Introduction

Microstructured metamaterials have been enabling the realization of exotic mechanical responses and deformation functionalities, including negative stiffness1, negative compressibility2,3, shape morphing4,5,6, and twist modes7. In terms of the fundamental metric determining mechanical deformations, Poisson’s ratio, metamaterials have been realized to support negative values based on auxetic deformation patterns, via bendable or buckled ligaments8,9,10,11,12, or rotatable nodes13,14,15. The range of admissible Poisson’s ratios for isotropic materials is [−1, 0.5]. Anisotropy may expand this range, but based on the orthotropic constitutive law, thermodynamics predicts a general bound on Poisson’s ratios in the linear and stable elastic regime16, namely 0 ≤ vijvji < 1 (Fig. 1f), where vij and vji are Poisson’s ratios in two orthogonal directions (i = 1, 2, 3; j = 1, 2, 3; i ≠ j) and micropolar elasticity is not considered17. This bound includes positive Poisson’s ratios (Fig. 1a) in ordinary materials and negative Poisson’s ratios (Fig. 1b) in auxetic metamaterials, but there is a large range of unexplored space out of this bound (Fig. 1f). Designing Poisson’s ratios surpassing the thermodynamic limit may realize extraordinary deformation patterns and substantially extend the functional applications of modern mechanical devices, such as soft robots18 and biomedical equipment19. For instance, the Poisson’s ratios v12v21 < 0 (Fig. 1c) break the symmetry of constitutive tensors that originates from Maxwell–Betti reciprocity theorem20,21, which predicts non-reciprocal transmission of the displacement field (Supplementary Note 1). Reciprocity stems from microscopic reversibility and time­reversal symmetry, which is a fundamental property in many physical systems, including electromagnetic22, acoustic23, elastodynamics24, and elasticity25. Breaking static reciprocity associated with Poisson’s ratios surpassing the thermodynamic limit may offer brand new opportunities to enrich the functionalities of mechanical systems, in the same way as acoustic non­reciprocity has enabled the creation of modern acoustic devices, including one-way mirrors and circulators23,26. The thermodynamic limit is derived from the linear elastic regime with the assumption of infinitesimally small strain, yet the introduction of nonlinearities has so far failed to produce Poisson’s ratios surpassing the limit27,28.

Fig. 1: Deformation patterns associated with different Poisson’s ratios and tailored microstructures.
figure 1

a, b Traditional deformation patterns of positive and negative Poisson’s ratios. The gray dashed boxes indicate undeformed structures of length L. The blue and red arrows represent the load directions, and u1 and u2 are displacements in the x1 and x2 directions, respectively. c–e Unusual deformation patterns associated with non-reciprocal, ultra-large, and step-like Poisson’s ratios, respectively (Supplementary Notes 1–3). u2→1 (u1→2) is the output displacement in the x1 (x2) direction under the load F2 (F1). f Comparison between ultra-large, non-reciprocal Poisson’s ratios and the thermodynamic limit 0 ≤ vijvji < 1, and comparison between step-like Poisson’s ratios and invariant vij under compression or tension in the linear elastic regime. g Displacement amplification of two different levers. The single-mode lever with one fulcrum can only amplify the input displacement on the left, namely, u1→2 > u1 and u2→1 < u2. The multi-mode lever with two detachable fulcrums can amplify the input displacements on both the left and right by changing different fulcrums, namely, u1→2 > u1 and u2→1 > u2.

Full size image

Like a single-mode lever with one fixed fulcrum (Fig. 1g), the thermodynamic limit (0 ≤ vijvji < 1) only allows displacement amplification in one direction with a Poisson’s ratio larger than 1. However, a multi-mode lever with two detachable fulcrums can overcome this limit and realize bidirectional displacement amplification by changing rotation modes via different fulcrums (Fig. 1g). In terms of internal connectivity, the lever system is reconfigured due to the transition between the connection and separation states of the fulcrums and the lever, and then the lever shows different rotation modes.

