Strong Conformable Structure via Tension Activated Kirigami

Kirigami, the cutting and folding of sheets, creates three-dimensional shapes from flat material. Known for


Introduction
The art of creating three-dimensional structures from flat sheets of material has captured the imagination and creativity of people for centuries.Origami, or the folding of sheets, has been used to turn continuous flat materials into complex shapes 1,2,3 .Origami approaches have been used to create durable, adaptable, bi-stable and inflatable structures 4,5,6 .A diversity of robotic systems have been built with origami components for actuation or locomotion 7 .Kirigami, which involves both cutting and folding, increases the design space and has been widely studied [8][9][10][11][12] .One area of study seeks to increase the conformability of flat sheets or controls their expanded shape by adding slits [13][14][15][16][17][18][19] .Much of the work in the realm of kirigami, true to the ancient artform, involves the controlled folding of the flat material.This folding process can be performed by actuators 20,21 , by machines 22 or by hand.We find that the folding process can be tedious and inconsistent, creating a potential impediment to broader adoption of kirigami.To bypass the folding process, we have investigated kirigami patterns that deploy from their flat state into their expanded state when tension is applied.The most basic of these Tension Activated Kirigami (TAK) patterns, the single slit, has been widely studied [23][24] and used in a solar tracker 25 and with non-straight slits as a shoe grip 26 .
We have discovered a particularly useful class of TAK patterns that expand quickly to produce significant wall structures that are orthogonal to the original plane of the flat material.We show that this new class of expanded Folding-Wall TAK structures are stronger, more voluminous, and can interlock with themselves, all improvements over the basic patterns most widely in use today.We demonstrate the value of these Folding-Wall TAK structures with two applications: as a replacement for honeycomb cores and as an improved cushioning wrap.
The advantages of honeycomb structures have been widely discussed 27 .Learning from naturally occurring structures in beehives and cork trees, honeycombs are widely used today when a strong, stiff structure is required with minimal weight 28,29 .Today the cores for honeycomb panels are made through a variety of processes taking multiple steps 30 .We propose that the creation of cores using self-deploying Folding-Wall TAK sheets could offer multiple benefits over current solutions.Three such benefits include lower cost, simplified manufacturing processes, and the creation of field deployable structures where the core is stored flat until the structure is assembled.
Even without top and bottom layers, a deployable honeycomb-like structure is valuable.One application for Folding-Wall TAK structures is as a protective wrap for objects during storage or shipment.The pattern is cut into a sheet such as paper and occupies minimal space until needed.When tension is applied to the material it will deploy and reveal the honeycomb-like shapes.The expanded material occupies over 50 times the volume of the undeployed material and is highly conformable so it can be wrapped around oddly shaped objects.The expanded material interlocks with itself when multiple layers are adjacent, allowing it to stay in place without tape, adhesive, or string.Additionally, modifications to the pattern and materials can enhance the volumetric expansion and the interlocking strength.
We begin by reviewing two configurations of the single slit TAK pattern 8,[23][24][25][26] and then introduce the Folding-Wall TAK pattern.We describe the deployment mechanics along with experimental results showing: 1) The stiffness and strength of the Folding-Wall TAK structure compared to single-slit TAK structures and a constructed honeycomb core of the same material.
2) The energy absorbing and volume expansion capability of the Folding-Wall TAK structure compared to single-slit TAK structures in the same material.

