Metamaterials with amplitude gaps for elastic solitons

We combine experimental, numerical, and analytical tools to design highly nonlinear mechanical metamaterials that exhibit a new phenomenon: gaps in amplitude for elastic vector solitons (i.e., ranges in amplitude where elastic soliton propagation is forbidden). Such gaps are fundamentally different from the spectral gaps in frequency typically observed in linear phononic crystals and acoustic metamaterials and are induced by the lack of strong coupling between the two polarizations of the vector soliton. We show that the amplitude gaps are a robust feature of our system and that their width can be controlled both by varying the structural properties of the units and by breaking the symmetry in the underlying geometry. Moreover, we demonstrate that amplitude gaps provide new opportunities to manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton splitters and diodes.

F ollowing John Scott Russell's observation of nonlinear water wave packets propagating with stable shape and constant velocity in the Union Canal in Scotland 1 , the unique properties of solitons have been studied and exploited in many areas of science and engineering [2][3][4] . Focusing on mechanical systems, granular crystals have been found to provide an effective platform for the propagation of highly nonlinear solitary waves [4][5][6][7] and have enabled the design of impact mitigation layers 8 , lenses 9 , switches 10 , and non-destructive detection techniques 11 . However, the solitons observed in granular media are of scalar nature and lack the multiple polarizations typical of elastic waves propagating in solid materials.
Polarization is an important property of vector waves like electromagnetic and elastic waves. The ability to control the polarization of light has enabled a broad range of applications, including optical communications, spectroscopy, and microscopy 3,12,13 . Moreover, a broad range of new functionality has been observed in elastic systems with architecture designed to manipulate both the longitudinal and shear polarizations of linear elastic waves 14,15 . While the field initially focused on linear elastic vibrations, it has been recently shown that highly deformable mechanical metamaterials can support elastic vector solitons with two polarizations-one translational and one rotational 16 , but the potential of such solitary waves in applications is unknown and remains to be explored.
Here, we combine experimental, numerical, and analytical tools to demonstrate that elastic vector solitons provide unique opportunities to manipulate the propagation of large amplitude vibrations. Specifically, we show that in mechanical metamaterials based on rotating rigid units, both the amplitude of the propagating waves and the symmetry of the underlying building blocks can be used to significantly alter the coupling between the two polarizational components of the vector solitons. We find that such control of the coupling strength results in the emergence of a new phenomenon: the formation of amplitude gaps for solitons. Notably, this new effect can be exploited to realize devices capable of controlling and manipulating the propagation of large amplitude vibrations in unprecedented ways, as demonstrated by the design of soliton splitters and diodes.

Results
Metamaterial design and characterization. Our system consists of a long chain of 2 × 50 rigid crosses made of LEGO bricks 17 with arm length l a = 19 mm connected by thin and flexible hinges made of polyester plastic sheets (Artus Corporation, NJ) with length l h = 4 mm and thickness t h = 0.127 mm, resulting in a spatial periodicity a = 2l a + l h = 42 mm ( Fig. 1a-see Supplementary Note 1 and Supplementary Movie 1 for details on fabrication). To investigate the propagation of elastic pulses in the system, we place the chain (supported by pins to minimize frictions) on a smooth horizontal surface and use an impactor excited by a pendulum to hit the mid-point at its left end (see Fig. 1b-top and Supplementary Movie 2). We apply different input signals to the chain by varying both the initial height of the striking pendulum and the distance traveled by the impactor and find that all of them initiate simultaneous rotation and displacement of the rigid units, with each pair of crosses in a column sharing the same displacement and rotating by the same amount, but in opposite directions (i.e., if the top unit rotates by a certain amount in clockwise direction, then the bottom one rotates by the same amount in counter-clockwise direction, and vice versa). To monitor the displacement, u i , and rotation, θ i , of the i-th pair of crosses along the chain as the pulse propagates, we use a high speed camera (SONY RX100V) and track markers via digital image processing (see Supplementary Note 2).
In Fig. 1c, we report the evolution of the rotation and longitudinal displacement of the second and fortieth pairs of crosses as a function of time during two different experiments. We find that when the amplitude of the input signal is large (A 2 = max(θ 2 (t)) = 13°in experiment #1) the pulse that propagates through the system conserves its amplitude and shape in both degrees of freedom. Differently, for inputs with small amplitude (A 2 = 5°in experiment #2), the output signal is severely distorted compared to the input one (see Supplementary Movie 3). While in Fig. 1c we focus on two representative experiments, all our experimental results are summarized as triangular markers in Fig. 1e, f, where we present the measured transmission, A 40 /A 2 (with A 40 = max(θ 40 (t))), and cross-correlation of θ 2 (t) and θ 40 (t) as a function of the amplitude of the input signal, A 2 . We find that if A 2 ≳ 7°both the transmission and the cross-correlation approach unity, suggesting that for large enough input signals, the system supports the propagation of elastic vector solitons. However, for input amplitudes below~7°a transition occurs and both the transmission and the cross-correlation significantly and systematically decrease. This indicates that our system might only support the propagation of solitary waves with amplitude above a certain threshold, manifesting a gap in amplitude for 0°≲ A 2 ≲ 7°.
Discrete and continuum models. To better understand these experimental results, we establish a discrete model in which the crosses are represented as rigid bodies of mass m and rotational inertia J. Guided by our experiments, we assume that the system has an horizontal line of symmetry and assign two degrees of freedom (u i and θ i ) to the top unit of the i-th pair of crosses (Fig. 1d). As for the flexible hinges, they are modeled using a combination of three linear springs: their stretching is captured by a spring with stiffness k l ; their shearing is described by a spring with stiffness k s ; their bending is modeled by a torsional spring with stiffness k θ . Under these assumptions, the dimensionless equations of motion for the i-th top unit are given by (see Supplementary Note 3) where U i = u i /a, T = t ffiffiffiffiffiffiffiffiffiffi k l =m p , K θ = 4k θ = k l a 2 ð Þ, K s = k s /k l and α = a ffiffiffiffiffiffiffiffiffiffiffiffiffiffi m=ð4JÞ p are all non-dimensional parameters, which in our system are measured as K s = 0.02, K θ = 1.5 × 10 −4 , and α = 1.8. Note that, as shown in Fig. 1d, to facilitate the analysis in our model we define the positive direction of rotation alternatively for neighboring crosses (i.e., if for the i-th top unit a clockwise rotation is positive, then for the (i − 1)-th and (i + 1)-th ones counterclockwise rotation is positive).
