Phase transitions in 2D multistable mechanical metamaterials via collisions of soliton-like pulses

In recent years, mechanical metamaterials have been developed that support the propagation of an intriguing variety of nonlinear waves, including transition waves and vector solitons (solitons with coupling between multiple degrees of freedom). Here we report observations of phase transitions in 2D multistable mechanical metamaterials that are initiated by collisions of soliton-like pulses in the metamaterial. Analogous to first-order phase transitions in crystalline solids, we observe that the multistable metamaterials support phase transitions if the new phase meets or exceeds a critical nucleus size. If this criterion is met, the new phase subsequently propagates in the form of transition waves, converting the rest of the metamaterial to the new phase. More interestingly, we numerically show, using an experimentally validated model, that the critical nucleus can be formed via collisions of soliton-like pulses. Moreover, the rich direction-dependent behavior of the nonlinear pulses enables control of the location of nucleation and the spatio-temporal shape of the growing phase.

One particular class of mechanical metamaterial obtains its nonlinear properties from the rotation of periodic internal features, such as squares connected at their hinges.Systems based on the rotating-squares mechanism have long been studied due to their interesting static properties (i.e., their auxetic characteristicis) [18,19].More recently, it has been observed that they are also capable of propagating a variety of nonlinear waves [11,12,20].A notable example is the propagation of vector solitons, which have coupled translational and rotational degrees of freedom (DOFs) and can display distinct solitary modes for different propagation directions [21].Interactions of these nonlinear waves have also been investigated, albeit mostly for one-dimensional systems.Due to the coupling between different DOFs, which is less often considered in Hertzian granular media [22][23][24][25][26], the collision of vector solitons has been shown to exhibit anomalous phenomena, including repelling, destruction, etc., in addition to classical soliton collisions [27].
Recently, the dynamics of multistable versions of these systems have also been studied.For example, multistability can be achieved by introducing permanent magnets [20,28], which produces multiple energy minima, each associated with equilibrium angles that the squares * Electronic address: raney@seas.upenn.educan snap between.If squares are rotated from one stable angle to another, it is possible for this reconfiguration to propagate throughout the structure in the form of a transition wave.In addition, the collision of transition waves of incompatible type can cause formation of stationary domain walls, which can be exploited for the design of reconfigurable metamaterials [20].
Here, we investigate collisions of nonlinear, soliton-like pulses in 2D multistable systems of rotating squares, and how these collisions can be used to remotely nucleate phase transitions at arbitrary locations.As a first step, we experimentally and numerically show how phase transitions can be initiated via quasistatic rotation of a "critical nucleus" of squares, analogous to nucleation during first-order phase transitions [29,30].Note, in this work, the phase transitions are enabled by multistability, which is achieved by embedding magnets in the squares.This is in contrast with other work [31][32][33], in which phase transitions are induced by applying static precompression to the entire system, or by dynamic recoil [34].Second, we investigate the criteria necessary for collisions of solitonlike pulses to induce this phase transition.Finally, we describe how the anisotropy associated with the symmetry of the system produces direction-dependent nucleation and propagation of the phase transition.These fundamental behaviors could enable new insights for the design of reconfigurable, shape-transforming, and deployable mechanical metamaterials.

