The phenomenon of continuous mode conversion of Lamb waves in CFRP plates C. Willberg 1,a and G. Mook 2,b and U. Gabbert 1,c and J.

Transcription

1 Key Engineering Materials Vol. 518 (2012) pp Online available since 2012/Jul/12 at (2012) Trans Tech Publications, Switzerland doi: / The phenomenon of continuous mode conversion of Lamb waves in CFRP plates C. Willberg 1,a and G. Mook 2,b and U. Gabbert 1,c and J. Pohl 3,d 1 Institute for Mechanics, Otto-von-Guericke-University Magdeburg, Germany Magdeburg 2 Institute for Materials and Joining Technology, Otto-von-Guericke-University, Germany Magdeburg 3 Anhalt University of Applied Science, Germany K othen. a christian.willberg@ovgu.de, b gerhard.mook@ovgu.de, c ulrich.gabbert@ovgu.de, d j.pohl@emw.hs-anhalt.de Keywords: mode conversion, Lamb waves, SHM, CFRP, fabric Abstract. The effect of mode conversion of Lamb waves is a well known phenomenon. Lamb waves occur in multiple modes, which can transform into each other under special conditions. Typically mode conversion takes place at discrete positions inside a structure, e.g. damages and edges. However, we observed a continuous mode conversion in a multi-layer composite plate. The symmetric S0-mode converts continuously into the A0-mode without passing a macroscopic discontinuity. In the paper this phenomenon of continuous mode conversion is investigated experimentally as well as numerically. Introduction Lamb wave based structural health monitoring systems are one promising approach to observe thin walled structures, e.g. aeroplanes [2, 4]. As ultrasonic waves used for non-destructive testing [6], Lamb waves interact with damages. They are also characterized by a low geometrical attenuation (1/ r) [10]. Therefore they are interesting for monitoring large areas of light weight CFRP (composite fibres reinforced plastics) structures, e.g. wings of airplanes, because it is possible to monitor great distances or surfaces with a relatively low number of sensors and actuators. However, Lamb waves have complex properties particularly in CFRP material. They are dispersive and exist in at least two basic modes, a symmetric (S0) and simultaneously an antisymmetric (A0) mode, and for higher frequencies higher order modes can occur, which are noted as A1-, S1-, A2 -, etc. mode [11, 13]. Under special conditions the modes can convert into each other. For a conversion between the symmetric and anti-symmetric mode the wave has to travel through a discontinuity, which is not symmetric to the center plane of the wave and visa versa [1, 16]. However, modes can only convert into other modes which exist at the same frequencies [3]. In our research on Lamb wave propagation in CFRP-structures we observed a continuous mode conversion phenomenon. The phenomenon is described first in Willberg et al. [15]. The paper presents the effect of continuous mode conversion (CMC) in detail and tries to explain the effect with the help of numerical models. Some new findings are presented, which helps to better understand the effect of CMC. The work is structured as follows. First, the experimental setup is shown. Then, the effect of CMC is shown and the properties are explained. The proof of CMC is given and the identification of the modes is done using B-scans. Finally, experimental and numerical models are applied to explain the phenomenon. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, (ID: /07/12,02:12:02)

