Mechanical and Microscopic Properties of the Reversible Plastic Regime in a 2D Jammed Material

At the microscopic level, plastic flow of a jammed, disordered material consists of a series of particle rearrangements that cannot be reversed by subsequent deformation. An infinitesimal deformation of the same material has no rearrangements. Yet between these limits, there may be a self-organized plastic regime with rearrangements, but with no net change upon reversing a deformation. We measure the oscillatory response of a jammed interfacial material, and directly observe rearrangements that couple to bulk stress and dissipate energy, but do not always give rise to global irreversibility.

Figure 1

Material and apparatus. (a) View from above of bulk material: mutually repulsive polystyrene microspheres adsorbed at oil-water interface. (b) Interfacial stress rheometer apparatus. The interfacial material is pinned on glass walls; a needle is embedded in the material between them, and is magnetically forced. Velocity profile is sketched.

Figure 2

Local deformation in a plastic event. (a),(b) Total (i.e., peak to peak, blue) and irreversible (i.e., stroboscopic, overlaid in red) $D_{\min }^2$ for cycles (a) 8 and (b) 20 of shear at ${\gamma }_0=0.020$, showing clusters of nonaffine deformation. One reversible cluster in (b) is boxed, and shown in (c)–(f). The magnetic needle is at the top of the image; the fixed wall is at the bottom. (c) Detail of a reversible cluster, showing the $D_{\min }^2$ of individual particles. Color scale is the same as in (a),(b). (d) Local relative displacement of particles in (c) at the minimum (red) and subsequent maximum (blue open circles) of $\gamma$, subtracting motion of neighbors within $10a$. (e) Streamlines computed from displacements (subtracting motion of neighbors within $40a$); square outline is region of (c),(d),(f). The hyperbolic character of the displacements is evident in the far field. (f) Micrograph with particle centers (small dots) and the centroid of the 4 particles in each T1 rearrangement (large dots) marked. (g),(h) Sequence illustrating a T1 rearrangement. Particles 3 and 4 begin as nearest neighbors but are separated in (h).

Figure 3

Rearrangements in a portion of material at ${\gamma }_0=0.02$ during transient (cycle 15; chosen to obtain more events than in steady state). Each particle is shown as a solid dot with size representing the extent of local crystalline ordering $|{\psi }_6|$; minimum and maximum size signify $|{\psi }_6|\simeq 0.1$ and 1. Color is solely to show differences in lattice director. The centroids of total T1 events [see Fig. 2] are shown as large open circles. Inset: Histograms of $|{\psi }_6|$ in reversible steady state (cycle 20). Curve: all particles. Shaded bars: particles involved in T1 events (555 out of $5.6\times {10}^4$). Dot positions and ${\psi }_6$ are for $\gamma \simeq ⟨\gamma ⟩$.

Figure 4

Hysteresis of rearrangements. (a) Global strain $\gamma$ at which individual particles rearrange ($D_{\min }^2\ge 0.015$) and reverse, in steady state for ${\gamma }_0=0.01$ ($+$) and ${\gamma }_0=0.02$ (opened circle). Shaded regions show limits of $\gamma$. The farther an event falls from the diagonal line, the more hysteretic it is. (b) Histogram of hysteresis at ${\gamma }_0=0.04$ (open, black curve), 0.02 (shaded, red), 0.01 (solid, blue). (c) Particle trajectories break time-reversal symmetry. A rearranging portion of the system is shown for one cycle in the steady state (${\gamma }_0=0.02$). Colors distinguish the particles. The effect of microscope vibration is reduced by subtracting average $y$ (here, vertical) motion of the entire visible system. (d) Number of particles $N_{\text{pl}}$ in plastic rearrangements in steady state, as a function of ${\gamma }_0$. Points show the number (out of $5.6\times {10}^4$) with total ($+$) or irreversible (filled circle) $D_{\min }^2\ge 0.015$. Hysteretic particles (opened circle) have ${\gamma }_{\text{on}}-{\gamma }_{\text{off}}$ exceeding the largest change in $\gamma$ between video frames. (e) Oscillatory rheology. (triangle): Estimated enhancement of $G^{\prime \prime }$ above zero-plasticity level (dashed line), based on the microstructure (see text).