|Journal||Journal of Mechanical Engineering – Problemy Mashynobuduvannia|
|Publisher||A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
|ISSN||0131-2928 (Print), 2411-0779 (Online)|
|Issue||Vol. 22, no. 1, 2019 (March)|
|Cited by||J. of Mech. Eng., 2019, vol. 22, no. 1, pp. 16-24|
This paper solves the problem of determining the stress state near cracks in an infinite hollow cylinder of arbitrary cross section during longitudinal shear oscillations. We propose an approach that allows us to separately satisfy conditions both on the cracks and boundaries of a cylinder. The problem reduces to the equations of motion in a flat domain with the defects bounded by arbitrary smooth closed curves under anti-plane deformation conditions. The solution scheme is based on the use of discontinuous solutions to the equations of motion of an elastic medium with displacement jumps on the surfaces of defects. Displacements in a cylinder with defects are represented both as a sum of discontinuous solutions constructed for each defect and an unknown specific function ensuring that the conditions of a harmonic load on the body boundaries are met. This function is sought as a linear combination of linearly independent solutions to the equations of the theory of elasticity in the frequency domain with unknown coefficients. The constructed representation makes it possible to separately satisfy the boundary conditions on the surfaces of defects, which results in a set of systems of integral equations that differ only in their right-hand sides and do not depend on the body boundary shape. The resulting systems of integral equations can be solved by the method of mechanical quadratures. After that, the conditions on the boundaries of the cylindrical body are satisfied, from which the unknown coefficients of the introduced specific function are determined by a collocation method. Using the approach proposed, the stress intensity factors in the vicinity of defects were calculated. With the help of those calculations, we investigated the effect of the frequency and location of the defects on the stress intensity coefficient values.
Keywords: hollow cylinder, harmonic oscillations, stress intensity factors, system of cracks.
Full text: Download in PDF
- Popov, V. G. (1995). Sravnitelnyy analiz difraktsionnykh poley pri prokhozhdenii uprugikh voln cherez defekty razlichnoy prirody [Comparative analysis of diffraction fields during the passage of elastic waves through defects of different nature]. RAN Mekhanika tverdogo tela – Mechanics of Solids, no. 4, pp. 99–109 (in Russian).
- Ang, D. & Knopoff, L. (1964). Diffraction of scalar elastic waves by a finite strip. Proc. Math. Sci. USA, vol. 51, no. 4, pp. 593–598. https://doi.org/10.1073/pnas.51.4.593
- Mykhas’kiv, V., Zhbadynskyi, I., & Zhang, (2010). Elastodynamic analysis of multiple crack problem in 3-D bi-materials by a BEM. Int. J. Num. Meth. Biomed. Eng., vol. 26, no. 12, pp. 1934–1946. https://doi.org/10.1002/cnm.1285
- Popov, V. G. (1999). Vzaimodeystviye ploskikh uprugikh voln s sistemami radialnykh defektov [Interaction of plane elastic waves with systems of radial defects]. RAN Mekhanika tverdogo tela – Mechanics of Solids, no. 4, pp. 118–129 (in Russian).
- Chirino, F. & Domingues, J. (1989). Dynamic analysis of cracks using boundary element method. Engineering Fracture Mechanics, vol. 34, no. 5–6, pp. 1051–1061. https://doi.org/10.1016/0013-7944(89)90266-X
- Bobylev, A. A. & Dobrova Yu. A. (2003). Primeneniye metoda granichnykh elementov k raschetu vynuzhdennykh kolebaniy uprugikh tel konechnykh razmerov s treshchinami [Application of the boundary element method to the calculation of forced vibrations of finite-sized elastic bodies with cracks]. Vestnik Khark. nats. un-ta – Bulletin of Kharkov National University, 590, iss. 1, pp. 49–54 (in Russian).
- Zhang, (2002). A 2D hypersingular time-domain traction BEM for transient elastodynamic crack analysis. Wave Motion, vol. 35, no. 1, pp. 17–40. https://doi.org/10.1016/S0165-2125(01)00081-6
- Poruchikov, V. B. (1986). Metody dinamicheskoy teorii uprugosti [Methods of the dynamic theory of elasticity]. Moscow: Nauka, 328 p. (in Russian).
- Popov, V. G. (1993). Sravneniye poley peremeshcheniy i napryazheniy pri difraktsii uprugikh voln sdviga na razlichnykh defektakh: treshchina i tonkoye zhestkoye vklyucheniye [Comparison of displacement fields and stresses in diffraction of elastic shear waves at various defects: crack and thin rigid inclusion]. Dinamicheskiye sistemy − Dynamical Systems, iss. 12, pp. 35–41 (in Russian).
- Vekua, N. (1948). Novyye metody resheniya ellipticheskikh uravneniy [New methods for solving elliptic equations]. Moscow: OGIZ, 296 p. (in Russian).
- Belotserkovskiy, S. M. & Lifanov, K. (1985). Chislennyye metody v singulyarnykh integralnykh uravneniyakh [Numerical methods in singular integral equations]. Moscow: Nauka, 253 p. (in Russian).
- Kyrylova, O. I. & Mykhaskiv, V. V. (2015). Ploska dynamichna zadacha dlia tsylindrychnoho tila dovilnoho pererizu z tonkym zhorstkym vkliuchenniam [A plane dynamic problem for a cylindrical body of arbitrary section with a thin hard inclusion]. Visnyk Kyivskoho natsionalnoho universytetu. Seriia: fizyko-matematychni nauky −Bulletin of Kyiv National University. Series: physical and mathematical sciences, no. 5, pp. 167−173 (in Ukrainian).
- Kyrylova, O. I. & Popov, V. H. (2017). Napruzhenyi stan u neskinchennomu tsylindri dovilnoho pererizu z tunelnoiu trishchynoiu pry kolyvanniakh v umovakh ploskoi deformatsii [Stressed state in an infinite cylinder of arbitrary cross section with a tunnel fracture under oscillations under conditions of flat deformation]. Visnyk Kyivskoho natsionalnoho universytetu. Seriia: fizyko-matematychni nauky −Bulletin of Kyiv National University. Series: physical and mathematical sciences, no. 3, pp. 71−74 (in Ukrainian).
Received 11 September 2018
Published 30 March 2019