Shape Morphing of Tubular Structures with Tailorable Mechanical Properties

Tubular structures are in high demand in robotics, medicine, and electronics. They are expected to match different shapes under loading and simultaneously exhibit certain mechanical behavior, e.g., specific radial strength and local flexibility, that poses a complex engineering design task. Herein, strategies to achieve programmable shape morphing in patterned tubular structures are explored, whose mechanical properties for all types of deformation modes can be tailored on demand. The general design problem is formulated, and the programmable response for fundamental—expansion, bending, and twisting—modes activated by tension and pressurization is demonstrated. The design problem for expansion modes is solved analytically; the numerical results agree well with the experimental data for stereolithography three‐dimensional printed cellular tubes. The effects of loading and boundary conditions on the deformed shapes and the structural mechanical response are analyzed. Algorithm‐based design strategies are proposed to achieve quantitative and automatic design for complex deformation modes. The possible use of the proposed structures is also discussed with respect to several applications. The findings pave the way for multifunctional tubular structures by exploring the pluridimensional space of the geometric parameters of metamaterial patterns.


Introduction
[16] In flexible electronics, recent findings report cylindrical inflatable catheters that combine versatile mechanical operation with local ablation of tissue and accurate sensory feedback on flow, tactile, and other data. [1,17]These and other developments have been facilitated by recent advances in additive manufacturing technology and constantly decreasing production costs that stimulate further search for more complex patterns capable of expanding current limits on structural mechanical properties. [2,4,11,18]However, clear design guidelines on how to mimic different shapes and concurrently to control the mechanical behavior of tubular structures are still missing. [19]Furthermore, existing designs often require different external loads to activate different deformation modes, which is impractical in most application scenarios.
One of the promising approaches to increase the number of deformation modes and control mechanical behavior relies on the combination of auxetic (negative Poisson's ratio) and conventional (positive Poisson's ratio) unit cells in a single structure.[21] They are often developed for flat structures that undergo in-plane uniaxial tension or compression loading.24][25] Patterned tubular designs under pressurization can be inflatables with reinforcing fibers [26,27] or kirigami. [23]The compliant behavior of inflatables enables high conformability, while selective manipulation of the fiber arrangement or geometric parameters of kirigamis ensures accurate shape morphing. [23,25,26]ubstantial progress in the shape morphing of tubular inflatables was achieved for curved stents, ranging from the structures that undergo a single curvature [28] to patient-specific stents with optimized deformation behavior. [29,30]The shape morphing strategy often relies on modifying the local mechanical properties of unit cells by changing the geometry of the elements in constituent unit cells.The accuracy of shape fitting can be increased by applying optimization algorithms. [29,31]n this work, we demonstrate programmable shape morphing for tubular lattice structures that can be activated by tensile loading or pressurization.The shape morphing is obtained for a so-called rose-shaped pattern [18] formed by auxetic and conventional unit cells, in which a single design parameter enables the variation between a positive and negative Poisson's ratio.The rose-shaped pattern is easy to manufacture and has sufficient freedom (in terms of the number of geometric parameters) to tailor the structural mechanical behavior-flexibility, conformability, stress distribution, etc.-as compared to pure auxetic, [32] wavy, [33,34] and ring-and-link [29,35] designs analyzed previously.Therefore, our approach effectively combines the advantages of shape-morphing designs for flat and tubular structures.
As the mechanical behavior of tubular structures is qualitatively different and quantitatively richer than that of their flat counterparts, [11] we formulate the general design problem theoretically in terms of cylindrical coordinates and then narrow it down to the analysis of fundamental-expansion, bending, and twisting-deformation modes.For the expansion modes, we derive an analytical solution that provides useful insights into the deformation process and suggests an theory-based design approach for programmable shape morphing.The analytical results are in excellent agreement with the numerical estimations.
We also analyze the differences in the deformation modes induced by tensile and pressure loading and fixed and free boundary conditions.The bending and twisting modes are studied numerically; however, the structure of their governing equations enables us to apply the intuition-based design approach to these modes.
Next, we propose an optimization-based design approach that can be used to program any fundamental or mixed deformation mode of a patterned tubular structure.We show that this approach provides good results for expansion and bending modes, while accurate shape morphing of more complex modes that can be relevant for specific applications is out-of-scope in this study because of high computational costs.
To validate our theoretical and numerical results, we propose a stereolithography three-dimensional (SLA 3D) printing as an accessible and versatile manufacturing technique for patterned tubular structures.It allows the preservation of a circular undeformed shape and uniform stress distribution within a tube in contrast to gluing or using connectors that suffer from distorted forms and concentrated stresses at joints. [36]he agreement between the programmed shapes predicted numerically and those of the pressurized samples is very good, confirming promising perspectives for our design approaches and rose-shaped patterns in morphing complex shapes.
][39] This extends the application potential of this study.

