Extreme Mechanics Letters 64 (2023) 102084

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Kirigami layer jamming
Robert Baines, Bilige Yang, Luis A. Ramirez, Rebecca Kramer-Bottiglio ∗Yale University, United States of America

ARTICLEINFO

Keywords:
Soft robotics
Laminar jamming
Variable stiffness

ABSTRACT
Variable stiffness materials bridge the desirable qualities of soft and rigid robotic components, forming thebasis of new machines that contextually adapt their structural properties. Recent developments have led to thecreation of jamming materials: Collections of media that undergo pressure-dependent kinematic and frictional
interactions to radically, rapidly, and reversibly change in stiffness. Among the types of jamming, laminarjamming harnesses an ensemble of sheets for stiffness changes. Most implementations of laminar jamming
employ unaltered, continuous sheets, though the vast design space of mechanical anisotropy programmed via
structured cuts invites a new jamming paradigm: kirigami laminar jamming. Through analytical, experimental,
and numerical studies, we show how intentional topological augmentation of laminar jamming sheets throughshell, void, and cut geometric primitives can elicit bespoke mechanical responses. In particular, we optimize
jamming specimens to enhance the jammed to unjammed flexural stiffness ratio across a range of deflections,which unlocks new move-and-hold capabilities. We also demonstrate pressure-induced shape morphing andboth sharp-edged and doubly-curved surface matching with kirigami patterns. Results hold broad significancefor robotics, morphing structures, and the systematic design of variable stiffness materials.

The boundaries between traditional material paradigms inrobotics – soft and rigid – are beginning to dissolve: Variable stiffness
materials grant machines seemingly incongruous compliance and loadbearing properties that may be switched on demand to adapt them tomultifarious environments and tasks [1–5]. A wide array of variable
stiffness materials employed in robotics harnesses changes in intrinsic
material properties, geometry, and sometimes both in tandem, to elicit
programmed stiffness differentials. Examples include shape memory
alloys [6,7], low melting-point alloys [8,9], electrostatic clutches [10,11], and jamming systems [12,13].
The past few years have witnessed jamming systems emerge as aparticularly effective solution for variable stiffness robot components,
due to their high stiffness differentials over rapid timescales, ease of
manufacture, lack of toxic and leaky components, and amenability tovarious form factors [13–17]. Jamming systems typically employ a vacuum to trigger increases in kinematic and frictional interactions within
a closed domain of material, resulting in stiffness changes [18]. There
are several classes of jamming systems, including granular, fiber, tensilefiber, laminar, and stretchable laminar. Each class is distinguished bya distinct form factor and the configuration of its constituent materials. Different deformation mechanics, theoretical stiffness differential
ranges, and favorable loading conditions have therefore been observed
among the classes of jamming [12].
Laminar (layer) jamming harnesses stiffness changes that result
from the compression of a collection of sheets into a cohesive beam-likecross section [19,20]. Among all classes of jamming, laminar jamming
offers the benefits of a thin profile, highly reversible stiffness transitions, and favorable scaling of the stiffness differential by increasing
the number of layers [21]. However, the unjammed flexural stiffness
increases as the cube of the sheets’ thicknesses and proportionallyto the number of sheets. The restoring bending moment and shear
force through the stack of sheets therefore become higher for a given
deflection. For assemblies composed of many thick sheets, move-andhold ability, describing shape retention of the jammed structure (i)after its initial forming load is released, and (ii) upon applicationof external loads, may consequently be compromised, because theassembly of sheets will tend to spring back flat rather than retain adeflected configuration. As a concrete example, consider bending a
rectangular jamming specimen about its short axis 180◦ into a smootharc. With design parameters of fifteen neat (unaltered) Polyethylene
Terephthalate (PET) sheets, each dimensioned 0.25 mm thick, 50 mm
wide, and 150 mm long, the specimen is unable to hold the desired
shape when jammed (Video S1). Another limitation of traditional laminar jamming is that move-and-hold operations are limited to smooth,
single-curvature bending deformations.
Several methods have been proposed to tune the mechanics oflaminar jamming structures and expand their versatility. For instance,
jamming sandwich structures use lightweight, compliant material as∗ Corresponding author.

E-mail address: rebecca.kramer@yale.edu (R. Kramer-Bottiglio).
https://doi.org/10.1016/j.eml.2023.102084
Received 6 July 2023; Received in revised form 25 August 2023; Accepted 23 September 2023
Available online 4 October 2023
2352-4316/© 2023 Elsevier Ltd. All rights reserved.

