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On the Theory of Direct and Inverse Problems in the Elastic Rectangle: Antiplane Case

In this article, we study the antiplane deformation of the boundary surface of a rectangular domain in the presence of a void and a shear force on the outer boundary surface. For a formulated inverse problem, we develop some analytical results and use them to solve the problem numerically for various elliptic geometrical configurations. The analytical method allows us to give an efficient representation for Green’s function in the rectangular domain. Then we derive the same Green’s function by an alternative method based on Fourier series expansions. Finally, for a number of configurations, we demonstrate the comparison between real and reconstructed defects.

1. I. G. Scott. Basic Acoustic Emission. Gordon and Breach, New York (1991). 2. J. Krautkra¨mer and H. Krautkra¨mer. Ultrasonic Testing of Materials. Springer-Verlag, Berlin (1983). 3. D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. John Wiley, New York (1983). 4. A. O. Vatulyan and A. N. Soloviev. Identification of the size of defects in a compound elastic solid. Russian J. NDT 5:298–304 (2004). 5. A. S. Saada. Elasticity: Theory and Applications, 2nd ed. Krieger, Malabar, Florida (1993). 6. A. Friedman and M. Vogelius. Determining cracks by boundary measurements. Indiana Univ. Math. J. 38:527–556 (1989). 7. G. Alessandrini, E. Beretta, and S. Vessella. Determining linear cracks by boundary measurements: Lipschitz stability. SIAM J. Math. Anal. 27:361–375 (1996). 8. R. Courant and D. Hilbert. Methods of Mathematical Physics, vol. 1. Interscience, New York (1953). 9. N. I. Muskhelishvili. Some Basic Problems of the Mathematical Theory of Elasticity. Kluwer, Dordrecht (1975). 10. B. Riemann and K. Hattendorf. Schwere: Elektrizita¨t und Magnetismus. Hannover (1880). 11. K. D. Cole and D. H. Y. Yen. Green’s functions, temperature and heat flux in the rectangle. Int. J. Heat Mass Transfer 44:3883–3894 (2001). 12. I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 5th ed. Academic Press, New York (1994). 13. H. Hardy. Divergent Series. Oxford University Press, London (1956). 14. M. Bonnet. Boundary Integral Equations Methods for Solids and Fluids. John Wiley, New York (1999). 15. A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-Posed Problems. Winston, Washington DC. (1977). 16. A. B. Abda, M. Kallel, J. Leblond, and J. P. Marmorat. Line segment crack recovery from incomplete boundary data. Inverse Probl. 18:1057–1077 (2002). 17. S. Andrieux and A. B. Abda. Identification of planar cracks by complete overdetermined data: Inversion formulae. Inverse Probl. 12:553–563 (1996). 18. T. Bannour, A. B. Abda, and M. Jaoua. A semi-explicit algorithm for the reconstruction of 3D planar cracks. Inverse Probl. 13:899–917 (1997). 19. A. Friedman and M. Vogelius. Determining cracks by boundary measurements. Indiana Univ. Math. J. 38:527–556 (1989). 20. B. B. Guzina, S. N. Fata, and M. Bonnet. An elastodynamic BIE approach to underground cavity detection. Electronic Journal of Boundary Elements 2:223–230 (2002, BETEQ 2001). 21. L. V. Kantorovich and G. P. Akilov. Functional Analysis. Pergamon Press, Oxford (1982). 22. P. E. Gill, W. Murray, and M. H. Wright. Practical Optimization. Academic Press, London (1981). 23. M. Corana, et al. Minimizing multimodal functions of continuous variables with the simulated annealing algorithm. ACM Trans. Math. Software 13:262–280 (1987).
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