.4% strain was obtained. This corresponds to the softening of the epoxy matrix. Lagace [7] reported that the non-linear behavior of the epoxy matrix became significant when the CFRP laminates were under shear stresses. He performed tensile tests for various angle-ply laminates and demonstrated that [ ± 20]S laminates exhibited a virtually linear response because the stiffening of the fibers was balanced by the softening of the matrix. Furthermore, the softening of the matrix became dominant for higher angle-ply laminates, so the amount of softening peaked for [ ± 45]S laminates. Remarkably, these laminates exhibited softening behavior at strains higher than the failure strain obtained in this study. From the minor softening of the matrix observed here, it can be concluded that the failure occurred at a low strain level before the matrix exhibited significant non-linearity. 3.3. Strain measurement in cross-ply laminates using Lamb waves Because the in situ YoungÕs moduli for the unidirectional laminates were accurately measured using the Lamb wave velocity, we now propose a new method
to measure the tensile strain in the cross-ply laminates using Lamb waves. An acoustic emission test was performed in parallel with a microscopic observation during the tensile tests for [0/90]2S and ½0=903 =0S cross-ply laminates to investigate the strain at which transverse cracks were initiated. The transverse cracks were initiated in the 90 plies and extended across the width of the specimen. The transverse cracks could be accurately counted by scanning the polished edge of the specimen using an optical microscope with an x–z transition stage. Acoustic emissions due to the transverse cracks were detected beyond a 0.9% strain for [0/90]2S laminate and 0.8% strain for ½0=903 =0S laminate. Thus, the dependence of YoungÕs modulus on strain for both cross-ply laminates was investigated within those strain ranges to eliminate the effect of the transverse cracks on the wave velocity. The experimental results of the normalized YoungÕs modulus as a function of strain for both laminates are shown in Fig. 8. Three specimens for each laminate were investigated. These laminates clearly exhibited the stiffening behavior, as expected from the stress–strain responses shown in Fig. 2, and the relationships between 1.25
Normalized Young's modulus
the fiber volume fraction variations or fiber misalignments. In addition, the stiffening of the carbon fibers can cause additional errors. Thus, it is best to use the YoungÕs modulus of the 0 ply determined at low enough strain levels in such calculations.
1.20
#1 #2 Predicted
1.10
1.05
1.00 0.0
0.2
#1 #2 #3
Normalized Young's modulus
Young's modulus (GPa)
0.4
0.6
0.8
1.0
0.8
1.0
Strain (%) 1.25
10.0
9.0
8.5
8.0 0.0
#3
1.15
(a)
9.5
2513
0.1
0.2
0.3
0.4
Fig. 7. Variations of YoungÕs modulus with strain for three [90]8 specimens.
(b)
#3
1.15
1.10
1.05
1.00 0.0
0.5
Strain (%)
1.20
#1 #2 Predicted
0.2
0.4
0.6
Strain (%)
Fig. 8. Experimental and predicted normalized YoungÕs modulus as a function of strain for (a) [0/90]2S and (b) ½0=903 =0S specimens.
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the normalized YoungÕs moduli and strain were very similar among the three specimens for both laminates. It was confirmed in the previous section that the YoungÕs modulus of the [90]8 laminate was much smaller than that of the [0]8 laminate. Furthermore, the change in the YoungÕs modulus with strain was very small. The effect of the softening of the 90 plies on the cross-ply laminates was thus neglected. The rule of mixture can be applied (1 m12m21 1) to predict the YoungÕs moduli of these cross-ply laminates as E1 ðeÞ ¼
0.8
#1 #2 #3
0.6 0.4 0.2
90
t EL ðeÞ þ t ET t0 þ t90
ð6Þ
where ET is the YoungÕs modulus of the 90 ply, 9.16 GPa (averaged value at zero strain for three [90]8 specimens), and t0 and t90 are the thicknesses of 0 and 90 plies. Fig. 8 also presents the predicted normalized YoungÕs modulus as a function of strain. Eqs. (5) and (6) were used for the prediction. The dependence of the normalized YoungÕs modulus on strain was effectively predicted for both laminates using the rule of mixture. Based on the above results, the strain can be evaluated by solving the following quadratic equation, which is obtained from Eqs. (1), (3), (5) and (6), with respect to e 2 E1 ðeÞ tR ð0Þ t0 ¼ ð1 þ eÞ ð7Þ E1 ð0Þ tR ðeÞ t0 The smaller value of the two solutions is the desired strain. Knowing the arrival times at the receiver at zero strain, tR(0), and zero distance, t0, the strain can be obtained by just measuring the arrival time at the receiver, tR(e). Fig. 9 compares the strains obtained from the strain gauge and this method for all specimens. Reasonable accuracy was obtained for all specimens, confirming the validity of this method. This method derived the strain from the in-plane stiffness along the passage of the Lamb wave. Thus, the strain corresponds to the averaged strain between the transducers, while the strain gauge measures the surface strain. In this study, the distance between the transducers was set to 100 mm. The strain can be measured for the greater distances as long as the arrival time of the wave can clearly be detected. In [0/90]2S laminate, for example, the transducers were placed at each end of a 280 mm long plate, and the leading part of the reflected wave could be clearly detected with enough signal-to-noise ratio after three traversals of the plate (over 840 mm). Therefore, the range may be extended to at least 1 m using this system. The change in the arrival time with strain is proportional to the distance between the transducers, so measuring the strain becomes easier and more precise for the longer ranges. However, it becomes more difficult for shorter ranges because the error in measuring the arrival time causes a greater effect on evaluating the strain. Lamb wave testing has primarily been used for damage detection in CFRP laminates through the changes in the detected
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Strain by strain gauge (%)
(a)
1.0
Strain by Lamb wave (%)
0
1.0
Strain by Lamb wave (%)
2514
0.8
#1 #2 #3
0.6 0.4 0.2 0.0 0.0
(b)
0.2
0.4
0.6
0.8
1.0
Strain by strain gauge (%)
Fig. 9. Comparison of the strain measured by strain gauge and Lamb waves for (a) [0/90]2S and (b) ½0=903 =0S specimens.
