Accepted Manuscript. Numerical simulation of the guided Lamb wave propagation in particle reinforced

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1 Accepted Manuscript Numerical simulation of the guided Lamb wave propagation in particle reinforced composites Ralf Weber, Seyed Mohammad Hossein Hosseini, Ulrich Gabbert PII: S (12) DOI: Reference: COST 4621 To appear in: Composite Structures Please cite this article as: Weber, R., Hosseini, S.M.H., Gabbert, U., Numerical simulation of the guided Lamb wave propagation in particle reinforced composites, Composite Structures (2012), doi: j.compstruct This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Numerical simulation of the guided Lamb wave propagation in particle reinforced composites Ralf Weber (corresponding author, Phone: / , Mobile: / ), Seyed Mohammad Hossein Hosseini Ulrich Gabbert University of Magdeburg, Institute of Mechanics, Universitätsplatz 2, Magdeburg, Germany, Phone: / , Fax: / Keywords: Lamb wave, particle reinforced composite, homogenization, FEM, SAFE Abstract: This paper deals with the investigation of the Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. A three-dimensional finite element method (FEM) modeling approach is set up to perform parameter studies in order to better understand how the Lamb wave propagation in particle reinforced composite plates is affected by change of central frequency of excitation signal, volume fraction of particles, size of particles and stiffness to density ratio of particles. Furthermore, the influence of different arrangements is investigated. Finally, the results of simplified models using material data obtained from numerical homogenization are compared to the results of models with heterogeneous build-up. The results show that the Lamb wave propagation properties are mainly affected by the volume fraction and ratio of stiffness to density of particles, whereas the particle size does not affect the Lamb wave propagation in the considered range. As the contribution of the stiffer material increases, the group velocity and the wave length also increase while the energy transmission reduces. Simplified models based on homogenization technique enabled a tremendous drop in computational costs and show reasonable agreement in terms of group velocity and wave length. 1. INTRODUCTION 1.1 Lamb Wave based Structural Health Monitoring (SHM) Of the various SHM approaches available, the use of Lamb waves excited by thin piezoelectric patches is a very interesting technique due to its low cost, online monitoring and 1

3 high sensitivity [1]. Lamb waves are guided plate waves which remain confined inside the walls of a thin structure and travel long distances with only little energy loss and are therefore attractive for use in SHM applications. They can be excited and sensed by piezoelectric patches which are bonded to the surface of the structure or by embedded piezoelectric ceramics [2], [3], [4]. There are two basic varieties of the modes: symmetric mode and antisymmetric mode. Both modes are dispersive, i.e. frequency dependent. In the case of linear elastic, isotropic and homogenous material Rayleigh-Lamb-relation can be derived for the simultaneously occurring symmetric tan( pd) ( k q ) = 2 (1) tan( qd) 4k pq and antisymmetric mode: 2 tan( pd) 4k pq = 2 2 tan( qd) ( k q ) 2. (2) With the Rayleigh-Lamb relation dispersion curves for group velocity and phase velocity can be calculated. However, the Rayleigh-Lamb relation can be hardly derived for complex geometries and materials, therefore different numerical approaches, among them FEM based approaches are in use to study Lamb wave propagation in complex structures and materials. Such numerical studies revealed that the lamb wave propagation in structures like Honeycomb structures or heterogeneous materials like fiber-reinforced composites is a complex issue and an on-going concern [1], [5], [4]. 1.2 Particle reinforced Metal Matrix Composites (MMC) Particle reinforced composites consist usually of a lightweight metal alloy matrix such as aluminium or magnesium alloys and reinforcing particles made of ceramics like silicium 2

