COMPARISON OF MODELING AND EXPERIMENTS OF LAMB WAVES AS APPLIED TO STRUCTURAL HEALTH MONITORING

Transcription

1 COMPARION OF MODELING AND EXPERIMENT OF LAMB WAVE A APPLIED TO TRUCTURAL HEALTH MONITORING Nicoleta A. Apetre, Jennifer E. Michaels, Massimo Ruzzene, Thomas E. Michaels and Pranaam Haldipur chool of Aerospace Engineering, Georgia Institute of Technolog, Atlanta, Georgia, UA chool of Electrical & Computer Engineering, Georgia Institute of Technolog, Atlanta, Georgia, UA ABTRACT. This paper eplores the effectiveness of the Finite Element Method (FEM) to model wave propagation in a two dimensional (D) structure. In particular, the capabilities of the FEM to model permanentl attached piezoelectric transmitters and receivers are illustrated. The validit of the FEM results is eamined b comparing times and amplitudes of direct and A arrivals to those obtained from eperimental measurements on a plate. Recommendations are made as how to best use FEM simulations to epedientl represent real sensors. Kewords: Ultrasonics, tructural Health Monitoring, Modeling, Guided waves, Lamb waves, Finite Element Analsis PAC: Zc, 8.7.Cv, 46.7.De, 43..Mv,.7.Dh INTRODUCTION Guided waves, such as Lamb waves, show sensitivit to a variet of damage tpes and have the abilit to travel relativel long distances within the structure under investigation. For this reason, guided ultrasonic waves are particularl suitable for structural health monitoring (HM) applications, where the objective is to identif possible anomalies or damages such as cracks, delaminations and corrosion in structures. Man dnamics-based HM techniques have been developed over the ears and valuable reviews of the state-of-the-art can be found in [-3]. The eisting techniques var on the basis of the tpe of dnamic response signals used for the analsis and the features or parameters considered as damage indicators. Accelerometers, ultrasonic transducers, piezoelectric sensors, and non-contact laser sensors have been emploed to record the structure s response over ranges of frequencies that should be optimized to provide optimal sensitivit to damage. Among the dnamics based techniques proposed, guided wave methods provide a good compromise in terms of sensitivit to defects and etent of the area that can be monitored in a timel fashion [3-5]. For this reason, significant efforts are being devoted to the development and improvement of defect characterization 69

2 procedures that support guided waves measurements. For eample, advanced signal processing techniques are being emploed to highlight signal features which are sensitive to the presence of damage, and which can be used for its classification and for the estimation of its etent [4]. Analtical and numerical models that incorporate changes in the structural dnamic characteristic parameters due to simulated damage are ver useful tools for the development of algorithms for damage localization and sizing. Literature offers several models for Lamb waves propagation in simple or comple structures. pectral finite element method [6,7], boundar element method [8,9], finite element method [,], semi-analtical finite element method [,3] or higher-order plate theor [4] are successfull used to demonstrate wave propagation phenomena and validate eperimental measurements. The present paper uses a finite element model to stud guided wave propagation in D structures. The paper is organized as follows. The brief introduction presented in this section is followed b a description of eperimental and finite element approaches. The comparison of the results obtained through the two methods is presented to discuss the influence of various modeling parameters. The paper concludes with summar and future recommendations. MEAUREMENT In the eperiments, described in detail in [5], four transducers are mounted on an aluminum (Young s modulus E = 7 GPa and densit = 75 kg/m 3 ) plate measuring 6 mm 6 mm.8 mm. According to the schematic of Figure (a), the transducers are arranged in a square pattern and are used as sources and receivers. Their location is defined b the following coordinates epressed in mm: T (7,355), T (355,43), T 3 (43,4) and T 4 (4,65). The transducers were constructed at Georgia Tech with longitudinall polarized,.5 MHz PZT disks, mm in diameter, backed with epo. A conventional ultrasonic pulser receiver was used for spike mode transducer ecitation and waveform amplification, and a multipleer was used to switch between the four transducers on both transmit and receive, resulting in si unique transmit-receive pairs. Waveforms were digitized with a sampling rate of 5 MHz using a Tektroni TD534 digital oscilloscope, and each recorded waveform was the average of 5 signals. 6 P (, 6) 6 5 P (, ) 5 Group/Phase speed of [mm] 4 3 T T T 3 peed [mm/ sec] 4 3 Phase speed of A Group speed of A T 4 Transducers (d T = ) [mm] Frequenc* Thickness [MHz-mm] (a) FIGURE. (a) chematic of the plate with four transducers. Dispersion curves for the plate. 7

