Lamb modes are commonly used for ultrasonic, nondestructive

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1 Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials Michel Castaings a) and Bernard Hosten b) Laboratoire de Mécanique Physique, Université Bordeaux 1, UMR CNRS , Cours de la Libération, Talence Cedex, France Received 20 February 2001; revised 1 October 2002; accepted 14 January 2003 The propagation of Lamb-like waves in sandwich plates made of anisotropic and viscoelastic material layers is studied. A semi-analytical model is described and used for predicting the dispersion curves phase velocity, energy velocity, and complex wave-number and the through-thickness distribution fields displacement, stress, and energy flow. Guided modes propagating along a test-sandwich plate are shown to be quite different than classical Lamb modes, because this structure does not have the mirror symmetry, contrary to most of composite material plates. Moreover, the viscoelastic material properties imply complex roots of the dispersion equation to be found that lead to connections between some of the dispersion curves, meaning that some of the modes get coupled together. Gradual variation from zero to nominal values of the imaginary parts of the viscoelastic moduli shows that the mode coupling depends on the level of material viscoelasticity, except for one particular case where this phenomenon exists whether the medium is viscoelastic or not. The model is used to quantify the sensitivity of both the dispersion curves and the through-thickness mode shapes to the level of material viscoelasticity, and to physically explain the mode-coupling phenomenon. Finite element software is also used to confirm results obtained for the purely elastic structure. Finally, experiments are made using ultrasonic, air-coupled transducers for generating and detecting guided modes in the test-sandwich structure. The mode-coupling phenomenon is then confirmed, and the potential of the air-coupled system for developing single-sided, contactless, NDT applications of such structures is discussed Acoustical Society of America. DOI: / PACS numbers: Mr DEC I. INTRODUCTION Lamb modes are commonly used for ultrasonic, nondestructive testing NDT of industrial structures since they allow fast and efficient control to be done. A wide range of publications can be found on the propagation of Lamb waves in various media such as plates or pipes made of either single or stratified, isotropic or anisotropic, elastic or viscoelastic material layers. A selection of papers dealing with plate modes is given in Refs These works help in understanding the behavior of Lamb waves, thus making it possible to optimize the choice of particular modes for detecting specific defects in tested plates. 7,8 Numerical predictions and/or experiments have demonstrated the potential of Lamb waves for detecting flaws However, among all the media of investigation, sandwichlike plates, made of two skins plus a thick core in between, have rarely been considered These types of structures are used more and more in aircraft design for their high mechanical performances and low weights. These assembles are usually made of anisotropic, viscoelastic materials with strongly different acoustic impedances. Moreover, skins do not necessarily have the same thickness, thus making the whole structure not have the mirror symmetry. The choice of Lamb waves for testing such structures has been motivated because of their high potential a Electronic mail: castaings@lmp.u-bordeaux.fr b Electronic mail: hosten@lmp.u-bordeaux.fr for quickly testing quite a long range of the specimen. However, before a NDT process gets set up, it is necessary to fully understand the propagation of Lamb-type modes in nonsymmetric sandwich plates made of viscoelastic materials. This is the purpose of this paper, which presents both numerical predictions and experiments carried out on a test sandwich plate made of two glass-epoxy composite skins of different thicknesses, plus a rigid foam core. A general model 2-D or 3-D model based on the wellknown transfer matrix method 20,21 has been developed for predicting the propagation of guided waves in stratified plates made of anisotropic and viscoelastic material layers, whatever their number and stacking sequence are. The current study is limited to a two-dimensional, plain strain problem since the plane of propagation is supposed to coincide with planes of symmetry of the anisotropic material layers constituting the test sandwich plate. The dispersion curves phase velocity, energy velocity and complex wave-number versus frequency and the through-thickness mode shapes displacement, stress and energy flow distributions are plotted, showing that guided waves in this sandwich structure are very different than classical Lamb modes. First of all, the mode shapes are neither symmetric nor antisymmetric with respect to the middle plane of the sample because this does not have mirror symmetry. Moreover, the viscoelastic material properties imply complex roots of the dispersion equation to be found that show coupling between some modes in specific zones of the dispersion curves. Gradual variation 2622 J. Acoust. Soc. Am. 113 (5), May /2003/113(5)/2622/13/$ Acoustical Society of America

2 FIG. 1. a Schematic of test sandwich plate; b coordinate axis used for defining the viscoelastic moduli of individual elements. from zero to nominal values of the imaginary parts of the viscoelastic moduli shows that the mode-coupling effect generally depends on the level of material viscoelasticity. Focus is made on a specific coupling zone, showing that dispersion curves for two modes get connected together under the effect of viscoelasticity, and that both strong attenuation and energy flow normal to the skin-foam interfaces are associated to this phenomenon. However, around a particular frequency, a mode coupling is shown to exist whether the medium is viscoelastic or not. In order to physically understand this observation, focus is made on this specific zone by investigating the complex wave-numbers, as well as the through-thickness energy flow distributions, and by comparing the analytical predictions to those obtained using finite element software 22 that models the propagation in structures made of elastic, anisotropic materials. This analysis showed the existence of a mode that is attenuated in the purely elastic plate, since it has a complex wave-number, but that does not carry energy along the plate. Experiments have also been made using ultrasonic, aircoupled transducers for generating and detecting guided modes in