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Experimental and theoretical characterization of acoustic emission transients in composite laminates
Composites Science and Technology 61 (2001) 1367–1378 www.elsevier.com/locate/compscitech
Experimental and theoretical characterization of acoustic emission transients in composite laminates Mikael Johnson, Peter Gudmundson* Department of Solid Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 26 September 2000; accepted 7 March 2001
Abstract Transient wave propagation resulting from transverse matrix cracking in cross-ply composite tensile test specimens are investigated both theoretically and experimentally. In the experiments, broad-band transducers and a fast data-acquisition system enable measurements of transients in a frequency interval up to 1 MHz. The theoretical predictions are based on a model which utilizes a finite-element discretization of the cross-section of the specimen and a Fourier representation of the axial and time dependence. The comparisons between experimental and theoretical results are encouraging. Weak points in the experimental technique and in the theoretical model are identified. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Matrix cracking; C. Finite-element analysis; C. Laminates; D. Acoustic emission
1. Introduction The acoustic emission (AE) technique is a well-known non-destructive evaluation technique [1], both in research and for industrial applications. It is mainly used to monitor the onset of cracking processes in materials and components. One of AE’s great advantages is the ability to scan a large structure with only a few sensors and to localize the source causing the captured AE transient. The first systems used resonant transducers, having a resonant peak in the frequency range of 150–500 kHz. The state of development 10–15 years ago is well described in a number of review papers [2–5]. The rapid development of computers with increasing speed and extended memory capacity have, however, enabled digital acquisition using broadband transducers. These transducers are sensitive in a wider frequency range, typically from 50 kHz up to 1.5 MHz, see Gorman [6] for a comparison between resonant and broadband transducers. The use of composite material as a construction material is increasing and can, nowadays, be found in
* Corresponding author. Tel.: +46-8-790-7548; fax: +46-8-4112418. E-mail addresses: [email protected] (P. Gudmundson), mikael@ hallf.kth.se (M. Johnson).
various components. A particular feature of composite laminates is that micro-structural damage generally develops prior to final failure. The damage itself is generally not critical for the structural integrity, but it influences certain macroscopic properties such as stiffnesses [7]. Typical damage modes are matrix cracking, delamination, debonding and fibre breakage. Matrix cracking usually develops at an early stage and it may act as a crack initiator for other mechanisms. Many investigations have been presented where conventional AE has been applied as a monitoring tool for crack surveillance in composite materials. Bhat et al. [8] use a pattern-recognition technique to discriminate different failure mechanisms in fatigue tests on unidirectional GRP composites. Through analysis of the AE amplitude distribution, Barre´ and Benzeggagh [9] identify different damage mechanisms in short-glass-fibre-reinforced polypropylene and Ely and Hill [10] characterize longitudinal splitting and fiber breakage in carbon/epoxy laminates. The use of broad-band transducers will, however, provide more extensive information in terms of frequency content in the AE signal. Prosser et al. [11] studied matrix cracking in a carbon/epoxy composite and concluded that dispersion and attenuation will influence the detected amplitude, depending on the distance of propagation. On the basis of signals measured by broadband transducers, researchers have tried to identify
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00036-7
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different wave modes in the recorded signals, so-called modal acoustic emission (MAE). Surgeon and Wevers [12] used MAE to determine the amplitude ratio between extensional and flexural modes in CFRP laminates for different crack mechanisms. A wider frequency spectrum will also give a possibility to make more realistic and relevant comparisons between experimentally measured and numerically calculated signals. Numerical simulations will give further insight into the influence of various parameters on the recorded AE-signal. Examples of parameters which have an effect on the transients are: source mechanism and orientation, crack propagation velocity and direction, lay-up configuration. Few quantitative comparisons between experimentally and numerically obtained transients have been presented in the past. One investigation is presented by Guo [13], who modelled the surface response in composite laminates due to different microfracture modes and compared with experiments on composite laminates with artificial defects. However, there are still uncertainties related to the transducers and the whole acquisition system, which are of essential and crucial importance to the comparisons. In the presented work, numerical modelling of AE wave propagation due to matrix cracking will be presented and compared to measured signals resulting from tensile tests of cross-ply glass/epoxy composite specimens. Problems concerning the wavelength dependence of the transducer sensitivity are also addressed.
2. Experimental set-up and procedures 2.1. Specimen preparation In the experiments, specimens with two types of layup configurations were used, [02, 903]S and [902, 03]S. They were both produced from cross-ply laminates manufactured by Saab Military Aircraft, Sweden, using epoxy/glassfiber prepregs. The angles refer to the loading direction and the unidirectional properties, presented in Table 1, were experimentally estimated and presented by Lundgren [14]. All specimens were 300 mm in length and had a thickness of about 1.39 mm. The specimen edges were wet sanded to a width of 25 mm, in order to prevent damage initiations from edge defects due to the cutting process and also to enable the use of replica technique. To reduce noise coming from the gripping region, soft aluminium tabs were glued on both sides of
the specimen at each end. The tabs were manufactured with a 18 mm long bevel and with a total length of 50 mm. The width and thickness were 25 and 3 mm, respectively. 2.2. Acoustic emission system For monitoring of the specimen during the tensile tests, a Fracture Wave Detector F4012 (FWD) system, manufactured by Digital Wave Corporation, Denver, USA, was used. The acoustic emission (AE) transients, generated due to micro cracking in the specimen, were captured using DWC B1025 piezoelectric transducers. These are of a broad-band type with a diameter of the sensing area of about 8 mm. The manufacturer claims that the transducers should have an approximately flat frequency response in the range of 50 kHz–1.5 MHz. A sensitivity in terms of displacement or velocity is, however, not given. The transducer has to be calibrated for each particular application. The calibration in the present applications will be described below. 2.3. Calibration of the AE system When an acoustic emission is picked up by the piezoelectric crystal the signal must pass through the whole acquisition system, which is a chain of filters, amplifiers and an A/D-converter. Thereby the measured signal will be distorted and it will result in a signal which may not directly be interpreted as the corresponding motion of the specimen surface. Therefore, to be able to make relevant comparisons between experimentally measured and numerically calculated AE signals, the transfer function of the whole acquisition system must be investigated. To explore the frequency response and to determine the transfer function of an AE system, several procedures have been suggested, see for example Hatano and Watanabe [15]. A method to determine the in-plane and out-of-plane sensitivity of a piezoelectric sensor has been proposed by Simmons et al. [16]. The calibration method used in this paper is mainly based on the method described in the ASTM standard [17]. The basic idea is to generate a step load, i.e. to suddenly remove a point force, on a big steel block. The generated emission is captured by a transducer with a known distance to the source. This signal can then be compared to the elastic solution for a point force on a half space, which was derived in closed form by Pekeris [18] for a Poisson’s
Table 1 Unidirectional stiffness properties in the [02 , 903 ]S and [902 , 03 ]S laminates EL (GPa)
ET (GPa)
LT
GLT (GPa)
(kg/m3)
Fibre volume fraction (%)
46.2
16.9
0.31
7.7
1942
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ratio of 0.25. For other values of the Poisson ratio, a solution has been presented by Mooney [19]. In this work, a steel block with the dimensions 1600 800150 mm was used. By breaking a fused silica capillary on the surface of the steel block, a sudden release of the force was achieved. The inner and outer diameter of the capillary was 0.18 and 0.34 mm, respectively. This capillary was placed on the surface of the steel block with a glass rod transversally aligned above the capillary, see Fig. 1. To break the capillary, it is being squeezed between the surface of the steel block and the glass rod, using a loading screw. When the capillary breaks, the sudden release of force is an approximate step function with a fast rise time and the generated transient is captured by the transducer undergoing calibration. The transducer was mounted at two different locations, 30 and 50 mm from the source, using vacuum grease. During the calibration procedure, the sampling frequency of the acquisition system was set to 10 MHz and a record length of 1024 sample points was used, enabling a time window of 102.4 ms. The low pass and high pass filters were set to 20 kHz and 4 MHz, respectively. The dashed curve in Fig. 2a shows a measured signal due to breaking a capillary 30 mm from the transducer. As a reference signal, the time response due to a vertical point force on a half space was calculated, using the solution presented by Mooney [19]. Mooney based his work on the closed-form solutions given by Pekeris and extended it to be applicable for arbitrary values of Poisson’s ratio. The signal shown in Fig. 2a, as the solid curve, has been calculated for a half space having an elastic modulus of 209 GPa, Poisson’s ratio 0.3 and a density of 7800 kg/m3 . The response is given for a distance of 30 mm and for an applied force of 10 N, which was obtained as the average value for breaking the capillary. The analytical signal is calculated as the velocity time history and as can be seen in Fig. 2a the experimentally measured signal agrees well with this calculated signal for the initial part of the transient. Both signals have been low pass filtered using a fifth order Butterworth
filter with a cut off frequency at 300 kHz. The filtering process must be provided since the AE transducer has a limit in sensitivity depending on the wavelength, i.e. when the wavelength is of the order of the diameter of the sensing area of the transducer, the sensitivity will decrease. The peak amplitudes of the measured and the analytical signal were compared and the sensitivity of the transducer was estimated to be 55.5 V/(m/s). In Fig. 2b the frequency spectra are shown and the sharp peak around 80 kHz for the experimental signal corresponds to the oscillations after 25 ms for the signal in Fig. 2a. Hence, from comparisons of the time histories and the frequency spectrum it was concluded that the AE transducers approximately were sensing the surface velocity.
Fig. 1. Sketch of the set-up for the AE transducer calibration.
Fig. 2. Calibration time signal (a) and frequency spectrum (b).
