# Hidden Asymptotic Symmetry in Long Elastic Beams on Softening Foundations

## TL;DR

This paper investigates hidden asymptotic symmetry in long elastic beams on softening foundations, revealing that phase-shifts of solutions differ by exponentially small terms as beam length increases, supported by asymptotic analysis and numerical validation.

## Contribution

It introduces the concept of hidden asymptotic symmetry in elastic beams, linking phase-shifts to exponentially small differences and providing analytical and numerical evidence.

## Key findings

- Phase-shifts differ by exponentially small terms in long beams.
- Asymptotic results via amplitude equations are validated numerically.
- Solution orbits exist for sufficiently long beams.

## Abstract

Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions, cf. Healey, et. al. [J. Nonlinear Sci., 23 (2013), pp.~777--805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis -- a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations, that are valid as a certain non-dimensional beam length becomes sufficiently large. We deduce that any two phase-shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as "hidden asymptotic symmetry".