  In the present work we focus on the morphology of well abraded, almost
perfectly ellipsoidal natural pebbles. Flat, oblate and prolate ellipsoidal
pebbles are expected to be characterized by qualitatively different shape
evolution due to being unequally prone to specific rolling and sliding motions
in natural environments. The three typical motions are considered as sliding on
the flattest side and rolling around the longest and shortest axes. First, we
assume by mechanical considerations that each of these motions alone affects
the shortening of only one or two principal axes, then we formulate a
mathematical expression associating the willingness as a probability to each
motion of a given ellipsoid with arbitrary axis ratios. As a consequence, any
of the motions is capable of suppressing the other two when the pebble is
either ideally flat, oblate or prolate. We present through a numerical analysis
that our model predicts a self-excited shape evolution and curiously suggests
the existence of an unstable equilibrium (repeller) in the vicinity of the
naturally dominant average axis ratios found in the literature. Due to the
self-excited evolution, pebbles diverging away from the proximity of the
repeller are expected to rapidly become sufficiently thin or flat to easily
break up into smaller fragments, which are no longer considered relevant in our
investigation. Ultimately, our model outlines the possibility that the dominant
axis ratios might be due to a well-balanced mixture of the motions and the
corresponding slow evolution around the repeller. Besides the theoretical
predictions, we present experimental investigations of the dominant motions of
various 3D-printed ellipsoidal shapes over an inclined planar surface and
compare the results with the predictions of our model.
