Composite Structures 97 (2013) 387–400

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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, Universitätsplatz 2, 39016 Magdeburg, Germany Institute of Process Engineering, Department of Thermal Process Engineering, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39016 Magdeburg, Germany c Institute of Computational Physics, Zurich University of Applied Sciences, Wildbachstrasse 21, 8401 Winterthur, Switzerland b

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 and analyzed numerically with a three-dimensional finite element simulation. The influence of geometrical properties of the foam sandwich plates (such as the irregularity of the foam structure, the relative density or the cover plate thickness) on the wave propagation is investigated in a parametric study. Open-cell and closed-cell structures are found to exhibit similar wave propagation behavior. In addition to the finite element model with fully resolved microstructure, a simplified, computationally cheaper model is also considered – there the porous core of the sandwich panel is approximated by a homogenized effective medium. The limitations of this homogenization approach are briefly pointed out. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The use of Lamb waves in structural health monitoring (SHM) of composite structures is a novel technology in modern industries such as aviation and transportation. Piezoelectric actuators and sensors are used to trigger and receive Lamb waves in modern micro-structured composite materials [1,2]. Compared to other recent SHM approaches used to detect damage in composite structures, the benefits of the SHM technique based on Lamb waves are the low cost of the required equipment, the possibility of online monitoring, as well as its high sensitivity [1]. Among the novel light-weight structures, metallic foam sandwich plates can also be subject to SHM using Lamb waves. Metallic foams are cellular materials which have been studied since the 1970s [3,4]. An excellent stiffness-to-weight ratio has been reported for steel foams under flexural load [5]. It has been shown that foam panels have a higher bending stiffness than solid steel sheets of the same weight [6]. The benefits of metallic foams compared to conventional materials are in the weight, stiffness, energy dissipation, mechanical damping, and vibration frequency, and these materials are used in the mechanical, aerospace, and automotive industry [7–11]. Based on their pore structure, solid foams are classified into closed-cell and open-cell foams – see Fig. 1 for an illustration.

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

The application of ultrasonic wave propagation in SHM of foam sandwich plates has been addressed in several recent publications. An experimental study of ultrasonic wave propagation in watersaturated cellular aluminum foams using the pulse transmission method was presented in [13]. The fast and slow longitudinal waves were identified and it has been shown that the measured propagation velocities agree with the predictions of Biot’s 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 to the nonlinear material behavior caused by the presence of damage, harmonics and sidebands are generated from the interaction between a low-frequency and a high-frequency harmonic excitation signal. By monitoring these harmonics and sidebands, one can detect structural changes in the material. The capability of the proposed method to detect impact damage was demonstrated. SHM based on Lamb wave propagation is a relatively new method for foam sandwich structures and has been studied only in a small number of research publications: damage detection was achieved using anti-symmetric low-frequency Lamb waves generated by thin piezoelectric discs bonded on the skins of a foam sandwich plate with glass fiber skins [16]. In addition to the experimental test, finite element modeling was used to optimize the identification procedure. Furthermore, an experimental test using a non-contact laser doppler vibrometer (LDV) was performed to scan the panel while exciting it with a surface-bonded piezoelectric actuator [17]. The capability of Lamb waves to detect damage was confirmed by the reflected wave fringe pattern obtained

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2.1. Geometry generation

5 mm

5 mm

(a)

(b)

Fig. 1. Digital photos of cellular aluminum foam structures with: (a) closed-cells and (b) open-cells [12].

from the LDV scan. The experimental study was supported by theoretical 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 essential for the design of efficient SHM systems [18]. The dependence of the 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 is investigated. The influence of the sandwich plate thickness and of the loading frequency on the wave propagation behavior is also shown. In addition to a fully resolved finite element model of the porous microstructure a computationally cheaper, homogenized model (involving effective mechanical properties) of the porous layer is also employed, and the quality of this approximation is assessed. 2. Finite element modeling A foam sandwich panel consists of two skin plates and a core layer 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. Compared to higher-order shell elements, these 2-D elements are less computationally expensive. However, these elements are not very sensitive to distortion and accurate results require high local resolution and relatively small elements. Linear straight trusses with constant cross section are used to model the open-cell foam structures. Implicit time integration is used to simulate the wave propagation.

Top cover plate

We use 3-D Voronoi tessellations to represent the geometry of the cellular material. These tessellations are passed to the finite element code, where the faces (for closed-cell foams) or the edges (for open-cell foams) are replaced by plate and beam elements for the computation. We follow the approach described in [19]: N > 0 points (called nuclei) xi 2 R3 ; i ¼ 1; . . . ; N, are placed randomly inside a cubic box with edge length a > 0. We require a minimum distance jxi  xjj P d P 0 between any two points xi,xj,i – j. This is achieved by generating the point positions one by one and discarding a new point if it is too close to any of the already accepted points. Obviously, a solution to this packing problem exists only if the three parameters N, a and d are somehow related. In practice, if no solution is found for the given parameters, we repeat the random point generation process with a slightly smaller value of d, until a solution is found. The Voronoi cell Ci associated to the nucleus xi is defined as the set of points in 3-D which are closer to xi than to any other nucleus xj,j – i:

