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

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.


Introduction
The aim of the present paper is mathematical modeling the dynamics of a massive body of composite material under harmonic oscillations. The investigation of the stressed and deformable state for such a body is of great interest for theoretical and practical analysis of strength of materials and reliability problems of technical constructions under long exploitation both in hard industry enterprises and agricultural machinery ones. In part, the problem in question appears when analyzing construction elements by nondestructive testing as well. Problems enumerated have been investigated by number of authors [1][2][3][4][5][6][7][8] at al. Аnalogous problems arise in seismic prospects when analyzing the wave propagation in the skew-layered medium near the earth crust surface. The problems mentioned above are reduced to mixed dynamic boundary value problems for the elastic wedge-shaped composite body. Investigation of such problems for the homogeneous medium has been usually based on Kontorovich-Lebedev integral transform techniques: where k is the wave number, ‫ܭ‬ ିఛ (kr) is McDonald function [9], infinite contour ߁ belongs to a neighborhood of the real axis ܴ ଵ and satisfied Summerfield radiation principle. However, the use of such a techniques by classic way for piecewise homogeneous medium dynamics deduces to the additional auxiliary integral equations complicating the problems in question essentially when satisfying media interface conditions on the dividing media boundary . Below it is offered a new method based on distributions' integral transforms technique permitting to exclude the mentioned additional integral equation from the consideration.
Let us consider steady oscillations arising in the wedge-shaped medium ߗ = ߗ ଵ ∪ ߗ ଶ under antiplane deformation, one being composed by two wedge-shaped elastic composants ߗ ଵ,ଶ of span angles ߙ ଵ,ଶ with the common vertex, mechanical densities ‫ܦ‬ ଵ,ଶ and shear modules ߤ ଵ,ଶ (Fig.1). Generators of harmonic oscillations ‫݁)ݎ(݂‬ ିఠ ௧ with circular frequency ߱ are located in the domain (ܽ, ܾ) on the upper boundary of the medium ߗ, the rest of the boundary being assumed to be unloaded The lower boundary of ߗ is stiffly connected. We state the problem of working out the method of reconstructing the wave field in the heterogeneous medium ߗ described above.
We will construct the matrix (named in the sequel as "matrix-propagator") connecting Kontorovich-Lebedev transforms of displacements and stresses on boundaries of the wedgeshaped medium of angle ߶.
By means of direct transformations using (2) it's easy to establish the correlation: where ‫ݑ‬ (߬), ߪ(߬) are Kontorovich-Lebedev transforms of displacements and stresses respectively . The lower index 10 means that function value in (3) is considered as the limit while point of observation ‫,ݎ(‬ ߶) tends in the direction to the line L0 within domain 1. Going over to the Kontorovich -Lebedev transforms in the correlations (3) we can write down its for the wedge-shaped domain of angle ߶ ଵ by the matrix form as follows: The lower index 11 means that function value in (4) The lower index 21 means that function value (5) is considered as the limit while point of observation tends to the boundary ‫ܮ‬ ଵ within domain 2 in the direction to the domain 1.

Method
To satisfy interface conditions (1) of the boundary value problem for composite wedge by the classic way in connection with the use of the Kontorovich-Lebedev transform, to investigate the re-expansion integral ‫,ݐ(ܬ‬ ߬) between systems of McDonald functions described in the monograph by Watson, G. H. [9] with different value of the wave number (ߢ ଵ , ߢ ଶ ) as follows: The latter fulfills the isomorphism ‫ܦ‬ ା ᇱ to ܼ ା ᇱ . Transformation of integrals (10), operations under distributions and use the correlation (6) lead to (8).
The proof of the sufficiency of the equality (9) points out by its direct substitution in (8) and subsequent using the transform (10).

Results
To solve the boundary value problem (1), to put ߢ ଵ,ଶ > 0 temporary. Obeying the interface conditions along the line ‫ܮ‬ ଵ dividing the media 1, 2 we establish the displacement and stress transformations have the saltus via the ‫ܮ‬ ଵ on the strength of the Theorem 2.2, the next equalities being taken place : The equality (11) may be transformed by means of (4), (5) to the form: 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 investigate the solvability problems for the BIE system the next theorem is established. Furthermore it may be used the method [11] and to obtain the solvability condition for BIE (13) as follows ∫ ‫|)ݖ(݂|‬ ଶ ஶ ‫ܭ‬ ିଵ (‫)ݖ‬dz < ∞ , It points out the existence of the unique solution ‫ݍ‬ ∈ ܹ ଶ ିଵଶ (ܽ, ܾ) for any right hand side ݂ ∈ ܹ ଶ ଵଶ (ܽ, ܾ) and the imbedding ‫,ܽ(ܪ‬ ܾ) ⊂ ܹ ଶ ିଵଶ (ܽ, ܾ). This result is in accordance with well-known ones about boundary properties of functions belonging to Sobolev space ܹ ଶ ଵ (ܽ, ܾ) in which the solution of boundary value problems of the dynamic elasticity is searched . The passage to the required case may be fulfilled by the analytical continuation principle used above [11,12].
Both the system and its solution are not submitted there because of their awkwardness. Solution (14) has the well-known power singularity on boundaries of the interval (ܽ, ܾ) which is typical for elasticity contact problems.
The reconstruction the wave field in the whole heterogeneous elastic wedge-shaped medium ߗis obtained by means of formulae (3) .Consideration of inverse problems for this medium may be fulfilled on the base of methods having been worked out in [15,16].