The Stress State in an Elastic Disk Due to a Temperature Variation in One Sector

Consider a linear elastic infinite disk, a sector of which, of arbitrary opening angle $2β$⁠, is subjected to a uniform temperature increase $ΔT$ with respect to the complementary portion. An analytical solution is sought, imagining that the disk is first cut along the interface with the heated sector, now free to expand; then the two parts are re-joined and the thermal mismatch is annihilated by arrays of glide dislocations, distributed along the interfaces. A sequence of approximate solutions is found as the length of the arrays of reconciling dislocations is increased, characterized by a logarithmic stress singularity at the sector tip. However, modulo the particular case $2β=π$⁠, the stress grows unboundedly when the length of the dislocation arrays tends to infinity. This is in agreement with the predictions from dimensional analysis, because for the infinite disk problem there is no internal length scale. If the disk is finite in size, its radius $R$ represents an additional length scale enriching the class of solutions, but the analytical treatment results much more complicated. Therefore, we propose to correlate the solution of this problem with that of an infinite disk for which the length of the arrays of reconciling dislocations is finite and depends upon $R$⁠. An excellent agreement with numerical experiments in abaqus is thus found. An approach of this type can be useful in many engineering problems for which the limit condition of infinite body, though leading to analytic simplifications, could imply spurious results.