An Undulating Kirigami Pattern With Enhanced Tear Strength

Useful structures can be created from sheet materials. Origami, or folding of sheets, turns continuous sheet materials into complex shapes [1–3]. Structures that are durable, adaptable, bi-stable, and inflatable have been created by origami [4–6]. Many robotic systems have also been built with origami components for actuation or locomotion [7]. Strong origami constructions have also been studied, such as the Miura-Ori pattern [8] and those made with curved-crease folding approaches [9,10]. Kirigami, which includes both cutting and folding of sheets, has also been widely studied [11–15]. One area of study seeks to increase the conformability of flat sheets or prescribes their expanded shape by adding slits or removing material [16–23]. Much of the work on kirigami and origami, like the ancient artforms, involves the controlled folding of flat material. This folding process can be performed by actuators [24,25], by machines [26], or by hand. Kirigami patterns also exist that deploy from a flat state into an expanded state when tension is applied. The most basic of these tension-activated kirigami (TAK) patterns, the single-slit pattern, has been widely studied [27–29]. A strong deployable structure is created from the folding-wall kirigami pattern [30,31]. These patterns have been used to create paper-based sustainable packaging products.

The multi-slit TAK pattern is a novel family of kirigami slit pattern that creates ribbons of material that undulate along the transverse axis while remaining parallel to the tension axis. These undulating ribbons enable unique features for the deployed structure.

One practical challenge for many tension-activated kirigami constructions is a low tear strength. A unique approach, called multi-beams, is shown to increase the maximum tension force that a multi-slit kirigami pattern can withstand before it tears.

The most widely studied and used tension-activated kirigami pattern is the single-slit pattern. It is so ubiquitous that many papers on the topic refer to it as simply “the kirigami pattern.” The basic single-slit pattern is shown in Fig. 1 along with the primary tension axis T that will expand the flat pattern into a deployed state. The deployed state will comprise some combination of two states, parallel and alternating (or anti-symmetric and symmetric) [29], as shown in Fig. 2.

It has been shown that the configuration of the deployed structure can be largely predicted [29]. In practice, especially with large arrays and low-modulus materials, most single-slit sheets will deploy into a combination of parallel and alternating configurations with many inversions where a row switches from one configuration to the other.

The single-slit kirigami pattern has been used or proposed for many useful applications, from expanded metal panels to solar trackers [32] and with non-straight slits as a shoe grip [33]. The pattern has also been used to create an expanding paper cushioning wrap [30].

For some applications, the single-slit pattern has weaknesses. The top and bottom surfaces of the deployed structure comprise only edges, so interfacing with adjacent surfaces can be challenging. For applications such as an expanding wrap for cushioning, it is desirable for multiple layers of the deployed material to interlock with adjacent layers to hold their position once wrapped around an object. The single-slit pattern has been found to interlock only weakly with adjacent layers. Finally, it is desirable in many applications for the kirigami pattern to have a high tear strength, so the material does not rip when being deployed near its maximum strain configuration.

A new tension-activated kirigami pattern, called the multi-slit pattern, is proposed that has advantages over the single-slit pattern. The multi-slit pattern can be conceived as a duplication of each slit in the single-slit pattern with an identical slit that is spaced along the tension axis. The most basic multi-slit pattern is the double-slit pattern, as shown in Fig. 3.

The dimensions defining the double-slit pattern are similar to the single-slit pattern with the addition of a variable, R, to define the distance between adjacent duplicate slits. Figure 4 shows a deployed double-slit kirigami structure where the duplicate slits have now created an undulating loop or ribbon of material that moves in and out of the original plane of the untensioned sheet of material. The double-slit pattern of Figs. 3 and 4 comprises alternating strips of undulating ribbons and strips of connecting material.

The loops of material provide enhancement over the single-slit patterns. The undulations can be used as soft cushioning elements. They can also interlock with adjacent materials. These undulations were found to interlock well with adjacent layers when multiple layers of deployed multi-slit kirigami were wrapped around an object. Further advantages of the undulating ribbons are being investigated.

The deployment kinematics of the multi-slit kirigami can be approximated by considering the path of the highest stress through the kirigami sheet as tension is applied, or the tension lines [30]. Figure 5(a) shows the basic double-slit kirigami pattern with the tension lines shown by dotted lines and a single unit cell region defined by shading (hatch) lines. Figure 5(b) shows the pattern cut into paper in a mostly deployed state with the tension lines overlaid on the image. Some tearing is seen at the corners of the tension lines suggesting that this material cannot support the force to fully straighten the tension lines.

This expansion ratio can also be used to calculate the density of the deployed structure [30].