3 Figure shows dispersion curves of the plate in the frequenc range of interest. The center (and dominan frequenc of the received signals is about. MHz, and this point on the dispersion curve is shown with a dotted line. In this low frequenc range onl the first smmetric ( ) and first antismmetric (A ) Lamb modes are present. At.MHz, the group velocit for the A mode is approimatel.4 mm/s, while that of the mode is 5.35 mm/s. Referring to the square pattern of the transducers, the four waveforms from the edge transducer pairs are used for comparison with the FE results. The signals shown in Figure (a) correspond to pairs (T - T ), (T - T 4 ), (T - T 3 ) and (T 3 - T 4 ). The averaged signal for the four pairs is presented in Figure, which also shows the arrival times calculated from the dispersion curves. The arrival times for the and A modes are A tdirect 38 and t direct 94 s, respectivel. In addition, the arrival time corresponding to the first boundar reflection of the mode is calculated b identifing the point P on the boundar located at a minimum distance between T and T. Two points, one on the left boundar ( P (, ) ) and one on the top boundar ( P (,) ) are considered (Figure (a)), and it is found that P(78,6) minimizes this distance. The minimum distance (T P + PT ) is calculated as 473 mm and the arrival time of the first wave reflected from the boundar is calculated as t boundar 88 s. Note that this time is similar for all four edge pairs because of the smmetr of the transducer pattern and the plate. These arrival times are useful for interpreting the measured signals of Figure. FEM ANALYI A detailed FE model of the plate is developed using the commercial software ABAQU. The first step of the FEM modeling of wave propagation, which is also the objective of the current work, is finding a useful wa to describe sources and receivers. To accomplish this goal, the current work considers a D FEM model in order to describe a beam with two identical sensors one acting as transmitter and the other as receiver. This is a simplified model that does not properl take into account the reflections from the plate boundaries, and the comparison with eperimental data should take into account onl the ver first direct waves. As presented in Figure 3, a clamped-clamped configuration is assumed, and the beam has length L = 658 mm and thickness h =.8 mm. The length is calculated as the length of a segment passing thorough transducers T and T and intersecting the plate boundar. The beam has same material properties as the plate described in the previous section. Amplitude pair T -T pair T -T 4 pair T -T 3 Averaged amplitude pair T 3 -T Time, [s] Time, [s] (a) FIGURE. (a) Waveforms from transducer pairs (T T ), (T T 4 ), (T T 3 ) and (T 3 T 4 ). Averaged waveform of the four pairs with arrival times: t 38, A t 88, 94 s direct boundar t direct 7

4 , i, U T T, j, U d d FIGURE 3. chematic of the beam with two identical transducers, one acting as transmitter (T ) and the other as receiver (T ). The transmitter T is modeled as a product between a time ecitation and a combination of in-plane and out-of-plane loads distributed over the length d of the transmitter, F i F j Q( ) F(, P( ) Q( t, () where denotes the in-plane coordinate d,, t is the time variable, and i and j are in-plane and out-of-plane unit vectors. Figure 4 shows distributions in space and time of the in-plane and out-of-plane loads considered here: Figure 4(a) shows the functions F () and F () whereas Figure 4 shows the function Q( and its Fourier transform. The equations of motion are solved b the eplicit central-difference time integration rule, which is suitable for large models with short dnamic response. The centraldifference operator is conditionall stable, and in order to have an accurate solution, the integration time step, t, is chosen such that t =.5/f*, where f* is the highest frequenc of interest, measured in Hz. The element size, l, is based on the minimum wavelength of elastic waves propagating in the beam, and l.5c l /f*, where C l is the longitudinal wave speed. Based on these rules, the time step and spatial discretization in this eample are selected as t =. s and l =. mm, respectivel, which corresponds to eight 4-node plane-strain linear elements through the beam thickness and a total of approimatel 53, elements throughout the entire beam. In-plane load Out-of-plane load Uniform Parabola.5 Hann Gauss Concentrated.5 linear quadratic 4 th degree polnomial Longitudinal coordinate under the transducer, [mm] Load [N] Frequenc Amplitude Time [ s] Frequenc [MHz] (a) FIGURE 4. (a) pace and time and frequenc distributions of the in-plane and out-plane loads. Note that the magnitudes of the in-plane loads are shown; the sign is negative for <. 7

