The two-dimensional stationary problems of heat conduction and thermoelasticity for a semi-infinite elastic body containing an inclusion and a crack are considered. For this purpose, mathematical models of these two-dimensional problems in the form of a system of singular integral equations (SIEs) of the first and the second kinds are constructed. The numerical solution of the system of integral equations in the case of a half plane containing an inclusion and thermally insulated crack due to local heating by a heat flux is obtained using the method of mechanical quadratures. We present graphical dependencies of stress intensity factors (SIFs), which characterize the distribution of intensity of stresses on the tops of a crack, on the elastic and thermoelastic characteristics of an inclusion and a matrix, as well as on a relative position of a crack and an inclusion. The obtained results are subsequently used to determine the critical values of a heat flux at which a crack starts to grow. This model is the development of known models of two-dimensional stationary problems of heat conduction and thermoelasticity for piecewise-homogeneous bodies with cracks.

- Sekine H. Thermal stress singularities at tips of a crack in a semi–infinite medium under uniform heat flow. Engineering Fracture Mechanics.
**7**(4), 713–729 (1975). - Sekine H. Thermal stresses near tips of an insulated line crack in a semi–infinite medium under uniform heat flow. Engineering Fracture Mechanics.
**9**(2), 499–507 (1977). - Tweed I., Lowe S. The thermoelastic problem for a half-plane with an internal line crack. International Journal of Engineering Science.
**17**(4), 357–363 (1979). - Konechnyj S., Evtushenko A., Zelenyak V. The effect of the shape of distribution of the friction heat flow on the stress-strain state of a semispace. Trenie i Iznos.
**23**, 115–119 (2002). - Matysiak S. J., Evtushenko A. A., Zelenyak V. M. Frictional heating of a half–space with cracks. I. Single or periodic system of subsurface cracks. Tribology International.
**32**(5), 237–242 (1999). - Zelenyak V. M., Kolyasa L. I. Thermoelastic state of a half plane with curvilinear crack under the conditions of local heating. Materials Science.
**52**, 315–322 (2016). - Konechny S., Evtushenko A., Zelenyak V. Heating of the semispace with edge cracks by friction. Trenie i Iznos.
**22**, 39–45 (2001). - Matysiak S., Evtushenko A., Zelenyak V. Heating of a half space containing an inclusion and a crack. Materials Science.
**40**, 467–474 (2004). - Hasebe N., Wang X., Saito T., Sheng W. Interaction between a rigid inclusion and a line crack under uniform heat flux. International Journal of Solids and Structures.
**44**(7–8), 2426–2441 (2007). - Kit G. S., Krivtsun M. G. Plane thermoelasticity problems for bodies with cracks. Kiev, Naukova dumka (1983), (in Russian).
- Kit H. S., Chernyak M. S. Stressed state of bodies with thermal cylindrical inclusions and cracks (plane deformation). Materials Science.
**46**, 315–324 (2010). - Chen H., Wang Q., Liu G., Sun J. Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method. International Journal of Mechanical Sciences.
**115**–**116**, 123–134 (2016). - Choi H. J. Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer. Acta Mechanica.
**225**, 2111–2131 (2014). - Savruk M. P. Two-dimensional elasticity problem for bodies with cracks. Kiev, Naukova dumka (1981), (in Russian).
- Erdogan F., Gupta G. D., Cook T. S. Numerical solution of singular integral equations. In: Sih G. C. (eds) Methods of analysis and solutions of crack problems. Mechanics of fracture, vol. 1. Springer, Dordrecht. 368–425 (1973).
- Podstrigach Ya. S., Burak Ya. Y., Hachkevych O. R., Chernyavskaya L. V. Thermoelasticity of electrically conductive bodies. Kiev, Naukova Dumka (1977), (in Russian).
- Panasyuk V. V., Savruk M. P., Datsyshin A. P. Stress distribution around cracks in plates and shells. Kiev, Naukova Dumka (1976), (in Russian).