Shape morphing is one of the most attractive functionalities of materials that are desired in many applications, including robotic grippers, medical stents, wearable electronics, and so on. Shape morphing can be implemented by using mechanical metamaterials that combine building blocks with properly designed mechanical or material properties. The design approaches are, however, mostly *ad hoc* or require materials with special properties. This work proposes two automated design strategies for programmable shape morphing and validates them on structures 3D-printed from a widely available commercial Stereolithography Durable resin. We proposed a so-called rose-shaped metamaterial with reduced stress concentration due to the absence of sharp corners and with a large range of tailorable Poisson’s ratios, from −0.5 to 0.9, governed by a single design parameter. We programmed the shape of the rose-shaped metamaterial sheets aiming at high shape comfortability or uniform effective stiffness. The shape-morphing performance is demonstrated in the linear (0.1% strain) and non-linear (20% strain) deformation regimes, and it agrees well with the tensile test results. Our findings show the potential to develop complex practical metamaterial structures at comparatively low costs.

## I. INTRODUCTION

Morphing of structures with initial simple shapes into programmed configurations is an emerging trend in materials engineering, offering a possibility to create complex two-dimensional (2D) and three-dimensional (3D) shapes.^{1,2} This functionality is highly relevant in medical applications,^{3–5} deployable and self-folding structures,^{6} flexible electronics,^{7} wearable devices,^{8,9} and soft robotics.^{10} It can be implemented by using a precisely tailored material structure in combination with the tuned mechanical properties of constitutive materials.^{11} Thus, mechanical metamaterials—rationally designed man-made materials with the properties governed by structural geometries and composition of building blocks^{12,13}—are ideal candidates to program shape,^{8} movement,^{14} or morphogenesis.^{10,15}

Shape morphing is achieved by exploiting the topological or space–time properties of metamaterials^{16} that are actuated by external stimuli, e.g., force,^{8} moisture,^{17} temperature,^{18} electric^{19} or magnetic^{20} fields, and so on. An overview of various programming strategies for 2D and 3D metamaterials, including buckling, multi-material tessellations, anisotropic bilayers, sequential shape-morphing, and so on, can be found elsewhere.^{11} Here, we classify these strategies into two groups based on the underlying mechanisms to achieve shape-morphing behavior. The first group relies on the special properties of materials or microstructures. For instance, van Manen *et al.*^{21} proposed a shape-shifting technique that exploits memory-polymer materials as constituents and enables combining 3D printing and programming shape in a single-step production process. Choi *et al.*^{22} used regular kirigami tessellations to reproduce target 2D or 3D shapes. These methods have a large degree of freedom in terms of shape-morphing forms while lacking systematic design approaches by relying mainly on trial-and-error. However, the programmed metamaterials in this group typically require special base materials, e.g., shape-memory polymers^{18} and/or a non-trivial assembly.^{23} The second group uses combinations of different building blocks (unit cells) or gradient designs to program a spatial distribution of desired structural properties. For example, Mirzaali *et al.*^{8} programmed the shape of deformed cellular structures by regulating the arrangement of auxetic and non-auxetic unit cells. Wenz *et al.*^{24} designed the shape morphing behavior of structures composed of re-entrant and hexagonal unit cells by manipulating a geometric gradient as well as the global stiffness gradient. The corresponding metamaterials are usually easier to fabricate by commercial 3D printing techniques,^{25} as their working mechanisms rely on relatively regular geometric gradients. For 2D metamaterials, non-additive manufacturing methods also work, e.g., laser cutting^{26} and mold casting.^{8} It should be noted that this group of methods delivers relatively low programmability in terms of deformation modes, while the design principles can be transformed into an on-demand automatic algorithm. The development of such algorithms for deformation modes of different complexity is one of the key goals of this work. To achieve this, we will consider combinations of auxetic and non-auxetic unit cells.

