Downloaded by guest on September 30, 2021 a ( Bar-Sinai Yohai metamaterials elastic of two-dimensional elasticity nonlinear and charges Geometric www.pnas.org/cgi/doi/10.1073/pnas.1920237117 these of mechanics underlying nontrivial their the from elasticity. emerge how metamaterials instability), cen- is elastic A at perforated mechanism. then strain nonlinear question the critical of tral origin the universal (e.g., a nonlinear qualitatively implying quantitatively only underlying not also an this independent, but Interestingly, of material (6). largely sign instability is telltale response an a for responsible is break- mechanism symmetry 1C), spontaneous (Fig. This of pattern load. Rosetta the ing of a direction or lat- a the “zig-zag” triangular on a of depending either in to formation whereas leads it the orientations, alternate tices taking vertical in In holes and plateaus. results elongated horizontal stress the instability with the loads this pattern and higher checkerboard lattice instability at square an because of a develops experimentally, direction system see the the to to is difficult 4, with Fig. perpendicular loads) ically ellipses, e.g., small oriented into (see, (at holes axes compression circular solid major the the of their of com- deformation (2D) uniaxial uniform response Under a two-dimensional linear 1A. Fig. such the (5). in pression, for shown holes is of curve metamaterials array elastic stress–strain elas- periodic an typical embedded an of by A with consists that of matrix metamaterials example shown tomeric prominent mechanical systems been A nonlinear these (4). has such functionalities instabilities it exotic and achieve recently can lin- deformations more the large 3), in embracing (2, properties regime mechanical ear unusual with metamaterials (2–4). physics emergent and the elements of constituting richness between the interactions the the between contrast of high simplicity the a to form due class metamaterials interesting mechanical particularly systems, such ever- of the list Within increasing well-understood elements. material of between out interactions phenomena simple collective of emergence the T metamaterials mechanical the for work metamaterials this elastic in porous discussed. 2D developed are disordered and Pos- tools ordered structures. of the study 2D of Apart prototypical applications several state. full sible reproduces of postinstability quantitatively simulations mechan- the also numeric formalism of of the onset aspects intuition, the from qual- and response, instability, framework, linear conceptual ical the charges. intuitive elastic explaining an of itatively elas- interaction offers (2D) nonlinear approach of two-dimensional This that the we into devel- elasticity, maps problem recent of that tic formulation formalism on geometric a building nonlinear develop and a in electrostatics, Using lacking. opments with largely ana- are analogy frameworks successful, an conceptual are and approaches geo- tools nontrivial due numeric lytic highly of complex While presence and effects. the and and rich metrical mechanics are metamaterials, nonlinear general, their 2019) in to mechanical 17, solids November review porous flexible for (received deformable 2020 23, of March approved and Problems IL, Evanston, University, Northwestern Rogers, A. John by Edited 91904 Israel Jerusalem, Jerusalem, colo niern n ple cecs avr nvriy abig,M 23;and 02138; MA Cambridge, University, Harvard Sciences, Applied and Engineering of School hl nta tde oue ntedsg fmechanical of design the on focused studies initial While .W nesni i ae Mr sdfeet 1,is (1), different” is “More paper his by in described as Anderson physics, W. condensed-matter P. of hallmark he | lsi charges elastic ) a .Ti eomto styp- is deformation This Right). areeLibrandi Gabriele , | olna elasticity nonlinear a ai Bertoldi Katia , doi:10.1073/pnas.1920237117/-/DCSupplemental at online information supporting contains article This to available a are as results, 1 well theoretical at the as GitHub of paper, on derivation reader this the detailed in a discussed contains that data notebook numerical All deposition: Data is ulse pi 9 2020. 29, April published First BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article This interest.y competing paper. no y the declare G.L., wrote authors Y.B.-S., M.M. The tools; and Y.B.-S. and reagents/analytic and K.B., data; new G.L., analyzed contributed Y.B.-S., M.M. and research; M.M. designed research; M.M. performed and M.M. K.B., Y.B.