In this work, inspired by the opportunities stemming from the topological reconfiguration of detachable fulcrums, we use predefined slits in the continuum to mimic detachable fulcrums. We develop a powerful inverse design framework that combines predefined slits and topology optimization, realizing self-bridging mechanical metamaterials that synthesize multi-mode microscale levers and exhibit Poisson’s ratios surpassing the thermodynamic limit. In terms of internal connectivity, the designed mechanical metamaterials respond to different loads with changed topological configurations induced by the self-bridging of predefined slits via self-contacts. The rotation behaviors of microscale levers in the metamaterials are then changed, and effective constitutive tensors of the metamaterials no longer obey the symmetry and invariance under different load scenarios. We can therefore achieve self-bridging metamaterials with Poisson’s ratios surpassing the conventional limits, enabling inaccessible deformation patterns, including one-way displacement amplification with broken static reciprocity (non-reciprocal Poisson’s ratios, vijvji < 0, in Fig. 1c, f), orthogonally bidirectional displacement amplification (ultra-large Poisson’s ratios, vijvji > 1, in Fig. 1d, f), and transverse expansion under both longitudinal tension and compression (step-like Poisson’s ratios, compressive vij > 0 and tensile vij < 0, in Fig. 1e, f). Particularly, the nonlinearity of self-contacts that is stronger than geometric and material nonlinearities can be active under small strains by setting zero initial distance between self-contact surfaces. Thus, the aforementioned deformation patterns can be activated by small strains and maintained under finite strains.

Results

Design, simulations, and experiments

We developed an inverse design framework to generate the freeform microstructural configurations of self-bridging mechanical metamaterials with target Poisson’s ratios surpassing the thermodynamic limit (see Methods section “Design method of self-bridging metamaterials”). In this framework, the design problem is described by the optimization problem in Eq. (7), where the target Poisson’s ratios are realized by constraining the error between actual and prescribed values. Slits are predefined in the unit cell (Fig. 2a, b), and then the topological configuration of each mechanical metamaterial is independently designed by setting target Poisson’s ratios in the optimization formulation. In the developed approach, elemental density variables are introduced to describe the material distribution29,30,31,32,33,34,35. The independently designed region in the approach is indicated in gray, and orange “F” is used to indicate the symmetry in the cell (Fig. 2b). The cell has four-fold rotational symmetry in the entire domain, and mirror symmetry exists in its quarter at each corner. The slits are distributed along the axis of mirror symmetry. Two slits (c2 and c4) are predefined for the design of non-reciprocal Poisson’s ratios, while four slits (c1, c2, c3, and c4) are predefined for the design of ultra-large and step-like Poisson’s ratios. The geometrical parameters of the slits are provided in Supplementary Fig. 1. Figure 2c–e show the designed microstructures with target Poisson’s ratios \({v}_{21}^{\ast }=1.4\) and \({v}_{12}^{\ast }=-0.5\), \({v}_{21}^{\ast }={v}_{12}^{\ast }=3\), and tensile \({v}_{12}^{\ast }=-0.8\) and compressive \({v}_{12}^{\ast }=0.8\), respectively. The locally enlarged views of the slits in the designed metamaterials are shown in Supplementary Fig. 1. The distributions of self-contacts induced by predefined slits and the topology of the microstructures are significantly different from those in other design frameworks using self-contacts6,11, which are built for different properties. Numerical simulations with 2 × 2 unit cells (Fig. 2c–e) were performed to evaluate Poisson’s ratios using finite-element models with periodic boundary conditions (Supplementary Fig. 2) and zero initial distance between self-contact surfaces in COMSOL Multiphysics 6.0. In this work, periodic boundary conditions are constructed based on orthotropy. More general periodic boundary conditions can be found in the studies36,37,38. We arranged the designed unit cells periodically and fabricated hyperelastic samples by molding (see Methods section “Measurement”). Poisson’s ratios of these samples are evaluated by uniaxial compression and tension tests (Supplementary Fig. 3).

Fig. 2: Design method, simulation, and comparison.
figure 2

a Initial unit cell with predefined slits (c1, c2, c3, and c4). The locally enlarged view of c3 is shown on the right. b Topology optimization and symmetry. c–e Designed microstructures and simulations for non-reciprocal, ultra-large, and step-like Poisson’s ratios, respectively. Each microstructure consists of 2 × 2 unit cells with lattice constant a, one of which is boxed by red dashed lines. Prescribed slits are labeled c1, c2, c3, and c4. Each color bar shows the output displacement field in the x1 or x2 direction. The rotation modes and fulcrums are denoted by arrows and balls, respectively.

