Based on origami self-locking or interlocking mechanisms, multiple stiffness programming of metamaterials can be achieved using geometric, structural, and external force modulation (assisted) strategies. For example, geometric parameter programming realizes stiffness conversion, origami layered structure programming realizes stiffness grading, and on-demand strain regulation programming realizes tunable stiffness. Figure 2A shows a tubular metamaterial constructed based on Waterbomb origami (Mukhopadhyay et al., 2020). Applying force/displacement to the end can cause rigid origami motion (near-zero stiffness) and non-rigid origami motion (high stiffness) (Mukhopadhyay et al., 2020). Among them, the critical transition point L c r is generated by the stop of rigid motion (self-locking mechanism) due to the contact of vertices inside the metamaterial Figure 2A(i) (Mukhopadhyay et al., 2020). This phenomenon has a significant stiffness transition, and thus, by controlling the occurrence of a critical point L c r during motion through geometric parameters ( m , n , α )and external forces/displacements, a sudden and abrupt increase in stiffness can be programmed (Figure 2A(ii)) (Mukhopadhyay et al., 2020); Figure 2B presents a metamaterial constructed based on a nonflat-foldable degree-4 vertex origami (single-collinear, SC) (Fang et al., 2018). Combined with the parametric tuning of sector angles γ 1 , γ 2 (Figure 2B(i), especially the concatenation of SC unit cells with different collinear crease thicknesses into geometrically heterogeneous multilayer structures (see Section 5 for hierarchical programming), Four stiffness segments (Figure 2B(ii)) and eight target stiffnesses with locking configurations (Figure 2B(iii)) based on FPR (flexible photosensitive resin) substrate materials can be implemented programmatically (Fang et al., 2018); Figure 2C presents a design of an on-demand deployable tunable stiffness metamaterial consisting of Triangulated cylinder patterns (Figure 2C(i) (Zhai et al., 2018). By manually unfolding or folding the geometry as needed to adjust the angles of α and β , the metamaterial can be converted between near-zero and high stiffness (Figure 2C(ii-iii)) (Zhai et al., 2018).

Other metamaterials with graded stiffness (Figure 2D(i) (Ma et al., 2018) (Figure 2D(ii)) (Yuan et al., 2020), tunable stiffness (Figure 2D(iii)) (Schenk and Simon, 2013) (Figure 2D(iv)) (Filipov et al., 2015) (Figure 2D(v)) (Sengupta and Li, 2018) can also be realized by using Miura-ori and applying the above strategy.

Based on origami geometry, metamaterials can be programmed to achieve tunable Poisson’s ratio properties. Such as geometric and structural parameter programming to realize “+” and “-”Poisson’s ratio conversion, manual strain programming to realize adjustable Poisson’s ratio. Y.L. He et al. demonstrates a three-dimensional metamaterial constructed from curved-crease origami (CCO) (He et al., 2020). The initial angle α = 65 ° remains unchanged, and when the aspect ratio c 1 : c 2 is lower than about 2 : 3 , the Poisson’s ratio V H B of metamaterials changes from “+” to “-” (He et al., 2020). Similarly, the Poisson’s ratios ( V H B , V H L , V H W ) of metamaterials can be positively or negatively interconverted in different directions in increasing or decreasing form through the modulation of geometrical parameters (e.g., initial angle α , layered incremental angular difference ∆ α ) (He et al., 2020). Figure 3A(i) presents a two-dimensional metamaterial constructed from Miura-ori origami (Figure 3A(ii)), exhibiting equal but opposite sign in-plane and out-of-plane Poisson’s ratios (Wei et al., 2013). When stretched along the x-axis, the material demonstrates a negative Poisson’s ratio behavior (Figure 3A(iii)), whereas when subjected to symmetric bending moments at the boundaries x = ± L , it shows a positive Poisson’s ratio behavior (Figure 3A(iv)) (Wei et al., 2013). Therefore, the metamaterial can realize “+” and “-” Poisson’s ratio tuning by external driving force.

Numerous metamaterials with the same properties are also realized using the above strategy, for example, those constructed by using Rigid-foldable square-twist crease pattern (Lyu et al., 2021), Tachi-Miura polyhedron (Figure 3B(i) (Yasuda and Yang, 2015), Zigzag strips (Figure 3B(ii)) (Eidini and Paulino, 2015), and Re-entrant hexagonal honeycomb structure (Figure 3B(iii)) (Wang Hairui et al., 2020), respectively.