5 (a).5 FEM data A mode mode Wavenumber [rad/mm] Frequenc [MHz] (c) (d) FIGURE 5. FEM results for a linear in-plane loading profile, a Hann-windowed out-of-plane loading profile, and a 5-ccle,. MHz, Hann-windowed source time function. Displacements as a function of time and longitudinal coordinates: (a) longitudinal displacement and (c) transverse displacement. Comparison of analtical and finite element dispersion curves for A and modes: longitudinal displacement and (d) transverse displacement. The first eample models the source as a combination of a linear in-plane load and a Hann distribution out-of-plane load (Figure 4). These two loads are distributed over the length d of the transmitter T according to Eq. (). The considered time ecitation is a fiveccle sinusoidal burst at. MHz, modulated b a Hann window. The resulting in-plane and out-of-plane displacements are plotted both in time and space in Figures 5(a) and 5(c). Both in-plane and out-of-plane displacements shows the presence of both A and, with the mode propagating faster than the A mode and in a non-dispersive manner. Note that there is a significant difference in amplitude for the two modes with being mostl in-plane while A is mostl out-of-plane; or one could sa that the in-plane displacement has a larger contribution whereas the out-of-plane displacement has a larger A contribution. Figures 5 and 5(d) compare analtical (dotted and broken lines) and finite element (contour plots) dispersion curves for the A and modes. Agreement is ver good, which serves to validate the finite element model. COMPARION OF EXPERIMENTAL AND FEM REULT In order to compare eperimental and FE data, a procedure is developed to average the FE response over the receiving transducer. The spatial distribution of the applied load, given b Eq. (), is assumed to be a product between constants c and c and weight functions w () and w (), 73

6 F i F jq( c w i c w j Q( ) F(, P( ) Q( t, () where the in-plane coordinate is d, (Fig. 3). Then, the total displacement under the receiver T is the vector sum of the resulting inplane displacement U (, and out-of-plane displacement U (,, where the in-plane coordinate is d u (, U (, i U (, j (3), (Fig. 3). An ad hoc representation for receiver T is to compute an averaged displacement b appling the same in-plane and out-of-plane spatial weight functions as transmitter T, epressing received signals as, U FEM d c w U (, cw U (, ( d. (4) Here the unknown scale factor is set to unit. Eight cases for the weight functions shown in Figure 4(a) are considered; the are summarized in Table. For all of the in-plane loads, the functions w are chosen such that the load is zero at the middle point of the transducer, = + d/, and varies in a linear, quadratic, or 4 th degree polnomial fashion (with a sign change at the origin). The 4 th degree polnomial approaches the case of two impulses applied at the boundaries of the transducer. The out-of-plane loads have the maimum value of one at the middle point of the transducer, and the are zero at the boundaries of the transducer, = and = + d. For all the cases considered, the averaged displacement is computed based on Eq. (4) with c = c = and plotted in Figure 6. In order to compare eperimental and FE results, the displacements are normalized with respect to their maimum amplitudes. TABLE. ummar of loads for the various FEM cases considered ( d, ). Case In-plane load Out-of-plane load None Concentrated: w ( ( d / )) None Uniform: w 3 ( / ) Linear: d w Hann: ( d / ) w cos d / d / 4 Linear Gauss: ( d / ) w ep d / Quadratic: 5 ( d / ) Concentrated w sgn ( d / ) d / 6 Quadratic Hann 7 4 th degree polnomial: 4 ( d / ) w sgn ( d / ) d / Quadratic: 8 4 th degree polnomial Uniform w ( d / ) d / 74

