Stretching Hookean ribbons part II: from buckling instability to far-from-threshold wrinkle pattern

Area of Science:

• Materials Science
• Solid Mechanics
• Physics of Soft Matter

Background:

• Wrinkling in stretched elastic sheets is a complex phenomenon.
• The Cerda-Mahadevan model describes wrinkle patterns in a "far-from-threshold" regime.
• Previous hypotheses suggested vanishing transverse strain drives wrinkle evolution.

Purpose of the Study:

• To investigate the theoretical underpinnings of wrinkle pattern formation in Hookean sheets under specific loading conditions.
• To elucidate the role of stress and strain in the development of fully formed wrinkles.
• To explore the relationship between sheet properties, load, and wrinkle characteristics.

Main Methods:

• Utilized Surface Evolver simulations for analyzing wrinkle patterns.
• Investigated a range of sheet thicknesses and tensile loads.
• Simulated a Hookean sheet with finite stretching modulus but no bending rigidity to compute the compression-free stress field.

Main Results:

• Wrinkle evolution is accompanied by the collapse of transverse compressive stress, not vanishing transverse strain.
• The stress field asymptotically approaches a compression-free limit, consistent with tension field theory.
• The singular limit of zero bending rigidity captures the geometric nonlinearity of wrinkle amplitude-wavelength ratios.

Conclusions:

• The study refines the understanding of wrinkle formation in elastic sheets, particularly in the "far-from-threshold" regime.
• Transverse stress collapse is identified as the key mechanism driving wrinkle development.
• The findings provide insights into the balance of bending and stretching energies governing wrinkle wavelength.