We start by numerically solving Eq. (1) using the Runge-Kutta method (the code implemented in MATLAB is available online). In our numerical analysis we consider a chain comprising 150 pairs of crosses, apply a longitudinal displacement with the form U input = b + b tanh[(T − T 0 )/w] to the mid-point at its left end (see Supplementary Note 3), and implement free-boundary conditions at its right end. In Fig. 1e, f we report as gray circular markers the results of 480 analyses in which we systematically change the applied displacement U input (with b ∈ [0, 0.75], w ∈ [50, 100] and T 0 = 400). In good agreement with our experimental data, we find that for A 2 ≲ 6°the signal does not preserve its amplitude and shape as it propagates through the structure. As such, these numerical results also point to the existence of an amplitude gap for solitons.
Amplitude gaps for solitons. To confirm the existence of such amplitude gap, we further simplify Eq. (1) to obtain an analytical solution. To this end, we assume that the wavelength of the propagating waves is much wider than the cell size and that θ ( 1, take the continuum limit of Eq. (1) and retain nonlinear terms up to the third order, obtaining (see Supplementary Note 4) where X = x/a (x denoting the initial position along the chain) and U(X, T) and θ(X, T) are two continuous functions of X and T.
It is easy to show that Eq. (2) admits an analytical solution in the form of an elastic vector soliton with two components 18 where c is the pulse velocity, and A and W are the amplitude and width of the solitary wave, which can be expressed in terms of c and the structural parameters as (see Supplementary Note 4) At this point it is important to note that, since the width W needs to be real-valued Condition in Eq. (6) clearly indicates that our system has an amplitude gap for solitons, since only solitary waves with amplitude greater than A upper = 6.55°are physically admissible   20 and so-called "sonic vacuum" found in not precompressed granular chains 21 . The amplitude gaps for solitons reported in our work are robust features of the system, intrinsically determined by its architecture and its ability to support elastic vector solitons. Looking into the mechanism behind the emergence of this amplitude gap, it is important to note that the propagation of vector solitons requires a strong coupling among different polarizations 22,23 . However, Eq. (2) show that the coefficients of the coupling terms in our structure are proportional to θ, so that large enough rotations are needed in order to activate them and enable the propagation of vector solitons (see Supplementary Note 5). Finally, to further verify the validity of our continuum model, in Fig. 1g we compare the relation between A and c as predicted by our analysis (magenta line) and measured in our experiments (triangular markers) and numerical direct simulations (circular markers) and find good agreement among all three sets of data.
Enhanced tunability via symmetry breaking. Equation (6) indicates that in our system the width of the amplitude gap can be controlled by changing K s , K θ , and α (see Supplementary Fig. 14). More excitingly, the tunability and functionality of the proposed mechanical metamaterial can be further enhanced by breaking the symmetry in each rigid cross to alter the coupling strength between the two polarizational components. In our system this is achieved by shifting neighboring horizontal hinges by a tan φ 0 in vertical direction (see Fig. 2a and Supplementary Movie 4). While for the chain with all horizontal hinges aligned (for which φ 0 = 0°) the energy cost to rotate any unit in clockwise and counter-clockwise directions is identical, the hinges shifting (i.e., φ 0 ≠ 0°) introduces a disparity between the two directions of rotation. Under compression in longitudinal direction, for all units of the shifted chain with the left hinge higher than the right one it is energetically more favorable to rotate in clockwise direction, while for the ones with a lower left hinge rotations in counter-clockwise direction are preferred ( Fig. 2a and Supplementary Fig. 6). By extending our analytical model to units with φ 0 ≠ 0°and assuming for each unit the positive direction of rotation to be the one that is naturally induced by compression, we find that such disparity introduced by the asymmetry is reflected in the amplitude gap (see Supplementary Note 5). For the aligned chain (i.e., for φ 0 = 0°) the upper (A upper ) and lower (A lower ) limits of the amplitude gap are identical in magnitude (i.e., A lower = −A upper , so that in Fig. 1c, d we only show A upper ). By contrast, as a result of the bias introduced by the hinges shifting, when φ 0 increases, A lower j jand A upper become larger and smaller, respectively (see Supplementary Note 5). We also find that a critical angle φ cr 0 exists at which A upper vanishes. In structures with φ 0 >φ cr 0 all solitons that induce an energetically favorable rotation at the i-th pair of crosses can propagate through the system, regardless of their magnitude. The validity of our analysis is confirmed by experiments and numerical simulations conducted on a chain comprising 2 × 50 and 2 × 150 crosses characterized by φ 0 = 5°, respectively. For such system our continuum model predicts A upper = 0°and A lower = −20.91°. In agreement with this analytical prediction, the amplitude transmission ratio and signal shape cross-correlation between the input, θ 2 (t), and the output, θ 40 (t), measured in both experiments and discrete simulations significantly drop when the input   (Fig. 2d, e). Moreover, the numerical and experimental data also closely match the amplitude-velocity relation predicted by our continuum model (Fig. 2f), confirming that the hinges shifting induces an asymmetric gap.
