Transformable topological mechanical metamaterials
TTransformable topological mechanical metamaterials
D. Zeb Rocklin, Shangnan Zhou, Kai Sun, and Xiaoming Mao
Department of Physics, University of Michigan, Ann Arbor MI 48109-1040 (Dated: October 22, 2015)Mechanical metamaterials are engineered materials that gain their remarkable mechanical proper-ties, such as negative Poisson’s ratios, negative compressibility, phononic bandgaps, and topologicalphonon modes, from their structure rather than composition. Here we propose a new design prin-ciple, based on a uniform soft deformation of the whole structure, to allow metamaterials to beimmediately and reversibly transformed between states with contrasting mechanical and acousticproperties. These properties are protected by the topological structure of the phonon band of thewhole structure and are thus highly robust against disorder and noise. We discuss the generalclassification of all structures that exhibit such soft deformations, and provide specific examples todemonstrate how to utilize soft deformations to transform a system between different regimes suchthat remarkable changes in their properties, including edge stiffness and speed of sound, can beachieved.
If a material has the ability of tuning its mechanicalproperties, such as stiffness, in real time, there will bebroad potential applications. For example, we can imag-ine a reusable launch system for space exploration madeof such a material, where the space vehicle is rigid dur-ing takeoff and in orbit, but during landing the rigid sur-face of the vehicle transforms into a soft cushion layerto absorb the impact. However, this is highly challeng-ing, because stiffness is an intrinsic property. Traditionaltechniques to change stiffness of a material are either ir-reversible, e.g., photo-polymerization that dentists use torigidify dental fillings, or involve significant stress in thematerial, e.g., tightening a guitar string. It is not untilrecently there have been proposals of mechanical meta-materials with tunability [1–9].In this Article, we propose a new design principlefor smart mechanical metamaterials, which we name“transformable topological mechanical metamaterials”(TTMM), whose mechanical and acoustic properties canbe easily tuned by orders of magnitudes without the needto disassemble/reassemble the system. Our design uti-lizes soft deformations, also known as mechanisms or floppy modes , which change configurations of the materialwith little energy cost. Structures with floppy modes areubiquitous in natural and engineered systems, e.g., a syn-ovial joint in human body or a door hinge. Our design,as shown in Fig. 1, involves periodic structures consistingof rigid building blocks (polygons or struts) connected byflexible hinges, which can be created using existing tech-nologies, like 3D printing [10] or self-assembly [11–14].These structures can exhibit soft deformations involvingchanging angles between building blocks at hinges with-out deforming any building blocks. These soft deforma-tions are either uniform, i.e., all repeating units twist inthe same way (soft deformations of this type has beencalled “Guest modes” [15, 16]), or spatially varying, i.e.,blocks at different locations show different twistings. Asshown in the Video in the Supplementary Information(SI) [17] , the uniform soft deformations, which we name“uniform soft twistings”, can be easily manipulated by asimple expansion of the lattice, and they serve as tun- ing knobs that control the mechanical properties of thesystem, transforming the edge of the system from soft torigid.Our design principles for the TTMM are based on thefollowing findings. First, utilizing elastic theory, we provethat if a two-dimensional (2D) structure exhibits one uni-form soft twisting, a series of spatially varying floppymodes must also exist. A proof of this using the lin-earized strain tensor has been given in Ref. [16], and herewe show that the same general conclusion holds for fullynonlinear strain. Then, using this theorem, we find thatall 2D structures with uniform soft twistings can be clas-sified into two categories: dilation dominant and sheardominant. Systems in these two regimes show very dif-ferent elastic properties. In the dilation dominant regime,the bulk of the material is a rigid 2D solid, but all edgesare soft due to floppy modes that reside on edges [18]. Inthe shear dominant regime, however, floppy modes arisein the bulk, while the properties of the edges can dif-fer sharply. Depending on the architecture, some of theedges may become rigid, while others remain soft. Fi-nally, we further show that these two different regimescan be realized in the same mechanical structure: thesoft twisting can reversibly shift it from one regime tothe other and hence alter various properties, includingedge stiffness and sound speed by orders of magnitude.In addition to this controllability, mechanical proper-ties of these systems show extraordinary robustness. Forexample, as mentioned above, a system in the dilationdominant