Simple and Broadly Applicable Definition of Shear Transformation Zones

Plastic deformation in amorphous solids is known to be carried by stress-induced localized rearrangements of a few tens of particles, accompanied by the conversion of elastic energy to heat. Despite their central role in determining how glasses yield and break, the search for a simple and generally applicable definition of the precursors of those plastic rearrangements—the so-called shear transformation zones (STZs)—is still ongoing. Here we present a simple definition of STZs—based solely on the harmonic approximation of a glass’s energy. We explain why and demonstrate directly that our proposed definition of plasticity carriers in amorphous solids is more broadly applicable compared to anharmonic definitions put forward previously. Finally, we offer an open-source library that analyzes low-lying STZs in computer glasses and in laboratory materials such as dense colloidal suspensions for which the harmonic approximation is accessible. Our results constitute a physically motivated methodological advancement towards characterizing mechanical disorder in glasses, and understanding how they yield.

Figure 1

Pseudoharmonic modes represent STZs in glasses, ranging from ultrastable [$T_p=0.2$, panels (a)–(d)] to poorly-annealed [$T_p=0.7$, panels (e)–(h)]. (a) The squared frequencies of a destabilizing PHM $\mathit{\pi }$ (solid red line), cubic mode ${\mathit{\pi }}_3$ (solid purple line), and vibrational mode $\mathit{\psi }$ (solid black line) of an ultrastable glass of $N=10\mathrm{K}$ particles in two dimensions, subjected to athermal quasistatic shear (see details in the Supplemental Material [36]). The horizontal dashed line indicates the first shear wave frequency ${\omega }_{\mathrm{ph}}=2\pi c_s/L$, with $L$ and $c_s(T_p)$ denoting the box length and (glass-history-dependent) shear-wave speed, respectively. Panels (b)–(d) show $\mathit{\psi }$ (top row), $\mathit{\pi }$ and ${\mathit{\pi }}_3$ (2nd row) at the strains indicated in panel (a), corresponding to strain differences away from the instability of $\mathrm{\Delta }\gamma ={10}^{-1}$, ${10}^{-2}$, and ${10}^{-5}$, from left to right. These data show that the firstly activated STZ under shear is present in the as-cast glass, in the form of a PHM. Panels (e)–(h) are the same as (a)–(d), but measured in a poorly annealed glass of $N=40\mathrm{K}$ particles, with $\mathrm{\Delta }\gamma =8\times {10}^{-4},2\times {10}^{-4}$, and ${10}^{-5}$, from left to right.

Figure 2

Pseudoharmonic modes in various computer glasses: (a) a hard-disk glass, (b) a glass of Hertzian soft spheres, (c) a Stillinger-Weber tetrahedral glass with three-body interactions [49], and (d) a CuZr bulk metallic glass [57].

Figure 3

Pseudoharmonic modes (versus harmonic vibrational modes: (a) Scaled participation ratio $Ne$ as a function of frequency $\omega$, calculated in an ensemble of computer glasses in three dimensions [36]. The convergence of the energy [panel (b)] and structure [panel (c)] of PHMs to those of their ancestral harmonic modes. In every panel the dashed line indicates the lowest phonon frequency.

Figure 4

Residual plastic strength map of a well-annealed Lennard-Jones binary glass at zero strain $\gamma =0$ (details about the model can be found in Ref. [59]). Black circles indicate the loci of the 10 first plastic instabilities. PHMs are extracted from the as-cast configurations by mapping the critical mode at each plastic event.