2.4. Testing procedure Quasi-static tensile tests, using a MTS servo hydraulic device, were conducted on composite laminate specimens with lay-up configurations [02, 903]S and [902, 03]S. In all tests, the cross-head speed was set to 0.01 mm/s. For monitoring of the AE transients due to damage evolution in the specimen, six AE transducers were arranged as
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schematically depicted in Fig. 3. Two transducers were placed near the tabs, at a distance of a=40 mm from the centre line, which made it easy to decide whether or not the captured emission occurred due to sliding or other spurious events in the gripping region. The other four transducers were located in pairs of two, with one transducer on the front side and the other on the back side of the specimen and at a distance of b=25 mm. This enables the possibility to add or subtract the measured signals on the two opposite transducers to decide whether the wave field in the specimen is symmetric, anti-symmetric or a combination thereof. A symmetric wave field will result in equally recorded signals on the two opposite transducers in a pair, while an anti-symmetric wave field will result in signals having opposite signs, see [20] for more details. The load and the displacement, measured by two displacement gauges, one on each side of the specimen, were recorded by the FWD unit so that a captured event could be correlated to the actual stress and strain in the specimen. In all tests the sampling frequency was set to 10 MHz and the record length to 2048 points. The pretrigger level, the level that determines the percentage of the captured waveform that occurs before receiving a valid trigger signal, was set to 12.5%. Also here, the low pass and high pass filters of the acquisition system were set to 20 kHz and 4 MHz respectively.
Fig. 3. The principal arrangement of AE transducers on test specimens.
3. Theoretical model In the present paper, the theoretical model developed by A˚berg [21] will be utilized in the numerical calculations. The model has been modified to include matrix cracking in the central layer and to capture the present lay-up configurations. Here, only a brief discussion of the model will be presented. More details are found in the Appendix. An infinitely long laminate of width 2b and thickness 2h under a uniaxial tensile loading defined by the strain "1 is considered, see Fig. 4. It is assumed that a transverse matrix crack is initiated at the edge of the laminate ðx1 ¼ 0; x2 ¼ bÞ at time t ¼ 0. The crack is then running over the width of the laminate with a velocity v. It is desired to predict the displacement or velocity response at an arbitrary point (x1 ; x2 ; x3 ) as a function of time. This solution will also be valid for finite length specimens up to a time defined by the arrival of the first wave reflection. The numerical model is based on a finite element discretization of the cross-section of the specimen. The displacements ui in direction i of the specimen can then be represented as ui xj ; t ¼ UTi ðx1 ; tÞ’ðx2 ; x3 Þ
ð1Þ
Fig. 4. The coordinate system and the matrix crack geometry for the [02, 903]S specimen (a) and the [902, 03]S specimen (b).
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where T denotes the transpose of a vector or matrix, Ui ðx1 ; tÞ represents the nodal point displacements and jðx2 ; x3 Þ is a vector of corresponding shape functions. Standard finite element shape functions can be used in this representation. According to Burridge and Knopoff [22], a displacement discontinuity can be represented by equivalent volume forces. Hence, if the crack opening displacements of the growing matrix crack are known as functions of time and position, equivalent volume forces fi can be computed that result in the same displacement solution. Assuming for a moment that the volume forces are known, the governing equations for the unknown nodal point displacements Ui ðx1 ; tÞ may be derived by use of the Hamilton principle. This system of differential equations in time and axial coordinate x1 may formally be solved by use of Fourier transforms in time and x1 . The resulting transformed equation takes the following form: 2 K2 þ iK1 þ K0 !2 MTot U^ ¼ F^ ð2Þ The matrices K0 ; K1 ; K2 ; MTot are all defined in the Appendix. The Fourier transformed nodal point displacements are collected in the vector U^ and the Fourier transformed volume forces enter the equation as nodal forces F^ . The transform variables corresponding to x1 and time are represented by and ! respectively. The imaginary unit is denoted by i. It was shown by A˚berg [21] that the matrices K0 ; K2 ; MTot are symmetric and that K1 is antisymmetric. Hence, for vanishing nodal forces Eq. (2) represents a Hermitian eigenvalue problem with eigenvalues !2n for a given . The solution for the Fourier transformed nodal displacements may then be expressed as a modal superposition. It remains to calculate the inverse Fourier transforms in order to find the displacements or velocities as functions of position and time. In A˚berg [21], this is done in two steps. First, the inverse transform with respect to is found by use of residue calculus. Secondly, the inverse transform with respect to ! is numerically determined by the fast Fourier transform (FFT). More details about the formulation of Eq. (2) and the inversion of the transforms are presented in the Appendix. A critical part in the above described procedure is an accurate model for the crack opening displacements, which in turn defines the equivalent volume forces and thereby F^ . It was argued by A˚berg [21] that except for a small region close to the fracture process of the dynamically advancing matrix crack, the crack opening could be well described by the corresponding static solution. Accurate estimates of static crack opening displacements are easily found for matrix cracks, see [23]. In the present paper, the analysis presented by A˚berg [21] has in this way been extended to include also matrix cracks advancing in the 90 layer of a [02,903]S laminate. Further details are presented in the Appendix.
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4. Results and discussion To generate experimental AE emissions due to matrix cracking events under tensile loading, a [02, 903]S specimen was used to study symmetric damage evolution and a [902, 03]S specimen for asymmetric matrix cracking. The specimens were not loaded until final failure as only matrix cracking was of interest. Signals from the tensile tests are compared to numerical signals, which are calculated as the vertical velocity response of the specimen surface. Calculations are made for two different observation points, corresponding to recording of the AE transient at two different located transducers. 4.1. Matrix cracking in the [02, 903]S specimen This specimen was loaded until a strain value of 2.5% and before unloading replicas were taken on both edges. During the test, the specimen was visually observed by naked eye and matrix cracks were observed transverse to the loading direction. For this lay-up configuration, almost all cracks extended over the whole width of the specimen and microscope inspections of the replica tapes confirmed matrix cracking in the 90 layer, i.e. transverse matrix cracking in the mid layer. It was also seen that these cracks extended through the whole thickness of the layer. 4.1.1. Experimental signal In Fig. 5, acoustic emission signals due to matrix cracking at a strain value of 0.9% are shown as the dashed curves. The signal in Fig. 5a was captured by transducer 4 and that in Fig. 5b by transducer 2. To determine the location of the event, the propagation velocity of the first arriving peak was estimated using the time difference between one transducer pair (4, 6) and the transducer located near the tabs (2) and the known distance separating these. The velocity was then taken as an average from the waves propagating in the two opposite directions and it was estimated to be 3930 m/s. The location was calculated as an average using the time difference between the two transducer pairs (3, 5 and 4, 6) and the time difference between the transducers closest to the tabs (1 and 2) and their known distances. The location of the source for the captured AE event, i.e. the matrix crack, was estimated to originate in between the two transducer pairs and at a distance of 45.5 mm from transducer 2. 4.1.2. Numerical signal As comparisons to the experimentally measured signals, the solid curves in Fig. 5 are numerically calculated signals, due to matrix cracking in the mid layer. The calculations were performed using Eq. (A23), see Appendix, up to fmax ¼ 310 kHz and the corresponding angular frequency range was discretized in !max =1000.
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Fig. 5. Acoustic emission signals shown as dashed curves and numerical calculations as solid curves on channels 4 (a) and 2 (b) for the [02, 903]S specimen.
