Before we dig into the concept of curved-crease origami, first the concept of the curvature and folding of paper should be introduced. We briefly introduce the concepts that are crucial for understanding surface curvature in general and its relation to origami.

The principle curvatures of a given point on the surface, κ ₁ and κ ₂, are the maximum and minimum values of all the normal curvatures at that point, with the directions of corresponding curvatures denoting the principle directions [62], as shown in Figure 2. The principle curvatures and the direction normal describe how the surface curves in the vicinity of that given point, and on special occasions, when all these curves of intersection are of equal curvature (e.g., a point on a sphere or a plane), this is an umbilic point, and in this case, the principal directions cannot be uniquely defined. Except umbilic, the principle directions of other regular points are orthogonal. To quantitatively determine the curvatures, two measures are given: the Gaussian curvature K and the mean curvature H, as shown in Eq. (1) and Eq. (2). K = κ 1 κ 2 (1) H = κ 1 + κ 2 2 (2)

The Gaussian curvature was introduced in Gauss’s landmark paper, which is considered to be the most important theorem in differential geometry, and the names of points are determined as parabolic, elliptic, and hyperbolic when the Gaussian curvature is zero, positive, and negative, respectively. The mean curvature is an extrinsic measure of the surface that is dependent on the surroundings; on the contrary, the Gaussian curvature is an intrinsic property of the surface that shows the property of the surface and is independent of the surroundings [63].

The difference between these two types of curvature is important, as a surface could have a zero Gaussian curvature but a non-zero mean curvature, which shows that a surface could remain intrinsically flat while being intrinsically curved, i.e., bending flat paper and warping it into a cylindrical shape. Because of this property, the plane with zero Gaussian curvature is called a “deployable surface,” since a flat plane could be bent into a cylindrical or a cone shape without stretching, tearing and gluing, the cylinder or the cone are both deployable surfaces with zero Gaussian curvature. In other words, the Gaussian curvature, K, is a bending invariant; for every point on the surface x, there is a mapping that transforms a simple connected surface x to another simple connected surface y that preserves the Gaussian curvature. Surfaces that are related to this mapping that preserve the Gaussian curvature are called isometric, one could refer to the work of Paul [65] for a vivid visualisation. Callens et al. provided a comprehensive review of the curvature of the surface regarding origami structures [19]. So it is the same when one says that if a surface is locally isometric to the plane, it is a developable surface (one cannot stretch paper!). Such surfaces fall into the theory of Euler, where a developable surface is a ruled surface, as shown in Figure 2E, the ruled surface consists of a family of straight lines, which are called rulings. The rulings are tangent to the edge of regression. The rulings of a developable surface are the envelope of a family of tangent planes where all the tangent planes along the rulings of a developable surface coincide. One can put a ruler (as the tangent plane along the rulings) that envelopes a developable surface. In some cases, those rulings are also called generators.

Since the ruling is a straight line, a ruled surface is defined to describe how a point on the straight line travels in space; specifically, a ruled surface is a curved surface that is made by continuously changing the location of a point on the straight line in space by one parameter, as shown in Figure 3A. The point of the straight line with a unit directional vector of e(s) travels along the parametric curve P(s), so any position of the point X(s, t) on the straight line is expressed as a parameterized form, as shown in Eq. (3): X s , t = P s + t e s (3) where t determines the position of the point on the straight line. Hence, t and s are two parameters that represent the parameterized form of a ruled surface. Note that not all the ruled surfaces could be made with paper, as shown in Figure 3B. Detailed qualitative properties of curved folding are shown in the work of Demaine et al. [66], Liu et al. [67] and Mundilova’s dissertation [68].

The study of curved-crease origami usually starts with understanding the basic principle of folding a simple curved line on paper. One of the first and most influential analyses of curved crease origami structures was performed by Huffman, where the local folding behaviour is examined by using Gauss’ spherical representation [69]. The geometry of curve folding has been further studied by Duncan and Duncan [70] and Fuchs and Tabachnikov [64] using the differential geometric approach, where the concepts of geodesic curvature and geodesics are used.

