Complex pathways and memory in compressed corrugated sheets

In our design, the corrugations lead to spatially localized instabilities upon compression which act as mechanical hysterons, the overall curvature prevents global buckling of the sheet, and the open cylindrical structure allows us to limit the number of grooves and facilitates both observation and manufacture. Our experimental protocol involves sweeping the axial compression of a groovy sheet while filming the sample and measuring the compressive force F (Fig. 1 A–E, Materials and Methods, and Movies S1–S3). We observe that our samples exhibit sequences of well-defined steps, seen as sharp jumps in the force F, and find that each event is associated with a localized (un)snapping event in a vertical groove, similar to those seen in tape springs (28) (Fig. 1 C–E, Materials and Methods, and Movies S1–S3). These transitions are hysteretic, and we observe that each groove can be in two distinct states—snapped and unsnapped—so that each groove acts as a mechanical hysteron. We refer to the hysteron transitions as “up” (from the unsnapped to the snapped state) and “down” (snapped to unsnapped) and denote the corresponding compression thresholds by the switching fields U⁺ and U^–(Fig. 1 F–H).

Repeated compression loops yield highly reproducible pathways with virtually identical force curves and sequences of hysteron flipping, evidencing the irrelevance of creep, plasticity, or aging (SI Appendix). A wide range of groovy cylinders responds similarly to cyclical compression (Materials and Methods), and while we focus on systems with three grooves/hysterons, which is the minimal number for scrambled pathways (defined in Tunable Pathways and Strong Interactions), we have observed similar phenomena in larger systems (SI Appendix). As we will show below, modifications of the sheet’s shape and boundary conditions allow us to geometrically tune the properties and interactions of their hysterons, making this system a viable platform to study reproducible, directly observable, and tunable pathways.

We map the full pathway by submitting the sample to a series of well-chosen compression/decompression cycles (Fig. 1I). Different driving cycles induce different pathways, which together form an intricate web of linked hysteresis loops (15, 16) which connect distinct states. These states and their transitions can be collected into a t graph, a directed graph that captures the response to any sequence of increases and decreases of the global driving field U (4, 5, 13, 14, 20, 29, 30). To experimentally map the t graph we systematically visit all states and determine all transitions, while tracking the state of each hysteron s_i, where s_i = 1 (0) refers to a snapped (unsnapped) state. A state S is characterized by the hysteron states $\{s_1,s_2,\dots \}$. For each collective state—with the exception of the ground state $\{0\dots 0\}$ and saturated state $\{1\dots 1\}$ —increased or decreased compression yields up and down transitions at critical switching fields $U_i^{+}(\mathbf{S})$ and $U_i^{-}(\mathbf{S})$. To determine all transitions, we first determine the main loop, the sequence of transitions that, under monotonic compression or decompression, connect the ground state and saturated state (14). We then check whether there are states for which there are undetermined transitions and determine these, and whenever we obtain a state that has not been previously visited we also determine its transitions, repeating these operations until no new states are found. Collecting all states and transitions we obtain the t graph in which the nodes represent the mesostates, and the directed edges, labeled by the values of their respective switching fields, represent the transitions (Fig. 1J). We stress that in our systems, all states that we consider are mutually reachable because there always is a specific driving protocol whereby any state is reachable from any other state (14).

For the simple case shown in Fig. 1 I and J, starting out at the uncompressed state {000} and monitoring the force and images, we find that continued compression yields a sequence $\left\{000\right\}\text{}\to \text{}\left\{001\right\}\text{}\to \text{}\left\{011\right\}\text{}\to \text{}\left\{111\right\}$; decompression starting at the saturated state yields a sequence $\left\{111\right\}\text{}\to \text{}\left\{011\right\}\text{}\to \text{}\left\{001\right\}\text{}\to \text{}\left\{000\right\}$. For this specific example, a simple compression/decompression cycle (Fig. 1D) is enough to obtain the full set of transitions. A more complex protocol (Fig. 1 I, Inset) yields a force response with three linked hysteresis loops (Fig. 1I) that illustrates the different pathways the system can follow and hence the importance of the loading history. The corresponding t graph also features three subloops embedded in the main loop and is spanned by four nodes and six edges; the material bits simply switch on and off when the driving is swept up and down.