4 carbide (SiC) or metals such as titanium [6]. Due to their superior material properties and high wear resistance particle reinforced composites are used in many applications including disc brakes for cars, pistons and cylinder liners in combustion engines [6],[7]. The size of the particles in particle reinforced MMCs is normally between 10 and 25 micrometers and the volume fraction of the particles primarily varies between 5% and 30% [6], [7]. The geometry of the particles is complex and changes from particle to particle. However, for the purpose of simplicity in many studies on the properties of particle reinforced composites using computational methods the particles are often considered as spheres [8], [9]. Application of Lamb wave based SHM systems in heterogeneous materials and structures requires a fundamental knowledge of Lamb wave behavior in such materials [1], [10]. Knowing the properties of the Lamb wave which propagates in a specific material one can design more efficient SHM systems choosing the appropriate excitation signal [3], [5]. In addition, this basic knowledge is crucial for the signal processing to determine possible damages which can be detected by the propagating wave [11]. To address this issue the paper provides an allembracing, FEM based parametric study of Lamb wave propagation in particle reinforced composite plates. 2. MODELING AND SOLVER SETTINGS 2.1 Overview of various Particle reinforced Plates modeled Three different types of regular arrangements of spherical particles in plates based on crystallography arrangements have been studied in order to highlight the influence of particle arrangements on wave propagation. Body centered cubic (BCC), face centered cubic (FCC) and square arrangement are taken into account. Figure 1 depicts the representative volume elements (RVEs) of the various regular arrangements studied in this paper. Besides regular arrangements randomly distributed particles with spherical and cubic particles has been 3

5 considered. In the case of cubic formed particles, each particle is represented by one single finite element, cf. Figure Modeling Approaches A representative model is depicted in Figure 3. The simulations have been performed with the commercial FEM code Abaqus. For complex geometries like spherical particles and their surrounding second-order tetrahedral elements have been used. Models with cubic particles, piezoelectric transducers and homogenized models were meshed with 8-node brick elements with additional internal degrees of freedom. In order to study Lamb wave propagation the basic system of equations in linear elastic FEM needs to be solved with respect to time. Therefore lumped mass matrices and an explicit solver based on central difference scheme have been applied due to its superior properties in simulating high-speed events. Mesh tie constraints have been applied to ensure a perfect bond between the actuator and the plate as well as between sensors and the plate. In addition to avoid reflections of the Lamb waves from boundaries which disturb the signal received at the sensors, a damping area with gradually increasing damping factors has been installed. This approach has been proposed by Liu et al. [12]. In order to decrease the model size symmetry conditions are applied so that only a quarter of the volume of the plate has to be considered, cf. [1]. Due to the isotropic overall behavior of particle reinforced composites [8], this has also been done in the models with randomly distributed particles. The actuator is excited with a three and a half-cycle sinus band tone burst modulated with hanning window, cf. [1], [5]. The geometrical properties of the models used are summarized in Table 1 and illustrated in Figure 3. 4

6 The distance between actuator and sensor is 40 mm. All matrix or reinforcement materials are considered as linear elastic and isotropic. The material for the piezoelectric sensors and actuators is Lead zirconate titanate (PZT). The material properties of PZT are taken from [1] and refer to the coordinate system shown in Figure 3. Further material properties are summarized in Table SIGNAL POSTPROCESSING 3.1 Evaluation of group and phase velocity As only very thin plates are considered, the different modes are separated by simply adding (or subtracting) signals from the top and bottom at the same position of the plate and dividing by two. The group velocity of symmetric mode and antisymmetric mode has been calculated by determining the arrival time of the propagating wave package. Therefore the signals received from sensors have been evaluated. The arrival time is defined by a special threshold that the signal is obliged to exceed. Knowing the distance Δs between actuator and sensor, the group velocity will then be c g Δs = t arr, (3) where t arr is the arrival time. To identify the phase velocity and subsequently the wave length, the same peak of the propagating wave has to be considered at two different locations in the model in order to determine the running time Δt (so-called B-scan method), cf. Figure 4. This has involved analysis of nodal displacement histories on the connecting line between actuator and sensors, cf. [13]. 5

7 The wave length can then be calculated by dividing the distance between the two locations by the running time Δt, as follows: λ = x2 x1 Δt f c. (4) The aim in future SHM systems is to install systems, which permanently reuse the converted energy from sensing for the generation of new wave packages. Hence, the greater the energy transmission, the less additional energy is required to drive a structural health monitoring system. Energy transmission is defined within this paper as the integral over the squared signal s(t): tend E = s 2 ( t) dt trans tstart. (5) 4. RESULTS AND DISCUSSION In this section influence of different parameters of the MMC plate on the wave propagation has been studied, using a parametric approach. The method of modeling described in previous sections has been verified by comparison with results from semi-analytical finite element method (SAFE, cf. [14]) for an isotropic test case. The results obtained from both methods reveal a good compliance. 4.1 Influence of different Particle Sizes The influence of particle size has been studied on models with randomly distributed cubic particles with a constant volume fraction of 10% SiC particles in an aluminium matrix. Figure 5 and Figure 6 show, that the size of the particle has no significant influence on the 6

