Geometrically controlled snapping transitions in shells with curved creases

Considering isometric deformations, the only contribution to the elastic energy density of folding comes from bending energy $E_B\sim B(H^{\pm }{)}^2$, where the mean curvature of the surface is $H^{\pm }=\frac{1}{2}Tr\left\{I{I}^{\pm }/I\right\}$ and $B$ is the bending rigidity of the shell (see Methods). For surfaces of normal curvature zero, these shells may be folded continuously. However, for ${\kappa }_N\ne 0$, as the surface is folded the normal curvature on one side of the crease must pass through zero, and we find that the mean curvature $H^{\pm }\sim 1/\tan \psi$. In order to fold the surface through $\psi =0$ the surface must develop corners or kinks along the crease so that the mean curvature diverges, similar to stress focusing phenomena seen in other curved surfaces arroyo2003nonlinear (); witten2007stress (); pauchard1998contact (); vaziri2008localized (); vaziri2009mechanics (); nasto2013localization (); datta2012delayed (). Since bending energy scales as the mean curvature squared, the barrier between the folded and unfolded state is infinite for isometric deformations. In any real material, as the shell bends the energy will reach a scale where stretching becomes favorable. Thus, geometry alone gives a prediction for a design rule: by introducing a crease with finite ${\kappa }_N$ an energetic barrier is created between a pair of locally isometric, bi-stable states. For any finite thickness shell, such transitions will require stretching of the surface and often lead to violent snaps.

To place a crease of zero ${\kappa }_N$ on a surface, the surface must have non-positive Gaussian curvature. On a helicoid, examples of such creases are the straight “construction lines” that are used to generate the surface. For these cases the fold angle can be changed continuously and no snapping transition is observed (2a, insets i and ii). For deformation along any other arbitrary crease with finite ${\kappa }_N$, we observe a snap (Fig. 1c, 2a insets 1 and 2, Movie M1).

Creases with finite ${\kappa }_N$ on a cylinder can be created by intersecting the surface with a plane at an oblique angle, as shown in Fig. 1c. According to our hypothesis, we expect the cylinder to undergo a snapping transition when deformed along this crease. Remarkably, despite the introduction of a crease, the cylinder displays global bending deformation instead of snapping, in line with results from other studies on un-creased cylinders boudaoud2000dynamics (); vaziri2008localized (); vaziri2009mechanics () (Fig. 2b). Such global deformations arise since cylinders have $K=0$, and thus can easily bend without stretching, so that there is a pathway accessible to the shell that costs less energy than snapping but still satisfies the constraint imposed by the indenter. These pathways can be eliminated by imposing rigid boundary conditions on the free ends of the cylinder by inserting a rigid cylindrical plug. Without access to these bending modes, the cylinder undergoes a snapping transition that first involves an antisymmetric mode, followed by a full snap (Fig. 2b, insets, Movie M2).

Mechanisms involving pure bending are avoided in spherical shells, since surfaces with $K>0$ naturally require stretching for almost any deformation of the surface. Thus we expect that intersecting a spherical surface with a plane to create a crease (Fig. 1c) with finite ${\kappa }_N$ will result in a snap. We create hemispheres with different crease radius $R_t$ and sphere radius of curvature $R_s$, and define the normalized crease radius as $\alpha =R_t/R_s$. The force displacement curve for these surfaces are shown in (Fig. 2c). For an un-creased shell ($\alpha =0$), we observe a monotonically increasing behavior, similar to previous studiesgupta1999axial (); vaziri2008localized (); nasto2013localization (). Surprisingly, we find that for small $\alpha$ the load displacement curve develops a local minimum, but the “snapped” state remains unstable. For higher values of $\alpha$ the folding pathway leads first to an unstable, non-axisymmetric snap, soon followed by a well-defined stable snap (Fig. 2c, insets, Movie M3).

To understand how stability depends on $\alpha$, we consider the classical problem of indenting a spherical shell (Fig. 3a,b). There is a well-known nearly isometric deformation of a sphere, the Pogorelov ridge, which is seen for displacements larger than the thickness but smaller than the crease size landau1959course (). This deformation regime is characterized by an inverted bulge of radius $r$ and bounded by a ridge of size $\ell \sim \sqrt{t{R}_s}$ (Fig. 3b). The energy for this state has a bending energy contribution from the inverted bulge that scales as $E_B\sim B(r/R_s{)}^2$, while the Pogorelov ridge, which contains all the stretching energy, scales as $E_P\sim Y(t/R_s{)}^{5/2}{r}^3$, with $Y$ the Young’s modulus of the material. Hence the total energy for deformation scaled by $B$ can be expressed as

In the case of a creased sphere, we assume that the deformation of the shell retains the same structure as this classical solution, but now the thickness of the sphere in the Pogorelov ridge is a function of the bulge radius $r$. Since the crease is localized to a small region on the surface (Fig. 2b), we define $t\equiv t_{0}g(r-R_t)$, where $t_0$ is the regular thickness of the shell and the function $g\left(x\right)$ is of the generic localized form such that $g\left(0\right)=ϵ\ll 1$ and $g(x\to \infty )=1$. With this assumption, the energy for the modified Pogorelov solution becomes

