Nonlinear mechanics of rigidifying curves

Area of Science:

• Solid Mechanics
• Geometric Mechanics
• Materials Science

Background:

• Thin shells exhibit high stretching cost relative to bending.
• Isometries of the shell's midsurface are crucial for mechanics.
• Curves with zero normal curvature significantly influence isometry behavior.

Purpose of the Study:

• Investigate the role of zero normal curvature curves in shell isometries.
• Explain the paradox of easier folding across rigidifying curves.
• Incorporate nonlinear strain to understand shell mechanics.

Main Methods:

• Analysis of linear and nonlinear isometries in thin shells.
• Derivation of explicit solutions for nonlinear isometry equations.
• Utilizing coordinate-free geometric arguments for validation.

Main Results:

• Presence of zero normal curvature curves decreases the number of linear isometries.
• Nonlinear strain analysis explains easier folding across these curves.
• Demonstrated folding isometries with explicit mathematical solutions.

Conclusions:

• Zero normal curvature curves have a dual effect on shell mechanics: rigidifying yet facilitating folding.
• Nonlinear strain theory is essential for accurately modeling shell folding behavior.
• Geometric arguments provide a robust framework for analyzing shell isometries.