0
Research Papers

A Hermite Spline Layerwise Time Domain Spectral Finite Element for Guided Wave Prediction in Laminated Composite and Sandwich Plates

[+] Author and Article Information
C. S. Rekatsinas

Department of Mechanical
Engineering and Aeronautics,
University of Patras,
Rion-Patras GR-26500, Greece

D. A. Saravanos

Professor
Department of Mechanical
Engineering and Aeronautics,
University of Patras,
Rion-Patras GR-26500, Greece
e-mail: saravanos@mech.upatras.gr

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 7, 2016; final manuscript received December 20, 2016; published online April 13, 2017. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 139(3), 031009 (Apr 13, 2017) (14 pages) Paper No: VIB-16-1390; doi: 10.1115/1.4035702 History: Received August 07, 2016; Revised December 20, 2016

A new time domain spectral plate finite element (FE) is developed to provide fast numerical calculations of guided waves and transient phenomena in laminated composite and sandwich plates. A new multifield layerwise laminate theory provides the basis for the FE, which incorporates cubic Hermite polynomial splines for the approximation of the in-plane and transverse displacement fields through the thickness of the plate, enabling the modeling of symmetric and antisymmetric wave modes. The time domain spectral FE with multi-degrees-of-freedom (DOF) per node is subsequently formulated, which uses integration points collocated with the nodes to yield consistent diagonal lumped mass matrix which expedites the explicit time integration process. Numerical simulations of wave propagation in aluminum, laminated carbon/epoxy and thick sandwich plates are presented and validated with an analytical solution and a three-dimensional (3D) solid element; moreover, the capability to accurately and rapidly predict antisymmetric and symmetric guided waves is demonstrated.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Diaz Valdes, S. H. , and Soutis, C. , 2000, “ Health Monitoring of Composites Using Lamb Waves Generated by Piezoelectric Devices,” Plast. Rubber Compos., 29(9), pp. 475–481. [CrossRef]
Giurgiutiu, V. , 2005, “ Tuned Lamb Wave Excitation and Detection With Piezoelectric Wafer Active Sensors for Structural Health Monitoring,” J. Intell. Mater. Syst. Struct., 16(4), pp. 291–305. [CrossRef]
Raghavan, A. , and Cesnik, C. E. S. , 2005, “ Finite-Dimensional Piezoelectric Transducer Modeling for Guided Wave Based Structural Health Monitoring,” Smart Mater. Struct., 14(6), pp. 1448–1461. [CrossRef]
Crawley, E. F. , and Lazarus, K. B. , 1991, “ Induced Strain Actuation of Isotropic and Anisotropic Plates,” AIAA J., 29(6), pp. 944–951. [CrossRef]
Wang, C. S. , and Chang, F. K. , 2000, “ Diagnosis of Impact Damage in Composite Structures With Built-in Piezoelectrics Network,” Proc. SPIE 3990, pp. 13–19.
Diamanti, K. , Soutis, C. , and Hodgkinson, J. M. , 2007, “ Piezoelectric Transducer Arrangement for the Inspection of Large Composite Structures,” Composites Part A, 38(4), pp. 1121–1130. [CrossRef]
Velichko, A. , and Wilcox, P. D. , 2007, “ Modeling the Excitation of Guided Waves in Generally Anisotropic Multilayered Media,” J. Acoust. Soc. Am., 121(1), p. 60. [CrossRef]