2 Key Engineering Materials Vol Experimental Setup The experimental investigations are performed with help of a 1D (PSV 300) as well as 3D (PSV400 3D) scanning laser vibrometer from Polytec. Laser scanning vibrometry is widely used for the experimental investigation of Lamb waves [5, 7, 9]. The scanning laser vibrometer has a high spatial resolution, does not need coupling media and measures the velocity of a measurement point in the direction of the laser beam using the Doppler shift. With three lasers of the PSV 400 3D the components of x 1, x 2 and x 3 can be measured resulting in a threedimensional wave field. Fig. 1: Experimental setup. Fig. 1 displays the experimental setup. The 3D laser scanning vibrometer scans the top surface of a CFRP plate (1 m 1 m 2.02 mm), which is positioned on foam. The stacking sequence of the investigated CFRP plate is given in Tab. 1. The center layer is made of a plain fabric (see Fig. 6a) and the top and bottom surface are made of a twill fabric (see Fig. 6b). The average distances between the lasers and the plate are 0.62 m and 0.94 m, respectively. Table 1: Layer setup of the CFRP plate (1 m 1 m 2.02 mm) [8]. layer Orientation [ ] type layer thickness [mm] 1 0/90 twill fabric UD layer UD layer /90 plain fabric UD layer UD layer /90 twill fabric 0.4 The Lamb waves are excited using a piezoceramic actuator of 20 mm diameter and a thick- ness of h = 1 mm made from the material Marco FPM To allow a reversible coupling between actuator and plate the source is attached with paraffin at the center of bottom surface. The edges are damped by silicon to reduce the amplitude of reflected waves. The top surface is coated by a retroreflective layer to enhance the signal-to-noise-ratio of the measurement signals. As excitation signal a 5 cycle sinus burst amplified by a NF-HSA-4011-amplifier is fed to the actuator. During the measurement the sinus burst is repeated for each measurement point. After scanning each point a C-scan is created. 4 marco Systemanalyse und Entwicklung GmbH, see: Oct

3 366 Structural Health Monitoring II The Phenomenon of Continues Mode Conversion The Description of the Phenomenon Fig. 2 shows the C-scan of an undamaged CFRP plate for an excitation frequency of f = 200 kh z. As illustrated in Fig. 1 the source is applied in the center at the bottom surface of the plate. The x 1 -axis corresponds to the zero degree orientation of the CFRP plate. It can be seen that two primary modes occur, a fast S 0 -mode (long wavelength) and a slower A 0 -mode (short wavelength). Fig. 2: Lamb wave propagation in a CFRP plate at f = 200 kh z, excited with a actuator in the center (300 mm 250 mm). Due to their different velocities the modes shown in Fig. 2 are already separated. The term primary means that both modes are excited directly by the piezoelectric actuator in the center of the plate. Reflected or converted modes are named in a different manner. For all presented scans only the out-of-pane displacements are shown. The in-plane components measured by the 3D laser scanning vibrometer are used to get a better identification of the S 0 -mode. However, no additional information could be obtained by it. Inside the S 0 -mode not primarily excited waves occur. These new waves are characterized by plane wave fronts being nearly parallel to each other. The orientation of the wavefront of the new modes depends on the region where the modes arise. In the bottom left of Fig. 2 the different orientation between two wavefronts is shown. In addition, the new waves do not occur everywhere in the plate. For 200 kh z for the upper left and the lower right region no new modes can be observed. Nevertheless, these regions change if Lamb waves are excited by another frequency. All investigated frequencies show that the orientation of the modes remain similar. The experimental data have been recorded between 100 kh z and 350 kh z with a frequency increment of 25 kh z. Measuring the presented CFRP plate, the new mode has been observed first at 150 kh z. Smaller frequency steps have been applied to figure out more accurately, when the new mode appears first time. This was at a frequency of 138 kh z. The continuous mode conversion (CMC) takes place at all investigated frequencies from kh z. In the following we try to clarify what type of mode arises in the S 0 -wave field in Fig. 2. Fig. 3 shows the dispersion curves of the presented CFRP plate. Lamb waves exhibit velocity dispersion and the dispersion curves illustrate the dependency between the frequency times thickness and the wave velocity. Furthermore, the existence of higher order modes and their