Full size image

Non-reciprocal Poisson’s ratios

The metamaterial in Fig. 2c expands and contracts under transverse and longitudinal compressions (strains of −10%), respectively, which results in positive and negative Poisson’s ratios (Fig. 3a), respectively. In the numerical simulations, all microstructural slits open under transverse compression. However, when the microstructure is longitudinally compressed, the self-bridging of slits at c2 and c4 reconfigures the microstructural topology with a change in internal connectivity. Then, the microscale levers show two different rotation modes: (1) for transverse compression, the microstructure rotates about two fulcrums; (2) for longitudinal compression, the arms at c2 and c4 drive the rotation of the microstructure. These two different rotation behaviors result in positive and negative Poisson’s ratios, v21 > 0 and v12 < 0, respectively (Fig. 3a). We found the experimental deformation patterns (Fig. 4a) and Poisson’s ratios (Fig. 3a) agree well with the simulated ones. These Poisson’s ratios violate the lower bound of the thermodynamic limit (vijvji ≥ 0) that is derived from static reciprocity (Supplementary Note 1). With positive elastic moduli (E1 > 0 and E2 > 0 in Fig. 3b), E2v21 > 0 > E1v12 (Fig. 3c), which demonstrates that the designed metamaterial breaks the symmetry of its constitutive tensors (Ejvj = Eivij), stemming from static reciprocity. Hence, we can predict the emergence of a static non-reciprocal bulk mode (F1u2→1 > 0 > F2u1→2), different from the one in “fishbone” metamaterials designed by breaking geometrical symmetry and introducing geometrical nonlinearity25. The Poisson’s ratios v21 > 1 and 0 > v12 > −1 (Fig. 3a) predict that the metamaterial can amplify the input displacement field unidirectionally. Owing to the tiny slits and strong nonlinearity of self-contacts, a huge difference between E2v21 and E1v12 emerges under small strains, as shown in Fig. 3c. Hence, small strains may activate the non-reciprocal transmission of displacement fields in the designed metamaterial.

Fig. 3: Poisson’s ratio, elastic moduli, and non-reciprocity evaluations.
figure 3

a Non-reciprocal Poisson’s ratio. “FEA (finite-element analysis),” “Exp. (experiment),” “Eqv. (equivalent),” and “Eff. (effective)” correspond to simulations of designed unit cells, experiments about designed unit cells, simulations of equivalent models, and effective material parameters, respectively. b Elastic moduli E1 and E2 of the metamaterial with non-reciprocal Poisson’s ratios in two orthogonal directions. c Products of elastic moduli and Poisson’s ratios. The shaded area corresponds to the difference between E2v21 and E1v12. d Output displacement u1→2 (u2→1) and input displacement u1 (u2) for different values of F1 (F2). The shaded area corresponds to the difference between u1→2 and u2→1. e Ultra-large Poisson’s ratios. f Step-like Poisson’s ratios. The two shaded areas correspond to compression and tension strains, respectively.

Full size image
Fig. 4: Experiments.
figure 4

a–c Samples of designed metamaterials and snapshots of the deformation patterns for non-reciprocal (Supplementary Video 1), ultra-large (Supplementary Video 2), and step-like (Supplementary Video 3) Poisson’s ratios, respectively. d Experiments about non-reciprocity under transverse and longitudinal loads. The structure is supported by rollers that are compressed using an aluminum alloy plate. In Fig. a–c, undeformed samples have the same scale bar of 50 mm, and the deformed samples have the same scale bar of 20 mm. In Fig. 4d, the scale bar of 20 mm applies to all structures.

