numerical simulation of lamb wave propagation in metallic foam a _ a parametric study

8/10/2019 Numerical simulation of Lamb wave propagation in metallic foam a _ a parametric study

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Numerical simulation of Lamb wave propagation in metallic foam sandwich

structures: a parametric study

Seyed Mohammad Hossein Hosseini a,, Abdolreza Kharaghani b, Christoph Kirsch c, Ulrich Gabbert a

a Institute of Mechanics, Department of Numerical Mechanics, Otto-von-Guericke-University Magdeburg, Universittsplatz 2, 39016 Magdeburg, Germanyb Institute of Process Engineering, Department of Thermal Process Engineering, Otto-von-Guericke-University Magdeburg, Universittsplatz 2, 39016 Magdeburg, Germanyc Institute of Computational Physics, Zurich University of Applied Sciences, Wildbachstrasse 21, 8401 Winterthur, Switzerland

a r t i c l e i n f o

Article history:

Available online 9 November 2012

Keywords:

Lamb wave propagation

Metallic foam structure

Finite element method

Parametric study

a b s t r a c t

The propagation of guided Lamb waves in metallic foam sandwich panels is described in this paper andanalyzed numerically with a three-dimensional finite element simulation. The influence of geometrical

properties of thefoam sandwich plates (such as theirregularity of thefoam structure, therelative densityor the cover plate thickness) on thewave propagation is investigated in a parametric study. Open-cell and

closed-cellstructures arefound to exhibit similar wave propagation behavior. In additionto thefinite ele-ment model with fully resolved microstructure, a simplified, computationally cheaper model is also con-

sidered there the porous core of the sandwich panel is approximated by a homogenized effectivemedium. The limitations of this homogenization approach are briefly pointed out.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

The use of Lamb waves in structural health monitoring (SHM) ofcomposite structures is a novel technology in modern industries

such as aviation and transportation. Piezoelectric actuators andsensors are used to trigger and receive Lamb waves in modern mi-cro-structured composite materials [1,2]. Compared to otherrecent SHM approaches used to detect damage in composite struc-

tures, the benefits of the SHM technique based on Lamb waves arethe low cost of the required equipment, the possibility of onlinemonitoring, as well as its high sensitivity [1].

Among the novel light-weight structures, metallic foam sand-

wich plates can also be subject to SHM using Lamb waves. Metallicfoams are cellular materials which have been studied since the1970s [3,4]. An excellent stiffness-to-weight ratio has been re-ported for steel foams under flexural load [5]. It has been shown

that foam panels have a higher bending stiffness than solid steelsheets of the same weight[6]. The benefits of metallic foams com-pared to conventional materials are in the weight, stiffness, energydissipation, mechanical damping, and vibration frequency, and

these materials are used in the mechanical, aerospace, and auto-motive industry[711]. Based on their pore structure, solid foamsare classified into closed-cell and open-cell foams see Fig. 1 for an

illustration.

The application of ultrasonic wave propagation in SHM of foamsandwich plates has been addressed in several recent publications.

An experimental study of ultrasonic wave propagation in water-saturated cellular aluminum foams using the pulse transmission

method was presented in [13]. The fast and slow longitudinalwaves were identified and it has been shown that the measuredpropagation velocities agree with the predictions of Biots theory.Ultrasonic guided waves were used to detect sub-interface damage

in foam core sandwich structures [14].In another study, nonlinear elastic wave spectroscopy was used

to detect damage in an aircraft foam sandwich panel [15]. Due tothe nonlinear material behavior caused by the presence of damage,

harmonics and sidebands are generated from the interaction be-tween a low-frequency and a high-frequency harmonic excitationsignal. By monitoring these harmonics and sidebands, one can de-tect structural changes in the material. The capability of the pro-

posed method to detect impact damage was demonstrated.SHM based on Lamb wave propagation is a relatively new meth-

od for foam sandwich structures and has been studied only in asmall number of research publications: damage detection was

achieved using anti-symmetric low-frequency Lamb waves gener-ated by thin piezoelectric discs bonded on the skins of a foamsand-wich plate with glass fiber skins [16]. In addition to the

experimental test, finite element modeling was used to optimizethe identification procedure. Furthermore, an experimental testusing a non-contact laser doppler vibrometer (LDV) was performedto scan the panel while exciting it with a surface-bonded piezo-

electric actuator [17]. The capability of Lamb waves to detect dam-age was confirmed by the reflected wave fringe pattern obtained

0263-8223/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2012.10.039

Corresponding author.