Based on origami geometry, metamaterials with programmable deformation properties can be realized. For example, angular parameters are programmed to control the deformation shape, and numerical optimization algorithms are programmed to control the curvature deformation. Figure 4A reveals a metamaterial consisting of a complex geometric extruded polyhedra (Figure 4A(i) (Johannes et al., 2016). Eqs 1–4 quantify the geometric deformation of the metamaterial in terms of vectors and internal volumes, respectively: P 1 = L , 0 , 0 , (1) P 2 = L ⁡ cos γ 3 , L ⁡ sin γ 3 , 0 , (2) P 3 = L ⁡ cos γ 2 , L δ , L 1 − ⁡ cos 2 γ 2 − δ 2 , (3) v int = P 3 ∙ P 1 × P 2 + 2 L P 3 × P 1 + P 1 × P 2 + P 2 × P 3 . (4)

Here, δ = cos γ 1 − ⁡ cos γ 2 cos γ 3 ∕ ⁡ sin γ 3 , therefore, changing the angles γ 1 , γ 2 , γ 3 can regulate its deformation state (Figure 4A(ii)) (Johannes et al., 2016). Among them, state #1 deformation is the smallest and state #4 is the largest (Figure 4A(iii)) (Johannes et al., 2016). At the same time, various deformations such as states #one to three can also be realized by applying an external inflatable load in a distributed driving manner (Figure 4A(iv-v)) (Johannes et al., 2016). Figure 6B displays a metamaterial constructed based on Miura-ori origami (Figure 4B(i) (Dudte et al., 2016). Using Kawasaki’s theorem and a numerical optimization algorithm based on this theorem, metamaterials can achieve deformations with controlled curvature properties (Figure 4B(ii)) (Dudte et al., 2016).

Using the above strategies, metamaterials with programmable force and displacement properties (Figure 4C(i) (Ting-Uei et al., 2021), reconfigurable deformation properties (Figure 4C(ii)) (Overvelde et al., 2017), the programmable behavior of a mechanical bit used for robots (Treml et al., 2018) and the fast shape changing and instantaneous shape locking (Novelino et al., 2020) were also realized based on Miura-ori pattern, Complex geometric extruded polyhedral and Bistable origami, respectively.

Metamaterials with various mechanical multistable properties can be realized through origami geometric parameters and unit cell arrangement (structure) strategies, for example, programmable bistable, tristable, and global multistable metamaterials. Figure 5A presents a metamaterial constructed based on Miura-ori strings (Figure 5A(i), which can be programmed to achieve unit cells (Figure 5A(ii)) with monostable, semi-bistable and bistable states by controlling the angles α 1 and α 2 (Kamrava et al., 2019). The number of Miura-ori (n) in an angle chord is also a critical programming parameter for it (Figure 5A(iii)). Figure 5A(iv) offers a configuration of this metamaterial: each orthogonal direction contains 12 star-shaped cells ( n = 4 ) in series (36 in total), with 15 design parameters, programmable to achieve three Behavior (Kamrava et al., 2019). In addition, based on Origami bellows geometry, the degree-four vertex geometry and hypar origami also realizes metamaterials with programmable bistable properties (Reid et al., 2017), global multistable properties (Scott et al., 2015) and multistable properties (Liu et al., 2019), respectively.

In addition to stiffness, Poisson’s ratio, deformation, and multi-stability properties, other programmable mechanical properties can be achieved using origami geometric and structural strategies. Figure 5B(i-iii) shows that based on Miura-ori origami, Rigid-foldable square-twist crease pattern, bistable Miura-ori unit cells can respectively achieve graded compressive strength (Li et al., 2021), comprehensive mechanical properties (Ma et al., 2021), and compressive modulus with reversible adjustment (Silverberg et al., 2014).

Using kirigami geometry, multiple programmable deformations of metamaterials can be achieved. For example, geometric parameters are used to program the deformation, unit cells (with the same mechanical properties but different deformation) are arranged to program special patterns, and optimization algorithms are used to program the deformation. A metamaterial constructed from the “Louvres” Krigami pattern is designed in Figure 6A (Tang et al., 2017). Figure 6A(i-ii) exhibits that due to the compressive force P and the notch breaking the original geometric symmetry, the center of gravity shifts down and is not aligned with the neutral plane (dotted line), and the generated bending moment M can guide different buckling behaviors to make the unit cell locally bend clockwise and counterclockwise in a controlled manner (Tang et al., 2017). Thus, the change in kiri-kirigami morphology can be programmed by controlling the homogeneous or heterogeneous tilt direction of the unit cells (Figure 6A(iii)) (Tang et al., 2017). Figure 6B presents a metamaterial constructed based on Hierarchical Kirigami Sheets (Figure 6B(i) (Ning et al., 2020). By adjusting the geometric parameters (e.g., δ 1 , δ 2 , l , t ), multiple sets of unit cells can be made to have a similar stress-strain response but different fractions of the void area due to buckling, for example, the combination: δ 1 1 / l = 0.1375 , δ 2 1 / l = 0.075 , γ 1 = 0 ° and δ 1 2 / l = 0.15 , δ 2 2 / l = 0.075 , γ 2 = 90 ° (Figure 6B(ii)) (Ning et al., 2020). Therefore, metamaterials can be programmed to achieve complex deformable patterns, such as text and flowers, through unit cell arrangement programming (Figure 6B(iii)) (Ning et al., 2020). Figure 8C shows a metamaterial with an array of notches embedded in an elastic polyester plastic sheet (Figure 6C(i) (Jin et al., 2020). In combination with δ 1 and H parameter modulation and heterogeneous unit cell programming configurations, simple macroscopic deformation of metamaterials can be achieved under external aerodynamic loading (Figure 6C(ii)) (Jin et al., 2020). However, more complex shape simulations can be completed by the programming of the Nelder–Mead simplex algorithm and the Melder–Nelson algorithm (Figure 6C(iii) (Jin et al., 2020).

