Few remarks on the use of Love waves in non destructive testing

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April 2016, 9(2): 427-444. doi: 10.3934/dcdss.2016005

Few remarks on the use of Love waves in non destructive testing

Philippe Destuynder ^1, and Caroline Fabre ^2,

1.	Département d'ingénierie mathématique, Conservatoire National des Arts et Métiers, 292, rue saint Martin, 75003 Paris, France
2.	Laboratoire de mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

Received January 2015 Revised October 2015 Published March 2016

Abstract
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This paper concerns a theoretical study on the possibility of using Love waves for non destructive testing. A mathematical model is presented and analyzed. Several numerical tests are given in order to show the mechanical behaviour of this model.

Keywords: wave equation, transparent boundary conditions, Non destructive testing, inverse problems., fracture mechanics.

Mathematics Subject Classification: Primary: 35R30, 35L20, 35P05, 35B40; Secondary: 65N06, 65N1.

Citation: Philippe Destuynder, Caroline Fabre. Few remarks on the use of Love waves in non destructive testing. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 427-444. doi: 10.3934/dcdss.2016005

References:

[1]	M. Amara, Ph. Destuynder and M. Djaoua, On a finite element schem for plane crack problems, Numer. Meth. in Frac. Mech., D.R.J. Owen and A.R. Luxmoore, Pinridge Press, Swansea, (1980), 41-50.
[2]	H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.
[3]	H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1979.
[4]	P. G. Ciarlet, The Finite Element Mehod for Elliptic Problems, Elsevier, Amsterdam, 1978.
[5]	Ph. Destuynder and C. Fabre, Singularities occuring in multimaterials with traPHDCF3nsparent boundary conditions, to appear in Quaterly of Applied Maths, (2016).
[6]	Ph. Destuynder and C. Fabre, On the Detection of Cracks in Linear Elasticity, research report CNAM, 2015.
[7]	Ph. Destuynder and M. Djaoua, Sur une interpretation mathématique de l'intégrale de Rice en mécanique de la rupture fragile, Mathematical Methods in the Applied Sciences, 3 (1981), 70-87. doi: 10.1002/mma.1670030106.
[8]	G. Diot, A. Kouadri-David, L. Dubourg, J. Flifla, S. Guegan and E. Ragneau, Mesures de Défauts par Ultrasons Laser Dans Des Soudures D'alliage D'aluminium, Publications du CETIM, 2014.
[9]	M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981.
[10]	J.-C. Dumont-Fillon, Contrôle non Destructif Par Les Ondes de Love et Lamb, Editions Techniques de l'ingénieur, 2012.
[11]	A. Galvagni and P. Cawley, The reflection of guided waves from simple supports in pipes, AIP Conf. Proc., 105 (2011), p1335. doi: 10.1063/1.3591845.
[12]	E. Holmgren, Über systeme von linearen partiellen differentialgleichungen, Öfversigt af kongl, Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103.
[13]	M. J. S. Lowe, Characteristics of the reflection of Lamb waves from defects in plates and pipes, Review of Progress in Quantitative NDE, DO Thompson and DE Chimenti (eds), Plenum Pr ess, New-York, 17 (2002), 113-120. doi: 10.1007/978-1-4615-5339-7_14.
[14]	S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999.
[15]	P. M. Marty, Modelling of Ultrasonic Guided Wave Field Generated by Piezoelectric Transducers, Thesis at Imperial college of science, technology and medecine, university of London, (2002), http://www3.imperial.ac.uk/pls/portallive/docs/1/50545711.PDF
[16]	J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, (1965).
[17]	P. A. Raviart and J. M. Thomas, Approximation des Équations aux Dérivées Partielles, Masson, Paris, 1986.
[18]	R. Ribichini, F. Cegla, P. Nagy and P. Cawley, Study and comparison of different EMAT configurations for SH wave inspection, IEEE Trans.UFFC, 58 (2011), 2571-2581. doi: 10.1109/TUFFC.2011.2120.
[19]	G. Strang and G. Fix, Analysis of the Finite Elements Method, Prentice Hall; Edition: First Edition, 1973.
[20]	A. N. Tychonoff, Solution of incorrectly formulated problems and the regularization method, Soviet Math Dokl, 4 (2011), 1035-1038; English translation of Dokl Akad Nauk SSSR, 151 (1963), 501-504.
[21]	D. Zagier, The Dilog function, http://maths.dur.ac.uk/~dma0hg/dilog.pdf (2007).

