3D Finite Element Modeling of Absorbing Regions for Guided Wave Scattering Problems in Composite Materials

Transcription

1 3D Finite Element Moeling of Absorbing Regions for Guie Wave Scattering Problems in Composite Materials Ching-Tai Ng Abstract This paper presents a three-imensional (3D) finite element (FE) moel of the absorbing regions for guie wave (GW) scattering in composite materials. The moel is aime at proviing an efficient an practical numerical moeling technique for analyzing GW propagation an scattering in infinitely long composite structures. The reporte stuy sought to fin the optimal configuration of the absorbing regions to not only minimize the wave reflection from bounaries, but also the computational cost. The moel with the optimal absorbing regions was then applie to investigate the GW scattering characteristics for ifferent types an sizes of efects in laminate composite beams. Inex Terms guie wave; absorbing region; finite element; scattering; composite material; amage etection I. INTRODUCTION nfrastructure safety has receive significant attention in the Ilast ecae, with a variety of amage etection methos evelope by a number of researchers in an effort to ensure structural safety an reuce maintenance costs [-3]. Their research suggests that guie wave (GW) offers outstaning etection capabilities, incluing large area inspection, high sensitivity to most types of amage an high efficiency in etecting small an subsurface efects [4]. The research investigations can be ivie into two major areas scattering characteristics stuies [5] an evelopments of amage etection techniques [6-8]. The use of laminate composite materials has steaily increase in ifferent engineering inustries, such as civil, builing an aerospace. However, elamination is a serious weakness in structures mae with composites as it can cause a reuction of up to 6% in the resiual compressive strength of structures [9] an is visually unetectable. The growing use of composite materials, however, has meant that researchers continue to evelop methos for etecting hien elamination in orer to ensure structural safety. The successful evelopment of GW base amage etection techniques requires an unerstaning of GW propagation an scattering characteristics at efects. Analytical solutions of GW Manuscript receive April 7, 23. This work was supporte in part by the Australian Research Council uner Grant DE326. The author is with the School of Civil, Environmental & Mining Engineering, The University of Aelaie, Aelaie, South Australia, Australia ( alex.ng@aelaie.eu.au). scattering at efects o not exist for laminate composite materials because of their anisotropic nature an multilayer characteristics. On the other han, a number of stuies [-5] have been carrie out using FE simulations to provie improve insight into the scattering phenomena of GWs at ifferent types of efects in laminate composite materials. Simulations of GWs in structures are computationally expensive as they require an enough number of FE noes per wavelength of GW to guarantee the accuracy of the simulation results. But the wavelength of GW is usually in the orer of millimeters. To improve the computational efficiency of GW simulations, therefore, ifferent FE moeling strategies have been evelope for stuying the scattering effects. Liu an Quek [6], for example, propose a non-reflecting bounary for wave propagation in an infinitely long two-imensional (2D) isotropic plate using a graually ampe artificial bounary; while Drozz et al. [7] suggeste a metho of moeling GW propagation with a FE metho using frequency omain analysis an absorbing regions for isotropic materials. A technique for preicting the far-fiel scattering behavior of GWs at efects in anisotropic materials evelope by Velichko an Wilcox [8] has prove both robust an efficient. The current stuy investigates 3D FE moeling of GW propagation with ifferent types of efects in laminate composite beams, as explaine in Section II. The effect of using graually increasing amping in the absorbing