Universally bistable shells with nonzero Gaussian curvature for two-way transition waves

Multi-welled energy landscapes arising in shells with nonzero Gaussian curvature typically fade away as their thickness becomes larger because of the increased bending energy required for inversion. Motivated by this limitation, we propose a strategy to realize doubly curved shells that are bistable for any thickness. We then study the nonlinear dynamic response of one-dimensional (1D) arrays of our universally bistable shells when coupled by compressible fluid cavities. We find that the system supports the propagation of bidirectional transition waves whose characteristics can be tuned by varying both geometric parameters as well as the amount of energy supplied to initiate the waves. However, since our bistable shells have equal energy minima, the distance traveled by such waves is limited by dissipation. To overcome this limitation, we identify a strategy to realize thick bistable shells with tunable energy landscape and show that their strategic placement within the 1D array can extend the propagation distance of the supported bidirectional transition waves.

which suggests that: There are two features of this result that I think are interesting: Firstly, this result shows that there is a minimum input energy for wave propagation, *%& = $√$ , ( ) ) "/$ $ . This result seems plausible given the results of the discrete analysis presented in figures 4 c and d: there is indeed a minimum energy for wave propagation and this minimum seems to be strongly dependent on H (fig 4c) and more weakly dependent on the 'air stiffness' k (see fig. 4d). Secondly, this result shows that the speed of propagation increases very rapidly for input energies just above the minimum but then starts to level off a little (also as seen in figs 4c and d). I think it is an important test of the continuum theory to compare the results above (together with the corresponding result for w) against the discrete results (as well as their numerical solution of the continuum problem).
I think that the results above are important for explaining features of the discrete model but, at least as far as I could tell, are not presented in either the main text or the SI.
I also have a number of minor comments that I think the authors should consider.
Minor comments: -I found the way that the single shell and double-glued shell were presented was confusing -given that the text builds up the expectation that the shell presented here will be "bistable for any thickness" figure 1c is a bit of a surprise. On further reading, it is clear that this is just a preliminary figure, and should be contrasted with the behavior of the double-glued shell presented in figure 2. It would be helpful if there could be clearer contrast between figures 1b and 2e and 1c and 2f in the text, and more anticipation that the single shell structure is not the one used throughout the paper. One way to do this might be to combine the relevant panels together in a single figure and present them side-by-side. Alternatively, better sign-posting in the text would help.
-The discussion at the bottom of the second page regarding the inflation of an individual double shell is unclear. Was the shell first attached to a rigid cylinder into which the water was pumped? Was the water pumped back between the constituent shells?
-I am confused about the speed of the transition wave presented in fig. 3d (and elsewhere in the SI). From fig. 3b and c, it seems that the wave snaps 10 shells in a period of 0.2s. That would make the average wave speed around 50 shells/s, which is significantly smaller than the 200-300 shells/s reported in the y-axis of fig. 3d. Is there an issue here?
-The authors mention repeatedly that the wave speeds up again as it approaches the free end. However, I could not see a clear discussion of how this softening enters into the discrete model (though it is clearly there since fig. 3d shows the same phenomenon in the discrete model results).
-The speed cs is only defined after eqn [11], even though it appears in eqn [9]. It would make sense to define it earlier.
-The results in figure 4 (and the discussion after eqn [13]) are, I think, for the case of zero dissipation. This should be specified clearly in the caption of figure 4.
-The authors state that their experiments indicate that the transition wave velocity increases with Ein but I could not see this experimental data in either the main text or SI. The only indication is the limited data (two different pressures) in fig. 3. It would be good to see more experimental evidence of this -the model is clear, but since this is a main finding of the theory, it is important to compare this back to as much experimental data as possible.
Reviewer #2 (Remarks to the Author): This paper describes a process by which symmetrically-bistable thick shells are fabricated from asymmetrically-bistable shells by first displacing them to a flat deformed configuration and then adhering them. The resulting symmetrically-bistable structures are then assembled in 1-D arrays at it is shown that by snapping through one bistable shell the resulting pressure change in the adjacent cavity causes the adjacent cell to itself snap through and so on until dissipation effects start to dominate. In a final step a concept is presented in which the shells can be made asymmetrically bistable. This asymmetry allows energy showed in the less preferential stable state to be released as useful work during its transition to the more preferential stable state with the result that the propagation can be made to extend further.
I should say first of all that I very much enjoyed reading this paper. However, I have the following comments for the authors' consideration.
The generation of symmetrically bistable structures via the coupling of asymmetrically-bistable structures is clever, and I believe this is a novel and useful contribution. I have concerns about the description of such structures as universal, however. I have considered the behaviour of coupled von Mises trusses with biasing springs as a useful simplified analogue (see attached figure). In this case it can be seen that as the degree of asymmetric bistability of the sub-structures increases, and beyond when the second stable state has been annihilated, the locations of the stable states of the coupled system move further away from the neutrally-stable point and the stiffness greatly increases. If we consider that this behaviour will be replicated by the coupled shells we can see that the theoretically-stable locations move to a degree of displacement which cannot be comfortably attained by the structure (at least without significant higher-order deformations). I do not know what would happen to the stability landscape in this case -it would be interesting to investigatebut I suspect that in the practical limit the adhered configuration ( Fig. 2c) will become the preferred stable configuration. It is also the case that there is practical limit on how far the initial thick shell can be compressed without the formation of local buckling etc. At best the universality can surely only be claimed for the theoretical and not the practical response of the structure.
The analysis of the response of the array is well carried out and the results are plausible and well validated. It is interesting to see the effect of the propagation of the instability. From a fundamental perspective the behaviour seems quite straightforward (essentially it is summed up by Fig. 4b) although I agree there is value to investigating a practical implementation.
I am not persuaded that the final section adds much to the story. It has been shown in literature (e.g. https://doi.org/ 10.1115/1.4000417) that asymmetric bistability can be utilized for unilateral high-frequency actuation. The practical implementation that is presented is interesting but it seems that it is one of several possible techniques that can be used to add a bias to the energy landscape.
Minor points: In the abstract it is stated that increased bending stiffness causes a stable energy state to annihilate as the thick-ness becomes larger. I think this is over-simplified -although the increased thickness leads to increased bending stiffness, increasing bending stiffness alone does not necessarily cause stable states to disappear.
Also in the abstract the phrase "bistable for any thickness" falls under my first point above.
Double curved and doubly curved appear to be used interchangeably.
In the conclusion the phrase "in vivo" would usually be restricted to operations carried out on living entities In this document, we provide a copy of the comments and points raised by each reviewer and address them one at a time. Pages R1-R14 address the comments raised by Reviewer 1, whereas pages R15-R21 address the comments raised by Reviewer 2. A copy of the reviewer's text is provided for each comment. Changes to the main text or Supporting information are highlighted in blue. All modified figures are included as part of our response for completeness.

