Modelling of in-plane wave propagation in a plate using spectral element method and Kane-Mindlin theory with application to damage detection *

Transcription

1 Modelling of in-plane wave propagation in a plate using spectral element method and Kane-Mindlin theory with application to damage detection * Magdalena Rucka Department of Structural Mechanics and Bridge Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology Narutowicza 11/1, Gdańsk, Poland ABSTRACT. This paper presents results of experimental and numerical analyses of in-plane waves propagating in a 5 mm-thick steel plate in the frequency range of khz. For such a thickness-frequency ratio extensional waves reveal dispersive character. To model inplane wave propagation taking into account the thickness-stretch effect, a novel D spectral element, based on the Kane-Mindlin theory, was formulated. An application of in-plane waves to damage detection is also discussed. Experimental investigations employing a laser vibrometer demonstrated that the position and length of a defect can precisely be identified by analyzing reflected and diffracted waves. Keywords: wave propagation; spectral element method; Kane and Mindlin theory; damage detection 1. Introduction Material damage is a potential threat to proper operation of civil, mechanical or aerospace infrastructure. Various damage detection and structural health monitoring methods have thus been investigated and developed to improve reliability and safety, and to solve maintenance problems of engineering structures. Guided wave-based damage detection methods are widely used in non-destructive testing and they have attracted many researchers interest [4, 7, 5, 6]. * Preprint submitted to Archive of Applied Mechanics February 3, mrucka@pg.gda.pl 1

2 Modelling of wave propagation in structural elements and structures is a subject of intensive investigations. One of methods is the frequency-based spectral finite element method (SFEM) developed by Doyle [6] and then extended by Gopalakrishnan et al. [8] to anisotropic media. A different approach can be found in Refs. [16] and [17], in which Lee and Staszewski proposed the local interaction simulation approach (LISA) for wave propagation in damage detection applications. Modelling of wave propagation can also be performed by the finite element method (FEM) [1, 0]. The advantage of the finite element technique is the availability of numerous commercial FEM codes (Ansys 5.3 [0] or ABAQUS EXPLICIT [1]) and its great ability to analyse structures with complicated geometry. The use of the FEM to model wave propagation requires very fine mesh; it is recommended to use more than 0 nodes per the shortest wavelength [0]. An expansion of the FEM is the time domain spectral element method (SEM). The main idea of the SEM is the use of an interpolating polynomial of high degree. In the SEM, Lagrange polynomials are applied at Gauss-Legendre-Lobbatto nodes [3]. In comparison with the classical FEM, an important property of the SEM is that the mass matrix is exactly diagonal, which allows to reduce significantly the algorithm cost [1]. Moreover, in the SEM, the required number of nodes per the shortest wavelength is of order of 10 or even less. In this paper, in-plane wave propagation in the form of a wave packet in a steel plate is studied. In previous work, Żak et al. [7] presented an analysis of in-planes waves in terms of the spectral element method based on the plane stress theory resulting in non-dispersive waves. They considered an aluminium plate of thickness 1 mm and an excitation signal of frequency 100 khz. Within such frequency-thickness range, an extensional wave is practically non-dispersive and the plane stress theory provides correct results. For many civil engineering structures (for example, steel bridge plate girders) plates are thicker and dispersion occurs. To solve this problem, Peng et al. [1] proposed a three-dimensional (3D) spectral element which makes the analysis of dispersive waves possible. Such a 3D model is useful in analysing structures of complicated geometries, but for dispersive wave propagation in plates, an alternative solution could be a D plate model, which results in a substantial reduction of the computational time in comparison with a 3D model. For flexural waves, the spectral finite element based on the Mindlin plate theory was developed in Ref. [15], but there is no higherorder plate spectral element for in-plane waves. One of refined plate theories is the Kane- Mindlin extensional theory [11] and it is equivalent to the Mindlin-Herrmann rod theory [19], for which a spectral element formulation was recently developed by Rucka [, 3]. The Kane-Mindlin theory includes the out-of-plane stress component and retains the simplicity of

