The present work is a collection of results relating to the application of the integral equations method to the case of prismatic bars subjected to pure torsion. The specific risk of fracture functions are independently obtained for volume and surface brittlenesses, and when the material of the bar simultaneously exhibits the two types of brittleness, both can be separated. In the cases of a square bar and a bar having a regular polygon cross section of n sides, a simple stress field is used. The particular cases of equilateral triangle, square, and circular cross section are explicitly indicated in the formulas. On the other hand, the cases of a round, a rectangular, and an elliptical bar are treated, knowing exactly the stress field acting in the bar’s cross section. The particular cases of a square, a thin rectangular, and a round bar are also studied. The purpose of this review is to emphasize that the specific risk of fracture function can be obtained without knowing previously its analytical form by solving an integral equation.

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