Full size image

To interpret the relation between the self-bridging feature and broken static reciprocity, we built two equivalent models in Supplementary Fig. 4a, b, where the slits closed under compression are replaced by solid connections. Each equivalent model is only valid for a special load case indicated by a gray arrow, which is determined by the mimicked metamaterial under the same load case (Supplementary Fig. 4). The internal connectivities of the equivalent models are different owing to the additional solid connections. The equivalent models were numerically simulated without defining self-contacts. The Poisson’s ratios and elastic moduli of the equivalent models are almost the same as those of the designed metamaterial (Fig. 3a, b). Thus, the designed metamaterial (Fig. 2c) can be substituted by these equivalent models. To explicitly distinguish these equivalent models, we calculated the effective compliance matrix of each equivalent model using the representative volume element method (Supplementary Note 4)39 based on the linear orthotropic constitutive law. The evaluated effective compliance matrices of our equivalent model are significantly different (Supplementary Equations (13, 14)), demonstrating topological reconfiguration of the metamaterial induced by self-bridging slits. Thus, the self-bridging feature of the metamaterial enables the variation of constitutive tensors under different load scenarios, which breaks the symmetry Ejvji = Eivij in static reciprocity and enables surpassing the lower bound (vijvji ≥ 0). In mechanical systems under small strains, general geometric and material nonlinearities may vanish25, but self-contacts can occur via zero initial distance between self-contact surfaces. The effective Poisson’s ratios and elastic moduli of the equivalent models agree well with those of the designed metamaterial under small strains (Fig. 3a, b), which demonstrates that the designed self-bridging metamaterial can break the static reciprocity under small strains.

The broken static reciprocity associated with Poisson’s ratios was tested in the designed metamaterial (Fig. 4d). Figure 4d shows the experimentally observed non-reciprocal deformation patterns of the metamaterial, which was supported by rollers to create sliding boundary conditions, and a force F1 = − 60 N (F2 = − 60 N) was applied at its right (top) side. Under the same force, u1→2 > 0 > u2→1, and |u1→2| is an order of magnitude greater than |u2→1|, as shown in Fig. 4d, which is evidence of non-reciprocal transmission of the displacement fields. In addition, the output displacement |u1→2| is large than the input displacement |u1|, while the output displacement |u2→1| is smaller than the input displacement |u2|. Thus, there is a unidirectionally amplified non-reciprocal displacement field, and the amplification performance well fits the prediction of designed Poisson’s ratios. Poisson’s ratio is a fundamental property that directly quantifies the relation between input (ui) and output (ui→j) displacement. Hence, our designed self-bridging metamaterial may provide an explicit and programmable way to manipulate non-reciprocal transmission of the displacement field via designing Poisson’s ratios.

Ultra-large Poisson’s ratios

The metamaterial in Fig. 2d shows large expansibility under compressions (strains of −15%) in both orthogonal directions. The output displacement is much larger than the input, which results in ultra-large Poisson’s ratios (with a maximum value of approximately 3) in both orthogonal directions (Fig. 3e). The product of Poisson’s ratios in two orthogonal directions is much larger than 1, namely \({v}_{12}{v}_{21} > 1\). We found the experimental deformation patterns (Fig. 4b) and Poisson’s ratios (Fig. 3e) agree well with the simulated ones. Hence, this metamaterial achieves stable and ultra-large Poisson’s ratios surpassing the thermodynamic limit (\({v}_{ij}{v}_{ji} < 1\)) (Supplementary Note 2). In the numerical simulations, the locations of rotation fulcrums (self-contacts) under transverse compression are c1 and c3, while the locations change to c2 and c4 under longitudinal compression (Fig. 2d). In terms of internal connectivity, different locations of self-contacts make the microstructure exhibit different topological configuration. The length of microscale levers between two fulcrums increases under compression, which amplifies the input displacement, and then the changed rotation fulcrums lead to amplification in both orthogonal directions. The bidirectional displacement amplification mechanism of the metamaterial is similar to a multi-mode lever with two detachable fulcrums, where input displacements on both the left and right can be amplified by changing the fulcrum (Fig. 1g).

The observed ultra-large Poisson’s ratios are analyzed by effective material parameters. We built two equivalent models (Supplementary Fig. 4c, d) to mimic the metamaterial (Fig. 2d) under different load scenarios, which have different internal connectivities owing to the additional solid connections. The Poisson’s ratios of the equivalent models are almost the same as those of the designed metamaterial (Fig. 3e). Thus, the designed metamaterial can be well substituted by these equivalent models. The upper bound vijvji < 1 is deduced by assuming an invariant positive-definite matrix to ensure a positive strain energy density. However, in the effective compliance matrices of these two equivalent models (Supplementary Equations (15, 16)), the values of the first and second elements on the principal diagonal are swapped, which demonstrates that the microstructural topology of the metamaterial is reconfigured as the locations of the self-bridging slits change. The constitutive tensors of the designed metamaterial vary under different load cases, and thus the upper bound can be violated by the self-bridging metamaterial.