Functional devices based on amplitude gaps. Having discovered the existence of amplitude gaps in our system, we next focus on how such effect can be exploited to design new functional devices to control mechanical signals, and, therefore, to provide new opportunities for phononic computing and mechanical logic. In analogy to beam splitters 24 , which are widely used in photonics to split an incident light beam into two or more beams, we start by taking advantage of amplitude gaps to design a soliton splitter-a device capable of splitting an incoming elastic vector soliton into a transmitted and a reflected ones. Remarkably, this can be achieved by simply introducing a pair of stiffer hinges within an aligned chain with φ 0 = 0°. To demonstrate the concept we take our 2 × 50 sample with φ 0 = 0°and introduce two stiffer hinges (made of polyester sheets with thickness t d h = 0.635 mm) to connect the 24th and the 25th pairs of units (Fig. 3a). We find that, if the amplitude of the input signal is large enough to be outside the amplitude gap, the excited solitary wave is split into two pulses by the pair of stiffer hinges (see Fig. 3b). To better understand the nature of the transmitted and reflected waves, we integrate Eq. (1) to simulate the response of a chain comprising 1000 units with φ 0 = 0 and a pair of stiffer hinges (with stiffness K d s and K d θ ) connecting the 500th and 501st rigid crosses. The numerical results for a chain with K d s =K s = K d θ =K θ = 30 are shown in Fig. 3c, d. We find that the pair of stiffer hinges split the incoming soliton into two pulses that propagate with stable shape and constant velocity and that no trains of solitons are generated. As for the radiation of linear waves, we find that the interaction between the incoming solitary wave and the stiffer pair of hinges generates only translational vibrations, since the frequency content of our solitons overlaps with a low-frequency band gap for rotation. However, we estimate that only 4% of the energy carried by the incoming soliton is finally transferred to translational linear vibrations (see Supplementary Notes 8-10). As such, these results clearly indicate that our simple structure acts as a splitter for solitons. Note that this behavior is remarkably different from that previously observed in unloaded granular chains, where heterogeneities have been found to split the propagating solitary wave into trains of solitons and to generate stress oscillations localized near the impurities 25 . Such difference is due to the presence of the amplitude gap, which in our mechanical metamaterial prevents fragmentation of the propagating pulse by suppressing the propagation of small amplitude solitons. To demonstrate this important point, we simulate the response of a 2 × 1000 chain with φ 0 = 5°and a pair of stiffer hinges in the middle. Note that this structures enables the propagation of solitary waves of any amplitude if they induce energetically favorable rotations (since A upper = 0°and there is no amplitude gap for such waves). We find that, when such waves are excited and hit the pair of stiffer hinges, trains of pulses are generated (see Fig. 3e, f), confirming the important role played by the amplitude gap. Finally, our numerical results also indicate that our soliton splitter is a robust device, since the ratio between the energy carried by the transmitted and reflected solitons only depends on the ratio K d s =K s = K d θ =K θ and not on the amplitude of the input signal, with the amount of reflected energy monotonically increasing with the stiffness ratio (see Supplementary Fig. 23). Further, we design a mechanical diode 26 for solitary waves-a system that is transparent to solitons incoming from one direction but blocks those propagating in the other one. While such nonreciprocal wave transmission has been previously reported for periodic waves [27][28][29] , irreversible transition waves 30 and wave packets 31 , here we extend the concept to solitary wave pulses. To achieve this, we introduce a few pairs of crosses with φ 0 ≠ 0 within a chain with φ 0 = 0°. More specifically, our diode comprises two external sections with 2N and (2N + 1) pairs of crosses characterized by φ 0 = 0 and a central portion consisting of 2N a pairs of crosses with φ 0 = 5° (Fig. 4a). Experiments conducted on a sample with N = 12, N a = 3 and the section with 2N + 1 units placed on the left show that a pulse initiated at the left end propagates through the entire structure (Fig. 4b, c), while a solitary wave excited at the right end is completely reflected by the boundary between the regions with φ 0 = 0°and 5° (Fig. 4d, e). This remarkable behavior is induced by the asymmetric amplitude gap of the region with φ 0 = 5°. Solitons excited at the left end of the chain induce energetically favorable rotations at the units with φ 0 = 5°and, since there is no amplitude gap for such waves (i.e., A φ 0 ¼5 upper = 0), they are able to propagate through the entire chain (Fig. 4f). In contrast, solitons initiated at the right end of the chain result in energetically unfavorable rotations for the crosses with lower , they are almost completely blocked by the boundary between aligned and shifted crosses (Fig. 4f). To explore the performance of our j j , as a function of the input amplitude, A 40 j j, for pulses excited at the right end of the chain mechanical diode, we apply different input signals at the left and right end of the chain in both experiments and discrete simulations. We find that for all pulses with amplitude larger than A φ 0 ¼0 upper ¼ 6:55 initiated at the left end of the system the transmission, A 40 j j= A 10 j j, approaches unity (Fig. 4g). Differently, when the excitation is applied at the right end of the chain, the transmission, A 10 j j= A 40 j j, is close to zero even if the amplitude of the input signal is outside the gap of the region with φ 0 = 0 (i.e., upper ). However, as typically observed in electronic and thermal diodes 32 , if the amplitude of the pulses becomes too large, the diode experiences a condition known as breakdown. As a result, solitary waves with amplitude larger than A br ≈ 15°p ropagate through the diode (i.e., if A 40 j j>A br ≈ 15°, then A 10 j j= A 40 j j~0.6-see Supplementary Note 11 for a detailed numerical study on the dependency of A br on φ 0 and N a ).

Discussion
In this study, we have experimentally observed, numerically simulated, and mathematically analyzed the existence of amplitude gaps for elastic vector solitons in highly deformable mechanical metamaterials consisting of rigid units and elastic hinges. First, we have shown that such amplitude gaps can be tuned by altering both the structural parameters and the symmetry of the crosses. Then, we have demonstrated that amplitude gaps can be exploited to design clean splitters and diodes for highly nonlinear solitary waves. In recent years, many strategies have been proposed to manipulate the propagation of elastic waves 29,33 , enabling a wide range of applications such as spatial guiding [34][35][36] , frequency filtering 37-39 , noise/impact mitigation 40,41 , and non-reciprocal transmission 14,42 . However, the vast majority of devices focus on small-amplitude vibrations and take advantage of spectral gaps in frequency 29,33,43 . As such, our study on amplitude gaps for highly nonlinear solitary waves adds a whole new dimension to our ability to design structures and materials with tailored dynamic behavior and open avenues for potential technological breakthroughs.