region is then iscusse in etail in Section III. Section IV emonstrates how the evelope 3D FE moel can be employe in absorbing regions to stuy the scattering characteristics of the funamental anti-symmetric moe (A ) GWs at ifferent types of efects in laminate composite beams. Finally, conclusions are rawn in Section V. II. 3D FINITE ELEMENT SIMULATIONS A. Moeling of guie waves in laminate composite beams GW propagation an scattering in laminate composite beams is escribe by a governing equation as a general structural ynamic problem Mu Cu Ku P () where u, u an u are acceleration, velocity an isplacement vectors. M, C, K an P are the mass matrix, amping matrix, stiffness matrix an external force vector. Commercial FE software ABAQUS/Stanar can be use to solve the DOI:.576/225-37_ GSTF

2 ynamics problems. ABAQUS/Stanar employs an implicit FE metho using the Newmark integration scheme an the Hilber-Hughes-Taylor metho [9]. A [-45/45//9] S laminate composite beam was selecte as the specimen for the current stuy. Each lamina was assume to be a Cycom 97/T3 uniirectional carbon/epoxy prepreg tape. The tape was.2 mm thick, an the material properties were: Young s moulus E = GPa, E2 E3= 8.35 GPa; shear moulus G2 G3 = 4.47 GPa, G 23 = 2.9 GPa; Poisson s ratio 2 3 =.33, 23 =.44. The ensity was 57 kg/m 3. The fiber volume fraction was.55. Each lamina was moele by 8-noe, 3D reuce integration soli brick elements (C3D8R) with hourglass control; an orthotropic material coul be assume, given the material properties of the prepreg tape. The key parameters of the laminate composite beam were beam length L = 2 mm; with b = 4. mm; an height h =.6 mm (Fig. ). z y Hanning winowe toneburst x L l i 8-noe 3D brick element Absorbing region st 2 n i th N th L D Fig.. Schematic iagram of the 3D FE laminate composite beam moel The excitation signal was a 2 khz narrow-ban eight-cycle sinusoial toneburst pulse moulate by a Hanning winow. The A GW was generate by applying the out-of-plane isplacement to the noes in the rectangular area at the beam en as shown in Fig.. The A GW at particular locations was obtaine by monitoring the out-of-plane isplacement of noes locate at the mi-thickness of the laminate composite beam. This ensure that only the A GW was etecte since the S an funamental shear horizontal (SH ) GW exhibit zero out-of-plane isplacement of the noes locate at mi-thickness. The phase velocity, group velocity an wavelength of the excite A GW were m/s, m/s an 6.3 mm. The in-plane imension of the soli brick elements was.4.4 mm 2 an the thickness was.2 mm. This guarantee that there were at least 6 FE noes per wavelength to ensure the accuracy of the simulation results. The out-of-plane isplacements were measure at x = 5 mm, 7 mm, 9 mm, mm an 3 mm; with y = -2. mm an z = mm. The FE calculate group velocity (5.42 m/s) was close to the velocity (529.9 m/s) calculate by the first orer shear eformation theory with coupling between axial-flexural-shear motion. Two types of amage notches an elaminations were consiere in the current stuy. The notches were create by removing elements from the FE moel. Delaminations across the full with of the laminate composite beams were moele L b h by creating uplicate noes at the elamination interface. Therefore, the coinciing noes in the elaminate region ha a separate ientity. Similar moeling techniques were employe in [,3] an there was goo agreement between the FE simulation results an experimental measurements. B. Moeling of absorbing regions using graually increase amping Graually increase amping can be use to moel the absorbing regions for the bounaries of GWs in laminate composite beams. Rayleigh amping, which is a combination of mass- an stiffness-proportional amping, was use in the current stuy to form the amping matrix C in () as C M K (2) where an are Rayleigh coefficients use for amping in the lower an higher frequencies, respectively. For GW problems, amping layers with graually increasing values of are use to construct the absorbing regions. As shown in