Problem Statement and Structural Design
We consider a tubular structure as a three-dimensional cylindrical anisotropic domain Ω composed of lattice unit cells Ω ða,cÞ (Figure 1) with the indices a ¼ 1, : : : , A and c ¼ 1, : : : , C referring to the position of a unit cell along the axial and circumferential directions, respectively.
If structural thickness h is small compared to the radius of curvature r, i.e., h=r ( 1=20, this structure can be considered as a thin shell with stress and strain fields independent of the radial coordinate. [40]It enables representing the displacements as continuous functions of θ and z u r ¼ u r ðθ, zÞ, u θ ¼ u θ ðθ, zÞ, u z ¼ u z ðθ, zÞ where r, θ, and z denote radial, circumferential, and axial coordinates, respectively.Assuming constant effective mechanical parameters for each unit cell Ω ða,cÞ , the structural mechanical properties can be described using piecewise constant functions over Ω. [41] Then, at the unit cell level, the strain-displacement relations have the conventional form for infinitesimal strains [42] ε ða,cÞ r Using Equation ( 1), these relations are simplified as follows For orthotopic unit cells, the stress-strain relations are [43] σ r σ θ σ z τ θz τ zr τ rθ where the superscripts ða, cÞ are taken out of the brackets for compact representation.The equilibrium equations for the r-independent stresses [42] 1 r can be combined with Equation ( 3) and ( 4) that results in the governing equations describing the equilibrium state of a patterned tubular shell where the superscripts ða, cÞ for elasticity moduli C ij and displacements u r , u θ , u z are omitted for brevity.In this study, the elasticity moduli of a unit cell, C ða,cÞ ij , are estimated numerically, as described in Section 2.1.
The governing Equation ( 6) must be solved under specified boundary conditions.To imitate tensile loading, we assume constant displacement u 0 at the one end of the tube and fix the other end along the axial (z) direction, i.e. u ðA,cÞ z ðθ, LÞ ¼ 0, u ð1,cÞ z ðθ, 0Þ ¼ 0 (7)   where A refers to the most right row of the unit cells, and L is the length of the tube.For the pressure loading, the pressure of magnitude p 0 acts at the interior of the tube counteracting the normal radial structural stress, i.e. σ ða,cÞ r 0 ðθ, zÞ ¼ Àp 0 (8)   while the ends of the structure are stress free.For the left end of the structure, the stress-free conditions are written as follows The same expressions are valid on the right end if "1" in the superscript is replaced by "A".
Finally, note that the circumferential and axial displacements and stresses are continuous at the interfaces of the unit cells, i.e.
The solutions to Equation (6) subject to boundary conditions (7) or ( 8) and ( 9) and continuity conditions (10) fully describe the mechanical behavior of a patterned tubular structure.An analytical solution for this general case cannot be obtained due to the complexity of the problem. [40]Therefore, we proceed by first specifying the unit cell designs and then considering separately the fundamental expansion, bending, and twisting deformation modes [26,44] with the aim to simplify the problem formulation and obtain analytical solutions, if possible.

Rose-Shaped Pattern and Its Mechanical Properties
We use rose-shaped designs as unit cells that reveal a low stress concentration due to the absence of sharp corners and a wide range of Poisson's ratios, from À0.5 to 0.9, governed by a single design parameter. [18]An exemplary structure with a rose-shaped pattern is shown in Figure 2a.The rose-shaped unit cell is formed by a four-leaf rose curve and four straight-line segments.The rose curve can be described as r ¼ l 2 ⋅ nÀcos 4θ nþ1 , where l=2 is the amplitude, r and θ are the polar coordinates, and n is the shape parameter.As n increases, the shape of the curve changes from four-leaf (n > 1) to circular (n > 30) (Figure 2b).For n ≤ 1, the curve has self-intersecting leaves irrelevant to our designs and thus not considered here.
The geometric parameters-length l, out-of-plane thickness h, in-plane thickness of the curved elements t c , and shape parameter n (Figure 2b)-can be varied to tune the effective mechanical properties of the unit cells, e.g., stiffness and Poisson's ratio.In the following, we discuss the influence of these parameters on the tunability of the mechanical properties considering that the unit cells form a tubular structure.
The length l is inherently related to the diameter D of a tube.Therefore, l should be small enough to ensure a circular structural shape and, simultaneously, large enough to guarantee a proper resolution of the geometric characteristics of the unit cell, which can be limited, for example, by a manufacturing technique.Therefore, the values of l are mainly governed by structural dimensions and production conditions and provide limited freedom to tune the structural mechanical behavior.
The thickness h influences the structural stiffness.Large values of h result in thick tubes and require large load magnitudes to induce shape morphing.Thick tubular structures are stiff, which is often disadvantageous in shape-morphing applications, and thus are not considered here.Hence, we fix the values of l ¼ 10 mm and h ¼ 1 mm to comply with our manufacturing conditions (see Section 3.5 for details) and to ensure a small tube thickness compared to its radius.
To estimate the influence of the parameters n and t c , we numerically study the mechanical behavior of a single unit cell and a collection of unit cells.Finite element analysis (FEA) is performed using the cell periodicity interface of the structural mechanics module in COMSOL Multiphysics.The material properties are those of isotropic durable resin (Formlabs Inc., U.S.): Young's modulus E mat ¼ 1 GPa, Poisson's ratio ν mat ¼ 0.35, elongation at break of 55%, and mass density 1.08 g cm À3 . [45]We used a free tetrahedral mesh of a predefined "finer" mesh size that contains around 160 000 solid finite elements for a single unit cell.For the parameter values within t c ∈ ½0.025l, 0.0375l and n ∈ ½1.2, 30 with steps Δt c ¼ 0.005l and Δn ¼ 0.1, we extract the complete elasticity matrix C for each specific unit cell, which corresponds to that of an orthotropic material owing to the symmetry of the rose-shaped unit cells.The effective Poisson's ratio ν and Young's modulus E can be derived from the relations between the compliance matrix (S ¼ C À1 ) and the engineering constants. [42]lternatively, the effective Poisson's ratio ν and Young's modulus E can be estimated by analyzing the mechanical behavior of a set of loaded unit cells.For instance, we considered a ring formed by 12 identical unit cells, which is clamped at one side and loaded by an axial loading f z at the other side.The effective Poisson ratio ν and Young's modulus E are then calculated as follows where u θ1 and u θ2 are the displacements of the two ends of a unit cell along the circumferential direction, u z and f z are the displacement and reaction force along the axial direction.The calculations show that the values of the mechanical characteristics obtained by these two approaches are consistent.The variations of the effective mechanical moduli shown in Figure 3a reveal that the rose-shaped unit cells behave auxetically-i.e., expand transversely when longitudinally stretched (negative Poisson's ratio)-for 1.3 < n < 2.42 and conventionally-i.e., contract transversely under stretching (positive Poisson's ratio)-for 2.42 < n < 30.The normalized Young's modulus decreases up to 75% for increasing n and increases up to 100% when t c increases from 0.025l to 0.035l.
The observed dependences can be compactly described by ratio k ¼ Àν=ðE=E mat Þ called here as a design parameter.From a physical point of view, k corresponds to the transverse strain in a unit cell of a specific stiffness when it is stretched longitudinally.The dependence of k on n and t c is shown in Figure 3b and can be used as a guide to control the structural deformation modes.