C i ¼ fx 2 R3 jjx  xi j < jx  xj j 8j–ig;

i ¼ 1; . . . ; N:

ð1Þ

Eight laterally translated copies of the cubic box, as well as regular top and bottom layers of nuclei are attached before the Voronoi tessellation is computed. Voronoi cells associated to these additional nuclei are removed after tessellation. This leads to a bounded structure with periodic lateral boundaries and regular top and bottom boundaries, as shown in Fig. 2. The Voronoi tessellation is computed with the free software Qhull [20]. It takes the nuclei positions as an input and returns vertex coordinates, as well as N lists of vertex indices associated to each Voronoi cell. Cell faces and edges are computed from this information in a post-processing step. To define the regularity of a 3-D tessellation, it is compared with the so-called Kelvin structure, which is a uniform, space-filling tessellation made from truncated octahedra (tetradecahedra). In the Kelvin structure, the minimum distance between nuclei is given by

pffiffiffi  1=3 6 a3 pffiffiffi d0 ¼ : 2 2N

ð2Þ

The regularity a P 0 for an arbitrary structure obtained from the generation process described above is now defined as

a¼

d : d0

ð3Þ

It satisfies a = 1 for the Kelvin structure and a = 0 for a completely random configuration of nuclei (d = 0). 2.2. Homogenization

Nodal load

The Lamb wave propagation is also studied in a simplified model, in which the porous core layer is described with 3-D solid elements with effective material properties, cf. Fig. 3.

7.5 mm

Nodal load

Top cover plate Z

Z Y

X

X

Ho

gen

ize

Bottom cover plate

(a)

dl

aye

r

(b) Bottom cover plate

Fig. 2. Foam sandwich plates: (a) closed-cell model; and (b) open-cell model. A single nodal load is used for Lamb wave generation.

Z

Nodal load

mo

X

Fig. 3. Simplified model with homogenized core layer.

Y

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Increment: 100 Time: 2.2e-5 (s)

Original size Reduced size

Dashpots

Dashpots

Direction of wave propagation

Nodal load (actuator) A0

S0

Reading point ( sensor)

Displacement (m)

Y -0

20

19 -0 0e 28 2. 19 -0 0e 79 1. 19 -0 0e 30 1. 20 -0 7e 09 8. 20

4e

20

-0

-0

9e

1e

0 .7

19 3.

-1

19

19

-0

-0

2e

1e

4 .6

5 .1

1 .6 -6

-1

9

19

01

-0

e-

2e

22 .6

3 .1

-1

-2

-2

X

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

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

  Ereps ¼ C open  Es  q2rel and Ereps ¼ C closed  Es  0:5  q2rel þ 0:3  qrel ; ð4Þ respectively, where Ereps denotes the representative homogenized Young’s modulus, Es Young’s modulus of the base material, and qrel the relative mass density. Note that the relative density of the foam (qrel) is the primary dependent variable for all foam mechanics. Copen 2 [0.1, 4] and Cclosed 2 [0.1, 1] (typical ranges) describe effects from the material morphology and manufacturing process. The expressions (4) are valid for a certain range of relative densities. The Gibson–Ashby expressions (4) were compared with experimental data for compressive yield stress and Young’s modulus [7]. It was concluded that despite of the poor agreement of the exact values for some foam structures with special morphology or manufacturing process (e.g. steel foams with unusual anisotropy, special heat treatments, and unusually thin-walled hollow spheres) the experimental results for typical foam structures remain within the established bounds of Gibson and Ashby [7]. Therefore, the Gibson–Ashby expressions provide reasonable effective mechanical properties for most common foam structures. The shear modulus for both open-cell and closed-cell structures is stated as [7]

Greps ¼

3  Ereps : 8

2.3. Wave propagation modeling A single nodal load is used to generate the Lamb waves. The load 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. 2 and 6 (F denotes the amplitude of the excitation signal, t denotes time, fc denotes the central frequency and H denotes the Heaviside step function):

     3:5 2pfc t sinð2pfc tÞ: 1  cos F in ðtÞ ¼ F HðtÞ  H t  fc 3:5

The amount of spurious reflections from artificial boundaries will depend on the model size. Consequently, a larger model has a greater attenuation effect on spurious reflected waves. A system of dashpot elements is used to reduce the influence of modes reflected from the outer boundaries, cf. Fig. 4. Various parameters including the damping factor, the direction and number of dashpot elements were examined to design an effective non-reflecting boundary. Fig. 5 illustrates the effect of a nonreflecting boundary made of dashpot elements on the wave propagation in a reduced-size open-cell foam sandwich structure. The attenuation of spurious reflected waves helps to identify the propagating modes in subsequent signal processing for SHM applications [23]. The major benefit of using dashpot elements is the ability to reduce the model size. The reflection of waves from artificial outer boundaries shows less dependency on the model size if dashpots are used. This phenomenon can be explained by the fact that

ð5Þ

Assuming an isotropic behavior for the foam structures one can calculate the Poisson’s ratio. The relative mass density is defined as

q ; qs

ð6Þ

where q and qs are the mass density of the cellular and solid material, respectively. Since the masses of the solid and cellular material itself are identical (if we assume vacuum between the solid), the relative mass density is equal to the relative volumetric density [22],

qrel ¼

Vs V sample

:

ð7Þ

The right-hand side of Eq. (7) is the relative solid volume, where Vs is 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. the sandwich plate core).

