  The evolution of massive stars even on the main sequence is not yet well
understood. Due to the steep mass-luminosity relation, massive main sequence
stars become very luminous. This brings their envelopes very close to the
Eddington limit. We are analysing stellar evolutionary models in which the
Eddington limit is reached and exceeded, and explore the rich diversity of
physical phenomena which take place in their envelopes, and investigate their
observational consequences. We use the grids of detailed stellar models by
Brott et al. (2011) and Koehler et al. (2015), to investigate the envelope
properties of core hydrogen burning massive stars. We find that at the stellar
surface, the Eddington limit is almost never reached, even for stars up to 500
Msun. When an appropriate Eddington limit is defined locally in the stellar
envelope, most stars more massive than 40 Msun actually exceed this limit, in
particular in the partial ionization zones of iron, helium or hydrogen. While
most models adjust their structure such that the local Eddington limit is
exceeded at most by a few per cent, our most extreme models do so by a factor
of more than seven. We find that the local violation of the Eddington limit has
severe consequences for the envelope structure, as it leads to envelope
inflation, convection, density inversions and possibly to pulsations. We find
that all models with luminosities higher than 4*10^5 Lsun i.e. stars above 40
Msun show inflation, with a radius increase of up to a factor of about 40. We
find that the hot edge of the S Dor variability region coincides with a line
beyond which our models are inflated by more than a factor of two, indicating a
possible connection between S Dor variability and inflation. Furthermore, our
coolest models show highly inflated envelopes with masses of up to several
solar masses, and appear to be candidates to produce major LBV eruptions.