Auxetic materials have a negative Poisson’s ratio that governs their counterintuitive behavior: under a tensile load, these materials expand rather than contract laterally.^{25,27–30} This feature enables one to program the shape-morphing of a deformed structure by combining auxetic unit cells with non-auxetic ones in a predefined way. This needs to be done at two geometric levels. The first level is the level of a unit cell that has a high-dimensional design space determined by a (typically large) number of geometric parameters and mechanical properties of constitutive materials. However, not all configurations of unit cells can readily be combined into a single structure^{5} without violating the structural integrity. This means that the unit cells should be compatible in terms of geometry and comparable in terms of mechanical properties. The second level is the structural level, where the approach of how to use the design space matters. Specifically, it is important to consider the relative degree of deformation for the hierarchical levels constituting a structure^{16} that governs the overall shape-morphing behavior.

To summarize, an efficient approach to program shape-morphing behavior should be able to consider the two geometric levels in a predictable way. This means dealing methodologically with a large number of degrees of freedom at the unit cell level and to avoid tedious trial-and-error procedures and excessive experimentation at the structural level.^{31} In addition, the manufacturability of a designed structure also needs to be taken into account for practical applications.

In this work, we propose such an approach and illustrate its possibilities by programming the shape-morphing of 2D metamaterial sheets composed of so-called rose-shaped cellular unit cells. A rose-shaped unit cell is represented by a four-leaved rose curve and four connecting straight lines and has variable effective mechanical characteristics depending on the values of geometric parameters. An important advantage of the rose-shaped design is the absence of sharp edges and thus a smooth shape reducing stress concentration. We have developed two strategies to implement the programmable shape-morphing design, allowing us to control the arrangement of different unit cells and, additionally, achieve the desired distribution of effective stiffness. The shape-morphing behavior has been validated numerically and experimentally on 3D-printed specimens. To illustrate programmable designs, we used a comparatively cheap commercial Stereolithography (SLA) 3D printer and manufactured the metamaterial samples from an inexpensive flexible resin.

## II. RESULTS

### A. Mechanical metamaterial based on rose curves

#### 1. Geometry

*r*=

*a*cos(

*mθ*), where

*m*is an angular frequency,

*a*is an amplitude, and

*r*and

*θ*are polar coordinates. The graphical representation of a rose curve contains 2

*m*(

*m*) leaves for even (odd)

*m*, which are responsible for the name of the curve, and retains common symmetric and periodic properties of sinusoids.

^{32}Figure 1(b) shows four-leaved rose curves as described by the following equation with

*m*= 4:

*n*. The minimum value of

*n*should be larger than 1. When

*n*= 1, the rose curve self-intersects in the center and forms closed leaves that are not desirable for our designs. Therefore, to ensure the leaves remain open, we consider

*n*> 1. In this way, there remain two parameters to control the shape of the curve: shape parameter

*n*and amplitude

*a*. As

*n*increases, the shape of the rose curve changes from a star-like (

*n*> 1) to a circular shape (

*n*> 30). The amplitude

*a*can be regarded as a positive size factor of rose curves. For any combination of the parameters, the curve remains within circle

*r*=

*a*. The two vertical and two horizontal straight ligaments are added for structural integrity.

To get a solid model of a rose-shaped unit cell [Fig. 1(c)], we expand the constructed curve [Fig. 1(b)] with two different thicknesses, *t*_{c} (the thickness of the rose curve) and *t*_{s} (the thickness of the straight connections), and then extrude it by height *h* in the out-of-plane direction. Hence, we have a total of five parameters to control the overall geometry of the rose-shaped unit cells: *n*, *l* = 2*a*, *t*_{c}, *t*_{s}, and *h*. Among them, the key parameter is *n*, which controls the shape of the rose-shaped unit cells. In this work, the unit cell size *l* is fixed to 10 mm to ensure structural manufacturability by means of an affordable commercial 3D printer (see Sec. IV B for details). The curve thickness *t*_{c} has an influence on the admissible range of *n*. The larger the value of *t*_{c}, the larger the minimum value of *n* due to the intersection of the leaves of the rose curve. Here, the range of *t*_{c} is set to be 0.05*a* to 0.07*a*, corresponding to the values of *n* [1.3, 30]. Thickness *t*_{s} is set to be a constant, 0.06*a*. Height *h* is also constant, 0.6*a*, as we focus on the in-plane behavior. We have numerically verified that larger values of *h* do not influence the in-plane mechanical properties of the designed metamaterial.

