IEEE TRAiSSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 39. NO. 3, MAY 1992 381

The Interaction of Lamb Waves with Defects David N. Alleyne and Peter Cawley

Abstruct-The interaction of individual Lamb waves with a va- riety of defects simulated by notches has been investigated using finite-element analysis, the results being checked experimentally. Excellent agreement has been obtained between the numerical and experimental results and it has been shown that a two- dimensional (2-D) Fourier transform method that the authors have developed may be used to quantify Lamb wave interactions with defects in both experimental and numerical investigations. The results have shown that the sensitivity of individual Lamb waves to particular notches is dependent on the frequency- thickness product, the mode type (symmetric or antisymmetric), the mode order, and the geometry of the notch. The sensitivity of the Lamb modes a], UO, and SO to simulated defects in different frequency-thickness regions has been predicted as a function of the defect depth to plate thickness ratio (hj2d) and the results have indicated that Lamb waves may be used to find notches when the wavelength to notch depth ratio is of the order of 40. It has also been shown that provided the notch width is small compared to the wavelength, the transmission and reflection amplitudes are insensitive to changes in width, so the ratio of the depth of the notch to the plate thickness is the controlling parameter. Both the numerical and experimental results have shown that the transmission ratios of Lamb waves across defects are highly frequency dependent, particularly at higher frequency- thicknesses. The positions of the maxima and minima in the transmission curves are a function of notch depth that suggests that monitoring the change in transmission ratio with frequency may provide a means of defect sizing.

I. INTRODUCTION

C ONVENTIONAL ultrasonic inspection of large struc- tures is very time-consuming because the transducer

needs to be scanned over each point of the structure that must be tested. The use of Lamb waves is potentially a very attrac- tive solution to this problem since they can be excited at one point on the structure and can be propagated over considerable distances. If a receiving transducer is positioned at a remote point on the structure, the received signal contains information about the integrity of the line between the transmitting and receiving transducers. The test therefore monitors a line rather than a point and considerable savings in testing time may potentially be obtained. Since Lamb waves produce stresses throughout the plate thickness (though with some stress nodes that must be carefully considered) the entire thickness of the plate is interrogated, which means that it is possible to find defects initiating at either surface, and also to detect internal defects.

Unfortunately, however, Lamb wave testing is complicated by the existence of at least two modes at any given frequency.

Manuscript received July 18, 1991; revised November 4, 1991; accepted December 19, 1991.

The author is with the Department of Mechanical Engineering, Imperial College, London SW7 2BX UK.

IEEE Log Number 9107533.

In practice, it is difficult to generate a single, pure mode, particularly above the cut off frequency-thickness of the a1 mode. Therefore, the received signal generally contains more than one mode, and the proportions of the different modes present is modified by mode conversion at defects and other impedance changes. The modes are also generally dispersive, which means that the shape of a propagating wave changes with distance along the propagation path. This makes interpre- tation of the signals difficult and also leads to signal-to-noise problems since the peak amplitude in the signal envelope decreases rapidly with distance if the dispersion is strong. These difficulties have tended to reduce the attractiveness of Lamb wave testing.

Viktorov has published extensively on the use of Rayleigh and Lamb waves in nondestructive testing and monitoring applications, and his book [l] has become a standard text in the field. However, it was probably Worlton [2], [3] who first recognized the advantages of using Lamb waves to nondestructively test plates. Since then there has been a great deal of interest in these and other guided waves in NDT applications.

A large number of workers have recognized the advantages of using Lamb waves for fast inspection, where some reduction in sensitivity and resolution compared with that obtained in standard, high-frequency ultrasonic inspection can be toler- ated. For example, Lamb waves have been used to carry out coarse, quick inspection on a variety of different strips and plates by Mansfield [4] and Ball and Shewring [ 5 ] . Silk and Bainton [6] investigated the use of guided waves to locate defects in boiler and heat exchanger piping and Rokhlin [7] has reported several studies on the sensitivity of Lamb waves to elongated delaminations. Rokhlin and Bendec [8] have also studied the interaction of Lamb waves with spot welds and have shown that the transmission of the first symmetric mode through a spot weld may be linearly related to the cube of the diameter of the spot weld. Rose et al. [9] have reported investigations using Lamb waves to globally inspect K-joints in off-shore structures and the Welding Institute in the United Kingdom (see, for example, Bartle [lo]) have been developing an acoustic pulsing technique to monitor crack growth in large plate-like structures.

It is also possible to use Lamb waves in localized nondestructive testing applications as an alternative to conventional ultrasonic testing using bulk waves. Here, the dispersive nature of Lamb waves is not so problematic, as the propagation distances are relatively small and reasonable signal-to-noise ratios (SNR) may be maintained in most frequency-thickness regions.

Lamb waves can also be used to determine the elastic properties of materials. For example, Nayfeh and Chimenti

0885-3010/92$03.00 0 1992 IEEE

z t

(b) Fig. 1. Schematic diagram of the notch geometry used in the finite-element

models.

[l11 and Mal and Bar-Cohen [l21 have carried out work to determine the elastic constants of composites and Okada [l31 has used Lamb wave techniques to measure anisotropy of cold rolled metals.

The key problem in Lamb wave testing is the measurement of the amplitudes of the individual modes present in a multimode, dispersive signal. If this could be achieved, the relative amplitudes of the different modes generated by mode conversion at a defect could be measured, leading to possibilities not only of defect detection, but also of defect sizing. The authors have recently developed a two-dimensional (2-D) Fourier transform technique [l41 that uses time records from a series of equally spaced points along a plate to produce a three-dimensional (3-D) plot of amplitude versus frequency and wavenumber, so enabling the required information to be obtained.

Before implementing a Lamb wave testing technique, it is necessary to decide which mode(s) and frequency-thickness region(s) to use. A study of the interaction of Lamb waves with defects has been carried out in order to assess the sensitivity of different modes in different frequency-thickness regions, and so to enable the best testing regime for a particular type of defect to be determined. Since analytical solutions can only be found for a very limited class of Lamb wave transmission and reflection problems, a numerical study of the reflection and transmission of Lamb waves in plates with and without defects using the finite-element method has been carried out, the amplitudes of the different modes being monitored by the 2-D Fourier transform technique. This study is presented here, together with the results of a set of experiments designed to check the numerical predictions.

11. THE FINITE-ELEMENT MODEL

In real structures or components, defects are arbitrary in

0 1 I I I I I G . ‘0

B

Region 1

0.0 Frequency-thickness [MHz-mm] 3 .O

Fig. 2. Lamb wave group velocity dispersion curves for steel showing the frequency-thickness regions in which the finite-element modeling was carried out.

geometry, size, orientation and position within the plate. However, in the finite-element studies reported here, straight sided notches in steel plates were investigated as they are a reasonable idealization of a type of defect (cracks) commonly found in engineering structures. Schematic diagrams of the finite-element model used and details of the notch geometry are shown in Fig. 1. The model allows variation of notch depth, width, position and angle with respect to the normal to the plate surface. However, in a large number of applications very thin cracks such as fatigue cracks, which are initiated at free surfaces and grow in a direction normal to the surface, are of primary importance. Therefore, surface breaking, straight sided notches of a constant width running normal to the plate surface were used in most of the finite-element predictions reported here. However, some predictions showing the effect of varying the width and orientation of the notch are also presented.

