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\begin{document}

\title{Development of Self-folded Corrugated Structures Using Automatic Origami
Technique by Inkjet Printing~}



\author[1]{Yuki Fukatsu}%
\author[1]{Hiroki Shigemune}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
The origami technique realizes unique mechanical properties of sheet
materials without additional parts. In this study, a self-folded
corrugated structure (SCS) is developed based on the reinforcing
properties of the origami technique. The corrugated structures are
employed as the core material for a high-strength, open-channel sandwich
structure. Research on self-folded core materials is scarce; thus, a
design concept is proposed, and the mechanical properties of the SCS are
evaluated. First, the structural parameters of the SCS fabricated by
changing the printing parameters (e.g., linewidth and number of
lines/creases), to derive the structural model are determined. The model
facilitates the design of an SCS with the desired structure. Thereafter,
the mechanical properties of the SCSs are evaluated by conducting
three-point bending tests to determine the essential design parameters
corresponding to high stiffness. Moreover, SCSs can be stacked without
occupying space, thus leading to improved strength. These SCSs
fabricated using self-folding paper by ink-jet printing are low cost and
eco-friendly. Moreover, they are specialized for rapid design and
fabrication, depending on the application. This paper proposes the use
of SCS as a novel smart core because it exhibits a high transportation
efficiency and stiffness without additional components.

\par\null

Corresponding author(s) Email: {ae18081@shibaura-it.ac.jp} ,
{hshige@shibaura-it.ac.jp}%
\end{abstract}%



\sloppy




\section*{ToC Figure}

{\label{188996}}\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-11-ToC/2022.2.9-ToC}
\caption{{\textbf{ToC Figure.~~}A corrugated structure is created by the
self-folding of a smart material. Strength is freely designable
according to the printing pattern, and the flexibility of the structure
allows for stacking to improve strength. The stacked structure can
withstand a load of about 4 kg. A promising new smart core made by
inkjet printing is proposed in this paper. ~
{\label{222182}}%
}}
\end{center}
\end{figure}

\par\null

\section*{Introduction}

{\label{957297}}

Origami is a technique for creating complex three-dimensional (3D)
structures by defining mountain and valley crease patterns on a
two-dimensional (2D) sheet. The defined crease patterns impart various
properties to the formed 3D structures. One of the properties imparted
is foldability and expandability to compactly~store the deployable
structures. These properties allow for the fabricated 3D structure to be
folded into a 2D shape and redeployed at the destination. These
properties have led to applications in medical stents
~\hyperref[csl:1]{[1]} , automotive airbags ~\hyperref[csl:2]{[2]} ,
shelters~\hyperref[csl:3]{[3]}, and space structures~\hyperref[csl:4]{[4]}. In
addition, the crease patterns have properties that increase the
structural strength of 3D structures. Specifically, the crease patterns
increase the stiffness of the entire structure by increasing the second
area moment. The stiffness can be increased by three orders of magnitude
by adding straight creases while using the same amount of
material~\hyperref[csl:5]{[5]}~\hyperref[csl:6]{[6]}. In addition, the
microstructure folded with lithographic patterning can withstand a load
greater than its weight by a factor of
7700~\hyperref[csl:7]{[7]}.\textsuperscript{~}Therefore, this technique has
been used to manufacture core materials for sandwich structures, as they
are lightweight and exhibit high stiffness.

The sandwich structure consists of two-face sheets sandwiching the core
material. A honeycomb core is widely used as one of the core materials
\hyperref[csl:8]{[8]}. However, the manufacture of honeycomb cores is
complex and costly. Therefore, extensive research has been conducted on
honeycomb cores based on the origami technique \hyperref[csl:9]{[9]}. In
this regard, the origami technique enables the manufacturing of
honeycomb cores without considerable processing. Folded cores based on
origami are currently developed as alternative core materials. Examples
include Miura-ori, which is commonly used to shape folded cores
\hyperref[csl:10]{[10]}, and creases similar to Ron Resch patterns
\hyperref[csl:11]{[11]}. Folded cores can be manufactured from various sheet
materials by a simple process and tailored into various shapes that meet
specific functional requirements \hyperref[csl:12]{[12]}. Thus, the origami
technique has been applied to core materials for sandwich structures.

On the contrary, corrugated cores are stiff, perpendicular to the
corrugation, flexible in the parallel direction, and the structures
exhibit anisotropy \hyperref[csl:13]{[13]}. This property is leveraged in
load-bearing applications such as corrugated cardboards, roofs, and
walls, in addition to morphing wings \hyperref[csl:14]{[14]}. In contrast,
the folded core is stiff in the perpendicular and parallel directions.
The stiff direction of a corrugated core with linear creases is stiffer
than that of the folded core by a factor greater than 75
\hyperref[csl:5]{[5]} ~\hyperref[csl:6]{[6]}~. The closed-cell structures lead
to problems of air and humidity retention, which can increase the total
weight and degrade core properties \hyperref[csl:15]{[15]}. Conversely,
corrugated cores have open channels in one direction, which minimizes
these problems.

Corrugated, honeycomb, and folded cores can be readily fabricated using
the origami technique. Origami structures can be fabricated by manual
folding or by using robotic devices
~\hyperref[csl:16]{[16]}\textsuperscript{~}. However, these methods are not
efficient for mass production at small or large scales and for remote
applications. Mass production can be realized using manufacturing
equipment; however, large machines require space allocation and initial
investment, and the compression process consumes large amounts of
energy. Moreover, it compromises the specialized feature of creating
structures with desired mechanical properties within a short time.

