The linear stability of a single component axi-symmetrical solid in contact with its melt is theoretically investigated with respect to the development of longitudinal sinusoidal fluctuations of wavelengths lambda < 2 pi R-e, with R-e its radius, in the solidification and melting regime. In the solidification regime, the applied stress adds to the temperature gradient to favor the development of the surface fluctuations. In the melting regime where the temperature gradient has a smoothing effect, the applied stress is the main destabilizing source for the cylinder that has been found to undergo morphological instability below a critical radius. (C) 2014 Elsevier B.V. All rights reserved.