1.5·10-10 S0 mode

Open-cell

Reflections

1·10-10

Displacement (m)

qrel ¼

ð8Þ

0.5·10-10 0·10-10 -0.5·10-10 -1·10-10 -1.5·10-10

fc = 200 kHz ρrel = 0.168 (-) α = 0.11 (-) tp = 1 mm 0.5·10-4

With dashpot Without dashpot

A0 mode 1·10-4

1.5·10-4

2·10-4

Time (s) Fig. 5. Propagated Lamb wave (nodal displacement signal) in a open-cell foam sandwich plate with non-reflecting boundary (solid line) and without nonreflecting boundary using dashpot elements (dashed line).

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

Reading point (sensor)

3.0e-09 2.1e-09 1.2e-09 3.0e-10

Top Surface

(a)

-6.0e-10

Bottom surface

-1.5e-09

Nodal load (actuator)

-2.4e-09

Z

-3.3e-09

X

-4.2e-09

Y

-5.1e-09 -6.0e-09

(c)

(b)

3·10-9 2·10-9

Displacement (m)

1·10-9 0·10-9 -1·10-9 -2·10-9 -3·10-9 -4·10-9 -5·10-9 -6·10-9 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Arc length (m)

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).

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 numerical results agree with the experimental results reported in [1,25]. To illustrate the accuracy of the time integration, the residual force at a randomly chosen node on a traction-free surface may be considered [25]:

€ ðtÞ þ KuðtÞ: F res ðtÞ ¼ M u

ð9Þ

In (9), M denotes the mass matrix, K the stiffness matrix, u denotes € the nodal acceleration. the nodal displacement in Z-direction, and u

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

Avergae value of (Fres(t)/Finertia(t)) (%)

by decreasing the model size in Y direction, Fig. 4, the number of dashpot elements in the direction of wave propagation, X, does not effectively change. Therefore, the attenuation of reflected waves would not change effectively. The simulation results presented here were obtained by using the commercial finite element package ANSYSÒ 11.0.

120

fc = 200 kHz

Total time: 0.22 ms

ρrel = 0.175 (-)

100

α = 0.11 (-) tp = 1 mm

80 60 40

The chosen value

20 Closed-cell

0 0

250

500

750

1000

1250

1500

1750

Total number of time steps (-) Fig. 7. Results of time step refinement analysis.

2000

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Top cover plate

Truss elements

simulated wave properties propagated on the honeycomb sandwich plate including the group velocity and the wave length values remained below 5% for various values of the central loading frequency in the range of 50–400 kHz.

Solid elements

Y X

3. Methodology

Bottom cover plate

(a)

(b)

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

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 small enough as the residual force tends close to zero and at each time step it is one order of magnitude below the elastic (Ku(t)) and iner€ ðtÞÞ forces. Another indication that this is a reasonable tia ðM u choice for the time step size is given by the following convergence test (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. 7 illustrates that the use of more than 1000 time steps (the value used for the following computations) will not decrease the residual significantly any further. The application of 3-D hexahedral solid elements to simulate the wave propagation have been proved in [1,25]. Therefore, in order to validate the accuracy of the truss elements to model the wave propagation in open-cell structures, the wave propagation in a lattice block sandwich plate with truss elements, Fig. 8a, is compared with a model using 3-D solid elements, Fig. 8b. The material properties of elements are steel with a cross section of 1 mm2. The average differences between measured wave properties in both models including the group velocity and the wave length values are 5.65%, 5.61% and 7.70% respectively, for various values of the central loading frequency in the range of 50–400 kHz. Furthermore, in order to validate the model of wave propagation in cellular materials using shell elements presented in the previous sections, the numerical results for the wave propagation in a standard honeycomb sandwich plate with closed-cells modeled by shell elements (which was available in our laboratory – CELLITE silver standard 69 sandwich plate, Axson GmbH) made of aluminum are compared with experimental measurements from a scanning laser vibrometer [26,27] (cf. Fig. 9 for an illustration of the experimental setup). The differences between measured and

