Theoretical search for heterogeneously architected2D structuresWeizhu Yanga, Qingchang Liua, Zongzhan Gaoa,b, Zhufeng Yueb, and Baoxing Xua,1

aDepartment of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904; and bSchool of Mechanics, Civil Engineering andArchitecture, Northwestern Polytechnical University, Xi’an, 710072 Shaanxi, China

Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved June 26, 2018 (received for review April 19, 2018)

Architected 2D structures are of growing interest due to theirunique mechanical and physical properties for applications instretchable electronics, controllable phononic/photonic modula-tors, and switchable optical/electrical devices; however, the un-derpinning theory of understanding their elastic properties andenabling principles in search of emerging structures with well-defined arrangements and/or bonding connections of assembledelements has yet to be established. Here, we present twotheoretical frameworks in mechanics—strain energy-based theoryand displacement continuity-based theory—to predict the elasticproperties of 2D structures and demonstrate their application in asearch for novel architected 2D structures that are composed ofheterogeneously arranged, arbitrarily shaped lattice cell structureswith regulatory adjacent bonding connections of cells, referred toas heterogeneously architected 2D structures (HASs). By pattern-ing lattice cell structures and tailoring their connections, the elasticproperties of HASs can span a very broad range from nearly zeroto beyond those of individual lattice cells by orders of magnitude.Interface indices that represent both the pattern arrangements ofbasic lattice cells and local bonding disconnections in HASs are alsoproposed and incorporated to intelligently design HASs with on-demand Young’s modulus and geometric features. This study of-fers a theoretical foundation toward future architected structuresby design with unprecedented properties and functions.

lattice structures | heterogeneously architected 2D structures |elastic properties | deterministic assembly | interface

Rational design of architected structures with well-definedorganizations has yielded many unique properties, including

ultrahigh specific stiffness, strength, and toughness (1–5),negative Poisson’s ratio (6–9), and shape reconfiguration (10–13). These extraordinary properties usually are independentof composition materials and are governed by structures.Therefore, the architected structures, in particular architected2D structures with the prosperity of low-dimensional materials,have attracted tremendous interest for applications in flexibleand stretchable electronics (14–16), mechanically controllablethermal structures (17–19), and structurally tunable optical andphononic devices (20–23). Most existing architected structuresdesigned by either shaping lattice cells at multiscales (1, 4, 5, 10)or utilizing origami/kirigami deformation mechanisms (11, 12,18, 22) are composed of the same unit architectures with peri-odic spatial arrangements, often referred to as mechanicalmetamaterials. By introducing unit cell diversity, a few me-chanical metamaterials are designed to achieve programmablemechanical performance (24–27). In parallel with the assemblyof unit cell architectures, the design concept of regulating net-work connections in architected structures, in particular mesh-like architected structures, also provides an alternative approachto achieve enhanced properties (28) and even new functionalitiessuch as allosteric behaviors (29, 30). Local modifications tobuilding unit cells or network connections will introduce het-erogeneous characteristics in architected structures and enrichthe design strategies of functional structures. In essence, theproperties of architected structures stem from both assembly of

unit cell structures and their bonding connections, and an in-telligent design with both integrated factors may open a newroute toward the search of heterogeneous superstructures withmultiple synergistic functions for widespread engineering appli-cations, beyond the capabilities of existing mechanical meta-materials (24, 31).Here we introduce a type of heterogeneously architected 2D

structures (HASs) that are composed of arbitrary distinct basiclattice structures in both geometric shape and mechanicalproperties with regulatory bonding connections. Two mechanicstheories—strain energy-based (SEB) theory and displacementcontinuity-based (DCB) theory—are established to quantita-tively predict the elastic properties of HASs by elucidating thedesign role of unit cell pattern arrangements and the bondingconnections between adjacent unit cells. The theoretical anal-yses indicate that the designed HASs yield a wide range ofdesired elastic properties including Young’s modulus andPoisson’s ratio far beyond those of individual lattice cells byorders of magnitude. The heterogeneous arrangements of unitlattice cells and their bonding connections are further in-corporated into two interface indices by design that highlightthe role of deformation mismatch and stress/strain informationtransfer among unit lattice cells in HASs to offer a direct guidefor on-demand search of HASs. Comprehensive computationalvalidations of the proposed HASs and their elastic propertiesindicate their potential for applications in practical engineeringsystems and also lay a theoretical foundation for searching

Significance

The bottom-up assembly of deterministic structures by latticecell structures for surpassing properties of individual compo-nents or their sums by orders of magnitude is of critical im-portance in materials by design. Here, we present a theoreticalstrategy in the search and design of heterogeneously archi-tected 2D structures (HASs) by assembling arbitrarily shapedbasic lattice structures and demonstrate that an extremelybroad range of mechanical properties can be achieved. Thisstrategy allows designing HASs through interface properties ofclose relevance to assembly patterns and bonding connectionsbetween basic lattice structures. Studies using extensive nu-merical experiments validate the robust, reliable, and lucrativestrategy of searching and designing HASs and offer quantita-tive guidance in the discovery of emerging 2D superstructures.

Author contributions: B.X. designed research; W.Y. and B.X. performed research; W.Y.,Q.L., Z.G., Z.Y., and B.X. analyzed data; and W.Y., Q.L., Z.G., Z.Y., and B.X. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence should be addressed. Email: bx4c@virginia.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1806769115/-/DCSupplemental.

Published online July 16, 2018.

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emerging architected 2D superstructures with unprecedentedproperties and functions.