To overcome these shortcomings, self-folding techniques have been
studied for the automatic fabrication of origami structures
\hyperref[csl:17]{[17]}. Self-folding generally uses smart materials that
respond to external stimuli, such as shape memory alloy (SMA)
\hyperref[csl:18]{[18]}, shape memory polymer (SMP) \hyperref[csl:19]{[19]},~and
hydrogel\hyperref[csl:20]{[20]}.~ The crease pattern is pre-defined for
these smart materials, and the application of heat or light triggers
automatic folding. Given that external stimuli automatically form the
origami structure, rapid manufacture and autonomous assembly at remote
sites can be achieved without human intervention. Moreover, given that
3D structures can be constructed by changing the 2D crease patterns, a
variety of origami structures can be created. Liu et al. proposed
self-folding SMP sheets using light. They defined crease patterns to
fold SMP sheets by 90\selectlanguage{ngerman}° and fabricated a rectangular shape. By changing
the crease patterns, the fold angle was adjusted to 60°, and a
tetrahedron structure was formed \hyperref[csl:21]{[21]}.

~\textsuperscript{~} Besides the origami technique, core materials for
sandwich structures are manufactured by compression molding
~\hyperref[csl:22]{[22]},~ hot-press molding ~\hyperref[csl:23]{[23]},~ and
embossing with rollers, among other methods ~\hyperref[csl:24]{[24]}.
~However, these manufacturing methods require molds and rollers for each
design requirement, which increases the cost, time, and energy required
to form the structure. Therefore, core materials fabricated with 3D
printing technology, which can be used to design complex structures at
low cost, have been
developed\hyperref[csl:25]{[25]}\hyperref[csl:26]{[26]}\hyperref[csl:27]{[27]}.
Furthermore, the development of core sandwich structures using smart
active materials, which are referred to as smart cores, was
investigated~\hyperref[csl:28]{[28]}. Tolley et al. fabricated a
self-folding origami consisting of a three-layer shape memory composite
with SMP as the intermediate layer to form a Miura-ori structure
\hyperref[csl:29]{[29]}. Evans et al. fabricated a micro-scale self-folding
origami structure consisting of three layers of photo-cross-linkable
copolymers with a thermosensitive hydrogel as the intermediate layer.
They realized a Miura-ori structure by controlling the fold angle
\hyperref[csl:30]{[30]}. However, limited studies were conducted on the
mechanical properties of the smart core sandwich structures. Hubbard et
al. fabricated corrugated structures with rectangular, square,
trapezoidal, and triangular waveforms using stimuli-responsive
thermoplastic sheets \hyperref[csl:31]{[31]}.~They conducted compression and
tensile tests by varying the wavenumbers of the fabricated corrugated
structures, and investigated the effects of the wavenumbers on the
compressive and tensile properties of the structures. However,
mechanical properties were not investigated with respect to the
amplitude and wavelength, which are important structural parameters of
the corrugated structure when using the tailor-made feature of the
self-folding methods. By investigating the mechanical properties with
respect to the amplitude and wavelength, the desired mechanical
properties can be imparted to the structure and specialized
characteristics can be realized. Therefore, we conducted experiments to
investigate the mechanical properties of corrugated structures with
respect to the amplitude and half-wavelength, which are the two most
important structural parameters of the corrugated structure.

In this study, we developed a corrugated structure made of a smart
material. A design theory of the structure was proposed, and its
mechanical properties were evaluated. We denote the developed structure
as a self-folded corrugated structure (SCS). Based on the reaction
between a smart material and inkjet printer ink, the structure was
formed autonomously without applying external energy. First, we proposed
a structural model to predict the final shape of the SCS from the
printed pattern. By calculating the fold angle from the proposed model,
a linear relationship between the printed linewidth and the fold angle
was obtained, similar to the conventional method
\hyperref[csl:32]{[32]},~which validated the proposed model. The design
parameters of the SCS were the printed line width and number of printed
lines. Self-folded corrugated structures with various amplitudes and
wavelengths were formed by varying the parameters. Second, the
mechanical properties of the SCS were evaluated by conducting a
three-point bending test. The results revealed that the SCS exhibited
anisotropic stiffness, which is one of the typical properties of
corrugated structures. In addition, with an increase in the amplitude of
the SCS, the second area moment increased, as expressed by the
theoretical equation of the second area moment of corrugated structures.
As a characteristic of the SCS, we confirmed that, for a large
wavelength range, the structure collapses due to the flexibility of the
paper. Furthermore, we found that the SCS is sufficiently flexible for
easy stacking after self-folding. The structural strength of the SCS can
be improved linearly by stacking structures. Thus, the fabrication
system of the SCS provided stiffness to a sheet of paper, which is
innately flexible to be applied as a core material. Given that the only
fabrication method is the application of solution by inkjet printing,
the corrugated structure with the desired stiffness can be created
rapidly according to the desired applications. Moreover, the SCS can be
stacked without occupying a significant amount of space, thus minimizing
energy consumption during transportation and reducing the transportation
costs. Therefore, the SCS will be developed as a novel smart core for
future applications owing to its high stiffness, digital manufacturing
ability, and high transportability.

