Self-locking and stiffening deployable tubular structures

Lee, Ting-Uei; Lu, Hongjia; Ma, Jiaming; Ha, Ngoc San; Gattas, Joseph M.; Xie, Yi Min

doi:10.1073/pnas.2409062121

Self-locking and stiffening deployable tubular structures

Ting-Uei Lee https://orcid.org/0000-0002-4003-8172, Hongjia Lu https://orcid.org/0000-0003-2777-3321, Jiaming Ma https://orcid.org/0000-0003-4330-1623, +2 , Ngoc San Ha https://orcid.org/0000-0003-4718-3935, Joseph M. Gattas https://orcid.org/0000-0002-0878-583X [email protected], and Yi Min Xie https://orcid.org/0000-0001-5720-6649 [email protected]Authors Info & Affiliations

Edited by Glaucio H. Paulino, Princeton University, Princeton, NJ; received May 7, 2024; accepted August 30, 2024 by Editorial Board Member John A. Rogers

September 27, 2024

121 (40) e2409062121

https://doi.org/10.1073/pnas.2409062121

Commentary

November 11, 2024

Unfurling packed-flat tubes into self-locked stiff structures

Damiano Pasini

PDF/EPUB

Significance

Origami-inspired mechanisms can transform flexible sheets into lightweight and efficient structures. Inspired by curved-crease origami, here, we design a deployable tubular structure featuring geometric self-locking and stiffening effects using simple shell components. This tube challenges the fundamental trade-off between expandability and load-bearing capacity in deployable structures. Despite its high material deformability, the deployed state provides high stiffness against external loading through internal stiffeners. These stiffeners, which bend and undergo pseudofolding, are confined during deployment. They induce elastic shell buckling and create an energy barrier that facilitates a snap-through transition. Our results demonstrate that the proposed tube opens different possibilities for improving existing deployable mechanisms and creating innovative structural designs, impacting diverse applications where deployable tubular structures are employed.

Abstract

Deployable tubular structures, designed for functional expansion, serve a wide range of applications, from flexible pipes to stiff structural elements. These structures, which transform from compact states, are crucial for creating adaptive solutions across engineering and scientific fields. A significant barrier to advancing their performance is balancing expandability with stiffness. Using compliant materials, these structures achieve more flexible transformations than those possible with rigid mechanisms. However, they typically exhibit reduced stiffness when subjected to external pressures (e.g., tube wall loading). Here, we utilize origami-inspired techniques and internal stiffeners to meet conflicting performance requirements. A self-locking mechanism is proposed, which combines the folding behavior observed in curved-crease origami and elastic shell buckling. This mechanism employs simple shell components, including internal diaphragms that undergo pseudofolding in a confined boundary condition to enable a snap-through transition. We reveal that the deployed tube is self-locked through geometrical interference, creating a braced tubular arrangement. This arrangement gives a direction-dependent structural performance, ranging from elastic response to crushing, thereby offering the potential for programmable structures. We demonstrate that our approach can advance existing deployment mechanisms (e.g., coiled and inflatable systems) and create diverse structural designs (e.g., metamaterials, adaptive structures, cantilevers, and lightweight panels).Weanticipate our design to be a starting point to drive technological advancement in real-world deployable tubular structures.

Deployable tubular structures possess a remarkable ability to transform from a highly compact state to a large functional form. This feature has been widely adopted in engineering and scientific applications, including temporary structures (1), deployable booms (2), medical devices (3), and soft robotics (4). A correspondingly wide range of deployment mechanisms have been developed to achieve the system transformation. These can be broadly categorized as rigid or compliant. Rigid mechanisms transform through the translation and rotation of undeformed components, for example, sliding motion in telescopic tubes (5) and hinge rotation in rigid origami tubes (6). In contrast, compliant mechanisms rely on deformations of materials to transform, for example, soft material coiling in rollable systems (7), membrane expansion in inflatable systems (8), and elastic shell deformation in nonrigid origami tubes (9). Notably, compliant mechanisms, compared to rigid mechanisms, promote simpler, lighter, and more efficient deployable tubular structures due to their inherent material deformability.

Designing deployable structures comes with the challenge of balancing conflicting performance requirements: They must be expandable, and once deployed, they need to become stiff and stable (10). An ideal feature of deployable structures is self-locking, which ensures that a specific shape is locked in place without extra mechanisms or external interventions. Two common design strategies for self-locking are collision and energy barrier methods. Collision methods utilize the geometric or kinematic interaction of components to block further motion beyond a specific point (11, 12). This ensures stability when deployed, while allowing for easy collapsibility when the deployment direction is reversed. Energy barrier methods create snap-through transitions to switch between stable states, often induced by large deformations that overcome a minimum energy threshold (13–15). Once stable, the energy barrier provides resistance in the deployment direction to maintain a specific shape.

For deployable tubular structures, high expandability is achieved through compliant deformations of materials, such as “coiling with bistable tape springs” (16) and “inflating through growth” (17). However, attaining enhanced stiffness against wall loading while preserving a high degree of material deformability, as suggested in ref. 10, is not only challenging but fundamentally conflicting. Nevertheless, some studies have incorporated origami patterns, such as Kresling (18) and Yoshimura (19) patterns, into the tube walls to improve stiffness in stable states, suggesting that origami-inspired techniques could enhance the performance of deployable tubular structures. From a different perspective, introducing internal stiffeners in tubular structures is the most straightforward method to achieve high stiffness against wall loading (20). However, such tubes are designed for static applications and are not yet deployable.