ResultsHASs and Mechanics Theory for Elastic Properties. Fig. 1A presents aHAS composed of a series of basic lattice cell structuresarranged periodically with connections of adjacent cells by theirshared nodes (highlighted by red dots). These basic cell struc-tures, as shown in Fig. 1B, can be three-node, four-node, or six-node cells with rationally designed architectures from latticetrusses to connected-star systems to variant honeycombs thatpossess a wide variety of elastic properties, like stiffness fromnearly zero to theoretical upper limit and Poisson’s ratio from−1 to 1, as shown in SI Appendix, Figs. S1 and S2. When a HAS issubjected to a uniaxial tensile stress σx, due to the deformationdiversity among the component cells which may be contractile orauxetic, nonuniform stresses will arise in the cells to coordinateboth local and global deformation.

SEB theory. We first focus on the mechanics model that can beutilized to extract elastic properties of HASs composed of arbi-trary basic cell structures. Starting with a HAS consisting of Ncbasic cells in one repeated unit cell (RUC) subjected to a uni-axial tensile stress in the x direction, σx, we partition the RUCinto M×N blocks, analogous to meshing structures in finite ele-ment method, as illustrated in Fig. 2A. Note that each block mayinclude multiple basic cells, and Nc is not necessarily equal toM×N. Based on the symmetric features and mechanical valida-tion of basic cells (SI Appendix, Figs. S1 and S2), we simplifythem to isotropic cells, as shown by two adjacent cells (the ith andjth cells) around the (m, n) block with corresponding elasticconstants (Ei, vi, Gi) and (Ej, vj, Gj). At a free deformation state

RUC

3-noded ea b c f

4-node

d ea b c f

6-node

d ea b c f

A

B

Fig. 1. HASs and their component basic lattice cells. (A) A periodic HAScomposed of diverse basic lattice structures. Each two adjacent cells inHASs are connected by their shared node (red dots). The blue dashed boxrepresents an RUC of the HAS. (B) Schematics of various basic latticestructures with three, four, and six connection nodes and their variants (a,b, and c, star-shaped cell; d, missing rib cell; e, horseshoe cell; and f, auxeticchiral cell). The star-shaped cells (a, b, and c) will be mainly employedin the present HASs as representatives and their elastic properties can bewell tuned through their geometric features characterized by the cornerangle of θ.

A

B

Fig. 2. SEB theory in search for HASs. (A) Mechanics model of an RUC in aHAS subjected to a uniaxial tensile stress along the x direction, σx. The RUCis divided into M×N blocks, analogous to meshing in FEA. Each basic cellstructure is simplified into an isotropic material element with Young’smodulus, Poisson’s ratio, and shear modulus of E, v, G, respectively; thesubscripts i and j represent the cell number. (B) Comparison of normalizedstrain energy with respect to strain in the x direction for three typical HASsobtained from FEA and SEB theory. The insets highlight the RUCs of theHASs composed of three-node (black), four-node (blue), and six-node (red)basic cells, respectively. Ac is the area of a basic cell in each HAS, and Es isthe intrinsic Young’s modulus of the solid material component. In eachHAS, the same type of basic unit cells but different corner angles are used,and the corner angles θ in the three-node, four-node, and six-node unitcells are [20°, 40°, 60°, 90°], [10°, 40°, 70°, 170°], and [20°, 60°, 90°, 120°],respectively.

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of HASs, all component unit cells are assumed to be only sub-jected to a longitudinal stress along the loading x direction,where the associated strain energy is referred to as Wfree. Inpractice, the constraints between adjacent cells in HASs willlead to transverse and shear stresses due to dissimilarity in ge-ometry and/or elastic properties and reduce the longitudinalstrain along the loading direction. Consequently, the resultantreal strain energy will become lower than that in free de-formation state under the same applied stress σx, and the en-ergies released for the shear and transverse deformation of unitcells in deformed HASs need to be included to make themequivalent to that of the free deformation state. Therefore, thestrain energy W will satisfy

W +Wtrans +Wshear =Wfree, [1]

where Wtrans and Wshear are the strain energies due to effects oflocal transverse (perpendicular to loading direction) and shearstresses, respectively. Assume that in the free deformation statethe longitudinal stress in each row is uniform, and its magnitudedepends on the overall Young’s modulus of the row; thus, thestrain energy Wfree is

Wfree =12  Lx

XMm=1

Lmy

�σRowm

�2ERowm

, m= 1,2, ...,M, [2]

where σRowm = σxLyERow

m =PM

k=1ðERowk LkyÞ is the longitudinal

stress applied to the mth row. ERowm is the overall Young’s mod-

ulus of the mth row and can be calculated by ERowm =

Lx=PN

n=1ðLnx=EBlockmn Þ; EBlock

mn is the Young’s modulus of the(m, n) block determined from the elastic moduli of basic latticecells overlapped with the block. Lx and Ly are the total lengthand height of the RUC in x and y directions, respectively; Lnx andLmy are the length of the nth column in the x direction and theheight of the mth row in the y direction, respectively.Similarly, assume that the transverse stress in each column is

uniform, and with σColn in the nth column, Wtrans can be written as

Wtrans =12Ly

XNn=1

Lnx

�σColn

�2EColn

, n= 1,2, ...,N, [3]

where EColn is the overall Young’s modulus of the nth column,

similar to ERowm , and can be calculated via ECol

n =Ly=PM

m=1ðLmy=EBlockmn Þ. The determination of σColn can be performed by

eliminating the difference among the transverse strain of eachcolumn in the free deformation state and it is σColn =PN

k=1EColk Lkxð« free

y,k − «freey,n Þ=PN

k=1ðEColk Lkx=ECol

n Þ, where «freey,n is theoverall transverse strain of the nth column in the free deforma-tion state. It is worth noting that the transverse stresses on col-umns must be self-balanced because the entire structure is onlysubjected to a uniaxial loading along the x direction.Further, let the shear stresses be τmn,1, τmn,2, τmn,3, and τmn,4 for

the left, top, right, and bottom edge of the (m, n) block, re-spectively, and Wshear can be expressed as

