Theoretical search for heterogeneously architected 2D structures

Weizhu Yang, Qingchang Liu, Zongzhan Gao, Zhufeng Yue, and Baoxing Xu
PNAS July 31, 2018 115 (31) E7245-E7254; first published July 16, 2018 https://doi.org/10.1073/pnas.1806769115

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

Significance

The bottom-up assembly of deterministic structures by lattice cell structures for surpassing properties of individual components or their sums by orders of magnitude is of critical importance in materials by design. Here, we present a theoretical strategy in the search and design of heterogeneously architected 2D structures (HASs) by assembling arbitrarily shaped basic lattice structures and demonstrate that an extremely broad range of mechanical properties can be achieved. This strategy allows designing HASs through interface properties of close relevance to assembly patterns and bonding connections between basic lattice structures. Studies using extensive numerical experiments validate the robust, reliable, and lucrative strategy of searching and designing HASs and offer quantitative guidance in the discovery of emerging 2D superstructures.

Abstract

Architected 2D structures are of growing interest due to their unique mechanical and physical properties for applications in stretchable electronics, controllable phononic/photonic modulators, and switchable optical/electrical devices; however, the underpinning theory of understanding their elastic properties and enabling principles in search of emerging structures with well-defined arrangements and/or bonding connections of assembled elements has yet to be established. Here, we present two theoretical frameworks in mechanics—strain energy-based theory and displacement continuity-based theory—to predict the elastic properties of 2D structures and demonstrate their application in a search for novel architected 2D structures that are composed of heterogeneously arranged, arbitrarily shaped lattice cell structures with regulatory adjacent bonding connections of cells, referred to as heterogeneously architected 2D structures (HASs). By patterning lattice cell structures and tailoring their connections, the elastic properties of HASs can span a very broad range from nearly zero to beyond those of individual lattice cells by orders of magnitude. Interface indices that represent both the pattern arrangements of basic lattice cells and local bonding disconnections in HASs are also proposed and incorporated to intelligently design HASs with on-demand Young’s modulus and geometric features. This study offers a theoretical foundation toward future architected structures by design with unprecedented properties and functions.

Rational design of architected structures with well-defined organizations 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 independent of composition materials and are governed by structures. Therefore, the architected structures, in particular architected 2D structures with the prosperity of low-dimensional materials, have attracted tremendous interest for applications in flexible and stretchable electronics (14⇓–16), mechanically controllable thermal structures (17⇓–19), and structurally tunable optical and phononic devices (20⇓⇓–23). Most existing architected structures designed 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 periodic spatial arrangements, often referred to as mechanical metamaterials. By introducing unit cell diversity, a few mechanical metamaterials are designed to achieve programmable mechanical performance (24⇓⇓–27). In parallel with the assembly of unit cell architectures, the design concept of regulating network connections in architected structures, in particular mesh-like architected structures, also provides an alternative approach to achieve enhanced properties (28) and even new functionalities such as allosteric behaviors (29, 30). Local modifications to building unit cells or network connections will introduce heterogeneous characteristics in architected structures and enrich the design strategies of functional structures. In essence, the properties of architected structures stem from both assembly of unit cell structures and their bonding connections, and an intelligent design with both integrated factors may open a new route toward the search of heterogeneous superstructures with multiple synergistic functions for widespread engineering applications, beyond the capabilities of existing mechanical metamaterials (24, 31).

Here we introduce a type of heterogeneously architected 2D structures (HASs) that are composed of arbitrary distinct basic lattice structures in both geometric shape and mechanical properties with regulatory bonding connections. Two mechanics theories—strain energy-based (SEB) theory and displacement continuity-based (DCB) theory—are established to quantitatively predict the elastic properties of HASs by elucidating the design role of unit cell pattern arrangements and the bonding connections between adjacent unit cells. The theoretical analyses indicate that the designed HASs yield a wide range of desired elastic properties including Young’s modulus and Poisson’s ratio far beyond those of individual lattice cells by orders of magnitude. The heterogeneous arrangements of unit lattice cells and their bonding connections are further incorporated into two interface indices by design that highlight the role of deformation mismatch and stress/strain information transfer among unit lattice cells in HASs to offer a direct guide for on-demand search of HASs. Comprehensive computational validations of the proposed HASs and their elastic properties indicate their potential for applications in practical engineering systems and also lay a theoretical foundation for searching emerging architected 2D superstructures with unprecedented properties and functions.

Results

HASs and Mechanics Theory for Elastic Properties.

