Guided wave excitation and propagation in damped composite plates

Transcription

1 Original Article Guided wave excitation and propagation in damped composite plates Structural Health Monitoring 1 25 Ó The Author(s) 218 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: 1.11/ journals.sagepub.com/home/shm Hanfei Mei and Victor Giurgiutiu Abstract Guided wave attenuation in composites due to material damping is strong, anisotropic, and cannot be neglected. Material damping is a critical parameter in selection of a particular wave mode for long-range structural health monitoring in composites. In this article, a semi-analytical finite element approach is presented to model guided wave excitation and propagation in damped composite plates. The theoretical framework is formulated using finite element method to describe the material behavior in the thickness direction while assuming analytical expressions in the wave propagation direction along the plate. In the proposed method, the Kelvin Voigt damping model using a complex frequencydependent stiffness matrix is utilized to account for anisotropic damping effects of composites. Thus, the existing semianalytical finite element approach is being extended to include material damping effect. Theoretical predictions are experimentally validated using scanning laser Doppler vibrometer measurements of guided wave propagation generated by a circular piezoelectric wafer active sensor transducer in a unidirectional carbon fiber reinforced polymer composite plate. The proposed method achieves good agreement with the experimental results. Keywords Structural health monitoring, material damping, attenuation, guided wave propagation, composite plates, predictive modeling, piezoelectric wafer active sensors Introduction The extensive use of composite materials in aerospace structures has posed new challenges for implementing effective structural health monitoring (SHM) techniques due to the general anisotropic behavior and complicated guided wave features in composites. 1 In composites, guided waves propagate at different velocities in different directions. 8 Besides, guided wave attenuation due to material damping is strong and anisotropic and, thus, cannot be neglected. 9 Compared to metals, composite materials show strong and anisotropic damping effects from the fiber and matrix constituents, which attenuate the amplitude responses of guided waves and shrink their effective interrogation range. 1 Moreover, material damping has different effects on the amplitude response of various guided wave modes. This aspect brings considerable challenges for SHM techniques based on the wave amplitude change. 11,12 Many computational methods for guided wave propagation in composite plates have been developed. 13 However, the modeling of guided wave propagation in composites, by itself, is a very challenging task. The difficulties come from the multimode, dispersive, and direction-dependent features of guided wave propagation in composites. A conventional finite element model containing the detailed information of each lamina may become computationally intensive and it is difficult to accommodate the accuracy requirements for highfrequency and short wavelength guided waves over long propagation distances. 14 Several highly efficient techniques have been developed for modeling guided wave propagation in composite plates such as the global matrix method (GMM), 15,16 local interaction simulation approach Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Corresponding author: Hanfei Mei, Department of Mechanical Engineering, University of South Carolina, 3 Main Street, Columbia, SC 2928, USA. hmei@ .sc.edu