Step-like Poisson’s ratios

The metamaterial in Fig. 2e expands transversely under both longitudinal tension (a strain of 15%) and compression (a strain of −15%), which results in step-like Poisson’s ratios (compressive v12 > 0 and tensile v12 < 0, Fig. 3f). Such Poisson’s ratios are inaccessible for materials in the linear elastic regime owing to the assumption of invariance of the constitutive tensors (Supplementary Note 3). The metamaterial is designed in the continuum instead of a discrete rigid body15, and it realizes the expansibility under both tension and compression, different from the contractility induced by bending beams40,41. In the numerical simulation under compression, in terms of internal connectivity, the microstructural topology is reconfigured by self-bridging slits at c2 and c4 in the microstructure (Fig. 2e). Then, the microscale levers show different rotation modes: (1) Under tension, all slits are pulled apart, and the microstructure rotates with anti-chirality; (2) Under compression, c1 and c3 open while c2 and c4 close, and the microstructure rotates about fulcrums at c2 and c4. Only the first rotation mode activates the auxeticity, and thus the metamaterial exhibits step-like Poisson’s ratios. We found the simulated deformation patterns agree well with the experimental ones (Fig. 4c).

We built two equivalent models (Supplementary Fig. 4e, f) to mimic the metamaterial (Fig. 2e) under different load scenarios. The Poisson’s ratios of the equivalent models are almost the same as those of the designed metamaterial (Fig. 3f). Thus, these equivalent models can accurately describe the metamaterial. The calculated effective compliance matrices of these two equivalent models are significantly different (Supplementary Equations (17, 18)). Because the constitutive tensors of the designed metamaterial are no longer invariant under different loads, the self-bridging metamaterial can show different Poisson ratios under different loads.

Customization

These unusual Poisson ratios can be tailored by adjusting the target values in the inverse design framework. More designed metamaterials and corresponding deformation patterns are shown in Supplementary Fig. 5. These metamaterials are designed based on the target non-reciprocal Poisson’s ratios \({v}_{21}^{\ast }=1.8\) and \({v}_{12}^{\ast }=-0.5\), ultra-large Poisson’s ratios \({v}_{21}^{\ast }={v}_{12}^{\ast }=2\), and step-like Poisson’s ratios \({v}_{12}^{\ast }=-0.4\) under tension and \({v}_{12}^{\ast }=0.4\) under compression (Supplementary Fig. 6), respectively. This shows that the realized deformation patterns can be customized by designing purely passive microstructures.

Discussion

Mechanical metamaterials surpassing the thermodynamic limit of Poisson’s ratios were designed and realized through inverse design based on predefined slits for self-contacts and topology optimization. Symmetry and invariance of constitutive tensors are basic assumptions for the theoretical limit of Poisson’s ratios in the linear elastic regime. In terms of internal connectivity, the microstructural topologies of the designed mechanical metamaterials are reconfigured by the self-bridging slits, driving different rotation behaviors of microscale levers. Then, under different loads, the constitutive tensors of the self-bridging metamaterials no longer obey symmetry and invariance, enabling Poisson’s ratios that surpass the conventional limits.

These metamaterials possess inaccessible deformation patterns, including one-way displacement amplification with broken static reciprocity, orthogonally bidirectional displacement amplification, and transverse expansion under both longitudinal tension and compression. These deformation patterns are active under both large and small strains by setting zero initial distance between self-contact surfaces, and can be tailored by the topological design of purely passive microstructure. Based on non-reciprocal Poisson’s ratios, we demonstrate the emergence of a bulk mode that breaks static reciprocity, which opens new avenues to manipulate non-reciprocal transmission of the displacement field via designing Poisson’s ratios. Our work may substantially extend the applications of Poisson’s ratios in modern devices, for example, mechanical energy harvesters42 with one­way amplified displacement field, strain sensors43 with bidirectional ultra-large displacement amplification, civil protection equipment44, and biomedical stents45 with expansibility of tension and compression, and soft robots18 with flexible structural deformation.

Methods

Design method of self-bridging metamaterials

The strain energy density \({W}_{M}\) of the two-term Mooney–Rivlin model is expressed as

$${W}_{M}={A}_{10}({I}_{1}-3)+{A}_{01}({I}_{2}-3)$$
(1)

where \({I}_{1}\) and \({I}_{2}\) are the first and second invariants of the right Cauchy–Green deformation tensor, respectively. \({A}_{10}\) and \({A}_{01}\) can be determined by fitting the relation between engineering strain and stress in uniaxial tension tests of the base material (Supplementary Fig. 7).