A wide variety of more complex t graphs can be observed by tilting one of the boundaries of the sample (Figs. 1B and 2 A and B and Materials and Methods). Due to the spatial separation of each hysteron, applying such global gradients in the driving modifies the relation between global and local compression, leading to the smooth tuning of the switching fields $U_i^{\pm }$ (Fig. 3A and SI Appendix). As the relative order of the switching fields determines the order in which hysterons flip, tilting allows us to visit different states and/or sequences, thus modifying the topology of the t graphs. By sweeping α, we observe seven distinct responses in sample A (Fig. 2A). We characterize the order of the switching fields of the main loop by the corresponding sequence of hysteron flips, P⁺ and P^–(Fig. 2A). For example, in the main loop of sample A in regime ii, the second hysteron flips first ($\left\{000\right\}\text{}\to \text{}\left\{010\right\}$), followed by the third hysteron ($\left\{010\right\}\text{}\to \text{}\left\{011\right\}$) and finally the first one ($\left\{011\right\}\text{}\to \text{}\left\{111\right\}$), yielding $P^{+}=[2,3,1]$. Similarly, during decompression, the first hysteron unflips first ($\left\{111\right\}\text{}\to \text{}\left\{011\right\}$), then the second ($\left\{011\right\}\text{}\to \text{}\left\{001\right\}$), and finally hysteron number three ($\left\{001\right\}\text{}\to \text{}\left\{000\right\}$), leading to $P^{-}=[1,2,3]$. Many of the t graphs exist on a large angle span, while others can only be observed for a limited range of tilt angles α (Fig. 2B). We note that during the tilting process, two switching fields can become extremely close, such that their respective (un)snapping events become indistinguishable [e.g., in Fig. 2B the two data points between A(ii) and A(iii) refer to such a case]. This degeneracy may cause an avalanche. However, we can distinguish degeneracy-driven avalanches from true avalanches by their response to changes in the tilt angle: degeneracy-driven avalanches quickly disappear when the angle is modified, while true avalanches persist over a significant range of α.

We note that in most cases, the difference between neighboring t graphs is associated with a single permutation of the upper or lower transitions; for example, compare graph A(i), where $P^{+}=[3,2,1]$, to A(ii), where $P^{+}=[2,3,1]$, corresponding to a swap of the snapping order of hysterons 2 and 3. Such permutation is consistent with the notion that tilting smoothly tunes the switching fields. Plots of the switching fields directly evidence their smooth variation with the tilt angle α and show that the ranges of α where each t graph occurs are consistent with the crossing of two switching fields; for example, the change from graph A(i) to A(ii) corresponds to the crossing of $U_2^{+}$ and $U_3^{+}$ (Fig. 3B and SI Appendix). We have obtained a similar range of t graphs for a sample C which contains four hysterons (SI Appendix). We conclude that global driving gradients are a powerful tool to systematically elicit a multitude of t graphs from a single sample.

Different pathways can be classified according to the topology of their t graphs, and in this paper we introduce the first three steps of complex pathways that arise when interactions become increasingly important. A close inspection of the topology of the t graphs of sample A reveals the first two distinct classes of graphs: Preisach graphs A(i), A(ii), A(iii), A(v), A(vi), and A(vii) and scrambled graphs A(iv).

To understand this distinction, consider the relation between the flipping of the individual hysterons and the state transitions. For n hysterons with states s_i, forming a collective state S, the up (down) transitions are set by the hysteron in state s = 0 (s = 1) which has the lowest up (highest down) switching field:

Here i₀ (i₁) runs over the hysterons that are in state 0 (1), $U^{\pm }(\mathbf{S})$ are the global switching fields for the collective states S, and $u_i^{\pm }$ are the state-dependent switching fields of individual hysterons.