waves. This study could expand the potential use of Lamb wave testing, not only for damage detection but also for measuring strains. This can be important for the structural health monitoring of composite structures. 3.4. Effect of transverse cracks on in situ YoungÕs modulus in cross-ply laminate Finally, the effect of transverse cracks on the in situ YoungÕs modulus in the cross-ply laminate was investigated. Modulus reduction due to transverse cracks depends on the ratio of the thicknesses of the 0 and 90 plies. The reduction is higher as this ratio becomes lower. A ½0=903 =0S laminate was therefore selected for this investigation. The wave velocity was first measured between zero and 0.75% strain during the first loading by the previously mentioned procedures and only two transverse cracks were initiated between the transducers. Thus, the wave velocity was first obtained as a function of strain for this laminate in an almost intact condition. The specimen was then unloaded to the strain-free condition, and it was confirmed that al-
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
most no permanent strain due to the transverse cracks remained. Using a knife, artificial flaws were then manually introduced on the 90 plies at an edge of the specimen between the transducers with intervals of about 1 mm. The authors [10] have been using this technique to introduce a larger number of transverse cracks to more closely investigate the effect of transverse cracks on the YoungÕs modulus. A tensile load was applied again to the flawed specimen, the wave velocity was measured, and the corresponding number of transverse cracks was counted on the intact side of the specimen as descried in the previous section. Fig. 10 depicts the measured wave velocity during the first (intact) and the second (flawed) loading for the ½0=903 = 0S specimen. An almost identical dependence of the wave velocity on strain for the intact and the flawed specimens was obtained between zero and 0.45% strain. However, a distinct discrepancy was observed beyond that strain level, i.e., the wave velocity of the flawed specimen was lower than that of the intact specimen at the same strain. Transverse crack density as a function of strain during the second loading is also shown in Fig. 10. Crack density was defined as the number of cracks located between the transducers per centimeter. This laminate had two layers consisting of three 90 plies, thus the crack density was calculated by the total number of transverse cracks divided by 20 cm. The number of the cracks during the second loading increased to 157 at 0.75% strain, which proved the effectiveness of this technique. The crack density suddenly increased beyond 0.5% strain, which corresponded to the strain at which the discrepancy in the wave velocity between the two loadings had occurred. We confirmed from these results that the decrease in the wave velocity during the second loading was due to the effect of the transverse cracks. Consequently, this determined the measurable strain range of the proposed method as between zero and the first cracking strain. The strain be-
8
6
6400 6300
4
6200 2 6100 6000 0.0
0.2
0.4
0.6
0 0.8
Strain (%)
Normalized Young's modulus Crack density 8
1.00
6 0.99 4 0.98
0.97 0.0
Crack density (/cm)
6500
10
1.01
Normalized Young's modulus
6600
yond the range should be underestimated due to the effect of transverse cracks. The wave velocity began to increase again after a slight decrease at 0.5% strain, even though the cracks continued to be initiated. The crack reduced the wave velocity, while the carbon fibers simultaneously increased it with increasing strain. Thus, this characteristic behavior of wave velocity was caused by the combination of their effects. This study was likely the first to investigate the effect of transverse cracks in CFRP laminate on the wave velocity under a tensile load. We found that the wave velocity increased to values exceeding those under stress-free conditions, even though several cracks were initiated because the stiffening effect of the carbon fibers was dominant. It should be noted that this may cause an error in the AE source location for the transverse cracks in the cross-ply CFRP laminates [17,18]. Both the stiffening effect of the carbon fibers and the decreasing effect of the transverse cracks on the wave velocity should be taken into account for the accurate AE source location. The equivalent YoungÕs modulus during the second loading was calculated and normalized by the YoungÕs modulus during the first loading at the same strain. Fig. 11 depicts the normalized YoungÕs modulus and the crack density as functions of strain. Fig. 11 clearly reveals the effect of the transverse cracks on the in situ YoungÕs modulus. As expected, the YoungÕs modulus decreased with increasing crack density. However, the decrease of the YoungÕs modulus even for the high crack density of 7.7/cm was only 2.6%. In spite of such a small decrease in the YoungÕs modulus, this method was able to clearly detect the change corresponding to the increase in crack density. It is very difficult to measure such slight changes using the stress–strain curve. This demonstrates that the Lamb wave method can be a powerful tool for investigating the effect of damage on the YoungÕs modulus even in stiffening CFRP laminates.