All the modeling was carried out assuming plane strain in the 2 2 plane shown in Fig. 1. Finel, a finite-element package developed at Imperial College [15], was used with a uniform square mesh of four-noded quadrilateral elements with more than 10 nodes per wavelength, which proved to be adequate. An explicit central difference scheme was employed to produce the time marching solution, the time step being chosen to be less than the time taken for the fastest wave (in this case the longitudinal wave) to travel between two adjacent nodes. The bulk wave velocities, cl and c2 used in the finite- element models were 5960 and 3260 mis respectively and the density, p was 8000 kg/m3. The sensitivity of individual Lamb modes to different defects was studied separately, by launching individual modes from the edge of the plate.

Datta et al. [16], [l71 have employed a hybrid approach to the numerical solution of the problem of Lamb wave interaction with defects. They modeled the region containing the defect by finite-element analysis and calculated the motion in the rest of the plate by a modal expansion method, the solutions being matched at the boundaries between the regions. The amplitudes of the different propagating modes may then readily be obtained from the modal expansion. This method is probably more computationally efficient than the finite- element procedure and 2-D Fourier analysis employed here, but the solutions obtained should be identical. The pure finite-element method, rather than hybrid, was used in this

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS 383

1.1 , 1

1.1 1

- 1 . 1 ~ " " " " " " " ' 0 Time [p S] 50

(b)

Fig. 3. Predicted time history at .I' = l50 mm in a 3.0-mm-thick plate when the center frequency was 0.45 MHz showing the first passage of the no mode and the reflection from (a) a 0.5-mm-deep notch and (b) a 2.0-mm-deep notch.

investigation because the code was readily available and had been tested [14]. Extraction of the amplitudes of the different propagating modes via the 2-D Fourier technique is advantageous because the same method can be applied to experimental data.

In order to excite a pure Lamb wave, two conditions have to be simultaneously satisfied. Firstly, the frequency of the excitation must be appropriate for the desired mode and secondly, the variation of the excitation with z at the excitation position ( x = 0 in the tests reported here) must correspond to the exact mode shape of the Lamb wave being excited. Assuming a single frequency input, the required excitation f ( 2 . t ) is of the form

f(z. t ) = q z . (1)

where (a( z . W ) describes the displacement of the plate in the .E

and z directions in the mode of interest at the desired frequency as a function of z .

However, in almost all modeling applications, single fre- quency excitation is not desirable or possible. For example, in explicit time marching finite-element methods the dura- tion of the input signal has to be finite. It is therefore very advantageous to be able to excite single modes over a significant frequency bandwidth. This can be achieved by ensuring that the distribution of the imposed displacement through the thickness of the plate at the excitation position exactly matches the mode shape of the desired Lamb mode. In earlier work [14], the waveform applied to the plate, g(t), was a toneburst enclosed in a Hanning window and the

Fig. 4. Normalized 3-D plot of the 2-D FFT results of the case given in (a)-Fig. 3(a) when the signal reflected from the notch was gated out. (b) Fig. 3(b) when the incident wave was gated out.

variation of the imposed displacement through the thickness was calculated from the mode shape of the desired mode at the center frequency of the toneburst. However, the frequency dependence of the mode shapes meant that the excitation signal was only appropriate at the center frequency, so other modes could be excited at the other frequencies present in the toneburst, and careful examination of the results revealed the presence of these modes.

The problem can be solved by summing the required inputs over a range of frequencies. For a single frequency component, W , the required input is given by (l), so if all the significant energy components in the excitation signal are over a range of frequencies from LU', to W6 then the required input is given by the integral of (1) over this range:

f (z . t ) = J' @ ( W . z)A(iu)ei"t dW

Wb

( 2 ) d a

3x4 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS. AND FREQUENCY CONTROL. VOL. 39, NO. 3, MAY 1992

1.1

4 x E ." Y

1.1 Frequency-thickness [MHzmm] 1.6

(c)

P * 0.0 1.1 Frequency-thickness [MHzmm] 1.6

1.1 Frqucncy-thickness [MHzmm] 1.6

( 4 Fig. 5. Predicted reflection ratios as a function of frequency-thickness in a 3.0-mm-thick plate when the ((0 mode is incident after

interaction with (a) a 0.5-mm-deep notch, (h) a 1.0-mm-deep notch, (c) a 1.5-mm-deep notch and (d) a 2.0-mm-deep notch.

v.v 8

1.1 Frequency-thickness [MHmm] 1.6

Fig. 6 . Predicted normalized reflection ratios corresponding to Fig. 5@).

Hence, A(&) is the complex amplitude of the Fourier trans- form of the waveform applied to the plate, g@), at frequency W . The required forcing function at each node through the thickness of the plate may therefore be calculated from the form of the excitation, g ( t ) , and the mode shape of the mode to be excited, @(x). This mode shape may readily be obtained analytically as discussed by, for example, Viktorov [l]. In the tests reported here, g ( t ) was a 10 cycle toneburst modified by a Hanning window. This produced excitation over a narrow range of frequencies 0.Sjo 5 f 5 1.20fo (40 dB down points), where fo is the center frequency of the toneburst.

The majority of the tests discussed here were carried out to investigate the sensitivity of the antisymmetric and symmetric modes 01, no, and S O to notches of varying depths. The principal reason for using the a0 and SO modes was that they can be readily excited and received experimentally at low frequency-thickness products where, provided very low frequency-thicknesses are avoided, the group velocity of a, is less than half that of S, and both modes are only slightly dispersive. The a 1 mode was chosen in order to obtain

predictions for a nonzero order mode in a dispersive region. The finite-element studies were conducted in two frequency- thickness regions, which are labeled on the group velocity dispersion curves for steel shown in Fig. 2, from which it may be seen that the modeling was carried out away from the cut-off frequency-thickness value of a1 where dispersion is a maximum. The finite-element predictions are presented in terms of the Lamb wave amplitudes as a function of the frequency-thickness product and the Lamb wave amplitudes as a function of notch depth at particular frequency-thicknesses. The time histories of the response of the finite-element models in the z direction on the plate surface are recorded, as this is the component of displacement measured when immersion or grease coupling is used.

111. NUMERICAL RESULTS

A. a0 and SO at 1.35 MHz-mm

The first set of tests was carried out to determine the sensitivity of the first symmetric and antisymmetric modes, 00 and SO, respectively, to notches of varying depths in region 1 of the dispersion curves shown in Fig. 2. In this frequency- thickness region, the mode shapes of a0 and S O are essentially those of flexural and extensional waves respectively. The majority of the finite-element models were 395 mm long and 3.0 mm thick, the spatial sampling interval was 1 mm and the sampling frequency was 3.125 MHz, the center frequency of the excitation toneburst being 0.45 MHz. In the reflection tests the notch was located at , x = 250 mm and in the transmission tests the notch was located at z = 150 mm, the location being chosen to minimize the length of the model, and hence the run time, while ensuring that the waves of interest could be captured without interference from reflections from the ends of the plate. The notch width was 0.5 mm in all these tests.

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS 385

1 1

1.1 Frequency-thickness [MHz-mm] 1.6

0.0 -I 1.1 Frequency-thickness [MHz-mm] 1.6

Fig. 8. Predicted transmission ratios as a function of frequency-thickness in a 6.0-mm-thick plate when the S O mode was incident after interaction with (a) a 0.5-mm-deep notch and (b) a 7.5-mm-deep notch.

o \ p - (b)

Fig. 7. (a) Predicted time history in a 3.0-mm-thick plate 65 mm after interaction with a 2.0-mm-deep notch, when the input signal was designed to excite only no. (h) Normalized 3-D plot of the 2-D FFI results for the case given in (a).

1024 point Fourier transforms were used in both the time and the spatial domains, the length of the spatial records being increased from the 64 measured points by zero padding.

