Identification of the geometry and elastic properties of rigid inclusions in thin plate

Abstract

The presence of inclusions in thin-walled structural elements in the course of their operation under mechanical and thermal loads gives rise to the material damage. In publications on this subject, the problem of defect identification is considered in two directions – the location and shape of the defect or the physical properties of rigid inclusions are determined. The paper proposes a method for simultaneous determination of the location of rigid inclusions and their elastic properties (Young's modulus) in a thin plate. The mathematical model of a plate with the inclusion is built in the framework of the linear theory of elasticity. For quantization of unknown functions, the finite element method is used. The geometric inverse problem is formulated in the conditional-correct statement. The parameters of rigid inclusions are determined by minimizing the quality functional. Additional conditions are attached to the functional using the Lagrange multipliers. The results of the identification of one or several inclusions of different sizes by the proposed method are presented. In solving the problem for a plate with one rigid inclusion, the error of restoration of the inclusion location does not exceed 3 %, the error of determination of the Young's modulus of this inclusion – 2 %. The error of determination of the Young's modulus for a group of several rigid inclusions is in the range of 4–7 %. Comparative analysis of numerical experiment results shows that the proposed method allows identifying the location and properties of several rigid inclusions of various sizes. Solution of the problem of identifying the inclusions, determining their size and elastic properties allows more accurate estimation of the strength characteristics of elements and prediction of the service life of structures.