On a mathematical model of dynamics of the elastic wedge-shaped medium with radiating defect

Abstract

In the paper the mixed boundary value problem of antiplane vibrations is considered in the elastic wedge-shaped medium containing the radiating defect J2. Radiating generators are assumed to be located on defect boundaries and on the interval J1 of the wedge free boundary as well. The problem of reconstructing the wave field in the whole wedgeshaped region with its boundary is stated. A number of problems of analyzing acoustic emission signals by radiating defect are reduced to the problem considered in connections with using non-destructive testing elements of the technological equipment under exploitation. The problem in question is reduced to studying the solvability problems of the equivalent boundary integral equation system both for stress saltus on the defect J2and contact stresses on the interval J1 of the upper plane of the wedge.

On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

“…Going over from displacement and stress transformations in (12) to its originals by the inverse Kontorovich-Lebedev transform it may be obtained the BIE of the mixed boundary value problem for the two-components composed elastic wedge stiffly connected by its lower boundary: , ߣ = ߤ ଶ ߤ ଵ ⁄ , ߢ ଶ = −ik ଶ All assertions have been provided above under assumption ߢ ଵ,ଶ > 0 and then the passage to the initial case ߢ ଵ,ଶ = −ik ଵ,ଶ is provided by the analytical continuation principle since all functions are analytical with respect to ߢ in the domain Reߢ ≥ 0, ߢ ≠ 0 of the complex plane ߢ, where, in part, the points ߢ ଵ,ଶ = −ik ଵ,ଶ are located [11].…”

“…To prove the theorem, to put ߢ > 0 temporary. Then the operator ‫ܭ‬ is positively defined and induces the space ‫,ܽ(ܪ‬ ܾ) of generalized solutions of the equation ( 13) by the norm: Furthermore it may be used the method [11] and to obtain the solvability condition for BIE (13) [11,12].…”