Reflection Tests with a0 Incident: Fig. 3(a) shows the time history of the response of the top

surface of the plate in the z direction at x = 150 mm when the notch depth was 0.5 mm, the signal applied at z = 0 was appropriate to excite only a g , and the duration of the test was long enough to include the response of the plate after reflection from the notch, but not from either end of the plate. Fig. 3(b) shows the corresponding plot for a notch depth of 2 mm. Fig. 4(a) was obtained by carrying out a 2-D Fourier transform on the time histories of 64 equally spaced positions from J: = 120 to x = 183 mm, when the reflected signal (t 2 80 ks, see Figs. 3(a) and (b)) was gated out. At each frequency of Fig. 4(a), the amplitude reaches a maximum at a single wavenumber that corresponds to that of the a0 mode, thus confirming that the incident wave is a pure a0 mode. The maximum amplitude of the response is at 0.45 MHz, the center frequency of the input signal.

Fig. 4(b) shows the result of carrying out a 2-D Fourier transform on the time histories of the same positions when the defect depth, h, was 2.0 mm, and the incident signal (t 5 80 PS, see Fig. 3(b) was gated out. Again the maximum amplitude

of the response is at 0.45 MHz. However, at each discrete frequency in Fig. 4(b) there are two distinct wavenumbers, relating to a0 and SO, at which the amplitude is a maximum. The vertical scale on the 3-D plots is linear and has been omitted to improve the clarity of the plots. Each 3-D plot has been normalized to a maximum amplitude of unity so the amplitudes of the a0 mode in Figs. 4(a) and 4(b) cannot be compared directly; quantitative data was obtained from other plots and is presented in the following.

Figs. 5(a) to 5(d) show the reflection ratios of a0 and S O

for h = 0.5, 1.0, 1.5 and 2.0 mm respectively, and were obtained by dividing the 2-D FFT results from the reflected response signals by the input 2-D FIT shown in Fig. 4(a). Fig. 5 therefore shows the ratio of the amplitude of the surface motion in the z direction in the two modes in the signal reflected from the defect to the amplitude of the z direction surface motion in the a0 mode in the incident signal. It should be noted that the vertical scales on Figs. 5(a) and (b) are different from those used in Figs. 5(c) and (d). It may be seen from Fig. 5(a) that with the 0.5-mm-deep notch, there is little variation of the modal amplitude with frequency, but Figs. S(b)-(d) show that significant variation with frequency is obtained with the deeper defects, indicating that by measuring over a wide frequency range, it may be possible to size the defects. However, this has not yet been investigated.

The plots of Fig. 5 show the amplitudes of the surface motion in the z direction in the different modes after reflection from the notch compared with that of the incident mode. Greater weight is therefore given to modes that have large

386 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 39, NO. 3, MAY 1992

1.25 MHzmm - 1 35 MHrmm -)-- 1 . 4 5 M H z m

0 4 l 0.0 hJ2d

0.8

(b)

Fig. 9. Predicted transmission ratios as a function of h / 2 d at 1.25, 1.35, and 1.45 MHz-mm when ,S() is incident (a) for the .m mode; (b) for (10 produced by mode conversion at the notch.

amplitudes of surface motion in the z direction. A more accurate picture of the true proportions of the different modes present may be obtained by normalizing the reflection ratios. Fig. 6 shows normalized reflection ratios corresponding to the case of the l-mm-deep notch shown in Fig. 5(b). The normalized reflection ratio, Qn, was defined by

Q Qn = -ui (4)

where Q is the unnormalized reflection ratio and U,, and ui are the z direction displacement components on the plate surface in the mode being normalized and the incident mode respectively. Hence, for the incident mode QTL and Q are equal. Comparing Fig. 6 with Fig. 5(b), i t may be seen that the normalization increases the apparent amplitude of S O because in this frequency-thickness region, the surface motion in the z direction in this mode is only around 37% of the maximum motion (which occurs in the IC direction at the middle of the plate), whereas in ao, the motion in the z direction at the surface is close to the maximum motion in this mode.

11 n

Transmission Tests with a0 Incident: Fig. 7(a) shows the finite-element prediction of the time

history of the response of a 3-mm-thick plate at .7: = 215 mm, when the input at x = 0 was appropriate to excite only ao, the defect depth, h, was 2.0 mm and the defect was located at z = 150 mm; the duration of the test was not long enough to include reflections from either end of the plate. After interaction with the notch, more than one propagating mode is present (a0 is mode converted), but in the time domain the modes are superimposed and their amplitudes may not

-1.11 , , , , , , , , , I I I , , 1 0 Time [p] 150

(a)

1.1 , 1

Fig. 10. Predicted time history in a 3.0-mm-thick plate when the center frequency of the input toneburst was 0.75 MHz and was designed to excite only ( 1 0 , (a) 105 mm before a 1.5-mm-deep notch and (b) 65 mm after the notch.

TABLE I PREDICTED REFLECTION (R) AND TRANSMISSION (T) RATIOS OF

"0 AS A FUNCTION OF THE RATIO OF NOTCH DEPTH TO PLATE THICKNESS, h / 2 d IN REGION 1 OF Flc; 2

Frequency-thickness h / 2 d (MHz-mm) 116 113 112 213

1.25 (R) 0.11 0.11 0.51 0.81 1.35 (R) 0.11 0.11 0.63 0.8 1 1.45 (R) 0.10 0.14 0.74 0.76 1.25 (T) 0.98 0.90 0.66 0.18 1.35 (T) 0.98 0.89 0.54 0.22 1.45 (T) 0.97 0.88 0.33 0.35

be determined from the ti.me histories of the responses of the plate. Fig. 7(b) shows the result of carrying out a 2-D Fourier transform on 64 equally spaced signals from .G = 152 to IC = 215 mm (i.e., from 2 to 65 mm beyond the notch), from which the two propagating modes a0 and SO (so is caused by mode conversion at the 2.0-mm-deep notch), may be identified and measured.

Transmission Tests with SO Incident: In order to verify that the interaction of individual Lamb modes with defects is frequency-thickness dependent, tests were carried out with SO incident on a 3-mm-thick plate with an excitation toneburst center frequency of 0.45 MHz, and on a 6-mm-thick plate with an excitation toneburst center frequency of 0.225 MHz. Notch depths of 0.5, 1.0, 1.5, and 2.0 mm were investigated in the 3-mm-thick plate, and depths of 0.5, 1.0, 1.5, 2.5, 3.5, and 5.0 mm were considered in the 6-mm-thick plate. It was found that the transmission ratios as a function of frequency-thickness

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS 387

(b)

Fig. 11. Normalized 3-D plot of the 2-D FFI results of the case given in (a) Fig. lO(a) when the signal reflected from the notch was gated out. (b) Results of the case given in Fig. lO(b).

calculated for the l-mm-deep notch in the 6-mm-thick plate were identical to those for the OS-mm-deep notch in the 3 mm plate, confirming that it is the ratio of notch depth to plate thickness and the frequency-thickness product, rather than the absolute notch depth and frequency that are the controlling parameters in Lamb wave testing.

The amplitudes of the transmission ratios as a function of the frequency-thickness product at notch depths of 0.5 mm and 1.5 mm in the 6-mm-thick plate are shown in Figs. 8(a) and (b) respectively. The wavelength of the SO mode in this frequency-thickness region is about 24 mm and from Fig. 8(b), a 1.5-mm-deep notch produces a reduction of around 5% in its transmission ratio, indicating that it is possible to detect defects of dimensions greater than about 6% of the wavelength if changes in amplitude of this order can reliably be measured. However, from Fig. 8(a), for a 0.5-mm-deep notch the amplitude of the a0 mode produced at the notch is about 10% of that of the incident SO mode. This indicates that in this case it is possible to detect defects of dimensions less than 2.5% of the wavelength of the incident wave if the amplitude of the wave produced by mode conversion at the notch is measured.

