Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 3, pp. 807-823, Warsaw 2011 CHARACTERISTICS OF ELASTIC WAVE PROPAGATION IN THICK BEAMS – WHEN GUIDED WAVES PREVAIL? Haikuo Peng Shanghai Jiao Tong University, State Key Laboratory of Mechanical System and Vibration, Shanghai; and Shanghai Institute of Satellite Engineering, Shanghai, China e-mail: penghaikuo@gmail.com Lin Ye University of Sydney, Laboratory of Smart Materials and Structure (LSMS), School of Aerospace, Mechanical and Mechatronic Engineering, Australia Guang Meng, Kai Sun, Fucai Li Shanghai Jiao Tong University, State Key Laboratory of Mechanical System and Vibration, Shanghai, China Characteristics of wave propagation in thick beams are analyzed using a three-dimensional (3-D) spectral element method (SEM) for the purpose of damage detection. Analysis of wave propagation in beams of different thick- ness under excitationswith different central frequency reveals thatwhen the thickness of the beam is comparable to the wavelength of the elastic wave, a local wavemode, besides quasi-symmetric and quasi-anti-symmetricmodes, exist simultaneously in the beam. In particular, when the wavelength ismo- re than two times the beam thickness, the local wavemodes are suppressed and the wavemodes in the beam can be regarded as traditional guided wa- ves, i.e., Lambwaves. It is demonstrated that the central frequency of wave signals can be selected according to the dimensions of the beam to obtain simple wave modes like those in thin beams. The characteristics of wave propagation in an intact beam and beams with a lateral crack are analyzed and the results are also validated by experiments, where wave propagation signals in thick steel beams are activated and captured using PZT elements. Keywords:wavepropagation, spectral elementmethod, thick beam, damage detection 1. Introduction For structures in aerospace, civil and infrastructure applications, abrupt im- pact or growth of fatigue defects during service life can result in catastrophic failure. It is therefore essential to develop techniques of surveilling the integrity 808 H. Peng et al. of these structures and improving their safety and reliability. Various struc- tural health monitoring (SHM) methods have been developed (Inman et al., 2005; Staszewski et al., 2009; Su and Ye, 2009). Due to the long-range propa- gation potential of elastic waves and their sensitivity to a variety of damage, elastic-wave-based techniques for damage detection received a considerable at- tention in the past decades (Giurgiutiu, 2003; Kessler et al., 2002; Raghavan and Cesnik, 2007; Su et al., 2006). For elastic-wave-based damage detection techniques, it is essential to un- derstand the wave propagation characteristics in structures to be examined. A number of numerical methods have been applied to analyze the propa- gation of elastic waves in various types of structures (Lee and Staszewski, 2003a,b). The orthogonal polynomials (e.g., Legendre and Cheybyshev poly- nomials) -based SEM, proposed by Patera (1984), which take advantage of the accuracy of the spectral method and conserves the flexibility of FEM, are much more suitable for analyzing wave propagation in a medium with complex geometry than the conventional FEM. This method has been suc- cessfully applied in many areas, such as fluids (Canuto et al., 1988; Fischer et al., 2002; Karniadakis and Sherwin, 2005), seismology (Komatitsch et al., 2002; Komatitsch andVilotte, 1998; Seriani, 1998) and acoustics (Seriani and Priolo, 1994). In the past few years, the SEM was used to simulate wave propagation in structures for the purpose of damage detection. For example, wave propagation in 1-D structures such as rods (Sridar et al., 2006) and be- ams (Kudela et al., 2007b), 2-D structures such as isotropic (Zak et al., 2006) and composite plate (Kudela et al., 2007a) was investigated using SEM.Wave propagation in 2-D plate structures using three-dimensional SEM for damage detection was also discussed in Peng et al. (2009). PZT transducers were mo- deled using SEM to generate and receive Lambwaves (Ha andChang, 2010b; Kim et al., 2008; Kudela andOstachowicz, 2009) and the effect of the adhesi- ve layer between a PZT and a host structure was discussed (Ha and Chang, 2010a). Thick beams play important roles in 3-D structures. Unlike the beam structures that are commonly considered, where the wavelength of the ela- stic waves is much greater than the thickness of the beam, the cross-section size of thick beams is comparable to the wavelength. In this paper, thick be- ams are modeled using the Legendre-polynomials-based 3-D SEM and ela- stic wave propagation characteristics are analyzed. The characteristics of elastic wave propagation in beams with and without a crack are evalu- ated. Subsequently, the simulation results are validated with experimental results. Characteristics of elastic wave propagation... 