Introduction

Truss metamaterials1,2 are widely known for their extensive range of programmable properties and exceptional functions. These unique properties arise from the architectural design in periodic patterns or geometric networks of repeating building blocks rather than intrinsic base materials. They have achieved remarkable merits, including the ultra-lightweight3,4, ultra-stiffness5,6, multi-stability7,8, negative thermal expansion9,10, and negative Poisson’s ratio11, etc. Accordingly, truss metamaterials have been broadly customized to fulfill various engineering applications, such as shape-morphing devices12, flexible skin13, sensors14, and mechanical cloaks15. The design strategy, which dominantly determines how the architectures and periodic patterns are constructed, is the kernel of developing preeminent truss metamaterials. However, current truss metamaterials are discretely inspired by several traditional strategies, imitating crystal2, molecular16,17, and biological18,19 architectures. The most prevalent architectures of two-dimensional (2D) truss metamaterials are typically constrained to the pure triangle, honeycomb, square, star, butterfly, and their variants, detrimentally leading to the corresponding mechanical properties, such as Young’s modulus and Poisson’s ratio, being also confined to a narrow range. Consequently, pursuing a large mapping database from architectures to properties represents a challenging yet indispensable endeavor in fulfilling the urgent quest for modern truss metamaterials.

Indeed, due to the highly complex mapping from architectures to properties, mining the full potential of truss metamaterials is still fraught with challenges. Hence, design strategies based on metamaterial databases have emerged. Nevertheless, due to the lack of generic database design rules dedicated to data acquisition, the current metamaterial database only comprises a limited set of architecture-property pairs, based merely on a few discrete metamaterials. Although the emerging data-driven strategy, such as machine learning, can leverage metamaterial databases for forward property prediction and inverse architecture design, the limited database still leads to the outputs of properties being circumscribed only by a narrow space20. Concurrently, the mapping from architectures to properties in machine learning is just regarded as an opaque black box21, where the natural unavailability of causality gives rise to further obstacles for mining extreme properties20. Moreover, even in the manual and topological optimization strategies, the limited and sparse databases of existing metamaterials lead to unclear mapping with ambiguous mechanical interpretation. This lack of clarity makes it difficult to provide effective initial design recommendations or predict achievable properties. Thus, a large database, covering the extensive architecture-property space, exactly containing unexplored architectures and extreme properties, will contribute to mining extreme properties and to exploiting the potential of the truss metamaterials.

Herein, to surmount the aforementioned obstacles, we systematically generate, interpret, and mine an exceptional architecture-property space by establishing a large 2D truss metamaterial database, which showcases progress in: (i) generating numerous architectures and (ii) expanding the corresponding property space, as well as (iii) mining extreme properties approaching theory bounds. This instantly available database22 contains 1,846,182 metamaterials with various architectures for mining mechanical properties, such as extreme Young’s modulus and Poisson’s ratio, providing diverse candidates for scientific research and engineering applications. Our graph-based design method for generating this large metamaterial database converts complex architectural topologies and graphical transformations into a compact encoding with basic mathematical languages. This framework can be transferred and adapted for advanced functional designs, further building other metamaterial databases, such as kirigami patterns23, origami folds24, hydrogel skeletons25, and multi-material devices26,27. Leveraging the generated metamaterial database, we demonstrate the mapping of extreme mechanical properties and anisotropy, originating from the low symmetry of the repeating building blocks, which is the fundamental mechanical description for achieving extreme Young’s modulus and Poisson’s ratio. Furthermore, we propose a definition of mechanical isomerism, elucidating the impact of subtle architectural variations on mining extreme properties. Notably, four kinds of metamaterials with isotropic bi-mode have been created, benefiting from the mechanical isomerism. Accordingly, we demonstrate that the architectural diversity in the large truss metamaterial database is a prerequisite for mining extreme properties, which will predictably support the emerging data-driven design in the future. Overall, the metamaterial database in this work can serve as a new paradigm for materials design, which will inspire the exploration of more anomalous and exceptional functionalities in interdisciplinary research, such as electromagnetic28, acoustic29, optical30, and thermodynamic31 metamaterials.

Results

Generating architecture-property database

Crystallography offers a rigorous and succinct description for characterizing periodic materials and structures, of which plane crystallographic groups (Supplementary Note 1) introduce a mathematical classification of 2D crystals, based on the symmetries of repeating unit cells in the plane32. Since 2D truss metamaterials are also constructed by repeating building blocks, an analogy can be established between crystallography and metamaterial mechanics. Accordingly, any 2D truss metamaterial conforms to a specific plane crystallographic group, comprising periodic tessellation of Representative Volume Elements (RVEs), which are fundamentally constructed from symmetry operations of the building blocks. Hence, the core of establishing a 2D truss metamaterial database is to sufficiently explore the flexible architectures of the building blocks, governed by the solid struts between certain nodes. Here, we propose a graph-based mathematical design framework to generate large metamaterial databases, as shown in Fig. 1. This systematic framework begins with the compact encoding of three basic architectural features within the building blocks, including boundary nodes, inside nodes, and struts. Through a series of architectural design rules, a large number of feasible building blocks are generated. Then, the RVEs are constructed by these building blocks through the graphical transformations, i.e., symmetric operations described in computer graphics33. Ultimately, the 2D truss metamaterials in the large database are constructed through the periodic tessellation of the corresponding RVEs.

