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Identification of an ellipsoidal defect in an elastic solid using boundary measurements. (English) Zbl 1236.74097
Summary: Elastostatic problem of identification of an ellipsoidal cavity or inclusion (rigid or linear elastic) in an isotropic, linear elastic solid is considered. The reciprocity gap functional method is used for solving the problem. It is shown that the parameters of the ellipsoidal defect (coordinates of its center, the directions and magnitudes of the semiaxes and elastic moduli in the case of isotropic, linear elastic inclusion), located in an infinite elastic solid are expressed by means of the values of the reciprocity gap functional. The values of the reciprocity gap functional can be calculated if the loads and displacements corresponding to uniaxial tension (compression) of an infinite solid are known on the closed surface containing the defect inside. Applications of the results to the problem of ellipsoidal defect identification in a bounded body are discussed. A number of numerical examples showing the efficiency of the developed identification method are considered.

##### MSC:
 74G75 Inverse problems in equilibrium solid mechanics 74B05 Classical linear elasticity
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##### References:
 [1] Alves, C. J. S.; Ben Abdallah, J.; Jaoua, M.: Recovery of cracks using point-source reciprocity gap function, Inverse problems in science and engineering 12, 519-534 (2004) [2] Anderson, O. L.: Determination and some uses of isotropic elastic constants of polycrystalline aggregates using simple crystal data, Physical acoustic: principles and methods, 43-95 (1965) [3] Andrieux, S.; Ben Abda, A.: Identification of planar cracks by complete overdetermined data: inversion formula, Inverse problems 12, 553-563 (1996) · Zbl 0858.35131 · doi:10.1088/0266-5611/12/5/002 [4] Andrieux, S.; Ben Abda, A.; Bui, H.: Reciprocity principle and crack identification, Inverse problems 15, 59-65 (1999) · Zbl 0920.35165 · doi:10.1088/0266-5611/15/1/010 [5] Asaro, R. J.: Somigliana dislocations and internal stresses; with application to second phase hardening, International journal of engineering science 13, 271-286 (1975) · Zbl 0294.73078 · doi:10.1016/0020-7225(75)90035-X [6] Avril, S.: Overview of identification methods of mechanical parameters based on full-filled measurements, Experimental mechanics 48, 381-402 (2008) [7] Bannour, T.; Ben Abda, A.; Jaoua, M.: A semi-explicit algorithm for the reconstruction of 3D planar cracks, Inverse problems 13, 899-917 (1997) · Zbl 0882.35128 · doi:10.1088/0266-5611/13/4/002 [8] Bonnet, M.; Constantinescu, A.: Inverse problems in elasticity, Inverse problems 21, R1-R50 (2005) · Zbl 1070.35118 [9] El Badia, A.: Inverse source problem in an anisotropic medium by boundary measurements, Inverse problems 21, 1487-1506 (2005) · Zbl 1086.35133 · doi:10.1088/0266-5611/21/5/001 [10] El Badia, A.; Ha-Duong, T.: An inverse source problem in potential analysis, Inverse problems 16, 651-663 (2000) · Zbl 0963.35194 · doi:10.1088/0266-5611/16/3/308 [11] Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion and related problems, Proceedings of the royal society 241, 376-396 (1957) · Zbl 0079.39606 · doi:10.1098/rspa.1957.0133 [12] Fisher, E. S.; Renken, C. J.: Single-crystal elastic moduli and the hcp-bcc transformation in ti, zr, and hf, Physical review 135, 482-494 (1964) [13] Goldstein, R. V.; Shifrin, E. I.; Shushpannikov, P. S.: Application of invariant integrals to the problems of defect identification, International journal of fracture 147, 45-54 (2007) · Zbl 1237.74035 [14] Shifrin, E. I.: Symmetry properties of the reciprocity gap functional in the linear elasticity, International journal of fracture 159, 209-218 (2009) · Zbl 1273.74019 [15] Shifrin, E. I.: Identification of ellipsoidal defect in an elastic solid using results of one uniaxial tension (compression) test, Izvestiya RAN mechanics of solids, No. 3, 131-142 (2010) [16] Shifrin, E. I.; Shushpannikov, P. S.: Identification of a spheroidal defect in an elastic solid using a reciprocity gap functional, Inverse problems 26, 055001 (2010) · Zbl 1277.74028
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