8 propagation of the Lamb waves. This corresponds to the results of the homogenization method where the size of the particles only has a negligible influence on the elastic properties of particle reinforced MMC [8]. These results do not however exclude that special interaction between waves and particles, such as mode conversions due to special periodicity of particle arrangement occur. Figure 7 indicates, that the the energy transmission of A 0 -mode is only slightly influenced by the particle size, whereas energy transmission of S 0 -mode is nearly unaffected by the particle size within the frequency range considered. 4.2 Influence of different Volume Fractions in BCC Arrangement In Figure 8 a comparison is shown for group velocities of S 0 -mode for isotropic plates and the different volume fractions of BCC arrangements of SiC particles. The results of the isotropic aluminium plate can be considered as a lower bound and those of the isotropic SiC plate as an upper bound. The results obtained lie within these bounds and the group velocities increase with the volume fraction of the stiffer material (SiC). This corresponds to the results obtained using the homogenization technique where the elastic properties such as Young s modulus also increase with the volume fraction. The same trend for A 0 -mode has been reported in [15]. The differences observed in wave length due to a changing volume fraction are comparably small but there is still a clear trend that the higher the volume fractions, the longer the wave length, cf. Figure 9. Contrary to wave length and group velocity the energy transmission generally decreases with a higher volume fraction, cf. Figure 10. This can be explained by the fact that the stiffer material absorbs more energy and therefore less energy will be transmitted to the sensor if the volume fraction of SiC particles increases. 7

9 4.3 Influence of different Arrangements Influence of different regular arrangements Lamb wave propagation in different regular arrangements (BCC, FCC, and Square) has been compared with a constant volume fraction of 30% of SiC particles. The group velocities are nearly unaffected by the different arrangements, cf. Figure 11. Generally, the results for group velocities correspond to the predictions of homogenization techniques since different regular arrangements only slightly affect overall Young s modulus [8]. However the wave lengths of S 0 -mode differ for lower excitation frequencies but tend towards the same value when frequency rises, whereas wave length of A 0 -mode remains unaffected by a change in arrangement, cf. Figure 12. Furthermore slight differences can be observed in energy transmission for the different arrangements, cf. Figure 13. The energy transmission of A 0 -mode is less affected by different arrangements than energy transmission of S 0 -mode. However, for both modes, square arrangement has the highest energy transmission on average Comparison of regular arranged and randomly distributed particles In this section, randomly distributed particles with a volume fraction of 10% and 20% are compared to two different regular arrangements of particles with similar volume fractions. The randomly distributed particles have a radius of 0.8 mm whereas in the regular arrangement, the particles have a radius of 0.35 mm. The group velocity dispersion curves (cf. Figure 14) and the wave length dispersion curves (cf. Figure 15) show very good correlation between regular arrangement of particles and randomly distributed particles, especially for S 0 - mode. This corresponds directly to the results of homogenization, where randomly distributed 8

10 particles have only minor influences on overall elastic properties compared to regular arrangements of particles. 4.4 Influence of different Material Properties of Particles Lamb wave propagation in plates with different stiffness to density ratios of particle has been studied and compared with that of isotropic aluminium. The matrix material was constantly aluminium. The different particle materials are SiC, steel and a cellular arrangement, i.e. the particles with a diameter of 1 mm have been removed from the model. The results shown refer to BCC arrangement and a volume fraction or volume loss of 31.03%. Both the group velocity of S 0 -mode and the group velocity of A 0 -mode increase from cellular build-up to steel particles to SiC particles, but the group velocity of steel particle reinforced aluminium is clearly below the group velocity of isotropic material, although the ratio of Young s modulus to density of the entire model hardly changed with steel particles, cf. Figure 16. For an excitation frequency of 400 khz no antisymmetric mode could be detected in the model with cellular build-up. This fact could be used as an indicator to detect detached particles, although the wave length is more than 5 times larger than the holes in the model. For the wave length the same trend observed as for the group velocity applies here, i.e. the wave length increases from cellular build-up to steel-reinforced plate to isotropic plate to SiC reinforced plate, although the wave length of S 0 -mode of isotropic aluminium is nearly equal to the wave length of steel reinforced composite plate, cf. Figure 17. The energy transmission of S 0 -mode for cellular build-up is on average higher than for steel and SiC reinforced composite, cf. Figure 18. Generally, it is clear that as the stiffness to density ratio of particles increases, the energy transmission decreases. For energy transmission of A 0 -mode, a less clear trend can be observed. 9