Fig. 1: A design framework for generating a 2D truss metamaterial database.
Fig. 1: A design framework for generating a 2D truss metamaterial database.
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The framework follows the sequence of building blocks, representative volume elements (RVEs), and truss metamaterials. A1 Locating boundary nodes in a design area. Each boundary node has one degree of freedom to traverse along its edge, and the locations are constrained by a set of rules. A2 Uniformly distributing the optional positions of inside nodes. The distribution area is enclosed by the boundary nodes. A3 Connecting struts between nodes. The feasible building blocks are generated with the validity of the architectural topology and the capability of mechanical loading. A4 Large building block database with various architectures. B1 ~ B4 Constructing RVEs from building blocks by four fundamental graphical transformations: translation, rotation, reflection, and glide. There are eleven combinations of these transformations for the building block of square shape32, corresponding to eleven plane crystallographic groups. C1 ~ C4 Metamaterials constructed from the periodic tessellations of RVEs along the horizontal and vertical directions.

In detail, as shown in Fig. 1(A1), boundary nodes are located on each edge of a design area (e.g., square shape) of which the relative locations are governed by the coefficients, kAi. These coefficients are constrained to specific intervals by the layout rules (Supplementary Table 2, Supplementary Note 2) to avoid coincidence in one building block and to ensure the accurate connection of adjacent building blocks (Supplementary Fig. 2). This design thought of flexible boundary nodes significantly expands the architecture space, contrary to the fixed node locations derived from the direct imitations of crystal, molecular, or biological architectures. Subsequently, inside nodes are located within the distribution area enclosed by the boundary nodes (Fig. 1(A2)). The optional positions of inside nodes are calculated by the uniform distribution rules (Supplementary Fig. 3, Supplementary Note 3), which is beneficial to achieving diverse architectural topologies by sufficiently covering the design area. This rule can also adapt to the building block with an arbitrary polygon shape comprising various numbers of boundary and inside nodes (Supplementary Fig. 4). Furthermore, as illustrated in Fig. 1(A3), analogous to the undirected graph concept in computer graphics, connections between nodes are determined using the adjacency matrix34, which serves as a lookup table for struts and base materials (Supplementary Note 4). We further supplement the adjacency matrix with additional characteristics that efficiently identify and exclude unfeasible building blocks erroneously comprising isolated inside nodes, intersectant struts, and disconnected architectures (Supplementary Fig. 5). Accordingly, comprehensive distributions of struts and materials, and optimal mechanical properties can be guaranteed in this process. As a result, a large database composed of various feasible building blocks (as exemplified in Fig. 1(A4)) can be generated by the above three steps. The generated building blocks showcase diverse node positions, different numbers of struts, various connections, and architectural topologies from regular to asymmetric. Subsequently, Fig. 1(B1 ~ B4) exemplifies four basic graphical transformations, including translation, rotation, reflection, and glide. RVEs are accordingly generated from building blocks through the combinations of these graphical transformations (Supplementary Table 1) corresponding to the plane crystallographic groups. The 2D truss metamaterials (Fig. 1(C1 ~ C4)) are ultimately generated by periodic tessellations of RVEs along the horizontal and vertical directions in the plane. Universally, this design framework can also generate 2D truss metamaterial databases based on arbitrary shapes of building blocks (Supplementary Fig. 4) or multiple base materials (Supplementary Fig. 5). The large number of building blocks in this database is expected to provide a wealth of options for burgeoning irregular metamaterial designs35. The development of architectures for other functional metamaterials23,24,26,27 can also refer to our design framework.

Without loss of generality, a series of parameters is specified to generate a large architecture database that contains 1,846,182 metamaterials (Supplementary Table 3 presents detailed generation parameters of eleven groups, and Supplementary Data22 provides the node coordinates and adjacency matrices). Each metamaterial is assigned a name in a format exemplified as pmm_0241_4Eq_5Strut_6. This nomenclature is separated by underlines, successively including (i) a plane crystallographic group, (ii) a position label of four boundary nodes, (iii) the number of equipartitions in uniform distribution, (iv) the number of struts, and (v) an ordinal. Figure 2 depicts the examples of the generated truss metamaterials (represented by RVEs).

Fig. 2: Examples of architectures in the 2D truss metamaterial database (represented by RVEs).
Fig. 2: Examples of architectures in the 2D truss metamaterial database (represented by RVEs).
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A Views of classic metamaterials reproduced in the database. Metamaterials c1 ~ c6 are derived from the crystal lattices37,38 (c1, pure triangle; c2, pure square; c3, Kagome; c4, double-braced square; c5, semi single-braced square; and c6, semi double-braced square). Metamaterials c7 ~ c12 possess the negative Poisson’s ratio40,41,42,43,44,45; (c7, double arrow; c8, re-entrant honeycomb; c9, star; c10, butterfly; c11, rotational square; and c12, anti-chiral cross). B Diverse examples of eleven metamaterial groups produced from the database. Extreme properties can be sufficiently mined through the various architectures characterized by flexible node positions, various strut connections, and complete graphical transformations.

The resulting database reproduces a series of classic 2D truss metamaterials reported in the literature (Fig. 2A). Without loss of generality, twelve kinds of representative classic metamaterials with the best-known designs2,36 are listed here. They are regarded as the metamaterials with exceptional mechanical properties. In detail, classic metamaterials37,38 (insets c1 ~ c6) derived from existing crystal networks, such as the pure triangle (c1) and Kagome (c3), have been elaborately explored due to their stretching-dominated behavior, to improve the stiffness and strength2,39. Besides, relatively intricate classic metamaterials with negative Poisson’s ratio are also reproduced, including double arrow (c7)40, re-entrant honeycomb (c8)41, star (c9)42, butterfly (c10)43, rotational square (c11)44, and anti-chiral cross (c12)45. Their bending-dominated behavior is proven to facilitate excellent energy absorption46. These reproductions of classic architectures demonstrate that our framework is expected to efficiently substitute classic design strategies of crystal, molecular, or biological imitations to achieve identical metamaterials and properties. Besides, the classic metamaterials are only constructed by the individual rotation or reflection transformations, i.e., merely employing the groups p4, pm, pmm. On the contrary, the flexible node positions, various strut connections, and complete graphical transformations can contribute to this large architecture database with numerous metamaterials that present more intricate architectures (Fig. 2B), which has the potential to further expand the property space.

