Introduction

What is your favorite bistable structure? For those who use various bistable structures in their daily lives, it may be difficult to choose just one favorite. In this article, I will introduce my favorite bistable structure.

A bistable structure is one that exhibits two different stable states (bistability). In a previous article, we introduced bistable structures using a hinge cap as an example. I also mentioned that a bistable structure becomes stable when the elastic energy is at a minimum and transitions to the other stable state by overcoming an energy peak.

Slap bands

My preferred bistable structure is the slap band.

Also known as a snap bracelet or one-touch band[1]. Many people use them as reflective materials to be visible from a distance at night (or as a toy to enjoy bistability).

The slap band wraps around the arm or other parts of the body when struck, making it easy to attach. This behavior is made possible by the bistability of slap bands, which are stable in both the stretched and rounded states. When the slap band is struck, it overcomes the elastic energy peak, releasing the elastic energy stored in the stretched state, and instantly transitions to the other stable state (rounded state).

When the cover of the slap band is removed and its contents are examined, a thin steel plate with bistability is found inside. From now on, this plate itself will be referred to as the slap band, and its characteristics will be explained.

Curved Surface Shape

What makes slap bands unique compared to other bistable structures is their curved surface shape. The curved surface of a slap band can be expanded into a flat surface without stretching or cutting.

In contrast, most bistable structures with curved surfaces require stretching, cutting, or bending to be flattened in 3-D space. Examples are shown in the following pictures.

\(\scriptsize{ 
\text{Examples of bistable structures utilizing non-developable(*1) surfaces: }\\
\text{(Left) A toy called the bouncing ball(*2), which features a double curved surface (hemispherical). }\\
\qquad\text{         When dropped on the ground in an inverted state, it releases stored elastic energy and bounces high.}\\
\text{(Middle) A collapsible bucket, purchased for 550 yen at a 100-yen shop.}\\
\qquad\text{              The gray rubber part stabilizes in an inverted state, allowing the volume to be reduced for storage.}\\
\text{(Right) A hairpin, where a double curved surface(*3) is created by connecting flat plates at the ends.}
}\)

The bistability of such a curved surface is created by the balance between the restoring force due to bending stiffness and the restoring force due to in-plane stiffness. (Because of in-plane expansion and contraction during deformation, large deformation by hand requires the use of soft materials such as rubber, or even metal with small dimensions, such as hairpins.)

As mentioned above, slap bands are curved surfaces that can be deformed into a flat surface. Since such a curved surface does not expand or contract in-plane during bending deformation, simply creating a curved surface does not make a bistable structure. So, how is bistability created?
\(

\)

\(\tiny{
\text{Notes:}\\
\text{(*1) In the classification of surfaces in diagrammatics, developable surfaces are ruled surfaces and planes, which allow cutting and bending. Here, "non-developable surfaces" are described as surfaces that are not globally-developable.}\\
\text{(*2) The bouncing ball is also referred to by various names, including "Bowl," "Ball," and "Hip Hop Bounce Fidget Toy."}\\
\text{(*3) Strictly speaking, the hairpin does not fall under the definition of a curved surface since it is not continuous. However, for simplicity, it is considered a double curved surface on a macroscopic scale. }
}\)

Residual Stress

The key to the bistability of slap bands lies in the residual stresses. When the slap band is flat, there are residual stresses in orthogonal directions on the front and back.

These stresses create a restoring force in each direction that stabilizes the slap band in its bent state. However, for a surface that can be expanded into a plane (a single curved surface) to have curvature in both directions at the same time (i.e., to become a double curved surface), the surface must expand or contract. Therefore, a slap band deforms in a way that has curvature in only one direction (a single curved surface).

When the slap band has curvature in one direction, the residual stress alone is not enough to keep it from bending due to the larger second moment of area with respect to the other direction. Bistability is created because this stabilization occurs in both directions.

Let's Make It

Let's try to make a craft using familiar materials to easily understand the principle.

■Preparations

• Rubber bands
• Scissors
• A flat plate with moderate elasticity against bending
  \(\small{〇\> \text{A document file or thick paper would be good. In this case, I used a business card.} }\)
  \(\small{〇\> \text{Homogeneous isotropic materials are more suitable to achieve the intended deformation.} }\)
  \(\small{〇\> \text{Materials that do not creep (stress relaxation) are better for long-term use.} }\)

■ Procedure

1. Cut slits in the corners of the prepared flat plate.
   
2. Insert rubber bands as shown in the figure. Adjust the tension of the rubber bands to make them even. If the tension is not enough, you can attach several rubber bands.
   