For each value of ! the 32 smallest eigenvalues in were computed and the ones meeting the demands of being in the upper complex half-plane or laying on the real axis having positive group velocity, were included in the mode sum. The contribution from higher eigenvalues is negligible, since a large imaginary part corresponds to a strong attenuation and a large real part to a short wavelength for which the transducers show a very small sensitivity. The wavelength dependence of transducer sensitivity has further consequences for the interpretation of the experimentally obtained signals, as will be discussed later. Before the resulting spectrum was inverted using MATLAB’s FFT algorithm, the spectrum was expanded to include the complex conjugate values. The final spectrum resulted in a time history for the velocity with a maximum time of Tmax ¼ 3:2 ms. To compare the experimental and numerical time histories, the numerical signal was time shifted to give the same
starting points. A 102 ms time window of the numerical signal was taken to match the experimental signal. In this time interval the numerical signal had less sample points compared to the experimental signal and, therefore, the number of sample points was expanded through linear interpolation. In doing so the sampling frequency of both the numerical and experimental signal corresponded to 10 MHz, i.e. vectors having 1024 sample points. Thereafter both signals could be manipulated using the same MATLAB procedures, i.e. filtering processes. : The numerical signals are presented as the 3 component at the surface of the specimen, having the material properties given in Table 1. The signals were calculated for a strain value of 0.9% and the crack propagation velocity was set to ¼ 1024 m/s. In Fig. 5a the signal has been calculated for a distance of 30.5 mm between the source and the observation point, corresponding to the location of transducer 4. The solid curve in Fig. 5b is the calculated signal at transducer 2, at a distance of 45.5 mm from the source. Both the experimentally measured and the numerically calculated signal have been filtered using a fifth order Butterworth filter. The high pass and low pass cut off frequencies were set to 80 and 300 kHz, respectively. The low pass filter was motivated by the limitation in transducer sensitivity at lower frequencies and the high pass filter was selected in accordance with the maximum frequency in the numerical calculations. To compare the experimental and numerical signals in Fig. 5, the experimental signals had to be scaled to be in a comparable amplitude range to the numerical signals. This resulted in a sensitivity factor of 140 and 180 V/(m/s) for transducers 4 and 2 respectively. The overall agreement between the measured and the numerical signals in Fig. 5 is acceptable. The amplitudes seem to be in phase, although there are discrepancies in the peak values, especially on transducer 2 (Fig. 5b). However, the general appearance of the measured signals is captured in the computations, which has only included the extensional mode. There is no part in the signal originating from the complaint flexural (will be called flexural mode), torsional or stiff flexural mode, i.e. bending around the 3-axis, which are described in [21] as other possible modes in a specimen like this. The flexural mode is being neglected because the studied matrix crack is assumed to be symmetric and therefore ideally no flexural mode will be present. The torsional and stiff flexural mode are not taken into account due to the transducers location at the centre of the specimen, i.e. at x2 ¼ 0. If it is assumed that the AE transient will be reflected at the tabs where the specimen is being clamped, the estimated propagation velocity of the first arriving peak and the calculated source location will predict the reflections to arrive just after 80 ms in Fig. 5a and b. Reflections are not accounted for in the numerical model and
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therefore only the reflection free parts of the transients are considered in the comparisons between theory and experiments. 4.2. Matrix cracking in the [902,03]S specimen This specimen was loaded up to a maximum strain of 1.9% and it was also visually observed during the test. Before unloading, replicas were taken on both edges of the specimen, which showed matrix cracking in the two surface layers, i.e. the 90 -layers. From the visual observations, matrix cracks were observed transverse to the loading direction and after the finished test, it could be concluded that not all cracks had transversed the entire width of the specimen. Many cracks extended from one edge and arrested within the specimen. Some cracks initiated within the specimen and did not reach the edge. It was also observed that if a crack had initiated from one edge and arrested before reaching the other edge, it acted as a crack arrest mechanism for later appearing cracks from the other edge with a path parallel and close to the already arrested crack. This could be explained by the screening effect which results for two closely spaced crack tips. Tsamtsakis et al. [24] reported on differences in amplitudes for stable and unstable matrix cracking and suggested that unstable crack growth would show larger peak amplitudes. It is believed that the observed differences in crack lengths are reflected in the AE transients picked up by the transducers. 4.2.1. Experimental signal The captured AE transients due to matrix cracking, at a strain value of 1.7%, are represented by the dashed curves shown in Fig. 6. The signal in Fig. 6a was captured by transducer 4 and in Fig. 6b by transducer 2. The propagation velocity of the first arriving peak was estimated, in the same way as described above, to be 4320 m/s. This results in a source location of the captured event to be 45.7 mm from transducer 2 and in between the two transducer pairs. 4.2.2. Numerical signal It was found that the numerically determined flexural response was much larger than the extensional one. Comparisons to measurements showed however that the predicted flexural response had to be reduced by a factor of 50 in order to achieve a good agreement. The reason for this substantial reduction is primarily related to the wavelength dependence of the transducers, as will be further discussed in Section 4.3. A major part of the flexural response in the considered frequency band corresponds to wavelengths of the order of the transducer diameter or less. After the reduction in the numerically determined flexural response, a reasonable agreement was found between numerical and experimental signals as is seen in Fig. 6a and b. The transducer sensitivity
Fig. 6. Acoustic emission signals shown as dashed curves and numerical calculations as solid curves on channels 4 (a) and 2 (b) for the [902, 03]S specimen.
which was applied to transducer 4 and 2 was 200 and 50 V/(m/s) respectively. Before comparing the signals they were also time shifted and extended as described for the [02, 903]S specimen. The velocity responses in Fig. 6 were calculated for a strain value of 1.7% and also for this specimen the crack propagation velocity was set to ¼ 1024 m/s. The numerical signal in Fig. 6a, corresponding to transducer 4, was computed for a distance of 30.7 mm separating the source and the observation point. In Fig. 6b, the distance between the source and the observation point was 45.7 mm, corresponding to the location of transducer 2. The experimental and numerical signals were filtered using the same Butterworth filter as was described for the [02, 903]S specimen. The numerical calculations showed that the extensional mode was estimated to arrive about 14 ms before the flexural mode at transducer 4 and 20 ms at transducer 2.
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As the sensitivity of the transducers are much smaller for the flexural mode compared to the extensional mode, there is less accuracy in the signal components which are due to the flexural modes. This is perhaps an explanation for the larger discrepancies between the measured and numerical signals in Fig. 6, compared to the signals in the [02, 903]S specimen. Another contribution to the discrepancies can be that the extensional and the flexural mode will show differences in wave attenuation. This was observed in a work by Prosser [25], where the peak amplitude attenuation with increasing propagation distance was studied. It was shown that the attenuation per unit length was stronger for the flexural mode compared to the extensional mode in quasi-isotropic carbon/epoxy composite plates. The attenuation is not included in the theoretical model. A certain compensation for this attenuation effect can, however, be included in the scaling factor for the flexural modes. As can be seen in Fig. 6b, the trends in the measured signal are captured by the numerical signal, although there are differences in the phase for the amplitudes, especially around 80 ms. It was found that the assumed crack propagation velocity played an important role on the appearance of the numerically calculated signal. As described above, many matrix cracks in this specimen did not transverse the entire width of the specimen, as the majority of the cracks did in the [02, 903]S specimen. In the calculations, a crack propagation velocity of 1024 m/s was used and it was assumed that the crack transversed the entire width. These are the same conditions as were used for the [02, 903]S specimen. The uncertainties in crack propagation conditions may also give rise to dissimilarities in the comparisons to experimental results.
propagate from the source at a minimum group velocity. As an example, the [902,03]S specimen will be considered with an AE event occurring at a distance of 30.7 mm from transducer 4, corresponding to the signals shown in Fig. 6a. Frequency components having smaller group velocity than 512 m/s have been marked by circles in Fig. 7a and b. These components will arrive 50 ms or later after the first peak has arrived at the transducer and therefore they will not contribute to considered time window. Frequency components propagating faster than this will arrive earlier in the recorded signal. In Fig. 7, three different limit curves can be identified, giving the window for possible wave mode components that the AE system have a potential to record. A minimum (80 kHz) and a maximum (310 kHz) frequency are defined by the low pass and high pass filters. A third limit curve corresponds to a wavelength that is of the order of the diameter of the transducer. The transducer
4.3. Transducer sensitivity The signals recorded in the [902,03]S specimen were studied in detail. By adding and subtracting signals from two opposite transducers, 3-5 and 4-6 in Fig. 3, it was concluded that the response could be divided between an extensional wave which arrived before a flexural wave. The experimentally determined extensional and flexural signal amplitudes were of the same order. The numerically obtained signals indicated however a much larger flexural amplitude in comparison to the extensional amplitude. It was, therefore, concluded that the transducer sensitivity was larger for the extensional mode compared to the flexural mode. In order to explain this discrepancy in transducer sensitivity, the dispersion curves were analysed for the [902,03]S specimen. The dispersion curves for the extensional modes are presented in Fig. 7a and those for the flexural modes in Fig. 7b. To have frequency components arriving at a given time in the signal, they must
Fig. 7. Dispersion curves for the extensional modes (a) and the flexural modes (b) in the [902, 03]S specimen.
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is sensitive over a finite area and the resulting voltage will be some average taken over this area. This means that the transducer sensitivity generally will decrease for wavelengths smaller than the diameter of the transducer. The diameter of the sensing area of the transducer is about 8 mm which gives a limit in wavenumber at k ¼ 785 rad/m, but it is reasonable to believe that the sensitivity of the transducer starts to decrease before this approximate limit. From these given limits and the fact that only wave components propagating at a minimum group velocity will appear in the time window of the recorded signal, it is clear that the practical window for which the transducers are sensitive is larger for the extensional mode compared to the flexural mode. From Fig. 7b it is also clear that the three lowest flexural modes will give a very small contribution to the recorded signals. The major flexural contribution is expected from the lowest mode. Since a large part of this mode corresponds to wavelengths smaller than 8 mm in the considered interval, it is not surprising that a low transducer sensitivity for flexural waves are recorded.
5. Conclusions In the presented work, acoustic emission transients due to evolution of matrix cracks in cross-ply composite laminate tensile test specimens have been compared to numerically calculated surface responses. The frequency content of the signals was limited in the range of 80–300 kHz, due to limitations in the transducer sensitivity at lower frequencies and due to the highest considered frequency in the numerical calculations. The comparisons indicate a close resemblance between experimentally measured and numerically calculated signals. This is encouraging for further investigations and also for the AE technique as a useful tool in the study of different parameter’s influence on the wave field generated by different micro failures in composites. There are, however, several uncertainties still to be explored. As highlighted in this work, the transducers need to be improved and reliable calibration methods must be developed, which not only takes the frequency characteristics into account but also the wavelength dependence on transducer sensitivity. Ideally, the transducers should be as small as possible, to minimize wavelength effects. They should have a high sensitivity and a flat frequency response. There is certainly scope for further improvements in these respects. In the theoretical modelling of the source, a more accurate model for the crack propagation velocity is desired. Damping in polymer materials do influence wave attenuation and it can be expected that it may affect different wave modes in different manners. This effect was not included in the present model. It would however be possible to modify the present numerical methodology to include suitable damping models.
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Acknowledgements The financial support from the Swedish Research Council for Engineering Sciences (TFR) is gratefully acknowledged.