Wshear =12

XMm=1

XNn=1

X4e=1

  Amn,eτ2mn,e

GBlockmn

, e= 1,2,3,4, [4]

where GBlockmn is the shear modulus of the (m, n) block. Amn,e

represents the area affected by the shear stress τmn,e on the cor-responding edge and can be estimated via Amn,e =LmyLnx=4. Thesubscript e denotes the edge of the block. To ensure connectiv-ity of adjacent blocks along the edge, for instance, between the(m, n) block and the (m, n − 1) block, the shear stress τmn,1 mustsatisfy

τmn,1

Lnx

2GBlockmn

+Ln−1,x

2GBlockm,n−1

!=Xmk=1

Lky

×

σColn

EBlockmn

−vBlockmn

EBlockmn

σRowm −

σColn−1

EBlockm,n−1

+vBlockm,n−1

EBlockm,n−1

σRowm

!, [5]

where vBlockmn is the Poisson’s ratio of the (m, n) block. Similarly,τmn,2, τmn,3, and τmn,4 can also be determined from the connec-tivity of their corresponding blocks.With Eqs. 1–4, the strain energy W of the RUC can be

obtained, and the Young’s modulus Ex and Poisson’s ratio vxy ofthe HAS are

Ex =ARUCσ2x2W

,   and  vxy =−

σColn

EColn

+ « freey,n

!ARUCσx2W

, [6]

where ARUC is the total area of the RUC. For vxy in Eq. 6, ncan be taken as any value from 1 to N because of the sametransverse strain for all columns. In addition, the overall strainalong the loading x direction can be easily calculated and is«x = 2W=ðσxARUCÞ.To verify this developed mechanics theory, referred to as SEB

theory, in the determination of Young’s modulus and Poisson’sratio, we constructed a series of HASs by patterning a number ofdifferent basic cells with three-, four-, or sixfold of rotationalsymmetry and performed finite element analyses (FEA). Theelastic properties of these basic cells exhibit a wide variety withrespect to their geometric feature such as θ, the angle of thecorner of the star-shaped unit, as shown in both theoretical andFEA in SI Appendix, Fig. S2 and Note A. Fig. 2B presents theelastic energy W of HASs obtained from both SEB theory andFEA, and the insets show the three typical HASs composed ofthree-node (black dashed box), four-node (blue dashed box),and six-node (red dashed box) basic cells. The good agreementindicates that the elastic properties of HASs can be well esti-mated by the SEB theory. By changing the geometry of unit cellsand their arrangements, we further performed analysis of thestrain energy on a series of other HASs. These calculations in-cluding the determination of elastic properties of each block arepresented in Materials and Methods and are detailed in SI Ap-pendix, Note B. The comparisons, as shown in SI Appendix, Fig.S3, further confirm the agreement between FEA and SEBtheoretical predictions.DCB theory. The SEB theory provides a general solution to extractthe elastic properties of HASs composed of arbitrary basic cells.However, it does not take into account the details of local unitlattice structures (which are not required before estimation) bymeshing the HASs into blocks and may lead to an inaccuracy.For example, Fig. 3A presents the strain energy of four-node-cellassembled HASs, where three selected arrangement patterns aregiven as representatives. Pattern 1 (the same as the one used inFig. 2) is shown in SI Appendix, Fig. S4A, and pattern 2 andpattern 3 are given in SI Appendix, Fig. S4 B and C, respectively.Significant differences between FEA and SEB theoretical pre-dictions are found in HASs with both pattern 2 and pattern 3. Asa consequence, the resulting theoretical calculations of Young’smoduli Ex and Poisson’s ratios vxy(Fig. 3B) show an obviousdifference from those from FEA, although the difference is verysmall for some HASs. To figure out the origin of such deviationsbetween FEA and SEB theory, we investigate the local stress andstrain (normalized by the applied stress and global strain ac-cordingly) for each basic cell in patterns 1–3. For example, uponloading, FEA in Fig. 3C indicates that the third row in pattern3 is subjected to smaller longitudinal stress than other rows. Bycontrast, the SEB theory shows that it has a larger stress because

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of larger overall Young’s modulus associated with the integrationof more numbers of unit cell A (θ = 10°). This inconsistencysuggests that the assumption on the determination of longitudi-nal stress in each row in the SEB theory is no longer satisfied.Similar findings are also obtained in pattern 2 (SI Appendix, Fig.S4B). Besides, the stress distribution among basic cells in a rowor column is not uniform. In comparison, the stress and straindistributions in pattern 1 (SI Appendix, Fig. S4A) only show asmall difference between FEA and SEB theoretical predictions,which is consistent with a slight difference in the correspondingstrain energy, Young’s modulus, and Poisson’s ratio (Fig. 3 A andB). Generally, the SEB theory will provide exact solutions for

HASs with simple layered patterns such as the HASs presentedin SI Appendix, Fig. S5.Given the local structures of HASs (i.e., if geometric shape of

unit cells and their assembled arrangements are known prior), anew theory is needed so as to provide an accurate estimation oftheir elasticity. This new theory model will be developed on thebasis of continuity of displacements at the connection nodesbetween adjacent unit cells, here referred to as DCB theory. Inthe DCB theory, each basic cell with a specific architecture isalso simplified into isotropic material elements, similar to that inthe SEB theory. Under this circumstance, the forces exertedon each node, including normal and tangential forces and a

