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Research Papers

Velocity-Stress Equations for Waves in Solids of Hexagonal Symmetry Solved by the Space-Time CESE Method

[+] Author and Article Information
Lixiang Yang

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210yang.1130@osu.edu

Yung-Yu Chen

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210chen.1352@osu.edu

Sheng-Tao John Yu1

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210yu.274@osu.edu

1

Corresponding author.

J. Vib. Acoust 133(2), 021001 (Feb 17, 2011) (13 pages) doi:10.1115/1.4002170 History: Received December 10, 2009; Revised June 02, 2010; Published February 17, 2011; Online February 17, 2011

This paper reports an extension of the space-time conservation element and solution element (CESE) method to simulate stress waves in elastic solids of hexagonal symmetry. The governing equations include the equation of motion and the constitutive equation of elasticity. With velocity and stress components as the unknowns, the governing equations are a set of 9, first-order, hyperbolic partial differential equations. To assess numerical accuracy of the results, the characteristic form of the equations is derived. Moreover, without using the assumed plane wave solution, the one-dimensional equations are shown to be equivalent to the Christoffel equations. The CESE method is employed to solve an integral form of the governing equations. Space-time flux conservation over conservation elements (CEs) is imposed. The integration is aided by the prescribed discretization of the unknowns in each solution element (SE), which in general does not coincide with a CE. To demonstrate this approach, numerical results in the present paper include one-dimensional expansion waves in a suddenly stopped rod, two-dimensional wave expansion from a point in a plane, and waves interacting with interfaces separating hexagonal solids with different orientations. All results show salient features of wave propagation in hexagonal solids and the results compared well with the available analytical solutions.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

The Cartesian coordinates and the lattice structure of solids of hexagonal symmetry

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Figure 2

Rotations of the Cartesian coordinate system. (a) Rotation with respect to the x3 axis. (b) Rotation with respect to the x¯2 axis. (c) Overall rotation.

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Figure 3

A schematic of the CESE method in one spatial dimension. (a) Zigzagging SEs. (b) Integration over a CE to solve ui and (ux)i at the new time level.

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Figure 4

The CEs and SEs of the two-dimensional CESE method.

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Figure 5

CEs and SE associated with the mesh point (j,n). (a) CEr(j,n),r=1,2,3. (b) SE(j,n).

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Figure 6

Four snapshots of simulated wave propagation along a compression direction in a block of cadmium sulphide, which is suddenly stopped. Calculated solutions are marked with × symbol, while the analytical solutions are plotted as solid lines. The maximum CFL number (ν) is about 0.95. (a) t=0 μs, (b) t=40 μs, (c) t=200 μs, and (d) t=280 μs.

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Figure 7

The snapshot for simulated wave propagation in a block of cadmium sulphide at t=40 μs with (a) original variables and (b) characteristic variables

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Figure 8

The snapshot for simulated wave propagation in a block of cadmium sulphide at t=120 μs with (a) original variables and (b) characteristic variables

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Figure 9

The snapshot for simulated wave propagation in a block of cadmium sulphide at t=280 μs with (a) original variables and (b) characteristic variables

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Figure 10

Two-dimensional mesh (2.2×106 triangular elements). (a) Close look at the mesh around origin. (b) Decomposed 16 subdomains.

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Figure 11

Comparison between the analytical solutions of the group velocities (in SH, qS, and qL polarizations) and the calculated energy profiles for beryl at t=91 μs, including (a) the analytical solution calculated from Eq. 6 and (b) the calculated energy density profiles

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Figure 12

Plots for the normalized total energy density (e/max(e)) at two times: (a) t=26 μs and (b) t=130 μs

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Figure 13

The computational domain is divided into three regions. The orientations of the solids in the central, left, and right regions are θ=0 deg and ϕ=0 deg, θ=0 deg and ϕ=60 deg, and θ=0 deg and ϕ=30 deg, respectively.

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Figure 14

Waves propagation in a heterogeneous domain. The total energy density (e/max(e)) are plotted at different times: (a) t=72 μs, (b) t=96 μs, (c) t=108 μs, and (d) t=180 μs. Δt=60 ns and CFL number=0.95.

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