In addition, based on Modular Kirigami geometry (Figure 6D(i) (Li et al., 2021), Algorithmically Optimised krigami geometry (Figure 6D(ii) (Gary et al., 2019), Open honeycombs (Figure 6D(iii)) (Neville et al., 2016), Cylindrical kirigami shells (Ahmad et al., 2019b), programmable deformation features are also implemented separately.

Other mechanical properties can be achieved with programmable kirigami metamaterials. For example, References (Yang et al., 2018) demonstrates a metamaterial constructed based on a paper-cut geometry similar to Figure 6A (Yang et al., 2018). Using the bistability property, this metamaterial can achieve a symmetric configuration transition of stiffness from 100% to 0% by adjusting the unit cell geometry parameters l c ∕ l x , l c ∕ l y (Yang et al., 2018). In addition, based on Hierarchical Kirigami Sheets (Cai and Abdolhamid, 2021), Kirigami MGs (Chen et al., 2019), and Layered hinge geometry (Tang et al., 2015), it is possible to achieve metamaterials with programmable Poisson’s ratio, programmable stress-strain relation.

Poisson’s ratio can be programmed based on lattice geometry strategies, such as geometric parameter tuning Poisson’s ratio, topological shape programming Poisson’s ratio, and topologically optimized geometry or structure programming Poisson’s ratio. Jianxing Liu et al. produces a triangular, honeycomb, square metamaterial constructed based on Wavy filamentary microgeometry (Liu and Zhang, 2018). In triangular metamaterials, when the slope tan ⁡ θ 0 remains unchanged, by changing the width ratio w 2 / w 1 and length ratio L 2 / L 1 + L 2 , the Poisson’s ratio can be controlled transition between −0.2 and −1.0 (Liu and Zhang, 2018). Similarly, Poisson’s ratio can be programmed to switch between positive and negative by manipulating tan ⁡ θ 0 , w 2 / w 1 , L s a m p l e / H s a m p l e of honeycomb and square metamaterials (Liu and Zhang, 2018). Figure 7A presents a metamaterial constructed based on beams of sinusoidal shape (Figure 7A(i) (Chen et al., 2017). By programming half wave-length n (Figure 7A(ii)), especially the topological geometry (Figure 7A(iii)), the metamaterial Poisson’s ratio can be adjusted as needed (Chen et al., 2017). Figure 7B represents a digital system metamaterial (Anders et al., 2015). Topological optimization of its microstructure by nonlinear geometric modeling can lead to different linear and nonlinear Poisson ratios (Figure 7B(i) (Anders et al., 2015). Figure 7B(ii) reveals that the Poisson’s ratio can be tuned between −0.8 and 0.8 using this method (Anders et al., 2015).

In addition, using the above construction methods, the metamaterials designed based on rectangular and spherical geometries (Figure 7C(i)) (Ren et al., 2018), cubic crystal systems (i.e., simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc)) (Figure 7C(ii)) (Babaee et al., 2013), bent beams (Figure 7C(iii)) (Li et al., 2017), and triangular geometry (Ling et al., 2020), respectively, also realized the programming regulation of Poisson’s ratio.

The lattice geometry strategy can realize the deformation control of metamaterials, for example, geometry and structure parameters can be programmed to realize shape controllability, geometric parameters combined with a self-locking mechanism can be programmed to realize multi-step path deformation, and heterogeneous voxel structure programming can realize controllable local patterns. Figure 8A displays a soft metamaterial (Architected Soft Machines, ASM) based on Voronoi tessellation (Goswami et al., 2019). Metamaterials can be programmed to bend 45° or 90°, respectively, when the thickness of the beam is distributed with a 0.5–1.0 mm lateral gradient (Figure 8A(i) or an orthogonal gradient (Figure 8A(ii)), respectively (Goswami et al., 2019). Similarly, by programming beam orientation (Figure 8A(iii)), unit cell size (Figure 8A(iv)), complex deformation behavior similar to a hand can be achieved (Goswami et al., 2019). Figure 10B(i) presents a metamaterial constructed based on freely hinged squares (Coulais et al., 2018). Setting the thickness t α = 1.0 m m , t β = 2.0 m m of the hinge beam can realize two-step path deformation (Coulais et al., 2018). First, the α hinge bucks at 4% strain under compression, triggering the first-step path deformation, while the thicker β hinge does not buckle (Coulais et al., 2018). Up to 49% strain, the metamaterial forms self-contact, completing the first deformation (Figure 8B(ii)) (Coulais et al., 2018). Then, further compression leads to buckling of the β hinge, starting the second step of path deformation until the structure is fully compressed (Figure 8B(iii-iv)) (Coulais et al., 2018). Therefore, based on the self-locking mechanism, by controlling the hinge thickness t α , t β and the number of hinges, metamaterials can be programmed to achieve more complex multi-step paths (Coulais et al., 2018). Figure 8C designs a metamaterial based on anisotropic cubic building voxels (Figure 8C(i) (Coulais et al., 2016). Its voxels have axially aligned soft deformation patterns that can lead to elongated (+) or flattened (−) shapes (Coulais et al., 2016). Thus, a “smiley face”-like patterned texture (Figure 8C(iii)) can be achieved by a programmed arrangement of voxels (Figure 8C(ii)) (Coulais et al., 2016). In addition, the metamaterial shown in Figure 8D is also programmed based on computational models to achieve controllable shape matching properties (Mirzaali et al., 2018); Metamaterial with pattern transformation also can be programmed by geometric parameters (Chen and Jin, 2018).