Sikdar, S. , and Banerjee, S. , 2016, “ Guided Wave Propagation in a Honeycomb Composite Sandwich Structure in Presence of a High Density Core,” Ultrasonics, 71, pp. 86–97. [CrossRef] [PubMed]
Santoni-Bottai, G. , and Giurgiutiu, V. , 2012, “ Exact Shear-Lag Solution for Guided Waves Tuning With Piezoelectric-Wafer Active Sensors,” AIAA J., 50(11), pp. 2285–2294. [CrossRef]
Yu, L. , Bottai-Santoni, G. , and Giurgiutiu, V. , 2010, “ Shear Lag Solution for Tuning Ultrasonic Piezoelectric Wafer Active Sensors With Applications to Lamb Wave Array Imaging,” Int. J. Eng. Sci., 48(10), pp. 848–861. [CrossRef]
Raghavan, A. , and Cesnik, C. E. S. , 2007, “ 3-D Elasticity-Based Modeling of Anisotropic Piezocomposite Transducers for Guided Wave Structural Health Monitoring,” ASME J. Vib. Acoust., 129(6), p. 739. [CrossRef]
Banerjee, S. , and Pol, C. B. , 2012, “ Theoretical Modeling of Guided Wave Propagation in a Sandwich Plate Subjected to Transient Surface Excitations,” Int. J. Solids Struct., 49(23–24), pp. 3233–3241. [CrossRef]
Ahmad, Z. A. B. , Vivar-Perez, J. M. , and Gabbert, U. , 2013, “ Semi-Analytical Finite Element Method for Modeling of Lamb Wave Propagation,” CEAS Aeronaut. J., 4(1), pp. 21–33. [CrossRef]
Barouni, A. K. , and Saravanos, D. A. , 2016, “ Layerwise Semi-Analytical Method for Modeling Guided Wave Propagation in Laminated and Sandwich Composite Strips With Induced Surface Excitation,” J. Aerosp. Sci. Technol., 51, pp. 118–141. [CrossRef]
Bartoli, I. , Marzani, A. , Lanza di Scalea, F. , and Viola, E. , 2006, “ Modeling Wave Propagation in Damped Waveguides of Arbitrary Cross-Section,” J. Sound Vib., 295(3–5), pp. 685–707. [CrossRef]
Marzani, A. , Viola, E. , Bartoli, I. , Lanza di Scalea, F. , and Rizzo, P. , 2008, “ A Semi-Analytical Finite Element Formulation for Modeling Stress Wave Propagation in Axisymmetric Damped Waveguides,” J. Sound Vib., 318(3), pp. 488–505. [CrossRef]
Datta, S. K. , Shah, A. H. , Bratton, R. L. , and Chakraborty, T. , 1988, “ Wave Propagation in Laminated Composite Plates,” J. Acoust. Soc. Am., 83(6), pp. 2020–2026. [CrossRef]
Nanda, N. , Kapuria, S. , and Gopalakrishnan, S. , 2014, “ Spectral Finite Element Based on an Efficient Layerwise Theory for Wave Propagation Analysis of Composite and Sandwich Beams,” J. Sound Vib., 333(14), pp. 3120–3137. [CrossRef]
Chrysochoidis, N. A. , and Saravanos, D. A. , 2009, “ High-Frequency Dispersion Characteristics of Smart Delaminated Composite Beams,” J. Intell. Mater. Syst. Struct., 20(9), pp. 1057–1068. [CrossRef]
Mullen, R. , and Belytschko, T. , 1982, “ Dispersion Analysis of Finite Element Semidiscretizations of the Two-Dimensional WaveEquation,” Int. J. Numer. Methods Eng., 18(1), pp. 11–29. [CrossRef]
Doyle, J. F. , 1997, Wave Propagation in Structures, Springer, New York.
Zak, A. , and Krawczuk, M. , 2010, “ Assessment of Rod Behaviour Theories Used in Spectral Finite Element Modelling,” J. Sound Vib., 329(11), pp. 2099–2113. [CrossRef]
Zak, A. , and Krawczuk, M. , 2012, “ Assessment of Flexural Beam Behaviour Theories Used for Dynamics and Wave Propagation Problems,” J. Sound Vib., 331(26), pp. 5715–5731. [CrossRef]
Ostachowicz, W. M. , 2008, “ Damage Detection of Structures Using Spectral Finite Element Method,” Comput. Struct., 86(3–5), pp. 454–462. [CrossRef]