-S., contributions: Author mle httecneto neatn oe a omtebasis the form success can 8) holes describ- interacting (7, to of Kamien’s limited concept and theory the Matsumoto a that state, implies as buckled However, the the and it. response ing on linear holes preinstability of the critical effects the or disloca- either instability of predict at cannot pair strain and modes, a qualitative buckled is as the approach captures modeled this successfully buckled is work their the hole While that each tions. if showed holes, of they configurations who interacting works energy-minimizing with the- 8), their consistent linear are In (7, the patterns on elasticity. Kamien based of and holes ory between Matsumoto interactions by the studied to taken was solutions nomenon analytical insightful of description, valid lack problem. a the the is emphasizes theory elasticity materials but elastic nonlinear perforated that of simula- mechanics confirms elastic (FE) the effective finite-element predicting of in the success tions and The (6). strain, accessi- etc. critical are coefficients, the the equations reproduce fields, accurately elastic deformation which models, nonlinear finite-element fully using ble the the direct However, of satisfying perspective. analytic seem- solutions while an edges—a from holes’ task elasticity the hopeless on ingly of conditions boundary equations free multiple nonlinear the ing owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To ttsia ehnc n yaia systems. the dynamical and tackling mechanics from for statistical techniques powerful approach non- importing allows of This naturally problem that charges. into problem interacting elastic conceptual linearly the accurate maps quantitatively we which elasticity and formalism, geometrical intuitive of an theory develop modern elec- a from with structures ideas an Combining trostatics novel ones. as existing of of and optimization design question and regarding scientific signif- challenge fundamental great of a engineering problem as a both is predictive icance, mechanical metamaterials, a elastic tunable, developing of i.e., theory and the response, affects unusual, mechanical structure hole emergent their the how to Understanding properties. due matter class important of an ligaments—form and engineered holes of an micropattern with solids metamaterials—stretchable Elastic Significance rtatmttwr hoeia xlnto o hsphe- this for explanation theoretical a toward attempt first A solv- requires problem elastic the of analysis theoretical A a n ihe oh ( Moshe Michael and , b aa nttt fPyis h erwUiest of University Hebrew The Physics, of Institute Racah PNAS https://github.com/yohai/elastic | a 2 2020 12, May . y raieCmosAttribution-NonCommercial- Commons Creative | . y o.117 vol. https://www.pnas.org/lookup/suppl/ charges ) | b,1 o 19 no. metamaterials | 10195–10202 Mathematica . y

PHYSICS field, leading to the buckling instability that creates the patterns shown in Fig. 1. Using the language of singular image charges to simplify cal- culations is common in field theories governed by the Laplace (or bi-Laplace) equation. Besides the well-known electrostatic example, this technique was used in analyzing low Reynolds number fluid dynamics (refs. 10–14, among many others), flux pinning in superconductors (15), capillary action (16), linear elasticity (17, 18), and relativity (19). Below we use the language of electrostatics, which we assume is familiar to the reader, to give a pedagogical analogy for the corresponding problem in elasticity.

Electrostatic Analogy. Consider a circular conductive shell in the presence of a uniform external electric field. Solving for the resultant field requires a solution of Laplace’s equation with specific boundary conditions on the conductive surface. One par- ticularly insightful method to solve this equation, introduced in elementary physics classes, is the method of image charges. The trick is that placing “imaginary” charges outside the domain of interest (i.e., inside the shell) solves by construction the bulk equation, and wisely chosen charges can also satisfy the bound- ary conditions. Indeed, the problem is solved exactly by placing a pure dipole at the shell center. From the perspective of an observer outside the shell, the presence of the conductive surface is indistinguishable from that of a pure dipole. Thus, the concept Fig. 1. (A) A sketch of a typical stress–strain curve for periodically per- of image charge not only opens an analytic pathway for solving forated elastic metamaterial. (B and C) Metamaterials composed of an the problem, but also provides intuition about the solution and elastomer with a square lattice (Left column) or triangular lattices at two specifically on the physical effect of boundaries. different orientations (Center and Right columns). The materials are shown We note two properties of the solution which will have exact in undeformed (B) and postbuckling deformed (C) configurations, under analogs in elasticity: First, the imaginary charge is a dipole, not a uniaxial compression. In the square lattice the instability is reflected as a monopole. Electrostatic monopole image charges are disallowed checkerboard pattern of horizontal and vertical hole shapes whereas in because they are locally conserved. That is, the net charge in a the triangular lattice, due to frustration, the unstable mode forms either given region can be completely determined by a surface integral a zig-zag or a Rosetta pattern, depending on the direction of loading. on the region’s boundary (Gauss’s theorem). Second, the mag- nitude of the dipole moment turns out to be proportional to the external field and to the circle’s area (in 2D). for an effective “lattice” theory of elastic metamaterials with How are the correct image charges found? A common strat- periodic arrays of holes. egy is to find them by enforcing the boundary conditions directly. In this work we derive a formalism that bridges the gap This works only in cases where the image charges can exactly between the successful “microscopic theory” (nonlinear elas- solve the problem. An alternative approach is via energy min- ticity) and the macroscopic effective theory. As we will show, imization, which gives an approximate solution when the exact this formalism provides an insightful and intuitive description of one cannot be represented by a finite number of image charges. perforated elastic metamaterials without losing the quantitative In fact, a potential φ that satisfies the bulk equation and its capabilities of the microscopic theory. While the algebra might boundary conditions is also a minimizer of the energy be somewhat technical, the qualitative picture that emerges from Z I it is clean and elegant. Therefore, we structure the paper as 1 ~ ext 2 F = 2 |∇φ − E | dS − ρ φ dl, [1] follows: First, we qualitatively derive the main results of our anal- Ω ∂Ω ysis, using an analogy to a well-known problem in electrostatics (Qualitative Picture). Then, we describe the full formalism (The where Eext is the imposed external field, Ω is the problem domain 2 Method) and finally we quantitatively compare its predictions to (e.g., R with a circle taken out), and ∂Ω is its boundary. For full numerical calculations (Results). simplicity, here we work in units where the permittivity of space is unity. The function ρ is a Lagrange multiplier enforcing a con- Qualitative Picture stant potential on the conducting boundary (SI Appendix). For There are two major challenges in writing an analytical the- the problem described above, after guessing a solution in the ory: the multiple boundary conditions imposed by the holes and form of a single dipole, its magnitude can be found by minimiz- the nonlinearity. As shown below, both these challenges can be ing the energy Eq. 1 with respect to the dipole vector and ρ. The tackled with the language of singular elastic charges. In what result satisfies the boundary conditions exactly. follows we demonstrate that the phenomena can be approxi- Consider now a harder problem: an array of conducting cir- mately, but quantitatively, described in terms of interacting elas- cular shells in an external electric field, introducing the com- tic charges with quadrupolar symmetry, located at the center of plication of multiple boundary conditions. In contrast with the each hole. These are image charges, much like the image charges single-shell problem, guessing a finite number of image charges that are used to solve simple electrostatic problems (9). When that will balance boundary conditions is impossible: The image the loading is weak (linear response), the interaction of the charges are now reflections of the external field, but also of all charges with the external field dominates and the quadrupoles other image charges. Therefore, in general, the image charge in align perpendicularly to the direction of compression. At each shell is composed of an infinite number of multipoles. While higher stresses, due to geometrical nonlinearities, the interaction an exact solution is hard to guess, by minimizing the energy we between charges dominates their interaction with the external can nonetheless obtain an approximate solution. Each circular

10196 | www.pnas.org/cgi/doi/10.1073/pnas.1920237117 Bar-Sinai et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 a-ia tal. et Bar-Sinai ansatz an guess we Specifically, solution. form the the improve of of would account- shell an accuracy and each reflects fields, inside the dipoles the multipoles higher-order of of the for variability of terms ing spatial scale in the the solution on on a assumption spatially guessing vary Thus, that shell. fields corre- balancing multipoles charge to higher-order circle, balance image spond a can dominant on dipoles fields imaginary the electric While and uniform dipolar: polarized, is shell be each to inside going is shell iemgiue om ilcto 2) ial,aquadrupolar a oppo- Finally, and (24). equal dislocation of a disclinations forms of an magnitude, pair dipole a site A as i.e., 25). interpreted charges, (24, disclination elastic be a of as singularity can manifested monopole is it the above materials, described fields, crystalline elastic In charge. of elastic source singular the at a singularity delta-function a state, is reference cone 25). the a (24, of of apex case curvature the incompatibility Gaussian in The the which state. by reference quantified flattened conical is the the of and confined embed- space state flat the surface ding stressed between incompatibility conical The geometric a a plane. thin reflects is, cone Euclidean a flat curva- consider that the Gaussian example, curvature; to of an distribution Gaussian As singular is ture. a asso- is charges quantity charge physical elastic monopolar The with charges. elastic ciated of nature ematical applied the ( in to area and charges? proportional elastic hole’s quadrupole are the is a to is magnitude and charge its stress The theory 22). linear (21, the center