  Recently, the existence and properties of unbounded cavity modes, resulting
in extensive plastic deformation failure of two-dimensional sheets of amorphous
media, were discussed in the context of the athermal Shear-Transformation-Zones
(STZ) theory. These modes pertain to perfect circular symmetry of the cavity
and the stress conditions. In this paper we study the shape stability of the
expanding circular cavity against perturbations, in both the unbounded and the
bounded growth regimes (for the latter the unperturbed theory predicts no
catastrophic failure). Since the unperturbed reference state is time dependent,
the linear stability theory cannot be cast into standard time-independent
eigenvalue analysis. The main results of our study are: (i) sufficiently small
perturbations are stable, (ii) larger perturbations within the formal linear
decomposition may lead to an instability; this dependence on the magnitude of
the perturbations in the linear analysis is a result of the non-stationarity of
the growth, (iii) the stability of the circular cavity is particularly
sensitive to perturbations in the effective disorder temperature; in this
context we highlight the role of the rate sensitivity of the limiting value of
this effective temperature. Finally we point to the importance of the form of
the stress-dependence of the rate of STZ transitions. The present analysis
indicates the importance of nonlinear effects that were not taken into account
yet. Even more importantly, the analysis suggests that details of the
constitutive relations appearing in the theory can be constrained by the modes
of macroscopic failure in these amorphous systems.
