  Molecular Dynamics (MD) computer simulations of a polydisperse soft-sphere
model under shear are presented. Starting point for these simulations are
deeply supercooled samples far below the critical temperature, $T_c$, of mode
coupling theory. These samples are fully equilibrated with the aid of the swap
Monte Carlo technique. For states below $T_c$, we identify a life time
$\tau_{\rm lt}$ that measures the time scale on which the system can be
considered as an amorphous solid. The temperature dependence of $\tau_{\rm lt}$
can be well described by an Arrhenius law. The existence of transient amorphous
solid states below $T_c$ is associated with the possibility of brittle
yielding, as manifested by a sharp stress drop in the stress-strain relation
and shear banding. We show that brittle yielding requires on the one hand low
shear rates and on the other hand, the time scale corresponding to the inverse
shear rate has to be smaller or of the order of $\tau_{\rm lt}$. Both
conditions can be only met for large life time $\tau_{\rm lt}$, i.e.~for states
far below $T_c$.