E-mail addresses: [email protected](S.M.H. Hosseini),abdolreza.kharaghani@

ovgu.de (A. Kharaghani), [email protected] (C. Kirsch), ulrich.Gabbert

@ovgu.de(U. Gabbert).

Composite Structures 97 (2013) 387400

Contents lists available atSciVerse ScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t
http://dx.doi.org/10.1016/j.compstruct.2012.10.039mailto:[email protected]:abdolreza.kharaghani@%20%20ovgu.demailto:abdolreza.kharaghani@%20%20ovgu.demailto:[email protected]:ulrich.Gabbert%20%20%[email protected]:ulrich.Gabbert%20%20%[email protected]://dx.doi.org/10.1016/j.compstruct.2012.10.039http://www.sciencedirect.com/science/journal/02638223http://www.elsevier.com/locate/compstructhttp://www.elsevier.com/locate/compstructhttp://www.sciencedirect.com/science/journal/02638223http://dx.doi.org/10.1016/j.compstruct.2012.10.039mailto:ulrich.Gabbert%20%20%[email protected]:ulrich.Gabbert%20%20%[email protected]:[email protected]:abdolreza.kharaghani@%20%20ovgu.demailto:abdolreza.kharaghani@%20%20ovgu.demailto:[email protected]://dx.doi.org/10.1016/j.compstruct.2012.10.039


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from the LDV scan. The experimental study was supported by the-oretical evaluation.

In the present paper, the propagation of Lamb waves in foam

sandwich panels is investigated in a parametric simulation study a fundamental understanding of this phenomenon is essentialfor the design of efficient SHM systems[18]. The dependence ofthe wave propagation behavior on different geometrical properties

of the foam sandwich panels, such as the cover plate thickness,irregularity of the foam cell distribution, and relative density isinvestigated. The influence of the sandwich plate thickness andof the loading frequency on the wave propagation behavior is also

shown. In addition to a fully resolved finite element model of theporous microstructure a computationally cheaper, homogenizedmodel (involving effective mechanical properties) of the porouslayer is also employed, and the quality of this approximation isassessed.

2. Finite element modeling

A foam sandwich panel consists of two skin plates and a corelayer filled with either open or closed cells, cf.Figs. 1 and 2. Foam

sandwich plates are commonly made of the aluminum alloy T6061[1]. In this study, 2-D bilinear thin-triangular shell elements are

used to model the closed foam cells and cover plates. Comparedto higher-order shell elements, these 2-D elements are less compu-tationally expensive. However, these elements are not very sensi-tive to distortion and accurate results require high local

resolution and relatively small elements. Linear straight trusseswith constant cross section are used to model the open-cell foamstructures. Implicit time integration is used to simulate the wavepropagation.

2.1. Geometry generation

We use 3-D Voronoi tessellations to represent the geometry of

the cellular material. These tessellations are passed to the finiteelement code, where the faces (for closed-cell foams) or the edges(for open-cell foams) are replaced by plate and beam elements forthe computation. We follow the approach described in[19]: N> 0points (called

nuclei) x

i 2R

3;

i1;. . . ;

N, are placed randomly in-

side a cubic box with edge length a> 0. We require a minimum dis-tance jxi xjjP dP 0 between any two points xi,xj,ij. This isachieved by generating the point positions one by one and discard-

ing a new point if it is too close to any of the already acceptedpoints. Obviously, a solution to this packing problem exists onlyif the three parameters N,aand dare somehow related. In practice,if no solution is found for the given parameters, we repeat the ran-

dompoint generation process with a slightly smaller value ofd, un-til a solution is found. The Voronoi cellCiassociated to the nucleusxi is definedas the set of points in3-D which are closer toxithan toany other nucleus xj,j i:

