This section introduces the differences between 1-DOF and multi-DOF deformable systems in terms of their ability to prescribe arbitrary responses. Multi-DOF deformable metamaterials behave like underactuated systems, where the number of degrees of freedom exceeds the number of actuators. In these systems, deformation is governed by elastic forces, inertial forces, and boundary constraints, with the kinematic path in the energy landscape following the principle of minimum energy gradient. However, 1-DOF systems behave like fully actuated systems, where the number of degrees of freedom equals the number of actuators. The deformation is completely determined by the 1-DOF kinematic base, allowing for arbitrary design of the kinematic path without being constrained by the energy gradient principle. To explore this limitation, this section analyzes the energy landscape of both linear and nonlinear multi-DOF systems and 1-DOF systems under displacement loading, providing insight into their ability to achieve arbitrary responses.

Firstly, consider a 2-DOF deformable system consisting of 2 linear springs, as shown in Fig. 2A. The blue spring, representing the soft spring, has a stiffness of 0.1 \mathrm{N} {\mathrm{mm}}^{-1}, while the red spring, representing the stiff spring, has a stiffness of 0.2 \mathrm{N} {\mathrm{mm}}^{-1}. Both springs have a compression limit of 10 mm. The energy landscape during the compression process of the linear series springs is shown in the middle of Fig. 2A. The horizontal and vertical axes represent the deformations of the soft and stiff springs, respectively, and the color indicates the total elastic energy stored in the 2 springs. The black solid lines represent equal-energy contours, while the −45° black dashed line represents the deformation compatibility constraint. The solid red line with arrows indicates the kinematic path traced by changing the total displacement d_{\text{linear}}^{2\mathrm{DOF}} from 0 \mathrm{mm} to 20 \mathrm{mm}. The kinematic path intersects the point of tangency between the deformation compatibility constraint and the equal-energy contours at nonboundary regions. When the loading displacement reaches 15 mm, the soft spring reaches its compression limit of 10 mm. The stiffness of the 2 series-connected springs is 0.067 \mathrm{N} {\mathrm{mm}}^{-1} (see the right side of Fig. 2A), which is lower than the stiffness of the soft spring ( 0.1 \mathrm{N} {\mathrm{mm}}^{-1}). After the soft spring reaches its stroke limit, only the stiff spring deforms, and the overall stiffness of the series-connected springs becomes equal to the stiffness of the stiff spring, 0.2 \mathrm{N} {\mathrm{mm}}^{-1} (see Section S3.1 and Movie S1). The kinematic path of the 2-DOF linear deformable system remains linear, inhibiting the generation of nonlinear responses.

Another example worth considering is the system of 2 nonlinear bistable units with different stiffnesses, connected in series, as shown in Fig. 2B. These bistable nonlinear units are assumed to follow 2 different sinusoidal functions (detailed information is provided in Section S3.2, Movie S1, and Code S1). The force–displacement curve of the series-connected nonlinear units demonstrates that the valleys of the total force–displacement curve do not decrease, as the stiff unit snaps and pulls the soft unit in an impulsive manner from state P_1 to P_2, instantly releasing elastic energy. P_2 is the minimum energy point of the deformation compatibility constraint at d_{\text{nonlin}}^{2\mathrm{DOF}} = 15 \mathrm{mm}. This reveals the challenge of achieving multi-stage softening in multi-DOF metamaterials, as the minimum energy gradient principle governs the deformation in multi-DOF deformable systems.

A 1-DOF deformable system, consisting of 2 linear springs coupled with 4 rotating rectangles, is introduced in Fig. 2C. The system's kinematic path is a customizable curve that can traverse the energy landscape arbitrarily, free from deformation compatibility constraints and not bound by the principle of minimum energy gradient. As a result, 1-DOF systems exhibit a stronger ability to prescribe arbitrary responses compared to multi-DOF systems.

To address the problem shown above, this paper employs a 1-DOF kinematic frame to couple elastic components to eliminate the room for the energy gradient to take effect in the deformation sequence. The conceptual design named 1DOFmat is shown in Fig. 3A. The kinematic base consists of hinged quadrilateral units arranged in a 6 × 6 grid. Each unit is a rectangular hinged quadrilateral measuring 20 mm in length, 10 mm in width, and 10 mm in thickness. In addition, each small hinged quadrilateral unit has a rectangular hole of 10 mm length and 5 mm width in the center of the block (Section S1). Rectangular units are joined by flexure hinges, printed using thermoplastic polyurethane (TPU) elastic material with dimensions of 1 mm in length, 0.4 mm in thickness, and 10 mm in width. The inner part of the rectangular block is made of a rigid plastic, polylactic acid (PLA), while the outer part is covered with a layer of TPU. This dual material approach [60] is designed to achieve the desired mechanical properties, with the TPU layer providing low stiffness for the flexible hinge and the underlying PLA structure providing robust support for the connected elastic components.