809 2. Formulation of the spectral element method As in the conventional FEmethod, SEMfirst requires the domain Ω in three dimensions to be decomposed into a number of non-overlapping elements Ωe. Each element in the physical domain is mapped to a reference domain Λ = [−1,1]3 with local coordinates using an invertible local mapping func- tion f.Onedifference between the SEMandFEMis the distribution of nodes. A set of nodes is defined as ξi ∈ [−1,1], i =1, . . . ,N +1 in the domain Λ. In SEM, these Gauss-Lobatto-Legendre (GLL) points are the (N +1) roots of (Canuto et al., 1988) (1−ξ2)P ′N(ξ)= 0 (2.1) where P ′N(ξ) is the derivative of the Legendre polynomial of the degree N. Those nodes are irregularly distributed, while the nodes are uniformly spaced in FEM. In a 3-D spectral finite element, the basis function can be written as ψijk = hi(ξp)hj(ηq)hk(γr)= { 1 i = p, j = q, k = r 0 otherwise (2.2) hm(ξ) denotes the m-th1-DLagrange interpolation at the (N+1)GLLpoints. As an example, a 125-node spectral element in the local coordinate system is shown in Fig.1. Fig. 1. A 125-node spectral finite element in the local coordinate system Therefore, the integrals for the element matrices, Me, Ke and Fe, are calculated numerically in the local coordinate 810 H. Peng et al. M e = ρ n1 ∑ i=1 ωi n2 ∑ j=1 ωj n3 ∑ k=1 ωk[Ψ e(ξi,ηj,γk)] ⊤[Ψe(ξi,ηj,γk)]detJ ijk e K e = n1 ∑ i=1 ωi n2 ∑ j=1 ωj n3 ∑ k=1 ωk[B e(ξi,ηj,γk)] ⊤ C[Be(ξi,ηj,γk)]detJ ijk e (2.3) F e = n1 ∑ i=1 ωi n2 ∑ j=1 ωj n3 ∑ k=1 ωk[Ψ e(ξi,ηj,γk)] ⊤ P(ξi,ηj,γk)detJ ijk e where ρ is the mass density, C is the material stiffness matrix, and P is the distributed load. Ψe are the shape functions based on the Legendre polyno- mials. The matrix Be is the strain-displacement matrix. Je is the Jacobian matrix associated with themapping function f from the physical domain Ωe to the reference domain Λ. ωi is the weights of the GLL quadrature (Koma- titsch et al., 2000; Pozrikidis, 2005). ni, i = 1,2,3 are the numbers of GLL points in each direction in the local coordinate. If we let Ue denote the elementary vector of an unknown displacement in the medium for wave propagation, the ordinary dynamic equation can be written as (Pozrikidis, 2005) M eÜ e +KeUe =Fe (2.4) where Me denotes the elementarymassmatrix, Ke is the elementary stiffness matrix, and Fe is the vector of time-dependent excitation force. In a structural healthmonitoring system, the PZT transducer can be used as an actuator as well as a sensor. The converse piezoelectric effect of piezo- electricmaterials, which converts an externally applied electric field into indu- ced strain, and the direct piezoelectric effect, which generates an electric field under deformation, can be adopted for actuator and sensor correspondingly. According to constitutive equations for piezoelectricmaterials, the elementary governing equations of motion (2.4) can be rewritten as follows (Kudela and Ostachowicz, 2009; Wang, 2004) M e Ü e +KeUe−Keuφφ e =Fe (2.5) K e φuU e+Keφφ e =Qe The piezoelectric couplingmatrices Keuφ and K e φu and dielectric permitti- vity matrix Keφ can be calculated as Characteristics of elastic wave propagation... 811 K e uφ =(K e φu) ⊤ =− ∫ Ωe B ⊤e⊤BφdΩe = =− n1 ∑ i=1 ωi n2 ∑ j=1 ωj n3 ∑ k=1 ωk[B(ξi,ηj,γk)] ⊤e[Bφ(ξi,ηj,γk)]detJ ijk e (2.6) K e φ = ∫ Ωe B ⊤ φgBφdΩe = = n1 ∑ i=1 ωi n2 ∑ j=1 ωj n3 ∑ k=1 ωk[Bφ(ξi,ηj,γk)] ⊤g[Bφ(ξi,ηj,γk)]detJ ijk e where e is the piezoelectric constantmatrix; g the dielectric constantmatrix, Qe the nodal externally applied charge vector. According to the displacement and traction boundary on the surface of both the host structure and piezoelectric transducer as well as electric po- tential and charge boundary on the surface of the piezoelectric transducer, the elementary governing equations of motion can be solved using a central difference time integration scheme (Peng et al., 2009). 