the test-sandwich structure. It has been found that some modes can be launched and received if the two transducers are used at one side of the specimen, and not at the other side, and vice versa. The predicted, nonsymmetric mode shapes of these particular waves explain this phenomenon. Measurements of the phase velocity have been made in order to confirm the mode-coupling phenomenon. Finally, the phase velocity and through-thickness displacement and stress distributions of some of the guided modes are shown to be not sensitive to the viscoelastic material properties of the sandwich plate, thus making the FE software appropriate for modeling the interaction of these modes with defects, for further studies. In this study, the test plate has been supplied by an industrial company. It is a sandwich structure made of two anisotropic viscoelastic material skins, separated by a rigid, isotropic, viscoelastic core, designed for a specific aerospace purpose. The choice of this structure for studying the modecoupling phenomenon was motivated by the NDT ultrasonicbased application which is to set up. However, from the scientific point of view, the plate does not necessarily have to be a sandwich medium, neither of the skins have to be made of anisotropic materials, for the mode-coupling phenomenon to be observed. This trend comes from an attenuation mechanism due to viscoelastic losses that makes the wave-number complex so that the otherwise hidden complex branches become unmasked and exhibit their influence. Similar results have been observed for a single layer made of a highly viscoelastic material. 23 One originality of the present paper compared to these previous works is the low level of viscoelasticity required for the mode-coupling to be observed in the case of the sandwich plate. Even when this is modeled as an elastic medium, the mode-coupling exists, thus meaning that the geometry of the sandwich structure is an essential cause of this phenomenon. Another originality is the focus on the physical behavior of the guided waves when the mode-coupling occurs, by investigating the variations with the level of viscoelasticity of the phase velocity, of the attenuation and also of the power flow transfers through the internal interfaces of the sandwich plate. II. DESIGN OF SANDWICH STRUCTURE The test-sandwich plate used in this study is made of a 1.3-mm-thick skin, a 7.4-mm-thick core, and a 4-mm-thick skin, so it does not have mirror symmetry, as shown in Fig. 1 a. The two skins are made of 2D woven, glass fibers and epoxy matrix plies, and the core is made of a rigid foam. These materials are supposed to be quadratic and isotropic, respectively. The stiffness tensor that links stresses to strains according to Hooke s law 24 is represented by the following matrix, using the usual contracted notation for the indices: C12 C C 22 C C C ij C C symmetric - - C C 66. To describe the viscoelastic properties of these materials, the C ij matrix is complex. Its elements are noted C ij C ij ic ij, where C ij represents the material stiffness, and C ij the material viscosity. They are defined in the coordinate axis shown in Fig. 1 b. Seven complex moduli have been measured on a 3.1-mm-thick glass-epoxy material plate, immersed in water, using an ultrasonic, through-transmission technique described in Ref. 25. Two complex moduli have been measured on an 8.2-mm-thick foam material plate, placed in air, using an ultrasonic, through-transmission technique described in Ref. 26. Extra measurements using conventional, contact transducers have also been made to estimate values of C ij, which could not be identified with the water-immersion or air-coupled techniques. The whole set of C ij thus obtained confirms that the glass-epoxy material possesses the quadratic symmetry plane P 12 identical to plane J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials 2623

3 TABLE I. Measured thickness in mm, density in kg/m 3 and viscoelastic moduli in GPa for the glass epoxy and foam materials constituting the sandwich structure. Material Thickness Density C 11 C 22 C 12 C 66 Glass epoxy i i i i0.14 Foam i i0.01 P 13 ) and that the foam is an isotropic material. Some of the measured C ij will be used as input data for the numerical models described in the next section. These are given in Table I. III. NUMERICAL MODELS A. Semi-analytical model The model used for predicting the dispersion curves is based on the transfer matrix method 20,21 that was initially intended for modeling wave propagation through stratified media. This method has already been adapted by Hosten and Castaings 27 for modeling heterogeneous plane waves through stratified plates made of anisotropic and viscoelastic layers, and for predicting the reflection and transmission coefficients of an incident plane wave. The purpose of the current project is to extend this model for simulating the propagation of leaky, guided modes Lamb, SH, Rayleigh,... along any direction of stratified plates made of anisotropic, viscoelastic materials. This work has begun with consideration of propagation of Lamb and SH modes along homogeneous anisotropic material plates. 6 In this paper, restriction is made to the case of Lamb waves propagating in planes of symmetry of the material layers. The direction of propagation chosen here is direction x 2, so the plane of propagation is plane P 12 formed by the pair of axis (x 1,x 2 ). According to the measured values of C ij, for the skins and for the foam, this plane is identical to plane P 13 formed by the pair of axis (x 1,x 3 ). Four complex C ij are therefore necessary for modeling the propagation along the skins (C 11, C 22, C 12, and C 66 ) and two for the core (C 11 and C 66 ). These are given in Table I. From a general point of view, the stratified plate is made of n solid layers. A transfer matrix that links the displacement and stress components from one surface of each layer j (1 j n) to the other is written according to the following relation: one quasi-longitudinal QL wave and one quasi-shear QT wave, propagating downward d and upward u in the layer. The two components of the total displacement vector, at any position (x 1,x 2 ) in the plane of propagation, and at a given time t, are the sum of the displacement components of each partial wave, so written as follows: where u A m P m e i( t km "x) is the displacement vector corresponding to each partial heterogeneous plane wave, using the following notations: A m is the complex amplitude, P m (P m 1,P m 2 ) is the polarization vector, is circular frequency, k m (k m 1,k m 2 ) is the complex wave vector, x (x 1,x 2 ) is the spatial coordinate vector, and m is the type of plane wave (m 1, 2, 3, or 4 for modes QL d, QL u, QT d or QT u, respectively. The stress components are then obtained from the generalized Hooke s law. 24 According to the boundary conditions at the internal solid solid interfaces of the plate, i.e., the continuity of displacements and stresses, the transfer matrix, noted A, of the whole plate is then ob- 3 2 using the usual contracted notation for the indices, 24 i.e., 1 11 and The routine for calculating the elements A pq (p,q 1,2,3,4) is given in Ref. 27. The displacement and stress components are defined in the coordinate axis shown in Fig. 2. This figure also illustrates that the whole displacement and stress fields in each layer result from the superposition of four bulk heterogeneous plane waves: FIG. 2. Schematic of the superposition of partial plane waves in each layer forming guided modes along the stratified plate J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials

4 tained by multiplying together all individual transfer matrices A j : 1 A j n A j. This matrix links the displacement and stress components at one surface of the plate top to those at the other surface bottom. In the actual study, the plates are always placed in air, which can be considered as a nonviscous fluid. Therefore, the boundary conditions at the two surfaces of the plate imply the normal displacement component u 1 and the stress components 1 and 6 to be continuous. Moreover, since the acoustic impedance of the air is negligible compared to that of the tested material plates, 28 a simplification that considers that leakage in air is equivalent to that in vacuum, i.e., null, can be made. These simplified boundary conditions lead to the following system of equations: The resolution of this system leads to the dispersion equation, which depends on the material layers properties density, thickness, orientation and complex elastic moduli and two variables, f and k 2, where f is the frequency, and k 2 k 2 ik 2 is the complex wave-number of a guided mode propagating along the direction x 2 : 4 5 V e h/2 h/2 Pdx 1 h/2 T 0 Edx 1 dt, 7 h/2 where P is the time-averaged Poynting vector defined by Eq. 8c, E is the total energy density in the plate, T is the temporal period and h is the plate thickness. However, if the materials are viscoelastic and/or the plate is coupled to a fluid, the definition of the group velocity is shown to have no physical meaning. Experimental data indicated that the use of the above energy velocity definition is valid for such situations. 23 This remark is of importance here since the sandwich plate is made of viscoelastic materials. Therefore, the component along direction x 2 of the energy velocity vector will be plotted later for the test-sandwich structure, whether the viscoelasticity is considered or not. Solutions ( f,k 2 ) are also used for plotting the throughthickness displacement, stress and energy flow distributions, at a given frequency, i.e., components U 1 (x 1 ) and U 2 (x 1 )of the displacement vector given by Eq. 8a, components 1 (x 1 ), 2 (x 1 ) and 6 (x 1 ) of the stress vector given by Eq. 8b, and components P 1 (x 1 ) and P 2 (x 1 ) of the Poynting vector given by Eq. 8c. This latest quantity represents the average power flow carried by the mode over a temporal period, at each position through the plate thickness, and for a guide supposed to have a unit width in the direction normal to the plane of propagation, i.e., along x 3, where is the stress tensor and v u/ t is the particle velocity in the plane of propagation * is the complex conjugate symbol : 8a 6 Solutions to this equation are couples, noted ( f,k 2 ). They are obtained using a numerical process based on the dichotomy method to find initial solutions at specific frequencies, plus the Newton Raphson method used to follow these roots as the frequency changes. The real part, k 2,ofk 2, is related to the phase velocity of the mode by the classical relation V ph 2 f /k 2, while the opposite of the imaginary part, i.e., k 2, represents its attenuation. It is important to note that stable solutions cannot be obtained for large frequency-thickness products, due to the well-known numerical instabilities of the transfer matrix method. 29 To avoid this problem, the transfer matrices in 2 and 4 are decomposed in 2 by 2 subdeterminants using the Delta operator method. 30 The dispersion equation in 6 is recognized as a2by2subdeterminant of the global transfer matrix A, which is then always accurately calculated. Thus, sought solutions ( f,k 2 ) can be reliably obtained. These solutions are used in turn for plotting the dispersion curves, i.e., the phase velocity, the energy velocity, the real wave-number and/or the attenuation versus frequency. In a recent publication, 23 it is shown that for nonattenuated waves propagating in elastic media, the group velocity V g / k is identical to the component along the plate of the energy velocity vector defined by 8b 8c The displacement and stress vectors expressed in Eq. 8 are actually normalized by the power flow carried by the mode through the whole plate thickness along direction x 2 ), for a unit width of the guide in direction x 3, and averaged over a temporal period. The Poynting vector is normalized by the square of this flow. This insures comparison to be possible between the various displacement, stress or energy fields of J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials 2625

5 FIG. 3. Schematic of the finite element modeling of the sandwich plate. any modes and for any frequencies. This normalization factor is expressed by Eq. 9 : P n h/2 h/2 P"x2 dx 1. 9 This model takes into account the attenuation of guided modes due to the anisotropic viscoelastic properties of the material layers. The choice of number of layers and stacking sequence is very flexible and the plate can either have mirror symmetry or not. The model can also consider propagation out of planes of symmetry and/or leakage in a surrounding perfect fluid, but this is not the purpose of the present paper. It is considered as a semi-analytical model since as many as possible closed form solutions are expressed, like for example the elements of the transfer matrices, and the throughthickness fields. However, solutions like the roots of the dispersion equation require numerical packages to be used. B. Finite element model This section describes a numerical analysis package, which was developed at Imperial College in London, UK, to model the propagation of plate, waves and their interaction with defects. 