ИСТИНА

This study is aimed at stress reconstruction in spacious elastic media. Stress identification is of great interest in different applications such as contact (indentation) problems, functionally graded materials, rock strata and others. Two problems are considered. The first one deals with the determination of contact stresses on a part of the boundary of elastic half-space by measured data of displacements on the rest of the stress-free boundary. This type of inverse problems belongs to the class of conditionally ill-posed problems with pronounced dependence of the solution from small perturbations in measured data. The 3D problem formulation is based on spatial harmonic functions and a Trefftz-type method is used to determine the sought harmonic functions based on the radial basis functions to solve the system of integral equations. The second problem deals with stress identification in layered structures. It is accepted that the displacement and stress vectors are continuous on the interfaces between all layers, which makes it possible to formulate an inverse elastic problem for stresses reconstruction in each layer by employing a minimum number of measurements. The following cases have been analysed. A) the values of tangential stresses are known in one layers; B) one of the tangential stresses is known in two layers; C) one of the tangential stresses is known in a layer and the average stress over the whole strata is also known; D) both averaged tangential stress over the whole strata are known. It is shown that all these formulations are equivalent and allow one to perform stress reconstruction in all layers provided that the normal stresses are piecewise linear functions of the coordinate normal to the layers. Several synthetic examples are presented to illustrate the proposed approach.