3. Completed!
   

In the crafting above, the restoring force due to residual stress is represented using rubber bands. However, the slap band discussed in this article uses residual stress caused by plastic deformation. There is a video demonstrating the fabrication of a bistable slap band by bending a convex metal strip until it reaches plastic deformation[3].

Similar bistability can also be achieved using CFRP (Carbon Fiber Reinforced Plastics) by carefully designing the stacking configuration. During curing, the thermal shrinkage anisotropy generates orthogonal residual stresses on the front and back sides [4-6]. (For example, the stacking configuration should be an asymmetric cross-ply, such as \([0_n/90_n]\).)

Similar Structures

The bistability mechanism of the slap band is also utilized in other structures. In this article, we will introduce two bistable structures that use a similar mechanism.

Venus Flytraps

Venus Flytraps are believed to use the same bistable mechanism as slap bands [7-9]. When the sensory hairs on the inner surface of the bivalve-like leaf are stimulated, the cells contract, causing the leaf to close. At this moment, the elastic energy stored in the open state is released, allowing the leaf to close quickly and capture prey.

\(\scriptsize{\text{Adapted from [7]}}\)

Bistable STEM

There is a type of extensible boom used in space deployable structures called a STEM (Storable Tubular Extendible Member) [10]. It is stored in a coiled form during transportation to save space, and then extended in space for use.

\(\scriptsize{\text{Adapted from [11]}}\)

Some STEMs are bistable, meaning they can remain stable in both the wound and extended states [11,12]. These bistable STEMs are manufactured with CFRP. Bistable STEMs do not require additional mechanisms to counteract reaction forces in the wound-up state, thus simplifying the storage mechanism. They can also be self-extensible, transitioning to an extended state when partially deformed (opposite to a slap band, which tends to retract). Thanks to their stowability and self-extensibility, these STEMs are sometimes used for antennas and tripods on the ground [13,14].

In a slap band, the convex surface inverts between the two stable states. In contrast, in a bistable STEM, the convex surface remains on the same side even when the shape changes. This is achieved not only through residual stress but also by utilizing the properties of CFRP. The details of this mechanism will be described later in this article.

Conditions for Bistability

You might be curious about the conditions under which a slap band exhibits bistability. I will explain this using equations.

We will derive the conditions under which multiple local minima of strain energy exist for a singly curved surface with residual stress, resulting in stable states.

Energy Calculation

When a single curved surface with uniform curvature deforms without in-plane deformation, the surface state can be described by two curvatures \(\kappa\) and its direction \(\theta\). As shown in the figure, when the curvature \(\kappa\) has an axis with the angle \(\theta\>(0\le\theta＜\pi)\) direction, the curvature in the xy coordinate is expressed as follows. (derived using the strain transformation matrix)

\([\kappa]=
\begin{bmatrix}
c^2 & s^2 & sc\\
s^2 & c^2 & -cs\\
-2sc & 2sc & c^2-s^2\\
\end{bmatrix}
\begin{bmatrix}
0\\ \kappa\\ 0\\
\end{bmatrix}
=\kappa \begin{bmatrix}
\sin^2\theta\\ \cos^2\theta\\ 2\sin\theta\cos\theta\\
\end{bmatrix}
=\kappa/2 \begin{bmatrix}
1-\cos2\theta\\ 1+\cos2\theta\\ 2\sin2\theta\\
\end{bmatrix} \tag{1}\)

Let it be assumed that the radius of curvature is \(R(> 0)\) initially and that moment loads \(m_0\) are generated due to residual stresses. Assume that the curvature is in the y-direction and the moment load is applied in the x-direction.

\([\kappa_0]=
\begin{bmatrix}
0\\ 1/R\\ 0\\
\end{bmatrix},
[m_0]=
\begin{bmatrix}
m_0\\ 0\\ 0\\
\end{bmatrix}\tag{2}\)

In this case, the change in curvature from the initial condition \([\Delta\kappa]\) and the moment load per unit dimension \([m]\) are as follows

\([\Delta\kappa]=[\kappa]-[\kappa_0]=\kappa/2
\begin{bmatrix}
1-\cos2\theta\\ 1+\cos2\theta\\ 2\sin2\theta\\
\end{bmatrix}-\begin{bmatrix}
0\\ 1/R\\ 0\\
\end{bmatrix}\\\tag{3}\)

\([m]=[D][\Delta\kappa]+[m_0]\tag{4}\)

where \([D]\) is the bending stiffness matrix under plane stress state, and for isotropic materials it is expressed as

\([D]=D\begin{bmatrix}
1 & \nu & 0 \\
\nu & 1 & 0\\
0 & 0 & (1-\nu)/2\\
\end{bmatrix}\tag{5}\)

The coefficient \(D\) is \(D=\frac{t^3E}{12(1-\nu^2)}\) assuming a central neutral plane for a flat plate of thickness \(t\). (Derived from the balance of moments when integrated in the thickness direction. If you want to know more details, please refer to the classical lamination theory mentioned in a previous article.)