regime displays rigid bulk and soft surface.Although similar mechanical properties can be achievedusing composite materials, e.g., by covering a stiff solid(e.g., a metal) with a soft cushion layer (e.g., rubber),there is one key difference between the two: robust-ness. The composite structure lacks robustness, i.e., ifthe outer cushion layer peels off, the protection will begone. However, the soft surface layer always exists in thesmart mechanical metamaterial we design. Because thewhole structure is built from the same building blocks, a r X i v : . [ c ond - m a t . s o f t ] O c t FIG. 1. (a) Uniform soft twisting of a deformed kagome lattice. Two types of triangles (red and blue) are connected by freehinges at their corners, forming a deformed kagome lattice with primitive vectors a , a . The angle θ between the trianglesdefines the twisting coordinate. The blue curve shows det ˜ (cid:15) [defined in Eq. (1)] as a function of θ . The 3 white dots on the θ axis represent 3 critical angles ( θ , θ , θ ) where sides of the triangles form straight lines (yellow stripes on the lattices)and topological polarization R T (shown as black arrows above the axes) changes. (b) Uniform soft twisting of a deformedsquare lattice constructed of 4 struts of different lengths (4 different colors), with each primitive unit cell contains 2 hinges.The spatially varying floppy modes in the deformed square lattice when det ˜ (cid:15) < k x , k y ) is shown on the right, where the zerofrequency phonon modes are shown in red, the two green dashed lines show the two zero speed of sound directions (1 , λ ± ) givenby Eq. (C8), and the yellow dots show reciprocal lattice sites. after the outer layers peels off, the newly exposed surfaceswill become soft . Such robustness is of critical importancefor devices working under severe conditions, whose pro-tecting layer may wear out due to extreme temperatureor friction, etc.The origin of this extraordinary robustness lies in topo-logical protection . As we will explain below, in additionto control the elastic properties, in certain systems, uni-form soft twistings can also trigger topological transi-tions, where phonon modes (sound waves) change theirtopological structure. As discovered recently, mechani-cal systems may exhibit different topologies [10, 19–36],in strong analogy to topological states in electronic sys-tems, e.g., topological insulators [37–39]. One key featureof topological states is the bulk-edge correspondence , inwhich edge properties of a topological system are dic-tated by the bulk of the system, and are independentof surface conditions or microscopic details. This bulk- edge correspondence offers strong protections to the edgeproperties in these systems against noise and local per-turbations, e.g., peeling off surface layers or insertingimpurities. The only possible way to modify the edgeproperties is by changing the topological structure of thewhole system, which requires changing globally the entirestructure, e.g., a uniform soft twisting discussed above.This is the root of the robustness in these systems andwhy we can control their elastic properties through softtwisting. Elastic theory and general classification
We start the analysis by considering an arbitrary 2D elas-tic system that exhibits (at least) one uniform soft twist-ing, while how to design such a structure will be discussedlater. Generally, deformations of an elastic medium canbe described using the left Cauchy-Green strain tensor.Here, we use ˜ (cid:15) to denote the strain tensor for the uniformsoft twisting ˜ (cid:15) = (cid:18) ˜ (cid:15) xx ˜ (cid:15) xy ˜ (cid:15) xy ˜ (cid:15) yy (cid:19) , (1)which is independent of position. As proved in the SI,utilizing the fact that any elastic deformation in flat spacemust have zero curvature, the existence of the zero energyuniform deformation ˜ (cid:15) leads to two families of spatiallyvarying floppy modes described by strain tensors (cid:15) + ( r ) = ˜ (cid:15) f + ( x + λ + y ) , (cid:15) − ( r ) = ˜ (cid:15) f − ( x + λ − y ) (2)where r = ( x, y ) is the coordinate, f ± ( w ) are two arbi-trary scalar functions and λ ± are two constants deter-mined by ˜ (cid:15) λ ± = (˜ (cid:15) xy ± √− det ˜ (cid:15) ) / ˜ (cid:15) xx , (3)where det ˜ (cid:15) = ˜ (cid:15) xx ˜ (cid:15) yy − (˜ (cid:15) xy ) is the determinant of ˜ (cid:15) .As shown in the SI, the elastic energy of these floppymodes ( E ) vanishes at the leading order, i.e., E ∼ O ( (cid:15) ),which is much lower than that of a typical elastic defor-mation with E ∼ O ( (cid:15) ). This is why they are dubbed asfloppy modes or soft modes.These floppy modes lead to the existence of soundwaves with zero velocity, i.e., soft phonon modes. It isknown that the speed of sound is proportional to thesquare root of the corresponding elastic constant. Here,because our floppy modes have vanishing elastic energyto O ( (cid:15) ), their corresponding elastic constants and soundvelocities vanish. Below, we show that these soft phononmodes can be either bulk or edge phonon modes.The characteristics of these floppy modes are dictatedby the sign of det ˜ (cid:15) , which distinguishes two differentregimes: the dilation dominate regime, det ˜ (cid:15) > (cid:15) <
0. In general, the uni-form soft twisting may contain both dilation (˜ (cid:15) xx and ˜ (cid:15) yy )and shear deformation (˜ (cid:15) xy ). If the uniform soft twistingis dominated by dilation (shear), we have ˜ (cid:15) xx ˜ (cid:15) yy > ˜ (cid:15) xy (˜ (cid:15) xx ˜ (cid:15) yy < ˜ (cid:15) xy ), which gives a positive (negative) det ˜ (cid:15) .It is worthwhile to emphasize here that det ˜ (cid:15) measuresthe intrinsic property of the uniform soft twisting and itis independent of the choice of coordinates. In addition,structures in the dilation dominant regime are necessar-ily auxetic [6] because they have ˜ (cid:15) xx ˜ (cid:15) yy > (cid:15) > f ± into Fourier series f ± ( w ) = (cid:80) k φ ± ( k ) e ikw so that the functions in Eq. (2) turn into f ± ( x + λ ± y ) = (cid:88) k φ ± ( k ) e ikx + iλ ± ky . (4) For any real number k , along the x direction, the ex-ponential factor e ikx describes a plane wave with wavenumber k x = k . However, along y , because λ ± is com-plex for det ˜ (cid:15) >
0, its imaginary part, Im λ ± , yields afactor e − κy with κ = k Im λ ± , so that the amplitude ofthis deformation decays exponentially along the y axis.If the system has an open edge parallel to the x -axis, thisis a plane wave along the edge whose amplitude decaysexponentially from the edge into the bulk of the system,i.e., an edge mode with zero sound velocity. The decayrate for this edge mode is proportional to the wavevector, κ ∝ k . Because the x -direction here is chosen arbitrarily,the same conclusion applies to arbitrary edge directionsand thus floppy modes arise on all edges. Because theelastic theory shows no bulk floppy modes, the bulk isin general rigid and has no floppy mode except the uni-form soft twisting, which we assumed from the begin-ning. One special case in the dilation dominant regime,the twisted kagome lattice, was discussed in Ref. [18],where the uniform soft twisting is a pure dilation ˜ (cid:15) xy = 0and ˜ (cid:15) xx = ˜ (cid:15) yy . For that special case, the system hasan emergent conformal symmetry and the floppy edgemodes are conformal deformations. As we prove here,the same qualitative properties shall always arise as longas det ˜ (cid:15) > (cid:15) <
0, the floppymodes are bulk plane waves along two special directions.This can be seen directly from Eq. (4). With negativedet ˜ (cid:15) , λ + and λ − are both real, and thus f + and f − bothdescribe bulk plane waves along the two directions of k y = λ + k x and k y = λ − k x . For bulk sound waves alongthese two special directions, the sound velocity vanishes,which is the key signature of the shear dominant regime.On the edge of the system, our general elastic theoryneither requires nor prevents the existence of floppy edgemodes, implying that the fate of the edge is not universaland relies on the architecture of the lattice. Generally ina solid, surface or edge sound waves, known as Rayleighwaves, could arise and the frequencies of these Rayleighwaves are lower than those of waves in the bulk (surfacewaves can also have frequencies located in a phonon bandgap, but because we only focus on low-frequency phononmodes, this case will not be considered here) [41]. For ourstructures with uniform twistings, similar Rayleigh wavesmay arise for certain edges. Because their frequencies arelower than the bulk ones, including the floppy bulk planewaves with zero sound velocity, these surface waves shallalso be soft and have zero sound velocity. At long wavelengths (small k ), these floppy edge modes have decayrate κ ∼ k and penetrate much deeper into the bulk,in comparison to the floppy edge modes in the dilationdominant regime discussed above, which has κ ∼ k .Finally, it is worthwhile to point out that the abovegeneral discussions are all based on the existence of asoft uniform deformation ˜ (cid:15) , without assuming any micro-scopic structures. Knowledge of the microscopic struc-tures provides more information on what form thesefloppy modes take. In particular, in periodic structures FIG. 2. General classification of lattices with uniform soft twistings. The spatially varying floppy modes are expressed interms of the wave number in the y direction when a plane wave of wave number k propagates in the x direction [see discussionsafter Eq. (4)]. The bulk phonon spectra show example phonon frequency contour plots as a function of k x , k y . The examplelattices are shown as rigid polygons (triangles or parallelograms) connected by free hinges at their corners [40], and they can bedirectly mapped into strut-hinge frames by replacing the triangles by 3 connected struts on their edges and the parallelogramsby 5 connected struts with 4 on edges and 1 on the diagonal to make it rigid. Thus the structure consists of triangles (deformedkagome lattices as defined the text) have (cid:104) z (cid:105) = 4 = 2 d and the structure consists of parallelograms (deformed checkerboardlattice) have (cid:104) z (cid:105) = 5 > d . built from struts and flexible hinges, as we discuss later,if the structure satisfies the Maxwell lattice condition (cid:104) z (cid:105) = 2 d (where (cid:104) z (cid:105) is the mean number of struts con-necting to one hinge and d is the spatial dimension), theaforementioned floppy modes may become of exactly zeroenergy. In contrast, this is not guaranteed in more con-nected structures with (cid:104) z (cid:105) > d . We summarize thisclassification in the table in Fig. 2. Transformations of structures and topologicalfloppy modes
