  Let $k$ be an algebraically closed field of positive characteristic $p$. We
first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and
then we study stratifications defined by inner isomorphism classes of these
$D$-truncations. This generalizes previous works of Kraft, Ekedahl, Oort,
Moonen, and Wedhorn. As a main tool we introduce and study Bruhat
$F$-decompositions; they generalize the combined form of Steinberg theorem and
of classical Bruhat decompositions for reductive groups over $k$.