10
First loading Second loading Crack density
Crack density (/cm)
Wave velocity (m/s)
6700
2515
2
0.2
0.4
0.6
0 0.8
Strain (%) Fig. 10. Experimental results of wave velocity during the first and the second loading as a function of strain for a ½0=903 =0S specimen. Crack density during the second loading is also depicted.
Fig. 11. Normalized YoungÕs modulus and crack density as functions of strain for the ½0=903 =0S specimen.
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4. Conclusions This study was undertaken to comprehensively investigate the non-linear elastic behavior of unidirectional and cross-ply CFRP laminates using Lamb waves. Following this experimental investigation, a new method to measure average tensile strains over lengths up to 1 m was proposed. Furthermore, the effect of transverse cracks on the in situ YoungÕs modulus under tensile loads was investigated for cross-ply laminates. The following conclusions were drawn. 1. Measurements of Lamb wave velocity as a function of strain for unidirectional laminates indicated a significant stiffening effect of the carbon fibers on [0]8 laminate and a slight softening effect of the epoxy matrix on [90]8 laminate. 2. The dependence of YoungÕs modulus on strain for [0/ 90]2S and ½0=903 = 0S laminates was accurately predicted by the rule of mixture using the YoungÕs modulus of the 0 ply as a quadratic function of strain. 3. The relation between the tensile strain and the corresponding arrival time of the Lamb wave was formulized for cross-ply laminates. The strains obtained from this method were consistent with those from the strain gauge. 4. The wave velocity increased due to the stiffening of the carbon fibers despite the initiation of many transverse cracks. This method could clearly detect even a slight change in the YoungÕs modulus due to transverse cracks.
References [1] Curtis GJ, Milne JM, Reynolds WN. Non-hookean behavior of strong carbon fibres. Nature 1968;220:1024–5. [2] Hughes JDH. Strength and modulus of current carbon-fibres. Carbon 1986;24:551–6.
[3] Shioya M, Hayakawa E, Takaku A. Non-hookean stress–strain response and changes in crystallite orientation of carbon fibres. J Mater Sci 1996;31:4521–32. [4] van Dreumel WHM, Kamp JLM. Non hookean behavior in the fibre direction of carbon-fibre composites and the influence of fibre waviness on the tensile properties. J Compos Mater 1977;11:461–9. [5] Ishikawa T, Matsushima M, Hayashi Y. Hardening non-linear behavior in longitudinal tension of unidirectional carbon composites. J Mater Sci 1985;20:4075–83. [6] Stecenko TB, Stevanovic MM. Variation of elastic moduli with strain in carbon/epoxy laminates. J Compos Mater 1990; 24:1152–8. [7] Lagace PA. Nonlinear stress–strain behavior of graphite/epoxy laminates. AIAA J 1985;23:1583–9. [8] Dayal V, Kinra VK. Lamb waves in an anisotropic plate. II: Nondestructive evaluation Leaky of matrix cracks on fiberreinforced composites. J Acoust Soc Am 1991;89:1590–8. [9] Seale MD, Smith BT, Prosser WH. Lamb wave assessment of fatigue and thermal damage in composites. J Acoust Soc Am 1998;103:2416–24. [10] Toyama N, Noda J, Okabe T. Quantitative damage detection in cross-ply laminates using Lamb wave method. Compos Sci Technol 2003;63:1473–9. [11] Toyama N, Takatsubo J. Lamb wave method for quick inspection of impact-induced delamination in composite laminates. Compos Sci Technol 2004;64:1293–300. [12] Datta SK, Ledbetter HM, Shah AH. Ultrasonic velocity studies of composite and heterogeneous materials. In: Duke Jr JC, editor. Acousto-ultrasonics theory and application. New York: Plenum; 1988. p. 93–109. [13] Karim MR, Mal AK, Bar-Cohen Y. Inversion of Leaky Lamb wave data by simplex algorithm. J Acoust Soc Am 1990;88: 482–91. [14] van Otterloo DL, Dayal V. How isotropic are quasi-isotropic laminates. Composites, Part A 2003;34:93–103. [15] Prosser WH, Gorman MR. Plate mode velocities in graphite/ epoxy plates. J Acoust Soc Am 1994;96:902–7. [16] Prosser WH, Seale MD, Smith BT. Time-frequency analysis of the dispersion of Lamb modes. J Acoust Soc Am 1999;105:2669–76. [17] Prosser WH, Jackson KE, Kellas S, Smith BT, McKeon J, Friedman A. Advanced waveform-based acoustic emission detection of matrix cracking in composites. Mater Eval 1995; 53:1052–8. [18] Surgeon M, Wevers M. One sensor linear location of acoustic emission events using plate wave theories. Mat Sci Eng A: Struct 1999;265:254–61.