3 the D model [14]. In earlier papers, the Kane-Mindlin theory was used in analytical studies of static and dynamic fracture problems of cracked plates [9, 10, 13, 14]. Wang and Chang [4] presented a study of plate waves scattered by a cylindrical inhomogeneity and compared analytical solutions based on the Kane-Mindlin theory with experiments performed on a plate of thickness 1.0 mm with an added mass. McKeon and Hinders [18] applied the Kane- Mindlin theory to derive analytical solutions for the scattering of symmetric Lamb waves at a circular inclusion. This paper presents results of experimental and numerical analyses of in-plane waves propagating in a 5 mm-thick steel plate in the frequency range of khz. For such a thickness-frequency ratio, extensional waves reveal dispersive character. To model in-plane wave propagation accounting for the thickness-stretch effect, a novel D spectral element, based on the Kane-Mindlin theory, is formulated. Finally, an application of in-plane waves to damage detection is discussed.. Formulation of Kane-Mindlin plate spectral finite element.1. Kane-Mindlin plate theory In-plane wave propagation in plates governed by the equations of plane stress theory is non-dispersive [7, 7]. An improvement of the plane stress theory can be achieved by including the thickness-stretch effect. In 1956 Kane and Mindlin [11] developed a higherorder plate theory taking into account coupling between extensional motion and the first mode of thickness vibration. Consider a plate lying in the xy plane bounded by planes z h. The components of displacements (in-plane displacements u, v and out-of-plane displacement w ) in the Kane- Mindlin theory are u( x, y, z, t) u( x, y, t), v ( x, y, z, t) v( x, y, t), w( x, y, z, t) w( x, y, t) z h, (1) and strains corresponding to the above deformations are u v w u v z w z w x, y, z, xy, xz, yz. x y h y x h x h y () The kinetic T and strain U energies are 3

4 T hρu hρv hρw dxdy (3) 1 ( 3), B x y h 1 u v U h G h G G w dxdy B 1 u v u v h w wdxdy B x y x y 1 u u v v hg w hg w + hg 4hG hg dxdy, B y y x x 3 x 3 y (4) where and G are the Lamé constants. Substituting T and U into Hamilton s principle, the governing equations of the Kane-Mindlin theory can then be derived u v w v u h G h hg hg hu p, x x xy x xy y v u w u v h G h hg hg hv p, y y xy y xy x w w u v 3 x 3 y h x y 3 Gh Gh G w hw, (5) where p x and p y are the external forces in x and y directions, respectively. The constant was inserted in the expression for the strain energy (4) to compensate the approximation of displacement fields given in Eqs. (1). There are different rules to set up this parameter, but due to approximate character of the Kane-Mindlin theory, neither approach can be considered better than the other. Kane and Mindlin chose the value of 1 by equating the frequency of pure thickness vibration obtained from the plate equation of motion with the corresponding frequency obtained from three-dimensional equations [11]. In this study, in which experimental investigations are reported, the constant was chosen to give the best compatibility with the experimental data within the frequency range of interest... Spectral element formulation Formulation of the spectral element method is analogous to that of the classical finite element method []. Following classical steps, the weak formulation is obtained in the form T T T δq μqdxdy δε Eε dxdy δq p dxdy 0, (6) B B B 4

5 where q and ε are the displacements and strains, q and ε are the corresponding virtual displacements and virtual strains, p is the external force, E is the stress-strain matrix and μ is the mass density matrix. In the SEM, the domain B can be approximated as a sum of n el nonoverlaping elements, i.e., B e1 B( e). For the standard the displacement field q ( ξ,, t) in a typical finite element B ( e) is n el 0 C elements, interpolation of q(,, t) q(,, t) N (, ) q ( t), ( e) ( e) q ( e) q1() t () t q () t, qn() t where the matrix of interpolation functions N ( ) ( ) ξ, is e ui() t q i( t) vi( t) (7) wt i() ( e) 1 N ( ξ, ) N ( ξ, ) N ( ξ, )... N ( ξ, ), (8) and the matrix of interpolation functions for a node i is Ni(, ) 0 0 N i(, ) 0 Ni(, ) 0, Ni(, ) N p( ) Nq( ). (9) 0 0 Ni(, ) In the above, tilde denotes the approximated quantity, Ni(, ) are the Lagrange type interpolation polynomials, ξ, [ 1, 1] is the parent domain, the index i ( i 1,,..., n) denotes nodal values and n np nq is the number of element nodes, where n p denotes the number of nodes in direction whereas n q in direction. Strains in the Kane-Mindlin plate can be interpolated through the relation: ε(,, t) ε(,, t) B( e) (, ) q ( e) ( t), ( e) ( e) In the above, the differential operator matrix D is given by n B (, ) DN (, ). (10) 5