Fig. 1A presents a HAS composed of a series of basic lattice cell structures arranged periodically with connections of adjacent cells by their shared nodes (highlighted by red dots). These basic cell structures, as shown in Fig. 1B, can be three-node, four-node, or six-node cells with rationally designed architectures from lattice trusses to connected-star systems to variant honeycombs that possess a wide variety of elastic properties, like stiffness from nearly 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 is subjected to a uniaxial tensile stress σx, due to the deformation diversity among the component cells which may be contractile or auxetic, nonuniform stresses will arise in the cells to coordinate both local and global deformation.

Fig. 1.
Fig. 1.

HASs and their component basic lattice cells. (A) A periodic HAS composed of diverse basic lattice structures. Each two adjacent cells in HASs are connected by their shared node (red dots). The blue dashed box represents an RUC of the HAS. (B) Schematics of various basic lattice structures 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, auxetic chiral cell). The star-shaped cells (a, b, and c) will be mainly employed in the present HASs as representatives and their elastic properties can be well tuned through their geometric features characterized by the corner angle of θ.

SEB theory.

We first focus on the mechanics model that can be utilized to extract elastic properties of HASs composed of arbitrary basic cell structures. Starting with a HAS consisting of Nc basic cells in one repeated unit cell (RUC) subjected to a uniaxial tensile stress in the x direction, σx, we partition the RUC into M×N blocks, analogous to meshing structures in finite element method, as illustrated in Fig. 2A. Note that each block may include multiple basic cells, and Nc is not necessarily equal to M×N. Based on the symmetric features and mechanical validation of basic cells (SI Appendix, Figs. S1 and S2), we simplify them to isotropic cells, as shown by two adjacent cells (the ith and jth cells) around the (m, n) block with corresponding elastic constants (Ei, vi, Gi) and (Ej, vj, Gj). At a free deformation state of HASs, all component unit cells are assumed to be only subjected to a longitudinal stress along the loading x direction, where the associated strain energy is referred to as Wfree. In practice, the constraints between adjacent cells in HASs will lead to transverse and shear stresses due to dissimilarity in geometry and/or elastic properties and reduce the longitudinal strain along the loading direction. Consequently, the resultant real strain energy will become lower than that in free deformation state under the same applied stress σx, and the energies released for the shear and transverse deformation of unit cells in deformed HASs need to be included to make them equivalent to that of the free deformation state. Therefore, the strain energy W will satisfyW+Wtrans+Wshear=Wfree,[1]where Wtrans and Wshear are the strain energies due to effects of local transverse (perpendicular to loading direction) and shear stresses, respectively. Assume that in the free deformation state the longitudinal stress in each row is uniform, and its magnitude depends on the overall Young’s modulus of the row; thus, the strain energy Wfree isWfree=12 Lx∑m=1MLmy(σmRow)2EmRow,m=1,2,...,M,[2]where σmRow=σxLyEmRow/∑k=1M(EkRowLky) is the longitudinal stress applied to the mth row. EmRow is the overall Young’s modulus of the mth row and can be calculated by EmRow=Lx/∑n=1N(Lnx/EmnBlock); EmnBlock is the Young’s modulus of the (m, n) block determined from the elastic moduli of basic lattice cells overlapped with the block. Lx and Ly are the total length and height of the RUC in x and y directions, respectively; Lnx and Lmy are the length of the nth column in the x direction and the height of the mth row in the y direction, respectively.

Fig. 2.
Fig. 2.

SEB theory in search for HASs. (A) Mechanics model of an RUC in a HAS subjected to a uniaxial tensile stress along the x direction, σx. The RUC is divided into M×N blocks, analogous to meshing in FEA. Each basic cell structure is simplified into an isotropic material element with Young’s modulus, Poisson’s ratio, and shear modulus of E, v, G, respectively; the subscripts i and j represent the cell number. (B) Comparison of normalized strain energy with respect to strain in the x direction for three typical HASs obtained from FEA and SEB theory. The insets highlight the RUCs of the HASs 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 is the intrinsic Young’s modulus of the solid material component. In each HAS, 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 unit cells are [20°, 40°, 60°, 90°], [10°, 40°, 70°, 170°], and [20°, 60°, 90°, 120°], respectively.