#### 2. Mechanical properties

We systematically studied the effective mechanical properties of the developed rose-shaped unit cells. First, we analyzed relative density *ρ*. As the rose curve is described by closed-form Eq. (1), one can derive an analytical expression for the relative density [see the supplementary material, Eq. (S1)]. When shell thickness *t*_{c} or *t*_{s} increases, the relative density increases linearly. When *n* increases, the relative density decreases. For instance, for *t*_{c} = *t*_{s} = 0.06*a*, the relative density decreases from about 0.2 to 0.1 when *n* increases within [1.3, 10]. The theoretical estimations provide slightly larger values of the density compared to the numerical results due to the simplification of the geometry (more details in the supplementary material, Sec. 1).

Second, we estimated the dependence of the effective Poisson’s ratio and stiffness on the geometric parameters of the unit cell (see Sec. IV A for the analysis details). The effective Poisson’s ratio of a unit cell can be defined as a negative number of the ratio of transverse strain to longitudinal strain. When stretching a unit cell longitudinally (along the *x* axis), the unit cells with *n* < 2.42 [e.g., *n* = 1.3 in Fig. 2(b)] expand transversely (along the *z* axis), while the ones with *n* > 2.42 [e.g., *n* = 10.0 in Fig. 2(b)] contract transversely. The design with *n* = 2.42 has a zero Poisson’s ratio and is not deformed transversely when stretched. It should be noted that the uniform distribution of the deformation field within the unit cells is shown in the insets of Fig. 2(c). This feature was observed previously for some optimized honeycomb designs with uniform deformability,^{33–36} and it has an advantage as compared to star-shaped designs.^{37} The parametric study reveals that the effective Poisson’s ratio changes from −0.5 to 0.9 as *n* increases within [1.3, 30], meaning that modifying *n* causes a change from the auxetic to non-auxetic behavior. When *n* increases, the effective relative stiffness *E*/*E*_{s}, where *E*_{s} is the stiffness of a base material, first increases, reaches a peak value, and then monotonically decreases (see Sec. 2 in the supplementary material for the explanation). The overall trend (without the peak) is consistent with the shape of the relative density curve (see the supplementary material, Fig. S1). Parameter *t*_{c} has no effect on the effective Poisson’s ratio [Fig. 3(a)] but influences the effective stiffness [Fig. 3(b)]. As *t*_{c} increases, the effective stiffness has larger values for all *n*. This suggests that *t*_{c} can be used as an auxiliary tool to tune the stiffness distribution within the metamaterial.

An increase in the material Poisson’s ratio from 0.3 to 0.45 does not influence the effective Poisson’s ratio of the rose-shaped metamaterial for *n* ∈ [1.3, 30], while the effective relative stiffness increases by up to 8.5%. As Young’s modulus of the base material increases from 1 to 2 GPa, the effective Poisson’s ratio and effective relative stiffness of the metamaterial remain unchanged. This indicates that the effective mechanical characteristics of the rose-shaped metamaterial are almost insensitive to the base material properties.

Finally, we have explored the effects of large deformation and geometric nonlinearity. Under 20% strain, for instance, the range of the effective Poisson’s ratio for n ∈ [1.3, 10] increases from around [−0.5, 0.7] to [−0.4, 0.8] and the range of the effective stiffness increases from around [0.20, 0.27] to [0.21, 0.30], as compared to the 0.1% strain (linear regime). This indicates that geometric nonlinearities weaken the auxetic effect in the rose-shaped metamaterial, as will be further discussed (Sec. II C).

To summarize, the proposed rose-shaped metamaterials have two notable advantages. The first advantage is the absence of sharp edges that enable reducing stress concentration and improve the structural stability due to a uniform deformation field within unit cells. The second advantage is a broad range of tailorable Poisson’s ratios, including negative and positive values, that are governed by a single design parameter *n*.

