Fig. 3: Modelling buckling.

a Force F as a function of the engineering strain ε predicted numerically from Eqs. (7) to (9); (blue) solid and (black) dotted lines correspond to the buckled solution branches with and without undulations near the can edges, respectively. Can deformations w/R that correspond to points labelled with i-v are shown in (b–e), respectively, with \({w}_{\max }/R\) indicated in d with the dashed line: b ε = 0.0815 and F = 3300.5 N for (blue) solid profile and ε = 0.134 and F = 3403.1 N for (red) dotted profile, c ε = 0.531 and F = 3387.4 N, d ε = 0.776 and F = 3234.6 N, e ε = 0.578 and F = 3387.4 N. The dashed lines indicate significant quantities, which include the critical strain \({\varepsilon }_{cr}^{pred}\) and force \({F}_{cr}^{pred}\), respectively, which are measured at the point when the localised solutions appear and compared to the experimental onset of buckling in (f), and the maximum strain \({\varepsilon }_{\max }\) for which solutions have been found. (f) Variation in the predicted critical values of the engineering strain \({\varepsilon }_{cr}^{pred}\) and force \({F}_{cr}^{pred}\) with the multiplier γ₂ from Eq. (5), where \({\gamma }_{2}^{f}=2.96\) is the fitted value of γ₂ and \({\sigma }_{{\gamma }_{2}}=0.43\) is the corresponding uncertainty in this fitted parameter, see Supplementary Note 4. The corresponding experimental values from Fig. 2a are indicated using dashed lines. g Variation of the maximal computed strain \({\varepsilon }_{\max }\), as defined in (a), and the maximal amplitude of undulations \({w}_{\max }\) with γ₂.