22 Simulated plates are considered as free plates i.e., placed in vacuum, and can be made of one or several different layers made of either isotropic or anisotropic elastic materials. This tool is based on the finite element FE method and includes an explicit central difference routine for producing a time marching solution. Therefore it is possible to vary the characteristics of the exciting temporal signal, i.e., the center frequency, the number of cycles and the envelope. The excitation can be produced at any point of the mesh, as displacements or forces. The response of the plate to various types of excitation is modeled by calculating the displacements at every point of the spatial mesh that defines the plate, as a function of time. Specific points of the mesh can be monitored, thus showing the time response at particular locations in/on the plate. Monitoring a series of points along the plate is usually done for processing a twodimensional Fourier transform which converts the timeposition, predicted data into a frequency-wave-number diagram. 31,32 Such a diagram is very useful since it allows the phase velocity and amplitude or decay in amplitude, i.e., attenuation of various modes propagating along the plate to be plotted versus the frequency, whether these modes are pure or superimposed. This processing will also be applied to our experimental data for measuring the phase velocities of modes produced in the test-sandwich plate. The FE code is also associated to specific interfaces used for modeling the excitation and/or the reception by finite transducers, the angle, the diameter, the spatial distribution and the position above the plate of which can vary. More details about these options can be found in Refs. 12 and 13. In the current study, the sandwich plate is modeled by three layers, which are meshed and joined together by their sides according to Fig. 3. Their thicknesses are 1.3, 7.4, and 4 mm for the first skin, the core and the second skin, respectively. They all have the same length of 500 mm, which is usually high enough for avoiding any reflection from the hand sides, at the monitoring points except for one particular case that will be mentioned later. The plate is considered as infinitely wide in the direction x 3 normal to the plane of propagation, i.e., plane strain condition is considered. The plane of propagation is formed by the axis x 1 and x 2. It coincides with a plane of symmetry for all layers. Therefore, the two skins properties are modeled by four real elastic moduli (C 11, C 22, C 12, and C 66 ) and the core properties by two real elastic moduli (C 11 and C 66 ), the values of which are given in Table I. The excitation of guided modes was produced in two main different ways. The first type of excitation was a through-thickness displacement distribution, applied at one end of the structure. This excitation was either a wide-frequency-band, uniform distribution for launching several guided modes, or a narrow-frequency-band, single mode-shape distribution for launching a pure mode. In both cases, monitoring series of points along the plate allowed two-dimensional Fourier transforms to be processed, so leading to sets of phase velocity versus frequency data that were compared to the dispersion curves obtained from the analytical solutions. The second type of excitation was simulating a finite air-coupled transducer placed at one surface of the specimen, as that used in the experiments see next section. This transmitter is modeled by the pressure distribution that it is supposed to locally apply at one plate surface, depending on its angular orientation and frequency excitation, which were changed according to the pure mode, which was suited to be launched. The incident beam was supposed to be collimated. 12 In-plane (x 2 direction and normal-to-plane (x 1 direction displacements were monitored at points located in the through-thickness of the plate. These were used for plotting the mode shapes at a given frequency, which were compared in turn to through-thickness displacement distributions predicted using Eq. 8a. The various options for forcing 2626 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials

6 FIG. 4. Experimental setup. and/or monitoring points of the FE mesh are presented on the schematic of Fig. 3. IV. EXPERIMENTS Air-coupled ultrasonic transducers are used for generating and detecting guided modes in the test-sandwich plate. These transducers are electrostatic elements with a 45-mm diameter, circular cross-section. Their frequency bandwidth is centered at 200 khz with 15 db points at 50 and 400 khz. More details can be found on such prototypes in Ref. 33. As shown in Fig. 4, one transmitter and one receiver are FIG. 5. Real solutions of analytical model lines compared to FE predictions for guided modes along the elastic sandwich plate; a positive and negative --- phase velocities and b energy velocities corresponding to positive phase velocities only. placed at one side of the specimen, and are oriented at angles optimised for selective coupling with guided modes that can propagate in the plate. The analytical model described in previous section predicts these angles. The receiver is moved away the transmitter, by a motorized translation stage, so varying the distance of propagation of the guided modes in order to measure a series of temporal signals visualized on a scope. As mentioned in the previous section, a twodimensional Fourier transform is applied to each set of experimental waveforms for quantifying the phase velocities of modes produced in the testsandwich plate. V. RESULTS A. Real solutions for elastic materials First of all, the simulation of guided modes propagating along the sandwich structure without taking into account the material viscoelastic properties is considered. This was first done by searching real solutions of Eq. 6 and by considering real elastic moduli as input data. Figure 5 presents the phase and energy velocities thus obtained. It is interesting to note that these dispersion curves are quite different than classical dispersion curves for Lamb modes. This is due to the fact that guided modes propagating in the test-sandwich plate are neither symmetric nor antisymmetric modes because the plate does not have mirror symmetry. Then, the FE software that models purely elastic materials was used. To launch several modes at the same time, a uniform, through-thickness, displacement distribution was applied in both directions, at the left-hand side of the meshed plate. The exciting signal was a two-cycle, 100-kHz center frequency, Hanningwindowed burst, having a frequency bandwidth comprised between 10 and 180 khz, at 20 db. Three series of 167 points were then monitored along 250-mm-long lines parallel to the plate, every 1.5 mm, at positions x 1 0, 8, and 12.7 mm, respectively, in the plate thickness. Therefore, the first series was at the thin skin surface, the second series was roughly in the middle of the core, and the last series was at the thick skin surface. The location of these monitored points to the right of the excitation was a way to model modes having positive energy velocities in the x 2 direction only. Processing two-dimensional Fourier transforms from these data led to a big set of phase velocity values in the frequency range of investigation. This technique allows modes with positive or negative phase velocities to be selected since the Fourier transform is defined the whole space, of J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials 2627

