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# Journal Club Theme of July 2015: Reconfigurable metamaterials -- putting the holes in the right place

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**Reconfigurable metamaterials -- putting the holes in the right place**

**Shu Yang**1 and **Jie Yin**2

1Department of Materials Science and Engineering, University of Pennsylvania, E-mail: shuyang@seas.upenn.edu

2Applied Mechanics of Materials Laboratory, Department of Mechanical Engineering, Temple University, E-mail: jieyin@temple.edu

Reconfigurable metamaterials that can reversibly change the size, shape, and symmetry are of interest in the design of flexible electronics, color displays, smart windows, actuators, sensors, and photonic/phononic devices. Often this is achieved through continuous structural reconfiguration induced by simple yet controllable mechanical deformation, e.g. expansion and collapse, fold and transformation, in local structural elements.

“The art of structure is how and where to put holes,” said structural engineer and architect, Le Ricolais. Here, we would like to initiate the discussion on the design of introduced prescribed holes, cuts, and folds to generate reconfigurable metamaterials, which is highly conformable, stretchable, deployable, and foldable, as well as for broad potential applications, including flexible electronics, tunable photonic and phononic bandgap materials, and reconfigurable soft robotics etc.

**1. Directed deterministic pattern transformation by prescribing asymmetry**

Rich pattern transformation in soft membranes with periodic hole arrays has been realized through buckling instabilities1-7, where the bending of interpore ligaments triggers reversible pore closure and opening2,7. However, buckling often undergoes a sudden structural change and symmetry breaking deformation5-7, which leads to an undetermined pattern transformation6-7, especially at large deformation when the porous membrane is collapsed to fully close the pores7.

One possibility to address the nondeterministic issue is to break the deformation symmetry of ligaments with prescribed pre-twisted asymmetric ligaments. For example, after introducing a pre-twisted angle, the collapsing mode of the kagome lattice with triangular structure units underwent no abrupt buckling but continuous pattern transformation8, which leads to deterministic and directed bending of asymmetric ligaments and thus complete compaction (Video S1, Video S2 and S3).

Figure 1: (a) Pattern transformation in Kagome porous structures without (buckling in Structure A) and with pre-twist (no-buckling in Structure B and C). Images edited from Ref. [8].

Kagome lattices offer an ideal model for the production of large deformations in response to relatively small external perturbations, a highly attractive feature for applications of mechanical metamaterials. In addition to pre-twist method, other pre-set symmetry breaking methods could be applied to generate non-frustrated and deterministic pattern transformation in lattice structures.

**2. Engineering fractal cuts for super-conformable metamaterials**

Design of expandable and shape-shifting structures can be considered as the reverse process of compacting porous structures as discussed above, where the kagome and checkerboard porous structures can be generated through stretching a thin sheet with simple prescribed line cuts (Figure 2a)9. Upon stretching, the cut triangular and square units undergo rotation, leading to the bi-directional expansion of the structure and thus the line cuts evolve into the kagome and checkerboard porous structures, respectively. The cut concept can be applied to any other shape of the rotating cut unit as well as more complex cut patterns such as fractal cuts9, 10, which allows for design of shape-shifting structures with much richer structural reconfiguration through simple prescribed cut patterns9, 10.

Figure 2: (a) Generation of expandable structures through rigid rotating of triangular and square units after introduction of line cuts, (b) Pattern transformation in stretching hierarchically cut structures by repeating the line cuts (represented by red lines) in corresponding subunits from level 1 to level 3 cuts. Images edited from Ref. [9].

The advantage of the hierarchical cut concept is that stretching occurs only by unit rotation without deformation of individual units in the hierarchical structures, which could in principle be applied to any materials. The concept of hierarchical cuts opens a new avenue for exploring design of soft metamaterials as a pluripotent material that can reproduce any shape of considerable complexity with tunable mechanical properties, as well as extremely expandable and shape-shifting metamaterials9. They can be used as scaffolds to integrated conventional rigid devices without sacrificing device performance during collapsing or stretching for conformable and stretchable curved electronics9. They can be exploited for highly tunable optics and acoustics11. They can also be extended to 3D for foldable and deployable materials by precisely controlled materials stiffness and mechanical response.

**3. Lattice kirigami toward pluripotent foldable materials**

In addition to realize superconformable materials through the combination of cut hierarchy and cut motifs of a 2-D sheet of rigid units, we can buckle rigid surfaces into desired 3-D shapes by making the prescribed cuts and identifying edges, so called *lattice* kirigami, thus, imposing localized points of Gaussian curvature on a surface12.

Different from conventional origami design through folding mountains and valleys, cuts in kirigami to remove part of a lattice are allowed. Therefore, cuts take away unnecessary parts, minimizing waste, and allow for more complex structures from fewer, simpler folds without stretching or shrinking the lattice’s edges. For example, when a cut surface with folds shown in Figure 3a is folded into a 3D stepped surface configuration, the original cut hexagonal pore will be closed. The two regions labeled “P” can independently pop up or pop down relative to their initial configuration, which leads to four allowed configurations (Figure 3b-e)13. Pluripotent kirigami blueprints are highly desired that can accommodate many different target structures through local fold reassignments and one example is a triangular lattice of sixons, which is open for investigation.

