Simulation of shear strain at arbitrary angles as a probe of packing instabilities

Abstract

Disordered solids distort and fail as particle contacts become unstable and rearrange under sufficiently large shear strains. Such instabilities can occur at different locations and, because of their proximity, can interact with one another. We develop a tool for simulations with periodic boundary conditions that allows strains to be applied at a continuously variable angle, $θ$. We show that instabilities can persist over a broad angular ranges of applied shear to form instability lines in phase space. By applying strain at different $θ$, we examine the correlations between the instabilities encountered at different angles and different positions in the sample. We find instabilities that pass through one another, others that change continuously as the angle is varied, and yet others that end by smoothly decreasing their magnitudes to zero as the instability fades away. Examining hysterons, i.e., instabilities that undo themselves upon reversing the direction of shear, we find that as $θ$ is varied towards the point where the instability disappears, the separation between the forward and backward instabilities shrinks to zero so as to produce an enhanced number of very small hysterons.

Figures (4)

• Figure 1: (a) Definition of lattice vectors in $d=2$, which define box shape and corresponding periodicity. (b) Pure and (c) simple shear along the chosen $\theta=0^{\circ}$ for each case. For simple shear this corresponds to Lees-Edwards boundary conditions lees1972computer. (d) Strain $\gamma_{1} (\theta)$ at which the first rearrangement event is encountered when shear is imposed along different directions $\theta$ ($N=1021$, $d=2$, $\Phi=0.87$). (e) Normalized correlation $C(\theta_1, \theta_2)$ between particle motions of instabilities encountered upon shear in different directions $\theta_1$ and $\theta_2$, with red indicating a high level of correlation and blue indicating negative correlation. (f) The same data as shown in (d) plotted in polar coordinates; color of data points indicates groupings determined from (e) with a threshold of $C>0.9$. All rearrangements out to $\gamma=0.002$ are included.
• Figure 2: (a) Part of a polar plot showing an example of two rearrangement lines ($C=0.076$ when comparing the circled rearrangements) that cross ($N=13$, $d=2$, $\Phi=0.9$). The finite step size causes the rearrangements to occur in the same strain step for two of the angles tested; this is not well-correlated with either the blue or red rearrangement individually (hence its distinct color) but is very well correlated ($C=0.99$) with the sum of the two crossing rearrangements. (b,c) Real space image of rearrangement events highlighted in (a) with arrows indicating the direction and relative magnitudes of displacements.
• Figure 3: (a) An example ($N=11$, $d=2$, $\Phi=0.9$) of a rearrangement that appears to end abruptly (teal rearrangement, end occurs around $\theta = 170^\circ$). The orange path shows a trajectory that starts at the origin, goes radially outward crossing the teal line of rearrangements, azimuthally to $\theta=180^{\circ}$, and then returns to the origin. (b) Strain values of rearrangements encountered in both directions $\gamma^+$ and $\gamma^-$ (top) and the difference between them $\gamma_h$ (bottom) versus angle. (c) Energy drop, $\Delta E$, (top) and particle displacements, $\Delta r$, (bottom) versus hysteresis $\gamma_h$ for each point shown in (b) (teal circles) as well as the equivalents for two other hysterons measured (grey squares: $N=11$, $\Phi=0.9$; red triangles: $N=17$, $\Phi=0.87$). Average of the three best-fit exponents are shown as guides to the eye ($\Delta E$: $1.55 \pm 0.04$; $\Delta r$: $0.57 \pm 0.05$).
• Figure 4: Lines of instabilities for various system sizes: (a) $N=7$, (b) $N=127$, and (c) $N=1021$ for $d=2$, $\Phi=0.87$, and polydispersity $P=0.2$. (d) Similar behavior is found in three dimensions with $N=13$, $d=3$, $\Phi=0.75$ and $P=0.02$.