7 FIG. 6. Effect of material viscoelasticity rate on complex solutions lines of analytical model, plus real solutions, for mode C around 110 khz; a phase velocity and b imaginary part of wave-number. FIG. 7. Through-thickness distribution of unit-power Poynting vector for various viscoelasticity levels for mode C at 110 khz; a component in direction x 1 and b component in direction x 2. wave-numbers. The upper part of Fig. 5 a positive phase velocities shows that the FE predictions are in good agreement with the dispersion curves plotted from the real solutions of the dispersion equation. This correlation gives a general good confidence in both the analytical and the finite element codes, used for modeling guided waves in stratified plates made of elastic, anisotropic materials. However, if careful attention is paid on specific zones of these curves, some differences can be seen between the FE and the analytical predictions. For instance, real positive solutions of Eq. 6 show that the phase velocity of the lowest-order dilatational mode mode B is multiple-valued around 30 khz where it sharply drops from 3.2 mm/ s to about 1.0 mm/ s. This result is not confirmed by the set of positive phase velocities obtained from the FE plus 2D-FFT process, which supplies positive phase velocities decreasing from 3.2 to 2.6 mm/ s, up to 29.5 khz, no positive solutions between 29.5 khz and 31.2 khz, and then solutions around 1.0 mm/ s above 31.2 khz. However, if negative phase velocities are selected by the 2D-FFT technique applied to the FE results, then very good agreement with the dispersion curves plotted from negative real solutions of Eq. 6 is obtained, as shown by the lower part of Fig. 5 a. This result means that the branch between 29.5 and 31.2 khz corresponds to a different mode than those obtained below 29.5 khz and above 31.2 khz. Figure 5 b presents the energy velocity curves that correspond to positive phase velocities only. It is clear that where the positive phase velocities form a positive slope between 29.5 and 31.2 khz, the energy velocity is negative. From a physical point of view, this is equivalent to having a negative slope formed by negative phase velocities and a positive energy velocity, i.e., dots in the lower part of Fig. 5 a. The multiple-valued curve mode B plotted in the upper part of Fig. 5 a is then an artifact due to the resolution of the dispersion equation 6 in the real positive plane. This is very similar to the case of the s 1 mode that propagates along free, elastic, isotropic plates. Indeed, it is shown in Ref. 24 that the dispersion curve for this particular mode is actually composed of two branches of real positive wave-numbers, the energy velocity being positive for one branch and negative for the other one. This latest is actually identified as the s 2 mode propagating in the opposite direction as that of s 1. In fact, the connection of the lower real branch of positive energy velocity to the upper real branch of negative energy velocity is an artifact due to the resolution of the dispersion equation in the real plane. For the same reasons, similar results are obtained for mode D around 34.5 khz and mode F around 114 khz. B. Complex solutions for viscoelastic materials Complex solutions of Eq. 6 are now sought in order to quantify the effect of material viscoelasticity on wave propagation. The material viscoelastic rate is gradually increased by varying the imaginary parts of the viscoelastic moduli, from 0.001% to 100% of the nominal values given in Table I. From a numerical point of view, in the case of the lowest level of viscoelasticity, it is necessary to input C ij values not exactly equal to zero in order to ensure convergence on complex solutions. However, for such small values of the C ij, the sandwich plate can be considered as an elastic medium. From a general point of view, the viscoelasticity generates a mode coupling effect demonstrated by connections between some of the dispersion curves together. As an example, Fig. 6 shows the coupling of two modes, labeled D and F in Fig. 5, between 80 and 140 khz. As expected when the viscoelasticity level is negligible, i.e., when the C ij are equal to 0.001% of their nominal values, the dispersion curves phase velocity and attenuation are unchanged whether real or complex solutions are sought. When the C ij are equal to 10% of their nominal values, the phase velocity of the mode labeled F in Fig. 5 does no longer follow a vertical asymptote, but it drops off while its attenuation gets very high, and the mode labeled D in Fig. 5 has an unchanged phase velocity but a nonzero attenuation. In this situation, still two independent modes exist. However, as seen in Fig. 6 a, as soon as the C ij are equal to or greater than 25% of their nominal values, the two phase velocity curves of modes D and F get connected together, so that a single curve is obtained forming a squabble shape that is less pronounced as the viscoelasticity rate increases. Figure 6 b shows that this phenomenon is accompanied by a relatively strong attenuation that diminishes from to mm 1, i.e., from 4.7 to 2.1 db/cm, as the C ij increase from 25% to 100% of their nominal values, at 110 khz. However, below 104 khz and above 114 khz, i.e., on both sides of the frequency at which the mode coupling occurs, the attenuation gets stronger as the C ij values increase. The two components of the Poynting vector have also been plotted for this mode, using Eq. 8c, at frequency 110 khz, and for several rates of the viscoelasticity. Figure 7 a shows that the flow of energy in direction normal to the plate 2628 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials

8 FIG. 8. Dispersion curves for guided modes propagating along the sandwich plate; a phase velocity, b energy velocity, c attenuation; analytical model for elastic --- or viscoelastic and symbols plate and experiments. direction x 1 ) is null if the structure is made of elastic materials, i.e., the energy flow is strictly parallel to direction x 2 at any position through the plate thickness. However, as soon as the viscoelasticity level is high enough for the modecoupling to occur, i.e., for C ij equal to or greater than 25% of their nominal values, energy is transferred through the skinfoam interfaces. The amount of this normal power flow in direction x 1 ) tends to diminish as the viscoelasticity level increases. Simultaneously, it is observed that the mode gets less attenuated since the ratio of imaginary to real parts of the wave-number drops off by a factor of 2 when the C ij increase from 25% to 100% of their nominal values. Figure 7 b shows that the power flow in direction x 2 is essentially carried both by the thin skin and by the foam if the materials have a low level of viscoelasticity, and that it gets more concentrated in the thin skin as the viscoelastic material properties increase. Similar phenomena as those described above have been observed at other frequencies where mode coupling occurs. For example, as seen in Fig. 8 that presents dispersion curves up to 200 khz, both for the elastic plate dashed lines and for the full level of viscoelasticity plain lines, modes labeled B and C in Fig. 5 get coupled around 120 khz, modes A and B around 160 khz, modes C and D around 104 khz and around 183 khz, etc. It can therefore be concluded that the material viscoelastic properties produce, at specific frequencies, a mode coupling effect that implies connections between some of the dispersion curves, and that is accompanied by a strong attenuation, and by transfer of energy through the skin-core interfaces. The viscoelasticity also makes some parts of the real dispersion curves disappear, and crossing points between the new curves possible. New labels are used for the various modes since the dispersion curves are strongly different than those obtained from real solutions. For instance, the new mode formed by the connection of curves D and F of Fig. 5, between 80 and 140 khz, is labeled mode C. Non-null normal components in direction x 1 ) of the Poynting vector have already been observed for the mode s 3 propagating with high attenuation in a single, viscoelastic, HPPE material layer, at low frequency. 23 As shown in this reference, this mode has a phase velocity curve that exhibits connections between several dispersion curves obtained from real solutions, due to an attenuation mechanism coming from either the material viscoelasticity or leakage into a surrounding fluid. The phenomenon is therefore very similar to that studied in the present paper, except that in the case of our viscoelastic, free sandwich plate many more connections occur even for a very low rate of viscoelasticity. Indeed, when this rate is chosen equal to 25% of the nominal level, the attenuation coefficients L and T in Np/wavelength for the individual components of the sandwich plate are about two to ten times smaller than those given in Ref. 23. The geometry of the sandwich plate is therefore likely to be one reason for the mode-coupling occurrence, i.e., in favor of exchanges of energy through the internal interfaces. This will be confirmed later in Sec. V C. In the same way, it has been checked that no mode-coupling occurred in the case of the propagation along the individual components of the sandwich plate skin or core, although the nominal level of viscoelasticity was considered in the model. This means that the material viscoelasticity is not the only reason for the modecoupling to exist in the sandwich plate. Figure 8 a compares the phase velocity curves predicted for the elastic structure dashed lines and for the fully viscoelastic one plain lines to experimental data measured using the air-coupled transducers, as described in Sec. IV. In order to check whether these measurements confirm the mode-coupling phenomenon or not, three zoomed regions are presented in Fig. 9. Concerning the new mode C mentioned above, Fig. 9 c shows that the tendency of the experimental data agrees much more with the complex root solutions solid line than with the real ones dashed line. Figure 9 b also shows that the experimental phase velocities confirm the coupling between modes B and C of Fig. 5 around 120 khz, between modes A and B around 160 khz, and between modes C and D around 183 khz, since none of the measured data follow the dashed line representing the uncoupled modes. The mode-coupling phenomenon is therefore quite well confirmed by these measurements, thus indicating that the complex roots of the dispersion equation correspond to realistic solutions for guided waves propagating J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials 2629

9 FIG. 11. Effect of material viscoelasticity on transfer of energy for mode C; a through-thickness distribution of component in direction x 1 of unitpower Poynting vector at 33 khz and b component in direction x 2 of energy velocity around 33 khz. indicating that this phenomenon is likely to be associated to strong dispersion of the propagation. The attenuation is obtained from the imaginary part of the complex wave-number roots, converted from mm 1 to db/cm, as a more convenient unit for NDT applications. It is confirmed that the mode coupling is systematically associated to an increase of attenuation, as said before. The attenuation of the various modes supposed to propagate along the sandwich plate is often quite high, i.e., greater than 0.5 db/cm 50 db/m, even in the low-frequency range of investigation. Since aeronautic structures are likely to be about one or several meters long rather than of a few centimeters, a careful choice of guided modes with low attenuation will be needed in real applications. FIG. 9. Zooms on phase velocity dispersion curves presented in Fig. 8 a ; a zoom 1: mode C, khz, b zoom 2: modes A and B, khz, c zoom 3: mode C, khz, and symbols complex solutions of analytical model, --- real solutions of analytical model and experiments. along this viscoelastic sandwich plate. The energy velocity and the attenuation of these waves are also presented by Figs. 8 b and 8 c, respectively. Sharp variations of the energy velocity are visible for most of the coupling zones, thus FIG. 10. Effect of material viscoelasticity rate on complex solutions of analytical model compared to FE predictions, for mode C around 33 khz; a phase velocity and b imaginary part of wave-number. C. Particular attenuated mode An interesting particular case is now