An investigation of non-linear elastic behavior of CFRP laminates and strain measurement using Lamb waves N. Toyama *, J. Takatsubo Research Institute of Instrumentation Frontier, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Received 10 September 2003; received in revised form 10 May 2004; accepted 12 May 2004 Available online 6 July 2004
Abstract This paper investigates the non-linear elastic behavior of unidirectional and cross-ply CFRP laminates and proposes a new method to measure tensile strain using Lamb waves. YoungÕs modulus was measured as a function of strain in situ using Lamb wave velocity during a tensile test. The stiffening effect of the carbon fibers on [0]8 specimens and the softening effect of the epoxy matrix on [90]8 specimens were accurately evaluated. YoungÕs modulus of the 0 ply was obtained as a quadratic function of strain. Using the function and the rule of mixture, the dependence of YoungÕs modulus on strain was accurately predicted for cross-ply laminates. Based on the results, the tensile strain was quantitatively correlated with the corresponding arrival time of the Lamb waves. The strains obtained from the proposed method agreed well with those from the strain gauge. Finally, the effect of transverse cracks on the in situ YoungÕs modulus of the cross-ply laminate under a tensile load was investigated. This method clearly detected even a small decrease in the YoungÕs modulus due to the transverse cracks in stiffening cross-ply laminate. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Carbon fibres; A. Polymer matrix-composites (PMCs); B. Non-linear behaviour; C. Elastic properties; D. Non-destructive testing
1. Introduction The tensile modulus of carbon fiber increases with increasing strain [1–3]. Curtis et al. [1] first demonstrated that the dynamic YoungÕs modulus of a single carbon fiber measured using an ultrasonic method significantly increased as the tensile load increased. They found that this phenomenon was completely reversible and was due to the improvement in orientation of imperfectly aligned crystallites with an increasing tensile load. This non-linear stress–strain response is also observed for carbon fiber products such as unidirectional and multidirectional CFRP laminates. The stiffening behavior of unidirectional CFRP laminates has been extensively investigated. *
Corresponding author. Tel.: +81-29-861-3025; fax: +81-29-8613126. E-mail address: [email protected] (N. Toyama). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.05.007
van Dreumel et al. [4], Ishikawa et al. [5], and Stecenko et al. [6] obtained more than a 20% increase in the YoungÕs modulus of unidirectional CFRP laminates between zero and failure strain. Ishikawa et al. [5] also proposed an empirical equation relating YoungÕs modulus and strain. Furthermore, Lagace [7] investigated the effects of the non-linearity caused by the stiffening fibers and softening resin matrix on the overall behavior of the numerous angle-ply CFRP laminates. These studies measured YoungÕs modulus by the stress–strain curves, and a relatively large scatter was observed. Ultrasonic Lamb waves have significant potential for large-area non-destructive evaluation for composite laminates because they can propagate across a long distance. However, the Lamb wave evaluation is more complicated than the conventional ultrasonic method due to its dispersive nature, i.e., the wave velocity depends on the frequency and plate thickness. To
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overcome this difficulty, a fundamental symmetric (S0) mode with a low frequency-thickness product below 1 MHz mm has often been used because it is almost non-dispersive in that range. The stiffness of composite laminates decreases with the initiation of damage such as transverse cracks, and the Lamb wave velocity is directly related to in-plane stiffness. Consequently, the cracks can be detected through a decrease in the Lamb wave velocity [8–10]. Furthermore, the authors [10,11] detected delamination and quantitatively evaluated its length in cross-ply CFRP laminates through an increase in the wave velocity. These studies utilized the high sensitivity of the Lamb wave velocity for the relatively small change in the in-plane stiffness caused by damage. Besides the detection of damage, Lamb waves have also been used for stiffness measurement in plates [12–14]. However, these studies have been conducted under strain-free conditions, and, as far as we know, there have been few reports measuring YoungÕs modulus of CFRP laminates in situ using Lamb waves under loads. In this study, we measured the velocity of the S0 mode for unidirectional and cross-ply CFRP laminates in situ under tensile loads, and investigated their non-linear elastic behavior in detail. Following this experimental investigation, a change in YoungÕs modulus with strain for cross-ply laminates is predicted using the in situ YoungÕs modulus of the 0 ply and the rule of mixture, and a new method to measure the tensile strain using Lamb waves is proposed. Furthermore, the effect of the transverse cracks on the in situ YoungÕs modulus is investigated for a cross-ply laminate. This research demonstrates that the Lamb wave velocity is a powerful tool for measuring tensile strain for long range, sensitive detection of transverse cracks in loaded CFRP laminates.
2. Experimental procedures 2.1. Materials The materials studied were unidirectional and crossply CFRP (T800H/3631, Toray Industries, Inc.) laminates with lay-ups of [0]8, [90]8, [0/90]2S and ½0=903 = 0S . The composite plate (300 mm · 300 mm) for each lay-up was fabricated by hand lay-up of unidirectional prepregs. The laminates were cured by a hotpress machine in accordance with the manufacturerÕs recommended processes. The average thickness of each ply after the curing was 0.135 mm. A sample from each plate was weighed and measured, and the average density obtained was 1530 kg/m3. A tensile coupon 210 mm long and 15 mm wide for each laminate was cut from the plate using a diamond-wheel saw, and 30 mm GFRP end tabs were bonded at both ends of the specimens.
