Interfacial metric mechanics: stitching patterns of shape change in active sheets

A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as phase-changing liquid crystal elastomers (LCEs), swelling gels and inflating baromorphs. Here, we show how to combine multiple patterns in a sheet by stitching regions of different shape changes together piecewise along interfaces. This approach allows simple patterns to be used as building blocks, and enables the design of multi-material or active/passive sheets. We give a general condition for an interface to be geometrically compatible, and explore its consequences for LCE/LCE, gel/gel and active/passive interfaces. In contraction/elongation systems such as LCEs, we find an infinite set of compatible interfaces between any pair of patterns along which the metric is discontinuous, and a finite number across which the metric is continuous. As an example, we find all possible interfaces between pairs of LCE logarithmic spiral patterns. By contrast, in isotropic systems such as swelling gels, only a finite number of continuous interfaces are available, greatly limiting the potential of stitching. In both continuous and discontinuous cases, we find the stitched interfaces generically carry singular Gaussian curvature, leading to intrinsically curved folds in the actuated surface. We give a general expression for the distribution of this curvature, and a more specialized form for interfaces in LCE patterns. The interfaces thus also have rich geometric and mechanical properties in their own right.

authors demonstrate several designs of compatible patterns. The authors also discussed actuated surfaces with concentrated Gaussian curvature of LCE-like sheet. The problem studied in this work is surely of interest to community of active materials. I suggest the acceptance of this manuscript after minor revision. Some specific questions are as follows: (1) The discussion in the paper could be applied to certain kinds of active mode instead of certain active materials, like gel and LCE mentioned in the paper. The LCE shows anisotropic contraction and expansion, while the gel usually exhibits both expansion in x and y directions. Could the theory work for both LCEs and gels? I suggest discussing their similarities and differences with LCEs and swelling gels in the manuscript.
(2) It could be better to give some comparison between theory and experiment (from literature) of a particular active material.
(3) In section 3, computational method in shell energy is introduced. The authors should elaborate more on the energy formula. And a and b in Eq. S1 should be given explicitly to make the paper self-contained.
(4) The authors may elaborate a bit more on the prefactors of the stretching and bending energy. Is it a ratio of the penalty between stretching and bending? If yes, the sheet in Fig 3d may not always form a continuous surface even in simulation.
(5) The out-of-plane shape changes within a patterned domain can be suppressed by increasing the sheet thickness. In the current model, the compatible stitched interface is achieved by designing the patterns around it. Is it possible to suppress the incompatible interface by increasing the sheet thickness?
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Best wishes Raminder Shergill proceedingsa@royalsociety.org Proceedings A on behalf of Professor Yihui Zhang Board Member Proceedings A Reviewer(s)' Comments to Author: Referee: 1 Comments to the Author(s) This paper provides a mathematical model to study the general condition for the formation of geometrically compatible interfaces in planar soft active sheets, with spatially patterned material properties, undergoing spontaneous deformation under an external stimulus. The paper highlighted the very interesting discrepancy between an LCE system and a gel system, where the former with anisotropic shape morphing has many more possible compatible interfaces when actuated, than the latter with isotropic shape change. This paper is very well written. The message from the mathematical model is well presented, clear, and highly interesting. The novelty by providing different possibilities to form geometrically compatible interfaces will inspire new experimental discoveries and related applications. Overall I am delighted to read this paper and recommend its acceptance without further revision. There is only one side question out of curiosity: the authors are clearly familiar with the works of Plucinsky and Bhattacharya on introducing small twists during the director pattern to form a biased pyramid of LCE (e.g., 2018 Soft Matter). What will be the effect of such "imperfections" on the general condition of compatible interfaces? Some "stability analysis" due to such "noise" seems interesting.
Referee: 2 Comments to the Author(s) This paper studies the flat active sheet programmed with different patterns in different regions. Once actuated, the flat sheet will morph into a curved surface. Specifically, the authors focus on the interface between regions with different patterns. The authors describe the geometrically compatible condition of the interface and applies the general rule to materials with different active modes, including LCE-like, gel-like and active/passive materials. For LCE-like materials, authors demonstrate several designs of compatible patterns. The authors also discussed actuated surfaces with concentrated Gaussian curvature of LCE-like sheet. The problem studied in this work is surely of interest to community of active materials. I suggest the acceptance of this manuscript after minor revision. Some specific questions are as follows: (1) The discussion in the paper could be applied to certain kinds of active mode instead of certain active materials, like gel and LCE mentioned in the paper. The LCE shows anisotropic contraction and expansion, while the gel usually exhibits both expansion in x and y directions. Could the theory work for both LCEs and gels? I suggest discussing their similarities and differences with LCEs and swelling gels in the manuscript.