The large amplitude of the mode converted a0 mode is partly due to the relative amplitudes of the z and z components of displacement in the mode shapes at the surface. If the z

TABLE I1 PREDICTED REFLECTION ( R ) AND TRANSMISSION ( T )

DEPTH TO PLATE THICKNESS, hjZd I N REGIOV 2 OF FIG 2 Frequency-thickness h j 2 d (MHz-mm)

RATIOS OF (10 AS A FUNCTION OF THE RATIO OF NOTCH

116 113 112 213 2.25 (R) 0.06 0.41 0.52 0.52 2.55 (R) 0.08 0.66 0.74 0.76 2.25 0 ) 0.93 0.59 0.29 0.25 2.55 (T) 0.92 0.23 0.20 0.24

direction displacement had been monitored rather than the z displacement, the reduction in amplitude of the SO mode would probably have been larger than the measured amplitude of the a0 mode. This shows the importance of monitoring modes that have a large proportion of their deflections at the surface in the z direction, as in most NDT applications it is the z direction surface displacement that is measured.

Dependence OfRefrection and Transmission on Notch Depth: Table I shows the relationship between the reflection and transmission ratios of the a0 mode when this is the incident mode at different frequencies and notch depths obtained from plots of the form of Fig. 5. It can be seen that with a0 incident, the relationship between reflection amplitude and notch depth is complicated, and is a strong function of frequency. For example, at a notch depth of 1.5 mm (h/2d = 1/2) the transmission ratio is 0.66 at 1.25 MHz-mm, but changes to 0.33 at 1.45 MHz-mm.

Fig. 9 shows the transmission amplitude ratios as a function of h /2d at three frequencies over the bandwidth of the input signal when the SO mode is incident. As the defect depth increases, the transmission of the SO mode decreases monotonically, while that of the uo mode, which is produced by mode conversion at the notch, initially rises as the notch depth increases, but then reduces toward zero as the notch depth approaches the plate thickness. This is as expected since there can be no transmission of either mode when the notch has completely severed the plate. In this case, there is only modest frequency dependence of the transmission ratios.

The results indicate that the wavelength of the Lamb waves is not the only factor that affects the sensitivity. For example, at 1.35 MHz-mm the wavelengths of SO and a0 are around 12 mm and 5.7 mm respectively, but the amplitudes of their transmission ratios across a 0.5-mm-deep notch were almost identical.

B. a0 and a1 at 2.25 MHz-mm The second set of tests was carried out to determine the

sensitivity of antisymmetric modes in region 2 of Fig. 2 to notches of varying depths. In this frequency-thickness region, a0 is essentially nondispersive, but a1 is grossly dispersive. The majority of the finite-element models were 410 mm long and 3 mm thick, the notches being located at z = 210 mm in the reflection tests and at z = 180 mm in the transmission tests. The notches were 0.5 mm wide and notch depths of 0.5, 1.0 1.5, and 2.0 mm were tested. In all the tests the spatial sampling interval was 1 mm and the sampling frequency was 4.167 MHz, the center frequency of the excitation toneburst being 0.75 MHz. Again, 1024 point Fourier transforms were

388 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 39, NO. 3. MAY 1992

13 Frequency-tbickm ~ H z m r n ] 2.6 1.9 Frequency-thickness [MHzmrn] 2.6

(c) ( 4

Fig. 12. Predicted transmission ratios as a function of frequency-thickness in a 3.0-mm-thick plate when the a0 mode was incident after interaction with (a) a 0.5-mm-deep notch, (b) a 1.0-mm-deep notch, (c) a 1.5-mm-deep notch, and (d) a 2.0-mm-deep notch,

0.5 1 I

d - l

0.5 1 l

z ] x

-0.54 , , , , , , , , , , , , , , 1 0 Time [PS] 150

(b)

Fig. 13. Predicted time history in a 3.0-mm-thick plate when the center frequency of the input tone burst was 0.75 MHz and was designed to excite only a l . (a) 2 mm after a 1.0-mm-deep notch. (b) 65 mm after the notch.

used in both the time and the spatial domains, the length of the spatial records being increased by zero padding.

a0 Incident: Fig. lO(a) shows the predicted time history of the response

of the plate in a reflection test 105 mm before a 1.5-mm-

deep notch and Fig. 10(b) shows the predicted time history of the response of the plate in a transmission test 65 mm after a notch of the same size. In each case the excitation at z = 0 was appropriate to launch only Q. The first wave packet seen in Fig. lO(a) is the a0 mode that was launched at z = 0 and its shape has hardly changed from the input 10 cycle toneburst in a Hanning window, confirming that the wave is essentially nondispersive in this frequency-thickness region. After interaction with the notch, a0 is mode converted and the response shown in Fig. lO(a) for t > 80 PS and in Fig. 10(b) beyond the notch indicate the presence of more than one mode, the increased duration of the signals indicating velocity dispersion.

The reference 2-D FFT of the a0 mode shown in Fig. 1l(a) was the result of carrying out a 2-D Fourier transform on the time histories of 64 equally spaced positions from z = l05 to 168 mm, when the signal due to reflection from the slot seen in Fig. lO(a) for t 2 80 PS was gated out. Fig. l l (b ) shows the amplitude versus wavenumber and frequency-thickness product information obtained by carrying out a 2-D Fourier transform of the response of 64 spatial positions from 2 to 65 mm after the notch in the transmission test of Fig. 10(b) when h was 1.5 mm. Three propagating modes, a1 and a. and SO are present, the a1 and SO modes being caused by mode conversion at the notch.

The transmission ratios of each mode as a function of frequency-thickness are shown in Fig. 12 for each of the notch depths and Table I1 gives the reflection and transmission amplitude ratios for the incident mode, ao, as a function of h/2d at at 2.25 and 2.55 MHz-mm.

Comparing the results presented here with those obtained with a0 incident at 1.35 MHz-mm, it may be seen that as expected at this higher frequency-thickness product, the sensitivity of a0 to defects is improved. For example, after

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES w m DEFECTS 389

f g # 0.0

- 1.1i ' - 1.1 a Y

P ii

Y

UJ 1 l.

2

t a E 4

Frequency-thickness [hpHunml 2.6 1.9 1.9 Frequency-thickncsq [MHzmml 2.6

1 4

.- W I ,

- .- - - 2

.- W

0.0 1 0.0 t

(c) ( 4 Fig. 14. Predicted transmission ratios as a function of frequency-thickness in a 3.0-mm-thick plate when the a1 mode was incident after interaction with (a) a 0.5-mm-deep notch, (b) a 1.0-mm-deep notch, (c) a 1.5-mm-deep notch, and (d) a 2.0-mm-deep notch.

interaction with a 1.0-mm-deep notch in a 3-mm-thick plate, the amplitudes of the reflected and transmitted a0 mode at 1.35 MHz-mm shown in Table I were 0.11 and 0.89 respectively, whereas at 2.25 MHz-mm the corresponding amplitudes shown in Table I1 were 0.41 and 0.59 respectively. However, the wavelength of the Lamb waves does not appear to be the only factor affecting sensitivity. For example, at 1.35 MHz-mm the wavelength of a0 is around 5.7 mm and the amplitude of its transmission ratio across a 2.0-mm-deep notch ( h / 2 d = 2/3) was 0.22, whereas the transmission ratio for a0 at 2.25 MHz- mm where its wavelength is 3.7 mm was 0.25. Therefore, the smaller wavelength has not produced a significant shift in the transmission ratio, indicating that appropriate mode selection can sometimes remove the need to go to higher frequencies where the waveform could be more complicated.