The 6 × 6 hinged quadrilaterals have an inverted trapezoidal block with upper and lower connections sliding left and right relative to the upper and lower strip plates, both of which are fitted with 4 bearings to ensure friction-free energy changes during compression of the 1-DOF kinematic base (Section S2). The elastic component consists of spring steel and 2 elastomeric couplers, which are fixed to the through holes of the rectangular block of the kinematic base. The elastomeric coupler is designed with a pin securely attached at one end to the through-hole of the rectangular unit in 1-DOF kinematic bases. At the other end, a TPU flex hinge is firmly connected to the spring steel. Under displacement loading, the 1-DOF rectangular unit undergoes rotation, leading to a variation in the span ( l_s) between the 2 end points of the spring steel.

Each elastic component is defined by 2 integer variables and 6 real variables; displacement loading of the metamaterial induces rotation in the 1-DOF kinematic base, altering the span l_s between the spring steel end points. Consequently, the spring steel bends more or less, causing the energy and reaction force to change. Adjusting the design variables of each elastic component allows precise control of steel deformation and bending, thereby controlling the reaction force. The inverse design of reaction forces is detailed in the next section and in Section S4.

We present 2 different 1DOFmats with distinct force–displacement curves (see Fig. 3B and Movie S2). The blue line represents a positive stiffness curve (monotonic increasing curve), while the red line illustrates a partially negative stiffness curve (nonmonotonic curve). The blue and red dashed lines show how the span ( l_s) between the 2 end points of the spring steel changes under displacement loading for positive and partially negative stiffness 1DOFmats, respectively. The positive stiffness of the 1DOFmat spring steel exhibits a linear decrease in the distance between its 2 end points, indicating stored energy. In contrast, the partially negative stiffness 1DOFmat spring steel initially undergoes an increase in distance followed by a decrease, suggesting both energy storage and subsequent release. The spring steel for positive and partially negative stiffness 1DOFmats shares characteristics such as Young's modulus E = 205 GPa, initial length L_0 = 115 mm, width w = 10 mm, and thickness b = 0.3 mm. The deformation diagrams corresponding to loading displacements of d = 0 mm, d = 35 mm, and d = 74 mm are presented in Fig. 3C. The metamaterial transforms energy primarily through the rotation of the flexible hinge in the 1-DOF kinematic base and the bending deformation of the spring steel in the elastic assembly. By determining the response of the 1-DOF kinematic base to displacement loading and adjusting the spring steel specifications, we can control the trend of change in span ( l_s) between the 2 end points of the spring steel. This control allows for the precise regulation of elastic energy changes during loading displacement, which affects the force–displacement curve. Ultimately, this adjustment enables the creation of metamaterials with customized stress–strain curves, offering versatility in designing materials with desired mechanical characteristics.

Each kinematic base contains 36 holes. The relative coordinates of the first elastic component support in terms of the local coordinates of the rectangular block at index N_1 are denoted as u_1 and v_1, while u_2 and v_2 refer to the second elastic component support at index N_2 (Sections S4 and S5). The rest of the spring steel on the elastic component is L_0, with w representing its width. Considering the rigidity of the 3D-printed kinematic base used in our 1DOFmat, we selected spring steel with a thickness of 0.3 mm to achieve an optimal balance between flexibility and rigidity. Each rectangular block undergoes rotational movement, which causes the attached elastic components to bend accordingly.

The fundamental process of inverse design involves the following steps:

Preliminary selection: The required number of elastic components N_{\text{elas}} and 1-DOF kinematic bases N_{\text{base}} are pre-selected based on the complexity of the target curve. A specific number of sample points N_{\text{test}} is selected from the target curve for fitting.

Parameter expression: The parametric representation of the elastic components expresses the energy of the elastic potential in accordance with the established kinematic relationships. The elastic energy stored in the components is a function of their geometric properties and the coupling with the rectangular through-holes. The reaction force F_{1\text{DOFmat}} is determined as the sum of the base kinematic force F_{\text{base}} (derived from compression experiment) and the elastic force F_{\text{elas}}, derived from the derivative of the elastic potential energy E_{\text{elas}}^{\text{total}} of the elastic components with respect to displacement loading d_{\text{disp}}:

Minimization: The objective is to minimize the squared sum of the differences between the reaction force fitted at the sampling points and the target reaction force. Inequality constraints are imposed on the span between the 2 end points of the spring steel, ensuring that its bending remains within the elastic range. The design problem for the elastic components can be formulated as a constrained optimization problem:where N_{\text{test}} is the number of sample points, F_{\text{target}}^j is the target response curve, and L_{\text{plastic}}^i is the distance of plastic deformation in the spring steel of elastic components. \mathit{lb} and \mathit{ub} represent the lower and upper bounds of the variables. The optimization problem can be solved using the fmincon function in MATLAB.