TAK Structures
Kirigami patterns can be used to expand materials into three-dimensional shapes when tension is applied.The most well-known TAK pattern is the single-slit pattern shown in Figure 1A.Straight slits are cut in offset columns with a distance between columns of H, an uncut gap between slits of W, and an overlap distance of L between adjacent slits.When tension is applied along axis T, the slits open and the material buckles forming the structure of Figure 1B or 1C. Figure 1B shows each column of the pattern buckling the same direction creating a parallel deployed state.Figure 1C shows an alternating configuration where columns buckle in opposite directions.We found that materials deployed with an alternating configuration have increased stiffness compared to the parallel configuration.One sheet of material can have some parallel and some alternating columns when deployed.It is also possible to have one column change from parallel to alternating, we call this an inversion.In this study we took care to avoid inversions which generally reduce the strength of the deployed material.The single-slit patterns have surfaces that rotate out of plane, but the rotation of those surfaces is limited.
We developed the Folding-Wall TAK family to produce stronger deployed structures with surfaces that rotate fully orthogonal to the original plane of the sheet.The planar form of the pattern is shown in Figure 2A along with the primary tension axis T. Figure 2B shows the structure formed when that pattern is cut into kraft paper and tension is applied along the tension axis.We find that the Folding-Wall TAK deploys quickly into the honeycomb pattern visible in Figure 2B when tension is applied.Additional tension causes the hexagonal regions to narrow and elongate but the total area occupied by the deployed structure remains quite constant (Supplemental S1A-4).3A.When tension is applied along axis T, the rectangular region experiences opposed forces represented by the arrows in Figure 3B.These forces cause the region to buckle and rotate out of plane, nominally along the Pivot Axis shown in Figure 3B.As the region rotates out of plane it also folds, creating the stable Folding-Wall structure shown in Figure 3C.The regions directly above or below the tabs fold to become part of the continuous folded vertical wall, although if the tab width W is very small these wall sections are more curved than flat wall regions.Figure 3C shows an idealized model of the deployed Folding-Wall section where the walls are orthogonal to the original plane of the rectangle represented by the dashed rectangle.The tabs largely remain parallel to the original plane.This folded wall section comprises a very strong structure that can stand on its own and resist significant compressive forces.A continuous array of these folded wall sections is created by connecting the tabs in rows of adjacent rectangular sections, as shown in the Folding-Wall TAK pattern of Figure 2A.It is important to note that if the material used for any TAK is too thick relative to the scale of the pattern, or if the material is too compliant, then the tension forces may not cause the material to buckle out of plane.For example, a soft foam sheet thicker than the longest slit would likely compress and deform in the original plane of the sheet and not buckle out of that plane.In addition, materials that are too weak or brittle may tear before they fully deploy.Predicting the material and geometry relationships that govern the boundary between the two responses is subject of future work.

Structural Strength
We evaluated the structural strength of Folding-Wall TAK samples against both configurations of single-slit TAK and honeycomb constructions using 3 different materials: paper, plastic, and aluminum.The dimensions and materials are summarized in Table 1 The TAK samples created using paper and plastic retracted when tension was released.To allow for consistent testing conditions, we placed the samples on a 12.7mm thick aluminum plate and taped the ends holding them in the deployed state.The forces introduced by the tape could not be easily measured and were not accounted for in the analysis.The constructed honeycombs and aluminum TAK structures did not require tape to maintain their deployed state.Six samples were created for each of the twelve scenarios in Figure 4. We placed the aluminum plate with each sample in a load frame and compressed the sample with a custom aluminum circular platen of diameter 76.2mm.Force and displacement were measured versus time for each sample and converted to stress and strain.Figure 5 shows typical stress vs. strain curves collected for constructions made of paper.The Folding-Wall TAK and constructed honeycomb samples show a well-defined elastic region with a peak force before the initial onset of buckling.The singleslit TAK patterns, both parallel and alternating, did not have a clear peak so we did not report the peak stress, or strength, for these.The absence of a clear peak for the single slit TAK can be attributed to the distributed nature of the contact points, differences in the undulation heights and the lower buckling resistance of undulations as opposed to a vertical wall.The general shape of the plots was consistent across all three materials: paper, plastic, and aluminum.The slope of the stress-strain curve was estimated to find the elastic modulus, or stiffness of the sample.We also measured the thickness of the deployed samples from the curves and combined with geometric calculations to estimate the density of the sample in the deployed state (Supplemental S1).The elastic modulus, or stiffness, was divided by the density to produce a specific stiffness.The peak stress (a.k.a.strength) before buckling was also extracted from the curves.This strength was divided by the density to produce a specific strength.The averages for each of the types of sample are summarized in Table 2.These results show that single-slit TAK structures deployed in the alternating configuration are stiffer than single-slit TAK deployed in the parallel configuration.They also show that Folding-Wall TAK structures are significantly stronger than either of the single-slit TAK configurations.Although the Folding-Wall TAK samples are shown to be weaker than the constructed honeycomb samples, the difference is only around a factor of two when the amount of material is considered as shown by the specific stiffness and specific strength.We conclude from this data that the Folding-Wall TAK samples create useful honeycomb-like structures with superior strength compared to single-slit TAK samples.They may also be useful replacements for some full honeycomb structures, eliminating the need for folding, arranging strips, and bonding adjacent rows while also providing high conformability that enables use in non-planar applications.

