Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-based Cylindrical Structures

Origami structures enrich the field of mechanical metamaterials with the ability to convert morphologically and systematically between two-dimensional (2D) thin sheets and three-dimensional (3D) spatial structures. In this study, an in-plane design method is proposed to approximate curved surfaces of interest with generalized Miura-ori units. Using this method, two combination types of crease lines are unified in one reprogrammable procedure, generating multiple types of cylindrical structures. Structural completeness conditions of the finite-thickness counterparts to the two types are also proposed. As an example of the design method, the kinematics and elastic properties of an origami-based circular cylindrical shell are analysed. The concept of Poisson’s ratio is extended to the cylindrical structures, demonstrating their auxetic property. An analytical model of rigid plates linked by elastic hinges, consistent with numerical simulations, is employed to describe the mechanical response of the structures. Under particular load patterns, the circular shells display novel mechanical behaviour such as snap-through and limiting folding positions. By analysing the geometry and mechanics of the origami structures, we extend the design space of mechanical metamaterials and provide a basis for their practical applications in science and engineering.

after folding determines completeness of this type of thick Miura-oir.

Geometric relationships, compatibility and Poisson's ratio of OCSs
The 4-crease origami pattern studied in this paper has single DOF. With zero-thickness models, four fold lines of the OCSs meet at one vertex, acting as revolute joints to make the origami act as spherical linkages (Fig. S3). According to spherical trigonometry 1 , Geometric relationships between the line angles and dihedral angles can be described as: For Miura-ori pattern, line angles i  are either equal or supplementary to each other, that is: Because of 1  and 2  ,  and  change alternately along the circumferential direction, while the dihedral angles  keeps constant, as can be seen in Fig. 3a Expressing all the variables in terms of  , equations (5-6) are obtained.
To maintain geometric compatibility, the shortest length of all the quadrilateral sides in OCSs should be positive (see Figs. 2c and 3c), which requrires: Together with the constraint Fig. S4 are obtained according to different ratios 1 2 l l . Figure Circumferential strain   is verified by the expression of strain in the cylindrical polar coordinates 2 : In Eq. (S11), r u , u  are radial and circumferential displacement, respectively.

Mechanics of OCS: rigid folding/unfolding
Analytical expressions of both the radial and axial balanced force are obtained with the method of minimum potential energy (equation (10)). Different load patterns and boundary conditions (Figs. S5a and S5b) determine the corresponding mechanical responses of folding/unfolding process (As a comparison, illustration of the line force and boundary conditions of inhomogeneous elastic deformation is also present in Fig. S5c). Both of the processes are implemented using the 9 5  OCS model. According to equation (10), external work induced by the isometric force is: For the axial force, the positive direction is defined in the compressive/folding direction, so the displacement: For the radial force, the positive direction is also defined in the compressive direction, but note that a compressive load leads an unfolding process in this condition. Simple supported boundary condition is applied on the endpoints of 1 R , so the radial displacement: where H is the rise of the shell structure (Fig. 3b), which can be described as: Finally we obtain the non-dimensional radial force: (S19) Figure S5d shows the predicted r F versus  of a 9 5  Fig. S5d). Excellent agreement between the analytical and FEM predictions is obtained. Figure

FEM models of rigid folding and inhomogeneous deformation
For the rigid folding/unfolding process of an OCS, models consisting of rigid plates and linear elastic hinges are adopted. To realize different folding or deformation modes of the OCS, proper load patterns and boundary conditions need be carefully adopted, which are shown in Fig. S5. Shear deformable shell elements are used in the FEM. The equivalency is ensured by assuming that an OCS and its EHCS have the same amount of material, in addition to the same radius 1 R and central angle 1  , i.e.,