In this study, we develop a deployable tubular structure that achieves a high stiffness in its deployed state using an innovative self-locking mechanism that combines features of both collision and energy barrier methods (Fig. 1A). The mechanism is designed using simple shell components, including two external skins and internal diaphragms, as distinct from current approaches using complex patterned sheets (1, 18, 19). During deployment, the diaphragm motion is confined by collision with the external skin, inducing elastic shell buckling and the consequential energy barrier. This barrier is surpassed when the tube is fully deployed, as contact conditions ease and the diaphragms can unbuckle to attain a braced tubular arrangement locked against all directions, including the deployment direction (Movie S1). In this arrangement, diaphragms function as internal stiffeners, partitioning and stabilizing the circular tube, while the collision of components prevents reverse motion, thereby securely maintaining the tubular form. Interestingly, this tubular form has high stiffness in a specific radial direction and variable stiffness in other directions. Therefore, the deployment process is not only self-locking but also self-reinforcing (stiffening). Given these useful features, our proposed design has significant potential applications across various fields where deployable tubular structures are widely employed, including biomedical, aerospace, construction, and robotics.

Fig. 1.

Design and characteristics of the proposed deployable tubular structure. (A) Physical demonstration of the deployment process from a flat, undeployed state into a locked, tubular deployed state. (B) The skin component undergoes elastic shell bending when boundary edges are compressed. (C) A compliant mechanism is formed by integrating a diaphragm component into the skin, enabling the diaphragm to bend and rotate around the hinge line. Note that the skin and diaphragm axes are parallel in the flat state and become perpendicular in the end state. (D) The self-locking mechanism is created by introducing an opposing skin into the compliant mechanism, fundamentally altering the elastic strain energy behavior by introducing an energy barrier. (E) Without the opposing skin, the diaphragm is unconfined and unbuckled (*Left*). However, under a confined boundary condition created by the two skins, the diaphragm undergoes elastic shell buckling (*Right*). (F) The idealized motion of the compliant mechanism (pure bending) shows that geometric interference is created due to the changing confinement, influencing the extent of buckling. (G) While curved-crease origami is created from a continuous sheet with a crease (*Left*), our design (*Right*), inspired by such origami, includes an additional hinge layer to facilitate a pseudofolding motion, making it “curved-crease origami-inspired.”

Movie S1.

Transformation process of the proposed deployable tubular structure. The undeployed state consists of flattened elastic shells. Upon compression of the boundary edges, the internal stiffeners exhibit motion inspired by curved-crease origami and undergo elastic shell buckling in a confined condition. This action leads to a snap-through transition to a tubular deployed state. Remarkably, the deployed state is self-locked, allowing it to maintain its shape even after the boundary compression is released.

Self-Locking Mechanism with Geometric Interference

The proposed self-locking mechanism relies on geometric interference between skin and diaphragm components. The skin, if deployed from compression of its boundary edges, undergoes elastic shell bending (Fig. 1B). A diaphragm is connected to the skin at an edge hinge line to form a compliant mechanism inspired by curved-crease origami (Fig. 1C). Specifically, this mechanism causes motion similar to the folding behavior observed in curved-crease origami; when the skin is bent, the diaphragm bends and rotates around the hinge line. The end state is set when the diaphragm axis is perpendicular to the skin axis. Subsequently, attaching an opposing (second) skin gives a circular tube representing the deployed (target) state of the proposed self-locking mechanism (Fig. 1D), internally partitioned by the diaphragm.

The diaphragm curvature and “curved folding motion” of the idealized compliant mechanism (Fig. 1 E and F) are initially simulated as the geometric reflection of the skin, following the mirror reflection method (21), detailed in SI Appendix. The proposed self-locking mechanism cannot be strictly classified as “curved-crease origami” because it is not made from a continuous sheet with creases. Instead, it is a “curved-crease origami-inspired” compliant mechanism (22). As shown in Fig. 1G, an additional hinge layer is used to connect the discrete diaphragms to the continuous skins, where its translational fixity and rotational freedom facilitate a pseudofolding motion in the diaphragm when the skin is being bent. During the motion, the skin and diaphragm axes are first parallel (flat state), and then, the diaphragm axis is rotated until the two axes are perpendicular (deployed state). In this process, the diaphragm height

h_{d}

(the closest distance from the skin axis to the tip of the diaphragm) consistently exceeds the overall depth created by the two bent skins

2 h_{s}

,

h_{d} > 2 h_{s}

. At the flat and deployed states only,

h_{d} = 2 h_{s}

.

In a physical tube, the diaphragm must be confined within the closed boundary condition of

2 h_{s}

, meaning that

h_{d}

cannot exceed

2 h_{s}

(Fig. 1E). This boundary condition thus induces geometric interference, causing elastic shell buckling of the diaphragm (double-curved) rather than pure bending in the idealized compliant mechanism (single-curved). The extent of buckling varies as the boundary condition continuously adjusts; in Fig. 1F, the shaded area represents the required diaphragm confinement, corresponding to changing the geometric interference condition that influences buckling.

Once the deployed state is reached, the geometric interference between components eases. The diaphragm becomes unconfined, allowing it to unbuckle to the single-curved form of the idealized compliant mechanism. Consequently, collisions between components lock the deployed state and no external compressive loads can reverse the diaphragm buckling. Further compression along the deployment axis (

x

) simply causes additional bending following the compliant mechanism. Compression along the perpendicular axis (

z

) induces axial loading in the diaphragm, enabling it to act as an internal stiffener.