Volume Expansion and Energy Absorption
In addition to strength or stiffness of individual layers, we found other advantages of the TAK structures compared to a full honeycomb structure.The TAK samples are flexible, deploy easily to occupy space and can absorb energy when compressed.These characteristics make them good candidates for use as an expanding cushioning wrap.
We laser cut single-slit and Folding-Wall TAK patterns into long sheets of paper, extended them, and wrapped them around a thin, flat frame producing the samples shown in Figure 6A.Six wrapped samples were made of each type.Details are summarized in We found that adjacent layers of the Folding-Wall TAK interlocked with each other, holding the pad in its final shape.The single-slit TAK naturally deployed into the parallel configuration then tended to unwrap, so we applied tape to hold the pad in its wrapped shape.We learned that the single-slit pattern took significantly more force to extend than the Folding-Wall TAK.All pads of material were compressed between a flat plate and a 76.2mm diameter platen in a load frame at 1mm per second to a maximum force of 8896N.We recorded the resistance force and displacement during compression.The total loft of wrapped layers was defined as the distance traveled from first detecting resistance until the maximum force of 4448N was reached.The total energy absorbed by the pad was calculated by integrating the force vs distance curve from the onset of resistance to 4448N.The results are shown in Table 4.
Both the TAK patterns became narrower when extended, so their area changes with strain.To calculate the total volume change when expanded, we used a geometric analysis to calculate the Area Expansion Ratio, or the ratio of the deployed area to the area of flat, undeployed material required to fill that area (Supplemental S1).This Area Expansion Ratio is multiplied by the loft per layer to find the volume expansion ratio for the two samples, where  = The results show that the Folding-Wall TAK is superior to the single-slit TAK in terms of energy absorption and volumetric expansion, suggesting that it might comprise an improved sustainable cushioning product.The strong interlocking of adjacent layers adds additional value to the Folding-Wall TAK since in a wrapped form it does not require a tape or adhesive to hold its shape.

Conclusions
We have introduced a new kirigami pattern that can be activated by tension into a deployed state with large Folding-Wall regions that are orthogonal to the original plane of the sheet.The resulting structure is strong relative to the amount of material used, with a high specific stiffness and specific strength.It also expands to occupy much greater volume in the deployed state, absorbs energy when compressed, and is very easy to deploy.In addition, we found that multiple layers of this pattern interlock with each other.The combination of these features suggests that the Folding-Wall Tension Activated Kirigami could have significant value for many applications.
Supplementary Information is available for this paper.
Correspondence and requests for materials should be addressed to Tom Corrigan.
Reprints and permissions information is available at www.nature.com/reprints.

Methods
Fabricating samples: We cut paper and plastic samples into respective substrates using a Model XLS 10.150D laser cutter (obtained from Universal Laser Systems, Inc., Scottsdale, AZ).Aluminum samples were formed using a Mitsubishi Fiber Optic Laser: ML 3015 eX-F40 with the EX-F M800/LC30EF Control. Patterns: The following patterns were used to produce the testing samples of Figure 4, with results shown in Table 2.We used these patterns for the roll testing shown in Figure 6, with results in Table 2. Paper: brown kraft paper from Uline, Pleasant Prairie, WI under trade designation "S-7051".It is 100% recycled paper and has a thickness of about 0.0075" (0.19 mm) when measured according to test method TAPPI T411 om-10.The basis weight was measured as 126.5 grams per square meter according to test method TAPPI T410 om-13.The density can be calculated as the basis weight divided by the caliper = .666g/cm 3 White Plastic (PET): 7mil polyester from Mitsubishi under the name Hostaphan™ W54B Film.Vendor specifies the density per ASTM D1505 to be 1.42 g/cm 3 .
Test Preparation:

1) Single Slit Tension Activated Kirigami
We prepared single slit TAK samples by taping one end of the sample to one edge of an aluminum plate (.5" thick, 12"x16").The sample was then stretched to the target length of 133% (13" for 9.75" long samples), and a single sheet of 9"x11" 3M Wetordry TM 100 grit sandpaper [431Q] was placed underneath the sample to avoid slipping during compressions.A 3" diameter piece of the sandpaper was also fastened to the top platen.
Using the laser cut samples and plate described, the edge of the sample was placed ½" from the edge of the plate and taped down with a 5" strip of tape.The sample was then deployed and expanded to the target length of 133% and taped down with another 5" strip of tape.For all samples, 3M Scotch Duct Tape [920-P-GRN-C] was used to secure the sample.For aluminum samples, no adhesive was needed to secure the samples as the aluminum held its shape after deployment.
We used a ruler to deploy the single slit pattern in the parallel alignment.While one hand pulled the edge of the sample to expand it at a slight upwards angle from the plate, the other hand applied light pressure to the undeployed section of the sample to ensure an even and parallel deployment.
To deploy the single slit pattern in the alternating alignment, the sample was gently folded prior to deployment.This fold is made by folding each ½" segment on top of one another, alternating direction like an accordion.At each level, we made a gentle crease to train each segment to deploy in the correct direction.The occasional region may invert despite folding but can be fixed by hand once the sample is fully deployed and taped down.

