Non-reflecting boundary condition for Lamb wave propagation - OvGU

Composites: Part B 54 (2013) 1–10

Contents lists available at SciVerse ScienceDirect

Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

Non-reflecting boundary condition for Lamb wave propagation problems in honeycomb and CFRP plates using dashpot elements Seyed Mohammad Hossein Hosseini ⇑, Sascha Duczek, Ulrich Gabbert Institute of Numerical Mechanics, Department of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39016 Magdeburg, Germany

a r t i c l e

i n f o

Article history: Received 25 January 2013 Received in revised form 8 April 2013 Accepted 15 April 2013 Available online 29 April 2013 Keywords: Non-reflecting boundary condition D. Ultrasonics A. Honeycomb A. Carbon fibre C. Finite element analysis (FEA)

a b s t r a c t The paper’s objective is to introduce a new non-reflecting boundary condition using dashpot elements. This is an useful tool to efficiently simulate Lamb wave propagation within composite structures, such as honeycomb and CFRP plates. Due to the steadily increasing interest in applying Lamb waves in modern online structural health monitoring techniques, several numerical and experimental studies have been carried out recently. The proposed boundary condition poses the advantage of reducing the computational costs required to simulate the wave propagation in heterogenous materials. Different parameters which can influence the functionality of such an artificial boundary are discussed and several applications are presented. Finally, the results are also experimentally validated. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Structural health monitoring (SHM) in composite structures using guided Lamb waves is a new technology in modern industries such as aviation and transportation. Piezoelectric (PZT) actuators and sensors are used to excite and to receive waves within complicated composite structures [1,2]. This approach for SHM applications is an interesting technique because of the low costs of the required equipment, the possibility of an online monitoring and the high sensitivity to detect small structural damages [1]. 1.1. Lamb wave propagation in composite plates Lamb wave propagation in composite plates has been studied in several Refs. [1,3–12]. guided waves were used to detect sub-interface damages in foam core sandwich structures in [3]. A suitable frequency was found that offers the highest sensitivity to detect skin/foam core delaminations and to facilitate the interpretation of the measured waveforms. Finally, delaminations were located and characterized using an adapted signal processing. The numerical simulations were validated experimentally. Wave propagation in light-weight plates with truss-like cores was investigated in [4]. It has been shown, that the vibrational behavior can be reduced to equivalent plate models in the low frequency region where global plate waves are dominant. An application example of a train floor ⇑ Corresponding author. Tel.: +49 3916711723. E-mail addresses: [email protected] (S.M.H. Hosseini), sascha.duczek@st. ovgu.de (S. Duczek), [email protected] (U. Gabbert). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.04.061

section was tested to validate the theoretical dispersion characteristics. The Lamb wave propagation in particle reinforced composite plates was studied in [5]. It has been reported that the volume fraction and the stiffness to density ratio of the particles are the main parameters to affect the Lamb wave propagation properties in such materials. In addition, a homogenization method was used to simplify the models. A reasonable agreement between the complex model, incorporating many details of the real structure and the simplified model in terms of the group velocity and the wavelength has been observed, while tremendous savings in computational costs were archived. Localized phase velocities in the frequency range of 5–50 kHz were measured in honeycomb plates in [6]. It has been reported that the proposed method is suitable to detect delamination between the cover plate and the core in honeycomb sandwich panels. In another study, a homogenization technique was formulated for the analysis of vibration and the wave propagation problems in a honeycomb-like slender skeleton [7]. In addition, the effect of the cell size on the overall dynamic behavior of a composite solid was characterized. The effect of the geometry of the unit cells on the dynamics of the propagation of elastic waves within the structure was studied in [8] using a two-dimensional finite element model and the theory of periodic structures. The desired transmissibility levels in specified directions were investigated for an optimal design configurations to obtain efficient vibration isolation capabilities. Debonding in sandwich CF/ EP composite structures with a honeycomb core was detected using the anti-symmetric (A0) Lamb mode in [9] and the finite element modeling approach was validated with experimental results. In a similar study theoretical, numerical and experimental

2

S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

approaches were used to detect the delamination between the skin and the honeycomb of a composite helicopter rotor blade in [10]. In addition, the Lamb wave propagation in honeycomb sandwich panels was studied using a three-dimensional finite element approach in [1,11]. The results were compared with a simplified model, where the core layer was replaced by a heterogeneous material with a simple cubic geometry. The received results were validated by experimental results in [1]. Continuous mode conversation of the Lamb wave propagation in CFRP composites was studied in [12] using numerical and experimental approaches. These studies show the steadily growing research activities in the field of structural health monitoring and highlight the need for computationally efficient numerical tools to gain a deeper understanding of the underlying physics of Lamb wave propagation in heterogenous materials.

boundary condition can be applied using all commercial finite element packages. In the following sections numerical and experimental results are presented. Different parameters are considered to design an efficient non-reflecting boundary condition designed by dashpot elements. Afterward, the capability of the proposed method is compared with the approach of gradually damped artificial boundaries. Thereafter, the ability of dashpot non-reflecting boundary condition is demonstrated to reduce the computational costs. To this end, the model size is reduced in such a way that only the minimum required solid medium between actuator and the sensor is considered for the Lamb wave propagation analysis in a structure. Finally, the results are validated numerically as well as experimentally.

