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

In the paper the mixed boundary value problem of antiplane vibrations is considered in the elastic wedge-shaped medium containing the radiating defect . Radiating generators are assumed to be located on defect boundaries and on the interval 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 and contact stresses on the interval of the upper plane of the wedge.


Introduction
The aim of the present paper is mathematical modeling of a pre-fracture state of the construction unit representing the junction of angular elastic elements. It is investigated correctness problems of applying mathematical modeling method for the wave process arising in angular elements examined by non-destructive testing methods both in hard industry enterprises and in ones of agricultural machinery. Under long dynamic exploitation of the technological equipment it appears the stress singularity at the angular point. In its neighborhood there arises the defect growing to the angular point (stress concentration) and generating the acoustic radiation (acoustic emission -AE). Non-destructive testing methods are worked out in details in [1][2][3][4][5][6]. At the paper the pre-fracture state is considered, provided the appearance of the radiating defect takes place only in one of angular elements. The angular element is modeling by elastic body of wedge-shaped medium, one of its planes is stiffly connected with other angular elements, radiating defect is modeling by the linear radial cut of finite length, antiplane vibration generators being located on the cut boundaries. The contact interaction zone is modeling by the harmonic oscillating punch under antiplane deformation (Fig.1).
1. The boundary value problem is formulated for the dynamic elasticity equations in the domain , presenting the angle of span with cut , simulating the defect located on the segment , of the line . Oscillating coherent generators of the antiplane shear displacements ( ) of the equal intensity are located on the banks of ± .
| +, = 0, 0 ± = ( , 2), where ', ) are density and shear modules of the wedge material, ± is the left and right cut boundaries of the cut = ∪ 4 respectively . No radiation sources are assumed to be on the vertex = 0, displacements vanish at infinity and Somerfield's radiation conditions principle take place: 78 7/ − #& = 9 : ; The lack of radiation sources on the vertex = 0 provides the fulfilment the Saint-Venant principle and the existence of the solution in Sobolev space @ ( ), the norm being given by the traditional way.
Solving the problem stated above is based on its reducing to the equivalent boundary integral equations (BIE) about the unknown (dimensionless) contact stresses amplitude ) . /+ 0 +, = A ( ), < < on the upper plane and the unknown amplitude saltus of (dimensionless) stresses C) . /+ D0 The statement of the problem in question leads to the next boundary value problem in the domain . To solve the problem there fulfils the construction of Green function ( , |E, F) in the wedge-shaped domain without defect. Green function obeys the nonhomogeneous Helmholtz equation and boundary conditions as follows: where I( ) is Dirac's function, n is the external normal to the boundary. Green function method is worked out in details [7][8][9] when solving static problems. The same method permits to solve boundary value problems of the dynamic elasticity and reconstruct the wave field generated by all vibration sources in the whole wedge-shared medium considered. As in the statement of the main problem (1), (2) no radiation sources are assumed to be on the vertex = 0 and the same conditions at infinity take place.
To construct Green function obeying non-homogeneous Helmholtz equation and boundary conditions (4) In the formulas (7) contour g is the part of the circumference of the finite radius R (with the center in the wedge vertex), closing the angular domain, g , g are parts of , truncated by g , n is external normal to the boundary, ± are banks of the cut . By virtue of coherence and equal intensity of vibration sources on the cut banks, boundary conditions, vanishing conditions for displacements and Green function as h → ∞, as well as radiation conditions (3)  Let us consider the auxiliary BIE, constructed on the base of (9), (10) and written in the abbreviated vector-matrix form 1,2 1,2 min , max In the correlation (11) } , ( ) are results of extending of functions , ( ) to the interval( , ), matrix function r(N) is real both on the imaginary axis and on the real one where r(N) is positively defined.
2. To investigate the solvability problems for the BIE system the next theorem is established.

Theorem
Operator & of the left hand side (11)  The result described is in accordance with known results on boundary properties of functions belonging to Sobolev spaces @ ( , ) in which the solution of boundary value problems is searched by the dynamic elasticity methods.
The passage to the initial case Q = −#r is provided by the analytical continuation principle [12] since all functions are analytical with respect to Q in the domain h Q ≥ 0, Q ≠ 0 of the complex plane, where, in part, the point Q = −#r is located.
It permits to ascertain the unique solvability of the initial boundary value problem (1), (2) in the Sobolev space @ ( ) for the whole wedge-shaped domain and there results the inequality : , C= const meaning the correct solvability of the problem in question permitting to apply varies analytical (for example, methods in [12]) and numerical methods to approach sought-forfunctions A , A as solutions of the BIE system (9), (10) .
The consequent use of described results to narrowing of functions A , (# = 1,2) from the domain ( , ) to the initial ones ( , ) ⊂ ( , ) leads to the unique resolution of the initial BIE system (9) The reconstruction of displacement wave field in and in the boundary may be fulfilled by representation (8) which is presented by means of displacements' amplitude ( , 2) when radiating AE from defect boundaries. The displacement wave field may be considered as the base to the statement of the inverse problem of reconstructing the displacements' amplitude ( ) on the defect by means of direct displacement measurements. It may be done when constructing the displacements' amplitude frequency reply on the unloaded part of the boundary and the sequel application of the least square method [14,15].