Lamb waves propagate along elastic plates with two different mode shapes, denoted by S and A: the plate displacements are symmetric with respect to the center plane for the S mode and anti-symmetric for the A mode (cf. Fig. 10). Both modes are dispersive, i. e. their velocities depend on the frequency. Both Lamb wave modes propagate across the whole thickness of the plate that is why the damages on the surfaces and the internal damages can be found and located using the Lamb waves. To avoid the analysis of the multi-modal Lamb wave signal and to simplify the signal processing, only the symmetric and anti-symmetric modes which arrive first, denoted by S0 and A0, 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 velocities of the S0 and A0 modes are higher at the top surface (where the excitation takes place) than at the bottom surface. Therefore, the signal from the bottom sensor cannot be used to identify different modes in this case. In this paper modes are identified in thick foam sandwich panels by observing changes in the amplitudes of the arrival signals. The velocity of different modes along the structure is called group velocity [31]. The phase velocity is associated with the phase difference between the vibrations observed at two different points during the passage of the wave [32]. The phase velocity is used to calculate the wave length of each mode. The wave length is an important factor to show the sensitivity of a Lamb wave to detect damage [31]. In this paper the group velocities are evaluated by transforming the nodal displacement signal in the vertical direction u(t) using the continuous wavelet transform (CWT) based on the Daubechies wavelet D10. As indicated in Eq. (10), the CWT is defined as the time integral of the signal u multiplied by the wavelet function w (here the Daubechies wavelet D10). The scale parameter b is inversely proportional to the frequency of the signal (the bar indicates complex conjugation) [2,33].

1 WTða; bÞ ¼ pffiffiffi a

Z

þ1

1

  ta dt: uðtÞw b

ð10Þ

3-D laser scanning vibrometer Silicon for damping

Retro

-refle

ctive

Actuator Position

layer

Normalized displacement (-)

1.00 0.75

Top plate sensor Bottom plate sensor

0.50 Symmetric mode (S0) 0.25 0.00 -0.25 -0.50 -0.75

Anti-symmetric mode (A0)

-1.00

te

osi

mp

Co

la te p

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

0.4·10-4

0.6·10-4

0.8·10-4

1·10-4

Time (s) Fig. 10. The first symmetric (S0) and anti-symmetric (A0) modes are plotted in the time domain using a normalized nodal displacement obtained from top and bottom surface of a thin aluminum plate. The central frequency of the loading signal is 100 kHz.

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40

Maximum value of the CWT coefficients

S0 35

A0

Scale (-)

Corresponding to the loading frequency of 200 (kHz)

30 25 20 15 10

Time of the flight for S0

150

200

250

300

Time of the flight for A0 350

400

450

500

Time increment (-) Fig. 11. The contour plot of the absolute values of the CWT coefficients based on the Daubechies wavelet D10 is shown. The signal is obtained from a Lamb wave propagating in a honeycomb sandwich panel. The central frequency of the loading signal is 200 kHz.

The time of flight for each Lamb wave mode is given by the location of the maxima of the CWT coefficients, cf. Fig. 11. By dividing the distance between the sensors by the time of flight one can calculate the group velocity for each mode [33]. Furthermore, a fast Fourier transform algorithm is used to obtain the phase function / [32], from which the phase velocity and wave length of each mode can be computed:

! b F 2 ðxÞ /ðxÞ ¼ arctan ; b F 1 ðxÞ

ð11Þ

where Fb denotes the Fourier transform of the vertical displacement signal u, with imaginary part b F 1 and real part b F 2 . The phase velocity is a function of angular frequency:

xL tðxÞ ¼ : /ðxÞ  /0

2pfL ; /ð2pf Þ  /0

ð13Þ

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

4000

Etrans ¼

Z

t end

uðtÞ2 dt:

ð15Þ

The post-processing calculations described in this section have been performed in MATLABÒ. 4. Results 4.1. Influence of frequency Fig. 12 shows that in the chosen range of loading frequencies the group velocity of the S0 mode is independent of the loading frequency. However, the group velocity of the A0 mode increases as the loading frequency increases. The wave lengths of both modes decrease as the loading frequency increases (Fig. 13). Fig. 14 indicates that the A0 mode transmits more energy on the surface in the lower frequency range. As the frequency exceeds the so-called

4000

Open-cell

3500

Group velocity (m/s)

Group velocity (m/s)

ð14Þ

:

t start

Closed-cell

3000 2500 2000 1500

S0 top

1000

A0 top

0

f

The energy transmission rate of the wave describes the leaky behavior of the propagating waves within a sandwich panel [30]. The magnitudes of the energy transmission rate at the top and bottom surfaces indicate how deep waves can travel inside a sandwich panel. The integral over the squared received displacement signal, u(t), is used to define the energy transmission proportion within this paper [33]:

3500

500

tðf Þ

ð12Þ

Considering the relation between the angular frequency x and the linear frequency f, x = 2pf, the phase velocity can be expressed in terms of linear frequency as

tðf Þ ¼

kðf Þ ¼

S0 bottom

ρrel = 0.168 (-) α = 0.11 (-), tp = 1 mm 50

100

150

A0 bottom

3000 2500 2000 1500 1000 500 0

200

250

Frequency, fc (kHz)

300

350

400

50

100

150

200

250

Frequency, fc (kHz)

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

300

350

400

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0.07

0.07

Closed-cell

A0 top

0.04

S0 bottom A0 bottom

0.03 0.02 0.01

Wave length (m)

S0 top

0.05

Wave length (m)

Open-cell

0.06

0.06

50

100

150

0.04 0.03 0.02 0.01

ρrel = 0.168 (-) α = 0.11 (-), tp = 1 mm

0

0.05

0 200

250

300

350

400

50

100

150

Frequency, fc (kHz)

200

250

300

350

400

Frequency, fc (kHz)

Closed-cell 10-25 10-28 10-31 S0 top

10-34

A0 top

ρrel = 0.168 (-) 10-37

S0 bottom

α = 0.11 (-) tp = 1 mm

10-40

50

100

A0 bottom 150

200

250

300

350

Energy transmission proportion (J.m/N)

Energy transmission proportion (J.m/N)

Fig. 13. The wave length vs. the central frequency fc of the loading signal, with constant geometry.