Deployment of the multi-slit kirigami differs from the single-slit kirigami because each row of connecting material of height H is now decoupled from other rows of connecting material by the “flat” region of height R. In the single-slit kirigami, each row of material is highly coupled to the adjacent row.

If more than one duplicate slit is included in the pattern, then a higher-order multi-slit can be created. Examples of triple-slit and quadruple-slit designs are shown in Fig. 6. For these examples, the additional slits were evenly spaced within the width of the ribbon, R. Higher-order multi-slit patterns create additional undulations or loops which can increase the interlocking. These additional loops tend to spread away from each other (Figs. 6(c) and 6(d)) because of the stress in the deployed structure, which might improve the interlocking even further.

The interlocking strength of various patterns was evaluated qualitatively by expanding and wrapping a section of the pattern around a 75 mm cylindrical tube so at least two layers of deployed material were overlapped. If the material held onto the tube, it was considered interlocking. The material was then slid off the tube and its resistance to disentangling the layers provided a qualitative evaluation of its interlocking strength. A quantitative evaluation method is being developed where one expanded layer is fixed and a second deployed layer is dragged across the first while force and displacement are recorded. This new method will be the subject of an upcoming publication.

The deployed multi-slit pattern of Fig. 3 produces loops of material that interlock with adjacent layers of deployed material, or other materials. The interlocking strength of that basic kirigami pattern can be increased by adjusting the shape of the slits to include protrusions that can more easily interlock. Examples of protrusions include cusps, as shown in Fig. 7, zigzags, flaps, or almost any excursion from a straight line. Having these protrusions on the undulating ribbons makes them more likely to interlock with adjacent materials which may include another layer of the same deployed kirigami pattern.

The multi-slit kirigami pattern has unique capabilities. However, like many kirigami patterns its ultimate tensile strength can be limited. Additional slits were added in the primary tension-carrying region of the pattern. The purpose of these additional slits was to create multiple parallel tension-carrying beams, or multi-beams. Figure 8 shows an example of a typical double-slit pattern in a single-beam design alongside the same double-slit pattern enhanced with slits to create a two-beam and three-beam pattern.

The tensile performance of the double-slit pattern was investigated for a design with one, two, and three beams, as shown in Fig. 8. We chose to use a kraft paper to investigate sustainable applications such as a paper-based cushioning wrap. The virgin kraft paper used had a thickness of 0.113 mm, and a basis weight of 67.5 g/m².

The patterns shown in Fig. 8 were laser cut into samples of paper with the machine direction of the paper aligned with the tension axis. The basic double-slit pattern followed Fig. 3 with values of H = 6.4 mm, R = 9.5 mm, L = 19.0 mm, and W = 6.4 mm. The multi-beam versions added slits uniformly distributed as shown in Fig. 8. The spot size of the laser, or the kerf, was carefully measured and the patterns were adjusted to account for the loss of material. For example, the total height of the two beams in the double-beam design equals the height H in the single-beam design after laser cutting.

Initial samples were clamped continuously along two edges, but the samples failed prematurely at the corner of the clamped region. The clamps were constraining the transverse strain, which according to Eq. (4) has a maximum theoretical limit of −0.75. The samples would tear due to transverse forces before they experienced maximum axial loads. Axial loading was the subject of this investigation, so the transverse constraints were removed.

A fixturing approach was introduced that allows the edges to move transversely while still applying force along the tension axis. Brackets were created to hold 25 mm diameter rods which were populated with clips normally used to hold shower curtains. The clips have balls so they slide easily along the rod. A simple experiment showed that the clips had a frictional coefficient of about 0.15 rolling on the rod. The testing patterns were modified to include individual tabs centered on the land region of a row of undulating ribbons. The test fixture and the layout for a test pattern are shown in Fig. 9.

The samples were attached to the clips on the brackets which were mounted in a load frame (MTS Criterion Model C43 104E). Force and axial displacement were recorded as the frame was moved at 10 mm/s. The zero-displacement position for this tensile test was defined to follow the last negative value in the force sensor.

A finite element analysis (FEA) model that can simulate multi-slit and multi-beam kirigami pattern expansion mechanics was developed to serve as a comparison with experimental findings as well as to predict and evaluate the mechanical behavior of multiple different designs in an accelerated manner.