2 2 Structural Health Monitoring () (LISA), 1,1 elastodynamic finite integration technique (EFIT), 18 spectral finite element method (SFEM), 19,2 and semi-analytical finite element (SAFE) method. 21 Glushkov et al. 22 proposed a Green s matrix-based method to investigate the guided wave excitation and diffraction by surface obstacles in composite plates. All the presented methods have shown the capability to model guided wave propagation in composite plates. Among the existing methods, the SAFE approach is a suitable candidate for modeling guided wave propagation in composite plates with material variation in the thickness direction. In this method, the material variation along the thickness direction is described using FEM, while analytical complex-valued exponential functions are used in the wave propagation direction. This method exploits the benefits of numerical and analytical approaches. A SAFE method for waveguides of arbitrary cross-section was demonstrated for the first time in Since then, the SAFE approach is mainly used to obtain dispersion curves of isotropic and composite plates. More complex waveguides, such as rods, 24 bonded joint, 25 and railroad tracks, 26 were also investigated. Recent works demonstrate the SAFE application to damped steel pipe. 2 As in conventional FEM applications, convergence study has to be considered to guarantee the simulation accuracy. 28 Simulations of the time-transient response of the plate due to an external force and to a piezoelectric wafer active sensor (PWAS) excitation have been investigated using the SAFE approach in two-dimensional (2D) 29,3 and three-dimensional (3D) cases. 21,31,32 The main advantage of the SAFE approach is the high computational efficiency. The SAFE method also offers wavemode separation. Guided waves typically propagate as a combination of multiple dispersive wave packets; it is very challenging to separate the individual modes when using brute-force FEM techniques. However, the SAFE approach does separate the wave modes during the thickness-wise FEM analysis. Thus, the complicated guided wave feature of each mode can be investigated individually. In the case of fiber reinforced polymer (FRP) composites, one of the main complexities encountered in modeling guided wave propagation is the anisotropic damping effect. The guided wave damping attenuation is essentially due to viscoelastic properties of the fiber and matrix constituents. 33 This attenuation is a critical parameter in mode selection for long-range SHM of laminated composites. However, attenuation due to material damping is often neglected in guided wave propagation analysis because of modeling complexities. In order to investigate the guided wave attenuation, numerous studies on the measurement of damping attenuation in structural materials have been conducted These studies indicate that attenuation due to material damping has a significant effect on the amplitude response of guided waves propagating in composite plates. If the material damping effect is incorporated in the predictive model, then SHM techniques based on amplitude change can be understood better and implemented more effectively. For this reason, the predictive modeling of guided wave excitation and propagation in damped composite plates is a critical issue and needs to be developed in coordination with experimental validations. Several researchers have studied damping models to describe the anisotropic damping effect of composites, including hysteretic damping model, 36 Rayleigh damping model, 9,3 and Kelvin Voigt damping model. 1 Gresil and Giurgiutiu 9 developed a predictive model for studying the attenuated guided wave propagation in a woven carbon fiber reinforced polymer (CFRP) composite plate using Rayleigh damping. Shen and Cesnik 1 extended LISA to model the anisotropic damping effects on guided wave propagation in composite plates by adopting the Kelvin Voigt damping model. All these research studies indicate that damping models have the capability to capture anisotropic damping effects in composites. In this article, a SAFE approach is presented to model guided wave excitation and propagation in damped composite plates. The theoretical framework is formulated using (a) FEM to describe the material variation along the thickness direction and (b) analytical expressions in the wave propagation direction. In our proposed method, which is an improvement over the existing SAFE approach, the Kelvin Voigt damping model using a complex frequency-dependent stiffness matrix is utilized to account for anisotropic damping effects of composites. To validate the proposed method, experimental damping attenuation is measured through scanning laser Doppler vibrometer (SLDV) line scans in different propagation directions. Then, the damping attenuation is incorporated in the predictive model using loss coefficient matrix that is obtained through an iterative updating to achieve an acceptable match with the measured damping attenuation. Finally, the improved predictions with anisotropic damping effect are experimentally validated using SLDV measurements of guided wave propagation in a unidirectional CFRP composite plate excited by a circular PWAS transducer. The proposed method achieves good agreement with the experimental results. This article contains the analytical developments and the experimental measurements.

3 Mei and Giurgiutiu 3 Figure 1. SAFE model of guided wave propagation in an infinite composite plate and its top view. Mathematical framework SAFE approach Consider an infinite composite plate as shown in Figure 1. To model guided wave propagation in this compositeplate,weusethesafeapproachinwhich the guided waves propagate analytically in the r, f directions, while in the thickness direction z the domain is discretized with the finite element method (FEM). Hence, we only need one-dimensional (1D) FEM mesh to discretize the thickness direction. The displacement field u(r, f, v) calculated at a reasonably far distance from the wave source (r=h 1, h is the thickness of the plate) can be expressed as ur, f, vþ = i XM ½ F m e irj m u m j m u m Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2prbu m Þ Þcosu m fþ+ p=4š ^V L m P ^V L m B u mþ^v R m 1Þ where j m is the mth wavenumber, and according to Appendix 1, ^V L m and ^V R m are the left and right eigenvectors, F m is the upper part of ^V R m, B(u m) is the righthand matrix of the eigenvalue problem, and P is the excitation. In addition, according to Appendix 2 bu m Þ= ½j mu m u m = f + arctan j mu m Þ j m u m Þ Þ j m u m ÞŠcosu m fþ 2j u m 2Þ Þsinu m fþ 3Þ Coupling between PWAS and the host structure PWAS transducers are convenient enablers for generating and receiving guided waves in structures for SHM applications. 38 Figure 2(a) shows the interaction between the PWAS and the host structure. In this study, a circular PWAS transducer is used as the guided wave excitation source. The effect of the actuator on the substrate can be accurately represented by shear tractions along the edges of the transducer on the substrate s surface. In the case of ideal bonding, the shear stress in the bonding layer is assumed to be concentrated at the ends and the pin-force model is utilized to represent the PWAS excitation. The forcing functions t x and t y expressed in the global coordinate are introduced in Figure 2(b) to represent the excitation of a circular PWAS of radius a t x = t dr aþcos f t y = t dr aþsin f 4Þ where t represents the traction amplitude, d denotes the Dirac delta function, and f is the energy velocity direction. Fourier transforms of the forcing functions can be calculated as reported in Nadella et al., 39 that is ~t x = iat J 1 jaþcos u ~t y = iat J 1 jaþsin u 5Þ where j is the azimuthal wavenumber, J 1 represents the Bessel function of the first kind, and u is the phase velocity direction. To simulate the guided wave propagation due to the excitation by a surface-mounted circular PWAS in

