April 27, 2016
Ochanomizu University
Japan Science and Technology Agency (JST)Director General for Science, Technology and Innovation,
Cabinet Office, Government of Japan

Simple equations that governs the deformation physics of Kirigami
promising for engineering applications and regenerative medicine
Summary
Professor Ko Okumura (Ochanomizu University) with a student, Midori Isobe, discovered
simple equations that governs the physics of deformation of Kirigami, a form of Japanesetraditional art to realize three-dimensional structure using a sheet with a certain cut
pattern. They uncovered that the high stretchability of Kirigami originates from thetransition from in-plane (two-dimensional) to out-of-plane (three-dimensional)
deformation, and showed that this transition is precisely determined by remarkably
simple laws of physics. Kirigami (“Kiri” and “Gami” are Japanese words standing for “cut”
and “paper,” respectively) has received rapidly increasing attention as a key structure forvarious engineering applications such as flexible electrodes or solar batteries. The
simplicity of the equations thus established allows us to provide simple guiding principles
for the development of applications, e.g., for cell sheets in regenerative medicine, bydesigning the Kirigami structure.
BackgroundFigure 1：Examples of Kirigami for daily use in Japan. (a) Shock absorbing Kirigami sheet
for protecting a bottle of wine. (b) “Airvase” (Toraf Architect, Japan) sold worldwide inmuseum shops. Photographs are from the article, Sci Rep, 2016.
http://www.nature.com/articles/srep24758. （CC BY 4.0）

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The group of Okumura has investigated fundamental issues concerning the toughnessof polymer materials in Prof. Ito’s project*, with his research background on toughbiological materials such as spider webs. It is a prerequisite for future automobiles totoughen and thin down porous sheets of polymers integrated in Lithium-ion batteries. Innature, there are biological materials that are reinforced by introducing many pores inthe structure, which include the soft bone of sea cucumbers. Okumura thus started a
project on the mechanical properties of Kirigami in Ito’s project. In fact, inspired by daily
examples of Kirigami (see Fig.1), Okumura initiated a preliminary project with a student,Midori Isobe, who joined Okumura’s Group about two and half years ago.
Quite recently, there has been a surge of engineering utilizations of the Kirigami
structure [1-4]. Examples include flexible electrodes or solar batteries, and applications
using graphene: softened Kirigami graphene can be an element for ultra-micro actuators.
However, no studies have been performed to clarify the physical principles of thedeformation.
Content of the study
Okumura and Isobe performed a systematic experiments and theoretical investigation
using a simplified pattern (see Fig. 2 (a)). As a result, they noticed that the highstretchability of Kirigami originates from the transition from in-plane (two-dimensional)
deformation to out-of-plane (three-dimensional) deformation (see Fig. 2 (b)). On the basisof this observation, they developed a theoretical framework that describes the transition.
As a result, they derived two equations (1) and (2) that determine the rigidity (elasticity)of the Kirigami sheet for small and large deformations (corresponding to in-plane andout-of-plane deformation, respectively). In addition, they derived another equation (3)that describes the condition of the crossover between the two regimes of small and softdeformation. Furthermore, they showed that the three equations are consistent with thedata. In particular, it is shown that the equations for the in-plane deformation (1) and forthe transition (3) hold with remarkable precision. The three equations expressing thelaws of physics are quite simple (see Fig. 3) so that they could be useful as guiding
principles for the development of various applications.

Figure 2：(a) Kirigami pattern used in the present study, which is simplified without
losing the physical essence. (b) Kirigami under a small tension (top) and illustration for
the in-plane deformation without rotation (bottom). (c) Kirigami under a large tension-2-(top) and illustration for the out-of-plane deformation with rotation.

Figure 3：Three simple equations describing the physical essence of the high stretchabilityof Kirigami established in the present study. “Elasticity” is a measure of flexibility and“strain” corresponds to the elongation. The parameters, d, h, and w, are the ones specified
in Fig. 2(a), whereas the parameter E is the elasticity of the original material, such as
paper (,, and  are numerical coefficients).
Perspectives
This study is being developed into a physics-engineering collaboration to seek more
possibilities of applications of the present study. This project has been performed in thephysics group of Okumura. Recently, Atsushi Takei (Ochanomizu University) has joined
the group with his research background in engineering. As a result, they started a more
application-oriented project, confirming that the equations are universally valid forpolymer materials such as rubber and plastics, as well as paper, and have applied for aJapanese patent.
Example of applications include the utilization of the Kirigami structure for cell sheets
promising for regenerative medicine. One may increase the possibility of the success ofregenerative surgery by adjusting the elasticity of the cell sheet to that of the plant regionof the patient. Applications in cylindrical geometry (see Fig. 1(a)) may include personallycustomizable elastic supporter for medicine and sport. The in-plane to out-of-planetransition can be easily detected because the transition is from two-dimensional tothree-dimensional deformation, which can be exploited for the detection of force (forcesensor). The Kirigami force sensor could be suitable for detecting and measuring
extremely small force if, for example, the sensor is made of graphene.
* This research was performed in the research project led by the program manager,Professor Kohzo Ito (The University of Tokyo), funded by ImPACT Program of Council
for Science, Technology and Innovation (Cabinet Office, Government of Japan). Prof.
Ito’s Project aims at developing various kinds of “Shinayaka” polymers (“Shinayaka” isa Japanese word meaning, tough and flexible). The research group of Prof. Okumura
in Ito’s project has investigated fundamental issues for toughening polymers, which
include porous thin polymers for Lithium-ion batteries indispensable for future
automobiles to obtain the present results.

References[1] M. K. Blees et al., Graphene kirigami, Nature 524 (2015) 204.
[2] T. C. Shyu et al., A kirigami approach to engineering elasticity in nanocomposites
through patterned defects, Nature Materials 14 (2015) 785.
[3] A. Lamoureux et al., Dynamic kirigami structures for integrated solar tracking,Nature Communications 6 (2015) 8092.

-3-[4] Z. Song et al., Kirigami-based stretchable lithiumion batteries, Scientific Reports 5(2015) 10988.
Original paper
Name of the journal: Scientific Reports (Nature PG)
Title of the paper: Initial rigid response and softening transition of highly stretchable
kirigami sheet materials
The Authors of the paper: Midori Isobe and Ko Okumura （Graduate School of
Ochanomizu University, Ochanomizu University）
Correspondence
Ko Okumura
Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
Tel：03-5978-5321
Fax : 03-5978-5321 E-mail: okumura@phys.ocha.ac.jp-4-