We solve the problem of determination of the dynamic stress intensity factors for a crack in the form of a three-link broken line. The crack is located in an infinite elastic medium with propagating harmonic longitudinal shear waves. The initial problem is reduced to a system of three singular integrodifferential equations with fixed singularities. A numerical method is proposed for the solution of this system with regard for the true asymptotics of the unknown functions.
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A. V. Andreev, “Direct numerical method for solving singular integral equations of the first kind with generalized kernels,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 126–146 (2005); English translation: Mech. Solids, 40, No. 1, 104–119 (2005).
B. A. Afyan, “On the integral equations with fixed singularities in the theory of branching cracks,” Dokl. Akad. Nauk Arm. SSR, 79, No. 4, 177–181 (1984).
R. V. Duduchava, Integral Equations of Convolution with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities, and Their Applications to Problems of Mechanics [in Russian], Proc. Razmadze Math. Inst., 60 (1979).
V. I. Krylov, Approximate Calculation of Integrals, Dover, New York (2006).
P. N. Osiv and M. P. Savruk, “Determination of stresses in an infinite plate with broken or branching crack,” Prikl. Mekh. Tekh. Fiz., No. 2, 142–147 (1983); English translation: J. Appl. Mech. Tech. Phys., 24, No. 2, 266–271 (1983).
V. G. Popov, “Diffraction of elastic shear waves on an inclusion of complex shape located in an unbounded elastic medium, in: Hydroaeromechanics and Elasticity Theory: Numerical and Analytic Methods of Solution of Problems of Hydroaerodynamics and Elasticity Theory [in Russian], Dnepropetrovsk State University, Dnepropetrovsk (1986), pp. 121–127.
V. G. Popov, “Investigation of the fields of stresses and displacements in the case of diffraction of elastic shear waves on a thin rigid separated inclusion,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 139–146 (1992).
V. G. Popov, “Stressed state near two cracks leaving the same point under harmonic longitudinal shear oscillations,” Visn. Kyiv. Shevchenko Nats. Univ. Ser. Fiz.-Mat. Nauk., Issue 3, 205–208 (2013).
G. Ya. Popov, Concentration of Elastic Stresses near Punches, Notches, Thin Inclusions, and Reinforcements [in Russian], Nauka, Moscow (1982).
M. P. Savruk, Two-Dimensional Problems of Elasticity for Bodies with Cracks [in Russian], Naukova Dumka, Kiev (1981).
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., New York (1959).
M. Isida and H. Noguchi, “Stress intensity factors at tips of branched cracks under various loadings,” Int. J. Fract., 54, No. 4, 293–316 (1992).
V. Vitek, “Plane strain stress intensity factors for branched cracks,” Int. J. Fract., 13, No. 4, 481–501 (1977).
X. Yan, “Stress intensity factors for asymmetric branched cracks in plane extension by using crack-tip displacement discontinuity elements,” Mech. Res. Comm., 32, No. 4, 375–384 (2005).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 58, No. 1, pp. 112–120, January–March, 2015.
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Popov, V.G. A Crack in the Form of a Three-Link Broken Line Under The Action of Longitudinal Shear Waves. J Math Sci 222, 143–154 (2017). https://doi.org/10.1007/s10958-017-3288-5
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DOI: https://doi.org/10.1007/s10958-017-3288-5