A

C

D E

B

Fig. 3. DCB theory in search for HASs. (A) Comparison of normalized strain energy of HASs with three different patterns (insets) obtained from SEB theory,DCB theory, and FEA, where pattern 1 is the same as that of the four-node-cell assembled HAS in Fig. 2, and both patterns 2 and 3 are assembled by two four-node cells (θ = 10° and 170°) but with different arrangements. (B) Comparison of normalized Young’s modulus and Poisson’s ratio obtained from SEB theory,DCB theory, and FEA in a broad range of defined HASs, where each pair of data represents one HAS, and the HASs with patterns 1, 2, and 3 are highlighted incircles. (C) HAS with pattern 3: schematics of its pattern structures (Left) and local stresses and strains among the basic cells obtained from SEB theory, DCBtheory, and FEA (Right). (D) Typical basic cell architecture and its simplification to a linearly elastic isotropic model (Top) in DCB theory and FEA verification ofthe simplification in terms of cell deformation (Bottom). In the DCB theory, the forces at the connection nodes (highlighted in circles) in the basic cell aresimplified to stresses acted on the edges of the simplified model. In FEA, the dark blue area represents the simplified model. (E) Illustration of displacementcomputation paths (the curved dashed arrows in yellow) between two nodes (1 and 2) in the development of continuity equations of displacements in fouradjacent cells A–D. The connection nodes associated with each basic cell are marked as a–d in the clockwise direction.

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moment, will reduce to normal and tangential stresses on theedges of the cell via the connection nodes (red dots), as illus-trated in Fig. 3D. With this simplification, the connection amongunit cells can be considered a hinge connection, which is vali-dated by the good consistency of displacements at the connectionnodes between deformation of a unit cell (e.g., θ = 170° frompattern 3) and its corresponding simplified model, as demon-strated in the bottom-right inset of Fig. 3D.Now, let us consider a four-node-cell assembled HAS with M

rows and N columns of basic cells in RUC. From the analyses inFig. 3C and SI Appendix, Fig. S4, when the HAS is subjected toan applied stress σx, the local normal (σ i, j

nx and σ i, jny) and tangential

(σi,jtx and σi, jty ) stresses for the basic cells may differ from each otherdue to the mechanical difference between the cells. As a conse-quence, 4MN unknown stresses need to be solved to obtain theYoung’s modulus and Poisson’s ratio of the HAS. Self-balance ofstresses for each local cell and the overall HAS in both x and ydirections will yield (2MN + 2) independent equilibrium equationsof stress, and details can be found in SI Appendix, Note C. Theother 2MN − 2 independent equations will be established by uti-lizing continuity of displacements in connections of cells. For ex-ample, consider four adjacent cells marked by A to D (Fig. 3E);the connection nodes associated with each cell are marked as a–d in the clockwise direction. The yellow dashed arrows show twopaths to compute the displacements between node 1 and node 2.The resultant displacements from these two computation pathsmust be the same, and we have

uAdc + uCcb = uBda + uDab, [7]

where uAdc = uAd − uAc is the displacement between node c andnode d in cell A and the others are similar. Eq. 7 must holdfor both x and y directions, and thus the displacement continuitycondition for the entire structure of HASs leads to a total of(2MN − 2) independent displacement equations. Detailed expla-nations can be found in SI Appendix, Note C. For simplicity,assume the stress components vary linearly across each basic cell,which is validated in SI Appendix, Fig. S6; the displacements inEq. 7 can be expressed in terms of the unknown local stresses(see Materials and Methods and details in SI Appendix, Note D).Therefore, the 4MN unknown stresses can be determined. Sub-sequently, the Young’s modulus Ex and Poisson’s ratio vxy of theHAS will be

Ex = σxLxPNj=1

Ljx

hσ1, jnx + σ

1, j+1nx

2E1j−v1jðσ1, jny + σ

2, jny Þ

2E1j

i,  

and  vxy =−LxPMi=1

Liy

hσi,1ny + σi+1,1ny

2Ei1−vi1ðσi,1nx + σi,2nxÞ

2Ei1

iLyPNj=1

Ljx

hσ1, jnx + σ

1, j+1nx

2E1j−v1jðσ1, jny + σ

2, jny Þ

2E1j

i. [8]

In general, if the HAS is composed of basic cells with κðκ≥ 3Þconnection nodes, in the determination of Young’s modulusand Poisson’s ratio the number of unknown local stresses inits RUC with Nc basic cells will be κNc, where 2Nc + 2 equationscan be obtained directly from the equilibrium of stress for eachbasic cell and overall structures, and ðκ− 2ÞNc − 2 equations canbe established by correlating the unknown stresses with conti-nuity of displacements. SI Appendix, Fig. S7 and Note E furtherdiscuss HASs with three-node (κ= 3) and six-node (κ= 6)basic cells.As examples, the strain energies and elastic properties of four-

node (κ= 4)-based HASs in Fig. 3 A and B are computed againfrom DCB theory via Eq. 8 and W =ARUCEx«

2x=2. Further, local

stresses and strains obtained from DCB theory are also given in

Fig. 3C and SI Appendix, Fig. S4. The comparisons show goodagreement between results from DCB theory and FEA, in-dicating an enhanced prediction to elastic properties of a broadrange of HASs from DCB theory over SEB theory. Undercertain circumstances, note that although inaccuracy of pre-dictions may arise in the SEB theory, it does not require inputinformation of each unit cell prior and provides a more generalway to predict the elastic properties of HASs, in particular forHASs composed of arbitrarily shaped cells and/or their as-sembled complex manners. In addition, for some simply layeredpatterns like the one illustrated in SI Appendix, Fig. S5, thesolutions obtained from both DCB and SEB theories arethe same.