A design method with density-based topology optimization is built to design the unit cell, in which a set of element-wise design variables, \(\xi \in [0,\,1]\), is introduced to describe the material distribution. The unit cell is constrained by periodic boundary conditions (Supplementary Fig. 2). To avoid a checkerboard pattern and enhance boundary discreteness, the design variables are first filtered with the Helmholtz filter24 and then projected using the tanh function25 via

$$-{\left(\frac{{r}_{f}}{2\sqrt{3}}\right)}^{2}\nabla \tilde{\xi }({{{{{\bf{r}}}}}})+\tilde{\xi }({{{{{\bf{r}}}}}})=\xi ({{{{{\bf{r}}}}}}),\,{r}_{f} > 0,\,{{{{{\bf{r}}}}}}=\left\{{x}_{1},{x}_{2}\right\},\,{{{{{\bf{r}}}}}}\in {\varOmega }_{D}$$
(2)
$$\bar{\xi }=\frac{\tanh \left(\beta \eta \right)+\,\tanh \left(\beta (\tilde{\xi }-\eta )\right)}{\tanh (\beta \eta )+\,\tanh \left(\beta (1-\eta )\right)}$$
(3)

where \({r}_{f}\) denotes the desired spatial filtering radius, \({r}_{f}=0.05a\) in this work. \(\xi\) is a continuous representation of the unfiltered design field. \(\tilde{\xi }\) and \(\bar{\xi }\) denote the filtered and projected design variables, respectively. \(\beta\) increases with iteration to enhance boundary discreteness. SIMP interpolation24 is used to establish the relation between the projected variables and material parameters:

$${A^{\prime\prime} }_{\!\!\!\!10}(\bar{\xi })={A^{\prime} }_{\!\!\!10}+{\bar{\xi }}^{3}({A}_{10}-{A^{\prime} }_{\!\!10})$$
(4)
$${A^{\prime\prime} }_{\!\!\!\!01}(\bar{\xi })={A^{\prime} }_{\!\!01}+{\bar{\xi }}^{3}({A}_{01}-{A^{\prime} }_{\!\!01})$$
(5)

where \({A^{\prime} }_{\!\!10}={10}^{-9}{A}_{10}\) and \({A^{\prime} }_{01}={10}^{-9}{A}_{01}\) Then, in Eq. (1), \({A}_{10}\) and \({A}_{01}\) are replaced by \({A^{\prime\prime} }_{10}\) and \({A^{\prime\prime} }_{01}\), respectively.

For accessible feature size in fabrication, a three-case robust formulation25 is used in which eroded, normal, and dilated manufacturing processes are mimicked with \(\eta={\eta }_{e}\), \({\eta }_{i}\), and \({\eta }_{d}\), respectively. \({\eta }_{e}\), \({\eta }_{i}\), and \({\eta }_{d}\) are set to 0.55, 0.5, and 0.45, respectively.

To enhance numerical stability under large compression strain, we adopt the energy interpolation form26

$$W={W}_{M}(\gamma \,{{{{{\bf{u}}}}}})+(1-{\gamma }^{2}){W}_{L}({{{{{\bf{u}}}}}})$$
(6)

where \({W}_{L}\) is the strain energy density of the linear elastic model in small-deformation theory, \({{{{{\bf{u}}}}}}\) is the displacement field, and \(\gamma\) is the interpolation factor.

The stiffness of the metamaterial is maximized using a formulated density-based topology optimization model

$$\begin{array}{c}\mathop{\max }\limits_{{{{{{\boldsymbol{\xi }}}}}}}\,\mathop{\min }\limits_{\eta }\,W\left(\bar{{{{{{\boldsymbol{\xi }}}}}}},\eta,{\varepsilon }_{1}^{t}\right)\hfill \\ s.t.\,\;\;\;\;\;\;\;\;\;\;\mathop{\max }\limits_{{\varepsilon }_{j}^{t}}\left({\left(v\left(\bar{{{{{{\boldsymbol{\xi }}}}}}},{\eta }_{i},{\varepsilon }_{j}^{t}\right)-{v}^{\ast }\left({\varepsilon }_{j}^{t}\right)\right)}^{2}\right)\, < \,\zeta,\,j=1,\,{{{{\mathrm{..}}}}}.,\,n\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{{{{\bf{s}}}}}}}^{{{{{{\rm{T}}}}}}}\bar{{{{{{\boldsymbol{\xi }}}}}}}({{{{{\boldsymbol{\xi }}}}}},{\eta }_{d})}{{a}^{2}}\le {s}^{\ast }\hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,{{{{{\bf{0}}}}}}\le {{{{{\boldsymbol{\xi }}}}}}\le {{{{{\bf{1}}}}}}\hfill \end{array}$$
(7)