2.2. Test procedures Fig. 1 illustrates the experimental setup for Lamb wave generation and detection during a tensile test. A broadband longitudinal transducer (M5W, Fuji Ceramics Corporation) with a diameter of 5 mm was used as a transmitting transducer. Another longitudinal transducer (M304A, Fuji Ceramics Corporation, 300 kHz) with a diameter of 4 mm was used as a receiving transducer. These transducers were glued normally to the surface of the specimen and their positions were on the center line. The distance between the transducers was 100 mm. A pulse generator (Model 5077PR, Panametrics, Inc.) generated pulses of 500 kHz to excite the transmitting transducer. The pulse generator also transmitted a trigger signal to a digital oscilloscope (TDS7054, Tektronix, Inc.) to set the initial time. The detected signals were amplified (A1002, Fuji Ceramics Corporation) and acquired at a sampling rate of 2.5 GS/s by the digital oscilloscope, which averaged 100 samples to improve the signal-to-noise ratio. The Lamb wave velocity under strain-free conditions was measured before the tensile test. The thin specimen combined with the low frequency used to generate the Lamb wave yield a frequency-thickness product of 0.5 MHz mm. Only the S0 and the fundamental anti-symmetric (A0) modes propagate in this region, and the velocity of the S0 mode is much higher than that of the A0 mode (e.g., about seven times higher for the [0]8 laminate). Therefore, the leading part of the detected wave was easily identified as the S0 mode, which is almost non-dispersive and more sensitive to the change in the in-plane stiffness of the laminate than the A0 mode. First, in order to measure the differences in arrival times of the S0 mode for different propagating distances, only the transmitter was glued and the receiver was placed on the specimen surface via coupling water 20–120 mm from the transmitter in steps of 20 mm. A least-squares fit from a plot of arrival time and distance was performed to obtain the velocity of the S0 mode (termed wave velocity) and the arrival time at zero distance. The arrival time of the S0 mode at the receiver was
Fig. 1. Schematic illustration of the experimental setup.
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516 1600
0.3
0.1
Voltage (V)
[0/90]2S 800
0.0 -0.1
[0/903/0]S
400
0 % 0.4% 0.8%
0.2
[0]8
1200
Stress (MPa)
2511
-0.2 0 0.0
0.2
0.4
0.6
0.8
-0.3
1.0
8
10
12
Strain (%) Fig. 2. Typical stress–strain curves for [0]8, [0/90]2S, and ½0=903 =0S laminates. The broken line indicates the initial linear slope.
determined to be the zero-cross point after the first positive peak, as illustrated below. The tensile test was performed at room temperature with a cross-head speed of 0.5 mm/min using a universal testing machine (AG-I, Shimadzu Corporation). The tensile strain was measured with a strain gauge with a gauge length of 5 mm attached to the center surface of the specimen, and the tensile load was measured with a 50 kN load cell. Fig. 2 presents the typical stress–strain responses of [0]8, [0/90]2S and ½0=903 = 0S laminates. The broken line indicates the initial linear slope. The stiffening effects of the carbon fibers on these laminates were clearly observed. Since the central focus was on the non-linear elastic behavior of the undamaged CFRP laminates, the strain range from zero up to 1.0% was investigated.
14
16
18
Time (µs) Fig. 3. Leading parts of the detected waveforms at different strains during the tensile test for the [0]8 specimen, #1. The dot indicates the arrival time used in this study.
0.05% step of strain exceeded 30 ns. Thus, it was accurately and easily measured using this method. The wave velocity at strain e can be calculated as V ðeÞ ¼
Lð1 þ eÞ tR ðeÞ t0
ð1Þ
where L is the initial distance between the transducers at zero strain, and tR(e) and t0 are the arrival times at the receiver for strain e and at zero distance. Fig. 4 also depicts the calculated wave velocity as a function of strain. The wave velocity increased to 10,953 m/s at 1.0% strain, which was 1.12 times as high as that under strain-free conditions. The wave velocity in the low frequency-thickness product regime for the principal axis of the orthotropic laminate is expressed as [8] sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11 E1 V ¼ ¼ ð2Þ qh qð1 m12 m21 Þ
3. Experimental results 3.1. Non-linear elastic behavior of [0]8 laminate 12.8
11200
10800
12.4 12.2
10400 12.0 11.8
10000
Wave velocity (m/s)
Arrival time Wave velocity
12.6
Arrival time (µs)
Fig. 3 illustrates the leading parts of the typical waveforms detected by the receiver at different strains during the tensile test for the [0]8 specimen, #1. Although the distance between the transducers increased with increasing strain, the entire waveform apparently shifted left and its amplitude decreased. In order to investigate this phenomenon in detail, the arrival times were measured in 0.05% increments of strain, from zero to 1.0%. The dot in Fig. 3 indicates the arrival time defined in this study. Fig. 4 depicts the experimental results of the variations in arrival time with strain. The arrival time decreased non-linearly with increased strain. The signal-to-noise ratio of the detected signal was sufficiently high, and the change in the arrival time for each
11.6 11.4 0.0
0.2
0.4
0.6
0.8
9600 1.0
Strain (%) Fig. 4. Experimental results of arrival time and wave velocity as functions of strain for specimen #1.