6 0 0 x 0 0 y h D 0 y x h x h y x = y 1 J (11) where the symbol 1 J denotes the inverse of the Jacobian matrix. Substitution of Eqs. Błąd! Nie można odnaleźć źródła odwołania. and (10) into Eq. (6) provides that following set of equations which holds on the local element level M q K q p, (1) ( e) ( e) ( e) ( e) ( e) where K ( e ) and M( e ) are the element matrices, p ( e) is the load vector. To evaluate the element matrices, numerical integration is employed, and the element matrices are integrated using the Gauss-Lobatto-Legendre (GLL) quadrature [3]. The formulae for the stiffness matrix, the mass matrix, and the load vector are n ( ) p n q T e ww p q ( e) p q ( e) p q p q p1 q1,, det, K B EB J, (13) n ( ) p n q T e ww p q ( e) p q ( e) p q p q p1 q1,, det, M N μ N J, (14) n ( ) p n q T e ww p q ( e) p q p q p q p1 q1,, det, p N p J, (15) where the stress-strain matrix E and the mass density matrix μ are h G h h h G G h E (16) Gh Gh Gh 3 6

7 h 0 0 μ 0 h 0. (17) 0 0 h 3 In the GLL integration quadrature, the element nodes ξ p and q are the same as the integration points and they are obtained as the roots of the following equations [3] (1 ξ ) Pn p 1( ξ) dξ 0, (1 ) Pn q 1( ) d 0, (18) where Pn p 1 and Pn q 1 are the Legendre polynomials of degree ( np 1) and ( nq 1), and the associated weights w p and w q are w p n ( n 1)( P ( ξ )) p p np 1 p, wq nq ( nq 1)( Pn 1( )) q q. (19) The system of equations of motion is then built using standard aggregation of element matrices and vectors n A el e 1 ( e) A n K K, el n M e 1 M ( e), p el e 1 p ( e), (0) yielding the global equation of equilibrium Mq Kq p. In the SEM approach, the element nodes are irregularly distributed (Fig. 1), in contrast to the classical FEM with uniformly distributed element nodes. Due to the application of the GLL rule, interpolation carried out over the GLL nodes leads to the diagonal mass matrix, and temporal integration of the global equation of motion can thus be efficiently conducted. A Fig. 1. An 81-node spectral finite element in the parent domain and selected shape function N (, ) 5 7

8 3. In-plane wave propagation in a steel plate 3.1. Experimental setup Wave propagation experiments were performed on a steel plate of dimensions 1000 mm 1000 mm and thickness h 5 mm (Fig. ). The experimentally determined mass density was found to be 787 kg/m 3. The modulus of elasticity E and the Poisson s ratio were also determined experimentally in a force-displacement test using two strain gauges attached to the specimen of cross-section 0 mm 5 mm in both longitudinal and transverse directions, and their values were identified as E = GPa and = 0.8 GPa. The plate lied on the flat surface and it was supported by four blocks of Plexiglas. The supporting blocks had no influence on the registered signals. All four edges of the plate were free but no rigid movement occurred since excited waves had low amplitudes. Two plates were taken into investigations: the intact plate and the plate with damage. The rectangular defect of length 50 mm, width 1.5 mm and depth.5 mm, obtained by machine cutting, was introduced at the position shown in Fig. a. Such a defect can represent corrosion damage which often occurs in civil engineering structures subjected to environmental conditions. Fig.. Experimental setup for measurements of in-plane waves in a steel plate: (a) geometry of tested plate and measurement points; (b) photograph of hardware and the plate with damage The experimental setup is shown in Fig. b. The piezoelectric (PZT) plate actuator Noliac CMAP11 of dimensions 5 mm5 mm mm was bonded at the edge of the plate, at x = 0 and y = 500 mm, to excite in-plane waves. The Tektronix function generator with the amplifier created an excitation signal in the form of a five-peak sine modulated with a Hanning window. The Hanning window provided smoothed tone burst in order to reduce excitation of side frequencies [7]. Velocity signals were detected and registered in 17 points evenly distributed along the left edge of the plate (Fig. a) by the scanning head PSV-I-400 of the Polytec Scanning Laser Vibrometer PSV-3D-400-M. 8