Geodesic curvature belongs to the world of intrinsic geometry. Let us now consider a surface x with an in-plane curve ξ (the plane is in blue colour, as shown in Figures 4A, B, the curvature vector of the curve ξ at a given on-curve point P points from P to the centre of the curvature of ξ, therefore, this vector could be manually decomposed by projecting the curvature vector onto two orthogonal planes: the normal plane of the curve ξ at the given point P that contains the normal vector N _P and the tangent plane of the curve ξ at the given point P, as shown in Figure 4B. The curvature of the projected curve on the normal plane denotes κ _n , while the geodesic curvature, denoted κ _g , equals the curvature of the projected curve ξ on the tangent plane at the given point. When the geodesic curvature is zero, there is no projection of the curve ξ onto the tangent plane, so the curvature of the curve ξ is identical to the normal curvature κ _n . A curve with geodesic curvature being zero everywhere is called a geodesic, and a geodesic connects two points on the surface with the shortest distance. Since the normal vectors along the geodesic curves now coincide with the normal vectors of the curves, there is an infinite number of geodesic curves passing through every given point on the surface (the readers are recommended to recall the process of deriving the principle curvatures where the intersecting curves could be generated by rotating the normal plane about the normal vectors, and those intersected curves are geodesic curves).

As the geodesic curves are definitely in-plane curves due to the definition, to describe a space curve, torsion is introduced here for measuring the bending of curves. A nonplanar space curve exhibits both curvature and torsion; the torsion of a curve describes the pitch and the degree of non-planarity of the curve. For a curve lying on a surface, it is general to use the geodesic torsion to measure the local bending of this curve, which complements the normal curvature κ _n and the geodesic curvature κ _g . The geodesic torsion of a given point at a given direction on a surface curve equals the torsion of the geodesic through that point in that direction. We could define a set of triple orthogonal vectors at a given point of the curve on the surface, the normal vector to the surface at the given point, n, the tangent vector to the surface at the given point, t, and the geodesic normal vector u (in some work, it is also known as the binormal vector and denoted as b) which is the crossed product of the n and t, u = t ×n. The changing rate of the normal vector n with respect to the arc length s projecting on the tangent vector equals the normal curvature, while the changing rate of the normal vector n with respect to the arc length s projecting on the vector u equals the geodesic torsion t _g , hence, geodesic torsion complements both the normal and the geodesic curvatures, yet note that it does not belong to the world of intrinsic geometry.

Now, we will introduce the fundamentals of differential geometry of curved folds from the work of Ducan and Ducan [70]. As shown in Figures 4C, D, consider a curved crease line C on the deployable surface, it divides the surface into two sections, the line PG is the generator, and the left surface S1 bends along the generator. Now, the curve C becomes a space curve, at a given point P, the moving triad of vectors (the normal vector n, the tangent vector t and the binormal vector b) could be determined. The space curve C is determined by the principle curvatures κ(s) along the curve s and torsion τ(s) using the Serret-Frenet formula: d t d s = κ n d n d s = − κ t + τ b d b d s = − τ n (4)

Note that this description is independent of any surface that contains the curve. However, with the assistance of a new set of triad vectors N, t and u in the tangent plane, the geodesic properties of the space curve could be described. As shown in Figure 4E, at the on-curve point P, the tangent plane is in grey colour, N is the unit normal vector of the tangent plane at the point P, the unit vector u is in the tangent plane and it is orthogonal to the unit tangent vector t. The projection of the space curve C at point P is C′, then, the curvature of the C′ on the tangent plane, by definition, is the geodesic curvature, κ _g of the surface curve, hence, the geodesic curvature vector denotes κ _g u, the normal curvature of the surface at point P is in the direction of the vector N, which is κ _N , the changing rate of the rotation of the normal vector N along the curve is the geodesic torsion denotes τ _g . The geodesic properties could be related using the Bonnet-Kovalevsky formulae, as shown in Eq. (5): d t d s = κ g u + κ N N d u d s = − κ g t − τ g N d N d s = − κ N t + τ g u (5)