Phase transitions in multistable metamaterials
We start by experimentally and analytically characterizing the energy landscape of the building block of the mechanical system, i.e., a 2 × 2 set of squares.To experimentally measure the behavior of such a system, we fabricate an elastomeric building block, following a conventional molding-casting process.Specifically, we design and 3D print a mold (MakerGear M2, polylactic acid).
We then pour silicone precursor (Dragon Skin 10) into the mold and allow it to cure.The squares have side length 12 mm and are connected by thin hinges of thickness 1.5 mm.Permanent magnets are inserted into each square (SI Appendix, Section 1 for fabrication details).A schematic of the building block is shown in Fig. 1A.The competition between the strain energy of the hinge and the interaction of the magnets gives the squares three stable angles [20].Each of these corresponds to a local minimum in the potential energy landscape (Fig. 1B).Then, in order to quantify the effects of different design parameters, we introduce a discrete model capable of capturing the multistable energy landscape.Each square, assumed to be a rigid body with mass M and moment of inertia J, has two translational degrees of freedom (u and v) and one rotational degree of freedom (θ).Each hinge is modeled by three springs (Fig. 1A): a linear longitudinal spring with stiffness K l , a linear shear spring with stiffness K s , and a nonlinear torsional spring with potential energy E θ (∆θ) expressed as where K θ is the linear torsional spring constant, θ 0 is the initial equilibrium angle, ∆θ is the relative angle of the hinge, and V Morse is the Morse potential, which is used to empirically describe the nonlinear magnetic interactions between squares.In Eq. 2, A and α define the depth and width of the Morse potential, respectively, and θ M determines the equilibrium points.To obtain these parameters for the numerical simulations, we conduct experimental tensile tests using a commercial quasistatic test system (Instron model 68SC-5) with custom fixtures (SI Appendix, Fig. S2 and S3).These are designed to allow the squares to rotate during the tests.Then, Eq. 1 gives the energy landscape of the building block, which exhibits three distinct phases (labeled as Phase L, C, and R), as shown in the inset of Fig. 1B.Before considering whether collisions of impulses can induce a phase transition in the system, we first seek to understand the threshold for nucleation of a phase transition more generally.We assume that the system is initially in Phase C, and that a small number of squares are forced to rotate to the new Phase R; we then experimentally and numerically observe whether this forced rotation nucleates a new phase, which can propagate throughout the rest of the structure.To confirm this experimentally, we fabricate a larger prototype of size 10 × 10 squares, following the same procedures described earlier (note, to reduce the effect of the boundaries on the behavior of the mechanical system, magnets are not embedded in the exterior squares).Nucleation is induced by quasistatically forcing a 2 × 2 building block at the center of the specimen to undergo the transition.The entire specimen is observed to subsequently undergo a phase transition, as shown in the optical images of Fig. 1C, obtained via a high-speed camera (Photron FASTCAM Mini AX; Movie S1 and SI Appendix, Fig. S4 and S5).
Next, we perform numerical simulations to investigate the nonlinear dynamics of the multistable system over a wider ranger of parameters.Based on the discrete model, we derive the equations of motion (EOMs) of each square in the system.By introducing the following normalized parameters: where a is the distance between the centers of two neighboring squares), we obtain the dimensionless EOMs (SI Appendix, Section 3).We quantify the dynamic response of the system by numerically solving the EOMs, using the fourth order Runge-Kutta method.In the numerical simulations we consider a system of 30 × 30 squares.To trigger a nucleation, we apply rotation θ in to the 2 × 2 squares at the center, similar to the experiments.We find that, given a proper set of parameters (e.g., in this case θ 0 = 25 • , K 1 = 0.2, K 2 = 0.0306, β = 3.0556), there exists a critical angle θ c .When θ 0 + θ in ≥ θ c , the nucleation of a new phase occurs, with the 2 × 2 squares transforming from the initial Phase C to the new Phase R.This transition propagates outward throughout the rest of the metamaterial in the form of a transition wave with some directional dependence (i.e., it travels faster along the x and y axes than along the diagonals; SI Appendix, Fig. S6).Snapshots from the numerical simulations are displayed in Fig. 1D for normalized times T = 0, 40, 75, and 100, showing qualitative agreement with the experimental observations (Movie S2).
The existence of the critical angle θ c suggests that there is an energy threshold E c .To understand the origin of this threshold, we characterize the phase transition observed in our mechanical system from the energy perspective.Analogous to classical first order phase transitions, there is a "critical nucleus size" that is required for the new phase to be stable [35,36].This results from the competing effects of energy terms that favor the transition (e.g., the energy released by moving from Phase C to Phase R in Fig. 1B) and terms that do not favor it (e.g., the interface energy between the new phase and the old phase).For the specific system investigated above, we find that the energy threshold is E c = 1.06 × 10 −2 (normalized by Ē = K l a 2 ; see also SI Appendix, Fig. S7).It is worth noting that the critical nucleus size depends on the choice of parameters.For a different set of parameters, it is possible to obtain a critical nucleus size other than 2 × 2 squares (SI Appendix, Fig. S8).

