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- 1 37(1) Swati Agrawal, V K Gupta, P K Kankar.pdf
- 1. Introduction
- Structure of the problem. A combination of two horn shaped single walled carbon nanotubes, attached together with the help of elastic medium (polymer like polyethylene or epoxy), is studied for free vibration analysis in this paper. Both the nanotubes...
- 8__37(1) N.V. Boyev.pdf
- ; . (1)
- Fig.1. Triple-periodic system of 3 ( 3 ( 3 = 27 (М = 3) equal spherical obstacles, located inside cube with elastic matrix: the isometric view.
- First step. The resulting field of the propagating elastic longitudinal waves, over the receiving face, is a sum of the through-transmitted rays, which can be of one of the following types: (i) the rays passed across the system of spherical obstacles ...
- The rays, passed through the system of inclusions without diffraction, are the rays, which propagate from the point sources of the spherical waves in the horizontal layers of thickness b between the neighbor layers of the spherical obstacles.
- In the case of the gently sloping incidence to the side obstacles, which are tangential to the upper and the lower boundary planes of the elastic layer, there is a possibility to reflect only once; after that the reflected longitudinal wave falls to t...
- When constructing the trajectories of multiple reflections, the diffraction field is very complex. This is connected with the fact that on the boundary surfaces of the solid inclusions and of the cube’s faces, additionally to the reflections of the lo...
- Second step. On this step the solution to the problem for the displacements can be written out explicitly, at the registering point, for each of the three types of waves discussed above.
- 1. For the rays passing across the cube without diffraction, in the high-frequency regime of oscillations, as and , along ray direction , let us write out the leading asymptotic term of the representation for the wave at the receiving point:
- , (2)
- where is the projection of the point source at point on the direction of the incident wave , is the mass density, , are the Lame coefficients, is the frequency of the oscillations, , , are wave numbers and waves speeds, for the longitudinal an...
- 2. In [13], on the base of the physical Kirchhoff diffraction theory, there is developed an integral Somiljana’s representation [14] for the displacements along the direction , in the longitudinal wave, at the wave one-time reflected from arbitrary sm...
- Here is the Gaussian curvature, is the average curvature of the surface at the point of mirror reflection , and is the curvature of the normal cut of the reflecting surface by the plane of the ray . The curvature is defined by Euler’s formula: (...
- 3. The multiple diffraction of the high-frequency longitudinal wave is studied in frames of an integral representation. The base for this integral representation for the radial displacement of N times reflected elastic wave, forms a modification of th...
- If the disposition of the boundary surfaces is such that the trajectory of the ray results in reflection, then the amplitude of the radial displacement in N times reflected ray at point , relatively the local spherical coordinate system at point o...
- , (5)
- ,
- ,
- , .
- Let us write out the leading asymptotic term for the radial displacement, which is obtained by the asymptotic estimate, as , of the diffraction 2N-fold integral (5) by the multidimensional (2N -fold) stationary phase method [14]:
- (6)
- In expression (6), are the reflection coefficients of wave at points of mirror reflection , . Parameter takes the values +1 or – 1, depending on the factor, in which part of the spherical surface the point of mirror reflection is located. Following [...
- If the exfoliations are totally absent in the specimen, then the deviation of the theoretical prediction from the practical measurements is insignificant. In the case of a significant difference of these results, to find spatial location of the exfoli...
- 10_37(1) L.A. Igumnov, A.A. Belov, A.A. Ipatov.pdf
- 3. Boundary-element approach
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