Can We Hear the Echos of Cracks?
- First Online:
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.
KeywordsMathematical modelling Non destructive testing Wave equation Fracture mechanics
Mathematics Subject Classification35C07 65M15 35M12 65T60
- 6.Dobrowolski, M.: Numerical Approximation of Elliptic Interface and Corner Problems. Habilitationsschrift, Bonn (1981) Google Scholar
- 7.Dumont-Fillon, J-C.: Contrôle non destructif par les ondes de Love et Lamb. Editions Techniques de l’ingénieur (2012) Google Scholar
- 11.Grisvard, P.: Singularity in Domains with Corner. Pitman, London (1988) Google Scholar
- 12.Holmgrem, E.: Über Systeme von linearen partiellen Differentialgleichungen. Öfvers. K. Vetensk.-Akad. Förh., vol. 58, pp. 91–103 (1901) Google Scholar
- 14.Kato, I.: Spectral Theory of Linear Operators. Springer, Berlin (1966) Google Scholar
- 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.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). http://www3.imperial.ac.uk/pls/portallive/docs/1/50545711.PDF
- 18.Necas, J.: Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1965) Google Scholar
- 19.Raviart, P.A., Thomas, J.M.: Approximation des équations aux dérivées partielles. Masson, Paris (1986) Google Scholar