1.2. Non-reflecting boundary condition

A widely accepted approach is to use PZT patches to generate the Lamb waves within a structure, cf. Fig. 1. As time dependent excitation signal (Vin) a three and half-cycle narrow banded tone burst [1] is applied as given in the following equation:

The application of a non-reflecting boundary condition was studied in several references in order to reduce the computational efforts [13–18]. A non-reflecting boundary condition was described for the scalar wave equation in [13]. It has been mentioned that this method is only applicable for wave propagation when uniform medium at the boundary (any inhomogeneities should be avoided) exist. In addition, the proposed method offers few choices for the shape of an artificial non-reflecting boundary. Infinite elements in Abaqus were used to design a non-reflecting boundary in [14]. However, it has been shown in [15] that this method is not satisfactory. In [15] a finite element approach for the analysis of the wave propagation in an infinitely long plate was presented. To avoid any spurious reflections generated by the finite boundary of the finite element model a non-reflecting boundary condition with a gradually damped artificial boundary was designed. The length of the damping section was considered to be long enough for gradual changes of the damping factor to avoid any spurious reflection from any sudden damping. The proposed method was implemented using the available finite element packages. In addition, the results in a plate with a horizontal crack were compared with the strip element method and a good agreement has been reported. A similar design of non-reflecting boundary was introduced in [16] using frequency domain analysis and absorbing regions. In this paper the longest wavelength was suggested as a measure for the length of the absorption region. It has been indicated in [15] that the proposed non-reflecting boundary condition in [15,16] can be computationally expensive depending on the model being analyzed. Furthermore, the perfectly matched layer (PML) is introduced as a flexible and accurate method to simulate the wave propagation in unbounded structures [17]. However, it is mentioned that this approach can be computational expensive [18]. Many additional unknowns insert in the standard PML formulations because the required wave equations stated in their standard second-order form to be reformulated as first-order systems. Additionally, it has been mentioned in [18] that local absorbing boundary conditions among the available non-reflecting boundary conditions are known as the simplest and most flexible approach with a reasonable computational cost. They do not require any special functions and are capable to be coupled with standard finite difference or finite element methods. Therefore, the aim of the present paper is to introduce a novel local non-reflecting boundary condition using dashpot elements which can reduce the computational efforts tremendously. The finite element method is known to be a versatile and efficient numerical tool for a vast variety of engineering problems. Thus we decided to employ FEM to model the guided wave propagation in heterogenous medium instead of utilizing other numerical approaches such as finite differences or the boundary element method [19,20]. The proposed non-reflecting

2. Finite element modeling

2pfc t sin 2pfc t: V in ¼ V½HðtÞ Hðt 3:5=fc Þ 1 cos 3:5

ð1Þ

There t is the time, fc is central frequency and H(t) is the Heaviside step function. A zero voltage is applied to the bottom surfaces of the sensors and the actuator. Symmetric boundary conditions are applied to the inner borders of the plate to reduce the model size and the computational costs, cf. Fig. 1. The piezoelectric sensor is attached parallel to the boundaries and located at a distance of 180 mm from the actuator. Dashpot elements are used to damp the wave reflections from borders of a structure, cf. Fig. 1. The viscous behavior of dashpots in which the damping force (F) is proportional to the velocity, provides the ability of energy dissipation during cyclic loading [21].

F ¼ Cðu_2 u_1 Þ;

ð2Þ

F represents the force generated by the dashpot, C is the damping factor, u_ 1 and u_ 2 are the velocity of two ends of the dashpot element (in our case u_ 1 ¼ 0). To apply the new non-reflecting boundary condition, dashpot elements are connected only to one row and column of the outer elements as a primary choice, cf. Fig. 1. In this paper, each numerical model includes an ‘‘inner part’’ (shown by dashed lines in Fig. 1) which represents and captures the main features of the micro- and macrostructure of the propagating medium. The sensor is considered to be attached to the plate in this region. The rest of the structure (apart from the inner part) is considered as outer borders of the plate. The definition of the ‘‘frame’’ is used to indicate number of outer rows and columns which are used in the border of a numerical model. A sketch is shown in Fig. 1, each frame of elements includes a set of rows and columns of elements from the outer borders of the plate. However, each frame may consist of several number of rows and columns (in this study number of rows and columns in each frame is considered to be equal to the number of rows in the inner part). The frame definition is also used to describe the model size and the gradually increasing damping boundary condition. As an example Fig. 1 shows a model with three columns and rows as frame size. One can see the outer element frames (shown by gradually changing colors) in a numerical model in Fig. 2. It has to be mentioned that each frame consists of all the nodes and elements across the structure. For instance, in order to damp the waves in a honeycomb sandwich plate, dashpot elements which are applied on each frame are connected to all the nodes over the entire thickness of the sandwich structure including the nodes on the cover plates and the nodes in the honeycomb core.


3

Fig. 1. A schematic representing of dashpot elements connected to a three frame-size plate. For the sake of visualization, the number of elements shown in the sketch of the model has been reduced. Each element in the figure represents four elements in x and y directions for the original FE-model.

spring-damper option is provided by COMBIN14 elements. These uniaxial tension–compression elements are available with up to three degrees of freedom (i.e. x, y and z directions) at each node.