Search for HASs via Assembly of Unit Lattice Structures. By taking10 four-node basic cells with θ = 10°, 20°, 30°, 40°, 50°, 60°, 70°,90°, 120°, and 170° as fundamental building blocks, whoseYoung’s modulus and Poisson’s ratio (black open circles) aregiven in Fig. 4, we will demonstrate the design of HASs withdifferent elastic properties by assembling and patterning theminto HASs. RUCs of the studied HASs which consist of 4 × 4, 5 ×5, or 6 × 6 basic cells are taken as representatives. Approximately4 × 107 HASs in total with relative density from 5.3 to 10.6% areconstructed and their elastic properties are calculated using theDCB theory (SI Appendix, Fig. S8). For comparison, FEA areconducted for 53 randomly selected HASs as representatives.Fig. 4 indicates that the relative Young’s modulus Ex=Es, whereEs is the intrinsic Young’s modulus of the solid material, will varyfrom 7.0 × 10−5 to 2.3 × 10−3, beyond the range of known ma-terials in Ashby’s modulus-density map (32). Besides, the Pois-son’s ratios of HASs range from −1.0 to 1.7, beyond that of thebasic cells (−0.8 to 1.0). Such a broad range can also cover elasticproperties of the popular 2D lattice metamaterials that haverelative densities similar to HASs (SI Appendix, Fig. S9). Forheterogeneous composite materials, the Reuss (33) and Voigt(34) bounds are often employed to search the lower and upperbounds of elastic properties and can be determined under theassumption of a uniform stress and strain field throughout ma-terials, respectively. Given the similarity of structures betweenHASs and composite materials, where each basic lattice cell in

Fig. 4. Search for super-HASs via the pattern arrangement of unit latticestructures. Map of Young’s modulus Ex and Poisson’s ratio vxy of HASs withRUCs composed of 4 × 4, 5 × 5, and 6 × 6 four-node basic cells. The design ofHASs and all theoretical calculations (∼4 × 107 cases in total) are based on arandom selection and arrangement of basic cells from a pool of 10 four-node basic cells (black open circles), and FEA are performed on 53 randomlyselected HASs.

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HASs is analogous to an individual material constituent in com-posites, Voigt and Reuss bounds for HASs are extracted as ref-erences and also plotted in Fig. 4. In comparison, the calculationsof elasticity in both FEA and DCB theory of HASs indicate anupper bound, much larger than the Voigt bound, and this boundcan be deduced from the present DCB theory. The significantimprovement of both Young’s modulus and Poisson’s ratio withthe bound in HASs is attributed to the deformation mismatch ofunit cells and can be probed by examining local stress/strain dis-tributions (SI Appendix, Fig. S10). More explanations of thebounds of material properties can be found in SI Appendix, Fig.S11 and Note F. Only a few data close to the upper bound in Fig. 4are presented for highlighting clear comparisons between FEA andtheoretical predictions, and in principle the whole region enclosedby the bounds can be completely filled by assembling the basic cellsinto different patterns in the design of HASs, as demonstrated inSI Appendix, Fig. S12. In addition, it is worth noting that, by solelychanging the Poisson’s ratios of basic cells while keeping their

Young’s modulus, which can be achieved through typological op-timization to cell shapes (35), the upper bound of the Young’smodulus can be as large as three orders of magnitude higher thanthat of the basic cells, and the Poisson’s ratio of HASs can also varyfrom −15 to 15, as illustrated in SI Appendix, Fig. S8F.

Search for HASs via Bonding Connections of Adjacent Unit LatticeStructures. Bonding connections of adjacent lattice cells inHASs are of great importance and can also be utilized to designHASs with on-demand elastic properties. Similar to bonds ofmolecules in biological systems and artificial low-dimensionalmaterials (36, 37), we here consider a spring-like connectionmodel between adjacent unit lattice cells in HASs, as illustratedin Fig. 5A, where kn and kt are the normal and tangential springstiffness, respectively. When kn and kt are infinitely large, therewill be no relative displacement between the adjacent latticecells, and the connection nodes via this spring will reduce to ahinge; when both kn and kt are equal to zero, no connections

A

B

D

C

E

Fig. 5. Search for super-HASs via the regulation of bonding connections between unit lattice structures. (A) Schematic illustration of HAS with springconnections (Left) and highlighted two adjacent basic cells connected with a spring model (Right). a–d denote the connection nodes in each basic unit cell; A–D denote four adjacent basic unit cells, and the curved dashed arrows in yellow illustrate the computational paths between two nodes (1 and 2). kn and kt arethe normal and tangential stiffness of the spring model, respectively. Variation of (B) Young’s modulus and (C) Poisson’s ratio of HAS with pattern 1 (Inset). Theshaded areas represent the range of elastic moduli of basic lattice cells. Effect of disconnected nodes on (D) Young’s modulus and (E) Poisson’s ratio of the HAS withpattern 1 (Inset). Numbers 1–4 in the inset show the location of disconnected nodes, and the number sets in the operator [ ] represent the specific disconnected nodes.

