Abstract
In this paper, we discuss the uniqueness in determining cavities (i.e., nonrectilinear cracks) in a heterogeneous isotropic elastic medium in two dimensions. Our main result asserts that there is at most one cavity in the elastic medium which yields the same surface displacements and stresses on an arbitarily small portion of the boundary. The boundaries of cavities are assumed to be piecewise smooth and admit edges where no net force is exerted. The key of the proof is the unique continuation for the isotropic Lamé system and geometric considerations.
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References
G. Alessandrini, Stable determination of a crack from boundary measurements. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 497-516.
G. Alessandrini, Examples of instability in inverse boundary-value problems. Inverse Problems 13 (1997) 887-897.
S. Andrieux, A. Ben Abda and H.D. Bui, Sur l'identification de fissures planes via le concept d'écart à la réciprocité en élasticité. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1431-1438.
D.D. Ang, M. Ikehata, D.D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients. Comm. Partial Differential Equations 23 (1998) 371-385.
D.D. Ang, D.D. Trong and M. Yamamoto, Unique continuation and identification of boundary of an elastic body. J. Inverse Ill-posed Problems 3 (1995) 417-428.
D.D. Ang and M.L. Williams, Combined stresses in an orthotropic plate having a finite crack. J. Appl. Mech. 28 (1961) 372-378.
E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data. SIAMJ. Math. Anal. 30 (1998) 220-232.
A.L. Bukhgeim, J. Cheng and M. Yamamoto, Stability for an inverse boundary problem of determining a part of a boundary. Inverse Problems 15 (1999) 1021-1032.
A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38 (1989) 563-579.
A. Friedman and M. Vogelius, Determining cracks by boundary measurements. Indiana Univ. Math. J. 38 (1989) 527-556.
S. Kubo, Requirements for uniqueness of crack identification from electric potential distributions. In: Inverse Problems in Engineering Sciences. Springer, Tokyo (1991) pp. 52-58.
M. McIver, An inverse problem in electromagnetic crack detection. IMA J. Appl. Math. 47 (1991) 127-145.
L. Rondi, Uniqueness and stability for the determination of boundary defects by electrostatic measurements. preprint, Ref: S.I.S.S. A. 73/98/AF, SISSA ISAS, Trieste, Italy (July 1998).
I.N. Sneddon and M. Lowengrub, Crack Problems in the Classical Theory of Elasticity. Wiley, New York (1969).
S.P. Timoshenko and J.N. Goodier, Theory of Elasticity. McGraw-Hill, New York (1970).
M.L. Williams, On the stress distribution at the base of a stationary crack. J. Appl. Mech. 24 (1957) 109-114.
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Ang, D.D., Trong, D.D. & Yamamoto, M. Identification of Cavities Inside Two-Dimensional Heterogeneous Isotropic Elastic Bodies. Journal of Elasticity 56, 199–212 (1999). https://doi.org/10.1023/A:1007661505879
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DOI: https://doi.org/10.1023/A:1007661505879