Influence of the cover plate thickness on the Lamb wave propagation in honeycomb sandwich panels

Transcription

1 CEA Aeronaut J (2013) 4:69 76 DOI /s ORIGINAL PAPER Influence of the cover plate thickness on the Lamb wave propagation in honeycomb sandwich panels. M. H. Hosseini U. Gabbert R. Lammering Received: 11 April 2012 / Revised: 19 November 2012 / Accepted: 13 December 2012 / Published online: 28 December 2012 Ó Deutsches Zentrum für Luft- und Raumfahrt e.v Abstract Within this paper, the guided Lamb wave propagation in thin honeycomb sandwich panels is studied. The Lamb waves are excited by thin piezoelectric (PZT) patch actuators glued to the surface of the plate, and the signals are received by similar PZT sensor patches. uch actuator and sensor systems can be used for a cost-effective online health monitoring of structures. In homogeneous plates, Lamb waves propagate with symmetrical and antisymmetrical modes. However, the propagation in heterogeneous media is not as clear and depends on the exciting frequency, the material properties, and the geometry of the structure. In this paper, the influence of the geometrical properties of honeycomb plates on the Lamb wave propagation is studied. For this purpose detailed 3-D finite element calculations are performed, which result in very time consuming computations. Consequently, also simplified models are developed to reduce the computing time without losing the quality of the results. To this end, the honeycomb core material is replaced by a homogeneous layer with orthotropic mechanical properties. The homogenized properties are calculated numerically using a homogenization technique based on the representative volume element method. The comparison of the results received with the two different approaches has shown that the simplified. M. H. Hosseini (&) U. Gabbert Institut für Mechanik, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany rsg.931@gmail.com U. Gabbert ulrich.gabbert@ovgu.de R. Lammering Institut für Mechanik, Helmut chmidt Universität Hamburg, Hamburg, Germany rolf.lammering@hsu-hh.de models are in a good agreement with the extended models for a certain range of exciting frequencies and geometrical properties only. The wave propagation on the top and bottom surfaces is also compared in order to show how deep the waves can travel inside the structure. Keywords Guided waves sandwich plate Finite element method 1 Introduction The application of high frequency guided Lamb waves in thin-walled structures is a challenging technique in industry to receive information about the health state of a structure (structural health monitoring HM). uch Lamb waves can be simply excited and received by a network of thin piezoelectric (PZT) patches glued to the surface of the structure. In recent years a lot of papers have been published dealing with the interaction of Lamb waves with different types of damages in metallic structures as well as in composite materials [2]. The high sensitivity of ultrasonic waves with respect to small structural changes and the low costs of health monitoring systems built from piezoelectric patches, make such systems very attractive for industrial applications. The wave fields are definitely changed by structural damages. But, unfortunately, there are a lot of open questions, especially regarding the application of ultrasonic waves in layered composite structures. In such structures creeping mode conversions, small reflections at inner boundaries and at small thickness changes, amplitude reductions due to material damping, etc., have been observed also in undamaged structures. Consequently, in such cases it is complicated to estimate structural changes (e.g. the type, the size, and the position