Lattice geometry programming can also realize stiffness regulation, such as geometric parameters to regulate stiffness, artificial intelligence algorithm programming stiffness, self-locking mechanism programming to achieve adjustable stiffness. Figure 8E exhibits a metamaterial constructed from Schwarz’ unit cell, diamond, and Schoen’s gyroid structures (Figure 8E(i) (Lee et al., 2016). The metamaterial can be controlled to display different E/S ratios (ratio of Young’s modulus to shear modulus) when the aspect ratio α is at 1.0, 2.0, 10, respectively (Figure 8E(ii)) (Lee et al., 2016). Similarly, the E/S ratio can be adjusted by programming the volume fraction φ (Figure 8E(iii)) (Lee et al., 2016). Figure 8F(i) presents a lattice structure designed using an artificial intelligence-based optimization algorithm (Anthony et al., 2021). To begin with, the approach entails training a Convolutional Neural Network (CNN) to forecast the effective stiffness and wave speed of the unit cell design. Subsequently, a Genetic Algorithm (GA) is employed, using the CNN as its evaluation function, to produce metamaterials exhibiting exceptional effective stiffness and wave speed (depicted in Figure 8F(ii)) (Anthony et al., 2021). This methodology showcases the effectiveness of combining Machine Learning (ML) and GA in attaining optimal stiffness designs for metamaterials, even in scenarios with intricate nonlinear constraints (Anthony et al., 2021). Figure 8G demonstrates a metamaterial constructed based on interlocking octahedral particles (Figure 8G(i)) (Wang et al., 2021). By applying variable compression at the boundary to trigger interlocking between unit cells, metamaterials can be controlled to switch between low stiffness (Figure 8G(ii)) and high stiffness (Figure 8G(iii)) (Wang et al., 2021).

The lattice geometry also offers great ability in programming energy absorption. Xiaojun Tan et al. presents a unit cell constructed based on negative stiffness (NS) geometry (Tan et al., 2019a). In this metamaterial, the energy absorption per volume W V , energy absorption per mass W m , and energy absorption efficiency η can be adjusted with increasing or decreasing controlled programming of h / L (vertex height to length ratio of the bending beam) and t / L (in-plane thickness to length ratio), respectively (Tan et al., 2019a).

In addition, energy absorption can also be programmed by lattice density (Yuan et al., 2019) and volume (Wang et al., 2019). At the same time, programmable comprehensive mechanics capability can also be achieved based on flexible porous geometry and deflated continuation algorithm (Medina et al., 2020).

The tensegrity programming strategy can achieve tunable Poisson’s ratio. Xu Yin et al. shows a metamaterial constructed based on truncated regular octahedral tensegrities (TROTs) (Yin et al., 2020). The metamaterial Poisson’s ratio v or stress-strain can be programmed by the initial twist angle α 0 , the residual chord prestress q s [ (Yin et al., 2020)]. In addition, Reference (Micheletti et al., 2023) proposed two tensegrity structures: the “six-node” and the “eight-node” units. By adjusting its prestress, bistable and multistable reprogramming characteristics can be achieved. Meanwhile, programmable Poisson’s ratio properties were also achieved based on Auxetic tubular structure (Ren et al., 2016), Ancient geometric motifs (Ahmad and Pasini, 2016) and other tensioned monolithic structures (Liu et al., 2019).

Figure 9A(i) presents a metamaterial unit cell with programmable displacement behavior based on adaptive hexagonal geometry with hinges (Wenz et al., 2021). By manipulating the parameters α , β , t 2 , this unit cell can realize the transformation of Poisson’s ratio from positive to negative values, accompanied by the deformation characteristics of expansion and contraction (Figure 9A(ii)) (Wenz et al., 2021). Therefore, based on the series and parallel connection of unit cells, local protrusions of metamaterials can be achieved by controlling the local deformation behavior (Wenz et al., 2021). This phenomenon offers the possibility of realizing complex nonlinear system behavior similar to earthworm movement (Figure 9A(iii)) (Wenz et al., 2021). In addition, programmable shape features are also implemented based on tensegrity (Figure 9B) (Lee et al., 2020), Cylindrical geometric (Yang and Ma, 2020a), and One-DOF reconfigurable module (Liu et al., 2021).