In absence of hysteron interactions, the switching fields of all hysterons are independent of the state of the other hysterons ($u_i^{\pm }(\mathbf{S})=u_i^{\pm }$), and the topology of the t graph is thus fully determined by the ordering of the up and down switching fields of individual hysterons. We refer to t graphs whose topology is consistent with a unique ordering of the upper and lower switching fields of the hysterons, and which do not contain avalanches, as Preisach graphs [recall that collections of noninteracting hysterons are referred to as the Preisach model (30)]. We note that the topology of a Preisach graph is fully determined by the order of the transitions of the main loop, encoded in P⁺ and P^–. We stress here that t graphs with a Preisach topology do not require the strict absence of interactions but only that interactions do not introduce differences between the ordering of the transitions in the main loop and the rest of the t graph.

Strikingly, we also observe a scrambled graph which features pairs of transitions that are not consistent with a unique, state-independent ordering of the switching fields—we call these transitions scrambled. In particular, t graph A(iv) contains the pair of transitions $\left\{111\right\}\to \left\{101\right\}$ and $\left\{110\right\}\to \left\{010\right\}$, which is not compatible with a unique, state-independent ordering of the switching fields $U_2^{-}>U_1^{-}$; rather these transitions imply that $U_2^{-}(111)>U_1^{-}(111)$ and $U_2^{-}(110)<U_1^{-}(110)$ respectively (Fig. 2A). Such state-dependent ordering of the switching fields is incompatible with a Preisach graph and demonstrates a dependence of the switching field of one hysteron on the state of another hysteron, thus directly evidencing hysteron interactions. We note here that scrambled pairs of transitions have recently been observed in numerical simulations, e.g., the scrambled pair of transitions $\left\{000\right\}\text{}\to \text{}\left\{100\right\}$ and $\left\{001\right\}\text{}\to \text{}\left\{011\right\}$ in figure 5b of ref. 20 and the scrambled pair of transitions $\left\{1101\right\}\text{}\to \text{}\left\{1100\right\}$ and $\left\{1111\right\}\text{}\to \text{}\left\{0111\right\}$ in figure 2 of ref. 21; see also ref. 22. Intuitively, scrambling implies that bit-flip sequences depend on the starting state, allowing for a far larger space of potential t graphs than Preisach graphs. Additional evidence of interactions can be obtained from the state dependence of the switching fields; for example, we observe a systematic difference of $U_2^{+}(000)$ and $U_2^{+}(100)$ (Fig. 3 C and D and SI Appendix). We note that the sign of the interactions is not constant throughout the samples—ferromagnetic and antiferromagnetic generically both occur, and different pairs of hysterons can feature different signs of interactions. Moreover, interactions are not reciprocal, and the sign and strength of interactions for the upper and lower switching fields may also be different. We stress here that the interaction strength necessary to obtain scrambling is directly proportional to the differences between the switching fields of different hysterons. Hence, as tilting allows us to make these differences arbitrarily small near crossings, our strategy is eminently suited to observe non-Preisach behavior, even if hysteron interactions are weak. We conclude that hysteron interactions can yield types of pathways and t graphs.

To further study the effect of interactions between hysterons we require samples with stronger coupling. To create these, we note that for typical parameters and $R\mathrm{\gtrsim }5$ cm, the (un)snapping of one hysteron triggers the (un)snapping of all hysterons, which we interpret as strong interactions, while for smaller R the snapping events occur in sequence. Indeed, radius of curvature serves as a proxy for the strength of interactions—the larger the radius, the stronger the interactions. We thus introduce sample B with R = 2 cm (the radius of curvature for sample A was 1 cm) and three grooves. We have verified that in comparison to sample A, the interactions as measured by the dependence of the switching fields on the state are indeed stronger (SI Appendix). We orient the sample and apply a shim to the boundaries so that the crossings of the switching fields as function of α are optimally separated.

