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IssuesArchive of Issues2002-6pp.77-84

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T. I. Belyankova, V. V. Kalinchuk and S. Yu. Ustinova, "On the dynamic rigidity of a two-layered prestressed half-space," Mech. Solids. 37 (6), 77-84 (2002)
Year 2002 Volume 37 Number 6 Pages 77-84
Title On the dynamic rigidity of a two-layered prestressed half-space
Author(s) T. I. Belyankova (Rostov-on-Don)
V. V. Kalinchuk (Rostov-on-Don)
S. Yu. Ustinova (Rostov-on-Don)
Abstract In the framework of the linearized theory of superposition of small strains on finite strains, we develop a method for the investigation of the dynamics of contact interaction between a rigid oscillating punch and an inhomogeneous layered prestressed body which consists of the layer |x1|, |x2|≤∞, 0≤x3h rigidly fixed to the half-space x3≤0. The punch is subject to steady-state vibrations with frequency ω and its horizontal projection coincides with a domain Ω.

We give formulas describing the wave field in the layered inhomogeneous prestressed half-space. The problem of vertical vibrations of a punch is used as a model for studying how the dynamic rigidity of a medium is affected by its structure and the initial stress localization (in the layer or the half-space). We examine the effect of various types of the initial strain state (ISS) of the inhomogeneous half-space on the dynamics of a two-mass inertial system which is in contact with the half-space and consists of a massive punch M2 connected to a massive body M1 by means of an elastic element.

Previously [1, 2], the properties of contact interaction between a rigid punch and a homogeneous prestressed half-space were investigated with [1] or without [2] the mass of the punch being taken into account. It was shown that there is a set of frequency values (minimal influence frequencies) for which one of the components (real or imaginary) of the dynamic rigidity of the half-space does not depend on the initial strain.

As shown in [2], there is another set of frequency values (maximal influence frequencies) for which the initial strain of the half-space has a maximal effect both on its dynamic rigidity and the vibration amplitude of the massive punch in contact with the half-space. These frequencies do not depend on the form of the ISS (uniaxial, biaxial, etc.).

The dynamic problem of contact interaction between a massive punch and a prestressed layer whose lower surface is rigidly fixed to an undeformable base was studied in [3]. It was shown that in this case also, there are two sets of frequency values for one of which the vibration amplitudes of the massive body do not depend on the initial stresses (minimal influence frequencies), and for the other set, the effect of the initial stress variation is maximal (for the same type of ISS). However, within the layer, the latter frequencies depend on a specific ISS.
References
1.  T. I. Belyankova and V. V. Kalinchuk, "On the interaction of an oscillating punch and a prestressed half-space," PMM [Applied Mathematics and Mechanics], Vol. 57, No. 4, pp. 123-134, 1993.
2.  T. I. Belyankova and V. V. Kalinchuk, "Dynamics of a massive body interacting with a prestressed half-space," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 83-94, 1994.
3.  T. I. Belyankova and V. V. Kalinchuk, "Dynamics of a massive body oscillating on the surface of a prestressed layer," Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 89-101, 1998.
4.  A. I. Lur'e, Nonlinear Theory of Elasticity [in Russian], Nauka, Moscow, 1980.
5.  I. I. Vorovich and V. A. Babeshko, Dynamical Mixed Problems of Elasticity in Nonclassical Domains [in Russian], Nauka, Moscow, 1979.
6.  V. V. Kalinchuk, "An effective method for the investigation of coupled dynamical mixed problems in electro-elasticity and thermo-elasticity," in Modern Problems in Continuum Mechanics Proceedings of the IV-th International Conference. Rostov-on-Don. Volume 2 [in Russian], pp. 35-39, Izd-vo SKNTs VSh, \hbox{Rostov-on-Don}, 1999.
7.  V. V. Kalinchuk and T. I. Belyankova, "On the investigation of mixed dynamical problems in electro-elasticity and thermo-elasticity for a layered inhomogeneous half-space," Izv. Vuzov. Sev. Kavkaz. Region. Estestv. Nauki, No. 3, pp. 72-74, 2000.
8.  V. A. Babeshko and O. D. Pryakhina, "A method of fictitious absorption in plane dynamic problems," PMM [Applied Mathematics and Mechanics], Vol. 44, No. 3, pp. 477-484, 1980.
9.  I. I. Vorovich, V. A. Babeshko, and O. D. Pryakhina, Dynamics of Massive Bodies and Resonance Phenomena in Deformable Media [in Russian], Nauchnyi Mir, Moscow, 1999.
Received 2 July 2001
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