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Research Papers

Controlling Out-of-Plane Elastic Shear Wave Propagation With Broadband Cloaking

[+] Author and Article Information
M. Liu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland,
Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland,
Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Manuscript received March 5, 2018; final manuscript received April 13, 2018; published online May 21, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(8), 081002 (May 21, 2018) (11 pages) Paper No: JAM-18-1128; doi: 10.1115/1.4040017 History: Received March 05, 2018; Revised April 13, 2018

A major challenge in designing a perfect invisibility cloak for elastic waves is that the mass density and elasticity tensor need to be independent functions of its radius with a linear transformation medium. The traditional cloak for out-of-plane shear waves in elastic membranes exhibits material properties with inhomogeneous and anisotropic shear moduli and densities, which yields a poor or even negative cloaking efficiency. This paper presents the design of a cylindrical cloak for elastic shear waves based on a nonlinear transformation. This excellent broadband nonlinear cloak only requires variation of its shear modulus, while the density in the cloak region remains unchanged. A nonlinear ray trajectory equation for out-of-plane shear waves is derived and a parameter to adjust the efficiency of the cylindrical cloak is introduced. Qualities of the nonlinear invisibility cloak are discussed by comparison with those of a cloak with the linear transformation. Numerical examples show that the nonlinear cloak is more effective for shielding out-of-plane elastic shear waves from outside the cloak than the linear cloak and illustrate that the nonlinear cloak for shear waves remains highly efficient in a broad frequency range. The proposed nonlinear transformation in conjunction with the ray trajectory equation can also be used to design nonlinear cloaks for other elastic waves.

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Figures

Grahic Jump Location
Fig. 1

Path trajectories of out-of-plane elastic shear waves in the cylindrical cloak

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Fig. 2

Computation scheme of the cloak for out-of-plane elastic shear waves; the lower right corner D under the cloak is a given integration region for measuring scattering in Secs. 2.2 and 3

Grahic Jump Location
Fig. 3

Snapshots of out-of-plane displacement fields of the elastic membrane with the nonlinear cloak with (a) ω=15 and (d) ω=30, and the linear cloak with (b) ω=15 and (e) ω=30; those without a cloak with ω=15,30 are shown in (c) and (f), respectively

Grahic Jump Location
Fig. 4

(a) Out-of-plane displacements of the elastic membrane along the line AB with the nonlinear (solid line) and linear (dotted line) cloaks, and without a cloak (dashed line), when ω=15 and (b) an enlarged view of the displacements in (a)

Grahic Jump Location
Fig. 5

(a) Out-of-plane displacements of the elastic membrane along the line AB with the nonlinear (solid line) and linear (dotted line) cloaks, and without a cloak (dashed line), when ω=30 and (b) an enlarged view of the displacements in(a)

Grahic Jump Location
Fig. 6

Scattering energies in the inclusion under the nonlinear and linear cloaks versus the nondimensional radian excitation frequency

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Fig. 7

Snapshots of out-of-plane displacement fields of the elastic membrane with the nonlinear cloak with (a) ω=50, (b) ω=60, and (c) ω=70, and the linear cloak with (d) ω=50, (e) ω=60, and (f) ω=70

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Fig. 8

Scattered energies in the lower right corner D under the nonlinear and linear cloaks versus the nondimensional radian excitation frequency

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Fig. 9

Displacement fields of the elastic membrane with different interior boundary conditions of the nonlinear cloak: (a) the Dirichlet boundary condition and (b) Neumann boundary condition. (c) The corresponding displacements along the line AB; the dashed line corresponds to (a) and the dotted line corresponds to (b). (d) An enlarged view of the displacements in (c).

Grahic Jump Location
Fig. 10

Out-of-plane displacement fields of the elastic membrane excited by two unit point sources (a) without and (b) with the nonlinear cloak. The corresponding displacements along the line AB are shown in (c); the dashed line corresponds to (a) and the dotted line corresponds to (b).

Grahic Jump Location
Fig. 11

Out-of-plane displacement fields of the elastic membrane excited by four unit point sources (a) without and (b) with the nonlinear cloak. The corresponding displacements along the line AB are shown in (c); the dashed line corresponds to (a) and the dotted line corresponds to (b).

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Fig. 12

Out-of-plane displacement fields of the elastic membrane with the nonlinear cloak and different impact parameters δ: (a) δ=1 and (b) δ=2. The corresponding displacements along the line AB are shown in (c); the dashed line corresponds to (a) and the dotted line corresponds to (b).

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