Modal Folding: Discovering Smooth Folding Patterns for Sheet Materials using Strain-Space Modes

TL;DR

Modal Folding proposes a nonlinear extension of elastic eigenmodes, termed strain-space modes, to automatically discover low-energy folding patterns for thin sheets. By inducing per-element rest-curvature changes and reconstructing shapes via energy minimization, the method achieves large bending deformations with minimal stretching, addressing the limitations of linear modes for folding. The approach supports multi-dimensional subspaces, periodic tilings, and inverse design, with extensive simulations and physical prototypes (fabric, paper, copper) validating the patterns. This framework enables systematic, material-aware design of smooth folding transformations with potential impact on deployable architectures, textiles, and metamaterials.

Abstract

Folding can transform mundane objects such as napkins into stunning works of art. However, finding new folding transformations for sheet materials is a challenging problem that requires expertise and real-world experimentation. In this paper, we present Modal Folding -- an automated approach for discovering energetically optimal folding transformations, i.e., large deformations that require little mechanical work. For small deformations, minimizing internal energy for fixed displacement magnitudes leads to the well-known elastic eigenmodes. While linear modes provide promising directions for bending, they cannot capture the rotational motion required for folding. To overcome this limitation, we introduce strain-space modes -- nonlinear analogues of elastic eigenmodes that operate on per-element curvatures instead of vertices. Using strain-space modes to determine target curvatures for bending elements, we can generate complex nonlinear folding motions by simply minimizing the sheet's internal energy. Our modal folding approach offers a systematic and automated way to create complex designs. We demonstrate the effectiveness of our method with simulation results for a range of shapes and materials, and validate our designs with physical prototypes.

Figures (9)

• Figure 1: Our method automatically generates a diverse set of folding patterns using strain-space modes. Here we show a particular folding transformation for a square sheet of fabric and its physical prototype (bottom right).
• Figure 2: Periodic folding patterns obtained using Modal Folding (a-e) with reflection boundary conditions and their corresponding physical prototypes (right) manufactured using copper sheets and 3D-printed press dies.
• Figure 3: Comparing internal energy and maximum in-plane strain along the first non-rigid mode computed using linear eigenmodes, nonlinear compliant modes (NCM), rotation-strain modes (RSM), and our strain-space modes.
• Figure 4: Modal folding patterns for a square sheet.
• Figure 5: Symmetric modal folding patterns for a disc-shaped sheet.