Figure 3: A flat cut surface with hexagonal cut-out and folds (a) can fold into four closed allowed 3-D structures (b-e) through kirigami designs combining origami-based folds and cuts. Images edited from Ref. [13].

The concept of simple line cuts, cut-outs, folds opens a new paradigm of generating 2-D and 3-D reconfigurable and pluripotent metamaterials, which is open for further studies on its mechanics as well as their broad applications in mechanical engineering, material sciences, and biomedical engineering.

References

1. Mullin, T.; Deschanel, S.; Bertoldi, K.; Boyce, M. C., "Pattern transformation triggered by deformation". Phys. Rev. Lett. 2007, 99 (8), 084301.

2. Zhang, Y. Matsumoto, E. A., Peter, A., Lin, P., Kamien, R. D., and Yang, S., “One-step nanoscale assembly of complex structures via harnessing of an elastic instability”, *Nano Lett*. 2008**, **8 (4), 1192-1196

3. Overvelde, J. T. B.; Shan, S.; Bertoldi, K., "Compaction Through Buckling in 2D Periodic, Soft and Porous Structures: Effect of Pore Shape". Adv. Mater. 2012, 24 (17), 2337-2342.

4. Shim J, Shan S, Kosmrlj A, Kang SH, Chen ER, Weaver JC, Bertoldi K. “Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials”. Soft Matter. 2013, 9, 8198-8202

5. Kang, S. H.; Shan, S.; Noorduin, W. L.; Khan, M.; Aizenberg, J.; Bertoldi, K., "Buckling Induced Reversible Symmetry Breaking and Amplification of Chirality Using Supported Cellular Structures". Adv. Mater. 2013, 25 (24), 3380-3385.

6. Kang, S. H., Shan, S.; Kosrli, A, Noorduin, W. L., Shian, S, Weaver, J. C, Clarke, D, Bertoldi, K., "Complex Ordered Patterns in Mechanical Instability Induced Geometrically Triangular Cellular Structures". Phys. Rev. Lett. 2014, 112 (9), 098701

7. Wu, G. X., Xia. Y., and Yang, S., “Buckling, symmetry breaking, and cavitation in periodically micro-structured hydrogel membranes”, *Soft Matter*, 2014, 10 (9), 1392 - 1399.

8. Wu, G., Cho, Y., Choi, I.-S., Ge, D., Li, J., Han, H. N., Lubensky, T., and Yang, S., "Directing the deformation paths of soft metamaterials with prescribed asymmetric units", *Adv. Mater*. 2015, 27 (17), 2747–2752. DOI

9. Cho, Y., Shin, J.-H., Costa, A., Kim, T. A., Kunin, V., Li, J., Lee, S. Y., Yang, S., Han, H., N., Choi, I.-S., Srolovitz, D., "Engineering the Shape and Structure of Materials by Fractal Cut", *Proc. Nat. Acad. Sci. USA*, 2014, 111 (49), 17390–17395. DOI

10.Gatt R, Mizzi, L, Azzopardi, J. I. Azzopardi, K. M. Attard D, Casha, A, Briffa, J. Grima, J., “Hierarchical auxetic mechanical metamaterials”, *Sci. Rep.* 2015, 5, 8395

11. Tang Y, Lin G, Han L, Qiu S, Yang, S, Yin J, “Design of hierarchically cut hinges for highly stretchable and reconfigurable metamaterials with enhanced strength”, 2015, under review.

12. Castle, T., Cho, Y., Gong, X. T., Jung, E., Sussman, D. M., Yang, S. and Kamien, R. D., "Making the Cut: Lattice Kirigami Rules", *Phys. Rev. Lett., *2014, 113, 245502. DOI

13. Sussman, D. M.; Cho, Y.; Castle, T.; Gong, X.; Jung, E.; Yang, S. and Kamien, R., D., “Algorithmic Lattice Kirigami: A Route to Pluripotent Materials”, *Proc. Nat. Acad. Sci. USA*, 2015. Early View. DOI

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## Comments

## Dear Shu,

Dear Shu,

Thank you very much for your timely and inspiring post as there are active works in the fields on reconfigurable metamaterials. Your works on superconformable metamaterials and kirigami are quite interesting and inspring.

As you describe kirigami, you mentioned that "Different from conventional origami design through folding mountains and valleys, cuts in kirigami to remove part of a lattice are allowed. Therefore, cuts take away unnecessary parts, minimizing waste, and allow for more complex structures from fewer, simpler folds without stretching or shrinking the lattice’s edges."

1) Is there a principle or an alogorithm to figure out unnecessary parts?

