Topology of the energy landscape of sheared amorphous solids and the irreversibility transition

Recent experiments and simulations of amorphous solids plastically deformed by an oscillatory drive have found a surprising behavior—for small strain amplitudes the dynamics can be reversible, which is contrary to the usual notion of plasticity as an irreversible form of deformation. This reversibility allows the system to reach limit cycles in which plastic events repeat indefinitely under the oscillatory drive. It was also found that reaching reversible limit cycles can take a large number of driving cycles and it was surmised that the plastic events encountered during the transient period are not encountered again and are thus irreversible. Using a graph representation of the stable configurations of the system and the plastic events connecting them, we show that the notion of reversibility in these systems is more subtle. We find that reversible plastic events are abundant and that a large portion of the plastic events encountered during the transient period are actually reversible in the sense that they can be part of a reversible deformation path. More specifically, we observe that the transition graph can be decomposed into clusters of configurations that are connected by reversible transitions. These clusters are the strongly connected components of the transition graph and their sizes turn out to be power-law distributed. The largest of these are grouped in regions of reversibility, which in turn are confined by regions of irreversibility whose number proliferates at larger strains. Our results provide an explanation for the irreversibility transition—the divergence of the transient period at a critical forcing amplitude. The long transients result from transition between clusters of reversibility in a search for a cluster large enough to contain a limit cycle of a specific amplitude. For large enough amplitudes, the search time becomes very large, since the sizes of the limit cycles become incompatible with the sizes of the regions of reversibility.

Figure 1

(a) Illustration of the construction of a catalog of mesostates starting from the reference state $O$ at generation $g=0$. Transitions in black/gray (red/orange) designate $\mathbf{U}$ ($\mathbf{D}$) transitions. Under $\mathbf{U}$ and $\mathbf{D}$ transitions, we obtain $2,4$, and 5 new meostates at generations $g=1,2$, and 3, respectively. Transitions leading to new mesostates at each generation have been highlighted. (b) A mesostate transition graph generated from an initial configuration $O$ (marked as a red dot) with several strongly connected components (SCCs) highlighted in different colors. The largest six SCCs have sizes: 929 (green), 222 (brown), 115 (yellow), 90 (orange), 37 (cyan), and 20 (purple). Transitions within an SCC correspond to reversible plastic events, since for any deformation path connecting two states in an SCC, by definition there is also a reverse path. Irreversible plastic events are transitions between states belonging to different SCCs. (c) The inter-SCC graph is a compressed representation of the graph in (b), showing the SCCs as squares with colors that correspond to the colors in (b). The arrows connecting the SCCs are the irreversible plastic events and the inter-SCC graph is therefore acyclic.

Figure 2

(a) Normalized distributions of energies at the onset of reversible (blue dots) and irreversible (red squares) plastic events (transitions). (b) Normalized distributions of the energy drops during reversible (blue dots) and irreversible (red squares) plastic events (transitions). The results in this figure combine data sampled from all eight catalogs.

Figure 3

(a), (b) Density plot of stress versus the energy at the onset of reversible (a) and irreversible (b) plastic events. The overall parabolic shape of the scattered points corresponds to the bulk elastic response of the samples. (c), (d) The stress drop after a plastic event versus the stress at the onset of the plastic event for reversible (c) and irreversible (d) transitions. The color bars to the right depict the color coding of the density from low (dark/blue) to high (bright/yellow). The results in this figure combine data sampled from all eight catalogs.