### B. Programmable design of rose-shaped metamaterials

#### 1. Design space

Structural deformations of the metamaterial and its overall shape can be programmed by manipulating the geometric properties of rose-shaped unit cells. The key parameters are *n* and *t*_{c}, which have a notable influence on the effective Poisson’s ratio and effective stiffness [Figs. 3(a) and 3(b)], respectively. We use *n* as the main parameter and *t*_{c} as the secondary parameter.

*k*corresponds to lateral expansion, while a negative value corresponds to lateral contraction. Moreover, the larger the absolute value of

*k*, the larger the deformation degree. The relationship between

*k*, and

*n*and

*t*

_{c}can be derived and is shown in Fig. 3(c). We can also build a reference design space represented by a curved surface (see the supplementary material, Fig. S4). The constructed design space can be explored in different ways, by defining a programmable design strategy. Before considering possible strategies, we formulate the ultimate design objective for a programmable functionality.

#### 2. Design objective

We aim to design the shape-morphing of a flat sheet composed of *p* × *q* rose-shaped unit cells [Fig. 4(a)]. In this work, we only focus on a quasi-one-dimensional arrangement of the unit cells, i.e., the metamaterial parameters can vary only in the longitudinal direction and remain constant in the transverse direction. This enables to represent the arrangement of the unit cell columns by a 1D sequence of varying parameters [Fig. 4(a) in the middle]. Specifically, we program the shape of the horizontal lateral edges that vary under a horizontal tensile loading to match the shape of a predefined curve [Fig. 4(a) at the bottom] by finding the proper values for a parameter sequence.

^{8}

*p*indicates the number of the unit cell columns,

*a*

_{1},

*a*

_{2}, and

*a*

_{3}specify the shape. We normalize the

*y*values to [0, 1] and discretize the

*x*values into

*p*points. The coordinates of each point with the column number as an

*x*-value and a lateral deformation degree as a

*y*-value act as an input for the programmable design. It should be noted that the objective shape is, in general, not limited to the Fourier-like series-type curves. As long as

*p*is large enough, the objective shape can be represented by any sequence of discrete points describing a continuous curve.

#### 3. Strategy 1 for the programmable design

In strategy 1, we use parameter *n* to form the parameter sequence for the programmable design, as illustrated in Fig. 4(b). Considering the limitations on the manufacturability of metamaterial specimens (see Sec. IV B for details), the total number of the unit cells is set to 12 × 7, and the length and thickness are *l* = 2*a* = 10 mm and *t*_{c} = *t*_{s} = 0.06*a* = 0.3 mm, respectively. The objective curve, Eq. (3), has parameters *a*_{2} = 1, *a*_{1} = *a*_{3} = 0 and is denoted as C2 for brevity. The *x*-value range is discretized to form a sequence of 12 points. We extract the *k* curve for unit cells with *t*_{c} = *t*_{s} = 0.06*a* from Fig. 3(c) and normalize it to [0, 1] as the ready-to-use design reference [Fig. 4(b), left bottom]. Then, we generate the required values of *n* by searching on the design reference curve using the normalized *y* values. Next, we can build the geometric model of the programmed flat structure.

To facilitate experimental testing, we add two solid material blocks to the lateral sides along the longitudinal direction that act as clamping ends. In the finite-element simulation, we apply a prescribed displacement equal to 20% of the longitudinal length of the rose-shaped part. As can be seen in Fig. 4(b), the predicted shape of the deformed structure matches the objective shape, confirming the programmed shape-morphing functionality. It should be noted that geometric nonlinearities are not considered here. As can be expected, the clamping ends partly constrain structural deformations, leading to a deviation from the target shape. To reduce this effect, a possible solution is to add a repeating extra column of the unit cells between the structure and the clamping ends.

The proposed strategy assumes a fixed value of *t*_{c} that results in different effective stiffness values for adjacent columns [Fig. 3(b), right]. Different stiffnesses result in different deformation degrees that, in turn, can increase stress concentration and even cause local failures (we indeed observed such failures during the experiment for some of the manufactured samples). Therefore, it seems promising to tune the effective stiffness to have similar values for different columns, which leads to a more uniform deformation field.