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Fig. 1. Kirigami layer jamming. a. Augmenting sheets of laminar jamming systems with various types of structured cuts enables new and improved variable stiffness material
functionality, including discontinuous shape-matching, self-morphing, and enhanced load-bearing. b. Summary of initial flexural stiffness ratio amplification due to kirigami layer
jamming. 𝑋-axis categorical labels represent layer jamming specimens with design parameters that are evaluated in this manuscript. Parameters 𝑐∕𝑏, , and 𝑤 are defined inFig. 2a, Fig. 3a, and Fig. 4a, respectively. The jammed to unjammed initial stiffness ratio is normalized by that of the neat specimen. Chart plots a mean of five samples; error
bars indicate one standard deviation from the mean. c. A canonical three-point bending plot of a neat laminar jamming specimen, characterized by three regimes: cohesive, inwhich the sheets collectively act as a solid beam, transition, in which the sheets begin to slip relative to one another, and fully-slipped, during which constant sliding of sheets
and substantial energy dissipation occurs. The dashed line juxtaposes a hand-drawn hypothetical performance of an ideal laminar jamming system that maintains high stiffness atlarge deflections, and exhibits substantially less unjammed stiffness than jammed stiffness. We step toward the ideal system by evoking kirigami techniques.

the interior layers and stiffer material as the outer layers to amplify
stiffness ratios [22]. The same work also suggests increasing porosityof the center materials to further increase stiffness ratios. Jamming
structures using honeycomb [23,24] or cellular [25] composites employ
a single block of material with patterned cuts sandwiched by a numberof elastic sheets. These architectures allow for distributed curvaturesand enhanced jamming ratios, yet occupy a large volume compared todiscrete sheet assemblies. Other work demonstrates how identical cuts
through all layers can create jammed forms with anisotropic stiffness
properties [12,26]. Woven and folded laminar jamming structures are
likewise amenable to creation of jammed faceted shapes [26], andstretchable laminar jamming – cutting a single sheet into multiple discretely anchored, interleaved sheets – presents a way to accommodate
doubly-curved surfaces [27]. Despite recent progress, no work to-datehas systematically analyzed the design space of cut contiguous sheetsthat are easy to manufacture, require only one type of material, andamplify flexural stiffness changes compared to discretely anchored orneat sheet constituents.
Harnessing the mechanical anisotropy of kirigami [28,29], we introduce kirigami layer jamming : strategic removal of material throughstructured cuts to modify contiguous sheets’ second moment of area,
contact mechanics, and stretchability, and induce internal curvatures
when the constituents are under vacuum (Fig. 1a). Although historically a technique for creating forms from folded and cut paper, researchcommunities have adopted the term ‘‘kirigami’’ to describe use of cutsin tuning the stiffness of materials [30]. We study kirigami layer jamming via minimal analytical models, experiments, and finite element
analysis (FEA) to quantify the influence of key design parameters on
the mechanical response under loading in a three-point bending setup.
We find that kirigami primitives including shells, voids, and cuts allow
programming of specimens’ jammed and unjammed flexural stiffnesses
over wide deflection ranges, enabling new mechanical functionality andsignificant amplification of stiffness ratios relative to neat specimens(Fig. 1b). Demonstrations employing kirigami layer jamming attest
to its robotic and structural utility compared to neat layer jammingfor discontinuous shape matching and move-and-hold load-bearing
operations. Moreover, we demonstrate how topological augmentations
can give rise to pressure-induced shape-morphing, or stretching toapproximate Gaussian-curved topography, expanding the repertoire offunctionality in layer jamming systems. This work not only presents a
rapid, predictable, and scalable solution for variable stiffness robotic
components, but provides a rigorous theoretical and empirical foundation for how geometry influences performance in variable stiffness
material design.
1. Results and discussion
1.1. Motivation
To contextualize the effects of kirigami augmentations to laminar jamming systems, we start by considering the force–displacement
behavior of neat jammed layers during a three-point bending test.
The behavior can be split into three regimes: pre-slip, transition, andfully-slipped (Fig. 1c) [19]. In the pre-slip regime, the layers remain
cohesive. Consequently, the jammed flexural stiffness, 𝑘𝑗 , is relatively
high. During the transition regime, the longitudinal shear stress begins to exceed the maximum possible shear stress determined by thepressure and friction between the layers. A decrease in stiffness thus
occurs in the force–displacement relations at larger deflections. The
fully-slipped regime is characterized by constant movement of layers
relative to one another and a flexural stiffness akin to that of theunjammed state, 𝑘𝑢𝑗 .
For variable stiffness materials, it is desired to modulate between
extremes of rigidity and compliance, in order to reap the benefitsof both load-bearing and shape-shifting in robotics applications. An
ideal laminar jamming system would therefore not only accommodate2Extreme Mechanics Letters 64 (2023) 102084

