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Research Papers

Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method

[+] Author and Article Information
A. B. M. Tahidul Haque

Department of Civil, Structural and
Environmental Engineering,
University at Buffalo,
116 Ketter Hall,
Buffalo, NY 14260

Ratiba F. Ghachi, Wael I. Alnahhal

Department of Civil and
Architectural Engineering,
Qatar University,
Doha 2713, Qatar

Amjad Aref

Department of Civil, Structural and
Environmental Engineering,
University at Buffalo,
235 Ketter Hall,
Buffalo, NY 14260

Jongmin Shim

Department of Civil, Structural and
Environmental Engineering,
University at Buffalo,
240 Ketter Hall,
Buffalo, NY 14260
e-mail: jshim@buffalo.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 26, 2016; final manuscript received March 28, 2017; published online June 28, 2017. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 139(5), 051010 (Jun 28, 2017) (13 pages) Paper No: VIB-16-1480; doi: 10.1115/1.4036469 History: Received September 26, 2016; Revised March 28, 2017

The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated.

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Figures

Grahic Jump Location
Fig. 1

(a) Geometry of a multilayered periodic composite, whose unit cell is spanned by primitive lattice vectors a1 and a3 in a two-dimensional (2D) coordinate space. Note that a1=||a1|| and a3=||a3||. (b) The corresponding wavevector space, where the topmost rectangle delineated by a thick solid line represents first Brillouin zone. Note that the aliasing paths are denoted by thick dotted lines.

Grahic Jump Location
Fig. 2

(a) (Left) A unit cell of the infinitely periodic three-layered composite having a1/a3 = 2.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 2.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 2(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 2(b) and 2(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Grahic Jump Location
Fig. 3

(a) (Left) A unit cell of the infinitely periodic three-layered composite having a1/a3 = 1.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 1.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 3(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 3(b) and 3(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Grahic Jump Location
Fig. 4

(a) (Left) A unit cell of the infinitely periodic four-layered composite having a1/a3 = 2.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ – X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 2.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 4(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 4(b) and 4(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Grahic Jump Location
Fig. 5

(a) (Left) A unit cell of the infinitely periodic four-layered composite having a1/a3 = 1.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 1.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relation obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 5(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed modes in Figs. 5(b) and 5(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Grahic Jump Location
Fig. 6

(a) Analytical dispersion relation obtained from Eq. (19) for the infinitely periodic three-layered composite: (a-1) κ3 = 0 and (a-2) κ3 = π/a3. (b) Analytical dispersion relation obtained from Eq. (19) for the infinitely periodic four-layered composite: (b-1) κ3 = 0 and (b-2) κ3 = π/a3. The dotted lines denote the approximated linear dispersion relations obtained from the effective modulus theory (41). Note that the κ1 axis is intentionally normalized by a3 because the periodic length a3 is the common characteristic length of the considered periodic composites.

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