3. Wave propagation analysis in a thick beam The wave propagation characteristics in steel beams of 34mm (thickness) ×25mm (width) ×300mm (length), shown in Fig.2, are investigated. The steel density ρ is 7900kg/m3, Young’smodulus E is 200GPa, andPoisson ra- tio ν is 0.3. The beam ismodeled using 4 (thickness)×5 (width)×30 (length) elements with 5×5×5 nodes, as depicted in Fig.1. In this model, damping is not considered and the boundaries are free. Fig. 2. Geometric configuration of the steel beam [mm] 3.1. Elastic wave propagation in an intact thick beam First, an intact beam is excited by a single shear force applied at the point A on the upper surface at the left-hand end of the beam, as shown in Fig.2. A 812 H. Peng et al. Hanning-windowed3.5-cycle sinusoidal toneburstwith the central frequency of 35kHz is usedas the excitation signal.Thewaveformof the excitation signal in the time domain is shown in Fig.3. The displacement components of response in the beam in the xdirection at the times of 0.050ms, 0.079ms, 0.106ms and 0.141ms are shown in Fig.4. It can be seen that the elastic waves propagating in the beam experience repeated reflections at the upper and lower surfaces alternately, and then they are reflected by the right-hand end of the beam. Wave propagation in the mid-plane in the width direction is illustrated in Fig.5.When the beam is excited, themain longitudinal wave andmain shear wave with reflections between the upper and lower surfaces can be observed. Superpositionof their reflections and thepossible converted longitudinalwaves and shear waves constitute the formulation of quasi-symmetrical (quasi-S) wavemode associated with the longitudinal mode and quasi-anti-symmetrical (quasi-A) wave mode associated with the flexible mode in the thick beam. Fig. 3.Waveform of the excitation signal in the time domain Two points, named B and B′, which are symmetric with respect to the mid-plane in the thickness direction, as shown in Fig.2, are selected to inve- stigate wave propagation in the beam. The displacement components in the x direction at the points B and B′ are shown inFig.6. The incidentwave and the reflected wave from the right-hand end of the beam can be observed in the responses of B and B′. As in the thin beam, both symmetrical and anti- symmetrical modes can be excited simultaneously by the single shear force of the input signal. However, the amplitudes of the displacement at both points are neither exactly identical nor opposite, which is the reason why these wave modes are called quasi-Smode and quasi-Amode in the present study. Under the excitation with a central frequency of 35kHz, the group velocities of the quasi-Smode and quasi-Amode are about 5210m/s and 3068m/s, respective- ly, calculated using the arrival time (captured at point B) of the wave packets Characteristics of elastic wave propagation... 813 Fig. 4. Displacement component in the x direction of the intact beam at (a) 0.050ms; (b) 0.079ms; (c) 0.106ms; (d) 0.141ms Fig. 5. Reflections of elastic waves in the thick beam Fig. 6. Displacement components in the x direction of the intact beam at points B and B′ under the excitation of a single shear force 814 H. Peng et al. reflected from the right-hand end of the beam.As expected, the quasi-Smodes travel more quickly than the quasi-Amodes. In addition, the quasi-Amode is dominant in wave propagation, with its normalized magnitude (0.543) being clearly greater than that of quasi-S mode (0.148). 3.2. Effects of the excitation frequency and thickness In the beamwith a large thickness, the characteristics of wave propagation become more complex when the excitation frequency is increased. For exam- ple, an excitation with a central frequency of 56kHz is used. Under such an excitation, the local wave mode can be observed as shown in Fig.7. The re- ason for the occurrence of the local wavemode is that the elastic waves reflect directly from the upper and lower surfaces. In this case, the reflected waves of the local wave mode overlap with the reflected wave from the right-hand end of the beam.The response at point B in Fig.8 clearly shows the signals of the local wave mode immediately after the incident wave. Fig. 7. Displacement component in the x direction of the intact beam at 0.078ms under the excitation with a central frequency of 56kHz Fig. 8. Displacement component in the x direction at the point B under the excitation with a central frequency of 56kHz Characteristics of elastic wave propagation... 