Appendix A coordinate system (x1 ; x2 ; x3 ) according to Fig. 4 is used to describe the matrix cracking event. The crack is initiated at the edge (x1 ¼ 0; x2 ¼ b) at time t ¼ 0 and it is advancing in the x2 -direction. The equivalent volume forces fi (Burridge and Knopoff [22]) which correspond to the crack opening displacements may according to A˚berg [21] be estimated as fi ¼ CT11ij
@ Jm ðx2 ; tÞ ustat ðx3 Þ ðx1 Þ @xj
ðA1Þ
where C11ij is the stiffness tensor of the ply containing the crack, ustat denotes the static crack opening displacement resulting from the static loading far away from the crack and represents the Dirac delta-function. The function Jðx2 ; tÞ takes the value one if the crack has reached the position x2 at time t, otherwise it is zero. This condition may be expressed as 0 t<0 ðA2Þ Jm ðx2 ; tÞ ¼ Hðx2 b þ tÞ t50 where denotes the crack propagation velocity and H the Heaviside step-function. For edge cracks, A˚berg [21] has presented accurate estimates for the static crack opening displacement based on solutions in Wu and Carlsson [26]. If this expression is modified to the present lay-up [902, 03]S, it may be expressed as
u
stat
pffiffiffi 2h 1 2TT 90 1 ð Þ ¼ 5ET pffiffiffi
4:486 0:7635 þ 0:3453 2 þ 0:0456 3 ðA3Þ
where is a dimensionless crack coordinate ð04 41Þ which is defined from h x3 ¼ ð2 þ 3Þ; 5
3h 4x3 4h 5
ðA4Þ
In Eq. (A3), ET denotes the transverse modulus in the cracking layer, TT the Poisson ratio in the transverse direction and 90 1 the axial stress in the 90 layer far away from the crack. Following standard laminate theory, the stress 90 1 may be related to the average strain " 1 as,
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90 1 ¼
ET 52LT ET 1 " 1 ð1 LT TL Þ 3ET þ 2ET
ðA5Þ
The static crack opening displacement for a central matrix crack in a [02, 903]S laminate may to a good accuracy be estimated from the corresponding solution for a crack in an infinite medium [23], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6h 1 2TT 90 x23 1 stat
u ðx3 Þ ¼ 1 ðA6Þ 5ET ð3h=5Þ2
for the conSimilarly to Eq. (A5), the stress 90 1 sidered lay-up is related to the average strain " 1 as
90 1
ET 52LT ET 1 ¼ " 1 ð1 LT TL Þ 2ET þ 3EL
ðA7Þ
The plies are assumed to be transversely isotropic. Symbolically, the constitutive equation is written as 2
3 2 C11 C12 11 6 22 7 6 C12 C22 6 7 6 6 33 7 6 C13 C23 6 7 6 6 23 7 ¼ 6 0 0 6 7 6 4 13 5 4 0 0 0 0 12 2 3 "11 6 "22 7 6 7 6 "33 7 6 7 6 7 6 2"23 7 4 2"13 5 2"12
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
ui xj ; t ¼ UTi ðx1 ; tÞ’ðx2 ; x3 Þ
ðA12Þ
The transformed nodal point displacements U^ and transformed nodal point forces Fˆ*aredefinedas 2 3 2 3 U^ 1 F^ 1 6 7 6 7 U^ ¼ 4 U^ 2 5 and F^ ¼ 4 F^ 2 5 ðA13Þ ^ ^ U3 F3
ðb ðh fi ’dx3 dx2
Fi ¼
ðA14Þ
b h
The matrices K0 ; K1 ; K2 ; MTot are determined from 2
ðA8Þ
K00 11 K2 ¼ 4 ½0 ½0
2 3 3 ½M ½0 ½0 ½0 00 ½0 K55 ½0 5; MTot ¼ 4 ½0 ½M ½0 5 ½0 ½0 ½M ½0 K00 66 ðA15Þ
ðA9Þ
where T denotes the transpose of a vector or matrix, Ui ðx1 ; tÞ represents the nodal point displacements and ’ðx2 ; x3 Þ is a vector of corresponding shape functions. Application of the Hamilton principle and Fourier transformation with respect to time t and axial coordinate x1 , results in algebraic equations for the transformed displacements U^ [21]. 2 K2 þ iK1 þ K0 !2 MTot U^ ¼ F^
ð1
Vðx1 ; tÞei!t dt; 1 ð 1 1 Vðx1 ; tÞ ¼ V ðx1 ; !Þei!t d! 2 1
V ðx1 ; !Þ ¼
where nodal point forces are derived from the equivalent volume forces according to Eq. (A1) and the shape functions ’ as
3 0 0 7 7 0 7 7 0 7 7 0 5 C66
where the stiffness coefficients CIJ may be identified in the usual way. The numerical model is based on a finite element discretization of the cross-section of the specimen. The displacements in direction i of the specimen can then be represented as
In the Fourier representation of the governing equations presented in Eq. (A10), the following transform pairs have been applied, ð1 Vðx1 ; tÞeix1 dx1 ; V^ ð; tÞ ¼ 1 ðA11Þ ð 1 1 ^ Vð; tÞeix1 d Vðx1 ; tÞ ¼ 2 1
ðA10Þ
2
33 K22 55 þ K66
6 ½0 K0 ¼ 6 4 ½0
3 ½0 ½0 22 33 23 T 23 7 K22 þ K44 K44 þ K23 7 5 T 23 22 K23 K33 44 þ K23 33 þ K44 ðA16Þ
2 ½0 6 6 02 T K02 K1 ¼ 6 55 K12 6 4 03 T K03 66 K13
T 02 K02 55 K12 ½0 ½0
T 03 3 K03 66 K13 7 7 7 ½ 0 7 5 ½ 0 ðA17Þ
where the stiffness and mass matrices are calculated as h i ðb ðh Cij ’’T dx3 dx2 ; K00 ¼ ij b h ðA18Þ h i ðb ðh @’@’T nm Kij ¼ Cij dx3 dx2 @xn @xm b h
M. Johnson, P. Gudmundson / Composites Science and Technology 61 (2001) 1367–1378
h i ðb ðh @’T K0n Cij ’ dx3 dx2 ; ¼ ij @xn b h ðb ðh ½M ¼ ’’T dx3 dx2
References ðA19Þ
b h
and the parameter denotes the density. Since the homogeneous version of Eq. (A10) represents a Hermitian eigenvalue problem for a given real value of , the solution may be expressed as a modal superposition of all modes with eigenvalues !2n ðÞ. U^ ð; !Þ ¼
3N X WT ðÞF^ ð; !Þ n
n¼1
!2n ðÞ !2
V n ð Þ
ðA20Þ
In Eq. (A20), Vn ðÞ and Wn ðÞ represent right and left eigenvectors respectively. It is here presumed that the eigenvectors have been normalized as
WTm ½MTot Vn ¼
1 0
n¼m n 6¼ m
1377
ðA21Þ
In the inversion of Eq. (A20), a positive coordinate x1 is considered. Formally, the inverse Fourier transform with respect to may be written as 3N ð 1 1 X WTn ðÞF^ ð; !Þ Vn ðÞeix1 d U ðx1 ; !Þ ¼ 2 n¼1 1 !2n ðÞ !2
ðA22Þ To evaluate the integral, is extended into the complex plane. A semi-circle of infinite radius in the halfplane ImðÞ50 is added to the integration path. For positive x1 , this contribution will vanish due to the exponential x1 -dependence. The integral can then be evaluated by residue calculus. Poles on the real axis corresponding to travelling waves are included if their group velocity cg is positive. If instead of displacements, nodal point velocities are considered, the inverse transform takes the form ! X WTp p F^ p ; ! : U ðx1 ; !Þ ¼ Vp p eip x1 2cgp p ðA23Þ where p denotes the poles in the residue calculus and cgp the group velocity. The inverse Fourier transform with respect to ! is then numerically determined by use of the inverse Fast Fourier Transform (FFT). Further details on the formulation of the theoretical model are presented by A˚berg [21].
[1] Acoustic emission testing. In: Miller RK, McIntire P, editors. Nondestructive testing handbook, vol. 5. Columbus: American Society for Nondestructive Testing, 1987. [2] Hamstad MA. A review: acoustic emission, a tool for compositematerials studies. Experimental Mechanics 1986;26:7–13. [3] Roget J. Acoustic emission: valuable applications and future trends. In: Farley JM, Nichols RW, editors. Non-destructive testing, Proc. 4th European Conference, London, vol. 4, 1987. [4] Arrington M. Acoustic emission. In: Summerscales J, editor. Non-destructive testing of fibre-reinforced plastics composites, vol. 1. Amsterdam: Elsevier, 1987. [5] Raj B, Jha BB. Fundamentals of acoustic emission. British Journal of NDT 1994;36(1):16–23. [6] Gorman MR. Plate wave acoustic emission. Journal of the Acoustical Society of America 1991;90(1):358–64. [7] Adolfsson E, Gudmundson P. Matrix crack induced stiffness reductions in 0m =90m =þp = q ÞS composite laminates. Composites Engineering 1995;5:107–23. [8] Bhat MR, Majeed MA, Murthy CRL. Characterization of fatigue damage in unidirectional GFRP composites through acoustic emission signal analysis. NDT&E International 1994;27(1):27– 32. [9] Barre´ S, Benzeggagh ML. On the use of acoustic emission to investigate damage mechanisms in glass-fibre-reinforced polypropylene. Composites Science and Technology 1994;52:369– 76. [10] Ely TM, Hill EvK. Longitudinal splitting and fiber breakage characterization in graphite epoxy using acoustic emission data. Materials Evaluation 1995;53(2):288–94. [11] Prosser WH, Jackson KE, Kellas S, Smith BT, McKeon J, Friedman A. Advanced waveform-based acoustic emission detection of matrix cracking in composites. Materials Evaluation 1995;53(9):1052–8. [12] Surgeon M, Wevers M. Modal analysis of acoustic emission signals from CFRP laminates. NDT&E International 1999;32:311– 22. [13] Guo D. Lamb waves from microfractures in composite plates. PhD thesis, University of California, Los Angeles: Mechanical Engineering, 1996. [14] Lundgren J-E. On damage and moisture in composite laminates. PhD thesis, Royal Institute of Technology, Department of Solid Mechanics, Stockholm, 2000. [15] Hatano H, Watanabe T. Reciprocity calibration of acoustic emission transducers in Rayleigh-wave and longitudinal-wave sound fields. Journal of the Acoustical Society of America 1997;101(1):1450–5. [16] Simmons J, Turner C, Wadley H. Vector calibration of ultrasonic and acoustic emission transducers. Journal of the Acoustical Society of America 1987;82(4):1122–30. [17] ASTM Designation: E 1106-86, American Society for Testing and Materials. Standard method for primary calibration of acoustic emission sensors, 1986. [18] Pekeris CL. The seismic surface pulse. Proceedings of the National Academy of Sciences of the United States of America 1955;41:469–80. [19] Mooney H. Some numerical solutions for Lamb’s problem. Bulletin of the Seismological Society of America 1974;62(2):473– 91. [20] Johnson M, Gudmundson P. Broad-band transient recording and characterization of acoustic emission events in composite laminates. Composites Science and Technology 2000;60(15): 2803–2818. [21] A˚berg M. Wave propagation and damage in composite laminates. PhD thesis, Royal Institute of Technology, Department of Solid Mechanics, Stockholm, 1999.