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where A11 is the longitudinal in-plane stiffness for the entire plate, q is the density, and h is the thickness of the plate. E1 is the longitudinal YoungÕs modulus, and m12 and m21 are the major and minor PoissonÕs ratios. For the CFRP laminates investigated in this study, 1 m12m211 and hence Eq. (2) reduces to
mens was similar. The plotted data were averaged and found to represent the relationship between YoungÕs modulus and strain for the 0 ply. Ishikawa et al. [5] proposed its relation as the quadratic expression
E1 ¼ qV 2
where EL0 is the YoungÕs modulus at zero strain, and EL1 and EL2 are two empirical coefficients that may be referred to as higher-order YoungÕs moduli. The quadratic relation of Eq. (4) was fitted with the averaged data using the least-squares method, and the following equation for the YoungÕs modulus was obtained as a function of strain
ð3Þ
Fig. 5 presents the calculated YoungÕs modulus using Eq. (3) deduced from the results of the wave velocity measurement and the tangent modulus derived from the stress–strain curve for specimen #1. The leastsquares fit from a plot of strain and stress between e ± 0.05% was performed to obtain the tangent modulus at strain e. The YoungÕs moduli obtained by the different methods exhibited similar values, but the scatter in the results from the stress–strain curve was larger, especially in the lower strain level. This was primarily due to the poor sensitivity of the load cell under the lower loads. In contrast, the YoungÕs moduli obtained from the wave velocity increased progressively with increasing strain. Based on these results, YoungÕs modulus could be measured more accurately using the Lamb wave velocity. Lagace [7] reported that the unloading path followed the original loading path in the stress–strain curve for unidirectional CFRP laminates. Thus, YoungÕs modulus during unloading was also measured. Values identical to those of the loading case were obtained, and it was confirmed that the dependence of YoungÕs modulus on strain was completely reversible within the strain range. Variations of YoungÕs modulus with strain for four additional specimens were measured and the results are also summarized in Fig. 5. Even though the YoungÕs modulus of each specimen varied to some extent, which may have been due to misalignments of the fibers and/or differences in the fiber volume fractions, the non-linear dependence of YoungÕs modulus on strain for all speci-
EL ðeÞ ¼ EL0 þ EL1 e þ EL2 e2
ð4Þ
EL ðeÞ ¼ 144:07 þ 4133:6e 70331e2 ðGPaÞ
ð5Þ
Fig. 6 depicts the normalized YoungÕs modulus as a function of strain for five [0]8 specimens. The YoungÕs modulus was normalized by the measurement at zero strain. Fig. 6 also shows a fitted curve deduced from Eq. (5). The YoungÕs modulus exhibited an almost similar dependence on strain, especially at the lower strain levels, consistent with the fitted curve even for the different specimens. This fact is the key to accurate measurement of the strain for the cross-ply laminates, as will be described later. An increase of 24% of the YoungÕs modulus from zero to 1.0% strain was obtained from Eq. (5), and this is similar to the magnitude reported by van Dreumel and Kamp [4], Ishikawa et al. [5], and Stecenko and Stevanoic [6] in various unidirectional CFRP laminates. It should be noted that this remarkable increase in YoungÕs modulus with strain may cause the disparity between the measured and calculated Lamb wave velocity. In fact, some authors [14–16] have reported that the calculated Lamb wave velocity of CFRP laminates using YoungÕs modulus derived from the stress–strain curve was higher than that measured under strain-free conditions. They explained that the discrepancy was due to
180
#1 #2 #3 #4 #5 #1 Stress-strain
1.30
Normalized Young's modulus
Young's modulus (GPa)
190
170
160
150
140 0.0
0.2
0.4
0.6
0.8
1.0
Strain (%) Fig. 5. Strain dependence of YoungÕs modulus derived from the wave velocity for five [0]8 specimens. The data from the stress–strain curve is also shown for specimen #1.
1.25
#1 #2 #3 #4 #5 Least-squares fit
1.20 1.15 1.10 1.05 1.00 0.0
0.2
0.4
0.6
0.8
1.0
Strain (%) Fig. 6. Experimental and fitted normalized YoungÕs modulus as a function of strain for five [0]8 specimens.
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
3.2. Non-linear elastic behavior of [90]8 laminate The same experiments were performed on the three [90]8 specimens. The failure strains for this laminate were between 0.4% and 0.5%. Ranges exceeding this strain could not be investigated. Fig. 7 depicts the variations of YoungÕs modulus with strain for three specimens. An average decrease of about 2.5% of the YoungÕs modulus between zero and 0.4% strain was obtained. This corresponds to the softening of the epoxy matrix. Lagace [7] reported that the non-linear behavior of the epoxy matrix became significant when the CFRP laminates were under shear stresses. He performed tensile tests for various angle-ply laminates and demonstrated that [ ± 20]S laminates exhibited a virtually linear response because the stiffening of the fibers was balanced by the softening of the matrix. Furthermore, the softening of the matrix became dominant for higher angle-ply laminates, so the amount of softening peaked for [ ± 45]S laminates. Remarkably, these laminates exhibited softening behavior at strains higher than the failure strain obtained in this study. From the minor softening of the matrix observed here, it can be concluded that the failure occurred at a low strain level before the matrix exhibited significant non-linearity. 3.3. Strain measurement in cross-ply laminates using Lamb waves Because the in situ YoungÕs moduli for the unidirectional laminates were accurately measured using the Lamb wave velocity, we now propose a new method
to measure the tensile strain in the cross-ply laminates using Lamb waves. An acoustic emission test was performed in parallel with a microscopic observation during the tensile tests for [0/90]2S and ½0=903 =0S cross-ply laminates to investigate the strain at which transverse cracks were initiated. The transverse cracks were initiated in the 90 plies and extended across the width of the specimen. The transverse cracks could be accurately counted by scanning the polished edge of the specimen using an optical microscope with an x–z transition stage. Acoustic emissions due to the transverse cracks were detected beyond a 0.9% strain for [0/90]2S laminate and 0.8% strain for ½0=903 =0S laminate. Thus, the dependence of YoungÕs modulus on strain for both cross-ply laminates was investigated within those strain ranges to eliminate the effect of the transverse cracks on the wave velocity. The experimental results of the normalized YoungÕs modulus as a function of strain for both laminates are shown in Fig. 8. Three specimens for each laminate were investigated. These laminates clearly exhibited the stiffening behavior, as expected from the stress–strain responses shown in Fig. 2, and the relationships between 1.25
Normalized Young's modulus
the fiber volume fraction variations or fiber misalignments. In addition, the stiffening of the carbon fibers can cause additional errors. Thus, it is best to use the YoungÕs modulus of the 0 ply determined at low enough strain levels in such calculations.