Based on the geometric hierarchical structure, the Poisson’s ratio of metamaterials can be programmed and controlled, such as the hierarchical arrangement of heterogeneous unit cells to achieve a tunable Poisson’s ratio. Figure 10A presents a metamaterial based on hexagonal honeycomb (Figure 10A(i) (Mousanezhad et al., 2015). Figure 10A(ii) shows that by varying the first level hierarchy parameter γ 1 , the Poisson’s ratio can be controlled to switch between positive and negative at different strains (Mousanezhad et al., 2015). In contrast, smaller Poisson’s ratios and more complex regulation of Poisson’s ratios can be achieved through the combined regulation of the first- and second-level hierarchical parameters γ 1 , γ 2 (Figure 10A(iii)) (Mousanezhad et al., 2015). In addition, tunable Poisson’s ratio properties were also achieved based on the auxetic hexagonal honeycomb (Sun and Nicola, 2013).

Programmable deformation characteristics can be achieved based on geometric hierarchy, such as the hierarchical arrangement of heterogeneous unit cells to achieve programmable deformation sequences, and hierarchical structures with different unit cell parameters programmed to control multi-step deformation paths. Figure 10B exhibits a metamaterial constructed from bistable unit cells (Figure 10B(i-ii)) (Che et al., 2017). By programming the beam parameters ( t = 0.96 m m , 1.15 m m , 1.08 m m , 1.02 m m , 0.90 m m ) of different layers of unit cells, controllable and deterministic complex deformation sequences can be realized under different axial strains (Figure 10B(iii)) (Che et al., 2017). Xiang Li et al. presents a metamaterial based on Hierarchical rotating structures, which can achieve a controlled multi-step deformation path by manipulating the angular parameters θ and α (Xiang et al., 2021). It shows a two-step path deformation ( θ = 25 ° , α = 35 ° ) with the first step deformation occurring at a strain of about 0.17, where θ is closed, and α is open (Xiang et al., 2021). When θ closes, the stress increases abruptly and, as compression continues, α closes at a strain of approximately 0.31 and a second deformation step topology occurs (Xiang et al., 2021).

The geometric hierarchical structure strategy can also realize the programming control of the stiffness of metamaterials. Such as fractal structure to realize elastic modulus control, and hierarchical structure programming to realize stiffness control. Figure 10C designs a metamaterial based on hexagonal honeycomb Figure 10C(i) (Oftadeh et al., 2014). It can achieve a wide range of effective elastic modulus by adjusting the relative density p and the layered betweenness n (Figure 10C(ii)) (Oftadeh et al., 2014). At the same time, the highest in-plane stiffness at a given weight ratio can be achieved by optimizing the structural configuration (Oftadeh et al., 2014). Figure 10D presents a metamaterial composed of post-buckled elements Figure 10D(i) arranged in a layered structure (Figure 10D(ii)) (Jiao, 2020). By adjusting the geometric ratios R g t = g / t and R l w = L / W , it is possible to tune the deformation configuration of the beams, leading to either an increase or decrease in tensile and compressive stiffness ( K T and K C ) (Jiao, 2020). Specifically, when the values of t and W are kept constant, increasing L and g results in a decrease in stiffness, whereas decreasing L leads to an increase in stiffness (Jiao, 2020).

Furthermore, using the above strategies, metamaterials with programmable shape memory (Figure 10E(i) (Matthew, 2018), multi-stability (Figure 10E(ii)) (Zhang Hang et al., 2021), and shock energy absorption properties (Figure 10E(iii-iv)) (Frenzel et al., 2016; Chen and Jin, 2021) were also realized.

By employing a composite layered design of the substrate materials, it becomes feasible to regulate the thermal expansion coefficient of the metamaterial. For example, by combining two layers of substrate materials with distinct thermal expansion coefficients and adjusting geometric parameters, the metamaterial can be programmed to exhibit positive and negative thermal expansion coefficient conversion. Yong Peng et al. illustrates a metamaterial unit constructed using such substrate materials (Peng et al., 2021). By controlling the ratio of thermal expansion coefficients α 1 / α 2 of the two layers, as it increases from 0.2 to 10, the thermal expansion coefficient ∆ S of the metamaterial can be changed from positive to negative (Peng et al., 2021). Meanwhile, In this metamaterials, based on the regulation of α 1 / α 2 , combined with the angle φ 1 and the height ratio S 1 / S 2 , a broader range of positive and negative thermal expansion coefficient adjustments can be achieved (Peng et al., 2021). Figure 11A(i-ii) also depicts a metamaterial composed of two layers of base materials with different thermal expansion coefficients (Wang et al., 2016). However, unlike the previous case, the initial compositions of these two base materials are the same, and the variation in thermal expansion coefficients is achieved by introducing different volume concentrations of copper nanoparticles into one of the materials (Figure 11A(iii)) (Wang et al., 2016). Meanwhile, geometric parameters such as beam width and thickness can also program the thermal expansion coefficient of this metamaterial (Figure 11A(iv)) (Wang et al., 2016). In addition, references (Wu et al., 2016; Jia et al., 2016; Boatti et al., 2017; Ni et al., 2019; Yang and Ma, 2020b; Chen et al., 2021; Wei et al., 2021) also implement programmable thermal expansion coefficients using the above strategy.