Notably, our framework enables the direct integration of three design hierarchies—building blocks, RVEs, and metamaterials—for the systematic creation of a large database. The resulting database extends the design flexibility of 2D truss metamaterials and ensures architectural integrity and feasibility, which will provide the foundation for mining extreme mechanical properties. Subsequently, the mechanical properties of every metamaterial are calculated using the numerical homogenization method47 (Supplementary Note 5), thus generating the corresponding property database. Accordingly, the mechanical properties of a 2D truss metamaterial can be exactly computed by its RVE with the relative displacement boundary condition48. Here, this boundary condition is employed to consider the relative displacement between the boundary nodes at the edges of RVEs (Supplementary Fig. 6), which facilitates the effective substitution of metamaterials by RVEs. The numerical homogenization method is based on the beam element, neglecting the local stress concentrations at the strut nodes. The cross-section size of the struts is fixed to 0.1 mm × 0.1 mm. The dimensions of building blocks are scaled uniformly with the same relative density, \(\bar{\rho}\) = 0.01, to obtain normalized properties for fair comparison and subsequent discussion. The Young’s modulus Es and Poisson’s ratio νs of a virtual base material are set to 1.0 MPa and 0.3, respectively. The numerical homogenization method is validated by the experimental results (see Method section and Supplementary Note 6). Afterward, we computed and developed a property database of 1,846,182 metamaterials, comprising effective Young’s moduli along the x and y directions (Ex and Ey), Poisson’s ratios (νyx and νxy), shear moduli (G), and anisotropy index (ASU) (see Supplementary Note 5 for property calculation and Supplementary Data22 for results). The Young’s moduli and shear moduli of the metamaterials are automatically normalized by the base Young’s modulus of Es = 1.0 MPa. Eventually, we exploit the architecture-property database to mine the superior designs and to identify the architectures responsible for the extreme properties.

Extreme Young’s Modulus and Poisson’s ratio

Young’s modulus and Poisson’s ratio, as fundamental mechanical properties, are critical for truss metamaterials, particularly in the exploration of their extreme values to fulfill engineering applications. Figure 3A illustrates Young’s moduli on a logarithmic scale of Cartesian coordinates along the x and y directions, covering wide magnitudes across 10−9 to 10−2 (results of eleven groups are provided in Supplementary Note 7). Considering the inherent anisotropy of the truss metamaterials in the database, the Voigt bound49 is more appropriate for evaluating the directional modulus rather than the Hashin-Shtrikman bound, which is only suitable for isotropic ones50. For instance, previous literature on anisotropic metamaterials has consistently used the Voigt bound as a benchmark5,51,52. The Voigt bound (red dashed-dotted lines) indicates the ideal upper limit of Young’s modulus for 2D truss metamaterials as a function of relative density, which is calculated as EVoigt = Es⋅(ρ/ρs) = 10−2 in this database. Under a consistent generation framework, the classic metamaterials (triangle scatters) only cluster at the bottom left and top right in Fig. 3A, leaving a big gap outside the reported space, especially in the unexplored space (blue area) between the reported bound (red solid lines) and Voigt bound. Besides, the gap between 10−6 to 10−4 limits their applications (Supplementary Fig. 8 provides the accurate distribution by a density plot), such as shape-morphing structures53, which require a specific balance between stiffness and flexibility to effectively prevent excessive deformation and catastrophic instability. Fortunately, in Fig. 3A, our database demonstrates that the extensive Young’s modulus space, including both extreme and intermediate moduli, can be achieved by the elaborate construction of a diverse metamaterial database. The coverage of property gaps also addresses the issue of ‘data cascade’ where distributional bias in property space usually restricts the exploration of expected mechanical properties20, thereby providing a robust foundation for data-driven metamaterial design.

Fig. 3: Mining extreme Young’s modulus and Poisson’s ratio from the resulting database composed of 1,846,182 truss metamaterials.
Fig. 3: Mining extreme Young’s modulus and Poisson’s ratio from the resulting database composed of 1,846,182 truss metamaterials.
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A The range of effective Young’s moduli Ex and Ey along the horizontal and vertical directions is visualized with extreme examples marked. Compared with the reported range composed of classic metamaterials (triangle scatters), the number of new metamaterials with superior effective Young’s moduli (star scatters) is up to 2939. The reported range refers to the properties of the classic metamaterials37,38,40,41,42,43,44,45 (Fig.2, insets c1 ~ c12), which are generated by the design method and included in the database22. B The range of Poisson’s ratios is visualized with extreme examples highlighted. νxy and νyx characterize the lateral contraction for loading along the x and y directions, respectively. The scales of the scatter represent the magnitude of Poisson’s ratios. C Mechanism for enhancing effective Young’s moduli. The diamond with reinforced struts presents the stretching-dominated behavior, which contributes to the superior effective Young’s moduli. D Mechanism for extreme Poisson’s ratio. The larger displacements of the struts on either side of the virtual fulcrum are responsible for the extreme Poisson’s ratio. E1 ~ E4 Trade-off bound and extreme examples for different couples of mechanical properties. Extreme modulus and Poisson’s ratio are inherent contradictions. The generated database can effectively mine the bounds of coupling properties. Source data are provided in the Source Data file.

The core principle underlying the high modulus in truss metamaterials is the stretching-dominated theory39, which maximizes the stiffness by efficiently transferring loads along the axial direction of the struts. Classic metamaterials (insets c1 ~ c6 in Fig. 2) are designed from crystal-inspired architectures to realize stretching-dominated behavior. Among them, anisotropic moduli (in the classic metamaterials c1 and c3) are commonly considered to approach the Voigt bound more readily. However, they only achieve up to 50% (5 × 10−3) of the Voigt bound (Supplementary Table 6 provides the detailed values) due to their specified symmetry and regularity. Therefore, directly designing stretching-dominated architectures only through intuitive or manual approaches is still highly challenging. Further investigation is required to elucidate the relationship between architectures and extreme modulus.