Similarly, assume that the transverse stress in each column is uniform, and with σnCol in the nth column, Wtrans can be written asWtrans=12Ly∑n=1NLnx(σnCol)2EnCol,n=1,2,...,N,[3]where EnCol is the overall Young’s modulus of the nth column, similar to EmRow, and can be calculated via EnCol=Ly/∑m=1M(Lmy/EmnBlock). The determination of σnCol can be performed by eliminating the difference among the transverse strain of each column in the free deformation state and it is σnCol=∑k=1NEkColLkx(εy,kfree−εy,nfree)/∑k=1N(EkColLkx/EnCol), where εy,nfree is the overall transverse strain of the nth column in the free deformation state. It is worth noting that the transverse stresses on columns must be self-balanced because the entire structure is only subjected 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, respectively, and Wshear can be expressed asWshear=12∑m=1M∑n=1N∑e=14 Amn,eτmn,e2GmnBlock,e=1,2,3,4,[4]where GmnBlock is the shear modulus of the (m, n) block. Amn,e represents the area affected by the shear stress τmn,e on the corresponding edge and can be estimated via Amn,e=LmyLnx/4. The subscript e denotes the edge of the block. To ensure connectivity of adjacent blocks along the edge, for instance, between the (m, n) block and the (m, n − 1) block, the shear stress τmn,1 must satisfyτmn,1(Lnx2GmnBlock+Ln−1,x2Gm,n−1Block)=∑k=1mLky×(σnColEmnBlock−vmnBlockEmnBlockσmRow−σn−1ColEm,n−1Block+vm,n−1BlockEm,n−1BlockσmRow),[5]where vmnBlock is the Poisson’s ratio of the (m, n) block. Similarly, τmn,2, τmn,3, and τmn,4 can also be determined from the connectivity 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 of the HAS areEx=ARUCσx22W, and vxy=−(σnColEnCol+εy,nfree)ARUCσx2W,[6]where ARUC is the total area of the RUC. For vxy in Eq. 6, n can be taken as any value from 1 to N because of the same transverse strain for all columns. In addition, the overall strain along the loading x direction can be easily calculated and is εx=2W/(σxARUC).

To verify this developed mechanics theory, referred to as SEB theory, in the determination of Young’s modulus and Poisson’s ratio, we constructed a series of HASs by patterning a number of different basic cells with three-, four-, or sixfold of rotational symmetry and performed finite element analyses (FEA). The elastic properties of these basic cells exhibit a wide variety with respect to their geometric feature such as θ, the angle of the corner of the star-shaped unit, as shown in both theoretical and FEA in SI Appendix, Fig. S2 and Note A. Fig. 2B presents the elastic energy W of HASs obtained from both SEB theory and FEA, and the insets show the three typical HASs composed of three-node (black dashed box), four-node (blue dashed box), and six-node (red dashed box) basic cells. The good agreement indicates that the elastic properties of HASs can be well estimated by the SEB theory. By changing the geometry of unit cells and their arrangements, we further performed analysis of the strain energy on a series of other HASs. These calculations including the determination of elastic properties of each block are presented in Materials and Methods and are detailed in SI Appendix, Note B. The comparisons, as shown in SI Appendix, Fig. S3, further confirm the agreement between FEA and SEB theoretical predictions.

DCB theory.

The SEB theory provides a general solution to extract the elastic properties of HASs composed of arbitrary basic cells. However, it does not take into account the details of local unit lattice structures (which are not required before estimation) by meshing the HASs into blocks and may lead to an inaccuracy. For example, Fig. 3A presents the strain energy of four-node-cell assembled HASs, where three selected arrangement patterns are given as representatives. Pattern 1 (the same as the one used in Fig. 2) is shown in SI Appendix, Fig. S4A, and pattern 2 and pattern 3 are given in SI Appendix, Fig. S4 B and C, respectively. Significant differences between FEA and SEB theoretical predictions are found in HASs with both pattern 2 and pattern 3. As a consequence, the resulting theoretical calculations of Young’s moduli Ex and Poisson’s ratios vxy(Fig. 3B) show an obvious difference from those from FEA, although the difference is very small for some HASs. To figure out the origin of such deviations between FEA and SEB theory, we investigate the local stress and strain (normalized by the applied stress and global strain accordingly) for each basic cell in patterns 1–3. For example, upon loading, FEA in Fig. 3C indicates that the third row in pattern 3 is subjected to smaller longitudinal stress than other rows. By contrast, the SEB theory shows that it has a larger stress because of larger overall Young’s modulus associated with the integration of more numbers of unit cell A (θ = 10°). This inconsistency suggests that the assumption on the determination of longitudinal 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 row or column is not uniform. In comparison, the stress and strain distributions in pattern 1 (SI Appendix, Fig. S4A) only show a small difference between FEA and SEB theoretical predictions, which is consistent with a slight difference in the corresponding strain energy, Young’s modulus, and Poisson’s ratio (Fig. 3 A and B). Generally, the SEB theory will provide exact solutions for HASs with simple layered patterns such as the HASs presented in SI Appendix, Fig. S5.

