Stiff Composite Cylinders for Extremely Expandable Structures

The realization of concurrently largely expandable and selectively rigid structures poses a fundamental challenge in modern engineering and materials research. Radially expanding structures in particular are known to require a high degree of deformability to achieve considerable dimension change, which restrains achievable stiffness in the direction of expanding motion. Mechanically hinged or plastically deformable wire-mesh structures and pressurized soft materials are known to achieve large expansion ratios, however often lack stiffness and require complex actuation. Cardiovascular or drug delivery implants are one example which can benefit from a largely expandable architecture that is simple in geometry and intrinsically stiff. Continuous shell cylinders offer a solution with these properties. However, no designs exist that achieve large expansion ratios in such shells when utilizing materials which can provide considerable stiffness. We introduce a new design paradigm for expanding continuous shells that overcomes intrinsic limitations such as poor deformability, insufficient stiffness and brittle behaviour by exploiting purely elastic deformation for self-expandable and ultra-thin polymer composite cylinders. By utilizing shell-foldability coupled with exploitation of elastic instabilities, we create continuous cylinders that can change their diameter by more than 2.5 times, which are stiff enough to stretch a confining vessel with their elastic energy. Based on folding experiments and analytical models we predict feasible radial expansion ratios, currently unmatched by comparable cylindrical structures. To emphasize the potential as a future concept for novel simple and durable expanding implants, we demonstrate the functionality on a to-scale prototype in packaging and expansion and predict feasible constellations of deployment environments.


Folding stiff composite cylinders into instability-inspired folding patterns
Although the structure is manufactured from carbon fiber reinforced polymers, its low thickness allows it to be folded by hand. However, at such low scale, an apparatus was needed to fold the structure without violating the minimum bending radii requirements. Since the folding process is mostly in the post-buckling regime of the composite ring, 3D-printable PLA rigs ( Figure S1) are sufficient to carry the reaction forces during folding. The second folding step follows a circle involute path winding the lobes around the radius R 2 (s describing the parametrization variable).
The folding rig can be seen in Figure S1. Once pinched by hand and inserted into the 3D-printed pins, the metallic cylinders are rotated around the eccentric axis following the path of the baseplate until they are fully enclosed by the crimped radius R c . The cylindrical PLA-sleeve can then be strapped around the composite. The rig can then be pulled out from the inside of the PLAcylinder until the structure is free from internal constraints and equilibrates. The geometrical model for the prediction of optimal packaging efficiency is based on a circlepackaging problem with several tangency requirements ( Figure S2). For the simpler case of a structure with t S =0 with given continuity of the structure, the fully packaged state can be assembled from circular arcs of radii R 1 ,R 2 and R 3 , each of them tangent to its adjacent radius and enclosed by the sleeve radius R c . We obtain the maximum packaging efficiency by maximizing the arc length of the structure packaged to a given radius, R c . Therefore we maximize the area of the lens created by the arc of R 1 and R 2 by differentiating it with respect to the radius R 1.
(2) and, with, and a describing the apothem of a regular polygon, which contains a number of sides N (equivalent to the number of lobes of the folded structure), With t S = 0, maximization of the area yields R 1 = R C . The two smaller radii inside the (now symmetrical) lens are R 2 = R 3 given the tangency requirements at their contact point as well as to their confining lens. The value can be derived based on the chosen number of lobes N and correlates with the crimped radius R C , Given all radii that assemble the geometric representation, the structure is fully defined yielding the overall arc-length which directly results in the unpackaged diameter and hence the packaging efficiency.
For t S > 0 we can follow Equations 2-5 to also derive the packaging efficiency for non-zero thickness structures. The symmetry regarding the radii R 2 and R 3 can not be maintained resulting from a now asymmetric lens L1-L2 ( Figure S2). The asymmetry results from the increased thickness at the contact points. Nevertheless we can use the area maximization of the asymmetric lens to obtain the radius R 1 , which, using this approach, is not dependent of R 2 and R 3 . Tangency requirements are still valid at contact points and between R 2 and R 3 , which are now shifted by t S /2 or 3t S /2 respectively. Assuming perfect contact between pinching bodies R 1 and a tightly compressed shell (2t S ), we can derive the optimal radius R 2 using the apothem of the polygon and the Pythagorean Theorem: When solved for R 2, we can use the condition that the sum of the angles α, γ and φ needs to be π to create the regular polygon with N sides and internal angle θ fold resulting from the circle packaging of the first folding stage. The angles can be calculated using the law of cosine between tangent circles, allowing to solve the resulting equation ( , 2 , 3 , ) + ( , 2 , ) + ( , 2 , 3 , ) − = 0 (8) numerically for R 3 using MATLAB. For a prescribed ratio of = and a given lobe count N, this results in the desired folding radii as well as the resulting geometric packaging efficiency of the structure.

Derivation of non-dimensional parameters for deployment
Based on the assumption that the bending stiffness of the confinement is small and the structures bending stiffness significantly lower than its hoop stiffness, we identify the ratio between critical buckling pressure of the deploying structure and hoop stiffness of the constraint as driving factor for deployment. This ratio can be split into two non-dimensional parameters that allow to decouple conveniently tunable geometrical parameters (Oversizing Ω) from stiffness driven parameters (Ψ). For a perfect cylindrical composite we can calculate the critical buckling pressure given its layup and radius [1] and the vessel hoop stiffness is approximated using thin shell theory. It shall be noted that for radius to thickness values of human arteries, thin shell theory is drastically simplified, however it enables easier comparison and understanding of the relations during the deployment process.
, ℎ , (9) Figure S3 shows the deployment test set-up of the deployment test. All fixtures where printed from PLA. The silicone sleeve was manufactured by casting Altropol Protosil RTV 245 Silicone in 3D printed molds. Therefore, the length was chosen to be at least 4x larger than the structure length so edge effects of the clamping do not influence the deployment. The silicone was circumferentially glued onto PLA vessel fixations (red in Figure S3) which then where screwed into the PLA fixtures. The center axis of the camera was aligned with the silicone vessel. Simultaneous release of the structure was achieved by pushing the structure out of its constraint. Movie S1. Supplementary Movie S1 shows the recording of a deployment process of a fully folded structure which is released from its sleeve seen in Figure S3. The high-speed camera recorded with 8500FPS. The edges of the structure were painted in white shortly before the experiment to facilitate contrast. The white particles in the video show excessive white ink accelerated by the inertial forces during deployment.