al Incident: Figs. 13(a) and (b) show the predicted time histories of the response of the plate at z = 182 and 245 mm respectively, after interaction with a 1.0-mm-deep notch located at IC = 180 mm, when the excitation at z = 0 was appropriate to launch only the a1 mode. The shape of the response is radically different from the input 10 cycle toneburst in a Hanning window, indicating that the response signal is multimode and/or dispersive. It may be seen that the amplitude at z = 182 mm is smaller than that at IC = 245 mm. This is due to the destructive and constructive superposition of the individual modes in the response wave packet produced after the incident a1 mode has interacted with the notch. This plot shows clearly why it is not reliable to use time domain methods to measure the amplitudes of Lamb waves at frequency-thickness products above the cut-off value of the a1

mode. The amplitudes of the transmission ratios as a function of the frequency-thickness product are presented in Fig. 14 and the transmission ratios are given as a function of h / 2 d at 2.25 and 2.55 MHz-mm in Table 111.

TABLE rrr PREDICTED TRANSMISSION RATIOS OF a1 AS A FUNCTION OF THE RATIO OF NOTCH DEPTH TO PLATE THICKNESS, h / 2 d . Frequency-thickness h / 2 d (MHz-mm) 116 113 112 213

2.25 0.91 0.58 0.35 0.23 2.55 0.82 0.03 0.61 0.75

Comparing Fig. 12 with Fig. 14, it may be seen that the transmission amplitude of all the response modes when a1

was incident (Fig. 14) is far more oscillatory with frequency- thickness, especially for notches where h / 2 d is bigger than 0.5, than when the a0 mode was incident (Fig. 12). This effect has not been studied further here but is probably due to the dispersive nature of the a1 mode in this frequency- thickness region, where its wavelength and mode shape are changing rapidly. It may also be seen from Tables I1 and I11 that at 2.25 MHz-mm and notch depths of 0.5 and 1.0 mm (h/2cl = 1/6 and 1/3), the sensitivity of the a0 and a1 modes are comparable, the higher order mode being slightly more sensitive. For example, at 2.25 MHz-mm the transmission ratios of the a1 and a0 modes after interaction with a 1.0 mm notch are 0.58 and 0.59 respectively. However, the wavelength of the a1 mode is more than twice that of the a0 mode. The transmission amplitudes of the modes differ markedly at 2.55 MHz-mm and it may be seen from Figs. 12 and 14 that this is due to the pronounced dips and peaks of the transmission curves.

C. Lamb Wave Interaction with Notches of Varying Width

One set of tests was carried out to check the effect of notch width on Lamb wave propagation. The incident wave was the SO mode in a 3.0-mm-thick plate in region 1 of the dispersion curves shown in Fig. 2. The notch was 1.0 mm deep

390 lEEE TRANSACTIONS ON ULTRASONICS, FERROELEmRICS. AND FREQUENCY CONTROL. VOL. 39, NO. 3, MAY 1992

- 1.25MHmun

1.45 MHzmm - 1.3JMHmyn

Fig. 15. Predicted transmission ratio of the SO mode as a function of w / 2 d at 1.25, 1.35, and 1.45 MHz mm.

and the tests were carried out using 5 different notch widths, W = 0.25, 0.75, 1 .O, 2.0, and 4.0 mm. Again, four-noded quadrilateral elements were used, the spatial sampling interval was 1 mm and the sampling frequency was 3.125 MHz, the center frequency of the excitation toneburst being 0.45 MHz. The plate was 350 mm long, the notch was located at z = 150 mm and time histories of the response of the plate at 64 equally spaced positions from z = 170 to 233 mm were captured in order to perform the 2-D FFT analysis.

Fig. 15 shows the transmission ratios as a function of w/2d at three frequency-thickness values. The results indicate that the amplitude of the transmitted SO mode is not sensitive to the notch width. The wavelength of S O in this frequency- thickness region is about 12 mm and the maximum defect width investigated was 4 mm. Therefore, the lack of sensitivity to notch width is not unexpected. However, the wavelength of uo in this frequency-thickness region is about 5.7 mm and some frequency dependence of the transmitted amplitude of the a0 mode was observed at the higher notch widths. These results confirm that when the notch width is small compared to the wavelength, as is usually the case with real cracks, the important parameter is the defect depth.

D. Lamb Wave Interaction with an Inclined Notch

In many practical situations notch like defects are oriented at an arbitrary angle to the surface of the plate as shown in Fig. l(b). One set of tests was carried out to determine the sensitivity of the SO mode in region 1 of the dispersion curve shown in Fig. 2 to a 1.0-mm-deep surface breaking notch lying at an angle of 45' relative to the plate surface (i.e, h = 1.0 mm and 0 = 45" in Fig. l(b). The plate was 3.0-mm-thick and 350 mm long and a four-noded 0.25-mm-square element was used. The center frequency of the 10 cycle excitation toneburst used was 0.45 MHz, the spatial sampling interval was 1 mm and the sampling frequency was 3.125 MHz.

Modeling a notch inclined at an angle using finite-elements represents a problem if wave propagation is to be modeled accurately, as a propagating wave will be reflected from any boundary across which there is a change in impedance, and any changes in shape or size of the element will cause a change in the effective impedance of the element and result in spurious reflections from the element boundaries. Hence, the 45" inclined notch was modeled by removing pairs of elements

0.0 -1 1 1.1 Frequency-thickness [MHz-mm] 1.6

1.2 1

0.6 4 1.1 Frequency-thickness [MHz-mm] 1.6

(b)

Fig. 16. Predicted transmission ratios as a function of frequency-thickness in a 3.0-mm-thick plate when the S O mode is incident (a) with an angled (453 to the plate surface) 1.0-mm-deep notch and (b) with a 1.0-mm-deep notch normal to the surface.

GPIB

1

4 Receiver Transmitter i

Plate Propagating wave

Fig. 17. Schematic diagram of experimental setup

in a "staircase" pattern. However, this created an angled notch with stepped edges, the width of the step being 0.25 mm (one element).

Fig. 16(a) shows the transmission ratios of the a0 and SO

modes as a function of the frequency-thickness product. The trend of the transmission ratio of the S O mode in Fig. 16(a) is slightly different from that for a l-mm-deep notch normal to the plate surface shown in Fig. 16(b), the amplitude in the case of the inclined notch increasing slightly at higher frequency thickness values. However, the average transmission ratio of

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS 39 1

Moveable transducer holder Immersion Transducer 7

Fixed transducer holder /------- / A

' Dial gauge / Fig. 18. The Lamb wave test rig.

1

0.0 Frequency-thickness [MHz-mm] 12

Fig. 19. Lamb wave dispersion curves plotted as a function of incidence angle in water.

so in region 1 is approximately 0.85 in both cases. The amplitude of the a0 mode produced by mode conversion

at the notch in the two cases is significantly different both in absolute value and in the form of its variation with frequency. The reasons for this have not been investigated, though it may be connected with the length of plate over which the notch runs. This may be thought of as an effective notch width, which in the case of the inclined notch is of the same order as the wavelength of no in this frequency-thickness region.