4 Key Engineering Materials Vol cut off frequencies can be taken from Fig. 3. The dispersion curves have been calculated with the semi-analytical finite element method (SAFE), published by Ahmad [1]. This approach allows a low cost calculation of dispersion curves for layered structures. It has been proven that the numerical estimated dispersion curves are in excellent agreement with experimental results. Fig. 3: Dispersion curves for the quasi-isotropic CFRP plate in 0 -direction. The dispersion curves for the investigated plate show that for frequencies lower than 205 kh z only three modes exist, the A 0 -, SH 0 - and the S 0 -mode. Above this frequency a fourth mode appears, the A 1 -mode. At frequencies higher than 280 kh z the symmetric S 1 -mode also occurs. In Fig. 2 the anti-symmetric A 0 -mode is the one of shorter wavelength, its wave field is next to the source and the wave front has a smaller diameter than that of the S 0 -wave field. The waves occurring inside the S 0 -mode have the wavelength similar to the wavelength of the primary A 0 -mode. The C-scan example correlates to the entry 0.4 M H z mm in the dispersion curves in Fig. 3. For the presented frequency two higher order modes could occur. Nevertheless, the phase velocities of the A 1 - and S 1 -mode (1) are drastically higher in comparison to the A 0 -mode. Because the wavelengths of the new mode are close to the A 0 -mode it is assumed that a dominant conversion between the S 0 - and the A 0 - mode occurs. However, the mode conversion does not take place at a specific discontinuity, but continuously while the S 0 -mode travels through the plate. The Identification of CMC between S0- and A0-mode In the following section argu- ments are gatherd to identify the converted mode as a A0-mode. The A0-mode is slower than the S0-mode. Therefore, the existence of the anti-symmetric mode inside the S0-wave field gives a strong evidence for CMC. Fig. 4: Analysis of the new mode at f = 200 kh z.

5 368 Structural Health Monitoring II Fig. 4a shows a strip of the Lamb wave C-scan. The source is located at the left side and the primary modes are already separated from each other. In the space between both primary modes a number of oblique waves can be seen. These waves travel at the same velocity as the A0-mode. Using the center line perpendicular to the wave front of the primary modes the C-scan strip can be transformed into a B-scan. The time-amplitude data of each point of the center line of Fig. 4a is plotted side by side as illustrated in Fig. 4b. The heights of the amplitudes at a specific time and position are visualized by the gray scale. Dark gray illustrates negative and bright gray positive amplitude values. If a Lamb wave is excited by a burst signal, at least two primary groups of waves travel through the plate which may separate from each other. The groups propagate with different velocities. The B-scan displays the movement of these groups by oblique parallel lines. The inclinations of the lines correspond to the group velocities (see Fig. 3b) of the excited modes. The dominant lines with the highest amplitudes correspond to the central frequency of the burst signal. Smaller angles α between the lines and the x1-axis conform with a higher velocity. Therefore, the upper lines corresponds to the S0-mode, whereas the lower lines belong to the anti-symmetric A0 - mode. If we start at the origin of the coordinate system corresponding to the source and use a line which has the inclination of the A0-mode we get the position corresponding to the primary A0 - mode in the B-scan. It must be noted that the A0-mode is highly dispersive. The group velocity of the A0-mode varies for the different frequencies. In reality a mono-frequent excitation is not feasible and A0-modes with different group velocities are excited. In dependence on the bandwidth of the excitation signal multiple A0 -modes are included in one group. Some of these modes travel faster than the group velocity of the chosen excitation frequency. However, these faster parts of the group are created inside the original A0-mode group and have a higher velocity with a lower angle α. However, the B-scan shows lines between the primary A0- and S0 -modes, which start inside the S0-mode lines. The inclination of these lines is equal to the inclination of the line corresponded to A0 -mode. Therefore this mode has to be created continuously by a mode conversion from S0- to A0-mode. The B-scans of other frequencies are analyzed in Willberg et al. [15] and show quite similar effects. Interpretation of CMC Analysis of the Problem After presenting the problem, we try to explain the reasons of the continuous mode conversion (CMC). The conversion between two modes occurs if a Lamb wave mode hits a discontinuity in such a way, that parts of the particle movement activate another mode. Normally mode conversion happens locally at holes, damages, etc. A mode conversion takes place, e.g. if a symmetric out-of-plane displacement of the particles inside the plate causes a anti-symmetric out-of-plane displacement by the adjacent particles and visa versa. In the following part we try to explain why CMC takes place in the presented CFRP plate. Fig. 5 shows a strip of a Lamb wave C-scan with a length of l = 64 mm. In the background of the measured data a photograph of the plate surface is displayed. The top layer is made of twill fabric displayed in Fig. 6b. A pattern can be seen which can be identified as the texture of the twill fabric of the top and bottom layer of the CFRP plate. The drawn lines are an extension of the fabrics texture. The distance between two lines is 2 mm. It can be seen that the wave front of the continuously converted mode corresponds to this direction. We have figured out that this texture induces the mode conversion, which can be seen in the measured wave field. It is already known that a wave traveling in a periodic structures can show an unexpected behaviour such as the dispersion of a longitudinal wave in a continuum [12].

