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THE ALTERNATIVE KIRCHHOFF APPROXIMATION IN ELASTODYNAMICS WITH APPLICATIONS IN ULTRASONIC NONDESTRUCTIVE TESTING

Part of: Waves Dynamical problems Special subfields of solid mechanics

Published online by Cambridge University Press:  23 April 2020

L. J. FRADKIN Open the ORCID record for L. J. FRADKIN [Opens in a new window] ,
A. K. DJAKOU Open the ORCID record for A. K. DJAKOU [Opens in a new window] ,
C. PRIOR Open the ORCID record for C. PRIOR [Opens in a new window] ,
M. DARMON Open the ORCID record for M. DARMON [Opens in a new window] ,
S. CHATILLON Open the ORCID record for S. CHATILLON [Opens in a new window]  and
P.-F. CALMON Open the ORCID record for P.-F. CALMON [Opens in a new window]
L. J. FRADKIN*
Affiliation:
Sound Mathematics Ltd., CambridgeCB4 2AS, UK; e-mail: l.fradkin@soundmathematics.com
A. K. DJAKOU
Affiliation:
Sound Mathematics Ltd., Cambridge CB4 2AS, UK. Now at GEFCO, Courbevoie Cedex, France; e-mail: noelleaudrey@yahoo.fr
C. PRIOR
Affiliation:
Sound Mathematics Ltd., Cambridge CB4 2AS, UK. Now at Department of Mathematical Sciences, Durham University, Durham, UK; e-mail: christopher.prior@durham.ac.uk
M. DARMON
Affiliation:
CEA, LIST, Department of Imaging and Simulation for NDT, F-91191Gif-sur-Yvette, France; e-mail: Michel.DARMON@cea.fr, Sylvain.CHATILLON@cea.fr, Pierre.CALMON@cea.fr
S. CHATILLON
Affiliation:
CEA, LIST, Department of Imaging and Simulation for NDT, F-91191Gif-sur-Yvette, France; e-mail: Michel.DARMON@cea.fr, Sylvain.CHATILLON@cea.fr, Pierre.CALMON@cea.fr
P.-F. CALMON
Affiliation:
CEA, LIST, Department of Imaging and Simulation for NDT, F-91191Gif-sur-Yvette, France; e-mail: Michel.DARMON@cea.fr, Sylvain.CHATILLON@cea.fr, Pierre.CALMON@cea.fr
*
e-mail: l.fradkin@soundmathematics.com
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Abstract

The Kirchhoff approximation is widely used to describe the scatter of elastodynamic waves. It simulates the scattered field as the convolution of the free-space Green’s tensor with the geometrical elastodynamics approximation to the total field on the scatterer surface and, therefore, cannot be used to describe nongeometrical phenomena, such as head waves. The aim of this paper is to demonstrate that an alternative approximation, the convolution of the far-field asymptotics of the Lamb’s Green’s tensor with incident surface tractions, has no such limitation. This is done by simulating the scatter of a critical Gaussian beam of transverse motions from an infinite plane. The results are of interest in ultrasonic nondestructive testing.

Keywords

nondestructive testingultrasoundhigh-frequency asymptoticsLamb’s Green’s tensorcritical Gaussian beam

MSC classification

Primary: 74H15: Numerical approximation of solutions
Secondary: 74H10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) 74J05: Linear waves 74J10: Bulk waves 74J15: Surface waves 74J20: Wave scattering 74L99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 4 , October 2020 , pp. 406 - 422
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Copyright
© 2020 Australian Mathematical Society

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References

Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th edn (US Government Printing Office, Washington, DC, 1972).Google Scholar
Borovikov, V. A., Uniform stationary phase method (IEE, London, 1994).Google Scholar
Cěrvenỳ, V., Seismic ray theory (Cambridge University Press, Cambridge, 2001).Google Scholar
Darmon, M., Leymarie, N., Chatillon, S. and Mahaut, S., “Modelling of scattering of ultrasounds by flaws for NDT”, in: Ultrasonic wave propagation in non homogeneous media, Volume 128 of Springer Proc. Phys. (Springer, Berlin–Heidelberg, 2009) 61–71; doi:10.1007/978-3-540-89105-5_6.Google Scholar
Deschamps, G., “Gaussian beam as a bundle of complex rays”, Electron. Lett. 7 (1971) 684–685; doi:10.1049/el:19710467.Google Scholar
Fradkin, L., Darmon, M., Chatilion, S. and Calmon, P., “A semi-numerical model for near-critical angle scattering”, J. Acoust. Soc. Am. 139 (2016) 141–150; doi:10.1121/1.4939494.Google Scholar
Fradkin, L. J. and Kiselev, A. P., “The two-component representation of time-harmonic elastic body waves in the high- and intermediate-frequency regimes”, J. Acoust. Soc. Am. 101 (1997) 52–65; doi:10.1121/1.417970.Google Scholar
Graft, K. F., Wave motion in elastic solids (Dover, New York, 1975).Google Scholar
Gridin, D., “A fast method for simulating the propagation of pulses radiated by a rectangular normal transducer into an elastic half-space”, J. Acoust. Soc. Am. 104 (1998) 3199–3211; doi:10.1121/1.423960.Google Scholar
Huet, G., Darmon, M., Lhemery, A. and Mahaut, S., “Modeling of corner echo ultrasonic inspection with bulk and creeping waves”, in: Ultrasonic wave propagation in non homogeneous media, Volume 128 of Springer Proc. Phys. (Springer, Berlin, Heidelberg, 2009) 217–226; doi:10.1007/978-3-540-89105-5_19.Google Scholar
Miller, G. F. and Pursey, H., “The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid”, Proc. R. Soc. 223 (1954) 521–541; doi:10.1098/rspa.1954.0134.Google Scholar
Pott, J. and Harris, J. G., “Scattering of an acoustic Gaussian beam from a fluid–solid interface”, J. Acoust. Soc. Am. 76 (1984) 1829–1838; doi:10.1121/1.391483.Google Scholar
Schmerr, L. W., Fundamentals of ultrasonic non-destructive evaluation: a modeling approach (Plenum Press, New York, 1998).Google Scholar
Zernov, V., Fradkin, L., Gautesen, A., Darmon, M. and Calmon, P., “Wedge diffraction of a critically incident Gaussian beam”, Wave Motion 50 (2013) 708–722; doi:10.1016/j.wavemoti.2013.01.004.Google Scholar