Mechanics |
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2007, No. 4
This volume contains papers submitted
to the Russian Session of Solid Mechanics Council
of the Russian Academy of Sciences
“Fundamental and Applied Problems of Solid Mechanics”
(Samara, Samara State University, July, 2-6, 2007)
Y.N. Radayev
TO PROFESSOR A.V. MANZHIROV 50th ANNIVERSARY
Alexander V. Manzhirov, D.Sc. (Phys.&Math.), Ph.D., is a
noted scientist in the fields of mechanics and applied mathematics, integral
equations, and their applications.
After graduating with honors from the Department of Mechanics and
Mathematics of Rostov State University in 1979, A.V. Manzhirov attended
postgraduate courses at Moscow Institute of Civil Engineering. He
received his Ph.D. Degree in 1983 at Moscow Institute of Electronic Engineering
Industry and D.Sc. Degree in 1993 at the Institute for Problems
in Mechanics of the Russian (former USSR) Academy of Sciences. Since
1983, A.V. Manzhirov has been working at the Institute for Problems in
Mechanics of the Russian Academy of Sciences. Currently, he is head of
the Laboratory for Modeling in Solid Mechanics at the same institute.
Professor Manzhirov is also head of a branch of the Department of
Applied Mathematics at Bauman Moscow State Technical University, professor
of mathematics at Moscow State University of Engineering and
Computer Science, vice-chairman of Mathematics and Mechanics Expert
Council of the Higher Certification Committee of the Russian Federation,
vice-chairman of Solid Mechanics Scientific Council of the Russian Academy
of Sciences, and expert in mathematics, mechanics, and computer
science of the Russian Foundation for Basic Research. He is a member
of the Russian National Committee on Theoretical and Applied Mechanics
and the European Mechanics Society (EUROMECH), and member of
the editorial board of the journal Mechanics of Solids and the international
scientific-educational Website EqWorld — The World of Mathematical
Equations (http://eqworld.ipmnet.ru).
Professor Manzhirov has made important contributions to the fields
of accreted solids mechanics, contact mechanics and tribology, viscoelasticity
and creep, theory of integral equations and their applications. He
is the author of eleven books (including Contact Problems in Mechanics
of Growing Solids [in Russian], Nauka, Moscow, 1991; Handbook of
Integral Equations, CRC Press, Boca Raton, New York, 1998; Handbuch
der Integralgleichungen: Exacte Lösungen, Spektrum Akad. Verlag, Heidelberg,
Berlin, 1999; Contact Problems in the Theory of Creep [in Russian],
National Academy of Sciences of Armenia, Erevan, 1999; Handbook of
Mathematics for Engineers and Scientists, Chapman & Hall/ CRC, Boca
Raton, New York, 2006), nearly 100 research papers, and two patents.
Professor Manzhirov is a winner of the First Competition of the Science
Support Foundation 2001, Moscow.
V.M. Alexandrov, A.V. Mark
FLAT INDENTOR MOTION WITH STEADY SPEED ON THE BOUNDARY OF A VISCOELASTIC HALF-PLANE
A Hertzian problem of interaction between a rigid flat indentor and the boundary of a viscoelastic half-plane is considered. It is proposed that the indentor have steady speed at the boundary and is indented by the constant pressure. Friction in the contact zone is neglected.
B.D. Annin
EXACT SOLUTION OF THE PLASTICITY EQUATIONS FOR INHOMOGENEOUS MEDIA
By the Ovsyannikov method a partially-invariant solution of the perfect plasticity Tresca equations in the case of full plasticity for inhomogeneous media (the yield limit is dependent on a coordinate) is found. For this solution the tangential and normal components of the stress vector have a constant value on a plane.
V.I. Astafiev, G.D. Fedorchenko
A SELF-SIMILAR SOLUTION OF HYDRAULIC FRACTURE CRACK GROWTH PROBLEM
V.A. Babeshko, O.V. Evdokimova, O.M. Babeshko
ON THE THEORY OF BLOCK AND NANO STRUCTURES
V.G. Bazhenov, N.S. Dyukina, S.V. Zefirov
NUMERICAL MODELLING OF SOIL-STRUCTURE DYNAMICAL INTERACTION UNDER SEISMIC LOADING
In this paper a mathematical model and numerical procedure for modelling such processes are considered. A computer-aided procedure for reconstructing the pulse load values applied to the entire lower boundary of the multi-layer soil domain from the known seismic diagram on the half-space surface is developed. An effective finite-element method for a modeling of half-space by introducing special boundary conditions is proposed.
A.V. Belokon, A.V. Nasedkin, A.S. Danilenko
The finite element technologies for piezoelectric devices modelling with electric circuits and acoustic load are considered. For finite element package ACELAN the basic finite elements of electric circuits and symmetric forms of finite element matrices and algorithms are constructed. The harmonic and transient operating regimes of piezoelectric devices are considered.