The elastic strain energies of the idealized compliant mechanism (bending only) and simulated self-locking mechanism (bending and buckling) are shown in Fig. 1 C and D, respectively. With the attachment of the second skin, diaphragm energy is sharply increased due to buckling. However, the interaction between the reducing geometric interference and increasing shell curvature relaxes the diaphragm buckling, ultimately snapping through to the unbuckled state. Consequently, the total elastic strain energy creates an energy barrier, establishing two stable states: prebuckling (undeployed) and postbuckling (deployed).

Deployment to Target Tubular Form

In early-stage experiments, we utilized flat components and observed that the deployment occasionally failed to achieve the target tubular form (Fig. 2A). This failure arises from the kinematic bifurcation in the compliant mechanism, as the flat diaphragm possesses bending instability (14). When the failure occurs, the diaphragms bend in the same direction as the skins. To prevent this, components can be precurved to set the intended bending direction such that undeployed and deployed states become stressed and stress-free, respectively. Here, the stress-free curvature of the component surfaces is a single-curved “arc” designed for the deployed state, with fabrication details provided in Materials and Methods.

Fig. 2.

Deployment to target tubular form and simulation of successful deployment. (A) The deployment may fail using flat components, attributed to the kinematic bifurcation in the compliant mechanism induced by bending instability. Using precurved components is an effective strategy to achieve the target tubular form. (B) The EXP deployment test involves compressing undeployed tubes between two rigid plates. Notably, of the 16 tests conducted (with different tubes), 15 achieved asymmetric buckling in all three diaphragms (S-S-S), while the remaining one exhibited symmetric buckling in the central diaphragm (S-V-S). (C) The two fundamental buckling modes from free edge indentation in the diaphragm are denoted as V and S, respectively, based on their shape. (D) The triggering conditions for the buckling modes are determined by the confinement: Symmetrical confinement results in the V mode, while asymmetrical confinement leads to the S mode. Here, V-V-V, S-V-S, and S-S-S buckling combinations represent “idealized,” “possible,” and “expected” diaphragm behaviors, respectively. (E) The assembly (Steps 1 and 2) and deployment (Step 3) processes are simulated using a multistep finite element (FE) analysis. (F) FE simulation results of buckling combinations (V-V-V, S-V-S, and S-S-S). These combinations are achieved following the same Steps 1 and 2, with different symmetrical conditions applied in Step 3. Combining (I) the elastic strain energy with (II) the frictional dissipation energy gives (*III*) the total internal energy.

Using precurved components, we fabricated and compressed 16 different undeployed tubes between rigid plates in an experimental (EXP) deployment test (see Materials and Methods for setup details), linking the applied force to the generated deformation (Fig. 2B). Each tube featured three diaphragms; adjacent diaphragms were alternately attached to the top and bottom skins, with three diaphragms selected as the minimum number necessary to provide periodic confinement to the central diaphragm. We observed that diaphragms can form both asymmetric and symmetric buckling modes during a successful deployment (Fig. 2C). These modes (denoted as S and V in our study, respectively, based on their shape) are similar to those observed as fundamental phenomena in cylindrical shell edge buckling from free edge indentation (23). A distinction in the buckling progression of the diaphragm is that the edge loading and hinged boundary conditions, from skin contact and attachment, respectively, change during deployment with the varying skin curvature.

Among these EXP tests, 15 exhibited asymmetric buckling in all three diaphragms, denoted as an S-S-S buckling combination. One showed an S-V-S buckling combination, whereby the central diaphragm buckled symmetrically. We find that the triggering conditions of the V or S buckling modes in the diaphragm are principally controlled by the confinement conditions created by the skins. Specifically, symmetrical confinement leads to the V mode, while asymmetrical confinement results in the S mode (Fig. 2D).

In a physical tube deployment, numerous factors can disrupt the symmetrical condition, including imperfections in fabrication (e.g., misalignment in the hinge layer) and actuation (e.g., uneven compression at boundary edges). To understand the relationship between diaphragm confinement, buckling mode manifestation, and mechanism energy barrier, this study numerically investigates V-V-V, S-V-S, and S-S-S buckling combinations, corresponding to “idealized,” “possible,” and “expected” diaphragm behaviors, respectively.

Simulation of Assembly and Deployment

A multistep finite element (FE) analysis is performed to simulate the proposed deployable tubular structure with precurved components (Fig. 2E). This sequential approach aligns with physical procedures, spanning from assembly (Steps 1 and 2) to deployment (Step 3). Step 1 flattens the compliant mechanisms through pulling; Step 2 combines the deformed parts, triggering diaphragm buckling through contact; Step 3 deploys the tube under displacement control (see Materials and Methods for details).

By controlling the symmetrical conditions in FE simulations (Materials and Methods), we have successfully simulated V-V-V, S-V-S, and S-S-S (Movie S2). As shown in Fig. 2F, these combinations are achieved following the same assembly process to assume equivalent predeployment, starting from

b = 30

mm. Different symmetrical conditions are applied only during deployment, resulting in the same target state and returning to

b = 30

mm. Our simulations reveal that these combinations produce (I) elastic strain energy (from shape change) and (II) frictional dissipation energy (from geometric interference). V-V-V and S-S-S show upper and lower energy behaviors, respectively, because of their strict and relaxed symmetrical conditions, with S-V-S showing an intermediate energy behavior. All combinations preserve two stable states, evidenced by the energy barrier in elastic strain energy.