11 4.5 Investigation of Models based on homogenized Material Data In order to possibly decrease the effort involved in modeling and simulation for future investigations, models based on homogenized material data are compared to those with heterogeneous build-up. The main aim of the homogenization technique is to find the properties of a homogenous material which is capable of storing the same strain energy as the considered heterogeneous material under any arbitrary load [16]. There is a wide range of analytical approaches available, e.g. Voigt and Reuss bounds, Hashin-Shtrikman bounds, the self-consistent method, the Mori-Tanaka method and Torquato s third-order approximation [8], [17]. The use of an RVE in combination with FEM is another powerful technique for predicting the overall elastic properties of nearly arbitrary heterogeneous materials [17], [8]. A RVE also known as unit cell can be considered as a model, which captures the main features of the microstructure. The unit cell or RVE is loaded in different load cases with special boundary conditions in order to determine all elements of Hooke s matrix. The volume averages of stresses and strains in the RVE are treated as the effective stress and strain in the homogenized RVE: 1 σ ij = V 1 ε ij = V V V σ dv ε dv ij ij (6) Using Hooke s law together with the effective stress and strain component of homogenized RVE, one can evaluate the related element of the Hooke s matrix, cf. [8] and [16]. Homogenized material data used in the studies presented here are taken from Kari [8]. The models with heterogeneous build-up have a BCC arrangement of SiC particles with a volume fraction of 10.64% and 31.03%, respectively. 10

12 The group velocity dispersion curves display good correlation for both modeling approaches, cf. Figure 19. Slightly smaller group velocities have been obtained with the simplified models but the deviations are negligible compared to the computational costs that have been saved by applying this modeling approach. The average deviation of group velocity of S 0 -mode is less than two per cent. The dispersion curves of wave length are plotted in Figure 20. Similar to results shown in previous sections the wave length of S 0 -mode appears to be relatively sensitive to changes in the model and thus deviations between simplified models and models with heterogeneous build-up can be observed. However when frequency rises (>200 khz) good correlation can be achieved for S 0 -mode and for A 0 -mode in general. Figure 21 indicates that the energy transmission is on average higher in the simplified models. Nevertheless, similar to the observations for comparison of BCC arrangement with different volume fractions, energy transmission increases with lower volume fractions for both simplified models and models with heterogeneous build-up. Finally, it is clear that the trend of the wave propagation properties is similar for simplified models and models with heterogeneous build-up for a volume fraction up to 30%. 4.6 Summary of Results and Conclusions Table 3 and Table 4 provide a general overview of the influence of the investigated parameters on Lamb wave propagation in particle reinforced composites. Similar to the results of homogenization, the results pertaining to group velocity show a high dependency on volume fraction. Furthermore, the stiffness to density ratio of particles also influences group velocity. Particle size does in fact not influence group velocity. 11

13 From Table 3 it is generally clear that both group velocity and wave length curves have the same trend depending on MMC plate. However, where group velocity and wave length increase the energy transmission decreases. In general when the contribution of stiffer material (SiC) in the plate increases (where the volume fraction increasing or for a higher ratio of stiffness to density), the group velocity and wave length rise but less energy will be transmitted to the sensors. Table 4 generally, indicates that for the same volume fraction the simplified models have a minor influence on the group velocity and the wave length. The energy transmission is on average higher in the simplified models. However, the simplified models show the same trends as the models with heterogeneous build-up, even if the absolute values do not totally agree. This is a great advantage for further studies, because the computational cost can be dramatically reduced with simplified models. The different arrangements (randomly and regular) have also only slight influences on the group velocity and wave length. Contrary to the trends of energy transmission, in general the trends for group velocity and wave length are similar. 5. NUMERICAL VERIFICATION Using FEM is an experimentally validated approach to study Lamb wave propagation in heterogeneous materials and structures [1], but the condition of at least 10 nodes per wave length should be fulfilled [1], [4]. This condition is satisfied all-out within performed parametric study. Besides local resolution, the accuracy of time integration must be taken into account when simulating high-speed events such as propagation of ultrasonic waves. The better the accuracy of time integration the better the balance of forces is fulfilled. Considering a region of traction-free surfaces residual forces can be calculated as follows: 12