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exist between adjacent unit cells in HASs. With the analyticalprocedure similar to that in DCB theory, consider cells A, B, C,

and D; the displacement continuity equation will becomeEq. 9 will be the same as Eq. 7 at an infinitely large kn and kt.Using Eq. 9, we recalculate the elastic properties of HASs withpatterns 1–3 given in Fig. 3A by using a series of spring constantsfrom 10−6Es to Es for all bonding connections. Fig. 5 B and Cpresents the obtained Young’s moduli and Poisson’s ratios forthe HAS with pattern 1. These plots show that both Young’smodulus and Poisson’s ratio of HASs can be tuned continuouslyfrom near zero to those with perfect connections. Besides, thenormal stiffness kn plays a dominant role in the elastic propertiesof HASs in comparison with the tangential stiffness kt, whichindicates a small contribution of the shear constraints betweenadjacent cells to the elastic properties of HASs. The associateddeformation morphologies are given SI Appendix, Fig. S13 A–Cand show that the basic cells themselves are barely deformedwhen spring constants are taken as 10−4Es. By contrast, as theyincrease to Es, the deformation of basic cells will be very similarto that in HAS with perfect bonding connections. SI Appendix,Fig. S13 D–G gives more results on HASs in patterns 2 and3 with spring connections. More importantly, all theoreticalcalculations from Eq. 9 agree well with FEA.For a local disconnection between adjacent unit cells with kn and

kt of zero, the displacement constraints at the nodes are completelyreleased, and displacement continuity will not be required in both xand y directions. Besides, there will be no transfer of local normaland tangential stresses between unit cells. As a result, the reductionsin the number of unknown stresses and displacement continuityequations are the same, and the solution remains self-contained.Consider local bonding disconnections in the HAS with pattern 1;Fig. 5 D and E gives both theoretical and FEA results on elasticproperties of HASs. Both Young’s modulus and Poisson’s ratio willdecrease with the increase of numbers of bonding disconnections.Additionally, they depend on the locations of bonding disconnec-tions, and accordingly the deformation modes are also different, asshown in SI Appendix, Fig. S14 A–C. Similar results are also foundfor HASs with patterns 2 and 3 in SI Appendix, Fig. S14 D–G.

Search for On-Demand HASs via Interface Properties. Either pat-terning unit lattice cells or tailoring their bonding connections inHASs indicates that the load transfer and deformation mismatchamong unit cells is essential for overall elastic properties andthus the assembly interfaces can also be utilized to design HASs.To incorporate the deterministic interfaces of relevance to pat-tern arrangements of unit cells into the elasticity of HASs, byconsidering a 4 × 4 RUC where two four-node basic cells (θ=10° and 170°) are used and each basic cell occupies eight blocksof the RUC, we define a pattern-dependent interface index γas

γ = αγv + βγh, [10]

where γv and γh reflect the similarity between the arrangedpattern and the pattern with only vertical (refer to SI Appendix,

Fig. S9A) and horizontal (refer to SI Appendix, Fig. S9C) interfacesin the RUC, respectively; α and β are the weight coefficients to the

enhancement of Young’s modulus and represent the contribution ofvertical and horizontal interfaces, respectively. Details for γv, γh, α,and β are given in Materials and Methods and SI Appendix, Note G.Fig. 6A shows the relative Young’s moduli obtained from bothDCB theory and FEA of randomly selected HASs with respectto γ, indicating an approximately linear variation in semilog coor-dinate system. Fig. 6B illustrates the evolution of patterns in HASswith γ. The calculations on more HASs (SI Appendix, Fig. S15)further confirm this dependence of elasticity on γ. Besides, whenthe connections between basic lattice cells become imperfect, theexponential variation will remain, but the effect of interface willbecome smaller with the decrease of spring stiffness kn in bondingconnections due to a weakened deformation mismatch and stress/strain transfer across the interfaces among unit lattice cells.In addition to the regulation of pattern arrangement associ-

ated interfaces among unit cells, the elasticity of HASs can betuned by controlling the number and locations of interfacialdisconnections among unit cells. Assuming the total number ofdisconnected interfaces in HASs is ndis, similar to the patternindex γ, we define a disconnection index η,

η=Yndis

k=1ηk, [11]

where ηkrepresents the effect of a single bonding disconnection,located at (I,J), on the elastic modulus of HASs and can beformulated by considering the local normal stress (i.e., σ I, J

nx orσ I, Jny ) at the interface before disconnected. Details for ηkare given

in SI Appendix, Note G.Take pattern 2 with γ = 1.15 as an example; by randomly dis-

connecting bonding interfaces from one to three, Fig. 6C shows thevariation of Young’s modulus of the locally disconnected HASs incomparison with that of the perfect structures, Ex,perfect/Ex. Ex,perfect/Exshows an approximately linear dependence on η for each combinationmode of disconnecting vertical and/or horizontal interfaces and is alsoconfirmed by FEA. Besides, given the same η, Ex,perfect/Ex increasesgradually as more vertical bonding disconnections occur. Meanwhile,a large difference in Ex,perfect/Ex can be obtained even with the samenumber of disconnections but different locations in HASs. Fig. 6Dgives the pattern structures of HASs with disconnection interface lo-cations highlighted. For other patterns of RUC, SI Appendix, Fig. S16further confirms the linear dependence of Ex,perfect/Ex on η in HASs.