Programmable Design of Deformation Modes
The arrangement of auxetic and conventional unit cells controls the deformation modes of tubular structures. [21,36]To achieve programmed deformations, we explore how to achieve fundamental deformation modes, i.e., expansion, bending, and twisting modes.We also analyze differences in deformed states originating from loading and boundary conditions.

Expansion Modes
The expansion deformation implies symmetric stress-stain fields along the cylindrical axis.The induced displacements are thus independent of θ, i.e.
Equation ( 6) are then transformed into ordinary differential equations with the equation for u θ uncoupled from the other two equations.Hence, one can solve the two sets of equations separately.The coupling of the equations for u r and u z indicates that the expansion or contraction of a tubular structure is coupled with its extension or shortening along the axial direction.

Analytical Solution for Tensile Loading Conditions
To solve Equation ( 13) analytically, we further simplify them by assuming that a normal to the median surface of an undeformed tube remains straight and normal to that of a deformed surface, according to the shell theory. [46]As a result, transverse shearing strains vanish, i.e., ε rθ ¼ ε zr ¼ 0.
The general solution to these equations is obtained straightforwardly with M ða,cÞ , N ða,cÞ , P ða,cÞ , Q ða,cÞ being unknown integration constants and r 0 indicating the radius of a tube.The integration constants are omitted here as corresponding to rigid body motions.Note that the expressions for displacements u z and u θ have identical forms with linear dependence on z.We focus on analyzing the first two solutions with two unknown integration constants P and Q as relevant for shape morphing.The unknown constants in the expression for the circumferential displacement can be derived from the solution for the radial displacement according to the relationship between the circumference and the structural radius.
The displacement boundary conditions (7) can be rewritten as follows (as shown in Figure 2c)  Therefore, for a tubular structure formed by a Â c unit cells, we have 2ac solutions (16) containing 2ac unknown integration constants, P ða,cÞ , Q ða,cÞ ða, c ¼ 1, 2, : : : , NÞ.These constants can be found by solving Equation ( 17) and 2ða À 1Þc Equations ( 18) that provides a complete description of the displacement field in a deformed tube for a defined arrangement of the unit cells.The full set of these equations can be found in Section S1, Supporting Information.
The solution ( 16) also allows us to qualitatively describe the deformation process in expansion modes.When a tube elongates or contracts along its axis, it simultaneously and proportionally contracts or expands in the circumferential direction, as indicated by the opposite signs for u r and u z .The amount of induced deformations depends on the pattern and the loading.Although the cross-sectional diameter of the deformed structure varies along the length, the axis of the tube remains in place, reflecting the radial symmetry of the deformation.

Theory-Based Programmable Design
The described deformation process suggests a possible design approach to programming the activation of expansion modes schematically illustrated in Figure 4a.Specifically, expansion deformations are obtained if the unit cells differ along the axial direction (columns a 1 , a 2 ,…) and remain identical along the circumferential direction (rows c 1 , c 2 , …), so that a pattern is formed by a combination of rings formed by identical unit cells.By expanding a tube into a sheet, it is possible to transform the design of a tubular structure under tensile loading into the design of a flat structure.Subsequently, the parameter denoted as k (discussed in Section 2.1) can be used to guide the programmable design, which aligns with our analytical solution.A positive value of k signifies the lateral expansion of a flat structure, specifically the contour expansion of a tubular structure, while a negative value represents the lateral contraction of a flat structure, namely the contour contraction of a tubular structure.In addition, the higher the absolute value of k, the more pronounced the degree of deformation. [18]Consequently, the programmable design for expansion modes can be accomplished through the following steps: normalizing the prescribed contour shape to the interval of [0,1], discretizing the continuous shape into a input values, and ultimately generating the corresponding values of n by conducting a search within the normalized k design space.

Numerical Analysis
To verify the programmable shape morphing obtained by the theory-based approach, we numerically studied the deformation modes of the tubes formed by 12 Â 24 unit cells along the circumference and the axis, respectively, with specific values of n and t c .We used "finer" mesh, and the prescribed displacement along the axial direction was equal to 20% of the total length L of the tube.
Figure 4b shows two examples of the n sequences corresponding to two expansion modes.Specifically, the E m ðm ¼ 1, 3Þ modes represent the curved shapes defined by y ¼ sinðmωðx À 1ÞÞ, where ω ¼ π=ðq À 1Þ and q indicate the number of columns of the unit cell.The values n in the sequence c are given in Table S1, Supporting Information.As can be seen, the shape morphing occurs as expected.The same structural designs were analyzed analytically.The subgraphs above the deformed shapes show the structural displacements evaluated analytically (red) and numerically (blue) and reveal excellent agreement proving the accuracy of the analytical solutions.

