Buckling of an Elastic Ridge: Competition between Wrinkles and Creases

We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $\varphi$ below a critical value ${\varphi }^{\star }\approx 90°$, experiments reveal an extended sinusoidal mode, while for $\varphi$ above ${\varphi }^{\star }$, we observe a series of creases progressively invading the lateral faces starting from the ridge. A numerical linear stability analysis is set up using the finite-element method and correctly predicts the sinusoidal mode for $\varphi \le {\varphi }^{\star }$, as well as the associated critical strain ${\epsilon }_c(\varphi )$. The experimental transition at ${\varphi }^{\star }$ is found to occur when this critical strain ${\epsilon }_c(\varphi )$ attains the value ${\epsilon }_c({\varphi }^{\star })=0.44$ corresponding to the threshold of the subcritical surface creasing instability. Previous analyses have focused on elastic crease patterns appearing on planar surfaces, where the role of scale invariance has been emphasized; our analysis of the elastic ridge provides a different perspective, and reveals that scale invariance is not a sufficient condition for localization.

Figure 1

Sketch of the experimental setup. To set the prism in axial compression, we first stretch the substrate, then glue the prism to the substrate while keeping the substrate in tension, and finally release the substrate. This induces buckling of the prism: extended wrinkling (top row, $\varphi =40°$) and localized creasing (bottom row, $\varphi =120°$) are observed, depending on the value of $\varphi$. Insets: Experimental pictures.

Figure 2

Experimental results. (a)–(c) Top views in the ($x$, $z$) plane. Each set of pictures is for increasing axial strain $\epsilon$. (a) $\varphi =40°$, $h=10\mathrm{mm}$, and $L_0=100\mathrm{mm}$; (b) $\varphi =90°$, $h=5\mathrm{mm}$, and $L_0=100\mathrm{mm}$; (c) $\varphi =120°$, $h=10\mathrm{mm}$, and $L_0=100\mathrm{mm}$. Black arrows highlight the creases visible on the faces. (d) Critical strain ${\epsilon }_c(\varphi )$. The thin, hand-drawn curves reveal the trend of the experimental data points. Inset: Rescaled wavelength of the extended mode $\lambda /h$ for $\varphi =30°$.

Figure 3

Antisymmetric wrinkling (AW). (a) Phase diagram ${\epsilon }_c(\varphi )$ from experiments (open circles), simulations (disks), and analytical model (dashed curve). (b) Numerical buckling mode for $\varphi =40°$, $h=5\mathrm{mm}$, shown with an arbitrary amplitude. The two color maps show the amplitude of the lateral displacement (left) and of the incremental hoop strain $E_{\theta \theta }^1$ (right).

Figure 4

(a) Full bifurcation diagram, comparing the modes predicted by the linear bifurcation analysis (AW and SW), Biot’s threshold, the nonlinear creasing threshold, and experiments. (b) Numerical linear buckling mode for $\varphi =120°$, $h=0.5$. Sketch of the deformed prism superimposed with a color map of the amplitude of the vertical displacement ${\xi }_y$. Color map of the amplitude of the hoop strain $E_{\theta \theta }^1$. (c) Sketch of the experimental surface creasing (SC) mode for $\varphi =120°$. (d) Sketch of the subcritical bifurcation curve $\mathcal{A}(\epsilon )$ for creasing.