Initiating phase transitions via collisions of soliton-like pulses
Now that we have characterized the energy criteria necessary to induce a phase transition quasistatically, we next consider how a transition could be nucleated by colliding vector solitons.Here, we have intentionally chosen design parameters that produce the smallest critical square nucleus, i.e., 2 × 2 squares.We consider a circular-shaped system with 30 squares along its diagonal.We impact the sample at different squares along its circumference to initiate pulses that propagate along different directions.Specifically, the impacts are displacement profiles in the form where A 0 and W are parameters that alter the impact amplitude and shape, respectively.To avoid triggering a nucleation directly at the impacted squares, in the simulations we impose θ = 0 to all squares on the boundary.

Head-on collisions of two pulses with same rotation
We first investigate head-on collisions of pulses by applying impacts at the left and right boundary.In Fig. 2A, we show snapshots of the wavefields at T = 15, 28.7, 35, and 60, demonstrating that a phase transition is induced where the two pulses collide (Movie S3).By sweeping the impact amplitude A 0 , we identify a critical amplitude A c = 0.306, below which a nucleation is not induced by the colliding pulses (see Fig. 3A and Movie S4).When A 0 ≥ A c , the collision of the two pulses can lead to the formation of a critical nucleus.In that case, the new phase propagates outward to the rest of the structure via a transition wave.In Fig. 2B, we plot the normalized energy of the squares at the nucleation site (i.e., the squares in the inset of Fig. 2A  for A 0 = A c .We observe that there also exists an energy threshold E nu c = 3.84 × 10 −2 during the collision process.Comparing this energy threshold E nu c with its counterpart in the previous quasistatic analysis, we note that E nu c is much larger than E c , a result of the fact that not all of the energy in the propagating pulses will be directed toward forming a new phase during the collision (e.g., some energy is lost in the form of scattered waves).Fig. 2C shows a spatiotemporal plot that provides the angle of the squares along the propagation direction (x axis) as a function of time and position.We also note that the location of nucleation can be changed simply by introducing a time delay ∆T for the initiation of the impulse on the left with respect to the initiation of the impulse on the right.In Fig. 2D, we demonstrate this by showing snapshots of the simulations for ∆T = 10 and 20 (Movie S5).

Head-on collisions of two pulses with opposite rotation
We also explore head-on collisions of pulses with different rotational directions (Fig. 3).In contrast with collisions between impulses with the same (positive) rotation (as was triggered by applying two compressive impulses at the left and right boundaries in Fig. 2A) Fig. 3B shows a collision of two pulses with opposite rotational directions.This is accomplished by changing the excitation at the right boundary from a compressive impact to a tensile impact.The two pulses pass through each other without inducing a nucleation for A 0 = A c (Movie S6).(B) Head-on collision of two pulses with the opposite rotational direction for A0 = Ac.(i) Snapshots of wavefields before collision at T = 15 and after collision at T = 28 and 35, showing that the pulses pass through each other.(ii) Spatiotemporal plot obtained from the numerical simulation, showing the angle θ for squares along the propagation direction as a function of time.(C) Kinetic energy of the whole structure as a function of time for a head-on collision of pulses for A0 = Ac with (i) same rotational directions and (ii) opposite rotational directions (the vertical dashed lines indicate the time when the two pulses collide); the former case exhibits a significant exchange between the transitional and rotational components of the kinetic energy.
To better understand this observation, we separate the kinetic energy into two components: one associated with translational motion and the other associated with rotational motion.The results are plotted in Fig. 3C (i-ii) with A 0 = A c for for same rotation and opposite rotation, respectively.We find that there is some energy exchange between the two kinetic energy components for the same rotation case, i.e., some portion of the translational kinetic energy is transferred to the rotational kinetic energy.However, this energy exchange is almost negligible for the opposite rotation case.This implies that the rotational kinetic energy gained during the collision process is critical for overcoming the energy barrier associated with nucleation.Another interesting scenario is collision of two pulses with negative rotation triggered by two tensile impulses.In this case, the energy exchange is negligible.As a result, nucleation cannot be initiated (SI Appendix, Fig. S9).