4 4 Structural Health Monitoring () (a) (b) Figure 2. Interaction between the PWAS and the host structure: (a) micrograph picture of an actual PWAS installed on a 1-mm plate 38 and (b) schematic model showing surface shear stress of ideal bonding. Figure 3. A pitch-catch configuration between PWAS excitation at origin and SLDV sensing at (r, f), as seen along the radial direction. composite plates, the component of P in equation (1) should be written as 2 P = ^F, ^F = 6 4 iat J 1 j m a 3 Þcos u m 5 iat J 1 j m aþsin u m. 6Þ configuration between PWAS excitation and SLDV sensing is shown in Figure 3. The predictive model can be constructed in the frequency domain using the following four steps: Step 1. Perform Fourier transform of the time-domain excitation signal V T (t), such as a tone-burst signal, to obtain the frequency-domain excitation spectrum ~V T (v) Time domain simulations Time domain simulations are required for direct comparison with experimental measurements. A pitch-catch ~V T vþ= FFTfV T Þ t g Þ Step 2. Calculate the frequency-domain structural transfer function G(r, f, v) from PWAS to SLDV in the propagation direction f. For SLDV measurements,

5 Mei and Giurgiutiu 5 the structural transfer function for the out-of-plane velocity at the top surface of the plate should be used. It can be calculated from the out-of-plane displacement expression in equation (1) by multiplying iv because of the time harmonic term e ivt Gr, f, v where d is the half-plate thickness. Þ= ivþu z r, f, vþj z = +d 8Þ Step 3. Multiply the structural transfer function by frequency-domain excitation signal ~V T (v) to obtain the frequency-domain signal ~V R r, f, vþ= ~V T vþgr, f, vþ 9Þ Step 4. Perform the inverse Fourier transform to obtain the time-domain guided wave signal at SLDV measurement point V R r, f, tþ= IFFT ~V R r, f, vþ 1Þ Guided wave attenuation in damped composite plates Attenuation mechanisms of guided waves Guided wave attenuation may be due to many different effects and mechanisms. Of the top four attenuation mechanisms mentioned by Pollock, 4 two major ones are of interest here, geometric spreading and material damping rffiffiffiffi r 1 A 2 = A 1 geometric spreading attenuationþ r 2 11Þ A 2 = A 1 e jim r 2 r 1 Þ material damping attenuationþ 12Þ where j Im is the damping attenuation coefficient. When guided waves propagate in damped composite plates, the wavenumber j of each mode becomes complex, as described below j = j Re + ij Im 13Þ where j Re and j Im are the real and imaginary part of the wavenumber, respectively. j Im is the damping attenuation coefficient given in equation (12) and the unit is Nepers per meter (Np/m). For a particular guided wave mode of interest propagating in composite plates, the wave amplitude at radial distance r and time t can be written as A(r, t)= p A ffiffi e r ijr ivt Þ 14Þ where A is the peak amplitude at source. Substituting equation (13) into equation (14) yields A(r, t)= p A ffiffi e r ijr ivt = p A ffiffi e jimr e ijre r ivtþ r Þ = p A ffiffi e i jre + ij Im r Þr ivt Þ 15Þ From equation (15), the peak wave amplitude at distance r can be identified as A(r)= p A ffiffi e jim r r 16Þ Comparing equation (16) with equations (11) and (12) reveals that j Im is responsible for the material p damping attenuation of guided waves, whereas 1= ffiffi r is responsible for geometric spreading attenuation. It can be noted that p the amplitude consists of the geometric spreading 1= ffiffi r and the material damping e j Imr. Because the wavenumber j = v=c is a function of frequency v, and since j = j Re + ij Im, it follows that the damping attenuation coefficient j Im is a function of frequency v. Hence, the material damping e jimr depends on both the distance r and the frequency v. Methods for measuring damping attenuation Experimental measurement is one of the convenient and effective ways to obtain the damping attenuation coefficient j Im. It can be determined from multiple measurements at different distances from the source. Using the magnitude of the signal received versus the distance, we propose to curve fit these values using geometric p spreading 1= ffiffi r and material damping e j Imr in equation (16). Then, a representative value of the damping attenuation coefficient j Im can be determined. Gresil and Giurgiutiu 9 experimentally measured the damping attenuation through curve fitting on a woven CFRP plate. The plate dimension is mm 3. The plate plies have the orientation [, 45,45,,, 45, 45, ]. A large number of PWAS disks (Steminc SM412, 9 mm diameter disks and.5 mm thick) were used for wave propagation analysis. A 2 V peak-to-peak three count tone-burst exciting signal at the frequency of 3 khz was used. Figure 4(a) shows the experimental received signal at different distances from the source. The magnitude of the received signal versus the distance is presented in Figure 4(b). p The fitted curve of the geometry spreading, that is, 1= ffiffi r, and the other due to material damping,