R. Baines et al.

diverse shape-morphing operations, but maximize the ratio of jammed
to unjammed flexural stiffnesses, i.e. 𝑚𝑎𝑥(𝑘𝑗 ∕𝑘𝑢𝑗 ), over a wide deflection
range. We can divide this ideal behavior into two testable design
objectives: (i) elicit high initial 𝑘𝑗 ∕𝑘𝑢𝑗 , and (ii) prevent a drop in
jammed stiffness during the transition regime. Scaling analysis provides
quantitative framing for these objectives. Treating the jamming systemin three-point bending as an Euler–Bernoulli beam with a rectangularcross section described by a width of 𝑏, individual layer height of ℎ,and number of layers 𝑁, the pre-slip regime stiffness ratio is given by(Note S1):
𝑘𝑗 ∕𝑘𝑢𝑗 ≈ 𝐼𝑗 ∕𝐼𝑢𝑗 = 𝑁 2

the emergence of topographical depressions in the shelled specimensthat increase in area with increasing 𝑐∕𝑏 (Fig. 2b–d). Profilometry slices
verify there is curvature induced by the top and bottom layers bowing
into the empty volume (Fig. 2e–g black lines). The curvature is notconstant and eventually vanishes at a flat contact interface formed by
the meeting of the top and bottom layers.
We developed a model that yielded qualitatively similar interface
shapes to actual specimens (Note S2; Fig. 2e–g blue lines). The model
projects that the second moments of area will decrease by 0.1%,
47%, and 70% relative to an un-deformed cross section for optimal
specimens with 𝑐∕𝑏 = 0.2, 0.6, and 0.8, respectively. FEA of jammed
shelled specimens completes the picture of a cross section, illustratingthat induced curvatures are symmetric about the midplane, regardlessof 𝑐∕𝑏 (Fig. 2h–j). As such, by purely geometric arguments, shelled
specimens’ cross sections decrease in their second moment of area somuch when jammed that their 𝑘𝑗 ∕𝑘𝑢𝑗 would appear less than that ofneat sheets (𝑁 2 ). Experimentally we see that such is not the case.
Subjecting the different shell designs to three-point bending and calculating initial stiffnesses via the slopes of the average force–displacement
plots between 0 mm and 0.1 mm deflection (in the pre-slip regime
for all samples), we discovered that all designs exceed 𝑘𝑗 ∕𝑘𝑢𝑗 of theneat control (Fig. 2k). Notably, 𝑐∕𝑏 scales inversely to the stiffness
differential, such that the shell with 𝑐∕𝑏 = 0.2 peaks at a 5.4 × greater
stiffness ratio than that of the neat specimen (Fig. 2k).
To explain the shelled designs’ amplified 𝑘𝑗 ∕𝑘𝑢𝑗 , we note the curvature-induced stiffness caused by in-plane strain in the thin sheets when
jammed [32,33]. The energetic cost to stretch thin sheets scales as 𝐿ℎ ,where 𝐿 is the length of the sheet [34]. Since a jammed shelled specimen comprises inextensible sheets which already have pre-curvatures,transverse loads must overcome the prominent energetic hurdle ofstretching the sheets to bend them about an axis misaligned with the
pre-curvature. Induced curvatures thereby partially compensate for the
stiffness lost through second moment of area decreases of the jammed
cross section. Moreover, it would seem that second moment of area
decreases impact stiffness more than the corresponding stiffness gains
from curvature past a value of 𝑐∕𝑏 = 0.2. Stiffness dependence on 𝑐∕𝑏
can be likened to work on discretely clamped laminar jamming, inwhich it was found that decreasing clamp area reduces stiffness [35].
Besides 𝑐∕𝑏, another factor that governs initial flexural stiffness isjamming pressure. Subjecting a shelled specimen to three-point bending
across a swath of vacuum values, we found that stiffness increases withthe magnitude of vacuum (Fig. S5). This behavior stands in contrast tothat of neat layers, which exhibit no stiffness change as a function of
vacuum [19]. The coupling of initial stiffness to pressure in the shelled
specimens can be attributed to the interplay of curvature-induced
stiffness and contact interface area.
Based on jamming-induced curvature’s impact on initial stiffness,
we thought that it could also play a role in explaining some shelled
specimens’ persisting high forces at high deflections. For instance,
whereas forces at high deflections for 𝑐∕𝑏 = 0.2 resemble those of theneat control specimen, 𝑐∕𝑏 = 0.6 and 𝑐∕𝑏 = 0.8 cases are substantiallyelevated (Fig. 2l–n). Also observed in the 𝑐∕𝑏 = 0.6 and 𝑐∕𝑏 = 0.8
cases are sudden drops in force corresponding to curvature reversalof the shells (at low magnitudes of vacuum, negligible curvature is
induced, and so no pronounced drop in force is evident; see Fig. S5).
To understand how the induced curvature might impact a transition
regime where slip occurs, we conducted FEA of jammed neat andshelled specimens in three-point bending (Note S2; Fig. S4). At thesame value of deflection, 3% of the specimen’s length, which is wellinto the fully-slipped regime for neat specimens, shelled simulations
exhibit slip that occurs in smaller areas and is of less magnitude thanthat exhibited by a neat simulation. Increasing 𝑐∕𝑏 sees slip occurring
toward the outside walls of the shell and slipped regions decreasing inmagnitude and area. Correspondingly, the contact pressure increaseson the edges of the shell, suggesting that curvature diminishes the rateof slip by tightly holding the sheets together at the neutral axis.