Fig., the value of i at the i-th amping layer in the absorbing region can be estimate by [2] as 2 li i 2 f for i,..., N (3) LD where l i is the istance from the centre of the i-th amping layer to the starting bounary of the absorbing region, as shown in Fig.. L D is the total length of the absorbing region. N is the total number of amping layers in the absorbing region. f (in Hz) is the central frequency of the excite GWs. L LD / N is the length of each iniviual amping layer. As inicate in (3) the value of i is graually increase from to N. Assuming the total length of the absorbing region L D is not fixe, there are funamentally two parameters N an L, controlling the configuration of the absorbing region. It is obvious that the larger the values of N an L the larger the size of the FE moel, increasing the computational cost of the simulations. Theoretically, there is a global optimal configuration of the absorbing region that can simultaneously minimize the wave reflecte from the bounaries an the computational cost of the simulations. However, the focus of the current stuy was not to etermine the global optimal configuration of the absorbing region. In this stuy, L was assume to be.8 mm in orer to etermine the optimal value of N that minimizes the wave reflecte from the beam en. III. VALIDATION OF THE ABSORBING REGIONS A. Optimal value of N for L =.8 mm A number of simulations were carrie out to etermine the optimal value of N. It was assume for 24 cases that N = 2 to 25. The values of i for each case are shown in Fig. 2a, an each curve in Fig. 2a represents one FE simulation. The out-of-plane isplacements were monitore at x = 4 mm, y = -2. mm an z = mm GSTF

3 i a) Reflection Ratio.5 x N = 2 N=25 b) Number of Damping Layers (N) Fig. 2. (a) Values of stiffness-proportional amping an (b) corresponing reflection ratios for ifferent N Fig. 2a shows the i for N values from 2 to 25. A series of parametric stuies was conucte using i to etermine the configuration of the absorbing region that coul minimize the wave reflecte from the bounaries. The absorbing region was then employe in the FE moel evelope in Section II, with the out-of-plane isplacements monitore at x = 4 mm, y = -2. mm an z = mm, the latter of which was 6 mm from the beam en. Normalize Amplitue (a) Normalize Amplitue Normalize Amplitue (b) Time ( s) Incient wave Reflecte wave from beam en.5 reuce Time ( s) Incient wave Incient wave Reflecte wave from beam en Reflecte wave is eliminate by absorbing region (c) Time ( s) Fig. 3. Normalize ampltiue of guie wave at location x = 4 mm, y 2. mm an z = mm for L =.8 mm, an (a) N = 2, (b) N = 8 an (c) N = 7. (soli lines: normalize wave signal, ashe lines: normalize wave envelopes calculate by the Hilbert transform) Figs. 3a, 3b an 3c show the calculate out-of-plane isplacements at the monitoring locations for the moels with N = 2, 8 an 7, respectively. All calculate out-of-plane isplacements were normalize with respect to their maximum isplacements at the monitoring location, an hence, the maximum amplitue is equal to. The normalize signals an the wave envelopes calculate by the Hilbert transform [3] are inicate as soli an ashe lines, respectively, in Fig. 3. Fig. 3a shows that the reflection ratio of the wave amplitue reflecte from the beam en is reuce to.372 by the absorbing region for which N = 2. The reflection ratio is the ratio of the incient wave amplitue to the reflecte wave amplitue at the monitoring location. Fig. 3b shows that the reflection ratio can be further reuce to.52 for the absorbing region for which N = 8. Fig. 3c shows that the reflection ratio for the absorbing region for which N = 7 exhibite the best performance. The figure shows that the amplitue of the wave reflecte from the beam en is almost completely eliminate by the absorbing region. Fig. 2b shows the calculate reflection ratio of the GW from the beam en for the absorbing layer with ifferent values of N. The figure inicates that the reflection ratio is minimize at N = 7, at which point, the reflection