Ci fx2R3jjxxij


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Mathematical models to describe the mechanics of metal foamswere initially developed by Gibson and Ashby[5,21]and are stillwidely used [7]. The homogenized elastic moduli for open-celland closed-cell foams are given by[7]

Ereps Copen Es q2rel and Ereps Cclosed Es 0:5 q2rel 0:3 qrel ;4

respectively, whereEreps denotes the representative homogenizedYoungs modulus, Es Youngs modulus of the base material, andqrel the relative mass density. Note that the relative density ofthe foam (qrel) is the primary dependent variable for all foammechanics. Copen2[0.1,4] and Cclosed2 [0.1,1] (typical ranges) de-scribe effects from the material morphology and manufacturingprocess. The expressions(4)are valid for a certain range of rela-

tive densities. The GibsonAshby expressions (4)were comparedwith experimental data for compressive yield stress and Youngsmodulus[7].

It was concluded that despite of the poor agreement of the ex-

act values for some foam structures with special morphology ormanufacturing process (e.g. steel foams with unusual anisotropy,

special heat treatments, and unusually thin-walled hollowspheres) the experimental results for typical foam structures re-main within the established bounds of Gibson and Ashby [7].Therefore, the GibsonAshby expressions provide reasonable

effective mechanical properties for most common foam struc-tures. The shear modulus for both open-cell and closed-cell struc-tures is stated as [7]

Greps38

Ereps: 5

Assuming an isotropic behavior for the foam structures one can cal-culate the Poissons ratio.

The relative mass density is defined as

qrelqqs

; 6

where qand qsare the mass density of the cellular and solid mate-rial, respectively. Since the masses of the solid and cellular materialitself are identical (if we assume vacuum between the solid), therelative mass density is equal to the relative volumetric density

[22],

qrel Vs

Vsample: 7

The right-hand side of Eq. (7) is the relative solid volume, whereVsis the volume of the solid material (e.g. the material of the cell

walls) and Vsample is the volume of the testing sample (e.g. thesandwich plate core).

2.3. Wave propagation modeling

A single nodal load is used to generate the Lamb waves. Theload signal of a three and half-cycle narrow band tone burst [1]

is applied as an excitation to the chosen node as shown in Figs. 2and 6(Fdenotes the amplitude of the excitation signal, tdenotestime,fcdenotes the central frequency and Hdenotes the Heavisidestep function):

Fint F Ht H t 3:5fc

1 cos 2pfct

3:5

sin2pfct: 8

The amount of spurious reflections from artificial boundaries

will depend on the model size. Consequently, a larger modelhas a greater attenuation effect on spurious reflected waves. Asystem of dashpot elements is used to reduce the influence ofmodes reflected from the outer boundaries, cf. Fig. 4. Various

parameters including the damping factor, the direction and num-ber of dashpot elements were examined to design an effective

non-reflecting boundary. Fig. 5 illustrates the effect of a non-reflecting boundary made of dashpot elements on the wave prop-agation in a reduced-size open-cell foam sandwich structure. Theattenuation of spurious reflected waves helps to identify thepropagating modes in subsequent signal processing for SHM

applications [23].The major benefit of using dashpot elements is the ability to re-

duce the model size. The reflection of waves from artificial outerboundaries shows less dependency on the model size if dashpots

are used. This phenomenon can be explained by the fact that

-2.622e-019

-2.132e-019

-1.642e-019

-1.151e-019

-6.611e-020

-1.709e-020

3.194e-020

8.097e-020

1.300e-019

1.790e-019

2.280e-019

Increment: 100

X

Y

A0 S Reading point ( sensor)

Nodal load (actuator)

Direction of wave propagation

Reduced size

Original size

0

DashpotsDashpots

Time: 2.2e-5 (s)

Displacement (m)

Fig. 4. A schematic representation of dashpot elements connected to a plate.