(1)

1
1
𝑏ℎ3 𝑁 3 and 𝐼𝑢𝑗 = 12
𝑏ℎ3 𝑁 are the jammed and unjammed
Here, 𝐼𝑗 = 12

cross sections’ second moments of area, respectively. The transverse
load at slip, 𝑉𝑚𝑎𝑥 , scales proportionally to the static coefficient offriction 𝜇, pressure 𝑃 , and area of the cross section 𝐴 = 𝑏𝑁ℎ.
1.2. Shelling

According to the above scaling arguments for neat layers, it is immediately clear that 𝑘𝑗 ∕𝑘𝑢𝑗 increases with 𝑁 [19,31]. Another approach toincrease 𝑘𝑗 ∕𝑘𝑢𝑗 – requiring no additional material – would be to directly
tune the second moments of area of the cross sections in their jammed
and unjammed states, such that additional terms remain besides 𝑁 2
when computing the ratio. Consider ‘‘shelling’’ the jamming structure
by removing an amount of material of width 𝑐 from the middle of alayer’s total width 𝑏. We impose the shelling operation to effect 𝑞 layers
and to be symmetric about the neutral axis of the cohesive system,
enforcing 𝑁, 𝑞 as both odd or even integers. We effectively augment the
sheets’ aggregate cross section to become a hollow rectangle (Fig. 2a).
The jammed stiffness becomes:
1 3 3
1 3 3
𝑏ℎ 𝑁 −𝑐ℎ 𝑞
12
12
The unjammed stiffness becomes:
𝐼𝑗 =

1 3
1 3
𝑏ℎ 𝑁 −𝑐ℎ 𝑞
12
12Their ratio is thus:

𝐼𝑢𝑗 =𝐼𝑗 ∕𝐼𝑢𝑗 = 𝑅 =𝑏𝑁 3 − 𝑐𝑞 3
𝑏𝑁 − 𝑐𝑞

(2)

(3)

(4)

Eqs. (1) and (4) indicate that 𝑅 > 𝑁 2 for all shelled specimens
with valid design parameters. In other words, shelling always increases
𝑘𝑗 ∕𝑘𝑢𝑗 relative to a neat specimen of the same boundary geometry. Extremizing 𝑅 with respect to 𝑞 yields shelled jamming design parametersthat satisfy 𝑚𝑎𝑥(𝑘𝑗 ∕𝑘𝑢𝑗 ) (Note S2; Fig. S1a). Based on this extremization,
we expect that specimens with 𝑐 toward the limit of 𝑏 and 𝑞 toward thelimit of 𝑁 will elevate 𝑘𝑗 ∕𝑘𝑢𝑗 .
The theoretical impact of shelling operations on the transition
regime can be understood by setting the maximum sustainable shear
stress equal to the longitudinal shear induced by loading, and solving
for the resultant force, 𝑉𝑠 , that would initiate slip. The equation for 𝑉𝑠
suggests increasing 𝑐 to the limit of 𝑏 and 𝑞 to the limit of 𝑁 will cause
slip to occur earlier (Note S2; Fig. S1b). Theoretical modeling thus
indicates that design choices for 𝑚𝑎𝑥(𝑘𝑗 ∕𝑘𝑢𝑗 ) may counteract design
choices for higher 𝑉𝑠 , and vice-versa.
In actuality, when a shelled specimen is jammed, it does not retain aperfect hollow rectangular cross section. Jamming pressure causes theunaltered top and bottom sheets to bow into the empty volume createdby shelling. To study the influence of design parameters on the resulting
cross section, we fabricated several laminar jamming specimens for
combinations of 𝑐∕𝑏 = 0.2, 0.6, and 0.8, and 𝑞 = 3, 5, and the optimal
number shelled layers based on the other dimensions and given by the
maximum of Eq. (4). We consider here the cases of varying 𝑐∕𝑏 with
fixed 𝑞 at the optimal values (See Note S2 and Fig. S2 for discussionof subsequent experiments varying 𝑞). Surface profilometry indicates3Extreme Mechanics Letters 64 (2023) 102084

