In this paper, we have studied the intensification of thermal stresses due to a transverse flow of heat in a fiber of prolate spheroidal shape embedded in an unbounded medium. The analysis shows that for a stiff fiber the shearing stress at the tip becomes unbounded as ξ−1.61 as the shape ratio ξ becomes small. The analysis has been extended to the case of a cavity and the results show that, as the cavity becomes a “needle-shaped” crack, there is no intensification of stress whatsoever. This result is at wide variance with either a two-dimensional or a “penny-shaped” crack.

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