The 21st International Congress on Sound and Vibration 13-17 July, 2014, Beijing/China
COMPUTATIONAL AND EXPERIMENTAL INVESTIGATION OF ULTRASONIC WAVE PROAGATION IN SANDWICH PANELS Ulrich Gabbert, Seyed Mohamad Hosseini and Sascha Duczek Institute of Mechanics, Otto-von-Guericke-University of Magdeburg, Germany E-Mail:
[email protected] The paper deals with the application of ultrasonic guided waves for damage detection in lightweight structures. Such waves are sensitive towards small damages, where reflections, refractions, and mode conversions may occur, which are distinct indications for the structural health state. But the application of such waves in more complex composite cellular structures, such as honeycomb, metallic foam, hollow sphere and particle rein-forced structures is still a demanding task. In the paper some insight in the propagation behavior of ultrasonic waves in cellular structures is presented. Numerical simulation approaches as well the possibilities of laser scanning vibrometer measurement are applied and discussed. In order to use ultrasonic waves to detect small damages, which are located within the core layer, their interaction with the micro- and macrostructure must be fully understood. Therefore, the interactions of ultrasonic waves with different cellular structures have been investigated and are discussed in paper.
1. Introduction The on-line monitoring of the structural integrity of thin-walled lightweight structures is of steadily growing interest.3, 7 Its main purpose can be seen in increasing the safety and in decreasing the maintenance costs. The application of ultrasonic guided waves is considered as an appropriate means for damage detection since their propagation is sensitive towards small damages, where reflections, refractions, and mode conversions may occur, which are distinct indications for the structural health state (see Fig. 1).2
Figure 1. Complexity of ultrasonic wave based damage detection in real structures
But the application of such waves in cellular lightweight sandwich panels (see Fig. 2) is still a demanding task.6, 12, 14 The wave behavior in such materials as well as their interaction with damages is not well understood, and, consequently, their application for health monitoring reasons is also not well accepted. It is also a great challenge to simulate the wave propagation in such complex structural systems, e.g. to design a proper structural health monitoring (SHM) system. Therefore, several numerical approaches have been developed and applied to reduce the computational cost, ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 such as the homogenization of the cellular material,11, 17 the combination of analytical und numerical approaches,15, 16 the application of semi-analytical finite elements,1 the development of special higher-order and spectral finite elements,4, 18 as well as the recent developments of the finite cell and the spectral cell method,5 which are of great advantage in the simulation of cellular materials. The application of non-reflecting boundary conditions is another way to reduce the computational effort by reducing the region which has to be modeled if the wave propagation between two points of interest has to be analyzed by a finite element approach.9
Figure 2. Different kinds of heterogeneous core materials
In addition, computer tomography images are shown to visualize the inner structure of the panel in a region of interest where strong wave interactions due to failures in the material can be observed.8
2. Finite element based wave analysis In the standard finite element approach the shape functions are introduced as polynomials in the following way u(x, t ) = N(x)U(t ) . (1) Here u(x, t) contains the displacements, x is the vector of co-ordinates, t is the time, N(x) is a matrix containing the shape functions of the finite element and U(t) is the vector of unknown parameters, which are functions of time t. In the standard commercial finite element software tools the shape functions are usually linear or quadratic polynomials. For the simulation of ultrasonic waves the application of higher order polynomials is recommended. The following three different types of polynomials, namely the spectral finite elements based on Lagrange polynomials with special supporting points (SEM), the Legendre polynomials (p-FEM) and the non-rational B-spline polynomials (N-FEM) have been investigated in detail and applied for wave analysis.18 In one dimension these functions can be written as N n ( x) =
Φ type n ( x ),
Lagrange : SEM type = Legendre : p − FEM NURBS : N − FEM
(2)
Based on the standard finite element procedure the semi-discrete form of the equation of motion is derived as + MÜ (t ) = F(t ) , (3) KU (t ) + CU with the stiffness matrix K, the damping matrix C, the mass matrix M, and the load vector F. The shape functions in two and three dimensions can be simply established as tensor product of the onedimensional polynomials Eq. (2). As an example Fig. 3 displays a typical result of a convergence study, which shows the great advantage of higher order finite elements. The investigations have shown, that the convergence rate is the faster, the higher the polynomial orders are. The best con-
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 vergence rate is received with NURBS based iso-geometric finite elements, which is due to the higher order continuity of these elements.
Figure 3. Convergence curves of the A0 mode for different “nodes” per wave length for the polynomial degrees of 3 and 4 in x1 and x2 direction, respectively; the ordinate shows the relative error in the group velocity18
3. Ultrasonic wave propagation in cellular composite sandwich panels The authors have performed several studies, numerically as well as experimentally, to better understand the wave propagation behavior in cellular sandwich panels.8, 9, 10 A brief discussion of the main results is given in the following. In the test examples the waves are excited in the cover plate only, with help of a small circular piezoelectric patch actuator, and in a given distance the waves are recorded on top as well as on bottom of the sandwich panel again with help of a thin piezoelectric sensor.
