  Variable-amplitude oscillatory shear tests are emerging as powerful tools to
investigate and quantify the nonlinear rheology of amorphous solids, complex
fluids and biological materials. Quite a few recent experimental and atomistic
simulation studies demonstrated that at low shear amplitudes, an amorphous
solid settles into an amplitude- and initial conditions-dependent dissipative
limit cycle, in which back-and-forth localized particle rearrangements
periodically bring the system to the same state. At sufficiently large shear
amplitudes, the amorphous system loses memory of the initial conditions,
exhibits chaotic particle motions accompanied by diffusive behavior and settles
into a stochastic steady-state. The two regimes are separated by a transition
amplitude, possibly characterized by some critical-like features. Here we argue
that these observations support some of the physical assumptions embodied in
the nonequilibrium thermodynamic, internal-variables based,
Shear-Transformation-Zone model of amorphous visco-plasticity; most notably
that "flow defects" in amorphous solids are characterized by internal states
between which they can make transitions, and that structural evolution is
driven by dissipation associated with plastic deformation. We present a rather
extensive theoretical analysis of the thermodynamic Shear-Transformation-Zone
model for a variable-amplitude oscillatory shear protocol, highlighting its
success in accounting for various experimental and simulational observations,
as well as its limitations. Our results offer a continuum-level theoretical
framework for interpreting the variable-amplitude oscillatory shear response of
amorphous solids and may promote additional developments.
