# Transition to stress focusing for locally curved sheets

Thomas Barois, Ilyes Jalisse  Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400
Talence, France  Loïc Tadrist  Aix Marseille Univ, CNRS, ISM, Marseille,
France  Emmanuel Virot  hap2U, 75 Avenue Gabriel Péri, 38400 Saint Martin
d’Hères, France

###### Abstract

A rectangular thin elastic sheet is deformed by forcing a contact between two
points at the middle of its length. A transition to buckling with stress
focusing is reported for the sheets sufficiently narrow with a critical width
proportional to the sheet length with an exponent 2/3 in the small thickness
limit. Additionally, a spring network model is solved to explore the thick
sheet limit and to validate the scaling behaviour of the transition in the
thin sheet limit. The numerical results reveal that buckling does not exist
for the thickest sheets and a stability criterion is established for the
buckling of a curved sheet.

##  I Introduction

The crumpling of a paper sheet is an usual outcome for someone trying to write
a deep and meaningful piece of text such as the introduction of a scientific
paper. After being crumpled, the paper is irreversibly damaged and its
topography may be viewed as a disordered network of ridges. The persistence of
such ridges is a structural memory of stresses focused during the forced
manipulation of the sheet.

Deformations without stretching should prevail for thin sheets due to the
large mismatch between thickness and typical sizeRayleigh (1888). However,
stretching deformations appear during crumpling because of stress
focusingWitten (2007). Stress focusing is observed with the indentation of a
thin plate against a circular contourCerda and Mahadevan (1998); Chaïeb  _et
al._ (1998); Cerda  _et al._ (1999). The resulting structure is called a
developable cone (or d-coneBen Amar and Pomeau (1997)) and it conveys the idea
that the plate adopts a conical shape to satisfy developability. Experimental
d-cones do not rigorously match with conical surfaces. A U-shaped scar is
found at the tip of a d-cone and its formation is the consequence of the sheet
avoiding diverging curvatures at the apex. This picture of stress focusing can
be generalized to crumplingLobkovsky  _et al._ (1995); Sultan and Boudaoud
(2006); Andresen  _et al._ (2007); Balankin  _et al._ (2010); Cambou and Menon
(2011); Croll  _et al._ (2019): a crumpled configuration is a state occupying
a small volume that combines regions approaching developability and localized
ridges focusing the stretching deformations.

In the previous situations reported, a confinement is imposed to a thin sheet
and stress focusing is compulsory because there are no developable
deformations that can be contained within the imposed volume. In this work, we
identify the existence of a controlled transition to stress focusing for a
thin sheet submitted to a bending deformation and without volume confinement.
The main outreach of this work is about the fundamental understanding of
crumpling dynamics with the appearance of stress focusing while smooth
solutions of pure bending satisfy the imposed condition. A second outcome is
the characterization of the buckling threshold. This has a practical interest
for the safe handling of a piece of thin material without inducing stress
focusing that would result in structural damagesGottesman  _et al._ (2015);
Chopin and Kudrolli (2016). The characterization of this transition is also of
interest in a context of exploiting buckling and crumpling for the realization
of functional materialsRogers and Huang (2009); Luo  _et al._ (2011); Mao  _et
al._ (2012); Ma  _et al._ (2012); Yan  _et al._ (2013); Reis (2015); Holmes
(2019).

The point of this work is to evidence a route to crumpling via the driving of
low-energy bending deformations, which can be viewed as a one direction
confinement. Stress focusing was previously observed for one direction
confinement but with peculiar boundary constraints such as an imposed 3-buckle
shapeSchroll  _et al._ (2011) or a pulling forceFuentealba  _et al._ (2015).
In other situations with frames, the transition to stress focusing can be
reversed and defects can appear at small confinement before
developableBoudaoud  _et al._ (2000) or smoothRoman and Pocheau (2012)
solutions establish for larger external loads. For drapingCerda  _et al._
(2004), a transition to stress focusing exists for a sheet bent under its own
weight but the supporting tip adds a localized stress.

Figure 1: (a) Schematics of a flat sheet of width $W$ and length $L$. The
square markers indicate the mid-length points. A picture shows the scars
formed after a self-contact procedure with an acetate sheet ($L=29.7$ cm,
$W=8$ cm, $t=0.2$ mm) (b) Rectangular paper strip ($L=29.7$ cm, $W=15$ cm,
$t=0.1$ mm) with mid-length points at contact and maintained by an adhesive
strip. (c) Strip with a smaller width ($L=29.7$ cm, $W=5$ cm, $t=0.1$ mm).