-1.510-10

-110-10

-0.510-10

010-10

0.510-10

110-10

1.510-10

Displacement(m)

0.510-4 110-4 1.510-4 210-4

Time (s)

S0mode

A0mode

Reflections

With dashpot

Without dashpot

rel= 0.168 (-)

= 0.11 (-)

Open-cell

tp= 1 mm

fc= 200 kHz

Fig. 5. Propagated Lamb wave (nodal displacement signal) in a open-cell foam

sandwich plate with non-reflecting boundary (solid line) and without non-reflecting boundary using dashpot elements (dashed line).

S.M.H. Hosseini et al. / Composite Structures 97 (2013) 387400 389


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by decreasing the model size in Ydirection,Fig. 4, the number ofdashpot elements in the direction of wave propagation, X, doesnot effectively change. Therefore, the attenuation of reflectedwaves would not change effectively.

The simulation results presented here were obtained by usingthe commercial finite element package ANSYS 11.0.

2.4. Model validation

2.4.1. Numerical validation

The choice of an element size of less than one tenth of the wave

length has resulted in a good numerical accuracy, and the numer-ical results agree with the experimental results reported in[1,25].To illustrate the accuracy of the time integration, the residual forceat a randomly chosen node on a traction-free surface may be con-

sidered[25]:

Frest Mut Kut: 9

In(9), Mdenotes the mass matrix, Kthe stiffness matrix,u denotesthe nodal displacement inZ-direction, and uthe nodal acceleration.

Accuracy of time integration may be expressed by how well theforces are balanced. The time step size is given by the minimum ele-ment size divided by the maximum group velocity (the group

0

20

40

60

80

100

120

Avergaevalueof(Fres

(t)/Finertia(t))(%)

0 250 500 750 1000 1250 1500 1750 2000

Total number of time steps (-)

The chosen value

fc= 200 kHz

tp= 1 mm

= 0.11 (-)

Closed-cell

rel= 0.175 (-)

Total time: 0.22 ms

Fig. 7. Results of time step refinement analysis.

Z

X Y

Nodal load

(actuator)

Increment: 50Time: 1.1e-005 (s)Displacement (m)Reading point

(sensor)

Bottom surface

Top Surface

(b) (c)

(a)

-610-9

-510-9

-410-9

-310-9

-210-9

-110-9

010-9

110-9

210-9

310-9

Displacement(m)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Arc length (m)

-6.0e-09

-5.1e-09

-4.2e-09

-3.3e-09

-2.4e-09

-1.5e-09

-6.0e-10

3.0e-10

1.2e-09

2.1e-09

3.0e-09

Fig. 6. (a) Wave propagation in an open-cell structure with 0.11 ms delay after the signal is excited actuating and measuring points are also indicated; (b) magnification of

the nodal load at the actuating point and (c) the displacements of the nodes located along the dashed line on the top surface (qrel= 0.168,tp= 1 mm, a = 0.108,fc= 200 kHz).

390 S.M.H. Hosseini et al. / Composite Structures 97 (2013) 387400


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velocity of S0 in the aluminum base material) serves as an initial

choice for the time step size. It is found that this time step is smallenough as the residual force tends close to zero and at each timestep it is one order of magnitude below the elastic (Ku(t)) and iner-tia Mut forces. Another indication that this is a reasonable

choice for the time step size is given by the following convergencetest (Fig. 7): we show the average value of Fres(t)/Finertia(t) at

T= 0.22 ms for different values of the time step size (corresponding

to certain numbers of time steps). Fig. 7illustrates that the use ofmore than 1000 time steps (the value used for the following compu-tations) will not decrease the residual significantly any further.