9 3.. Dispersion curves Group velocity dispersion curves were experimentally determined for the intact plate. A velocity signal was measured on the plate edge (at the same position as that of the source x = 0 and y = 500 mm) for frequencies varying from 10 to 300 khz with the increment of 10 khz. Figure 3 presents examples of registered signals for frequencies 10, 00 and 50 khz. The excitation force applied normal to the plate edge resulted in propagation of both in-plane waves: an extensional wave (P wave) and a shear horizontal wave (SH wave). Since the measurements were made on the plate edge, a strong Rayleigh wave (R wave) was also observed. In the measured signals, the first reflection was the non-dispersive Rayleigh wave while the second reflection was the first symmetric S 0 mode. Figure 3 reveals dispersive character of the measured S 0 mode. The SH wave was not directly registered on the plate edge. Based on the time-of-flight, the group velocities of the S 0 mode and R wave were determined. The group velocity of the SH-wave c SH was then calculated through the relation csh cr (1 ) / ( ), where c R is the group velocity of the Rayleigh wave. Fig. 3. Time history of the experimentally measured in-plane waves in the intact plate for the determination of dispersion curves: (a) 10 khz; (b) 00 khz; (c) 50 khz 9

10 Fig. 4 shows experimental and analytical dispersion curves (for both the plane stress theory and the Kane-Mindlin theory) for the considered 5 mm-thick steel plate. The plane stress theory captures two modes: fundamental extensional mode and fundamental shear horizontal mode. Shear horizontal mode is the SH 0 mode (non-dispersive for isotropic body), but the extensional mode approximates the S 0 mode only at low frequencies because it reveals non-dispersive character and in general the plane stress theory cannot model the dispersion behaviour of the S 0 mode properly. For the Kane- Mindlin theory three modes exist: the first and the second extensional modes and the fundamental SH 0 mode. The two extensional modes of the Kane-Mindlin theory correctly approximate the dispersion behaviour of S 0 and S 1 Lamb modes. The parameter in the Kane-Mindlin theory was chosen to give the best fit to the experimentally measured wave group velocity for the frequency range khz. It was determined by applying the method of least squares and its values was set to Note in Fig. 4 that the Kane-Mindlin analytical dispersion curve for the S 0 mode fits the experimental data, moreover it agrees with the exact Lamb mode in the selected frequency range khz. Fig 4. Dispersion curves for the considered 5 mm steel plate: experimental results and analytical solutions for the plane stress and the Kane-Mindlin theories 3.3. Numerical model of wave propagation for defect detection Numerical modelling of in-plane wave propagation in plates was performed by applying the time domain spectral element method. The plate was meshed to spectral finite elements, each element with GLL nodes (Fig. 1). The defect was modelled 10

11 using 0 elements with height reduced by.5 mm. The highest frequency used in numerical simulations was 50 khz and for this frequency the applied mesh guaranteed 9.1 nodes per the shortest wavelength. Damping was not considered in this model. Temporal integration was performed using the Newmark scheme with the time step set at t s. This algorithm uses accelerations as the primary variables and takes the advantage of the diagonal structure of mass matrix [5]. In Fig. 5, experimental signals for in-plane wave propagation are compared with numerical results for the plane stress and Kane-Mindlin theories. Amplitudes of experimental and numerical signals were normalized to 1 and only signal envelopes were plotted for clarity. The velocity signal v () 9 t was measured on the left edge of the plate at the same position as the actuator, i.e., x = 0 and y = 500 mm (Fig. a). When the plane stress theory is used, experimental data are not compatible with the numerical ones. It is visible in Fig. 5 that reflections (from defect or from plate edge) of extensional waves in the numerical signal are delayed with respect to reflection of extensional waves in the experimental signal. Considering the Kane-Mindlin plate theory, we note that numerical simulations are in good agreement with experimental data so that this theory guarantees better approximation for the S 0 mode than the plane stress theory. Fig. 5. Propagation of in-plane wave of frequency 50 khz comparison between experimental and numerical velocity signals: (a) intact plate; (b) plate with damage 11