We find that as a function of the tilt angle α, sample B yields nine distinct t graphs, occurring on well-separated ranges of α (Fig. 4). We distinguish a number of distinct features. First, t graphs $B(i$), B(ii), B(iii), and B(ix) are all Preisach graphs. Second, t graphs B(iv) and B(v) feature an avalanche $\left\{100\right\}\to \left\{111\right\}$ over a significant range of tilt angles α. While we cannot rule out that this avalanche is caused by inertial effects or degeneracies, we note the avalanche occurs over a significant range of α and that hysteron interactions can also cause such avalanches; if $U_3^{+}(110)<U_2^{+}(100)<U_3^{+}(100)$, state {100} transitions to {110} at $U=U_2^{+}(100)$, and since state {110} is unstable at this value of U, it transitions to a stable state {111}. Third, t graphs B(v), B(vi), B(vii), and B(viii) all contain scrambled transitions—see Table 1 for the pairs of scrambled transitions. Hence, manipulating the overall geometry of our corrugated sheets allows us to increase the magnitude of interactions to obtain a variety of robust, non-Preisach pathways.

To further classify the t graphs’ topologies, we consider the recent definition of loop-RPM (l-RPM) (14, 30). l-RPM requires that all loops within the t graph are absorbing. In essence, a loop is defined by a pair of nodes S_m and S_M, where the system evolves from S_m to S_M (and vice versa) by a series of up (down) transitions, and the intermediate states are defined as the up (down) boundaries; l-RPM then requires states S_m (S_M) to be accessible by a sequence of down (up) transitions from any up (down) boundary. All t graphs of sample A and all t graphs of sample B—with the exception of B(viii), which we will discuss in detail in this section—are consistent with l-RPM.

We now focus on sample B in regime viii. To see that its t graph violates l-RPM, consider the loop with $S_m=\left\{000\right\}$ and $S_M=\left\{011\right\}$. State {010} is then part of its down-boundary, but starting from this state and increasing the driving never reaches $S_M=\left\{011\right\}$: hence, this loop is not absorbing, and the t graph violates l-RPM. Hence, in this sample, hysteron interactions are sufficiently strong to observe a next step in the hierarchy of t-graph complexities that yields t graphs with topologies that violate l-RPM.

We stress here that it is known that ferromagnetic interactions, where one hysteron switching from zero to one promotes the switching of others from zero to one and vice versa, preserve l-RPM (18). Hence, our observation of weak and strong RPM breaking indicates the presence of antiferromagnetic interactions in our system, which clearly can cause a wide range of t graphs (4, 20–22).

To understand the new qualitative feature corresponding to this specific case we compare the response to cyclic driving in regime (vii), where the t graphs have l-RPM, and regime (viii), focusing on the loop that connects states $S_m=\left\{000\right\}$ and $S_M=\left\{011\right\}$ (Fig. 5 A–D). For cyclic driving, specified by U_m and U_M, and a specific starting state S, the system must reach a periodic orbit since there are a finite number of states (14). The system may require τ cycles before reaching its final orbit, and this orbit may be subharmonic with perioditicity T—simple cycles or absorbing states correspond to T = 1 (14, 20, 21). In regime vii, once the state {011} is reached, cycling U between $U_m(>U_2^{-}(010))$ and $U_M(<U_1^{+}(011))$, the system follows the same {011} to {010} loop repeatedly (Fig. 5 A and B). In contrast, applying a similar driving to the sample for α in regime viii yields state {011} at first maximal driving, but the second and subsequent maxima produce state {110} (Fig. 5 C and D). Hence, the transition $\left\{010\right\}\to \left\{110\right\}$ erases the memory of the {011} state and brings the system to a new subloop. The system thus reaches a simple periodic orbit (T = 1) after a training of τ = 2 driving cycles. We refer to orbits where $\tau >1$ and T = 1 as accumulators: during a transient of τ cycles, the system visits unique states from which the number of driving cycles can be deduced; after more driving cycles, the system visits the same orbit, which encodes that the number of driving cycles is larger or equal than τ. Such orbits can be seen as a concrete realization of a long training time pathway with τ = 2, as presented numerically in ref. 20. This behavior, where the sample remembers how often it is driven to a certain maximum, is different from classical forms of memory (29), and we suggest that t graphs containing accumulator orbits may underpin this behavior, although we note that here we specifically focus on the subcase where T = 1. We stress that the observed accumulator behavior can also be seen as an elementary form of information processing: counting to two. From this standpoint, we suggest that our observation of accumulator behavior is a first step toward the realization of systems with complex t graphs that encode information processing. We moreover stress that this accumulator behavior (31) can be observed on a robust range of tilt angles (Fig. 4B). Finally, we note that in our system, where all states are mutually reachable, the existence of an accumulator orbit necessitates the breaking of l-RPM (14).*