#### 4. Strategy 2 for the programmable design

Strategy 2 relies on tuning two parameters *n* and *t*_{c} to implement the programmable shape-morphing behavior, where *t*_{c} can be varied to modify the effective stiffness of the rose-shaped unit cells. To demonstrate the design process, the structural geometric parameters are set to be 12 × 7, *l* = 2*a* = 10 mm, and *t*_{s} = 0.06*a* = 0.3 mm, respectively.

Figure 5(b) illustrates the programmable design procedure for the C2 curve. We first tune the effective relative stiffness of unit cells with different *n* to be the same by modifying their *t*_{c}. The tuning target is set as $E/Es\u0304$, the mean value of the effective relative stiffness of unit cells with *t*_{c} = *t*_{s} = 0.06*a*. After tuning, we can get unit cells with similar effective relative stiffness by varying *t*_{c}, as shown in Fig. 5(b) at the left top. The *t*_{c} curve of the tuned unit cells has a valley that is opposite to the peak in the effective relative stiffness curve with constant *t*_{c} [Fig. 3(b)]. The first step of strategy 2 is the same as strategy 1—discretizing and normalizing the C2 curve as the input. Different from strategy 1, the design reference for strategy 2 is obtained by normalizing the curve of $k=\u2212\nu E/Es\u0304$ to [0, 1]. The required sequences of *n* and *t*_{c} can be generated by searching the design reference curve using the normalized *y* values. Next, the geometric model of the programmed metamaterial sheet can be built [Fig. 5(b), right bottom].

By analyzing the von Mises stress distributions under the same prescribed displacements for the designs delivered by Strategies 1 [Fig. 4(b)] and 2 [Fig. 5(b)], we see that the maximum value of internal stress in the structure provided by strategy 2 is about 15% smaller compared to that by strategy 1. The maximum value of local strain (first principle strain) in the structure provided by strategy 2 is 0.056, which is about 14% smaller than the value of 0.065 by strategy 1. This confirms that strategy 2 enables more uniform structural deformations. However, the deformation amplitude of the structure provided by strategy 2 is smaller than that by strategy 1, as shown in Fig. 6. Furthermore, we can see that the rose-shaped unit cells with *n* > 20 have low programmability as the corresponding *k* values remain almost unchanged. Thus, the range of *n* can be reduced to [1.3, 20]. It should be noted that the variation of *t*_{c} in strategy 2 is very small due to that the stiffness is sensitive to the change of *t*_{c}. The subtle difference in the thickness values requires a high-precision printer to fabricate a sample, which complicates experimental validation.

### C. Experimental verification

The designed metamaterial structures were fabricated using a commercial SLA 3D printer and experimentally tested. We focus on the designs proposed by strategy 1, as the thickness variance in the programmed structures used by strategy 2 is too small to satisfy the manufacturing constraints. The fabricated structures have the same dimensions as in the numerical simulations, i.e., 12 × 7 unit cells, *l* = 2*a* = 10 mm, and *t*_{c} = *t*_{s} = 0.06*a*. We consider three specimens denoted as C1 (*a*_{1} = 1, *a*_{2} = *a*_{3} = 0), C2 (*a*_{2} = 1, *a*_{1} = *a*_{3} = 0), and C3 (*a*_{3} = 1, *a*_{1} = *a*_{2} = 0), as shown in Fig. 7. Overall, we observe a very good agreement between the predicted and measured structural shapes. It should be noted that this agreement can further be improved by considering geometric nonlinearities in the numerical simulations (see the bottom graphs in Fig. 7). This effect can be explained by the reduced deformation degree due to the geometric nonlinearities, which is consistent with the influence of nonlinearity on the mechanical properties of unit cells (see Sec. II A 2 for details). The videos for the shape-morphing process of the three specimens are provided [see the supplementary material, Videos S7(a)–S7(c)], revealing its reversibility—an advantageous feature of the proposed programmable behavior implemented in the elastic and flexible material.

