*Purpose*. A two-dimensional mathematical model for the problem of elasticity theory on joined dissimilar elastic half-planes containing rectilinear crack under the action of mechanical efforts on the shores of a crack 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 (SIF) at the tops of the crack, which are subsequently used to determine critical values of the normal pressure on the shores of the crack. Therefore the aim of present work is to determine the two-dimensional elastic state in joined dissimilar half-endless plates containing a rectilinear randomly-oriented crack under conditions of power load on the shores of the crack. This will make it possible to determine critical values of mechanical load on the shores of a crack 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 stationary elasticity are reduced to a SIF of the first kind, a numerical solution which was obtained by the method of mechanical quadratures.

*Findings.* In this paper, two-dimensional mathematical model in the form of singular integral equations on the contours of cracks in order to determine perturbed power stresses due to the presence of cracks are obtained; numerical solutions to singular integral equations of the problem of elasticity theory for a specified region under the action of normally distributed pressure on the shores of the crack are found; stress intensity factors at the tops of a crack and to detect the effects of mechanical character are identify and explored. Graphical dependences of SIF, which characterize distribution of the intensity of stresses at the tops of a crack, on the angle of crack inclination and elastic characteristics of half-planes is 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 normal pressure on the shores of the crack at which the growth of the crack starts, as well as the local destruction of the body. It is shown that the proper selection of elastic characteristics of the components of joined dissimilar half-planes can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIF 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 ( joined dissimilar half-endless plates containing a rectilinear randomly-oriented crack ) under the action of normally distributed pressure on the shores of the crack 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 is obtained for preventing the growth of a crack through the appropriate selection of composite's components with the corresponding mechanical characteristics.

[1] 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].

[2] M. P. Savruk, V. M. Zelenyak, “Plane problem of thermal conductivity and thermal elasticity for two joined dissimilar half-planes with curved inclusions and cracks”, *Soviet Materials Science*, vol. 24, no. 2,

pp. 124–129, 1988. https://doi.org/10.1007/BF00736348

[3] V. Zeleniak, and 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].

[4] I. P. Shatskyi, T. M. Daliak, “Vzaiemodiya trishchyny z kolinearnoiu shchilynoiu za zghynu plastyny” [“Interaction of crack and collinear slot in plate bending”], *Visnyk Zaporizkoho natsionalnoho universytetu [Bulletin of Zaporizhzhіa National University]*, no. 1, pp. 211–218, 2015. [in Ukrainian].

[5] 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, no. 7–8, pp. 1141–1153, 2011. https://doi.org/10.1016/j.ijsolstr.2010.12.017

[6] S. N. G. Chu, “Elastic interaction between a screw dislocation and surface crack”, *Journal of Applied Physics*, vol. 53, no. 12, pp. 8678–8685, 1982. https://doi.org/10.1063/1.330465

[7] Z. Ming-huan, T. Ren-ji, “Interaction between crack and elastic inclusion”, *Applied Mathematics and Mechanics*, vol. 16, no. 4, pp. 307–318, 1995. https://doi.org/10.1007/BF02456943

[8] 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, no. 3–4,pp. 602–616, 2009. https://doi.org/10.1016/j.ijsolstr.2008.09.005

[9] O. F. Kryvyy, “Interface circular inclusion under mixed conditions of interaction with a piecewise homogeneous transversally isotropic space”, *Journal of Mathematical Sciences*, vol. 184, no. 1, pp. 101–109, 2012. https://doi.org/10.1007/s10958-012-0856-6

[10] N. R. F. Elfakhakhre, N. M. A. Nik long, Z. K. Eshkuvatov, “Stress intensity factor for multiple cracks in half plane elasticity” in *Proc. of 2nd International Conference and Workshop on Mathematical Analysis 2016 (ICWOMA2016)*, Langkawi, Malaysia, August 2–4, 2016, pp. 020010-1–020010-8.

[11] S. Konechnyj, A. Evtushenko, V. Zelenyak, “The effect of the shape of distribution of the friction heat flow on the stress-strain state of a semispace”, *Trenie i Iznos* *[Friction and Wear]*, vol. 23, no. 2, pp. 115–119, 2002.

[12] S. Konechny, A. Evtushenko, V. Zelenyak, “Heating of the semispace with edge cracks by friction”, *Trenie i Iznos [Friction and Wear]*, vol. 22, no. 1, pp. 39–45, 2001.

[13] 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].

[14] 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].