Discontinuous Buckling of Wide Beams and Metabeams

We uncover how nonlinearities dramatically alter the buckling of elastic beams. First, we show experimentally that sufficiently wide ordinary elastic beams and specifically designed metabeams—beams made from a mechanical metamaterial—exhibit discontinuous buckling, an unstable form of buckling where the postbuckling stiffness is negative. Then we use simulations to uncover the crucial role of nonlinearities, and show that beams made from increasingly nonlinear materials exhibit an increasingly negative postbuckling slope. Finally, we demonstrate that for sufficiently strong nonlinearity, we can observe discontinuous buckling for metabeams as slender as 1% numerically and 5% experimentally.

Figure 1

Buckling of slender beams, wide beams, and metabeams. (a) Slender beams in their undeformed (top) and buckled (bottom) states. The force displacement curve for slender beams has a postbuckling slope $S=1/2$. (b) Discontinuous $(S<0)$ buckling of wide beams. (c) Slender metabeams consisting of a nonlinear elastic metamaterial can also exhibit discontinuous buckling.

Figure 2

Discontinuous buckling. (a) Force-displacement curves for beams with aspect ratios ranging from $t=0.034$ (lowest curve) to $t=0.284$ (highest curve) for experiments (solid lines) and finite elements simulations (dashed lines). Here, $\tilde{\sigma }/E≔-F/(Ewd)$ and $\widetilde{ϵ}≔-u/L$. (b) The postbuckling slope in experiments (orange diamonds) and simulations (blue crosses), $S$, increasingly deviates from the elastica limit $1/2$ for large $t$. Discontinuous buckling ($S<0$) occurs for $t>0.12$. (c) Force-displacement curves (rescaled by the buckling stress and strain, ${\sigma }_b$ and ${ϵ}_b$) for metabeams of $t=0.10$, $\ell =0.3$, and various values of $b/a$. Here ${\sigma }_b$ and ${ϵ}_b$ denote the values at buckling. Inset: geometry of our metamaterial. (d) Corresponding postbuckling slope $S$ vs $b/a$.

Figure 3

(a)–(c) Stresses in 3D slender (a) $L\times d=45\mathrm{mm}\times 35\mathrm{mm}$, $t=0.034$) and wide (b) same $L\times d$, $t=0.134$) beams and for a 2D metabeam [(c) $t=0.054$]. The green curves depict the stress-strain relation, the black dashed lines denote linear fits around $\widetilde{{ϵ}_b}$, and the red semicircles denote the range of stresses and strains throughout the beams for $\widetilde{ϵ}/\widetilde{{ϵ}_b}=1.2$. (d)–(f) Energies $E_c$ (dotted, blue line), $E_b$ (dot-dashed, green line), and $E_{NL}$ (dashed, red line) [Eqs. (3)–(5)] and total energy $E_t$ (black) vs. displacement $\widetilde{ϵ}$. A quadratic fit to $E_t$ in the postbuckling regime gives ${\partial }^2{E}_t/\partial {\widetilde{ϵ}}^2=1.6\times {10}^{-2}$ (a), $-1.1\times {10}^{-1}$ (b), and $-4.4\times {10}^{-2}$ (c).

Figure 4

Numerical simulations of 2D metabeams of 5.4% aspect ratio (${ϵ}_b=-0.0096$) and $6\times 111$ holes for the nonlinearity $\eta$ (a) and postbuckling slope $S$ (b) vs $e$ for $\ell =0.13$ (black diamonds), $\ell =0.23$ (pink squares), and $\ell =0.42$ (green circles). See Ref. [30] for movies.

Figure 5

Rational design of discontinuous buckling. (a) Boundaries between ordinary (right) and discontinuous (left) buckling in the ($e$, $\ell$, $t$) parameter space. For each pair of ($e$, $\ell$) values (crosses), simulations for a range of beam thicknesses have been performed. (b)–(d) Large metabeam ($L=520\mathrm{mm}$, $w=9.5\mathrm{mm}$, $d=16.0\mathrm{mm}$, $E=1.1\times {10}^6\mathrm{Pa}$) with 1980 holes with $e\approx 0.1$ and $\ell \approx 0.2$. (e) Experimental postbuckling slope $S$ vs aspect ratio $t$.