ratio is only.66. Therefore, the absorbing region for which N = 7 an L =.8 mm was selecte to investigate the GW scattering characteristics at efects in the laminate composite beam. IV. GUIDED WAVE SCATTERING ANALYSIS As shown in Fig. 4, the absorbing regions for which N = 7 an L =.8 mm are applie to both sies of the laminate composite beam in orer to avoi wave reflection from the beam ens. The configuration of the efficient scattering moel is shown in Fig. 4. The total length of each absorbing region at each beam en is 3.6 mm. As shown in Fig. 4, the measure out-of-plane isplacements at monitoring points an 2 are reflecte an transmitte waves from the efects, respectively. Fig. 4. Schmatic of the 3D finite element scattering moel A. Effect of notches with ifferent epths Once the absorbing regions were valiate, notch amage was investigate. As iscusse in Section IIA, the notches are moele by removing elements from the moel. Fig. 5 shows four typical contour snapshots of the FE simulate out-of-plane isplacement for A GW in the laminate composite beam with a notch. The first an secon snapshots in Fig. 5 show that the A GW is generate at monitoring point an then propagates to the notch at the left han sie an the absorbing region at the right beam en. As shown in the thir snapshot, the A GW interacts with the notch, which inuces a transmitte an reflecte wave in the laminate composite beam. Simultaneously, the excite A GW is absorbe by the absorbing region at the left beam en. The inuce transmitte an reflecte waves then propagate an GSTF

4 reach monitoring points an 2, as shown in the forth snapshot in Fig. 5. Fig. 7. Wave envelopes of the transmitte waves from notches.8 mm wie of varying epths Fig. 5. Typical contour snapshots of FE simulate out-of-plane ispalcment for a A guie wave in the laminate composite beam with a notch A series of FE simulations was performe, in which each simulation ealt with a notch of a ifferent size. The effect of varying notch epths from.2 mm to.2 mm in.2 mm step increments was first investigate. The with of all the notches was.8 mm. Fig. 6 shows the envelopes of the reflecte waves calculate at monitoring point. As the absorbing regions are use to eliminate the reflecte waves from the beam ens, the signals in the figure only contain the incient wave an the wave reflecte from the notches. In Fig. 6 the first wave peak represents the incient wave an the secon represents the reflecte wave from the notches. In the present stuy the maximum amplitue of the reflecte waves from the notches was a focus of the investigation. The reflecte wave amplitue increase with the increase in notch epth. However, the reflecte wave amplitue of the notch.4 mm in epth prove to be slightly less than that of the notch.2 mm eep. This phenomenon is consistent with the finings in [2], in which the reflecte wave amplitue ha a similar change with the increase in notch epth. B. Effect of notches with ifferent withs Notches with withs varying from.4 to 2.4 mm with.4 mm step increments were also investigate, as escribe in this subsection. The epth of all notches was maintaine at.8 mm. Figs. an show the envelopes of the reflecte an transmitte waves calculate at monitoring points an 2, respectively. In contrast to the results of the notches of varying epths in Section IVA, the reflecte an transmitte wave amplitue from the notches oes not increase or ecrease linearly in relation to notch with. The amplitue of the waves reflecte an transmitte from the notches follows either a sinusoial or cosinusoial pattern accoring to the with of the notch [2]. Fig. 8. Wave envelopes of the waves reflecte from notches.8 mm eep an of varying withs Fig. 6. Wave envelopes of the reflecte waves from notches with.8 mm with an varying epths Fig. 7 shows the envelopes of the transmitte waves at monitoring point 2. In contrast with Fig. 6, the signals shown in Fig. 7 only contain one wave pack because monitoring point 2 is locate after the notches, causing the incient wave an the wave inuce by the notch to overlap. As shown in Fig. 7, the amplitue of the transmitte wave envelopes ecreases with the notch epth as a result of the scattering of energy that converts transmitte waves into reflecte waves at the notches. Fig. 9.Wave envelopes of the waves transmitte from notches.8 mm eep an of varying withs GSTF

