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TECHNICAL PAPERS

Analysis of Wave Propagation in Beams With Transverse and Lateral Cracks Using a Weakly Formulated Spectral Method

[+] Author and Article Information
N. Hu

Department of Aeronautics and Space Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japanhu@ssl.mech.tohoku.ac.jp

H. Fukunaga, M. Kameyama

Department of Aeronautics and Space Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan

D. Roy Mahapatra, S. Gopalakrishnan

Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India

J. Appl. Mech 74(1), 119-127 (Jan 18, 2006) (9 pages) doi:10.1115/1.2188015 History: Received December 16, 2004; Revised January 18, 2006

In this paper, a novel numerical technique based on the global-local hybrid spectral element (HSE) method is proposed to study wave propagation in beams containing damages in the form of transverse and lateral cracks. The ordinary spectral element method is employed to model the exterior or far field regions, while a new type of element (HSE) is constructed to model the interior region containing damages. To develop this efficient new element for the damaged area, first, the flexural and the shear wave numbers are explicitly determined using the first-order shear deformation theory. These wave modes, in one of the two mutually orthogonal directions for two-dimensional transient elastodynamics, are then used to enrich the Lagrangian interpolation functions in context of displacement-based finite element. The equilibrium equation is then derived through the weak form in the frequency domain. Frequency-dependent stiffness and mass matrices can be accurately formed in this manner with a coarse discretization. The proposed method takes the advantage of using (i) a strong form for one-dimensional wave propagation and also (ii) a weak form by which a complex geometry can be discretized. Numerical verification is carried out to illustrate the effectiveness of the method. Finally, this method is employed to investigate the behaviors of wave propagation in beams containing various types of damages, such as multiple transverse cracks and lateral cracks.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 3

Deflections of the present element and the traditional FEM at two measurement points

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Figure 4

(a) Transverse velocities of the present element and the throw-off spectral element at two measurement points for a beam of thickness of 10mm, (b) transverse velocities of the present element and the throw-off spectral element at two measurement points for a beam of thickness of 5mm, (c) relative difference between velocities of the present element and the throw-off spectral element at two measurement points for a beam of thickness of 10mm, (d) relative difference between velocities of the present element and the throw-off spectral element at two measurement points for a beam of thickness of 5mm, and (e) convergence check of number of Gauss integration points along the x-axis for a beam of thickness of 10mm

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Figure 5

(a) Schematic diagram of a cantilever beam using the spectral element only and (b) schematic diagram of a cantilever beam using the hybrid approach

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Figure 9

Schematic diagram of a cantilever beam with four transverse cracks

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Figure 10

Transverse velocities of intact and cracked beams at the load point

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Figure 11

Schematic diagram of a cantilever beam with a lateral crack at midplane

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Figure 12

Transverse velocities of intact and cracked beams at the load point

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Figure 1

Schematic diagram of the problem geometry

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Figure 2

Schematic diagram of a 1D problem

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Figure 6

Transverse velocities of the spectral element only and the hybrid approach at the load point

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Figure 7

Schematic diagram of a cantilever beam with two symmetric transverse cracks

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Figure 8

(a) Transverse velocities of intact and cracked beams for h1=3.0mm at the load point and (b) transverse velocities of intact and cracked beams for h1=1.5mm at the load point

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