where \(v\) and \({v}^{\ast }\) are the actual and prescribed properties for a given target strain \({\varepsilon }_{j}^{t}\), respectively; \(n\) is the total number of target strains; \(\zeta\) is used to relax the constraint on the Poisson’s ratio, which is set to 0.052. \({a}^{2}\) is the area of the unit cell; s is the elemental area vector; the prescribed area fraction is \({s}^{\ast }=0.5\). The method of moving asymptotes27 is the optimizer used to solve the model, where the gradient of the objective and constraints are calculated via the adjoint method28. The update processes of the microstructural topologies are presented in Supplementary Fig. 8.

Sample fabrication

The samples with non-reciprocal, ultra-large, and step-like Poisson’s ratios consisted of 3 × 3, 6 × 6, and 6 × 8 unit cells with lattice constant a = 25 mm, as shown in Fig. 4a–c, respectively. The lattice constant was chosen according to the manufacturing limit. The samples of the designed unit cells were fabricated by molding. Before molding, all slits were connected with a solid. Because of the complexity of the shapes of the designed unit cells, the designed unit cell prototypes for molding were fabricated with a light-curing 3D printer (iSLA660). The processed prototypes were covered with a silica gel solution in boxes, and the air in the liquid silica gel was eliminated with vacuum equipment. The molds were obtained after prototypes were removed from the cured silica gel solution. Then, the molds were placed into a vacuum pouring machine (HZK1000) and injected with mixed polyurethane raw material (T0387). The molds were removed to obtain the polyurethane samples when the material was cured. In postprocessing, the samples were knifed to recover the slits. For the material, A10 and A01 of the two-term Mooney–Rivlin model were −0.17 and 2.5 MPa, respectively, which were obtained by least-squares fitting of the relation between engineering strain and stress in a uniaxial tension test (Supplementary Fig. 7).

Measurement

Tension and compression tests were performed using a HANDPI machine with a 500 N loading cell. For non-reciprocal Poisson’s ratios, the sample in Fig. 4a was tested via transverse and longitudinal uniaxial compression, where the sample was supported by rollers to create sliding boundary conditions (Supplementary Fig. 3a). The thickness of the sample in Fig. 4a was 25 mm, which was chosen to avoid out-of-plane buckling. For ultra-large Poisson’s ratios, the sample in Fig. 4b was tested via transverse and longitudinal uniaxial compression. For step-like Poisson’s ratios, the sample in Fig. 4c was tested via longitudinal uniaxial tension (Supplementary Fig. 3b) and compression (Supplementary Fig. 3c). The thickness of the samples in Fig. 4b, c was 10 mm. Under compressions, the samples in Fig. 4b, c were held vertically between two polymethyl methacrylate sheets, which were held 10.5 mm apart (Supplementary Fig. 3b). Under tension, the top and bottom 1×6 unit cells of the sample in Fig. 4c were fastened between two aluminum alloy plates (Supplementary Fig. 3c). We repeated all experiments five times for evaluating Poisson’s ratios. The tests were monitored using a camera (EOS M6 Mark II) with 3840×2160 pixels. The videos were analyzed with MATLAB Image Processing Toolbox. We obtained Poisson’s ratios for different strains by tracking the displacement vectors of red markers on the samples (Supplementary Fig. 3d). The Poisson’s ratios were approximated as \({\tilde{v}}_{12}={\bar{u}}_{1}/{\bar{u}}_{2}\) and \({\tilde{v}}_{21}={\bar{u}}_{2}/{\bar{u}}_{1}\), where \({\bar{u}}_{1}\) (\({\bar{u}}_{2}\)) is the average displacement difference between two columns of vertical red markers (two rows of horizontal markers).