Fig. 3.
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 in circles. (C) HAS with pattern 3: schematics of its pattern structures (Left) and local stresses and strains among the basic cells obtained from SEB theory, DCB theory, and FEA (Right). (D) Typical basic cell architecture and its simplification to a linearly elastic isotropic model (Top) in DCB theory and FEA verification of the simplification in terms of cell deformation (Bottom). In the DCB theory, the forces at the connection nodes (highlighted in circles) in the basic cell are simplified to stresses acted on the edges of the simplified model. In FEA, the dark blue area represents the simplified model. (E) Illustration of displacement computation paths (the curved dashed arrows in yellow) between two nodes (1 and 2) in the development of continuity equations of displacements in four adjacent cells A–D. The connection nodes associated with each basic cell are marked as a–d in the clockwise direction.

Given the local structures of HASs (i.e., if geometric shape of unit cells and their assembled arrangements are known prior), a new theory is needed so as to provide an accurate estimation of their elasticity. This new theory model will be developed on the basis of continuity of displacements at the connection nodes between adjacent unit cells, here referred to as DCB theory. In the DCB theory, each basic cell with a specific architecture is also simplified into isotropic material elements, similar to that in the SEB theory. Under this circumstance, the forces exerted on each node, including normal and tangential forces and a moment, will reduce to normal and tangential stresses on the edges of the cell via the connection nodes (red dots), as illustrated in Fig. 3D. With this simplification, the connection among unit cells can be considered a hinge connection, which is validated by the good consistency of displacements at the connection nodes between deformation of a unit cell (e.g., θ = 170° from pattern 3) and its corresponding simplified model, as demonstrated 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 in Fig. 3C and SI Appendix, Fig. S4, when the HAS is subjected to an applied stress σx, the local normal (σnxi,j and σnyi,j) and tangential (σtxi,j and σtyi,j) stresses for the basic cells may differ from each other due to the mechanical difference between the cells. As a consequence, 4MN unknown stresses need to be solved to obtain the Young’s modulus and Poisson’s ratio of the HAS. Self-balance of stresses for each local cell and the overall HAS in both x and y directions will yield (2MN + 2) independent equilibrium equations of stress, and details can be found in SI Appendix, Note C. The other 2MN − 2 independent equations will be established by utilizing continuity of displacements in connections of cells. For example, 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 two paths to compute the displacements between node 1 and node 2. The resultant displacements from these two computation paths must be the same, and we haveudcA+ucbC=udaB+uabD,[7]where udcA=udA−ucA is the displacement between node c and node d in cell A and the others are similar. Eq. 7 must hold for both x and y directions, and thus the displacement continuity condition for the entire structure of HASs leads to a total of (2MN − 2) independent displacement equations. Detailed explanations 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 in Eq. 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. Subsequently, the Young’s modulus Ex and Poisson’s ratio vxy of the HAS will beEx=σxLx∑j=1NLjx[σnx1,j+σnx1,j+12E1j−v1j(σny1,j+σny2,j)2E1j], and vxy=−Lx∑i=1MLiy[σnyi,1+σnyi+1,12Ei1−vi1(σnxi,1+σnxi,2)2Ei1]Ly∑j=1NLjx[σnx1,j+σnx1,j+12E1j−v1j(σny1,j+σny2,j)2E1j].[8]In general, if the HAS is composed of basic cells with κ(κ≥3) connection nodes, in the determination of Young’s modulus and Poisson’s ratio the number of unknown local stresses in its RUC with Nc basic cells will be κNc, where 2Nc+2 equations can be obtained directly from the equilibrium of stress for each basic cell and overall structures, and (κ−2)Nc−2 equations can be established by correlating the unknown stresses with continuity of displacements. SI Appendix, Fig. S7 and Note E further discuss 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 again from DCB theory via Eq. 8 and W=ARUCExεx2/2. Further, local stresses and strains obtained from DCB theory are also given in Fig. 3C and SI Appendix, Fig. S4. The comparisons show good agreement between results from DCB theory and FEA, indicating an enhanced prediction to elastic properties of a broad range of HASs from DCB theory over SEB theory. Under certain circumstances, note that although inaccuracy of predictions may arise in the SEB theory, it does not require input information of each unit cell prior and provides a more general way to predict the elastic properties of HASs, in particular for HASs composed of arbitrarily shaped cells and/or their assembled complex manners. In addition, for some simply layered patterns like the one illustrated in SI Appendix, Fig. S5, the solutions obtained from both DCB and SEB theories are the same.

Search for HASs via Assembly of Unit Lattice Structures.