investigated. It corresponds to mode B in Fig. 5 that was previously considered because its real solutions were multiple-valued around 30 khz. Figure 10 a shows that the phase velocity curve plotted from complex roots of the dispersion equation presents a continuous coupling occurring between this lowest-order dilatational mode and a higher-order mode labeled mode D in Fig. 5. As shown in Fig. 10 b, this coupling is accompanied by a strong imaginary part of the wave-number. The interesting point here is that this mode coupling even exists if the viscoelasticity level is negligible, i.e., when the C ij values are equal to 0.001% of their nominal values, indicating that the mode is attenuated even if the medium is elastic. In order to quantify the effect of the material viscoelasticity on the attenuation of the mode, the ratio between the imaginary and real parts of its wave-number has been calculated for several level of viscoelasticity, at frequency 33 khz. This ratio decreases from about 39% to 23% when the C ij values increase from 0.001% to 100% of their nominal values. The Poynting vector and the energy velocity have been calculated for this mode, at frequency 33 khz, and for several rates of viscoelasticity. Figure 11 a that presents the component in direction x 1 of the Poynting vector shows that energy is transferred through the skin-core interfaces, and that the level of these exchanges gets stronger as the viscoelasticity diminishes. Figure 11 b shows that the energy velocity component in direction x 2 ) tends towards zero, around 33 khz where the mode-coupling occurs, when the material is almost 2630 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials

10 elastic. This means that the mode does not carry energy along the platelike complex modes already identified in elastic free layers. 24 When the material viscoelastic properties are increased, the energy velocity increases, thus meaning that the nature of the mode is changing since it carries energy along the viscoelastic plate. To confirm that this attenuated mode exists if the sandwich plate is purely elastic, a specific FE calculation has been run by applying its exact, through-thickness displacement distribution, at the left-hand side of the meshed sandwich plate. The two components U 1 and U 2 of this displacement distribution have been calculated by injecting in Eq. 8a the complex wave-number obtained for the lowest level of viscoelasticity, at 33 khz. The temporal excitation in the FE model was then a five-cycle toneburst having a center frequency of 33 khz, thus producing energy in the khz bandwidth at 30 db. In-plane displacements in direction x 2 ) were monitored at points located 2 mm away from the surface of the thick skin, i.e., where other modes which are supposed to exist at the same frequency produce a negligible in-plane displacement compared to that created by mode C. These points were placed from 16 to 200 mm away from the excited edge. The scale in time was extended up to 1.4 ms and the plate length was fixed equal to 2 m, so that even the slowest components of mode C were properly modeled without being overlapped in time by the fastest components that could be twice reflected from the plate edges. The amplitude of these displacements is not presented here but it definitively reduces by 96%, at 33 khz, as the position along the plate gets from 16 to 100 mm away from the input. This latest position precisely corresponds to one wavelength, at this frequency, thus confirming that this mode does not carry energy along the plate. Then, the two-dimensional Fourier processing mentioned in Sec. III B was applied to this set of displacements. Dots in Fig. 10 correspond to the FE results. The very good agreement obtained between the FE and analytical predictions, both for the phase velocity and the attenuation, validate the existence of the attenuated mode in the elastic sandwich plate. Finally, experimental data of the phase velocity are also presented in Fig. 8 a, and a zoom of the mode C around 33 khz is shown in Fig. 9 a. The strong attenuation of this mode and the small normal-to-plate displacement that it produces at the skin surfaces prevented two air-coupled transducers to be used for its generation/detection. In order to insure a high level of the energy input to this mode, a piezoelectric contact transmitter was coupled to one end of the sandwich plate. A ten-cycle, 33-kHz toneburst was then applied to this element, so that it created a fairly uniform through-thickness in-plane displacement that was suitable for launching the mode C with a relatively strong energy. The amplitude of the normal-to-plate displacement produced by this mode at the thin skin surface was then high enough for measurements to be made using an air-coupled receiver as those presented in Sec. IV. However, 100-time averaging was necessary for increasing the signal-to-noise ratio, so that processing was possible. This receiver was moved along the plate from 20 to 200 mm away from the emitter, with a 10-mm step. The set of signals captured for each position TABLE II. Configuration data used for modeling finite transmitter excitation producing a pure mode Mode Incident angle degrees Center frequency khz Frequency bandwidth khz at 20 db A B D were processed using the 2D-FFT technique. As seen in Fig. 9 a, the quality of the phase velocity measurements is a bit poor compared to data measured for other modes, which are much easier to generate-detect. However, the reproducibility of these measured phase velocities and the elbow shape they form show a reasonable agreement with the predicted results. D. Through-thickness displacement and stress fields The overall good results obtained up to this point confirm the efficiency of the numerical and experimental tools used in this study. The potential of air-coupled transducers for launching and/or detecting guided modes in sandwich structures has been shown, so they may be used for developing single-sided, contactless, NDT applications which are aimed in this project. The choice of a particular mode for detecting specific defects will depend on the ability of the transducers to generate/detect it, on its level of attenuation according to the size of the sample to be tested, and on its sensitivity to various types of defects holes, delaminations, disbonds,..., based on the through-thickness shape displacement and stress that it produces when propagating. 