Open-cell 10-25 10-28 10-31 10-34 10-37 10-40

400

50

100

Frequency, fc (kHz)

150

200

250

300

350

400

Frequency, fc (kHz)

Fig. 14. The energy transmission vs. the central frequency fc of the loading signal, with constant geometry. A logarithmic scale has been used here.

3000

3000

Closed-cell

2500

2000 1500 S0 top

1000

A0 top

fc = 200 kHz 500 0

S0 bottom

α = 0.11 (-) tp = 1 mm 0.14

Group velocity (m/s)

Group velocity (m/s)

2500

Open-cell

2000 1500 1000 500

A0 bottom 0 0.16

0.18

0.20

0.22

0.24

Relative density, ρrel (-)

0.12

0.14

0.16

0.18

0.20

0.22

Relative density, ρrel (-)

Fig. 15. The group velocity vs. the relative density qrel, where tp, a and fc are constant.

critical frequency (which is 300 kHz in this case) the S0 mode transmits the same amount of energy as the A0 mode.

damping occurs as the thickness of the solid structure increases. A similar behavior is observed for open-cell and closed-cell structures.

4.2. Influence of geometrical properties 4.2.1. Relative density Figs. 15 and 16 indicate that the relative density does not play a significant role for the group velocity and wave length values. However, Fig. 17 shows that the energy transmission values decrease as the relative density increases. This indicates that more

4.2.2. Irregularity Figs. 18–20 show that the irregularity factor does not influence the wave propagation properties in open-cell and closed-cell foam sandwich panels. These results also imply that the wave propagation in the panels is independent of the grain size d0 in the range considered here (since the irregularity factor is proportional to

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0.018

Closed-cell

0.016

0.016

0.014

0.014

0.012 0.01 0.008 S0 top

0.006 0.004

fc = 200 kHz

0.002

α = 0.11 (-) tp = 1 mm

0

0.14

A0 top

Wave length (m)

Wave length (m)

0.018

Open-cell

0.012 0.01 0.008 0.006 0.004

S0 bottom

0.002

A0 bottom

0 0.16

0.18

0.20

0.22

0.24

0.12

0.14

Relative density, ρrel (-)

0.16

0.18

0.20

0.22

Relative density, ρrel (-)

Closed-cell 10-25 10-28 10-31 S0 top

10-34 10-37 10-40

A0 top fc = 200 kHz

S0 bottom

α = 0.11 (-) tp = 1 mm 0.14

A0 bottom 0.16

0.18

0.20

0.22

Energy transmission proportion (J.m/N)

Energy transmission proportion (J.m/N)

Fig. 16. The wave length vs. the relative density qrel, where tp, a and fc are constant.

Open-cell 10-25 10-28 10-31 10-34 10-37 10-40

0.24

0.12

0.14

Relative density, ρrel (-)

0.16

0.18

0.20

0.22

Relative density, ρrel (-)

Fig. 17. The energy transmission vs. the relative density qrel, where tp, a and fc are constant. A logarithmic scale has been used here.

3000

3000

Closed-cell

2000 1500 S0 top

1000 500 0

Open-cell

2500

A0 top

fc = 200 kHz ρrel = 0.17 (-)

S0 bottom

2000 1500 1000 500

A0 bottom

tp = 1.00 mm 0.2

Group velocity (m/s)

Group velocity (m/s)

2500

0 0.4

0.6

0.8

0.2

Irregularity factor, α (-)

0.4

0.6

0.8

Irregularity factor, α (-)

Fig. 18. The group velocity is plotted over the irregularity factor a, where tp, qrel and fc are constant.

d0, cf. Eq. (3)). The grain size of a typical foam structure, d0, does not exceed 1 mm [7]. Figs. 16 and 19 indicate that the average wave length is at least one order of magnitude larger than the typical grain size. If the wave length is very large compared to the characteristic length scale of the foam structure, the wave will not be influenced by variations in the microstructure, which explains why the irregularity has little influence on the wave propagation.

pendent of the cover plate thickness. However, the group velocity and the wave length of the S0 mode in open-cell structures and A0 mode in both open-cell and closed-cell structures increase slightly as the cover plate thickness increases. Fig. 23 shows that the energy transmission of the propagated wave increases significantly as the cover plate thickness increases. This phenomenon indicates that a thicker cover plate transmits more energy to the structure. 4.3. Cell morphology

4.2.3. Cover plate thickness It is shown in Figs. 21 and 22 that both the group velocity and the wave length of the S0 mode in closed-cell structures are inde-

