 | | 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 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| V. S. Nikishin, "Problems of elasticity for ring and circular cracks on the interface between a layer and a half-space," Mech. Solids. 36 (3), 112-121 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
3 |
Pages |
112-121 |
| Title |
Problems of elasticity for ring and circular cracks on the interface between a layer and a half-space |
| Author(s) |
V. S. Nikishin (Moscow) |
| Abstract |
Two related axisymmetric mixed problems of elasticity for a ring
crack, a<r<b, and a circular crack, 0<r<b, are exactly solved
in closed form.
The cracks are located on the interface between a layer of arbitrary
thickness H and a half-space. The outer surface of the layer and
the crack surfaces are subjected to arbitrary normal and tangential
loads. The mathematical statement of these problems is based on a
specific regularization of the solution of the first fundamental
boundary value problem of elasticity for a separate layer whose
boundary planes are subjected to arbitrary normal and tangential
loads. This solution is constructed by means of the Hankel
transform, providing that integrals for all stresses and
displacements converge. The mixed problems for the ring and
circular cracks are thus reduced to singular integral equations
(SIE) with the Cauchy kernel for complex functions of a real
variable. For these equations, we can completely use the results of
the Carleman-Vekua regularization method, which were obtained for
similar SIE in the problems of ring and circular punches indenting
a laminated half-space [1]. This regularization reduces the SIE to
the system of regular Fredholm integral equations of the third and
second kind for the ring and circular cracks, respectively. This
permits us also to determine and separate singularities of the
solution of the SIE at the ends of the integration interval and
singularities of the desired stresses at the crack boundaries. The
suggested approach to the statement of the mixed problems for
cracks or punches and a laminated half-space based on regularizing
the solution of the first fundamental boundary value problem was
previously used in [1-3] for the case of zero tangential stresses
at the layer interfaces. The main advantage of this approach is
that the complex problems for cracks are reduced to the equivalent
constitutive integral equations of a simple form.
The problem for a circular crack on a layer
interface in a laminated space was fist solved exactly in [4, 5].
The authors of these studies reduced the solution to the SIE with the
Cauchy kernel and then, by using the Carleman-Vekua regularization
method, transformed this SIE into the equivalent system of Fredholm
equations of the second kind, which allowed effective numerical
solution.
[The analytical solution of this problem was implemented numerically.
The results are presented in detail in Preprint No. 61 [in Russian], In-t
Problem Mekhaniki AN SSSR, Moscow 1975.] |
| References |
| 1. | V. S. Nikishin, Well-Posed Statement and Numerical Solution
of Fundamental and Mixed Problems of Elasticity for Multilayered
and Continuously Inhomogeneous Media. D. Sc. Thesis [in Russian],
VTs AN SSSR, Moscow, 1982. |
| 2. | V. S. Nikishin and G. S. Shapiro, "On the local
axisymmetric compression of an elastic layer weakened by a ring or
circular slot," PMM [Applied Mathematics and Mechanics], Vol. 38,
No. 1, pp. 139-144, 1974. |
| 3. | V. S. Nikishin and G. S. Shapiro, "Contact problem of
elasticity for a layer and a half-space locally pressed together,"
Izv. AN ArmSSR. Mekhanika, Vol. 29, No. 2, pp. 3-15, 1976. |
| 4. | V. M. Vainshel'baum and R. V. Goldstein, "Axisymmetric
problem for a crack at an interface between layers in a multi-layered
medium," Izv. AN SSSR. MTT [Mechanics of Solids], No. 2,
pp. 130-143, 1976. |
| 5. | R. V. Goldstein and V. M. Vainshel'baum, "Axisymmetric
problem of a crack at the interface of layers in a multi-layered
medium," Int. J. of Eng. Sci., Vol. 14, No. 4, pp. 81-95, 1976. |
| 6. | V. S. Nikishin and G. S. Shapiro, Three-Dimensional
Problems of Elasticity for Multilayered Media [in Russian], VTs AN
SSSR, Moscow, 1970. |
|
| Received |
12 October 1998 |
| << Previous article | Volume 36, Issue 3 / 2001 | Next article >> |
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