charge shell elastic imaginary the pure a at analog, to electrostatic equivalent the is like solution and, Inglis the function stress Airy equa- the biharmonic Math- for the stress. cavity tion solving remote to circular to amounts a problem subject a the medium, 1913: ematically, elastic be in 2D to (20) infinite happens an Inglis in problem by shell solved elastic example, conducting linear famous single The the theory. of elasticity analog to modifications, some with Problem. Elastic The known integration straightforward. calculating is of dipole form, a task of analytical potential explicit the Since in field. external the with and shells, different matrix The where charges unknown the written in be form can quadratic energy The a dipole. as electric single a of potential the the where † noteedtishere. details these into npicpe n hudas decompose also should one principle, In ic h asincraueo h eeec tt csas acts state reference the of curvature Gaussian the Since math- the uncovers (23) elasticity to approach geometric A i hcnutn hl) and shell), conducting th p i M steiaedpl etrlctdat located vector dipole image the is m ij M i = = unie neatosbteniaedplsin dipoles image between interactions quantifies I 2 1 ∂ F Z Ω φ Ω † l fteaoecnet a etranslated, be can concepts above the of All φ = hn iiiigteeeg Eq. energy the minimizing Then, . (x)=  m p X ∇φ (x ~ i ,j unie neatoso hs dipoles these of interactions quantifies − X M p (x x i ij i )ρ − p p .Btwhat But 2). section Appendix, SI φ ρ i i i x p ntrso h ioa ed.W ontgo not do We fields. dipolar the of terms in p · (x)dl i j φ ) stewl-nw ouinfor solution well-known the is −  p (x X ∇ ~ . i − φ m M p x (x i i ), p p and i − i , , x x j i ) m tecne of center (the  dS satrivial a is , 3 [4] [3] [2] is fteqarplswt h xenlfil n hi nearest their and field external the interaction with the minimize quadrupoles simultaneously the to quadrupoles; of impossible interacting is the it of those i.e., with incompatible buckled are the tice of 1). pattern Fig. (cf. the lattice exactly square next-nearest the is i.e., of this state diagonal, that square unit Note quadrupoles the neighbors. between of interaction sides opposing 2 the on minimizes Fig. also in it (as because quadrupoles vertical and and horizontal 2A, Fig. of in pattern (as of axis angle horizontal an the having to quadrupoles all condition configura- 1) energy-minimizing the quadrupoles: are distinct satisfy interactions two nearest-neighbors tions account, only into if lattice, taken minimizers. square a of For continuum one-dimensional a 2A is Fig. satisfying which configurations for 2, π/ minimized is energy This (33) is energy interaction the 2A), Fig. quadrupoles; For between medium. magnitude interaction elastic (θ of infinite the an quadrupoles understand in two quadrupoles to elastic need pure two quadrupole. we elastic an proceed, i.e., each dislocations, To opposite described of who pair phenomenological (7), a the as Kamien hole to and spirit Matsumoto in in close description very 1 is Fig. in quadrupoles, shown ing the cases the rotate that for to assume approximation free good also B are a us they fact and let in fixed is attempt, (this is first quadrupoles a all cen- of As the magnitude hole. at located each quadrupole of a ter of composed indeed is solution Quadrupoles. Interacting by imaginary derived dominated is, are be that fields charges, can elastic (22). nontopological quadrupoles load the lowest-order which external the for by given approx- solution the an a a with case, guessing for electrostatic and complex the solution other mul- the in each of imate As is, terms with field. That external in interacting imposed described shells. charges be conducting image electrostatic can tiple of their the holes create in array between also like interactions an charges holes, of these other but inside case hole, charges charges each image imaginary attack of own placing now center by of can solved the array be we in an can containing hand, problem metamaterials This other elastic holes. the 2D of on problem charges the elastic of all cept of derivation rigorous a 32. ref. For see 31). results, these (30, (Burgers’ dipole conserved the are and elastic- vector) vector) In (Frank’s conservation. monopole the by both local disallowed ity, electrostatics, is monopoles In of direct theorem. a creation conservation is quadrupole a a contractile of is problem consequence hole adherent the solves by that multipole e.g., applied substrate, dipole elastic force (29). cells a an biological is to quadrupole a locally 28). (27, of ellipse Stone- an realization into irre- Another domain the an circular i.e., a as inclusion, of Eshelby deformation elastic elliptic versible known the an theory as continuum is known of is Burgers context quadrupole lattices the opposite In hexagonal (26). and is defect equal problem, Wales in with hole which pair circular dislocation the vectors, a solves which as one realized the like charge, i nietesur atc,tesmere ftetinua lat- triangular the of symmetries the lattice, square the Unlike quadrupoles? of lattice a of configuration optimal the is What con- the and hand one on charges image of method the With lowest-order the that fact the case, electrostatic the in Like and smaue ihrsett h iecnetn h two the connecting line the to respect with measured is B, C, .Tecekror atr a oe energy lower a has pattern checkerboard The Bottom). rsnstosc pia configurations. optimal such two presents Left and E PNAS Center = Q e sasm o h oetta the that moment the for assume us Let π | 1 Q r .Ti itr,o neatn rotat- interacting of picture, This ). a 2 2020 12, May 2 θ o (2θ cos 1 + Q θ 1 2 , n )acheckerboard a 2) and Top) = Q 1 2 + 2 2 π/ | n orientations and o.117 vol. θ 2 ). o l neighboring all for | o 19 no. 4 π/ θ 1 relative | + θ 10197 θ 1 2 , [5] θ = A 2