2 70. M. H. Hosseini et al. of damage) reliably and with a sufficient accuracy. To overcome such problems a large amount of papers are dealing with different aspects of the computer-assisted design of reliable HM systems [1, 5]. The situation becomes even more complex and complicated if the wave propagation is studied in honeycomb sandwich panels. Only a few papers have been found dealing with this special case of a heterogeneous material system (see for instance [9, 11, 12]). In the paper by ong et al. [9] a 3-D finite element model is used to investigate the propagation of guided waves excited by a PZT actuator/sensor system in a honeycomb panel. In order to reduce the computational effort the authors have also used a simplified model with a homogenized core layer. A good agreement of the 3-D model and simplified model is obtained, if the central frequency of the exciting signals is relatively low (5 khz). In higher frequency ranges (from 40 until 90 khz) there is no agreement between both models. Interesting is that the authors found again a good agreement between both models if the exciting frequency is about 100 khz; only the amplitudes are different. The experimental and the simulation results are always in a good agreement. In another study by Hosseini and Gabbert [3] also a 3-D finite element method is used to study the wave propagation in honeycomb sandwich panels. In this study two different honeycomb sandwich panels with different geometrical properties are considered. It is shown that the ultrasonic waves propagate mainly either in the top layer or more in the core layer depending on the thickness relations and the properties of the respective materials. In the present paper, the main aim is to investigate the influence of changes of the cover plate thickness (t p ) (cf. Fig. 1) of the honeycomb sandwich structure on the wave propagation. In addition to a 3-D finite element model of the honeycomb structure, two other models are also considered. First, a simplified model with homogenized material properties in the core layer is evaluated, and the aim is to study the influence of geometry and material properties individually on the wave propagation as well as reducing the calculation time. And secondly, a single plate model is also considered; in this model the cover plate of the honeycomb structure is taken as the single plate and the rest of components of the sandwich structure are omitted. In this case, the influences of the core and bottom face are neglected. Comparing the results from the extended honeycomb model and single plate model one can see clearly the influence of the core layer and cover plate on the wave propagation. The paper is organized as follows: At first the finite element modeling of the sandwich panel is presented, and the data of the test cases are given; the test cases are aimed to study the influence of the changes in thickness of the cover plate and also changes in the central frequency of the exciton signal t p Height Cell size t h on the wave propagation in the honeycomb sandwich panel. Then the methodology to receive the phase and group velocity from the finite element results is discussed. In the next section the calculated results are given and general conclusions are drawn. The paper concludes with a summary and outlook to further investigations. 2 Lamb wave propagation in honeycomb sandwich plates 2.1 FEM modeling core layer Fig. 1 The sandwich panel design and geometrical properties (cf. Table 1) sandwich panels consist of two plates on top and on bottom with a core material in the middle. The thickness of the plates is normally much higher than the thickness of the honeycomb material. Thus, to be able to include the symmetrical as well as the asymmetrical wave modes it is obvious to model the cover plates with 3-D finite elements. 2-D finite shell elements are sufficiently accurate to model the honeycomb cell structures. For modeling the piezoelectric actuators and sensors, 3-D electromechanical coupled solid elements are applied. In addition, in case of the simplified model, the honeycomb core layer is also modeled with 3-D solid finite elements with orthotropic material properties. For a proper estimation of the orthotropic material parameters also a homogenization method based on representative volume element (RVE) is applied [8]. The RVE is a sample volume of a heterogenous material which is large enough to represent effectively all microstructural heterogeneities of the structure. After applying the periodic boundary conditions to the RVE model, several tensile tests are implemented to evaluate the mechanical properties of the RVE in different directions. In these cases the average stress is calculated, dividing the resulting tractions on the borders of RVE by the surface area [4]. X Z Y

