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Boundary problems for a circular insert bonded partially to the interior of an infinite medium of another material under an uniform heat flow are formulated and solved in closed form. The unbonded portions of the interface may be regarded as circular arcs of discontinuities or curved cracks. By application of the complex function theory dealing with sectionally holomorphic functions, the present problem is reduced to the solution of the problem of linear relationship or Hilbert problem. Exact solution is given for an example of a single circular‐arc crack. It is found that the temperature gradients or heat flux near the tips of a curved crack possess the characteristic inverse square‐root singularity in terms of the radial distance away from the crack tip which is the same as those obtained for a straight crack between dissimilar materials. Due to this singular behavior, the heat flux intensity factor is introduced to measure the thermal energy intensification cumulated in the vicinity of the crack tip. Numerical results for the temperature and heat flux intensity factor are provided in graphic forms. It is shown that the material of a circular insert having a lower heat conductivity would make the heat flux intensity factor smaller. Consequently, the thermal energy intensification is diminished.