As shown in Figure 5, at a point P in any surface, the normal curvatures in two orthogonal directions, 0 and 1, corresponding to the maximum and the minimum values of principle curvatures, κ ₀ and κ ₁ exist, the geodesic torsions, by definition, are zero in these directions. For the other direction x with an angle of γ with respect to P0, the properties are given by Euler’s formulae as follows. κ N x = κ 0 ⁡ cos 2 ⁡ γ + κ 1 ⁡ sin 2 ⁡ γ τ g x = 1 2 κ 0 − κ 1 sin ⁡ 2 ⁡ γ (6) therefore, as the direction of y is orthogonal to the direction x (κ ₀ sin² γ + κ ₁ cos² γ), there are invariants from Eq. (6), as shown in Eq. (7): κ N x + κ N y = κ 0 + κ 1 = 2 × κ mean κ N x κ N y − τ g x 2 = κ 0 κ 1 = κ Gauss (7) where κ _mean and κ _Gauss denote the mean curvature and the Gauss curvature respectively. As shown in Figure 5A, the surface S1 deforms about the generator PG and becomes a deployable surface (cylindrical shape), PG is a principle direction with the normal curvature κ ₀ = 0. In the tangent vector of t with an angle of γ to the direction of PG, the normal curvature and the geodesic torsion are found from Eq. (6), these could be further represented by a Mohr circle diagram (as shown in Figure 5C), the point T stands for the geodesic curvature κ _g and the normal curvature κ _N at the given point P in the tangential direction t.

Three curvatures, the geodesic curvature κ _g (describes the tangent plane curvature, i.e., the planar layout of the fold curve), the normal curvature κ _N (describes the degree of out-planarity relating to the current surface curvature in a tangential direction to the geodesic curvature) and the curvature κ, are related in the Eq. (8) using Meusnier’s theorem. κ n = κ g u + κ N N (8) this relates the curvatures to the fold angle 2ω, the torsion of the space curve τ, the changing rate of the fold angle with respect to the arc length dω/ds and the geodesic torsion τ _g using the theorem of Bonnet as follows (the proof is given in [70]), as shown in Eq. (9). τ = d ω / d s − τ g (9) the relations are derived to form the compatibility conditions under the assumptions that, for the two surfaces S1 and S2 sectioned by the fold curve, at any shared point, the normal curvatures of the surfaces have the same magnitude but opposite sign; the changing rate of the folding angle with respect to the arc length on two surfaces has the same magnitude but opposite sign, as shown in Eqs (10) and (11), respectively. κ N , 2 = − κ N , 1 (10) d ω 1 / d s = − d ω 2 / d s (11)

Then, the relationship between the geodesic torsion and the changing rate of the fold angle with respect to the arc length is shown in Eq. (12). τ g , 2 = τ g , 1 − 2 d ω / d s (12)

Now, the normal curvature of the surface is obtained by the orientation angle between the tangent vector and the generator γ, and the principal curvature κ ₁ from Eq. (6), as shown in Eq. (13) and Eq. (14). κ N = κ 1 ⁡ sin 2 ⁡ γ (13) τ g = κ 1 ⁡ cos ⁡ γ ⁡ sin ⁡ γ (14)

The fold angle of the on-curve points then forms the relationship between the curvature of the curve and the geodesic curvature κ _g , as shown in Eq. (15). sin ⁡ ω = κ g / κ (15)

Here, it is explained how the curvature properties of a surface influence the folding process and the resulting shape, which is crucial for designing structures with specific mechanical properties. It is important to determine the distribution assumptions of the ruling lines on both sides of the crease, so the distribution of these virtual creases could physically satisfy the minimized energy condition of the surface.

By now, the derivations of the differential geometric approach to curved folding have been completed. When analysing the curved-crease origami, the given conditions are important. Two special degenerative cases were considered.

Case I: the folded curve remains planar after the folding;

Case II: the folding angle along the curve is the same.

