Universal Wrinkling of Supported Elastic Rings

An exactly solvable family of models describing the wrinkling of substrate-supported inextensible elastic rings under compression is identified. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The analytical predictions agree with numerical continuation results to within numerical accuracy, for a large range of parameter values, up to the point of self-contact.

Figure 1

Possible regimes of interest. The blue curves represent the solution profiles $\mathbf{r}(s)$. Left panel: competition between pressure and substrate forces ($F>0$) leads to a nontrivial critical wrinkle wave number $m=5$. Right panel: the substrate-free ($F=0$) case leads to a buckled state with $m=2$. The problem variables are shown in the right panel.

Figure 2

Free-ring buckling for $F=0$ and $R_1=1$. Solutions with wave numbers $2\le m\le 7$ (black) bifurcate from the circle solution (gray) as $P$ increases starting with $m=2$. Sample solution profiles for $m=2$ (light blue) and $m=7$ (dark blue) are shown. Crosses represent points of self-contact.

Figure 3

Top: Numerical continuation for two values of ${\alpha }_2$ showing solution branches in the $(T,P)$ plane for $R_1=1,R_2=3.4108$ and wave numbers $m=3$ (red triangles), $m=4$ (purple squares), and $m=5$ (blue pentagons). Solid lines: ${\alpha }_2=0$ (see Fig. 2); dashed lines ${\alpha }_2=64$ (left) and ${\alpha }_2=576$ (right). Colored markers on the dashed lines map to the corresponding markers on the solid lines. Bottom: Color-coded solution profiles at points indicated in the top panels. The solid profiles show the analytical solution while the superposed orange dashed profiles are from numerical continuation (right half of each profile). The solutions agree to within numerical accuracy. In each case, the final profile corresponds to self-contact.

Figure 4

Analytical solutions ($n=4$ blue, $n=6$ green) overlaid with numerical solutions (orange) for ${\alpha }_{4,6}=500$ and $R_1=1,R_2=1$.

Figure 5

The compression $\mathrm{\Delta }\equiv 1-A/\pi$ for area $A$ plotted against maximal curvature ${\kappa }_{\max }$ for $m=6$ and $m=15$. Solutions for $n=2$ (dark red, ${\alpha }_2=10$; light red, ${\alpha }_2=1000$), $n=4$ (dark blue, ${\alpha }_4=10$; light blue, ${\alpha }_4=1000$), and $n=6$ (dark green, ${\alpha }_6=10$; light green, ${\alpha }_6=1000$). Solutions are shown at $A=0.4$ ($\mathrm{\Delta }\approx 0.87$) and $R_1=1$.

Figure 6

Secondary fold states for ${\alpha }_n=576$ and $R_1=1$ at the point of self-contact bifurcating from the first primary branch in each case: $n=2$ (red, bifurcates from $m=5$), $n=4$ (blue, bifurcates from $m=6$), and $n=6$ (green, bifurcates from $m=7$). (a) Fold states with intrusion. (b) Fold states with extrusion. The profiles are strongly dependent on the exponent $n$.