2) Folding-Wall Tension Activated Kirigami
We prepared Folding-Wall TAK samples by taping one end of the sample to one edge of an aluminum plate (.5" thick, 12"x16").The sample was then stretched the to the target length of 150% (15" for 10" long samples), and a single sheet of 9"x11" 3M Wetordry TM 100 grit sandpaper [431Q] was placed underneath the sample to avoid slipping during compressions.A 3" diameter piece of the sandpaper was also fastened to the top platen.
Using the laser cut samples and plate described, the edge of the sample was placed ½" from the edge of the plate and taped down with a 10" strip of tape to cover the complete end of the sample.The sample was then deployed and expanded to the target length of 150% and taped down with another 10" strip of tape.For all samples, 3M Scotch Duct Tape [920-P-GRN-C] was used to secure the sample.For aluminum samples, no adhesive was needed to secure the samples as the aluminum held its shape upon deployment.
We used two hands to extend the folding wall samples from the secured end.Prior to extension, the free end of the sample was also taped down for easier taping of the second edge.Most times there are singular segments that need to be flipped individually, but each row should all go the same direction and will be near perpendicular to the plate at full extension.

3) Constructed Honeycomb
The constructed honeycomb samples consist of 13 strips (9.5"x0.5")which are stuck together using Scotch ATG Adhesive Transfer Tape [924].We individually folded these strips with 18 consecutive 60-degree bends that are all ½" apart and form half-hexagons.The folds used to make these bends alternate every two folding locations which gives the hexagon shape.For the paper and plastic samples, we folded by hand.The aluminum samples were folded using a 3D printed tool for faster and safer bending.After all strips were folded, they were stuck together with ½"x ½" pieces of transfer tape to form a honeycomb structure.To aide in accurate testing, the adhesion of these strips must be done carefully making sure the strips are completely fastened together all at the same height, flush with the tabletop, and aligned properly for a complete honeycomb structure. .
The pattern of Figure S1 can be broken into a repeating rectangular pattern defined by the dot-dash rectangle shown.The width of the rectangle in the flat state is 2W+2L.As the tension lines straighten and each rows axial length is increased to H+δ, the width of the rectangle becomes 2 + 2√ 2 +  2 − ( + ) 2 When the tension lines become straight (aligned with the tension axis T) then , and at maximum extension is    =  (+) .
The expansion ratio at arbitrary extension δ, and corresponding axial strain (  ) is: Where the axial strain   =   .
The single slit TAK samples created for the Volume Expansion and Energy Absorption experiments had dimensions H=.125" (9.53mm), L=.125" (9.53mm), W=.25" (19.0mm).For these dimensions, the Expansion Ratios are the same as the previous result because the ratio of the lengths is the same.
The theoretical maximum axial strain can also be calculated using the equation above and the single-slit TAK dimensions.Since H=L for both single slit TAK designs,    = √ 2 + 2  = √ 2 + 2  = √2, this suggests a theoretical maximum extension of 41%.In practice, very large forces are required to reach this level, and the material often rips, which is why the single slit TAK samples were extended by 33% to achieve practical near full deployment.

2) Folding-Wall Tension Activated Kirigami
The basic pattern is repeated from Figure 3A.That basic pattern is further divided into a repeating unit cell, represented by the red dashed line.
The area of each rectangular unit cell in the flat state can be easily shown to equal: When that pattern is fully deployed, it produces the following geometric pattern when viewed from above (viewing angle is orthogonal to the original plane of the sheet).The thickness of the sheet is ignored.The pattern resembles a hexagon or octagon, depending on whether you count the width of the connecting Combining equations, we can calculate the expansion ratio as a function of axial strain: For the dimensions used above (L=.5",H=.5", W=0.1"), for a 50% extension (δ =.2,   = .5,  = 1.5) the expansion ratio is 1.33.
We can plot the Expansion Ratio as a function of the strain, from 0 to 1.The results below (S1-A-4) show that the total planar area of the expanded Folding-Wall TAK structure remains quite constant across a wide range of strain value, staying above 1.3 from 40% to 70% strain.The deployed thickness stays constantly at H for this entire range.

