 | | 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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| E.V. Mochalov and V.V. Sil'vestrov, "Problem of Interaction of Thin Rigid Needle-Shaped Inclusions Located between Different Elastic Materials," Mech. Solids. 46 (5), 739-754 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
5 |
Pages |
739-754 |
| DOI |
10.3103/S0025654411050086 |
| Title |
Problem of Interaction of Thin Rigid Needle-Shaped Inclusions Located between Different Elastic Materials |
| Author(s) |
E.V. Mochalov (Ul'yanov Chuvash State University, Moskovskii pr-t 15, Cheboksary, 428015 Russia)
V.V. Sil'vestrov (Gubkin Russian State University of Oil and Gas, Leninskii pr-t 65, Moscow, 119991 Russia, v-silvestrov@yandex.ru) |
| Abstract |
We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a two-sheeted Riemann surface. We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.
A piecewise-homogeneous elastic plane with a separate thin rigid rectilinear inclusion at the interface between the media in the case of perfect adhesion of the inclusion to the environment was studied in [1-3]. In the book [1], the case of absolutely smooth contact between the inclusion and the elastic medium was also studied, where, in both cases, the unknown contact stresses on the inclusion sides and the rotation angle of the inclusion were found depending on the normal force and moment of forces applied; the solution was obtained using the Fourier integral transform to reduce the problem to a system of singular integral equations on an interval. Explicit analytical formulas for the contact stresses on the sides of the rectilinear rigid inclusion perfectly adherent to the medium and the stress intensity factors near its tips, can be found in [2] and [3]; the formulas were obtained by the methods of integral equations and the Riemann boundary-value problem. The problem of interaction between several rigid rectilinear inclusions perfectly adherent to the medium and located between different elastic materials under the action of tensile stresses given at infinity was solved in [4, 5]. The results of studies of a homogeneous elastic plane with one or several thin rigid needle-shaped inclusions perfectly adherent to the medium and with cracks are contained in the book [6]. |
| Keywords |
thin interphase inclusions, mixed problem, separation, rotation, stresses |
| References |
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Concentration of Elastic Stresses Near Punches, Cuts, Rigid Inclusions, and Stiffeners
(Nauka, Moscow, 1982)
[in Russian]. |
| 2. | R. A. Ballarini,
"A Rigid Line Inclusion at a Bimaterial Interface,"
Engng Fract. Mech.
37 (1), 1-5 (1990). |
| 3. | X. Markenscoff, L. Ni, and J. Dundurs,
"The Interface Anticrack and Greens's Functions
for Interacting Anticracks and Cracks/Anticracks,"
Trans. ASME. J. Appl. Mech.
61 (4), 797-802 (1994). |
| 4. | K. C. Wu,
"Line Inclusions at Anisotropic Bimaterial Interface,"
Mech. Mater.
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| 5. | A. Asundi and W. Deng,
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| 6. | L. T. Berezhnitskii, V. V. Panasyuk, and I. I. Trush,
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[Strength of Materials (Engl. Transl.)
5 (7), 775-779 (1973)]. |
| 7. | J. R. Rice and G. C. Sih,
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32 (2), 418-423 (1965). |
| 8. | L. T. Berezhnitskii, V. V. Panasyuk, and N. G. Stazhchuk,
Interaction of Rigid Linear Inclusions and Cracks in Deformable Body
(Naukova Dumka, Kiev, 1983)
[in Russian]. |
| 9. | I. V. Simonov,
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49 (2), 275-283 (1985)
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49 (2), 207-214 (1985)]. |
| 10. | I. V. Simonov,
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in Mechanics of Contact Interactions
(Fizmatlit, Moscow, 2001)
[in Russian]. |
| 11. | G. P. Cherepanov,
Fracture Mechanics of Composite Materials
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[in Russian]. |
| 12. | N. I. Muskhelishvili,
Singular Integral Equations
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[in Russian]. |
| 13. | E. I. Zverovich,
"Boundary Value Problems in the Theory of Analytic Functions
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26 (1), 113-179 (1971)
[Russ. Math. Surv. (Engl. transl.)
26 (1), 117-192 (1971)]. |
|
| Received |
23 April 2009 |
| Link to Fulltext |
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