These two conditions are exclusive to each other, which leads to the varying folding angles in Case I and the folded curve being a spatial curve (non-planarity) in Case II. The geodesic torsions of both surfaces τ _g,1 = −τ _g,2 in Case I and τ _g,1 = τ _g,2 in Case II. Since the tangential vector t is known, the generators on both sides of the curve are now determined through Mohr’s diagram. Note that it is exactly because the generators specify the developable geometries of both surfaces. As shown in Figures 6A, B, the folded curves and their corresponding generators. To conclude, folding along a curved crease satisfies the developability of the sheet, a curved-folded origami consists of developable surfaces (i.e., a cylinder, a cone, or a tangent developable to a space curve) sectioned by the curve crease.

Several studies have extended the research on curved-crease origami using the differential approach. Demaine et al. developed basic tools for curved-crease origami in terms of the definitions and theorems in a mathematical way; specifically, the studies of definitions of rulings and smooth folding are comprehensively investigated, and they provided proof of five high-level properties in terms of the rule segments [72]. Tachi [71] used a discretized form to analyse the curved-crease origami structures; hence, the non-rigid curved-crease origami could be simulated like a rigid origami structure with at most one degree of freedom. The differential geometric equations are expressed in the form of a matrix, which facilitates the use of the differential geometric properties in the analysis of the mechanical behaviours in the later studies, as shown in Eq. (16), as shown in Figures 6C, D. d t / d s d n / d s d b / d s = 0 κ s 0 − κ s 0 τ s 0 − τ s 0 t s n s b s (16) where the normal vector n, the tangent vector t and the binormal vector b are defined in the same way above, note that this is the matrix form of Eq. (4). Based on the above studies, Honda et al. [73, 74] provided proof that for a given pair consisting of a crease and crease pattern, there are four distinct non-congruent curved folding.

The differential geometric method above, which is a conventional approach, could be seen as Eulerian from the perspective of continuum mechanics. From the deformed configuration of the structures, one identifies kinematic relationships and constraints (i.e., the dihedral angles between adjacent tiles, the distance between vertices), and then derives the relationship between these objects [75]. This could be further described from the perspective of the Lagrangian, where the relationship between the crease pattern in the flat reference configuration and the deformed configuration is built in a dynamic case, the work of Liu and James [76] has developed this Lagrangian method in the curved tile origami.

Ignoring any in-plane deformation of a flat sheet, the Gaussian curvature remains zero at each point regardless of arbitrary folding processes, ensuring that the resulting three-dimensional structure is developable. With this premise, we can skillfully design creases to realize our targeted three-dimensional shapes. Jun Mitani has introduced an innovative design approach that leverages a series of fundamental curved folding patterns [40], just as shown in Figure 7. By tessellating these basic units, a plethora of captivating three-dimensional forms can be crafted, showcasing the beauty and versatility of origami-inspired design.

Mirror reflection is a traditional method to design the crease pattern of both the straight and the curved crease origami structures [79]. Here, a variety of research methodologies are demonstrated, as shown in Figure 8. The mirror reflection technique in the origami design does not alter the intrinsic curvature of the surface [71], so the differential equations for developable surfaces still hold when using the mirror reflection technique.

Another technique is to design the curved-crease origami by discretizing the non-rigid origami into a rigid origami. Tachi [80] showed the discretized method could be extended to tessellated, cylindrical, and cellular structures, as the condition of flat-foldability originating from constant-angle folding is satisfied. The non-twisting property of the deployable surface makes sure that the adjacent rulings could form a planar quadrangle. Thus, it is feasible to discretize a curved folded surface and approximate it with a planar-quadrilateral (PQ) mesh. Later on, Tachi [81] showed that when the curvature and torsion functions are given, a space curve could be defined, and this method could be used for the geometric construction of the tubular structure. The curved folded tube origami structure under the discretized method shows a single-DOF rigid folding motion.

Similarly, Zhang et al. [84] investigated the problem of discretizing the curved developable surfaces that satisfy the equivalent surface curvature change discretizations using the Gauss map (to investigate the normal curvature change of the curved surface). After the discretization, the adjacent surface normal vectors could be derived under developability constraint. The Gauss spherical curves of different developable surfaces are set up under the Gauss map. The whole process contains two steps. First, spatially curved surfaces are mapped to spherical curves. Each point on the spherical curve represents the normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. Second, the Gauss map is linked back to the normal curvature of the developable surfaces by the error analysis. These developable curved surfaces are discretized into planar patches to acquire the geometric properties of adjacent ruling lines, hence, the geometric properties and constraints of the curved folding such as fold angle, folding direction, folding shape, and foldability are derived. This method is applicable to analysing curved folding on any generic developable surface.

