Stretch-Induced Bending of Soft Ribbed Strips

We show that ribbed elastic strips under tension present large spontaneous curvature and may close into tubes. In this single material architectured system, transverse bending results from a bilayer effect induced by Poisson contraction as the textured ribbon is stretched. Surprisingly, the induced curvature may reverse if ribs of different orientations are considered. Slender ribbed structures may also undergo a nontrivial buckling transition. We use analytical calculations to describe the evolution of the morphology of the ribbon and the transitions between the different experimental regimes as a function of material properties, geometrical parameters, and stretching strain. This scale-independent phenomenon may help the manufacturing of tubular textured structures or easily controllable grippers at small scale.

Figure 1

Stretching of textured ribbons. (a) Sketch of a portion of the ribbon of width $l$ (top). Cross section of the corrugated ribbed strip with membrane thickness $t$, walls of height $h$ and width $w$ separated by distance $d$ (bottom). (b) Experimental setup: a laser sheet intercepts the ribbon (stretched along the ${\mathbf{e}}_{\mathbf{y}}$ direction) in order to measure its curvature. (c) Ribbon with increasing imposed strain ${ϵ}_0$: the ribbon progressively bends along its transverse direction and eventually rolls itself into a tube.

Figure 2

Stretch-induced curvature. (a) Curvature $\kappa$ as a function of the imposed strain ${ϵ}_0$ for various strip geometries. Green dots: $t=2\mathrm{mm}$; wall density $\varphi =1/2$; $w=0.75\mathrm{mm}$; $0.5\mathrm{mm}\le h<3\mathrm{mm}$; $0\le {ϵ}_0\le 0.6$. Yellow to blue dots: $t=0.5\mathrm{mm}$; wall density $\varphi =1/2$; $0.5\mathrm{mm}\le w\le 1\mathrm{mm}$; $1\mathrm{mm}\le h\le 5\mathrm{mm}$; $0\le {ϵ}_0\le 0.9$. The ribbons are made indifferently of two different silicone elastomers (Elite Double 8 and 22 from Zhermack) of respective Young’s modulus $E=250\mathrm{kPa}$ and $E=650\mathrm{kPa}$, and Poisson ratio $\nu =0.5$. (b) Experimental versus theoretical dimensionless curvature $\tilde{\kappa }=\kappa t$. (c) Curvature-over-strain ratio $\tilde{\kappa }/{ϵ}_0$ [color with scale given by the ordinate axis in plot (d)] as a function of the relative wall height $\tilde{h}=h/t$ and wall density $\varphi =w/(d+w)$. The full line corresponds to the optimal value of $\tilde{h}$ as a function of $\varphi$. (d) Curvature-over-strain ratio as a function of $\tilde{h}$ for $\varphi =0.5$ [dashed line in (c)]. Triangles are obtained by a linear fit of the experimental curves $\kappa =f({ϵ}_0)$ and the black line is the theory (Eq. (1)).

Figure 3

Varying the wall orientation. (a) Ribbon with walls tilted by an angle ${\chi }_0$. (b) Stretched ribbons with walls of various orientations ${\chi }_0$ and thus different transverse curvature directions. (c) Dimensionless curvature $\tilde{\kappa }$ as a function of the applied strain ${ϵ}_0$ for various angles ${\chi }_0$ (colors according to the angle ${\chi }_0$ with the color bar presented in (d); $h=1\mathrm{mm}$; $t=1\mathrm{mm}$; $d=w=0.28\mathrm{mm}$). Solid lines correspond to the theoretical model with shear [Eq. (5)]. (d) Curvature-over-strain ratio at small strain as a function of ${\chi }_0$. Triangles are obtained by a linear fit of the first 6 experimental points in graph (c). The dashed line is the theoretical prediction [Eq. (4)] of the simplified model neglecting shear. The solid black line corresponds to the full model accounting for shear (see the Supplemental Material [29]). A quantitative agreement is obtained for all angles without any fitting parameter.

Figure 4

Wall buckling. (a) Walls for different geometries and stretching ${ϵ}_0$ in a nonbuckling ($a$) and a buckling ($b–d$) regime. The lettering corresponds to the lettered symbols in the diagram. (b) Diagram with the rescaled applied strain ${ϵ}_0/{\epsilon }_c$ as a function of the dimensionless height $\tilde{h}=h/t$ for $\varphi =1/2$. The black lines correspond to the expected buckling threshold [Eq. (6)]. Symbols represent the geometry of the walls: triangles ($h=3\mathrm{mm}$; $w=0.75\mathrm{mm}$); squares ($h=5\mathrm{mm}$; $w=1\mathrm{mm}$); circles ($h=1\mathrm{mm}$; $w=0.5\mathrm{mm}$).