  A steady shear flow can drive supercooled liquids into a non-equilibrium
state. Using molecular dynamics simulations under steady shear flow
superimposed with oscillatory shear strain for a probe, non-equilibrium
mechanical responses are studied for a model supercooled liquid composed of
binary soft spheres. We found that even in the strongly sheared situation, the
supercooled liquid exhibits surprisingly isotropic responses to oscillating
shear strains applied in three different components of the strain tensor. Based
on this isotropic feature, we successfully constructed a simple two-mode
Maxwell model that can capture the key features of the storage and loss moduli,
even for highly non-equilibrium state. Furthermore, we examined the correlation
functions of the shear stress fluctuations, which also exhibit isotropic
relaxation behaviors in the sheared non-equilibrium situation. In contrast to
the isotropic features, the supercooled liquid additionally demonstrates
anisotropies in both its responses and its correlations to the shear stress
fluctuations. Using the constitutive equation (a two-mode Maxwell model), we
demonstrated that the anisotropic responses are caused by the coupling between
the oscillating strain and the driving shear flow. We measured the magnitude of
this violation in terms of the effective temperature. It was demonstrated that
the effective temperature is notably different between different components,
which indicates that a simple scalar mapping, such as the concept of an
effective temperature, oversimplifies the true nature of supercooled liquids
under shear flow. An understanding of the mechanism of isotropies and
anisotropies in the responses and fluctuations will lead to a better
appreciation of these violations of the FDT, as well as certain consequent
modifications to the concept of an effective temperature.