Effects of propagation distance on nucleation
Since the pulses are triggered at the boundary and collide at the center of the structure, it is expected that the propagation distance can affect the wave interactions during the collisions, and therefore may affect the nucleation.We repeat the above analysis for systems with different sizes to examine this effect.The results, as reported in SI Appendix (Fig. S10), show that the critical amplitude A c increases significantly as the size increases.We observe dispersion, especially in the direction perpendicular to propagation, which is qualitatively similar to the expected 2D dispersion behavior observed previously [21].As a result, its amplitude spatially decays as it propagates through the media.In contrast, the critical energy barrier E nu c does not change in an appreciable way, which indicates that the energy barrier for inducing a nucleation is a local quantity, and therefore there is no statistically significant change to the energy barrier.

Collisions of pulses at other angles
Finally, we consider the effects of propagation direction on the ability of colliding pulses to nucleate a new phase (Fig. 4).The circular shape of the system allows facile excitation of pulses along arbitrary directions of propagation.For example, by applying impacts at the left and top boundary, the two pulses can propagate along both the x and y principal axes (i.e., the positive x direction and the negative y direction, respectively).As shown in Fig. 4B, the two pulses nucleate a new phase during their collision.In this case, the nucleation can be induced at impact amplitude A 0 = 0.292, which is lower than the critical amplitude of a head-on collision (replotted in Fig. 4A).In addition, we observe that, after nucleation, the new phase grows predominantly along the diagonal, at 45 • relative to the x and y axes.We refer to such pulses, traveling along the x or y axes, as mode I pulses.Another feasible propagation direction is along the diagonals (referred to as mode II pulses), a direction previously found to support the propagation of vector solitons in monostable systems of rotating squares [21].Fig. 4C shows a head-on collision between impulses propagating along this direction.Mode-I pulses travel much faster than mode-II pulses under the same impact amplitude, and the wave speeds of both modes slightly decrease as the impact amplitude increases (SI Appendix, Fig. S12).With the above observations from Fig. 4C, we demonstrate that the head-on collisions of two mode-II pulses can initiate a nucleation with impact amplitude A 0 = 0.278.Then, the new phase grows predominantly along the diagonal at −45 degrees.Fig. 4D shows collision of two mode-II pulses propagating along principal axes oriented to one another at 90 degrees for A 0 = 0.24.
Interestingly, we report in Fig. 4E that a mode-I pulse can collide with a mode-II pulse at nearly 135 degrees to initiate a nucleation for A 0 = 0.302 (note that the pulse of mode I is delayed by ∆T = 16 to compensate the speed difference between the two modes).Lastly, Fig. 4F shows collision of a mode-I pulse and a mode-II pulse propagating along directions oriented 45 degrees with respect to one another for A 0 = 0.314 and ∆T = 10.It is also worth noting that these various collisions can lead to nucleation with different shape, resulting in rich propagation characteristics of the phase transition as described above.

Conclusion
In conclusion, we have experimentally and numerically investigated phase transitions in macroscopic mechanical metamaterials, analogous to classical solid-solid phase transitions in crystals.First, we have experimentally confirmed and numerically corroborated the existence of phase transitions, which can propagate in the form of transition waves in 2D rotating-squares structures.We have identified the fundamental requirements for induc-ing nucleation, including the energy threshold and the critical nucleus size.More importantly, we have found a fundamentally new way to initiate these phase transitions, i.e., by colliding two soliton-like pulses.This allows nucleation to occur at arbitrary locations in the metamaterial, which may have significant utility in facile control of shape-morphing structures.Therefore, this work not only contributes fundamentally to the understanding of nonlinear waves, and particularly how collisions of one type of nonlinear wave can induce formation of another type, but could also open new doors for the design of tunable, shape-transforming, and deployable structures.
By fitting the experimental data using Eqs.S2-S4 (red lines in Fig. S3), we obtain the parameters for the hinge components: K j = 2.5 × 10 −4 for the linear torsional stiffness, and A = 2 × 10 −4 and α = 8.5 for the Morse potential.With these parameters, we can approximate the multistable energy landscape of the hinge as shown in Fig. 1(b) in the main text.