5 C. Effect of elaminations with ifferent lengths Investigations of elaminations of ifferent lengths are escribe in this subsection. Figs. an show the calculate envelopes of reflecte an transmitte waves at monitoring points an 2, respectively. The elaminations were locate between the 4 th an 5 th lamina, which are mi-plane ( z = ) of the laminate composite beam. The wave envelope amplitues were normalize so that the amplitue of the incient wave at monitoring point equale. As shown in Fig., the withs of the reflecte wave envelopes for 8 an mm long elaminations are wier than those for 4 an 6 mm elaminations. These results reflect the fact that the wavelength of the incient wave is shorter than the 8 an mm long elaminations, an therefore, multiple wave reflections can occur, increasing the with of the reflecte waves. The amplitue of the waves reflecte from the elaminations was usually smaller than the amplitue of the waves reflecte from the notches. Fig. shows the amplitue of the waves transmitte from the elaminations. As shown in Fig., the minimum transmitte wave amplitue is 75% of the incient wave amplitue at monitoring point (Fig. ). The change in the amplitue of the transmitte wave is smaller than the change in the amplitue of the wave emanating from the notches (Figs. 8 an 9). However, the transmitte wave is still consiere sensitive for amage etection. scattering characteristics. This subsection further investigates this phenomenon by consiering 8 mm long elaminations locate between the 4 th an 5 th, 3 r an 4 th, 2 n an 3 r, an st an 2 n lamina of the composite beam. The results of the reflecte an transmitte wave amplitues are shown in Figs. 2 an 3, respectively. These results are labele L4-5, L3-4, L2-3 an L-2. Although all the elaminations have the same length, the reflecte an transmitte wave amplitues are ifferent for the elaminations locate between the 4 th an 5 th, 3 r an 4 th, 2 n an 3 r, an st an 2 n lamina. This is mainly ue to the moe conversion effect, in which part of the reflecte an transmitte A wave energy is converte to other GW moes. Fig. 2. Wave envelopes of the reflecte waves from 8 mm long elaminations at ifferent through-locations Fig.. Wave envelopes of the reflecte waves from elaminations of varying lengths Fig. 3. Wave envelopes of the transmitte waves from 8 mm long elaminations at ifferent through-thickness locations Fig.. Wave envelopes of the transmitte waves from elaminations of varying lengths D. Effect of elaminations at ifferent through-thickness locations It is obvious that the elaminations at ifferent through-thickness locations have ifferent effects on the A GW V. CONCLUSION The current stuy has emonstrate the use of graually increasing amping to construct the absorbing regions for bounaries of GW problems in composite materials, thereby reucing the size of the FE moels an the computational cost of the simulations. The 3D FE metho together with the evelope absorbing regions were employe in an investigation of the scattering characteristics of the A GW at notches an elaminations in laminate composite beams. A series of case stuies was use to investigate the characteristics of the waves reflecte an transmitte from these efects. Notches with ifferent epths an withs, an elaminations of varying length an at ifferent through-thickness locations were also investigate. The results of this stuy provie improve insight into the scattering phenomena of A GW at efects in composite materials, insight which is essential to further avance the evelopment of GW amage etection techniques GSTF