3 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 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 Lamb wave propagation 71 symmetric boundary condition increasing damping factor actuator ties sensor bottom increasing damping factor sensor top Fig. 2 The orientation of the PZT elements in a single plate model. In addition, symmetric boundary condition and non-reflecting boundary condition are shown In the test example, the piezoelectric sensor is located in parallel to the actuator on both top and bottom layers. The sensors are glued to the structure in a distance of 180 mm from the actuator in the x direction (cf. Fig. 2). The bottom nodes of the PZT elements are considered as grounded. The exciting signal is an electric voltage in form of a half-cycle narrow band tone burst [9], which is applied to the top nodes of the actuator (t is time, f c is central frequency and H(t) is the Heaviside step function) as: V in ¼ V½HðtÞ Hðt 3:5=f c ÞŠ 1 cos 2pf ct sin 2pf c t: 3:5 ð1þ To represent the conductivity of the copper layer on the top layer of the sensor, all nodes on the top layer of the sensor are tied together. ymmetric boundary conditions are applied to reduce the model size. Also non-reflecting X Z Y factors of the elements are additionally increased gradually from non-damped elements to elements with high damping factors at the free borders, by applying an exponential function. These procedures make sure that the results are not influenced by the boundaries of the test specimen (cf. Fig. 2). The element size of the applied finite element mesh has been evaluated to guarantee numerical results with high accuracy. It has been proved that a mesh size smaller than one tenth of the wavelength results in solutions with sufficient accuracy. ong et al. [9] have shown that results calculated with this mesh size also match well with experimental results. In a case study several honeycomb sandwich panels with different thickness of the cover plate (t p ) from 0.5, 0.75, 1, 1.25, 1.5, 1.75 and 2 mm have been analyzed to investigate the influence of the cover plate thicknesses on the wave propagation. By increasing the thickness of the honeycomb core unit cells (t h ) more energy will transmit to the sensors, therefore, rather large value of 1.48 mm has been considered for t h in order to receive a clear response from the sensors. The rest of the geometrical properties are shown in Table 1 (see also Fig. 1). Table 2 shows the material properties of the skin plates and the honeycomb core materials. The material properties of the PZT actuators and sensors are presented in [9]. The dielectric matrix ½eŠ and the piezoelectric matrix [e], are, respectively, 2 3 6:450 0 ½eŠ ¼4 6: ðcv 1 m 1 Þ; ymmetry 2 3 5: : : :1 ½eŠ ¼ ðc m 2 Þ; : :7 0 and the stiffness matrix is :9 6:78 7: :9 7: : ½cŠ ¼ 6 3: ðpaþ: 7 4 2: ymmetry 2:56 boundary conditions [5], are applied to reduce the wave reflections from free borders of the plate. The damping The calculations are performed using the commercial finite element package ANY 11.0.

4 72. M. H. Hosseini et al. 2.2 Methodology The Lamb waves propagate along the media with different modes with different group velocities and wave-lengths. A continuous wavelet transform (CWT) based on the Daubechies wavelet D10 is used to calculate the time of flight for each mode [10, 12]. Using the time of the flight and knowing the distance between sensor and actuator (in our test cases 180 mm) one can calculate the group velocity for each mode [10]. The phase velocity and the wave length of each mode can be determined using a fast Fourier transform algorithm. The phase velocity can be expressed in terms of the frequency using the following equation: 2pfL tðf Þ¼ ð2þ ½/ðf Þ / 0 Š where f is the frequency, / 0 is the exciting phase function, /(f) is the received phase function, and L stands for the axial distance between the actuator and the sensor, [7]. Dividing the phase velocity by the frequency will give the wavelength [6]. It must be mentioned that a mode with a wavelength a is only able to determine damages bigger than a [6]. The evaluation of the finite element results is performed with help of the software package MATLAB Ò. 3 Results To show the influence of changes of the cover plate thickness (t p ) (cf. Fig. 1) of the honeycomb sandwich structure on the wave propagation, the results will be presented in the following order: Table 1 Geometrical properties of the sandwich panel (units are in mm) kin plate cell [9] PZT actuator/sensor [9] Length Width Cell size Height Radius Thickness Voltage and displacement based time signal: the electrically excited ultrasonic wave is received at the piezoelectric sensor again in form of an electric voltage signal measured at the top nodes of the piezoelectric finite elements, called voltage signal. In addition, the out of plane displacement (in z direction) of a single node located on the sensor is also evaluated, called nodal displacement signal (cf. Fig. 3). The voltage (or out of plane displacement) of a specific node for different time increments called time based signal. In this part different time signals based on nodal voltage and out of plate displacement are described and compared. 2. Wave field: to visualize the wave transformation in the structure, a 2-D and 3-D wave field snapshot is presented in this part. 3. Group velocity: as a first property of wave propagation in a structure the group velocity of different modes in different structures is presented. 4. Wave length: using wave lengths one can predict which kind of damage can be detected by each mode in a specific structure. In all mentioned parts, results from three different models are compared, a 3-D model of the honeycomb structure, a simplified model with a homogenized core layer and a single cover plate. In different models changes of the cover plate thickness (t p ) are applied. In addition, the influence of changes in the central exciting frequency is also evaluated. 3.1 Voltage and displacement based time signal In this part the signals based on voltage and out of plane displacement are compared. Figure 3 compares the sensor signals in the time domain calculated with different excitation frequencies from 5 to 150 khz. In the first row the responded voltage signals at the sensor patch, and in the second row the z-displacement at the reference node on top of the sensor patch are shown. It can be seen in the low frequency range that both results are not matching. Table 2 Material properties of the plate and honeycomb cells Young s modulus (GPa) Poisson s ratio Density (kg m -3 ) kin plate (aluminum alloy T6061 [9]) ,700 E x = E y (GPa) E z (GPa) t xy and t yz = t xz G xy (GPa) G yz = G xz (GPa) Density (kg m -3 ) cell (HRH-36-1/8-3.0) [8]