\par\null

\section*{Concept of the Self-Folded Corrugated
Structure}

{\label{120298}}

~\textbf{Figure 2}a presents the fabrication process of the SCS using
inkjet printing. In this experiment, A4 size paper (210 mm × 297 mm)
with a thickness of 80.5 \selectlanguage{greek}μ\selectlanguage{english}m (XDT Tracing FS, TAKEO Co.) was used. The
materials used were introduced in previous studies
~\hyperref[csl:32]{[32]}~\hyperref[csl:33]{[33]}~\hyperref[csl:34]{[34]} . By inkjet
printing on the A4 paper, which is then set aside, valley folds are
automatically formed along the printing lines. The self-folding process
of paper is as follows. The reaction between paper and ink causes the
fibers to expand and contract. During the expansion phase, the top
surface of the paper expands, causing the paper to fold into a mountain
fold against the printing surface. This is followed by a contraction
phase, in which the paper folds into valley folds against the printing
surface. As plastic deformation occurs, the origami structure is fixed
in the valley-folded state~\hyperref[csl:32]{[32]}. Figure 2b and c present
the process of SCS formation during the contraction phase. To form
mountain and valley folds, a printing pattern is defined on both sides.
In this study, we fabricated an SCS by inkjet printing under humidity
levels ranging from 55\% to 65\% (\selectlanguage{ngerman}± 3\% RH), and set it aside for
approximately 10 min. As the paper transformed into an SCS, it developed
stiffness anisotropy; thus, it increased in stiffness in the direction
perpendicular to the corrugation, and in flexibility in the parallel
direction. The fabricated SCS was placed in a sealed container with a
desiccant and stored in a low-humidity environment.

Figure 2b presents the self-folding process of the SCS with four
creases. Immediately after printing (Figure 2b(i)), mountain folds are
formed on the prior-printed side. Given that both sides cannot be
printed simultaneously, there is a time lag between the folding times of
the mountain folds and the valley folds due to the time required to set
the printer. After 60 s (Figure 2b(ii)), the side printed subsequently
starts to fold. After 600 s (Figure 2b(iii)), folding on both sides is
completed, and the SCS is formed with the mountain and valley folds
(\textbf{Figure}~\textbf{3}). Figure 2c presents the self-folding
process of the SCS with the number of creases increased to 10
(\textbf{Figure}~\textbf{4}).

Figure 2d presents a demonstration, wherein the paper is transformed
into an SCS with increased stiffness. The central part of the structure
is fixed to a wooden board with double-sided tape. As shown on the left
side of the figure, the sheet of paper without the SCS cannot withstand
its self-weight and bends downward. Conversely, as shown on the right,
the SCS structure can support its own weight without deflection. The SCS
improves the second area moment and stiffness of the paper. Furthermore,
a number of SCS sheets can be stacked to increase the stiffness without
occupying significant space. Figure 2e presents a demonstration of the
stacked SCS, wherein an SCS stacked with 10 sheets withstands a mass of
approximately 4 kg (\textbf{Figure}~\textbf{5}).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-11-Figure1/2022.2.2-Figure1}
\caption{{Self-folded corrugated structure (SCS). (a) Conceptual diagram of the
SCS fabrication process. The SCS is fabricated by defining the printing
pattern on both sides and inkjet printing. (b) Self-folding process of
the SCS with four creases and (c) 10 creases. The SCS required 600 s to
self-fold after printing. (d) Demonstration of the SCS. A single sheet
of paper deflects under its self-weight; however, the SCS can support
its self-weight by increasing the second area moment. e) Demonstration
of stacking the SCS. A stack of 10 sheets can withstand a mass of about
4 kg.
{\label{280918}}%
}}
\end{center}
\end{figure}



\par\null

\section*{}

{\label{958214}}

Rich media available at \url{https://youtu.be/khWODtu-12c}



Rich media available at \url{https://youtu.be/0wW2qcU-YH4}



Rich media available at \url{https://youtu.be/3kQbWAerjHk}

\section*{}

{\label{958214}}

\section*{Design of Self-Folded Corrugated
Structure}

{\label{958214}}

\textbf{\textbf{Effect of Design Parameters}}

This section discusses the design method of the SCS based on the
self-folding of paper. As shown in~\textbf{Figure 6}a(i), the
linewidth~\emph{L} and number of lines~\emph{n} are defined as the
design parameters. These two design parameters determine a line
interval~\emph{i}. With the design shown in Figure 6a(i), the
cross-sectional corrugation is formed in the horizontal direction
(length of 210 mm) of the paper. Thus,~\emph{i} is expressed by the
following equation: ~

\begin{equation}
i=\frac{210}{n\ }-L
\end{equation}

By design, as shown in Figure 6a(i), a cross-sectional corrugation with
amplitude~\emph{d} and half-wavelength~\emph{w~}is formed at each
crease, as shown in Figure 6a(ii). Moreover,~\emph{L} can then be
considered as an arc of the cross-sectional corrugation; and~\emph{n}
changes the number of creases, thus resulting in a change in the number
of half-wavelengths in the cross-sectional corrugation. In the
experiment, we ensured that~\emph{n} was equal to the number of half
wavelengths by setting the line interval as~\emph{i}/2 between both
edges. The remainder of this sub-section discusses the effects of the
design parameters,~\emph{L} and~\emph{n,} on the final shape of the
SCS.~

First, we investigated the effect of~\emph{L} on the shape of the SCS.
Figure 6b presents an example of an SCS designed with~\emph{n~}= 4
and~\emph{L~}= 5 mm, 10 mm, and 15 mm. As can be seen from the figure,
with an increase in the linewidth, the paper folded more, i.e., the arc
length increased, assuming that the curvature was constant for any
linewidth. Consequently, the gradient of the straight-line parts joining
each arc increased. Therefore, as~\emph{L} increased, the amplitude of
the corrugated structure~\emph{d} increased, and the
half-wavelength~\emph{w} decreased. Thus,~\emph{d} and~\emph{w} were
found to be inversely proportional.

