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Axisymmetric Stressed State of Uniformly Layered Space with Periodic Systems of Internal Disc-Shaped Cracks and Inclusions

Authors: Hakobyan V.N., Amirjanyan H.A., Kazakov K.Ye. Published: 22.04.2020
Published in issue: #2(89)/2020  
DOI: 10.18698/1812-3368-2020-2-25-40

 
Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics  
Keywords: mixed problem, disk-shaped crack, circular rigid inclusion

Using the Hankel integral transform, we construct discontinuous solutions for the problem of the axisymmetric stress state of a piecewise homogeneous, uniformly layered space, obtained by alternately connecting two heterogeneous layers of the same thickness. The space on the middle planes of the first heterogeneous layer contains a periodic system of circular disc-shaped parallel cracks, and on the middle planes of the second layer has a periodic system of circular disc-shaped parallel rigid inclusions. The determining system of equations is obtained in the form of a system of integral equations with kernels of the Weber --- Sonin type with respect to the crack extension and tangent contact stresses acting on the facial surfaces of rigid inclusions. With the help of rotation operators, the resulting determining system of equations is reduced to a system of integral equations of the second kind of Fredholm type. The equation solution is constructed by the method of mechanical quadratures. A numerical analysis was carried out and regularities were revealed in the variation of the intensity factors of rupture stresses, crack extension and contact stresses under the inclusions depending on the physical and mechanical and geometrical characteristics of the problem

The study was carried out with financial support of MESCS RA SC and RFBR as part of a joint research project SCS 18RF061 and 18-51-05012

References

[1] Erdogan F. Stress distribution in bonded dissimilar materials containing circular or ring-shaped cavities. J. Appl. Mech., 1965, vol. 32, iss. 4, pp. 829--836. DOI: https://doi.org/10.1115/1.3627323

[2] Willis J.R. The penny-shaped crack on an interface. Q. J. Mech. Appl. Math., 1972, vol. 25, iss. 3, pp. 367--385. DOI: https://doi.org/10.1093/qjmam/25.3.367

[3] Kassir M.K., Bregman A.M. The stress-intensity factor for a penny-shaped crack between two dissimilar materials. J. Appl. Mech., 1972, vol. 39, iss. 1, pp. 308--310. DOI: https://doi.org/10.1115/1.3422648

[4] Popov G.Ya. O kontsentratsii uprugikh napryazheniy vozle tonkogo otsloivshegosya vklyucheniya. Sovremennye problemy mekhaniki i aviatsii [On concentration of elastic stresses near delaminated thin inclusion. In: Modern problems of mechanics and aviation]. Moscow, Mashinostroenie Publ., 1980, pp. 156--162.

[5] Hakobyan V.N., Dashtoyan L.L. On some axis-symmetrical problem for compound space weakened by semi-infinite crack. Izv. NAN RA, Mekhanika [Mechanics --- Proceedings of National Academy of Sciences of Armenia], 2006, vol. 59, no. 1, pp. 3--10 (in Russ.).

[6] Hakobyan V.N., Mirzoyan S.T., Dashtoyan L.L. Axisymmetric mixed boundary value problem for composite space with coin-shaped crack. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2015, no. 3 (60), pp. 31--46 (in Russ.). DOI: http://doi.org/10.18698/1812-3368-2015-3-31-46

[7] Popov G.Ya. Izbrannye trudy. T. 1 [Selected works. Vol. 1]. Odessa, VMV Publ., 2007.

[8] Hakobyan V.N. Smeshannye granichnye zadachi o vzaimodeystvii sploshnykh deformiruemykh tel s kontsentratorami napryazheniy razlichnykh tipov [Mixed boundary value problems on interaction of continuum deformable bodies with different types of stress concentrators]. Yerevan, Gitutyun NAN RA Publ., 2014.

[9] Hakobyan V.N. [Axisymmetric stress state of piecewise homogeneous layered space with parallel coin-shaped cracks]. Tr. XVIII mezhdunar. konf. "Sovremennye problemy mekhaniki sploshnoy sredy" [Proc. XVIII Int. Conf. Modern Problems of Continuum Mechanics]. Rostov-on-Don, YuFU Publ., 2016, pp. 35--39 (in Russ.).

[10] Hakobyan V.N., Hakobyan L.V., Dashtoyan L.L. Discontinuous solutions of axisymmetric elasticity theory for a piecewise homogeneous layered space with periodical interphase disk-shape defects. Mekhanika kompozitnykh materialov [Mechanics of Composite Materials], 2019, vol. 55, no. 1, pp. 13--29 (in Russ.).

[11] Hakobyan V.N., Dashtoyan L.L., Murashkin E.V. [The axissymmetrical stress state of piecewise layered space with periodical inner disk-shaped cracks]. Sb. tr. 9 mezhdunar. konf. "Problemy dinamiki vzaimodeystviya deformiruemykh sred" [Proc. 9th Int. Conf. Problems of Interaction Dynamics of Deformable Media]. Goris, NUACA Publ., 2018, pp. 29--33 (in Russ.).

[12] Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products]. Moscow, Fizmatgiz Publ., 1963.

[13] Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i ryady [Integrals and series]. Moscow, Nauka Publ., 1981.

[14] Amirjanyan A.A., Sahakyan A.V. Mechanical quadrature method as applied to singular integral equations with logarithmic singularity on the right-hand side. Comput. Math. and Math. Phys., 2017, vol. 57, iss. 8, pp. 1285--1293. DOI: https://doi.org/10.1134/S0965542517080036

[15] Sahakyan A.V., Amirjanyan A.A. Method of mechanical quadratures for solving singular integral equations of various types. J. Phys.: Conf. Ser., 2018, vol. 991, art. 012070. DOI: https://doi.org/10.1088/1742-6596/991/1/012070