(2) It could be better to give some comparison between theory and experiment (from literature) of a particular active material.
(3) In section 3, computational method in shell energy is introduced. The authors should elaborate more on the energy formula. And a and b in Eq. S1 should be given explicitly to make the paper self-contained.

Is the paper of sufficient general interest? Excellent
Is the overall quality of the paper suitable? Excellent

Recommendation? Accept as is
Comments to the Author(s) Very nice work.

Review form: Referee 3
Is the manuscript an original and important contribution to its field? Good

Is the paper of sufficient general interest? Good
Is the overall quality of the paper suitable? Excellent Dear Dr Feng I am pleased to inform you that your manuscript entitled "Interfacial metric mechanics: stitching patterns of shape change in active sheets" has been accepted in its final form for publication in Proceedings A.
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Sincerely, Raminder Shergill proceedingsa@royalsociety.org Referee 1. Comments to the Author(s): This paper provides a mathematical model to study the general condition for the formation of geometrically compatible interfaces in planar soft active sheets, with spatially patterned material properties, undergoing spontaneous deformation under an external stimulus. The paper highlighted the very interesting discrepancy between an LCE system and a gel system, where the former with anisotropic shape morphing has many more possible compatible interfaces when actuated, than the latter with isotropic shape change. This paper is very well written. The message from the mathematical model is well presented, clear, and highly interesting. The novelty by providing different possibilities to form geometrically compatible interfaces will inspire new experimental discoveries and related applications. Overall I am delighted to read this paper and recommend its acceptance without further revision.
We thank Referee 1 for this entirely positive assessment and the recommendation for publication. There is only one side question out of curiosity: the authors are clearly familiar with the works of Plucinsky and Bhattacharya on introducing small twists during the director pattern to form a biased pyramid of LCE (e.g., 2018 Soft Matter). What will be the effect of such "imperfections" on the general condition of compatible interfaces? Some "stability analysis" due to such "noise" seems interesting.
In these boundaries the boundary is chosen to be compatible, as in the cases treated in our paper. However, along the boundary itself, a sliver of incompatible material is added later in the design process to bias the actuation. We agree that what happens in this situation is an interesting question, and not resolved in the present work. We speculate that the answer depends on the width of the sliver compared to sheet thickness, and also the emergent mechanical length scale of the curvature of the fold: if the width is large, we would expect (hierarchical?) wrinkling to accommodate the incompatibility, while if it is short, a 3D elastic treatment is required rather than a shell/plate theory. We have added a comment to the final paragraph of the paper mentioning such interfaces, but make no claim to predict how they will behave. Referee 2. This paper studies the flat active sheet programmed with different patterns in different regions. Once actuated, the flat sheet will morph into a curved surface. Specifically, the authors focus on the interface between regions with different patterns. The authors describe the geometrically compatible condition of the interface and applies the general rule to materials with different active modes, including LCE-like, gel-like and active/passive materials. For LCE-like materials, authors demonstrate several designs of compatible patterns. The authors also discussed actuated surfaces with concentrated Gaussian curvature of LCE-like sheet. The problem studied in this work is surely of interest to community of active materials. I suggest the acceptance of this manuscript after minor revision.
We thank Referee 2 for the positive comment and the recommendation for publication. Some specific questions are as follows: (1) The discussion in the paper could be applied to certain kinds of active mode instead of certain active materials, like gel and LCE mentioned in the paper. The LCE shows anisotropic contraction and expansion, while the gel usually exhibits both expansion in x and y directions. Could the theory work for both LCEs and gels? I suggest discussing their similarities and differences with LCEs and swelling gels in the manuscript.
