Research Papers

Multiple Scattering of Flexural Waves on Thin Plates

[+] Author and Article Information
Liang-Wu Cai

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

Stephen A. Hambric

Applied Research Laboratory,
Pennsylvania State University,
State College, PA 16804

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 15, 2015; final manuscript received August 31, 2015; published online October 26, 2015. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 138(1), 011009 (Oct 26, 2015) (10 pages) Paper No: VIB-15-1219; doi: 10.1115/1.4031535 History: Received June 15, 2015; Revised August 31, 2015

In this paper, the scattering of flexural waves on a thin Kirchhoff plate by an ensemble of through-thickness circular scatterers is formulated by using the concept of the T-matrix in a generalized matrix notation, with a focus on deterministic numerical computations. T-matrices for common types of scatterers, including the void (hole), rigid, and elastic scatterers, are obtained. Wave field properties in the multiple-scattering setting, such as the scattering amplitude, and scattering cross section, as well as properties of the T-matrix due to the energy conservation are discussed. After an extensive validation, numerical examples are used to explore the band gap formation due to different types of scatterers. One of the interesting observations is that a type of inclusion commonly referred to as the “rigid inclusion” in fact represents a clamped boundary that is closer to a riveted confinement than a rigid scatterer; and an array of such scatterers can block the wave transmission at virtually all frequencies.