The balance of these two energies yields a critical Pogorelov radius $r^{\ast }$ such that a local minimum will appear at a finite value of $r$. If $r^{\ast }<R_t$, this bi-stability is accessible to the shell. Solving Eq. (4) for $r^{\ast }$, and setting $r^{\ast }\equiv R_t$ yields the scaling law

To confirm this hypothesis, we use finite element simulations (ABAQUS, Dassault Systemes) to determine the conditions under which there is a stable snap. For linearly elastic materials this system is fully characterized by two dimensionless numbers, the reduced crease radius $\alpha$ and the Föppl-von Kármán number $\gamma$. We report the total energy for axisymmetric solutions with $\gamma ={10}^4$ (corresponding to the elastomeric hemispheres discussed here), as a function of the indenter displacement ($h$) and the normalized crease radius ($\alpha$) in Fig. 3c. We find that, beyond a critical crease radius, there is a bifurcation of stability and the energy curves develop a well-defined local minimum (solid) and maximum (dashed), with the region between these curves denoting a basin of attraction for the folded state.

By examining creased hemispherical shells over a wide range of $\gamma$, we identify how the critical crease radius necessary for generating a stable snap depends on shell thickness (Fig. 4). Through numerical simulations, we find that for increasing thickness larger values of the crease radius are required to create a stable snap. Moreover, the boundary between mono- and bi-stable regimes is well-captured by the scaling argument suggested by our modified Pogorelov ridge calculation. As a final test of our theory and simulations, we conduct a series of experiments on spherical shells with a range of $\gamma$ and $\alpha$, and identify the stability of the folded state. These too reveal a boundary between bi-stability and mono-stability, that is in excellent agreement with our numerical calculations and theoretical law. Moreover, for some samples we observe the presence of folded states that are temporarily stable (for times on the order of seconds); the proximity of these samples to the predicted phase boundaries further demonstrates the agreement between experiment, simulation, and theory.

3-dimensional models of different geometries were designed in a CAD software. The non-Euclidean geometries (helicoid and hemisphere) were fabricated using a commercial 3d printer (uDimensions, Stratys Inc.) to obtain two-part molds with embossed features to generate creases. The hemispherical shells were fabricated using poly(vinyl siloxane) by curing a commercially available two part base-catalyst mixture (Zhermack SpA Elite Double 32, Elastic modulus $\left(Y\right)\sim 100$ kPa). Prior to filling the mold, the 1.2:1 base:catalyst mixture was degassed for removing bubbles which may defect the sample. Helicoid samples were fabricated using poly(caprolactone) (Monomer-Polymer & Dajac Labs, 1258, $Y=353$ MPa), by first melting the polymer, filling and cooling down inside the molds. The hemispherical and helicoid shells studied were 1 mm thick, and the crease had a rectangular cross-section 0.75 mm deep ($ϵ=0.75$) and 1mm wide along the appropriate curve. Only the samples without any bubbles/ other structural defects were included for testing. Owing to their Euclidean nature, cylinders could be fabricated using a conventional two dimensional technique. Here, we used a commercial laser cutter (Zing Epilog 16) to score a poly(ethylene terephthalate) sheet (Grafix Dura-Lar®, 120 $\mu$m thick, $Y\sim 5$ GPa) with a curve. The shape of this plane curve is set to be sinusoidal such that when the sheet is wrapped to form a cylinder, the resulting space curve is the intersection between a plane and a cylinder at an oblique angle.

A custom-built force displacement device, combining a linear translation stage (Zaber Technologies Inc., T-LSM 100) and a load cell (Loadstar Sensors Inc., RPG-10), was used to perform strain controlled force measurements. Point indenters (radius ratio of indenter with respect to shell $\sim 0.05$) were 3D printed, and samples were indented at a strain rate of 5 mm/min. Data collection and analysis was performed using an in-house algorithm in MATLAB (The Mathworks), without any signal processing/ filtering.

The crease is characterized by a space curve that lies entirely on the surface of the shell (Fig. 1a), parametrized by a tangent vector $\hat{t}$. The derivative of the tangent along the curve defines a vector whose magnitude is the curvature $\kappa$, and whose direction is given by the Frenet unit normal ${\hat{N}}_F$. At any point on the surface, the vector $\hat{u}$ that is simultaneously tangent to the surface and orthogonal to $\hat{t}$ makes an angle $\psi$ with the Frenet normal (Fig. 1a, left), while the normal to the surface is defined by the vector $\hat{n}$. The surface itself is composed of two regions that are divided by the crease, each parametrized by a local orthonormal frame. In a frame of reference where one surface is fixed in space a local orthonormal frame $\{\hat{t},{\hat{u}}^{+},{\hat{n}}^{+}\}$ defines the two surfaces in the unfolded state. When the surface is folded, another frame $\{\hat{t},{\hat{u}}^{-},{\hat{n}}^{-}\}$ is used to signify the change from the undeformed state.