By taking 10 four-node basic cells with θ = 10°, 20°, 30°, 40°, 50°, 60°, 70°, 90°, 120°, and 170° as fundamental building blocks, whose Young’s modulus and Poisson’s ratio (black open circles) are given in Fig. 4, we will demonstrate the design of HASs with different elastic properties by assembling and patterning them into HASs. RUCs of the studied HASs which consist of 4 × 4, 5 × 5, or 6 × 6 basic cells are taken as representatives. Approximately 4 × 107 HASs in total with relative density from 5.3 to 10.6% are constructed and their elastic properties are calculated using the DCB theory (SI Appendix, Fig. S8). For comparison, FEA are conducted for 53 randomly selected HASs as representatives. Fig. 4 indicates that the relative Young’s modulus Ex/Es, where Es is the intrinsic Young’s modulus of the solid material, will vary from 7.0 × 10−5 to 2.3 × 10−3, beyond the range of known materials in Ashby’s modulus-density map (32). Besides, the Poisson’s ratios of HASs range from −1.0 to 1.7, beyond that of the basic cells (−0.8 to 1.0). Such a broad range can also cover elastic properties of the popular 2D lattice metamaterials that have relative densities similar to HASs (SI Appendix, Fig. S9). For heterogeneous composite materials, the Reuss (33) and Voigt (34) bounds are often employed to search the lower and upper bounds of elastic properties and can be determined under the assumption of a uniform stress and strain field throughout materials, respectively. Given the similarity of structures between HASs and composite materials, where each basic lattice cell in HASs is analogous to an individual material constituent in composites, Voigt and Reuss bounds for HASs are extracted as references and also plotted in Fig. 4. In comparison, the calculations of elasticity in both FEA and DCB theory of HASs indicate an upper bound, much larger than the Voigt bound, and this bound can be deduced from the present DCB theory. The significant improvement of both Young’s modulus and Poisson’s ratio with the bound in HASs is attributed to the deformation mismatch of unit cells and can be probed by examining local stress/strain distributions (SI Appendix, Fig. S10). More explanations of the bounds 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. 4 are presented for highlighting clear comparisons between FEA and theoretical predictions, and in principle the whole region enclosed by the bounds can be completely filled by assembling the basic cells into different patterns in the design of HASs, as demonstrated in SI Appendix, Fig. S12. In addition, it is worth noting that, by solely changing the Poisson’s ratios of basic cells while keeping their Young’s modulus, which can be achieved through typological optimization to cell shapes (35), the upper bound of the Young’s modulus can be as large as three orders of magnitude higher than that of the basic cells, and the Poisson’s ratio of HASs can also vary from −15 to 15, as illustrated in SI Appendix, Fig. S8F.

Fig. 4.
Fig. 4.

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

Search for HASs via Bonding Connections of Adjacent Unit Lattice Structures.

Bonding connections of adjacent lattice cells in HASs are of great importance and can also be utilized to design HASs with on-demand elastic properties. Similar to bonds of molecules in biological systems and artificial low-dimensional materials (36, 37), we here consider a spring-like connection model between adjacent unit lattice cells in HASs, as illustrated in Fig. 5A, where kn and kt are the normal and tangential spring stiffness, respectively. When kn and kt are infinitely large, there will be no relative displacement between the adjacent lattice cells, and the connection nodes via this spring will reduce to a hinge; when both kn and kt are equal to zero, no connections exist between adjacent unit cells in HASs. With the analytical procedure similar to that in DCB theory, consider cells A, B, C, and D; the displacement continuity equation will becomeudcA+ucbC+[LAy/knA,B00LAy/ktA,B]{σnxA,BσtyA,B}+[LAx/ktA,C00LAx/knA,C]{σtxA,CσnyA,C}=udaB+uabD+[LCy/knC,D00LCy/ktC,D]{σnxC,DσtyC,D}+[LBx/ktB,D00LBx/knB,D]{σtxB,DσnyB,D}.[9]Eq. 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 with patterns 1–3 given in Fig. 3A by using a series of spring constants from 10−6Es to Es for all bonding connections. Fig. 5 B and C presents the obtained Young’s moduli and Poisson’s ratios for the HAS with pattern 1. These plots show that both Young’s modulus and Poisson’s ratio of HASs can be tuned continuously from near zero to those with perfect connections. Besides, the normal stiffness kn plays a dominant role in the elastic properties of HASs in comparison with the tangential stiffness kt, which indicates a small contribution of the shear constraints between adjacent cells to the elastic properties of HASs. The associated deformation morphologies are given SI Appendix, Fig. S13 A–C and show that the basic cells themselves are barely deformed when spring constants are taken as 10−4Es. By contrast, as they increase to Es, the deformation of basic cells will be very similar to that in HAS with perfect bonding connections. SI Appendix, Fig. S13 D–G gives more results on HASs in patterns 2 and 3 with spring connections. More importantly, all theoretical calculations from Eq. 9 agree well with FEA.

