Home – Home – Jornals – Journal of Applied Mechanics and Technical Physics 2002 number 3

Main problems and trends in mathematical
modeling — a new line of research into
various processes and phenomena — are
formulated. The status and future
prospects are analyzed using as an
example the mechanics of continuous
media. Emphasis is on two stages of
modeling — the selection of
physicomathematical models of the
mechanics of continuous media and
numerical algorithms of solution.

The present paper considers the ultimate
(under heating conditions) kinematic
characteristics of composite solid
bodies accelerated by unsteady magnetic-
field pressure. The accelerated sheet
comprises two layers: a layer of a
composite material consisting of a
mixture of two materials with different
electrothermal properties and a
homogeneous material layer. The
electrical properties of the composite
layer with the coordinate are varied
along the coordinate by changing the
volume concentration of its constituent
materials. For an exponential magnetic
field rise, an analytical solution is
obtained for the problem of determining
the optimum distribution of the volume
concentration of the composite
constituents for which there is a
maximum increase in the ultimate
velocity of the sheet. Numerical
simulation showed that the distribution
of the volume concentration obtained
from analytical relations is also nearly
optimal for pulse shapes of the
accelerating magnetic field different
from exponential ones. The possibility
of considerably increasing the ultimate
velocity through the use of composite
layers compared to the ultimate
velocities for the homogeneous materials
constituting the composite is shown
analytically and numerically.

The breakdown channel pressure is
estimated for the initial stage of
breakdown of a KCl crystal using
experimental data on the fracture of KCl
crystals upon anode pulsed discharges,
the known strength of this crystal, and
solutions of the corresponding static
boundary-value problem of the
anisotropic theory of elasticity.

This paper considers the generation of a
longitudinal magnetic field between
rigid conducting layers of different
electrical conductivity which are in
shear motion at constant velocity across
the flux lines of the initial field. The
amplification limits for the generated
field due to the finite thickness of the
conducting layers are established. The
evolution of the magnetic field under
various conditions of generation was
described using the models of two layers
of finite thickness, a semi-infinite
layer and a layer of finite thickness,
and two semi-infinite layers.

This paper analyzes some physical
effects that occur on the electrode
surface in plasma-armature rail
launchers when the linear current
density is higher than critical value.
It is shown that under typical
experimental conditions, Rayleigh–Taylor
and Kelvin–Helmholtz instabilities and
magnetohydrodynamic  instabilities,
which arise from  the interaction of the
current with the self-magnetic field,
can develop over times much smaller than
the launcher operation time and can be
responsible for the entry of the
electrode material into the  discharge.
Flash radiography of the electrode
surface confirmed the presence of
inhomogeneities and ejection of the
material from the surface. Under certain
conditions, the emergence of conducting
metal jets from the electrode surface
was detected.

The problem of interaction of shock
waves of various types (completely
dispersed, frozen-dispersed, dispersed-
frozen, and frozen shock waves with a
two-front configuration) with a
motionless combined discontinuity in a
mixture of two condensed materials is
considered. The mathematical description
is based on the equations of mechanics
of heterogeneous media in a one-
dimensional isothermal approximation
with allowance for differences in
velocities and pressures of the
components. In the equilibrium
approximation in terms of velocities and
pressures of the components, the wave
configuration and flow parameters are
determined in all equilibrium states
behind the incident, transient, and
reflected shock waves.

The paper gives an exact solution of the
steady system of equations for stable
three-component diffusion in the entire
range of concentrations for a long
capillary under a controlled capillary
pressure differential. The solution
allows one to calculate the
distributions of component
concentrations and mixture density along
the capillary. It is shown that if the
diffusion coefficients are markedly
different, an extremum of mixture
density can arise inside the capillary.
In particular, if the density of the
mixture in the upper flask is higher
than that in the lower flask and the
stratification of the system is
generally stable, a region with a
reverse density gradient that is
unstable against gravity convection can
appear inside the capillary. A
comparison with experimental results
shows that the resistance to gravity
convection is disturbed when an extremum
of mixture density arises in the channel
during steady diffusion.

A partially invariant solution of the
Euler equations is considered, where the
vertical component of velocity is a
function of the vertical coordinate and
time, whereas the remaining components
of velocity and pressure are independent
of the polar angle in a cylindrical
coordinate system. Using the
classification of equations obtained by
analysis of an overdetermined system, we
consider two hyperbolic systems: the
first one describes the motion of a
cylindrical layer of an ideal
incompressible liquid under a punch, and
the second system allows obtaining
solutions in a half-cylinder with
singularities at the axis of symmetry. A
class of new exact solutions is
obtained, which describe vortex motion
of an ideal incompressible liquid,
including the motion with singularities
(sources of vortices) located along the
axis of symmetry.

Axially symmetrical waves on the surface
of a ferromagnetic viscous-fluid film
flowing down a cylindrical conductor
with alternating current are considered.
In this case, in addition to the
gravitational force, the film is
affected by a spatially nonuniform time-
dependent magnetic field. The film
thickness was assumed to be small
compared to the radius of the conductor.
In the long-wave approximation, a model
equation for the deviation of the film
thickness from its undisturbed value is
obtained. Some numerical solutions of
this equations are reported.

The regions of parametric resonance were
determined for a sphere immersed in water
and suspended from a string with varying
tension.