The application of 3-D hexahedral solid elements to simulatethe wave propagation have been proved in [1,25]. Therefore, in or-der to validate the accuracy of the truss elements to model the

wave propagation in open-cell structures, the wave propagationin a lattice block sandwich plate with truss elements, Fig. 8a, iscompared with a model using 3-D solid elements, Fig. 8b. Thematerial properties of elements are steel with a cross section of 1mm2. The average differences between measured wave propertiesin both models including the group velocity and the wave lengthvalues are 5.65%, 5.61% and 7.70% respectively, for various valuesof the central loading frequency in the range of 50400 kHz.

Furthermore, in order to validate the model of wave propaga-

tion in cellular materials using shell elements presented in the pre-vious sections, the numerical results for the wave propagation in astandard honeycomb sandwich plate with closed-cells modeled byshell elements (which was available in our laboratory CELLITE

silver standard 69 sandwich plate, Axson GmbH) made of alumi-num are compared with experimental measurements from a scan-ning laser vibrometer[26,27](cf. Fig. 9 for an illustration of theexperimental setup). The differences between measured and

simulated wave properties propagated on the honeycomb sand-wich plate including the group velocity and the wave length valuesremained below 5% for various values of the central loading fre-

quency in the range of 50400 kHz.

3. Methodology

Lamb waves propagate along elastic plates with two differentmode shapes, denoted by S and A: the plate displacements aresymmetric with respect to the center plane for the Smode and

anti-symmetric for theAmode (cf.Fig. 10). Both modes are disper-sive, i. e. their velocities depend on the frequency.

Both Lamb wave modes propagate across the whole thickness ofthe plate that is why the damages on the surfaces and the internal

damages can be found and located using the Lamb waves. To avoidthe analysis of the multi-modal Lamb wave signal and to simplifythe signal processing, only the symmetric and anti-symmetricmodes which arrive first, denoted by S0 andA0, respectively, are

usually considered in the literature on damage detection [29].To identify different modes in thin plates, one can use sensor

signals obtained at both the top and bottom surfaces of the plate[30]. In the case of thick sandwich plates however, the group veloc-

ities of the S0 andA0 modes are higher at the top surface (where theexcitation takes place) than at the bottom surface. Therefore, thesignal from the bottom sensor cannot be used to identify differentmodes in this case. In this paper modes are identified in thick foam

sandwich panels by observing changes in the amplitudes of the ar-rival signals.

The velocity of different modes along the structure is calledgroup velocity [31]. The phase velocity is associated with the phase

difference between the vibrations observed at two different pointsduring the passage of the wave[32]. The phase velocity is used tocalculate the wave length of each mode. The wave length is an

important factor to show the sensitivity of a Lamb wave to detectdamage[31].

In this paper the group velocities are evaluated by transforming

the nodal displacement signal in the vertical direction u(t) usingthe continuous wavelet transform (CWT) based on the Daubechieswavelet D10. As indicated in Eq. (10), the CWT is defined as thetime integral of the signal u multiplied by the wavelet functionw(here the Daubechies wavelet D10). The scale parameter b is in-versely proportional to the frequency of the signal (the bar indi-cates complex conjugation)[2,33].

WTa; b 1ffiffiffia

pZ 1

1utw ta

b

dt: 10

X

Y

(a) (b)

Top cover plateTruss elements Solid elements

Bottom cover plate

Fig. 8. A lattice block sandwich plate modeled with (a) truss elements and (b) 3-D

solid elements.

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Normalizeddisplacement

(-)

0.410-4 0.610-4 0.810-4 110-4

Time (s)

Top plate sensor

Bottom plate sensor

Symmetric mode (S0)

Anti-symmetric mode (A0)

Fig. 10. The first symmetric (S0) and anti-symmetric (A0) modes are plotted in the

time domain using a normalized nodal displacement obtained fromtop and bottom

surface of a thin aluminum plate. The central frequency of the loading signal is100 kHz.

noitisoProtautcA

3-D laser scanning

vibrometer

Retro-reflectivelayer

Comp

ositep

late

gnipmadrofnociliS

Fig. 9. Setup for experimental test[28].