A key result in this work is that the dilation and sheardominant regimes can be realized in the same structureand a transition between the two regimes can be achievedby a uniform soft twisting. Here we demonstrate thiswith two examples. The first example is a deformedkagome lattice [20, 40] constructed by connecting rigidtriangles with free hinges as shown in Fig. 1a (the term“deformed” refers to the fact that this lattice consists oftriangles of shapes that differ from those in the regularkagome lattice, and does not mean the lattice is strained). This structure has one uniform soft twisting, which uni-formly rotates the triangles and varies the angle θ markedon the figure. As shown in the SI video , this uniform softtwisting can be readily controlled by a simple expansionof the structure, which transforms the system betweendilation dominant and shear dominant regimes, and wefurther demonstrate dramatic changes in its mechanicalproperties.As we vary θ , the system goes through five transitionsat critical angles θ = θ , θ , . . . , θ respectively (Fig. 1a).For θ < θ or θ > θ , the system is in the dilation dom-inant regime det ˜ (cid:15) > θ < θ < θ , the system is in the shear dominantregime det ˜ (cid:15) <
0, where sound modes in the bulk alongtwo directions show zero velocity. The edge modes in theshear dominate regime show different characters for dif-ferent values of θ , according to which the shear dominantregime can be further classified into four sub-regimes,separated by critical angles θ , θ , θ . For θ < θ < θ ,floppy modes exist on all edges of the system. As θ → θ − edge modes on the bottom edge penetrate deeper anddeeper into the bulk and eventually become bulk modes(with zero decay rate κ ) at the θ = θ . Upon further FIG. 3. (a-c) show the evolution of a pair of floppy modes (red and black arrows) as the example deformed kagome latticeshown in Fig. 1a traverse its soft twisting coordinate θ across the critical angle θ where the lattice develops a topologicalpolarization. Periodic boundary condition is applied to left-right edges and open boundary condition to top-bottom edges. (d)Numerical results for the dramatic change of stiffness against local displacements at surfaces as θ changes, in a 60 ×
60 generickagome lattice of the structure shown in (a) with free hinges and fixed boundaries except the measurement edge (see Methods). increasing θ , these modes transform into edge modes onthe top edge, doubling the number of floppy modes at thetop edge. This evolution of floppy modes as θ increasesis illustrated in Fig. 3a-c. The transitions at θ and θ are of the same nature, where floppy modes shift fromcertain edges to edges on the opposite side of the system.These transitions lead to a dramatic change in the edgestiffness. We perform conjugate-gradient minimizationcalculations of the response to a point force on one edgeof a lattice with other edges held fixed, and find that theedge stiffness increases by orders of magnitude as floppymodes leave the edge (Fig. 3d).The edge properties in these four sub-regimes are dic-tated by the topological structure of the phonon band,which is characterized by a vector topological index called“topological polarization” ( R T ). As first discovered inRef. [20], this topological index points to an edge thatgains extra floppy edge modes. For the deformed kagomelattice discussed above, as θ crosses the three critical an-gles θ , θ , θ , the change of R T follows 0 → ( a − a ) → a → a and a are the unit vectors of the lat-tice marked in Fig. 1a) , so the two regimes θ < θ < θ and θ < θ < θ have nonzero R T (called topologicallypolarized) and stiff edges in the direction of − R T .The transitions at θ , θ , θ are called topological tran-sitions, because a topological index changes its valueacross the transitions. Topological transitions have beenwell studied for topological states in electronic systems.Our design of the TTMM offers a concrete platform to ex-plore these topological transitions in mechanical systems. At the transition, edges of the triangles form straightlines along certain direction, which is intimately relatedto the arise of bulk soft modes at the transition. As dis-cussed in Ref [16, 18, 20], straight lines in the bulk allowstates of self stress (possible ways to distribute of internalstress without net forces on any parts) such that floppybulk modes can arise.As another