  As if it has become a consensus that the slope of the granular heap is
straight, and from that, an angle of repose is defined. However, closer
inspection shows that the slopes are not truly straight, instead, more often
they have convex shapes, although with small convexity. Similar curvature is
observed on the sand dunes' surfaces. We derive equations together with
experiments on 10 types of granular materials to show that, in general, the
slope of the granular heap is convex while the straight slope is a critical
state that rarely occurs. The angle of repose is allowed to vary within a
narrow range around a critical value. If the angle of repose is slightly
greater (smaller) than the critical value, the slope is convex (concave). The
model also answers why the surfaces of granular heaps, including sand dunes,
are more often convex rather than concave or straight, and advocate the
simultaneous estimation of the coefficient of friction on the grain and the
type of packing in the heap. The results might open new understandings related
to granular material properties as well as sand transport and geological
records in several deserts.
