 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. A. Vakhonina and V. G. Popov, "Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion," Mech. Solids. 44 (2), 294-302 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
2 |
Pages |
294-302 |
| DOI |
10.3103/S0025654409020150 |
| Title |
Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion |
| Author(s) |
A. A. Vakhonina (Mykolayiv State Agrarian University, Parizhskoy Komunny 9, Mykolayiv, 54010, Ukraine, vakhonina-l@rambler.ru)
V. G. Popov (Odessa National Maritime Academy, Didrikhsona 8, Odessa, 65029, Ukraine, dr_popov@te.net.ua) |
| Abstract |
We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion.
Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions. |
| References |
| 1. | G. Ya. Popov, Concentration of Elastic Stresses Near Punches, Cuts, Rigid Inclusions, and Stiffeners (Nauka, Moscow, 1982) [in Russian]. |
| 2. | V.M. Alexandrov, B. I. Smetanin, and B. V. Sobol’, Thin Stress Concentrators in Elastic Bodies (Nauka, Fizmatlit, Moscow, 1993) [in Russian]. |
| 3. | V. G. Popov, "Studies of Stress and Displacement Fields under Shear Wave Diffraction on a Thin Rigid Separated Inclusion," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 139–146 (1992) [Mech. Solids (Engl. Transl.)]. |
| 4. | V. G. Popov, "The Diffraction of Plane Elastic Waves by a Delaminated Rigid Inclusion when There is Smooth Contact in the Delamination Region," Prikl. Mat. Mekh. 62(2), 290–296 (1998) [J. Appl. Math. Mech. (Engl. Transl.) 62 (2), 269–274 (1998)]. |
| 5. | L. V. Vakhonina and V. G. Popov, "The Stress Concentration near a Delaminated Thin Rigid Inclusion in the Interaction with a Torsional Wave," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 70–76 (2004) [Mech. Solids (Engl. Transl.) 39 (4), 55–60 (2004)]. |
| 6. | V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies (Naukova Dumka, Kiev, 1981) [in Russian]. |
| 7. | G. Ya. Popov, "Construction of a Discontinuous Solution to Dynamic Equations of the Elasticity Theory for a Layered Media with Interphase Defects," Dokl. Ross. Akad. Nauk 364(6), 769–773 (1999) [Dokl. Phys. (Engl. Transl.) 44 (2), 117–121 (1999)] |
| 8. | L. V. Vakhonina and V. G. Popov, "Interaction of Elastic Waves with a Thin Rigid Circular Inclusion in the Case of a Smooth Contact," Teor. Prikl. Mekh., No. 38, 158–166 (2003). |
| 9. | L. V. Vakhonina and V. G. Popov, "Torsional Vibrations of Space with a Thin Rigid Circular Inclusion in the Case of a Smooth Contact," in Theory and Practice of Crushing, Separation, Mixing, and Compacting Processes, Collection of Scientific Papers, No. 8 (Odessa, 2000), pp. 31–38. |
| 10. | S. M. Belotserkovskii and I. K. Lifanov, Numerical Methods in Singular Integral Equations and Their Applications in Aerodynamics, Elasticity, and Electrodynamics (Nauka, Moscow, 1985) [in Russian]. |
| 11. | V. I. Krylov, Approximate Computations of Integrals (Nauka, Moscow, 1967) [in Russian]. |
|
| Received |
13 July 2006 |
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