  Glassy solids may undergo a fluidization (yielding) transition upon
deformation whereby the material starts to flow plastically. It has been a
matter of debate whether this process is controlled by a specific time scale,
from among different competing relaxation/kinetic processes. Here, two
constitutive models of cage relaxation are examined within the microscopic
model of nonaffine elasto-plasticity. One (widely used) constitutive model
implies that the overall relaxation rate is dominated by the fastest between
the structural ($\alpha$) relaxation rate and the shear-induced relaxation
rate. A different model is formulated here which, instead, assumes that the
slowest (global) relaxation process controls the overall relaxation. We show
that the first model is not compatible with the existence of finite elastic
shear modulus for quasistatic (low-frequency) deformation, while the second
model is able to describe all key features of deformation of `hard' glassy
solids, including the yielding transition, the nonaffine-to-affine plateau
crossover, and the rate-stiffening of the modulus. The proposed framework
provides an operational way to distinguish between `soft' glasses and `hard'
glasses based on the shear-rate dependence of the structural relaxation time.