PHYSICS known as the elastic (or stiffness) tensor, which encodes mate- rial properties such as Young’s modulus and Poisson’s ratio (SI Appendix, Eq. S4). Although the energy is quadratic, the theory as presented above is still nonlinear due to the ∇dT · ∇d term in strain (Eq. 6). Neglecting it (assuming ∇d  1) yields the familiar theory of linear elasticity (30). That is, linear elasticity is obtained by performing two conceptually distinct linearizations: a rheological linearization, neglecting higher-order material properties (the Fig. 2. Interacting elastic quadrupoles, illustrated by the deformation fields O(u3) term in Eq. 7), and a geometrical linearization, neglecting they induce on the holes’ edges. (A) Two energy-minimizing configurations the quadratic term in Eq. 6. In the former, the neglected non- of quadrupoles of fixed magnitudes and free orientations. Top configura- linear behavior is rheological and therefore material specific. In tion shows θ1 = θ2 = π/4 while Bottom one shows θ1 = 0, θ2 = π/2. (B) An the latter, the neglected terms are geometrically universal and array of quadrupoles on a square lattice minimizing their interaction energy with nearest and next to nearest neighbors, as given by Eq. 5. The relative relate to rotational invariance. Since, as described above, the orientation of any nearest-neighbor pair is like that in A, Bottom, and that nonlinear mechanics of elastic metamaterials with arrays of holes of next-nearest pairs is like that in A, Top. (C) Like B, but for a triangular are largely material independent, it is reasonable to speculate lattice. that a suitable analytical description of the system is that of a nonlinear geometry with a quadratic (Hookean) energy. There- fore, we take Eqs. 6 and 7 to be the governing equations in neighbors. Direct minimization of nearest-neighbors interactions this work. energy with respect to quadrupoles orientations gives the pattern Numerical analysis has confirmed the applicability of these shown in Fig. 2C. As before, the quadrupole orientations are in equations in two respects: First, a full numerical solution of agreement with the observed unstable mode. The Rosetta pat- the governing equations accurately reproduces experimental tern observed in Fig. 1C, Right, however, is not captured by this results (6). Second, calculations show that even in the buck- simplified model, since in it the quadrupole magnitudes are not led state, which is clearly a nonlinear response, |∇d| is of uniform. order unity‡ due to almost-rigid rotations of the junctions between holes, invalidating the geometric linearization. How- Collecting the Pieces. The conclusion from the previous section ever, the nonlinear strain, Eq. 6, is small due to cancelation is that the unstable modes resemble a collection of interacting of the linear and quadratic terms, justifying the rheological quadrupoles. We suggest that rigorously describing the system as linearization in Eq. 6. From a theoretical perspective this a collection of interacting quadrupoles is a perturbative approx- observation suggests that a careful analysis of small nonlinear imation of the full solution: At low stresses, all quadrupoles strains should recover the phenomenology of perforated elastic are aligned with the external field. At higher stresses the elas- metamaterials. tic metamaterial buckles and, as we have just seen, the buckled states are consistent with a model of interacting quadrupoles. Bulk Energy. Similarly to Eq. 2, we express the total deformation This suggests that the postinstability response is dominated by in the system as induced by quadrupoles located at the centers of charge–charge interaction rather than interactions of charges the holes, with some charges placed in lattice sites immediately with the external load. outside the solid, as illustrated in Fig. 3. Using a recent general- We emphasize, however, that this picture does not have ization of the method of Airy stress function, which allows solving a (linear) electrostatic analog. In linear systems, the induced elastic problems with arbitrary constitutive relations, strain def- ext charges are always proportional to the external loading (E initions, or reference states (33, 35), we perform a perturbative in Eq. 1) and therefore the interaction between themselves expansion of the nonlinear quadrupolar fields§. Symbolically, the cannot, by construction, dominate their interaction with the displacement induced by a single charge qαβi located at xi is external field. The