Response to Referee #1
This article presents a combined theoretical and experimental study of the propagation of snap-through waves within a new kind of meta-material. The authors develop an experimental system that consists of several bistable elastic caps that influence one another by displacing the gas between them. This is possible because they have found a clever way to fabricate caps that remain bistable, even as the thickness changes. There are several good ideas in this paper and the analysis presented is convincing (though I have a couple of suggestions on this below). The one real weakness in the paper overall is that no compelling application of these ideas is presented. If more could be said about how this sort of system could be used in, for example, soft robotics, then it would be much more compelling for a journal like Nature Communications.
We thank the reviewer for the positive remarks and insightful suggestions. The point raised by the reviewer is a valid one and we modified the main text to illustrate potential applications of our strategy. More specifically, we have added the following text on page 8 of the manuscript "Even though in this study we used rigid chambers to connect adjacent shells, we envision the proposed strategy to provide a new route for soft robotic locomotion. By making the chambers unidirectionally stretchable, they would sequentially extend during the propagation of transition waves and emulate the rectilinear locomotion of snakes. Additionally, our system's unique property, namely the dependence of transition wave velocity to the input energy, could enable the design of smart energy absorption devices which effectively transfer energy but are able to avoid energy concentrations through dissipation. Further, systems based on our strategy could also serve as energy sensors, as the energy input can be determined by monitoring the effective transition wave velocity."

R1-R21
Expanding the RHS for c/c0 1, I find that, which suggests that, There are two features of this result that I think are interesting: Firstly, this result shows that there is a minimum input energy for wave propagation,