A solution is derived for the two-
dimensional unsteady problem of the
behavior of an elastic beam of finite
dimensions floating on the free surface
of water under external loading. It is
assumed that the fluid is ideal and
incompressible and its depth is well
below the beam length. The simultaneous
motion of the beam and the fluid is
considered within the framework of
linear theory, and the fluid flow is
assumed to be potential. The behavior of
the beam under various loadings with and
without allowance for the inertia of the
load is studied.

The paper studies the local variation of
the concentration of a neutral dilute
solution during its radial flow around a
spherical cavity in the approximations
of an adsorption layer and the Langmuir
adsorption kinetics. The authors used
the boundary-layer method and the method
of asymptotic series expansion of the
solution in a small parameter, which is
the ratio of the time of establishing an
adsorption equilibrium to the time of
establishing a steady diffusion layer
around the cavity. The equations
obtained for a zeroth approximation were
studied analytically and numerically. In
the case of high-frequency oscillations
of the cavity in the solution, a
solution of the problem was found that
corresponds to the process of
"straightened" adsorption or "pumping"
an admixture into the adsorption layer.

Stability of periodic solutions of a
non-self-similar nonlinear problem is
studied. The problem describes the
thermal state of an axial fluid flow
with continuously distributed sources of
heat. The flow experiences the action of
external low-amplitude perturbations
changing in time in accordance with
known periodic laws. The spectral
problem is solved by the method of
parametrix, and the critical conditions
of the thermal explosion are determined
in the linear approximation. Stability
of the periodic solution at the critical
point is evaluated using the known
theorem of factorization, which takes
into account the effect of nonlinear
terms of the heat-balance equation. The
calculation results show that the
periodic solution is stable if the total
action of external periodic
perturbations at the critical point is
directed to reduction of the fluid-flow
temperature.

A model governing a steady flow of a
viscoplastic material between coaxial
cylinders is proposed. Nonlinear
velocity sensitivity typical of
superplastic materials is taken into
account. An algorithm of calculating the
characteristics of the material is
developed. The algorithm is based on the
experimental data on moments and angular
velocities of the rotating coaxial
cylinders. The stability of the
algorithm to errors in the initial data
is estimated.

An isotropic linear-elastic
(viscoelastic) plane containing various
physically nonlinear elliptic inclusions
is considered. It is assumed that the
distances between the centers of the
inclusions are much greater than their
dimensions. The problem is to determine
the orientation of the inclusions and
the loads applied at infinity which
ensure a specified value of the
principal shear stress in each
inclusion. Necessary and sufficient
conditions of existence of the solution
of the problem are formulated for a
plane strain of an incompressible
inhomogeneous medium.

Direct and inverse problems of forming
of long-length profiles with double
curvature and a given angle of twisting
under conditions of creep are
considered. A finite-difference scheme
for the numerical solution is proposed.
Examples of solving problems with
different types of external actions for
a profile with a rectangular cross
section are given. Experimental and
numerical data are compared for twisting
of beams with square and circular cross
sections in the regime of creep at
temperatures of 725 and 740°C for St. 45
steel.

This paper considers a model of a
plastically compressible porous medium
with a cylindrical-type yield condition
and its associated constitutive
relations, which ensure independent
mechanisms of shear and compaction of
the porous material. This allows one to
use the well-known theorems of plastic
theory to analyze plastically
compressible media and obtain analytical
solutions for a number of boundary-value
problems, including those taking into
account conditions on strong-
discontinuity surfaces. Results from
full-scale studies of the structural
periodicity of noncompact materials
using wavelet analysis were employed to
choose a physical model for a porous
body and determine the properties and
dimensions of a representative volume.
The problem of extrusion of a porous
material through a conical matrix was
solved.

Fracture of a composite medium with a
brittle matrix is studied. The brittle
or plastic material of the reinforcing
elements is highly deformable. For
normal-rupture macrocracks, necessary
criteria of brittle strength and
sufficient criteria of quasibrittle
strength are proposed. Simple analytical
dependences of the macrocrack length on
the loading parameter, structural,
rigidity, and strength parameters of the
medium, and damage parameters of the
material of the components are obtained.
The critical loads for these criteria
may differ substantially even if the
reinforcement coefficient is small and
the material of reinforcing elements is
highly deformable. If the necessary
criterion is satisfied, crack extension
occurs and microcracks are formed in the
bonds of the structure located ahead of
the macrocrack tip. The number of
damaged bonds depends on the macrocrack
opening and characteristics of
postcritical deformation of the damaged
bonds.

Sufficient conditions of technical
stability of nonlinear dynamic states of
extended elastic flying systems in
controlled longitudinal vertical flight
are obtained. In these flying systems,
the effect of variation of their cross-
sectional area, transverse strains, and
oscillations is taken into account. The
formulated criteria of technical
stability depend on the basic parameters
of the process controlled, namely, on
the increment of the transverse load due
to the curvature of the axis of the
system and aerodynamic forces during
vertical flight.

The problem of interpretation of hot-
wire measurements of acoustic fields in
compressible flows is considered.
Relations between mass-flow and total-
temperature fluctuations registered by a
hot-wire anemometer and pressure and
velocity fluctuations are found. The
relations obtained are applicable in the
general case for measurement of
resultant acoustic fluctuations at some
point of the flow, which are generated
by arbitrary distributed sources of
sound with prior-unknown properties.