14 M v ( t) + K v ( t) FRe ( t) (7) n n = s For the model with largest time increment size, the residual force has been calculated and plotted for a node on the connecting line between actuator and sensor (cf. Figure 22). As the residual force tends close towards zero and as its magnitude is of five orders below the magnitude of elastic and inertia forces in each time step, time integration can be considered as accurate. 6. OUTLOOK This paper provides an all-embracing, parametric study of Lamb wave propagation in particle reinforced composites. However in order to provide a deeper understanding of Lamb wave propagation in particle reinforced composites several further investigations could be performed. Experimental results may help to further understand Lamb wave propagation in such materials. Furthermore, a deeper understanding of interactions between particles and Lamb waves could be provided by a two-scale FEM approach. On the other hand, methods which reduce computational costs are highly desirable and should be further extended to include heterogeneous materials such as particle reinforced composites. 7. ACKNOWLEDGMENT The work was partially supported by the German Research Foundation (GA 480/13-1). This support is gratefully acknowledged. 13

15 8. BIBLIOGRAPHY [1] Song, F., Huang, G. L. and Hudson, K., Guided wave propagation in honeycomb sandwich structures using a piezoelectric actuator/sensor system, Smart Materials and Structures, Volume 18, Number [2] Giurgiutiu, V. Structural Health Monitoring with Piezoelectric wafer active sensors, San Diego : Elsevier Inc., [3] Paget, C. A. Active Health Monitoring of Aerospace Composite Structures by Embedded Piezoceramic Transducers. PhD Thesis, Department of Aeronautics, Royal Institute of Technology, Stockholm [4] Zhongqing S., Lin Y. and Ye L. Guided Lamb waves for identification of damage in composite structures: A review. Journal Of Sound And Vibration, Volume 295, Issues , pp [5] Hosseini, S.H.M. and Gabbert, U. Analysis of Guided Lamb Wave Propagation (GW) in Honeycomb Sandwich Panels. PAMM. 2010, pp [6] Kaczmara, J.W., Pietrzakb, K. and Wlosinski, W, The production and application of metal matrix composite materials, Journal of Materials Processing Technology, Volume 106, Issues , pp [7] Miyajima, T. and Iwai, Y. Effects of reinforcements on sliding wear behavior of aluminum matrix composites. Wear, Volume 255, Issues , pp [8] Kari, S. Micromechanical Modelling and Numerical Homogenization of Fibre and Particle Reinforced Composites, Düsseldorf : VDI-Verlag,

16 [9] Lee, J.H., Maeng, D.Y., Hong, S.I. and Wona, C.W. Predictions of cracking mode and hardening behavior of MMC via FEM. Materials Science and Engineering, Volume 339, Issues , pp [10] Wang, L.; Yuan, F.G., Group velocity and characteristic wave curves of Lamb waves in composites: Modeling and experiments. Composites Science and Technology, Volume 67, 2007, pp [11] S. Mustapha, L., Ye, D., Wang, Y. Lu, Assessment of debonding in sandwich CF/EP composite beams using A0 Lamb, Composite Structures 93, 2011, pp [12] Liu, G.R. and Quek Jerry, S.S, A non-reflecting boundary for analyzing wave propagation using the finite element method, Finite Elements in Analysis and Design, Volume , pp [13] Köhler, B. Dispersion Relations in Plate Structures studied with a Scanning Laser Vibrometer, European NDT Conference [14] Vivar-Perez, J. M., Ahmad, Z. A. B. and Gabbert, U. Spectral analysis and semianalytical finite element method for Lamb wave, Proceedings of the Fifth European Structural Health Monitoring [15] Weber, R. Numerical simulation of the guided Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators, Master Thesis, University of Magdeburg [16] Berger, H., Kari, S., Gabbert, U., Rodríguez Ramos, R., Castillero, J. B. and Díaz, R. G. Evaluation of effective material properties of randomly distributed short cylindrical fiber composites using numerical homogenization technique, Journal of mechanics of materials and structures, Volume , pp