Figure 4

(a) The SCC size distribution taken from the full eight catalogs (in blue) exhibits a heavy tail. The solid line is a power-law exponent 2.67 and serves as a guide to the eye. Colors other than blue correspond to distributions derived from the same catalogs but only up to a maximal generation number of 24,28,32,36, demonstrating that the distribution of SCC sizes becomes stable for networks significantly smaller than the ones used to calculate the exponent. (b) Plastic deformation history leading from the initial state $O$ to a mesostate $A$ of the catalog after $g=40$ plastic events. Each vertical blue line is an intermediate mesostate $P$ with its stability range $({\gamma }^{-}\left[P\right],{\gamma }^{+}\left[P\right])$, while the horizontal line segments in black ($\mathbf{U}$) and red ($\mathbf{D}$) that connect adjacent mesostates indicate the strains at which the corresponding plastic events occurred. For each mesostate $A$ and deformation history, we can identify the largest and smallest strains under which a $\mathbf{U}$, respectively, $\mathbf{D}$ transition occurred, ${\gamma }_{\max }^{\pm }$, as illustrated by the extended horizontal lines. (c) Deformation path history dependence of $k_{\mathrm{REV}}$: Each dot represents a mesostate of catalog No. 1. The coordinates of each dot represent the largest positive and negative strains ${\gamma }_{\max }^{\pm }$, cf. panel (b), that were required to reach a specific mesostate, while their color represents how many reversible transitions $k_{\mathrm{REV}}=0,1$, or 2, go out of it, as indicated in the legend. The locations of the yield strain in both positive and negative directions have been marked by dotted vertical and horizontal lines. The region highlighted by the light blue triangle contains the set of all mesostates that can be reached without ever applying a shear strain whose magnitude exceeds $|{\gamma }_{\max }^{\pm }|=0.085$. The prevalence of mesostates with $k_{\mathrm{REV}}=2$ (blue dots) inside this region implies that mesostates reached by applying strains whose magnitudes remain below 0.085 undergo predominantly reversible transitions, i.e., lead to mesostates that are part of the same SCC. (d) Scatter plot of the mesostates with $|{\gamma }_{\max }^{\pm }|\le 0.085$ across the five catalogs with 40 or more generations. As was the case for the single data set shown in panel (c) of this figure, the region $|{\gamma }_{\max }^{\pm }|\le 0.085$ shows a high degree of reversibility across all five catalogs: the region contains 9298 mesostates out of which 7728 have $k_{\mathrm{REV}}=2$ and 1194 have $k_{\mathrm{REV}}=1$ outgoing irreversible transitions. Inset: Mean SCC size that a mesostate belongs to, given that it is stable at some strain $\gamma$ calculated from all eight catalogs. Error bars represent the standard deviation of fluctuations around the mean. The figure shows that mesostates stable at large strains tend to belong to small SCCs.

Figure 5

Large excerpt of the inter-SCC graph, cf. Fig. 1, obtained from catalog No. 1. Shown is the subgraph of 3228 SCCs (squares) that can be reached from the SCC of the initial state by at most 15 inter-SCC transitions. These SCCs contain a total of $12790$ mesostates. The size of the SCCs is indicated by the legend in the top left corner of the figure. The coloring scheme of the SCCs indicates the (average) number of yield events in the plastic deformation history of the mesostates constituting the SCC, i.e., the number of mesostate transitions in their deformation history that occur at stresses of magnitude 2.5 and higher. The figure shows patches of SCCs with the same number of yield events. Among these, the green patch of SCCs whose constituting mesostates have suffered no yield experience is dominant. Note that even for two (orange) or more yield events, there are relatively extended patches of SCCs of large sizes, such as the orange patch in the top left part of the graph. These findings suggest that the transition graph contains multiple reversibility regions, such as the one shown in Fig. 4, that differ only by the common history of past yield events of their constituent mesostates.

Figure 6

(a) The range of strain amplitudes of oscillatory shear of the form Eq. (2) that give rise to a limit cycle consisting of $\ell$ plastic events/mesostate transitions. Blue dots refer to average strain values for the corresponding cycle lengths. The solid black line is a power law with exponent 2.5. (b), (c) Note that mesostates forming a cyclic response must belong to the same SCC and, consequently, a limit cycle formed by $\ell$ mesostates can only be part of an SCC whose size $s_{\mathrm{SCC}}\ge \ell$. (b) Scatter of SCC size $s_{\mathrm{SCC}}$, against the strain amplitude $\gamma$ generating the limit cycle with largest $\ell$ contained in the SCC (red small dots). The bigger red dots connected by a dashed line outline the boundary of this region—they are the smallest SCCs that contain a limit-cycle of a strain amplitude $\gamma$. Blue dots with error bars replot the inset of Fig. 4, displaying the average sizes of SCCs that contain states stable at strain $\gamma$. (c) Scatter plot of the SCC sizes against the length ${\ell }_{\max }$ of the largest limit cycles, under oscillatory shear Eq. (2) that these contain. The red curve is a prediction of the Preisach model. Refer to text for further details. The results in this figure combine data sampled from all eight catalogs.