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Fig. 2. Effect of shelling on laminar jamming. a. Schematic of shelled laminar jamming cross section labeling variables of interest. 𝑁 is the number of layers, ℎ is an individual
layer’s height, 𝑏 is the width, 𝑐 is the shell width, and 𝑞 is the number of shelled layers. Below is an image of an actual jammed specimen with 𝑐∕𝑏 = 0.8. Scale bar: 20.5 mm.
b–d. Surface profilometry of a region of the specimens. Insets are to-scale illustrations of a shell layer, with dashed box indicating the section observed with surface profilometry.
e–g. Cross section elevation plots corresponding to pink vertical lines in above profilometry images. Black lines are experimental data of the cross section shape; blue lines aretheoretical predictions. h–j. FEA gives insight into the jammed specimens’ cross section shapes. Shown are cross-sectional views of the jammed structures, with top (black) andbottom (red) unaltered layers highlighted. k. Jammed to unjammed initial stiffness ratio normalized by the neat specimen. The neat specimen results are plotted at 𝑐∕𝑏 = 0. Error
bars represent one standard deviation from the mean of five samples. l–n. Three-point bending results of jammed and unjammed specimens with 𝑁 = 15 and 𝑏 = 50. Different
𝑐 of 10, 20, and 40 mm are made with optimal 𝑞 (derived from the maximum of Eq. (4)) of 9, 11, and 11, respectively. Solid lines indicate a mean value of five trials; clouds
indicate one standard deviation from the mean. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Overall, by tuning the design variables 𝑐 and 𝑞 of the shelling operations, we found a method to increase 𝑘𝑗 ∕𝑘𝑢𝑗 , thwart the rate of slip,
and thereby increase the high-deflection stiffness of laminar jamming
systems. These desirable properties came in part from the induced curvature of the upper and lower shell boundaries upon jamming. Despite
its advantages, the induced curvature poses issues for move-and-holdapplications. Indeed, jamming shelled specimens after bending themlength-wise into 180◦ arcs spontaneously forces them flat, owing to
the induced curvature opposite the bending direction and the fact thatthe sheets cannot accommodate double curvatures (Video S1). We werethus prompted to explore methods to achieve improved move-and-holdperformance while retaining the stiffness benefits of shelled layers.

we sought to isolate induced curvatures to locally shelled regions,i.e. distributed voids. Specimens with distributed voids contain two
distinct cross sections periodically spaced along their length: unalteredlayers, or shelled layers (Fig. 3a). Distributed voids can be oriented
with induced curvatures primarily along the same axis as bending
deformation, or perpendicular to it, by tuning the ratio between thevoid depth, 𝑑, and a constant cut dimension, 𝑐. For example, 𝑐∕𝑑 =1 parameterizes a square void; 𝑐∕𝑑 > 1 or 𝑐∕𝑑 < 1 accordingly
parameterize a rectangular void with its greater dimension along thewidth or length of the specimen, respectively. For designs abiding
by either of these inequality cases, we hypothesized that distributed
voids would (i) prevent curvature reversal propagation along the entire structure by serving as energetically stable regions; (ii) decrease
the magnitude of induced curvature in void boundaries during jamming, and thereby prevent large reductions in the second moment ofarea.