17 [17] Würkner, M., Berger, H. and Gabbert U, On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangements, International Journal of Engineering Science, Volume 49, Issue , pp

18 Figure 1: a), b), c) particles and d), e), f) particles in matrix of different regular arrangements (from left to right: BCC, FCC and Square arrangement). Figure 2: Finite Element Model of randomly distributed a) spherical and b) cubic particles; c) spherical and d) cubic particles inside matrix (volume fraction 10%). Figure 3: FE model of cubic particle reinforced composite plate with surfacebonded PZT patches. Figure 4: Schematic representation of propagating dispersive wave at two different locations with x 1 <x 2 Figure 5: Group velocity dispersion curves for randomly distributed cubic particles of different sizes (volume fraction 10%). Figure 6: Wave length dispersion curves for randomly distributed cubic particles of different sizes (volume fraction 10%). Figure 7: Energy transmission curves of top sensor for randomly distributed cubic particles of different sizes (Logarithmic scale). Figure 8: Group velocity dispersion curves of S 0 -mode for BCC arrangement with different volume fractions and isotropic plates made of aluminium and SiC. Figure 9: Wave length dispersion curves for BCC arrangement of SiC particles with different volume fractions. Figure 10: Energy transmission curves of top sensor for BCC arrangement of SiC particles with different volume fractions (Logarithmic scale). Figure 11: Group velocity dispersion curves for different regular arrangements of spherical SiC particles with constant volume fraction (30%). 17

19 Figure 12: Wave length dispersion curves for different regular arrangements of SiC particles with constant volume fraction (30%). Figure 13: Energy transmission curves of top sensor for different regular arrangements of SiC particles with constant volume fraction of 30% (Logarithmic scale). Figure 14: Group velocity dispersion curves for regular arranged and randomly distributed spherical particles Figure 15: Wave length dispersion curves for regular arranged and randomly distributed spherical particles Figure 16: Group velocity dispersion curves for plates with different reinforcing materials and isotropic plate made of aluminium (obtained using SAFE method, cf. [12]). Figure 17: Wave length dispersion curves for plates with different reinforcing materials and isotropic plate made of aluminium (obtained using SAFE method, cf. [12]). Figure 18: Energy transmission curves of top sensor for plates with different reinforcing materials. Figure 19: Group velocity dispersion curves for simplified models and models with heterogeneous build-up (BCC arrangement, spherical particles). Figure 20: Phase velocity dispersion curves for simplified models and models with heterogeneous build-up (BCC arrangement, spherical particles). 18

20 Figure 21: Energy transmission curves of top sensor for simplified models and models with heterogeneous build-up (BCC arrangement, spherical particles). Figure 22: Evaluated residual force in each time step (left) for considered node of the model (right) Table 1: Geometrical properties of the model (all units: mm). Table 2: Mechanical Properties of matrix and reinforcement materials Table 3: Influence of different parameters on Lamb wave propagation (Legend of symbols: strong increase, increase, unchanged, decrease, strong decrease) Table 4: Influence of different parameters on Lamb wave propagation (Legend of symbols: strong influence, no influence, slight influence) 19

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43 Table 1: Geometrical properties of the model (all units: mm). Plate Actuator/Sensor Length Width Thickness Damping Area Diameter Height ~60 ~ ~ / /0.38 Table 2: Mechanical Properties of matrix and reinforcement materials Material Young s modulus [GPa] Poisson s ratio [] Density [kg/m³] Aluminium SiC Steel Table 3: Influence of different parameters on Lamb wave propagation (Legend of symbols: strong increase, increase, unchanged, decrease, strong decrease) Group velocity Wave length Energy transm. Mode S 0 A 0 S 0 A 0 S 0 A 0 Central frequency Volume fraction Particle size Stiffness to density Table 4: Influence of different parameters on Lamb wave propagation (Legend of symbols: strong influence, no influence, slight influence) Group velocity Wave length Energy transm. Mode S 0 A 0 S 0 A 0 S 0 A 0 Arrangement Simplified model