  In this paper we focus on the TP-AGB evolution of intermediate-mass stars
experiencing envelope burning (M=4-5Mo). The approach we have adopted is a
semi-analytical one as it combines analytical relationships derived from
complete models of TP-AGB stars with sole envelope models in which the physical
structure is calculated from the photosphere down to the core. Our efforts are
directed to analyse the effects produced by envelope burning, such as: i) the
energy contribution which may drive significant deviations from the standard
core mass-luminosity relationship; and ii) the changes in the surface chemical
composition due to nuclear burning via the CNO cycle. Evolutionary models for
stars with initial mass of 4.0, 4.5, 5.0 Mo and two choices of the initial
chemical composition ([Y=0.28,Z=0.02] and [Y=0.25,Z=0.008]) are calculated from
the first thermal pulse till the complete ejection of the envelope. We find
that massive TP-AGB stars can rapidly reach high luminosities (-6>Mbol>-7),
without exceeding, however, the classical limit to the AGB luminosity of
Mbol~-7.1 corresponding to the Chandrasekhar value of the core mass. No carbon
stars brighter than Mbol~-6.5 are predicted to form (the alternative of a
possible transition from M-star to C-star during the final pulses is also
explored), in agreement with observations which indicate that most of the very
luminous AGB stars are oxygen-rich. Finally, new chemical yields from stars in
the mass range 4-5 Mo are presented, so as to extend the sets of stellar yields
from low-mass stars already calculated by Marigo et al. (1996). For each CNO
element we give both the secondary and the primary components.