Figure 4. Displacement perpendicular to the surface of a 2 mm thick honeycomb plate due to a traveling wave (a) low frequency range (5 kHz) and (b) higher frequency range (100 kHz).
It is shown that in the low frequency range the whole sandwich panel acts as a single plate, and no interaction of the waves with the cellular micro structure can be observed (Fig. 4a). Therefore, simplified models in which the core structure is replaced by an orthotropic homogenized layer can be applied. However, in the higher frequency range, where the wavelength is smaller than the plate thickness, more energy is contained in the wave packet propagating in the top plate in comparison to the bottom plate of the honeycomb sandwich panel (Fig. 4b). The propagation behavior deICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 pends on the frequency, the thickness of the cover plates and the core layer as well as their material properties. A high frequency wave excited in the top plate interacts with the core layer, and travels partially through the core layer, causing mode conversions. As the wave enters the core structure the wave energy is divided. The wave is partially reflected and partially transmitted into the core and converted into a different wave mode. Therefore, at least the two basic modes, the symmetric mode S0 as well as the anti-symmetric A0 mode are propagating in the bottom surface. However, as reported in parametric studies, less energy is transmitted to the bottom surface due to the energy lost in the conversion process.10 In addition, a delay in the time of flight to the bottom surface for both modes has been observed. Due to the interaction of waves with the core structure, damages, such as a small debonings in the connection area between the core structure and the cover plate can be recognized.
4. Simplifications of the cellular core layer To reduce the computational efforts some of the investigated simplified models for heterogeneous cellular core layers are reviewed in the following. The top and the bottom layers are always modeled with 3D finite elements (see chapter 2). But, it is not feasible to discretize also the core completely with volume elements. In the low frequency range the core layer can be modeled with 3D finite brick elements with homogenized orthotropic material properties.10 These properties can be calculated with help of a representative volume element approach.11 However, the investigations have shown, that the compact homogeneous core layer does not interact with the ultrasonic guided waves in a similar way as the real cellular structures do. Better results are received, if a closed cell media can be replaced by a mesh of shell type finite elements (see Fig. 5), whereas for an open cell media a beam type model (Fig. 6) is recommended as a good choice.10
Figure 5. Closed cell model; left: real shell type model, right: simplified shell type model
Figure 6. Open cell model: left: real beam type model, right: simplified beam type model
Additionally, it has been shown that a hollow sphere core layer glued together with epoxy can be replaced as a simplified cube type finite element model consisting of an equivalent volume fraction of the hollow spheres and the epoxy glue material.8 Parametric studies have revealed that the influence of the particle size and the particle arrangement is marginal. But, it has also been shown that high frequency ultrasonic waves with a wavelength in the size of the particle diameter cannot be modeled and analyzed with any simplified finite element model. In our opinion the only method which is accurate and powerful enough to solve such problem numerically is the finite cell method and its recent extension to the spectral cell method (SCM), which is especially suited for wave propagation problems.5 The SCM combines the fictitious domain approach with the spectral finite element approach. The main advantage the SCM is that the method avoids generation of a body ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 fitted mesh. The generation of body fitted meshes for cellular structures is an extreme time consuming procedure and results in a very high number of finite elements and degrees of freedom. In the SCM the finite cells do not need to conform to the physical boundaries. So, the integrand of the corresponding cell matrices is discontinuous. Computing an integral with a discontinuous integrand is, however, less troublesome than the mesh generation task and can be carried out automatically. The effort of mesh generation is obviously shifted to computing integrals with discontinuous integrands. The method is very convenient for problems with complex microstructures where the geometry can be obtained directly from computer tomography. Fig. 7 depicts a typical mesh, as used in the SCM for a foam-like structure given by a voxel model.
Figure 7. Application of the finite spectral cell method (SCM); computer tomography of an open cell structure (left), not body fitted cubic finite element mesh (mid) and calculated stresses (right)
5. Experimental investigations An experimental investigation of the wave propagation can be performed, e.g., with help of a laser-scanning vibrometer as shown in Fig. 8. This method is widely used to study wave propagation at the surface of structures. A great advantage is the application of three laser heads, which allow recording the three-dimensional wave field with a very high precision. The additional information about the in-plane components provide an important additional knowledge about the wave propagation especially in cellular sandwich panels.