1.3. Distributed voids and cuts
As a first attempt to mitigate spontaneous flattening of bent shelled
specimens upon jamming while retaining high flexural stiffness ratios,
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Fig. 3. Effect of distributed voids on laminar jamming. a. Schematic of the multiple cross sections associated with voided laminar jamming, labeling variables of interest.
𝑁 is the number of layers, ℎ is an individual layer’s height, 𝑏 is the width, 𝑐 is the shell width, 𝑑 is the shell depth along the length, and 𝑞 is the number of shelled layers.
Below is an image of an actual jammed specimen with  = 0.4. Scale bar: 20 mm. b–d. Surface profilometry of a region of specimens with different void numbers of  = 2, 4,
and 9. Insets are to-scale illustrations of an augmented layer. e–g. Cross section elevation plots corresponding to pink vertical lines in above profilometry images. Black lines areexperimental data of the cross section shape. h–j. FEA give insight into the jammed specimens’ cross section shapes. Shown are cross-sectional views of the jammed structures,with top (black) and bottom (red) unaltered layers highlighted. k. Jammed to unjammed initial stiffness ratio normalized by the neat specimen (plotted at number of voids = 0).
Error bars represent one standard deviation from the mean of five samples. l–n. Three-point bending results of jammed and unjammed specimens with 𝑁 = 15 and 𝑏 = 50. Solidlines indicate a mean value of five trials; clouds indicate one standard deviation from the mean. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

To ascertain the effect of 𝑑 and the corresponding axis of induced
curvature in furnishing stable surface regions, we conducted profilometry on specimens with a certain number of voids, , separated byunaltered sections of a fixed width. In particular, we studied  = 2,
4, and 9 cases. 𝑞 was held constant at 11 layers. For  = 2, induced
curvature mainly arises perpendicular to the axis of bending, as withshelled specimens. For  = 4 and 9, induced curvature mainly manifests
along the axis of bending. Profilometry of the jammed specimens
confirms that, through voids, we were able to both localize and reduce
the amount of curvature along target axes relative to purely shelled
specimens (Fig. 3b–d). For instance, the  = 4 specimen deflects a
maximum of 1.7 mm in 𝑧 when jammed, compared to 2.1 mm and
3 mm for the 𝑐∕𝑏 = 0.6 and 0.8 shells, respectively. Plotting thedeflection of the top layer in each specimen shows how curvature
decreases with increasing  (Fig. 3e–g). FEA likewise confirms that thecross-section shape morphs less with increasing  (Fig. 3h–j). Thus,
for high , we can reap the benefits of high stiffness differentials by
mitigating decreases in the second moment of area due to curvature.

We conducted three-point bending on the specimens and calculated stiffness from the slopes of the average force–displacement plots
between 0 mm and 0.1 mm deflection (in the pre-slip regime). The unjammed state of the specimens exhibited, on average, over 70% lower
stiffness than that of the neat control, effectively elevating 𝑘𝑗 ∕𝑘𝑢𝑗 of all
voided specimens relative to the neat control (Fig. 3k). Crucially, 𝑘𝑗 ∕𝑘𝑢𝑗
increases with , suggesting our strategy to mitigate second moment ofarea decreases by discretizing shells into voided sections was successful.
The force–displacement responses of the samples when jammed demonstrate similar qualitative behavior to the neat control—defined pre-slip,
transition, and fully-slipped regimes. More importantly, no rapid drop
in force associated with curvature reversal is seen (Fig. 3l–n).
Topologically augmenting the specimens to create voids proved
to be a viable method for enabling move-and-hold shape retention.
Conducting the same format of experiment in which we bent jamming
specimens into 180◦ arcs, we found that  = 4 and  = 9 retained
the shape (Video S1). The  = 2 specimen did not, however, partiallysnapping back to a flat configuration upon jamming. We attribute this5Extreme Mechanics Letters 64 (2023) 102084

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Fig. 4. Effect of offset cuts on laminar jamming. a. Schematic of the multiple cross sections associated with offset cutting operations, labeling variables of interest. 𝑁 is the∑number of layers, ℎ is an individual layer’s height, 𝑏 is the width, 𝑐 is the cut width, 𝜙 is the offset spacing between centers of cuts in every other layer, and 𝑞 is the numberof cut layers in a given cross section along the length. Below is an image of an actual jammed specimen with 𝑤 = 1 mm. Scale bar: 20 mm. b–d. Three-point bending resultsof jammed and unjammed specimens with 𝑁 = 15 and 𝑏 = 50. Different cut dimensions of 𝑐 = 0.5, 1, and 2 mm are shown. Insets are to-scale illustrations of an augmented
layer. Solid lines indicate a mean value of five trials; clouds indicate one standard deviation from the mean. e. Jammed to unjammed initial stiffness ratio normalized by the neat
specimen (plotted at 𝑐 = 0). Error bars represent one standard deviation from the mean of five samples.

