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

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.

Thank you very much for your helpful answers, Shu.

I am looking forward to more to come.

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,

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

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 lattice kirigami. 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

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.