Comparison of the wave behavior in open-cell and closed-cell structures is highlighted and discussed in this section. The ratio

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0.018

Closed-cell

0.016

0.016

0.014

0.014

Wave length (m)

Wave length (m)

0.018

0.012 0.01 0.008 S0 top

0.006 0.004

fc = 200 kHz

0.002

ρrel = 0.17 (-)

0

A0 top

0.012 0.01 0.008 0.006 0.004

S0 bottom

0.002

A0 bottom

tp = 1.00 mm

Open-cell

0

0.2

0.4

0.6

0.8

0.2

Irregularity factor, α (-)

0.4

0.6

0.8

Irregularity factor, α (-)

Closed-cell 10-25 10-28 10-31 S0 top

10-34

A0 top

fc = 200 kHz 10-37 10-40

S0 bottom

ρrel = 0.17 (-)

A0 bottom

tp = 1.00 mm 0.2

0.4

0.6

Energy transmission proportion (J.m/N)

Energy transmission proportion (J.m/N)

Fig. 19. The wave length is plotted over the irregularity factor a, where tp, qrel and fc are constant.

Open-cell 10-25 10-28 10-31 10-34 10-37 10-40

0.8

0.2

0.4

Irregularity factor, α (-)

0.6

0.8

Irregularity factor, α (-)

Fig. 20. The energy transmission is plotted over the irregularity factor a, where tp, qrel and fc are constant. A logarithmic scale has been used here.

4000

4000

Open-cell

Closed-cell 3500

Group velocity (m/s)

Group velocity (m/s)

3500 3000 2500 2000 1500

S0 top

1000

A0 top

500 0

fc = 200 kHz ρrel = 0.168 (-)

0.75

2500 2000 1500 1000

S0 bottom A0 bottom

α = 0.11 (-) 0.50

3000

500 Open cell 0

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Cover plate thickness, tp (mm)

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Cover plate thickness, tp (mm)

Fig. 21. The group velocity vs. the thickness of the cover plate tp, where a, qrel and fc are constant.

between different wave properties, Vclosed/Vopen, of the A0 mode at the top surface of the open-cell and closed-cell foam sandwich panel plotted over the loading frequency, cf. Fig. 24, and geometrical properties, cf. Figs. 25–27. Vopen denotes the value obtained from the open-cell model and Vclosed denotes the value obtained from the closed-cell model (logarithmic values are used to compare

energy transmission results). It is shown in Fig. 24 that the deviation between the wave properties in open-cell and closed-cell structures is minimal if the loading frequency is in the range of 150–200 kHz. However, Figs. 25–27 show that the geometrical properties have only a minor effect on the deviation between the wave properties in open and closed cell structures.

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0.018

0.018 0.016

0.014

0.014

Wave length (m)

Wave length (m)

Closed-cell 0.016

0.012 0.01 0.008 S0 top

0.006 0.004 0.002 0

A0 top

fc = 200 kHz ρrel = 0.168 (-)

0.50

0.75

0.01 0.008 0.006 0.004

S0 bottom

0.002

A0 bottom

α = 0.11 (-)

0.012

Open-cell

0 1.00

1.25

1.50

1.75

2.00

2.25

2.50

0.50

0.75

Cover plate thickness, tp (mm)

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Cover plate thickness, tp (mm)

Closed-cell 10-25 10-28 10-31 S0 top

10-34 10-37 10-40

A0 top fc = 200 kHz ρrel = 0.168 (-)

S0 bottom A0 bottom

α = 0.11 (-) 0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Energy transmission proportion (J.m/N)

Energy transmission proportion (J.m/N)

Fig. 22. The wave length vs. the thickness of the cover plate tp, where a, qrel and fc are constant.

Open-cell 10-25 10-28 10-31 10-34 10-37 10-40 0.50

0.75

Cover plate thickness, tp (mm)

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Cover plate thickness, tp (mm)

Fig. 23. The energy transmission vs. the thickness of the cover plate tp, where a, qrel and fc are constant. A logarithmic scale has been used here.

4.4. Homogenization Using a homogenized model is much cheaper computationally than the fully resolved model described in Section 2. As an example, an open-cell structure with a = 11 and qrel = 0.168 is considered. The size of the homogenized model is 2.6 times smaller than the fully resolved foam structure model and the computation

time is 1.4 times faster. The relative difference (in %) between the wave propagation properties obtained from the homogenized model and from the numerical simulation test with the foam sandwich plate have been calculated using (16) and plotted vs. the central frequency of the loading signal. Vhomogenized stands for the value obtained from the homogenized model and Vfoam stands for the value obtained by numerical simulation of the wave propagation in

1.3

1.3

1.2

Vclosed /Vopen (-)

Vclosed /Vopen (-)

1.2 1.1 1 0.9

1.1 1 0.9

Group velocity 0.8 0.7

top surface

Anti-symmetric mode

top surface

Anti-symmetric mode

ρrel = 0.168 (-) α = 0.11 (-), tp = 1 mm 50

100

150

0.8

Wave length Energy transmission

0.7 200

250

300

350

400

Frequency, fc (kHz) Fig. 24. The ratio between wave propagation properties in open-cell and closed-cell structures vs. the central frequency. The results are plotted using Spline approximation.