2) If one would like to buckle rigid surfaces into desired 3-D shapes using kirigami, are there ways to figure out how to make cuts and identify edges?

Thank you very much for your help in advance.

## Dear Sung Hoon,

Dear Sung Hoon,

Thanks for your interests and very good questions. The kirigami design is still in its infancy. So there are many ways of cuts and the purposes of cuts could be different.

what we suggested in Fig. 3 was to cut parts away, allowing more freedom to fold. But you can also do cutting as shown in Fig. 2, which doesn't take any part away. But the cuts will allow for stretching and possible folding as well.

1) Is there a principle or an alogorithm to figure out unnecessary parts?

It's a very open space. We are borrowing ideas from Physics on geometry and topological defects. What we shown in Fig. 3 is

latticekirigami. By starting with a periodic lattice (e.g. honeycomb), it'll be easier to find the relatioinship, where to place the cuts. Then using paper to visualize the concept.2) If one would like to buckle rigid surfaces into desired 3-D shapes using kirigami, are there ways to figure out how to make cuts and identify edges?

As shown in ref. 13, Fig 4, we first identify a target surface with different elevations, then project it to the triangulation (in the case of sixons) to determine the hight. Then assign the mountain or valley folds to the 2-D sixon sheet. Depending on the complexity of the 3-D shape, it could be tedious to assign the folds. So we will need someone in computer science to help program the folding if we want to get truely pluripotent structures.

Shu

## Thank you very much for your

Thank you very much for your helpful answers, Shu.

I am looking forward to more to come.

## Questions on Figure 2 and 3

Dear Shu,

Thank you very much for posting this fantastic thread. A really interesting and open field! I have two questions on Figure 2 and 3:

1. Is the expansion in Figure 2 isotropic or anisotropic? If you measure Poisson's ratios in two directions, are they equal to each other?

2. If we regard the shape transformation from Figure 3A to Figure 3B-E as a mechanical deformation process, how about the applied loading or boundary conditions for the different shape transformation? Can this process by modelled by the finite element method?

Thanks,

Jinxiong

## Jinxiong,

Jinxiong,

see my response to your questions.

1. Is the expansion in Figure 2 isotropic or anisotropic? If you measure Poisson's ratios in two directions, are they equal to each other?

A: The expansion is isotropic in Fig. 2. Since the unit is symmetric, the Poisson's ratios should be the same if the sheet is stretched equal-biaixially. Of course, if the sheet is stretched uniaxially or non-equal-biaxially, the Poisson's ratio should be different in the two directions.

You can design a stretcher to record the sheet expansion.

In ref. 9, mostly discussion was about finite element simulation and the design of the fractal cutting. In real materials, depending on materials properties, whether the units will be deformed during stretching or fractured or not will affect the actual expansion and Poisson's ration.

2. If we regard the shape transformation from Figure 3A to Figure 3B-E as a mechanical deformation process, how about the applied loading or boundary conditions for the different shape transformation? Can this process by modelled by the finite element method?

A: there is no deformation of the solid parts except at the folding lines. Whether it's a mountain fold or valley fold, the region P will pop up or down vs. region R. So, yes, depending on how the loading is applied and/or the boundry conditions, there will be different ups and downs. In the case of Fig. 3 shown here, we have one cut, so there are only four configurations of ups vs. downs. In ref. 13, we showed different rendering of cuts in a hexagon. Depending on how you place the cuts and folding lines, there are basic modes of the folding configurations in a hexagon. Then if you make an array of it, you will have more combination of different heights, which we defined as -2, -1, 0, 1, 2, and so on. If all the regions pop/fold in the same direction, you get a pyramid.

Yes, all of these can be modeled by finite element method as we show in ref. 13.

## Dear Shu,

Dear Shu,

Thank you so much for posting this insteresting and inpiring discussion.

As you mentioned, by making engineering fractal cuts for super-conformable metamaterials, the stretching of the structure occurs only by rotation of unit cells with trivial deformation in each unit cell. My concern is about the joints between neighboring unit cells. I would expect to see excessive deformation at the joints. Such excessive deformation may tear/fracture the joints, undermining the structural integrity of the metamaterial. I wonder if you observed any failure/fracture of joints in response to large deformation. Many thanks.

Look forward to your response.

## Hi Zheng,

Hi Zheng,

You are correct that unit rotation will cause severe stress concentration in the joints. In real materials, especially those less-stretchable materials, it will break the joints and thus the structure is less expandable, which is one of the limitaiton of cuts. With Shu, we have a paper under review discussing the mechannical and phononic behavior of the hierarhically cut metamaterials made of brittle and superelastic materials. In the paper, we do see the joints break in the first level joints in both materials. We have proposed two strategies to address the issue you mention. One is through design of the local cut shape and the other is design of global distribution of hinge width across hierarhical levles to distibute the load more evenly to each level. Hope I have answered your quesiton. Thanks.

Jie