6 6 Structural Health Monitoring () Figure 4. Experimental measurements of damping attenuation: (a) experimental received signal at different distances at 3 khz for S mode and (b) curve fitting of the peak-to-peak magnitude versus the distance. 9 that is, e jimr, fit very well the experimental data when the damping attenuation coefficient j Im is equal to 15. Damping models for composite materials In the theoretical SAFE formulation, the stiffness matrix may be real or complex. For composite materials, damping attenuation of guided waves is caused by the viscoelastic nature of the polymeric matrix and wave scattering from the fibers and matrix inhomogeneities. Damping can be modeled by assuming a complex frequency-dependent stiffness matrix. 25 This section reviews the linear viscoelastic models that may be used in the SAFE approach. Linear viscoelasticity can be modeled by allowing complex components in the material stiffness matrix, that is ~C = C ic 1Þ where C is the storage matrix corresponding to the elastic behavior and C is the loss matrix corresponding to the dissipative behavior. Two damping models, hysteretic and Kelvin Voigt, which are well established in ultrasonic nondestructive examination (NDE), are considered in this study. In the hysteretic damping model, the loss matrix is independent of frequency, that is Cpq = h pqcpq, where < h pq <1 is the loss coefficient. equation (1) can be written as ~C pq = 1 ih pq ÞCpq, p, q = 1,..., 6 hysteretic damping modelþ 18Þ However, in the Kelvin Voigt damping model, the loss matrix is frequency-dependent. If a reference frequency f is specified, then the loss matrix at a generic frequency f can be scaled as Cpq = f h pqcpq =f and equation (1) becomes ~C pq = 1 i f h f pq Cpq, p, q = 1,..., 6 Kelvin Voigt damping modelþ 19Þ It is apparent that at the reference frequency f, both models predict the same damping attenuation. In the case of unidirectional composites, the loss coefficient matrix h contains only diagonal terms and the damping effects in different directions are decoupled when the lamina orientation coincides with the global coordinates or with the loading axis. 41 Hence, the loss coefficient matrix h for unidirectional composites becomes 2 3 h 11 h 22 h 33 h = h h 55 5 h 66 unidirectional compositesþ 2Þ In general, h 11 6¼ h 22 = h 33 (related to dilatational/ extensional damping) and h 44 6¼ h 55 = h 66 related to distortional/shear damping). 1

7 Mei and Giurgiutiu (a) (b) Figure 5. Verification of damping attenuation for symmetric modes (S and SH) in the carbon-epoxy plate: (a) Neau 42 and (b) SAFE predictions. Verification of damping models using literature data To verify the damping models, a CFRP composite plate studied by Neau 42 is examined and the corresponding comparisons are made with original results obtained from an analytical model by Neau. 42 The specimen is a 3.6-mm-thick plate and the material density is 156 kg/m 3. The storage matrix and loss coefficient matrix are given as :6 9 6:4 9 13:5 6:8 C 6:4 6:8 14 = 2:2 GPa 6 4 4:6 5 4: 21Þ 2 3 8:66% 3:33% 9:3% 3:33% 4:44% 3:6% 9:3% 3:6% 2% h = 3:6% 6 4 2:95% 5 5:95% 22Þ The hysteretic damping model investigated by Neau 42 is utilized to calculate the damping attenuation coefficients. The comparisons between Neau s results and our SAFE predictions for the damping attenuation of S, SH, and A modes are shown in Figures 5 and 6, respectively. Our predictions are in very good agreement with Neau s. 42 These polar plots represent the damping attenuation as a function of the phase velocity direction u. It can be noted that the material damping effect on the composite plate is anisotropic. Effect of damping models on guided wave modes In this section, the effect of damping models on various guided wave modes is investigated. The hysteretic and Kelvin Voigt damping models are utilized to predict the damping attenuation of the composite plate used in section Verification of damping models using literature data. Figure shows the predicted damping attenuation of hysteretic and Kelvin Voigt damping models as a function of frequency. It can be observed that the attenuation of SH and A modes increases with frequency in this frequency range, while S mode first increases then decreases. In former studies using conventional ultrasonic interferometry, 42 the reference frequency f was selected as the frequency of the resonant transducer. In our study, which is more general and does not assume a resonant transducer, the reference frequency f was set in the middle of the frequency range under consideration (i.e. 25 khz in our case). It can be clearly noted that both the models are coincident at the reference frequency f = 25kHz and away from this frequency, the difference between the models becomes increasingly significant, with Kelvin Voigt damping model resulting in a smaller attenuation than hysteretic damping model below f, and in a larger attenuation above f.