5 Lamb wave propagation 73 Normalized Displacement (-) Normalized Voltage (-) (a) Time signal obtained based on nodal voltage of nodes on free surface of PZT sensor Fexciting = 5kHz 0.5 Fexciting = 100 khz Fexciting = 40 khz Fexciting = 150 khz (b) Time signal obtained based on out of plane nodal displaciment of nodes on free surface of PZT sensor Fexciting = 5kHz 0.5 Fexciting = 40 khz Fexciting = 150 khz Fexciting = 100 khz ingleplate implifiedmodel Time (s) Time (s) Time (s) Time (s) Fig. 3 Responded time domain voltage signals based on voltage and out plane displacement. The results are plotted for honeycomb (solid line), simplified model (big dashed) and single plate (small dashed). Inc: 200 Time: 4.4e e e e e e e e e e e e-010 (a) (b) Z X But with increasing central frequency of the exciting signal a better agreement can be observed for all models. 3.2 Wave field Y Fig. 4 The wave field of uz in sandwich panel structure, the exciting signal with central frequency of 100 khz has been generated. In the model a tp is 1.75 mm and in part b tp is 0.75 mm. In both cases th is 1.48 mm The thickness of the cover plate is (tp) 0.5 mm and the honeycomb thickness (th) is 1.48 mm. The sensor is 180 mm far from the actuator implified A 3-D snapshot of the wave field uz in the sandwich panel with the plate layer of 1.75 and 0.75 mm (tp) is shown in Fig. 4. It can be seen that the group velocity and the wave length is nearly the same for the mode. On the other hand, Fig. 5 shows the 2-D wave forms in a quarter of the plate calculated with the three different models; the central frequency of 150 khz and a cover plate thickness 0.5 mm (tp) is used. As the conclusion in this part it must be mentioned that in Fig. 4 one can see that for a different thickness of the skin plate (tp) the wave still can travel inside the structure ensor Actuator Inc: 250 Time: 5.500e e e e e e e e e e e e-010 Displacement Z (m) Y X Fig. 5 The uz wave field on top surface of honeycomb, simplified and single plate structures. Central frequency of the exciting signal is 150 khz, tp is 0.5 mm and th is 1.48 mm