Pressure vs Tensile Loading
The numerical analysis of the same structures under pressure loading of p 0 ¼ 30 kPa reveals deformation modes similar to those shown in Figure 4b (Figure S1, Supporting Information).However, the applied pressure prohibits the radial contractions of the tubes, similarly as a tensile load prohibits the axial contractions (see details in Figure S2, Supporting Information).This restricts shape matching under pressurization to smaller shape variations of deformed profiles or requires larger pressure values to achieve substantial shape variations.This limitation and the shortening of the length of a deformed tube must be taken into account in the design of practically relevant shape-morphing structures.

Effects of the Boundary Conditions
Boundary conditions also influence shape morphing.We consider their effects under pressure loading, as these are less studied in the literature.When both ends of a tube are fixed, the design E3 (Figure 4b) still deforms to the target shape under pressurization (Figure S1, Supporting Information), but the structural curvature is less pronounced because the tube cannot contract.When one end of the tube is fixed and the other end is free, the tubes formed by auxetic unit cells extend (Figure S3a, Supporting Information), and those with conventional unit cells contract (Figure S3b, Supporting Information) along the axial direction, improving shape morphing.For example, for design E1 (Figure S3c,d, Supporting Information, with the same geometry as E1 in Figure 4b) with one or two free ends, the unit cells near the ends expand rather than contract (Figure 4b) or remain unchanged (Figure S1, Supporting Information).This occurs because these unit cells have higher n values corresponding to lower stiffness values (Figure 3a) and thus are easier to expand under pressurization.
For the other deformation modes, we consider only the pressure loading with fixed ends of the tubes (Figure 2c), since this combination is favorable for shape morphing and relevant in practical scenarios.

Bending Modes
Bending deformation modes can be analyzed according to the Euler-Bernoulli theory for slender beams [47] by considering small deflections of originally straight circular tubes.This results in the key assumption that the plane cross sections of a tube remain plane in the deformed state without shear deformation across the cross sections.It implies that the axis of a patterned tube is curved, while the circumferential rings remain almost undeformed.Then, the displacements are independent of z, i.e.
that enables simplifying the governing Equation ( 6) as follows We have here two sets of ordinary differential equations: one uncoupled equation for u z and two coupled equations for u r and u θ , similarly to the case of expansion modes.The solution to the uncoupled equation is u , where C ða,cÞ 1 and C ða,cÞ 2 are unknown constants, i.e., has the same form as the corresponding displacement in Equation ( 16).The remaining two equations in (20) cannot, however, be directly solved analytically due to the strong coupling.The solution procedure is cumbersome, and thus we opt to continue with the numerical FEA.Another reason for this choice is that in practical relevant scenarios, required bending deformations can be large, violating the assumptions for slender beams that largely reduces the applicability of the analytical solution.

Intuition-Based Programmable Design
The qualitative description of the bending deformations and the partial separation of the governing Equation ( 20) suggest the possibility of extrapolating the programmable approach from Section 3.1.2to the case of bending modes.
Intuitively, it is straightforward that plane cross sections under a uniformly distributed loading can be obtained in tubular structures composed of identical rings.The arrangement of unit cells in a ring can then be chosen to induce axial bending, e.g., by introducing asymmetry along the circumferential direction. [26]Thus, if one varies the arrangement of unit cells along the circumferential direction (i.e., rows c 1 , c 2 , …) and keeps identical unit cells along the axial direction (i.e., columns a 1 , a 2 , …), as schematically shown in Figure 5a, the bending modes can be activated.
The bending direction, curvature, and level of deformation can be programmed by manipulating the number of "divergent" unit cells in the rings and the "contrast" in effective mechanical moduli of the unit cells.For example, the larger the difference in the k values for adjacent unit cells, the smaller the radius of the bending curve for the tubular axis.Figure 5b shows two intuition-based patterns for a tube to activate the first two fundamental bending deformation modes, B1 and B2.As can be seen, the intuition-based designs have expected deformed shapes that confirm the viability of the proposed approach.To program a specific bending mode more accurately, we solve an optimization problem as described in Section 3.4.1.

Numerical Analysis
The numerical validation of the described design approach is carried out by analyzing the deformed behavior of the tubes activated by pressure p 0 ¼ 5 kPa.We analyze tubes of the same dimensions and with comparable FE meshes as those described in Section 3.1.3.
Assuming that one end of a tube is fixed, and the other end is free, we first consider bending mode B1 of a uniform curvature.This mode can be obtained, e.g., in a homogeneous patterned tube with n ¼ 2.42, t c ¼ 0.03l, by replacing a single column of unit cells along the axial direction with the unit cells formed by more slender elements, e.g., n ¼ 2.42, t c ¼ 0.025l (Figure S4a, Supporting Information).The difference in the stiffness of the unit cells results in bending deformations.Alternatively, the B1 mode can be activated by varying Poisson's ratio and preserving identical stiffness in all the unit cells.For this, an axial column of unit cells with positive Poisson's ratio in a homogeneous tube (e.g., n ¼ 1.3, t c ¼ 0.0285l) can be replaced by those with negative Poisson's ratio (e.g., n ¼ 30, t c ¼ 0.032l).These two simple designs are suitable for inducing a large bending curvature corresponding to small bending deformations.
To increase the magnitude of the deformations, the two ends of the tubes can be fixed.The corresponding examples of the programmed designs for the B1 and B2 bending modes are shown in Figure 5b for t c ¼ 0.03l and the a sequences of n values specified in Table S2, Supporting Information.In this B1 design, the unit cells along the circumferential direction have monotonically increasing n values, corresponding to auxetic (n ¼ 1.3) and nonauxetic (n ¼ 30) behavior.In the B2 design, the majority of unit cells are auxetic (n ¼ 1.3) with two strips of nonauxetic unit cells (n ¼ 30) (see Table S2, Supporting Information, for more details).The bending deformation can be amplified by increasing the applied pressure magnitude.We note that a similar approach can be used to induce bending deformations under tensile load (Figure S4b, Supporting Information).