##  II Self-contact experiment

The present study is based on a simple self-contact procedure applied to a
thin sheet in order to characterize the transition to stress focusing. Figure
1 (a) represents an elastic sheet of length $L$ and width $W$. The thickness
$t$ is small compared to $W$ and $L$. The material coordinates of the sheet
$u,v$ verify $u=x$ and $v=y$ for the flat sheet. The origin for $xyz$ is the
center of the sheet. Figure 1 shows two elongated sheets for (b) a small
($L/W\approx 2$) and (c) a large aspect ratio ($L/W\approx 6$).

For the aspect ratio $L/W\approx 2$, the sheet is transversely curved without
stress focusing. The sheet is however not free of stretching deformation and
the presence of a small longitudinal stretching is responsible for the
symmetric opening of the sheet. This coupling between transverse bending and
longitudinal stretching related to a persistence length for the curved
regionLobkovsky and Witten (1997); Cerda and Mahadevan (2003); Vandeparre  _et
al._ (2011); Barois  _et al._ (2014); Pini  _et al._ (2016); Matsumoto  _et
al._ (2018); Taffetani  _et al._ (2019). The limit for a sheet free of
stretching would be obtained for a small length $L\ll W$. In this case, the
sheet profile would be invariant along $x$ and it would match with Euler’s
elasticaAudoly and Pomeau (2010) model in the projection plane $yz$. This
limit identifies a smooth solution with self-contact and free of stretching:
for any $L$, a developable surface can be obtained from an elastica profile in
the plane $yz$ invariant in the direction $x$.

For the large aspect ratio $L/W$, the sheet buckles and scars appear close to
the mid-line $u=0$. The picture in figure 1 (a) shows the two scars at the
apex of the d-cones formed after self-contact ($L=29.7$ cm, $W=8$ cm and
$t=0.2$ mm). For the paper sheet in figure 1 (c) four d-cones are formed. In
this case, $L/W$ is larger than for the acetate sheet. This points towards a
route to crumpling with an increasing disorder for the sheets of large ratio
$L/W$. Here, the experiments are performed close to the transition and the
post-buckling morphology does not show the usual ridges networkKramer and
Witten (1997) observed at high compaction.

The appearance of stress focusing for a sheet simply curved, even locally, is
counterintuitive because the bending modes have an elastic energy orders of
magnitude below the stretching modes. For a typical scale $W=5$ cm for the
sheet in figure 1 (c), the ratio bending-to-stretching energy with linear
coupling is of the order of $(t/W)^{2}\sim 4\times 10^{-6}$. Stress focusing
is even more surprising here because bending is a well-known strategy to
prevent collapse under external loads, according to the process of curvature-
induced rigidityPini  _et al._ (2016); Taffetani  _et al._ (2019).

##  III Scaling law for the buckling transition

The self-contact procedure could be sensitive to unintentional poking or
unwanted gravity load. It is therefore crucial to ensure that these
contributions can be ruled out. This is done by comparing experimental data to
a numerical model of the self-contact procedure. A set of experiments is
performed with acetate sheets ($t=0.2$ mm) to identify the buckling
transition. Figure 2 presents the state of a sheet at self-contact as a
function of $L/t$ and $W/t$. The numerical results are obtained with a spring-
network model that will be discussed further. A postcard and a metro ticket
are shown to get a sense of the orders of magnitude without the corresponding
experiments actually performed.

Figure 2: State of a thin sheet in self-contact ($L$: length, $W$: width). The
experiments are with acetate sheet (thickness $t=0.2$ mm) with squares for a
smooth opening and crosses for sheets with stress focusing. The dashed line is
a power-law with an exponent 2/3 and a numeric prefactor 3.15. The gray area
indicates a stress focusing region obtained from the simulation (blue dots
without stress focusing and red dots with stress focusing). A representation
of two simulated sheets ($W=200$) is presented with $L=500$ for the smooth
opening and $L=550$ for the opening with focused stresses. A paper sheet (A4
format) at self-contact remains smooth ($L/t\sim 3000$ and $W/t\sim 2000$,
outside the plot range).