1.20
#1 #2 Predicted
1.10
1.05
1.00 0.0
0.2
#1 #2 #3
Normalized Young's modulus
Young's modulus (GPa)
0.4
0.6
0.8
1.0
0.8
1.0
Strain (%) 1.25
10.0
9.0
8.5
8.0 0.0
#3
1.15
(a)
9.5
2513
0.1
0.2
0.3
0.4
Fig. 7. Variations of YoungÕs modulus with strain for three [90]8 specimens.
(b)
#3
1.15
1.10
1.05
1.00 0.0
0.5
Strain (%)
1.20
#1 #2 Predicted
0.2
0.4
0.6
Strain (%)
Fig. 8. Experimental and predicted normalized YoungÕs modulus as a function of strain for (a) [0/90]2S and (b) ½0=903 =0S specimens.
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
the normalized YoungÕs moduli and strain were very similar among the three specimens for both laminates. It was confirmed in the previous section that the YoungÕs modulus of the [90]8 laminate was much smaller than that of the [0]8 laminate. Furthermore, the change in the YoungÕs modulus with strain was very small. The effect of the softening of the 90 plies on the cross-ply laminates was thus neglected. The rule of mixture can be applied (1 m12m21 1) to predict the YoungÕs moduli of these cross-ply laminates as E1 ðeÞ ¼
0.8
#1 #2 #3
0.6 0.4 0.2
90
t EL ðeÞ þ t ET t0 þ t90
ð6Þ
where ET is the YoungÕs modulus of the 90 ply, 9.16 GPa (averaged value at zero strain for three [90]8 specimens), and t0 and t90 are the thicknesses of 0 and 90 plies. Fig. 8 also presents the predicted normalized YoungÕs modulus as a function of strain. Eqs. (5) and (6) were used for the prediction. The dependence of the normalized YoungÕs modulus on strain was effectively predicted for both laminates using the rule of mixture. Based on the above results, the strain can be evaluated by solving the following quadratic equation, which is obtained from Eqs. (1), (3), (5) and (6), with respect to e 2 E1 ðeÞ tR ð0Þ t0 ¼ ð1 þ eÞ ð7Þ E1 ð0Þ tR ðeÞ t0 The smaller value of the two solutions is the desired strain. Knowing the arrival times at the receiver at zero strain, tR(0), and zero distance, t0, the strain can be obtained by just measuring the arrival time at the receiver, tR(e). Fig. 9 compares the strains obtained from the strain gauge and this method for all specimens. Reasonable accuracy was obtained for all specimens, confirming the validity of this method. This method derived the strain from the in-plane stiffness along the passage of the Lamb wave. Thus, the strain corresponds to the averaged strain between the transducers, while the strain gauge measures the surface strain. In this study, the distance between the transducers was set to 100 mm. The strain can be measured for the greater distances as long as the arrival time of the wave can clearly be detected. In [0/90]2S laminate, for example, the transducers were placed at each end of a 280 mm long plate, and the leading part of the reflected wave could be clearly detected with enough signal-to-noise ratio after three traversals of the plate (over 840 mm). Therefore, the range may be extended to at least 1 m using this system. The change in the arrival time with strain is proportional to the distance between the transducers, so measuring the strain becomes easier and more precise for the longer ranges. However, it becomes more difficult for shorter ranges because the error in measuring the arrival time causes a greater effect on evaluating the strain. Lamb wave testing has primarily been used for damage detection in CFRP laminates through the changes in the detected
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Strain by strain gauge (%)
(a)
1.0
Strain by Lamb wave (%)
0
1.0
Strain by Lamb wave (%)
2514
0.8
#1 #2 #3
0.6 0.4 0.2 0.0 0.0
(b)
0.2
0.4
0.6
0.8
1.0
Strain by strain gauge (%)
Fig. 9. Comparison of the strain measured by strain gauge and Lamb waves for (a) [0/90]2S and (b) ½0=903 =0S specimens.