Movie S2.

Finite element simulation of the proposed deployable tubular structure with pre-curved components. The simulation follows a sequential process, spanning from assembly to deployment. In assembly, components are flattened through pulling and are subsequently combined, triggering diaphragm buckling through contact. The deployment is demonstrated by compressing the tube between rigid plates under displacement control. We control the symmetrical conditions to generate different buckling combinations in the three diaphragms, including S-S-S (expected), S-V-S (possible), and V-V-V (idealized).

The frictional dissipation energy shows nonlinear trends, confirming continuous changes in geometric interference during deployment. Notably, although both S and V modes are double-curved, the S mode has only one side of the diaphragm’s free edge contacting the skin (asymmetrical), while the V mode has both sides (symmetrical), resulting in lower and higher friction, respectively. At the target state, the diaphragm shape must become single-curved and symmetrical. Therefore, the S mode requires a larger shape transformation than the V mode, causing a sudden snap-through transition toward the late deployment stage. In contrast, the V mode undergoes a relatively smoother transition throughout the deployment. Consequently, S-V-S and S-S-S (i.e., combinations with the S mode) exhibit a delay in snap-through transition, a more rapid drop in strain energy, and a sudden increase in friction dissipation energy compared to V-V-V.

For (III) total internal energy, the barrier is amplified by frictional effects, increasing the threshold of the snap-through transition. The left-side minimum (square marker) represents the initial stable equilibrium, corresponding to a relaxed state postassembly (the undeployed state). In this state, components are near-flat, held together by friction, preventing self-relaxation toward their precurved forms. Additionally, the snap-through point, identified as the peak of the barrier (circle marker), shows a transition from energy storage to release. This transition drives the deployment toward the right-side minimum (star marker), which is the deployed state. The position of the snap-through point reveals that the expected S-S-S buckling behavior has the lowest energy transition of the investigated buckling combinations, consistent with EXP observations of S-S-S being the most common mechanism deformation behavior.

Comparison between EXP and FE Deployment Tests

Our EXP results are broadly consistent with FE simulations, where the similarity is first observed in the buckling progression of diaphragms (Fig. 3A) and is further extended to the force–displacement responses (Fig. 3 B and C).

Fig. 3.

Comparison of the experimental (EXP) deployment test with FE simulations. (A) The buckling progression of the central diaphragm across both EXP and FE corresponds well. (B) Force–displacement responses for the S-V-S buckling combination are presented with gray (EXP) and green (FE) curves. We reveal that the deployment features three zero-force stages. These stages, with the *Left* and *Right* sides of the snap-through point, correspond to energy storage and release, respectively. (C) Force–displacement responses for the S-S-S buckling combination are presented with gray curves (for the 15 EXP tests) and a red curve (for the FE result). (D) During energy release, simultaneous snap-through transitions of diaphragms show a single negative peak, indicating that all three diaphragms are transitioned simultaneously. Sequential snap-through transitions exhibit multiple negative peaks as diaphragms transition independently.

In Fig. 3B, we show that the deployment features three zero-force stages: initial stable equilibrium, unstable snap-through point, and stable deployed state. Their locations determine the durations of energy storage and release based on the Left and Right sides of the snap-through point, respectively. Response variations between FE simulations and EXP results can be observed on both sides (Fig. 3 B and C). On the Left (energy storage), EXP results show a lower positive peak, primarily due to minor misalignment and gaps at curved hinges and edge connections. These manual fabrication defects, which relax the confinement condition and the interaction between the skin and diaphragm components, collectively soften the tube. However, such defects are necessary to accommodate assembly tolerance and differ from the well-aligned and zero-gap assumptions used in FE simulations.

On the Right (energy release), the measured negative force suggests that the tube is pulling the rigid plates rather than being compressed by them, thereby confirming the existence of the snap-through transition. EXP results show variations in the number of negative peaks, based on either simultaneous or sequential buckling shape change in each of the three diaphragms (Fig. 3D). A single peak indicates simultaneous snap-through transitions of all three diaphragms; two peaks suggest that one diaphragm transitions separately from the other two; three peaks indicate that each diaphragm transitions independently. Conversely, all FE simulations possess only one negative peak, as the diaphragms are simultaneously deformed in an idealized computational environment. These variations show that the force magnitude required to actuate the tube mechanism is sensitive to imperfections, but the overall snap-through transition behavior is not.

Simplified Deployment Prediction

High-fidelity FE simulations need to account for a wide range of physical factors, including nonlinear deformations, geometric collisions, contact forces, frictional effects, and dynamic responses. However, the complexity and interdependence of these factors make it difficult to definitively identify key drivers of successful deployment using FE simulations. To better isolate these drivers, rather than progressively excluding a range of factors, we present a simplified analysis model using a spring network to simulate elastic buckling shape transitions (Fig. 4A), detailed in SI Appendix.

Fig. 4.

Deployment prediction using a simplified analysis model. (A) This model is employed to capture elastic buckling shape transitions, including both S and V modes. (B) It is developed based on different spring systems, assuming that skin deformation is single-curved, and diaphragm deformation is double-curved. (C) Simplified prediction results align closely with FE simulations, including V-V-V, S-V-S, and S-S-S buckling combinations. (D) The simplified model is used to perform a parametric study on different stress-free curvatures to examine their impact on the expected S-S-S deployment. We show that the diaphragms are the primary contributors to the energy barrier, regardless of how skins and diaphragms relieve prestress.