Notably, our database advances Young’s modulus of 2D truss metamaterials into the unexpanded space, identifying 2939 superior metamaterials (blue stars in Fig. 3A) that are closer to the Voigt bound than classic metamaterials. The insets a ~ f in Fig. 3A show examples of metamaterials with extreme Young’s modulus, incorporating basic architectures such as (i) straight struts aligned along the main axis and (ii) diamonds with a long diagonal, both accompanied by reinforced struts. Herein, we provide an intuitive pattern extracted from the database to highlight the architectural design principles underlying the extreme modulus. As illustrated in Fig. 3C, specific architectures are further analyzed with their effective moduli computed under unidirectional loading in the y direction. As a reference for comparison, the cross-architecture, constituted by orthogonal struts along the loading dircetions, corresponds to the highest modulus in classic metamaterials (inset c2 in Fig. 2). Notably, the modulus of various reinforced diamonds is observed to be enlarged with the increasing number of reinforced struts, reaching up to 2.4 times that of the cross, due to their stretching-dominated behavior. Accordingly, for example, the metamaterial pm_2034_4Eq_7Strut_6794 (inset b in Fig. 3A) possesses the most extreme Young’s modulus (7.27 × 10−3) and the other metamaterials (inset a and c ~ f in Fig. 3A) also possess the extreme Young’s moduli greater than 6.07 × 10−3, consistent with the results in Fig. 3C. In contrast to the reinforced diamonds, there is a sharp reduction of about 33 times in modulus when the reinforced struts are removed to just form a simple diamond (see metamaterials with extremely high and low moduli in Supplementary Table 7).

In the superior metamaterials (blue stars in Fig. 3A), struts are concentrated near one main axis, forming multiple stiff triangles. These concentrated struts focus on the enhancement of modulus in a certain loading direction. Although the pronounced voids in corresponding metamaterials created by this concentration might appear counterintuitive in the design paradigm for obtaining extreme modulus, our findings still confirm that the concentrated struts near the main axis are crucial for achieving extreme modulus. This finding challenges the conventional design strategy, which always pursues regular and symmetric architectural topology. It also highlights the importance of a diverse architecture database in mining extreme properties.

Figure 3B illustrates the entire space of the reciprocal Poisson’s ratios, νxy and νyx, for all metamaterials in the database (the density plot of the data points and scatter plots for eleven groups are provided in Supplementary Fig. 9). The top right subplot in Fig. 3B illustrates a certain range between −2.5 ≤ ν ≤ +2.5. The size of scatters increases in proportion to the maximum magnitude of νxy and νyx, namely, max(|νxy|, |νyx|). According to the Neumann’s principle, the range of Poisson’s ratios is strongly related to the symmetry of materials54. Isotropic materials exhibit high architectural symmetry with Poisson’s ratios bounded between −1.0 ≤ ν ≤ +0.5 55, as observed in most bulk materials. For orthogonally isotropic materials, medium architectural symmetry extends the range to −1.0 ≤ ν ≤ +1.0, such as classic chiral and star metamaterials (insets c9, c11, and c12 in Fig. 2). Furthermore, anisotropic metamaterials with six independent elastic constants exhibit low architectural symmetry in orthogonal directions and have no bound on the range of their Poisson’s ratios55,56. However, as shown in the subplots of Fig. 3B, classic metamaterials are confined to a quite narrower range of −2.0 ≤ ν ≤ +1.0 due to their high-symmetric architectures, despite the classic re-entrant metamaterials (insets c7, c8, and c10 in Fig. 2). The high-symmetric architectures are related to the reflection axis or rotation center, but the low-symmetric architectures possess none of them. Therefore, the preference of inspired manual design for obtaining high-symmetric architectures is exactly an obstacle to mining the extreme Poisson ratio. On the contrary, our database extends the range of Poisson’s ratio to −65.034 ≤ ν ≤ +69.602, far beyond ~30 times compared with the reported space. The diverse design results of anisotropic metamaterials in our database are derived from the generation of various building blocks with low architectural symmetry, which provide an important foundation for uncovering extreme Poisson’s ratios. Despite this considerable extension in the range of extreme Poisson’s ratios, all the metamaterials in the database still conform to the thermodynamic stability condition 0 < νxy ⋅ νyx < 1. This condition indicates the positive strain energy density and the positive definiteness of the stiffness matrix to ensure mechanical feasibility57. Consistent with this condition, the reciprocal Poisson’s ratios exhibit the same sign and are bounded within the area enclosed by the hyperbola, νyx = 1/νxy, as shown in Fig. 3B. Moreover, the bound metamaterials (blue circles in Fig. 3B) within the interval 0.8 < νxy ⋅ νyx < 1.0 account for 30% of the entire database, demonstrating the capacity to generate a large number of prospective architectures for mining extreme properties approaching the theory bound.

