Purpose. A two-dimensional mathematical model for the problem of elasticity theory in a three-component plate containing rectilinear crack due to the action of mechanical efforts is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks, we determined stress intensity factors (SIFs) at the tops of the crack, which are subsequently used to determine critical values of the tension. Therefore, the aim of present work is to determine the two-dimensional elastic state in plate containing an elastic two-component circular inclusion and crack under conditions of power load in the case of unidirectional tension of the plate perpendicular for the crack line. This makes it possible to determine the critical values of unidirectional tension in order to prevent crack growth, which will not allow the local destruction of the body. Methodology. The methods of studying two-dimensional elastic state body with crack as stress concentrators based on the function of complex variable method by which the problem of elasticity theory is reduced to singular integral equations (SIE) of the first and second kind, the numerical solution by the method of mechanical quadratures was obtained. Findings. In this paper two-dimensional mathematical model in the form of the system of two singular integral equations on closed contour (boundary of inclusion) and unclosed contour (crack) are obtained; numerical solutions of these integral equations were received by the method of mechanical quadratures; stress intensity factors at the tops of a crack are identify and explored to detect the effects of mechanical character. Graphical dependencies of SIFs, which characterize distribution of the intensity of stresses at the tops of a crack as function of elastic properties of inclusion and also as function of the distance between crack and inclusion are obtained. This makes it possible to analyze the intensity of stresses in the vicinity of a crack's tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of unidirectional tension of the plate perpendicular to the crack line at which the crack begins to grow and the body being locally destroyed. It is shown that the proper selection of elastic characteristics of the components of three-component plate can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIFs at the crack's tops. Originality. Scientific novelty lies in the fact that the solutions of the new two-dimensional problems of elasticity for a specified region (plate containing an elastic two- component circular inclusion and a rectilinear crack) under the action of unidirectional tension of the plate perpendicular to the crack line are obtained. Practical value. Practical value of the present work lies in the possibility of a more complete accounting of actual stressed-strained state in the piecewise-homogeneous elements of a structure with cracks that work under conditions of different mechanical loads. The results of specific studies that are given in the form of graphs could be used when designing rational operational modes of structural elements. In this case, the possibility for preventing the growth of a crack through the appropriate selection of composite's components with the corresponding mechanical characteristics is obtained.
[1] N. Gupta, G. Tagliavia, M. Porfiri, “Elastic interaction of interfacial spherical-cap cracks in hollow particle filled composites”, International Journal of Solids and Structures, vol. 48, issues 7–8, pp. 1141–1153, 2011. https://doi.org/10.1016/j.ijsolstr.2010.12.017
[2] S. N. G. Chu, “Elastic interaction between a screw dislocation and surface crack”, Journal of Applied Physics, vol. 53, issue 12, pp. 8678–8685, 1982. https://doi.org/10.1063/1.330465
[3] Z. Ming-Huan, T. Renji, “Interaction between crack and elastic inclusion”, Applied Mathematics and Mechanics, vol. 16, issue 4, pp. 307–318, 1995. https://doi.org/10.1007/BF02456943
[4] I. P. Shatskyi, T. M. Daliak, “Vzaiemodiya trishchyny z kolinearnoiu shchilynoiu za zghynu plastyny” [“Interaction of the crack with the collinear slit by bending the plate”], Visnyk Zaporizkoho natsionalnoho universytetu. Fizyko-matematychni nauky [Bulletin of Zaporizhzhya National University. Physical and mathematical sciences], vol. 1, pp. 211–218, 2015. [in Ukrainian].
[5] V. V. Mykhas’kiv, O. M. Khay, “Interaction between rigid-disc inclusion and penny-shaped crack under elastic time-harmonic wave incidence”, International Journal of Solids and Structures, vol. 46, issue 3–4,
pp. 602–616, 2009. https://doi.org/10.1016/j.ijsolstr.2008.09.005
[6] O. F. Kryvyy, “Interface circular inclusion under mixed conditions of interaction with a piecewise homogeneous transversally isotropic space”, Journal of Mathematical Sciences, vol. 184, issue 1, pp. 101–109, 2012. https://doi.org/10.1007/s10958-012-0856-6
[7] N. R. F. Elfakhakhre, N. M. A. Nik Long, Z. K. Eshkuvatov, “Stress intensity factor for multiple cracks in half plane elasticity”, AIP Conference Proceedings, vol. 1795, Article ID 020010, 2017. https://doi.org/10.1063/1.4972154
[8] V. Zeleniak, R. Martyniak, B. Slobodian, “Napruzhennia v spaianykh riznoridnykh pivploshchynakh z vkliuchenniam i trishchynoiu za dii roztiahu” [“Stress in soldered heterogeneous half-planes with inclusion and crack under the action of tension”], Visnyk Natsionalnoho universytetu «Lvivska politekhnika» [Bulletin of Lviv Polytechnic National University], vol. 625, pp. 54–58, 2008. [in Ukrainian].
[9] V. Zeleniak, B. Slobodian, “Modeliuvannia termopruzhnoho dvovymirnoho stanu dvokh spaianykh riznoridnykh pivploshchyn z vkliuchenniamy i trishchynamy” [“Modelling of thermoelastic two-dimensional state of bonded heterogeneous halfplanes with inclusions and cracks”], Fizyko-matematychne modeliuvannia ta informatsiyni tekhnolohiy [Physico-mathematical modeling and informational technologies], vol. 12, pp. 94–101, 2010. [in Ukrainian].
[10] M. P. Savruk, Dvumernye zadachi uprugosti dlya tel s treshchinami [Two-dimensional elasticity problems for bodies with cracks]. Kyiv, Ukraine: Naukova dumka Publ., 1981. [in Russian].
[11] V. V. Panasyuk, M. P. Savruk, A. P. Datsyshin, Raspredelenie napryazheniy okolo treshchin v plastinah i obolochkah [Distribution of stresses near cracks in plates and shells]. Kyiv, Ukraine: Naukova dumka Publ., 1976. [in Russian].