We are confused by this comment. Our discussion in Section 2(a) use on a general mathematical model of a spontaneous deformation, and is not limited to particular materials or types of modes. However, we already have discussed the specific cases of LCEs and gels in Section 2(b), highlighting, in the words of referee 1 "the very interesting discrepancy between an LCE system and a gel systems" -this appears to be exactly what the referee is asking for.
(2) It could be better to give some comparison between theory and experiment (from literature) of a particular active material.
We found this a very useful comment. We have added some comparison to the discussion section, including an example of arrays of cones [T. Guin et al., Nat. Commun., 9(1):1-7, 2018] and an example of lines of Gaussian curvature [D. Duffy et al., J. Appl. Phys., 129(22):224701, 2021]. We hope this clarifies that such interfaces are already a topic of active experimental research, and will allow the reader to explore the experimental systems if they so desire.
(3) In section 3, computational method in shell energy is introduced. The authors should elaborate more on the energy formula. And a and b in Eq. S1 should be given explicitly to make the paper self-contained.
We agree that it is a good idea to include some more details about the shell energy that underpins our simulation. We have therefore elaborated a little in the supplement, including giving definitions of the first and second fundamental forms (a and b), and discussing the physical origin of the energy. We have also added a reference to the supplement from the main text so that the interested reader can find these details. We do still think that these details belong in a supplement, as they are not new, and are ancillary to our main content which is focused on geometry rather than mechanics.
(4) The authors may elaborate a bit more on the prefactors of the stretching and bending energy. Is it a ratio of the penalty between stretching and bending? If yes, the sheet in Fig 3d may not always form a continuous surface even in simulation.
Yes, the h and h 3 prefactors in the energy represent the penalty for stretching and bending, as ubiquitous in shell and plate mechanics. We have clarified this origin in our extended explanation of the shell energy.
With regards to the reviewers question about continuity, in our simulation, we assume the sheet is always in the elastic regime, and no plasticity or fracture occurs. Therefore, the simulated surface is always continuous. We suspect the referee actually meant sharp (i.e. discontinuous gradient) rather than a discontinuous displacement. In fact, we see that the interfaces are always blunted by a stretch-bend trade-off, and only become sharp only in the limit of zero thickness. We intend to study these mechanical considerations in a future work.
(5) The out-of-plane shape changes within a patterned domain can be suppressed by increasing the sheet thickness. In the current model, the compatible stitched interface is achieved by designing the patterns around it. Is it possible to suppress the incompatible interface by increasing the sheet thickness?
This is a very interesting question, but one which we largely reserve for future work: our emphasis here is on the geometry of thin isometric sheets, not the mechanical stretch/bend trade offs that emerge in thick ones. We believe the actuated interface in a physical system is somewhat blunted by such mechanics, and would become progressively smoother as the thickness increases. There may well be a critical thickness past which the sheet remains planar despite having a non-flat metric, but at this point it cannot be regarded as thin or isometric.
Referee 3. This paper describes how to stitch together so-called active patterns at an interface. On actuation each such pattern has a different induced metric that constrains how the pattern morphs into a 3D configuration. Stitching together two such patterns, thus, requires that certain rank-one compatibility conditions be satisfied at the patterned interface prior to actuation. To begin, the authors characterize all possible compatible interfaces for all possible induced metrics. They then show how this characterization furnishes design rules to stitch together two active patterns. They do so in the context of well-known active sheet examples: Two swelling gels, an active and passive region, and, most thoroughly, two LCE patterns. Although this paper is primarily theoretical, the examples combine the design rules with numerical simulations in a way that opens the door for a lot of further exploration with experimental groups. Lastly, the authors develop rather elegant and general formulas for calculating the Gaussian curvature of an interface in these systems. This paper is clear, concise and authoritative. It asks and resolves interesting questions in mechanic about the nature of active sheet embeddings. The ideas also have the potential to inform the design of active sheets, which is an important emerging topic in materials science. The paper, in my opinion, is suitable for publication in PRSA, after the authors address my comments below.
We thank Referee 3 for the careful reading, constructive suggestions and encouraging comments. Major comment.
