 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| I. I. Il'ina and V. V. Sil'vestrov, "The problem of a thin interfacial inclusion detached from the medium along one side," Mech. Solids. 40 (3), 123-133 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
3 |
Pages |
123-133 |
| Title |
The problem of a thin interfacial inclusion detached from the medium along one side |
| Author(s) |
I. I. Il'ina (Cheboksary)
V. V. Sil'vestrov (Cheboksary) |
| Abstract |
We consider a piecewise-homogeneous elastic plane formed by two dissimilar
elastic half-planes. Between the half-planes, a finite thin rigid
sharp-cornered inclusion is located. One side of the inclusion is
completely attached to the medium and the other side contacts with the
medium in the friction-free slip mode (similar to the case of a smooth
punch). We study a plane stress state caused by the stresses prescribed at
infinity under the most common boundary conditions specified on the sides
of the inclusion. The boundary conditions on the inclusion are as follows:
the displacement vector is prescribed on one side and the shear component
of the stress vector and the normal component of the displacement are
prescribed on the other side.
Using the Riemann matrix boundary-value
problem method, we find explicitly the complex potentials for the composite
elastic plane and study the behavior of the stresses near the ends of the
inclusion. Depending on the elastic parameters of the composite plane,
at the ends of the inclusion the stresses have either a power-law
singularity of order from 1/2 to 1 or a power-law singularity of order
1/2 combined with an oscillating singularity. In the first case, the
stress intensity is determined by a single real coefficient and in the
second case by three coefficients.
The model of a piecewise-homogeneous medium with a partly separated
inclusion under consideration can be used for studying composite materials
with the stiffening elements in the form of thin rigid inclusions,
which have been separated from the medium along one side during the
process of usage of the material, for example, because of the difference
between the elastic properties of the media adjacent to the inclusion.
A thin rigid sharp-cornered inclusion between dissimilar
isotropic and anisotropic media in the case where one side of the
inclusion is completely attached to the medium and the other is
contact-free was considered in [1-8]. In these papers, using the methods
of singular integral equations, the generalized integral Fourier
transform, and the Riemann matrix boundary-value problem, the solution of
the corresponding mathematical problem is found and the behavior of the
stresses near the inclusion ends is studied. In the case of a homogeneous
medium, the solution of the problem mentioned was given in [9-13].
A thin rigid inclusion, with one side being contact-free and
the other being in contact with the medium like a smooth punch, was
studied in [3, 13, 14]. In [13, 14], the case of a
homogeneous medium was considered. In [15], for a thin
elastic inclusion in a homogeneous elastic medium, one side of which is
rigidly attached to the medium and the other is separated and is in
contact with the medium in the absence of shear stress, the reaction
stress at the points of the inclusion was obtained in the form of the
first-order Chebyshev polynomials. The corresponding antiplane problem for
a thin rigid inclusion in a homogeneous elastic layer was solved in [16].
Other models of a thin rigid inclusion separated from the medium and the
related mixed problems of elasticity were considered in [17-28]. In [17,
18, 21], the basic mixed problem of elasticity for a homogeneous plane
with collinear cuts was considered for different arrangements of the
points at which the type of the boundary conditions on the cut surface
changes.
In [19], the stress state of a homogeneous elastic plane
containing four identical alternate cracks was considered in
the case where on one surface of a typical cut the displacements are
prescribed and on another surface the stresses are specified. A similar
problem for an elastic half-plane with a boundary cut was solved in [28].
In [22], the problem of a thin rigid inclusion was considered in the case
where one side of the inclusion is attached to the medium, a certain inner
part of the other inclusion side is contact-free, and the inclusion parts
located near the ends are in slip contact with the medium.
The papers [20,
23-27] are devoted to the investigation of the process of separation of a
rigid line inclusion from the medium on the parts adjacent to the
inclusion ends. In all these papers, the medium was assumed to be
homogeneous. A thin rigid interfacial inclusion separated from the medium
was considered in [3]. |
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|
| Received |
09 September 2003 |
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