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[22] Burridge R, Knopoff L. Body force equivalents for seismic dislocations. Bulletin of the Seismological Society of America 1964;54(6):1875–88. [23] Gudmundson P, Zang W. An analytic model for thermoelastic properties of composite laminates containing transverse matrix cracks. International Journal of Solids and Structures 1993; 30(23):3211–31.
[24] Tsamtsakis D, Wevers M, De Meester P. Damage monitoring during monotonic tensile loading of quasi-isotropic carbon epoxy laminates. Materials Science Forum 1996;210-213:125–32. [25] Prosser W. Advanced AE techniques in composite materials research. Journal of Acoustic Emission 1996;14(3/4):1–11. [26] Wu W-R, Carlsson AJ. Weight functions and stress intensity factor solutions. Oxford: Pergamon, 1991.
Experimental and theoretical characterization of acoustic emission transients in composite laminates Mikael Johnson, Peter Gudmundson* Department of Solid Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 26 September 2000; accepted 7 March 2001
Abstract Transient wave propagation resulting from transverse matrix cracking in cross-ply composite tensile test specimens are investigated both theoretically and experimentally. In the experiments, broad-band transducers and a fast data-acquisition system enable measurements of transients in a frequency interval up to 1 MHz. The theoretical predictions are based on a model which utilizes a finite-element discretization of the cross-section of the specimen and a Fourier representation of the axial and time dependence. The comparisons between experimental and theoretical results are encouraging. Weak points in the experimental technique and in the theoretical model are identified. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Matrix cracking; C. Finite-element analysis; C. Laminates; D. Acoustic emission
1. Introduction The acoustic emission (AE) technique is a well-known non-destructive evaluation technique [1], both in research and for industrial applications. It is mainly used to monitor the onset of cracking processes in materials and components. One of AE’s great advantages is the ability to scan a large structure with only a few sensors and to localize the source causing the captured AE transient. The first systems used resonant transducers, having a resonant peak in the frequency range of 150–500 kHz. The state of development 10–15 years ago is well described in a number of review papers [2–5]. The rapid development of computers with increasing speed and extended memory capacity have, however, enabled digital acquisition using broadband transducers. These transducers are sensitive in a wider frequency range, typically from 50 kHz up to 1.5 MHz, see Gorman [6] for a comparison between resonant and broadband transducers. The use of composite material as a construction material is increasing and can, nowadays, be found in
* Corresponding author. Tel.: +46-8-790-7548; fax: +46-8-4112418. E-mail addresses: [email protected] (P. Gudmundson), mikael@ hallf.kth.se (M. Johnson).
various components. A particular feature of composite laminates is that micro-structural damage generally develops prior to final failure. The damage itself is generally not critical for the structural integrity, but it influences certain macroscopic properties such as stiffnesses [7]. Typical damage modes are matrix cracking, delamination, debonding and fibre breakage. Matrix cracking usually develops at an early stage and it may act as a crack initiator for other mechanisms. Many investigations have been presented where conventional AE has been applied as a monitoring tool for crack surveillance in composite materials. Bhat et al. [8] use a pattern-recognition technique to discriminate different failure mechanisms in fatigue tests on unidirectional GRP composites. Through analysis of the AE amplitude distribution, Barre´ and Benzeggagh [9] identify different damage mechanisms in short-glass-fibre-reinforced polypropylene and Ely and Hill [10] characterize longitudinal splitting and fiber breakage in carbon/epoxy laminates. The use of broad-band transducers will, however, provide more extensive information in terms of frequency content in the AE signal. Prosser et al. [11] studied matrix cracking in a carbon/epoxy composite and concluded that dispersion and attenuation will influence the detected amplitude, depending on the distance of propagation. On the basis of signals measured by broadband transducers, researchers have tried to identify
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00036-7
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different wave modes in the recorded signals, so-called modal acoustic emission (MAE). Surgeon and Wevers [12] used MAE to determine the amplitude ratio between extensional and flexural modes in CFRP laminates for different crack mechanisms. A wider frequency spectrum will also give a possibility to make more realistic and relevant comparisons between experimentally measured and numerically calculated signals. Numerical simulations will give further insight into the influence of various parameters on the recorded AE-signal. Examples of parameters which have an effect on the transients are: source mechanism and orientation, crack propagation velocity and direction, lay-up configuration. Few quantitative comparisons between experimentally and numerically obtained transients have been presented in the past. One investigation is presented by Guo [13], who modelled the surface response in composite laminates due to different microfracture modes and compared with experiments on composite laminates with artificial defects. However, there are still uncertainties related to the transducers and the whole acquisition system, which are of essential and crucial importance to the comparisons. In the presented work, numerical modelling of AE wave propagation due to matrix cracking will be presented and compared to measured signals resulting from tensile tests of cross-ply glass/epoxy composite specimens. Problems concerning the wavelength dependence of the transducer sensitivity are also addressed.
2. Experimental set-up and procedures 2.1. Specimen preparation In the experiments, specimens with two types of layup configurations were used, [02, 903]S and [902, 03]S. They were both produced from cross-ply laminates manufactured by Saab Military Aircraft, Sweden, using epoxy/glassfiber prepregs. The angles refer to the loading direction and the unidirectional properties, presented in Table 1, were experimentally estimated and presented by Lundgren [14]. All specimens were 300 mm in length and had a thickness of about 1.39 mm. The specimen edges were wet sanded to a width of 25 mm, in order to prevent damage initiations from edge defects due to the cutting process and also to enable the use of replica technique. To reduce noise coming from the gripping region, soft aluminium tabs were glued on both sides of
the specimen at each end. The tabs were manufactured with a 18 mm long bevel and with a total length of 50 mm. The width and thickness were 25 and 3 mm, respectively. 2.2. Acoustic emission system For monitoring of the specimen during the tensile tests, a Fracture Wave Detector F4012 (FWD) system, manufactured by Digital Wave Corporation, Denver, USA, was used. The acoustic emission (AE) transients, generated due to micro cracking in the specimen, were captured using DWC B1025 piezoelectric transducers. These are of a broad-band type with a diameter of the sensing area of about 8 mm. The manufacturer claims that the transducers should have an approximately flat frequency response in the range of 50 kHz–1.5 MHz. A sensitivity in terms of displacement or velocity is, however, not given. The transducer has to be calibrated for each particular application. The calibration in the present applications will be described below. 2.3. Calibration of the AE system When an acoustic emission is picked up by the piezoelectric crystal the signal must pass through the whole acquisition system, which is a chain of filters, amplifiers and an A/D-converter. Thereby the measured signal will be distorted and it will result in a signal which may not directly be interpreted as the corresponding motion of the specimen surface. Therefore, to be able to make relevant comparisons between experimentally measured and numerically calculated AE signals, the transfer function of the whole acquisition system must be investigated. To explore the frequency response and to determine the transfer function of an AE system, several procedures have been suggested, see for example Hatano and Watanabe [15]. A method to determine the in-plane and out-of-plane sensitivity of a piezoelectric sensor has been proposed by Simmons et al. [16]. The calibration method used in this paper is mainly based on the method described in the ASTM standard [17]. The basic idea is to generate a step load, i.e. to suddenly remove a point force, on a big steel block. The generated emission is captured by a transducer with a known distance to the source. This signal can then be compared to the elastic solution for a point force on a half space, which was derived in closed form by Pekeris [18] for a Poisson’s
Table 1 Unidirectional stiffness properties in the [02 , 903 ]S and [902 , 03 ]S laminates EL (GPa)
ET (GPa)
LT
GLT (GPa)
(kg/m3)
Fibre volume fraction (%)
46.2
16.9
0.31
7.7
1942
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ratio of 0.25. For other values of the Poisson ratio, a solution has been presented by Mooney [19]. In this work, a steel block with the dimensions 1600 800150 mm was used. By breaking a fused silica capillary on the surface of the steel block, a sudden release of the force was achieved. The inner and outer diameter of the capillary was 0.18 and 0.34 mm, respectively. This capillary was placed on the surface of the steel block with a glass rod transversally aligned above the capillary, see Fig. 1. To break the capillary, it is being squeezed between the surface of the steel block and the glass rod, using a loading screw. When the capillary breaks, the sudden release of force is an approximate step function with a fast rise time and the generated transient is captured by the transducer undergoing calibration. The transducer was mounted at two different locations, 30 and 50 mm from the source, using vacuum grease. During the calibration procedure, the sampling frequency of the acquisition system was set to 10 MHz and a record length of 1024 sample points was used, enabling a time window of 102.4 ms. The low pass and high pass filters were set to 20 kHz and 4 MHz, respectively. The dashed curve in Fig. 2a shows a measured signal due to breaking a capillary 30 mm from the transducer. As a reference signal, the time response due to a vertical point force on a half space was calculated, using the solution presented by Mooney [19]. Mooney based his work on the closed-form solutions given by Pekeris and extended it to be applicable for arbitrary values of Poisson’s ratio. The signal shown in Fig. 2a, as the solid curve, has been calculated for a half space having an elastic modulus of 209 GPa, Poisson’s ratio 0.3 and a density of 7800 kg/m3 . The response is given for a distance of 30 mm and for an applied force of 10 N, which was obtained as the average value for breaking the capillary. The analytical signal is calculated as the velocity time history and as can be seen in Fig. 2a the experimentally measured signal agrees well with this calculated signal for the initial part of the transient. Both signals have been low pass filtered using a fifth order Butterworth
filter with a cut off frequency at 300 kHz. The filtering process must be provided since the AE transducer has a limit in sensitivity depending on the wavelength, i.e. when the wavelength is of the order of the diameter of the sensing area of the transducer, the sensitivity will decrease. The peak amplitudes of the measured and the analytical signal were compared and the sensitivity of the transducer was estimated to be 55.5 V/(m/s). In Fig. 2b the frequency spectra are shown and the sharp peak around 80 kHz for the experimental signal corresponds to the oscillations after 25 ms for the signal in Fig. 2a. Hence, from comparisons of the time histories and the frequency spectrum it was concluded that the AE transducers approximately were sensing the surface velocity.