mechanism described above is manifestly expanded in powers of charges nonlinear and requires a generalization of the electrostatic arguments. The observed instability emerges from a geomet- X (1) 2 (2) 3 ric nonlinearity, which is inherent to elasticity and does not d(x) = qαβi dαβ (x − xi ) + qαβi dαβ (x − xi ) + O(q ), [8] have an electrostatic analog. Below we show how the frame- i,α,β work of interacting charges can be expanded to account for all these effects. (n) where dαβ is the displacement to the nth order associated with the charge q . Here Greek indices represent the different The Method αβi quadrupolar components and Latin indices represent the loca- The fundamental field in the theory of elasticity is the displace- tion of the image charge in the 2D lattice. A detailed derivation ment field d, which measures the spatial movement of material is given in SI Appendix, section 1 and explicit analytical forms elements from a reference position to its current one. Local of the geometrically nonlinear fields associated with small elastic length deformations are quantified by the strain tensor u (34), multipolar charges are given in an attached Mathematica note- 1   book (see Data Availability for details). In addition to image u = ∇d + ∇dT + ∇dT · ∇d . [6] charges at the hole centers, we also allow for uniform elastic 2 fields, which within the formalism are described as quadrupolar The elastic energy density, which results from local length charges located at infinity. changes, can be written as a function of u. Linear elasticity is a leading-order perturbation theory for small deformations and therefore E is written as a quadratic function of u, alias Hookean ‡e.g., with respect to the Frobenius norm. energy §In fact, a careful analysis of the elastic equations reveals that there are two distinct types 3 E = hu, ui + O(u ). [7] of elastic monopoles and consequently also two types of quadrupoles. For succinctness R 1 in the text we refer to quadrupoles in a general manner, but in the actual calculations Here hv, ui ≡ Ω 2 vAu dS is an integration over the domain Ω of we do take into account both types of quadrupoles in each hole. A detailed calculation the contraction of the tensor fields u, v with a 4-rank tensor A, is presented in SI Appendix.

10198 | www.pnas.org/cgi/doi/10.1073/pnas.1920237117 Bar-Sinai et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 Here, d where and Eq. ansatz the Combining index. l opnnso l hre,ete thl etr ra infinity, at or centers vector hole single at a either by charges, all of components all (nonlinear) small of limit the approx- in (32). controlled strains exact a is becomes latter which the and imation, system, on the depends of validity geometry whose the in approximation order uncontrolled quadratic an is the mer at expansion and the order truncating quadrupolar the 2) at expansion multipole the truncating imposed are conditions arrows. boundary The with signs. displacement marked + are black the with which marked are over charges ligaments image the subjected of locations size The uniform aries. with holes displacement circular external of to lattices triangular and square 3. Fig. a-ia tal. et Bar-Sinai is lattice–hole edges the external within ways of the on context introduced few conditions the boundary be a in imposing that, geometry, can are found conditions We there formalism. boundary and our these approach which different 1. in Fig. a system in requires finite as a This conditions, reality: boundary analyze in to displacement-controlled want encountered with we commonly work this most in case but the too, theory elastic the in term a by (E imposed term was energetic loading bulk external the where systems infinite with Loading. External work, this numerically. additional integrals In avoid the and research. evaluate approach we future elastic-charges approximations, of the test subject strictly the to is inter- which nearest-neighbor domain. only actions), perforated keeping under integrals (i.e., the these approximations calculate some over analytically to attempt expressions could One known explicitly of on. so Similarly, domain. the of M geometry the and account position into that relative taking matrix their charges, positive-definite between a interactions is pair It quantifies interpretation: simple a has 4, Eq. in implied is summation no that Note ( j k o oainlsmlct,i sese odnt h olcinof collection the denote to easier is it simplicity, notational For Eq. that note We acltn h neato matrices interaction the Calculating matrix The (3) ) eobtain we 7, nue yteiaecagsand charges image the by induced ecie h neatosbtentilt fcags and charges, of triplets between interactions the describes u h he rttpclltie tde nti ok hw are Shown work. this in studied lattices prototypical three The E ( j k ) = stesri eddrvdfo h ipaeetfield displacement the from derived field strain the is M M X ij M ijk (3) ij (2) M (2) = = elcn h he indices three the replacing Q, nteeetottceapeaoew dealt we above example electrostatic the In ij (2) iia oteeetottcanalog electrostatic the to similar , D D u u 8 d Q i (1) i (1) ext ext otistodsic prxmtos 1) approximations: distinct two contains i