example, we construct a deformed squarelattice using free hinges to connect rigid struts with dif-ferent lengths (Fig. 1b). This structure also has one uni-form soft twisting, which changes the angle θ uniformly.As θ increases, the system undergoes one transition fromthe dilation dominant regime to the shear dominant one.Agreeing with our elastic theory, the dilation dominantregime shows a rigid bulk and soft edges, while the sheardominant regime has floppy bulk modes. Interestingly,in contrast to the deformed kagome lattice, the deformedsquare lattice shows no floppy edge modes. Instead it hasbulk modes with exactly zero energy. These floppy bulkmodes follow the predicted directions (1 , λ ± ) at small k ,but deviate at larger k (zero frequency lines in Fig. 3bare curved).We emphasize that both of these two examples satisfythe Maxwell lattice condition (cid:104) z (cid:105) = 2 d (the deformedkagome lattice is a strut-hinge frame with z = 4, seeCaption of Fig. 2), and as a result floppy modes in thesestructures, either edge modes or bulk modes, are of ex-actly zero energy, although the general elastic theory dis-cussed above only requires the modes to be soft (i.e.,elastic energy scales as (cid:15) or higher).Finally, we provide one design principle, which canbe used to generate TTMM with many different struc-tures. As shown in Refs. [15, 16], 2D structures satisfying (cid:104) z (cid:105) = 2 d must have at least one uniform soft twisting (seeMethods). This explains why the deformed kagome andthe deformed square lattices we discussed above have uni-form soft twistings even with arbitrarily chosen shapesof triangles and strut lengths. In contrast, the deformedcheckerboard lattice (Fig. 2) has z = 5 > d and the uni-form soft twisting disappear when the shape of the paral-lelograms are changed into arbitrary quadrilaterals. Thusby choosing periodic structures with balanced degrees offreedom and constraints ( (cid:104) z (cid:105) = 2 d in the language ofstrut-hinge frames) the uniform soft twistings are guar-anteed to exist and the structures exhibit zero energyfloppy modes. On the other hand, over-constrained struc-tures with carefully chosen geometry (e.g., the deformedcheckerboard lattice) can also exhibit uniform soft twist-ings but their floppy modes in general are not of zeroenergy.A primary challenge in the fabrication of TTMMs isgenerating sufficiently flexible “hinges”. The rigidity ofrotations at the hinges must be much smaller than therigidity of deforming the building blocks. When thishinge rigidity is small but finite it determines proper-ties that would vanish with completely flexible hinges,such as the stiffness of the soft edges and sound velocitiesof floppy modes. In this perspective, self-assembly mayoffer a promising approach. If tip-to-tip attraction be-tween polygon (colloidal/nano) particles can be realized,an extended periodic 2D lattice may be self-assembled.Although the tip-to-tip attraction needs to be directionalto ensure stability of the open structure, it can be muchsofter than actually deforming the particles. Thus bind-ing sites at the tips can serve as flexible hinges for theassembled TTMM. Acknowledgments
We thank Tom C. Lubensky and Vincenzo Vitelli for use-ful discussions. DZR thanks NWO and the Delta Insti-tute of Theoretical Physics for supporting his stay at theInstitute Lorentz. This work was supported in part bythe ICAM postdoctoral fellowship (DZR) and the Na-tional Science Foundation, under grants PHY-1402971at the University of Michigan (KS).
Methods
Generalized Maxwell’s counting rule and uni-form soft twistings
The number of zero modes (modes of deformation whichcost no energy) N of a structure is determined by thenumbers of degrees of freedom N d.o.f. , constraints N c andstates of self stress (i.e., possible ways to distribute inter-nal stress without net forces on any parts) N ss throughthe generalized Maxwell’s counting rule [42, 43] N = N d.o.f. − N c + N ss . (5)One simple setup to demonstrate this relation is a frameconsisting of N c struts connected at N free hinges (e.g.,the structure in Fig. 1b). For a system with spatial di- mension d , each hinge needs a d -component coordinateto describe its location, so it has d degrees of freedomand N d.o.f. = N d . Each strut fixes the distance betweentwo hinges and thus enforces one constraint. It is worth-while to note that the constraints enforced by struts maynot be independent, i.e., some of the struts may be re-dundant and thus do not introduce new constraints. Asshown in Ref. [43], each redundant constraint contributesone state of self-stress (i.e., stress may be introduced ifthe length of the strut change), which is the last term inEq. (5). The term isostatic refers to the special marginalstate where N = d ( d + 1) / N ss = 0 where the structure isboth stable and stress-free. A critical mean coordina-tion number (cid:104) z (cid:105) = 2 d for isostaticity [44–46] follows from N d.o.f. = N c , which is a weaker condition of mechanicalstability that assumes all struts are