result to the fact that bending deformation must be sustained when
jammed mostly at the hinges separating the voids. A single hinge is
not sufficient to sustain the magnitude of bending deformation. The
primary performance difference between the  = 4 and  = 9 samples
was the continuity of held bending angles. Whereas  = 4 expectantly
created four discrete straight sections upon jamming,  = 9 allowed
for more continuous bending, manifesting not only at the hinges but
also at the extremities of voided portions, due to their diminished 𝑑and thereby lower curvature-induced stiffness.
Another avenue we explored to achieve high 𝑘𝑗 ∕𝑘𝑢𝑗 and enhance
the continuity of bending angles upon jamming relative to voided
specimens was offset slender cuts. To create a laminar jamming systemwith offset cuts, we used two types of layers, each having 20 incisions ofwidth 𝑤. The difference between the layers is a phase offset, 𝜙 between
every other layer’s cuts, giving rise to cross sections where gaps in thelayers alternate (Fig. 4a; Note S3).
Three-point bending tests show force–displacement relations comparable to those of specimens with voids. Stiffnesses were calculated
from the slopes of the average force–displacement plots between 0 mm
and 1 mm deflection (in the pre-slip regime). Especially for smaller𝑤, noise in the curves arises from sudden small drops and subsequent
increases in force due to stick–slip interactions between edges of cuts
(Fig. 4b–d). Due to no induced curvatures, the effect of curvature isdecoupled from initial stiffness calculations. Correspondingly, 𝑘𝑗 ∕𝑘𝑢𝑗
increases with increasing 𝑤, as would be predicted by theory assuming
an undeformed cross section (Fig. 4e). Move-and-hold performance
was also enhanced. All samples made with offset cuts were able tomove-and-hold to the prescribed bending angle of 180◦ (Video S1).

in contrast to neat sheets, whose thickness precludes folding to sharpedges and instead, enforces smooth curvatures. Under compression, thefaceted tetrahedron and cube exhibit higher global stiffnesses and areable to support 2× and 5× greater loads than their smooth counterparts,
respectively. Perforations allow faceted shapes to retain the relative
configurations of their faces, reinforcing structural stiffness.
Second, we harnessed kirigami alterations to enhance move-andhold operations for load-bearing. Tuning the second moment of area
while avoiding the instabilities of induced curvature, we achieved aradical increase in 𝑘𝑗 ∕𝑘𝑢𝑗 for cut and voided sheets relative to neat
sheets. Kirigami layer jamming specimens held an imposed bending of180◦ and supported weight 36 × greater than their own with negligible
deflection (Fig. 5b; Video S1). The analogous neat specimen, on theother hand, could not hold the bending deformation and collapsed
under the same applied weight.
Third, we leveraged a meander cut pattern to fabricate contiguous
jamming sheets with bi-axial strain relief. A stack of these sheets
was used to realize stretchable laminar jamming, and successfully
conformed to doubly-curved surfaces of a 3D-printed face and a toy
car (Fig. 5c, Video S3). The shape matching of the kirigami sheets
was substantially better than that of neat sheets, though the meander
pattern had limits in terms of the resolution of features it could accommodate. Details on the face are clearly evidenced in the kirigami
laminar jamming system, but more nuanced features of the toy car, suchas the ridges and gaps between the wheels and chassis, are lost. Further
shrinking the meander would offer a way to capture such fine details.
Kirigami techniques thus provide a scalable and rapid alternative tostretchable layer jamming, which normally requires tedious fabricationof discrete, interleaved sub-units [27].
Lastly, we created shape-morphing jamming structures by stacking
multiple voided specimens with varying induced curvature directions.
As one example, elastic frustration caused by the interface of stacked
opposite-curvature shells forces the assembly to assume a doublycurved midplane [36] (Fig. 5d; Video S4). The arising saddle shapedemonstrates how sheets of zero Gaussian curvature, when altered via
kirigami, can collectively equilibrate to Gaussian curved topography.
Compared to existing shape-morphing structures using vacuum [37,38], kirigami layer jamming allows for realization of doubly-curved
surfaces in addition to a high degree of stiffness tunability, boding
well for future robotics applications from manipulation to mechanicalcamouflage.

