Networks and Hierarchies: How Amorphous Materials Learn to Remember

We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions joined by occasional one-way transitions. The highly connected regions include hierarchically organized hysteresis cycles and subcycles. At small to moderate strains this organization leads to near-perfect return point memory. The transitions in the network can be traced back to localized particle rearrangements (soft spots) that interact via Eshelby-type deformation fields. By linking topology to dynamics, the network representations provide new insight into the mechanisms that lead to reversible and irreversible behavior in amorphous solids.

Figure 1

Mesostates and transition graphs. (a) Potential energy $\mathcal{U}$ of particle configurations associated with an initial mesostate (green segment) and the two mesostates it transits into when the applied strain $\gamma$ ($x$ axis) becomes ${\gamma }^{\pm }$. (b) The network generated starting from a zero-strain configuration $O$. The 1 cycle transient and limit cycle (between $X$ and $Y$) for $\gamma =0.05$ are marked in black and red.

Figure 2

State transition graph of the ${\gamma }_{\max }=0.05$ limit cycle. (a) Detailed view of the mesostate transitions associated with the 0.05 limit cycles depicted in Fig. 1. Transitions out of the end points $X$ and $Y$ are marked as green triangles and will be ignored. Regions of interest have colored backgrounds and refer to (b) and Fig. 3. (b) Network motifs involving one and two soft spots, (i) and (ii)–(iv), respectively. Soft spots are shown as black ellipses with states corresponding to their orientation. Motif background color and transition pattern highlighted in (a) coincide. (c) The particle displacements associated with the transitions of the avalanche motif in (iv) and (a). (d) Tree representation of the hierarchy of loops and subloops making up the limit cycle shown in (a). Refer to text for details.

Figure 3

Limit cycles as interacting soft-spot systems. (a) Location and label of the six soft spots in the sample that produce the plastic events of the cycle $(C^{\prime },Y)$ in Fig. 2. (b) Schematic description of particle displacements during a state change of a soft spot $0\to 1$ and back, $1\to 0$. (c) State transitions in the subcycles marked $[*\mathbf{0}]$ and $[*\mathbf{1}]$. The two cycles are topologically identical and the transitions in each are due to the same 5 soft spots. Transition from one cycle to the other occurs via the state change of a sixth soft spot. Labels next to the transitions in subcycle $[*\mathbf{0}]$ indicate the soft spots involved (same in subcycle $[*\mathbf{1}]$). Binary strings next to each mesostate indicate the individual states of soft spots. The transitions marked (23) and (234) are avalanches. (d) “Spectroscopic” plot showing the nonmonotonic changes in the switching strains ${\gamma }^{\pm }$ of soft spots 1–5, depending on the state of 6. The horizontal lines indicate the values of the switching strains along with the soft spots involved.

Figure 4

One-way rabbit-hole transitions. The transition out of the mesostate $Z$ inside the ${\gamma }_{\max }=0.05$ limit cycle leads to a set of states, from which there does not appear to be a path back.