8 8 Structural Health Monitoring () (a) (b) Figure 6. Verification of damping attenuation for anti-symmetric mode (A) in the carbon-epoxy plate: (a) Neau 42 and (b) SAFE predictions. Figure. Comparison of damping attenuation between hysteretic damping model and Kelvin Voigt damping model as a function of frequency. Experimental validation in a CFRP composite plate To validate the proposed model, experimental measurements and numerical predictions are conducted. The experimental damping attenuation is measured through SLDV line scans in different propagation directions. Then, the attenuation due to material damping is incorporated in the predictive model using Kelvin Voigt damping model. This section demonstrates that Kelvin Voigt damping model is capable of capturing the anisotropic damping effect on guided wave propagation in a unidirectional CFRP composite plate by comparing with experiments. Experimental setup for measuring guided wave propagation and attenuation In this experiment, a 1.8-mm-thick in-house unidirectional CFRP composite plate with stacking sequences of [] 8 was examined. The material properties were measured using the ultrasonic immersion technique, given in section Material properties measurement.

9 Mei and Giurgiutiu 9 Figure 8. Experimental setup of the hybrid SLDV-PWAS measurements. To excite guided waves in the unidirectional composite plate, a circular PWAS (Steminc SM412, 9 mm diameter disks and.5 mm thick) was bonded on the top surface as the excitation source. A laser vibrometer (Polytec PSV-4-M2) was adopted to measure the outof-plane velocity of the plate. Thus, a hybrid SLDV- PWAS system was established to measure the guided wave propagation and damping attenuation. Figure 8 shows the experimental setup of the hybrid SLDV- PWAS measurements. Under electrical excitation, the PWAS generates guided waves in the composite plate. The guided waves propagate with an out-spreading pattern, undergo dispersion, experience attenuation due to the material damping, and are finally picked up by SLDV. The experimental setup on the unidirectional composite plate is shown in Figure 9. Figure 9(a) presents the measurement of damping attenuation coefficient; this can be done by SLDV line scans at different distances from the PWAS along various angular directions. Reflective tape was bonded on the top surface along five different angular directions (, 3, 45, 6, and 9 ), which was used to enhance the SLDV signal quality. An Agilent 3312A function generator was used to generate three-count tone-burst signals at the central frequency ranging from 45 to 3 khz in steps of 15 khz. An HSA 414 power amplifier was utilized to amplify the excitation signal to 14 V peak to peak. Figure 9(b) shows the SLDV area scan for the measurement of guided wave propagation pattern in the composite plate. The excitation signal is a three-count tone-burst at the central frequency of 9 khz. In this experiment, the bottom surface of the plate was adopted as the SLDV scanning area. The SLDV area scan experiment was utilized to validate theoretical prediction in the following section. Experimental measurements of damping attenuation For measurement of damping attenuation in each propagation direction, totally 1 SLDV measurements were performed at various locations along the scanning line from 1 to 1 mm away from PWAS. The interval of two adjacent measurements is 1 mm. Typical received signals of 9 khz excitation in the fiber direction ( direction) at different locations are shown in Figure 1(a). It can be noted that the signal amplitudes attenuate with the propagation distance

10 1 Structural Health Monitoring () Figure 9. Experimental setup on the 1.8-mm-thick CFRP composite plate: (a) SLDV line scan measurements and (b) SLDV area scan measurements. and the wave packets become more dispersive. The curve fitting method was performed to obtain the damping attenuation coefficient from the SLDV signals. The experimental and fitted amplitudes versus the distance in the direction of a 9-kHz excitation are shown in Figure 1(b). The blue cross marker denotes the attenuation of experimental signal amplitudes and the red solid line represents the curve fitting using p equation (16), which includes geometry spreading 1= ffiffi r and material damping e jimr. The damping attenuation coefficient j Im can be determined as 1.6 Np/m. Following the same method, experimental damping attenuation coefficients of A and S modes at various frequencies along the and 9 directions can be determined, as shown in Figure 11. Because of the tuning effect for the out-of-plane velocity response of S mode, measurements could be made only at the frequency range from 21 to 3 khz in the direction and from 12 to 3 khz in the 9 direction. Figure 11(a) and (b) shows the measured damping attenuation coefficients of A and S modes in the direction, respectively. It can be observed that the damping attenuation of fundamental modes (A, S) increases with frequency, which is in agreement with the predictions of the Kelvin Voigt damping model. Besides, experimental results indicate that A mode experiences a much higher level of attenuation than S mode. Figure 11(c) and (d) presents the determined damping attenuation coefficients of A and S modes in the 9 direction, respectively. Similarly, the damping attenuation of A and S modes increases with frequency and A mode shows much larger attenuation. Comparing the damping attenuation in the and 9 directions, it can be noted that the damping attenuation is anisotropic, showing higher attenuation in the matrix dominated direction (9 direction). In Figure 11, the measured damping attenuation of A and S modes were fitted using the polynomial curve-fitting