Nanda, N. , and Kapuria, S. , 2015, “ Spectral Finite Element for Wave Propagation Analysis of Laminated Composite Curved Beams Using Classical and First Order Shear Deformation Theories,” Compos. Struct., 132, pp. 310–320. [CrossRef]
Chakraborty, A. , and Gopalakrishnan, S. , 2005, “ A Spectrally Formulated Plate Element for Wave Propagation Analysis in Anisotropic Material,” Comput. Methods Appl. Mech. Eng., 194(42–44), pp. 4425–4446. [CrossRef]
Chakraborty, A. , and Gopalakrishnan, S. , 2006, “ A Spectral Finite Element Model for Wave Propagation Analysis in Laminated Composite Plate,” ASME J. Vib. Acoust., 128(4), p. 477. [CrossRef]
Samaratunga, D. , Jha, R. , and Gopalakrishnan, S. , 2014, “ Wavelet Spectral Finite Element for Wave Propagation in Shear Deformable Laminated Composite Plates,” Compos. Struct., 108(1), pp. 341–353. [CrossRef]
Kudela, P. , Krawczuk, M. , and Ostachowicz, W. , 2007, “ Wave Propagation Modelling in 1D Structures Using Spectral Finite Elements,” J. Sound Vib., 300(1–2), pp. 88–100. [CrossRef]
Rekatsinas, C. S. , Nastos, C. V. , Theodosiou, T. C. , and Saravanos, D. A. , 2015, “ A Time-Domain High-Order Spectral Finite Element for the Simulation of Symmetric and Anti-Symmetric Guided Waves in Laminated Composite Strips,” Wave Motion, 53, pp. 1–19. [CrossRef]
Kudela, P. , Zak, A. , Ostachowicz, W. , Krawczuk, M. , Kudela, P. , Zak, A. , and Krawczuk, M. , 2007, “ Modelling of Wave Propagation in Composite Plates Using the Time Domain Spectral Element Method,” J. Sound Vib., 302(4–5), pp. 728–745. [CrossRef]
Zak, A. , Krawczuk, M. , and Ostachowicz, W. , 2006, “ Propagation of in-Plane Elastic Waves in a Composite Panel,” Finite Elem. Anal. Des., 43(2), pp. 145–154. [CrossRef]
Kim, Y. , Ha, S. , and Chang, F.-K. , 2008, “ Time-Domain Spectral Element Method for Built-In Piezoelectric-Actuator-Induced Lamb Wave Propagation Analysis,” AIAA J., 46(3), pp. 591–600. [CrossRef]
Ha, S. , and Chang, F.-K. , 2010, “ Optimizing a Spectral Element for Modeling PZT-Induced Lamb Wave Propagation in Thin Plates,” Smart Mater. Struct., 19(1), p. 15015. [CrossRef]
Komatitsch, D. , Barnes, C. , and Tromp, J. , 2000, “ Simulation of Anisotropic Wave Propagation Based Upon a Spectral Element Method,” Geophysics, 65(4), p. 1251. [CrossRef]
Li, F. , Peng, H. , Sun, X. , Wang, J. , and Meng, G. , 2012, “ Wave Propagation Analysis in Composite Laminates Containing a Delamination Using a Three-Dimensional Spectral Element Method,” Math. Probl. Eng., 2012, pp. 1–19.
Peng, H. , Meng, G. , and Li, F. , 2009, “ Modeling of Wave Propagation in Plate Structures Using Three-Dimensional Spectral Element Method for Damage Detection,” J. Sound Vib., 320(4–5), pp. 942–954. [CrossRef]
Rekatsinas, C. S. , and Saravanos, D. A. , 2016, “ A Time Domain Spectral Layerwise Finite Element for Wave SHM in Composite Strips With Physically Modelled Active Piezoelectric Actuators and Sensors,” J. Intell. Mater. Syst. Struct., (epub).
Plagianakos, T. S. , and Saravanos, D. A. , 2009, “ Higher-Order Layerwise Laminate Theory for the Prediction of Interlaminar Shear Stresses in Thick Composite and Sandwich Composite Plates,” Compos. Struct., 87(1), pp. 23–35. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Typical approximation of a field variable through the thickness of a DL

