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.
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