Gattas and You [83] proposed a new method for generating and parametrizing rigid-foldable, curved-crease geometries from Miura derivative prismatic base patterns. First, the ellipse is created, and then the curved crease is subdivided into straight-line segments. Through these two-step constructive geometric methods, a curved-crease Miura pattern could be generated based on the Miura-base pattern, and further, the curved-crease configurations of tapered Miura, Arc, Arc-Miura, and piecewise patterns are generated. Later on, Gattas [85] proposed a generative design method to specify a shape grammar (which comprises a shape and rules) for the assembly and interaction of geometric elements. The method to generate piecewise cylindrical surfaces based on curved-crease origami surfaces was presented, and the grammar utilises a parametric arc shape and end-tangent continuity condition for recursive shape addition and arc assembly. This “end arc addition rule” provides a design technique for the practical construction of curved-crease cladding.

Mundilova [82] investigated the smoothly deformed developable surfaces in a mathematical method, instead of using the traditional methods to form the approximations such as discretizing the surfaces or optimising for developability and isometry to a given patch, the proposed method could explicitly provide a smooth description of the obtained surfaces and their developments. Although the limitations are that it, unfortunately, cannot prevent surfaces from self-intersecting and crease curves from escaping to infinity, the proposed method gives simple formulas for the parametrization of crease curves. Using the properties of the geodesic and the ruled surface, design methods for all types of developable surfaces into cylinders and cones are provided.

Folding along the curved crease on the developable surface cannot change the intrinsic curvature of the surface. However, there are still two approaches identified that could alter the global intrinsic Gaussian curvature using the curved-crease origami technique: the curved-crease couplet [86] and folding along the concentric curve crease, which mostly could be categorized as the constructive geometric approach.

Curved-crease couplets were first proposed by Leong [86], which stands for combining pairs of curved and straight creases to create 3D origami with either positive or negative intrinsic Gaussian curvature that has been used by origami artists. Mitani proposed a design method and the software to generate the curved-crease pattern that could derive an origami structure with a rotational symmetric configuration [87]. Appropriate flaps are placed in between the polygonal faces to represent non-zero Gaussian curvature surfaces. The basic idea is to fold a flat sheet that could warp around the target object (a cylinder or a cone), and the extra parts are folded to form a flap, it is different from the design of Resch’s pattern, where the folds are hidden within the geometry [88]. A flap consists of a straight crease and a piecewise straight crease, which could be seen as a curved crease; hence, this forms a kind of curved-crease couplet. The proposed method mainly designs 3D origami structures with cylindrical and conical shapes. Later, Mitani proposed another method that could design 3D axisymmetric objects, the concept of “flap” evolved as the new feature of the method had 3D tucks with a triangular cross section [89].

Concentric pleating is folding the concentric shapes with the assignment of the mountain or valley of the crease pattern so that the geometries with negative Gaussian curvature are shown. This is the second technique that changes the global intrinsic curvature. As previously discussed, the flat sheet is a zero Gaussian curvature surface; hence, it cannot be folded into a non-zero Gaussian curvature surface. However, this could be realized when non-rigid folding is considered. By using the differential geometric constraints for the developable surface, Dias et al. showed that when the sheet is isometrically deformed everywhere except along the concentric fold itself, a relationship of constant curvature and oscillatory torsion is derived for a high stiffness of folds, whereas relatively softer folds result in oscillatory curvature and torsion with broken symmetry of the buckled shape [90]. Still, these works remain a field that is primarily reserved for artists.

