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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.

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Figures

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Fig. 1

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

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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

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Fig. 3

Outline of the developed 81-node Plate TDSFE

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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

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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

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Fig. 6

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

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Fig. 7

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

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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.

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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.

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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.

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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

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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

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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.

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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.

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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

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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

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