Fig. 1. Sketch of the set-up for the AE transducer calibration.
Fig. 2. Calibration time signal (a) and frequency spectrum (b).
2.4. Testing procedure Quasi-static tensile tests, using a MTS servo hydraulic device, were conducted on composite laminate specimens with lay-up configurations [02, 903]S and [902, 03]S. In all tests, the cross-head speed was set to 0.01 mm/s. For monitoring of the AE transients due to damage evolution in the specimen, six AE transducers were arranged as
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schematically depicted in Fig. 3. Two transducers were placed near the tabs, at a distance of a=40 mm from the centre line, which made it easy to decide whether or not the captured emission occurred due to sliding or other spurious events in the gripping region. The other four transducers were located in pairs of two, with one transducer on the front side and the other on the back side of the specimen and at a distance of b=25 mm. This enables the possibility to add or subtract the measured signals on the two opposite transducers to decide whether the wave field in the specimen is symmetric, anti-symmetric or a combination thereof. A symmetric wave field will result in equally recorded signals on the two opposite transducers in a pair, while an anti-symmetric wave field will result in signals having opposite signs, see [20] for more details. The load and the displacement, measured by two displacement gauges, one on each side of the specimen, were recorded by the FWD unit so that a captured event could be correlated to the actual stress and strain in the specimen. In all tests the sampling frequency was set to 10 MHz and the record length to 2048 points. The pretrigger level, the level that determines the percentage of the captured waveform that occurs before receiving a valid trigger signal, was set to 12.5%. Also here, the low pass and high pass filters of the acquisition system were set to 20 kHz and 4 MHz respectively.
Fig. 3. The principal arrangement of AE transducers on test specimens.
3. Theoretical model In the present paper, the theoretical model developed by A˚berg [21] will be utilized in the numerical calculations. The model has been modified to include matrix cracking in the central layer and to capture the present lay-up configurations. Here, only a brief discussion of the model will be presented. More details are found in the Appendix. An infinitely long laminate of width 2b and thickness 2h under a uniaxial tensile loading defined by the strain "1 is considered, see Fig. 4. It is assumed that a transverse matrix crack is initiated at the edge of the laminate ðx1 ¼ 0; x2 ¼ bÞ at time t ¼ 0. The crack is then running over the width of the laminate with a velocity v. It is desired to predict the displacement or velocity response at an arbitrary point (x1 ; x2 ; x3 ) as a function of time. This solution will also be valid for finite length specimens up to a time defined by the arrival of the first wave reflection. The numerical model is based on a finite element discretization of the cross-section of the specimen. The displacements ui in direction i of the specimen can then be represented as ui xj ; t ¼ UTi ðx1 ; tÞ’ðx2 ; x3 Þ
ð1Þ
Fig. 4. The coordinate system and the matrix crack geometry for the [02, 903]S specimen (a) and the [902, 03]S specimen (b).
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where T denotes the transpose of a vector or matrix, Ui ðx1 ; tÞ represents the nodal point displacements and jðx2 ; x3 Þ is a vector of corresponding shape functions. Standard finite element shape functions can be used in this representation. According to Burridge and Knopoff [22], a displacement discontinuity can be represented by equivalent volume forces. Hence, if the crack opening displacements of the growing matrix crack are known as functions of time and position, equivalent volume forces fi can be computed that result in the same displacement solution. Assuming for a moment that the volume forces are known, the governing equations for the unknown nodal point displacements Ui ðx1 ; tÞ may be derived by use of the Hamilton principle. This system of differential equations in time and axial coordinate x1 may formally be solved by use of Fourier transforms in time and x1 . The resulting transformed equation takes the following form: 2 K2 þ iK1 þ K0 !2 MTot U^ ¼ F^ ð2Þ The matrices K0 ; K1 ; K2 ; MTot are all defined in the Appendix. The Fourier transformed nodal point displacements are collected in the vector U^ and the Fourier transformed volume forces enter the equation as nodal forces F^ . The transform variables corresponding to x1 and time are represented by and ! respectively. The imaginary unit is denoted by i. It was shown by A˚berg [21] that the matrices K0 ; K2 ; MTot are symmetric and that K1 is antisymmetric. Hence, for vanishing nodal forces Eq. (2) represents a Hermitian eigenvalue problem with eigenvalues !2n for a given . The solution for the Fourier transformed nodal displacements may then be expressed as a modal superposition. It remains to calculate the inverse Fourier transforms in order to find the displacements or velocities as functions of position and time. In A˚berg [21], this is done in two steps. First, the inverse transform with respect to is found by use of residue calculus. Secondly, the inverse transform with respect to ! is numerically determined by the fast Fourier transform (FFT). More details about the formulation of Eq. (2) and the inversion of the transforms are presented in the Appendix. A critical part in the above described procedure is an accurate model for the crack opening displacements, which in turn defines the equivalent volume forces and thereby F^ . It was argued by A˚berg [21] that except for a small region close to the fracture process of the dynamically advancing matrix crack, the crack opening could be well described by the corresponding static solution. Accurate estimates of static crack opening displacements are easily found for matrix cracks, see [23]. In the present paper, the analysis presented by A˚berg [21] has in this way been extended to include also matrix cracks advancing in the 90 layer of a [02,903]S laminate. Further details are presented in the Appendix.
1371
4. Results and discussion To generate experimental AE emissions due to matrix cracking events under tensile loading, a [02, 903]S specimen was used to study symmetric damage evolution and a [902, 03]S specimen for asymmetric matrix cracking. The specimens were not loaded until final failure as only matrix cracking was of interest. Signals from the tensile tests are compared to numerical signals, which are calculated as the vertical velocity response of the specimen surface. Calculations are made for two different observation points, corresponding to recording of the AE transient at two different located transducers. 4.1. Matrix cracking in the [02, 903]S specimen This specimen was loaded until a strain value of 2.5% and before unloading replicas were taken on both edges. During the test, the specimen was visually observed by naked eye and matrix cracks were observed transverse to the loading direction. For this lay-up configuration, almost all cracks extended over the whole width of the specimen and microscope inspections of the replica tapes confirmed matrix cracking in the 90 layer, i.e. transverse matrix cracking in the mid layer. It was also seen that these cracks extended through the whole thickness of the layer. 4.1.1. Experimental signal In Fig. 5, acoustic emission signals due to matrix cracking at a strain value of 0.9% are shown as the dashed curves. The signal in Fig. 5a was captured by transducer 4 and that in Fig. 5b by transducer 2. To determine the location of the event, the propagation velocity of the first arriving peak was estimated using the time difference between one transducer pair (4, 6) and the transducer located near the tabs (2) and the known distance separating these. The velocity was then taken as an average from the waves propagating in the two opposite directions and it was estimated to be 3930 m/s. The location was calculated as an average using the time difference between the two transducer pairs (3, 5 and 4, 6) and the time difference between the transducers closest to the tabs (1 and 2) and their known distances. The location of the source for the captured AE event, i.e. the matrix crack, was estimated to originate in between the two transducer pairs and at a distance of 45.5 mm from transducer 2. 4.1.2. Numerical signal As comparisons to the experimentally measured signals, the solid curves in Fig. 5 are numerically calculated signals, due to matrix cracking in the mid layer. The calculations were performed using Eq. (A23), see Appendix, up to fmax ¼ 310 kHz and the corresponding angular frequency range was discretized in !max =1000.
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Fig. 5. Acoustic emission signals shown as dashed curves and numerical calculations as solid curves on channels 4 (a) and 2 (b) for the [02, 903]S specimen.