11 Mei and Giurgiutiu 11 Figure 1. Experimental measurements of damping attenuation: (a) experimental received signal at different distances in the direction and (b) curve fitting of the peak-to-peak magnitude versus the distance (A = 4.5,j I:6). function of MATLAB. The fitted experimental damping attenuation is used to compare with the predicted results of our damping model in section Loss coefficient adjustment for the prediction of damped guided wave propagation. To illustrate the anisotropic damping attenuation, A mode directivity plot of damping attenuation coefficient is shown in Figure 12. It can be observed that material damping is anisotropic and it has the highest influence on the 9 direction because, in this direction, a major contribution to composite stiffness is due to the matrix. Validation of guided wave excitation and propagation model for damped composite plates In order to validate the simulation, the proposed SAFE approach is used to predict the time-domain response signals measured on the CFRP composite plate and compare them with the SLDV area scan experiment. First, the stiffness properties are evaluated using ultrasonic immersion technique. 43 Next, dispersion curves in the composite plate were calculated using our SAFE formulation. Then, the SAFE approach is utilized to model the guided wave excitation and propagation in the composite plate under various damping assumptions and loss coefficient adjustment was performed to obtain a good match between experiment and prediction. Material properties measurement. The material properties of our in-house composite plate were measured experimentally using the ultrasonic immersion technique proposed in Barazanchy et al. 43 The retrieved storage matrix are given as :3 :2 :2 :2 15:3 :9 C :2 :9 15:3 = 3: GPa, 6 4 5: 5 5: r = 164kg=m 3 23Þ To validate the retrieved storage matrix, we used a comparison between the experimental frequency wavenumber dispersion curve and the dispersion curve predicted using the retrieved material properties. In this experiment, the time-space guided wave data was obtained from SLDV line scan. To obtain the frequency wavenumber dispersion curves, the measured timespace data u(t, x) was transformed into the frequencywavenumber domain by applying a 2D fast Fourier transform (FFT) 44 Uf, jþ= i jx 2pft ut, xþe Þ dtdx 24Þ

12 12 Structural Health Monitoring () Figure 11. Experimental damping attenuation coefficients versus frequency of A and S modes: (a) A mode, ; (b) S mode, ; (c) A mode, 9 ; and (d) S mode, 9. where U(f, j) is the resulting frequency-wavenumber representation in terms of the frequency variable f and the wavenumber variable j. The theoretical frequency wavenumber dispersion curve was calculated using the measured material properties and our SAFE formulation. Comparisons between the experimental and predicted frequency wavenumber dispersion curves in the and 9 directions are shown in Figure 13(a) and (b), respectively. Because of the tuning effect for the out-of-plane velocity response of S mode, measurements could be made only in the 9 direction. The red and black lines are the theoretical frequency wavenumber dispersion curves for A and S modes, respectively. A good match between the experiment and the prediction of measured material properties can be observed in both the directions. In addition, a larger A wavenumber in the 9 direction and a smaller one in the direction due to the influence of anisotropy can be observed. Prediction of dispersion curves in the composite plate. In the SAFE model, we use the 1D quadratic isoparametric element, which comprises three nodes per element. To

13 Mei and Giurgiutiu 13 Figure 12. A mode directivity plot of damping attenuation coefficient at 9 khz. ensure the convergence of the SAFE model, dispersion curves in terms of wavenumbers have been computed for various discretizations of the composite plate. Convergence is achieved when the dispersion curves do not change with an increasing number of elements. In this study, four elements per layer are used, which can ensure the convergence. Energy velocity dispersion curves of the CFRP composite plate were calculated using the SAFE approach (Figure 14). Figure 14(a) presents the energy velocity curves in the direction (along the fiber direction). It can be noted that at relatively low frequency, only three fundamental wave modes (S, SH, and A) exist. The S mode is not very dispersive at the frequency below 45 khz and has a large energy velocity. The A mode, on the other hand, is highly dispersive and has the lowest energy velocity. The SH mode is non-dispersive in this propagation direction and possesses an energy velocity between A and S. Figure 14(b) shows the energy velocity curves in the 9 direction. It should be noted that, compared with the direction, S mode becomes more dispersive and has a much lower energy velocity. The SH energy velocity does not change much, while the A energy velocity becomes much lower. Figure 15 shows the energy velocity directivity plot at 9 khz. The curves represent the spatial wave propagation pattern. It can be observed that both S and A have the highest energy velocity along the fiber direction at and 18, whereas in the transverse direction at 9 and 2 their energy velocities are much lower. Thus, if the waves are excited by a point source, then an elliptical wave crest would be obtained. The SH energy velocity curve shows the self-crossing behavior already reported in Glushkov. 45 Loss coefficient adjustment for the prediction of damped guided wave propagation. In order to achieve a good match between prediction and experiment, we need to determine the loss coefficient matrix h of equation (2). This is a challenging task. The available literature data for CFRP composite materials is limited. In this study, Figure 13. Comparison between experimental and predicted frequency wavenumber dispersion curves: (a) direction and (b) 9 direction.