Symmetric boundary condition

3. Methodology

Sensor top FEMAPMaterial2:asd material7 FEMAPMaterial2:asd_1 FEMAPMaterial2:asd_2 FEMAPMaterial2:asd_3 FEMAPMaterial2:asd_4 FEMAPMaterial2:asd_5 FEMAPMaterial2:asd_6 FEMAPMaterial2:asd_7 FEMAPMaterial2:asd_8 FEMAPMaterial2:asd_9 FEMAPMaterial2:asd_10 FEMAPMaterial2:asd_11 FEMAPMaterial2:asd_12 FEMAPMaterial2:asd_13 FEMAPMaterial2:asd_14 FEMAPMaterial2:asd_15 FEMAPMaterial2:asd_16 FEMAPMaterial2:asd_17 FEMAPMaterial2:asd_18 FEMAPMaterial2:asd_19 FEMAPMaterial2:asd_20 FEMAPMaterial2:asd_21 FEMAPMaterial2:asd_22 FEMAPMaterial2:asd_23 FEMAPMaterial2:asd_24 FEMAPMaterial2:asd_25 FEMAPMaterial2:asd_26 FEMAPMaterial2:asd_27

Sensor bottom FEMAPMaterial2:asd material7 FEMAPMaterial2:asd_1 FEMAPMaterial2:asd_2 FEMAPMaterial2:asd_3 FEMAPMaterial2:asd_4 FEMAPMaterial2:asd_5 FEMAPMaterial2:asd_6 FEMAPMaterial2:asd_7 FEMAPMaterial2:asd_8 FEMAPMaterial2:asd_9 FEMAPMaterial2:asd_10 FEMAPMaterial2:asd_11 FEMAPMaterial2:asd_12 FEMAPMaterial2:asd_13 FEMAPMaterial2:asd_14

Z

Actuator

X

Y

Fig. 2. Schematic representation of the outer element frames in numerical model. The PZT elements’ orientation and the symmetric boundary conditions are also shown. The PZT actuator and sensors are modeled by SOLID5, coupled field elements with displacement and voltage degree of freedoms in ANSYSÒ 11.0.

The Lamb waves propagate along the medium with different wave forms, which are known as modes. Each mode can be either a symmetrical (S) mode or an anti-symmetrical (A) mode. By subtracting (or adding) the signals on the top and bottom surfaces of a plate one can identify the different modes propagating inside the structure. However, this method is not suitable for thick sandwich panels, where the arrival of the modes on the top and the bottom surfaces differs. In this paper different modes and their reflections are identified based on differences in the group velocity, the wavelength and the amplitude [11]. In addition, to verify the mode splitting, using B-scan images is also an alternative method to identify different modes and reflections [22,11], cf. Fig. 3. The displacements of the nodes (located along the wave propagation direction) in the time domain are shown in B-scan. The energy transmission caused by the reflected and propagated waves is measured to show the functionality of the proposed non-reflecting boundary condition. The transmitted energy is defined within this paper as the integral over the squared signal [23].

Etrans ¼ Location (mm)

0.1

Symmetric mode

0 Reflections

-0.1 Anti-symmetric mode

-0.15 -0.2 0

0.05

tend

V 2 ðtÞ dt:

ð3Þ

t start

0.05

-0.05

Z

0.1

0.15

0.2

Time (ms) Fig. 3. The propagation of the Lamb modes and reflection waves are shown in a Bscan diagram. The Lamb wave propagation is considered in a honeycomb sandwich plate. The geometrical properties of the plate are presented in Table 2, the central frequency of the loading signal is 250 kHz.

To execute these simulations the commercial FEM software ANSYSÒ 11.0 has been used. In this software, the longitudinal

To show the efficiency of the non-reflecting boundary, the ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) is calculated. As an example, one can see the Lamb waves and the reflection in Fig. 5 which are separated with a dashed line. The Ereflected and ELamb stand for the energy transmission caused by reflections and the Lamb waves (including both S0 and A0 modes), respectively. In this study an equal period of time which is needed for propagation of the Lamb modes (which depends on the central frequency of the excitation wave) is considered as a minimum period to measure the reflected waves. The value of less than 1 for this ratio, shows an attenuation in reflected waves and therefore one can distinguish the reflected waves from the propagated Lamb modes. The value of bigger than 1 indicates that non-reflecting boundary is too weak and the reflected waves are sensed several times with sensor. As the ratio tends to zero a better non-reflection boundary condition is indicated. In this study, the ratio of 0.009 and less is considered as

4


an ideal non-reflecting boundary condition. The post-processing calculations described in this paper have been performed using MATLABÒ. 4. Design parameters Following parameters are considered to design an efficient boundary condition using dashpot elements to avoid reflection from borders. Damping factor. Direction of dashpot elements. Number of dashpot elements. 4.1. Damping factor Eq. (2) indicates that the magnitude of the generated force with the dashpot elements is related to the damping factor and the nodal displacement. Therefore, the damping factor multiplied by the group velocity (C Vg) is used to show the capabilities of the dashpot elements as non-reflecting boundary. Fig. 4 shows the energy transmission values of the reflected waves over C Vg. Finding an efficient value of C Vg and knowing the group velocity of the Lamb waves (which depends on the central frequency of the excitation signal, the material properties of the propagating medium and the thickness of the plate) one can indicate the efficient damping factor. Fig. 4 demonstrates that very small values of damping factor do not generate enough force to avoid propagation of the reflected waves effectively. On the other hand, the nodal displacements are not big enough to move dashpot elements with very high damping factors. A range of 28,000–280,000 of C Vg is found for the minimum value of the ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb). In this example an aluminum plate with the thickness of 2 mm is considered and the Lamb wave with central frequency of 200 kHz is generated. The lowest group velocity belongs to the A0 mode and is approximately 2800 m/s [11], subsequently a damping factor between 10 and 100 (N s/m) is suggested. 4.2. Direction of dashpot elements Different direction of dashpot elements according to Eq. (2) are considered. It is observed that the dashpot elements in direction of wave propagation have the best effect to reduce the reflection waves. In this case one side of the dashpots are connected to the