Accuracy assessment: The mean relative error (MRE) serves as the evaluation metric for the accuracy of the fitting of the customized curve:

If the MRE is below the specified error {\text{error}}_{\text{target}}, the design parameters of the elastic components are finalized. However, if it exceeds {\text{error}}_{\text{target}}, adjusting N_{\text{elas}} and N_{\text{base}} is required. The inverse resolution process for a simple curve example is described in detail in the Supplementary Materials (Section S4).

The proposed 1DOFmat integrates a 1-DOF kinematic base with elastic components, with each component attached to the base at 2 end points. The base features 36 independently rotating rectangular blocks, enabling these end points to provide the elastic components with extensive motion flexibility. This configuration allows the coupled system to exhibit a variety of motion behaviors, facilitating the realization of arbitrarily stress–strain curves (see Movie S3 and Code S2). To verify the generalizability of the 1DOFmat, 4 different force–displacement curves were generated. Each target curve included 10 sample points (Section S4), and errors were calculated by comparing the inverse solution theory curve and the experimental test curve with the target curve, as shown in Fig. 4 . The black, red, and blue lines represent the target curve, the inverse solution curve, and the experimental test curve, respectively. N_{\text{base}} and N_{\text{elas}} represent the number of kinematic bases and elastic components. The left side of Fig. 4 shows the force curves, while the right side presents snapshots of the experimental compression tests.

Firstly, the first target curve represents a common force curve for zero stiffness, which can be effectively achieved with a single elastic component. We formulate a force design using an elastic component and one kinematic base ( N_{\text{base}} = 2, N_{\text{elas}} = 1). The designed response proposed by the 1DOFmat model closely approximates the target response, resulting in an error of 2%, while the experimental fitting error is also very accurate at 5.3%. The zero stiffness curve demonstrates marked potential for various applications in frequency vibration reduction, where effective control of vibrations across a wide range of frequencies is crucial to enhancing performance and ensuring structural integrity.

Second, we considered a target response with a large negative stiffness curve ( N_{\text{base}} = 2, N_{\text{elas}} = 2), where the stiffness suddenly changes from positive to negative. The material can rapidly shift from “rigid” to “flexible”, making it suitable for applications that require rapid adjustments in stiffness. The matched error for the 1DOFmat-designed response is 4.4%, while the experimental fitting error is also low at 4.8%.

Third, we explore the design of a double gradient response curve ( N_{\text{base}} = 4, N_{\text{elas}} = 6), where the force is initially 30 N and transitions to 20 N in the second stage. This can be used in scenarios where multiple gradient switches are required for different load requirements. The optimized solution of the 1DOFmat model aligns closely with the target response curve, with an error of 6.5%, while the experimental adjustment error is 25.3%.

Fourth, we considered a multistage softening response ( N_{\text{base}} = 3, N_{\text{elas}} = 3), representing a curve of gradual softening. This type of response addresses a limitation where multi-DOF deformable metamaterials cannot be achieved. The error for the inverse solution of 1DOFmat is 9.1%, while the experimental fitting error is 31.5%. Due to the large energy variation required by multistage softening, the synchronization constraints on the base need to be improved, placing greater demands on the stiffness and torsional resistance of the base hinges.

We studied the relationship between the number of elastic components and the fitting error of the target curve, fixing the number of kinematic bases ( N_{\text{base}} = 2) and ensuring that the base reactions are consistent. The base configuration can be diversified, ensuring that the kinematic base retains a one degree of freedom. Figure 5 shows 5 target curves fits obtained by adjusting the number of elastic components; Fig. 5A to C shows a sinusoidal curve, a linear softening curve, and a cosine curve; Fig. 5D and E displays 2 distinct response curves with variations in peak and valley values. These 5 distinct target response curves illustrate the versatility of the 1DOFmat model in fitting complex response curves. Figure 5F shows the fitting errors for these target curves with different elastic components ( N_{\text{elas}} = 2, N_{\text{elas}} = 4, N_{\text{elas}} = 6), demonstrating that increasing the number of elastic components improves the fitting of the force–displacement curves, particularly when intricate stiffness variations are involved (see Code S3).

Active mechanical metamaterials and the active control of stimuli-responsive materials have garnered substantial research interest due to their potential for dynamic reprogrammability. The physical reprogrammability of pneumatically driven metamaterials typically requires exceptional sealing performance and a sophisticated pneumatic drive system [48,61,62]. Several approaches to reprogrammability are available, including SMA actuators, magnetic actuators, electromagnetic switches, and pneumatic actuators, each offering distinct advantages in terms of adaptability and control (see Section S1.3). In this study, choosing SMA allows for rapid reconfiguration of performance in response to external stimuli with minimal alteration, enabling the metamaterials to achieve enhanced properties.