14 14 Structural Health Monitoring () Figure 14. Energy velocity dispersion curves in the CFRP composite plate: (a) (fiber direction) and (b) 9 (transverse direction). Figure 15. Energy velocity directivity curves at 9 khz in the composite plate. the loss coefficients were obtained through an iterative updating in coordination with the experimental measurements of damping attenuation. To illustrate the effect of material damping on wave propagation patterns, Figure 16 presents the comparison between the SAFE prediction with and without damping at 9 khz. It shows that both predictions can depict the outward spreading wave propagation pattern with spatial dispersion. However, a difference appears in the 9 direction, where the wave amplitude is much more diminished due to material damping. The radial propagation comparison of waveforms at 1 mm away from PWAS along the and 9 directions is presented in Figure 1; the 9-kHz simulation signals with and without damping effect are plotted. Comparing the wave amplitude change in the and 9 directions, it can be noted that the damping attenuation is much stronger in the 9 direction than in the direction. In addition, it can be observed that A mode experiences higher attenuation than S mode from the comparison in the 9 direction, which is in agreement with the trends of the experimental damping attenuation in section Experimental measurements of damping attenuation. This is related to the motion characteristics of the respective mode shapes and corresponding wavelengths. To capture the anisotropic damping effect in the composite plate, loss coefficient adjustment was performed. This was done by comparing the attenuation predicted at various frequencies for the A and S modes with the experimentally measured attenuation. Figure 18 shows the damping attenuation comparison between prediction of Kelvin Voigt damping model with the obtained loss coefficient and the experimental results in the and 9 directions. The red line represents the curve fitting of the experimental damping attenuation obtained in section Experimental measurements of damping attenuation, while the blue cross markers indicate the values predicted by our model. It can be noted that the predicted damping attenuation of A mode matches well with the experiment, whereas the S mode shows some discrepancies between the prediction and the experiment. To obtain the results shown in Figure 18, we had to iteratively adjust the loss coefficient matrix of equation

15 Mei and Giurgiutiu 15 Figure 16. Predicted wave propagation pattern in the CFRP composite plate: (a) SAFE prediction without damping and (b) SAFE prediction with damping. Figure 1. Waveform comparison between SAFE prediction with and without damping effect: (a) (fiber direction) and (b) 9 (transverse direction). (2). This adjustment was done through iterative updating of the loss coefficient components in equation (2) while trying to best match the fitted experimental data. Since the material in a unidirectional composite plate is transversely isotropic, the loss coefficient matrix contains only diagonal terms made up of four independent values (h11, h22 = h33, h44, and h55 = h66 ). In the iterative updating process, these loss coefficients were initially taken as 1% (recall that loss coefficients in stiff solids are usually small,hpq\5%46). These values were then iteratively adjusted to bring the predicted damping attenuation closer to the experimentally measured damping attenuation (Figure 18). This process was done manually. (An automated optimization process could be devised in the future, but this did not make the object of present investigation.) Eventually, we found the most appropriate values for the Kelvin Voigt loss coefficient matrix h as 2 :3% 6 2:4% h= f = 25 khz 2:4% 3:1% 3, 5 2:5% 2:5% 25Þ

16 16 Structural Health Monitoring () Figure 18. Damping attenuation comparison between prediction and experimental results: (a) A mode, ; (b) S mode, ; (c) A mode, 9 ; and (d) S mode, 9. Note that h 22 and h 33 are equal since the material is transversely isotropic. These two elements are also much larger than h 11 because the damping in the transverse directions x 2 and x 3 is dominated by the polymeric matrix, which is viscoelastic, whereas the material behavior in the x 1 direction is dominated by the fibers, which are much stiffer and much less lossy. Summary of model-experiment comparison. To demonstrate the improvement of our extended SAFE approach, comparison between experiment and SAFE prediction with damping effect is presented in Figure 19. Kelvin Voigt damping model using the obtained loss coefficient matrix was utilized to capture the anisotropic damping effect. Wave propagation patterns of 9 khz in the composite plate were plotted. A good match between experiment and predicted wave pattern can be observed. The elliptical wave propagation patterns match quite well, though some small discrepancies still exist along the fiber direction ( and 18 ). This may be due to thickness variation in the composite plate. The waveform comparison between experiment and prediction at 1 mm away from PWAS in the and 9 directions is shown in Figure 2. In the direction, along the fiber direction, strong A mode can be observed, while the wave amplitude of S mode is very