Group velocity

fc = 200 kHz α = 0.11 (-)

Wave length Energy transmission

tp = 1 mm 0.14

0.16

0.18

0.20

0.22

0.24

Relative density, ρrel (-) Fig. 25. The ratio between wave propagation properties in open and closed-cell structures vs. the central frequency. The results are plotted using Spline approximation.

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1.3

1.3

top surface

Anti-symmetric mode

1.2

Vclosed /Vopen (-)

Vclosed /Vopen (-)

1.2 1.1 1 0.9 0.8 0.7

fc = 200 kHz

Group velocity

ρrel = 0.168 (-) tp = 1 mm

Wave length Energy transmission

1.1 1 0.9 0.8 0.7

0.2

0.4

0.6

0.8

the fully resolved foam sandwich plate (logarithmic values are used to compare the energy transmission results). A clear similarity can be observed for the group velocity and the energy transmission values for both open and closed cell structures in Figs. 28, 29, 32 and 33, respectively. However, despite the agreement in the low-frequency range, a clear difference in the computed wave lengths can be seen for higher frequencies in both open-cell and

100

1.00

1.25

1.50

1.75

2.00

2.25

2.50

closed-cell foam structures (Figs. 30 and 31). Further studies must be done to develop a more efficient simplification approach for a better comparison of wave length values

rel: error ¼ 100 

50

50

Difference (%)

75

25 0 Closed-cell ρrel = 0.168 (-)

V homogenized  V foam : V homogenized

ð16Þ

100

75

-75

0.75

Fig. 27. The ratio between wave propagation properties in open and closed-cell structures vs. the central frequency. The results are plotted using Spline approximation.

top surface

Group velocity

-50

Group velocity Wave length Energy transmission

Cover plate thickness, tp (mm)

Fig. 26. The ratio between wave propagation properties in open and closed-cell structures vs. the central frequency. The results are plotted using Spline approximation.

-25

fc = 200 kHz ρrel = 0.168 (-) α = 0.11 (-) 0.50

Irregularity factor, α (-)

Difference (%)

top surface

Anti-symmetric mode

bottom surface

25 0 -25 -50

S0 top

α = 0.11 (-) tp = 1 mm

S0 bottom

-75

A0 top

-100

A0 bottom

-100 50

100

150

200

250

300

350

400

50

100

Frequency, fc (kHz)

150

200

250

300

350

400

Frequency, fc (kHz)

Fig. 28. Relative error of the group velocity obtained from the homogenized model compared with the closed foam structure model. The values are plotted vs. the central frequency of the loading signal.

100 75

75

50

50

Difference (%)

Difference (%)

100

top surface

Group velocity

25 0 -25 -50 -75

Open-cell ρrel = 0.168 (-)

bottom surface

25 0 -25 -50

S0 top

α = 0.11 (-) tp = 1 mm

S0 bottom

-75

A0 top

-100

A0 bottom

-100 50

100

150

200

250

Frequency, fc (kHz)

300

350

400

50

100

150

200

250

300

350

400

Frequency, fc (kHz)

Fig. 29. Relative error of the group velocity obtained from the homogenized model compared with the open foam structure model. The values are plotted vs. the central frequency of the loading signal.

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100

S0 top

50

Difference (%)

bottom surface

75

Closed-cell A0 top

25

S0 bottom

50

Difference (%)

75

100

top surface

Wave length

0 -25

0 -25

-50

ρrel = 0.168 (-)

-50

-75

α = 0.11 (-) tp = 1 mm

-75

-100

A0 bottom

25

-100 50

100

150

200

250

300

350

400

50

100

150

Frequency, fc (kHz)

200

250

300

350

400

Frequency, fc (kHz)

Fig. 30. Relative error of the wave length obtained from the homogenized model compared with the closed foam structure model. The values are plotted vs. the central frequency of the loading signal.

100

bottom surface

75

Open-cell S0 top

50

A0 top

25

S0 bottom

50

Difference (%)

Difference (%)

75

100

top surface

Wave length

0 -25

0 -25

-50

ρrel = 0.168 (-)

-50

-75

α = 0.11 (-) tp = 1 mm

-75

-100

A0 bottom

25

-100 50

100

150

200

250

300

350

400

50

100

150

Frequency, fc (kHz)

200

250

300

350

400

Frequency, fc (kHz)

Fig. 31. Relative error of the wave length obtained from the homogenized model compared with the open foam structure model. The values are plotted vs. the central frequency of the loading signal.

100

75

50

50

Difference (%)

Difference (%)

100

top surface

Energy transmission 75

25 0 -25 -50 -75 -100

Closed-cell ρrel = 0.168 (-)

bottom surface

25 0 -25 -50

S0 top

α = 0.11 (-) tp = 1 mm

S0 bottom

-75

A0 top

A0 bottom

-100 50

100

150

200

250

300

350

400

Frequency, fc (kHz)

50

100

150

200

250

300

350

400

Frequency, fc (kHz)

Fig. 32. Relative error of the energy transmission values obtained from the homogenized model compared with the closed foam structure model. The values are plotted vs. the central frequency of the loading signal.

