Direct and Inverse Problems for Interface Crack Identification in Layered Media

Research contributions over the past 50 years on the theory and analysis of elastodynamics are reviewed in this paper. Major topics reviewed are: general theories, steady-state waves in waveguides, transient waves in layered media, diffraction and scattering, and one and two-dimensional theories of elastic bodies. A brief discussion on the direct and inverse problems of elastic waves completes this review.

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Integral equations in direct and inverse problems of elastostatics for crack detection in layered structures

Mathematics and Mechanics of Solids ◽

10.1177/1081286520903436 ◽

2020 ◽

Vol 25 (5) ◽

pp. 1140-1154

Author(s):

AN Galybin

Keyword(s):

Inverse Problems ◽

Boundary Conditions ◽

Integral Equations ◽

Crack Detection ◽

Fourier Transforms ◽

Interfacial Crack ◽

External Boundary ◽

Crack Identification ◽

Cauchy Type ◽

Direct And Inverse Problems

This study deals with both direct and inverse problems for interfacial crack identification in laminates. The main focus is given to the case of an elastic substrate coated by a film made of a different elastic material. It is assumed that delamination can be developed on the interface between these materials. It is modelled as a combined open-sliding interface crack or by a pure sliding crack (slip). Its position may not be specified. The boundary conditions on the interface assume continuity of the stress vector across the whole interface and continuity of the displacements outside the crack. In the case of the slip the normal displacements are assumed to be continuous. The inverse boundary value considered is of the Cauchy type; it assumes that both stress and displacement vectors are known on the external boundary of the structure. In the case of the slip the problem is overdetermined on a part of the boundary, where three conditions are imposed, and undetermined on its remainder where just one condition is imposed. It is further referred to as a semi-inverse formulation. Therefore, these problems are ill-posed with the specified boundary conditions. Inverse, semi-inverse and direct problems are reduced to integral equations derived from basic properties of holomorphic functions followed by applications of Fourier transforms. Analytical solutions are found for the formulations considered and difficulties of numerical implementation are discussed in brief.

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Mathematical Modeling of Direct and Inverse Problems of Dynamics of Thick Elastic Layer. Part I. Mathematical Modeling of the Field of Transverse Dynamic Displacements of Layer

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v48.i7.60 ◽

2016 ◽

Vol 48 (7) ◽

pp. 55-64

Author(s):

Vladimir Antonovich Stoyan ◽

Konstantin V. Dvirnychuk

Keyword(s):

Mathematical Modeling ◽

Inverse Problems ◽

Elastic Layer ◽

Direct And Inverse Problems ◽

Transverse Dynamic ◽

Inverse Problems Of Dynamics

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Identification of the string boundary conditions using natural frequencies

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2016.1.007 ◽

2016 ◽

Vol 11 (1) ◽

pp. 38-52

Author(s):

I.M. Utyashev ◽

A.M. Akhtyamov

Keyword(s):

Inverse Problems ◽

Power Series ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Natural Frequencies ◽

Boundary Value ◽

External Environment ◽

Direct And Inverse Problems ◽

Symmetric Potential ◽

Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

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