6 Key Engineering Materials Vol Fig. 5: Influence of twill fabric to the S 0 -mode wave front caused by CMC. Two different fabrics are included in the stacking sequence of the investigated CRFP plate. As illustrated in Fig. 6 both fabrics have a texture. The research on the plain fabric shows a weak evidence for mode conversion applying several numerical models, whereas the experimental investigation of a two layer plain fabric plate gives no indication for CMC [15]. Therefore, here only the twill fabric is investigated in more detail. In the following section the Lamb wave propagation in the fabric is studied, where experimental as well as numerical models are used for an interpretation of the mode conversion. (a) Plain fabric (b) Twill fabric Fig. 6: Fabric types in the CFRP plate. Analysis of the Twill fabric Experimental investigation The experiments are accomplished first by investigating a sin- gle layer twill fabric plate (1 m 1 m 0.3 mm). The measurements of the plate are performed several times after rotating the plate. We always get the same results for all measurements. The influence due to a prestressing bending of the plate under its own weight on the results can be excluded. In Figure 7a the C-scan for a frequency f = 20 kh z is plotted. The mode conversion occurs primary in 0 -, 90 -, and 270 -direction. Perpendicular and parallel to the texture of the twill fabric no mode conversion can be observed. The CMC arises at all mea- sured frequencies of f > 20 kh z. In Figure 7b the B- scan at 50 kh z in 0 -direction is plotted. The symmetric mode can not be seen by displaying only the out-of-plane components of the laser vibrometer measurements. However, the B-scan corresponding to the S 0 -mode is plotted by the connection of the starting point of the converted A 0 -mode lines. Besides the primary, a converted A 0 -mode occurs which starts inside the S 0 -mode.

7 370 Structural Health Monitoring II Fig. 7: Experimental investigation of an one layer twill fabric for f = 50 kh z. Numerical investigation After an experimental investigation of the twill fabric plate a numerical analysis is performed. The assumed material properties of the fibres as well as the matrix are given in Tab. 2. The fibres are assumed as a transversal isotropic and the matrix as an isotropic material. The twill fabric models are calculated with the commercial finite element programm ABAQUS. The Lamb waves are excited using a Hann-window modulated sinusoidal point force in x3-direction at the center of the plate. All edges are assumed to be free. Table 2: Material properties of the numerical models. parameter unit fibers matrix E 11 [10 9 N/m 2 ] E 22 [10 9 N/m 2 ] G 12 [10 9 N/m 2 ] ν ν ρ [kg/m 3 ] The Fig. 8 shows the two investigated twill fabric models. Fig. 8a illustrates the model without matrix material and Fig. 8b with matrix material. The lengths and the thicknesses of the squared twill fabric cell models are defined for the first model as lf = 16 mm and d = 2 mm and for the second one as lf = 32 mm and d = 2 mm. The higher thickness of the numerically investigated plate in comparison to the experimental plate (0.3 mm) is due to the requirement of undistorted linear finite elements to avoid the locking phenomenon. The original plate would result in an extremely large finite element model, which cannot be handled with the existing computer power. Our new developed higher order special finite elements successfully tested for wave propagation simulations [14] are now implemented on a parallel computer machine, which in close future will allow a detailed and accurate analysis of the wave propagation in real twill fabric plates. The material properties of the fibres and matrix are given in Tab. 2. The squared cells are copied and merged several times to create a complete plate. The twill fabric plate (0.48 m 0.48 m 2 mm) is modelled with a single layer.

