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The literature is rich in analysis and results related to thermallypulsing-asymptotic giant branch (TP-AGB) stars, but the problem of theinstabilities that arise and cause the divergence of models during the latestages of their evolution is rarely addressed. We investigate the physicalconditions, causes and consequences of the interruption in the calculations ofmassive AGB stars in the late thermally-pulsing AGB phase. We have thoroughlyanalysed the physical structure of a solar metallicity 8.5 solar mass star anddescribed 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 ofFe-group elements lead to the accumulation of an energy excess, to thedeparture of thermal equilibrium conditions at the base of the convectiveenvelope and, eventually, to the divergence of the computed models. For the 8.5solar mass case we present in this work the divergence occurs when the envelopemass is about 2 solar mass. The remaining envelope masses range betweensomewhat less than 1 and more than 2 solar mass for stars with initial massesbetween 7 and 10 solar mass and, therefore, our results are relevant for theevolution and yields of super-AGB stars. If the envelope is ejected as aconsequence of the instability we are considering, the occurrence ofelectron-capture supernovae would be avoided at solar metallicity.

The end of super AGB and massive AGB stars I. The instabilities that determine the final mass of AGB stars