this result seems plausible given the results of the discrete analysis presented in figures 4c and d: there is indeed a minimum energy for wave propagation and this minimum seems to be strongly dependent on H (fig 4c) and more weakly dependent on the 'air stiffness' k (see fig. 4d). Secondly, this result shows that the speed of propagation increases very rapidly for input energies just above the minimum but then starts to level off a little (also as seen in figs 4c and d). I think it is an important test of the continuum theory to compare the results above (together with the corresponding result for w) against the discrete results (as well as their numerical solution of the continuum problem)
. I also think that the above result would give a more compelling picture of how many snaps can be generated before the wave extinguishes itself. At present, eqn (S28) assumes each snapping shell dissipates the same amount of energy. In fact, eqn (S27) shows that this is a sensitive function of c and, given the above result, is sensitively dependent on the rate at which the energy decreases close to extinction. For example, close to extinction (i.e. using c/c0 1 one could write the energy prior to the (n + 1) st snapping event in terms of that prior to the n th by writing: This is a difference equation for En and can easily be solved using the discrete-continuum approximation already used by the authors. I find that: for some constant of integration n * (which corresponds to the number of shells after which extinction occurs). A simple estimate, then, is that extinction should occur after snapping shells. As a result, inputting more energy has limited efficiency at snapping more elements -since snap-through happens faster, the dissipation is also higher -as reflected by the scaling n * ∼ E 1/2 in I think that the results above are important for explaining features of the discrete model but, at least as far as I could tell, are not presented in either the main text or the SI.
We thank the reviewer for the suggestions and detailed derivations that he/she was willing to provide. We consider this discussion to be an extremely valuable addition to our work. To this end, we have updated both the main text and SI to include these derivations and the new analytical results. More specifically, we have modified Fig. 4 (also shown below as Fig. R1 for completeness) and added the following text to the manuscript to emphasize that an explicit expression for c can be obtained "Since in the absence of dissipation E is equal to the energy supplied to the first unit to initiate the pulse, Ein, we find that which we can numerically solve to obtain c for a given Ein. Further, to obtain an explicit expression for c as a function of Ein, we take a Taylor's series expansion of Eq. (R1) around c/c0 = 0 (since in our system c/c0 ∼ 0.2), while retaining terms up to the third order. This yields represents the minimum amount of input energy required to initiate the transition wave. Eq. (R2) confirms that the speed of the propagating transition waves can be tuned by modifying the amount of energy supplied to the system. Further, we have added a new panel to Fig. 5 (also shown below as Fig. R2 for completeness) and added the following text to describe how the number of units that the wave switches before stopping can be determined "By introducing Eq. (11), Eq. (6) can be rewritten as which, by taking a Taylor's series expansion around c/c0 = 0 and retaining terms up to the second order, can be further simplified to Finally, introduction of Eq. (R2) into Eq. (R5) yields where Ei denotes the energy carried by the transition wave when propagating through the i-th unit. .......
Next, we use our analytical model to predict the finite propagation distance in systems with a nonzero dissipation. Towards this end, we impose conservation of energy To solve Eq. (R7) and determine the number of units that the wave switches before stopping, Nstop, we take the continuum limit of Eq. (R7), dE c d e f where E(x) is a continuum function that interpolates Ei as By integrating both sides of Eq. (R8) we obtain In Fig. R2b we consider an array comprising 500 double shells with R = 25.4 mm, H = 15 mm, T total = 4 mm and report the evolution of Nstop as predicted by Eq. (R11) and by our discrete model for different values of β. We find excellent agreement between analytical and numerical results, with Nstop that monotonically increases as either the damping coefficient and the energy input become larger. "  figure 2. It would be helpful if there could be clearer contrast between figures 1b and 2e and 1c and 2f in the text, and more anticipation that the single shell structure is not the one used throughout the paper. One way to do this might be to combine the relevant panels together in a single figure and present them side-by-side. Alternatively, better sign-posting in the text would help This is another good point. Following the reviewer's suggestion we have combined relevant panels of Figs. 1 and 2 to present the single and double-glued shells side by side. Panels a, b and c of the new Fig. 1 (also shown as Fig. R3 for completeness) showcase the geometry and response upon inflation of the single doubly-curved thick shell. Panels d, e and f illustrate the step by step process to construct the universally bistable doubly-curved double shells. Finally panels g and h showcase the response of the doubly-curve universally bistable shells upon inflation, in direct contrast to panels b and c for the single shell.

I found the way that the single shell and double-glued shell were presented was confusing -given that the text builds up the expectation that the shell presented here will be "bistable for any thickness" figure 1c is a bit of a surprise. On further reading, it is clear that this is just a preliminary figure, and should be contrasted with the behavior of the double-glued shell presented in
Further, we modified Fig.2 of the main text to serve as the experimental validation of our numerical approach for modeling the behavior of the doubly-curved universally bistable shells. Panel a of the new Fig. 2 (also shown below as Fig. R4 for completeness) showcases a schematic of the testing apparatus used to inflate and deflate the shells in quasi-static conditions using a syringe pump. Panels b and c show a comparison between the experimental findings and numerical predictions for the pressure-volume and the pole displacement-volume relationships of the universally bistable shells.
Finally, we have added a new Figure (Figure S6 -also shown below as Fig. R5 for completeness) to clarify how the shells were inflated/deflated.

The discussion at the bottom of the second page regarding the inflation of an individual double shell is unclear. Was the shell first attached to a rigid cylinder into which the water was pumped? Was the water pumped back between the constituent shells?
In an effort to better explain our experimental process and avoid confusion we added panel a to Fig. 2 of the main text (also shown as Fig. R4 for completeness). In the panel we illustrate the experimental apparatus used to quasi-statically inflate our universally bistable shells. Further, we have also re-worded the text in page 2 of our main text to clearly illustrate our experimental process: "We then characterize its quasi-static response by attaching its boundaries to an enclosed rigid cylinder and supplying water with a syringe pump (Pump 33DS, Harvard Apparatus) at a constant rate of 30 mL/min to inflate it and deflate it (see Fig. R4a)." R8 | Vasios et al.