17 Mei and Giurgiutiu 1 Figure 19. Wave propagation patterns in the CFRP composite plate: (a) experiment and (b) SAFE prediction with damping effect. Figure 2. Waveform comparison between experiment and prediction with damping effect in the composite plate: (a) (fiber direction) and (b) 9 (transverse direction). small. On the other hand, in the 9 direction, much stronger S amplitude was measured. The S and A wave packets are clearly separated. Predicted waveforms by the SAFE approach with Kelvin Voigt damping model achieved good agreement with the experimental measurements. This shows that the extended SAFE approach can successfully capture the anisotropic damping effects in the composite plate. Summary, conclusion, and future work Summary This article presented a SAFE approach to the simulation of guided wave excitation and propagation in damped composite plates. First, the mathematical framework of our SAFE approach was presented. Next, guided wave attenuation in damped composite plates, including attenuation mechanisms, damping models, and experimental methods for measuring the damping attenuation, were discussed. Two different damping models, hysteretic and Kelvin Voigt damping models, were considered. The effect of damping models on the propagation of various guided wave modes was investigated. The code verification was done against literature data. For the hysteretic damping model, the available literature results were compared with our results and good agreement was found. However, no literature results were available for the Kelvin Voigt damping model.

18 18 Structural Health Monitoring () Model validation was done using experimental data. Experiments were performed on a unidirectional CFRP composite plate using SLDV measurements of guided wave propagation generated by a circular PWAS transducer. The attenuation of A and S modes due to material damping was determined experimentally at various frequencies and then curve-fitted. In a separate experiment, elastic properties of the CFRP plate were determined with ultrasonic immersion technique using a small coupon cut from the plate. Then the loss coefficient matrix of our SAFE model was obtained through iterative updating to achieve an acceptable match with the measured damping attenuation. It was found that the Kelvin Voigt damping model gave a better match than the hysteretic damping model. Finally, our SAFE approach using Kelvin Voigt damping model was utilized to model guided wave excitation and propagation in the CFRP plate. The predicted waveforms were compared with SLDV measurements. A good match between prediction and experiment was obtained. Conclusion Our extended SAFE approach for modeling guided wave excitation and propagation in damped composite plates achieved very good agreement with the experimental measurements. This is an important improvement over the conventional simulation of guided wave propagation in composite plates using an undamped SAFE approach. Experimental measurements demonstrated that the damping attenuation has a significant effect on amplitude attenuation of guided waves propagating in composite plates. It was found that the material damping in unidirectional composites is highly anisotropic, showing higher damping effect in matrix dominated transverse directions than in the fiber directions. Moreover, the A mode experiences a much higher damping attenuation than the S mode. SHM techniques based on amplitude change can be understood better and implemented more effectively if the material damping effect is included in the model. Future work An immediate extension of this work would be in the simulation and experimental validation of guided wave excitation and propagation in composite plates with different layups. It is expected that the layup sequence would strongly affect the damping attenuation effects on various guided wave modes. For thick composites, the higher modes other than the fundamental A and S should be also considered. A further extension of this work would be in detecting composite damage. The damages in composites are more critical than those in metallic materials. Various damage modes exist (e.g. delamination, fiber fracture, and matrix cracking) and their detection and characterization are rather difficult. It is desirable that the proposed approach should be extended to capture the guided wave interaction with composite damage using the hybrid global local (HGL) concept, in which a local FEM discretization only in the damage regions is deployed to capture wave damage interaction coefficients while the global domain wave propagation is solved using the very efficient SAFE approach. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Aeronautics and Space Administration (NASA) Grant No. NNL15A A16 C and by the Air Force Office of Scientific Research (AFOSR) Grant No. FA ORCID id Hanfei Mei References 1. Su Z, Ye L and Lu Y. Guided Lamb waves for identification of damage in composite structures: a review. J Sound Vib 26; 295(3): Park B, An YK and Sohn H. Visualization of hidden delamination and debonding in composites through noncontact laser ultrasonic scanning. Compos Sci Technol 214; 1: Mitra M and Gopalakrishnan S. Guided wave based structural health monitoring: a review. Smart Mater Struct 216; 25: Yeager M, Todd M, Gregory W, et al. Assessment of embedded fiber Bragg gratings for structural health monitoring of composites. Struct Health Monit 21; 16(3): Harb MS and Yuan FG. Barely visible impact damage imaging using non-contact air-coupled transducer/laser Doppler vibrometer system. Struct Health Monit 21; 16(6): Yuan S, Ren Y, Qiu L, et al. A multi-response-based wireless impact monitoring network for aircraft composite structures. IEEE T Ind Electron 216; 63(12):

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