Corner singularities and shape of stretched elastic sheets

TL;DR

This study combines experiments, numerical analysis, and asymptotic methods to explore how stretched elastic sheets develop corner singularities and shape features, revealing stress divergence and deformation patterns at boundaries.

Contribution

It provides a comprehensive analysis of corner singularities in stretched elastic sheets, integrating experimental measurements with numerical and asymptotic analysis to understand stress divergence.

Findings

Displacement near corners scales as r^α with α<1.

A diverging stress region exists at the corners.

Good agreement between experiments and finite element analysis except at corners.

Abstract

We investigate the deformation of a longitudinally stretched rectangular sheet which is clamped at two opposite boundaries and free otherwise with experiments, numerical analysis and asymptotic analysis of the biharmonic elastic equation governing their planar equilibrium configurations. The displacement field of the sheet is measured by tracking embedded fluorescent tracers with a digital image correlation (DIC) technique. The experiments and numerical finite element analysis (FEA) are found to be in overall good agreement except at the very corners where large deformations occur. We find that the deformed sheet can be broadly divided into a uniaxially stretched central region and two clamp dominated side regions. A subregion characterized by a diverging stress can be identified at each of the four clamped-free corners within the clamp dominated region. We postulate that the divergence at the corners is regularized by nonlinear elastic deformations occurring in this subregion at the very corners and provide a nontrivial scaling for its size. Within the intermediate corner dominated subregion, measured displacements grow with distance $r$ from the corners as $r^{\alpha}$, with power $\alpha < 1$ consistent with the development of stress singularities at the intersection of the free and clamped edges.