DiscussionThe HASs presented here indicate a rational design of archi-tected structures to achieve on-demand elastic properties byassembling arbitrary unit cell structures and/or controlling ad-jacent bonding connections. The established theories, validatedfrom extensive FEA, provide a fundamental guidance in searchof emerging HASs from a broad range (as high as three orders ofmagnitude) of elastic properties. Specifically, the elastic strainenergy theory (referred to as SEB theory) in search of HASs withon-demand elastic properties does not require detailed prior

uAdc + uCcb +�LAy�kA,Bn 0

0 LAy�kA,Bt

�(σA,Bnx

σA,Bty

)+�LAx�kA,Ct 0

0 LAx�kA,Cn

�(σ A,Ctx

σ A,Cny

)

= uBda + uDab +�LCy�kC,Dn 00 LCy

�kC,Dt

�(σC,Dnx

σC,Dty

)+�LBx�kB,Dt 0

0 LBx�kB,Dn

�(σB,Dtx

σB,Dny

). [9]

Yang et al. PNAS | vol. 115 | no. 31 | E7251

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information on comprised basic cells. When the basic cells andtheir patterns in HASs are known, the DCB theory developed onthe basis of displacement continuity throughout the local basiccells can be utilized to improve the search accuracy. Additionally,interface properties among the basic cells are implanted into theYoung’s moduli of HASs by introducing the pattern-dependentand bonding disconnection indices and offer a straightforwardapplication in search of HASs. Future optimization design thatcould integrate artificial intelligence with the proposed theoriescould be pursued to speed up the search of HASs with on-demandmechanical properties and geometric features, and the seamlessincorporation of these interface indices of close relevance togeometric parameters will be particularly helpful.Although the developed theories and underlying search of HASs

are performed with the simplification of basic lattice structures into

isotropic cells, they can be readily extended to architected struc-tures composed of anisotropic unit cells, as described in SI Ap-pendix, Fig. S17 and Note H. By taking into account spatial energybalance and/or displacement fields, the established theories of SEBand DCB could be easily extended to predict the elastic propertiesof existing 3D lattice structures (4, 5, 24, 38) and to explore newones with unprecedented properties. Besides, given the similarityin structures of HASs with atom-bond structures of molecularsystems (31, 39, 40), the proposed theories may also be useful inthe prediction and design of molecular networks by introducingfunctional groups into proper positions or trimming local molec-ular connections. In applications, HASs with different bondingconnection stiffness can be additively manufactured by eitherselecting different materials or designing on-purpose connectionmanners such as serpentine shapes (41). Through the integration

A

C D

B

Fig. 6. Search for and quantitative characterization of super-HASs via the interface properties between unit lattice structures. (A) Variation of Young’smodulus of HASs, Ex/Es, with the pattern-dependent interface index γ. (B) Pattern structures of RUCs in HASs with defined γ and Ex/Es from FEA. Each RUC inthe HASs consists of 4 × 4 basic lattice cells with two geometric features (θ= 10° and 170°). (C) Variation of relative Young’s moduli of locally disconnectedHASs composed of RUCs in pattern 2 (γ = 1.15), Ex,perfect/Ex, with the interface bonding disconnection index η. VI and HI represent the vertical and horizontalinterfaces, respectively, and a total of 5,232 cases were calculated by arbitrarily disconnecting one to three interfaces. (D) Illustrations of the locations ofdisconnected interfaces (highlighted with red crosses) for six typical disconnection cases, circled numbers 1–6, and their corresponding η and Ex,perfect/Ex fromFEA. These six cases in colored dashed boxes correspond to different combinations of VIs and HIs as marked in C.

E7252 | www.pnas.org/cgi/doi/10.1073/pnas.1806769115 Yang et al.

of stimulus-responsive materials with either unit cells or theirbonding connection, HASs capable of dynamically responding toexternal environments such as temperature (42), electric field (43),or pH value (44) are also possible. In addition, under an externallarge mechanical loading, the nonlinear behavior of HAS such asplastic deformation or buckling may occur and should be carefullydesigned so as to benefit the practical applications; for example,local strut buckling (SI Appendix, Fig. S18) is expected to be ofinterest to design functional structures with programmable per-formance for potential applications in phononic/photonic devices(45). In summary, it is envisioned that our reported results hereindicate a route in the prediction and design of emerging archi-tected structures with unprecedented mechanical properties andfunctionalities in a wide range.

Materials and MethodsFinite Element Modeling and Analysis. To obtain the elastic properties, in-cluding Young’s modulus E, Poisson’s ratio v, and shear modulus G, of thethree-node, four-node, and six-node basic cells and their assembled HASs,FEA in plane strain situation with a linear elastic material model were per-formed. Periodic boundary conditions were applied to simulate the periodicarrangement in the structures. The intrinsic Young’s modulus Es and Pois-son’s ratio vs of the solid material components in the basic cells were takenas 3.97 MPa and 0.495, respectively. The geometries of the basic cells werediscretized by linear quadrilateral elements (CPE4R) and the number of el-ements varied from 2,010 to 5,800 for the four-node basic cells dependingon different corner angles. A mesh sensitivity study was conducted to con-firm that the discretization of model is sufficient for extracting the me-chanical properties. In HASs, unless otherwise stated, the basic unit cellswere perfectly bonded at the connection nodes to ensure local continuity ofdisplacements and stresses.

SEB and DCB Theory. In the SEB theory, the RUC is uniformly divided intoM×Nblocks in the longitudinal (x direction) and transverse (y direction) direction.

For each block, its elastic properties, EBlockmn , vBlockmn , and GBlock

mn , can be directlyobtained from local experimental measurements in applications. However,for HASs with known geometry details like the examples studied in thiswork, since we assume that the stress in a block is uniform in the SEB theory,the elastic properties of the (m,n) block can be calculated using the Reussiso-stress model via

EBlockmn =

LnxLmyPNci=1Ai,mn

�Ei,   vBlockmn =

PNci=1viAi,mn

�EiPNc

i=1Ai,mn�Ei

,   and GBlockmn =

LnxLmyPNci=1Ai,mn

�Gi

, [12]

where Ai,mn is the overlap area between the ith cell and the (m,n) block.Here, the average longitudinal (x direction) and transverse (y direction)length of all of the cells are used to divide the RUC of HASs into rows andcolumns, respectively.