Twisting Modes
These modes imply the twist of a tube along its axis with resultant shear stresses in plane cross sections acting perpendicularly to the radius. [26,44]Even in the case of a simple twist, the displacements (1) preserve the dependence on the two coordinates, θ and z, so that the governing Equation ( 6) are strongly coupled.Because of this, we analyze twisting deformation modes numerically, which also helps avoid simplification assumptions restrictive for practical applications.
The mechanisms inducing the twist of a tube suggest that twisting deformations can be obtained by varying unit cells along both the circumferential and axial directions, e.g., in a spiral manner as shown in Figure 6a.This enables deriving an intuition-based design approach for twisting modes in the patterned tubes.Specifically, twisting deformations can be activated by manipulating the mechanical properties of the unit cells and changing the number and arrangement of the unit cells along a spiral (Figure 6a).The density of the spiral turns governs the order of a twisting mode.The subsequent steps of the design procedure are identical to those for the bending modes described in Section 3.2.1.
We illustrate the implementation of this procedure by numerically analyzing twisting deformations in tubular structures with two fixed ends under pressurization.The numerical model for the patterned tube is similar to that described for expansion modes in Section 3.1.3.Design T1 (Figure 6b), for example, contains a single spiral of non-auxetic unit cells (n ¼ 30) embedded in the homogeneous structure formed by auxetic unit cells (n ¼ 1.3).An inverse design, i.e., a non-auxetic spiral in the auxetic structure, is also possible (design T1' in Figure S5, Supporting Information) and enables a twist in the opposite direction.In these two designs, the structural stiffness governed by t c ¼ 0.03l remains unchanged.More complex twisting modes are obtained by increasing the number of spirals as illustrated by design T2.The structure T2 contains two parallel spirals of nonauxetic unit cells (n ¼ 30, t c ¼ 0.03l) embedded in the auxetic unit cells (n ¼ 1.3, t c ¼ 0.03l) (the values of n are given in  S2, Supporting Information.S5, Supporting Information, and also reveals the opposite rotation direction to T2.The fine-tuning of the twisting deformations can be implemented by adjusting the values of t c , in a similar way as for the bending modes.

Figure S6, Supporting Information). The corresponding inverse design T2' is given in Figure
The understanding of the mechanisms behind the deformations in the fundamental modes enables us to generalize the intuition-based design approaches in order to obtain more complex deformations.To demonstrate this, we show three exemplary designs (Figure S7, Supporting Information), which combine expansion and bending (M1), expansion and twisting (M2), and bending and twisting (M3).

Algorithm-Based Programmable Design Approach
The developed intuition-based programmable design approaches have illustrated the shape-morphing potential of tubular structures composed of rose-shaped unit cells.However, these approaches have limitations, especially for complex deformation modes, as they rely on manual adjustments or expert knowledge.To address these shortcomings, we propose an optimizationbased approach that is promising to become a universal design strategy for programmable shape morphing.This approach involves setting an objective function and using one of numerous available optimization algorithms [29,31,48] to quantitatively and automatically manipulate the arrangement of unit cells with the aim to fit a desired shape.
As the arrangement of unit cells for complex deformations, such as bending, twisting, or especially mixed modes with local shape variations, might be not evident and complex to predict following the intuition-based approach, it is valuable to employ intelligent optimization algorithms, such as the genetic algorithm (GA). [31]te that unlike conventional techniques that rely on gradient information, the intelligent algorithms operate based on computation, a factor directly related to the number of parameters involved.
To optimize efficiency and reduce computational costs, it is reasonable to tailor the parameters according to the characteristics of a target shape.For instance, for symmetric shapes, we can significantly cut down computational costs by exploiting structural symmetry, thus reducing the number of parameters at least by half.
In this study, we use both manual GA coding and Isight-a commercial software providing tools for parameter optimization, such as design of experiment (DOE), approximation, and optimization-to speed up the optimization procedure and the design process.The link between our numerical simulations in COMSOL and parametric optimization in Isight is implemented through MATLAB code, according to ref. [48] (Figure 7a).Specifically, the MATLAB code uses the parameters from Isight as input and transfers them to COMSOL to run the numerical simulations; the calculated results are then forwarded to Isight again using the MATLAB code.To obtain the shape fitting of a specific deformation mode, we consider a series of n values for each unit cell as optimization parameters for the objective function aimed to minimize the deviations between the shape of a deformed tubular structure and a target.