Thereafter, we investigated the effect of~\emph{n} on the shape of the
SCS. Figure 6c presents an example of the SCS with~\emph{L} = 15 mm
and~\emph{n} = 4, 6, 8, and 10. As shown in Figure 6c, the number
of~\emph{w} corresponds to~\emph{n}. In this study, all SCSs were
fabricated in A4 size. With an increase in the number of
half-wavelengths, the length of paper required to form one
half-wavelength decreased. Therefore, as~\emph{n} increased,~\emph{d}
and~\emph{w~}decreased. The length of the paper required to form one
half-wavelength~\emph{w}\textsubscript{max} is determined by the
following equation:~

\begin{equation}
w_{\max}=\frac{210}{n}
\end{equation}

The effects of the linewidth~\emph{L} and the number of lines~\emph{n}
on the amplitude~\emph{d} and half-wavelength~\emph{w} of the
cross-sectional corrugation were verified based on measured values. In
an experiment, 60 SCSs were fabricated by changing~\emph{L} by 1 mm in
the range of 1--15 mm for each structure with~\emph{n~}= 4, 6, 8, and
10.~Thereafter, the amplitude~\emph{d} and the half-wavelength~\emph{w}
of the cross-sectional corrugation were measured. Figure 6d presents the
results of the experiment, with the amplitude~\emph{d} as a circle plot
and the half-wavelength~\emph{w} as a cross plot. To improve legibility,
the horizontal axis was set as~\emph{i}, which is determined by~\emph{L}
and~\emph{n} from Equation (1). From the blue plot (\emph{n~}= 4),
as~\emph{i} decreased,~\emph{L} increased. Subsequently,~\emph{d}
increased, and~\emph{w} decreased. The trend was the same for all the
structures with~\emph{n~}= 6, 8, or 10. Additionally, as~\emph{n}
increased, the changes in the amplitude and half-wavelength with respect
to the linewidth decreased due to the decrease in the length of paper
required to form one half-wavelength~\emph{w}\textsubscript{max}.

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-5-Figure-2/2022.2.2-Figure-2}
\caption{{Design concept of the SCS. a) Definition of (i) design parameters and
their correspondence to (ii) structural parameters. b) Effect of
linewidth \emph{L} on the SCS formation. As \emph{L} increased, the
amplitude \emph{d} of the SCS increased, and the half-wavelength
\emph{w} decreased. c) Effect of \emph{n} on the SCS formation, where
\emph{n} corresponds to the number of half-wavelengths. d) Relationship
between the line interval \emph{i} and \emph{d} and \emph{w}, as
determined using 60 samples of the SCS. The circle plots represent the
amplitude, and the cross plots represent the half-wavelength.
{\label{880873}}%
}}
\end{center}
\end{figure}

\section*{}

{\label{103455}}

\textbf{Modeling}

As shown in \textbf{Figure 7}, the SCS was modeled based on the design
concept. The cross-sectional corrugations were modeled by approximation
as triangles, as shown in Figure 7a. Thereby, the SCS formation angle
2\selectlanguage{greek}\emph{α} \selectlanguage{english}and fold angle~\selectlanguage{greek}\emph{θ} \selectlanguage{english}are defined as follows:

\begin{equation}
2\alpha=2\tan^{-1}\frac{w}{2d}\left[\selectlanguage{ngerman}°\right]
\end{equation}

\begin{equation}
\theta=180-2\alpha\ \left[°\right]
\end{equation}

We substituted the measured values of the amplitude~\emph{d} and the
half-wavelength~\emph{w}, as obtained from Figure 6d, into Equation (3),
and then calculated the SCS formation angle 2\selectlanguage{greek}\emph{α}. \selectlanguage{english}The fold
angle~\selectlanguage{greek}\emph{θ~}\selectlanguage{english}was calculated from Equation (4). Figure 7b presents the
relationship between~\selectlanguage{greek}\emph{θ} \selectlanguage{english}and~\emph{L}. Based on a linear
approximation of the plot in Figure 7b,~\selectlanguage{greek}\emph{θ~}\selectlanguage{english}and~\emph{L} are
defined from the fold coefficient~\emph{m} by the following equation:

\begin{equation}
\selectlanguage{greek}θ\selectlanguage{english}=mL\ [\selectlanguage{ngerman}°]
\end{equation}

