Complete closed-form solutions to the problem of inextensional bending for surfaces of translation and origami tessellations

1. Problem Statement

The paper addresses the challenge of characterizing inextensional (isometric) deformations in thin shells and origami tessellations.

• Context: Thin shells generally possess six deformation modes: three high-energy membrane modes (stretching) and three low-energy bending modes (inextensional). For complex geometries like corrugated shells and origami, identifying which modes are truly inextensional (zero strain energy to leading order) is difficult.
• The Gap: While isometric deformations are well-understood for simple surfaces (e.g., developable, axisymmetric), finding closed-form solutions for surfaces of translation (surfaces swept by translating one curve along another) that include curved creases, straight creases, and smooth variations has been elusive.
• Specific Challenge: Existing methods often rely on discretizing panels into triangles (adding artificial diagonal creases) or fail to handle curved creases properly. Furthermore, for periodic origami patterns (like Miura-ori or Eggbox) that bend into doubly curved shapes, the fold angles must be non-periodic, making analytical characterization difficult.
• Goal: To derive complete, closed-form analytical expressions for the three infinitesimal inextensional deformation modes (stretching, bending, and twisting) for a broad class of surfaces of translation, valid for both smooth and piecewise-smooth (creased) geometries.

2. Methodology

The authors employ a direct construction approach based on differential geometry and partial differential equations (PDEs).

• Geometric Setup: The surface is defined as a surface of translation $r(x, y) = \alpha(x) + \beta(y)$, where $\alpha$ and $\beta$ are planar curves. This formulation allows for overhangs and curved creases, provided the curves lie in perpendicular planes and their tangents do not intersect the plane intersection.
• Isometric Constraints: The problem is formulated by requiring the infinitesimal change in the metric tensor ($\delta I = 0$). This leads to a system of three first-order linear PDEs for the displacement components $(u, v, w)$ in a specific adapted basis.
• Variable Separation Ansatz: The authors solve the PDE system by proposing specific ansatzes for the out-of-plane displacement component $w(x, y)$, separating variables to decouple the equations:
  1. Twisting: $w(x, y) = X(x)Y(y)$
  2. Stretching: $w(x, y) = X(x) + Y(y)$
  3. Bending: $w(x, y) = X(x)Y(y) + S(x) + T(y)$
• Integration and Correction: The authors integrate the resulting equations to find displacement fields. Crucially, they derive periodic correctors (integration constants that are periodic functions) to ensure the solutions are valid for periodic surfaces without introducing secular terms (unbounded growth) that would violate physical constraints.

3. Key Contributions

1. Closed-Form Solutions: The paper provides the first complete set of closed-form analytical expressions for the three inextensional modes (twisting, stretching, bending) for surfaces of translation. These solutions are valid for:
   • Smooth surfaces.
   • Surfaces with straight creases (e.g., standard Miura-ori).
   • Surfaces with curved creases (e.g., curved-crease origami).
   • Surfaces with overhangs.
2. No Artificial Discretization: Unlike previous approaches that required adding diagonal creases to triangularize panels, this method works directly on the continuous or piecewise-smooth surface geometry.
3. Effective Poisson's Effect: The derivation explicitly links the geometry of the surface to an effective Poisson's coefficient. It demonstrates how the ratio of normal curvatures in bending relates to the effective strain in stretching.
4. Completeness Proof: The work confirms that for simply connected periodic surfaces, there are exactly three linearly independent inextensional modes, and the derived family spans this complete space.

4. Key Results

The paper derives explicit displacement vectors for three distinct modes:

• Mode 1: Twisting

  • Describes a torsional deformation.
  • For periodic surfaces, the solution converges to the standard Kirchhoff-Love plate twisting mode ($xy$-dependence) plus periodic corrections.
  • Valid for both smooth and creased surfaces.

• Mode 2: Stretching (Effective Stretching)

  • Although the deformation is locally isometric (no stretching), the macroscopic behavior mimics stretching.
  • Poisson's Ratio Transition: The effective Poisson's ratio $\nu(\theta)$ is derived as a function of the inclination angle $\theta$ between the profile planes.
    • At $\theta = 0^\circ$ (Eggbox pattern), the material exhibits a positive Poisson's ratio (stretch in one direction, contract in the other).
    • At $\theta = 90^\circ$ (Miura-ori), the material exhibits a negative Poisson's ratio (auxetic) (contract in both directions).
    • A critical angle exists where the denominator of the strain expression vanishes, causing a sign switch in the Poisson's ratio.

• Mode 3: Bending

  • This is the most complex result. The authors identify that bending solutions are linear combinations of symmetric and permuted solutions derived from the stretching ansatz.
  • Curvature-Stiffness Link: The paper establishes a rigorous relationship: the ratio of the normal curvatures in the bending solution equals the negative of the effective Poisson's ratio found in the stretching solution ($\kappa_x / \kappa_y = -\nu$).
  • The solutions include specific periodic correctors that were previously only computable via numerical algorithms for simple cases. These correctors are essential for reconstructing local kinematics from global deformation.

• Tubular Structures: The authors extend the results to tubular surfaces (e.g., deployable booms), showing how the modes translate to torsion, cross-section changes, and bending, while highlighting topological constraints (dislocations) that arise in closed cross-sections.

5. Significance and Applications

• Benchmarking: The closed-form solutions serve as a rigorous benchmark for validating numerical methods (Finite Element Analysis) and reduced-order models (bar-and-hinge models) used in origami mechanics.
• Metamaterial Design: By providing explicit formulas for effective stiffness and Poisson's ratios, the work aids in the design of mechanical metamaterials with tunable properties (e.g., switching between auxetic and non-auxetic behavior by changing the angle $\theta$).
• Stiffness Prediction: Since inextensional modes dominate the response of thin shells ($h/R \ll 1$), these solutions allow for the precise prediction of the elastic stiffness of complex corrugated shells and origami structures without expensive simulations.
• Generalization: The methodology bridges the gap between discrete origami (faceted) and continuous differential geometry, offering a unified framework for analyzing structures with curved creases, which are increasingly important in soft robotics and deployable space structures.

In summary, this paper solves a long-standing geometric problem by providing a unified, analytical framework for the low-energy deformation modes of a wide class of origami and corrugated shells, revealing deep connections between surface geometry, effective material properties, and deformation kinematics.