Journal Journal of Mechanical Engineering
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 21, no. 4, 2018 (December)
Pages 41-48
Cited by J. of Mech. Eng., 2018, vol. 21, no. 4, pp. 41-48



Andrii S. Misharin, National University “Odessa Maritime Academy” (8, Didrikhson Str., Odesa, 65029, Ukraine), e-mail:

Vsevolod H. Popov, National University “Odessa Maritime Academy” (8, Didrikhson Str., Odesa, 65029, Ukraine), e-mail:, ORCID: 0000-0003-2416-642X



Modern elements of building structures and machine parts often contain structural elements or technological defects that can be considered as thin inclusions of high rigidity. Reinforcing elements of composite materials can also be thin rigid inclusions. But studies show that thin rigid inclusions cause a significant stress concentration in the environment, which can lead to the formation of cracks at the inclusion. The problems of determining the stress state in the vicinity of complex defects were solved, as a rule, in a static formulation and for the case of rectilinear defects. This is due to the difficulties that arise in the case of their solution by the common method of boundary integral equations, which consists in reducing such problems to singular integral or integro-differential equations with fixed singularities. Such equations require that special methods be created for their numerical solution. Recently, there has been a continuous growth in the number of papers where special quadrature formulas are used for singular integrals with fixed singularities, for example, for cracks or inclusions in the form of broken or branched defects. These works propose a collocation method that takes into account the real feature of the solution, and in order to calculate integrals with fixed singularities special quadrature formulas are used. The problems of determining the stress state around the defects, which are thin inclusions from whose edge a crack propagates at a certain angle, have been barely solved. The purpose of this paper is to study the stress state near the crack that initiates at the inclusion when subjected to a longitudinal shear wave. The formulated problem is reduced to a system of singular integro-differential equations with fixed singularities with respect to the unknown voltage surges and displacements on the surface of a defect. To solve this system, a similar collocation method is used. There have been shown dependences of the change in the dimensionless values of the stress intensity factors (SIF) on the dimensionless value of the wave number in the case of wave propagation at different angles. For numerical experiments, different values of the angle between the inclusion and crack were taken. In all cases, there was found the value of the dimensionless wave number at which SIFs for the crack reach their peaks. With an increase in the angle between the inclusion and crack, SIF values for the inclusion, up to certain oscillation frequency values, decrease. For the case when the defects are on the same straight line, SIF values for the inclusion are smallest. Conversely, when the angle between the defects increases, SIF values for the crack increase too. In general, as a result of the complexity of the wave field created by the reflection of waves from a defect, SIF dependence on frequency has significant maxima, whose magnitude and position are influenced by the configuration of the defect.


Keywords: stress intensity factors, singular integro-differential equations, harmonic oscillations, fixed singularity, inclusion, crack.


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  1. Sulym, H. T. (2007). Osnovy matematychnoi teorii termopruzhnoi rivnovahy deformivnykh tverdykh til z tonkymy vkliuchenniamy [Fundamentals of the mathematical theory of thermoelastic equilibrium of deformable solids with thin inclusions]. Lviv: Doslid.-vydav. tsentr NTSh, 716 p. (in Ukrainian).
  2. Berezhnitskiy, L. T., Panasyuk, V. V., & Stashchuk, N. G. (1983). Vzaimodeystviye zhestkikh lineynykh vklyucheniy i treshchin v deformiruyemom tele [Interaction of hard linear inclusions and cracks in a deformable body]. Kiyev: Naukova dumka, 288 p. (in Russian).
  3. Berezhnitskiy, L. T. & Stashchuk, N. G. (1981). Koeffitsiyenty intensivnosti napryazheniy okolo treshchiny na prodolzhenii lineynogo zhestkogo vklyucheniya [Stress intensity factors near a crack at the linear rigid inclusion]. Dokl. Ser. A. AN USSRProceedings of the USSR Academy of Sciences, Series: A, no. 11, pp. 30–46 (in Russian).
  4. Berezhnitskii, L. T., Stashchuk, N. G., & Gromyak, R. S. (1989). Determination of the critical dimension of macrocracks originating at the continuation of a linear rigid inclusion. Strength of Materials, vol. 21, iss. 2, pp. 217–220.
  5. Akopyan, V. N. & Amirdzhanyan, A. A. (2015). Napryazhennoye sostoyaniye poluploskosti s vykhodyashchim na granitsu absolyutno zhestkim vklyucheniyem i treshchinoy [The stress state of a half-plane with an absolutely rigid inclusion and a crack at the boundary]. Izv. NAN Armenii. Mekhanika − Proceedings of National Academy of Sciences of Armenia. Mechanics, vol. 68, no. 1, pp. 25–36 (in Russian).
  6. Popov, V. H. (2013). Napruzhenyi stan navkolo dvokh trishchyn, shcho vykhodiat z odniiei tochky pry harmonichnykh kolyvanniakh povzdovzhnoho zsuvu [The stressed state of two cracks emerging from one point with harmonic fluctuations of longitudinal displacement]. Visn. Kyiv. nats. un-tu im. Tarasa Shevchenka. Seria: Fizyko-mat. nauky – Bulletin of Taras Shevshenko Kyiv National University. Seria: Physical and Mathematical Sciences, iss. 3. pp. 205–208 (in Ukrainian).
  7. Popov, V. H. (2017). A crack in the form of a three-link broken line under the action of longitudinal shear waves. Journal of Mathematical Sciences, vol. 222, iss. 2, pp. 143–154.
  8. Lytvyn, O. V. & Popov, V. H. (2019). Interaction of harmonic longitudinal shear waves with V-shaped inclusions. Journal of Mathematical Sciences, vol. 240, iss. 2, pp. 113–128.
  9. Popov, V. G. (1986). Difraktsiya uprugikh voln sdviga na vklyuchenii slozhnoy formy, raspolozhennoy v neogranichennoy uprugoy srede [Diffraction of elastic shear waves on the inclusion of a complex shape, located in an unlimited elastic medium]. Gidroaeromekhanika i teoriya uprugosti: Chislennyye i analiticheskiye metody resheniya zadach gidroaerodinamiki i teorii uprugostiHydroaeromechanics and theory of elasticity: Numerical and analytical methods for solving problems of fluid dynamics and theory of elasticity, pp. 121–127 (in Russian).
  10. Andreyev, A. R. (2005). Pryamoy chislennyy metod resheniya singulyarnykh integralnykh uravneniy pervogo roda s obobshchennymi yadrami [Direct numerical method for solving singular integral equations of the first kind with generalized kernels]. Izv. RAN. Mekhanika tverdogo tela − Mechanics of Solids, no. 1, pp. 126–146 (in Russian).
  11. Krylov, V. I. (1967). Priblizhennoye vychisleniye integralov [Approximate calculation of integrals]. Moscow: Nauka, 500 p. (in Russian).


Received 11 September 2018

Published 30 December 2018