\emph{~}This linear relationship between the linewidth and fold angle
was confirmed in previous research ~ ~\hyperref[csl:32]{[32]} . Figure 7c
reveals that the coefficient of determination R\textsuperscript{2}
values were close to 1 for all numbers of lines; thus, the triangle
approximation can be considered appropriate. Although, for the same
linewidth, \selectlanguage{greek}\emph{θ} \selectlanguage{english}differs according to~\emph{n}. The value of~\emph{m}
decreases as~\emph{n} increases. This is due to the difference between
the ``folding'' and ``bending'' of the SCS creases, as discussed by Liu
et al~\hyperref[csl:35]{[35]}~. When the linewidth is low (\emph{L} = 5 mm),
as shown in Figure 7d, the error of the triangle approximation is small
because the creases are far from each other. When the linewidth is low
(\emph{L~}= 5 mm), as shown in Figure 7d, the creases are closer to the
``folding'' shape, and the error of the triangle approximation is
smaller because the creases are far from each other. However, as the
linewidth increases (\emph{L} = 15 mm) and shifts to a ``bending''
shape, the effect of the approximation increases, and~\selectlanguage{greek}\emph{θ}
\selectlanguage{english}decreases. As shown in Figure 7e, the more the number of lines, the
closer the folds, and the shorter the straight-line parts. Therefore,
with an increase in the number of lines, the creases shift toward a
``bending'' shape as the linewidth changes. Consequently, if the
distance between the lines is small, the change in the gradient of the
straight-line parts caused by the linewidth is difficult to observe, and
the observed value is smaller than the actual value. Thereafter, we
derived the relationships between~\emph{L} and~\emph{d} and~\emph{L}
and~\emph{w~}from the fold angle. The diagonal
side~\emph{i}/2+\emph{L}/2 of the right triangle shown in Figure 7a is
defined from Equation (1) as follows: ~

\begin{equation}
\frac{i}{2}+\frac{L}{2}=\frac{105}{n}
\end{equation}

Therefore, the amplitude \emph{d} and half-wavelength \emph{w} are
obtained from Equation (4) and (5).

\par\null

\begin{equation}
d=\frac{105\cos\alpha}{n}=\frac{105\cos[?](90-\frac{mL}{2})}{n}
\end{equation}

\begin{equation}
w=\frac{210\sin\alpha}{n}=\frac{210\sin(90-\frac{mL}{2})}{n}
\end{equation}

The measured values of~\emph{d} and~\emph{w} were plotted as shown in
Figure 7f and g, where Equation (7) and (8) are represented as solid
lines. The fold coefficient~\emph{m} required for the theoretical
equation is the value derived in Figure 7c. As shown in Figure 7f and g,
the measured values and the theoretical equations are in good agreement.
The parameters~\emph{d} and~\emph{w} can be derived from the design
parameters~\emph{L} and~\emph{n}. Using this model, we can design the
printing pattern for the structural parameters of the desired SCS
(\emph{d}~and~\emph{w}). This model equation can be applied to
structures with other values of~\emph{n}, and SCSs with the desired
number of half-wavelengths can be designed based on the application.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-5-Figure3/2022.2.9-Figure3}
\caption{{Modeling of the SCS. a) Triangle approximation of the corrugation. b)
The relationship between the linewidth~\emph{L} and fold angle \emph{\selectlanguage{greek}θ}.
\selectlanguage{english}c) The fold coefficient~\emph{m} and the coefficient of determination
R\textsuperscript{2}, as calculated from the linear approximation. d, e)
Verification of the triangle approximation. The transition of f)
amplitude~\emph{d} and g) half-wavelength~\emph{w} with respect
to~\emph{L}. The measured values and proposed model equation are in good
agreement. ~
{\label{585177}}%
}}
\end{center}
\end{figure}

\section*{Evaluation of SCS}

{\label{988336}}

\textbf{Evaluation Method}

We evaluated the stiffness of the SCS by conducting a three-point
bending test.~\textbf{Figure 8}a presents a schematic of the test. We
used a tension and compression testing machine (Shimadzu, AG-50kNX) to
conduct the test for both ends supported with a concentrated load. The
test conditions were as follows: the displacement speed was set at 5.00
mm/min, the load was applied until the stroke reached 30 mm, and the
data were recorded with a sampling period of one point per
second.\begin{CJK}{UTF8}{gbsn}　\end{CJK}\selectlanguage{english}Twelve types of SCSs with~\emph{n~}= 4, 6, 8, and 10
and~\emph{L~}= 5 mm, 10 mm, and 15 mm were tested. A load bar was
attached to the testing machine, such that the load could be applied
only to the center of the SCS (Figure 8a(i)). A wooden board was placed,
such that the distance between fulcrums was 280 mm, and the SCS was
fixed by pins inserted into the valley folds (Figure 8a(ii)). Figure
8a(iii) presents a photograph of the actual test. The valley fold was
not formed at one end of the SCS (Figure 8a(iv)).~\textbf{Figure
9}~presents a movie of the actual test.

Figure 8a(v) presents the beams in a three-point bending test of both
ends supported with a concentrated load. The relationship between the
load~\emph{P} and~deflection~\selectlanguage{greek}\emph{δ~}\selectlanguage{english}is expressed as the following
equation:

\begin{equation}
\delta=\frac{Pl^3}{48EI}
\end{equation}

Where \emph{l} is the distance between the fulcrums, \emph{E} is Young's
modulus, and \emph{I} is the second area moment, which can be obtained
from Equation (9).

\begin{equation}
I=\frac{Pl^3}{48\delta E}
\end{equation}

Using \emph{I,} the stiffness of the SCS can be evaluated. The distance
between the fulcrums \emph{l} was 280 mm, and Young's modulus of the
paper \emph{E} derived from the tensile test was 5.32 GPa
(\textbf{Figure S1}). We fabricated three of each 12 types of SCSs as
samples and conducted a total of 36 tests. The three test results
obtained for each structure were averaged.

