Accepted Manuscript

Numerical simulation of the guided Lamb wave propagation in particle rein‐

forced composites

Ralf Weber, Seyed Mohammad Hossein Hosseini, Ulrich Gabbert

PII: S0263-8223(12)00180-8

DOI: http://dx.doi.org/10.1016/j.compstruct.2012.04.020

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: http://dx.doi.org/10.1016/

j.compstruct.2012.04.020

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1

Numerical simulation of the guided Lamb wave propagation in particle reinforced composites

Ralf Weber (corresponding author, ralfweber84@gmx.de, Phone: +0049-621/39159324, Mobile: +0049-176/60818993),

Seyed Mohammad Hossein Hosseini (rsg.931@gmail.com),

Ulrich Gabbert (Ulrich.Gabbert@Masch-Bau.Uni-Magdeburg.de)

University of Magdeburg, Institute of Mechanics, Universitätsplatz 2, 39106 Magdeburg,

Germany, Phone: +0049-391/6718608, Fax: 0049-391/6712439

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

2

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

pqkqk

qdpd

2

222

4)(

)tan()tan( −−= (1)

and antisymmetric mode:

222

2

)(4

)tan()tan(

qkpqk

qdpd

−−=

. (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

3

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 all-

embracing, 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

4

considered. In the case of cubic formed particles, each particle is represented by one single

finite element, cf. Figure 2.

2.2 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.

5

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 2.

3. 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

arrg t

sc Δ=, (3)

where tarr 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].

6

The wave length can then be calculated by dividing the distance between the two locations by

the running time Δt, as follows:

cftxx

Δ−

=12

λ. (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):

∫=end

start

t

ttrans dttsE )(2

. (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

7

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 A0-mode is only slightly influenced by

the particle size, whereas energy transmission of S0-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 S0-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 A0-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.

8

4.3 Influence of different Arrangements

4.3.1 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 S0-mode differ for lower excitation frequencies but tend towards

the same value when frequency rises, whereas wave length of A0-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

A0-mode is less affected by different arrangements than energy transmission of S0-mode.

However, for both modes, square arrangement has the highest energy transmission on

average.

4.3.2 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 S0-

mode. This corresponds directly to the results of homogenization, where randomly distributed

9

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 S0-mode and the group velocity of A0-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 S0-mode of isotropic aluminium is nearly equal

to the wave length of steel reinforced composite plate, cf. Figure 17.

The energy transmission of S0-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 A0-mode, a less clear trend can be observed.

10

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:

dVV

dVV

Vijij

Vijij

∫

∫

⋅=

⋅=

εε

σσ

1

1

(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.

11

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 S0-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 S0-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 S0-mode and for A0-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.

12

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:

13

)()()( Re tFtvKtvM snn =⋅+⋅ (7)

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.

14

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

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Diego : Elsevier Inc., 2007.

[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. 2001.

[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 3-5.

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[5] Hosseini, S.H.M. and Gabbert, U. Analysis of Guided Lamb Wave Propagation (GW) in

Honeycomb Sandwich Panels. PAMM. 2010, pp. 11-14.

[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

1-3. 2000, pp. 58-67.

[7] Miyajima, T. and Iwai, Y. Effects of reinforcements on sliding wear behavior of

aluminum matrix composites. Wear, Volume 255, Issues 1-6. 2003, pp. 606–616.

[8] Kari, S. Micromechanical Modelling and Numerical Homogenization of Fibre and Particle

Reinforced Composites, Düsseldorf : VDI-Verlag, 2007.

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[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 1-2. 2003, pp. 175-182.

[10] Wang, L.; Yuan, F.G., Group velocity and characteristic wave curves of Lamb waves in

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analytical finite element method for Lamb wave, Proceedings of the Fifth European Structural

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[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. 2011.

[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

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[17] Würkner, M., Berger, H. and Gabbert U, On numerical evaluation of effective material

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17

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 surface-

bonded PZT patches.

Figure 4: Schematic representation of propagating dispersive wave at two

different locations with x1<x2

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 S0-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%).

18

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).

19

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)

20

Fig. 1.

21

Fig. 2.

22

Fig. 3.

23

Fig. 4.

24

Fig. 5.

25

Fig. 6.

26

Fig. 7.

27

Fig. 8.

28

Fig. 9.

29

Fig. 10.

30

Fig. 11.

31

Fig. 12.

32

Fig. 13.

33

Fig. 14.

34

Fig. 15.

35

Fig. 16.

36

Fig. 17.

37

Fig. 18.

38

Fig. 19.

39

Fig. 20.

40

Fig. 21.

41

Fig. 22.

Table 1: Geometrical properties of the model (all units: mm).

Plate Actuator/Sensor Length Width Thickness Damping Area Diameter Height

~60 ~20 1.5 ~15 6.35/3.75 0.76/0.38

Table 2: Mechanical Properties of matrix and reinforcement materials

Material Young’s modulus [GPa] Poisson’s ratio [] Density [kg/m³] Aluminium 70 0.3 2700

SiC 450 0.17 3210 Steel 210 0.3 7850

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 S0 A0 S0 A0 S0 A0

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 S0 A0 S0 A0 S0 A0

Arrangement Simplified model