  We have combined determinations of the epoch-dependent star formation rate
(SFR) function with relationships between SFR and radio (synchrotron and
free-free) emission to work out detailed predictions for the counts and the
redshift distributions of star-forming galaxies detected by planned Square
Kilometer Array (SKA) surveys. The evolving SFR function comes from recent
models fitting the far-infrared (FIR) to millimeter-wave luminosity functions
and the ultraviolet (UV) luminosity functions up to z=10, extended to take into
account additional UV survey data. We used very deep 1.4 GHz number counts from
the literature to check the relationship between SFR and synchrotron emission,
and the 95 GHz South Pole Telescope (SPT) counts of dusty galaxies to test the
relationship between SFR and free-free emission. We show that the SKA will
allow us to investigate the SFRs of galaxies down to few Msun/yr up to z=10,
thus extending by more than two orders of magnitude the high-z SFR functions
derived from Herschel surveys. SKA1-MID surveys, down to microJy levels, will
detect hundreds of strongly lensed galaxies per square degree; a substantial
fraction of them will show at least two images above the detection limits.