independent. Follow-ing the nomenclature of Ref. [16] we call periodic latticeswith (cid:104) z (cid:105) = 2 d “Maxwell lattices”.When the generalized Maxwell’s counting rule is ap-plied to periodic lattices, as shown in Refs. [15, 16],an interesting consequence follows that all lattices with (cid:104) z (cid:105) = 2 d (Maxwell lattices) must have d ( d − / ho-mogeneous deformations that are of zero energy . For 2Dlattices, the case this Article is mainly concerned with,Maxwell lattices have at least one such soft deformation(which we name the uniform soft twisting). These floppymodes have also been called “Guest modes” [15, 16].Certain lattices with (cid:104) z (cid:105) > d , such as the deformedcheckerboard lattice in Fig. 2, also possess uniform softtwistings, with these necessarily accompanied by statesof self stress.In addition, this type of counting rules and the result-ing floppy deformations apply equally to simple frameswith struts-hinges and more complicated structures, pro-vided that the degrees of freedom and constraints arecountable. For example, a sub-class of these floppy de-formations, the “rigid-unit-modes” (RUMs), has beenstudied in the context of crystals with the structure ofperiodic corner-touching polyhedra and argued to be re-sponsible for negative thermal expansion in some crys-tals [47, 48], as well as utilized to realize negative Pois-son’s ratio metamaterials [6, 49]. In this Article we dis-cuss more general situations which do not necessarily in-volve rigid polyhedra. Numerical calculation of edge stiffness
Systems of 60 ×
60 unit cells were generated. Three of thefour sides were held fixed, while one triangle from the freeside was pressed into the structure in the linear regime(qualitatively similar behavior was observed under non-linear deformations). The Conjugate Gradient methodwas used to obtain the minimum-energy configurationand the ratio of force to displacement was extracted asthe edge stiffness. Units were chosen such that the springconstant of the struts and the length of the strut that ishorizontal in Fig. 1a were both unity.The residual edge stiffness of the soft edge is due tofinite size effects as the sides of the lattice are clamped.Because the zero modes are exponentially localized to thesoft edge, the stiffness of this edge falls exponentially withsystem size. In real systems this soft edge stiffness will becontrolled by friction or bending stiffness at the hinges.In addition, the sharp rise in the edge stiffness of thesoft edge at θ is due to the fine-tuned geometrical effectof the line of struts being pulled taut in the transversedirection. Appendix A: Elastic deformations and the straintensor
In order to provide a self-contained discussion, here wefirst briefly review some basic concepts on elasticity.In an elastic system, if we focus on macroscopic phe-nomena at length scales much longer than the scale ofthe microscopic structure, we can ignore microscopic de-tails and treat the system as a continuous medium. Insuch a picture, each point in the elastic medium can belabeled by its coordinate r (here we use bold symbols torepresent vectors and tensors). Under deformation, thepoint r is now displaced to a new location with coordi-nate R . Such a deformation is described by a mapping r → R ( r ). In this language, the space that r lives in iscalled the reference space , i.e., the space before the de-formation. and the space that R lives in is dubbed the target space , i.e. the space after deformation.For a slowly varying displacement field, one can keeponly the first order derivative ∂ i R j = ∂R j ∂r i (where i, j areCartesian indices denoting x, y in 2D) in the elastic en-ergy and ignore higher order derivatives. This derivative, ∂ i R j , appears to be a rank-2 tensor. However, it is im-portant to realize that the two indices of this matrix livein two different spaces. The index i is from r , which livesin the reference space, but the other index j is from R ,which lives in the target space. Symmetry transforma-tions are independent in these two spaces (e.g., a rotationbefore deformation and the same rotation after deforma-tion result in different strains of the elastic medium). Toexpress the strain field as a true tensor one can contracteither the reference space or the target space indices. Aconvenient choice is the metric tensor g ij = ∂ i R k ∂ j R k , (A1)which is a tensor that lives in the reference space ( i, j here are both indices in the reference space, and indicesin the target space are contracted). Here we follow theEinstein summation convention, i.e. the repeated index k is summed over.It is easy to verify if there is no deformation, R ( r ) = r up to rigid translations and rotations, the metric tensoris the identity matrix. To describe the strain, the leftCauchy Green strain tensor is defined by subtracting theidentity matrix from the metric tensor, (cid:15) ij = 12 ( g ij − δ ij ) , (A2) where δ represents an identity matrix ( δ ij = 1 for i = j and δ ij = 0 otherwise). Appendix B: Elastic energy and zero energydeformations