815 For the purpose of damage detection using an elastic wave, simple wave modes with a long range of propagation are desired. To obtain the simple modes of wave propagation in a thick beam, the excitation frequency should be carefully selected to avoid the local wave mode. To evaluate the local wavemode in detail, three beamswith thicknesses of 28mm, 34mm and 40mm are activated using a shear force with frequencies from 25kHz to 70kHz with a step of 5kHz. Three curves of the normalized magnitudeof the localwavemodeat thepoint B as a functionof the excitation frequency are plotted in Fig.9a. It is evident that the normalized magnitude of the local wave mode is small at low frequencies. With an increase in the excitation frequency, the magnitude increases significantly from a particular frequency for the beams of different thickness, indicating that the existence of the local wave mode is dependent on the excitation frequency f and the thickness of the beam h. When the normalized magnitudes of the local wave mode are plotted as a function of Lw/h, as shown in Fig.9b, it is interesting to see that the three curves merge into one, where Lw is calculated as Lw = cL f (3.1) where cL is thewave speedof shearwave in the steelmaterial and f is the exci- tation frequency. One essential observation here is that if the wavelength Lw, of the shear wave is about more than two times the thickness of the beam h, the normalizedmagnitude of the local wavemode becomes very small, indica- ting that the local wavemode are almost invisible, and in particular, a simple wave mode (i.e. Lamb waves) like those in thin beams can be obtained. Fig. 9. Normalizedmagnitude of the local wavemode as functions of (a) excitation frequency f and (b) L w /h According to the above conclusion, to prevent the occurrence of local wa- ve mode, for beams with a thickness of 28mm, 34mm and 40mm, when 816 H. Peng et al. Lw/h ­ 2, the corresponding excitation frequencies should be less than about 57kHz, 47kHz and 40kHz, respectively. It is worthmentioning that when the excitation frequency is greater than 64kHz for beams with a width of 25mm, the wavelength of the shear wave Lw < 50mm, which is less than two times thewidth, and a local wavemode symmetric about themid-plane in thewidth direction is also formulated because the elastic waves are also reflected by the two side surfaces of the beam, similar to the case in the thickness direction. 3.3. Wave propagation under the excitation of PZT actuator In a structural healthmonitoring system, thePZTtransducer is commonly used to generate and capture Lamb wave signals. In this study, two PZT transducers are modeled using the spectral element method. The dimensions of the PZT transducer (PIC151) are 20mm (length) ×5mm (width) ×1mm (thickness). One PZT transducer acts as an actuator and another one act as a senor, as show in Fig.10. An excitation signal of Hanning-windowed 3.5- cycle sinusoidal toneburst with themagnitude of 30V is applied to the upper surface of the PZT actuator. Displacement responses of the structure and PZT transducer in the x direction at 0.058ms are shown in Fig.11. A large displacement can be observed in two ends of the PZT actuator. Fig. 10. Schematic diagram of the coupled PZT-structure excitationmodel Fig. 11. Displacement response of the structure and PZT transducer in the x direction at 0.058ms Characteristics of elastic wave propagation... 