## III. DISCUSSION

We developed two strategies to explore the parameter design space of rose-shaped mechanical metamaterials and validated them by showing the programmable behavior of flat sheets numerically and experimentally. The accuracy of the shape matching depends on the deformation degree and the effective stiffness of the unit cells. We have shown that the sequence of the *n* values determines the overall structural shape, and the sequence of the *t*_{c} values can be used to tune the deformation degree. Specifically, strategy 2 enables tuning of the effective stiffness of the unit cells with the aim to obtain a close-to-uniform stress distribution but at the price of the reduced deformation degree (see Figs. 4 and 5). If needed, the effective stiffness of the unit cells can be tuned to different values for a larger overall deformation degree.

We programmed and verified the shape morphing for three Fourier-like series curves (C1, C2, and C3) using the proposed strategies. It is also possible to match the shape of specific objects. For illustration purposes, we numerically analyzed the shape morphing of a wine bottle and a foot sole (see the supplementary material, Fig. S6). The proposed strategies can be further extended to program designs of more advanced metamaterial with different objectives in mind, e.g., to achieve a localized programmable behavior. Other examples include complex deformation modes, such as the shear or torsional deformation under tensile loading. This can be implemented by controlling a 2D arrangement of rose-shaped (or other) unit cells. Further work in this direction is required.

The approach proposed in this paper can be applied to 3D analogs of the rose-shaped unit cells. We show two possible configurations of the 3D rose-shaped unit cells in Figs. 8(a) and 8(b) and the corresponding SLA 3D-printed samples [Fig. 8(c)]. The performed numerical analysis reveals that the range of admissible effective Poisson’s ratios for the 3D unit cells is smaller than for the 2D counterparts. However, the shape of the 3D unit cells is also governed by a single parameter *n* that allows combining them in metamaterial structures with a programmable behavior, e.g., by applying strategy 1 extended to the 3D setting.

Finally, we emphasize that the transfer of the programmable design concept to practical applications requires overcoming various manufacturing issues of currently available technology.^{38} Due to complicated geometric features of metamaterials (e.g., thin suspended elements, non-trivial spatial arrangement, the presence of large voids, etc.), most conventional manufacturing methods are not suitable to fabricate cellular programmed metamaterials. Alternative methods, such as mold casting^{8} or laser cutting,^{39} can be inefficient. The mold casting for 2D designs can deliver final structures prone to micro-defects, which are caused by air bubbles unavoidable during the production process. Laser cutting is not applicable to the proposed 3D designs. Therefore, 3D printing remains the simplest and most efficient method to fabricate most of the mechanical metamaterials,^{30,40–44} as long as the technique and raw material are properly selected.^{9,45} Nevertheless, 3D printing techniques have their limitations. For instance, the developed designs are still challenging for commercial selective laser sintering (SLS) and multi jet fusion (MJF) printing^{46} and are difficult to scale to micro- or nano-size structures. This is nicely illustrated by the difficulty of implementing the found subtle variance in *t*_{c} for strategy 2 by the low-cost SLA 3D-printing. The printing efficiency (in terms of time) and material costs hinder practical applications of the programmable mechanical metamaterials. Further developments in the additive manufacturing techniques may eliminate the mentioned obstacles in fabrication, speeding up the practical implementation of the programmed metamaterials.

## IV. METHODS

### A. Finite element analysis

Numerical simulations were conducted by means of the finite-element analysis using COMSOL Multiphysics^{Ⓡ}. In the simulations, we assumed that the rose-shaped mechanical metamaterial has a perfect, defect-free geometry and used solid elements to reproduce the metamaterial models. The base material was isotropic with the following mechanical properties:^{42} Young’s modulus *E* = 1 GPa, Poisson’s ratio *ν* = 0.35, and mass density 1.08 g/cm^{3} that correspond to the properties of Durable resin (Formlabs, Inc., U.S.). We ignored eventual viscoelastic effects and assigned tensile strength *σ* = 28 MPa based on the material datasheet from the Formlabs company.^{47} We applied a swept triangular mesh with an extremely fine mesh size and performed the mesh convergence analysis. For the unit cell with *n* = 1.3 as an example, the fine mesh has 1052 elements for a single unit cell, and the extremely fine mesh has 23 340 elements and can provide more accurate results [see the supplementary material, Figs. S5(a) and S5(b)]. If the mesh size is not fine enough, the predicted effective stiffness can be larger than expected [see the supplementary material, Fig. S5(c)], and the stiffness curve is not smooth.