In this section, we prove that if there exists one uniformdeformation that does not cost any elastic energy, thesystem must also support a series of spatially varyingzero-energy deformations.In general, the energy cost for a elastic deformation,i.e., the elastic energy, is a functional of the strain tensor.To the leading order, the elastic energy is E = (cid:90) dr c ijkl (cid:15) ij ( r ) (cid:15) kl ( r ) , (B1)where c ijkl are elastic constants. We have assumed thatthe elastic medium has no internal stress. This formfor elastic energy is a standard description for an elas-tic medium. For an isotropic medium, these elastic con-stants reduces into two independent ones, bulk and shearmoduli. Here, because we are considering a generic sys-tem, we will maintain this general form and allow theelastic constants to be independent. Same as above, herewe adopt the Einstein summation convention, so all re-peated indices are summed over. The higher order terms,which are not shown in Eq. (B1), contain both higherorder terms of the strain tensor as well as spatial deriva-tives on the strain tensor. Here, we will first ignore thesehigher order terms and their contributions will be exam-ined in App. D.If an elastic medium has (at least) one uniform defor-mation, which can be written as a position-independentstrain tensor ˜ (cid:15) , that costs no elastic energy, we have E = (cid:90) dr c ijkl ˜ (cid:15) ij ˜ (cid:15) kl = 0 . (B2)Because ˜ (cid:15) is position independent, this indicates c ijkl ˜ (cid:15) ij ˜ (cid:15) kl = 0 . (B3)Next, we search for additional spatially varying zero-energy deformations in this system. It is easy to verifythat a deformation described by the following strain ten-sor (cid:15) ij ( r ) = ˜ (cid:15) ij φ ( r ) , (B4)where φ ( r ) is an arbitrary scalar function, has zero elasticenergy, E = (cid:90) dr c ijkl (cid:15) ij ( r ) (cid:15) kl ( r ) = c ijkl ˜ (cid:15) ij ˜ (cid:15) kl (cid:90) dr φ ( r ) = 0 , (B5)where we have used the fact that c ijkl ˜ (cid:15) ij ˜ (cid:15) kl = 0[Eq. (B3)]. Appendix C: Constraints on the function φ ( r ) fromcurvature It is important to point out that although the elas-tic energy [Eq. (B5)] vanishes for any arbitrary function φ ( r ), not every function φ ( r ) corresponds to an elasticdeformation. This is because the strain tensor is not anarbitrary rank-2 tensor. According to the definition ofthe strain tensor, in order to ensure that a strain ten-sor indeed describes a physical deformation, there has toexist a deformation R ( r ) such that (cid:15) ij ( r ) = ∂ i R k ∂ j R k − δ ij , (C1)is satisfied. This condition enforces strong strain con-straints on the function φ ( r ) and in this section we willfind the necessary and sufficient condition to guaranteea physical zero-energy deformation.For this purpose, it is more convenient to use the metrictensor instead, which relates to the strain tensor throughEq. (A2). The question now translates to finding thecriterion, under which a metric tensor corresponds to areal physical deformation, i.e. to decided whether or notthere exists a deformation R ( r ) exist such that g ij ( r ) = ∂ i R k ∂ j R k (C2)is satisfied. The answer to this question has been re-vealed in the study of differential geometry, where thesame question is known as the problem of flat (local)coordinates. According to Riemann’s Theorem , the nec-essary and sufficient condition for the existence of suchan R ( r ) is that the metric tensor must have a zero cur-vature. The proof of this statement can be found in liter-ature on Riemannian geometry or differential geometry.Here, instead of going through the full proof, we providea physical picture to demonstrate the origin of this zerocurvature condition. Because both our reference spaceand the target space (i.e. the material before and afterthe elastic deformation) are defined in a flat space , themapping between these two spaces, R ( r ), must not haveany nonzero curvature associated with it. Therefore, themetric tensor defined from this mapping must have zerocurvature [50].To determine the curvature for an arbitrary metric ten-sor g ij ( r ), we first define the Levi-Civita connection, i.e.the Christoffel symbols, using the derivative of g ij ,Γ kij = 12 ( ∂ j g ki + ∂ i g kj − ∂ k g ij ) . (C3)Then, by taking another derivative to the Levi-Civitaconnection, the Ricci curvature tensor is obtained, R ijkl = ∂ k Γ ilj − ∂ l Γ ikj + g mn Γ ikm Γ nlj − g mn Γ ilm Γ nkj , (C4)where g mn is the matrix inverse of the metric tensor g ij .For a physical deformation in a flat space, the Riccicurvature tensor must vanish, R ijkl = 0. For the zero energy deformations shown in Eq. (B4), the correspond-ing metric tenor is g ij ( r ) = ˜ (cid:15) ij φ ( r ) + δ ij . (C5)In 2D, generically, the function φ ( r ) depends on bothcoordinates x and y . However, the zero curvature condi-tion enforces a constraint on φ ( r ). Using Eq. (C4) it isstraightforward to verify that the curvature vanishes, ifand only if φ ( r ) takes one of the following two forms φ ( r ) = f + ( x + λ + y ) (C6)or φ ( r ) = f − ( x + λ − y ) (C7)Here, f + ( s ) and f − ( s ) are arbitrary functions of s . and λ + and λ − are two constants that are determined by thestrain tensor of the uniform zero-energy deformation λ + = (˜ (cid:15) xy ± √− det ˜ (cid:15) ) / ˜ (cid:15) xx , (C8) λ − = (˜ (cid:15) xy ± √− det ˜ (cid:15) ) / ˜ (cid:15) xx , (C9)where det ˜ (cid:15) is the determinant of ˜ (cid:15) . It is worth pointingout that this result is independent of the choice of thecoordinate. If the directions of x, y are chosen differently, λ ± will change accordingly, but the two directions givenby x + λ ± y are invariant.We have shown in Eq. (B5) that these deformationscost no elastic energy. Because the zero curvature condi-tion is the necessary and sufficient condition which guar-antees that the strain tensor defined in Eq. (B4) corre-sponds to a physical deformation, we conclude that thefollowing spatially varying deformations are all zero en-ergy modes of the system (cid:15) ij ( r ) = ˜ (cid:15) ij f + ( x + λ + y ) ,(cid:15) ij ( r ) = ˜ (cid:15) ij f − ( x + λ − y ) . (C10)Because we can choose arbitrary f + and f − , the numberof these zero energy deformations is infinite in the contin-uous theory. In a real system, with lattice structure andwith finite size, the number of zero modes scales with thelinear size of the system ∼ L/a , where L is the size of thesystem and a is the lattice constant. Thus the numberof these zero modes is sub-extensive .In summary, we prove here that for a 2D elastic system,as long as there exists one uniform zero-energy mode,which is described by a spatially independent strain ten-sor ˜ (cid:15) , there must exist two families of spatially varyingzero-energy modes, as shown in Eq. (C10). Appendix D: Higher order terms in the elasticenergy
In our analysis above, we ignored higher order terms inthe elastic energy. These higher order terms involve bothhigher powers in (cid:15) and higher order derivatives, such as ∂ (cid:15) .In the previous section we solved for modes that havezero elastic energy in the leading order theory. Restoringcontributions from higher order terms, the elastic energyof these modes is E = 0 + O ( (cid:15) ) + O ( ∂ (cid:15) ∂ (cid:15) ) , (D1)which is small when the strain is small and slowly vary-ing in space. Thus, strictly speaking, these zero modesshould be called floppy modes because they are not nec-essarily exactly zero energy.In addition, in the Article, we consider frequencies ofplane waves (in the bulk or on the surface) that belongto these two families of floppy modes with wave number k . Our theory then predicts that the frequency of thesewaves are ω = O ( k ) . (D2)Ordinary plane waves in stable elastic medium have ω = c k , where c is the speed of sound. In contrast, thesefloppy modes correspond to plane waves with zero speedof sound.This zero sound velocity is a key signature of the sys-tems with floppy uniform deformations that we studyhere. Regardless of the details of the system, these con-clusions hold universally.In special families of structures with uniform floppytwisting (e.g., Maxwell lattices), these floppy modes mayhave exactly zero elastic energy, even if higher orderterms are taken into account. This phenomenon is dis-cussed in our Article, where we show that the exact zeroelastic energy is protected by Maxwell’s counting rule.Nevertheless, it is worthwhile to emphasize that althoughin the general case (where there is no protection from thecounting rule) the elastic energy receives higher order cor-rections, the acoustic sound velocity for these modes willalways be zero. Appendix E: Additional information on the SI Video
The prototype is constructed of commercially availableplastic ”K’NEX” parts. A rigid triangle consists of three rods extending from a central white connector. There aretwo species of triangles of different shapes (red and blue,as shown in Fig. 4): those ending in blue hinge-parts andthose ending in black hinge-parts. Note that althoughthere are no direct connections between two hinge-partsin the same triangle the length between them is fixed bythe rods joining them to the central part (which cannotrotate relative to one another) so the triangles are rigid.Each pair of blue hinge-part and black hinge-part formone flexible hinge. Connected triangles are thus able torotate freely relative to one another. This is a realizationof the deformed kagome lattice described in the maintext.The frame consists of four metal rods connected to tri-angles on the edge of the structure and manipulated byhand. The triangles are free to slide along the lengthsof the rods so that the spacing between edge triangleschanges even as they remain collinear. The rods are ro-tated relative to one another, resulting in a uniform softtwisting as described in the main text that alters thelattice structure of the prototype.
FIG. 4. Illustration of the plastic prototype used in the SIVideo .[1] R. D. Kornbluh, H. Prahlad, R. Pelrine, S. Stanford,M. A. Rosenthal, and P. A. von Guggenberg, in