Shape Fitting of Expansion Modes
The general optimization-based design approach described above can be expressed mathematically as follows min FitV ¼ Here we aim to minimize fitness value FitV by finding an optimum distribution of n values.For the expansion modes, the FitV is calculated by summing up the square distances between the objective (target) diameter change, ObjD i , and the actual (individual) diameter change, IndD i , of the ith ring from i ¼ 1 to i ¼ A [31] with N in Equation ( 21) indicating the input n sequence generated by the algorithm.The continuous objective deformation is discretized in the FE model, and thus replaced by a piecewise variation of the diameter change for separate rings.The objective and actual shapes are normalized to [0,1] prior to calculations.To further reduce computational costs, we replace a 3D FE solid structural model of a patterned lattice tube with its shell counterpart.
Now we demonstrate the generation of the n sequence for algorithm-based programmable shape morphing of expansion mode E1.We consider a tube with 12 Â 12 unit cells, which implies a sequence of 12 n values in an optimized design.Given the mirror symmetry of the E1 shape with respect to the central cross section of a tube, we can reduce the number of design parameters from 12 to 6.Moreover, knowing the mechanical properties of the rose-shaped unit cells (Section 2.1), we set the n values for the unit cells at the start and end of the tube to be 30 and 1.3 in order to provide a good initial guess for the optimization algorithm.This further reduces the number of unknown n values to 4. Hence, we need to find the sequence ½n1, n2, n3, n4 that minimizes FitV in Equation (21).
There are two pathways to identify the optimal sequence of n values.One approach involves constructing a surrogate model utilizing the approximation component in Isight.We developed a quartic response surface model (RSM) based on a comprehensive database comprising over 1200 records of n sequences and their corresponding fitness values.Once the surrogate model was established, it yielded sequence ½n1, n2, n3, n4 ¼ ½10.0, 3.3, 3.0, 1.3 delivering a minimum fitness value of 0.07.Subsequently, we replicated the corresponding structural model in COMSOL and verified that the designed structure obtains the target E1 shape upon deformation, as depicted in Figure 7b.It is worth noting that this approach relies on a meticulously prepared database tailored to the specified objective shape.Additionally, the accuracy of the surrogate model is contingent upon the number and distribution of data points within the design space.
An alternative approach for generating the required n sequence involves the utilization of intelligent optimization algorithms within Isight, as depicted in Figure 7a.Here, we employ the multi-island genetic algorithm (MIGA) [31] to generate the four-value n sequence.Notably, the fitness value exhibits a steady decrease as the evolution process unfolds.To streamline computations and enhance efficiency, we constrain the range of n values within the interval [1.3, 10].This limitation is based on the observation that beyond n ¼ 10, the variations in Poisson's ratio and stiffness diminish significantly, as illustrated in Figure 3a.Therefore, restricting this range has limited impact on the shape-morphing effect.As a result, we successfully derive an n sequence [9.9, 2.4, 1.9, 1.4] with a notably low fitness value of 0.05.The deformed shape is presented in Figure 7c.
It is entirely feasible to extend these optimization algorithms to explore designs with a greater number of parameters, i.e., 12 or even more.However, enlarging the dimensionality of parameter space by including more parameters demands increased computational resources.It is also worth remembering that these optimization techniques may not consistently yield a globally optimal solution, as in general, they converge to a locally optimal solution.
Nonetheless, an advantage of the optimization-based approach is that even locally optimal solutions obtained by these methods can still fulfill design requirements to a significant extent.This versatility allows for a broader exploration of feasible designs within the given parameter space, despite potential limitations in finding an absolute global optimum.S3, Supporting Information.

Shape Fitting of Bending Modes
The goal of the optimization-based programmable design for bending modes is to find the sequence of n values minimizing fitness value FitV expressed as follows min FitV ¼ where FitV is calculated by summing up the square distances between the objective displacement ObjAx i and the individual displacement (IndAx i ) of the axis of the ith ring for i ¼ 1, : : : , A.
To illustrate the optimization process, we fit the shape of a deformed tubular structure with 12 Â 12 unit cells to that of the uniformly curved shape B1 (see the left of Figure 5b).To reduce computational costs, we again rely on the mode symmetry and the known mechanical properties of the unit cells and reduce the full sequence of 12 values for n to that with four values, ½n1, n2, n3, n4, similar to the case of the E1 mode analyzed previously.
We employed the radial basis function (RBF) model in insight, which utilizes a database comprising over 800 records, to construct a surrogate model.Upon completion of the surrogate model, we generated a sequence ½n1, n2, n3, n4 ¼½13.1, 9.2, 4.5, 4.0 with a corresponding fitness value of 0.15 for the objective shape B1.Next, we reproduced the corresponding arrangement of the unit cells in COMSOL and estimated the resulting deformed shape as shown in Figure 7d.The overall agreement between the calculated and target shapes is reasonable, confirming that the employed optimization approach can provide a viable solution for the quantitative and automated programmable shapemorphing design.A better fitting can be obtained by lifting the assumptions for the fixed values of n at the cost of increased computational resources.
The bending effect can be enhanced by replacing the consideration of an effective 1D structure formed by a collection of identical rings with the full-size design space for a 2D unfolded structure represented by a two-dimensional distribution of unit cells.
Shape fitting of twisting and mixed deformation modes with 3D deformation for the central axis requires a further increase in the design space and may require the analysis of longer tubes with a larger (than 12) number of unit cells along the axial direction.The corresponding studies are computationally heavy and go beyond the scope of this work.

