List of Recent Abstracts
A coupled plane-bending
problem is considered for an elastic Kirchhoff-Poisson plate containing a
through-the-thickness or (part-through) surface crack under closure.
The stress intensity factors at the ends of the crack are not zero.
Asymptotic solutions are derived for cases in which the ratio of the
crack length to its depth is large. As is shown, the width of the
contact strip decreases as the crack length increases; the limiting
contact force and moment distribution may be determined by considering an
edge-cracked strip with zero stress intensity factor in the thickness
direction. As is also shown, the crack surface interaction in-plane force
and bending moment can be derived directly from the initial force and
moment distribution acting in the intact plate on the prospective crack
line. The same result is valid for a collinear system of cracks. In
addition, the closure stress distribution is determined. For the latter,
the width of the contact strip asymptote is derived as a function of the
crack length coordinate, and the asymptotic stress distribution is found
as a product of the thickness averaged closure stress and a function of a
normalized plate coordinate. The latter stress distribution is unique and
universal for any slow curving crack or crack system under closure.
The exact solution to the nonlinear vector
equations governing an inextensible, flexible, helicoidal fiber
yields a one-parameter set of solitary waves.
Analytical expressions are found for the
displacements, internal force, axial and angular momenta and energy.
This solution represents an interesting example of three-dimensional
solitary waves propagating in a system devoid of potential energy.
Influence of layer thickness on the stress distribution in the vicinity of a
crack tip is examined, taking into account the fact that the conventional
stress intensity factor concept becomes invalid if the thickness of the layer
is not much more than the size of the fracture process zone. The
eigen-problem is considered which is characterized by two asymptotes. The
first is a near one; it is formed in a small vicinity of the crack tip in the
layer thickness scale. The second asymptote is a far one in the same scale.
The regions of validity of these asymptotes are determined and shown to
depend upon material
parameters and crack tip speed. The complete stress distribution in front of
the crack is determined, as well. Some conclusions are made concerning the
stress distribution and energy release rate for the general problem. Mode 3
crack propagation is considered in detail.
Progressive radial cracking of
a clamped plate subjected to crack-face closure is studied. The
material behavior is assumed to be elastic-brittle. The cracks are
assumed to be relatively long in the sense that the three dimensional
contact problem can be described via a statically equivalent
two-dimensional idealization. The number of cracks is supposed large
enough to permit a quasi-continuum approach rather than one involving
the discussion of discrete sectors. The formulation incorporates the
action of both bending and stretching as well as closure effects of the
radial crack face contact. Fracture mechanics is used to explore the
load-carrying capacity and the importance of the role of the
crack-surface-interaction. For a given crack radius, the closure
contact width is assumed to be constant. Under this condition, a
closed-form solution is obtained for the case of a finite clamped plate
subjected to a concentrated force. Crack growth stability
considerations predict that the system of radial cracks will initiate
and grow unstably over a significant portion of the plate radius. The
closure stress distribution is determined exactly in the case of narrow
contact widths and approximately otherwise.
The complete analytical solution
of the governing nonlinear vector equations is found which describes
periodical and solitary waves in a helicoidal fiber.
The solitary wave velocity is found to be proportional to the
square root of the amplitude of the internal force, and the effective wave
length is shown to be independent of the amplitude. The
essential influence of the rigid body rotation of the helix on the solitary
wave shape is shown.
Axial and angular momenta of the wave are determined as well.
A flexible
helicoidal fiber with an arbitrary, unspecified, stress-strain relations
is considered, and a
general, analytical, steady-state solution of the 3D nonlinear vector
equation is found. This solution describes a new class of spatial motion
of an elastic string which
is shown to be a waveguide for subsonic solitary waves.
The geometrically nonlinear problem for a linearly elastic fiber
is considered in more detail.
An infinite elastic-brittle plate subjected to a transverse impact is
considered. Many cracks, each of which extends through the full thickness
of the plate, are assumed to grow radially from the impact point.
Due to circumferential bending, the faces of these radial cracks tend to
close over a portion of the plate thickness. In this paper, the dynamics of the radial crack system within
the influence of a tangential unilateral constraint which prevents crack face interpenetration
is examined. The analysis is simplified by assuming that the faces interact
only over the lines where these faces meet a surface of the plate, rather than over some unknown area of the fracture surfaces. In other words, the local contact strain is neglected. The radial cracks are assumed to be
large in number and evenly distributed so that the deformation retains
axial symmetry. The governing equations are derived, and a transient,
self-similar solution of these equations is constructed, relating the
plate response to the impact force. The extension of the radial cracks
according to the principles of fracture mechanics is explored.