1.2

fc = 200 kHz ELamb = 7.44e-8 J.Ω

nodes on the plate and the other sides are fully fixed (with C Vg equal to 2.8e5 N). In a particular example, an aluminum plate of 0.5 mm thickness is considered as an initial model, where the nodes on the outer borders are connected to the dashpot elements in direction of wave propagation (x). The Lamb wave is excited with a central frequency of 150 kHz, cf. Fig. 1. In this case the transmitted energy by the reflected waves is 1.13 107 J X and the ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) is equal to 0.7. In the first case the direction of dashpot elements are changed and set to the y direction. Subsequently, the energy transmission of the reflected waves increase to 166% with the initial model (Ereflected/ELamb = 2.0). In the second case the dashpot elements are to be considered to the direction of z. It is observed that the energy transmission of the reflected waves increase by 240% in comparison to the initial model (Ereflected/ELamb = 2.6). Fig. 5 compares the reflected waves in models with dashpot elements in x and z directions. Furthermore, the combination of dashpot elements in all three directions (x, y and z) is considered and only 3% reduction of the energy transmission of the reflected waves (Ereflected/ELamb = 0.68) in comparison to the initial model is observed. These results can be explained by the fact that in a reduced-size model (cf. Fig. 1) there are more nodes which are connected to the dashpot elements in the direction of wave propagation (x) in comparison to the other directions of y and z.

4.3. Number of dashpot elements The influence of number of dashpot elements on the reflecting wave is also considered. In a particular example an aluminum plate with the thickness of 2 mm is considered and the Lamb wave with central frequency of 200 kHz is excited. Initially 617 dashpot elements (with C Vg equal to 2.8e5 N and the dashpots are in x direction) are used to be connected to the first outer element row and column from the outer borders of the plate (one frame of outer elements, cf. Fig. 1). In this case the transmitted energy by the reflected waves is 2.05 109 J X, where the energy of the Lamb waves is 70.6 109 J X (Ereflected/ELamb = 0.03). To show the influence of dashpot element number on the reflected waves, it is increased in two steps and the results are compared with the initial model. In the first case the number of dashpot elements is increased to 1234 to be connected to two outer rows and columns (two frames of outer elements). In the second case the dashpot elements are further increased to 2468 elements, which are connected to four outer rows and columns (four frames

aluminum plate thickness: 2 mm

0.15 0.10

Model size: 4 frames Dashpots: 1 frame

Dashpots: 1 frame 0.6

Model size: 4 frames

0.15

0.05 0.00

0.10

-0.05

0.05

-0.10

0.00 0·105

1·10

5

2·10

5

3·10

5

5

4·10

5

5·10

5

6·10

Damping factor times the group velocity, C·Vg (N)

aluminum plate thickness: 0.5 mm

A0

0.8

Voltage (V)

Ereflected / E Lamb (-)

1.0

S0 Lamb waves Reflections

Dashpot in x direction Dashpot in z direction

-0.15 0.5·10-4

1.0·10-4

fc = 150 kHz C · Vg = 2.8e5 N 1.5·10-4

2.0·10-4

Time (s) Fig. 4. The ratio between the energy transmission of the reflected waves and the Lamb waves for different damping factors. A model of an aluminum plate with thickness of 2 mm is considered, and the Lamb wave is excited with a central frequency of 200 kHz.

Fig. 5. Reflected waves in models with different dashpot element directions. A model of an aluminum plate with thickness of 0.5 mm is considered, and the Lamb wave is excited with a central frequency of 150 kHz.

5


6 Damping

Fig. 6. Increasing number of dashpot elements is shown schematically in a sixframe size model.

Ereflected / ELamb (-)

5

Dashpot

4

Top surface honeycomb C · Vg = 2.8e5 N

3 Dashpots: 1 frame

2

Model size: 4 frames 1

0.15

0 Model size: 6 frames

Voltage (V)

0.10


0.00

A0 617 dashpots (1 frame) 2468 dashpots (4 frames )

-0.15 0.5·10-4

1.0·10-4

200

250

300

350

400

Frequency (kHz) Fig. 8. The values of energy transmission of the reflected waves are plotted over the central frequency of the excitation signal. Two different kinds of non-reflecting boundary conditions are considered. First a non-reflecting boundary condition based on gradually damped artificial boundary with four frames of damping materials on the borders is considered, labeled damping. Secondly a model with dashpot elements on the borders is taken into account, labeled dashpot. A honeycomb sandwich panel is considered, cf. Table 2 for the geometrical properties and cf. Table 3 for the materials properties.

S0

-0.10

150

Lamb waves Reflections

0.05

-0.05

100

fc = 200 kHz C · Vg = 2.8e5 N 1.5·10-4

2.0·10-4

Time (s) Fig. 7. Reflected waves in models with different number of dashpot elements. A model of an aluminum plate with thickness of 2 mm is considered, and the Lamb wave is excited with central frequency of 200 kHz.

of outer elements). Fig. 6 represents increasing number of dashpot elements schematically. The ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) decreases in the first and the second cases to 0.012 and 0.011, respectively. For all three cases the obtained ratio between the transmitted energy of the reflected waves and the Lamb waves is in an acceptable range and fairly far from 1, cf. Section 3. Fig. 7 compares the reflected waves in a model with 617 dashpot elements and a model with 2468 dashpot elements. The results in Fig. 7 and the Ereflected/ELamb values indicates that the additional number of dashpot elements does not improve the efficiency of the proposed boundary condition significantly. In addition, considering the fact that the most of added dashpot elements (in the first and second cases) are connected to the nodes along the outer rows one can conclude that the most of reflected waves which are sensed with the sensor are reflected from the outer columns (shown on the right hand-side of the actuator in Fig. 1).