By designing multi-layered substrate materials, it is feasible to manipulate the Poisson’s ratio of the metamaterial (Ai and Gao, 2018; Janbaz et al., 2020). Young-Joo Lee et al. portrays a metamaterial constructed using two layers of substrate materials with different moduli (Young-Joo et al., 2019). This metamaterial has one combination (PDMS - soft, with TPU) where the Poisson’s ratio can be adjusted to switch between positive and negative values (Young-Joo et al., 2019). Furthermore, in this metamaterial, by adjusting geometric parameters such as rib width, unit width, and unit height, the Poisson’s ratio can also be significantly controlled (Young-Joo et al., 2019).

Programmable deformation properties can also be achieved through the programming strategies described above. Figure 11B(i–ii) demonstrates a metamaterial based on flexible and stiff materials (Janbaz et al., 2019). It allows for controlled deformation behavior through composite programming of dual material spatial distribution and geometric design (four-fold type one, four-fold type two, circular) (Figure 11B(iii) (Janbaz et al., 2019). In addition, more complex deformation programming can also be achieved through the layering and geometric design of hydrogels (Wu et al., 2013; Wei et al., 2020), LCE (Peng et al., 2021). Likewise, critical stress and strain (Janbaz et al., 2018), variable stiffness (Qi et al., 2021) can also be programmed to control.

Thermal stimuli-responsive materials, such as Shape Memory Polymers (SMPs), can undergo various deformation states, including bending, curling, and swelling, when subjected to temperature stimulation. By utilizing SMPs as substrate materials, real-time tunable Poisson’s ratio properties can be achieved by programming the external stimulus. Figure 12A showcases a metamaterial (Figure 12A(i) with the unique ability to adjust its Poisson’s ratio through geometric parameters, specifically the unit cell center angle (Figure 12A(ii)) (Xin et al., 2020). Consequently, employing SMPs programmed to tune the geometrical parameters enables this metamaterial to exhibit a wide range of Poisson’s ratios, following the outlined strategy (Xin et al., 2020):

• During the heating stage, when the metamaterial is subjected to an external tensile load and the temperature surpasses its glass transition temperature ( > T g ), the shape memory polymer (SMP) substrate material transitions from a glassy state to a highly elastic state. The random molecular chains (soft segments) elongate, leading to deformation in the metamaterial, and the central angle of the unit changes from 2 θ 2 α to 2 θ ′ 2 α ′ .

Therefore, the metamaterial temperature is raised above T g and stretched to the programmed setting of multiple deformations λ t a r g e t (Xin et al., 2020). Then, cooled and unloaded to fix various programmed configurations (mainly changes in geometric parameters 2 θ 2 α ), the new configuration can achieve a variety of different auxetic properties than the original configuration (Figure 12A(iii)) (Xin et al., 2020). Figure 12B(i) offers a metamaterial whose Poisson’s ratio can be tuned by a layered design of soft and hard substrate materials (Figure 12B(ii)) (Zhao et al., 2019). Specifically, a hierarchical design (SMPs are distributed at the unit cell marker ①, and rubber materials are distributed at the unit cell marker ②) based on SMPs (different elastic moduli of materials at different temperatures) and rubber materials (with constant material modulus) found that: Compression at 25°C has a positive Poisson’s ratio, while at 70°C, the Poisson’s ratio is negative (Figure 12B(iii)) (Zhao et al., 2019). This shows that the metamaterial can program the Poisson’s ratio in real-time by temperature (Zhao et al., 2019). In addition, Figure 12C(i-iv) also implements the ability of Poisson’s ratio to be programmed and tuned in real-time, respectively (Park et al., 2018; Lei et al., 2019; Gao et al., 2020; Jiao et al., 2021).