Dynamic testing
To experimentally demonstrate phase transformations, we use a 10-column by 10-row sample on a plastic surface (Fig. S4(a)).Quasistatic loading is applied to the two vertical hinges connecting the center four squares at the nucleation site.Note, the squares at the edges do not have magnets, to prevent unintended nucleation at the edges induced by boundary effects.Figure S4(b) and (c) show a detailed view of the center four squares and friction-reducing feet (MakerGear M2, PLA), respectively.The phase transformation is recorded using a high-speed camera (Photron FASTCAM Mini AX) at 6400 frames per second.Diamond-shaped markers are placed at the center of each square to allow tracking of the rotation and displacement of the squares, using a custom Python script (Fig. S4(b)).In Fig. S5, we plot the experimentally measured angles of the four squares highlighted in the inset of Fig. S5, showing the transition from the initial phase to the new phase (i.e., Phase R, with θ R ≈ 45 • ).

Equations of Motion
Based on the discrete model introduced in the main text, the Hamiltonian of a 2D rotating-squares system can be written as where L = a 2 cos θ0 is half of the diagonal length of the square.Then, Hamilton's equations read From Eq. S6 to Eq. S9, the equations of motion (EOMs) for the square at site (n, m) can be derived as where ∆θ n±1,m±1 = θ n,m + θ n±1,m±1 + 2(θ 0 − θ Lin ).Note, we define the positive direction of rotation with alternating sign for neighboring squares.
By introducing where TMorse = T M orse /(K l a 2 ).The dimensionless EOMs of the system can be obtained by considering Eqs.S13-S15 for all squares.Then, full-scale simulations can be conducted by numerically solving the system's EOMs using the fourth order Runge-Kutta method (via the Matlab function ode45 ).Based on preliminary numerical results, we observe that, after a transition wave is initiated, squares in the new phase can undergo large oscillations due to the energy release from the initial Phase C to the new Phase R. To account for the disspision observed in the experiments, we include damping in the simulations by introducing the following simple viscous damping terms in the EOMs: ∂T , and c θ = λ θ ∂θ ∂T , in which λ u and λ v are damping coefficients for translational motion in the x and y directions, respectively, and λ θ is the damping coefficient for rotational motion.Damping is added only after the new phase is formed in the simulations (put numbers).

Anisotropy of the 2D transition wave
As reported in the main text (Fig. 1(d)), a transition wave triggered at the center of our system propagates outward anisotropically.The wave fronts propagate along the diagonals of the system (i.e., ±45 • with respect to the x axis).To further corroborate this observation, we extract and plot in Fig. S6(b) the spatial profiles at T = 40 for all three degrees of freedom (i.e., displacement u and v, and angle θ) along the horizontal and diagonal directions, as indicated by the black and magenta dots in Fig. S6(a), respectively.In Fig. S6(c) and (d), we display the contour plots of the spatio-temporal data of the angles along the horizontal and diagonal directions, respectively.The transition wave propagates considerably faster along the horizontal direction.

Energy threshold for inducing a nucleation quasistatically
As discussed in the main text, the existence of the critical angle θ c suggests that there is an energy threshold E c .Once the energy threshold is reached, a critical nucleus can be formed (i.e., 2 × 2 squares in the new phase R).We consider four different, but concentric, square clusters A (this is where the rotations are applied), B, C, and D, as shown in Fig. S7.Then, we plot the dimensionless energy (normalized by Ē = K l a 2 ) as a function of angle θ 0 + θ in for the four clusters during the whole quasistatic loading process (i.e., until cluster A is fully rotated into the new Phase R). we note that, when the quasistatic loading is present, a phase transition cannot be induced before cluster A is fully in Phase R. We indicate the critical angle θ c by the vertical dashed line.Clearly, each cluster features an energy barrier E i c at a certain angle θ i c , in which i = 1, 2, 3, 4 correspond to cluster A, B, C, and D, respectively.Moreover, we note that the critical angle θ c is located between θ 2 c and θ 3 c , which implies that a nucleation can be triggered after cluster B overcomes its energy barrier E 2 c .Thus, the energy threshold for inducing a nucleation under quasistatic loading conditions is identified as E c = E 2 c .

