Multi-layered folding with voids

Area of Science:

• Solid Mechanics
• Materials Science
• Geophysics

Background:

• Layered materials deform due to competing mechanical properties and geometric constraints.
• Voids form with smooth layers; curvature singularities arise from close packing.
• External pressure influences deformation by penalizing voids, leading to complex patterns.

Purpose of the Study:

• To model geometrically nonlinear multi-layered structures under axial loading and pressure confinement.
• To analyze the trade-offs between mechanical properties and geometry in layered material deformation.
• To investigate the emergence of periodic patterns in compressed layered systems.

Main Methods:

• Constructed a simple model for multi-layered structures with non-interpenetration conditions.
• Characterized energy minimizers via fourth-order nonlinear differential equations and contact-force Lagrange multipliers.
• Numerically solved a fourth-order free-boundary problem to analyze solutions.

Main Results:

• Identified energy minimizers as solutions to a free-boundary problem.
• Demonstrated that external pressure forces a trade-off between voids and curvature.
• Observed striking periodic patterns resulting from these competing effects.

Conclusions:

• The model accurately captures the complex deformation behavior of layered materials.
• Periodic patterns are a consequence of the interplay between geometry, mechanics, and external pressure.
• Numerical solutions align with experimentally observed periodic patterns in layered systems.