How mass assembly occurs in galaxies and which process(es) contribute to this activity are among the most highly debated questions in galaxy formation theories. This has motivated our survey MASSIV of 0.9<z<1.9 star-forming galaxies selected from the purely flux-limited VVDS redshift survey. For the first time, we derive the relations between galaxy size, mass, and internal velocity, and the baryonic Tully-Fisher relation, from a statistically representative sample of star-forming galaxies. We find a dynamical mass that agrees with those of rotating galaxies containing a gas fraction of ~20%, perfectly consistent with the content derived using the Kennicutt-Schmidt formulation and the expected evolution. Non-rotating galaxies have more compact sizes for their stellar component, and are less massive than rotators, but do not have statistically different sizes for their gas-component. We measure a marginal evolution in the size-stellar mass and size-velocity relations in which discs become evenly smaller with cosmic time at fixed stellar mass or velocity, and are less massive at a given velocity than in the local Universe. The scatter in the Tully-Fisher relation is smaller when we introduce the S05 index, which we interpret as evidence of an increase in the contribution to galactic kinematics of turbulent motions with cosmic time. We report a persistently large scatter for rotators in our relations, that we suggest is intrinsic, and possibly caused by complex physical mechanism(s) at work in our stellar mass/luminosity regime and redshift range. Our results consistently point towards a mild, net evolution of these relations, comparable to those predicted by cosmological simulations of disc formation for at least 8Gyr and a dark halo strongly coupled with galactic spectrophotometric properties.
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