nEq. in , , ple ntelgmnsfrigtebound- the forming ligaments the on applied Q u u j j (2) j (1) + E E X .I spsil oicuesc a such include to possible is It 1). ijk δ jk 8 M ihteeatceeg Eqs. energy elastic the with + D ijk (3) δ u ij i (2) M Q steKoekrdelta. Kronecker the is 10. i , Q u novsintegration involves j (1) j Q α, E k δ β + ik , . i . . . q yasingle a by h for- The . M , fEq. of [10] [9] 6 hr o aeo oainw mte h superscripts the omitted we notation of M ease for where ranges that porosity with systems area analyze fractional the we from as Here defined holes. are porosity, and of respectively, their holes, by 77 and lat- characterized 81 triangular contain a and square lattices and The triangular lattice orientations. different square three two a along analyze compressed 3: to tice Fig. quadrupoles in image of shown method situations, the use we Here Results nonlinear the minimizing to Eq. amounts energy conditions boundary the vector the and N where are charges of the collection on That a ligament. constraints each the to for is, one translates charges, ligaments the average on of constraints given nonlinear set a a Imposing 4). on section displacement Appendix, (SI ligament the where t,mauigtesse’ opinefruixa od,ie,its i.e., loads, uniaxial for compliance system’s the measuring ity, and lattice square the boundary the on Focus- displacement 3. average a Fig. the by are in given them, is arrows mechanism of with one loading on marked the ing ligaments, bound- contact and of bottom actual set system The and plate. discrete the rigid top a The between by loaded 3: points are Fig. geometry lattice the in the of Consider depicted aries region. system, particular the a sat- of in be be average will can finite conditions on solution boundary a isfied the approximate with that demanding an fields by However, obtained relevant charges. the of expressing number when exactly isfied as conditions charges boundary unknown the the on treating constraints by done conveniently most bandi lsdfr;sedrvto in derivation see 4A form; modulus closed Young’s in obtained effective the example, For Eq. of terms Appendix, (SI by given 4 are section charges desired the and algebra, linear constraints mini- the linear we only is, limit, energy quadratic That this the considered. mize In are displacements. contributions small leading-order under system the Response. 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PHYSICS next-leading order. This technical calculation is done in detail in SI Appendix, section 4B. The charges in the linear solution, Q∗, are proportional to the imposed displacement dext; cf. Eq. 13. This means that the leading-order correction to the Hessian (the bracketed term in Eq. 14), as well as the correction to the displacement constraints, is also linear in dext. When the imposed displacement is large enough, the constrained Hessian can become singular; i.e., one of its eigenvalues can vanish. This is the onset of instability. We note that this calculation is in line with the intuitive pic- ture described above: For small loads (i.e., in the linear regime) the dominant interaction is that of the charges with the external loading and with themselves, quantified respectively by N (1) and M(2). In this regime the solution is linear in dext and given by Eq. 13. It is stable because M(2) is positive definite. For larger loads, the interaction of the induced charges with themselves, quanti- fied by M(3), becomes important and eventually destabilizes the linear solution. Fig. 5, Left column shows the critical strain, i.e., the strain at which the Hessian becomes singular, as a function of porosity for the three different lattices. Our method is in good quantitative agreement with the full numerical simulations, except possibly at very low porosities. This happens because smaller porosities lead to larger critical strains, making the image charge magni- tudes larger. Because our method is a perturbative expansion in the charge magnitude, its accuracy deteriorates when the charges Fig. 4. Comparison between the elastic-charges calculation and a direct are large. This effect is more noticeable in the triangular lattices fully nonlinear numerical solution in the linear regimes for three different (Fig. 5, Middle and Bottom rows). lattices. Left column shows the effective Young’s modulus as function of For each lattice, we also plot the unstable eigenmode asso- porosity (blue for finite element, orange for elastic charges). Each point ciated with the vanishing eigenvalue. Representative ones are represents the slope in the stress–strain curve of a system with the cor- plotted in Fig. 5, Center Left and Center Right columns. In two responding hole pattern and porosity. Center and Right columns show of the three cases shown, the unstable modes computed with our xy representative fields of the σ component of the stress-field distribution method agree with those found in finite-element simulations. In plotted on top of the strained configurations with porosity p = 0.3. the case shown in Fig. 5, Middle row, there is a discrepancy, which might come as a surprise because the formalism properly identi- fies the critical strain, i.e., the