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A continuum mechanics approach for the deformation of non-Euclidean origami generated by piecewise constant nematic director fields

Linjuan Wang, Fan Feng

TL;DR

This work develops a geometric continuum mechanics framework to design non-Euclidean origami by prescribing piecewise-constant nematic director fields in 2D LCE sheets. It establishes metric-compatibility across interfaces, proves the existence of compatible director patterns at a general $n$-fold vertex (two orthogonal duals in the generic case; a continuous family in degenerate cases), and derives explicit kinematic relations for actuated three-fold and four-fold vertices. Building on these, it presents two scalable designer patterns: a quadrilateral tiling with alternating actuated Gaussian curvature and a mixed three-fold/four-fold pattern governed by a folding-angle theorem, both compatible in reference and actuated states. The framework enables active metamaterials that can morph by folding or external stimulation, with potential applications in deployable structures and programmable shells, and points to future directions in plate theories and general compatibility analysis for non-Euclidean origami.

Abstract

We merge classical origami concepts with active actuation by designing origami patterns whose panels undergo prescribed metric changes. These metric changes render the system non-Euclidean, inducing non-zero Gaussian curvature at the vertices after actuation. Such patterns can be realized by programming piecewise constant director fields in liquid crystal elastomer (LCE) sheets. In this work, we address the geometric design of both compatible reference director patterns and their corresponding actuated configurations. On the reference configuration, we systematically construct director patterns that satisfy metric compatibility across interfaces. We prove the existence and uniqueness of compatible director fields at a vertex for the generic case, up to orthogonal duals. The Gaussian curvature of the actuated vertex is computed based on the compatible director fields. On the actuated configuration, we develop a continuum mechanics framework to analyze the kinematics of non-Euclidean origami. In particular, we fully characterize the deformation spaces of three-fold and four-fold vertices and establish analytical relationships between their deformations and the director patterns. Building on these kinematic insights, we propose rational designs of large director patterns: one based on a quadrilateral tiling with alternating positive and negative actuated Gaussian curvature, and the other combining three-fold and four-fold vertices governed by a folding angle theorem. Remarkably, both designs achieve compatibility in both the reference and actuated states. We also propose a design strategy for active metamaterials based on the periodic non-Euclidean origami. The active metamaterials can have two modes of motions by folding or stimulating.

A continuum mechanics approach for the deformation of non-Euclidean origami generated by piecewise constant nematic director fields

TL;DR

This work develops a geometric continuum mechanics framework to design non-Euclidean origami by prescribing piecewise-constant nematic director fields in 2D LCE sheets. It establishes metric-compatibility across interfaces, proves the existence of compatible director patterns at a general -fold vertex (two orthogonal duals in the generic case; a continuous family in degenerate cases), and derives explicit kinematic relations for actuated three-fold and four-fold vertices. Building on these, it presents two scalable designer patterns: a quadrilateral tiling with alternating actuated Gaussian curvature and a mixed three-fold/four-fold pattern governed by a folding-angle theorem, both compatible in reference and actuated states. The framework enables active metamaterials that can morph by folding or external stimulation, with potential applications in deployable structures and programmable shells, and points to future directions in plate theories and general compatibility analysis for non-Euclidean origami.

Abstract

We merge classical origami concepts with active actuation by designing origami patterns whose panels undergo prescribed metric changes. These metric changes render the system non-Euclidean, inducing non-zero Gaussian curvature at the vertices after actuation. Such patterns can be realized by programming piecewise constant director fields in liquid crystal elastomer (LCE) sheets. In this work, we address the geometric design of both compatible reference director patterns and their corresponding actuated configurations. On the reference configuration, we systematically construct director patterns that satisfy metric compatibility across interfaces. We prove the existence and uniqueness of compatible director fields at a vertex for the generic case, up to orthogonal duals. The Gaussian curvature of the actuated vertex is computed based on the compatible director fields. On the actuated configuration, we develop a continuum mechanics framework to analyze the kinematics of non-Euclidean origami. In particular, we fully characterize the deformation spaces of three-fold and four-fold vertices and establish analytical relationships between their deformations and the director patterns. Building on these kinematic insights, we propose rational designs of large director patterns: one based on a quadrilateral tiling with alternating positive and negative actuated Gaussian curvature, and the other combining three-fold and four-fold vertices governed by a folding angle theorem. Remarkably, both designs achieve compatibility in both the reference and actuated states. We also propose a design strategy for active metamaterials based on the periodic non-Euclidean origami. The active metamaterials can have two modes of motions by folding or stimulating.

Paper Structure

This paper contains 20 sections, 4 theorems, 59 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Deformation of constant director fields and metric compatibility condition
  3. Patterning a compatible $k$-fold director vertex
  4. Two patterning strategies
  5. Concentrated Gaussian curvature at the vertex
  6. Three-fold intersection
  7. Three-fold origami with angular deficit and surplus
  8. Connection to director fields
  9. Four-fold intersection
  10. Reference domain
  11. Compatibility condition and folding angle functions
  12. Calculations of $\gamma_2$, $\gamma_3$ and $\gamma_4$
  13. Calculation of $\gamma_1$
  14. Admissible domain to avoid self-intersection
  15. Connection to director fields
  16. ...and 5 more sections

Key Result

Lemma 3.1

Suppose a $k$-fold director field is compatible at each crease ($|{\bf t}_i \cdot {\bf n}_{i-1}| = |{\bf t}_i \cdot {\bf n}_i|$). Then the orthogonal dual is also compatible ($|{\bf t}_i \cdot {\bf R}_{{\bf e}_3}(\pi/2){\bf n}_{i-1}| = |{\bf t}_i \cdot {\bf R}_{{\bf e}_3}(\pi/2){\bf n}_i|$).

Figures (13)

  • Figure 1: (a) An active origami adapted from plucinsky2018patterning. (b) A generalized design composed of three-fold and four-fold vertices with no symmetry.
  • Figure 2: (a) A disc with constant director is deformed into an ellipse. (b) A sector angle $\alpha$ is deformed to $\tilde{\alpha}$. (c) A metric compatible interface with tangent ${\bf t}$ (or ${\bf t}'$) between two directors ${\bf n}_1$ and ${\bf n}_2$.
  • Figure 3: (a) An n-fold director field and its orthogonal dual for the generic case. The solution for ${\bf n}_i$ is unique up to orthogonal duals. (b) A degenerate case that has a continuous family of solutions for ${\bf n}_i$.
  • Figure 4: The Gaussian curvature of orthogonal duals (red and blue) for the degenerate case with (a) $k=6, r=0.9$, $(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=2\pi(0.2,0.13,0.07,0.2,0.23)$ and (b) $k=6, r=2/3$, $(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=2\pi(0.2,0.13,0.07,0.2,0.23)$. The Gaussian curvature of the degenerate case can be modulated within a wide range (positive or negative).
  • Figure 5: (a) A three-fold non-Euclidean origami constructed through cutting, deforming by ${\bf U}_{{\bf n}}$, and stitching. (b) A three-fold intersection with a deficit ($\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3<2\pi$). (c) A three-fold intersection with a surplus ($\tilde{\alpha}_1+\tilde{\alpha}_2+\tilde{\alpha}_3>2\pi$).
  • ...and 8 more figures

Theorems & Definitions (8)

  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof