Remote determination of mechanical parameters of thin flat inclusions of low contrast
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REMOTE DETERMINATION OF MECHANICAL PARAMETERS OF THIN FLAT INCLUSIONS OF LOW CONTRAST Ya. I. Kunets’,1,2 V. V. Matus,1 and M. D. Hrylyts’kyi3
UDC 539.3
We propose an algorithm for the remote determination of physicomechanical parameters of thin flat spatial elastic inclusions of variable thickness. We also establish the orientation of inclusions in an elastic matrix. The difference between the mechanical properties of the inhomogeneity and the matrix is insignificant as compared with the small thickness of an inclusion. The defect is sounded by elastic longitudinal waves, and the required parameters are determined from the amplitudes of scattering of these waves to the Fraunhofer zone. Keywords: inverse scattering problem, thin-walled elastic inhomogeneities, inclusions of low contrast.
Earlier studies in inverse problems of the dynamic theory of elasticity (the determination of mechanical and geometric parameters of thin-walled elastic inclusions from the data of wave scattering to a far-field zone) have mainly been carried out in a two-dimensional setting. In these studies, inhomogeneities of high or low stiffness were considered under the conditions of their perfect mechanical contact with the matrix [1, 2]. Algorithms for the solution of related inverse problems of the remote determination of geometric parameters of cracks and the stress-strain state in their neighborhood were proposed in [1, 3–5]. A necessary component of approaches to the solution of inverse scattering problems is the analysis of the corresponding direct problems. In this connection, we should mention the works [6–11], in which methods for studying problems of scattering of elastic waves by cracks and thin elastic inclusions of different stiffness are presented. For engineering applications, in particular, for the development of new methods of ultrasound diagnostics and nondestructive testing, of special importance are thin inhomogeneities of low contrast for which the difference between the elastic properties of the inclusion and the matrix is insignificant as compared with a small parameter that characterizes the relative thickness of the defect. The appearance of such defects in an elastic material may be a sign of its prefracture state. In the present work, we propose an algorithm for the identification of the material of a thin-walled elastic low-contrast flat spatial inhomogeneity of variable thickness, which is located in an elastic matrix under conditions of perfect mechanical contact, according to the amplitudes of longitudinal waves scattered to the Fraunhofer zone. Under similar assumptions, the inverse scattering problem was considered in [12] in the case of a longitudinal shear of an elastic system. Formulation of the Problem Assume that, in a homogeneous elastic unbounded medium with Lamé moduli λ and μ and density ρ , there is a thin-walled flat elastic inclusion with parameters λ 0 , μ 0 , and μ 0 under the conditions of perfect mechanical contact. The inclusion occupies a domain Wε = {x : (x1, x 2 ) ∈W0 , 2 x 3 ≤ h(x1, x 2 )
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