The FEA model was created, executed, and post-processed in the commercial software abaqus/standard 2023 (Dassault Systèmes Inc.). A planar computer-aided design (CAD) surface model of the unexpanded (or undeployed) kirigami pattern was created in Siemens NX and then imported as a STEP file into abaqus. Every simulated kirigami pattern was designed to ensure enough unit cells exist in row and column directions (7.5 and 5.5, respectively) to be representative and mitigate any edge effects. Further, to enable maximal similarity to the experimental setup, features such as tabs and hook holes in the laser cutting of designs were preserved in the CAD used for simulation. A typical undeployed kirigami pattern prior to import in abaqus is shown in Fig. 10.

Shell elements (S4R) were used to mesh the imported pattern and were assigned the same thickness as the kraft paper (0.113 mm). It should be noted that this thickness is significantly smaller (∼3 orders of magnitude) than the other two dimensions for all the designs considered and hence simulating this structure as a shell is justified. The material properties of the kraft paper in the simulation were modeled as linearly elastic with a density of 0.599 g/cc and Young's modulus of 2.03 GPa. The static analysis step was used for all the simulations presented in this work.

Figure 10 demonstrates the loading conditions used in the expansion simulation. The bottom edge of the design was constrained in the loading direction (Y) and a uniform Y-displacement was applied to the top edge, while keeping the other degrees-of-freedom unconstrained. In addition, the left and the right-side edges were also fully unconstrained to mimic the stretching mode shown in Fig. 9(a) as closely as possible. High nominal strains (>100%) were applied to the patterns for this study. An example of the expanded kirigami pattern at an intermediate stage is shown in Fig. 11.

The simulation results were post-processed to extract the displacement and reaction force at the loading edge and then converted to nominal stress and nominal strain using the known size of the pattern and the specified thickness of the shell. As this model did not explicitly incorporate any damage or fracture behavior, tearing and tear strength (nominal stress at the point of tearing) were identified from the post-processed results using a maximum elemental stress-based criterion. It's worth mentioning that, unlike the rounded slit corners created as a result of laser cutting in the experimental study, simplified sharp rectangular slit corners were used in the simulation. For the kirigami patterns studied in this work, the three base double-slit patterns (single, double, triple-beam) were used as representative and a maximum elemental stress failure threshold of 106.733 MPa was derived by directly correlating the simulations and experimental results. This same threshold was used to identify the tearing/failure for all simulated patterns thereafter.

A broad family of patterns was generated, including, but covering a bigger design space than the experimental portion of this study. They were generated by varying the length L, height H, width W, number of slits, and number of beams. Figure 12 shows typical nominal stress–strain curves for example patterns with the tearing point marked. Table 2 lists the different designs that were simulated.

Kirigami samples were created with a laser cutter using the method and dimensions described in the section “Experimental Method”. Five copies of each double-slit pattern were created as shown in Fig. 8 (single-beam, double-beam, and triple-beam). In addition, five copies of the single-slit kirigami pattern shown in Fig. 1 were created with the same values for H, L, and W, as a reference. All the samples were tensile tested as described in the section “Experimental Method”.

The force and displacement data were converted to stress and strain values. Typical stress–strain curves are shown in Fig. 13 for single, double, and triple-beam multi-slit patterns as well as the single-slit pattern. The results of the testing are summarized in Table 1. The tear was defined as occurring at the peak force, or ultimate tensile strength, although some amount of stretching and tearing may have occurred before that peak.

The results show a 26% increase in the tear strength of the kirigami pattern by changing from the single-slit to the double-slit pattern. An additional 25% increase in tear strength was found by changing from the single-beam to the double-beam version of the double-slit kirigami pattern. Changing to the triple-beam version produced another 87% increase in tear strength.

The single-slit kirigami pattern showed a much larger strain at tear. This can be explained by the absence of undulating ribbons which do not extend axially at all. All the double-slit patterns had significantly lower maximum strains. The triple-beam double-slit kirigami pattern showed a slightly higher maximum strain.

FEA simulations were carried out for each of the three double-slit patterns: single-beam, double-beam, and triple-beam as well as the single-slit pattern. The nominal stress and strain were obtained from the reaction force and displacement output in the simulation. Failure points were identified for each of these cases using the maximum stress criterion described in the section “Simulation Method”. The plots were found to have a similar qualitative shape but differed quantitatively, leading to significantly different failure points. The nominal stress rises slowly in the initial phase of the stretching but builds up very steeply at higher strains. This behavior is similar to the observations in the experiments. A summary of these simulations is presented in Table 2 and the stress–strain curves are shown in Fig. 14.

The failure threshold obtained in the section “Simulation Method” was further verified in the single-slit pattern. The predicted stress and strain at failure for the single-slit and double-slit single-beam pattern from the simulation match very closely with the experimental results and generally follow the experimental data for the double-slit double-beam and double-slit triple-beam results.

