Shear Yielding and Shear Jamming of Dense Hard Sphere Glasses

We investigate the response of dense hard sphere glasses to a shear strain in a wide range of pressures ranging from the glass transition to the infinite-pressure jamming point. The phase diagram in the density-shear strain plane is calculated analytically using the mean-field infinite-dimensional solution. We find that just above the glass transition, the glass generically yields at a finite shear strain. The yielding transition in the mean-field picture is a spinodal point in presence of disorder. At higher densities, instead, we find that the glass generically jams at a finite shear strain: the jamming transition prevents yielding. The shear yielding and shear jamming lines merge in a critical point, close to which the system yields at extremely large shear stress. Around this point, highly nontrivial yielding dynamics, characterized by system-spanning disordered fractures, is expected.

Figure 1

Inverse reduced pressure $d/p$ versus packing fraction $\widehat{\phi }=2^d\phi /d$ (both scaled to remain finite for $d\to \infty$) during a slow compression [44, 45]. The liquid EOS is $d/p=2/\widehat{\phi }$. The dynamical transition ${\widehat{\phi }}_d$ is marked by a black dot. We focus on a liquid slowly compressed up to ${\widehat{\phi }}_g=8$ (blue full circle). From that point on, the system is followed in a restricted equilibrium, confined to the glass state (full blue line). At high pressure, the glass state becomes marginally stable. Jamming is reached around ${\widehat{\phi }}_j\approx 10$. The thick lines indicate the specific glass we follow in this paper. Other glasses corresponding to different ${\widehat{\phi }}_g$ (different compression rates) are plotted with thinner lines.

Figure 2

Applying adiabatically a shear strain $\gamma$ on a glass prepared at equilibrium at ${\widehat{\phi }}_g=8$ and adiabatically compressed to $\widehat{\phi }\in [{\widehat{\phi }}_g,{\widehat{\phi }}_j]$. The black dots along the lines represent the Gardner transition. (a) Inverse reduced pressure $d/p\equiv {\widehat{p}}^{-1}$ vs $\gamma$. At lower $\widehat{\phi }$, the pressure is finite until the system yields at ${\gamma }_y(\widehat{\phi })$. At higher density, the pressure diverges at the shear jamming point ${\gamma }_j(\widehat{\phi })$. (b) Inverse of the reduced shear stress ${\widehat{\sigma }}^{-1}\equiv d/\sigma$ vs $\gamma$. The behavior is very similar to the pressure. At lower $\widehat{\phi }$, the stress overshoots before yielding. At higher $\widehat{\phi }$, the stress diverges at ${\gamma }_j$ without any overshoot. The inset shows the behavior of $\widehat{\sigma }$ vs $\gamma$ in log scale. (c) The glass MSD $\mathrm{\Delta }$ vs $\gamma$. At lower $\widehat{\phi }$, $\mathrm{\Delta }$ remains finite at yielding. At higher $\widehat{\phi }$, $\mathrm{\Delta }$ vanishes at shear jamming. (d) The MSD ${\mathrm{\Delta }}_r$ between the initial equilibrium configuration at ${\widehat{\phi }}_g$ and the one at $(\widehat{\phi },\gamma )$. At lower $\widehat{\phi }$, ${\mathrm{\Delta }}_r$ remains finite and displays a square-root singularity at yielding, such that $\mathit{d}{\mathrm{\Delta }}_r/\mathit{d}\gamma \to \infty$ for $\gamma \to {\gamma }_y$. At higher density, ${\mathrm{\Delta }}_r$ remains finite at shear jamming with no singularity.

Figure 3

Phase diagram of the glass prepared in equilibrium at ${\widehat{\phi }}_g=8$ and followed adiabatically at density $\widehat{\phi }>{\widehat{\phi }}_g$ and shear strain $\gamma$. The shear jamming line ${\gamma }_j(\widehat{\phi })$ and the shear yielding line ${\gamma }_y(\widehat{\phi })$ are plotted. The two lines merge at a critical point $({\widehat{\phi }}_c,{\gamma }_c)$. At this special point, yielding happens at infinite pressure or strain. For $\widehat{\phi }\lesssim {\widehat{\phi }}_c$, yielding happens at $\gamma \sim {\gamma }_c$ with extremely large pressure or strain. The vertical dotted line is the isochoric line of a glass prepared at $\widehat{\phi }>{\widehat{\phi }}_c$ and then strained at fixed volume until it shear jams. The dotted-dashed line represents the isobaric line of a glass prepared at the same initial packing fraction, but then strained at fixed pressure. The shear jamming transition is thus avoided and the glass yields for sufficiently high strains.