Contact interactions are managed purely by geometric constraints, excluding contact forces and the consequent frictional effects. To stabilize a compliant structure at rest without friction, where numerical simulations often necessitate artificial damping (24), our model uses sequential optimization. This approach identifies intermediate states that represent minimum energy configurations by treating each state as a separate optimization problem. With reference to Fig. 4B, skin deformation is simplified to a single-curved profile controlled by “green” springs, while diaphragm deformation, remaining double-curved, is managed by superimposed “red,” “blue,” and “gray” springs to control elastic deformations in different directions. Bending and stretching energies are generated from torsional and line springs, respectively, with spring properties calibrated based on their material and effective surface areas to reflect actual shell behaviors (25, 26). Collectively, our model is strategically designed to approximate the global behavior of elastic shape changes, reducing the sensitivity to the instabilities associated with high-resolution local deformations (27).

Despite the simplifications, our model still demonstrates good correspondence with FE simulations, effectively capturing key deformation (Movie S3) and energy characteristics (Fig. 4C). Given that our model primarily focuses on geometric aspects, this finding suggests that successful deployment is predominantly driven by geometrical interference. Furthermore, using the simplified model, we have conducted a parametric study on different stress-free curvatures to examine their impact on the expected S-S-S deployment (Fig. 4D). Our findings reveal that diaphragms are the primary contributors to the energy barrier, exhibiting consistent energy behavior due to geometrical interference, regardless of how skins and diaphragms relieve prestress. We also demonstrate that the magnitude of the energy barrier within the total elastic strain energy can be adjusted by altering the relative energy contributions from the skins and diaphragms, suggesting the potential for self-deployment through the strategic elimination of this energy barrier.

Movie S3.

Deployment simulation using a geometry-based simplified model grounded in a spring network. It successfully replicates the buckling combinations seen in finite element simulations (S-S-S, S-V-S, and V-V-V). Therefore, successful deployment is confirmed mainly attributed to geometric interference between the skin and diaphragm components.

Toward Innovative Deployable Tubular Structures

We suggest two innovative applications to advance existing deployment mechanisms. The first is a coiled system (Fig. 5A), whereby a flattened tube is gradually pulled through an annulus to easily deploy and lock into the final deployed state (Movie S4). Notably, the deployment process is self-reinforcing, as diaphragms lock and stiffen the circular tube automatically after passing through the annulus. This stands apart from traditional coiled systems with hollow tubes, which rely on material properties to achieve the necessary stiffness, meaning that their tube geometry is fundamentally limited in structural capabilities (28, 29). The second is an inflatable system (Fig. 5B), whereby a flattened tube is sealed with two rubber caps. Inflation air pressure is then used to fully deploy the tube, with the self-locking configuration maintaining the tube state when pressure is released (Movie S5). This is similar to the “inflatable shelter” approach shown in ref. 30 and differs fundamentally from traditional inflatable systems (e.g., inflatable playground), which rely on continuous internal air pressure to retain the deployed form (31).

Fig. 5.

Designing innovative deployable tubular structures. (A) Coiled system: The flat section is stored in a rolled state and gradually pulled through an annulus to achieve a locked and stiff deployed state. (B) Inflatable system: The tube shape is maintained by the self-locking property, even with excess air inflation, and without relying on continuous internal air pressure. (C) Type i represents loading the tube in its stiff direction, leading to a nonlinear buckling (crushing) response. Type ii represents loading in the deployment direction, exhibiting an elastic response arising from shell bending. Type iii is Type i without diaphragms. (D) EXP force–displacement results of crushing (Type i) and elastic (Types ii and iii) responses. (E) Combining multiple tubes to create a metamaterial with direction-dependent stiffness. (F) Optimizing the overall stiffness of an adaptive structure under given load and boundary conditions. The deployed tubes are inserted into a grid of sleeves and rotated according to our optimization result. (G) In a cantilever setup, Type i tube can significantly outperform conventional deployable booms (Type iii). (H) Lightweight load-bearing panel.

Movie S4.

Advancing an existing deployment mechanism, specifically the coiled system. A flattened tube is gradually pulled through an annulus to easily deploy and lock into the final deployed state. The deployment process is self-reinforcing, as diaphragms lock and stiffen the circular tube automatically after passing through the annulus.

Movie S5.

Advancing an existing deployment mechanism, specifically the inflatable system. A flattened tube is sealed with two rubber caps. Inflation air pressure is then used to fully deploy the tube, with the self-locking configuration maintaining the tube state when pressure is released.

Deployed tubes are loaded to examine their structural performance (Fig. 5C), with EXP results summarized in Fig. 5D (see Materials and Methods for setup details). We show that the tube may exhibit a nonlinear buckling response to wall loading (Type i), which also shares characteristics with the classic axial crushing behavior (32). This response is mainly due to the perpendicular arrangement of diaphragms. Specifically, each vertical slice of the diaphragm is straight and compressed under the skin confinement. As a result, components reinforce each other until the critical buckling point, after which they soften to absorb energy. Notably, when the tube is lightly loaded, the diaphragm’s main position and deformation do not change significantly, even if its free edge may move slightly. This is because each diaphragm is stabilized, with its position fixed by the hinge layer and its deformation restricted by the circular confinement. When the tube is loaded in the deployment direction (Type ii), vertical slices of the diaphragm become curved, taking on postbuckled shapes. Therefore, applying compressive forces on the tube continues to bend the components without triggering buckling, generating a nonlinear elastic response (22). For comparison purposes, Type i tubes are tested without diaphragms (Type iii), demonstrating an elastic response with notably lower stiffness. Collectively, we show that the deployed tube has a direction-dependent structural performance, which can be rotated to achieve either a crushing response (Type i) or an enhanced elastic response (Type ii) with diaphragms acting as internal stiffeners.