2. Applications
The three topological augmentation primitives characterized
above – shells, voids, and cuts – provide a toolbox of techniques
for programmed laminar jamming behavior with applications ranging
from deployable structures to robotics. As a simple testament to the
new shape-matching abilities afforded by linear cuts, we first createdperforated jamming systems that fold to match discontinuous threedimensional (3D) shapes, like a tetrahedron and a cube (Fig. 5a;Video S2). Perforations act as hinges at specified regions of the developed surface, allowing folding to sharp edges. This property stands
6Extreme Mechanics Letters 64 (2023) 102084

R. Baines et al.

Fig. 5. Demonstrations of kirigami layer jamming. a. In contrast to neat layers, perforations allow move-and-hold operations to match 3D shapes with discontinuities, such as atetrahedron and cube. Both neat and kirigami examples have the same number of layers 𝑁 = 10 and sheet thicknesses ℎ = 0.1 mm. Scale bars: 20 mm. Plots show force–displacement
responses of the 3D shapes under compressional loading. Solid lines represent an average of five trials; error clouds represent one standard deviation from the mean. Optical
micrograph inset scale bar: 2 mm. b. Bending specimens to 180◦ and jamming to assess move-and-hold performance. Load bearing over 36 × a specimen’s own weight can beachieved by kirigami sheets that use distributed cuts or voids. On the other hand, the restoring bending force of the neat specimen makes it unable to support the load, and theshelled specimen undergoes spontaneous curvature reversal, forcing it flat. Scale bar: 25 mm. c. Kirigami cuts provide bi-axial strain relief, allowing facile creation of stretchable
laminar jamming systems. Here, we approximate the doubly-curved surfaces of a face and toy car. Neat sheets are unable to capture the complexity of the surfaces. Scale bars:
13 mm (top); 19 mm (bottom). d. Curvature induced by voids can be harnessed to create elastically frustrated shape-morphing jamming structures. In this example, we use 𝑁 = 30.
Two unaltered layers on top and bottom sandwich 14 shelled layers tailored to induce curvature in either of two principal directions (red and black schematics of the shell layers
is indicated in figure). Scale bar: 15 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Concluding remarks

Further exploration led us to study offset cuts that completely removed undesired effects of induced curvature, enabling continuous
move-and-hold performance and retention of desirable stiffness ratios (See Note S4 for a tabulation of all designs’ stiffness values andjammed-to-unjammed ratios).
Different demonstrations employing laminar jamming systems withthe above kirigami primitives showcase new capabilities: discontinuous shape-matching, enhanced load-bearing, pressure-induced shapemorphing, and doubly-curved surface matching with contiguous sheets.
Looking forward, we envision kirigami layer jamming will prove especially useful in all manner of move-and-hold robotic applications
where energy consumption is a concern, such as untethered variable
stiffness continuum manipulators, robotic appendages, and assistive
exosuits. Topological augmentations can reduce the unjammed stiffnessof sheets, diminishing the required force of an actuator to move it to adesired configuration, yet simultaneously preserving jammed stiffness
requisite to load-bearing applications.
Beyond the scope of this manuscript, there are still many kirigami
patterns to evaluate and other aspects of jamming performance to

We presented kirigami layer jamming, a straightforward way totopologically augment sheets in laminar jamming systems to elicit desired mechanical properties. We focused on three augmentation primitives: shells, voids, and cuts. Each boasted their own performanceenhancing qualities. Shelling radically increased specimens’ jammedto-unjammed flexural stiffness ratios, and combatted slip at high deflections due to induced curvatures under jamming. However, thissame useful quality diminished the utility of shelling in move-and-holdapplications, because imposed bending deflections reversed the curvature upon jamming, forcing the system back flat. Since move-and-holdperformance is governed by the competition between a deformed specimens’ restoring spring force and the cohesion force due to jamming,
we explored other kirigami primitives to augment these quantities atdifferent rates. Voids presented a compromise between high jammed
to unjammed stiffness ratio and move-and-hold capability by mitigating curvature reversal through energetically stable regions of surface.
7Extreme Mechanics Letters 64 (2023) 102084

R. Baines et al.

optimize through geometry. For instance, we did not consider tensile or
torsional stiffnesses in the present work. Tuning the frictional qualitiesof the sheets through cutting, or creating jamming-induced interlocking
mechanisms with shells might increase tensile stiffness. Lastly, weforesee the geometric concepts employed herein to amplify laminarjamming systems’ properties translating to other jamming systems withdifferent constituent form factors. It may be possible, for example, toamplify the bending response of fiber jamming by inducing hierarchicalcurvatures within the fiber bundle. Overall, kirigami layer jamming
unveils an immense design space for variable stiffness materials that
we set the foundation for in this manuscript.

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Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgments
We thank Medha Goyal for assistance with mathematical derivations. This project was sponsored by the Office of Naval Research,
United States under award N00014-21-1-2417. Any opinions, findings,
and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of the
Office of Naval Research.
Appendix A. Supplementary data
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.eml.2023.102084.
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