Grahic Jump Location
Fig. 2

Approximation of displacement field and layerwise expansion of CSLT with a two DL configuration; continuity of displacements across layer interfaces is self-satisfied for the case of two distinctive consequent DLs: (a) the DLs before connection and (b) the DLs connected

Grahic Jump Location
Fig. 3

Outline of the developed 81-node Plate TDSFE

Grahic Jump Location
Fig. 4

Lamb waves generated by a point load on the free surface of an aluminum plate predicted by the present CSLT-TDSFE and an analytical solution

Grahic Jump Location
Fig. 5

(a) The symmetry supported one-quarter of the examined plate, as well the in-plane dimensions both of the full plate and the top left right of the plate and (b) the applied pair of symmetric forces along the out of plane direction at the at the bottom left corner of the symmetric top right quarter of the examined plate

Grahic Jump Location
Fig. 6

Wavelength versus excitation frequency of a [02/902]S carbon/epoxy plate [29]. Horizontal lines indicate excitation frequencies used in this work.

Grahic Jump Location
Fig. 7

Convergence evaluations: (a) Root-mean-square (RMS) error percentage per A0 wavelength

Grahic Jump Location
Fig. 8

Predicted normal displacements of the [016] carbon/epoxy composite plate excited by a tone burst of 150 kHz of unidirectional ((a) and (c)) and bidirectional ((b) and (d)) force pairs. (a) Contour plot of A0 wave at t = 0.08 ms, (b) contour plot of S0 wave t = 0.045 ms, (c) transverse displacement of A0 wave at x = 0.08 m, and (d) transverse displacement of S0 wave at x = 0.08 m.

Grahic Jump Location
Fig. 9

Predicted normal displacements of the [04/904]s carbon/epoxy composite plate excited by a tone burst of 150 kHz of unidirectional ((a), (c), and (e)) and bidirectional ((b), (d), and (f)) force pairs. (a) Contour plot of A0 wave at t = 0.085 ms, (b) contour plot of S0 wave t = 0.05 ms, (c) contour plot of A0 wave reflections at t = 0.187 ms, (d) contour plot of S0 wave reflections at t = 0.064 ms, (e) transverse displacement of A0 wave at x = 0.08 m, and (f) transverse displacement of S0 wave at x = 0.08 m.

Grahic Jump Location
Fig. 10

Predicted normal displacements of the [452/−452/902/02]s carbon/epoxy composite plate excited by a tone burst of 150 kHz unidirectional ((a) and (c)) and bidirectional ((b) and (d)) force pairs. (a) Contour plot of A0 wave at t = 0.085 ms, (b) contour plot of S0 wave t = 0.05 ms, (c) transverse displacement of A0 wave at x = 0.08 m, and (d) transverse displacement of S0 wave at x = 0.08 m.

Grahic Jump Location
Fig. 11

Simultaneous prediction of both fundamental modes (A0, S0), reflections, and interference; ((a) and (c)) S0 reflections from the boundaries at 0.0079 ms, contour plot and midline snapshot, respectively; ((b) and (d)) A0 reflections from the boundaries at 0.0182 ms, contour plot and midline snapshot, respectively

Grahic Jump Location
Fig. 12

Time responses of the transverse displacement at location (x, y) = (0.15, 0) m from the excitation point of a [02/902]S carbon/epoxy plate (a) of the first symmetric mode and (b) of the first antisymmetric mode excited by a central frequency of 400 kHz, ((c) and (d)) through the thickness normalized axial displacement fields at locations x = 0.1869 m and x = 0.185 for the first symmetric and antisymmetric modes, respectively, ((e) and (f)) through the thickness normalized transverse displacement at the same locations

Grahic Jump Location
Fig. 13

Time responses of a [02/902]s carbon/epoxy plate excited by antisymmetric load of 1.2 MHz central frequency. ((a) and (b)) Axial strain and transverse displacement at location (x, y) = (0.1,0) m from the excitation point; ((c) and (d)) through the thickness strain and displacement fields of A0 mode at x = 0.1089; ((e)–(f)) through the thickness strain and displacement fields of A1 mode at x = 0.2324 m.

Grahic Jump Location
Fig. 14

Time responses of a [02/902]s carbon/epoxy plate excited by symmetric load of 2.5 MHz central frequency. ((a) and (b)) axial strain and transverse displacement at location (x, y) = (0.1,0) m from the excitation point, ((c) and (d)) through the thickness strain and displacement fields of S0 mode at x = 0.1193, ((e) and (f)) through the thickness strain and displacement fields of S1 mode at 0.1487 m, and ((g) and (h)) through the thickness strain and displacement fields of S2 mode at 0.2179 m from the excitation.

Grahic Jump Location
Fig. 15

Contour plot of the normal displacement of a sandwich composite plate [(02/902)s/foam¯]s at t = 0.4 ms for the (a) A0 and (b)S0, respectively; Predicted transverse displacement time histories at x = 0.15 m from the excitation point for the (c) A0, and (d) S0 propagating modes, respectively

Grahic Jump Location
Fig. 16

Through the thickness approximation of the normalized axial and normal displacement fields of a sandwich composite plate [(02/902)s/foam¯]s, for the ((a) and (c)) A0; ((b) and (d)) S0 propagating modes, respectively

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In