Pattern Transitions in a Soft Cylindrical Shell

Instability patterns of rolling up a sleeve appear more intricate than the ones of walking over a rug on floor, both characterized as systems of uniaxially compressed soft film on stiff substrate. This can be explained by curvature effects. To investigate pattern transitions on a curved surface, we study a soft shell sliding on a rigid cylinder by experiments, computations and theoretical analyses. We reveal a novel postbuckling phenomenon involving multiple successive bifurcations: smooth-wrinkle-ridge-sagging transitions. The shell initially buckles into periodic axisymmetric wrinkles at the threshold and then a wrinkle-to-ridge transition occurs upon further axial compression. When the load increases to the third bifurcation, the amplitude of the ridge reaches its limit and the symmetry is broken with the ridge sagging into a recumbent fold. It is identified that hysteresis loops and the Maxwell equal-energy conditions are associated with the coexistence of wrinkle-ridge or ridge-sagging patterns. Such a bifurcation scenario is inherently general and independent of material constitutive models.

Figure 1

Experiments of soft shells sliding on rigid cylinders. (a) Patterns upon rolling up a sleeve. (b) An air-inflated latex balloon of thickness $h=0.2\mathrm{mm}$ with one end fixed on an acrylic rod of radius $R=3\mathrm{mm}$. With increasing compression from the right side, four distinguished states are observed: (c) initial undeformed configuration, (d) wrinkle, (e) ridge, (f) sagging ridge. (g)–(j) are representative sketches corresponding to (c)–(f), respectively.

Figure 2

Comparison of theoretical, numerical and experimental results at the critical wrinkling strain ${ϵ}_w$: (a) critical axial strain ${ϵ}_w$ as a linear function of dimensionless curvature $h/R$, (b) wrinkling wavelength ${\lambda }_w$ as a linear function of $\sqrt{Rh}$.

Figure 3

FEM simulations on pattern evolutions of a soft-shell on rigid-cylinder under axial compression: (a) initial undeformed state, (b) wrinkle, (c) ridge, and (d) sagging ridge. In simulations, we took $R=10\mathrm{mm}$ and $h=0.1\mathrm{mm}$.

Figure 4

Bifurcation diagram of dimensionless deflections $w/\sqrt{Rh}$ of the ridge peak (red curve) and its neighboring wrinkle peak (blue curve) as a function of overall compressive strain $ϵ$ ($R=10\mathrm{mm}$ and $h=0.1\mathrm{mm}$). Three successive bifurcations and four phase regions are distinguished: smooth (blue), wrinkle (yellow), ridge (red), and sagging ridge (green).

Figure 5

Comparison between loading and unloading stages ($R=10\mathrm{mm}$ and $h=0.1\mathrm{mm}$). The first line shows the deflections $w$ of the ridge-sagging peak and its neighboring wrinkle peak as a function of overall compressive strain $ϵ$ within hysteresis loops: (a) wrinkle-ridge transition, (b) ridge-sagging transition. (c) and (d) demonstrate the elastic energy differences between the loading and unloading states, $U_l-U_{ul}$, as a function of compressive strain $ϵ$, corresponding to (a) and (b), respectively. In the range ${ϵ}_{r\to w}<ϵ<{ϵ}_{w\to r}$ ($U_l-U_{ul}=U_w-U_r$) and ${ϵ}_{s\to r}<ϵ<{ϵ}_{r\to s}$ ($U_l-U_{ul}=U_r-U_s$), the Maxwell equal-energy conditions are defined by the points when $U_l-U_{ul}=0$.