Thermodynamic interpretation of soft glassy rheology models

Mesoscopic models play an important role in our understanding of the deformation and flow of amorphous materials. One such description, based on the shear transformation zone theory, has recently been reformulated within a nonequilibrium thermodynamics framework and found to be consistent with it. We show here that a similar interpretation can be made for the soft glassy rheology (SGR) model. Conceptually this means that the “noise temperature” $x$, proposed phenomenologically in the SGR model to control the dynamics of a set of slow mesoscopic degrees of freedom, can consistently be interpreted as their actual thermodynamic temperature. (Because such modes are slow to equilibrate, this generally does not coincide with the temperature of the fast degrees of freedom and/or heat bath.) If one chooses to make this interpretation, the thermodynamic framework significantly constrains extensions of the SGR approach to models in which $x$ is a dynamical variable. We assess in this light some such extensions recently proposed in the context of shear banding.