From equation (4), the strain energy per unit area \(U\) is

\(\begin{align}
U &= 1/2[\Delta\kappa]^T[D][\Delta\kappa]+[\Delta\kappa]^T[m_0]\\
 &= \frac{1}{2} \left\{
 \frac{D}{R^2}-\kappa(D\kappa+m_0)-
\kappa\left\{\frac{D(1+\nu)}{R}+\left(\frac{D(1-\nu)}{R}+m_0\right)\cos2\theta\right\}
\right\} \tag{6}
\end{align}\)

Determining Extremum

For the strain energy \(U\) to be a stopping point (Extremum or saddle point) with respect to \(\kappa\) and \(\theta\), the derivative with respect to each variable must be zero. Additionally, if the Hessian matrix at that point is positive definite, the point is a local minimum.

\(\frac{\partial U}{\partial\theta}=\frac{\partial U}{\partial\kappa}=0, \>|H_U(\theta,\kappa)| > 0\Rightarrow \text{Local minimum}\)

Focusing on \(\theta\), we find that \(\frac{\partial U}{\partial\theta}=0\) when \(\theta=0,\pi/2\).

(A) when \(\theta=0\),

With the initial condition \(\kappa=1/R\) ,\(\frac{\partial U}{\partial\kappa}=0\). At this point, the Hesse matrix becomes

\(H=\begin{bmatrix}
D &  0 \\
0 & \frac{2}{R}(\frac{D(1-\nu)}{R}+m_0)\\
\end{bmatrix}\tag{7}\)

Thus, this point is local minimum when

\(\begin{align}
detH=\frac{2D}{R}\left(\frac{D(1-\nu)}{R}+m_0\right) > 0\\
m_0 > -\frac{D(1-\nu)}{R}\tag{8}
\end{align}\)

(B) When \(\theta=\pi/2\),

the partial derivative with respect to \(\kappa\) is

\(\frac{\partial U}{\partial\kappa}=m_0+D\kappa-D\nu/R \tag{9}\)

and it satisfies \(\frac{\partial U}{\partial\kappa}=0\) when \(\kappa = \frac{\nu}{R}-\frac{m_0}{D}\). At this point, the Hessian matrix is as follows.

\(H=\begin{bmatrix}
D &  0 \\
0 & 2(\frac{\nu}{R}-\frac{m_0}{D})(-\frac{D(1-\nu)}{R}-m_0)\\
\end{bmatrix}\tag{10}\)

Thus, this point is local minimum when

\(\begin{align}
detH=2\left(\frac{D\nu}{R}-m_0\right)\left(-\frac{D(1-\nu)}{R}-m_0\right)>0\\
m_0＜-\frac{D(1-\nu)}{R}, \frac{D\nu}{R}＜ m_0\tag{11}
\end{align}\)

From equations (8) and (11), we find that when \(\frac{D\nu}{R}＜ m_0\), there are two points where the system is stable:

A. \(\theta=0, \kappa = \frac{1}{R}\)

B. \(\theta=\pi/2, \kappa = \frac{\nu}{R}-\frac{m_0}{D}\> (＜ 0)\)

In this case, we can see that the curvature at point B is negative, indicating that the convex surface inverts.

As an example, we visualize the strain energy for the case where \(D=1, \nu=0.3, R=1, m_0=0.5\). In the graph, the radial direction represents \(\kappa\> (-1.5＜\kappa＜1.5)\), and the circumferential direction represents \(2\theta\> (0\le\theta＜\pi)\).

From the graph, we can see that points A and B are local minima. Additionally, when considering point A as the rolled state and point B as the extended state in a slap band, the following observations can be made:

• The energy at B (extended state) is higher than at A (rolled state), with a larger transition range to A.
• The curvature at B (extended state) is closer to 0 than at A (rolled state), in other words, A is in a more planar state.

Moreover, the strain energy for cases where \(m_0\) does not meet the above bistable conditions is shown in the figure below. In these cases, there is either only one local minimum or none. (It is also interesting to note that when \(m_0=-\frac{D(1-\nu)}{R}\), as shown in the center of the figure, \(\frac{\partial U}{\partial\theta}=0\) holds true for all values, resulting in zero stiffness with respect to angular changes [15].)