show all references

References:

[1]	M. Amara, Ph. Destuynder and M. Djaoua, On a finite element schem for plane crack problems, Numer. Meth. in Frac. Mech., D.R.J. Owen and A.R. Luxmoore, Pinridge Press, Swansea, (1980), 41-50.
[2]	H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.
[3]	H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1979.
[4]	P. G. Ciarlet, The Finite Element Mehod for Elliptic Problems, Elsevier, Amsterdam, 1978.
[5]	Ph. Destuynder and C. Fabre, Singularities occuring in multimaterials with traPHDCF3nsparent boundary conditions, to appear in Quaterly of Applied Maths, (2016).
[6]	Ph. Destuynder and C. Fabre, On the Detection of Cracks in Linear Elasticity, research report CNAM, 2015.
[7]	Ph. Destuynder and M. Djaoua, Sur une interpretation mathématique de l'intégrale de Rice en mécanique de la rupture fragile, Mathematical Methods in the Applied Sciences, 3 (1981), 70-87. doi: 10.1002/mma.1670030106.
[8]	G. Diot, A. Kouadri-David, L. Dubourg, J. Flifla, S. Guegan and E. Ragneau, Mesures de Défauts par Ultrasons Laser Dans Des Soudures D'alliage D'aluminium, Publications du CETIM, 2014.
[9]	M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981.
[10]	J.-C. Dumont-Fillon, Contrôle non Destructif Par Les Ondes de Love et Lamb, Editions Techniques de l'ingénieur, 2012.
[11]	A. Galvagni and P. Cawley, The reflection of guided waves from simple supports in pipes, AIP Conf. Proc., 105 (2011), p1335. doi: 10.1063/1.3591845.
[12]	E. Holmgren, Über systeme von linearen partiellen differentialgleichungen, Öfversigt af kongl, Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103.
[13]	M. J. S. Lowe, Characteristics of the reflection of Lamb waves from defects in plates and pipes, Review of Progress in Quantitative NDE, DO Thompson and DE Chimenti (eds), Plenum Pr ess, New-York, 17 (2002), 113-120. doi: 10.1007/978-1-4615-5339-7_14.
[14]	S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999.
[15]	P. M. Marty, Modelling of Ultrasonic Guided Wave Field Generated by Piezoelectric Transducers, Thesis at Imperial college of science, technology and medecine, university of London, (2002), http://www3.imperial.ac.uk/pls/portallive/docs/1/50545711.PDF
[16]	J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, (1965).
[17]	P. A. Raviart and J. M. Thomas, Approximation des Équations aux Dérivées Partielles, Masson, Paris, 1986.
[18]	R. Ribichini, F. Cegla, P. Nagy and P. Cawley, Study and comparison of different EMAT configurations for SH wave inspection, IEEE Trans.UFFC, 58 (2011), 2571-2581. doi: 10.1109/TUFFC.2011.2120.
[19]	G. Strang and G. Fix, Analysis of the Finite Elements Method, Prentice Hall; Edition: First Edition, 1973.
[20]	A. N. Tychonoff, Solution of incorrectly formulated problems and the regularization method, Soviet Math Dokl, 4 (2011), 1035-1038; English translation of Dokl Akad Nauk SSSR, 151 (1963), 501-504.
[21]	D. Zagier, The Dilog function, http://maths.dur.ac.uk/~dma0hg/dilog.pdf (2007).

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