We conclude that our system allows us to observe three classes of topologically distinct t graphs. The first class consists of Preisach t graphs which are topologically equivalent to those of the Preisach model. The second class consists of t graphs that contain scrambled transitions, yet satisfy l-RPM. The third class consists of t graphs that violate l-RPM, and in our specific example the t graph encodes accumulator behavior, which may underpin transient memories (20). This classification, which is experimentally accessible, organizes the impact of hysteron interactions on the pathways.

The topological properties characterize that the system can reach certain states but do not restrict the corresponding values of the driving. However, as we demonstrate in this section, the precise ordering of the switching fields may impact the pathways, even if it does not impact the t graphs’ topology. Hence, the ordering of the switching fields provides a secondary characterization of pathways.

An important example of a property that involves a precise statement on the values of the switching fields is ordinary RPM, which requires that when the driving strength revisits a previous extremal value, the system revisits a previous extremal state (15, 16, 18, 29, 30). RPM implies l-RPM, as all extremal states are absorbing states when RPM is valid (14). The reverse, however, is not true: RPM implies conditions on the relative ordering of the switching fields beyond those captured by the topology of the t graph: a system can be l-RPM without strictly satisfying RPM.

To illustrate the importance of the ordering of the switching fields, consider a scenario where state {011} can be reached by up transitions from either {001} or {010} (Fig. 6A). First, increase U to a value U_M so that state {011} is reached, then decrease U to U_m so that state {010} is reached. Now consider the response to increasing U back up again to U_M. If the system satisfies RPM, it must then revisit state {011} (and not remain stuck at {010}), which implies the following condition on the ordering of the switching fields: $U_2^{+}(001)\ge U_3^{+}(010)$. For a Preisach system, the sequence of transitions on the main loop implies that $U_2^{+}>U_3^{+}$, so that this condition is satisfied. However, in the presence of hysteron interactions, the sequence of transitions on the main loop only implies that $U_2^{+}(000)>U_3^{+}(000)$ and $U_2^{+}(001)>U_3^{+}(000)$ but does not imply that $U_2^{+}(001)\ge U_3^{+}(010)$; indeed, for the same t-graph topology, RPM can be satisfied or violated, depending on the ordering of the switching fields.

A specific example of such a t graph that satisfies l-RPM but violates RPM is sample B in regime vii (Fig. 6). Here $U_2^{+}(001)\approx 1.47$ mm and $U_3^{+}(010)\approx 1.50$ mm. Hence, when we cycle the system (starting from {010}) between $U^M=1.476\hspace{0.17em}$ mm and $U^m=1.210\hspace{0.17em}$ mm, the system reaches state {011} in the first cycle but then remains stuck in the {010} state; reaching {011} again requires raising U beyond $U_3^{+}(010)$ (Fig. 6C). Hence, sample B(vii) does not satisfy ordinary RPM, because the condition $U_2^{+}(001)\ge U_3^{+}(010)$ is violated.

This is an example of the violation of the NP property. The topology of the t graph (Fig. 6B) implies the following ordering of the states: $\left\{000\right\}\prec \left\{001\right\}\prec \left\{011\right\}$ and $\left\{000\right\}\prec \left\{010\right\}\prec \left\{011\right\}$ (14). If NP were to hold, two orbits starting out at different states, e.g., {000} and {010}, must preserve this ordering (equalities allowed) when they occur at the same driving. However, starting from these states and ramping the driving to, e.g., $U^M=1.476$ mm, the orbit $\left\{000\right\}\text{}\to \text{}\left\{001\right\}\text{}\to \text{}\left\{011\right\}$ passes the other orbit that remains stuck at {010}.