For the experiments, buckling occurs in a region with sufficiently large value
of $L/t$ for a fixed $W/t$. The smallest width explored is $W=3$ cm which
corresponds to $W/t=150$. For this value, the bending deformation for self-
contact is such that the sheet reaches a plastic regime, which is out of the
scope of this work. The largest width explored is $W=9$ cm, slightly above the
width for the transition for a sheet of length 29.7 cm ($L/t=1485$, A4
format). The dashed line is $W_{c}/t=a(L_{c}/t)^{2/3}$ with the adjustment
$a=3.15$. The experimental transition is reproducible and the existence of a
scaling regime validates the idea that such crumpling events relate to a
controlled mechanical instability. The value of the exponent 2/3 will be
discussed below.

Two strategies were tested to impose the self-contact at the midpoints ($u=0$,
$v\pm W/2$): (i) using two adhesive strips (width 5 mm) and (ii) tightening of
a knotted wire (diameter 0.3 mm) between 2 holes close to the midpoints. Both
strategies resulted in the same transition which suggest that the detail of
the self-contact does not affect the transition. One could indeed wonder if
the point-like forces at the self-contact might not play the role of an
indenting tipWitten (2007); Cerda and Mahadevan (1998); Chaïeb  _et al._
(1998); Cerda  _et al._ (1999); Boudaoud  _et al._ (2000). This is not the
case here because the singularities are always formed away from the self-
contacted points.

##  IV Spring network model

For the experimental points below $L/t=500$, the transition deviates from the
scaling regime. To explore the thick sheet regime and the validity of the
scaling exponent, a spring network model is computed. Although crystalline
spring networks do not rigorously model the isotropic elasticity of continuum
mediaSeung and Nelson (1988); Curtin and Scher (1990); Kramer and Witten
(1997); Ostoja-Starzewski (2002); Omori  _et al._ (2011), such models are well
adapted to explore the scaling properties of thin sheet configurationsDiDonna
(2002). Figure 3 (a) presents the structure of the unit cell used in the
simulation. A thin sheet is obtained by 2D-replication of the unit cell to
form a bi-layer. First, a flat sheet is generated and then additional springs
are added and tuned to reach self-contact. The equilibrium positions of the
vertices are computed by iteration of the spring forces until a static
equilibrium is found (see appendix C.4).

The spring network model is computed for different sheet dimensions to
identify the transition between a smooth opening and a buckled sheet with
focused stresses. The criterion to identify a buckled sheet is the sign of the
transverse curvature. With a smooth opening, the transverse curvature is
always positive. For a buckled sheet, this curvature is partly negative near
$u=0$. Figure 2 represents by dots the simulations performed and a filled area
to identify the buckling region.

Stress focusing is not observed in simulations below $W/t=56$. For $W/t<56$,
the simulations indicate that the sheet can open smoothly for any length $L$.
In this regime, there is no risk of damages by stress focusing during
manipulation and self-contact. This regime is however difficult to explore
with thin elastic sheets. With paper ($t\approx 0.1$ mm), it would require to
use strips with $W$ of the order of 5 mm. For such narrow strips, it is not
possible to force a self-contact without plasticity.

The regime of focused stresses is observed for a larger range of parameters in
the experiments compared to the simulations. This systematic difference could
be explained by the limitations of the spring-network model or the difficulty
to realize controlled experiments with thin sheets for which mechanical and
physical properties are not perfectly uniform.

The simulation results are further processed to find a stability criterion for
the transition. Figure 3 (b) represents the longitudinal strain profile
$\epsilon_{uu}$ for a sheet of $1800\times 470\times 2$ vertices ($L/t=1184$,
$W/t=309$) with contact of the mid-length points. The strain profile is
obtained during the simulated dynamics of the sheet, just after the buckling
event. The strain profile combines regions of positive stretching, to allow
the sheet to open, and regions of negative stretching (compression). The
existence of longitudinal compression induced by transverse curvature was
proposed in a previous workBarois  _et al._ (2014) to satisfy a zero
longitudinal net force. Here, we uncover the structure of the stretching
strain and the 5-band pattern. There are four points of maximal compression
lying at the middle of the compression bands and longitudinally separated by a
distance $\delta$. For a post-buckled sheet, each point of maximal compression
is the source point of a d-cone structure. The time-resolved simulation
(video, see at [movie URL] for a representation of the strain map and the
sheet profile during a simulation of a sheet with 1801x471x2 vertices. )
clearly shows that the d-cones are not nucleated at the self-contact. This
proves that the self-focusing transition is not due to a local indentation by
the point-like force at the self-contact.