S.A. Bochkarev, V.P. Matveynko
RA METHOD OF AEROELASTIC STABILITY ANALYSIS OF SHELLS OF REVOLUTION
In this paper solutions to the problems of aeroelastic stability of revolution shells are found by a method based on direct integration of the system of differential equations. For calculation of complex eigenvalues it is proposed to use the parabola method (Muller’s method). The obtained results are compared with the known numerical solutions at different boundary conditions for shells of revolution subjected to external or internal supersonic gas flow.
R.V. Goldstein, N.M. Osipenko
LAMINATION OF COATINGS UNDER THERMOELASTIC STRESSES (BEAM APPROXIMATION)
The beam approximation is used in the fracture mechanics problems
on lamination of thin elastic coatings taking into account bonds on the
border of joint and the effect of transitional temperatures.
Edge zone of the laminating is modelled by an elastic beam loaded
by external bending moment and the force factors due to bonds of the
ground. Estimations of changes of effective crack growth resistance of
adhesive junction for various conditions are found. The obtained results
can be applied to technology of selection of adhesive junction parameters
providing its effective adhesive crack growth resistance.
S.A. Grishin
DYNAMIC PROBLEM FOR THE SHELLS TECHNICAL THEORY EQUATIONS IN MIXED FORM
The flutter problem for a cylindrical clamped-free shell under the effect of the supersonic gas flow ejected from it is studied. The shell is assumed to be elastic isotropic and homogeneous, modelled with the equations of the technical theory of shells in the mixed form. For such a model we have to find two scalar functions on the shell middle surface: the normal deflexion and the membrane-stress potential. The boundary conditions therefore also must be formulated in terms of these functions and their derivatives. It is extremely important that the mechanical system is conservative when no flow interacts with it.
A.O. Vatulyan
PROBLEMS OF SOLIDS HETEROGENEOUS PROPERTIES IDENTIFICATION
In the paper various settings of inverse problems of solids heterogeneous properties identification (the Lame module, prestressed state) are analyzed by the aid of coefficient inverse problems of elasticity theory solution. On the basis of generalized reciprocity relation iterative processes (the Fredholm integral equations of first type are solved on each step) are formulated.
V.N. Zimin
MECHANICS OF TRANSFORMABLE SPACE STRUCTURES
Ground tests of the large space structures under conditions simulating properly the space environment are very expensive and sometimes they can not be carried out at all. That is the reason why mathematical modeling and numerical experiments are so important for development and construction of such structures.
D.D. Ivlev
ON THE THREE DISCUSSIONS ON MECHANICS
I.I. Il’ina, V.V. Silvestrov
The stressed state of a piecewise-homogeneous elastic plane which is
formed of two different half-planes by glued them together and contains
thin rigid smooth inclusion on their line connection is studied. One side
of inclusion is perfectly connected to elastic body. Other side of inclusion
is detached from elastic body and contacts to body with friction. The
finite set of the concentrated forces and pairs of forces is enclosed to a
plane outside of inclusion.
The complex potentials describing the stressed state of the plane are
found in explicit form using the Riemann-Hilbert matrix boundary-value
problem. The behaviour of stresses near the inclusion tips is studied.
D.A. Indeitzev, V.N. Naumov, B.N. Semenov
DYNAMIC EFFECTS IN COMPOUND STRUCTURE MATERIALS
D.A. Kazenin, S.P. Karlov, B.G. Pokusaev, A.I. Zhurov, D.P. Aeschlimann
A convective dispersion problem for a solute in a flow through a nanoporous medium is considered; the solute injection is modelled by a point source and the medium is characterized by a linear volumetric absorption function. The exact solution to the problem is obtained and a one-parameter family of closed embedded isocontentration regions is constructed. The geometry of the regions is described and the lateral dispersion coefficient is evaluated. The results obtained are intended to be used in designing flow microbioreactors for (in vitro) cultivation of cell cultures.
K.E. Kazakov
CONTACT PROBLEMS FOR COVERED SOLIDS
When fabricating different elements of machines various coatings are often used. In some cases characteristics of such coatings are inhomogeneous and depend on surface coordinates that greatly effects the strain-stress state in contact region. Another significant and abundant case is interaction of solids with variable-thickness coatings. The thickness variability is possible to take place due to different modes of surface treatment and described by complicated experimentally identified function. In the paper either types of problems is considered, the corresponding integral equations are obtained, their solutions are constructed, some results of numerical computations are presented.
V.A. Kovalev, O.V. Taranov
FAR FIELD OF THE RAYLEIGH WAVE FOR ELASTIC CYLINDRICAL SHELL UNDER END LOADING
In the paper specific features of propagation of non-stationary waves in an elastic cylindrical shell generated by shock end load of a normal type are studied. The equations for finding the solution in a vicinity of conditional front of the surface Rayleigh waves generated by the normal shock load are constructed by the aid of asymptotic method.