The double-slit pattern has a 25% higher tear strength than the single-slit pattern. The strength was further increased by 91% for double-beam and another 10% for the triple-beam design. The strain at failure for the three double-slit patterns was found to be reduced significantly (by more than half) when compared to the single-slit pattern. All three double-slit patterns showed similar strains at tearing with the double and triple-beam cases slightly higher than the single-beam.

The design space for the double-slit double-beam pattern was investigated using FEA simulations which revealed the effect of design parameters L, H, and W. Double-slit double-beam designs with twice and half the values of L, H, and W were generated resulting in six new patterns in addition to the base pattern already discussed above. The results are summarized in Table 2 and the stress–strain curves are shown in Fig. 15.

The simulation demonstrated that slit length plays a significant role in the strength-strain response in the sense that doubling the slit length L while keeping other parameters the same can lead to 2.15 times higher strain at tearing but had 62% lower tear strength. This case appears to have similar behavior as the single-slit design which suggests that we start to approach single-slit behavior if the slit length gets very big. On the other hand, halving the slit length L can lead to a 38% higher tear strength value but at the cost of one-third the strain at tear.

Changing the height H was found to have a moderate effect on the tear strength but strongly affected the strain at tear. Doubling the height H led to 20% lower tear strength but almost halved the strain at tear. On the other hand, halving the height H led to a 36% reduction in tear strength but enabled the pattern to sustain 1.3 times the strain at tear. Additionally, comparing the nominal stress–strain curves for different H values (not shown here) did confirm that the pattern with double the H dimension was, in fact, stiffer during the expansion but then reached the stress threshold for failure much earlier thereby exhibiting lower tear strength than the base pattern.

The effect of the width W was found to be the least impactful on the performance in our study. Both double and half W patterns showed similar, albeit one-tenth lower, strain at tear properties when compared to the base case. The tear strength for both the cases is also very similar and about 35% lower than the base double-slit double-beam case.

Both the simulations and experiments demonstrated an increase in tear strength, or ultimate tensile strength, for the basic double-slit pattern over a similar single-slit pattern. The tear strength was then significantly increased by adding multi-beams to that basic double-slit pattern and this observation was also corroborated by the simulation results.

There were some differences in the simulation results for the double and triple-beam designs when compared with the experiments. We found that the strain at tear predicted by the FEA simulation was lower than the experimental value which can be attributed to the fact that the simulation does not have any damage/yielding mechanism and the cuts are modeled as sharp corners which could underpredict the strain at which maximum stress is achieved. We also observed that the double-beam failure point is much closer to the single-beam in the experiments, but it seems closer to the triple-beam results in the simulation. This difference could be due to the steep rise in stress close to the failure point which means that the predicted failure stress and strain values are highly sensitive to any errors/variations in the failure threshold.

A reduction in the maximum strain was also found in the double-slit patterns versus the single-slit pattern. This was predicted largely by the derivation of a simple maximum axial strain value in Eq. (3). An axial strain limit of 216% was predicted for the single-slit pattern (with R = 0), which the experimental tests did not reach with an average limit of 183%. An axial strain limit of about 86% was predicted for the double-slit patterns, but the experimental tests exceeded these values reaching over 100% axial strain. It is possible that the higher transmission of forces in the double-slit patterns allowed additional stretching and initial tearing of the sheets before a catastrophic tear occurred and the force dropped quickly.

The simulation-based parametric study of pattern length, width, and height showed that they affect the mechanics of extension and tearing differently, with the length having a significant effect and width changing the response to a much lesser extent.

The multi-slit kirigami pattern produces potentially useful undulating ribbons of material when deployed. The multi-beam design is an enhancement to the basic multi-slit pattern. Both experimental and simulation studies were performed on the tensile performance of multi-beam double-slit kirigami pattern and showed an increase in the tear strength with the double-slit design that could be increased further by adding second and third beams. The effect of slit length, height, and width on the mechanical behavior of the pattern was also investigated using FEA simulations.

This paper was presented at the 2024 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC-CIE2024).

• 3M Company.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

H =

height: distance between adjacent non-duplicate slits

L =

length: overlap distance between adjacent non-duplicate slits

R =

distance between adjacent duplicate slits

T =

primary tension axis

W =

width: gap between slits along the slit direction

A_flat =

unit cell area when flat (pre-deployment)

A_dep =

unit cell area when deployed

R_exp =

expansion ratio

δ =

unit cell displacement

δ_max =

maximum displacement

ε_ax =

axial strain

ε_tran =

transverse strain

ε_ax,max =

maximum axial strain

ε_tran,max =

maximum transverse strain