For each value of ! the 32 smallest eigenvalues in were computed and the ones meeting the demands of being in the upper complex half-plane or laying on the real axis having positive group velocity, were included in the mode sum. The contribution from higher eigenvalues is negligible, since a large imaginary part corresponds to a strong attenuation and a large real part to a short wavelength for which the transducers show a very small sensitivity. The wavelength dependence of transducer sensitivity has further consequences for the interpretation of the experimentally obtained signals, as will be discussed later. Before the resulting spectrum was inverted using MATLAB’s FFT algorithm, the spectrum was expanded to include the complex conjugate values. The final spectrum resulted in a time history for the velocity with a maximum time of Tmax ¼ 3:2 ms. To compare the experimental and numerical time histories, the numerical signal was time shifted to give the same
starting points. A 102 ms time window of the numerical signal was taken to match the experimental signal. In this time interval the numerical signal had less sample points compared to the experimental signal and, therefore, the number of sample points was expanded through linear interpolation. In doing so the sampling frequency of both the numerical and experimental signal corresponded to 10 MHz, i.e. vectors having 1024 sample points. Thereafter both signals could be manipulated using the same MATLAB procedures, i.e. filtering processes. : The numerical signals are presented as the 3 component at the surface of the specimen, having the material properties given in Table 1. The signals were calculated for a strain value of 0.9% and the crack propagation velocity was set to ¼ 1024 m/s. In Fig. 5a the signal has been calculated for a distance of 30.5 mm between the source and the observation point, corresponding to the location of transducer 4. The solid curve in Fig. 5b is the calculated signal at transducer 2, at a distance of 45.5 mm from the source. Both the experimentally measured and the numerically calculated signal have been filtered using a fifth order Butterworth filter. The high pass and low pass cut off frequencies were set to 80 and 300 kHz, respectively. The low pass filter was motivated by the limitation in transducer sensitivity at lower frequencies and the high pass filter was selected in accordance with the maximum frequency in the numerical calculations. To compare the experimental and numerical signals in Fig. 5, the experimental signals had to be scaled to be in a comparable amplitude range to the numerical signals. This resulted in a sensitivity factor of 140 and 180 V/(m/s) for transducers 4 and 2 respectively. The overall agreement between the measured and the numerical signals in Fig. 5 is acceptable. The amplitudes seem to be in phase, although there are discrepancies in the peak values, especially on transducer 2 (Fig. 5b). However, the general appearance of the measured signals is captured in the computations, which has only included the extensional mode. There is no part in the signal originating from the complaint flexural (will be called flexural mode), torsional or stiff flexural mode, i.e. bending around the 3-axis, which are described in [21] as other possible modes in a specimen like this. The flexural mode is being neglected because the studied matrix crack is assumed to be symmetric and therefore ideally no flexural mode will be present. The torsional and stiff flexural mode are not taken into account due to the transducers location at the centre of the specimen, i.e. at x2 ¼ 0. If it is assumed that the AE transient will be reflected at the tabs where the specimen is being clamped, the estimated propagation velocity of the first arriving peak and the calculated source location will predict the reflections to arrive just after 80 ms in Fig. 5a and b. Reflections are not accounted for in the numerical model and
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therefore only the reflection free parts of the transients are considered in the comparisons between theory and experiments. 4.2. Matrix cracking in the [902,03]S specimen This specimen was loaded up to a maximum strain of 1.9% and it was also visually observed during the test. Before unloading, replicas were taken on both edges of the specimen, which showed matrix cracking in the two surface layers, i.e. the 90 -layers. From the visual observations, matrix cracks were observed transverse to the loading direction and after the finished test, it could be concluded that not all cracks had transversed the entire width of the specimen. Many cracks extended from one edge and arrested within the specimen. Some cracks initiated within the specimen and did not reach the edge. It was also observed that if a crack had initiated from one edge and arrested before reaching the other edge, it acted as a crack arrest mechanism for later appearing cracks from the other edge with a path parallel and close to the already arrested crack. This could be explained by the screening effect which results for two closely spaced crack tips. Tsamtsakis et al. [24] reported on differences in amplitudes for stable and unstable matrix cracking and suggested that unstable crack growth would show larger peak amplitudes. It is believed that the observed differences in crack lengths are reflected in the AE transients picked up by the transducers. 4.2.1. Experimental signal The captured AE transients due to matrix cracking, at a strain value of 1.7%, are represented by the dashed curves shown in Fig. 6. The signal in Fig. 6a was captured by transducer 4 and in Fig. 6b by transducer 2. The propagation velocity of the first arriving peak was estimated, in the same way as described above, to be 4320 m/s. This results in a source location of the captured event to be 45.7 mm from transducer 2 and in between the two transducer pairs. 4.2.2. Numerical signal It was found that the numerically determined flexural response was much larger than the extensional one. Comparisons to measurements showed however that the predicted flexural response had to be reduced by a factor of 50 in order to achieve a good agreement. The reason for this substantial reduction is primarily related to the wavelength dependence of the transducers, as will be further discussed in Section 4.3. A major part of the flexural response in the considered frequency band corresponds to wavelengths of the order of the transducer diameter or less. After the reduction in the numerically determined flexural response, a reasonable agreement was found between numerical and experimental signals as is seen in Fig. 6a and b. The transducer sensitivity
Fig. 6. Acoustic emission signals shown as dashed curves and numerical calculations as solid curves on channels 4 (a) and 2 (b) for the [902, 03]S specimen.
which was applied to transducer 4 and 2 was 200 and 50 V/(m/s) respectively. Before comparing the signals they were also time shifted and extended as described for the [02, 903]S specimen. The velocity responses in Fig. 6 were calculated for a strain value of 1.7% and also for this specimen the crack propagation velocity was set to ¼ 1024 m/s. The numerical signal in Fig. 6a, corresponding to transducer 4, was computed for a distance of 30.7 mm separating the source and the observation point. In Fig. 6b, the distance between the source and the observation point was 45.7 mm, corresponding to the location of transducer 2. The experimental and numerical signals were filtered using the same Butterworth filter as was described for the [02, 903]S specimen. The numerical calculations showed that the extensional mode was estimated to arrive about 14 ms before the flexural mode at transducer 4 and 20 ms at transducer 2.
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As the sensitivity of the transducers are much smaller for the flexural mode compared to the extensional mode, there is less accuracy in the signal components which are due to the flexural modes. This is perhaps an explanation for the larger discrepancies between the measured and numerical signals in Fig. 6, compared to the signals in the [02, 903]S specimen. Another contribution to the discrepancies can be that the extensional and the flexural mode will show differences in wave attenuation. This was observed in a work by Prosser [25], where the peak amplitude attenuation with increasing propagation distance was studied. It was shown that the attenuation per unit length was stronger for the flexural mode compared to the extensional mode in quasi-isotropic carbon/epoxy composite plates. The attenuation is not included in the theoretical model. A certain compensation for this attenuation effect can, however, be included in the scaling factor for the flexural modes. As can be seen in Fig. 6b, the trends in the measured signal are captured by the numerical signal, although there are differences in the phase for the amplitudes, especially around 80 ms. It was found that the assumed crack propagation velocity played an important role on the appearance of the numerically calculated signal. As described above, many matrix cracks in this specimen did not transverse the entire width of the specimen, as the majority of the cracks did in the [02, 903]S specimen. In the calculations, a crack propagation velocity of 1024 m/s was used and it was assumed that the crack transversed the entire width. These are the same conditions as were used for the [02, 903]S specimen. The uncertainties in crack propagation conditions may also give rise to dissimilarities in the comparisons to experimental results.
propagate from the source at a minimum group velocity. As an example, the [902,03]S specimen will be considered with an AE event occurring at a distance of 30.7 mm from transducer 4, corresponding to the signals shown in Fig. 6a. Frequency components having smaller group velocity than 512 m/s have been marked by circles in Fig. 7a and b. These components will arrive 50 ms or later after the first peak has arrived at the transducer and therefore they will not contribute to considered time window. Frequency components propagating faster than this will arrive earlier in the recorded signal. In Fig. 7, three different limit curves can be identified, giving the window for possible wave mode components that the AE system have a potential to record. A minimum (80 kHz) and a maximum (310 kHz) frequency are defined by the low pass and high pass filters. A third limit curve corresponds to a wavelength that is of the order of the diameter of the transducer. The transducer
4.3. Transducer sensitivity The signals recorded in the [902,03]S specimen were studied in detail. By adding and subtracting signals from two opposite transducers, 3-5 and 4-6 in Fig. 3, it was concluded that the response could be divided between an extensional wave which arrived before a flexural wave. The experimentally determined extensional and flexural signal amplitudes were of the same order. The numerically obtained signals indicated however a much larger flexural amplitude in comparison to the extensional amplitude. It was, therefore, concluded that the transducer sensitivity was larger for the extensional mode compared to the flexural mode. In order to explain this discrepancy in transducer sensitivity, the dispersion curves were analysed for the [902,03]S specimen. The dispersion curves for the extensional modes are presented in Fig. 7a and those for the flexural modes in Fig. 7b. To have frequency components arriving at a given time in the signal, they must
Fig. 7. Dispersion curves for the extensional modes (a) and the flexural modes (b) in the [902, 03]S specimen.
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is sensitive over a finite area and the resulting voltage will be some average taken over this area. This means that the transducer sensitivity generally will decrease for wavelengths smaller than the diameter of the transducer. The diameter of the sensing area of the transducer is about 8 mm which gives a limit in wavenumber at k ¼ 785 rad/m, but it is reasonable to believe that the sensitivity of the transducer starts to decrease before this approximate limit. From these given limits and the fact that only wave components propagating at a minimum group velocity will appear in the time window of the recorded signal, it is clear that the practical window for which the transducers are sensitive is larger for the extensional mode compared to the flexural mode. From Fig. 7b it is also clear that the three lowest flexural modes will give a very small contribution to the recorded signals. The major flexural contribution is expected from the lowest mode. Since a large part of this mode corresponds to wavelengths smaller than 8 mm in the considered interval, it is not surprising that a low transducer sensitivity for flexural waves are recorded.