Competing interests: The authors declare no competing interests.
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Supplementary Note 2: Testing
To investigate the propagation of pulses in our samples, we place the chain on a smooth horizontal surface (see Supplementary Figure 3a Figure 3c). We find that all of them initiate simultaneous rotation and displacement of the rigid units, with each pair of crosses sharing the same displacement and rotating by the same amount, but in opposite directions (i.e. if the top unit rotates by a certain amount in clockwise direction, then the bottom one rotates by the same amount in counter-clockwise direction, and vice versa). To monitor the displacement, u i , and rotation, θ i , of i-th pair of crosses along the chain as the pulses propagate, we use a high speed camera (SONY RX100V) recording at 480 fps and track four markers placed on the external arms of each pair of crosses (see Supplementary Figure 4) via digital image correlation analysis [1]. More specifically, the longitudinal displacement u i and rotation θ i of the i-th pair of rigid units is obtained as i (0) are the coordinates of the γ-th marker placed on the i-th pair of rigid units at time t and that time t = 0 (i.e. before the impact), respectively.
Finally, we note that, while for the aligned chain the energy cost to rotate any unit in clockwise and counter-clockwise directions is identical, for the shifted chain there is a disparity between the two directions of rotation. Under compression in the longitudinal direction, for all units of the shifted chain with the left hinge higher than the right one, it is energetically more favorable to rotate in the clockwise direction, while for the ones with a lower left hinge, rotations in counter-clockwise direction are preferred. In our experiments, we can excite any cross of the shifted chain in either direction of rotation by changing the type of unit placed at the left end of the structure. To excite energetically favorable rotations (i.e. to rotate all crosses with the left hinge higher than the right one in clockwise direction and all crosses with the right hinge higher than the left one in counter-clockwise direction), we place a pair of crosses with ϕ 0 = 0 at the left end of the chain (see Supplementary Figure 5a

Supplementary Note 3: Discrete model
Our system consists of a long chain of 2×N crosses with center-to-center distance a connected by thin and flexible hinges (see Supplementary Figure 6). In our model we focus on the most general case in which neighboring horizontal hinges are shifted in vertical direction by a tan ϕ 0 (see Supplementary Figure 6-a). Note that the response of the configuration with all horizontal hinges aligned (see Supplementary Figure 6-b) can be then simply obtained by setting ϕ 0 = 0. Since in this work we focus on the propagation of longitudinal nonlinear waves along the chain, we assign two degrees of freedom to each rigid cross: the longitudinal displacement u and the rotation in the x−y plane θ. Moreover, guided by our experiments, we assume that each pair of crosses shares the same displacement and rotates by the same amount, but in opposite directions (i.e. if the top cross rotates by a certain amount in clockwise direction, then the bottom one rotates by the same amount in counter-clockwise direction, and vice versa). As such, two degrees of freedom are assigned to the i-th pair of crosses: the longitudinal displacement u i and the rotation θ i (see Supplementary Figure 6a and b). Moreover, to facilitate the analysis, we define the positive direction of rotation alternatively for neighboring units along the x-axis. Specifically, for each cross we assume the energetically favorable direction of rotation to be the positive one. As such, for the i-th top unit (for which the left horizontal hinge is higher than the right one -see Supplementary Figure 6a) a clockwise rotation is positive, while for the i − 1-th and i + 1-th top ones (for which the left horizontal hinges are lower than the right ones -see Supplementary Figure 6a), counterclockwise rotations are considered positive.
As for the hinges, we model them using a combination of three linear springs: (i) their stretching is captured by a spring with stiffness k l ; (ii) their shearing is governed by a spring with stiffness k s ; (iii) their bending is captured by a torsional spring with stiffness k θ (see Supplementary Figure 6a).
Under these assumptions, the equations of motion for the i-th pair of crosses are given by where m and J are the mass and moment of inertia of the rigid crosses, respectively. Next, we introduce the normalized displacement U i = u i /a, time T = t k l /m, inertia α = a/(2 cos ϕ 0 ) m/J and stiffness ratios K θ = 4k θ cos 2 ϕ 0 /(k l a 2 ) and K s = k s /k l . Eqs.

9
(S2) can then be written in dimensionless form as For a chain comprising N pairs of units Eqs. (S3) result in a system of 2N coupled differential equations, which can be numerically solved for a given set of initial and boundary conditions. In this study, we use the 4th order Runge-Kutta method (via the Matlab function ode45) to numerically solve Eqs. (S3) (the code implemented in MATLAB is available online). Moreover, as initial conditions we set U = 0 and θ = 0 for for all pairs of crosses. Finally, we enforce two different sets of boundary conditions: • First, to compare the numerical predictions to our experimental results and verify the relevance of our discrete model, we apply the experimentally extracted displacement, u 2 (t), and rotation, θ 2 (t), signals to the first pair of crosses at the left end of the chain, while implementing free-boundary conditions at the right end; • Second, to generate the numerical results reported in the main text, as in our experiments, we apply a displacement in the longitudinal direction to the mid-point at the left end of the chain (see Supplementary Figure 7). In our simulations we use b ∈ [0, 0.75], w ∈ [50, 100] and choose T 0 = 400 to ensure that U input → 0 at T = 0. Such applied displacement induces simultaneous translation and rotation of the first pair of crosses, which in turn results in a normalized translation of the excited mid-point equal to U 1 + 1/2 sin θ 1 . Finally, is is important to note that, to avoid numerical instabilities, we do not apply U input (t) directly to the mid-point at the left end of the chain, but to a linear spring with stiffness K input = 1 connected to it (note that the numerical results are not influenced by the particular value chosen for K input ). Therefore, U input (t) results in a longitudinal force and in a moment applied to the first pair of crosses. As such, the governing discrete equations for the first pair of crosses become Estimation of K θ , K s and α To connect the discrete model to our experimental sample, we need to estimate the mass of the cross-shaped units (m), their rotational inertia (J) and the spring stiffnesses (k l , k s and k θ ). where r denotes the distance from the center of mass. We find that for the rigid crosses in the aligned structure J = 605 g·mm 2 , while for the crosses in the shifted structure J = 615 g·mm 2 .
Spring stiffness k s and k l : To determine the stiffnesses k s and k l we build a small structure consisting of four parallel hinges connected at both ends to interlocked LEGO plates (see Fig Supplementary Figure 8c-e). We start by fixing one column of LEGO plates and applying a displacement u 1 to the other one in direction perpendicular to the hinges (see Supplementary  Figure 8d), so that the hinges are only subjected to shearing deformations. The stiffness k s is then obtained from the measured force F 1 as (S11) Next, we apply a displacement u 2 to one column of LEGO plates in direction parallel to the hinges (see Supplementary Figure 8e), with u 2 small enough so that the hinges are only compressed axially and do not buckle. The stiffness k l is then obtained from the measured force F 2 as Spring stiffness k θ : To determine the stiffness k θ of the torsional spring we connect an hinge to a fixed base and to a cross made of four LEGO brackets (see Supplementary Figure 8f). We then apply an initial rotation to the systems and let it free to vibrate. k θ can be obtained from the experimentally measured frequency of vibration f using where J = 3000 g·mm 2 is the rotational inertia of the cross with respect to the pivot point.

Numerical results
To verify the relevance of our discrete model, we apply the experimentally extracted input displacement and rotation to the first pair of crosses and then compare the signal recorded at the 40th unit in both experiment and simulations. In Supplementary Figure 9 we present such comparison for the two experiments shown in Fig. 1c of the main text. We find that the pulse profiles and amplitudes are well captured by the discrete model. When the amplitude of the input signal is large as in experiment #1 (for which A 2 = max(θ 2 (t)) = 13 • ), our numerical simulations reveal the propagation of a pulse that conserves its amplitude and shape in both degrees of freedom (see Supplementary Figure 9a). Differently, for inputs with small amplitude as in experiment #2 (for which A 2 = 5 • ), the numerical analyses predict an output signal that is severely distorted compared to the input one, in full agreement with our experimental results (see Supplementary Figure 9b

Supplementary Note 4: Continuum model
Although Eqs. (S3) can be numerically solved to obtain the response of the system, a deeper insight into the dynamics of the system can be achieved by simplifying the equations to derive analytical solutions. To this end, we introduce two continuous functions U (X, T ) and θ (X, T ) that interpolate the displacement and rotation of the i-th pair of crosses located at x i = i a as Assuming that the width of the propagating waves is much larger than the unit cell size, the displacement U and rotation θ in correspondence of the i + 1 and i − 1-th pairs of crosses can then be expressed using Taylor expansion as Substitution of Eqs. (S19) into Eqs. (S3) yields which represent the continuum governing equations of the system. Since these two coupled partial differential equations cannot be solved analytically, guided by our experiments, we further assume that θ ∼ ϕ 0 1, so that By substituting Eqs. (S21) into Eqs. (S20) and retaining the nonlinear terms up to third order, Finally, we introduce the traveling wave coordinate ζ = X − cT , c being the normalized pulse velocity (the dimensional pulse velocity is ca k/m), so that Eqs. (S22) become By integrating Eq. (S23) 1 with respect to ζ we obtain, where C is the integration constant. Since in this study we focus on the propagation of waves with a finite temporal support and do not consider periodic waves, we require that from which we obtain C = 0. Substitution of Eq. (S25) into Eq. (S23) 2 yields sin(2ϕ 0 ), which is the Klein-Gordon equation with quadratic and cubic nonlinearities [2]. Note that Eq. (S27) differs from the equation recently derived to study the propagation of solitary waves in structures comprising a network of squares connected by thin hinges [3]. While in the previous work only terms up to the second order were considered in the continuum model (see Eq. (8) in [3]), here also the third order terms are included. This enable us to investigate waves with amplitude equal or larger than ϕ 0 . The analytical solution of Eq. (S27) exists in the form of where D 1 , D 2 and W are solution parameters. Eq. (S29) defines a solitary wave with characteristic width W and amplitude Next, we determine D 1 , D 2 and W as a function of the geometry of the system and the pulse velocity c. To this end, we substitute the solution (S29) into Eq. (S27) and find that the latter is identically satisfied only if where C 1 , C 2 and C 3 are defined in Eqs. (S28). Finally, the solution for the displacement U is found by integrating Eq. (S25) with θ given by Eq. (S29), As recently found for structures comprising a network of squares connected by thin and highly deformable ligaments [3], Eqs. (S29) and (S32) reveal that our system supports the propagation of elastic vector solitons (i.e. solitary waves with two components -one translational and one rotational -that are coupled together and co-propagate without dispersion). In Supplementary Figure 11a-b we report the solution given by Eqs. (S29) and (S32), assuming that K s = 0.02, K θ = 1.5 × 10 −4 , α = 1.8, c = 0.1 and ϕ 0 = 5 • . We find that the two vector solitons supported by the system are both characterized by positive values of longitudinal displacement U (i.e. both waves are accompanied by a translation from left to rightsee Supplementary Figure 11b). As for the rotational component, one solution is characterized by positive θ and the other one by negative θ (see Supplementary Figure 11a). This means that in one case the top unit of the i-th pair of crosses rotates clockwise, while in the other it rotates counter-clockwise (in both cases neighboring units rotate in opposite directions). It is also important to note that, while the two solutions for U have comparable magnitude, the negative solution for θ (i.e. the one that involves counter-clockwise rotation for the top unit of the i-th pair of crosses) has much larger amplitude than the positive one. Such disparity is due to the asymmetry of the structure with ϕ 0 = 0, which makes rotation in clockwise direction for the i-th unit (for which the left horizontal hinge is higher than the right one) energetically more favorable. Finally, to verify the validity of our analytical solution, in Supplementary Figure  11c-d we compare the analytical solutions to numerical results obtained by direct integration of the full discrete model (Eqs. (S3)). Note that in this set of simulations we consider a chain with 2 × 150 crosses and input our theoretical solutions (i.e. Eqs. (S29) and (S32)) as boundary con-ditions for the first pair of crosses at the left end of the chain, while implementing free-boundary conditions at its right end. We find an excellent agreement between our analytical (lines) and numerical (markers) results, indicating that the assumptions we made, i.e., wave length unit length and θ ∼ ϕ 0 1, are appropriate.

Supplementary Note 5: Amplitude gaps for solitons
Eq. (S29) defines the solitary waves with stable profile that propagate in our system. However, inspection of Eqs. (S29) and (S31) reveals that such waves exist only if: since an imaginary W results in a periodic solution, which violates Eq. (S26); (ii) D 2 is a real number, yielding since we want the solution θ to be real; (iii) the denominator in Eq. (S29) is different from zero, yielding since we require the solution θ to be of finite amplitude. Note that, since D 2 > 0 (see Eq. By substituting Eqs. (S24) and (S28) into conditions (S33), (S34) and (S36) and assuming that c < 1 (note that in all our experiments and simulations c ∈ [0.05, 0.5]) and that ϕ 0 < 0.7137 = 41 • (so that 7 cos(2ϕ 0 ) > 1), we find that solitary waves exist in our system only if Conditions (S37) clearly show that there is an upper limit for the velocity of the propagating solitary waves, i.e., Substitution of condition (S38) into Eq. (S30) yields indicating that solitary waves can propagate in our system only if their amplitude is larger than A upper and lower that A lower . Therefore, conditions (S40) define an amplitude gap for solitons. By substituting Eqs. (S31) and (S28) into (S40), A upper and A lower can be expressed in terms of structural parameters as,  ( Supplementary Figure 12c). We find that, while for ϕ 0 = 0 • (i.e. when all horizontal hinges are aligned) A lower = −A upper , |A lower | and A upper become larger and smaller, respectively, as ϕ 0 increases. This is because the hinges shifting (i.e. ϕ 0 = 0 • ) introduces a disparity between the two directions of rotation. Under compression in the longitudinal direction, for all units of the shifted chain with the left hinge higher than the right one, it is energetically more favorable to rotate in the clockwise direction, while for the ones with a lower left hinge, rotations in the counter-clockwise direction are preferred. As such, pulses that excite energetically favorable rotations can propagate more easily than those inducing energetically unfavorable rotations. We also note that a critical angle exists at which A upper eventually vanishes. In structures with ϕ 0 > ϕ cr 0 all solitons that induce energetically favorable rotations can propagate through the system, irrespectively of their magnitude. In Supplementary Figure 12d we report the evolution of the angle ϕ cr 0 as a function of K s and K θ (assuming α = 1.8). We find that ϕ cr 0 increases for larger values of K θ , while it decreases as K s becomes larger.
Focusing on the mechanism behind the emergence of the observed amplitude gap for solitons, it is important to note that the propagations of vector solitons require a strong coupling among different polarizations [4,5]. As such, we expect such amplitude gaps to emerge when there is weak coupling between the two polarizational components. According to Eqs. (S22), the coupling terms in our system are Eqs. (S44) and (S45) clearly show that if ϕ 0 = 0 (i.e. for the aligned structure), the coefficients of both coupling terms are proportional to θ, so that large enough rotations are needed in order 22 to activate them and enable the propagation of vector solitons. Differently, if ϕ 0 = 0 (i.e. for the shifted structure), the coefficients are proportional to ϕ 0 + θ and the strength of the coupling depends on the direction of rotation. The coupling among U and θ is strong when the wave induces an energetically favorable rotation (since ϕ 0 and θ have the same sign and their effect sums up), resulting in lower values for A uppers as ϕ 0 increases. By contrast, if the wave induces an energetically unfavorable rotation (i.e. θ < 0), large rotations are required to make the coupling strong enough (i.e. θ < 0 has to go more negative to compensate for the bias induced by the positive ϕ 0 ), resulting in large |A lower |.
In Supplementary Figure 13a-b we plot the analytical solution given by Eqs. (S46), assuming K s = 0.0185, K θ = 1.5 × 10 −4 , α = 1.8 and c = 0.1. We find that the two solutions supported by the system are characterized by identical translational components. The rotational components are also identical, but have opposite direction. Note that the two solutions have the same magnitude because in the aligned chain the units do not have a preferential direction of rotation. Finally, in Supplementary Figure 13c-d, we compare the analytical solution to numerical results obtained by direct integration of the full discrete model (Eqs. (S3)). Note that in this set of simulations we consider a chain with 2 × 150 crosses and assign the displacement and rotation signals given by Eqs. (S46) to the first pair of units on the left , while keeping free boundary conditions at the right end. Again, we find an excellent agreement between our analytical (lines) and numerical (markers) results.
As for the amplitude gap, if ϕ 0 = 0, Eqs. (S41) and (S42) reduce to In Supplementary Figure 14 we report the evolution A upper = −A lower as a function of K s , K θ and α. The contour plots indicate that A upper can be tuned by varying either K s or K θ , while α has a more moderate effect.

Supplementary Note 7: Solitons excited by pulling
While in Supplementary Note 5, we consider the case in which condition (S36) is satisfied and the system supports two elastic vector solitons, here we investigate the response of the system when only one elastic vector soliton exists (i.e. when D 1 > D 2 or D 1 < −D 2 ). Let us consider the case D 1 < −D 2 . Existence of such solitary wave requires that Substitution of Eqs. (S24) and (S28) into conditions (S49) yields, Assuming that ϕ 0 < 0.7137 = 41 • (so that 7 cos(2ϕ 0 ) > 1), condition (S50) 3 is only satisfied if cos(2ϕ 0 )(21K θ + 8) − 2 cos(4ϕ 0 ) − 6 − 3K θ < 0 (S51) since positive values of the denominator always lead to c > 1. In Supplementary Figure 15a, we report the region of the ϕ 0 -K θ domain in which condition (S51) is satisfied as the shaded area. We find that this type of solitary waves is supported by the system only for large enough ϕ 0 .
Conditions (S50) 1, 2 clearly show that there is a lower limit for the velocity of the propagating solitary waves, i.e. c > c min (S52) where When constraint (S52) is satisfied, the system supports only one solution in the form model (Eqs. S3). Note that in this set of simulations we consider a chain with 2 × 150 crosses and input our theoretical solutions (i.e. Eqs. (S54)) as boundary conditions for the first pair of crosses at the left end of the chain, while implementing free-boundary conditions at its right end. We find an excellent agreement between our analytical (lines) and numerical (markers) results, confirming the validity of our continuum model. Finally in Supplementary Figure 15f, we present the c-A relation for solitary waves excited by pulling. Unlike the case of compressive solitons, which propagate faster for smaller amplitudes, we find that the ones with larger amplitude travel faster (i.e. c always increases when the amplitude A increases). This difference is due to the fact that under compression our system exhibits nonlinear strain softening, while under tension it exhibits strain hardening. Finally, we note that we expect pulling solitons to be difficult experimentally observe, since they require an excitation applied at a very high speed.

Supplementary Note 8: Energy carried by solitons
In this section we determine the energy carried by the elastic vector solitons supported by our system. To begin with, we focus on the i-th pair of crosses, whose total energy is given by the sum of the elastic energy stored in the hinges and the kinetic energy of two rigid crosses. As shown in Supplementary Figure 16, there are two types of hinges associated to the i-th pair of crosses: two horizontal hinges (whose deformation is characterized by δ i l H , δ i s H and δ i θ H ) and one vertical hinge (whose deformation is characterized by δ i l V , δ i s V and δ i θ V ). It follows that the total energy for the i-th pair of crosses can be expressed as, Eq. (S55) can be written in dimensionless form as where ∆ i * = δ i * /a are the normalized deflections Next, we introduce the continuum functions θ(X, T ) and U (X, T ) and assume that the width of the propagating waves is much larger than the unit cell size and that θ ∼ ϕ 0 << 1. The total energy of the i-th pair of crosses (Eq. (S57)) can then be approximated as which can be rewritten in terms of the traveling coordinate ζ = X − cT as Finally, the energy carried by the elastic vector solitons is given by where U and θ entering in Eq. (S60) are defined by Eqs. (S29) and (S32). In Supplementary Figure 17 we report the evolution of E S as a function of the amplitude A as predicted by Eq. (S61) for the aligned (i.e. ϕ 0 = 0) and shifted (i.e. ϕ 0 = 5 • ) chains. ϕ 0 = 0 Eq. (S67) can only be solved numerically to find ω (1) and ω (2) , for the special case ϕ 0 = 0 an analytical solution exists with the corresponding eigenvectors given bỹ In Supplementary Figure 18a and b we report the dispersion relation for a chain with ϕ 0 = 0 • and 5 • , respectively. Note that each point on the dispersion curves is color-coded to show the corresponding modal polarization, which for the k-th branch is defined as We find that the linear modes of the structure with ϕ 0 = 5 • contain both rotational and translational component (i.e., the two degrees of freedom are coupled). Differently, for the special case ϕ 0 = 0 • the two propagation modes are fully decoupled, each associated with a single degree of freedom (i.e. since η θ = 0 on the first branch and η θ = 1 on the second one, the corresponding modes are purely translational and rotational, respectively). We also find that for the case ϕ 0 = 0 • the purely rotational mode branch starts at a finite frequency (i.e. linear waves with rotational component are not supported for ω < α √ 6K θ ), indicating that the system possesses a linear low-frequency band gap for rotation (highlighted in green in Supplementary Figure  18a). To demonstrate the effect of such linear frequency gap, we numerically solve the ODEs given by Eqs. (S3) for a chain with 150 pairs of crosses excited by a longitudinal displacement U input = 10 −4 sin ωT applied to the mid-point at its left end. In Supplementary Figure 18c and d we report the numerically recorded transmittance for both rotational and translational component, max(θ 100 (T ))/ max(θ 2 (T )) and max(U 100 (T ))/ max(U 2 (T )). We find that for the case ϕ 0 = 0 the transmission of rotational modes significantly drops at low frequencies, confirming the existence of the low-frequency band gap for rotations. By contrast, no such drop in transmission is observed for the system with ϕ 0 = 5 • , a consequence of the absence of any type of gap.
Lastly, it is important to note that the frequency content of the solitary waves supported by our system overlaps with this low-frequency band gap for rotation (see Supplementary Figure  18e for a typical soliton characterized by A = 0.30). As discussed in the main text and in Supplementary Note 9, the behavior of our soliton splitters is affected by such overlap, which prevents radiation of rotational vibrations. In this Section, we report the results of additional numerical analysis and experiments conducted on our soliton splitter.

Experimental results
While in the main text we show experimental results for a soliton splitter with a pair of stiffer hinges made of polyester sheets with thickness t d h = 0.635 mm (so that K d s /K s = K d θ /K θ = 30), in Supplementary Figure 19 we present experimental results for our 2 × 50 sample with ϕ 0 = 0 and two stiffer hinges made of polyester sheets with thickness t d h = 0.508 mm ( Supplementary Figure 19a) and t d h = 0.635 mm ( Supplementary Figure 19b) and paper clips ( Supplementary Figure 19c), which connect the 24th and the 25th pairs of crosses. Note that these three pairs of stiffer hinges result in K d s /K s = K d θ /K θ = 18, 30 and 175, respectively. The results clearly indicate that the amount of reflected and transmitted energy can be controlled by varying the stiffness ratio. For the splitter characterized by K d s /K s = K d θ /K θ = 18, most of the energy of the incident pulse is carried by the transmitted soliton and only a small portion by the reflected one; for the splitter characterized by K d s /K s = K d θ /K θ = 30, as described in main text, the energy carried by the incident pulse is roughly equally split between the transmitted and reflected solitons; for the splitter characterized by K d s /K s = K d θ /K θ = 175, most of the energy of the incident pulse is carried by the reflected soliton.

