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For a field $F$ and a connected reductive group $G$ defined over $F$, wedevelop a theory of Kempf-Rousseau-Hesselink unipotent $F$-strata in $G(F)$that should allow us to attack open problems in positive characteristic. As anapplication, we use this theory to establish the fine expansion of theunipotent contribution to the (non-twisted) trace formula over a global fieldof characteristic $p>0$. The unipotent $F$-strata play here the role of theunipotent geometric orbits in Arthur's work over a number field. The expansionin terms of products of local distributions is not discussed here; it will bethe subject of further work.

D\'eveloppement fin de la contribution unipotente \`a la formule des traces sur un corps global de caract\'eristique p>0, I