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.
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