This example of NP and RPM violation, which does not affect the t graph’s topology, exemplifies the role of the precise ordering of the switching fields. We note that similar effects can also occur in more complex t graphs; for example, in the accumulator t graph, some cyclic driving protocols can remain stuck at {011} only when $U_2^{+}(001)<U_1^{+}(010)$. We finally stress that while both topology changes, such as scrambling, and secondary effects, such as RPM breaking, relate to ordering of switching fields, they are distinct. For example, scrambling requires specific differences in the ordering of two switching fields in two states and changes the topology of the t graph; secondary ordering effects do not impact the topology of the t graph and can be associated with single inequalities. Hence, by clearly distinguishing topological and nontopological impact of hysteron interactions, transition graphs and pathways can precisely be characterized and classified.

The fabrication of corrugated sheets starts by spin coating a liquid mixture of a two components silicone elastomer (Zhermack Elite double 32 Fast, Young’s modulus $E\approx 1$ MPa, Poisson’s ratio $\nu \approx 0.5$) on a surface with sinusoidal corrugations with pitch p and amplitude A₀. Rotation is maintained until complete curing of the polymer ($\approx 20\hspace{0.17em}$ min). The sheet is then peeled and rolled in an open cylinder; top and bottom ends are dipped in a liquid layer of the same polymer mixture to set the cylindrical shape and fix the boundary conditions. The resulting sample is characterized by its height H, thickness t, radius of curvature R, pitch p, number of corrugations N, and amplitude A₀. We have observed similar local and sequential snap-through behavior in over 10 samples, with the only limitation appearing to be that the corrugation amplitude A₀ is not too small and R is not too large—a natural scale to compare these to is the pitch p. Paint is splattered on the samples to enhance contrast and ease visualization.

The mechanical response of our samples is probed in a uniaxial testing device (Instron 3366) which controls the axial compression U better than 10 µm; we use a 5N sensor which accurately measures the force down to 10 mN with an accuracy of ${10}^{-4}\hspace{0.17em}$ N. We define U = 0 where the force during compression reaches the small value $F(U_0)=20\hspace{0.17em}$ mN. We use compression speeds of 1 mm/min and have checked that further lowering the compression speed by an order of magnitude does not affect the phenomenology, thus ensuring we operate in the quasistatic regime. We focus on the compression range (strain less than 5%) where grooves can snap but where no additional instabilities are observed.

We image the deformation of the groovy sheet during compression at a frame rate of 3Hz or faster, using a charge-coupled device camera (Basler acA2040-90um) mounted with a 50 mm objective. We calculate the mean squared differences in each local region Ω_i of the normalized digital image as ${\displaystyle {\sum }_{k,l\in {\mathrm{\Omega }}_i}{\mathrm{\Delta }}_i^2}k,l≔{(A_{k,l}^{t+\mathrm{\Delta }t}-A_{k,l}^t)}^2$, where k and l label the pixels, t is time, and $\mathrm{\Delta }t$ the time interval ($\mathrm{\Delta }t=100\hspace{0.17em}$ ms in Fig. 1E). Each region Ω_i targets a part of a single groove, chosen such that events in neighboring regions do not create secondary peaks.

The sample rests on a Thorlabs tilt stage that allows us to control the tilt angle α with an accuracy of ${3.10}^{-5}\hspace{0.17em}$ rad. We incrementally change α with steps ranging from ${3.10}^{-4}\hspace{0.17em}$ rad to ${2.10}^{-3}\hspace{0.17em}$ rad and for each tilt angle α measure the full t graph and the mechanical response. All transition graphs presented in this paper were obtained multiple times over the course of several weeks, and all angles were visited several times to ensure a good reproducibility (SI Appendix, Fig. S1 C and D). By exploring the reproducibility of the boundary between different t graphs, when two switching fields are essentially degenerate, we estimate our accuracy of the boundaries to be better than $\pm {2.10}^{-4}$ rad.

To determine the switching fields, each transition is probed between two and four times, and we report mean switching fields which have an SD typically smaller than the symbol size. We estimate small viscous relaxation effects to affect the switching field by at most 4% (SI Appendix, Fig. S1B), thus requiring larger differences to evidence interactions.