12 3.4. Wave propagation in damaged plate In this experiment, the wave packet was imposed along the x axis at the node 9, whereas the velocity responses were measured at nodes 1 to 17 evenly distributed along the left edge of the plate (Fig. a). The frequency of the incident wave was chosen as 50 khz. This frequency was found to be the most effective for the considered specimens and applied instrumentation. As the reference state, the intact plate was first examined. Experimental and numerical signals for this case are illustrated in Fig. 6 in the time and spatial domains. The fronts of the S 0 mode and Rayleigh waves were registered on the plate edge. To visualize wave patterns occurring on the plate edge, a C-scan was performed. The C-scan, based on the numerical velocity signals (Kane-Mindlin theory), provided a twodimensional xy plane view at the selected time instant t = 0.1 ms. The force applied perpendicular to the plate edge resulted in propagation of the S 0 and SH 0 modes with cylindrical fronts, as is indicated in Fig. 7. Moreover a Rayleigh surface wave (R wave) was visible on the plate edge. The second example concerned the plate with damage (Fig. 8). In both the numerical and experimental results, the fronts of the S 0 modes reflected form the damage were visible. Two fronts of the S 0 modes were caused by the 1st and nd reflection from the damage; they are marked by solid lines in Fig. 8. The first reflection in the numerical signal occurred at the time instant equal to ms. Knowing the plate geometry and the group velocity of the S 0 mode ( m/s) localization of defect can be identified as 86 mm. In the case of experimental signal, reflection occurred at the time instant equal to ms and the velocity of the S 0 mode was m/s, thus the identified position of damage was 86 mm. Two additional wave fronts, marked with dashed line in Fig. 8, were caused by the S 0 mode diffraction at the defect ends. This S 0 mode arose from a mode conversion upon interaction of the SH 0 mode with the defect, which is visible in Fig. 7b. The fronts of diffracted waves can be used to estimate the length of the defect. Moreover, in the experimental signals, an additional reflection appeared. It was the R wave reflected at the defect (dotted line in Fig. 8b). This reflection was identified as coming from imperfect work of the equipment. The amplifier created the additional wave packet (of amplitude about 0.01 of the incident wave) at the moment of arriving the S 0 mode reflected from defect. This wave packet created propagation of an additional R wave, which provide the supplementary indicator of damage existence. 1

13 Fig. 6. Propagation of in-plane wave of frequency 50 khz in the time and spatial domains in the intact plate: (a) numerical simulations based on Kane-Mindlin theory; (b) experimental results 13

14 Fig. 7. C-scan of numerical in-plane waves based on the Kane-Mindlin theory (at the time instant t = 0.1 ms): (a) intact plate; (b) plate with damage 14

15 Fig. 8. Propagation of in-plane wave of frequency 50 khz in the time and spatial domains in the plate with damage: (a) numerical simulations based on Kane-Mindlin theory; (b) experimental results 15

16 4. Conclusions In this paper, the spectral Kane-Mindlin finite element was successfully developed and applied to numerical simulations of in-plane wave propagation in a steel plate. Application of the Kane-Mindlin spectral finite element guarantees that the mass matrix is diagonal so that the temporal integration can be efficiently performed. Moreover, higher order Kane-Mindlin theory provides an accurate description of dispersive behaviour of the S 0 mode which was proved by the comparison with experimentally measured signals, and it also allows an analysis of the S 1 mode. The detection of damage was considered by analyzing wave speeds and reflection times in the recorded velocity signals. Modelling of in-plane wave propagation by the Kane- Mindlin spectral finite element predicted proper times of reflections from damage so that the numerical model used in structural health monitoring systems should employ the SEM formulation based on the Kane-Mindlin theory. Measurements of time velocity signals in several points (17 points in this study) provided information of wave propagation in timespatial plane. As a result, the interaction of waves with boundaries or potential discontinuities could be observed more precisely. These experimental investigations demonstrated that position and length of the defect could clearly be identified by the reflected and diffracted waves. Acknowledgments This work was partially supported by the project POIG / References 1. Bartoli, I., Lanza di Scalea, F., Fateh, M., Viola, E.: Modeling guided wave propagation with application to the long-range defect detection in railroad tracks. NDT&E International, 38, (005). Bathe, K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996) 3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1998) 4. Cawley, P., Alleyne, D.: The use of Lamb waves for the long range inspection of large structures. Ultrasonics, 34, (1996) 5. Chróścielewski, J., Rucka, M., Witkowski, W., Wilde, K.: Formulation of spectral truss element for guided waves damage detection in spatial steel trusses. Archives of Civil Engineering, 55, (009) 6. Doyle, J.F.: Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, nd edn. Springer, New York (1997) 7. Giurgiutiu, V.: Structural Health Monitoring with Piezoelectric Wafer Active Sensors. Academic Press, New York (008) 8. Gopalakrishnan, S., Chakraborty, A.,. Mahapatra, D.R.: Spectral Finite Element Method: Wave Propagation, Diagnostics and Control in Anisotropic and Inhomogeneous Structures. Springer, London (008) 16