817 Figure 12 shows the output voltage signal of sensor in the intact beam under the excitation of PZT actuator. Similar to the case under excitation of single shear force, there are threewave packets in the captured signal.Thefirst one is the incident wave; the second one is the quasi-Smode and the third one is the quasi-A mode. The magnitude of the quasi-A mode is obviously larger than the quasi-S mode. Fig. 12. Captured signal of the sensor in the intact beam under excitation of the PZT actuator with the central frequency of 35kHz As described before, when the excitation frequency is increased, a local wave mode will appear under the excitation of a point shear force. Under the excitation of the PZT actuator, when the central frequency of excitation is increased to 56kHz, the local wave mode can also be observed, as shown in Fig.13. The signal is very complex because of the local wave mode. Fig. 13. Captured signal of the sensor in the intact beam under excitation of the PZT actuator with the central frequency of 56kHz 818 H. Peng et al. 4. Elastic wave propagation in beams with a lateral crack Wave propagation in beams with a lateral crack located in the middle of the beam, shown in Fig.14, are investigated next. The degree of the damage is defined by d/h, where d is the depth of the crack and h is the thickness of the beam. Here, h = 34mm. Two cases with degrees of damage of 25% and 50% are considered. The crack in the beam is modeled by the nodes separation method, schematically shown in Fig.15. Initially, the structure is meshedwithout the crack.Thenodes exactly on the crackplane areduplicated and attached to the adjacent elements. Fig. 14. Crack in the beamwith degree of damage: (a) 25%; (b) 50% [mm] Fig. 15.Modeling of the crack: (a) mesh without crack; (b) mesh with crack Under the excitation with a central frequency of 35kHz, the captured si- gnals of the sensor are presented in Figs.16 and 17. The reflected signal from damage can be observed between the incident wave packet and the reflected wave packet from the right-hand end of the beam. It is mainly the reflection of the quasi-Amode for that the quasi-Amode dominates the wave propaga- tion under the excitation with the central frequency of 35kHz, as discussed previously. The normalized magnitude (0.25) of the wave reflected from the crack in the case of a 50% degree of damage is greater than that (0.20) for the 25% degree of damage. It is worth mentioning that the reflected wave of the Characteristics of elastic wave propagation... 819 quasi-S mode with a small magnitude from the right-hand end of the beam and the reflectedwave of the quasi-Amode from the crack overlap each other. Fig. 16. Captured signal of the sensor in the beamwith 25% damage degree under the excitation with a central frequency of 35kHz Fig. 17. Captured signal of the sensor in the beamwith 50% damage degree under the excitation with a central frequency of 35kHz 5. Experimental validation Three steel beams with dimensions shown in Fig.2 are used to validate the wave propagation characteristics in thick beams. One specimen is kept intact as a benchmark, and the other two are cut in the mid-span with a notch to simulate the crack. Thedepths of the notches are 8mmand 17mm, to simulate cracks of 25%and50%degrees of damage, respectively. Twopieces of thePZT (PI ROPIC151, PQYY-0586) element are surface-mounted on the upper surface 820 H. Peng et al. at the left-hand end of each specimen. One serves as the actuator to generate elastic waves while the other serves as the sensor to acquire the reflected wa- ve signals in the steel beams. The generation and acquisition of elastic wave signals are fulfilled using an active signal generation and data acquisition sys- tem developed on the VXI platform, consisting mainly of a signal generation unit (Agilent©E1441), signal amplifier (PiezoSys ROEPA-104), signal condi- tioner (Agilent©E3242A) and signal digitizer (Agilent©E1437A). 3.5-cycle sinusoidal tonebursts enclosed in a Hanningwindowwith amagnitude of 30V as used in simulation are generated and applied to the PZT actuator. Under theexcitationwithacentral frequencyof 35kHz, the captured signal in the experiments for intact beam is shown in Fig.12. The simulation results are in good agreement with the experimental results, in which both quasi- S mode and quasi-A mode can be observed. When the central frequency is increased to 56kHz, the captured signal is more complex, as shown in Fig.13, and it has a similar profile as the simulation signal, indicating the existence of the local wave mode. When there is a notch in the beam, the elastic wave is reflected from both the damage and the right-hand end of the beam, shown in Figs.16 and 17.When the degree of damage is 50%, the normalizedmagnitude of the reflection from the damage (0.31) is greater than that of the beam with degree of damage of 25% (0.15), as predicted by the simulation results, indicating that the magnitude of the reflected wave from crack can convey informationabout thedegree of damage in sucha3-D structure.Therefore, the effectiveness of the proposed SEMmodel is validated by the good agreement in general trends between the simulation and experiment results. 