A comparative analysis of numerical approaches to the mechanics of elastic sheets

TL;DR

This paper compares numerical methods for simulating wrinkling in thin elastic sheets, highlighting the advantages of dynamic relaxation over static finite element approaches in terms of robustness and predictive accuracy.

Contribution

The study provides a systematic comparison of static finite element and dynamic relaxation methods for modeling elastic sheet wrinkling, demonstrating the latter's robustness and wider applicability.

Findings

Static finite element methods are highly sensitive to initial imperfections.

Dynamic relaxation methods are less sensitive and more versatile.

Numerical results align well with analytical predictions.

Abstract

Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of wrinkles in these problems have important implications in design and is an area of increasing interest in the fields of physics and engineering. In this work, several numerical approaches previously proposed to model equilibrium deformations in thin elastic sheets are compared. These include standard finite element-based static post-buckling approaches as well as a recently proposed method based on dynamic relaxation, which are applied to the problem of an annular sheet with opposed tractions where wrinkling is a key feature. Numerical solutions are compared to analytic predictions, enabling a quantitative evaluation of the predictive power of the various methods. Results indicate that static finite element approaches are highly sensitive to initial imperfections, relying on \textit{a priori} knowledge of the equilibrium wrinkling pattern to generate optimal results. In contrast, dynamic relaxation is much less sensitive to initial imperfections and can generate solutions for a wide variety of loading conditions without requiring knowledge of the equilibrium solution beforehand.