waves. This study could expand the potential use of Lamb wave testing, not only for damage detection but also for measuring strains. This can be important for the structural health monitoring of composite structures. 3.4. Effect of transverse cracks on in situ YoungÕs modulus in cross-ply laminate Finally, the effect of transverse cracks on the in situ YoungÕs modulus in the cross-ply laminate was investigated. Modulus reduction due to transverse cracks depends on the ratio of the thicknesses of the 0 and 90 plies. The reduction is higher as this ratio becomes lower. A ½0=903 =0S laminate was therefore selected for this investigation. The wave velocity was first measured between zero and 0.75% strain during the first loading by the previously mentioned procedures and only two transverse cracks were initiated between the transducers. Thus, the wave velocity was first obtained as a function of strain for this laminate in an almost intact condition. The specimen was then unloaded to the strain-free condition, and it was confirmed that al-
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
most no permanent strain due to the transverse cracks remained. Using a knife, artificial flaws were then manually introduced on the 90 plies at an edge of the specimen between the transducers with intervals of about 1 mm. The authors [10] have been using this technique to introduce a larger number of transverse cracks to more closely investigate the effect of transverse cracks on the YoungÕs modulus. A tensile load was applied again to the flawed specimen, the wave velocity was measured, and the corresponding number of transverse cracks was counted on the intact side of the specimen as descried in the previous section. Fig. 10 depicts the measured wave velocity during the first (intact) and the second (flawed) loading for the ½0=903 = 0S specimen. An almost identical dependence of the wave velocity on strain for the intact and the flawed specimens was obtained between zero and 0.45% strain. However, a distinct discrepancy was observed beyond that strain level, i.e., the wave velocity of the flawed specimen was lower than that of the intact specimen at the same strain. Transverse crack density as a function of strain during the second loading is also shown in Fig. 10. Crack density was defined as the number of cracks located between the transducers per centimeter. This laminate had two layers consisting of three 90 plies, thus the crack density was calculated by the total number of transverse cracks divided by 20 cm. The number of the cracks during the second loading increased to 157 at 0.75% strain, which proved the effectiveness of this technique. The crack density suddenly increased beyond 0.5% strain, which corresponded to the strain at which the discrepancy in the wave velocity between the two loadings had occurred. We confirmed from these results that the decrease in the wave velocity during the second loading was due to the effect of the transverse cracks. Consequently, this determined the measurable strain range of the proposed method as between zero and the first cracking strain. The strain be-
8
6
6400 6300
4
6200 2 6100 6000 0.0
0.2
0.4
0.6
0 0.8
Strain (%)
Normalized Young's modulus Crack density 8
1.00
6 0.99 4 0.98
0.97 0.0
Crack density (/cm)
6500
10
1.01
Normalized Young's modulus
6600
yond the range should be underestimated due to the effect of transverse cracks. The wave velocity began to increase again after a slight decrease at 0.5% strain, even though the cracks continued to be initiated. The crack reduced the wave velocity, while the carbon fibers simultaneously increased it with increasing strain. Thus, this characteristic behavior of wave velocity was caused by the combination of their effects. This study was likely the first to investigate the effect of transverse cracks in CFRP laminate on the wave velocity under a tensile load. We found that the wave velocity increased to values exceeding those under stress-free conditions, even though several cracks were initiated because the stiffening effect of the carbon fibers was dominant. It should be noted that this may cause an error in the AE source location for the transverse cracks in the cross-ply CFRP laminates [17,18]. Both the stiffening effect of the carbon fibers and the decreasing effect of the transverse cracks on the wave velocity should be taken into account for the accurate AE source location. The equivalent YoungÕs modulus during the second loading was calculated and normalized by the YoungÕs modulus during the first loading at the same strain. Fig. 11 depicts the normalized YoungÕs modulus and the crack density as functions of strain. Fig. 11 clearly reveals the effect of the transverse cracks on the in situ YoungÕs modulus. As expected, the YoungÕs modulus decreased with increasing crack density. However, the decrease of the YoungÕs modulus even for the high crack density of 7.7/cm was only 2.6%. In spite of such a small decrease in the YoungÕs modulus, this method was able to clearly detect the change corresponding to the increase in crack density. It is very difficult to measure such slight changes using the stress–strain curve. This demonstrates that the Lamb wave method can be a powerful tool for investigating the effect of damage on the YoungÕs modulus even in stiffening CFRP laminates.
10
First loading Second loading Crack density
Crack density (/cm)
Wave velocity (m/s)
6700
2515
2
0.2
0.4
0.6
0 0.8
Strain (%) Fig. 10. Experimental results of wave velocity during the first and the second loading as a function of strain for a ½0=903 =0S specimen. Crack density during the second loading is also depicted.
Fig. 11. Normalized YoungÕs modulus and crack density as functions of strain for the ½0=903 =0S specimen.
2516
N. Toyama, J. Takatsubo / Composites Science and Technology 64 (2004) 2509–2516
4. Conclusions This study was undertaken to comprehensively investigate the non-linear elastic behavior of unidirectional and cross-ply CFRP laminates using Lamb waves. Following this experimental investigation, a new method to measure average tensile strains over lengths up to 1 m was proposed. Furthermore, the effect of transverse cracks on the in situ YoungÕs modulus under tensile loads was investigated for cross-ply laminates. The following conclusions were drawn. 1. Measurements of Lamb wave velocity as a function of strain for unidirectional laminates indicated a significant stiffening effect of the carbon fibers on [0]8 laminate and a slight softening effect of the epoxy matrix on [90]8 laminate. 2. The dependence of YoungÕs modulus on strain for [0/ 90]2S and ½0=903 = 0S laminates was accurately predicted by the rule of mixture using the YoungÕs modulus of the 0 ply as a quadratic function of strain. 3. The relation between the tensile strain and the corresponding arrival time of the Lamb wave was formulized for cross-ply laminates. The strains obtained from this method were consistent with those from the strain gauge. 4. The wave velocity increased due to the stiffening of the carbon fibers despite the initiation of many transverse cracks. This method could clearly detect even a slight change in the YoungÕs modulus due to transverse cracks.
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