Eq. (4) with Cclosed = 0.5 and Copen = 2.05 is used in this part. A variation of Cclosed and Copen within the typical range (cf. Section 2.2) had little effect on the wave propagation in the homogenized model.

4.5. Conclusion & outlook The Lamb wave propagation behavior depending on the loading frequency and geometrical properties (i.e. the relative density, the

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100 75

75

50

50

Difference (%)

Difference (%)

100

top surface

Energy transmission

25 0 -25 -50

Open-cell ρrel = 0.168 (-)

25 0 -25 -50

S0 top

α = 0.11 (-) tp = 1 mm

-75

bottom surface

S0 bottom

-75

A0 top

-100

A0 bottom

-100 50

100

150

200

250

300

350

400

50

100

Frequency, fc (kHz)

150

200

250

300

350

400

Frequency, fc (kHz)

Fig. 33. Relative error of the energy transmission values obtained from the homogenized model compared with the open foam structure model. 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.

Central loading frequency fc" Relative density qrel" Irregularity a" Cover plate thickness tp"

Group velocity

Wave length

Energy transmission

S0: – A0: " – – " (Closed-cell, S0: –)

;

Below critical frequency: S0", A0; Above critical frequency: ; ; – "

structural irregularity (grain size) and the cover plate thickness) of a foam sandwich plate has been studied via numerical simulation. The results are summarized in Table 1. A similar behavior of the wave propagation in closed-cell and open-cell structures has been observed. In order to reduce the computational effort a simplified homogenized model was used where the micro-structured foam network has been replaced by brick elements with effective material properties. It has been shown that the group velocity and the energy transmission by the Lamb waves are equal in the simplified and in the fully resolved model, whereas the wave lengths obtained for the S0 and A0 modes did not correspond. Further studies must be carried out in order to develop a better simplification approach. In addition, there are still some open questions on SHM based on Lamb waves for foam sandwich structures, e.g. finding the appropriate loading signal for SHM applications [24] as well as the determination of possible damage which can be detected by the propagating wave at the signal-processing level [31]. Furthermore, more studies must be carried out to prove the usefulness of Lamb waves in detecting invisible damage experimentally in light-weight cellular sandwich structures, including foam sandwich plates. Acknowledgment The authors acknowledge the German Research Foundation (DFG) for the financial support of this research (GA 480/13). References [1] Song F, Huang GL, Hudson k. Guided wave propagation in honeycomb sandwich structures using a piezoelectric actuator/sensor system. Smart Mater Struct 2009;18:125007–15. [2] Ungethuem A, Lammering R. Impact and damage localization on carbon-fibrereinforced plastic plates. In: Casciati MGF, editor. Proceedings 5th European workshop on structural health monitoring. Italy: Sorrento; 2010.

– – " (Closed-cell, S0: –)

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[23] Hosseini SMH, Gabbert U. Analysis of guided lamb wave propagation (GW) in honeycomb sandwich panels. In: PAMM, vol. 1; 2010. p. 11–4. [24] Mustapha S, Ye L, Wang D, Lu Y. Assessment of debonding in sandwich CF/EP composite beams using A0 lamb. Compos Struct 2011;93:483–91. [25] Weber R. Numerical simulation of the guided lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. Master’s thesis. Institut für Mechanik, Fakultät für Maschinenbau, Otto-vonGuericke-Universität Magdeburg, Germany; 2011. [26] Pohl J, Mook G, Szewieczek A, Hillger W, Schmidt D. Determination of lamb wave dispersion data for SHM. In: 5th European workshop of structural health monitoring; 2010. [27] Köhler B. Dispersion relations in plate structures studied with a scanning laser vibrometer. In: ECNDT. Berlin; 2006. [28] Willberg C, Mook G, Gabbert U, Pohl J. The phenomenon of continuous mode conversion of lamb waves in CFRP plates. Key Eng Mat 2012;518:364–74.

[29] Ahmad ZAB. Numerical simulations of lamb waves in plates using a semianalytical finite element method. Ph.D. thesis, Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Germany; 2011. [30] Weber R, Hosseini SMH, Gabbert U. Numerical simulation of the guided lamb wave propagation in particle reinforced composites. Compos Struct 2012;94(10):3064–71. [31] Paget CA. Active health monitoring of aerospace composite structures by embedded piezoceramic transducers. Ph.D. thesis, Department of Aeronautics Royal Institute of Technology, Stockholm, Sweden; 2001. [32] Sachse W, Pao Y. On the determination of phase and group velocities of dispersive waves in solids. J Appl Phys 1978;8:4320–7. [33] Song F, Huang G, Kim J, Haran S. On the study of surface wave propagation in concrete structures using a piezoelectric actuator/sensor system. Smart Mater Struct 2008;17:055024–32.