IV. EXPERIMENTAL SETUP

The instrumentation used in the experimental investigation is shown schematically in Fig. 17. A pulse from the pulse generator was used to simultaneously trigger the oscilloscope and an arbitrary function generator (Le Croy type 9101) that delivered a toneburst modified by a Hanning window function to the power amplifier. The use of a windowed toneburst rather than a simple signal comprising an integer number of cycles enabled the bandwidth of the signal to be limited, so assisting the generation of a pure mode. This is discussed in more detail in (181. The power amplifier delivered the input signal to the transmitting transducer with a gain of 50 dB. The Lamb waves excited by the transmitting transducer propagated along the plate and were received by the receiving transducer, the received signal being amplified and input to the oscilloscope (Le Croy type 9400) for digital capture and display. The signals from the oscilloscope were sent to the computer via the GPIB bus to be edited and then transferred to the spectrum analyzer (B&K type 2033). which was used to carry out the

0 5(

0.0 (1

-40 0 Frequency [MHz]

(b)

1 .o

Fig. 20. (a) The time history of the measured response 350 mm from the transmitter when the excitation was appropriate for S ( ) . (b) Amplitude spectrum of the time history shown in (a).

digital Fourier transformation. The resulting transformed data was then transferred back to the computer for redisplaying and editing.

In order to increase the SNR, 100 successive response sig- nals captured by the digital oscilloscope were averaged. This was achieved by retriggering the system once the response of the plate to the previous input had decayed to zero, the averaging being carried out by dedicated software within the oscilloscope.

The steel plates used in the experimental investigations were 3 mm thick and approximately 300 mm wide and 1 m long. Five plates were tested, one having no notch and the others having 0.5-mm-wide notches milled across the full plate width

392 IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. ANI) FREQUENCY CONTROL, VOL. 39, NO. 3. MAY 1992

Q)

m CI Q)

m 1l Y 2 el .!i d

-8 E!

-8 .r( .r( 42 E Ea

.S 5 Y

U U 0 Time [ p ] 50

(a)

c Q)

m Q)

m U v1

- 2 2 a .e Y

E

0 'p

2 .- ." E" E? U b.

0 Time [ p ] 50 0 Time [PS] 50 - (c ) ( 4

Fig. 21. The time history of the measured response 350 mm from the transmitter when the excitation was appropriate for ,SI]

and a notch was located 250 mm from the transmitter. (a) O.5-mrn-deep notch. (b) 1.0-mm-deep notch. ( c ) 1.5-mm-deep notch (d) 2.0-mm-deep notch.

normal to the transmission path used. Plates with notches 0.5, 1.0, 1.5, and 2.0 mm deep were tested. In all the tests, the propagation distance was restricted to under 500 mm in order to keep the SNR high.

Conventional wideband ultrasonic immersion transducers were used in all the experiments as previous work [l81 had shown that immersion coupling gave the most repeatable results and facilitated the generation of a pure mode. However, if the whole plate is immersed in a fluid, energy from the Lamb waves leaks into the fluid and severely limits the propagation distance. Therefore, local immersion coupling was employed, the test rig being shown in Fig. 18. It comprised a frame that maintained two transducer holders in line with one another. one holder being fixed, and the other moveable. The transducer holders were essentially blocks of aluminum with a hole drilled through at a chosen angle to the normal. A column of water in the transducer holders allowed acoustic energy transfer between the transducer and the plate, O-rings located in a groove on the bottom of the holders providing a seal. The immersion coupling area was therefore localized in the transducer holders (the coupling area was less than 10-3m'). Hence, only a small part of the plate was immersed so attenuation of the received signal due to leakage into the coupling fluid was minimized.

In order to carry out the 2-D Fourier analysis, time records from a series of equally spaced points along the plate surface were required. This was achieved by indexing the moveable transducer along the plate, a dial gauge measuring the travel to obtain an accurate spatial sampling interval. The resolution in positioning the transducer in the experiments was better than

4 . 0 2 mm. In the 2-D FFT tests reported here, the receiver was indexed from 350 to 413 mm from the transmitter, the sampling frequency was 5 MHz and a 1024 by 1024 point 2- D FFT was used (the 64 spatial points were padded with 960 zeros). The tests were carried out approximately in the middle of the plate so that any reflections from the edge of the plate were not included in the captured signals.

Individual Lamb waves were selectively excited by applying the coincidence principle (see, for example, Worlton [2] or Jitsumori et al. [19]). The angle of incidence H required for the excitation of the desired mode was calculated from H = sifl(cL/c) where C L is the phase velocity of a com- pression wave incident on the surface of the plate and c is the phase velocity of the Lamb wave to be excited. In all the tests reported here, the required coincidence angle was approximately 16" so transducer holders oriented at this angle were employed.

V. EXPERIMENTAL RESULTS

A. s o at 1.4 MHz-mm The first set of tests was carried out to measure the sensitiv-

ity of the symmetric, SO, Lamb wave in the frequency thickness region around 1.4 MHz-mm to notches of varying depths. A 12-cycle, 0.48-MHz toneburst modified by a Hanning window was employed giving a center frequency-thickness of 1.44 MHz-mm in the 3-mm-thick plate. The 16" angle of incidence from water to steel is appropriate for the excitation of S O in this frequency-thickness region as shown in the dispersion curves of Fig. 19.

ALLEYKE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS 393

(h)

Fig. 22, (a) The normalized 3-D plot of the 2-D FFT results corresponding to Fig. ?O(a). (b) The normalized 3-D plot of the 2-D FFT results correspond- ing to Fig. 21(b).

The response of the plate 350 mm from the transmitter in the absence of a defect is shown in Fig. 20(a) and the amplitude spectrum, shown in Fig. 20(b), that was obtained from the time history shown in Fig. 20(a) demonstrates that the signal is narrow band, the working range being from 0.41 MHz to 0.55 MHz, (20 dB down points). The wave packet seen in Fig. 20(a) was identified as the so mode from a measurement of its group velocity using the time of flight method. The shape of the response wave packet indicates that very little dispersion is present over the frequency-thickness interval of the input signal.

Figs. 21(a)-(d) show the response of the plate 350 mm from the transmitter after interaction with 0 . 5 , l .@, 1 . 5 , and 2.0- mm-deep notches respectively, which were located 250 mm from the transmitter, the excitation being the same as in Fig. 20. The signals in Figs. 2l(a)-(d) are dominated by the so mode, though some evidence of WO is seen, particularly with the deeper notches. The numerical predictions indicated that significant amplitudes of (L() could be generated by mode con- version. However, the observed amplitudes are small because the coincidence angle for (10 is 36' in this frequency-thickness

c

c2

.B L]

8

k m

Y

- .M

3 4

0 Wavenumber [rad/m]

Fig. 23. Normalized plot of measured amplitude versus wavenumber at 1.45 MHz-mm for different notch depths when the .SO mode was excited by the transmitter.

region (see Fig. 19) so it was inefficiently received by the transducer oriented at 16. The no mode is therefore partially "decoupled" by the orientation of the receiving transducer.

Fig. 22(a) shows a 3-D plot of amplitude versus frequency- thickness and wavenumber obtained by carrying out a 2- D Fourier transform on the response of the plate in the absence of a defect measured with the receiver placed at 64 equally spaced positions between 350 and 413 mm from the transmitter, the time history with the receiver 350 mm from the transmitter being shown in Fig. 20(a). At each frequency- thickness of Fig. 22(a), the amplitude reaches a maximum at a single wavenumber that corresponds to that of the S O mode, thus confirming that a pure SO mode has been successfully launched. Fig. 22(b) shows the result of carrying out a 2-D FFT analysis on the response of the plate over the same propagation distances after interaction with a 1.0-mm-deep notch located 250 mm from the transmitter, the time history 350 mm from the transmitter being shown in Fig. 21(b). As in Fig. 22(a), the maximum amplitude of the response is at 1.44 MHz- mm, which corresponds to the center frequency of the input toneburst. However, at each discrete frequency-thickness in Fig. 22(b) there are two wavenumbers at which the amplitude is a maximum. These correspond to the so mode and to the no mode, which was produced by mode conversion at the notch. The wavenumber of a0 is greater than that of the SO mode so it is partially hidden in the 3-D plot. Here, as in the time domain responses presented in Fig. 21, the magnitude of the mode is very small when compared with the finite-element predictions due to the decoupling effect discussed previously. The vertical scale on the 3-D plots is linear and has been omitted to improve the clarity of the plots. As with the numerical results, each 3-D plot has been normalized to a maximum amplitude of unity so the amplitudes of the S O mode in Figs. 22(a) and (b) cannot be compared directly; quantitative data was obtained from other plots and is presented in the following.