Programmable deformation properties can be achieved based on thermally stimulated responsive material programming. For example, thermally stimulated responsive materials are used as hinges to achieve deformation programming, and inhomogeneous temperature stimulated programming to control deformation. Nan Yang et al. designs a metamaterial based on ring-shaped origami unit cells, whose deformation mechanism is controlled by the local deformation angle θ (Yang et al., 2020). The deformation state (outward or inward) formed by each θ angle can be represented by “0,1″in this metamaterial (Yang et al., 2020). Meanwhile, this deformed state can be further controlled by thermal stimulation through the distribution of γ -shaped SMA hinges (Yang et al., 2020). Therefore, based on thermal stimulation, more complex shape programming can be achieved by controlling the “0,1” states (e.g., programmed to 100 or 000) of each θ angle of the quarter cells (Yang et al., 2020). Figure 12D proposes a metamaterial constructed based on the horseshoe lattice (Figure 12D(i) and the substrate material of SMPs (Wang et al., 2020). Its programming logic is similar to Figure 12A. But the difference is that the metamaterial can deform on demand (Figure 12D(iii)) by inhomogeneous heating (Figure 12D(ii)) (Wang et al., 2020). Moreover, applying the same strategy, programmable deformation properties are also achieved in Figure 13A(i-ii), respectively (Yuan et al., 2018; Tang et al., 2019).

The tunable stiffness properties are also achieved by the above methods. Figure 13B(i) illustrates a metamaterial based on SMPs and microlattice structure (Yang et al., 2019). Figure 13B(ii) displays that it has two states of high and low stiffness under temperature stimuli of 30°C and 90°C, which utilizes the two states of the substrate material (SMP) under temperature stimulation (glass and rubbery state) (Yang et al., 2019). Meanwhile, Figure 13B(iii) demonstrates that the metamaterial can program the shape by temperature stimulation as required while the stiffness remains constant (Yang et al., 2019). Furthermore, adjustable stiffness characteristics are also implemented respectively in Figure 13Bi (Rossiter et al., 2014). Moreover, applying a strategy similar to Figure 12A, Figure 13C(ii-iii) also achieves programmable stress-strain (Xin et al., 2022) and multistability properties (Tan et al., 2020a), respectively.

Magnetic materials controlled by a magnetic field can achieve different states. Therefore, employing magnetic materials as substrate materials or incorporating them into the substrate materials enables the geometric or structural changes of metamaterials to be controlled by external magnetic fields. Poisson’s ratio can be programmed by this method. Figure 14A(i) exhibits a metamaterial constructed by embedding an electromagnetic switch into a honeycomb structure (Haghpanah et al., 2016a). Figure 14A(ii-iii) demonstrates that active control of Poisson’s ratio can be achieved by electromagnetically switching (activating or deactivating) the mode of the unit cell geometry (Haghpanah et al., 2016a). Specifically, Figure 14A(iv) illustrates that by deactivating the electromagnets on the diagonal of the sample, the Poisson’s ratio of the metamaterial can switch from 0.15 to 0.5 under axial compression (Haghpanah et al., 2016a). In contrast, the Poisson’s ratio value will be close to one by deactivating all electromagnets (Haghpanah et al., 2016a). Besides, Figure 14B(i-ii) also achieves tunable Poisson’s ratio properties using similar strategies, respectively (Grima et al., 2013; Ma et al., 2021).

The geometric or structural changes of the metamaterial can be programmed by the composite configuration of magnetic and conventional materials, which can accomplish the tunable stiffness. Figure 14C proposes a metamaterial consisting of a physical binary element (m-bits) unit cell (Figure 14C(i) composed of magnetic materials and conventional materials (Chen et al., 2021). Figure 14C(ii) reveals that this unit cell can switch between two stable geometrical states (ON and OFF) under the influence of an electromagnetic field (Chen et al., 2021). Through this switching, the unit cell stiffness can be adjusted in a controlled manner (Figure 14C(iii)) (Chen et al., 2021). Also, the metamaterial stiffness can increase linearly with the percentage ”ξ” of the number of unit cell (ON), which demonstrates strong programmable stiffness properties (Figure 14C(iv)) (Chen et al., 2021). Figure 14D(i–iii) offers a metamaterial constructed by filling a magnetic fluid into a cuboctahedral lattice (Jackson et al., 2018). By programming domination of the external magnetic field, its stiffness can be varied by about 35 percent increase (Figure 14D(iv)) (Jackson et al., 2018). Furthermore, Figure 14B(iii) also implements programmable stiffness using the analogous design (Macrae Montgomery et al., 2021).

By the composite of magnetic materials and conventional materials, metamaterials can be adjusted by external magnetic fields to achieve multistable properties. Figure 14E(i) presents a metamaterial unit cell consisting of a system of magnets (Tan et al., 2019b). This metamaterial can transition between stable states through a load of the external magnetic field (Tan et al., 2019b). And the number of stable states is determined by the number of unit cells connected in series (Tan et al., 2019b). For example, Figure 14E(ii) exhibits that a single unit cell has two stable states, while Figure 14E(iii) shows that a metamaterial composed of multiple unit cells has multiple stable states (Tan et al., 2019b). Using this approach, Figure 14F(i–ii) also implement programmable multi-stable characteristics (Fang et al., 2020; Yasuda et al., 2020), respectively. Additionally, controlled deformation properties of metamaterials also have been achieved. Figure 15A(i) produces a magnetic ink composite metamaterial consisting of a silicone rubber soft material and neodymium-iron-boron (NdFeB) alloy (Kim et al., 2018). By programming the magnetic orientation of the ink, it can achieve controllable shape changes, such as complex three-dimensional deformation (Figure 15A(ii)) and motion (Figure 15A(iii)) (Kim et al., 2018).

