  The literature is rich in analysis and results related to thermally
pulsing-asymptotic giant branch (TP-AGB) stars, but the problem of the
instabilities that arise and cause the divergence of models during the late
stages of their evolution is rarely addressed. We investigate the physical
conditions, causes and consequences of the interruption in the calculations of
massive AGB stars in the late thermally-pulsing AGB phase. We have thoroughly
analysed the physical structure of a solar metallicity 8.5 solar mass star and
described the physical conditions at the base of the convective envelope (BCE)
just prior to divergence.
  We find that the local opacity maximum caused by M-shell electrons of
Fe-group elements lead to the accumulation of an energy excess, to the
departure of thermal equilibrium conditions at the base of the convective
envelope and, eventually, to the divergence of the computed models. For the 8.5
solar mass case we present in this work the divergence occurs when the envelope
mass is about 2 solar mass. The remaining envelope masses range between
somewhat less than 1 and more than 2 solar mass for stars with initial masses
between 7 and 10 solar mass and, therefore, our results are relevant for the
evolution and yields of super-AGB stars. If the envelope is ejected as a
consequence of the instability we are considering, the occurrence of
electron-capture supernovae would be avoided at solar metallicity.