7 It can be seen in Figure 6 that the direct arrival times of the fast mode and the slower A mode for the FEM data are in reasonable agreement with calculated times. The calculated times of the and A direct arrivals and the boundar reflection (for the D FEM model) are A t 38 s, direct tdirect 94 s and t boundar 8 s, respectivel; these times are shown as vertical lines in Figure 6. Eperimental data has a more complicated A arrival due to interference with reflections from the boundaries. Also, it has to be noted that in all cases the eperimental data shows a secondar echo that is not present in an of the FEM data. This is most likel due to a secondar ecitation function created b the transmitter. For the first two cases, when onl out-of-plane loads are applied, the FEM results show ver small amplitudes compared with A (Figure 6(a)). This is consistent with the results shown in Figure 5, which demonstrate that is mostl an in-plane mode. To improve the FEM results, an in-plane component of the load is added. When a combination of the in-plane quadratic distributed load and the out-of-plane concentrated load is used (case 5 of Figure 6(c)), the component is still much smaller than the A component, and the FEM result is not comparable with the eperimental data. A fairl good comparison for the mode is obtained when the in-plane load is a linear distributed load (cases 3 and 4 of Figure 6). Among all the cases considered, the combination of a 4 th degree polnomial for the in-plane load and a uniform out-of-plane load is considered to be the best match to eperimental data for both and A modes (case 8 of Figure 6(d)), although the eperimental boundar reflections make a quantitative comparison difficult FEM - case FEM - case pair T -T -.5 FEM - case 3 FEM - case 4 pair T -T Time, [s] (a) Time, [s] FEM - case 5 FEM - case 6 pair T -T Time, [s] -.5 FEM - case 7 FEM - case 8 pair T - T Time, [s] (c) (d) FIGURE 6. Comparison of eperimental data (dotted lines) and FEM results (cases as per Table ). (a) Cases and, cases 3 and 4, (c) cases 5 and 6, and (d) cases 7 and 8. Arrival times for the FEM data are shown as vertical lines: A t 38, t 94, 8 s. direct direct t boundar 75

8 UMMARY This paper presents a D finite element model for the propagation of waves in plates. A general method to model the transmitter and receiver using spatiall distributed weighting functions is presented that allows the weighting functions to be adjusted to best match measurements. Reasonable agreement to eperimental data is obtained with a distributed combination of in-plane and out-of-plane forces. Future improvements should emplo a three dimensional finite element model and quantitative comparisons between modeled and measured and A arrivals. REFERENCE. H. ohn, C. Farrar, M. Francois, D. Devin, W. Daniel and B. Nadler, A Review of tructural Health Monitoring Literature: Los Alamos National Laborator Report, LA-3976-M (3).. D. E. Adams, Health Monitoring of tructural Materials and Components: Methods and Applications. John Wile and ons, UK (7). 3. V. Giurgiutiu, tructural Health Monitoring with Piezoelectric Wafer Active ensors. Academic Press (7). 4. A. Raghavan and C.E.. Cesnik, Review of guided wave structural health monitoring, The hock and Vibration Digest, 39(), pp. 9 4 (7). 5. J. L. Rose, A Baseline and vision of ultrasonic guided wave inspection potential, Journal of Pressure Vessel Technolog, 4, pp (). 6. N. Apetre, M. Ruzzene,. Hanagud and. Gopalakrishnan, pectral and perturbation analsis of first order beams with notch damage, Journal of Applied Mechanics, 75(3), 39 (8). 7. U. Lee, J. Kim and A. Leung, The spectral element method in structural dnamics, The hock and Vibration Digest, 3(6), pp (). 8. J. Jin,. Quek and Q. Wang, Wave boundar element to stud Lamb wave propagation in plates, Journal of ound and Vibration, 88, pp (5). 9. Y. Cho and J. L. Rose, A boundar element solution for a mode conversion stud on the edge reflection of Lamb waves, Journal of the Acoustical ociet of America, 99(4), pp (996).. M. J.. Lowe, P. Cawle, J. Kao and O. Diligent, The low frequenc reflection characteristics of the fundamental antismmetric Lamb wave A from a rectangular notch in a plate, Journal of the Acoustical ociet of America, (6), pp Y. Lu, L. Ye, Z. u and C. Yang, Quantitative assessment of through-thickness crack size based on Lamb wave scattering in aluminium plates, NDT&E International, 4, pp (8).. I. Bartoli, A. Marzani, F. Lanza di calea and E. Viola, Modeling wave propagation in damped waveguides of arbitrar cross-section, Journal of ound and Vibration, 95, pp (6). 3. O. Mukdadi and. Datta, Transient ultrasonic guided waves in laered plates with rectangular cross section, Journal of Applied Phsics, 93(), pp (8). 4. L. Wang and F. Yuan, Lamb wave propagation in composite laminates using a higher-order plate theor, Proceedings of the PIE, 653, pp. 653I.-. (7). 5. J. E. Michaels and T. E. Michaels, Enhanced differential methods for guided wave phased arra imaging using spatiall distributed piezoelectric transducers, Review of Progress in QNDE, 5B, D. O. Thompson and D. E. Chimenti (Eds.), AIP, pp (6). 76