Fig. 5.
Fig. 5.

Search for super-HASs via the regulation of bonding connections between unit lattice structures. (A) Schematic illustration of HAS with spring connections (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 are the 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). The shaded 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 with pattern 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.

For a local disconnection between adjacent unit cells with kn and kt of zero, the displacement constraints at the nodes are completely released, and displacement continuity will not be required in both x and y directions. Besides, there will be no transfer of local normal and tangential stresses between unit cells. As a result, the reductions in the number of unknown stresses and displacement continuity equations 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 elastic properties of HASs. Both Young’s modulus and Poisson’s ratio will decrease with the increase of numbers of bonding disconnections. Additionally, they depend on the locations of bonding disconnections, and accordingly the deformation modes are also different, as shown in SI Appendix, Fig. S14 A–C. Similar results are also found for HASs with patterns 2 and 3 in SI Appendix, Fig. S14 D–G.

Search for On-Demand HASs via Interface Properties.

Either patterning unit lattice cells or tailoring their bonding connections in HASs indicates that the load transfer and deformation mismatch among unit cells is essential for overall elastic properties and thus the assembly interfaces can also be utilized to design HASs. To incorporate the deterministic interfaces of relevance to pattern arrangements of unit cells into the elasticity of HASs, by considering a 4 × 4 RUC where two four-node basic cells (θ= 10° and 170°) are used and each basic cell occupies eight blocks of the RUC, we define a pattern-dependent interface index γasγ=α¯γv+β¯γh,[10]where γv and γh reflect the similarity between the arranged pattern and the pattern with only vertical (refer to SI Appendix, Fig. S9A) and horizontal (refer to SI Appendix, Fig. S9C) interfaces in the RUC, respectively; α¯ and β¯ are the weight coefficients to the enhancement of Young’s modulus and represent the contribution of vertical 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 both DCB theory and FEA of randomly selected HASs with respect to γ, indicating an approximately linear variation in semilog coordinate system. Fig. 6B illustrates the evolution of patterns in HASs with γ. The calculations on more HASs (SI Appendix, Fig. S15) further confirm this dependence of elasticity on γ. Besides, when the connections between basic lattice cells become imperfect, the exponential variation will remain, but the effect of interface will become smaller with the decrease of spring stiffness kn in bonding connections due to a weakened deformation mismatch and stress/strain transfer across the interfaces among unit lattice cells.

Fig. 6.
Fig. 6.

Search for and quantitative characterization of super-HASs via the interface properties between unit lattice structures. (A) Variation of Young’s modulus 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 in the 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 disconnected HASs 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 horizontal interfaces, respectively, and a total of 5,232 cases were calculated by arbitrarily disconnecting one to three interfaces. (D) Illustrations of the locations of disconnected interfaces (highlighted with red crosses) for six typical disconnection cases, circled numbers 1–6, and their corresponding η and Ex,perfect/Ex from FEA. These six cases in colored dashed boxes correspond to different combinations of VIs and HIs as marked in C.

In addition to the regulation of pattern arrangement associated interfaces among unit cells, the elasticity of HASs can be tuned by controlling the number and locations of interfacial disconnections among unit cells. Assuming the total number of disconnected interfaces in HASs is ndis, similar to the pattern index γ, we define a disconnection index η,η=∏k=1ndisηk,[11]where ηkrepresents the effect of a single bonding disconnection, located at (I,J), on the elastic modulus of HASs and can be formulated by considering the local normal stress (i.e., σnxI,J or σnyI,J) 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 disconnecting bonding interfaces from one to three, Fig. 6C shows the variation of Young’s modulus of the locally disconnected HASs in comparison with that of the perfect structures, Ex,perfect/Ex. Ex,perfect/Ex shows an approximately linear dependence on η for each combination mode of disconnecting vertical and/or horizontal interfaces and is also confirmed by FEA. Besides, given the same η, Ex,perfect/Ex increases gradually as more vertical bonding disconnections occur. Meanwhile, a large difference in Ex,perfect/Ex can be obtained even with the same number of disconnections but different locations in HASs. Fig. 6D gives the pattern structures of HASs with disconnection interface locations highlighted. For other patterns of RUC, SI Appendix, Fig. S16 further confirms the linear dependence of Ex,perfect/Ex on η in HASs.

