Shear-transformation-zone theory of yielding in athermal amorphous materials

Yielding transitions in athermal amorphous materials undergoing steady-state shear flow resemble critical phenomena. Historically, they have been described by the Herschel-Bulkley rheological formula, which implies singular behaviors at yield points. In this paper, I examine this class of phenomena using an elementary version of the thermodynamic shear-transformation-zone (STZ) theory, focusing on the role of the effective disorder temperature, and paying special attention to scaling and dimensional arguments. I find a wide variety of Herschel-Bulkley-like rheologies but, for fundamental reasons not specific to the STZ theory, conclude that the yielding transition is not truly critical. In particular, for realistic many-body models with short-range interactions, there is a correlation length that grows rapidly but ultimately saturates near the yield point.

Figure 1

Log-log plot of stress $\sigma$ in units of the yield stress ${\sigma }_y$ as a function of dimensionless shear rate $q$. The data points are from HL with rescaled values of $q$ as described in the text.

Figure 2

Semilogarithmic plot of the steady-state effective temperature $\widehat{\chi }$ as a function of dimensionless shear rate $q$. The data points are from HL with rescaled values of $q$ as described in the text.

Figure 3

Stress $\sigma$ in units of the yield stress ${\sigma }_y$ as a function of dimensionless shear rate $q$. The data points are from HL with rescaled values of $q$ as described in the text. The dashed line is the approximation given in Eq. (4.2) displaced upward by one unit of the yield stress.

Figure 4

Stress $\sigma$ in units of the yield stress ${\sigma }_y$ as a function of dimensionless shear rate $q$ for five different values of the dimensionless activation energy $A$. From bottom to top, $A=2.0,1.33,1.1,1.0,\mathrm{and}0.9$. The corresponding values of the HB exponent are $\beta =1.0,0.5,0.18,0,\mathrm{and}-0.22$.

Figure 5

Correlation length $\xi /a$ as a function of dimensionless shear rate $q$. The two sets of experimental data (blue triangles and red squares) are for the two larger forcings shown in Fig. 5 of Jop et al. [26], and the theoretical scales of length $a$ and shear rate $q$ are chosen here to be consistent with that data.