load where a specific eigenmode effective spring constant. It is defined by the ratio of the aver- becomes unstable, while the mode itself is not the right one. age compressive stress to the compressive strain. Comparison to A deeper investigation reveals that many eigenmodes become direct numerical simulations shows that the formalism quanti- unstable almost simultaneously, making it difficult to pinpoint tatively captures the coarse-grained response of the system. In the least stable one. This is clearly seen in Fig. 5, Right column, addition, in Fig. 4, Center and Right columns we plot the spa- xy where at the onset of instability many eigenvalues are densely dis- tial distribution of the shear stress field σ for a representative tributed close to the vanishing one. The zig-zag–like mode, like porosity and imposed strain, plotted on top of the deformed con- the one predicted by finite-element simulations and by the inter- figurations, showing a favorable agreement also in the detailed acting quadrupole model of Fig. 2, also becomes unstable at a spatial structure of the solution. We emphasize that the charge similar strain. While we are still not sure about the precise origin formalism has no free parameters to fit. of this inconsistency, we suspect that it is rooted in the difficulty A slight discrepancy in the deformation field is observed in of satisfying the boundary condition on the external boundary of one orientation of the triangular lattice, as shown in Fig. 4, Bot- the solid, i.e., a finite-size effect. tom row, reflecting the fact that quadrupolar charges cannot fully describe the solution. To capture these details, higher-order Summary and Discussion multipoles are needed. We introduced a formalism that identifies image elastic charges as the basic degrees of freedom of perforated elastic metama- Instability (Nonlinear Response). Encouraged by the success of the terials. The continuum elastic problem, which contains multiple image charge method in the linear regime, we now proceed to boundary conditions, is reduced to a simpler problem of a lattice study the nonlinear instability of the system. In particular, we are of nonlinearly interacting elastic quadrupoles. interested in the critical strain at the onset of instability and the While the focus of our work is on 2D elastic metamaterials, unstable modes. the concept of image charges is applicable in principle in any The stability of the system is determined by the Hessian of the (2) dimension, although its implementation may be rather compli- energy which in the linear response regime is simply 2M . It cated. For example, elastic charges are defined as singularities is guaranteed to be positive definite and the system is thus sta- ext of a curvature field. In 2D the curvature is a scalar, intro- ble. Expanding to the next order in d , we find that the Hessian ducing a significant level of simplicity. In higher dimensions reads curvature is no longer a scalar but a tensor, and in 3D elastic (2) ∗ h (3) (3) (3)i charges are singularities of a second-rank tensor field (the ref- Hij = 2Mij +2Qk Mijk + Mikj + Mkij , [14] erence Ricci curvature). Generalizing our theory to 3D elastic metamaterials requires a classification of 3D elastic charges and where no summation is intended on i and j . In addition, the calculation of their resulting elastic fields, a subject of ongoing displacement constraints of Eq. 12 should also be corrected to research.

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PHYSICS materials described by distributing induced electric dipoles, but of the resulting stress–strain curve. To characterize the criti- with a richer response. cal strain, we conducted a buckling analysis on the undeformed configuration (∗BUCKLE step in Abaqus). Theoretical and Finite-Elements Method. The commercial software Abaqus/Standard was used for our FE simulations. Each mesh Data Availability. All numerical data discussed in this paper, was constructed using six-node, quadratic, plane-stress elements as well as a Mathematica notebook that contains a detailed (ABAQUS element type CPS6) and the accuracy was checked derivation of the theoretical results, are available to the by mesh refinement. The material was modeled as an isotropic reader on GitHub at https://github.com/yohai/elastic charges linear elastic material with 2D Poisson’s ratio ν = 0.3 and 2D metamaterials. Young’s modulus Y = 1. In all our analyses the models were ext loaded by imposing a displacement d to the two opposite ACKNOWLEDGMENTS. M.M. acknowledges useful discussion with David R. horizontal edges, while leaving the vertical one traction-free Nelson, Mark J. Bowick, and Eran Sharon. Y.B.-S. acknowledges support from (Fig. 3). To characterize the linear response, we conducted the James S. McDonnell postdoctoral fellowship for the study of complex ∗ systems. M.M. acknowledges support from the Israel Science Foundation a static analysis assuming small deformations ( STATIC step (Grant 1441/19). K.B. acknowledges support from NSF Grants DMR-1420570 with NLGEOM=OFF in Abaqus) and defined Yeff as the slope and DMR-1922321.

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