5. Conclusions In the presented work, acoustic emission transients due to evolution of matrix cracks in cross-ply composite laminate tensile test specimens have been compared to numerically calculated surface responses. The frequency content of the signals was limited in the range of 80–300 kHz, due to limitations in the transducer sensitivity at lower frequencies and due to the highest considered frequency in the numerical calculations. The comparisons indicate a close resemblance between experimentally measured and numerically calculated signals. This is encouraging for further investigations and also for the AE technique as a useful tool in the study of different parameter’s influence on the wave field generated by different micro failures in composites. There are, however, several uncertainties still to be explored. As highlighted in this work, the transducers need to be improved and reliable calibration methods must be developed, which not only takes the frequency characteristics into account but also the wavelength dependence on transducer sensitivity. Ideally, the transducers should be as small as possible, to minimize wavelength effects. They should have a high sensitivity and a flat frequency response. There is certainly scope for further improvements in these respects. In the theoretical modelling of the source, a more accurate model for the crack propagation velocity is desired. Damping in polymer materials do influence wave attenuation and it can be expected that it may affect different wave modes in different manners. This effect was not included in the present model. It would however be possible to modify the present numerical methodology to include suitable damping models.
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Acknowledgements The financial support from the Swedish Research Council for Engineering Sciences (TFR) is gratefully acknowledged.
Appendix A coordinate system (x1 ; x2 ; x3 ) according to Fig. 4 is used to describe the matrix cracking event. The crack is initiated at the edge (x1 ¼ 0; x2 ¼ b) at time t ¼ 0 and it is advancing in the x2 -direction. The equivalent volume forces fi (Burridge and Knopoff [22]) which correspond to the crack opening displacements may according to A˚berg [21] be estimated as fi ¼ CT11ij
@ Jm ðx2 ; tÞ ustat ðx3 Þ ðx1 Þ @xj
ðA1Þ
where C11ij is the stiffness tensor of the ply containing the crack, ustat denotes the static crack opening displacement resulting from the static loading far away from the crack and represents the Dirac delta-function. The function Jðx2 ; tÞ takes the value one if the crack has reached the position x2 at time t, otherwise it is zero. This condition may be expressed as 0 t<0 ðA2Þ Jm ðx2 ; tÞ ¼ Hðx2 b þ tÞ t50 where denotes the crack propagation velocity and H the Heaviside step-function. For edge cracks, A˚berg [21] has presented accurate estimates for the static crack opening displacement based on solutions in Wu and Carlsson [26]. If this expression is modified to the present lay-up [902, 03]S, it may be expressed as
u
stat
pffiffiffi 2h 1 2TT 90 1 ð Þ ¼ 5ET pffiffiffi
4:486 0:7635 þ 0:3453 2 þ 0:0456 3 ðA3Þ
where is a dimensionless crack coordinate ð04 41Þ which is defined from h x3 ¼ ð2 þ 3Þ; 5
3h 4x3 4h 5
ðA4Þ
In Eq. (A3), ET denotes the transverse modulus in the cracking layer, TT the Poisson ratio in the transverse direction and 90 1 the axial stress in the 90 layer far away from the crack. Following standard laminate theory, the stress 90 1 may be related to the average strain " 1 as,
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90 1 ¼
ET 52LT ET 1 " 1 ð1 LT TL Þ 3ET þ 2ET
ðA5Þ
The static crack opening displacement for a central matrix crack in a [02, 903]S laminate may to a good accuracy be estimated from the corresponding solution for a crack in an infinite medium [23], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6h 1 2TT 90 x23 1 stat
u ðx3 Þ ¼ 1 ðA6Þ 5ET ð3h=5Þ2
for the conSimilarly to Eq. (A5), the stress 90 1 sidered lay-up is related to the average strain " 1 as
90 1
ET 52LT ET 1 ¼ " 1 ð1 LT TL Þ 2ET þ 3EL
ðA7Þ
The plies are assumed to be transversely isotropic. Symbolically, the constitutive equation is written as 2
3 2 C11 C12 11 6 22 7 6 C12 C22 6 7 6 6 33 7 6 C13 C23 6 7 6 6 23 7 ¼ 6 0 0 6 7 6 4 13 5 4 0 0 0 0 12 2 3 "11 6 "22 7 6 7 6 "33 7 6 7 6 7 6 2"23 7 4 2"13 5 2"12
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
ui xj ; t ¼ UTi ðx1 ; tÞ’ðx2 ; x3 Þ
ðA12Þ
The transformed nodal point displacements U^ and transformed nodal point forces Fˆ*aredefinedas 2 3 2 3 U^ 1 F^ 1 6 7 6 7 U^ ¼ 4 U^ 2 5 and F^ ¼ 4 F^ 2 5 ðA13Þ ^ ^ U3 F3
ðb ðh fi ’dx3 dx2
Fi ¼
ðA14Þ
b h
The matrices K0 ; K1 ; K2 ; MTot are determined from 2
ðA8Þ
K00 11 K2 ¼ 4 ½0 ½0
2 3 3 ½M ½0 ½0 ½0 00 ½0 K55 ½0 5; MTot ¼ 4 ½0 ½M ½0 5 ½0 ½0 ½M ½0 K00 66 ðA15Þ
ðA9Þ
where T denotes the transpose of a vector or matrix, Ui ðx1 ; tÞ represents the nodal point displacements and ’ðx2 ; x3 Þ is a vector of corresponding shape functions. Application of the Hamilton principle and Fourier transformation with respect to time t and axial coordinate x1 , results in algebraic equations for the transformed displacements U^ [21]. 2 K2 þ iK1 þ K0 !2 MTot U^ ¼ F^
ð1
Vðx1 ; tÞei!t dt; 1 ð 1 1 Vðx1 ; tÞ ¼ V ðx1 ; !Þei!t d! 2 1
V ðx1 ; !Þ ¼
where nodal point forces are derived from the equivalent volume forces according to Eq. (A1) and the shape functions ’ as
3 0 0 7 7 0 7 7 0 7 7 0 5 C66
where the stiffness coefficients CIJ may be identified in the usual way. The numerical model is based on a finite element discretization of the cross-section of the specimen. The displacements in direction i of the specimen can then be represented as
In the Fourier representation of the governing equations presented in Eq. (A10), the following transform pairs have been applied, ð1 Vðx1 ; tÞeix1 dx1 ; V^ ð; tÞ ¼ 1 ðA11Þ ð 1 1 ^ Vð; tÞeix1 d Vðx1 ; tÞ ¼ 2 1
ðA10Þ
2
33 K22 55 þ K66
6 ½0 K0 ¼ 6 4 ½0
3 ½0 ½0 22 33 23 T 23 7 K22 þ K44 K44 þ K23 7 5 T 23 22 K23 K33 44 þ K23 33 þ K44 ðA16Þ
2 ½0 6 6 02 T K02 K1 ¼ 6 55 K12 6 4 03 T K03 66 K13
T 02 K02 55 K12 ½0 ½0
T 03 3 K03 66 K13 7 7 7 ½ 0 7 5 ½ 0 ðA17Þ
where the stiffness and mass matrices are calculated as h i ðb ðh Cij ’’T dx3 dx2 ; K00 ¼ ij b h ðA18Þ h i ðb ðh @’@’T nm Kij ¼ Cij dx3 dx2 @xn @xm b h
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h i ðb ðh @’T K0n Cij ’ dx3 dx2 ; ¼ ij @xn b h ðb ðh ½M ¼ ’’T dx3 dx2
References ðA19Þ
b h
and the parameter denotes the density. Since the homogeneous version of Eq. (A10) represents a Hermitian eigenvalue problem for a given real value of , the solution may be expressed as a modal superposition of all modes with eigenvalues !2n ðÞ. U^ ð; !Þ ¼
3N X WT ðÞF^ ð; !Þ n
n¼1
!2n ðÞ !2
V n ð Þ
ðA20Þ
In Eq. (A20), Vn ðÞ and Wn ðÞ represent right and left eigenvectors respectively. It is here presumed that the eigenvectors have been normalized as
WTm ½MTot Vn ¼
1 0
n¼m n 6¼ m
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ðA21Þ
In the inversion of Eq. (A20), a positive coordinate x1 is considered. Formally, the inverse Fourier transform with respect to may be written as 3N ð 1 1 X WTn ðÞF^ ð; !Þ Vn ðÞeix1 d U ðx1 ; !Þ ¼ 2 n¼1 1 !2n ðÞ !2
ðA22Þ To evaluate the integral, is extended into the complex plane. A semi-circle of infinite radius in the halfplane ImðÞ50 is added to the integration path. For positive x1 , this contribution will vanish due to the exponential x1 -dependence. The integral can then be evaluated by residue calculus. Poles on the real axis corresponding to travelling waves are included if their group velocity cg is positive. If instead of displacements, nodal point velocities are considered, the inverse transform takes the form ! X WTp p F^ p ; ! : U ðx1 ; !Þ ¼ Vp p eip x1 2cgp p ðA23Þ where p denotes the poles in the residue calculus and cgp the group velocity. The inverse Fourier transform with respect to ! is then numerically determined by use of the inverse Fast Fourier Transform (FFT). Further details on the formulation of the theoretical model are presented by A˚berg [21].
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