5. Applications 5.1. Influence of different central frequencies The influence of changes in the central frequency of the excitation signal on the wave reflection in a honeycomb sandwich plate using a non-reflecting boundary condition with dashpot elements (labeled dashpot) is compared with the gradually damped artificial boundary which is introduced in [15] (labeled damping). These two approaches can be realized in commercial finite element software without difficulties. The geometrical properties and the materials properties are given in Tables 2 and 3, respectively. Fig. 8 shows that, as the central frequency increases less energy is transmitted by the reflected waves from the non-reflecting

boundary condition with gradually damped artificial boundary. A similar trend can be observed for the non-reflecting boundary with dashpot elements. This phenomenon can be explained by the fact that as the central frequency of the excitation Lamb wave increases, the energy which is transmitted by the Lamb waves, decreases. However, it is clear that the non-reflecting boundary with dashpot elements is less sensitive to the changes in central frequency of the exciting signal and it works enough good even in lower frequency ranges. Table 1 shows the absolute values of transmitted energy by the Lamb waves with different central frequencies. 5.2. Reduced-size model The major benefit to use dashpot elements is the possibility to reduce the model size. This reduction in the model size results in decreased computational costs. Within this study a non-reflecting boundary condition which results in an attenuation of the reflected waves is considered to be acceptable (Ereflected/ELamb < 1). This attenuation behavior will help to distinguish the reflected waves from the propagated modes in signal processing for structural heath monitoring applications [11]. Fig. 9 shows how the model reduction can effect the energy transmission of the reflected waves from the borders. It is clear that the attenuation of the reflected waves completely depends on the model size. This can be explained by the fact, that the damping factor of such a non-reflecting boundary is increasing gradually. Consequently, a bigger model has a greater attenuation effect on the reflected waves. However, the models with dashpots show less dependency on the model size (in this example only one frame of outer elements are connected to dashpot elements). This phenomena can also be explained by the fact that number of dashpot elements does not change the efficiency of the proposed boundary condition significantly, cf. Section 4.3 (by decreasing the model size, the number of dashpot elements is also decrease). 5.3. Comparison with an ‘‘ideal’’ non-reflecting boundary In order to verify the proposed non-reflecting boundary and model reduction, Fig. 10 compares the Lamb wave propagation in a model using an ‘‘ideal’’ non-reflecting boundary condition with

6


Table 1 Absolute ELamb values for different central excitation frequencies. Central excitation frequency (kHz)

100

ELamb (J X) (106)

0.060

Ereflected / ELamb (-)

2.5

150

0.046

200

250

0.036

0.014

300

350

0.0048

400

0.0017

Dashpots: 1 frame

Table 2 The geometrical properties of the honey comb sandwich panel and the PZT actuator/ sensor (units are in mm).

0.00057

Skin plate (28 frames)

Honeycomb cell

PZT actuator and sensor

Length Width Thickness

Height Thickness Core size

Radius Thickness actuator/sensor distance

290 124 2

15 0.5 4.8

Damping

2.0

Dashpot

1.5

Aluminum plate thickness: 2 mm

Honeycomb height

fc = 200 kHz C ·Vg = 2.8e5 N ELamb = 7.44e-8 J.W

1.0

Honeycomb core size Z

0.5

Honeycomb thickness

0.0 0

4

3.17 0.7 180

8

12

16

20

X

Y

Fig. 11. Schematic representation geometrical properties of a honeycomb sandwich panel, cf. Table 2.

Number of frames (model size) Fig. 9. The values of energy transmission of reflected waves plotted over the model size, cf. Fig. 1. Different non-reflecting boundaries including (a) damping materials and (b) dashpot elements are compared. An aluminum plate with thickness of 2 mm is considered, and the Lamb wave are excited with a central frequency of 200 kHz.

0.100 0.075

Young’s modulus (GPa)

Density (kg m3)

Poisson’s ratio (–)

Skin plate (Aluminum alloy T6061) 70 0.33

fc = 40 kHz

Top surface

C · Vg = 2.8e5 N

0.050

Voltage (V)

Table 3 Material properties of the plate and honeycomb cells [1].

0.025

2700

Honeycomb cell (HRH-36-1/8-3.0) Ex = Ey txy &tyz = txz (–) Ez (GPa) (GPa)

Gxy (GPa)

Gyz= Gxz (GPa)

Density (kg m3)

2.46

0.94615

1.154

50

3.4

0.3

0.000 -0.025 -0.050 -0.075


-0.100 0·10-4

1·10-4

Table 4 Material properties of the twill CFRP (0°/90°) plate [12].

Ideal non-reflecting boundary Dashpot non-reflecting boundary 2·10-4

3·10-4

4·10-4

Time (s) Fig. 10. Comparison of the Lamb wave propagation in a model using an ‘‘ideal’’ nonreflecting boundary condition with twenty damping frames of gradually damped artificial boundary (dashed line) and a reduced-size model with four frames and one frame of dashpot elements (solid line). The excited Lamb wave is propagating with a central frequency of 40 kHz in an aluminum plate of 2 mm thickness.

twenty damping frames of gradually damped artificial boundary (Ereflected/ELamb is equal to 0.0027) and a reduced-size model (with four frames) with one frame of dashpot elements (Ereflected/ELamb is equal to 0.17). The time-of-flight is the same for both signals. But the A0 mode in the model with dashpot elements has relatively higher amplitude which can be explained by the amplification effect of reflections from the borders in a reduced-size model.