Discussion

The HASs presented here indicate a rational design of architected structures to achieve on-demand elastic properties by assembling arbitrary unit cell structures and/or controlling adjacent bonding connections. The established theories, validated from extensive FEA, provide a fundamental guidance in search of emerging HASs from a broad range (as high as three orders of magnitude) of elastic properties. Specifically, the elastic strain energy theory (referred to as SEB theory) in search of HASs with on-demand elastic properties does not require detailed prior information on comprised basic cells. When the basic cells and their patterns in HASs are known, the DCB theory developed on the basis of displacement continuity throughout the local basic cells can be utilized to improve the search accuracy. Additionally, interface properties among the basic cells are implanted into the Young’s moduli of HASs by introducing the pattern-dependent and bonding disconnection indices and offer a straightforward application in search of HASs. Future optimization design that could integrate artificial intelligence with the proposed theories could be pursued to speed up the search of HASs with on-demand mechanical properties and geometric features, and the seamless incorporation of these interface indices of close relevance to geometric 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 structures composed of anisotropic unit cells, as described in SI Appendix, Fig. S17 and Note H. By taking into account spatial energy balance and/or displacement fields, the established theories of SEB and DCB could be easily extended to predict the elastic properties of existing 3D lattice structures (4, 5, 24, 38) and to explore new ones with unprecedented properties. Besides, given the similarity in structures of HASs with atom-bond structures of molecular systems (31, 39, 40), the proposed theories may also be useful in the prediction and design of molecular networks by introducing functional groups into proper positions or trimming local molecular connections. In applications, HASs with different bonding connection stiffness can be additively manufactured by either selecting different materials or designing on-purpose connection manners such as serpentine shapes (41). Through the integration of stimulus-responsive materials with either unit cells or their bonding connection, HASs capable of dynamically responding to external environments such as temperature (42), electric field (43), or pH value (44) are also possible. In addition, under an external large mechanical loading, the nonlinear behavior of HAS such as plastic deformation or buckling may occur and should be carefully designed so as to benefit the practical applications; for example, local strut buckling (SI Appendix, Fig. S18) is expected to be of interest to design functional structures with programmable performance for potential applications in phononic/photonic devices (45). In summary, it is envisioned that our reported results here indicate a route in the prediction and design of emerging architected structures with unprecedented mechanical properties and functionalities in a wide range.

Materials and Methods

Finite Element Modeling and Analysis.

To obtain the elastic properties, including Young’s modulus E, Poisson’s ratio v, and shear modulus G, of the three-node, four-node, and six-node basic cells and their assembled HASs, FEA in plane strain situation with a linear elastic material model were performed. Periodic boundary conditions were applied to simulate the periodic arrangement in the structures. The intrinsic Young’s modulus Es and Poisson’s ratio vs of the solid material components in the basic cells were taken as 3.97 MPa and 0.495, respectively. The geometries of the basic cells were discretized by linear quadrilateral elements (CPE4R) and the number of elements varied from 2,010 to 5,800 for the four-node basic cells depending on different corner angles. A mesh sensitivity study was conducted to confirm that the discretization of model is sufficient for extracting the mechanical properties. In HASs, unless otherwise stated, the basic unit cells were perfectly bonded at the connection nodes to ensure local continuity of displacements and stresses.

SEB and DCB Theory.

In the SEB theory, the RUC is uniformly divided into M×N blocks in the longitudinal (x direction) and transverse (y direction) direction. For each block, its elastic properties, EmnBlock, vmnBlock, and GmnBlock, can be directly obtained from local experimental measurements in applications. However, for HASs with known geometry details like the examples studied in this work, 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 Reuss iso-stress model viaEmnBlock=LnxLmy∑i=1NcAi,mn/Ei, vmnBlock=∑i=1NcviAi,mn/Ei∑i=1NcAi,mn/Ei, and GmnBlock=LnxLmy∑i=1NcAi,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 and columns, respectively.