8 Key Engineering Materials Vol Fig. 8: Numerical models of single layer squared twill fabric cell. A mode conversion can be observed as illustrated in Fig. 9 for both models. The converted modes for the first model without matrix material are oriented dominantly in 45 -direction corresponding to the texture of the twill fabric. The continuous mode conversion primarily occurs in 0 -, 90 -, and 270 -directions. The model qualitatively describes the behavior of the experimental investigation of the single layer twill fabric plate. The second model including matrix material also shows mode conversion. The orientation of the converted mode (between the black lines) differs from the experiments. However, the main regions of conversion are similar to the experiments. Nevertheless, the second model does not describe the real properties of the twill fabric plate as good as the model without matrix material. Fig. 9: Numerical results of a one layer twill fabric plate (0.48 m 0.48 m 2 mm) for f =250 kh z. Analysis of the Regions of Appearance of CMC In the previous section it is shown that the twill fabric plate causes CMC in the experiments as well as in numerical models. The dominant regions of conversion are concentrated in 0 -, 90 -, and 270 -direction. To understand the conversion of the S0- into the A0-mode a simple twill fabric strip is modelled. Because the inplane displacements in propagation direction of the S0-mode at a specific position are nearly constant over the thickness of the plate, a static numerical model is used to investigate the coupling between this displacements and the other two components.

9 372 Structural Health Monitoring II The solution and the boundary conditions are shown in Fig. 10. A one-dimensional displacementover the whole surface at the left and the right end in x1-direction is applied. The material equals the twill fabric model without matrix material, which is illustrated in Fig. 8a. After applying the displacement the deformation of the plate is calculated. Corrugations in 45 direction parallel to the pattern of the twill fabric surface can be seen. We assume that these corrugations are affecting the S0-mode. Because the corrugations are not symmetric to the center plane of the plate they can cause mode conversion. However, this theory does not explain why the mode conversion does not occur everywhere in the plate. u 1 -displacement u 1 -displacement x 2 x 1 orientation of the twill texture x 3 x 1 Fig. 10: One-dimensional static load test of a one layer twill fabric. In Fig. 11 the analog computation is done applying an in-plane displacement in the ±45 - direction. The resulting displacements as well as the scale to plot the deformations are equal for all three models. In Fig. 11 the corrugations exist too, but they are much weaker in comparison to the results of Fig. 10. This result correlates with the region of CMC in the experiments of a one-layer twill fabric plate. It seems that continuous mode conversion mainly takes place in regions where the coupling between the in-plane and out-of-plane displacement is strong. Assuming that the illustrated coupling effect (Fig. 10) is mainly responsible for the CMC effect it remains unclear why dependency on the frequency CMC occurs everywhere in the seven layer CFRP plate. The effects has to be researched in more detail. Fig. 11: One-dimensional static load test of a one layer twill fabric.