Our deployed tubes offer exciting opportunities for different structural designs. Combining multiple tubes, we can create a metamaterial (Fig. 5E) that exhibits both Type i and ii behaviors depending on the load direction (Movie S6). In Fig. 5F, tubes are inserted into a grid of sleeves with free rotation, resulting in an adaptive structure (33, 34). We have developed an optimization algorithm (SI Appendix) to orient these tubes for maximum stiffness under specified load and boundary conditions (Movie S7). In the cantilever example (Fig. 5G), we show that the Type i tube can significantly outperform conventional deployable boom structures (represented by the Type iii tube) as diaphragms enhance the overall stiffness. Finally, we demonstrate that the proposed tube is suitable for large-scale constructions (Fig. 5H), enabling load-bearing lightweight panels (Movie S8).

Movie S6.

Creating a metamaterial using multiple deployed tubes. This metamaterial can exhibit both stiff and flexible behaviours depending on the load direction.

Movie S7.

Creating an adaptive structure using multiple deployed tubes inserted into a grid of sleeves with free rotation. We have developed an optimization algorithm to orient these tubes for maximum stiffness under specified load and boundary conditions. The video also highlights the significance of this optimization; when the tubes are placed without the algorithm’ guidance, the structure exhibits a notable large deflection.

Movie S8.

Creating a load-bearing lightweight panel for large-scale constructions. The video demonstrates the impressive load-carrying capacity of the braced tubular arrangement, emphasizing that even though the tube is composed of thin shell components, it can still support substantial loads.

Discussion and Outlook

This study has presented a deployable tubular structure that successfully meets conflicting performance requirements of expandability and stiffness. Our design approach is straightforward, using only simple shell components and an additional hinge layer to make circular tubes deployable. We show that the key innovation is the use of internal diaphragms, whose deformation is controlled by the synergy of curved-crease origami like motion and elastic shell buckling, enabling self-locking and stiffening. Our findings reveal that the energy barrier causing the snap-through transition is primarily due to geometric interference. This happens when diaphragms exhibit complex elastic deformation under changing boundary conditions posed by the skins. We also show that the integration of diaphragms as both locking and stiffening elements can significantly expand the capabilities of existing deployment mechanisms (coiled and inflatable systems) and create different structural designs (metamaterials, adaptive structures, cantilevers, and lightweight panels).

Future research includes several promising areas: refining the design, incorporating advanced materials, enhancing manufacturing techniques, and improving optimization capabilities. First, extending the proposed self-locking mechanism to noncircular cross-sections or exploring structural performances beyond wall loading (e.g., shearing, bending, and twisting) could unlock further applications. Second, while current prototypes are achieved using subtractive manufacturing, producing smaller tubes (e.g., microscale) may require the precision of additive manufacturing to achieve the desired self-locking and stiffening mechanism behavior. Third, a key area of exploration is self-deployment, aiming to minimize manual intervention. Fourth, investigating alternative methods to avoid kinematic bifurcation, such as employing slightly curved diaphragms and flat skins, holds the potential to simplify manufacturing. Finally, the scope of the orientation optimization algorithm also presents an opportunity for expansion, such as considering other objective functions, multiple load cases, and variable boundary conditions. As the field of deployable structures evolves, the insights gained from this study are well positioned to drive technological advancement in real-world applications.

Materials and Methods

Sample Fabrication.

In SI Appendix, Fig. S1A, flat skin and diaphragm components for the three-diaphragm tube (Fig. 1A) are precisely laser-cut from polyethylene terephthalate glycol (PETG) sheets, with their final thickness (

t

) measured to be approximately 0.10 to 0.15 mm. We have also measured the linear elastic properties of the PETG material and obtained its Young’s modulus as

E = 2, 299.18

MPa and Poisson’s ratio as

ν = 0.37

.

Each flat skin has a length of

L = 150

mm and a width of

B = 15 π \approx 47.1

mm. In the target assembly (Fig. 1A), we employ two skins to create a circular tube with a diameter of

D = 30

mm, with the bent diaphragm shape fit seamlessly inside to partition the tube. The flat diaphragm thus has a

B \times D = 15 π

mm

\times

30 mm maximum design space to fill the skin width and the tube diameter, respectively. However, the diaphragm shape is deliberately designed to be slightly smaller than its maximum design space. Its curved edge is offset by 1.5 mm, resulting in the final flat size of 45.6 mm

\times

28.5 mm. This smaller size effectively ensures tolerance for manual assembly and eases contact conditions between components to avoid plastic deformations during deployment.

Diaphragm positions are lightly laser engraved on the skins to guide manual placements (SI Appendix, Fig. S1A). The spacing between diaphragms is 37.5 mm, ensuring that diaphragms interact only with the skins, not each other, during deployment. Components are connected using an extra hinge layer (tape) of polypropylene (PP) material, which is approximately 0.05 mm thick and 23 mm wide. In the final assembly (Fig. 1G), the stack order of a compliant mechanism is skin–diaphragm–hinge, with the hinge layer also connecting back to the skin. This arrangement creates a small gap between the skin and diaphragm components. We find that this gap, along with the slight misalignment of components due to manual placement, introduces an initial imperfection in the compliant mechanism, which is instrumental in controlling kinematic bifurcation.

