  We show that in monomeric supercooled liquids and glasses that are
plastically flowing at a constant shear stress $\sigma$ while being deformed
with strain rate $\dot{\epsilon}$, the microscopic structural relaxation time
$\tau_{\rm str}$ is given by the universal relation
$\sigma/G_\infty\dot{\epsilon}$ with $G_\infty$ a modulus. This equality holds
for all rheological regimes from temperatures above the glass transition all
the way to the athermal limit, and arises from the competing effects of elastic
loading and viscous dissipation. In macromolecular (polymeric) glasses,
however, the stress decouples from this relaxation time and $\tau_{\rm str}$ is
in fact further reduced even though $\sigma$ rises during glassy strain
hardening. We develop expressions to capture both effects and thus provide a
framework for analyzing mobility measurements in glassy materials.