• Metric compatibilty is a necessary conditions. I think more needs to be conveyed about the fact that you are characterizing a necessary condition for an embedding, not the necessary and sufficient conditions. You do so with the example in Fig 3d, but it feels kind of "swept under the rug" in my opinion. As I understand it, your hope is that there are a rich enough family of isometric deformations on the two sides of an interface such that a metric compatible interface implies an actual embedding. I basically think you should be upfront and transparent about this "hope" from the start. There does seem to be a reasonable sales pitch for this view, albeit it requires a thoughtful paragraph: In terms of design, metric compatibility is the only designable feature of the characterization of interfaces anyway. You also have access to simulations, which can test the validity of the designs obtained by your characterization, ect., etc. . . This feels like a discussion that needs to be had either at the beginning or end of Section 2(b).
We are pleased to add some discussion of this very interesting point, as the referee suggests. We agree that metric compatibility is a necessary condition, rather than a sufficient condition for an embedding. We are unaware of any embedding theorems that can definitively settle the question of sufficiency: for example, Nash embedding assumes smooth embeddings and smooth metrics. However, it is our numerical experience that well behaved embeddings with only a sharp ridge along the interface are generically available. We also agree with the referee's point that metric compatibility is the only intrinsic design tool available. We have elaborated on these points at the start of Section 2(b), as suggested.
Minor comments.
• Page 3, line 36: The phrasing "Even more relatedly" sounds awkward to me and is vague. I'd suggest something link "Beyond describing the physics of interfaces,. . . ".
We have done so.
• Page 4 line 23: Given your notation in this work, I think you should replace the identity I with I or I or Id for consistency.
We have used I to represent identity throughout the paper.
• Eq (2.3): The implied notation a i = U 2 i might be "too slick" for a typical reader. I think you should define it explicitly.
We have defined a explicitly below eq. (2.2) and pointed a i to it.
• Page 4, line 53: "One may solve algebraically, but a graphical approach is highly instructive for understanding when these different cases arise, which will transpire to determine the ease of constructing such stitched interfaces between patterns." 1 I feel like this sentence needs a reorganization. Maybe something like: "One may characterize the equation algebraically, but a graphical approach is highly instructive for understanding when these different cases arise and the relative ease of constructing such stitched interfaces between patterns." We have changed it to "One may characterize the equation algebraically, but a graphical approach is highly instructive for understanding these different cases and constructing such stitched interfaces between patterns." • For case (ii) on page 5, I'm confused. You have a critical angle. At this angle, there is only one possible interface? Even though this "1-solution" case is not generic, shouldn't it still be clearly stated in your characterization?
Yes, the critical angle corresponds to one possible interface. We have stated this more clearly.
• Page 6, line 58: "symmetrically" sounds awkward here. I guess the correct thing to say is something like "the actuations U1,2(x) are mirrors of each other across the interface".
We have changed it to "We call interfaces constructed along these directions twinning interfaces, as the directors are discontinuous at the interface but symmetrically satisfy the twinning condition |n 1 · t| = |n 2 · t|. Then the the deformations U 1,2 (x) are mirrors of each other across the interface." • Eq (4.2): Why is lA boldface in the integration domain, and not boldface in the text? We use boldface l A to represent a curve, and l A to represent its arclength. We have unified the notations.
• Eq (4.3): This is a very minor comment. You can save some space by simply replacing (∇ × n) with ∇ · n ⊥ . Also, in solid mechanics, the 2D curl is not used all that much. So I personally stumbled over ∇ × n for a bit thinking that (∇ × n)n ⊥ was a tensor, not a vector.
We have pointed out that the curl is taken in a 2D scalar sense in the original version. We prefer keeping this form to make the notation consistent with the related references.
• Page 13: I think you should be a bit more mathematically accurate about your integrals. I sympathize with the attempt to keep a concise notation, but, as you well know, it is not proper to have x(l) indicate the domain of integration, while also integrating over l. In addition, x(l) indicates both a point on a curve and the curve itself. My feeling is that there is too much "abuse of notation" here that could confuse a typical reader.
We have changed the domain of integration to [0,l] wherel is the total arclength of the curve.
• Eq (4.5): I think you should pick one of the a 's in the integrand, rather than use the implied notation √ t · a i · t = √ t · a · t. Maybe write dl A /dl = √ t · a 1 · t = √ t · a 2 · t as a reminder to the reader.
We have done so.
• Page 14, line 6: interface should be interfaces. We have corrected it.