6 74. M. H. Hosseini et al. distinguishably. However, a pervious study by Hosseini and Gabbert [3] has shown that for a weaker honeycomb core and thicker cover plate of the sandwich panel, the wave mostly propagates on the top plate. In addition, it has been shown in Fig. 5 that the wave form is similar in all three models. However, the group velocity of the traveling wave is higher in the single plate compare with the two other models. 3.3 Group velocity Figure 6 shows the influence of changes in the exciting frequency on the group velocity in models with specific geometrical properties. The results are calculated on the top and on the bottom plates for different frequency ranges. In Fig. 7 the influence of the cover plate thickness on the group velocity of the mode on the top and on the bottom layer is presented for all three different models. Only the mode is shown in the figure, because the mode is hardly seen in models with higher plate thicknesses. It must be mentioned that the presented results are fitted linearly. To summarize the results in this part, one can see in Fig. 6 that both and propagate slower on the bottom plate in all cases. As expected, the mode propagates faster than. In Fig. 7 it is obvious that the group velocity of the single plate model is not influenced by the plate thickness; the solutions calculated with the two other models show more significant influence of the thickness. All three models indicate that the wave propagates faster on the top layer than on the bottom layer. 3.4 Wave length Figure 8 shows the wave length of different modes calculated for the top and the bottom layer in dependence of the frequency. In Fig. 9 the wave length of the anti-symmetric modes calculated from the nodal and voltage signals of the top surface sensors are compared. The cover plate thickness of the honeycomb structure (t p ) is changing from 0.5 till 2 mm and the central frequency of the exciting signal is 100 khz. As the result Fig. 8 shows that the wavelength of the mode in all there models are in a good agreement. In Fig. 9 it can also be seen that the results from the voltage and nodal signals are in a same range. Also, it is clear that by increasing of t p, the wavelength is decreasing in both top and bottom surfaces in all models. Group velocity (m/s) t h = 1.48, t p = 0.5 mm t h = 1.48, t p = 0.5 mm Centeral exciting frequency (khz) Group velocity (m/s) Bottom surface implified model Centeral exciting frequency (khz) Fig. 6 The influence of changes in the central exciting frequency on the group velocity Group velocity (m/s) t h = 1.48 mm F exciting = 100 khz implified model Plate s thickness, t (mm) p Group velocity (m/s) t h = 1.48 mm F exciting =100kHz Bottom surface Plate s thickness, t (mm) p Fig. 7 Dependency of the group velocity on the cover plate thickness, which is changing from t p equal mm

7 Lamb wave propagation 75 Wave length (m) Centeral exciting frequency (khz) t h 0.14 Bottom surface = 1.48, t p = 0.5 mm t h = 1.48, t p = 0.5 mm implified model Wave length (m) Centeral exciting frequency (khz) Fig. 8 Wave length of different modes on the top layer (left) and on the bottom layer (right) based on changes of central exciting frequency from 5 to 400 khz Wave length (m) mode mode F exciting = 100 khz t h = 1.48 mm F exciting = 100 khz t h =1.48mm Voltage signal Plate s thickness, t (mm) p Wave length (m) Nodal signal implified model Plate s thickness, t (mm) p Fig. 9 Wave length as function of the plate thickness 4 ummary and outlook The Lamb wave propagation in honeycomb sandwich panels excited and received with piezoelectric actuators and sensors has been analyzed with three different 3-D finite element models. In the detailed 3-D model the honeycomb core layer is modeled with shell type elements. In the simplified model, the core layer has been homogenized to an orthotropic layer which is also modeled with 3-D finite elements. In the third model, the honeycomb plate is modeled as a single layer where the influences of the core and bottom face are refused. It has been shown that for the investigated parameter variations the thickness of the skin plates (t p ) does not play a significant role to prevent waves travel inside the structure. Also, it has been shown that the wave form in all three models is similar. However, by comparing the group velocity values in the honeycomb sandwich panel and single plate; it has been observed that the honeycomb core layer in the extended honeycomb model causes lower group velocity values on the bottom surface compared to the top surface, while in the single plate model group velocity on top and bottom layers are nearly the same. Furthermore, it has been figured out that the responded voltage signals and nodal displacements at the sensor are nearly the same. If the central frequency of the exciting signal is increasing, also a better agreement between the three different models could be observed. The comparison of the group velocities calculated with the three different models has shown that both the and the mode propagate slower on the bottom plate. For different thicknesses of the skin plate (t p ), the group velocities calculated with the different models are only slightly changing. The wave length of the different modes has also been calculated. It has been shown that when t p is increasing the wavelength is decreasing in both top and bottom surfaces in all models. The wavelengths are evaluated with the voltage signals of the sensor as well as with the normal displacements of a nodal point on top of the sensor. Both results are in a good agreement in all models. To understand better the energy transmission phenomena through the honeycomb structure further investigations are required to receive a global map for the energy transmission and the wave length changes inside the structure in dependence of the thicknesses t p /t h and the applied frequency range. In addition, further studies are also needed to evaluate the influence of the other geometrical properties on the wave propagation, such as cell size and cell height, as well as the sandwich panel material properties on the

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