A composite structure consisting of
doubly periodic
moderately wavy layers under tension is considered.
The periodic cell of the
composite is represented by two layers of different materials,
but having the same arbitrary initial shape.
Using a geometrical nonlinear Cosserat rod model
to describe the layers,
we take into account
extension, flexure and shear deformation in the layers.
The problem is reduced to
a boundary value problem for a scalar nonlinear ODE
of second order for the rotational degree of freedom.
The determination of the
deformed shape of the layers, as well as the relation between
the elongation of the periodic cell
and the applied tension force
is then reduced to nonlinear quadratures.
For the case of small initial waviness, an analytical solution
is obtained for an arbitrary initial shape of the layers.
For moderate initial waviness,
numerical results are presented relating stress, strain and
amplitude of waviness. The influence of material
and structural parameters is investigated and discussed.
In particular, as opposed to theories with full homogenization,
the thickness of the periodic cell with
respect to its length is not assumed to approach zero in the analysis.
The influence of this structural
parameter is shown to be especially significant %particularly
if one of the materials of the layers
is much weaker than the other.
Axial dynamic tension of a flexible helical fiber is found to lead to a
specific, extraordinary nonlinear wave consisting of two different portions.
The leading portion is a quasi-periodical propagating wave with rotation
opposite to the initial twist of the helix, while the rear portion
is a sequence of standing waves rotating in the reverse direction.
The interface is the origin of two angular momentum fluxes which,
being different in sign, fill up angular momenta of the leading and rear
waves. The phenomenon described presents an interesting example of a
zero-total-angular-momentum wave propagating in a twisted waveguide.
A plate with a pre-existent through crack is considered under the action of
a remote in-plane force. The problem statement is reduced to the solution
of two coupled integral equations with strongly singular kernels. The
independent variables in the latter equations are the closure displacement
and rotation angle. The corresponding closure force and moment
distribution, and the contact-crack opening boundary (the closure
perimeter), are found as functions of the remote bending-compression
ratio. The validity of previously stated analytical asymptotics for the
contact boundary is examined. The dependence of the extent of closure on
the crack length-to-thickness ratio is studied. Comparisons are made with
experimental results.
A plane, periodic, square-cell lattice is considered, consisting of point
particles connected by mass-less viscoelastic bonds. Homogeneous and
inhomogeneous problems for steady-state semi-infinite crack propagation
in an unbounded lattice and lattice strip are studied. Expressions for the
local-to-global energy-release-rate ratios, stresses and strains of the
breaking bonds as well as the crack opening displacement are derived.
Comparative results are obtained for homogeneous viscoelastic materials,
elastic lattices and homogeneous elastic materials. The influences of
viscosity, the discrete structure, cell size, strip width and crack speed
on the wave/viscous resistances to crack propagation are revealed.
Some asymptotic results related to an important asymptotic case of large
viscosity (on a scale relative to the lattice cell) are shown. Along with
dynamic crack propagation, a theory for a slow crack in a viscoelastic
lattice is derived.
An elastic space containing an elastic helical rod
subjected to both
axial and radial extension as well as torsion is
considered. Due to
translation-rotation helical symmetry, the resulting
elastic fields
in the matrix can be expressed in terms of a
two-dimensional
helix-associated coordinate system. In this problem,
a `helical elastic foundation' as a generalization of
the
Winkler foundation is determined by means of which the
interacting force
and
moment at the rod/matrix interface can be expressed in
terms of the rod
displacement. The matrix is assumed to be linear
elastic while the
geometric nonlinearity of the helical rod is taken into
account. Using
superposition of fundamental solutions for a
homogeneous elastic space
(in the absence of the rod), and the constitutive and
equilibrium equations for the rod, the internal forces
and
moments in the rod as well as the displacement and
elastic fields
in the matrix are obtained.
Along with the general results, two
asymptotic solutions are presented. The first,
corresponding to a small
curvature but not too small pitch, allows an analytical
integration of
the rod-matrix interaction over the rod
cross-section boundary. The second
corresponds to an almost straight helical rod:
the helix becomes a straight line, but in the limit the
main
normal to its axis
describes a screw surface as in the case of a `genuine'
helix. In this case,
the helical elastic foundation
has a closed-form parametric expression which is
valid for a rather large range of the helix parameters.