3) Constructed Honeycomb
The constructed honeycomb pattern requires the connecting of bent strips of material.The strips are H units wide then bent and assembled into a honeycomb pattern shown below, with regular hexagons.We assume that the material thickness is very small compared to the scale of H and L. We also ignored the thickness of the adhesive as well as its mass.Considering one unit area, shown by the red rectangle in Figure S5.The total length of material in the two strips that occupy a unit cell area is 8L so the total area in that region is 8LH, which is the area that the material from the unit cell would occupy in the flat state.The total area of the deployed cell unit area is (√3) (2 + 2 (  2 )) = 3√3 2 .The honeycombs are not tension deployable, and they were constructed as shown, so their expansion ratio is constant for this design.The ratio of the deployed area to the original area of the material can be calculated to be: [7]    For the dimensions used above (L=.5",H=.5") the expansion ratio is 0.65.

4) Expansion Ratio Plots
The expansion ratios versus strain are plotted below for both the Single-Slit TAK and Folding-Wall TAK.The plots utilize the dimensions of the samples created.
Figure S6: Expansion Ratio Plots

S1-B: Expanded Thickness (Loft)
The thickness of one layer of the expanded samples (the loft) was determined by using the data from the compression test.The loft was defined as the distance traveled from the top of the sample to the bottom of the sample while compressing it.
The top of the sample was defined as the point where the compression sensor first detects the material.That top was defined as the position where the sensor no longer displays noise as the upper platen is being lowered.Normally that noise it centered on zero, but sometimes the sensor has a little drift.The zero level of the sensor is defined as the average value of the first 500 readings (5 seconds).Contact with the top of the sample is then defined as the last position where the sensor last reads any value below that average zero level.All of the compression values are then trimmed to define that position as the zero position where compression first starts.It is also noted that some of the samples arch away from the platen slightly, and some small force is required to settle those samples against the platen.Those forces did not seem to dramatically affect the results so they can still be seen in the compression data.
The bottom of the sample was defined as the position reached when 1000lb (4448N) was applied.

S1-C: Density Calculations
The expansion ratio can be used to calculate the density of the expanded material.Since we know that the deployed area under the 76.2mm diameter compression head is   =  2 =  ( 76. 22 ) 2 = 4,560 2 .The flat area that would produce that expanded area for any sample can be calculated from the definition of Expansion Ratio as: .The mass of the material under the compression head can then be calculated based on the flat area, the material thickness, and the known density of the material:   ℎ =  = (  )( ℎ)( ) [8]    The density of the expanded structure can be determined by dividing the mass of the material under the compression head by the expanded volume of the material under the compression head: ) 2 () [9]    The results of the density calculations for all samples are shown in Table 1S.

S1-D: Stiffness to Weight Calculations
The stiffness to weight ratio, or specific stiffness, can be calculated by dividing the stiffness (measured during testing) by this density for a give material and structure. [10] S1-E: Density Calculation Summary Table 1S: Density Calculations

S1-F: Volume Expansion Ratio Calculations
The volume expansion ratio is the ratio of the volume of material after expansion (deployed) to the volume of material before expansion (flat).It can be calculated from the thickness (or loft) values before and after deployment and the expansion ratio.(1.33) = 55.8 [13]    Note: The Single-Slit TAK samples were held in their deployed position with tape, while the Folding-Wall TAK held its position without tape.Without the tape, the Single-Slit TAK were significantly less thick in the deployed state.

S1-G: Stress-Strain Curves for Compression Tests
The plots of the stress vs. strain curves as well as the linearization of the stiffness slope for the 6 sets of samples for all 12 of the cases shown in Figure 4 appear below.Compressive strain is shown as positive in the graphs.

Figure
Figure 1A: Single Slit TAK Pattern

Figure
Figure S1: Single Slit tension lines

s u n d e r h e a d ( g ) E x p a n d e d D e n s i t y ( g / c m 3
Fig Description E x p a n s i o n R a t i o l o f t ( i n ) l o f t ( c m ) F l a t A r e a ( m m 2 ) m a t e r i a l t h i c k n e s s ( m m ) m a t e r i a l d e n s i t y ( g / c m 3 ) M a s s u n d e r h e a d ( g ) E x p a n d e d D e n s i t y ( g / c m 3 Figure S7: Stress-Strain Plots for Paper Samples

Figure S10 :
Figure S10: Force vs. Displacement Plots for Rolled Samples

Table 1 :
. Sample materials and dimensions

Table 2 :
Stiffness and strength measurements for compressed samples.

Table 3 .
The single-slit dimensions were chosen to match several commercially available paper-wraps first developed in the 1970's 31 .

Table 3 :
Volume Expansion and Energy Absorption Test Materials and Dimensions

Table 4 :
Volume Expansion and Energy Absorption Results For the samples rolled and compressed, the average volume expansion ratios were calculated as: