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Supporting Information for

Elastic vector solitons in soft architected materials

B. Deng,1J. R. Raney,2V. Tournat,1, 3 and K. Bertoldi1, 4

1Harvard John A. Paulson School of Engineering and Applied Science, Harvard University, Cambridge, MA 02138

2Department of Mechanical Engineering and Applied Mechanics,

University of Pennsylvania, Philadelphia, PA 19104

3LAUM, CNRS, Universit´e du Maine, Av. O. Messiaen, 72085 Le Mans, France

4Kavli Institute, Harvard University, Cambridge, MA 02138

(Dated: 26th April 2017)

FABRICATION

We make use of an extrusion-based 3D printing technique known as direct ink writing to produce the structures

in this work. Unlike many conventional commercial 3D printers that rely on either temperature changes or photo-

polymerization, direct ink writing is an ambient process that relies on material rheology to produce a pattern that

maintains its shape [1]. Subsequent immobilization steps (thermal crosslinking, sintering, etc.) can then be taken

after the pattern is formed, in a materials-dependent manner. The advantage of this approach is the broader palette

of materials that is compatible with it. Polydimethylsiloxane (PDMS) is a well-behaved silicone rubber that possesses

the necessary elastomeric qualities for our structures. However, its conventional precursors are Newtonian ﬂuids

that do not maintain their shape after extrusion. A 3D-printable “ink” version of PDMS can be produced through

the addition of fumed silica to the resin, resulting in a non-Newtonian paste. Our ink was produced by blending

commercially-available PDMS materials (85 wt% Dow Corning SE-1700 and 15 wt% Dow Corning Sylgard 184) in a

mixer (Flacktek SpeedMixer). This results in a rheological proﬁle that includes both shear-thinning eﬀects as well as

viscoelastic yielding behavior (see SI of Ref. [2] for more details). A shear-thinning response, deﬁned by a decrease

in apparent viscosity with increasing shear rate, facilitates extrusion of the material through tapered nozzles (in this

case 0.84 mm diameter) during printing. The viscoelastic yielding behavior is characterized by a high storage modulus

(G’) when shear stress is low (such that the material maintains its shape and behaves like an elastic solid) and a

deﬁned yield stress above which the storage modulus suddenly drops (allowing ﬂowability).

The paste-like material therefore ﬂows well during extrusion, but maintains its shape when patterned in 3D.

Patterning is performed by a commercial 3D motion control system, which is controlled by G code commands which

we generated via python scripts. After the material is patterned, a cross-linking step (100 ◦C for approximately

30 minutes) produces the familiar hyperelastic mechanical response of PDMS. After curing, additional structural

features can be added, for example, through the addition of additional PDMS (Sylgard 184) and (optionally) Cu

cylinders that add nodal mass and facilitate motion tracking during subsequent experiments (as in Ref. [3]).

To characterize the response of the two cured PDMS variants (i.e. the standard cast and printed PDMS) used in our

structures, we performed dynamic mechanical analysis (DMA) using a TA Instruments RSA III in compression mode.

A nominal 100 kPa pre-stress was used, and oscillations of 0.001 strain were imposed up to approximately 90 Hz

at room temperature. As shown in Fig. S1, we measured the storage and loss moduli over the relevant frequency

range. There is negligible diﬀerence between the standard variety of PDMS (indicated as “PDMS (control)”) and the

silica-ﬁlled variety (indicated as “PDMS (printed)”) that we use as a 3D printing ink.

2

Figure S1: Storage and loss moduli of the cured PDMS materials used in this work. PDMS (control) refers to standard cast

PDMS (used in the centers of the squares) while PDMS (printed) refers to the silica-ﬁlled PDMS used during 3D printing of

the structure.

ADDITIONAL EXPERIMENTAL RESULTS

Input signals

In Fig. S2 we report the input displacement proﬁle, u1(t), and the corresponding velocity proﬁle, v1(t), for the

ﬁve experiments presented in Fig. 2(b). It should be noted that all proﬁles present similar features. Note that the

displacement proﬁle for the impact characterized by (umax

1, vmax

1) =(4.10 mm, 1166 mm/s) has a very similar shape

as tanh function and therefore produces the best solitary wave.

Figure S2: (a) Input displacement proﬁle u1(t) and (b) corresponding velocity proﬁle v1(t) for the ﬁve experiments presented

in Fig. 2(b).

Rotation of the squares

Movies S1 and S2 reveal that the squares not only move horizontally when the pulse propagates, but also rotate.

To capture the rotational waves propagating through the sample we conduct an additional set of experiments where

the camera is focused only on three squares located at two-thirds of the sample (i.e. the 20th, 21st and 22nd square),

as shown in Fig. S3(a) (see also Movie S3). To capture the rotational waves propagating through the sample, we track

the positions of two diametrically opposed markers on the copper cylinders, highlighted by red dots and labelled as

”top” and ”bottom” in Fig. S3(a). The rotation θjof the j-th square is then obtained as

3

θj(t)=(−1)jtan−1 xtop

j(t)−xtop

j(0)−xbot

j(t)−xbot

j(0)

ytop

j(t)−ytop

j(0)−ybot

j(t)−ybot

j(0)!,(S1)

where (xtop

j, ytop

j) and (xbot

j, ybot

j) (j= 20,21 and 22) denote the positions of the two markers. Furthermore, for

the same three squares we also monitored their horizontal displacement, by tracking the horizontal position of the

marker at the center of the copper cylinders (highlighted by a red dot and labelled as ”center” in Fig. S3(a))

In Figs. S3(b) and (c) we show the evolution of ujand θjas a function of time, respectively. The results conﬁrm

the simultaneous propagation of translational and rotational waves in our structure.

Figure S3: (a) Movie frame: three squares are analyzed (i.e. the 20th, 21st and 22nd squares). For each unit cell three markers

(red dots) are tracked. (b) Extracted horizontal displacements ujand (c) angles θjas a function of time.

ANALYTICAL EXPLORATION

To get a deeper understanding of the mechanical response of the structure, we analytically investigate its behavior.

We ﬁrst establish a discrete model and determine the governing equations. Then, we take the continuum limit and

derive analytical solutions.

Discrete model

Our structure consists of a network of square domains connected by thin ligaments (see Fig. 1 of the main text and

Figure S4-a), all made of elastomeric material (polydimethylsiloxane - PDMS). The squares have diagonal lengths

of 2lthat are rotated by an angle θ0with respect to the horizontal direction. In this study we are investigating

the propagation of plane waves along the x-direction. To eﬃciently model the system, we ﬁrst notice that when a

planar wave propagates through the system all deformation is localized at the hinges that bend in-plane, inducing

pronounced rotations of the squares. Therefore, the structure can be modeled as a network of rigid squares connected

by springs at their vertices (see Figure S4-b). More speciﬁcally, we model each hinge with two linear springs: (i) a

compression/tension spring with stiﬀness kand (ii) a torsional one with stiﬀness kθ.

Finally, we also ﬁnd that, when a planar wave propagates in the x-direction, (i) the squares do not move in the

y-direction; (ii) neighboring squares aligned vertically experience the same horizonal displacement and rotate by the

same amount but in opposite directions; and (iii) neighboring squares always rotate in opposite directions. Therefore,

since in this study we focus on the propagation of planar waves in the x-direction, each rigid square in our discrete

model has two degrees of freedom: the displacement in the x-direction, u, and the rotation about the z-axis, θ.

Moreover, focusing on the rigid [j, i]-th square (see Figure S4), we have

u[j, i]=u[j, i+1], θ[j, i]=θ[j, i+1] .(S2)

Note that, as indicated by the blue and red arrows in Fig. S4, we deﬁne positive direction of rotation alternatively for

neighboring squares (i.e., if for the [j, i]-th square a clockwise rotation is positive, then for [j, i −1]-th, [j, i + 1]-th,

[j+1, i]-th and [j−1, i]-th ones counterclockwise rotation is considered as positive). We found this choice to facilitate

our analysis.

4

Figure S4: (a) Picture of our structure. (b)-(c) Schematics of the system.

Governing equations of the discrete model

To determine the governing equations for the discrete model, we focus on the [j, i]-th rigid square, for whose

behavior is governed by

m[j, i]¨u[j,i]=

4

X

p=1

F[j, i]

p,

J[j, i]¨

θ[j, i]=

4

X

p=1

M[j, i]

p,

(S3)

where m[j, i]and J[j, i]are the mass and moment of inertia of the rigid square, respectively. Moreover, F[j, i]

pand

M[j, i]

pare the forces in horizonal direction and moments generated at the p-th vertex of the rigid square by the ten-

sion/compression and torsional springs, respectively. To calculate these forces and moments, we start by determining

the vectors r[j, i]

p(p=1, 2, 3, 4) that connect the center of the [i,j]-th rigid square to its four vertices (see Fig. S4-c),

r[j, i]

1(θ[j, i]) = lhcos(θ[j, i]+θ0)iex+lh(−1)jsin(θ[j, i]+θ0)iey,

r[j, i]

2(θ[j, i]) = lh−(−1)jsin(θ[j, i]+θ0)iex+lhcos(θ[j, i]+θ0)iey,

r[j, i]

3(θ[j, i]) = lh−cos(θ[j, i]+θ0)iex+lh−(−1)jsin(θ[j, i]+θ0)iey,

r[j, i]

4(θ[j, i]) = lh(−1)jsin(θ[j, i]+θ0)iex+lh−cos(θ[j, i]+θ0)iey,

(S4)

The deformation of the springs connected to the vertices of the rigid square can then be written as

∆l1[j, i]=u[j+1, i]−u[j, i]ex+hr[j+1, i]

3(θ[j+1, i])−r[j+1, i]

3(0)−r[j, i]

1(θ[j, i])−r[j, i]

1(0)i

∆θ[j, i]

1=θ[j, i]+θ[j+1, i]

∆l2[j, i]=hr[j, i+1]

4(θ[j, i+1])−r[j, i+1]

4(0)−r[j, i]

2(θ[j, i])−r[j, i]

2(0)i

∆θ[j, i]

2=θ[j, i]+θ[j, i+1]

∆l3[j, i]=u[j−1, i]−u[j, i]ex+hr[j−1, i]

1(θ[j−1, i])−r[j−1, i]

1(0)−r[j, i]

3(θ[j, i])−r[j, i]

3(0)i

∆θ[j, i]

3=θ[j, i]+θ[j−1, i]

∆l4[j, i]=hr[j, i−1]

2(θ[j, i−1])−r[j, i−1]

2(0)−r[j, i]

4(θ[j, i])−r[j, i]

4(0)i

∆θ[j, i]

4=θ[j, i]+θ[j, i−1]

(S5)

5

where ∆l[m, n]

pand ∆θ[m, n]

pdenote the changes in length and angle experienced by the tension/compression and

rotational springs on the p-th vertex of [m, n]-th rigid square, respectively. It follows that

F[j, i]

p= k∆l[j, i]

p+kθ∆θ[j, i]

p

l2ez×r[j, i]

p!·ex,

M[j, i]

p=−kθ∆θ[j, i]

p−kr[j, i]

p×∆lp[j, i].

(S6)

Substitution of Eqns. (S6) and (S2) into Eqns. (S3) yields

m[j, i]¨u[j,i]=ku[j+1, i]−2u[j, i]+u[j−1, i]−lcos(θ[j+1, i]+θ0) + lcos(θ[j−1, i]+θ0)

+kθ

lθ[j−1, i]−θ[j+1, i]sin(θ[j, i]+θ0),

J[j, i]¨

θ[j, i]=−kθθ[j+1, i]+ 6θ[j, i]+θ[j−1, i]−kl u[j+1, i]−u[j−1, i]sin(θ[j, i]+θ0)

+kl2sin(θ[j, i]+θ0)cos(θ[j+1, i]+θ0) + 6 cos(θ[j, i]+θ0) + cos(θ[j−1, i]+θ0)−8 cos(θ0).

+kl2cos(θ[j, i]+θ0)sin(θ[j+1, i]+θ0) + sin(θ[j−1, i]+θ0)−2 sin(θ[j, i]+θ0)

(S7)

which represent the governing equations for the discrete system.

Continuum limit

While Eqs. (S7) contains the full nonlinear and dispersive terms of the modeled system and can only be solved

numerically, a deeper insight into the system dynamics can be achieved by further simplifying them to derive analytical

solutions. To this end, we ﬁst introduce the normalized displacement U[j,i]=u[j, i]/(2lcos θ0), time T=tpk/m,

stiﬀness K=kθ/(kl2) and inertia α=lpm/J . Moreover, since in Eqs. (S7) only the displacements and rotations of

squares in the i-th appear, for the sake of simplicity we set Uj=U[j, i], and θj=θ[j, i]. The governing equations Eqs.

(S7) can be then be written in dimensionless form as

∂2Uj

∂T 2=Uj+1 −2Uj+Uj−1−1

2 cos(θ0)[cos(θj+1 +θ0)−cos(θj−1+θ0) + K(θj+1 −θj−1) sin(θj+θ0)]

∂2θj

∂T 2=α2−K(θj+1 + 6θj+θj−1)−2(Uj+1 −Uj−1) cos(θ0) sin(θj+θ0)

+ sin(θj+θ0)hcos(θj+1 +θ0) + 6 cos(θj+θ0) + cos(θj−1+θ0)−8 cos(θ0)i

+ cos(θj+θ0)hsin(θj+1 +θ0) + sin(θj−1+θ0)−2 sin(θj+θ0)i.

(S8)

Next , we introduce two continuous functions U(X) and θ(X), which interpolate the discrete variables Ujand θj

as

U(Xj) = Uj,and θ(Xj) = θj,(S9)

where Xj=xj/2lcos(θ0) denotes the normalized coordinate along the x-axis. Using Taylor expansion, the displace-

ment Uand rotation θin correspondence of the (j−1)-th and (j+ 1)-th squares can then be expressed as

U(Xj−1)≈U(Xj)−∂U

∂X X=Xj

+1

2

∂2U

∂X 2X=Xj

,

U(Xj+1)≈U(Xj) + ∂U

∂X X=Xj

+1

2

∂2U

∂X 2X=Xj

,

θ(Xj−1)≈θ(Xj)−∂θ

∂X X=Xj

+1

2

∂2θ

∂X 2X=Xj

,

θ(Xj+1)≈θ(Xj) + ∂θ

∂X X=Xj

+1

2

∂2θ

∂X 2X=Xj

,

(S10)

6

from which the derivatives of Uand θare obtained as

∂U

∂X X=Xj

≈1

2[U(Xj+1)−U(Xj−1)] ,

∂2U

∂X 2X=Xj

≈U(Xj+1)−2U(Xj) + U(Xj−1),

∂θ

∂X X=Xj

≈1

2[θ(Xj+1)−θ(Xj−1)] ,

∂2θ

∂X 2X=Xj

≈θ(Xj+1)−2θ(Xj) + θ(Xj−1).

(S11)

Moreover, to further simplify the equations, we assume that the rotation angle θis small, so that sin θ∼θand

cos θ∼1. It follows that

sin(θj+θ0)≈sin θ0+θjcos θ0,

cos(θj+θ0)≈cos θ0−θjsin θ0.(S12)

Finally, we substitute Eqs. (S11) and (S12) into the discrete governing equations (Eqs.(S8)) and retain the nonlinear

terms up to the second order as well as the dominant dispersion terms, obtaining

∂2U

∂T 2=∂2U

∂X 2+ (1 −K) tan(θ0)∂θ

∂X ,

∂2θ

∂T 2=α2(cos(2θ0)−K)∂2θ

∂X 2−2 sin(2θ0)∂U

∂X −42K+ cos2(θ0)∂U

∂X + 2 sin2(θ0)θ−4 sin(2θ0)θ2,

(S13)

which represent the continuum governing equations of the system.

Next, we introduce the travelling wave coordinate ζ=X−cT ,cbeing the normalized pulse velocity (the real pulse

velocity is c2lpk/m), so that Eqs. (S13) become

∂2U

∂ζ 2=−(1 −K) tan(θ0)

1−c2

∂θ

∂ζ ,(S14)

∂2θ

∂ζ 2= 2α2βsin(2θ0)∂U

∂ζ + 4α2βsin(2θ0)θ2+ 4α2β[2K+ cos2(θ0)∂U

∂ζ + 2 sin2(θ0)]θ, (S15)

where

β=1

α2(cos(2θ0)−K)−c2.(S16)

Note that the displacement Uand rotation θare now continuous functions of ζand T. Integration of Eq. (S14)

with respect to ζ, with the assumption of a zero integration constant (i.e. a wave with a ﬁnite temporal and spatial

support), yields

∂U

∂ζ =−(1 −K) tan θ0

1−c2θ, (S17)

which can then be substituted into Eq. (S15) to obtain

∂2θ

∂ζ 2+P θ +Qθ2= 0,(S18)

where

P=4α2β

(1 −c2)(2c2−1−K) sin2θ0−2(1 −c2)K,

Q=2α2β

(1 −c2)(2c2−1−K) sin(2θ0).

(S19)

7

Note that for θ0→0, Q→0 and Eq. (S18) becomes a linear equation. Therefore, the analytical solution derived

here is not valid when θ0→0, since the cubic term, which is omitted here, must be considered to properly describe

the propagation of nonlinear waves in such structures.

Eq. (S18) has the form of the well-known nonlinear Klein-Gordon equation with quadratic nonlinearity. When

P < 0 and Q > 0, analytical solutions of Eq. (S18) exist in the form of a ﬁnite amplitude solitary wave with a stable

proﬁle

θ=Asech2ζ

W,(S20)

where A,cand Wdenotes the amplitude, velocity and characteristic width of the wave (note that solutions for P < 0

and Q < 0 also exist, but are diverging for ζ→0). Moreover, by substituting Eq.(S20) into Eq. (S17) the solution

for the displacement is found as

U=A(1 −K)Wtan(θ0)

(1 −c2)1−tanh ζ

W.(S21)

Note that the pulse velocity cand width Wdepend both on the amplitude Aof the wave and the geometry of the

structure (i.e. α,Kand θ0). In fact, substitution of Eq. (S20) into Eq. (S18) yields

AP+4

W2sech2ζ

W+AAQ −6

W2sech4ζ

W= 0,(S22)

which is satisﬁed for any ζonly if

P+4

W2= 0,and AQ −6

W2= 0.(S23)

By substituting Eqs. (S19) into Eqs. (S23), we ﬁnally ﬁnd

c=s6K+ 3 (1 + K) sin2(θ0) + A(1 + K) sin(2θ0)

6K+ 6 sin2(θ0)+2Asin(2θ0),

W=1

αs(1 −c2)[α2(cos(2θ0)−K)−c2]

2(1 −c2)K+ (1 −2c2+K) sin2(θ0)

(S24)

In Fig. S5 we report the evolution of cand Was predicted by Eqs. (S24). In Figs. S5-a and -d we consider

K= 0.073, α= 1.70 and report the evolution of Wand cas a function of Aand θ0. Note that we consider

5◦< θ0<30◦. The lower limit for θ0is dictated by the fact that Eq. (S18) is not valid when θ0→0 (since the

quadratic term vanished in this case), while the upper limit is determined by noting that, for this particular choice of

Kand α, the characteristic width Wis an imaginary number for θ0>36.7◦(indicating that the solitons no longer

exist for θ0>36.7◦). In Fig. S5-b and -e we consider θ0= 25◦and α= 1.70 and report the evolution of Wand cas

a function of Aand K. Finally, in Fig. S5-c and -f we consider θ0= 25◦,K= 0.073 and report the evolution of W

and cas a function of Aand α. Note that the structure used in this study is characterized by θ0= 25◦,α= 1.70 and

K= 0.073.

The contour plots reveal that the pulse speed cis not signiﬁcantly aﬀected by the amplitude A. In contrast, A

has an important eﬀect on W, that is found to dramatically increase as the pulse amplitude decreases. In fact, the

results of Figs. S5-a, -b and -c indicate that W→ ∞ as A→0. Note that as A→0 the nonlinear response of

the system is weakly activated and Wneeds to be very large (a low frequency or long wavelength pulse) to ensure

a balancing weak dispersion. As such, solitary waves are expected to form only after long propagation distances,

even for excitations very close to the ideal ones. Experimentally, this requires very long samples, but then the pulse

would be subjected to strong damping, posing serious limitations to the observation and existence of small amplitude

solitary waves. Moreover, we ﬁnd that the pulse width Wcan also be tuned by changing the stiﬀness parameter K.

Our results indicate that cis aﬀected by changes in both θ0and K.

It is important to note that the existence of the solitary solution to the Klein-Gordon equation (Eq. (S18)) requires

that

P=4α2β

(1 −c2)(2c2−1−K) sin2θ0−2(1 −c2)K<0,

Q=2α2β

(1 −c2)(2c2−1−K) sin(2θ0)>0.

(S25)

8

By substituting Equation (S24)1into Equation (S25), we obtain two non-linear inequalities in A,α,Kand θ0.

The structure supports a soliton if these two inequalities are satisﬁed for all amplitudes A < π/4−θ0, where the

constraint is introduced to avoid contact between neighboring squares. We ﬁnd that the system supports a soliton

for α∈[1.09,∞), K∈[0,0.336] and θ0∈[0◦,36.7◦]. Note that the system considerd in this study is characterized by

α= 1.70, K= 0.073 and θ0= 25◦.

Figure S5: Contour plots showing the evolution of cand W. (a) Evolution of Was a function of Aand θ0(assuming K= 0.073

and α= 1.70). (b) Evolution of Was a function of Aand K(assuming θ0= 25◦and α= 1.70). (c) Evolution of Was a

function of Aand α(assuming θ0= 25◦and K= 0.073). (d) Evolution of cas a function of Aand θ0(assuming K= 0.073

and α= 1.70). (e) Evolution of cas a function of Aand K(assuming θ0= 25◦and α= 1.70). (f) Evolution of cas a function

of Aand α(assuming θ0= 25◦and K= 0.073).

1 2

5 10

15 20

[mm] [mm/s]

8.48 4191

6.59 2064

7.03 3923

5.46 2327

3.86 492

4.10 1312

3.39 1170

umax

1vmax

1

0 0.05 0.1

0

0.2

0.4

0.6

0.8

1

1.2

normalized displacement

normalized velocity

square number

Figure S6: Comparison between analytical solution (continuous line) and experimental results (markers). Experimental results

are reported for seven diﬀerent impacts characterized by diﬀerent combinations of umax

1and vmax

1.

Finally, we note that the maximum displacement and velocity induced by the pulse, Umax and Vmax , can be

9

obtained from Eq. (S21) as

Umax = max (U)=2A(1 −K)Wtan(θ0)

(1 −c2)

Vmax = max dU

dT =cA(1 −K) tan(θ0)

(1 −c2)

(S26)

so that

Umax

Vmax =2W

c=2

αs6K+ 6 sin2(θ0)+2Asin(2θ0)(1 −c2)[α2(cos(2θ0)−K)−c2]

6K+ 3 (1 + K) sin2(θ0) + A(1 + K) sin(2θ0)2(1 −c2)K+ (1 −2c2+K) sin2(θ0).(S27)

Eqn. (S27) deﬁnes a parametric representation of a curve, where Ais the parameter. Such a curve is plotted in Fig. 4

of the main text with results from ﬁve diﬀerent experiments and in Fig. S6 together with results from another seven

experiments. Note that the experimental data (markers) are obtained by monitoring the maximum displacement

and velocity experienced by the 1st, 2nd, 5th, 10th, 15th and 20th squares. Interestingly, we ﬁnd that all applied

excitations result in the propagation of a soliton. However, it is important to note that this observation is not general

and related to the limited variety of excited displacement proﬁles (all of them are reasonable closed to tanh - see Fig.

S2). When in our numerical simulations we use an input displacement proﬁle very diﬀerent from tanh,

U1(T) = sech T

B,(S28)

Bbeing a constant, solitons are not generated (see Fig. S7).

Figure S7: Response of the system for excited displacement proﬁle with the form of the sech function. The proﬁle displacement

is deﬁned as U1(T) = sech(T/B) with (a) B= 20, (b) B= 10 and (c) B= 5.

10

Propagation of small amplitude waves

As discussed above, for suﬃciently small amplitudes the propagating elastic waves do not excite the nonlinear

response of the system. As such, in this case we expect small amplitude dispersive waves and not stable solitary waves

to propagate through the structure.

To better understand how elastic waves with suﬃciently small amplitudes propagate through the system, we make

use of Eqs. (S12) and linearize the discrete governing equations (S8) to obtain

∂2Uj

∂T 2=Uj+1 −2Uj+Uj−1+1

2tan(θ0) (1 −K) (θj+1 −θj−1),

∂2θj

∂T 2=α2(cos(2θ0)−K) (θj+1 +θj−1)−21 + 2 sin2(θ0)+3Kθj−sin(2θ0) (Uj+1 −Uj−1).

(S29)

Eqs. (S29) can be written in matrix form as

M¨

Uj+X

p=−1,0,1

K(p)Uj+p= 0 (S30)

where

M=1 0

0 1,¨

Uj+p="∂2Uj+p

∂T 2

∂2θj+p

∂T 2#,Uj+p=Uj+p

θj+p,,K−1=−11

2(1 −K) tan(θ0)

−α2sin(2θ0)−α2(cos(2θ0)−K)

K0=2 0

0 2α2(1 + 2 sin2(θ0)+3K),K1=−1−1

2(1 −K) tan(θ0)

α2sin(2θ0)−α2(cos(2θ0)−K)(S31)

Next, we seek a solution in the form of a harmonic wave

Uj+p(T) = ˜

U(µ) exp i(µXj+p−ωT ) (S32)

where ωis the temporal frequency of harmonic motion, µis the wavenumber and ˜

Uis a complex quantity that deﬁnes

the amplitude of wave motion. Substitution of Eq. (S32) into Eq. (S30) yields

−ω2M¨

Uj+X

p=−1,0,1

K(p)epµ = 0 (S33)

which can be solved numerically for wavenumbers µ∈[0, π] to obtain the dispersion relation curves shown in Fig.

S8-a. Note that in this band structure the frequency ωis normalised by pk/m. It is important to point out that the

two degrees of freedom of the system are coupled, so that both dispersion curves have translational and rotational

components.

Finally, in Fig. S8-b we report the evolution of the group velocity (cg=dω/dk) and phase velocity (cp=ω/k) for

the lower branch as a function of the wavenumber. Both velocities are normalized by 2lpk/m.

Estimation of kand kθ

To connect the discrete model to our sample, we need to estimate the mass of the squares (m), their rotational

inertia (J) and the spring stiﬀnesses (kand kθ). The mass mcan be easily measured as 2.093 g and the rotational

inertia Jcan be calculated from the geometry of the squares to obtain J= 18.11 g·mm2, so that nondimensional

parameter αis determined as α=lpm/J = 1.70 (note that ldenotes the half length of the square diagonals, l=

5.517 mm). To estimate the spring stiﬀness k, we start by extracting from our experiments the group velocity of

the fastest travelling wave packets, ˜cmax

g. We ﬁnd that any applied excitation results in ˜cmax

g≈29 m/s. Since the

numerical results shown in Fig. S8-b indicate that the maximum normalized group velocity is cmax

g= 0.8670, it

follows that

cmax

g2lrk

m= 0.8670 ·2·0.005517rk

0.002093 ≈29 m/s (S34)

11

Figure S8: (a) Dispersion curves and (b) evolution of the group and phase velocity for the lower branch as a function of the

wavenumber. To generate the plots, we considered K=0.073, α=1.70 and θ0=25◦.

from which we obtain k= 19235 N/m.

Having determined k, we then use equilibrium considerations and Finite Element (FE) simulations to obtain kθ.

On the analytical side, since the structure is periodic, we focus on a single square and consider quasi static uniaxial

compression along the vertical direction (see Fig. S3-a). For this loading case, a force Fis applied to the top and

bottom hinges, while there are no forces on the left and right hinges (since the structure is stress-free in horizonal

direction). The moment generated by Fis therefore balanced by those generated by the four rotational springs, so

that

XMA=−8kθθ+ 2F l sin(θ0+θ)=0,(S35)

where the reference point Ais indicated in Fig. S3-a. It follows from Eq. (S35) that

F=4kθθ

lsin(θ0+θ).(S36)

Moreover, the resulting compressive strain can be written as

=F

2lk + (cos(θ0)−cos(θ0+θ))

=2Kθ

sin(θ0+θ)+ (cos(θ0)−cos(θ0+θ))

(S37)

where the ﬁrst term accounts for the compression of the linear springs and the second one for the rotation of the

square. Finally, Eqs.(S36) and (S37) can used to generate the force-strain (F-) curve, shown as a continuous line in

Fig. S9-b.

On the numerical side, we simulate the response of the structure under uniaxial compression using

ABAQUS/Standard. To reduce the computational costs and make sure the response of the system is not domin-

ated by boundary eﬀects, we consider a unit cell comprising a 2×2 array of squares with identical geometry as those

considered in the experiments and apply periodic boundary conditions. The unit cell is discretized with plane strain

triangular elements (ABAQUS element type: CPE6) and the material is modeled using an almost incompressible

Neo-Hookean material with initial shear modulus µ0= 0.32M P a [2] . The compressive force as a function of the

applied strain is then extracted from the simulation and compared to the analytical prediction. The best agreement

between the two curves is found for kθ= 0.0427 Nm/rad (see Fig. S3-b), so that we obtain K=kθ/kl2= 0.073.

12

Figure S9: (a) Schematic of an individual square. (b) Force-strain curve under uniaxial compression. Comparison between

analytical (continuous line) and numerical (markers) results.

13

MOVIE CAPTIONS

Movie S1 Experiment in which the impactor prescribes a displacement signal to the ﬁrst square characterized by

(umax

1, vmax

1)=(3.11 mm, 524 mm/s). Note that after the pulse is applied the squares near the impactor vibrate at

high frequency. This is because the applied impact results in a displacement signal that does not exactly match that

of the supported solitary wave. Therefore, not all the energy applied by the impactor goes into the soliton and some

activates vibrations of the squares near the impactor. It is important to note that these vibrations have frequencies

in the range of those of the upper branch of the dispersion relation shown in Fig. S8 (i.e. ∼1170 −1720 Hz - note

that the frequency in the plot is normalized by pk/m).

Movie S2 Experiment in which the impactor prescribes a displacement signal to the ﬁrst square characterized by

(umax

1, vmax

1)=(4.10 mm, 1166 mm/s). Note that after the pulse is applied the squares near the impactor vibrate at

high frequency. This is because the applied impact results in a displacement signal that does not exactly match that

of the supported solitary wave. Therefore, not all the energy applied by the impactor goes into the soliton and some

activates vibrations of the squares near the impactor. It is important to note that these vibrations have frequencies

in the range of those of the upper branch of the dispersion relation shown in Fig. S8 (i.e. ∼1170 −1720 Hz - note

that the frequency in the plot is normalized by pk/m).

Movie S3 Experiment with the camera focused only on four squares, located at two-thirds of the sample. This

experiment is conducted to capture the rotational waves propagating through the sample.

[1] J. A. Lewis, Adv. Funct. Mater. 16, 2193 (2006).

[2] S. Shan, S. H. Kang, J. R. Raney, P. Wang, L. Fang, F. Candido, J. A. Lewis, and K. Bertoldi, Adv. Mater. 27, 4296 (2015).

[3] J. R. Raney, N. Nadkarni, C. Daraio, D. M. Kochmann, J. A. Lewis, and K. B. Bertoldi, PNAS 113, 9722 (2016).