Figure 3: (a) Unit cell of the spring network model. Circles and squares
represent the lower and the upper surface. The linear springs of length $1$
and $\sqrt{2}$ are for the first neighbours and the second neighbours
respectively. (b) Normalized longitudinal strain $\epsilon_{uu}\times(W/t)$ of
a simulated sheet ($L=1800$, $W=470$) just after buckling. The dashed line is
the mid-line $u=0$. Positive values corresponds to a stretching and negative
values are for a compression. 4 points of stress focusing are present. (c)
Normalized minimal strain $\mbox{min}(\epsilon_{uu})(W/t)$ (compression) as a
function of the dimensionless width $W/t$. The circles are for simulated
sheets with smooth opening and square symbols are for sheets with stress
focusing. The horizontal line is at $-1.16$. The profile at mid-length $u=0$
illustrates the curvature sign change with buckling. The star symbol indicates
the sheet ($L=1800$, $W=470$) selected for the strain profile represented in
(b). $t=1.52$. (d) Horizontal spacing of the local minima for the longitudinal
strain. $\delta=0$ means one pair of d-cones at $u=0$ and $\delta>0$ two pairs
at $u\pm\delta/2$ as in (b).

Figure 3 (c) represents the value of the maximal compression strain
$\mbox{min}(\epsilon_{uu})$ for a sheet as a function of the width $W/t$. The
strain is normalized with the dimensionless factor $W/t$. Each data point
corresponds to a unique couple ($W,L$) and the length $L$ of the sheets is
indicated by a horizontal bar for each data point. The plot shows two families
of data points: the circle scatters are for the smooth sheets, without
curvature inversion, and the square or star scatters are for the post-buckled
sheet with curvature inversion of the transverse profile (see inserted
schematics). The data point with a star corresponds to the dimension for the
strain map in figure (b).

The plot in figure 3 (c) indicates a threshold value
$\mbox{min}(\epsilon_{uu})\times(W/t)=-1.16$ below which a sheet at self-
contact buckles. The buckling dynamics can be followed in more details with
the successive profiles of the sheet during the simulation. Figure 14 in the
appendix represents the evolution of the maximal compression strain during the
convergence of the mechanical equilibrium of the simulated sheet. The sheet
with $1700\times 470\times 2$ vertices ($L/t=1118$, $W/t=309$) converges to a
smooth solution with a minimal strain above the threshold $-1.16$. For the
simulation with 1750 and 1800 vertices ($L/t=1151$ and $L/t=1184$), the
minimal strain is sufficiently low to cross over the threshold value -1.16 and
a sudden drop of the minimal strain is observed as a signature of stress
focusing. For the small sheet with $W/t=35$ and $W/t=49$, there is no buckling
transition because the compression strain remains below the threshold
$1.16\times(t/W)$ for any $L/t$.

Figure 3 (d) represents the longitudinal spacing $\delta$ between the points
of minimum strain. A value $\delta=0$ means that there is a total of 2 local
minima for the compression strain located on the mid-length line $u=0$. With
$\delta>0$, there are 4 local minima for the compression strain as shown in
figure 3 (b). The number of local minima sets the profile of the first
buckling event. In the simulation with $1750\times 470\times 2$ vertices, the
stress focusing is first triggered at the 4 symmetric points of maximal
compression strain. Then, the 4 points travel towards the mid-length line and
an abrupt increase of the maximal compression is observed when the pairs of
stress focusing points merge.

##  V Analytical toy model

The transition to stress focusing identified in experiments and simulations
can be rationalized by a toy model based on (i) elastic energy minimization
Barois  _et al._ (2014) and (ii) a buckling criterion Hutchinson (2010).

To estimate the elastic energy, we assume that the sheet at self-contact is
described by its typical transverse curvature that takes the value $c_{0}\sim
1/W$ at $u=0$ and $c_{e}$ at the free ends at $u=\pm L/2$. By considering only
the first-order contributions, the total elastic energy scales as:

$\frac{\mathcal{E}(c_{e})}{ELWt}\sim
t^{2}({c_{0}}^{2}+{c_{e}}^{2})+t^{2}\frac{W^{4}}{L^{4}}(c_{0}-c_{e})^{2}+\frac{W^{8}(c_{0}-c_{e})^{4}}{L^{4}}$
(1)

with $E$ the Young modulus. The terms in the right hand-side stand for the
transverse bending energy, the longitudinal bending energy and the
longitudinal stretching energy (see appendix A). $c_{e}$ is the only free
parameter of the model and its value is obtained by the minimization of the
energy $\partial\mathcal{E}(c_{e})/\partial c_{e}=0$ that yields to an
equilibrium between two bending terms and one stretching term:

$t^{2}c_{e}-t^{2}\frac{W^{4}}{L^{4}}(c_{0}-c_{e})=2\frac{W^{8}(c_{0}-c_{e})^{3}}{L^{4}}$
(2)

We discuss here the solution of equation (2) in the thin sheet limit: with
$W\gg t$ and $L\gg t$, the transition is such that $L$ is larger than $W$ and
we can assume $L^{4}\gg W^{4}$. Consequently, the term $t^{2}c_{e}\sim
t^{2}/W$ dominates the left hand-side in equation (2) and the scaling for the
curvature difference is $c_{0}-c_{e}\sim t^{2/3}L^{4/3}/W^{3}$. According to
our previous workBarois  _et al._ (2014), the scaling for the compression
induced by a transverse bending is $\epsilon_{uu}\sim
W^{4}(c_{0}-c_{e})^{2}/L^{2}$. Considering the criterion
$|\epsilon_{uu}|(W_{c}/t)\sim 1$, we obtain

$W_{c}\sim t^{1/3}{L_{c}}^{2/3}$ (3)

The exact solution of equation (2) confirms that the scaling (3) is valid only
for large $W/t$ (see appendix A and figure 5). This exact solution also
predicts an absence of buckling transition for small $W/t$ as found in the
simulations.

Experiments were also performed with paper and the transition was found for
lower $L/t$ (not shown in figure 2 for clarity, see appendix B). This
difference is attributed to the anisotropy of paper Alava and Niskanen (2006).
By taking into account the anisotropy Matsumoto  _et al._ (2018) with a
smaller Young modulus for the transverse direction, we show in the appendix B
that the prefactor for the scaling relation is modified accordingly.

##  VI Conclusion

The present results have evidenced a simple scaling relationship for the
critical compressive strain of a sheet buckling with stress focusing, that is
$\epsilon_{c}\sim t/W$ with a prefactor close to unity, as shown in figure
3(c). This criterion is used in the toy model and the state diagram obtained
from the exact solution of equation (2) is consistent with experiments and
simulations. In our previous workBarois  _et al._ (2014), the existence of a
longitudinal compression for a transversely curved sheet was proposed to
satisfy internal mechanical equilibrium in a model. Its relevance was only
justified indirectly via a detailed analysis of the persistence length. Here,
we show that the induced compression strain has the same effect than an
external compression load applied to a shell with the same curvature: the
geometric stability criterion $\epsilon_{c}\sim t/W$, with $1/W$ the typical
curvature of the transversely curved sheet, is the equivalent of the classical
buckling of perfect cylindrical shells, where linear stability analysis
predicts $\epsilon_{c}=1/\sqrt{3(1-\nu^{2})}\times t/R$ Hutchinson (2010),
with $\nu$ denoting the Poisson coefficient of the shell and $1/R$ its
curvature. Notably, the above-mentioned formula infamously overestimates the
buckling load of cylindrical shells Peterson  _et al._ (1968), because those
systems are extremely imperfection-sensitive Singer  _et al._ (2002);
Gerasimidis  _et al._ (2018). This affects the reliability of thin shell
structures, even for precisely manufactured shells such as soda cans Virot
_et al._ (2017) and space rockets Peterson  _et al._ (1968). In contrast, the
present work identifies a strain threshold $\epsilon_{c}$ that leads to robust
scaling regimes observed in experiments and simulations. This hints that
buckling by self-contact of a curved sheet is relatively insensitive to
imperfections. We attribute this to a buckling threshold reached locally,
contrarily to cans for which buckling can be triggered at any weak spot.
Furthermore, while the buckling of cans by external loads leads to collapsing,
buckling by self-contact offers more control with a self-regulation of the
internal stress after the transition. This provides an interesting strategy to
reliably control buckling events in functional and smart materials by choosing
geometries close to the transition to take advantage of the fast morphology
switch.

##  Appendix A Toy model and theoretical state diagram

The sheet surface can be described by a curvature function $c(u)$ (see figure
4). For a simple scaling approach, we reduce the sheet kinematics to 2 degrees
of freedom for the transverse curvature:

  * -

$c_{0}$ for the curvature at mid-length, and

  * -

$c_{e}$ for the curvature at the free end $u=\pm L/2$.

To estimate the elastic energy of the sheet, we will consider the first order
contributions from the quadratic variables obtained from the 2 degrees of
freedom $c_{0}$ and $c_{e}$.

Figure 4: View of strip of width $W$ with the shape of a surface ruled by the
curvature function $c(u)$ with $-L/2<u<L/2$.