6. Conclusions Characteristics of wave propagation in thick beams are analyzed. It is shown that the characteristics of wave propagation in thick beams are complex, with the interaction between local wave modes, quasi-symmetric and quasi-anti- symmetric wave modes, when the thickness of the beam is comparable to the wavelength of the elastic wave. In particular, when the shearwavelength is two times the beam thickness, the local wave modes are suppressed and the wave modes in the beam can be regarded as traditional guided wave modes, i.e. Lamb waves. Therefore, the excitation frequency should be carefully selected when carrying out damage detection strategies for such thick beams.A lateral crack in thick beams is modeled using a node separation method, and the interaction of elastic wave with the lateral crack is analyzed. The simulation Characteristics of elastic wave propagation... 821 results ofwavepropagation in thickbeamsarevalidatedbyexperiments,where the wave propagation in a thick steel beam is activated and captured using PZT elements. Acknowledgements The authors are grateful for the support received from the National Natu- ral Science Foundation of China (NSFC No. 11072148, 11061160491), the National High Technology Research and Development Program of China (863 Program, No. 2009AA044800)andResearchProject of StateKeyLaboratory ofMechanical System andVibration (MSV201110). References 1. Canuto C., Quarteroni A., Hussaini M.Y., Zang T.A., 1988, Spectral Methods in Fluid Dynamics, Berlin: Springer 2. Fischer P., Kruse G., Loth F., 2002, Spectral element methods for trans- itional flows in complex geometries, Journal of Scientific Computing, 17, 1, 81-98 3. Giurgiutiu V., 2003, Lamb wave generation with piezoelectric wafer active sensors for structuralhealthmonitoring,Smart Structures andMaterials: Smart Structures and Integrated Systems, 5056, 111-122 4. HaS.,ChangF., 2010a,Adhesive interface layer effects inPZT-inducedLamb wave propagation, Smart Materials and Structures, 19, p.025006 5. 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Su Z., Ye L., Lu Y., 2006, Guided Lamb waves for identification of damage in composite structures: A review, Journal of Sound and Vibration, 295, 3/5, 753-780 28. Wang S., 2004, A finite element model for the static and dynamic analysis of a piezoelectric bimorph, International Journal of Solids and Structures,41, 15, 4075-4096 29. Zak A., Krawczuk M., Ostachowicza W., 2006, Propagation of in-plane waves in an isotropic panel with a crack,Finite Elements in Analysis and De- sign, 42, 11, 929-941 Charakterystyki propagacji fal sprężystych w belkach grubych – w jakich przypadkach fala prowadzona jest falą dominującą? Streszczenie Praca przedstawia problem detekcji uszkodzeń w belkach o znacznej grubości za pomocą trójwymiarowej metody elementów spektralnych (SEM) ze szczególnym uwzględnieniem charakterystyk propagacji fal w ośrodku. Analiza rozchodzenia się fal w belkach o różnej grubości poddanych wymuszeniom o różnej częstotliwości cen- tralnej ujawniła, że gdy grubość belki jest porównywalna z długością fali sprężystej, to oprócz quasi-symetrycznych i quasi-antysymetrycznychpostaci pojawia się lokalna postaćwłasna fali.Gdydługość fali przekracza conajmniej dwukrotnie grubośćbelki, postacie lokalne zanikają i postacie falmogązostaćuznane jako tradycyjne fale prowa- dzone, tj. fale Lamba. Pokazano, że centralna częstotliwość sygnałów falowychmoże zostać dobrana do konkretnych rozmiarów belki tak, aby otrzymać proste postacie własne fal przypominające kształt fal rozchodzących się w belkach cienkich.Wpracy zbadano charakterystyki propagacji fal w belce nieuszkodzonej i belce z pęknięciem poprzecznym. Rezultaty tych badań zweryfikowano doświadczalnie poprzez genero- wanie i rejestrację sygnałów w grubych stalowych belkach aktuatorami i czujnikami piezoelektrycznymi. Manuscript received March 7, 2011; accepted for print March 17, 2011