To compare the stiffness of the SMAs before and after heating with traditional spring steel, a 4-point bending test was performed, the specimen dimensions being 100 mm in length, 15 mm in width, and 0.3 mm in thickness (see Fig. 6A). The selected transition temperature for the SMA is 70 °C. When not heated, the SMA remains in the martensite phase at low temperatures, showing a “soft phase”, while upon heating it transitions to the austenite phase at high temperatures, showing a “stiff phase”. SMAs exhibit superelastic behavior at high temperatures in their austenitic state. The stiffness of heated SMA is 0.55 times of the spring steel chosen before; the stiffness of unheated SMA is 0.15 times of the spring steel. In the in situ rapid reconfigurable metamaterial experiment, a polyimide heating element was used to heat the SMA sheet, with a temperature sensor probe attached to its surface (Fig. 6B). A proportional-integral-derivative (PID) temperature controller, a feedback control system that continuously adjusts the control input based on the error between the setpoint and the measured process variable, was used to ensure that the SMA remained above its transition temperature of 70 °C, thereby maintaining the high stiffness characteristic of the stiff phase (see Movie S4).

The first experiment illustrates an example of “in situ rapid activation”, where a quasi-zero-stiffness vibration isolator adapts to varying loads with different plateaus. To meet diverse load requirements, the designed metamaterial features switchable force–displacement curves with tunable plateaus. Figure 6C displays the rapidly reconfigurable metamaterial consisting of 3 layers of SMA, where “0” denotes the soft state of the unheated SMA and “1” indicates the stiff state of the heated SMA. In the “001” state, the material shows a low load vibration reduction effect shown in red, while in the “101” state, it exhibits a high load vibration reduction effect depicted in blue. Furthermore, by modifying the size and number of SMA sheets, a wider range of target curves can be inversely designed, offering more options for various operating conditions and application scenarios. Figure 6D presents another set of rapid reconfiguration examples that used the same base and SMA sheets but designed different base-coupled components, resulting in 3 distinct response curves: a quasi-zero stiffness curve, a small-amplitude negative stiffness curve, and a large-amplitude negative stiffness curve. These innovative designs are particularly relevant for applications in virtual reality (VR) haptics, where they can effectively simulate various surface conditions, including clay–gravel layer [63], thin ice, and thick ice. By accurately replicating the tactile feedback associated with these different terrains, the proposed metamaterials enhance the immersive experience of VR environments. This advancement opens new avenues for VR applications, particularly in enhancing footstep interactions and providing users with a more realistic and engaging sensory experience [63–65]. Furthermore, these variations illustrate the flexibility and adaptability of the metamaterial design, allowing for tailored performance characteristics suited to specific applications. Beyond VR, this technology could be utilized in various fields such as soft robotics, adaptive structures, and biomechanical systems. One promising application is the development of variable stiffness materials that can actively change their rigidity in response to external stimuli, which could be used in impact-resistant materials, shock absorbers, or exoskeletons. For example, 3D polycatenated architected materials, which are designed with intricate, interconnected geometric structures, can enable these materials to switch between multiple stable states, offering a range of adjustable mechanical properties suitable for a variety of applications, including self-assembling systems, deployable structures, or even energy-efficient buildings.

The 1DOFmat model is capable of designing the force response not only in the Y direction but also in the X direction. Figure 7A illustrates the variation of the rectangular angle \theta and the unit height h during compression in the X and Y directions. State i is the initial state ( \theta = 1.86), states ii and iii indicate compression in the X direction, and states iv and v represent compression in the Y direction. The X-direction compression curve (Fig. 7B and C) exhibits a nearly linear force response with an elastic beam span changing very slightly, while the Y-direction compression curve (Fig. 7D and E) transitions from high stiffness to zero stiffness with a substantially reduced elastic beam span. Due to the different paths of the angle \theta changes corresponding to the compression of the X direction and the Y direction, representing different rotation modes of the rectangular cell, 2 distinct force curves can be designed (see Movie S5). Anisotropic responses can be arbitrarily customized, with distinct force–displacement curves in the X and Y directions, as discussed in Section S4.3. Unlike the design of force–displacement curves in a single direction, achieving anisotropic responses requires 2 different curves with the same set of elastic components. This increases the number of target curves for inverse design, necessitating more design parameters and computational resources. Furthermore, anisotropic responses can be made reconfigurable; by replacing the spring steels with smart materials, rapid reconfiguration of these responses can be achieved.