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The time of flight for each Lambwave mode is given by the loca-

tion of the maxima of the CWT coefficients, cf. Fig. 11. By dividing

the distance between the sensors by the time of flight one can cal-culate the group velocity for each mode[33].

Furthermore, a fast Fourier transform algorithm is used to ob-

tain the phase function / [32], from which the phase velocityand wave length of each mode can be computed:

/x arctanbF2xbF1x !

; 11

where bFdenotes the Fourier transform of the vertical displacementsignal u, with imaginary part bF1and real part bF2. The phase velocityis a function of angular frequency:

tx xL/

x

/0

: 12

Considering the relation between the angular frequency x and thelinear frequencyf, x = 2pf, the phase velocity can be expressed interms of linear frequency as

tf 2pfL/2pf /0

; 13

where /0 denotes the loading phase, / the measured phase and L

the distance between the actuator and the sensor in axial direction.Dividing the phase velocity by the loading frequency yields the

wave length as

kf tff

: 14

The energy transmission rate of the wave describes the leakybehavior of the propagating waves within a sandwich panel [30].The magnitudes of the energy transmission rate at the top and bot-

tom surfaces indicate how deep waves can travel inside a sandwichpanel. The integral over the squared received displacement signal,u(t), is used to define the energy transmission proportion withinthis paper[33]:

EtransZ tend

tstart

ut2 dt: 15

The post-processing calculations described in this section havebeen performed in MATLAB.

4. Results

4.1. Influence of frequency

Fig. 12shows that in the chosen range of loading frequencies

the group velocity of the S0mode is independent of the loading fre-quency. However, the group velocity of the A0 mode increases asthe loading frequency increases. The wave lengths of both modesdecrease as the loading frequency increases (Fig. 13).Fig. 14indi-

cates that the A0 mode transmits more energy on the surface inthe lower frequency range. As the frequency exceeds the so-called

Time increment (-)

Scale(-)

150 200 250 300 350 400 450 500

10

15

20

25

30

35

40S

0

A0

Time of the flight for A0Time of the flight for S

0

Correspondingtothe

loading

frequencyof200(kHz)

Maximum value of the

CWT coefficients

Fig. 11. The contour plot of the absolute valuesof theCWT coefficientsbased on theDaubechies wavelet D10 is shown. The signalis obtainedfrom a Lamb wave propagating

in a honeycomb sandwich panel. The central frequency of the loading signal is 200 kHz.

0

500

1000

1500

2000

2500

3000

3500

4000

Groupvelocity(m/s)

50 100 150 200 250 300 350 400

Frequency,fc(kHz)

S0top

A0top

S0bottom

A0bottomrel= 0.168 (-)

= 0.11 (-), tp= 1 mm

Closed-cell

0

500

1000

1500

2000

2500

3000

3500

4000

Groupvelocity(m/s)

50 100 150 200 250 300 350 400

Frequency,fc(kHz)

Open-cell

Fig. 12. The group velocity vs. the central frequency fcof the loading signal, with constant geometry.



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structural irregularity (grain size) and the cover plate thickness) ofa foam sandwich plate has been studied via numerical simulation.The results are summarized in Table 1. A similar behavior of thewave propagation in closed-cell and open-cell structures has been

observed. In order to reduce the computational effort a simplifiedhomogenized model was used where the micro-structured foamnetwork has been replaced by brick elements with effective mate-rial properties. It has been shown that the group velocity and the

energy transmission by the Lamb waves are equal in the simplifiedand in the fully resolved model, whereas the wave lengths ob-tained for theS0andA0modes did not correspond. Further studiesmust be carried out in order to develop a better simplification ap-

proach. In addition, there are still some open questions on SHMbased on Lamb waves for foam sandwich structures, e.g. findingthe appropriate loading signal for SHM applications [24]as wellas the determination of possible damage which can be detected

by the propagating wave at the signal-processing level [31]. Fur-thermore, more studies must be carried out to prove the usefulness

of Lamb waves in detecting invisible damage experimentally inlight-weight cellular sandwich structures, including foam sand-

wich plates.