Figure 8b presents a sample result of a load-deflection curve obtained
from a three-point bending test. In this section, an area of up to 10 mm
of the entire 30 mm stroke was excerpted to better understand the
analysis method. To derive the second area moment from the
load-deflection curve, an explanation of the three stages (i)--(iii)
shown in Figure 8b is presented. The load bar descends in the first
stage; however, it is not in contact with the SCS (Figure 8b(i)). Then,
as the load bar continues to descend, it contacts the SCS (Figure
8b(ii)). This point is regarded as the starting point of the three-point
bending test, and is the origin of the load-deflection curve. In this
study, the starting point was located within the stroke range of 0--5 mm
for all structures tested; however, there was variation among the
structures. Therefore, the test was conducted up to a stroke of 30 mm,
and the deflection~\selectlanguage{greek}\emph{δ~}\selectlanguage{english}was unified in the range of 0--25 mm in the
load-deflection curve. In the final stage, the mountain fold parts
collapsed as the load was applied to the SCS until its peak was reached
(Figure 8b(iii)). The stiffness~\emph{k} can be calculated from the
slope of the initial stage in the range from the test start point to the
peak. In this experiment, the increase in the deflection~\selectlanguage{greek}\emph{δ~}\selectlanguage{english}from 0
mm to 2 mm was defined as the slope of the initial stage.

\begin{equation}
I=\frac{Pl^3}{48\delta E}=\frac{kl^3}{48E}
\end{equation}

We confirmed the presence of stiffness anisotropy, which is one of the
typical properties of the corrugated structure, by conducting a
three-point bending test in two directions~(\emph{x} and and~\emph{y),
as shown in}~Figure 8c. Given that the tension and compression testing
machine (Shimadzu, AG-50kNX) did not allow the installation of wooden
plates in the~\emph{y}-direction, we used a desktop type tension and
compression testing machine (A\&D, MCT-1150) for the~\emph{y}-direction
compression. Figure 8d presents the load-deflection curves obtained from
the three-point bending tests conducted in both directions. The results
reveal that the stiffness was high in the~\emph{x}-direction and low in
the~\emph{y}-direction, which is a typical property of the corrugated
structure.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-5-Figure4/2021-12-5-Figure4}
\caption{{Stiffness evaluation of the SCS. a) Schematic of the three-point bending
test; i) front view; ii) top view; iii) photograph; iv) side view; v)
relationship between the load~\emph{P} and deflection~\emph{\selectlanguage{greek}δ}. \selectlanguage{english}b)
Load-deflection curve obtained by three-point bending test; i) before
contact; ii) start of the test; iii) loaded. c) Confirmation of
anisotropy of stiffness. d) Anisotropy test results. The load-deflection
curves are significantly different in the~\emph{x}-
and~\emph{y}-directions.~~
{\label{453746}}%
}}
\end{center}
\end{figure}

\par\null

Rich media available at \url{https://youtu.be/\_3phQd0zrmk}

\par\null

\textbf{Results and Discussion}

We investigated the influence of the design parameters,
linewidth~\emph{L}, and number of lines~\emph{n} on the stiffness of the
SCS. The second area moment of the corrugated structure is expressed by
the following equation~\hyperref[csl:36]{[36]}:

\par\null

\begin{equation}
I=\frac{d^2h}{2}\left[1-\frac{0.81}{1+2.5\left(\frac{d}{2w}\right)^2}\right]
\end{equation}

Equation (12) expresses that with an increase in the amplitude of the
corrugated structure, the second area moment increases. Thereafter, we
derived the equation for the relationship between the second area moment
and design parameters. By substituting Equation (7) and (8) into
Equation (12), the relationship between the second area moment and
design parameters is expressed by the following equation.



\(\begin{equation}
I=\frac{\left(105\cos\left(90-\frac{mL}{2}\right)\right)^2\ h}{2n^2}\left[1-\frac{0.81}{1+2.5\left(\frac{105\cos\left(90-\frac{mL}{2}\right)}{2\times210\sin\left(90-\frac{mL}{2}\right)}\right)^2}\right]
\end{equation}\)

Equation (13) expresses that with a decrease in the number of lines and
increase in the linewidth, the second area moment
increases.~\textbf{Figure~10}a presents the load-deflection curve
obtained from the SCS fabricated with~\emph{n~}= 4 and 10 and the second
area moment derived from the results. The SCS results fabricated
with~\emph{n~}= 6 and 8 are presented in Supporting information
(\textbf{Figure S2}). The translucent lines in the graph indicate the
results of the three measurements, and the opaque lines indicate the
averaged results. First, we evaluated the stiffness of the SCS
fabricated with~\emph{n~}= 10 (Figure 10a(i)). The upper left of the
photographs presents the SCS for~\emph{n~}= 10 with~\emph{L~}= 5 mm, 10
mm, and 15 mm. From the load-deflection curve, with an increase in the
linewidth, the slope of the initial stage increased, and the stiffness
increased. As a result, the second area moment increased. Therefore, as
expressed as Equation (12) and (13), with an increase in the linewidth
and amplitude of the SCS for~\emph{n~}= 10, the stiffness of the
structure with a higher second area moment increased. Moreover, as can
be seen from the load-deflection curve at~\emph{L~}= 15 mm, the first
peak load was 3.24 N, thus indicating that the structure can withstand a
load of approximately 330 g. This indicates that a sheet of paper with a
mass of 5.72 g can withstand a load greater than its mass by a factor of
57.7, thus acting as a SCS. ~The microstructure mentioned by Bassik et
al. can withstand a load greater than its mass by 7700; however, it has
the maximum load capacity of 0.883 g because the structure is
small~\hyperref[csl:7]{[7]}.\textsuperscript{~}Using the proposed
paper-printing self-folding technology, we successfully tailor-made a
large-scale corrugated structure with various stiffness properties and
high load capacity. Thereafter, we evaluated the stiffness of the
structure with~\emph{n~}= 4 (Figure 10a(ii)). For the structure
with~\emph{n} = 4, there was no increase in the slope of the initial
stage with an increase in the linewidth, and the stiffness results
exhibited a different trend from that expressed by Equation (12).
Furthermore, despite the decrease in the number of lines, the second
area moment was lower than~\emph{n~}= 10, which exhibited a different
trend from that expressed by Equation (13). In addition, although the
SCS was fabricated with the same linewidth, there was a significant
variation between the three tests. We found that the structural
characteristics can be typical for self-folded corrugated structures
such as the SCS.~