A.S. Kravchuk, A.S. Karlyshkov
NUMERICAL MODELLING OF THE DEFORMATION AND DESTRUCTION PROCESSES AT THE NANO–SCALE
In the paper a brief review of the works on the problem of studying the displacements, forces and breakdown of systems composed from the finite number of particles (atoms and molecules) interacting on the long distance is given. New results consist of the numerical solutions of one-dimensional systems – chains with Morse’s potential. A quasi-static problems are solved by the traditional Newton’s method. The Verle’s method is used for solution of some dynamic problem. An analysis of the numerical solutions which permits to propose a hybrid model is given. This model takes into account the non-linear phenomena near the boundaries and failure surfaces, and the continuum equations are used inside the domain.
S.A. Lychev
ON CONSERVATION LAWS OF MICROMORPHIC NONDISSIPATIVE THERMOELASTICITY
In this paper following Noether’s theorem from the variational symmetries of the action integral in micromorphic nondissipative thermoelasticity of the Green-Nagdy type new conservation laws associated with translational, rotational and scaling transformations of material, 3-spatial and physical manifolds are derived. Explicit corresponding of transformations of microstructure to macrostructure manifolds is of new results obtained in the paper.
E.V. Lomakin, B.N. Fedulov
PLASTIC DEFORMATION OF STRIPES OF STRESS-STATE-DEPENDENT MATERIAL PROPERTIES
The plastic deformation of medium of stress-state-dependent plastic properties are analyzed. On the base of the afterflow law associated with plasticity condition represented in generalized form and rigid-plastic model, the analytical solutions of the problems of notched stripes tension are obtained. The dependence of stress fields in plastic domains and the values of limit loads on the stress state sensitivity parameter is studied. The numerical simulation of plastic domains formation on the base of elastic-plastic model is performed for the same problems considered in analytical studies. The effect of elastic deformation on the shape of plastic domains is studied. The possibility of rigid-plastic scheme usage for the determination of the limit loads is analyzed for the solids sensitive to the stress state type.
A.A. Lukyanov, V.B. Pen’kov
A MATHEMATICALLY CORRECT MODEL OF INCOMPRESSIBLE ANISOTROPIC ASSOCIATED PLASTICITY
A mathematically correct associated incompressible anisotropic plastic flow model based on full decomposition of stress tensor into generalised spherical and deviatoric parts is studied in the paper. The concept of total generalised pressure is redefined for anisotropic materials. It is shown that total generalised pressure reduces to the classical hydrostatic pressure in the limit of isotropy. The formulation of anisotropic plasticity in the case of associated incompressible plastic flow does not depend from the generalised hydrostatic pressure. A mathematically consistent modification to the anisotropic Hill criterion is proposed.
A.V. Manzhirov, D.A. Parshin
MODELING OF THE DEFORMATION PROCESS OF ACCRETED CONIC SOLIDS
A lot of applied topical problems of technics, technology, civil engineering, biology, medicine, and many other areas of natural sciences require precise knowledge of stress-strain field evolution in growing (accreted) conic solids. Various shafts, bearings, columns, bones, biological soft tissues can be taken as particular examples of such solids. The deformation process of viscoelastic aging accreted conic solids under outer tensile or compressive end forces is studied. The general formulation of the problem is presented. Boundary value problems which arise on various stages of accretion process are formulated. Their closed solutions are obtained.
M.N. Mikhin
In the paper the theory of torsion problem of aging viscoelastic round shafts is studied. Two methods for problem setting are considered. The main stages of solid deformation are analyzed: before the beginning of growing, during the process and after the growing stage.
M.D. Novopashin, Ñ.Â. Suknev
ON A GRADIENT CRITERIA OF A LIMIT STATE
Y.N. Radayev
The mathematical model proposed by Hutchinson of stress distribution near a mode I crack tip in a perfectly plastic solid under plane stress conditions is considered. Exact formulae for the stress distribution within the plastic zone near a crack tip are obtained. Comparison with the numerical data obtained by Hutchinson for the stress distribution is given.
I.Y. Tsvelodub
A PRIMARY MULTI-MODULAR THEORY OF ELASTICITY OF ISOTROPIC MATERIALS
In the paper the variant of three-constant tensor-linear multi-modular elasticity theory is proposed. The stability problem and uniqueness of boundary-value problems solution are studied. Some illustrative examples are considered.
M.A. Sumbatyan, V. Zampoli, M. Vaccaro
BASE ISOLATION FROM SEISMIC WAVES BY A VISCOELASTIC LAYER
In the present paper we study harmonic oscillations of elastic rectangle above a viscoelastic layered half-space. The latter consists of an elastic half-space to which a viscoelastic layer is embedded at a certain depth. By combining Fourier integral transform in the half-space and series representation of the solution in the rectangle the problem is reduced to an integral equation over the base of the rectangle. By solving this integral equation we investigate the possibility of base isolation in dependence upon viscoelastic properties of the intermediate layer as well as upon geometrical and physical parameters of the materials.