Numerical determination of critical nucleus size
Fig. S8 shows how we determine the critical nucleus size via full-scale simulations for two other sets of parameters.Specifically, we sweep the size of squares that are quasistatically rotated into the new phase (Phase R), starting from 2 × 2 squares at the center, until a phase transition is triggered and then propagates.For (K 1 = 0.2, K 2 = 0.0336, β = 3.0568), we numerically determine the critical nucleus size as 6 squares with a rectangular shape as shown in Fig. S8(b).For (K 1 = 0.2, K 2 = 0.0428, β = 3.0593), we numerically determine the critical nucleus size as 12 squares with a "+" shape as shown in Fig. S8(d).We note that this numerical approach becomes inefficient in cases where a set of parameters leads to a large critical nucleus size.

Head-on collision of two pulses triggered by tensile impulses
We show in Fig. S9 the simulation result for a head-on collision of two pulses with same (negative) rotation, which is obtained with two tensile impulses for A 0 = 0.306.Fig. S9(a) displays snapshots of the wavefield before collision at T = 15, during collision at T = 28, and after collision at T = 35.Fig. S8(b) gives a spatiotemporal plot of the angle of the squares extracted along the propagation direction, and Fig. S9(c) gives the total kinetic energy of the system as a function of time.In this case, nucleation does not occur and the energy exchange between the the two components(i.e., translational and rotational) of the kinetic energy is negligible.

Effect of propagation distance on collision-induced nucleation
To explore the effect of propagation distance on collision-induced nucleation, we consider three circular systems with different diameters D = 24, 30, and 36 (note that D = 30 is the reference case studied in the main text).In Fig. S10(a)-(f), we report the snapshots of the wavefields and the energy of the nucleus highlighted in maroon for D = 24, 30, and 36.Based on the full-scale simulations, we numerically identify the critical energy barrier E nu c , the critical impact amplitude A c , and the critical total input energy E in c for the three cases, which are reported in Fig. S10(g).As expected, the critical impact amplitude A c and total input energy E in c increase as the diameter increases, because the nonlinear pulse spreads in the 2D domain, and therefore its amplitude spatially decays as it propagates through the media.In contrast, the critical energy barrier E nu c shows no statistically significant change (the small differences may be caused by inevitable numerical errors).
To further investigate the spreading of the pulses mentioned above, we consider the propagation of a single pulse.Fig. S11(a) shows snapshots from the numerical simulation of a single pulse propagation at normalized times T = 13.9, 20.8, 27.8, and 34.7, and the corresponding spatial profiles of the pulse along its propagation direction are given in Fig. S11(b).We observe dispersion, especially in the direction perpendicular to propagation, which is qualitatively similar to the expected 2D dispersion behavior observed previously [21].As a result, the amplitude of the pulse decreases as it propagates through the media.

Characterization of the anisotropic behavior of the nonlinear pulses
Similar to the previous discussion of transition waves in 4.1, we show in Fig. S12 the contour plots for mode-I and mode-II pulses using the spatiotemporal data of the angles for two different impact amplitudes (A 0 = 0.1 and 0.3).From these contour plots, we can approximately calculate the wave speed for each case, as reported in Fig. S12.The wave speed of mode I is much faster than that of mode II.Moreover, the wave speeds associated with both modes slightly decrease as the impact amplitude increases from A 0 = 0.1 to 0.3.Moreover, we reported in Fig. S13 the snapshots for impact angle of 30 • .We find that the wave separate into two modes with different wave speeds.Comaparing Fig. S13(a) and (b), we observe that this separation behavior is more pronounced in a larger structure.These findings are consistent with previous work on a monostable system of rotating squares [21].

3 FIG. 1 :
FIG.1:(A) Schematic of a four-square building block of the metamaterial and (B) its multistable potential energy landscape.(C) Optical snapshots of an experimental specimen with the center four squares subjected to quasistatic rotation via the application of the force F ; this causes the formation of a new phase and its eventual growth outward through the rest of the metamaterial.(D) Snapshots of quasistatic nucleation and growth observed via numerical simulation for a system comprising 30 × 30 squares.The positive rotational direction is defined in a way that rotates the squares from the initial Phase C to the new Phase R.