We further elucidate the underlying mechanism for this extreme Poisson’s ratio in view of architectural symmetry. Poisson’s ratio is defined as the ratio between the transverse strain (εt) and longitudinal strain (ε1) under unidirectional loading, i.e., ν = −εt/εl. Due to the orthogonal isotropic properties with close and constructed by four-fold rotation symmetry, chiral architectures exhibit only a narrow range of Poisson’s ratios, −1.0 ≤ ν ≤ +1.0, e.g., groups p4 and p4g (see Supplementary Fig. 9 and Supplementary Table 8). Thus, the extreme Poisson ratio requires a significant difference between εt and ε1, programmed by their architectural symmetry, which is consistent with the Neumann’s principle. In the intrinsic experience and intuition, the remaining metamaterial groups with various graphical transformations also seem to have high architectural symmetry (Supplementary Table 8 provides views). However, plenty of metamaterials in these groups indeed exhibit remarkable anisotropy (such as insets g ~ j in Fig. 3B), coupled with their Poisson’s ratios significantly beyond the reported space (Supplementary Fig. 9 provides the ranges). We attribute this counterintuitive result to an underlying mechanism, where the low architectural symmetry of building blocks, rather than that of RVEs or metamaterials, is the essential source of anisotropy and extreme Poisson’s ratios. As illustrated in Fig. 3D, even in group p1 with only translation transformation, the Poisson’s ratio of metamaterial p1_1133_6Eq_6Strut_5375 (inset k), which is constructed by a pair of mirror triangles with the reflection symmetry, exhibits orthotropic isotropy and is confined to −1.0. Conversely, when the building block is constructed by a couple of different triangles, the metamaterial cm_2120_4Eq_6Strut_23848 (inset l) exhibits a negative Poisson’s ratio, νyx, of -21.721. Case 1 and Case 2 (metamaterials pmm_3201_4Eq_6Strut_188338 and pmm_1101_4Eq_6Strut_57804) in Fig. 3D further elucidate that this phenomenon is attributed to the extremely low architectural symmetry. The low symmetry leads to the extreme Poisson’s ratios reaching the extreme positive and negative values of +69.602 (inset m) and -65.034 (inset k), respectively, through amplifying the transverse strain. In conclusion, we identify two conditions essential for achieving extreme Poisson’s ratios: (i) low-symmetric architectures within the building blocks, i.e., the absence of any reflection axis and rotation center, and (ii) graphical transformations except the four-fold rotation within the RVEs.

Figure 3(E1 ~ E4) demonstrates that our database also reaches the trade-off bounds of integrated properties (Supplementary Note 8 offers detailed explanations), such as the mutually exclusive nature of extreme Young’s modulus and extreme Poisson’s ratio (insets ① and ②). It is manifested that mining extreme mechanical properties and unexplored theory bounds is facilitated by the large architecture-property database generated by our design method, which includes various anisotropic metamaterials. In contrast to the conventional method in literature through optimization of specific architecture, our database directly offers a rich variety of ready-to-use metamaterials with extreme properties. Looking forward, the prospective enlargement of the architecture database is anticipated to further extend the range of extreme properties.

Extreme anisotropy and isotropy

Bulk materials, such as alloys and polymers, typically have isotropic properties, whereas truss metamaterials with anisotropic properties show significant advantages in achieving functional innovations58. The applications of anisotropy include mechanical flexibility and durability for wearable devices59, simulation of the bio-mechanical behavior for orthopedic implants60, etc. Here, the elastic anisotropy of 2D truss metamaterials is measured using the anisotropy index ASU 61, which is calculated by the stiffness and compliance matrices (Eq. S34, Supplementary Note 5). According to the shapes in polar plots (Fig. 4A) of the directional property, anisotropy and isotropy are specified as ASU > 10−3 and ASU ≤ 10−3, respectively. Along with the decrease of ASU, the scatters in Fig. 4A converge to the area, Ex/Ey = νxy/νyx = 1, exhibiting a funnel shape and gradually exhibiting isotropy. Supplementary Fig. 10 provides other detailed relationships for Young’s moduli, Poisson’s ratios, shear modulus, and ASU. Fig. 4B shows the ranges of ASU for eleven metamaterial groups and classic metamaterials. The corresponding range of metamaterials in the database far exceeds that of classic metamaterials that only possess anisotropy. In particular, groups pm and pg have the highest extreme anisotropy due to their strong architectural differences in the two orthogonal directions, exactly resulting from their unidirectional reflection and glide transformations. The extreme anisotropy and isotropy in eleven metamaterial groups are given in Supplementary Table 9.

Fig. 4: Mapping between extreme mechanical properties to anisotropy index.
Fig. 4: Mapping between extreme mechanical properties to anisotropy index.
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A Anisotropy index ASU versus Ex/Ey and νyx/νxy with their projections on three planes. The colors of the scatters represent the intervals of ASU. The conditions for anisotropy and isotropy are ASU > 10−3 and ASU ≤ 10−3, respectively. B Floating columns of the range of ASU for eleven groups and classic metamaterials. The metamaterials with the highest and lowest ASU in the groups pg and pm are identified. The classic metamaterials with the highest and lowest ASU are marked for comparison. C ASU versus shear modulus G. ASU is divided into three areas: (I) extreme, (II) intermediate, and (III) low ASU, to describe the aggregation along the vertical direction of the scatters. The metamaterials are aggregated into three clusters based on the (i) low, (ii) intermediate, and (iii) extreme values of G. The boundaries of these regions (dashed line) are at ASU = 106 and ASU = 10−3. D Examples of metamaterials with various combinations of anisotropy index and shear modulus. The double central reflection in the orthogonal directions notably enhances shear resistance, as demonstrated by the group cmm, while the single reflection results in extreme anisotropy and low G. E Anisotropy index versus νxy and νyx, and their projections on three planes. The range of νxy and νyx is limited to -0.490 ~ + 0.510 for mining isotropic bi-mode. F Examples of isotropic bi-mode metamaterials in groups p4g and pmm. Metamaterial, pmm_3044_4Eq_8Strut_8092, simultaneously exhibits outstanding effective Young’s moduli and isotropic bi-mode. Source data are provided in the Source Data file.