Acknowledgment

The authors acknowledge the German Research Foundation(DFG) for the financial support of this research (GA 480/13).

References

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[3] Brayl H. Eng Mater Des 1972;16:19.[4] Barton R, Carter FWS, Roberts TA. Chem Eng 1974;291:708.[5] Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG. Metal

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structures: a review of applications, manufacturing and material properties. JConstr Steel Res 2012;71:110.

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Cambridge; 1999. ISBN-10: 0521499119.[12] Curran D. Metal foam pictures .[13] Ji Q, Le LH, Filipow LJ, Jackson SA. Ultrasonic wave propagation in water-

saturated aluminum foams. Ultrasonics 1998;36:75965.[14] Terrien N, Osmont D. Damage detection in foam core sandwich structures

using guided waves. In: Leger MDA, editor. Ultrasonic wave propagation innon homogeneous media. Springer proceedings in physics, vol. 128. BerlinHeidelberg: Springer; 2009. p. 25160.

[15] Zumpano G, Meo M. Damage detection in an aircraft foam sandwich panelusing nonlinear elastic wave spectroscopy. Comput Struct 2008;86:483490.

[16] Osmont D, Devillers D, Taillade F. Health monitoring of sandwich plates basedon the analysis of the interaction of lamb waves with damages. In: Davis LP,editor. Smart structures and materials 2001: smart structures and integratedsystems, vol. 4327; 2001. p. 290301.

[17] Chakraborty N, Mahapatra DR, Srinivasan G. Ultrasonic guided wavecharacterization and damage detection in foam-core sandwich panel usingpwas and ldv. In: Health monitoring of structural and biological systems. SanDiego, California, USA; 2012.

[18] Wang L, Yuan F. Group velocity and characteristic wave curves of lamb wavesin composites: modeling and experiments. Compos Sci Tech2007;67:137084.

[19] Zhu HX, Hobdell JR, Windle AH. Acta Mater 2000;48:4893.[20] Information .[21] Gibson MF, Ashby LJ. Cellular solids: structure & properties. Oxfordshire Press;

1997.[22] chsner A, Hosseini S, Merkel M. Numerical simulation of the mechanical

properties of sintered and bonded perforated hollow sphere structures (phss).J Mater Sci Technol 2010;26:7306.

-100

-75

-50

-25

0

25

50

75

100

Difference(%)

50 100 150 200 250 300 350 400

Frequency,fc(kHz)

S0top

A0top

rel= 0.168 (-)

= 0.11 (-)

Open-cell

tp= 1 mm

top surfaceEnergy transmission bottom surface

S0bottom

A0bottom

-100

-75

-50

-25

0

25

50

75

100

Difference(%)

50 100 150 200 250 300 350 400

Frequency,fc(kHz)

Fig. 33. Relative error of the energy transmissionvalues obtained fromthe homogenizedmodel compared withthe openfoam structuremodel. The values are plotted vs. the

central frequency of the loading signal.

Table 1

Summary of results showing the dependency of the wave propagation under variations of the loading frequency and geometrical parameters of the foam sandwich plate. "

indicates an increase, indicates no change or slight changes and ; indicates there is a decrease in the respective values.

Group velocity Wave length Energy transmission

Central loading frequency S0: ; Below critical frequency:S0", A0;

fc" A0:" Above critical frequency:;

Relative densityqrel" ;Irregularitya" Cover plate thickness " " "

tp" (Closed-cell, S0: ) (Closed-cell, S0: )

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conversion of lamb waves in CFRP plates. Key Eng Mat 2012;518:36474.

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[30] Weber R, Hosseini SMH, Gabbert U. Numerical simulation of the guided lambwave propagation in particle reinforced composites. Compos Struct2012;94(10):306471.

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numerical simulation of lamb wave propagation in metallic foam a _ a parametric study

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