Due to the geometric complexity and the variety of the designs, researchers often use analytical methods to build their own models to describe the research question. Inspired by the ring-shape designs that are buckled or folded in tridimensional saddle shapes such as overstrained bicycle wheels, released bilayered microrings and strained cyclic macromolecules, Mouthuy et al. [95] proposed an analytical model using the minimization of the energy method of overcurved rings to capture the shape and buckling behaviour quantitatively. Here, the overcurvature O is defined by the contour length L and the curvature of the ring in the planar shape κ, O = L 2 π κ > 1 , and the bending energy is quantified by the geometry of the overcurved shape.

Regarding the energy method that applied in the compliant mechanism, there is another interesting topic intrigued by curved-crease origami or curve folding: elastic neutral stability. An elastic mechanism in neutral equilibrium is defined to only deform without load if the necessary energy for deformation is already stored in the system and redistributed upon reconfiguration [96]. This unique property was investigated by Guest et al. [97]. Since it requires the energetic state of the structure to remain unchanged during a deformation mode, the neutral stable mechanism features a continuous equilibrium as its behaviour resembles the fascinating class of statically balanced structures [98].

For certain specific structures, especially those with periodic or fractal characteristics in origami, it is possible to first obtain the deformation coordination relationship of the basic unit through discretization methods, and then derive the continuum equation for the entire structure. Dias et al. [99] developed recursion equations to describe the three-dimensional shape of concentric pleating, where folds have been inscribed. Using the differential geometric equations detailed in the previous section, the continuum equations (valid in the limit of vanishing spacing between folds) and the energy of the structures are derived, and the smooth surface intersecting all the mountain folds is described. Multiple folds are considered in their study, but the relationship between two consecutive folds needs to be predefined first. They explored the methods of designing the concentric pleated folds. Two types of folds are considered: closed concentric folds form a saddle-like configuration (which is also demonstrated in Section. 2.2) and open concentric folds result in helical shapes.

The bending facet during the folding of curved crease origami could trigger interesting effects such as the “snap-through” effect, where the facet could change its configuration swiftly during the deformation, or the “pop-through defect” [102, 103], where the local deformed vertex changes its equilibrated position and creates a self-locking effect or bistability of the fold. For thin materials, curvature and mechanics are intimately connected to each other. Inspired by the swift “snap-through” effect, Bende et al. [100] introduced the design rule to explain how to generate the snapping transitions on arbitrary surfaces. This design rule is independent of the material system and the length scale. When the curved surface is deformed to another position by a “pop-through” effect, it could be understood that the partial surface osculates along the fold (defined to have zero normal curvature κ _N ) about a certain angle ψ, and the type of the curved surface is determined by the Gaussian curvature and further determines the curvature properties along the fold. Through the geodesic properties, the bending energy is derived using the mean curvature of the shell near the fold. The finite element method is used to validate the case study.

Similar to the discrete method, the Gauss map method is an efficient tool to analyze the shell damage mechanism and bistable creased strips [104, 105]. As shown in Figure 11, the normal vectors of a closed contour in the vicinity of a given point could be transformed into a unit sphere. The Miura-patterned origami in unfolded state, the planar configuration has zero Gaussian curvature, after the transformation, the overlapping unit normal vectors form a zero surface area, indicating zero Gaussian curvature. After folding the miura pattern, the unit normal vectors of 4 surfaces transform into the Gauss map, unit normal vectors trace out arcs of great circles on the unit spherical surface, with lengths equal to the relative dihedral angles between the facets across hinge lines (the length of C in Figure 11D equals the dihedral angle ϕ ₁₂ between S1 and S2, as the radius of the unit sphere r = 1), the enclosed surface is formed by two same-area surface, but following the sequence from n ₁–n ₄ forms a clockwise and a counter-clockwise surface in Figure 11D, which gives a net-zero surface, indicating that bending does not change the metric of the sheet, the Gaussian curvature will remain zero at (nearly) all points on the folded sheet [106]. The signed area enclosed by these arcs Γ′ is equal to the angular defect at the vertex, which measures the solid angle and, hence, the Gaussian curvature of the vertex P. The Gauss mapping technique involves generalising an arbitrary number of hinge lines, which become the generators of a developable surface as the number increases. It is capable of describing the kinematics of a hinge and facet model, which forms a discrete version of the bistable creased strip. Walker and Seffen [104] studied the creased strip problem. By discretizing the single vertex on the strip into several sectors and deriving the energy function of both regions separately through the analytical method, the energy function could be solved by getting the minimum energy and the net zero Gauss mapping constraint. Although the limitation of this work is that the discrete model that is based on the Gauss map only predicts the final shape of real sheets well for small deflections, large deflections are often observed in experiments [107]. Still, this creased strip with a hole structure with bistability could be used as part of a multistable structure for future applications.