Ex Ey Ez

127.5 (GPa) 7.9 (GPa) 7.9 (GPa)

txy tyz txz

0.273 (–) 0.348 (–) 0.017 (–)

Gxy Gyz Gxz

5.58 (GPa) 2.93 (GPa) 2.93 (GPa)

an aluminum plate (thickness of 2 mm) from borders of the combined boundary increases by 23% in comparison to the model in which only dashpot elements are used (where the transmitted energy by the reflected waves is 2.05 109 J X). The ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) for the models with dashpot elements and combined boundary are 0.02 and 0.03, respectively. This increase can be explained due to the fact that the material damping results in a reduction in the nodal velocity and subsequently the dashpot force decreases. 5.5. Wave propagation in composite structures

5.4. Combination of damping materials and dashpot elements Furthermore, a combination of gradually damped artificial boundary [15] and a dashpot boundary is considered, labeled combined boundary. The energy transmission by the reflected waves in

In this section two examples of application of the non-reflecting boundary condition on the wave propagation in a honeycomb and a CFRP plates are presented. Finally, the experimental results are presented.

7


0.15

Top surface honeycomb

fc = 200 kHz C · Vg = 2.8e5 N

0.10

Maximum

Dashpots: 1 frame

Voltage (V)

Model size: 4 frames 0.05 0.00

S0 -0.05

3D laser sanning vibrometer

A0 Damping


-0.10

Dashpot -0.15 0·10-4

0.5·10-4

1·10-4

1.5·10-4

Silicon for damping Retro

-refle

2·10-4

Time (s)

ctive

Actuator Position

Fig. 12. Reflected waves are compared for a honeycomb reduced-size model with four frames of damping materials on borders, labeled damping. Secondly the same size model is considered with dashpot elements on borders, labeled dashpot. The graph shows the voltage signal received from the sensor on the top surface of the honeycomb model, the signal has a central frequency of 200 kHz, cf. Fig. 1 for definition of frames and the reduced model size.

layer

te

te osi

pla

mp

Co

Fig. 14. Setup for experimental test. -9

Displacement (m)

4·10

Maximum

fc = 150 kHz

3·10-9

C · Vg = 2.8e5 N

2·10-9

A0

1·10-9

Without dashpot With dashpot

S0

-9

0·10

-1·10-9

Model size: 4 frames

Twill CFRP (0/90°)

-4·10

0·10-4

0.5·10-4

1·10-4

1.5·10-4

2·10-4

Time (s) Fig. 13. Propagated Lamb wave (nodal displacement signal) in twill CFRP (0°/90°) plate (a) without non-reflection boundary and (b) with non-reflection boundary using dashpot elements. The excited Lamb wave is propagated with a central frequency of 150 kHz. The plate thickness is 1 mm, and the rest of geometrical properties are presented in Table 2, cf. Table 4, also cf. Fig. 1 for definition of frames and the reduced model size.

The geometrical properties of the honeycomb sandwich plate and the PZT transducers are presented in Table 2. Fig. 11 represents the geometrical dimensions of a honeycomb sandwich plate. The same dimension is used for the other plates in the following sections. The material properties of the honeycomb sandwich panel components and the aluminum plate are provided in Table 3, while the material properties of the twill CFRP (0°/90°) plate are shown in Table 4. It has been reported in [12] that the finite element model of a twill CFRP (0°/90°) plate without matrix provides results which are in a better agreement with the experimental results. The fibers are modeled using the material coordinate option in the finite element model. The dielectric matrix [e] and the piezoelectric matrix [e] are, respectively [1]:

6 ½e ¼ 4 0 0

5:2

3

0

and the stiffness matrix is Dashpots: 1 frame

-3·10-9

6:45

0

0 12:7

-2·10

2

0

60 0 5:2 7 7 6 7 6 60 0 15:1 7 7ðC m2 Þ; ½e ¼ 6 60 0 0 7 7 6 7 6 4 0 12:7 0 5


-9

-9

2

0 6:45 0

0

3

7 0 5109 ðC V1 m1 Þ; 5:62

2 6 6 6 6 ½c ¼ 6 6 6 6 4

13:9

3

6:78

7:43

0

0

0

13:9

7:43

0

0

0

0

0 7 7 7 0 7 10 710 ðPaÞ: 0 7 7 7 0 5

11:5 sym:

3:56

0 2:56

2:56 The mass density of the PZT is 7700 kg m3 [1] and the mass density of the twill CFRP (0°/90°) is 1550 kg m3 [12]. The skin plates of the honeycomb sandwich plate are modeled using cubic 3D solid elements while 2-D shell elements are used to model the honeycomb cells. Fig. 11 shows the connection of 2-D shell elements (core) and the 3-D elements (cover plate), however, the thickness of core structure is shown for a better visualization and explanation of geometrical dimensions. An accurate modeling approach is needed to consider the correct stresses and deformation in the connection region where the solids elements links to the shells which is often the weakest area. In the Lamb wave propagation study, where we are only interested in the displacement field of the plate, multi-point constraint equations are a reliable method to connect two portions of a structure using solids and shells [24]. Constraint equations are deployed to join the shell and solid elements within this study. 5.5.1. Honeycomb composite plate Fig. 12 compares effects of different non-reflecting boundary conditions (including artificial damping boundary and dashpot boundary) on the wave propagation in a reduced-size honeycomb model. In the reduced model four frames of outer boundary elements are used and the plate length is reduced to 220 mm and the plates width is reduced to 24 mm (the dimensions of the original model are given in Table 2 which represents a model with

8


40

Maximum value of the CWT coefficients

S0 35

A0

Scale (-)

30 25 20 15 10

Time of the flight for S 0 150

200

250

300

Time of the flight for A 0 350

400

450

500

Time increment (-) Fig. 15. Schematic representation of the absolute values of the CWT coefficients based on the Daubechies wavelet D10 in a contour plot. The Lamb wave is excited with a central frequency of 200 kHz on a honeycomb sandwich plate.

agation in reduced-size CFRP model (cf. the dashed line in Fig. 13). The ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) is equal to 1.2 (acceptable range is less than 1) for the model without non-reflecting boundary condition.