Although current strategies have certain logical operation capabilities (pre-programming: geometric or structural parameter programming) and autonomous feedback capabilities (external driving force programming), their level of intelligence is still low. This is specifically reflected in the following aspects.

First, the geometric or structural programming strategy only obtains the corresponding mechanical performance parameters by simply changing the geometric or structural parameters. And it also can be based on the unit cells with different mechanical performance parameters, they are arranged on demand to simply control the propagation of mechanical signals. This approach cannot create metamaterials with complex mechanical properties. For example, a piece of metamaterial can have multiple mechanical properties at the same time and has the property of converting mechanical properties in real time in the same dimension. Hence, the programming strategy should adopt more artificial intelligence-related technologies and methods. Artificial intelligence has a convoluted system that can imitate human intelligent thinking to make various behaviors and calculations, including various methods, such as language recognition, image recognition, and natural language processing (Guo et al., 2016). Although some metamaterials have used artificial intelligence as the strategies (Anthony et al., 2021), such methods are more about applying artificial intelligence to the generation process of geometry or structure rather than programming the process of building metamaterials. If artificial intelligence algorithms are applied to the programming and construction of metamaterials, can metamaterials be as smart as robots in the future? People can achieve many functions and effects like robots just from the perspective of material design.

Second, geometric, or structural programming can also be called pre-programming; once fabricated, the mechanical properties cannot be regulated by adjusting its geometric or structural parameters. It can only operate according to pre-programmed logic and exhibit corresponding mechanical properties. It cannot achieve the real-time programming. External driving force programming can achieve real-time programming through some methods. However, this is achieved by relying on external driving forces rather than the metamaterial itself. So, is it possible to look for some methods in the metamaterial itself that enables real-time programming? At present, we have not seen any relevant scholars put forward some feasible methods.

Third, external driving force programming means that the mechanical properties of metamaterials can be adjusted in real-time according to external stimuli. Through this strategy, metamaterials can realize simple interactive feedback of stimulation (Chen et al., 2021). However, its intelligence level is still some distance from 4D materials (Momeni et al., 2017), programmable electromagnetic metamaterials (Liu and Cui, 2017; Bao and Cui, 2019), programmable planar soft matter (van Manen et al., 2018), and programmable DNA materials (Zhang et al., 2018; Albrecht et al., 2023). Simultaneously, numerous external stimuli (e.g., pneumatics, actuator drives) require many complex devices to control, which make metamaterials sometimes look more like “machines” than “materials”. At present, no scholar has given a more precise explanation. Regardless of whether external driving forces will become the main development direction of real-time programming of programmable mechanical metamaterials. More types of external driving forces should be explored and discovered.

The general understanding in modern materials science about the composition of matter is as follows: Specific quantities and types of atoms can combine into molecules through certain bonding mechanisms, and a large number of atoms or molecules can come together in specific spatial arrangements to form various unique materials (Bohr, 1961). Due to the interactions between atoms within molecules, the physical and chemical properties of molecules depend not only on the types and numbers of constituent atoms but also on their structure (Bohr, 1961). The properties of materials are influenced not only by the types and structures of atoms or molecules but also by the arrangement and bonding of atoms or molecules (Van Melsen and Andrew, 2004). At the same time, electrons play a crucial role in determining the properties of materials. Their directional motion generates electric current, and under the influence of external electric and magnetic fields, their trajectories and states can be altered as needed (Van Melsen and Andrew, 2004).

Therefore, from the perspective of materials science, the three main construction strategies (geometric programming, structural programming, and external driving force programming) of programmable mechanical metamaterials are also derived from the above basic cognitive logic. Although more and more new mechanical metamaterials have been proposed in recent years (Bai et al., 2022; Farzaneh et al., 2022; Ma et al., 2022; Mehboob et al., 2022; Pagliocca et al., 2022; Wagner et al., 2022; Yang et al., 2022; Zhang et al., 2023; Liang et al., 2023; Sundararaman et al., 2023; Tian et al., 2023; Wang et al., 2023), the construction of metamaterials is currently unable to reach the atomic or molecular scale. At present, most metamaterials are still large in scale and more like “building structures” and “products” and cannot be used as “materials” for the design and manufacture of various applications. This is a question worthy of consideration and breakthrough in the future.

Although Table 1 and 2 lists numerous geometric or structural types for achieving various programmable mechanical properties, it is far from enough. Need to explore more geometry or structure (such as hypercube) to develop more prosperous programmable mechanical properties (such as programmable tensile strength and compressive strength).