Figure 4C illustrates the anisotropy index versus the shear modulus. Based on shear modulus, the metamaterials are aggregated into three clusters: (i) low, (ii) intermediate, and (iii) extreme G. Moreover, ASU is divided into three areas: (I) extreme, (II) intermediate, and (III) low ASU, to describe the aggregation along the vertical direction of the scatters. Our database reveals that only specific combinations of areas (I + i, II+ii, II+iii, and III+ii) exist within the property space, exemplified by the typical metamaterials in Fig. 4D (see Supplementary Table 10 for details). These findings highlight inherent contradictions: extreme G versus isotropy, extreme G versus extreme anisotropy, and low G versus isotropy. Contrary to the literature that suggests a correlation between negative Poisson’s ratio and shear resistance (high shear modulus)1,62, counterexamples (Fig. 4D) are identified from our database: the metamaterial cmm_3200_4Eq_6Strut_34640 has a positive Poisson’s ratio (+0.718) but exhibits high shear modulus (2.65 × 10−2). Besides, pm_4123_4Eq_5Strut_55890 has a negative Poisson’s ratio (−3.931) but displays a low shear modulus (5.13 × 10−15). Although the Poisson’s ratios vary from negative to positive, the area (i) with extreme G consists of metamaterials only from the group cmm, whereas the area (iii) with low G comprises metamaterials only from the groups pm and pg (Fig. 4C). This deviation demonstrates that the superior shear resistance is indeed governed by the graphical transformations of various architectures, rather than the values of Poisson’s ratio suggested in the literature. Notably, we uncover that superior shear resistance can also be achieved through double-centered reflection transformation (in group cmm), whereas individual reflection or glide transformations (in group pm or pg) significantly weaken the shear resistance.

We further search for the extreme bi-mode metamaterials, of which the elastic properties approximate those of liquids, featured by an isotropic Poisson’s ratio of +0.5 and a near-zero shear modulus63. This unique property has attracted emerging applications, such as mechanical64,65 and acoustic66 cloaks. However, achieving bi-mode behavior imposes rigorous design requirements, giving rise to a significant challenge for conventional metamaterial research. Traditionally, two-dimensional bi-mode metamaterials were primarily limited to the modifications of hexagonal honeycombs67,68. In contrast, our database achieves the bi-mode design by flexibly designing truss metamaterials within the square building blocks. In detail, within an interval ASU ≤ 10−3 in Fig. 4E, we have identified four types of bi-mode metamaterials (represented by tetrahedrons) with Poisson’s ratios close to 0.5, accompanied by near-zero shear moduli, G ≤ 7.17 × 10−4. As shown in Fig. 4F, polar plots of their isotropic Young’s modulus and Poisson’s ratio demonstrate their rigidity under hydrostatic pressure (see Supplementary Table 11 for details). From the architectural perspective, this isotropic bi-mode is strongly linked to the double reflection and glide transformations in groups p4g and pmm. Notably, the metamaterial pmm_3044_4Eq_8Strut_8092 (Fig. 4F) exhibits a higher Young’s modulus, elevated by four orders of magnitude compared with those in group p4g, concurrently combining bi-mode with stretching-dominated behavior. These findings highlight the potential of our database in mining exceptional mechanical properties. For future research, isotropic bi-mode is also expected to be achieved in groups cm and pgg that also incorporate double reflection or glide transformations, and anisotropic bi-mode metamaterials with even lower G can be uncovered in the area (I + i) in Fig. 4C.

Mechanical isomerism for extreme properties

According to the parameter sensitivity analysis (see Supplementary Fig. 11, Supplementary Note 10), the plane crystallographic groups and the combination of boundary nodes are directly related to the variations in mechanical properties. However, the effect of individual architectural features is not apparent. The underlying mechanism for achieving extreme performance should be further investigated.

In the concept of isomers in chemistry, small structural changes among isomers will lead to significant variations in physicochemical properties69,70. The macroscopic properties of metamaterials are traditionally considered to be attributed to their primary architectural skeletons. However, the influence of the subtle architectural alterations on the macroscopic properties is invariably ignored and remains underexplored. Herein, in analogy with chemical isomerism, we define mechanical isomerism as the phenomenon where two or more metamaterials (named mechanical isomers) share identical constitutive features in the name format but differ in their architectures (arrangements of nodes and struts in building blocks), thus resulting in distinct mechanical properties.

We manifest that mechanical isomerism is an underlying principle for mining extreme mechanical properties. For example, Fig. 5A highlights a series of consecutively numbered mechanical isomers, p4_1331_6Eq_7Strut_1204 ~ 1230. Within these isomers, Young’s modulus of the metamaterial p4_1331_6Eq_7Strut_1224 significantly exceeds that of p4_1331_6Eq_7Strut_1204 by four orders of magnitude, just owing to the subtle architectural alternation of the inside node in their building blocks, even though the primary windmill shape skeleton remains unaltered. Similarly, as illustrated in Fig. 5B, the metamaterial with extreme Young’s modulus in the group pmg is excavated, based on a series of isomers, pmg_1242_6Eq_5Strut_18893 ~ 18919. Their Young’s moduli exhibit an exceptional enhancement, from 2.88 × 10−6 to 6.16 × 10−3, through delicate programming of inside nodes in the building blocks (Fig. 1B and Supplementary Note 3).

Fig. 5: Mechanical isomerism for mining extreme properties.
Fig. 5: Mechanical isomerism for mining extreme properties.
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A Effective Young’s Moduli, Ex = Ey, versus a variety of isomers, p4_1331_6Eq_7Strut_1204 ~ 1230. Subtle variations significantly enhance Young’s Modulus beyond four orders of magnitude. B Variation of Young’s modulus along the x direction, Ex versus a series of isomers, pmg_1242_6Eq_5Strut_18893 ~ 18919. Enhancement beyond three orders of magnitude for the modulus is beneficial to discover the unexplored extreme modulus. C Effect of isomers, cm_0001_4Eq_5Strut_1288 ~ 1314, on Poisson’s ratio along the y direction, νyx. With the same boundary node locations, various arrangements of the inside node and strut connectivity program extreme Poisson’s ratios across from negative to positive. D Examples of mining a metamaterial with extreme bi-mode by isomers, p4g_2213_6Eq_4Strut_7065 ~ 7092. The extensive architecture space in the large metamaterial database contributes to efficiently mining the rare isotropic bi-mode. Source data are provided as a Source Data file.