Comment 1D
I am confused about the speed of the transition wave presented in fig. 3d (and elsewhere in the SI). From fig. 3b and c, it seems that the wave snaps 10 shells in a period of 0.2s. That would make the average wave speed around 50 shells/s, which is significantly smaller than the 200-300 shells/s reported in the y-axis of fig. 3d. Is there an issue here?
We thank the reviewer for pointing out this inconsistency between the wave speeds reported in Fig. 3d and those that one can compute from the data reported in Fig. 3b and Fig. 3c. The reason for this inconsistency was that velocities reported in Fig. 3d were computed by determining the value of c that best fits the experimental data for the pole displacement of the shells using the analytical solution [R12] However, for our highly dissipative system, the value of c obtained using Eq. (R12) (which inherently neglects any dissipation effects) seems to overestimate the actual transition wave velocity. As such, we have recomputed all transition wave velocities reported in Fig. 3d by estimating the time ti at which shell i is half-inverted at u(ti) = H (i.e. this is done by determining the zero of the tanh argument in Eq. (R12)). Then, the transition wave velocity (in shells/s) between shells i and i + 1 is computed as In Fig. 3d we now report the transition wave velocities for all experiments and discrete model simulations obtained using Eq. R13. We find that the recomputed wave velocities are much closer to ∼ 50 shells/s. For completeness we include the modified Fig. 3 below. We also modified the main text to indicate how such velocities are being calculated "To better characterize these elastic waves, in Fig. 3d we report the evolution of their velocity (calculated by monitoring the time at which u pole,i = H) during propagation." Finally, we note that in the absence of dissipation (or for systems with small dissipation) the two methods described above for determining the transition wave velocity (i.e. from a linear squares fit using Eq. (R12) or by using Eq. (R13)) produce almost identical predictions.   fig. 3. It would be good to see more experimental evidence of this -the model is clear, but since this is a main finding of the theory, it is important to compare this back to as much experimental data as possible.

The authors state that their experiments indicate that the transition wave velocity increases with Ein but I could not see this experimental data in either the main text or SI. The only indication is the limited data (two different pressures) in
We thank the reviewer for this comment. In Fig. S22 of the revised SI (also shown below as Fig. R7 for completeness) we report experimental results and numerical predictions for 8 experiments in which pulses were initiated using 10, 15, 20 and 25 psi of pressure.

Response to Referee #2
This paper describes a process by which symmetrically-bistable thick shells are fabricated from asymmetrically-bistable shells by first displacing them to a flat deformed configuration and then adhering them. The resulting symmetrically-bistable structures are then assembled in 1-D arrays at it is shown that by snapping through one bistable shell the resulting pressure change in the adjacent cavity causes the adjacent cell to itself snap through and so on until dissipation effects start to dominate. In a final step a concept is presented in which the shells can be made asymmetrically bistable. This asymmetry allows energy showed in the less preferential stable state to be released as useful work during its transition to the more preferential stable state with the result that the propagation can be made to extend further.I should say first of all that I very much enjoyed reading this paper. However, I have the following comments for the authors' consideration.
We thank the reviewer for his/her comments. We are happy to see that the reviewer enjoyed reading our paper.

Comment 2A
The generation of symmetrically bistable structures via the coupling of asymmetrically-bistable structures is clever, and I believe this is a novel and useful contribution. I have concerns about the description of such structures as universal, however. I have considered the behaviour of coupled von Mises trusses with biasing springs as a useful simplified analogue (see attached figure).
In this case it can be seen that as the degree of asymmetric bistability of the substructures increases, and beyond when the second stable state has been annihilated, the locations of the stable states of the coupled system move further away from the neutrally-stable point and the stiffness greatly increases. If we consider that this behaviour will be replicated by the coupled shells we can see that the theoretically-stable locations move to a degree of displacement which cannot be comfortably attained by the structure (at least without significant higher-order deformations). I do not know what would happen to the stability landscape in this case -it would be interesting to investigate -but I suspect that in the practical limit the adhered configuration (Fig.  2c) will become the preferred stable configuration. It is also the case that there is practical limit on how far the initial thick shell can be compressed without the formation of local buckling etc. At best the universality can surely only be claimed for the theoretical and not the practical response of the structure.
We thank the reviewer for this comment. We believe that the our shells are bistable as long as the resulting geometry can still be identified as a thick shell. Increasing the thickness arbitrarily, at some point, results in a structure that can no longer be referred to as a shell but rather resembles a 3D solid. In such cases, our findings do not apply. From a theoretical standpoint, any thick shell constructed using our approach is bistable. However, it is true that for practical applications extremely thick shells are challenging to fabricate and test.