In the DCB theory, the relation between the displacement field and localstress components within a basic cell is pivotal to solve the overall elasticity ofthe HASs. Here, for simplicity, assume that all stress components vary linearlyacross the unit cell structure; such a linearized stress field will automaticallysatisfy the compatibility equations. Under this circumstance, the stress field inthe (i, j) cell can be expressed as

σi, jxxðx, yÞ=1Ljx

�σi, jnx�Ljx − x

�+ σi, j+1nx x

�+1−

2yLiy

�σi, jty − σi, j+1ty

�[13a]

σi, jyyðx, yÞ=1Liy

hσi, jnyy + σ i+1, j

ny

�Liy − y

�i+1−

2xL jx

�σi, jtx − σi+1, jtx

�[13b]

σi, jxyðx, yÞ=1Liy

hσi, jtx y + σi+1, jtx

�Liy − y

�i−

1L jx

hσi, jty�Ljx − x

�+ σi, j+1ty x

i. [13c]

Note that the tangential stresses σi, jty and σi, j+1ty at the left and right sides of

the cell not only affect the shear stress field σi, jxyðx, yÞ but will also result in abending moment to the cell, leading to the appearance of the second termin the right side of Eq. 13a. Similarly, the second term in the right side of Eq.

13b results from the tangential stresses σi, jtx and σi+1, jtx . Given the stress field,the strain field can be obtained via a linear stress–strain relation, and thedisplacement field and strain field are related by

«i, jxx =∂ui, j

x

∂x,   «i, jyy =

∂ui, jy

∂y,   «i, jxy =

12

∂ui, j

x

∂y+∂ui, j

y

∂x

!. [14]

Given the linear variation of stress across the basic cell, the displacements inboth x and y directions can be written as quadratic functions of x and y:

ui, jx ðx, yÞ=a1x2 + a2xy + a3y2 + a4x +a5y +a6 [15a]

ui, jy ðx, yÞ=b1x2 +b2xy +b3y2 +b4x +b5y +b6, [15b]

where the coefficients a1 −   a5 and b1 −   b5 can be obtained by substitutingEq. 15 into Eq. 14, and detailed expressions can be found in SI Appendix,Note D. Besides, the constants a6 and b6 in Eq. 16 can be determined byfixing an arbitrary point (i.e., reference point) in the cell that will not affectthe relative displacement between the connection nodes. SI Appendix, Fig.S6 shows a comparison of displacement field from theoretical analysis andFEA, and good agreement is observed. Once the expression of displacementfield in a basic cell is obtained, the displacement continuity equations can beeasily written in terms of linear equations of the unknown stresses.

Interface Property Indices γ and ηk. In the expression of pattern-dependentindex γ, γv and γh represent the similarity of a HAS pattern with two typi-cal patterns: a vertically layered pattern (VLP) and a horizontally layeredpattern (HLP), respectively. They can be expressed as

γv = γv,corr

�1+

nrow

M

�,   and  γh = γh,corr

�1+

ncol

N

�, [16]

where γv,corr is the correlation coefficient between the design matrices of theHAS pattern and the VLP; γh,corr is that between the HAS pattern and theHLP. For each specific pattern, one basic cell represents −1 in its corre-sponding design matrix, and the other one represents 1 in the design matrix.nrow is the number of rows in the HAS that are the same with rows in theVLP, and ncol is the number of columns in the HAS that are the same with

columns in the HLP. The two weighted coefficients α and β are

α= EVLP=ðEVLP +EHLPÞ,   and  β= EHLP=ðEVLP + EHLPÞ, [17]

where EVLP and EHLP are Young’s moduli of VLP and HLP in HASs,respectively.

In terms of the interface disconnection index ηk, we aim to define it to belinearly related to the change of Young’s modulus of the locally discon-nected HASs in comparison with that of the perfect structures, referred to asEx,perfect/Ex. With the applied stress σx, ηk can be related to the change in theoverall strain of HASs along the x direction «x in comparison with that of theperfect structure «x,perfect. With Eq. 8, we will have

ηk ∝Ex,perfect

Ex=

«x«x,perfect

=

PNj=1

Ljx

�σ1, jnx + σ1, j+1nx

2E1j− v1 jðσ1, jny + σ2, jny Þ

2E1j

�

PNj=1

Ljx

24σ1, j

nx,perfect+ σ1, j+1

nx,perfect

2E1j−

v1j

�σ1, jny,perfect

+ σ2, jny,perfect

�2E1j

35. [18]

When a vertical interface located at (I, J) is disconnected, the original normalstress σI, Jnx along the x direction at the interface will be transferred to unit cells in

other rows, leading to the change of both σi, jnx and σi, jny among the unit cells inthe HASs. Assume that such stress transfer does not change the distribution of

σi, jnx in other rows, for a disconnected vertical interface,ηk can be obtained via

ηk =σxLy

σxLy − LIyσI,Jnx. [19]

Besides, when a horizontal interface located at (I, J) is disconnected, the releaseof the original transverse stress σI, Jny along the y direction at the interface will

cause the redistribution of σi, jny in the adjacent columns and will also affect the

local stresses in the x direction σi, jnx. For a disconnected horizontal interface, ηk is

ηk = 1+ σ

I,Jny

σx

. [20]

More details can be found in SI Appendix, Note G.Note that the local stresses will redistribute after introducing a discon-

nection; as a consequence, the interplay between disconnections at differentlocations needs to be considered when computing η via Eq. 11, and ηk for the

Yang et al. PNAS | vol. 115 | no. 31 | E7253

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latter disconnections should be calculated using local stresses of the HASwith former disconnected interfaces.

ACKNOWLEDGMENTS. This work was supported by the University ofVirginia.

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