Furthermore, the extreme Poisson’s ratios of the isomers, cm_0001_4Eq_5Strut_1288 ~ 1314 (Fig. 5C), achieve a remarkable sign reversal from an extremely negative value of -54.994 to a positive value of +19.299. This finding for mining extreme Poisson’s ratio benefits from the elaborate arrangement of strut connectivity when designing the building blocks (Fig. 1C and Supplementary Note 4). The mechanical isomerism can also be observed in all eleven groups within the database (Supplementary Data), which also reiterates the pivotal role played by the diverse architectural topologies of the building blocks for mining extreme properties, rather than the symmetry induced by graphical transformations. Finally, mechanical isomerism also addresses the design of extreme metamaterials with isotropic bi-mode. In Fig. 5D, the isotropic metamaterial with ASU = 2.17 × 10−4 is uncovered from a variety of isomers, p4g_2213_6Eq_4Strut_7065 ~ 7092, comprising an identical primary architecture skeleton (star with two ribs) but different inside node positions. This subtle difference prompts the metamaterial p4g_2213_6Eq_4Strut_7085 to be highlighted from the isomers, as it is subsequently confirmed to be a valuable isotropic bi-mode metamaterial. This isotropic bi-mode, identified exclusively in groups p4g and pmm, indicates that the overall symmetry of the metamaterial mainly contributes to the achievement of isotropic properties.

The mechanisms of Young’s modulus and Poisson’s ratio in Fig. 3C, D provide a theoretical backing for mechanical isomerism. In Fig. 5B, from the isomer 18905 to 18900, the highlighted struts in the isomers gradually become parallel to the horizontal direction. Eventually, the struts concentrate near one main axis, and the metamaterial pmg_1242_6Eq_5Strut_18900 exhibits stretching-dominated behavior and extreme Young’s modulus, which is consistent with the analysis in Fig. 3C. As illustrated in Fig. 5C, the highlighted structure in the isomer 1305 is similar to the low-symmetric architecture with an extremely negative Poisson’s ratio in Fig. 3D. On the contrary, the isomer 1289 exhibits a lower positive Poisson’s ratio due to its highlighted convex angle.

In conclusion, mechanical isomerism, discovered from our large architecture-property database, is available for programming the mechanical response and holds profound prospects for mining extreme or abnormal properties. This sensitivity to subtle architectural variations has not been sufficiently paid attention to in previous metamaterial mechanics, as they may be easily ignored in sparse databases. We expect this work will attract more attention to the metamaterial isomers and delicate architectural design for mining the extreme properties of various metamaterials in mechanics, acoustics, optics, thermodynamics, electromagnetism, and beyond.

Discussion

In summary, we uncover the extreme properties of truss metamaterials through a large database with diverse architectures, which is elaborately generated by a compact coding of architectural features and graphical transformations. The resulting database contains 1,846,182 unique entries, comprising a large number of metamaterials exhibiting extreme elastic properties, which could serve as an instantly available directory to facilitate further scientific research and engineering applications. We show the necessity of a large database for mining the extreme property space, which is essential not only for obtaining customized properties based on prior knowledge and experience but also for exceeding the property ranges. The advantages in terms of mining extreme properties include: (i) extreme Young’s modulus approaching the Voigt bound, outperforming the reported range by 40%, (ii) programmable Poisson’s ratio spanning extremely negative to positive values, proving a 30-fold amplification, and (iii) exceptional isotropic bi-mode similar to the liquid performance. Notably, the specified low relative density of 0.01 will make the bend-dominated architecture inefficient within the metamaterial, and the influence of different relative densities on extreme mechanical properties deserves further exploration.

We provide intuitive architectures and mechanical interpretations associated with these extreme properties, revealing the significant influence and role of the low symmetry in underlying building blocks, rather than the symmetric operations in constructing metamaterials. Moreover, according to the mechanical isomerism we defined, we further demonstrate that the core principle of mining extreme properties is various building blocks with low-symmetric architectures, while the graphical transformations are supplementary to the construction of metamaterials and mainly contribute to isotropy.

Consequently, our work addresses the challenges of the absence of design principles and the mapping of architecture to extreme properties. Researchers and engineers can directly utilize our database to develop other desired truss mechanical metamaterials. As brief selection criteria, truss metamaterials with extremely high Young’s moduli can be applied to connectors, rocket main load-bearing structures, and rotatory conical structures in aerospace71,72,73. Applications in shape-morphing74, protective engineering75,76,77, soft robots78,79, biomedical industry80,81,82, etc., can query our database for metamaterials with extremely positive and negative Poisson ratios. Besides, the main applications of bi-mode metamaterials are acoustic64 and mechanical15 cloaks. Our design framework and extensive database for modeling of metamaterials also provide substantial candidate architectures for the further development of data-driven design, shape optimization, and advanced manufacturing, to fully acquire and mine the intricate potential introduced by architectural diversity.

Several directions can be improved for further development. When the relative density deviates from 0.01, the ranges of properties will be significantly affected. The influence of specific geometric parameters on the ranges is worthy of exploration. Due to the diversity and complexity of the architectures, the minimum feature size, sharp corners, and residual stress will affect the additive manufacturing of these metamaterials. The extension from 2D to 3D design method necessitates optimized traversal and computational strategies for high-dimensional tensors.

