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Research Papers

Three-Dimensional Multiple Scattering of Elastic Waves by Spherical Inclusions

[+] Author and Article Information
Zunping Liu1

Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506lzping@anl.gov

Liang-Wu Cai2

Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506cai@ksu.edu

1

Present address: X-ray Science Division, Argonne National Laboratory, Argonne, IL 60439.

2

Corresponding author.

J. Vib. Acoust 131(6), 061005 (Nov 04, 2009) (11 pages) doi:10.1115/1.4000425 History: Received December 03, 2007; Revised June 10, 2009; Published November 04, 2009; Online November 04, 2009

This paper extends the scatterer polymerization methodology to three-dimensional multiple scattering of elastic waves by spherical inclusions. The methodology was originally developed for analyzing multiple scattering of elastic antiplane shear waves in two-dimensional spaces. The analytically exact solution of multiple scattering is reformulated by using this methodology, which is verified by using different ways, with or without scatterer polymerization, to solve physically the same multiple scattering problem. As an application example, the band gap formation for elastic wave propagating in a cubic lattice of spherical scatterers is observed through a series of numerical simulations. These simulations also demonstrate the capability of the present computational system for simulating three-dimensional multiple scattering of elastic waves.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 2

Five ways to solve the multiple scattering problem of eight scatterers: (a) NP, (b) P1, (c) Px, (d) Py, and (e) Pz

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Figure 3

Modulus of uz of scattered waves at the y=0 plane for the multiple scattering problem of a longitudinal plane wave (κa=0.5) by eight scatterers of low-carbon steel spheres in epoxy matrix. The problem is solved in five ways designated as (a) NP, (b) P1, (c) Px, (d) Py, and (e) Pz, which are specified in Sec. 4. Wave information inside an abstract scatterer is lost and uz inside an abstract scatterer is set to zero when the scatterer polymerization method is used. The color bar serves all five subfigures.

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Figure 1

Local coordinate systems belonging to Scatterers I and J and arbitrary field point P. Dotted arrows represent projections of position vectors on horizontal planes.

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Figure 4

Relation of truncation number, significant figures, the ratio dIJ/rJ and three orientations of vector dIJ during coordinate transformation. The wave number is κrJ=1.5. (a) Translation along the X-axis, (b) translation along the Y-axis, and (c) translation along Z-axis.

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Figure 5

Relation of wave number and significant figure after transformation between two abstract scatterers. The truncation number is 14. dIJ/rJ=1.66756.

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Figure 6

Sixteen actual scatterers in a square arrangement in a layer. (a) Three-dimensional view: each sphere in light green color represents an abstract scatterer, each sphere in green color represents an actual scatterer, and an abstract scatterer encloses four actual scatterers. (b) Two-dimensional view: each dashed circle represents an abstract scatterer and each solid circle represents an actual scatterer.

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Figure 7

Displacement |uz| and stress |σzz| in the y=0 plane of total waves at κa=1.2: (a) one layer, |uz|; (b) one layer, |σzz|; (c) two layers, |uz|; (d) two layers, |σzz|; (e) four layers, |uz|; and (f) four layers, |σzz|. Wave fields inside black circles are not computed during scatterer polymerization procedure and set to zero. The axes are normalized by scatterer’s radius. The color bar at left column serves subfigures (a), (c), and (e), while the color bar at right column serves subfigures (b), (d), and (f).

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Figure 8

Comparison of |uz| at D/a=8 for different layers

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Figure 9

Forward spectrum map at z=8a. The discrete texture is due to numerical steps.

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