Manufacturing and Experiments
To verify the shape-morphing capability of the rose-shaped tubular structures, we manufactured several samples from durable resin (E mat ¼ 1 GPa, ν ¼ 0.35), [45] which has excellent impact and wearable resistance, using a commercial SLA 3D printer Form3 (Formlabs Inc., U.S.).Each sample is composed of 12 Â 12 unit cells along the axial and circumferential directions, respectively.The size of the unit cell is l ¼ 10 mm; the wall thickness and out-of-plane thickness are t c ¼ 0.3 mm and h ¼ 1 mm, respectively.The structural dimensions are thus D ¼ 38.4 mm and L ¼ 130 mm complying with the size of the building platform of the printer (145 mm).
To maintain a circular shape in the undeformed state, the samples were 3D printed with the support of the same durable resin with a layer thickness of 0.05 mm (Figure S10, Supporting Information).The raw 3D printed samples are heated in a heating box at 60 °C for 60 min, and then the support is manually removed.To induce internal pressure loading, we used inflatable balloons placed inside the samples (Figure 8).
We also fabricated the same structures by 3D printing flat sheets, rolling them, and gluing the interface, similarly to the manufacturing procedure proposed in other works. [36]For this, we used an FDM 3D printer Ultimaker 3 (Ultimaker, the Netherlands) with PLA and TPU filaments.It appeared that the PLA flat sheets could not be rolled into tubes of a circular cross section due to the high rigidity of PLA (Figure S10d, Supporting Information).The TPU samples were flexible enough (Figure S10e, Supporting Information) to be rolled in a tube; however, the limited build area and the printing resolution did not allow a sufficient amount of unit cells to achieve the desired shape-morphing performance.For both materials, the rigid glued connections restrict deformations and influence the shape of the deformed samples, substantially limiting or even disabling shape morphing.Besides, these connections can easily break, even for T-shape designs with an enlarged contact area (Figure S10d-e, Supporting Information).Therefore, the one-step manufacturing solution with 3D printing of a tubular sample, i.e., SLA 3D printing with support, is preferred as ensuring good quality of the samples with required uniform mechanical properties and a circular crosssectional shape of a final structure.
The fixed structural ends are imitated by adding solid rings of 5 mm thickness at the ends of the samples (Figure 2c and 8).It enabled to avoid clamping [49] or gluing the ends, thus simplifying the testing procedure.
We made three samples with the E2, B1, and T1 patterns corresponding to the expansion, bending, and twisting deformation modes under pressurization (Figure 2c), respectively.(The specific parameters of the unit cell are given in Table S4 and Figure S6, Supporting Information, and the actual measured dimensions are given in the Supplementary Information.)One can see an excellent agreement between the predicted and observed shapes, considering the difference between the two fixed ends in the numerical models and not fully fixed ends in the experiments (see the bottom of Figure 8).This validates the applicability of the rose-shaped pattern to achieve shape morphing and the proposed design approaches developed to program pressure-induced deformations.

Discussion and Conclusion
In this work, we investigate the underlying mechanisms and propose programmable design approaches for patterned rose-shaped lattice tubes composed of auxetic and conventional rose-shaped unit cells.Our study encompasses the three fundamental deformation mode families and focuses on the practical applicability of shape-morphing structures, particularly under tensile loading and pressurization.We also estimate the effects of boundary conditions on the resulting deformed shapes.
We first formulate the general problem theoretically and then apply assumptions to simplify the analysis for each mode family.
Our analytical results are validated in numerical FE simulations and agree very well with those for the SLA 3D printed resinpatterned tubes deformed on inflatable balloons.We propose theory-based, intuition-based, and optimization-based design approaches for qualitative and quantitative shape morphing, respectively.We speculate that the optimization-based strategy can be applied to fit an arbitrary complex shape with local features or fully 3D deformations in different parts of a tubular structure if sufficient computational resources are available.

Unit Cell Geometry and Design Space
Through simulations, we showed that the deformed shape of a patterned tube can be controlled by varying a single design parameter of constituent unit cells, governing the transition from non-auxetic to auxetic behavior and vice versa, while local stiffness can be tuned by changing the in-plane dimensions of the unit cell elements.An opposite design approach, i.e., adjusting the local stiffness for shape morphing and varying Poisson's ratio for fine-tuning, is also possible for this pattern, as shown by our several examples. [18]nvoking more design parameters expands the design space and requires more advanced, as compared to the intuition-based, approaches for programmable designs, e.g., the optimization-based approach, which, however, requires high computational costs.
The use of the other parameters-unit cell size l and/or out-ofplane thickness h-for shape morphing is out of the scope of this work due to the limitations imposed by the chosen manufacturing technique, the base material, and the experimental tests.In particular, smaller unit cells require a finer printing resolution; larger unit cells result in either impractically short tubular structures or large samples that cannot be fitted on the build platform of our printer.Thicker tubular structures cannot be deformed by our inflated balloons.
Even for the limited part of the design space analyzed in this work, we have demonstrated the feasibility of obtaining diverse expansion, bending, and twisting modes, as well as combinations of them.The literature proposes various methods to explore multiparameter design space and to generate programmable designs. [29,31,48,50]In this work, the approach based on the analytical solution is proven to be the most efficient in terms of fitting accuracy and computational costs; however, it is applicable only to the simplest (expansion) deformation modes.The intuition-based approach offers straightforward guidelines for a programmable design but heavily relies on the experience and trial-and-error adjustments.The optimization-based approach is applicable in any scenario, yet accurate shape fitting can be obtained only at substantial computational costs.In practice, a suitable approach can be chosen depending on application requirements and available computational resources.

Manufacturing Techniques
Our shape-morphing design strategies are, to some extent, scalable and material independent because the effective mechanical behavior of rose-shaped patterns is mainly governed by the geometric features of the unit cells (Figure 3a) rather than structural sizes or material properties.This means that the proposed and other programmed designs can be translated into applications on different scales by exploiting various manufacturing techniques.For example, melt electro-writing, [19] direct laser writing, [51] two-photon lithography, [52] and micro-pSLA 3D printing can be used to produce structures on the micro-and even nanoscales, which can be used, e.g., in scaffolding [11,19] or light-weight load-bearing applications. [53]The superior printing resolution of these techniques can also improve shape-morphing functionality by scaling down the unit cell size and increasing the number of unit cells.

