  (shortened) It has been proposed that the envelopes of luminous stars may be
subject to substantial radius inflation. The inflation effect has been
discussed in relation to the radius problem of WR stars, but has yet failed to
explain the large observed radii of Galactic WR stars. We wish to obtain a
physical perspective of the inflation effect, and study the consequences for
the radii of WR stars, and LBVs. For WR stars the observed radii are up to an
order of magnitude larger than predicted by theory, whilst S Doradus-type LBVs
are subject to humongous radius variations, which remain as yet ill-explained.
We use a dual approach to investigate the envelope inflation, based on
numerical models for stars near the Eddington limit, and a new analytic
formalism to describe the effect. An additional new aspect is that we take the
effect of density inhomogeneities (clumping) within the outer stellar envelopes
into account. Due to the effect of clumping we are able to bring the observed
WR radii in agreement with theory. Based on our new formalism, we find that the
radial inflation is a function of a dimensionless parameter W, which largely
depends on the topology of the Fe-opacity peak, i.e., on material properties.
For W>1, we discover an instability limit, for which the stellar envelope
becomes gravitationally unbound, i.e. there no longer exists a static solution.
Within this framework we are also able to explain the S Doradus-type
instabilities for LBVs like AG Car, with a possible triggering due to changes
in stellar rotation. The stellar effective temperatures in the upper HR diagram
are potentially strongly affected by the inflation effect. This may have
particularly strong effects on the evolved massive LBV and WR stars just prior
to their final collapse, as the progenitors of SNe Ibc, SNe II, and long GRBs.
