Journal of Elasticity

, Volume 130, Issue 1, pp 25–58 | Cite as

Can We Hear the Echos of Cracks?



The detection of cracks in mechanical engineering is mainly based on ultrasonic testing and Foucault currents. But even if they are efficient tools, this technology requires an important handling and is limited to the detection of cracks which are close to the source. Recently, several searchers have discussed the possibility of using waves as Lamb waves, for thin plates and shells, but also Love waves for bimaterials. In both cases the structure works as a wave guide and enables a long range propagation which is a promising possibility for detecting a crack quite far from the source. In this paper, we discuss the observability property of a small crack inside an open set using Love waves (the goal is to detect the beginning of the growth). It is proved that an adapted selection of these waves is necessary in order to avoid a black out which can occur for particular frequencies. This phenomenon is due to the fact that the crack has two extremities which can cancel their influence in a detection criterion. The main contribution of this paper is to discuss this point from a mathematical point of view, using an energy criterion requiring measurements quite far from the defect which should be detected.


Mathematical modelling Non destructive testing Wave equation Fracture mechanics 

Mathematics Subject Classification

35C07 65M15 35M12 65T60 


  1. 1.
    Avila-Pozos, O., Mishuris, G., Movchan, A.: Bloch-Floquet waves and localisation within a heterogeneous waveguides with long cracks. Contin. Mech. Thermodyn. 22, 545–553 (2010) ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blum, H., Dobrowolski, M.: On finite element methods for elliptic equations on domains with corners. Computing 28, 53–63 (1982) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Destuynder, Ph., Djaoua, M.: Sur une interpretation mathématique de l’intégrale de Rice en mécanique de la rupture fragile. Math. Methods Appl. Sci. 3, 70–87 (1981) ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Destuynder, Ph., Fabre, C.: Singularities occurring in multimaterials with transparent boundary conditions. Q. Appl. Math. 3, 443–463 (2016) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Destuynder, Ph., Fabre, C.: Few remarks on the use of Love waves in non destructive testing. Discrete Contin. Dyn. Syst., Ser. S 9, 427–444 (2016) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dobrowolski, M.: Numerical Approximation of Elliptic Interface and Corner Problems. Habilitationsschrift, Bonn (1981) Google Scholar
  7. 7.
    Dumont-Fillon, J-C.: Contrôle non destructif par les ondes de Love et Lamb. Editions Techniques de l’ingénieur (2012) Google Scholar
  8. 8.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part 1: General Theory. Wiley Classic Library. Wiley New York (1988) MATHGoogle Scholar
  9. 9.
    Friedman, A., Vogelius, M.: Determining cracks by boundary measurements. Indiana Univ. Math. J. 38, 527–556 (1989) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Galvagni, A., Cawley, P.: The reflection of guided waves from simple supports in pipes. J. Acoust. Soc. Am. 129, 1869–1880 (2011) ADSCrossRefGoogle Scholar
  11. 11.
    Grisvard, P.: Singularity in Domains with Corner. Pitman, London (1988) Google Scholar
  12. 12.
    Holmgrem, E.: Über Systeme von linearen partiellen Differentialgleichungen. Öfvers. K. Vetensk.-Akad. Förh., vol. 58, pp. 91–103 (1901) Google Scholar
  13. 13.
    Hutchinson, J.W., Mear, M.E., Rice, J.R.: Cracks paralleling an interface between dissimilar materials. J. Appl. Mech. 109, 828–832 (1987) CrossRefGoogle Scholar
  14. 14.
    Kato, I.: Spectral Theory of Linear Operators. Springer, Berlin (1966) Google Scholar
  15. 15.
    Lions, J.L., Magenès, E.: Problèmes aux limites non homogènes, T.1. Dunod, Paris (1968) MATHGoogle Scholar
  16. 16.
    Lowe, M.J.S.: In: Thompson, D.O., Chimenti, D.E. (eds.) Characteristics of the Reflection of Lamb Waves from Defects in Plates and Pipes. Review of Progress in Quantitative NDE, vol. 17, pp. 113–120. Plenum, New York (2002) Google Scholar
  17. 17.
    Marty, P.M.: Modelling of ultrasonic guided wave field generated by piezoelectric transducers. Thesis at Imperial College of Science, Technology and Medicine, University of London (2002).
  18. 18.
    Necas, J.: Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1965) MATHGoogle Scholar
  19. 19.
    Raviart, P.A., Thomas, J.M.: Approximation des équations aux dérivées partielles. Masson, Paris (1986) Google Scholar
  20. 20.
    Ribichini, R., Cegla, F., Nagy, P., Cawley, P.: Study and comparison of different EMAT configurations for SH wave inspection. IEEE Trans. UFFC 58, 2571–2581 (2011) CrossRefGoogle Scholar
  21. 21.
    Rice, J.R.: A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968) ADSCrossRefGoogle Scholar
  22. 22.
    Rice, J.R.: Two general integrals of singular crack tip deformation fields. J. Elast. 20(2), 131–142 (1988) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Département d’ingénierie mathématique, laboratoire M2NConservatoire National des Arts et MétiersParisFrance
  2. 2.Laboratoire de mathématiques d’Orsay, Univ Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

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