Hidden Asymptotic Symmetry in Long Elastic Beams on Softening Foundations

TL;DR

This paper studies how long elastic beams on nonlinear softening foundations exhibit an emergent continuous-like symmetry in their buckling patterns, despite a finite underlying symmetry group. By combining Crandall–Rabinowitz bifurcation theory with a two-scale amplitude analysis, the authors derive an amplitude equation and show that, for very long beams, phase-shifted wrinkle patterns differ only by exponentially small terms. They obtain explicit asymptotics: the primary bifurcations occur at $\lambda=2$ with lengths $L_s=(2n-1)\pi$ and $L_a=2n\pi$, and secondary bifurcations follow from an associated amplitude equation; the long-beam limit yields an asymptotic envelope $A^*(X)=2\sqrt{2/3}\,\mathrm{sech}(X/2)$ and a family of solutions $w_{\infty,\phi}=2\sqrt{2/3}\,\mathrm{sech}(X/2)\sin(x-\phi)$. Finite-element computations validate these results, showing excellent agreement and, for sufficiently long beams, an apparent closed orbit of phase-shifted solutions. This provides a rigorous mechanism for the phase degeneracy observed in wrinkling of membranes and links it to an emergent asymptotic symmetry in a tractable beam model.

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".

Figures (10)

• Figure 1:
• Figure 2: $\xi=\text{max}|w|$ vs. $\lambda=$ load
• Figure 3: Comparison for Anti-symmetric Deformation modes
• Figure 4: Comparison for Symmetric Deformation modes
• Figure 5: Amplitude Envelope for $L_a=40\pi$ and $L_s=45\pi$