  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
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T. I. Belyankova, V. V. Kalinchuk and S. Yu. Ustinova, "On the dynamic rigidity of a twolayered prestressed halfspace," Mech. Solids. 37 (6), 7784 (2002) 
Year 
2002 
Volume 
37 
Number 
6 
Pages 
7784 
Title 
On the dynamic rigidity of a twolayered prestressed halfspace 
Author(s) 
T. I. Belyankova (RostovonDon)
V. V. Kalinchuk (RostovonDon)
S. Yu. Ustinova (RostovonDon) 
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 x_{1}, x_{2}≤∞, 0≤x_{3}≤h
rigidly fixed to the halfspace x_{3}≤0. The punch
is subject to steadystate 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 halfspace. 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 halfspace).
We examine the effect of various types of the initial strain state (ISS)
of the inhomogeneous halfspace on the dynamics of a twomass
inertial system which is in contact with the halfspace
and consists of a massive punch M_{2} connected to
a massive body M_{1} by means of an elastic element.
Previously [1, 2], the properties of contact interaction
between a rigid punch and a homogeneous prestressed halfspace 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 halfspace 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 halfspace has a maximal effect
both on its dynamic rigidity and the vibration amplitude of
the massive punch in contact with the halfspace. 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 halfspace,"
PMM [Applied Mathematics and Mechanics], Vol. 57, No. 4, pp. 123134, 1993. 
2.  T. I. Belyankova and V. V. Kalinchuk,
"Dynamics of a massive body interacting with a prestressed halfspace,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 8394, 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. 89101, 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 electroelasticity and thermoelasticity," in Modern Problems
in Continuum Mechanics Proceedings of the IVth International
Conference. RostovonDon. Volume 2 [in Russian], pp. 3539, Izdvo SKNTs
VSh, \hbox{RostovonDon}, 1999. 
7.  V. V. Kalinchuk and T. I. Belyankova,
"On the investigation of mixed dynamical problems in electroelasticity
and thermoelasticity for a layered inhomogeneous halfspace," Izv. Vuzov.
Sev. Kavkaz. Region. Estestv. Nauki, No. 3, pp. 7274, 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. 477484, 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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