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IssuesArchive of Issues2005-3pp.123-133

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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].
References
1.  G. P. Cherepanov, "On the stress state in an inhomogeneous plane with cuts," Izv. AN SSSR. Mekhanika i Mashinostroenie, No. 1, pp. 131-137, 1962.
2.  F. Erdogan, "Stress distribution in bonded dissimilar materials containing circular or ring-shaped cavities," Trans. ASME. Ser. E. J. Appl. Mech., Vol. 32, No. 4, pp. 829-836, 1965.
3.  G. Ya. Popov, Concentration of Elastic Stresses Near Punches, Cuts, Thin Inclusions, and Reinforcements [in Russian], Nauka, Moscow, 1982.
4.  T. C. T. Ting, "Explicit solution and invariance of the singularities at an interface crack in anisotropic composites," Intern. J. Solids and Structures, Vol. 22, No. 9, pp. 965-983, 1986.
5.  T. A. Homulka and L. M. Kerr, "A mathematical solution of a special mixed-boundary value problem of anisotropic elasticity," Quart. J. Mech. and Appl. Math., Vol. 48, No. 4, pp. 636-658, 1995.
6.  R. Ballarini, "A certain mixed boundary value problem for a bimaterial interface," Intern. J. Mech. Solids and Structures, Vol. 32, No. 3-4, pp. 279-289, 1995.
7.  V. N. Akopyan, "On a mixed problem for a composite plane weakened by a crack," Izv. NAN Armenii, Vol. 48, No. 4, pp. 57-65, 1995.
8.  X. Markenscoff and L. Ni, "The debonded interface anticrack," Trans. ASME. J. Appl. Mech., Vol. 63, No. 3, pp. 621-627, 1996.
9.  D. I. Sherman, "A mixed problem of the theory of potentials and elasticity theory for a plane with a finite number of straight cuts," Dokl. AN SSSR, Vol. 27, No. 4, pp. 330-334, 1940.
10.  N. I. Muskhelishvili, Some Basic Problems of the Theory of Elasticity [in Russian], Nauka, Moscow, 1966.
11.  L. M. Kerr, "Mixed boundfary value problems for a penny-shaped cut," J. Elasticity, Vol. 5, No. 2, pp. 89-98, 1975.
12.  E. E. Gdoutos, C. G. Kourounis, M. A. Kattis, and D. A. Zacharopoulos, "A partially unbounded rigid fiber inclusion in an infinite matrix," Advances in Fracture Mechanics, pp. 223-227, 1989.
13.  G. A. Morar', The Discontinuos Solution Methods in Solid Mechanics [in Russian], Shtinitsa, Kishinev, 1990.
14.  S. V. Bosakov, "The solution of a contact problem for a plane with a slot," Prikl. Mekhanika, Vol. 13, No. 7, pp. 127-129, 1977.
15.  S. V. Bosakov, "The calculation of buried anchor plates of finite stiffness," Prikl. Mekhanika, Vol. 16, No. 3, pp. 81-87, 1980.
16.  V. M. Alexandrov, B. I. Smetanin, and B. V. Sobol', Thin Stress Concentrators in Elastic Bodies [in Russian], Nauka, Fizmatlit, Moscow, 1993.
17.  G. P. Cherepanov, "The solution of a linear boundary value Riemann problem for two functions and its application to some mixed problems of plane elasticity theory," PMM [Applied Mathematics and Mechanics], Vol. 26, No. 5, pp. 906-912, 1962.
18.  E. I. Zverovich, A mixed problem of elasticity theory for a plane with cuts on the real axis, In Proc. Sympos. on Continuum Mechanics and Related Problems of Analysis [in Russian], Vol. 1, pp. 103-114, Metsnierba, Tbilisi, 1973.
19.  V. N. Akopyan and A. V. Saakyan, "The stress state of a homogeneous elastic plane with a star-shaped crack under mixed boundary conditions on the crack faces," Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 106-113, 1999.
20.  V. V. Zakharov and L. V. Nikitin, "The influence of friction on the process of delamination of dissimilar materials," Mekhanika Kompositnykh Materialov, No. 1, pp. 20-25, 1983.
21.  S. M. Mkhitarian, "On one class of mixed problems of elasticity," in Continuum Mechanics and Related Problems of Analysis: Proc. Intern. Symp. Tbilisi, pp. 163-171, 1993.
22.  Yu. A. Antipov, "A detached inclusion in the case of adhesion and slip," Prikl. Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 4, pp. 669-680, 1996.
23.  N. M. Kundrat, "Local fracture in a composite containing rigid line inclusions," Mekhanika Kompozitsionnykh Materialov i Konstruktsii, Vol. 4, No. 4, pp. 115-127, 1998.
24.  N. M. Kundrat, "A limiting equilibrium of a composite containing a rigid inclusion under tension by lumped forces," Mekhanika Kompozitsionnykh Materialov i Konstruktsii, Vol. 6, No. 1, pp. 103-112, 2000.
25.  N. M. Kundrat, "Investigation of mechanisms of fracture in a composite containing a rigid inclusion under tension by lumped forces," Mekhanika Kompozitsionnykh Materialov i Konstruktsii, Vol. 6, No. 3, pp. 333-342, 2000.
26.  N. M. Kundrat, "The detachement of a rigid inclusion in an elastoplastic matrix under tension by lumped forces," Mekhanika Kompozitsionnykh Materialov i Konstruktsii, Vol. 7, No. 1, pp. 107-113, 2001.
27.  N. M. Kundrat, "The separation of a rigid line inclusion under static tension," Fiz.-Khim. Mekhanika Materialov, Vol. 37, No. 1, pp. 37-40, 2001.
28.  V. N. Akopyan and A. V. Saakyan, "The stress state of an elastic half-plane containing a thin rigid inclusion," Izv. RAN. MTT [Mechanics of Solids], No. 6, pp. 76-92, 2002.
29.  J. R. Rice and G. C. Sih, "Plane problems of cracks in dissimilar media," Trans. ASME. Ser. E. J. Appl. Mech., Vol. 32, No. 2, pp. 418-423, 1965.
30.  G. P. Cherepanov, Mechanics of Fracture of Composite Materials [in Russian], Nauka, Moscow, 1983.
31.  N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow, 1968.
32.  P. Lankaster, Theory of Matrices [Russian translation], Nauka, Moscow, 1978.
33.  F. D. Gakhov, Boundary Value Problems [in Russian], Nauka, Moscow, 1977.
34.  M. P. Savruk, Stress Intensity Factors in Bodies with Cracks [in Russian], Nauk. Dumka, Kiev, 1988.
35.  J. R. Rice, "Elastic fracture mechanics concepts for interfacial cracks," Trans. ASME. Ser. E. J. Appl. Mech., Vol. 55, No. 1, pp. 98-103, 1988.
36.  J. R. Willis, "Fracture mechanics of interfacial cracks," J. Mech. and Phys. Solids, Vol. 19, No. 6, pp. 353-368, 1971.
37.  A. Asundi and W. Deng, "Rigid inclusions on the interface between dissimilar anisotropic media," J. Mech. and Phys. Solids, Vol. 43, No. 7, pp. 1045-1058, 1995.
38.  M. L. Williams, "The stress around a fault or crack in dissimilar media," Bull. Seismol. Soc. America, Vol. 49, No. 2, pp. 199-204, 1959.
39.  R. Ballarini, "A rigid line inclusion at a bimaterial interface," Eng. Fract. Mech., Vol. 37, No. 1, pp. 1-5, 1990.
Received 09 September 2003
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