As opposed to the discretized method, Lee et al. [55, 101] presented an analytical geometric construction method for curved-crease origami by combining a 1D elastica solution for large elastic bending deformation with a straight-crease origami projection and mirror reflection process; the proposed method could accurately capture the principal curvature and developability characteristics of the surfaces of the elastically-bent curved-crease origami. Also, by defining the conditions for the appearance of distortion (distortion will occur in the primary surface if secondary pattern creases are parallel to primary pattern rulings), the multistate origami, defined as the structure that could achieve multiple design configurations or objects, could be designed [108]. Elastica curves are the elastically-deformed shapes of a straight, slender beam; therefore, the analytical solution of the “elastica” curve could be derived; however, when used as the curved-crease of the origami, the curved-crease unit cell constructed would have a surface with non-zero Gaussian curvature. Still, this construction technique can generate an equilibrated minimum-energy surface [55]. The compliant folding behaviour of an “elastica” curved-crease origami metamaterial is studied in Ref. [109]. When using the “elastica” curve to build the curved-crease origami metamaterials by tessellating a single unit cell, due to the definition of the curve, it could not be differential at the shared node of two adjacent curves. Since the analytical solution of the “elastica” curve could be derived, the energy function could be generated through methods detailed in Refs. [110, 111]. For a given in-plane curve, such as the case of “elastica,” with the analytical expression f = y(x), the curvature of the in-plane curve could be given by structural mechanics, as shown in Eq. (18). κ = y ″ x 1 + y ′ x 3 2 = M E I (18) where y′(x) and y″(x) are the first and second derivatives of the curve function, respectively, with moment and flexural rigidity values denoting M and EI [55]. Then, the energy function could be written as Eq. (19): U = E I 2 ∫ s κ − κ ′ 2 ⁡ d s (19) where s is the arc length of the curve, and κ and κ′ are the curvatures before and after the deformation at the given point along the curve. The elastica surface generation method in a later experimental and manufacturing study can be concluded to be an effective surface design method to specify the actual shape of an elastically-deformed curved-crease surface [112]. Using a similar method to calculate the energy of the curved-crease origami and the geodesic property that the curve remains in-plane if the folding angle at any given on-curve point is the same, various researchers derive the energy of the structure by deriving the analytical form of the in-plane (after deformation) curve and integration. Sun et al. [42] provide the analytical form of the energy of the curved-crease origami metamaterials. Du et al. [113] proposed analytical models based on differential and integral methods for predicting the compressive stiffness and strength of curved-crease origami foldcores made of composite material, and a three-dimensional failure mechanism map was constructed, where the buckling theory of cylindrically curved plates will be used to estimate the critical stress. This study shows that the dominant failure mode of the curved-crease origami foldcores changed from buckling to crushing under the compressive loading condition. Note that the analytical form of the sandwich structure with curved-crease foldcore is first derived.

In practice, many applications are constrained to situations in which origami structures are made from assemblies of flat, rigid plates connected by hingelike creases. In such situations, the geometric configurations and the kinematics are fully determined by the crease network, while the structural response is a result of the crease network and the crease mechanics. By contrast, when considering the elastic response of the plates (i.e., bending, stretching, and twisting), a variety of new behaviours may emerge. In this case, the elastic response of the structure is determined by the competition between the flexural stiffness of the panels and the torsional stiffness of the creases. This raises interest in studying the mechanical properties of origami structures, the frustrated shaping [114], where the effective curvature of the deployed origami structures in equilibrium could be reached by a crease pattern, and the effects of crease stiffness and sheet thickness on mechanical behaviour [115].