Thermoforming Curved Components.

The thermoforming process utilized in this study is identical to the one described in ref. 22, which was developed previously by the authors. Here, skin and diaphragm components are first joined into a compliant mechanism. This mechanism is then “folded” and placed into an acrylonitrile butadiene styrene (ABS) 3D printed mold (SI Appendix, Fig. S1 B and C). The entire mold assembly is heated in an oven at a constant temperature of 75

^{°}

C for two hours. This heating process allows for thermal annealing and stress relief in bent PETG materials without impacting the ABS mold. Following this, the compliant mechanism is removed from the mold and allowed to cool at a room temperature of approximately 15

^{°}

C for two hours. This cooling process is critical to ensure that the components are fully cured into their target curved shape before any elastic deformations are applied (SI Appendix, Fig. S1D). Notably, two different mold sets are used to produce the required compliant mechanisms. One mold is specifically designed for a single diaphragm, while the other is tailored for two diaphragms. Both molds feature a single-curved, arc curvature with

D = 30

mm. Directly assembling the thermoformed results of these two molds gives the target tubular form with three diaphragms (SI Appendix, Fig. S1E).

FE Simulation.

This study uses Rhino 7 Grasshopper software to create the computer-aided design (CAD) model of the proposed deployable tubular structure in its deployed (target) state. Curved skins, curved diaphragms, and flat rigid plates (SI Appendix, Fig. S2A) are included to replicate the EXP setup shown in Fig. 2B. Notably, skin and diaphragm components are imparted with “soft boundaries” (initial subdivision) to ensure high-quality meshing later.

The CAD model is imported into the commercial finite element analysis software, Abaqus 2018, to perform a nonlinear (large deformation), muti-step finite element (FE) simulation (SI Appendix, Fig. S2B). The PETG material is set to be isotropic and linearly elastic, assigned with a thickness of

t = 0.15

mm and a typical mass density of

ρ = 1, 380

kg/m

^{3}

. Thickness directions of component surfaces are carefully assigned to avoid components initially penetrating each other (SI Appendix, Fig. S2C). Specifically, the input skin surfaces of the CAD model represent the inner wall of the tube, and the diaphragm surfaces represent the convex side of themselves. All surfaces, including the rigid plates, are joined by hinges using “ties” in Abaqus, enabling connection and free rotation during motion (SI Appendix, Fig. S2A). The “general contact” algorithm is adopted to simulate the self-contact behavior, with the friction coefficient set to be 0.2 (35). Skin and diaphragm surfaces are meshed using fully integrated, quadrilateral S4 shell elements; rigid plates are discretized using R3D4 rigid elements. After conducting the mesh convergence test (SI Appendix, Fig. S3 A and B), a final mesh size of 2 mm is selected, ensuring stable results while controlling computational time.

We perform the simulation using one initialization step and three subsequent steps (SI Appendix, Fig. S2B). All steps are subjected to displacement control using the explicit dynamic solver (with double precision) to handle large deformation geometric nonlinearity. A constant velocity of 1 mm/s is set to allow for quasi-static simulation, minimizing dynamic effects, as verified in SI Appendix, Fig. S3C. First, the initialization step breaks the target tubular configuration into two compliant mechanisms through linear translation. Each mechanism is moved 5 mm away from the other, resulting in a total separation of 10 mm. Notably, the initialization step sets up the simulation by reversing from the target shape. Such a shape allows for correctly applying ties between components, ensuring connected components can be effectively controlled by rigid plates to achieve their desired motion later. Skipping this step could lead to numerical errors when assembling the compliant mechanisms, causing small gaps at boundary hinges and affecting the control of the rigid plates.

To accurately represent the undeployed state, Step 1 simulates the flattening of the compliant mechanisms. Both sides of the boundary edges are pulled by 8 mm, resulting in a motion similar to curved-crease origami and changing

b

from 30 mm to 46 mm (SI Appendix, Fig. S2B). The selection of

b = 46

mm is based on the averaged measurements taken from six undeployed physical samples, ranging from

b =

45.4 mm to 46.9 mm. To capture the initial diaphragm buckling from skin contact, Step 2 simulates the combining of the flattened compliant mechanisms. Each flattened mechanism is moved 5 mm toward the other, closing the separation. Notably, the assembled result from Step 2 does not yet represent a stable undeployed tube. This is due to relaxation that may occur with the release of boundary pulling, potentially leading to a stable equilibrium (undeployed) state with

b < 46

mm (Fig. 2F). Finally, Step 3 simulates the snap-through transition to the final deployed state. To better mimic the imperfect compression observed in physical samples, one rigid plate is fixed while the other compresses the tube by 16 mm.

In our FE simulation, we use rotational constraints on the curved hinges to ensure that the diaphragms do not bend in the same direction as the skins. This is important because, after flattening the diaphragms in Steps 1 and 2, their bending direction in Step 3 becomes uncertain. Therefore, the constraints are key in guiding the diaphragm to bend in the correct direction. To achieve this, we restrain the rotation component UR3 in the curved hinges during Steps 1 and 2 (SI Appendix, Fig. S2D). This helps maintain the “curved folding motion” (primarily governed by UR1 and UR2) while preventing the diaphragms from becoming completely flattened, effectively introducing an initial imperfection for Step 3. With the UR3 constraint remaining activated in Step 3, we find that the diaphragms undergo a V buckling mode due to the symmetrical conditions. Conversely, if UR3 is activated only on half of the curved hinge in Step 3, an S buckling mode emerges, attributed to the asymmetrical conditions.