In the DCB theory, the relation between the displacement field and local stress components within a basic cell is pivotal to solve the overall elasticity of the HASs. Here, for simplicity, assume that all stress components vary linearly across the unit cell structure; such a linearized stress field will automatically satisfy the compatibility equations. Under this circumstance, the stress field in the (i, j) cell can be expressed asσxxi,j(x,y)=1Ljx[σnxi,j(Ljx−x)+σnxi,j+1x]+(1−2yLiy)(σtyi,j−σtyi,j+1)[13a]σyyi,j(x,y)=1Liy[σnyi,jy+σnyi+1,j(Liy−y)]+(1−2xLjx)(σtxi,j−σtxi+1,j)[13b]σxyi,j(x,y)=1Liy[σtxi,jy+σtxi+1,j(Liy−y)]−1Ljx[σtyi,j(Ljx−x)+σtyi,j+1x].[13c]

Note that the tangential stresses σtyi,j and σtyi,j+1 at the left and right sides of the cell not only affect the shear stress field σxyi,j(x,y) but will also result in a bending moment to the cell, leading to the appearance of the second term in the right side of Eq. 13a. Similarly, the second term in the right side of Eq. 13b results from the tangential stresses σtxi,j and σtxi+1,j. Given the stress field, the strain field can be obtained via a linear stress–strain relation, and the displacement field and strain field are related byεxxi,j=∂uxi,j∂x, εyyi,j=∂uyi,j∂y, εxyi,j=12(∂uxi,j∂y+∂uyi,j∂x).[14]

Given the linear variation of stress across the basic cell, the displacements in both x and y directions can be written as quadratic functions of x and y:uxi,j(x,y)=a1x2+a2xy+a3y2+a4x+a5y+a6[15a]uyi,j(x,y)=b1x2+b2xy+b3y2+b4x+b5y+b6,[15b]

where the coefficients a1​− a5 and b1​− b5 can be obtained by substituting Eq. 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 by fixing an arbitrary point (i.e., reference point) in the cell that will not affect the relative displacement between the connection nodes. SI Appendix, Fig. S6 shows a comparison of displacement field from theoretical analysis and FEA, and good agreement is observed. Once the expression of displacement field in a basic cell is obtained, the displacement continuity equations can be easily written in terms of linear equations of the unknown stresses.

Interface Property Indices γ and ηk.

In the expression of pattern-dependent index γ, γv and γh represent the similarity of a HAS pattern with two typical patterns: a vertically layered pattern (VLP) and a horizontally layered pattern (HLP), respectively. They can be expressed asγv=γv,corr(1+nrowM), and γh=γh,corr(1+ncolN),[16]

where γv,corr is the correlation coefficient between the design matrices of the HAS pattern and the VLP; γh,corr is that between the HAS pattern and the HLP. For each specific pattern, one basic cell represents −1 in its corresponding 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 the VLP, 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¯ areα¯=EVLP/(EVLP+EHLP), and β¯=EHLP/(EVLP+EHLP),[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 be linearly related to the change of Young’s modulus of the locally disconnected HASs in comparison with that of the perfect structures, referred to as Ex,perfect/Ex. With the applied stress σx, ηk can be related to the change in the overall strain of HASs along the x direction εx in comparison with that of the perfect structure εx,perfect. With Eq. 8, we will haveηk∝Ex,perfectEx=εxεx,perfect=∑j=1NLjx[σnx1,j+σnx1,j+12E1j−v1j(σny1,j+σny2,j)2E1j]∑j=1NLjx[σnx,perfect1,j+σnx,perfect1,j+12E1j−v1j(σny,perfect1,j+σny,perfect2,j)2E1j].[18]

When a vertical interface located at (I, J) is disconnected, the original normal stress σnxI,J along the x direction at the interface will be transferred to unit cells in other rows, leading to the change of both σnxi,j and σnyi,j among the unit cells in the HASs. Assume that such stress transfer does not change the distribution of σnxi,j in other rows, for a disconnected vertical interface,ηk can be obtained viaηk=σxLyσxLy−LIyσnxI,J.[19]

Besides, when a horizontal interface located at (I, J) is disconnected, the release of the original transverse stress σnyI,J along the y direction at the interface will cause the redistribution of σnyi,j in the adjacent columns and will also affect the local stresses in the x direction σnxi,j. For a disconnected horizontal interface, ηk isηk=1+|σnyI,Jσx|.[20]

More details can be found in SI Appendix, Note G.

Note that the local stresses will redistribute after introducing a disconnection; as a consequence, the interplay between disconnections at different locations needs to be considered when computing η via Eq. 11, and ηk for the latter disconnections should be calculated using local stresses of the HAS with former disconnected interfaces.

Acknowledgments

This work was supported by the University of Virginia.

Footnotes

  • ↵1To whom correspondence should be addressed. Email: bx4c{at}virginia.edu.

  • 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. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

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

Published under the PNAS license.

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Theoretical search for heterogeneously architected 2D structures
Weizhu Yang, Qingchang Liu, Zongzhan Gao, Zhufeng Yue, Baoxing Xu
Proceedings of the National Academy of Sciences Jul 2018, 115 (31) E7245-E7254; DOI: 10.1073/pnas.1806769115
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