Difference in group velocity (%)

Honeycomb top surface: twill CFRP (0/90°) 20 10 0 -10

S0 mode

-20

A0 mode 75

100

125

150

175

200

Frequency, fc (kHz) Fig. 16. The group velocity values which are obtained from the experimental tests are compared with simulation results at all tested frequencies. A honeycomb sandwich plate (the honeycomb cell height is 8 mm and the cover plate is made of twill CFRP (0°/90°) with 1 mm thicknesses, cf. Table 2 for the rest of geometrical properties, cf. Table 3 and 4 for the material properties) is investigated.

twenty-eight frames of outer boundary elements, also cf. Fig. 1 for definition of frames and the reduced model size). The attenuation behavior of the reflected waves in the model with non-reflecting boundary using dashpot elements is clearly shown (cf. the solid line in Fig. 12). One can observe the amplification and attenuation effects of the reflected waves on the wave propagation in reduced-size models with the gradually damped artificial boundary. It is clear that the reflected waves in the model with gradually damped artificial boundaries have almost the same amplitude as the propagated A0 which can cause difficulties to identify modes in a later performed signal processing (cf. the dashed line in Fig. 12). The ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb) is equal to 1.1 (the acceptable range is less than 1) for the model with gradually damped artificial boundary. 5.5.2. CFRP composite plate Fig. 13 presents an application example of using dashpot elements as a non-reflecting boundary in a twill CFRP (0°/90°) plate as a complicated composite structures. The plate thickness is 1 mm and the rest of geometrical properties are presented in Table 2 and cf. Table 4 for material properties. It is clearly shown how dashpot elements can reduce the amplitude of the reflected waves from the borders (cf. the solid line in Fig. 13). In addition, it is shown that the reflected waves may influence the wave prop-

5.5.3. Experimental validation In addition to the numerical verification, results are also validated experimentally. The experimental setup is shown in Fig. 14. The velocity of the nodes on the retro-reflective layer is measured with scanning a laser vibrometer to evaluate the wave properties [22,12]. The flight velocity of the propagated waves are calculated in reduced-size numerical models and compared with experimental results. The nodal displacement signal in the vertical direction u(t) is transformed using the continuous wavelet transform (CWT) based on the Daubechies wavelet D10 to evaluate the flight velocities (the bar indicates complex conjugation).

1 WTða; bÞ ¼ pffiffiffi a

Z

þ1

1

ta dt: uðtÞw b

ð4Þ

The location of the maxima of the CWT coefficients gives the timeof-flight for each Lamb wave mode, cf. Fig. 15. Knowing the distance and the time-of-flight one can calculate the group velocity [11]. The time-of-flight is measured between the excitation and the arrival at two different points on the top and the bottom surface, in order to show the influence of the sandwich plate thickness and the core material on the flight velocity of the wave propagation. Therefore, the flight velocity on the bottom surface includes the effect of plate thickness and differs from the flight velocity on the top surface which is known as the group velocity. In this paper we use ‘‘group velocity’’ to describe the flight velocity for both top and bottom surfaces. Because of the combination of the propagated modes and the reflected waves in the reduced-size models without non-reflecting boundary the location of the maxima of the CWT coefficients may change up to 20% (in comparison to experimental results). Fig. 13 shows how the group velocity of the A0 mode changes in a model without non-reflecting boundary condition. One can observe that the maximum amplitude of the A0 mode occurs approximately 0.2 ms earlier in the model without non-reflecting boundary which can be explained with amplification and attenuation effects of the reflected waves on the wave propagation in the reduced-size model. A similar illustration is given in Fig. 12 where a weak nonreflecting boundary with gradually damped artificial in reduced size model may effect the group velocity of the A0 mode. Therefore,

9

15

top surface

Aluminum honeycomb

10 5 0 -5 -10 -15 100

150

200

250

300

350




bottom surface 15 10 5 0 -5 -10 S0 mode A0 mode

-15 100

150

Frequency, fc (kHz)

200

250

300

350

Frequency, fc (kHz)

Fig. 17. The group velocity values which are obtained from the experimental tests are compared with simulation results for different excitation frequencies. The group velocities are compared on both sides of a honeycomb plate which is made of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cell size is 6.4 mm and the honeycomb wall thickness is 0.0635 mm.

the agreement between the group velocities in the simulation and experimental tests is considered to show the capabilities of the proposed reduced-size model using non-reflecting boundary based on dashpot elements. In the first example, the excited Lamb wave is propagated in a honeycomb sandwich plate, where the honeycomb cell height is 8 mm and the cover plate is made of twill CFRP (0°/90°) with 1 mm thickness, cf. Table 2 for the rest of geometrical properties, cf. Table 3 for the material properties of the honeycomb cells, and cf. Table 4 for the material properties of the CFRP plate. Fig. 16 compares the group velocity values of different modes which are obtained experimentally and numerically, where the central frequency of the excitation signal is increased from 50 kHz to 220 kHz. The relative difference in percent is calculated as

