  We investigate the morphology of thin discs and rings growing in the
circumferential direction. Recent analytical results suggest that this growth
produces symmetric excess cones (e-cones). We study the stability of such
solutions considering self-contact and bending stress. We show that, contrary
to what was assumed in previous analytical solutions, beyond a critical growth
factor, no symmetric \textit{e}-cone solution is energetically minimal any
more. Instead, we obtain skewed e-cone solutions having lower energy,
characterized by a skewness angle and repetitive spiral winding with increasing
growth. These results are generalized to discs with varying thickness and rings
with holes of different radii.