A
Figure 8. Principal of a scanning laser measurement (left); a three dimensional laser measurement (right) with the Polytec PSV 400 3D applied to a composite plate
With help of laser scanning measurements several simulated panels have also be experimentally investigated, such as aluminum and steel plates, plates manufactured from carbon reinforced ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 plastics, sandwich plates with core layers made from honeycomb and hollow spheres, and other particle reinforced materials.19 Fig. 9 shows the measured Lamb wave propagation in three different CFRP plates with different types of obstacles (damages). The main feature of Fig. 9 is the mode conversion from the fast S0-mode into the slow A0 mode, which can be recognized at sensor positions. A similar plate has also been numerically investigated with help of the developed higher order finite elements (see Fig. 10). A circular piezoelectric patch was used to excite a Lamb wave at the right hand side of the plate in form of sinus burst signal. Figs. 9, 10 show pictures where the fast symmetric wave S0 with the larger wave length has nearly reached the left hand side of the plate, and the slow anti-symmetric wave A0 has not yet reached the obstacle in the middle of the plate. But, in the surrounding of the obstacle already waves with the same small wave length as the A0 mode can be observed. At the obstacle a part of the symmetric S0 wave has changed into the antisymmetric A0 wave. This mode conversion from one wave form into another one is a very characteristic phenomenon if an ultrasonic wave travels over an obstacle (damage, failure in the material or another structural change). But, such mode conversion can only be observed in a thin plate if the obstacle shows no symmetry with respect to the plate.1 If, e.g., the disturbance would be a bore hole through the plate no mode conversion could be seen.
Figure 9. Mode conversion at different types of damages in the middle of the plate (from left: circular thickness reduction, elliptic thickness reduction, and circular thickening); the plate is excited at the right hand side on top with a burst sin signal; the pictures show a snap shot, were the fast S0 wave mode has nearly reached the left hand side of the plate and the slow A0 mode has not reached the damage in the middle of the plate; but, around the damage the S0 mode is partially converted into the A0 mode
Figure 10. Finite element simulation of the wave progation in a plate with a partially circular thickness reduction (see Fig. 9, left hand side)
Finally, the wave propagation in a plate is presented to underline the advantage of higher order finite elements for wave propagation analysis. The test plate is made from aluminium (E = 70 GPa, ν = 0.33) with the dimensions of 0.2 m x 0.2 m x 0.002 m. The surface-bonded piezoelectric transducer has the dimensions 0.005 m x 0.005 m x 0.001 m. The wave is excited using a three-cycle windowed sine-burst with a centre frequency of 200 kHz. Due to the symmetry of the structure only a quarter of the plate has to be modelled. The discretization consists of 793 hexahedral p-finite elements with mechanical properties and 9 coupled field higher order p-type finite eleICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 ments (p-FEM) with a total number of 69312 degrees of freedom. To account for the small out-ofplane dimension an anisotropic polynomial degree template is utilized with a polynomial degree of 3 in the in-plan direction and 4 in thickness direction.4, 18 The wave propagation results are depicted as a snap shot in Fig. 11. The faster symmetric mode (S0) is almost non-dispersive in this frequency range and thus consists of exactly three cycles, whereas the anti-symmetric mode (A0) is drastically influenced by dispersion and already consists of five cycles after the presented time of flight. A simulation using the aforementioned discretization resolves the behaviour of Lamb waves quite well. With respect to the group velocity of the wave packet a relative error compared to the analytic solution constitutes 5.6% for the anti-symmetric and 0.1% for the symmetric mode. All higher order finite element schemes presented in chapter 2 exhibit similar convergence behaviour and accuracy. For a comparable accurate solution with conventional linear finite elements about 350000 degrees of freedom are required, which again highlights the superiority of higher order finite element approaches.
Figure 11. Finite element simulation of the wave propagation in a plate
Figure 12. Wave field of the plate in Fig. 10 measured with a laser-scanning vibrometer
6. Conclusion and outlook The paper presents some recent results for applying ultrasonic waves (Lamb waves) for health monitoring purposes in modern lightweight structures, including carbon reinforced structures and sandwich panels with heterogeneous core layers, such as honeycomb, open and closed form cells, foams, particle reinforced materials etc. It is shown that higher order finite elements, especially spectral finite elements, are the only approach to efficiently solve the ultrasonic wave propagation in real structure, and to numerically design and optimize structural health monitoring system. The application of ultrasonic waves for health monitoring Figure 13. Airplane structure covered with a net of in cellular sandwich panels is possible, but, repiezoelectric units acting as actuators and sensors quires wave length which are able to interact with the cellular microstructure as well as with failures, meaning that the wavelength should be in the order of the size of the failure. The application of simplified numerical models is possibly, but, clearly depends on the objective of the simulation. Some simplified models are suggested in the paper for different sandwich panels with cellular core layers including honeycomb, hollow sphere, and open-cell and closed-cell foam structures. The development of SHM systems is an ongoing research process. Future SHM systems may consist of a network of surface-bonded encapsulated piezoelectric actuators and sensors (Figure 13). Each sensor and actuator themselves may constitute a single, autonomous and intelligent subsystem, ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014 which communicates with other subsystems via the structure, e.g. by applying special ultrasonic waves. Local energy harvesting units, integrated into the subsystems, autonomously provide the subsystems with the required energy. The local intelligence at the SHM subsystems requires reduced local models, control units13 as well as strategies for compensating failures of the units automatically. The development of SHM based on ultrasonic waves promises a vast potential for the reduction of maintenance costs in aeronautic, wind energy and other fields of engineering. Acknowledgement: The authors like to thank the German Research Foundation (DFG) and all partners of the joint project PAK357 for their support (GA 480/13-3).
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