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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

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 & 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. Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.  Google Scholar

[3]

H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1979. Google Scholar

[4]

P. G. Ciarlet, The Finite Element Mehod for Elliptic Problems, Elsevier, Amsterdam, 1978.  Google Scholar

[5]

Ph. Destuynder and C. Fabre, Singularities occuring in multimaterials with traPHDCF3nsparent boundary conditions, to appear in Quaterly of Applied Maths, (2016). Google Scholar

[6]

Ph. Destuynder and C. Fabre, On the Detection of Cracks in Linear Elasticity, research report CNAM, 2015. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981. Google Scholar

[10]

J.-C. Dumont-Fillon, Contrôle non Destructif Par Les Ondes de Love et Lamb, Editions Techniques de l'ingénieur, 2012. Google Scholar

[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.  Google Scholar

[12]

E. Holmgren, Über systeme von linearen partiellen differentialgleichungen, Öfversigt af kongl, Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103. Google Scholar

[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.  Google Scholar

[14]

S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999. Google Scholar

[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 Google Scholar

[16]

J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, (1965). Google Scholar

[17]

P. A. Raviart and J. M. Thomas, Approximation des Équations aux Dérivées Partielles, Masson, Paris, 1986. Google Scholar

[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.  Google Scholar

[19]

G. Strang and G. Fix, Analysis of the Finite Elements Method, Prentice Hall; Edition: First Edition, 1973.  Google Scholar

[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. Google Scholar

[21]

D. Zagier, The Dilog function, http://maths.dur.ac.uk/~dma0hg/dilog.pdf (2007). Google Scholar

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. Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.  Google Scholar

[3]

H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1979. Google Scholar

[4]

P. G. Ciarlet, The Finite Element Mehod for Elliptic Problems, Elsevier, Amsterdam, 1978.  Google Scholar

[5]

Ph. Destuynder and C. Fabre, Singularities occuring in multimaterials with traPHDCF3nsparent boundary conditions, to appear in Quaterly of Applied Maths, (2016). Google Scholar

[6]

Ph. Destuynder and C. Fabre, On the Detection of Cracks in Linear Elasticity, research report CNAM, 2015. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981. Google Scholar

[10]

J.-C. Dumont-Fillon, Contrôle non Destructif Par Les Ondes de Love et Lamb, Editions Techniques de l'ingénieur, 2012. Google Scholar

[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.  Google Scholar

[12]

E. Holmgren, Über systeme von linearen partiellen differentialgleichungen, Öfversigt af kongl, Vetenskaps-Academien Förhandlinger, 58 (1901), 91-103. Google Scholar

[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.  Google Scholar

[14]

S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999. Google Scholar

[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 Google Scholar

[16]

J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, (1965). Google Scholar

[17]

P. A. Raviart and J. M. Thomas, Approximation des Équations aux Dérivées Partielles, Masson, Paris, 1986. Google Scholar

[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.  Google Scholar

[19]

G. Strang and G. Fix, Analysis of the Finite Elements Method, Prentice Hall; Edition: First Edition, 1973.  Google Scholar

[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. Google Scholar

[21]

D. Zagier, The Dilog function, http://maths.dur.ac.uk/~dma0hg/dilog.pdf (2007). Google Scholar

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