Loading and Boundary Conditions
We have shown that loading and boundary conditions influence deformed shapes.Different expansion, bending, and twisting modes can be activated by specific arrangements of unit cells when a tube is subjected to tensile or pressure loading.The loading and boundary conditions can be adjusted to achieve desired levels of deformation in practically relevant scenarios.For example, if significant deformation is required in the radial direction combined with unrestricted axial extension, tensile loading is preferred.For pressure-activated deformations, e.g., in ballooninflated medical stents, [33] proper shape matching can be achieved by selecting boundary conditions and tuning in-plane thickness t c of unit cells. [18]To activate bending and twisting modes, we have mainly considered pressure-induced deformations, which were mainly studied for inflatables, [23,26] and linked the mechanics of these modes to the mechanical properties of the unit cells that was previously done only for tensile-loaded designs. [11,21]he combination of shape morphing and tailorable mechanical properties is beneficial for practical applications, e.g., in the design of medical stents and air-pumped robotic fingers, [34,54] where shape morphing must be accompanied by sufficient structural rigidity to enable required functionalities.In the future, it is worth analyzing the influence of additional stimuli, e.g., temperature, that are of importance for practical scenarios and can be enabled by emerging 4D printing techniques. [10,55]

Application Potential
Practical use of the proposed and other programmable designs can be implemented by considering specific application scenarios and complying with additional requirements on the mechanical behavior of patterned tubes.For example, clinical use of medical stents requires their high stretchability, bending flexibility, and radial strength. [56]Such stents, aimed to keep the structural shape resistant to vascular restenosis, are made of metals and can be deployed using balloon pressurization or tension. [57]any current designs of the stents have straight ligaments, [24,38,58] which restricts their flexibility and complicates maintaining proper comfortability and navigation during placement. [34]The conceptual design of the stents, e.g., by employing the proposed rose-shaped pattern (Figure S11a, Supporting Information), could offer additional programmability by preserving the mentioned characteristics.The actual design of patientspecific stents is, however, beyond the scope of this work.Note that other medical applications of programmable tubular structures include bilayer 3D cannula aimed at improving their clinical applicability. [59]nother promising application of patterned tubular structures is the fingers of soft robotic grippers, where shape matching must be accompanied by sufficient structural stiffness to enable grasping functionality.This can be achieved, e.g., by programming the bending deformation mode for our tubular design, when one structural end is fixed and the other end is free.The bending deformation can be actuated by the pressure difference between the internal and external surfaces, or by distributed expansion actuators like shape memory wires [60] As a starting point, one can use, e.g., the B1 design for the first bending mode shown in Figure S11b, Supporting Information, and further modify it to adjust the mechanical properties to desired values.Note that in this application scenario, fine-tuning the shapemorphing functionality is not required, as the objects to be grasped can have different shapes, varying dimensions, or orientations.Therefore, the final designs of the grippers can have locally compliant behavior to improve the grasping of objects with dimensions within a predefined range.
Twisting deformation modes have the potential in compact pressure sensors. [61]Combinations of fundamental deformation modes are promising in the fields of advanced medical stents and soft robotics, where highly localized deformations or other mechanical responses are required.
Finally, note that there is still a large gap between the possibilities of conceptual designs of patterned tubular structures and their practical use that can be bridged by overcoming multiple manufacturing issues of current technologies through further developments and improvements in additive manufacturing and reducing related production costs. [3,62]

Figure 1 .
Figure 1.Schematics of a patterned tubular structure of radius r 0 and the relation between the global (the structural level) cylindrical coordinates and the local (the unit cell level) Cartesian coordinates.

Figure 2 .
Figure 2. Geometry of a tubular structure and schematics of the considered loading conditions.a) A tubular structure with a random arrangement of roseshaped unit cells.b) Unit cells with n ¼ 1.3 and n ¼ 30.c) The loading and boundary conditions analyzed in this work (loading directions are shown in blue).

Figure 3 .
Figure 3. a) The effective Poisson's ratio and relative stiffness of the rose-shaped unit cells forming a tubular structure vs n and t c .b) The variation of design parameter k with respect to n and t c .

Figure 4 .
Figure 4. Theory-based programmable design approach and examples of expansion deformation modes for rose-shaped patterned tubes: a) The schematics for the design strategy-the unit cells vary along the axial direction and are identical along the circumferential direction.b) The examples of two expansion modes predicted theoretically and numerically under 20% strain (the deformation scaling factor is 1).The corresponding sequences of the n values can be found in TableS1, Supporting Information.

Figure 5 .
Figure 5. Intuition-based programmable design approach and examples of bending deformation modes for rose-shaped patterned tubes.a) The schematics of the design strategy-the unit cells are identical along the axial direction and vary along the circumferential direction.b) Two bending deformation modes activated by pressure loading of 5 kPa (deformation scaling factor of 1).The corresponding sequences of the n values for B1 and B2 can be found in TableS2, Supporting Information.

Figure 6 .
Figure 6.Intuition-based programmable design approach for twisting deformation modes for rose-shaped patterned tubes.a) The schematics for the design strategy-the unit cells are identical everywhere except for the diagonal lines.b) Two twisting modes activated by pressure p 0 ¼ 5 kPa (deformation scaling factor of 1): T1 with a single twist and T2 with a double twist.The corresponding unit cell arrangement can be found in Figure S6, Supporting Information.(The deformation modes T1 0 and T2 0 with the inverse unit cell arrangement to T1 and T2 are shown in Figure S5, Supporting Information).

Figure 7 .
Figure 7. Optimization-based programmable design approach for shape fitting of an arbitrary deformation mode.a) Schematic diagram of the approach.b) Example of an optimized shape fitting for expansion E1 mode yielded by the established RSM surrogate model.c) Example of an optimized shape fitting for E1 mode generated by the MIGA optimization algorithm.d) Example of an optimized shape fitting for bending deformation B1 yielded by the established RBF surrogate model.The corresponding sequences of the n values for these designs can be found in TableS3, Supporting Information.

Figure 8 .
Figure 8. 3D printed durable resin samples of the rose-shaped patterned tubular structures.a) Balloons are put inside to inflate the samples (The parts of the balloons outside the structures are not shown.)b-d) The simulation vs experimental results for the fundamental deformation modes E2, B1, T1, also shown in Figure 4-6.