To analyze the mechanical properties of a single unit cell, we fixed its one edge by assigning zero displacements and applied a prescribed displacement (equal to 0.1% of unit cell length) at the opposite edge along the *x*-axis direction. The vertical edges were subject to periodic boundary conditions. Here, the effective Poisson’s ratio of the unit cell can be defined as a negative number of the ratio of transverse strain to longitudinal strain. Then, the effective Poisson’s ratio can be calculated as $\upsilon =\u2212(w1\u2212w2)u$, where *u* is the applied displacement, and *w*_{1} and *w*_{2} are the displacements of the top and bottom ends in the *z*-axis direction. The effective stiffness can be calculated as $E=FlAu$, where *F* is the reaction force along the *x*-axis direction and *A* is the projected area along the *x*-axis direction.

For the metamaterial structures, the displacement applied in the longitudinal direction is equal to 20% of the total length of the rose-shaped part. The lateral strain of each deformed column is calculated by $\epsilon iz=wi,1\u2212wi,\u20092Lz$, where *w*_{i,1} and *w*_{i,2} are the displacements of the top and bottom ends of the column, and *L*_{z} is the column length.

### B. Fabrication and experiments

Commercial SLA 3D printer Form3 (Formlabs, Inc., U.S.) was used to fabricate the designed metamaterials. As a base material, we used Durable resin that has a high impact and wear resistance and can bear large deformations before breaking (the elongation at break can reach 55%).^{47} The metamaterial samples were 3D-printed with supporting structures made of the same resin that was removed manually by cutting. The raw samples were cured at 60 °C in a heating box for 60 min that allowed them to reach the expected mechanical properties. It should be noted that the specimens were deformed due to stretching during the removing and curing processes. To measure the final dimensions of the printed samples, we used a digital caliper. The length, width, and height of specimen C1 are 129.3, 71.0 ± 1.0, and 3.0 ± 0.1 mm, respectively [Fig. 7(a)]. The length, width, and height of specimen C2 are 123.7, 70.0 ± 0.5, and 3.0 ± 0.1 mm, respectively [Fig. 7(b)]. The length, width, and height of specimen C3 are 127.0, 70.0 ± 0.8, and 3.0 ± 0.1 mm, respectively [Fig. 7(c)].

The programmable response of the 3D-printed samples was analyzed in tensile tests. For this, we used a UniVert testing system (CellScale, Canada) with a 100 N load cell and applied a prescribed displacement at a speed of 5 mm/min. Each sample was stretched to develop up to 20% strain, i.e., 24 mm displacement. Subtracting the deformation before the tests, the actual displacements applied to C1, C2, and C3 were 14.7, 20.3, and 17.0 mm, respectively. The deformation process was recorded by a camera (Logitech, Switzerland) integrated into the UniVert system at the rate of 5 photos/s. At the maximum displacement (achieved when the sample length reached 120 mm), a holding time of 2 s was set to capture the maximum deformation. The photos at the maximum displacement were exported into the open-source software ImageJ to measure the displacements of the unit cell columns. Finally, the lateral strain was calculated using the measured displacements.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the following: Section 1 describes the relative density of rose-shaped unit cells. Section 2 explains the peak in the stiffness curves. Section 3 shows the design space of k. Section 4 reveals mesh convergence analysis. Section 5 demonstrates shape-morphing design for specific objects. Section 6 shows the videos of the shape-morphing processes of specimens.

## ACKNOWLEDGMENTS

We would like to thank Dr. A. G. P. Kottapalli and Dr. D. Ribas Gomes from the Engineering and Technology Institute Groningen for their technical support and access to the devices to conduct experiments. This study was supported by a fellowship from the China Scholarship Council.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Z. Zhang**: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). **A. O. Krushynska**: Conceptualization (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## REFERENCES

*Mathematical Models*