Erel ¼ 100

Gsim Gexp ½%: Gexp

ð5Þ

Gsim represents the group velocity values which are obtained from the simulation test, and Gexp shows the group velocity values which are obtained from the experimental investigations. For the proposed example we achieve a good agreement of the results. The average of absolute differences (jErelj) is 2.30% and the absolute maximum is 8.01%. In another example the group velocity values of the propagated waves in a honeycomb sandwich palate are compared on both sides of the structure, cf. Fig. 17. The honeycomb plate is fully made of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cell size is 6.4 mm and the honeycomb wall thickness is 0.0635 mm. The relative error is not exceeding 11% and the average of absolute differences is 4.11%. 6. Summary A non-reflecting boundary condition using dashpot elements is introduced in order to reduce the reflections of the Lamb waves at boundaries. It has been shown that by applying the proposed boundary condition the computational costs for wave propagation simulations can be reduced significantly. Different parameters including the damping factor, the direction of dashpot elements and the number of dashpot elements were examined to design an efficient non-reflecting boundary. The influence of each parameter is summarized as follows: Damping factor: It has been shown that very small damping factors do not generate enough force to suppress the propagation of the reflected waves effectively. On the other hand, the nodal displacements are not large enough to move dashpot elements

with very high damping factors. A range of 28,000–280,000 of C Vg (damping factor times group velocity) is found as an appropriate choice to significantly reduce the propagation of the reflected waves. Direction of dashpot elements: It has been observed that the dashpot elements in direction of wave propagation are most effective in reducing the reflections. Number of dashpot elements: To apply the new non-reflecting boundary condition, dashpot elements are connected only to one row and column of the outer elements as a primary choice. It has been indicated that the additional number of dashpot elements does not improve the efficiency of the proposed boundary condition significantly. The application of the proposed method is demonstrated for the Lamb wave propagation within different heterogenous materials including a honeycomb sandwich plate with twill CFRP (0°/90°) cover plate and an aluminum honeycomb sandwich plate. In addition, results were validated experimentally. Furthermore, combination of dashpot elements with gradually damped artificial boundary were considered and it has been shown that this combination has only a minor effect on reflection of the waves from the borders and causes extra efforts in the modeling process.

7. Conclusions The advantages of the proposed non-reflecting boundary condition using dashpot elements to reduce the computational costs together with the possibility of implementing the proposed scheme in commercial FEM packages, provide an efficient tool for researchers to numerically investigate the interaction of the Lamb waves within complicated structures such as CFRP and honeycomb composites even with ordinary personal computers. These investigations are very important to design further health monitoring systems using Lamb waves for composite structures. However, further investigations are required to extend the application of the proposed non-reflecting boundary condition to study wave propagation in other heterogenous structures.

Acknowledgments By means of this, the authors acknowledge the German Research Foundation for the financial support (GA 480/13). We would also like to thank C. Willberg for that was invaluable in the execution of the experimental tests.

10


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 F, Giordano M, editors. Proceedings 5th european workshop on structural health monitoring. Sorrento, Italy; 2010. [3] Terrien N, Osmont D. Damage detection in foam core sandwich structures using guided waves. In: Leger MDA, editor. Ultrasonic wave propagation in non homogeneous media, springer proceedings in physics, vol. 128. Berlin, Heidelberg: Springer; 2009. p. 251–60. [4] Kohr T, Petersson BAT. Wave beaming and wave propagation in lightweight plates with truss-like cores. J Sound Vib 2009;321:137–65. [5] Weber R. Numerical simulation of the guided Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. Master’s thesis, Institute of Numerical Mechanics, Department of Mechanical Engineering. Germany: Otto-von-Guericke-University Magdeburg; 2011. [6] Thwaites S, Clark NH. Non-destructive testing of honeycomb sandwich structures using elastic waves. J Sound Vib 1995;187(2):253–69. [7] Wierzbicki E, Wozńiak C. On the dynamic behavior of honeycomb based composite solids. Acta Mech 2000;141:161–72. [8] Ruzzene M, Scarpa F, Soranna F. Wave beaming effects in two-dimensional cellular structures. Smart Mater Struct 2003;12:363–72. [9] 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. [10] Qi X, Rose JL, Xu C. Ultrasonic guided wave nondestructive testing for helicopter rotor blades. In: 17th World conference on nondestructive testing. Shanghai, China; 2008. [11] Hosseini SMH, Gabbert U. Numerical simulation of the Lamb wave propagation in honeycomb sandwich panels: a parametric study. Compos Struct 2012 [Accepted for publication].

[12] Willberg C, Mook G, Gabbert U, Pohl J. The phenomenon of continuous mode conversion of Lamb waves in CFRP plates. Key Eng Mater 2012;518:364–74. [13] Alpert B, Greengard L, Hagstrom T. Nonreflecting boundary conditions for the time-dependent wave equation. J Comput Phys 2002;180:270–96. [14] Lysmer J, Kuhlemeyer R. Finite dynamic model for infinite media. J Eng Mech Div ASCE 1969;95:859–77. [15] Liu GR, Jerry SQ. A non-reflecting boundary for analyzing wave propagation using the finite element method. Finite Elem Anal Des 2003;39:403–17. [16] Drozdz M, Moreau L, Castaings M, Lowe MJS, Cawley P. Efficient numerical modelling of absorbing regions for boundaries of guided waves problems. In: AIP conference proceeding (820); 2006. p. 126. [17] Bérenger JP. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 1994;114:185–200. [18] Sim I. Nonreflecting boundary conditions for time-dependent wave propagation. Ph D thesis, faculty of science. Switzerland: University of Bassel; 2010. [19] Eymard RHR, Gallouët T. Handbook of numerical analysis Vk. Ch. finite volume methods. North Holland; 2000. p. 713–1020. [20] Cheng AH-D, Cheng DT. Heritage and early history of the boundary element method. Eng Anal Bound Elem 2005;29:268–302. [21] Bower A. Applied mechanics of solids. CRC Press; 2010. [22] 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. [23] 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. [24] Suranat K. Transition finite elements for three-dimensional stress analysis. Int J Numer Meth Eng 1980;15:991–1020.