Fig. 23 shows the measured amplitude versus wavenumber information at 1.45-MHz-mm for the different notch depths. The reference curve in Fig. 23 was obtained from the 2-D FFT results obtained in the test without a defect shown in Fig. 22(a), the other curves being from the 2-D FFT results obtained in the tests in which the SO mode propagated across one of the notches. The variation of the amplitude with wavenumber may clearly be seen in Fig. 23 and it is possible to measure

394 IEEE TRANSACTIONS ON ULTRASO?JlCS. FERROELEnRICS, AND FREQUENCY CONTROL, VOL. 39, NO. 3, MAY 1992

TABLE IV TRANShllSSlON RATIO OF so AT 1.45 MHz-mm

h / 2 d 116 113 112 213 FE predictions 0.97 0.81 0.57 0.36 Measured in time domain 0.98 0.82 0.59 0.39 Measured by 2-D FFT 0.98 0.82 0.58 0.37

I 0 Time [PS] 50

0.5 Frequency [MHz] 1.5 (b)

El I dj l

0 Time [IS] 50

Fig. 24. (a) The time history of the measured response 350 mm from the transmitter when the excitation was appropriate for a l . (b) Amplitude spectrum of the time history shown in (a). (c) Time history of the measured response 350 mm from the transmitter when a 1.0-mm-deep notch was located 250 mm from the transmitter and the excitation was the same as in (a).

the reduction in the amplitude of the SO mode produced by transmission across the different notch depths, and the growth in amplitude of the a0 mode. However, it must be remembered that the a0 amplitudes should not be compared with the SO

amplitudes as the receiver was "focussed" on the SO mode. In this particular case, the amplitude of the SO mode decreases monotonically with increasing notch depth, as was predicted by the finite-element results.

Table IV gives transmission ratios for the SO mode that were obtained from the finite-element predictions by the 2- D FFT method, together with corresponding results from the experimental 2-D FFT measurements and time domain measurements obtained by dividing the maximum amplitudes of the signals shown in Fig. 21 by the maximum amplitude of the reference signal shown in Fig. 20(a).

.- m

5 0 Wavenumber [rad/m] 2000

Fig. 25. Normalized plot of the measured transmission amplitude versus wavenumber at 2.5 MHz-mm for different notch depths when the ( 1 1 mode was excited by the transmitter.

The agreement between the predictions using the finite- element method in conjunction with the 2-D FFT technique and the measured results using the 2-D FFT method is ex- cellent, the largest difference being 0.01. The time domain measurements gave very similar results, though with slightly larger differences at the larger notch depths. From Figs. 21(a) to 21(d) it may be seen that the amplitude of the (LO mode, which has partially separated from SO and is seen when t > 25 ,m is significant. Therefore, the difference between the finite- element predictions and the measured results is probably due to the peak value in the time domain being a summation of the amplitude of the SO mode together with a small contribution from the a0 mode that was produced by mode conversion from SO at the notch. The 2-D FFT technique separates the two modes, so giving a true measurement of the amplitude of SO. The results in the time domain would be more severely in error if the group velocities of the two modes were more similar, or if the a0 mode was less efficiently decoupled by the coincidence effect.

B. a1 at 2.5 MHz-mm

The second set of tests was designed to check the finite- element predictions in a frequency-thickness region where more than two modes can propagate. The same set of 3- mm-thick steel plates was used, the center frequency of the excitation toneburst being 0.83 MHz, giving a center frequency-thickness of 2.49 MHz-mm. Fig. 19 shows that in this frequency-thickness region, a coincidence angle of 16" is appropriate for excitation and reception of the a1 mode. The length of the excitation toneburst was increased to 25 cycles for this set of tests because the a1 mode is highly dispersive in this frequency-thickness region so it was desirable to limit the excitation bandwidth by employing a larger number of cycles, as well as enclosing the toneburst in a Hanning window.

Fig. 24(a) shows the response measured 350 mm from the transmitter in the absence of a defect. The wave packet was identified as the a1 mode by measuring the time of flight of the leading edge of the signal. The amplitude spectrum of the time record of Fig. 24(a) is shown in Fig. 24(b), from which it may be seen that acoustic energy is available in the frequency range between 0.78 MHz and 0.88 MHz, (20 dB down points). Fig. 24(c) shows the response of the plate 350

ALLEYNE AND CAWLEY. THE INTERACl3ON OF LAMB WAVES WITH DEFECTS 395

0.0 J I 2.3 Frequency-thickness [MHz-mm] 2.6

0.0 I 2.3 Frequency-thickness [MHz-mm] 2.6

(c)

0.0 1 Q

2.3 Frequency-thickness [MHz-mm] 2.6

(b)

n n l 1 "._ . 2.3 Frequency-thickness [MHz-mm] . 2.6

( 4

Fig. 26. Transmission ratio of the n l mode as a function of frequency-thickness after interaction with (a) a 0.5-mm-deep notch, (h) a 1.0-mm-deep notch, (c) a 1.5-mm-deep notch, and (d) a 2.0-mm-deep notch. 0 indicates experimental results and the solid rule indicates finite-element predictions.

mm from the transmitter after interaction with a 1.0-mm-deep notch, which was located 250 mm from the transmitter, when the excitation was the same as in Fig. 24(a). In this frequency- thickness range three propagating modes (al, ao, and so) are possible, though since the incidence angle of the receiving transducer was appropriate for the a1 mode, the measured amplitudes of the a0 and SO modes are significantly less than their true values. The shape of the response time history in Fig. 24(c) indicates the presence of Lamb waves with similar group velocities since the wave packets have not separated. Therefore, it is not possible to measure the amplitude of the a1 mode in the time domain unless the propagation distance after the notch is considerably increased so that the modes can separate.

Fig. 25 shows the measured transmission amplitudes of the a l , so, and a0 modes versus wavenumber at 2.5 MHz-mm. These curves were obtained from the 2-D FFT results from tests on plates without a notch (the reference case) and with 0.5-, 1.0-, 1.5, and 2.0-mm-deep notches. The curve having the largest amplitude at the wavenumber relating to the a1 mode in Fig. 25 was obtained in the reference test in which the transmitted wave was a pure a1 mode. However, the amplitude of the a1 mode does not decrease monotonically as the notch depth increases. For example, the transmission amplitude is considerably smaller after interaction with the 1- mm-deep notch (h/2d = 1/3, the case shown in Fig. 24(c)) than after interaction with the 1.5 mm (h /2d = 1/2) or 2.0 mm (h/2d = 2/3) deep notches.