17 9. Jin, Z.H., Batra, R.C.: A crack at the interface between a Kane-Mindlin plate and a rigid substrate. Engineering Fracture Mechanics, 57, (1997) 10. Jin, Z.H., Batra, R.C.: Dynamic fracture of a Kane-Mindlin plate. Theoretical and Applied Fracture Mechanics, 6, (1997) 11. Kane, T.R., Mindlin, R.D.: High-frequency extensional vibrations of plates. Journal of Applied Mechanics, 3, (1956) 1. Komatitsch, D., Martin, R., Tromp, J., Taylor, M.A., Wingate, B.A.: Wave propagation in -D elastic media using a spectral element method with triangles and quadrangles. Journal of Computational Acoustics, 9, (001) 13. Kotousov A., Wang CH.: Three-dimensional stress constraint in an elastic plate with a notch. International Journal of Solids and Structures, 39, (00) 14. Kotousov A.: Fracture in plates of finite thickness. International Journal of Solids and Structures, 44, (007) 15. Kudela, P., Żak, A., Krawczuk, M., Ostachowicz, W.: Modelling of wave propagation in composite plates using the time domain spectral element method. Journal of Sound and Vibration, 30, (007) 16. Lee, B.C., Staszewski, W.J.: Modelling of Lamb waves for damage detection in metallic structures: Part I. Wave propagation. Smart Materials and Structures, 1, (003) 17. Lee, B.C., Staszewski, W.J.: Modelling of Lamb waves for damage detection in metallic structures: Part II. Wave interactions with damage. Smart Materials and Structures, 1, (003) 18. McKeon, J.C.P., Hinders, M.K.: Lamb wave scattering from a through hole. Journal of Sound and Vibration, 4, (1999) 19. Mindlin, R.D., Herrmann, G.: A one dimensional theory of compressional waves in an elastic rod. Proceedings of First U.S. National Congress of Applied Mechanics, (1950) 0. Moser, F., Jacobs, L.J., Qu J.: Modeling elastic wave propagation in waveguides with the finite element method, NDT&E International, 3, 5-34 (1999) 1. Peng, H., Meng, G., Li, F.: Modeling of wave propagation in plate structures using three-dimensional spectral element method for damage detection. Journal of Sound and Vibration, 30, (009). Rucka M.: Experimental and numerical studies of guided wave damage detection in bars with structural discontinuities. Archive of Applied Mechanics, 80: (010) 3. Rucka M.: Experimental and numerical study on damage detection in an L-joint using guided wave propagation. Journal of Sound and Vibration, 39, (010) 4. Wang, C.H., Chang, F.-K.: Scattering of plate waves by a cylindrical inhomogeneity. Journal of Sound and Vibration, 8, (005) 5. Yang, Y., Cascante, G., Polak, M.A.: Depth detection of surface-breathing crack in concrete plates using fundamental Lamb modes. NDT&E International, 4, (009) 6. Yu, L., Giurgiutiu, V.: In situ -D piezoelectric wafer active sensors array for guided wave damage detection. Ultrasonics, 48, (008) 7. Żak, A., Krawczuk, M., Ostachowicz, W.: Propagation of in-plane waves in an isotropic panel with a crack. Finite Elements in Analysis and Design, 4, (006) 17