Methods

Generation parameters

By specifying four boundary nodes, one inside node, and one base material, we utilize the established design framework to construct a sample database, containing eleven metamaterial groups with square building blocks. Parameters kAi(i = 1,2,3,4) are selected from five fixed values: 0.00, 0.25, 0.50, 0.75, and 1.00. In fact, kAi is flexible as long as they conform to the layout rules (see Supplementary Table 2). The inside node traverses the uniform positions based on four or six equipartitions. In addition to reproducing this work, the number of inside nodes can also be changed to achieve more design results. Subsequently, the adjacency matrices are generated by all binary digits with a length of ten,  segmentally folded into upper triangular matrices without the main diagonal. Notably, the number of struts should not be predetermined in the inputs, and all feasible building blocks are composed of struts that count between four and eight, automatically generated by the rules of the adjacency matrix (see Supplementary Note 4). Detailed generation parameters of the database in this work are listed in Supplementary Table 3.

Fabrication

The specimens of truss metamaterials were fabricated by an Ultimaker S5 fused deposition modeling (FDM) 3D printer using the base material, red Acrylonitrile Butadiene Styrene (ABS). Four types of classic metamaterials were selected as test subjects, including c1, c2, c8, and c11 in Fig. 2, and c8 was tested in both horizontal and vertical (x and y) directions. Their CAD models, constructed using SolidWorks 2021, are shown in Supplementary Fig. 7A and provided in a Source Data file. At least 3 specimens were fabricated for each metamaterial. All specimens contain 3 × 3 RVEs. The size of these truss metamaterials without grips was 120 mm × 120 mm, with struts measuring 2 mm in both thickness and width. All necessary process parameters of the additive manufacturing are listed in Supplementary Table 4.

Experimental validation

As shown in Fig. 6A, the effective Young’s modulus and Poisson’s ratio were measured by tensile experiments, using an MTS-E45.105-B electronic universal testing machine. The loading force and displacement were recorded by the built-in sensor. The deformation measurement was captured by Image Correlation Theory83, using an industrial CCD and camera (BASLER-a2A4504, 4512 × 4512 pixels). The digital image correlation (DIC) analysis of displacement was conducted using the open-source MATLAB software, Ncorr84. As shown in Fig. 6B, speckle stickers for DIC analysis were located on the surface of the center RVE. The speckle pattern was generated by the open-source software, Glare85. The diameter of the speckles was 0.3 mm, and the spacing between the speckles was 1.0 mm.

Fig. 6: Experiment validation of the numerical homogenization method using additively manufactured classic truss metamaterials.
Fig. 6: Experiment validation of the numerical homogenization method using additively manufactured classic truss metamaterials.
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A The testing environment for the tensile test. B The standard specimen for testing the reference Young’s modulus of the base material. The grips engage with the clamps during the tensile test. All other specimens were also composed of the 3 × 3 RVEs in the middle and the grips on both sides. All speckle patterns of the center RVE were generated for the displacement analysis with DIC. C Stress-strain curves of pmm_0000_4Eq_4Strut_106 specimens in elastic regions (Hookean regions). D Views of specimens for validation. These specimens are related to the classic truss metamaterials in Fig. 2. E Examples of the displacement contours calculated by DIC analysis. F Stress-strain curves in the elastic region (Hookean region) for Young’s modulus validation. The curves were derived from the forces and displacements measured by the testing machine. G Transverse-tensile strain curves in the elastic region (Hookean region) for Poisson’s validation. The curves were derived from the DIC analysis. The experimental results are the average values and standard deviations of three repeated specimens. The simulation results are marked for comparison. Source data are provided as a Source Data file.

As reported in the literature, the printing quality is greatly influenced by the printing approach of ABS86. However, the standard tensile samples have different printing traces from the metamaterial specimens. Consequently, the well-known square truss metamaterial (c2 in Fig. 2) was fabricated in place of the standard samples to calculate the reference modulus of the base material ABS, as shown in Fig. 6B. The value of the normalized effective modulus E/Es of the square truss metamaterial is equivalent to half of the relative density 37. Thus, with a relative density of 0.2 and the measured Young’s modulus of 108.570 ± 1.391 MPa, the reference modulus of the ABS was calculated as 1085.700 ± 13.910 MPa through the test results in Fig. 6C. To calculate the simulation results, this modulus served as the material property in numerical homogenization.

Figure 6D illustrates the test specimens for the experimental validation, which are related to the classic truss metamaterials in Fig. 2. Their CAD models are shown in Supplementary Fig. 7A and provided in the Source Data files. Examples of the DIC analysis results are illustrated in Fig. 6E. The images were acquired at a frequency of one frame per 0.1 mm of tensile displacement. For each specimen, Fig. 6E shows two of the series of images with DIC results. As illustrated in Fig. 6F, G, Young’s moduli and Poisson’s ratios can be derived from the slope of the linear fitted lines. The engineering stress was calculated by dividing the force by the cross-sectional area of the overall specimen. The engineering strain was calculated by dividing the displacement by the initial length of the metamaterial (composed of three RVEs). The tensile and transverse strains were obtained by the average displacement difference across opposite boundaries of the center RVE, normalized by the initial RVE length. The simulation results were recalculated by the same numerical homogenization method (Supplementary Note 5) using the physical geometric dimensions of the CAD models (2 mm in both thickness and width of the struts, and 30 mm for the length of RVEs) and the reference modulus of ABS (1085.7 MPa). The numerical homogenization is conducted by Abaqus/CAE 2021. The Abaqus/standard analysis was conducted using the B21 beam element (2-node linear beam). The boundary conditions, result output, and property calculation are also described in Supplementary Note 5.

Notably, the reproducibility of the results is unaffected by variations in geometric dimensions of the specimen and grip, or properties of the base material, since the simulation values should be recalculated using the numerical homogenization based on the actual geometric and material parameters. Detailed experimental and simulation results are listed in Supplementary Table 5, and the comparison is illustrated in Supplementary Fig. 7. Potential error sources include: (i) defects inherent to additive manufacturing, (ii) misalignment between the two clamps, and (iii) force sensor noise. For reproducibility, we recommend using a base material with a higher Young’s modulus to further mitigate these error sources.