Mechanical Testing.

In the EXP deployment test (Fig. 2B), the undeployed tube undergoes a uniaxial quasi-static compression in a universal testing machine. The tube is first connected to the edges of two thin, rigid plates using additional hinge materials (PP tape) to allow free rotation. Subsequently, the tube is compressed at a 2 mm/min loading rate until a total displacement of 18 mm is reached, with force and displacement measured at the crosshead. EXP data for the 16 tubes (Fig. 3 B and C) can be found in SI Appendix, Fig. S4.

In the EXP structural performance test (Fig. 5 C and D), tubular structures are compressed between the inner surfaces of two thick, rigid plates. Tested tubes are simply placed between the plates without additional boundary constraints. The compression is carried out at a 60 mm/min loading rate until a total displacement of 20 mm is achieved, reducing the cross-section height of tubes from 30 mm to 10 mm, with force–displacement responses recorded.

Data, Materials, and Software Availability

All relevant files and data are provided on GitHub (https://github.com/CISM-tech/DeployableTube2024) (36), including the Grasshopper script (gh) file for modeling the idealized tube in Rhino CAD software, the Abaqus input (inp) file for reproducing the numerical simulations, the raw experimental data, simplified simulation results, and the Matlab code for orientation optimization.

Acknowledgments

We gratefully acknowledge the financial support provided by the Australian Research Council (Australian Laureate Fellowship grant FL190100014).

Author contributions

T.-U.L., H.L., J.M., N.S.H., J.M.G., and Y.M.X. designed research; T.-U.L., H.L., and J.M. performed research; T.-U.L. and H.L. analyzed data; and T.-U.L., H.L., J.M.G., and Y.M.X. wrote the paper.

Competing interests

The authors declare no competing interest.

Supporting Information

Appendix 01 (PDF)

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15.03 MB

Movie S1.

Transformation process of the proposed deployable tubular structure. The undeployed state consists of flattened elastic shells. Upon compression of the boundary edges, the internal stiffeners exhibit motion inspired by curved-crease origami and undergo elastic shell buckling in a confined condition. This action leads to a snap-through transition to a tubular deployed state. Remarkably, the deployed state is self-locked, allowing it to maintain its shape even after the boundary compression is released.

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2.20 MB

Movie S2.

Finite element simulation of the proposed deployable tubular structure with pre-curved components. The simulation follows a sequential process, spanning from assembly to deployment. In assembly, components are flattened through pulling and are subsequently combined, triggering diaphragm buckling through contact. The deployment is demonstrated by compressing the tube between rigid plates under displacement control. We control the symmetrical conditions to generate different buckling combinations in the three diaphragms, including S-S-S (expected), S-V-S (possible), and V-V-V (idealized).

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3.18 MB

Movie S3.

Deployment simulation using a geometry-based simplified model grounded in a spring network. It successfully replicates the buckling combinations seen in finite element simulations (S-S-S, S-V-S, and V-V-V). Therefore, successful deployment is confirmed mainly attributed to geometric interference between the skin and diaphragm components.

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7.03 MB

Movie S4.

Advancing an existing deployment mechanism, specifically the coiled system. A flattened tube is gradually pulled through an annulus to easily deploy and lock into the final deployed state. The deployment process is self-reinforcing, as diaphragms lock and stiffen the circular tube automatically after passing through the annulus.

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1.69 MB

Movie S5.

Advancing an existing deployment mechanism, specifically the inflatable system. A flattened tube is sealed with two rubber caps. Inflation air pressure is then used to fully deploy the tube, with the self-locking configuration maintaining the tube state when pressure is released.

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1.05 MB

Movie S6.

Creating a metamaterial using multiple deployed tubes. This metamaterial can exhibit both stiff and flexible behaviours depending on the load direction.

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1.87 MB

Movie S7.

Creating an adaptive structure using multiple deployed tubes inserted into a grid of sleeves with free rotation. We have developed an optimization algorithm to orient these tubes for maximum stiffness under specified load and boundary conditions. The video also highlights the significance of this optimization; when the tubes are placed without the algorithm’s guidance, the structure exhibits a notable large deflection.

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1.85 MB

Movie S8.

Creating a load-bearing lightweight panel for large-scale constructions. The video demonstrates the impressive load-carrying capacity of the braced tubular arrangement, emphasizing that even though the tube is composed of thin shell components, it can still support substantial loads.

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2.20 MB

References

1

E. T. Filipov, T. Tachi, G. H. Paulino, Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proc. Natl. Acad. Sci. U.S.A. 112, 12321–12326 (2015).

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Abstract

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Self-Locking Mechanism with Geometric Interference

Deployment to Target Tubular Form

Simulation of Assembly and Deployment

Comparison between EXP and FE Deployment Tests

Simplified Deployment Prediction

Toward Innovative Deployable Tubular Structures

Discussion and Outlook

Materials and Methods

Sample Fabrication.

Thermoforming Curved Components.

FE Simulation.

Mechanical Testing.

Data, Materials, and Software Availability

Acknowledgments

Author contributions

Competing interests

Supporting Information

References

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