In this article we study the reconstruction of the geometry of elliptic voids located in the
elastic wedge, in frames of an antiplane two-dimensional problem. The wedge domain is
bounded by a pair of straight boundaries. The first boundary is horizontal, and the second
one is inclined at some angle. We assume that a known point force is applied to the horizontal
boundary surface of the wedge, whose other face is fixed or is free from load. It is also
assumed that the defect is unknown and that we can measure the shape of the surface over
a certain finite-length interval of the same side where the outer force is applied. Such measurements
can in principle be performed experimentally, but for our numerical experiments
we carried out them by solving the respective direct problem, with the use of boundary
element method (BEM). Then it is shown that an explicit form of Green’s function can be constructed
when the interior angle of the wedge is an integer part of 180. For these cases, we
construct an algorithm to restore the position, the size, and the orientation of the elliptic void.
Some numerical examples demonstrate the good stability of the proposed algorithm.
1. A. Friedman and M. Vogelius. Indiana Univ. Math. J. 38:527–556 (1989).
2. G. Alessandrini, E. Beretta, and S. Vessella. SIAM J. Math. Anal. 27:361–375 (1996).
3. A. B. Abda, M. Kallel, J. Leblond, and J. Marmorat. Inverse Problems 18:1057–1077 (2002).
4. S. Andrieux and A. B. Abda. Inverse Problems 12:553–563 (1996).
5. T. Bannour, A. B. Abda, and M. Jaoua. Inverse Problems 13:899–917 (1997).
6. M. Ciarletta, G. Iovane, and M. A. Sumbatyan. Inverse Probl. Sci. Eng. in press.
7. M. A. Sumbatyan, V. Tibullo, and V. Zampoli. Math. Probl. Eng. article ID 49797, 12 pages (2006).
8. A. Pompei, A. Rigano, and M. A. Sumbatyan. Far East J. Appl. Math. in press.
9. J. W. Nunziato and S. C. Cowin. Arch. Ration. Mech. Anal. 72:175–201 (1979).
10. S. C. Cowin and J. W. Nunziato. J. Elasticty 13:125–147 (1983).
11. A. S. Saada. Elasticity. Theory and Applications, (2nd ed.). Krieger, Malabar, FL (1993).
12. M. Bonnet. Boundary Integral Equations Methods for Solids and Fluids. John Wiley, New York (1999).
13. J. R. Barber. Elasticity, 2nd ed. Kluwer, Dordrecht (2002).
14. O. D. Kellogg. Foundations of Potential Theory. Dover, New York (1953).
15. M. Corana, M. Marchesi, C. Martini, and S. Ridella. ACM Trans. Math. Software 13:262–280 (1987).
16. M. M. Kogan. Internat. J. Control 68:1437–1448 (1997).
17. M. Ciarletta and S. Chirita. Ser. Adv. Math. Appl. Sci., 2001, 62:31–41 (2002). Scientific World,
New Jersey, London, Hong Kong.
18. M. Ciarletta. J. Thermal Stresses 25:969–984 (2002).
19. J. F. V. Vasconcellos, A. J. Silva Neto, and C. C. Santana. Inverse Probl. Sci. Eng. 11:391–408 (2003).
ELLIPTIC VOIDS IN THE ELASTIC WEDGE 119
20. V. I. Rimlyand, A. V. Kazarbin, and M. B. Dobromyslov. Research in Nondestructive Evaluation
21. T. Erling Unander. Research in Nondestructive Evaluation 15:119–148 (2004).
22. M. J. Berrocal Tito, N. C. Roberty, A. J. Silva Neto, and J. Bravo Cabrejos. Inverse Probl. Sci. Eng.
23. G. E. Khalil, F. Kimura, A. Chin, M. Ghandehari, R. Wan, W. Shinoki, M. Gouterman, J. B. Callis, and
L. R. Dalton. Research in Nondestructive Evaluation 16:119–130 (2005).