In Bistable STEMs

As mentioned above, when a curved surface forms a bistable structure, the convex surface inverts between the two stable states. However, in the bistable STEMs introduced as a similar structure, the convex surface remains on the same side in both stable states. This behavior is achieved by utilizing the anisotropy of FRP. The bending stiffness matrix of an orthotropic material can be described as follows.

\([D]= \frac{t^3}{12}
\begin{bmatrix}
\frac{E_X}{1-\nu_{XY}\nu_{YX}} & \frac{\nu_{YX}E_Y}{1-\nu_{XY}\nu_{YX}} & 0\\
\frac{\nu_{YX}E_Y}{1-\nu_{XY}\nu_{YX}} & \frac{E_Y}{1-\nu_{XY}\nu_{YX}} & 0\\
0 & 0 & G_{XY}\\
\end{bmatrix}
=
\begin{bmatrix}
D_{11} & D_{12} & 0\\
D_{12} & D_{22} & 0\\
0 & 0 & D_{66}\\
\end{bmatrix} \tag{12}\)

Using this matrix, we will consider the energy in the initial state with a curvature radius \(R(> 0)\) and a moment load \(m_0\), employing the same method as for the isotropic material mentioned above.

\(U=1/2[\Delta\kappa]^T[D][\Delta\kappa]+[\Delta\kappa]^T[m_0]\)

Focusing on \(\theta\), \(\frac{\partial U}{\partial\theta}=0\) when \(\theta=0,\pi/2\).

(A) When \(\theta=0\),

As with the isotropic material, \(\kappa=1/R\) satisfies \(\frac{\partial U}{\partial\kappa}=0\). At this point, the Hessian matrix is as follows.

\(H=\begin{bmatrix}
D_{22} &  0 \\
0 & \frac{2}{R}(\frac{2D_{66}}{R}+m_0)\\
\end{bmatrix} \tag{13}\)

When the determinant is positive, this point is local minimum.

\(\begin{align}
detH=\frac{2D_{22}}{R}\left(\frac{2D_{66}}{R}+m_0\right) > 0\\
m_0 > -\frac{2D_{66}}{R} \tag{14}
\end{align}\)

(B) When \(\theta=\pi/2\),

\(\frac{\partial U}{\partial\kappa}=m_0+D_{11}\kappa-D_{12}/R \tag{15}\)

and \(\kappa = \frac{D_{12}}{D_{11}R}-\frac{m_0}{D_{11}}\) satisfies \(\frac{\partial U}{\partial\kappa}=0\) (critical point).
At this point, the Hessian matrix is as follows.

\(H=\begin{bmatrix}
D_{11} &  0 \\
0 & \frac{2}{D_{11}^2R^2}(D_{12}-m_0R)\{-D_{11}D_{22}+(D_{12}+2D_{66})(D_{12}-m_0R)\}\\
\end{bmatrix}\tag{16}\)

When the determinant is positive, this point is local minimum.

\(\begin{align}
detH= \frac{2}{D_{11}R^2}(D_{12}-m_0R)\{-D_{11}D_{22}+(D_{12}+2D_{66})(D_{12}-m_0R)\}>0\\
m_0＜\frac{1}{R}(D_{12}-\frac{D_{11}D_{22}}{D_{12}+2D_{66}}),\frac{D_{12}}{R} ＜ m_0 \tag{17}
\end{align}\)

From equations (14) and (17), the system is stable at the following two points when either \((\alpha)\> \frac{D_{12}}{R} ＜ m\), or \((\beta)\> -\frac{2D_{66}}{R} ＜ m ＜ \frac{1}{R}(D_{12}-\frac{D_{11}D_{22}}{D_{12}+2D_{66}})\):

A. \(\theta=0, \kappa = \frac{1}{R}\)

B. \(\theta=\pi/2, \kappa = \frac{D_{12}}{D_{11}R}-\frac{m_0}{D_{11}}\)

In case of (\(\alpha\)), as with the isotropic material, \(\kappa ＜ 0\) at point B, resulting in the inversion of the convex surface.

On the other hand, in case of (\(\beta\)), \(\kappa> 0\) at point B, resulting in the convex surface remaining on the same side. In this case, the following condition must be satisfied.

\(\begin{align}
-\frac{2D_{66}}{R} < \frac{1}{R}(D_{12}-\frac{D_{11}D_{22}}{D_{12}+2D_{66}})\\
D_{11}D_{22} < (D_{12}+2D_{66})^2  \tag{18}
\end{align}\)

Additionally, since \(-\frac{2D_{66}}{R} ＜ 0\), if the following condition is met, the system can be bistable even when \(m_0=0\), that is, in the absence of a moment load due to residual stress.

