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Research Papers

An Improved Technique for Elastodynamic Green's Function Computation for Transversely Isotropic Solids

[+] Author and Article Information
Samaneh Fooladi

Department of Mechanical Engineering,
Penn State University,
The Behrend College,
Erie, PA 16510;
Department of Aerospace and Mechanical Engineering,
University of Arizona,
Tucson, AZ 85721
email: sxf567@psu.edu

Tribikram Kundu

Department of Aerospace and Mechanical Engineering;
Department of Civil and
Architectural Engineering and Mechanics,
University of Arizona,
Tucson, AZ 85721
e-mail: tkundu@email.arizona.edu

Manuscript received April 3, 2019; final manuscript received April 22, 2019; published online May 21, 2019. Special Editor: Wieslaw Ostachowicz.

ASME J Nondestructive Evaluation 2(2), 021005 (May 21, 2019) (7 pages) Paper No: NDE-19-1017; doi: 10.1115/1.4043605 History: Received April 03, 2019; Accepted April 22, 2019

Elastodynamic Green's function for anisotropic solids is required for wave propagation modeling in composites. Such modeling is needed for the interpretation of experimental results generated by ultrasonic excitation or mechanical vibration-based nondestructive evaluation tests of composite structures. For isotropic materials, the elastodynamic Green’s function can be obtained analytically. However, for anisotropic solids, numerical integration is required for the elastodynamic Green's function computation. It can be expressed as a summation of two integrals—a singular integral and a nonsingular (or regular) integral. The regular integral over the surface of a unit hemisphere needs to be evaluated numerically and is responsible for the majority of the computational time for the elastodynamic Green's function calculation. In this paper, it is shown that for transversely isotropic solids, which form a major portion of anisotropic materials, the integration domain of the regular part of the elastodynamic time-harmonic Green's function can be reduced from a hemisphere to a quarter-sphere. The analysis is performed in the frequency domain by considering time-harmonic Green's function. This improvement is then applied to a numerical example where it is shown that it nearly halves the computational time. This reduction in computational effort is important for a boundary element method and a distributed point source method whose computational efficiencies heavily depend on Green's function computational time.

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References

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Fooladi, S., and Kundu, T., 2017, “Ultrasonic Field Modeling in Anisotropic Materials by Distributed Point Source Method,” Ultrasonics, 78, pp. 115–124. [CrossRef] [PubMed]
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Figures

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Fig. 1

Integration over the unit hemisphere for computing the regular part of the dynamic Green's function: (a) 3D view and (b) top view

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Fig. 2

Two unit vectors n(1) and n(2) on the unit sphere with the same azimuthal angle

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Fig. 3

The rotation of the coordinate system from (x1, x2, x3) to (x1′, x2′, x3′)

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Fig. 4

The components GR(1, 1) and GR(1, 2) from the regular part of Green's function

Grahic Jump Location
Fig. 5

The derivative of the components GR(1, 1) and GR(1, 2) from the regular part of Green's function with respect to coordinate x1

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Fig. 6

Ratio of time of computation for quarter-sphere to that of hemisphere

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