Cusp-Shaped Elastic Creases and Furrows

The surfaces of growing biological tissues, swelling gels, and compressed rubbers do not remain smooth, but frequently exhibit highly localized inward folds. We reveal the morphology of this surface folding in a novel experimental setup, which permits us to deform the surface of a soft gel in a controlled fashion. The interface first forms a sharp furrow, whose tip size decreases rapidly with deformation. Above a critical deformation, the furrow bifurcates to an inward folded crease of vanishing tip size. We show experimentally and numerically that both creases and furrows exhibit a universal cusp shape, whose width scales like $y^{3/2}$ at a distance $y$ from the tip. We provide a similarity theory that captures the singular profiles before and after the self-folding bifurcation, and derive the length of the fold from finite deformation elasticity.

Figure 1

The interface of a soft gel is deformed by slowly reducing the volume of a liquid inclusion (initial radius $R$), located at an initial distance $\ell$ below the free surface. First, a localized furrow with tip curvature $\kappa$ forms; at larger deformations the furrow bifurcates into a crease that folds the free surface onto itself over a length $L$.

Figure 2

Bifurcation between the “furrow” and the “crease” (axisymmetric cavity, $\mu =1\mathrm{kPa}$, $R=2.1\mathrm{mm}$, $\ell =2.4\mathrm{mm}$). (a) Tip radius of curvature ${\kappa }^{-1}$ in the image plane and (b) fold length $L$, as a function of the deformation amplitude $d$. Arrows indicate the course of the experiment. Increasing the deformation (filled symbols) beyond $d_b$ nucleates a fold of finite length. Decreasing the deformation again (open symbols), the fold disappears continuously at $d_0$. The solid lines represent Eqs. (1) and (2) for the curvature and the fold length, respectively. The inset demonstrates the scaling law (1) in a double logarithmic plot (here, $R=1.9\mathrm{mm}$, $\ell =2.4\mathrm{mm}$).

Figure 3

Self-similar evolution of free surface profiles for 3D and 2D experiments, and 2D simulations, prior to the creasing instability. Left: measured (simulated) profiles. Right: profiles rescaled according to Eq. (4), and superimposed with the similarity solution $\mathrm{\Phi }$ (red).

Figure 4

Numerical simulation of the creased state, with a self-contacting fold of length $L$; the tip of the fold is at $T$; the self-contact ends at $C$ (the tip of the cusp). (a) The deformed state $\mathbf{x}$, showing the solid pressure as a color plot; the reference state $\mathbf{X}$ is shown in the inset. (b) Surface profile of the crease above the fold ($x$ and $y$ measured relative to $C$; red line, fit with a $2/3$-power law). The inset shows the $1/2$-power law of the normal (contact) traction near $C$.