Figure 10b presents the stiffness distribution of the fabricated SCS.
Due to the reaction between the paper and printed solution, the
stiffness of the crease parts increased in the SCS. However, the
stiffness of the straight-line parts connecting the creases remained low
because no new components were added, and no hard compression was used
to process it. Therefore, with a decrease in the number of lines, there
were more straight-line parts with low stiffness; thus, there was a
higher probability of deformation during the three-point bending test.
Figure 10c presents the structure with~\emph{n~}= 4 and 10 immediately
after the end of the three-point bending test. Compared with the
structure with~\emph{n~}= 10, the structure with~\emph{n~}= 4 underwent
with significant deformation. Therefore, we considered that the increase
in stiffness with an increase in amplitude could not be confirmed for
the structure~with \emph{n~}= 4. Thus, increasing the linewidth and
amplitude is not sufficient to design an SCS with high stiffness;
considering that the stiffness distribution over the entire SCS is
required.

~Furthermore, the results revealed that the load repeatedly increased or
decreased in all the load-deflection curves. The results may indicate
that the SCS buckle repeatedly during the three-point bending test.
First, at the peak of the load, the mountain folds buckled, and the load
decreased. After the buckling of the mountain folds, the residual part
of the SCS received additional compressive load, which caused the
buckling and increase or decrease of the load. This process was repeated
during the test, thus causing the load to increase or decrease
repeatedly. We attributed this to the flexibility of the paper material.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-5-Figure-5/2022.1.29-Figure-5}
\caption{\selectlanguage{ngerman}{Stiffness evaluation of the SCS with respect to the linewidth \emph{L}
and number of lines \emph{n}. Results of three-point bending tests with
a) i) \emph{n} = 10 and ii) \emph{n} = 4. Error bars indicate ± 1 SD. b)
Stiffness distribution of the SCS. The crease parts where the ink was
printed exhibited high stiffness, whereas the straight-line parts
exhibited low stiffness. c) The SCS cross-section immediately after the
completion of the three-point bending test. The SCS fabricated with
\emph{n} = 4 was significantly deformed.
{\label{714906}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\section*{Comparison with CAD Software and Stacking Performance
~}

{\label{239029}}

We designed equivalent corrugated models using CAD software (SolidWorks)
and derived the second area moment. The values obtained from the
experiment and CAD software were compared using~SCSs fabricated
with~\emph{n~}= 10 (\textbf{Figure~11}a). We created the SCS models
using the measured amplitude~\emph{d} and half-wavelength~\emph{w}, as
shown in Figure 6d and the equation shown in Figure 11a. Figure 11b
presents the results of the comparison.{~}The simulation results were
consistent with the experimental results, and the second area moment
increased with an increase in amplitude. However, there were significant
differences in the values.~Furthermore, the amplitude dependence of the
SCS stiffness exhibited a linear scaling relationship, which is
different from the quadratic scaling expressed by Equation (12) and
previous studies ~\hyperref[csl:31]{[31]} . This is one of the
characteristics of SCSs fabricated from flexible sheet materials and may
be because the structure has not been deformed with maintaining the
corrugated cross section. Figure 11c presents the mountain fold parts
after the three-point bending test. The deformation of the mountain fold
parts can be observed. ~

Due to the flexibility of the developed SCS, we found that the SCS can
be stacked and strengthened without occupying space. Stacking a large
number of sheets allows for the realization of the same strength as that
of a sheet of thickness that would not inherently undergo self-folding.
We fabricated 2, 3, and 10 sheets of the SCS with~\emph{n~}= 10 and
linewidth~\emph{L~}= 15 mm.~\textbf{Figure S3} presents the comparison
between a one-sheet SCS and a 10-sheet SCS. Given that the SCS is thin
and flexible, 10 sheets could be stacked without occupying a significant
amount of space. We conducted a three-point bending test on the 2, 3,
and 10 sheets of the stacked SCSs. Figure 11d and e present the results.
As shown in Figure 11d (load-deflection curves), the slope of the
initial stage increased, and the stiffness increased as the number of
sheets stacks increased. Furthermore, the second area moment increased
linearly with the number of sheets (181\% for 1--2 sheets, 452\% for 3
sheets, and 1461\% for 10 sheets), as shown in Figure 11e. Hence, the
stiffness of an SCS is dependent on the design parameters of the
corrugated structures and the number of stacked sheets. Moreover, the
load-deflection curve for 10-sheet stacking did not increase or decrease
the load, as shown in Figure 11d. Based on this result, we inferred that
only the mountain-folded part of the sheet buckled when 10 sheets were
stacked, and the residual part of the SCS does not buckle. This can be
attributed to the improvement of material strength by stacking 10
sheets.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/2021-12-5-Figure6/2022.2.2-Figure6}
\caption{\selectlanguage{ngerman}{~a) Comparison between the photograph and the 3D model when \emph{n} =
10 and \emph{L} = 5, 10, 15 mm. b) Comparison of the second area moment
derived from the simulation and obtained from the experiment. c)
Mountain fold parts of the SCS after the three-point bending test.
Deformation occurred at the crease parts, and uniform deformation of the
cross-sectional structure did not occur. d) Three-point bending test
results of the stacked SCS (1, 2, 3, and 10 sheets) and e) the
calculated second area moment. As stacking increased, the slope of the
initial stage increased, and the second area moment increased linearly.
The error bars indicate ± 1 SD.
{\label{364082}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\section*{}

{\label{379597}}

\section*{Conclusion}

{\label{379597}}

In this study, we developed an SCS that self-folds into high-strength
structures from a single sheet of paper by inkjet printing. The concept
of the SCS is based on the properties of the origami technique for
enhancing structural strength. The SCS has the following design
parameters: printing line width \emph{L} and number of printing lines
\emph{n}. By changing these parameters, we fabricated an SCS with
various amplitudes \emph{d} and half-wavelengths \emph{w}. Specifically,
with an increase in the linewidth, the amplitude increased, and the
half-wavelength decreased. Additionally, with an increase in the number
of lines, the amplitude decreased, and half-wavelength changed with
respect to \emph{L}. This relationship was validated by measuring
\emph{d} and \emph{w} for a total of 60 SCSs. Furthermore, by
approximating the SCS cross-section as a triangle, we derived a linear
approximation relationship between \emph{L} and the fold angle \selectlanguage{greek}\emph{θ}.
\selectlanguage{english}The R\textsuperscript{2} values of the derived approximations were all
close to one, which were consistent with the linear relationship between
\emph{L} and \selectlanguage{greek}\emph{θ.}\selectlanguage{english}~Based on this approximation, we derived a model
for \emph{d} and \emph{w} of the SCS, which allowed for the design of an
SCS with the desired shape. Thereafter, we developed a methodology to
design the SCS of the desired shape by self-folding a paper.

Furthermore, we conducted 36 three-point bending tests on 12 types of
SCSs with different design parameters to evaluate their mechanical
properties. By evaluating the mechanical properties, we established the
design theory of an SCS with high stiffness. Corrugated structures have
higher second area moments at higher amplitudes \hyperref[csl:36]{[36]}
.~However, the three-point bending test results revealed that the SCS
with the highest amplitude deformed significantly under loading, and the
second area moment decreased. This is because the SCS exhibited a
non-uniform stiffness of printed and unprinted parts in its self-folded
structure. To fabricate an SCS with higher strength, it is not
sufficient to simply design a higher amplitude. In particular, it is
necessary to consider the stiffness distribution throughout the SCS.
Moreover, we evaluated the mechanical property of the SCS stacked with
2, 3, and 10 sheets to verify the strength-enhancing effect of the
stacking sheets. As observed, the second area moment of 10 sheets was
1461\% higher than that of one sheet.

The proposed SCS could be designed and fabricated rapidly to achieve the
desired shape and stiffness. As the material is paper, the SCS is
inexpensive and recyclable. Additionally, the flexibility of the paper
material allows for it to be stacked without occupying a significant
amount of space, even after forming the SCS. Moreover, it improves the
stiffness without requiring additional components. Hence, the proposed
SCS is a promising smart core material that self-folds while maintaining
the stiffness required by the core material of the sandwich structure.
Moreover, it does not create space before or after self-folding, and
demonstrates a high transport performance. Based on the stiffness
investigation, it was confirmed that the SCS exhibits anisotropic
stiffness. Hence, the SCS can be used in applications that require
load-bearing, such as the core material of sandwich structures; in
addition to anisotropy, e.g., as morphing wings and the skeleton of
pneumatic actuators~\hyperref[csl:37]{[37]}.~ ~More complex and robust
structures can be created by filling the printer with various chemicals
and ordering the folding sequence ~\hyperref[csl:33]{[33]}. By printing
electrodes on the same paper surface, the application of intelligent
core materials with integrated actuators and sensors can be realized in
the future~\hyperref[csl:34]{[34]}.

~

\section*{\texorpdfstring{Supporting Information
\emph{}}{Supporting Information }}

{\label{186000}}

The supporting information file related to our article is hosted on
Authorea.

\par\null

\section*{Conflict of Interest}

{\label{401086}}

The authors declare no conflict of interest.

\section*{Acknowledgements}

{\label{376091}}

This work was supported by JSPS KAKENHI, grant numbers 18H05895 and
19K20377. This research is supported by Adaptable and Seamless
Technology transfer Program through Target-driven R\&D (A-STEP) from
Japan Science and Technology Agency (JST) Grant Number JPMJTM20CK.~ This
research is supported by SIT International Research Center for Green
Electronics. We would like to thank Editage
(\href{http://www.editage.com/}{www.editage.com}) for English language
editing.

\selectlanguage{english}
\FloatBarrier
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