The
foundation stiffness is found as a function of the
helix pitch and the
rod radius;
the problem thus is reduced to a system of finite,
nonlinear equations.
The related questions `how to avoid oscillations under
an impact' and
`why a crack or phase-transition wave can/cannot
propagate slowly' are
discussed. The underlying phenomenon is the dynamic
overshoot which can show itself in deformation of a
body
under a load suddenly applied. The manifestation of
this
phenomenon in a unit cell of the material structure is
shown to trigger
a fast crack in fracture as well as a fast wave in
phase transition.
Two ways for the elimination of the overshoot, to
obtain a
static-amplitude response (SAR), are examined. The
first is a proper
control of the load in an initial portion of the
loading time.
This is illustrated by means of an example of elastic
collision.
In the case of fracture, such control can be envisioned
as provided by a
proper post-peak tensile softening of the material.
Secondly, the SAR can be achieved under the influence
of viscosity.
In this connection, the following transient problems
are considered:
a viscoelastic-spring oscillator under a step
excitation, a square-cell
viscoelastic lattice with a crack and a two-phase
viscoelastic chain as the phase-transition waveguide.
For each problem,
in the space of viscosity parameters, the SAR domain is
separated from
the dynamic-overshoot-response (DOR) domain.
In the SAR domain, in contrast to the DOR domain, a
slow crack or a slow
phase-transition wave can exist. A structure-associated
size
effect in the SAR/DOR domains separation is noted.
In the considered lattice structure, crack propagation
is caused by
feeding waves, carrying energy to the crack
front, and
accompanied by dissipative waves
carrying a part of this energy away from the front (the
difference
is spent on the bond disintegration).
The feeding waves differ by
their wavenumber. A zero feeding wavenumber corresponds
to a
macrolevel-associated solution with the classical
homogeneous-material
solution as its long-wave approximation. A nonzero
wavenumber
corresponds to a genuine
microlevel solution which has no analogue on the
macrolevel.
In the latter case, on the crack surfaces and their
continuation,
the feeding wave is located behind (ahead) the
crack front
if its group velocity is greater (less) than the
phase velocity. Dissipative waves, which appear in both
macrolevel-associated and microlevel solutions, are
located in accordance
with the opposite rule. (Wave dispersion is the
underlying
phenomenon which
allows such a wave configuration to exist.)
In contrast to a homogeneous material model,
both these solutions
permit supersonic crack propagation. Such feeding
and dissipative waves and other lattice phenomena
are characteristic of dynamic phase transformation as
well.
In the present paper, mode III of crack propagation in
a
square-cell elastic lattice is studied.
Along with the lattice model, some simplified
one-dimensional
structures are considered allowing one to retrace
qualitatively
(with no technical
difficulties) main lattice phenomena.
Dscrete and homogeneous models of a structured material
are considered
to resolve difficulties in the analysis of dynamic
phase transition.
The discrete model is a chain consisting of particles
connected
by mass-less bonds, while the continuous model is
represented by a
partial differential equation with higher than the
second order of
coordinate-derivatives. The macrolevel constitutive law
is
represented by a bi-linear stress-strain relation, such
that the
transition from the first, stiffer phase to the second
one is
irreversible. Solutions of two types,
macrolevel-associated and
microlevel, are derived. The first type
solution is characterized by a macrolevel feeding wave
(the wave
delivering energy to the phase-transition
front is of a zero wavenumber), while the microlevel
solutions
correspond to
a nonzero feeding wavenumber. Subsonic, intersonic and
supersonic
phase-transition waves are described. For the
homogeneous model it is
shown that the contradiction between the limiting
stress and energy
criteria, inherent for the macrolevel formulation of
the problem,
is eliminated if and only if the phase transition does
not concern the
highest-order modulus. Total structure- and
speed-dependent dissipation,
as the energy carried by
microlevel waves away from the phase-transition front,
as well as
parameters of each dissipative wave are determined. For
the
fourth-order partial differential equation, the
existence of the
Maxwell type, dissipation-free, subsonic
phase-transition wave is shown.
In this case, the microstructure plays the role of a
catalyst. Common
and distinctive properties of the discrete and
homogeneous models
are discussed.
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Last modified: 12 13:59:40 1998