Numerical results
While in Fig. 3 of the main text we present results only for the rotational component θ, in Supplementary Figure 21 we show also the evolution of U as the solitary wave propagates in a 2 × 1000 chain characterized by ϕ 0 = 0 with a pair of stiffer hinges (with K d s /K s = K d θ /K θ = 30) connecting the 500th and 501st rigid crosses. The spatio-temporal plots for both θ (Supplementary Figure 21a) and U (Supplementary Figure 21c) clearly indicate that the pair of stiffer hinges splits the incoming soliton into two pulses that propagate with stable shape and constant velocity and that no trains of solitons are generated. As for the radiation of linear waves, we find that the interaction between the soliton and the stiffer hinges generate only translational vibrations (see small fluctuations in Fig Supplementary Figure 21c), since the overlap between the frequency content of the solitary waves supported by our system and the low-frequency band gap for rotation (see Supplementary Figure 18) prevents radiation of small amplitude rotational waves. We also note that the energy transferred to the translational vibrations can be estimated by comparing the energy carried by the incident soliton to that carried by the reflected and transmitted ones. Since in the aligned chain (i.e. for ϕ 0 = 0) the amplitudes of the incident, reflected and transmitted solitons are measured to be A i = 17.2 • , A r = 13.75 • and A t = 14.89 • (see Supplementary Figure 21a and b), using Eq. (S61) and the results of Supplementary Figure 17 we find that the normalized energy carried by them is E S, i = 0.0465, E S, r = 0.0184 and E S, t = 0.0262, respectively. Hence, since in our splitter (E S, r + E S, t )/E S, i = 0.96, only 4% of the energy carried by the incoming soliton is transferred to translational linear vibrations. By contrast, for the shifted structure (for which ϕ 0 = 5 • ) we measure A i = 16.45 • , A r = 5.45 • and A t = 12.08 • (see Supplementary Figure 21e and f), so that E S, i = 0.1575, E S, r = 0.008 and E S, t = 0.064. As such, in the shifted structure, which lacks both an amplitude gap for solitons and a low-frequency band gap for linear rotational vibrations, ∼ 54% of the energy carried by the incoming solitary wave is transferred to small amplitude solitons and linear vibrations. Finally, it is important to point out that the group velocity of the longitudinal vibrations supported by the chain characterized by ϕ 0 = 0 is one magnitude larger than the typical velocity of solitons (i.e. the solitons propagating in the chain have velocity c ∼ 0.1, whereas the velocity of the longitudinal vibration in the long wavelength limit is found from the dispersion relation of Supplementary Figure 18a to be ∼ 1). As such, the linear waves propagating in our splitter are very fast and do not interfere with the transmitted and reflected solitary waves, leading to a clean splitter for solitons. While the results reported in Supplementary Figure 21 are for a chain with a stiffer hinge, we also conducted an additional numerical simulation on a chain that embeds a softer hinge.
More specifically, we considered a chain comprising 1000 units and a pair of softer hinges (with K d s /K s = K d θ /K θ = 1/30) connecting the 500th and 501st rigid crosses. The results shown in Supplementary Figure 22a-d indicate that in the system with ϕ 0 = 0 the fragmented oscillations are wiped out due to the amplitude gap and that only a small amount of purely translational vibrations are generated by the interaction between the soliton and the softer pair of hinges. Differently, for the shifted chain (ϕ 0 = 5 • ) due to the absence of both the amplitude gap for solitary waves (i.e. A upper = 0 • for the system) and the low-frequency linear gap for rotational vibrations, trains of pulses and linear vibrations are generated when the propagating soliton hits the pair of softer hinges.

Numerical results
In the main text, we have investigated the response of a mechanical diode with 2N and (2N +1) pairs of crosses characterized by ϕ 0 = 0 and a central portion consisting of 2N a pairs of crosses with ϕ 0 = 5 • , assuming N = 12 and N a = 3 (see Fig. 4 of the main text). We found that for all pulses with amplitude larger than A ϕ 0 =0 upper = 6.55 • initiated at the left end of the system the transmission, |A 40 |/|A 10 | approaches unity. Differently, when the excitation is applied at the right end of the chain, the transmission |A 10 |/|A 40 | is close to zero, even if the amplitude of the input signal is outside the gap of the region with ϕ 0 = 0 (i.e. |A 40 | > A ϕ 0 =0 upper ). However, as typically observed in electronic [6] and thermal [7] diodes, if the amplitude of the pulses becomes too large, the diode experiences a condition known as breakdown. As a result, solitary waves with amplitude larger than A br ≈ 15 • propagate through the diode (i.e. if |A 40 | > A br ≈ 15 • , then |A 10 |/|A 40 | ∼ 0.6). It is important to note that the breakdown amplitude A br is analogous to the breakdown voltage of electronic diodes and determines the reliability of the device, since it defines the largest amplitude of a soliton that the diode can block. To study the effect of ϕ 0 and N a on A br , we conduct a set of discrete simulations in which we excite pulses of different amplitude at the right end of the chain and measure the transmission |A 10 |/|A 40 |. In Supplementary Figure 25a we report the measured transmission as a function of the amplitude of the input signal for diodes characterized by different values of N a and ϕ 0 = 5 • (while keeping N =12). We find that as N a increases, A br becomes larger, indicating a longer shifted section enhances the performance of our diode. We also note that for N a = 1, the transmission is always close to unity and our structure no longer functions as a diode. In fact, every incident soliton large enough to propagate through the aligned chain tunnels through such a diode.
In Supplementary Figure 25b we report the measured transmission as a function of the amplitude of the input signal for diodes characterized by N a = 3, ϕ 0 = 3 • and 7 • , and N =12. We find that lower values of ϕ 0 make the diode less reliable, since A br decreases (A br ≈ 15 • for ϕ 0 = 5 • and A br ≈ 10 • for ϕ 0 = 3 • ). Differently, for ϕ 0 = 7 • , all pulses considered in the simulations are blocked by the diode and no solitons tunnel through.
Finally, in Supplementary Figure 25c we summarize all of our numerical results and report the evolution of A br as a function of N a and ϕ 0 .