\(\begin{align}
0 ＜ D_{12}-\frac{D_{11}D_{22}}{D_{12}+2D_{66}}\\
D_{11}D_{22} ＜ D_{12}(D_{12}+2D_{66}) \tag{19}
\end{align}\)

Here, looking at the stiffness matrix of the orthotropic material shown in equation (12), we can see that the conditions in equations (18) and (19) are satisfied if the Poisson's ratio and shear stiffness are large. As an example of such a metamaterial, we consider the following lattice structure.

When the above shape is created using a homogeneous isotropic material with a Poisson's ratio of 0.3, the in-plane stiffness matrix calculated using the finite element method is as follows.

\([A]= \begin{bmatrix}
\frac{E_X}{1-\nu_{XY}\nu_{YX}} & \frac{\nu_{YX}E_Y}{1-\nu_{XY}\nu_{YX}} & 0\\
\frac{\nu_{YX}E_Y}{1-\nu_{XY}\nu_{YX}} & \frac{E_Y}{1-\nu_{XY}\nu_{YX}} & 0\\
0 & 0 & G_{XY}\end{bmatrix}
= \begin{bmatrix}
E & 0.96E & 0\\
0.96E & E & 0\\
0 & 0 & 0.88E \end{bmatrix}\)

Using this structure, the strain energy graphs for (\(α\)) \(m_0=1.5\) and (\(β\)) \(m_0=0\) are as follows.

As shown in the graph on the right, using anisotropic materials allows the creation of a bistable structure without inversion of the convex surface.

In this example, we used a lattice structure from the perspective of metamaterials. However, in actual bistable STEMs, the same bistability is achieved by using CFRP with fibers oriented at ±45 degrees to the longitudinal direction.

Intuitively, this can be understood as bistability being achieved by maintaining a curved state in the ±45 degree directions.

In conclusion, by considering the conditions under which the bistability of a slap band is established, we were able to understand its characteristics:

• When using isotropic materials, the convex surface always inverts between the two stable states.
• By using orthotropic materials, it is possible to achieve bistability without inverting the convex surface. In such cases, residual stress may not be necessary.

Conclusion

Thank you for reading until the end. The bistability of slap bands and STEMs has been explained with both practical examples and mathematical equations. I hope I was able to convey the interesting aspects of these structures. I encourage you to explore your favorite bistable structures further.

Reference

[1] Wikipedia. Slap bracelet. (Ref. May 2024)

[2] 舘知宏. 複曲面／ねじれ面／単曲面. 図学研究 第51巻. 2017年7月

[3] youtube.com. Make SNAP BANDS from Unusual Materials. (Ref. May 2024)

[4] Michael W. Hyer. Calculation of the room-temperature shapes of unsymmetric laminates. 1981.

[5] Michael W. Hyer. Some Observations on the Cured Shape of Thin Unsymmetric Laminates. Journal of Composite Materials. 1981, 15(2).

[6] Min Sun et al. A Unified High-Order Semianalytical Model and Numerical Simulation for Bistable Polymer Composite Structures. Polymers 2022, 14(4), 818

[7] Esser et al. Artificial Venus Flytraps: A Research Review and Outlook on Their Importance for Novel Bioinspired Materials Systems. Frontiers in Robotics and AI. 2020, 7(25).

[8] Dario Lunni et al. Plant‐Inspired Soft Bistable Structures Based on Hygroscopic Electrospun Nanofibers. Asvanced Materials Interfaces. 2020, 7, 1901310

[9] Anna S. Westermeier et al. How the carnivorous waterwheel plant (Aldrovanda vesiculosa) snaps. Proceedings of the royal society B. 2018, 285(1878).

[10] 宇宙科学研究所, 超軽量展開宇宙構造物 (Ref. May 2024)

[11] Alex Brinkmeyer et al. Effects of Long-Term Stowage on the Deployment of Bistable Tape Springs. Journal of Applied Mechanics. 2016, Vol.83

[12] 秋人渡邊. 双安定性ブーム. JP6590218B2. 2016

[13] https://revolve-tec.com/ (Ref. May 2024)

[14] https://www.rolatube.com/ (Ref. May 2024)

[15] SIMON D. GUEST et al, Journal of Mechanics of Materials and Structures. 2021, 6, 1-4

[16] Evripides G Loukaides. Expanding the Build Plate: Functional Morphing 3D Printed Structures Through Anisotropy. Frontiers in Materials. 2020, 7, 159.