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Pao, Y. H. , 1962, “ Dynamic Stress Concentration in an Elastic Plate,” ASME J. Appl. Mech., 29(2), pp. 299–305. [CrossRef]
Pao, Y. H. , and Mow, C. C. , 1973, Diffraction of Elastic Waves and Dynamic Stress Concentration, Crane Russak, New York.
Norris, A. N. , and Vemula, C. , 1995, “ Scattering of Flexural Waves on Thin Plates,” J. Sound Vib., 118(1), pp. 115–125. [CrossRef]
Squire, V. A. , and Dixon, T. W. , 2000, “ Scattering of Flexural Waves From a Coated Cylindrical Anomaly in a Thin Plate,” J. Sound Vib., 236(2), pp. 367–373. [CrossRef]
Matus, V. V. , and Emets, V. F. , 2010, “ T-Matrix Method Formulation Applied to the Study of Flexural Waves Scattering From a Through Obstacle in a Plate,” J. Sound Vib., 329(14), pp. 2843–2850. [CrossRef]
Pao, Y. H. , and Chao, C. C. , 1964, “ Diffraction of Flexural Waves by Cavity in an Elastic Plate,” AIAA J., 2(11), pp. 2004–2010. [CrossRef]
Vemula, C. , and Norris, A. N. , 1997, “ Flexural Wave Propagation and Scattering on Thin Plates Using Mindlin Theory,” Wave Motion, 26(1), pp. 1–12. [CrossRef]
Hu, C. , Han, G. , Li, F.-M. , and Huang, W.-H. , 2008, “ Scattering of Flexural Waves and Boundary-Value Problem in Mindlins Plates of Soft Ferromagnetic Material With a Cutout,” J. Sound Vib., 312(1–2), pp. 151–165. [CrossRef]
Grahn, T. , 2003, “ Lamb Wave Scattering From a Circular Partly Through-Thickness Hole in a Plate,” Wave Motion, 37(1), pp. 63–80. [CrossRef]
Wang, C. H. , and Chang, F.-K. , 2005, “ Scattering of Plate Waves by a Cylindrical Inhomogeneity,” J. Sound Vib., 282(1–2), pp. 429–451. [CrossRef]
Cegla, F. B. , Rohde, A. , and Veidt, M. , 2008, “ Analytical Prediction and Experimental Measurement for Mode Conversion and Scattering of Plate Waves at Non-Symmetric Circular Blind Holes in Isotropic Plates,” Wave Motion, 45(3), pp. 162–177. [CrossRef]
Peng, S. Z. , 2005, “ Flexural Wave Scattering and Dynamic Stress Concentration in a Heterogeneous Plate With Multiple Cylindrical Patches by Acoustical Wave Propagator Technique,” J. Sound Vib., 286(4–5), pp. 729–743. [CrossRef]
Chao, H. , Xueqian, F. , and Wenhu, H. , 2007, “ Multiple Scattering of Flexural Waves in a Semi-Infinite Thin Plate With a Cutout,” Int. J. Solids Struct., 44(2), pp. 436–446. [CrossRef]
Fang, X.-Q. , and Wang, X.-H. , 2009, “ Multiple Scattering of Flexural Waves From a Cylindrical Inclusion in a Semi-Infinite Thin Plate,” J. Sound Vib., 320(4–5), pp. 878–892. [CrossRef]
Chan, K. L. , Smith, B. , and Wester, E. , 2009, “ Flexural Wave Scattering in a Quarter-Infinite Thin Plate With Circular Scatterers,” Int. J. Solids Struct., 46(20), pp. 3669–3676. [CrossRef]
Lee, W. M. , and Chen, J. T. , 2010, “ Scattering of Flexural Wave in a Thin Plate With Multiple Circular Inclusions by Using the Null-Field Integral Equation Approach,” J. Sound Vib., 329(8), pp. 1042–1061. [CrossRef]
Lee, W. M. , and Chen, J. T. , 2011, “ Scattering of Flexural Wave in a Thin Plate With Multiple Circular Inclusions by Using the Multipole Method,” Int. J. Mech. Sci., 53(8), pp. 617–627. [CrossRef]
Waterman, P. C. , 1965, “ Matrix Formulation of Electromagnetic Scattering,” Proc. IEEE, 53(8), pp. 805–812. [CrossRef]
Waterman, P. C. , 1969, “ New Formulation for Acoustic Scattering,” J. Acoust. Soc. Am., 45(6), pp. 1417–1429. [CrossRef]
Cai, L.-W. , and Williams, J. H., Jr ., 1999, “ Large Scale Multiple-Scattering Problems,” Ultrasonics, 37(7), pp. 453–462. [CrossRef]
Parnell, W. J. , and Martin, P. A. , 2011, “ Multiple Scattering of Flexural Waves by Random Configurations of Inclusions in Thin Plates,” Wave Motion, 48(2), pp. 161–175. [CrossRef]
Graff, K. F. , 1975, Wave Motions in Elastic Solids, Dover Publications, New York.
Abramowitz, M. , and Stegun, I. , 1965, Handbook of Mathematical Functions, Dover Publications, New York.
Watson, G. N. , 1944, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, UK.
Cai, L.-W. , and Williams, J. H., Jr. , 1999, “ Full-Scale Simulations of Elastic Wave Scattering in Fiber Reinforced Composites,” Ultrasonics, 37(7), pp. 463–482. [CrossRef]
Twersky, V. , 1961, “ Elementary Function Representations of Schlömilch Series,” Arch. Ration. Mech. Anal., 8(1), pp. 323–332. [CrossRef]
Cai, L.-W. , 2006, “ Evaluation of Layered Multiple-Scattering Method for Antiplane Shear Wave Scattering From Gratings,” J. Acoust. Soc. Am., 120(1), pp. 49–61. [CrossRef]


Grahic Jump Location
Fig. 1

Geometries in multiple scattering: (a) geometry for Graf's addition theorem for Bessel functions and (b) geometric relations between two local coordinate systems

Grahic Jump Location
Fig. 2

Amplitude of deflection in thin plate embedded with four different circular inclusions, from quadrant I–IV: void, rigid, rubber, and steel. For the cases of no wave fields inside the inclusion (void and rigid), the deflection is set to zero.

Grahic Jump Location
Fig. 3

Distribution of the deflection amplitude along the two diagonal lines of Fig. 2. Dashed vertical lines demarcate the interfaces between host plate and scatterers.

Grahic Jump Location
Fig. 4

Configuration of ensemble of inclusions forming a rectangular lattice arrangement. The line segment to the right indicates the location where deflection is averaged.

Grahic Jump Location
Fig. 5

Spectra of averaged deflection amplitude in forward direction for sonic crystals of four different types of inclusions

Grahic Jump Location
Fig. 6

Amplitude distribution near sonic crystals at f = 213 kHz (ka = 1.3077). Left: void inclusions and right: rubber inclusions.

Grahic Jump Location
Fig. 7

Amplitude distribution near sonic crystals at f = 1 MHz (ka = 2.8334). Left: void inclusions and right: rubber inclusions.




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