Figs. 26(a) to (d) show the transmission ratios of the a1

mode obtained from the 2-D FIT results as a function of the frequency-thickness product after interaction with the 0.5-, 1.0- , 1.5, and 2.0-mm-deep notches respectively. The measured transmission amplitude ratios are represented by squares and

the numerical predictions are shown as continuous lines. It may be seen that the trend of the experimental results follow the finite-element predictions very well, but that the predictions are shifted to higher frequency-thicknesses by about 2% (0.05 MHz-mm). A possible cause of these differences is that the numerical predictions assumed that the plates were isotropic, the assumed longitudinal and shear wave speeds being very similar to those measured normal to the plane of the plate. However, the plates used in the experiments were cold rolled mild steel, in which significant anisotropy is often observed, the bulk velocities in the plane of the plate in the rolling direction being different from those normal to the plane of the plate. These errors were not seen in the results for the transmission of the SO mode at 1.45 MHz-mm, because in this case the transmission ratios are almost constant for all the notches over the frequency-thickness range tested so the shift would not be seen.

It is particularly interesting to note that the minima in the transmission curves predicted in Fig. 26 are reproduced in the experiments, confirming that this effect is readily measurable and so could be used for defect sizing.

VI. CONCLUSION

The interaction of individual Lamb waves with a variety of defects simulated by notches has been investigated using finite- element analysis, the results being checked experimentally. Excellent agreement has been obtained between the numerical and experimental results and it has been shown that the 2-D FFT method may be used to quantify Lamb wave interactions with defects in both experimental and numerical investigations.

The results have shown that the sensitivity of individ- ual Lamb waves to particular notches is dependent on the frequency-thickness product, the mode type (symmetric or

396 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, &VD FREQUENCY CONTROL, VOL. 39, NO. 3, MAY 1992

antisymmetric), the mode order, and the geometry of the notch. The sensitivity of the Lamb modes a l , a0 and SO to simulated defects in different frequency-thickness regions has been predicted as a function of the defect depth to plate thickness ratio (h/2d) and the results have indicated that Lamb waves may be used to find notches when the wavelength to notch depth ratio is of the order of 40.

Preliminary numerical predictions with an inclined notch have indicated that the reflection and transmission of Lamb waves is largely dependent on the notch depth (i.e., distance of penetration normal to the plate surface) rather than overall notch length. The finite-element results on notches of differ- ent widths have indicated that provided the width is small compared to the wavelength, the transmission and reflection amplitudes are insensitive to changes in width so the ratio of notch depth to plate thickness ( h / 2 d ) is the controlling parameter.

The experiments reported here were carried out using 32- mm-diameter transducers oriented at the appropriate coinci- dence angle for the excitation and reception of a particular mode. Quantitative comparisons have therefore only been presented for the transmission of this mode since the measure- ments underestimate the amplitudes of the other propagating modes. However, the results of the numerical predictions indicate that monitoring the amplitude of the modes generated by mode conversion at a defect can be a very sensitive means of detecting the presence of defects. This is partly because if a pure wave is launched, these modes are only seen in the presence of a defect (provided the plate is uniform over the transmission path) and also because if, for example, the SO mode is launched at low frequency-thicknesses, the a0 mode generated by mode conversion has a much larger amplitude of motion in the z direction (normal to the plate) at the surface and so is more easily detected by practical transducers using immersion or grease coupling that are only sensitive to motion in this direction.

If the technique is to be applied in industrial NDT, it will not generally be practical to index a single transducer over the plate surface. Instead, a linear array transducer will be employed and the elements of the array will give a good approximation to point measurements of the surface amplitudes. The decoupling due to the coincidence principle seen in the results presented here will therefore not be observed and measurements of the relative amplitudes of the different modes will be possible.

The results of Table IV show that below the cut-off fre- quency of the a1 mode (1.61 MHz-mm in steel) where only the a0 and SO modes can propagate, satisfactory measurements of the transmission amplitudes can be obtained in the time domain. This is because the two modes have very different group velocities so the wave packets corresponding to the two modes separate after a short propagation distance. However, in higher frequency-thickness regions, the different propagating modes frequently have similar group velocities so time domain measurements of their relative amplitudes are much more difficult.

The experimental results have confirmed the predictions of strong frequency dependence of the Lamb wave transmission

ratios after interaction with notches, particularly in higher frequency-thickness regions. The positions of the maxima and minima in the transmission curves are a function of notch depth that suggests that monitoring the change in transmission ratio with frequency may provide a means of defect sizing.

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[l11 A. H. Nayfeh and D. E. Chimenti, “Propagation of guided waves in fluid- coupled plates of fiber-reinforced composites,” J. Acoust. Soc. Am., vol. 83, pp. 17361743, 1988.

[l21 A. K. Mal and Y. Bar-Cohen, “Characterization of composite laminates using combined LLW and PBS methods,” Rev. Progress in Quantitative NDE, vol. 10B, D. 0. Thompson and D. E. Chimenti, Eds. New York: Plenum, 1991, pp. 1555-1560.

[l31 K. Okada, “Ultrasonic measurement of anisotropy in rolled materials using surface wave,” Jpn. J. Appl. Phys., vol. 25, suppl. 25-1, pp.

[l41 D. N. Alleyne and P. Cawley, “A two-dimensional Fourier transform method for the measurement of propagating multimode signals,” J. Acoust. Soc. Amer., vol. 89, pp. 1159-1168, 1991.

vol. 6. pp. 138-145, 1973.

J. , vol. 23, pp. 358-364, 1983.

197-199, 1986.

[l51 D. Hitchings, “FE77 user manual,” Imperial College int. rep., 1987. (161 S. K. Datta, Y. AI-Nassar, and A. H. Shah. “Lamb wave scattering by

a surface-breaking crack in a plate,” Rev. of Progress in Quantitative NDE, vol. loa, D. 0. Thompson and D. E. Chimenti, Eds. New York Plenum, 1991. pp. 97-104.

[l71 Y. AI-Nassar, S. K. Datta, and A. H. Shah, “Scattering of Lamb waves by a normal rectangular strip weldment,” Ultrasonics, vol. 29, pp. 125-132, 1991.

[l81 D. N. Alleyne and P. Cawley, “The optimisation of Lamb wave

[l91 A. Jitsumori, S . Inoue, T. Maekawa, and T. Inari, “Generation of Lamb inspection techniques,” in preparation.

waves using linear array probe and its application to flaw detection,” Jap. J. Appl. Phys. Supp. (Proc. 6th Symp. Ultrason. Electron.), Tokyo, vol. 25, pp. 200-202, 1986.

David Alleyne was born in February 1959. He left school in 1975 to become an engineering apprentice at a major food manufacturing company. From 1975 he carried on his education on a part-time basis, and in 1986 he received the B.Sc. degree in mechanical engineering from Kingston Polytechnic, Surrey, England. In 1987 he became a full-time student at Imperial College, London, England, and received the Ph.D. degree in 1991. The major achievement of his research was the development of quantitative methods of applying ultrasonic Lamb waves in the nondestructive testing of plates and plate-like structures. His current research interests are in the application of Lamb waves in fast long- range inspection, signal processing methods and the finite-element modelling of propagating Lamb waves and their interactions with defects.

ALLEYNE AND CAWLEY: THE INTERACTION OF LAMB WAVES WITH DEFECTS

Peter Cawley was born in 1953. He received the B.Sc. and Ph.D. degrees in 1975 and 1979, respectively, from the University of Bristol, Bristol, UK.

He then worked in industry for two years before joining Imperial College, London. England, in 1981 where he is now a senior lecturer in the Department of Mechanical Engineering. His main research area is sonic and ultrasonic methods of nondestructive testing and he has published more than 60 papers in this and related fields. His current projects include the development of techniques for the inspection of adhesive and diffusion bonded joints and the application of Lamb waves for the inspection of pipework and for the rapid detection of impact damage in composite materials.

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