7,8 The FE software is an interesting feature for this latest point since it allows the interaction of guided modes with defects to be modeled. However, since the present model is restricted to simulate purely elastic materials, it is necessary to carefully select zones in the dispersion curves where the viscoleasticity has negligible effects. Figure 8 a clearly shows that some parts of the phase velocity curves remain unchanged when the viscoelastic properties of the materials are taken into account. This is the case for modes A and B at 100 khz, and mode D at 140 khz, for example. However, to fully guarantee that these modes are insignificantly affected by the viscoelastic properties, it is necessary to carefully look at the through-thickness, displacement and stress field distributions that they produce when propagating. In this purpose, these fields have been predicted using the analytical model with and without taking into account the material viscoelastic properties, for modes A and B at 100 khz, and for mode D at 140 khz. Then, the FE software has been used to predict the through-thickness displacements that are generated when the excitation is produced by a virtual, finite, air-coupled transmitter, oriented at an angle and excited in a local frequency region, together corresponding to one of the three previous specific zones of the dispersion curves where the viscoelasticity has little effect. For each run, the transmitter was simulated as a line strip source 2-D model of 30 mm long, and the excitation was a ten-cycle, hanning-windowed toneburst. Table II summarizes extra data specific to these various cases. J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials 2631

11 FIG. 12. Through-thickness unit-power displacement components a, c, e and unit-power stress components b, d, f without viscoelasticity --- and with viscoelasticity, plus FE data ; a and b mode A at frequency 100 khz, c and d mode B at frequency 100 khz and e and f mode D at frequency 140 khz. Displacements in both directions x 1 and x 2 were monitored at points distributed through the sandwich thickness, located 100 mm away the excitation area. These were normalized so that the normal component (U 1 ) at the thin skin surface be equal to that predicted by the analytical solution, which uses a unit-power normalization factor. In this way, the fields predicted by the two models have comparable order of magnitude. Figures 12 a f confirm that modes A and B at 100 khz, and mode D at 140 khz are very slightly sensitive to the viscoelastic properties, since the changes in the analytical predictions of both displacement and stress distributions are very small whether the viscoelasticity is taken into account or not. The FE through-thickness displacements are then in good agreement with both of these analytical predictions. Therefore, if one of these modes is considered as incident on a defect, then the FE model is likely to simulate properly the local interaction phenomenon, i.e., the amplitudes in the vicinity of the defect of the various modes produced by mode conversion. However, since such modes would probably have different attenuations, specific terms like e k m x 2, where k m is the attenuation of a given scattered mode m and x 2 is a distance away from the defect, would be needed for predicting the diffracted field at any position along the structure. This may be useful for positioning transducers when preparing NDT applications. Ideally, it would be more suitable to implement a viscoelasticity option in the FE software, so that no correction of the diffracted-mode amplitudes would have to be made. Moreover, such an option would allow the interaction of any incident mode with defects to be modeled, even modes having phase velocity and/or through-thickness shapes sensitive to the material viscoelasticity. This work is being done and should be efficient soon. Figures 12 b and d show that modes A and B propagating at frequency 100 khz are skin modes producing high levels of stress in the thin or thick skins, respectively. These kinds of modes are therefore suitable for detecting defects in the skins, which can be caused by external impacts. The experimental phase velocity data measured for these two modes see Fig. 8 a have been obtained by disposing the two air-coupled transducers either at the thin or thick skin side, respectively. This is due to the normal-to-plate displacement component U 1 which is continuous at the air-plate interfaces, and which is high at the thin skin side and small at the thick skin side for mode A, and vice versa for mode B. The attenuation is about 1.5 and 1 db/cm, around 100 khz, for modes A and B, respectively, thus making them suitable for short or average range testing, depending on the power of the transmitter. Figures 12 e and f show that mode D at frequency 140 khz is not a skin mode, but a plate mode producing roughly the same amount of stresses through the two thins than through the core. This kind of mode is therefore suitable for detecting disbondlike defects at the skin-core interfaces, but also defects in the core, like holes or local collapses of the foam. This mode has been experimentally generated and detected by disposing the two air-coupled transducers at the thin-skin side, where it produces a normal-to-plate displacement component U 1 that is higher than that at the other plate surface. With no averaging, the signal to noise ratio at the scope was about 2, thus making the use of two air-coupled transducers not suitable for efficient industrial tests of the skin-foam interfaces or of the foam core. It would be more judicious to produce such modes using another generation system like, for example, a contact PZT transmitter that could be permanently fixed in the sandwich plate, an aircoupled receiver being moved above the specimen for wave detection. Moreover, the very high attenuation level of this mode about 3.2 db/cm at 140 khz makes a high level of input energy required for testing real structures. VI. CONCLUSIONS A semi-analytical model based on the Thomson Haskell method 20,22 has been developed for predicting the dispersion curves and the through-thickness, displacement, stress and energy flow distributions of guided modes propagating along stratified plates made of anisotropic, viscoelastic material layers. Numerical predictions have been carried out for guided waves in a test-sandwich plate used in aircraft industry. Results show that guided modes very different than Lamb modes propagate in the sandwich specimen. Phase velocity measurements have been made using air-coupled transducers placed at the same side of the individual skin and core samples, thus aiming the contactless and single-sided 2632 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 M. Castaings and B. Hosten: Guided waves in viscoelastic materials