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IssuesArchive of Issues2004-1pp.28-35

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F. V. Loveikin and A. F. Ulitko, "Stress distribution in an elastic space weakened by two wedge-shaped cuts lying in one plane," Mech. Solids. 39 (1), 28-35 (2004)
Year 2004 Volume 39 Number 1 Pages 28-35
Title Stress distribution in an elastic space weakened by two wedge-shaped cuts lying in one plane
Author(s) F. V. Loveikin (Kiev)
A. F. Ulitko (Kiev)
Abstract The problem of equilibrium is considered for an elastic space with two wedge-shaped cuts issuing from the same vertex, lying on the same plane, and having a common bisectrix. Stress distribution near the cuts is studied on the basis of homogeneous solutions of the corresponding problem. These solutions are constructed in the spherical coordinates with the help of transformations of Meler-Fock type [1, 2]. The analysis of these homogeneous solutions allows us to describe the behavior of stresses near the common vertex of the cuts. It is shown that the stresses at the vertex have a local power-law singularity. We examine the dependence of the exponent of this singularity on the angle of the cuts. Special attention is given to the case of the classical root singularity at the vertex. Previously [3-5], equilibrium problems were solved for an elastic space with a single wedge-shaped cut and with two cuts - one along a half-plane and the other wedge-shaped. These problems are special cases of the problem considered here.
References
1.  N. N. Lebedev and I. P. Skal'skaya, "Integral expansions related to the Meler-Fock transformations," Diff. Uravnenia, Vol. 22, No. 9, pp. 1515-1523, 1986.
2.  D. N. Parfenenko and F. A. Ulitko, "On a new integral transformation and its application in contact problems of elasticity," Doklady AN UkrSSR, Ser. A, No. 11, pp. 53-58, 1988.
3.  D. N. Parfenenko, "On singularities of the stress field in an elastic material with a plane wedge-shaped cut," in Mathematical Modelling of Dynamic Processes in Systems of Bodies with Fluids [in Russian], pp. 72-76, Izd-vo In-ta Matem. AN UkrSSR, Kiev, 1988.
4.  F. A. Ulitko and A. V. Loveikin, "Equilibrium of an elastic space weakened by two wedge-shaped cuts," in Mechanics of Fracture of Materials and Structures [in Ukranian], Vol. 2, No. 2, pp. 229-233, Kamenyar, Lviv, 1999.
5.  F. A. Ulitko and A. V. Loveikin, "Equilibrium problem for a space with two cuts- along a plane and wedge-shaped- that lie in one plane," Matem. Metodi ta Fiziko-mekhanichni Polya, Vol. 43, pp. 109-114, 2001.
6.  A. F. Ulitko, The Method of Vector-Valued Eigenfunctions in Spatial Problems of Elasticity [in Russian], Naukova Dumka, Kiev, 1979.
7.  N. N. Lebedev, Special Functions and Their Applications [in Russian], Fizmatgiz, Moscow, Leningrad, 1963.
8.  H. Bateman and A. Erdélyi, Higher Transcendental Functions. Volume 1, McGraw-Hill Book Co., New York, 1953.
9.  B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, London, 1958.
Received 17 May 2001
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