6 ACKNOWLEDGMENT The author thanks Mr. Yang Yi an Ms. Yueyang Chen, who have unertaken the numerical simulations in this paper as part of their unergrauate theses. This research is part of an ongoing structural health monitoring program at The University of Aelaie. REFERENCES [] W. J. Stazewski, C. Boller an G. Tomlinson, Health Monitoring of Aerospace Structures: Smart Sensor Technologies an Signal Processing, West Sussex, UK: Wiley, 24. [2] H. F. Lam, C. T. Ng an A. Y. T. Leung, Multcrack etection on semirigily connecte beams utilizing ynmaic ata, J. Eng. Mech. ASCE, vol.34, pp.9 99, 28. [3] C. T. Ng an M. Veit, A Lamb-wave-abse techniuqe for amage etection in compoiste laminates, Smart Mater. Struct., vol. 8(746), pp. 2, 29. [4] J. L. Rose, A baseline an vision of ultrasonic guie wave inspection potential, J. Pressure Vessel Technol., vol 24, pp , 22. [5] M. J. S. Lowe, P. Cawley, J. Y. Kao an O. Diligent, The low frequency reflecion characteristics of the funamental antisymmetric lamb wave A from a rectangular notch in a plate, J. Acoust. Soc. Am., vol. 2, pp , 22. [6] M. Veit, C. T. Ng an T. Wattingeer, Imaging laminar amage in plate using Lamb wave beamforming, Av. Mater. Res., vol. 33, pp.293 3, 29. [7] C. T. Ng, M. Veit an H. F. Lam, Guie wave amage characterisation in beams utilising probabilistic optimisation, Eng. Struct., vol. 3, pp , 29. [8] C. T. Ng, M. Veit an N. Rajic, Integrate piezoceramic transcuers for imaging amage in compsite lamiantes, Proc. of SPIE, vol. 7493(74932M), pp. 8, 29. [9] G. A. O. Davies an R. Olsson, Impact on composite structures, The Aeron. J., vol. 8, pp , 24. [] T. Hayashi an K. Kawashima, Multiple reflections of Lamb waves at a elamination, Ultrasonics, vol. 4, pp.93 97, 22. [] C. T. Ng an M. Veit, Scattering analysis of funamental anti-symmetric Lamb wave at elaminations in composite laminates, Aust. J. Mech. Eng., vol. 8, pp , 2. [2] M. Veit an C. T. Ng, Influnece of stacking sequence on scattering characteriestics of the funamental anti-symmetric Lamb wvae at through holes in composite laminates, J. Acoust. Soc. Am., vol. 29, pp , 2. [3] C. T. Ng an M. Veit, Scattering of the funamental anti-symmetric Lamb wave at elaminations in composite laminates, J. Acoust. Soc. Am., vol. 29, pp , 2. [4] C. T. Ng an M. Veit, Scattering characteristics of Lamb waves from ebonings at structural features in composite laminates, J. Acoust. Soc. Am., vol. 32, pp.5 23, 22. [5] C. T. Ng, M. Veit, L. R. F. Rose an C. H. Wang, Analytical an finite elmeent preiction of Lamb wave scattering at elaminations in quasi-isotropic composite laminates, J. Soun an Vib., vol. 33, pp , 22. [6] G. R. Liu an S. S. Quek, A non-reflecting bounary for analyzing wave propagation using the finite element metho, Fn. Elem. in Anal. Des., vol. 39, pp.43 47, 23. [7] M. Drozz, L. Moreau, M. Castaings, M. J. S. Lowe an P. Cawley, Efficient numerical moelling of absorbing regions for bounaries of guie waves problems, Rev. of Prog. Quant. Nonest. Eval., Vol. 25, pp.26 33, 26. [8] A. Velichko an P. D. Wilcox, A generalize approach for efficient finite element moeling of elastoynamic scattering in two an three imensions, J. Aoust. Soc. Am., vol. 28, pp.4 4, 2. [9] ABAQUS,ABAQUS Analysis User s Manual, Version 6.9, Dassault Systemes, 29. [2] A. Velichko an P. D. Wilcox, A geralize approach for efficient finite elment moeling of elastoynamic scattering intwo an three imensions, J. Acoust. Soc. Am. Vol. 28, pp.4 4, 2. [2] M. J. S. Lowe, P. Cawley, J. Y. Kao an O. Diligent, The low frequency reflection characterstics of the funamental antisymmetric Lamb wave a from a rectangular notch in a plate, J. Acoust. Soc. Am., vol. 2, pp , 22. Ching-Tai Ng was born in Hong Kong, China in 983. He receive B.Eng. an M.Phil. egrees from City University of Hong Kong, in 25 an 27, respectively, an then Ph.D. egree from The University of Queenslan, Australia, in 2. He joine The University of Aelaie as a Lecturer (Level B) in 2. His research interests inclue structural health monitoring, guie wave non-estructive evaluation, system ientification, engineering reliability an risk analysis an composite materials with applications to ifferent engineering areas GSTF