10 Key Engineering Materials Vol Conclusions The paper presents the effect of continuous Lamb wave mode conversion in a CFRP plate. The effect is observed first in a multi-layer composite plate, where 50% of the whole plate is made of fabrics. For the investigated CFRP plate the effect of CMC first occurs at about 138 kh z and is identified as mode conversion from S 0 - to A 0 -mode. For this identification the dispersion curves as well as B-scans are used. For all measured frequencies between 138 kh z and 350 kh z the CMC occurs. The B-scans have shown that at frequencies between 138 kh z and 175 kh z the new mode travels opposite to the parent S 0 -mode. Above 175 kh z both the parent and the converted A 0 -mode travel in the same direction. For a better understanding of the effect of CMC pure twill fabric plates are analyzed both numerically and experimentally. The experiments of the twill fabric plate have shown that the conversion arises for all investigated frequencies f > 20 kh z. The orientation of the converted A 0 -mode is parallel to the texture of the twill fabric. The converted mode occurs only in 0 -, 90 -, and 270 -direction. The numerical model without matrix material shows the same behavior in comparison to the experimental results. The second model including matrix material is not consistent with the measured wave field. Both models show qualitatively the phenomenon of CMC. A more detailed theoretical interpretation based on very accurate numerical simulations is under progress. One-dimensional numerical tensile tests show that in 0 -, 90 -, and 270 -direction a strong coupling between u 1 - and u 2 -displacements exists. In ±45 -direction this coupling is much weaker. Therefore the mode conversion mainly takes place in this regions. However, in the seven layer CFRP plate CMC occurs everywhere in the plate depending on the frequency. The effects are more complex and not fully understood yet. Acknowledgments The authors like to thank the Deutsche Forschungsgemeinschaft (DFG) and all project partners for their support (GA 480/13-1, MO 553/9-1). References [1] Z. A. B. Ahmad. Numerical Simulations of Lamb waves in plates using a semi-analytical finite element method. PhD thesis, Otto-von-Guericke-University of Magdeburg, Fortschritt- Berichte VDI Reihe 20, Nr. 437, Düsseldorf: VDI Verlag, ISBN: , [2] C. Boller, F.-K. Chang, and Y. Fijino. Encyclopedia of Structural Health Monitoring. JohnWiley & Sons, ISBN-10: , [3] Y. Cho. Estimation of ultrasonic guided wave mode conversion in a plate with thickness variation. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 47(3): , [4] V. Giurgiutiu. Structural Health Monitoring with Piezoelectric Wafer Active Sensors. Academic Press (Elsevier), ISBN-13: , [5] L. Mallet, B. C. Lee, W. J. Staszewski, and F. Scarpa. Structural health monitoring using scanning laser vibrometry: II. Lamb waves for damage detection. Smart Materials and Structures, 13: , [6] G. Mook, J. Pohl, and F. Michel. Non-destructive characterization of smart CFRP structures. Smart Materials and Structures, 12: , [7] J. Pohl, C. Willberg, U. Gabbert, and G. Mook. Theoretical analysis and experimental determination of the dynamic behaviour of piezoceramic actuators for SHM. Experimental Mechanics, 51(5), 2011.

11 374 Structural Health Monitoring II [8] D. Schmidt. Dokumentation CFK-Testplatten. Technical report, DLR, Institute of Composite Structures and Adaptive Systems, [9] H. Sohn, D. Dutta, J. Y. Yang, M. DeSimio, S. Olson, and E. Swenson. Automated detection of delamination and disbond from wavefield images obtained using a scanning laser vibrometer. Smart Materials and Structures, 20(4):045017, [10] Z. Su and L. Ye. Identification of Damage Using Lamb Waves: From Fundamentals to Applications. Springer, ISBN: , [11] I. A. Viktorov. Rayleigh and Lamb Waves. Plenum Press, [12] J. M. Vivar-Perez, U. Gabbert, H. Berger, R. Rodriguez-Ramos, J. Bravo-Castillero, R. Guinovart-Diaz, and F. J. Sabina. A dispersive nonlocal model for wave propagation in periodic composites. Journal of Mechanics of Materials and Structures, 4(5): , [13] P. D. Wilcox, M. J. S. Lowe, and P. Cawley. Mode and transducer selection for long range Lamb wave inspection. Journal of Intelligent Material Systems and Structures, 12: , [14] C. Willberg, J. M. Duczek, S.and Vivar-Perez, D. Schmicker, and U. Gabbert. Comparison of different higher order finite element schemes for the simulation of Lamb waves. Computer Methods in Applied Mechanics and Engineering, 2012 (in Review). [15] C. Willberg, S. Koch, G. Mook, U. Gabbert, and J. Pohl. Continuous mode conversion of Lamb waves in CFRP plates. Smart Materials and Structures, in review, [16] C. Willberg, J. M. Vivar-Perez, and U. Gabbert. Lamb wave interaction with defects in homogeneous plates. In International Conference on Structural Engineering Dynamics. ICEDyn Ericeira, Portugal Juni, 2009, 2009.

12 Structural Health Monitoring II / The Phenomenon of Continuous Mode Conversion of Lamb Waves in CFRP Plates /