Abstract
The identification of distributed elastic moduli of damaged materials or crack like defects in elastic media are inverse problems known as generalized elastic tomography. It consists of recovering the damaged zone or the crack in a 3D body using mechanical overdetermined boundary data. For 21 distributed unknowns which are small perturbations of elastic isotropy, a linear system of rank 5 may be derived directly from the observation equations which involves both domains and boundary integrals, with actual mechanical fields and the proposed adjoint fields. It is found that the generalization of Calderon’s method in elasto-statics provided a linear system of rank 5, hence identification problems for small symmetry up to 5 elastic moduli fields could be solved. Finally, the problem of identification of a plane crack in 3D elastic body illustrates the ability of the observation equation method to provide closed form solution for the identification of the crack plane and the crack geometry.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alessandrini, G. On the Identification of the Leading Coefficient of an Elliptic Equation. Bolletino U.M.I., Annalisi funzionale et applicazioni, Serie VI, IV-C:87–111, 1985.
Alessandrini, G. Stable Determination of Conductivity by Boundary Measurements. Appl. Analysis, 27:153–172, 1988.
Andrieux S. and Ben Abda A. Determination of planar cracks by external measurements via reciprocity map functionals: identifiability results and direct location methods, in H.D. Bui et al., editors Inverse Problems in Engineering, Balkema, Rotterdam/ Brookfield, 1994.
Andrieux S. Fonctionnelles d’écart la réciprocité généralisée et identification.. C. R. Acad. Sci. Paris, 1,315, 1992.
Andrieux, S. and Ben Abda, A. and Bui, H.D. On the Identification of Planar Crack in Elasticity Via Reciprocity Gap Concept. C. R. Acad. Sci. Paris, I, 324:1431–1438, 1997.
Bonnet, M. and Constantinescu A. Quelques remarques sur l’identification de modules élastiques l’aide de mesures sur la frontire. Actes du 11e Congres francais de Mécanique, Villeneuve d’Ascq, 6–10 Sept. 1993, Presse Univ. de Lille, 1993.
Bui, H.D. Introduction aux problmes inverses en mécanique des matériaux. Eyrolles, Paris, 1993 (English Version: Inverse Problems in the Mechanics of Materials. An Introduction, CRC Press, Boca Raton, 1994; Japanese Version: Shokabo, Tokyo, 1994).
Calderon, A. On an Inverse Boundary Value Problem. Seminar on numerical analysis and its application to continuum physics, Soc. Brasiliera de Matematica, Rio de Janeiro, pp. 65–73, 1980.
Constantinescu A. Sur l’identification des modules élastiques. PhD Thesis, Ecole Polytechnique, Palaiseau, France, 1994.
Constantinescu A. On the identification of elastic moduli from displacement-force boundary measurements. Int. J. of Inverse Problems in Engineering, 1:293–315, 1995
Cowin, S.C. and Mehrabadi, M. Eigen-tensors of Linear Elastic Materials. Quart. J. Mech. Appl. Math, 43:1, 1990.
Friedman, A. and Vogelius, M. Determining Cracks by Boundary Measurements. Indiana Univ. Math. J., 48(3), 1989.
Grediac M. and Toukourou, C. and Vautrin, A. Inverse Problem in Mechanics of Structures: A New Approach Based on Displacement Field Processing. Bui H.D. et Tanaka M., editors In Inverse Problems in Engineering Mechanics, IUTAM Symposium, Tokyo, Springer Verlag, 1992
Ikehata, M. Inversion for the Linearized Problem for an Inverse Boundary Value Problem in Elastic Prospection. SIAM J. Appl. Math., 50(6):1635, 1990.
Isaacson, D. and Isaacson E.L. Comments on Calderon’s Paper “On an Inverse Boundary Value Problem”. Math. Compt., 52:553, 1989.
Kohn, R.V. and Vogelius M. Determining Conductivity by Boundary Measurements. Coram. Pure Appl. Math., 27:289–298, 1984.
Kohn, R.V. and Vogelius M. Relaxation of a Variational Method for Impedance Computed Tomography. Comm. Pure Appl. Math., 15:745–777, 1987.
Lions, J.L. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris, 1968
Nakamura, G. and Uhlmann, G. Uniqueness for Identifying Lamé Moduli by Dirichlet to Neumann Map. Yamaguti M. editor, Inverse problems in Engineering Sciences, ICM-90 Satellite Conference Proceedings, Springer Verlag, Tokyo, 1991.
Sabatier, P. Basic Methods of Tomography and Inverse Problems. Malvern Physics Series, Adam Hilger, Bristol, 1987.
Sun, Z. and Uhlmann, G. Generic Uniqueness for Determined Inverse Problems in 2D. Yamaguti M. editor, Inverse problems in Engineering Sciences, ICM-90 Satellite Conference Proceedings, Springer Verlag, Tokyo, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Bui, H.D., Constantinescu, A. (2001). Applications of Boundary Integral Equations for Solving Some Identification Problems in Elasticity. In: Burczynski, T. (eds) IUTAM/IACM/IABEM Symposium on Advanced Mathematical and Computational Mechanics Aspects of the Boundary Element Method. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9793-7_3
Download citation
DOI: https://doi.org/10.1007/978-94-015-9793-7_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5737-2
Online ISBN: 978-94-015-9793-7
eBook Packages: Springer Book Archive