CFIG. 2 :
FIG. 2: Head-on collisions of two soliton-like pulses.(A) Snapshots of wavefields for A0 = 0.306 ≡ Ac: (i) before collision at T = 15, (ii) during collision at T = 28.7,(iii) nucleation at T = 35, and (iv) phase transition at T = 100.(B) Energy of the cluster at the nucleation site as a function of time, suggesting an energy barrier E nu c in the total energy curve.(C) Spatiotemporal plot obtained from the numerical simulation, showing the angle θ for squares along the propagation direction (x axis) as a function of time.(D) Control of the location of nucleation via timing of the impulses for (i) ∆T = 10 and (ii) ∆T = 20, where ∆T is the time delay of the impact on the left boundary with respect to the impact on the right boundary.

FIG. 3 :
FIG. 3: (A) Head-on collision of two pulses with the same rotational direction for A0 = 0.3 < Ac. (i) Snapshots of wavefields before collision at T = 15 and after collision at T = 35 and 40, resulting in no phase transition.(ii) Spatiotemporal plot extracted from the numerical simulation, showing the angle θ for squares along the propagation direction as a function of time.(B)Head-on collision of two pulses with the opposite rotational direction for A0 = Ac.(i) Snapshots of wavefields before collision at T = 15 and after collision at T = 28 and 35, showing that the pulses pass through each other.(ii) Spatiotemporal plot obtained from the numerical simulation, showing the angle θ for squares along the propagation direction as a function of time.(C) Kinetic energy of the whole structure as a function of time for a head-on collision of pulses for A0 = Ac with (i) same rotational directions and (ii) opposite rotational directions (the vertical dashed lines indicate the time when the two pulses collide); the former case exhibits a significant exchange between the transitional and rotational components of the kinetic energy.

FIG. 4 :
FIG. 4: Different collision scenarios.(A) Head-on collision of two mode-I pulses with A0 = 0.306.(B) Perpendicular collision of two mode-I pulses, with A0 = 0.292.(C) Head-on collision of two mode-II pulses along the diagonal, with A0 = 0.278.(D) Perpendicular collision of two mode-II pulses, with A0 = 0.24.(E) Collision of a mode-I pulse and a mode-II pulse propagating along directions oriented 135 • with respect to one another, with A0 = 0.302.(F) Collision of a mode-I pulse and a mode-II pulse propagating along directions oriented 45 • with respect to one another, with A0 = 0.314.
FIG. S2: (a) Tensile tests were conducted using an Instron model 68SC-5 equipped with a custom aluminum fixture.(b) Schematic of tensile test on a 2 × 2 unit without magnets (the rods are inserted through the holes of the squares).(c) Schematic of tensile test on a 2 × 2 unit with magnets (the rods with bases are attached to both sides of the squares).Green arrows indicate the direction of the applied force; orange arrows indicate the direction of rotation of each square.(d) Schematic of a 2 × 2 unit under tensile loading.
FIG. S4: (a) The experimental specimen rests on a support table.The orange circle indicates the nucleation site subjected to loading.(b) Top view of the center four squares, with diamond-shaped markers adhered to the center of each square.(c) Detailed view of a square with plastic feet (to reduce friction).
FIG. S5: Experimentally measured angles of the four squares indicated by the yellow dashed lines in the inset.The three vertical lines correspond to the times of the later three optical images displayed in Fig.1(c) in the main text.

1 FIG
FIG. S6: Anisotropy of the transition wave.(a) Snapshots of the wavefield at T = 0 (T is set to 0 when the 2 × 2 squares at the center are rotated to Phase R) and 40.(b) Spatial profiles along the horizontal (black dots) and diagonal (magenta dots) directions for angle θ, displacement u, and displacement v. Contour plots of the spatio-temporal data for the angle along the (c) horizontal and (d) diagonal directions.

S14
FIG. S9: Head-on collision of two pulses triggered by tensile impulses.(a) Snapshots of the wavefield before collision at T = 15, during collision at T = 28, and after collision at T = 35.(b) Spatiotemporal plot obtained from the numerical simulation, showing the angle θ for squares along the propagation direction as a function of time.(c) Kinetic energy of the whole structure as a function of time.