For the transverse bending energy, we assume that the energy comes from the
two contributions:

$\mathcal{E}_{b,\perp}\sim\alpha ELWt^{3}{c_{0}}^{2}+\beta
ELWt^{3}{c_{e}}^{2}$ (4)

in which $E$ is the young modulus, $L\times W\times t$ the physical dimensions
of the sheet and $\alpha,\beta$ two numeric prefactors that could be
determined with a more detailed analysis of the sheet’s state. In the
following, the numerical prefactors will be omitted, given that we are
discussing scaling relations here.

To estimate the longitudinal bending energy, we use the relation that gives
the vertical displacement of the sheet

$Z(u,v)=\frac{1}{2}c(u)v^{2}$ (5)

that gives the scaling for the typical vertical displacement $Z_{0}\sim
c_{0}W^{2}$ and $Z_{e}\sim c_{e}W^{2}$. In a first approximation, the
longitudinal curvature is

$\frac{1}{R_{long}}\sim\frac{Z_{0}-Z_{e}}{L^{2}}$ (6)

which gives an estimation for the longitudinal bending energy

$\mathcal{E}_{b,\parallel}\sim ELWt^{3}\frac{W^{4}}{L^{4}}(c_{0}-c_{e})^{2}$
(7)

Finally, the longitudinal stretching strain is obtained by the ratio

$\epsilon=\frac{\sqrt{L^{2}+(Z_{0}-Z_{e})^{2}}-L}{L^{2}}$ (8)

in which the square root estimates the excess of length at the edge of the
sheet because of the relative vertical displacement $|Z_{0}-Z_{e}|$. After a
Taylor expansion for $Z_{0}-Z_{e}\ll L$, the stretching energy is obtained
with

$\mathcal{E}_{s,\parallel}\sim ELWt\frac{W^{4}}{L^{4}}(c_{0}-c_{e})^{4}$ (9)

Using equations (4), (7), and (9), we get the scaling for the total energy
used in the article:

$\mathcal{E}\sim\mathcal{E}_{b,\perp}+\mathcal{E}_{b,\parallel}+\mathcal{E}_{s,\parallel}$
(10)

In figure 4, the sheet is represented for fairly low curvatures $c_{0}$ and
$c_{e}$ in order to visualize easily the sheet’s shape. For self-contact, the
curvature at mid-length scales as $c_{0}\sim 1/W$.

With the self-contact condition, the only free parameter in equation (4) is
$c_{e}$. The local optimal for the energy is obtained from the cancellation of
the derivative with respect of $c_{e}$ which reads:

$t^{2}{c_{e}}-t^{2}\frac{W^{4}}{L^{4}}(c_{0}-c_{e})-2\frac{W^{8}}{L^{4}}(c_{0}-c_{e})^{3}=0$
(11)

With the variable $\delta_{c}=c_{0}-c_{e}$, we get

$2\frac{W^{8}}{L^{4}}{\delta_{c}}^{3}+t^{2}\left(1+\frac{W^{4}}{L^{4}}\right)\delta_{c}-t^{2}{c_{0}}=0$
(12)

Equation (12) is a third-order polynomial that may be written
$\delta_{c}^{3}+p\delta_{c}+q=0$ to fits Cardano’s formula with

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{L^{4}t^{2}}{2W^{8}}\left(1+\frac{W^{4}}{L^{4}}\right)$
(13) $\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{L^{4}t^{2}}{2W^{9}}$ (14)

with the condition $c_{0}=1/W$. The solution of equation (12) is then:

$\delta_{c}=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$

The stretching energy involves region of positive (extension) and negative
(compression) strain with the same scaling magnitude

$\epsilon=\frac{W^{4}{\delta_{c}}^{2}}{L^{2}}$

obtained after the Taylor expansion of equation (8).

If we use a stability criterion found in the simulations and consistent with
the buckling of cans under axial load:

$\epsilon<\epsilon_{c}\mbox{, with }\epsilon_{c}=\alpha\frac{t}{W}$ (15)

with $\alpha=-0.43$, we find the phase diagram presented in figure 5.

The diagram $(L/t,W/t)$ shows a dark filled area (buckled I) for which the
buckling criteria (15) is satisfied. The buckled region buckled (I) has a
tongue-like structure. Away from this region, the sheet’s shape should be
smooth. This is the case almost everywhere in experiments and simulations,
except in the region identified by buckled II. Although the model does not
predict stress focusing for this region, we suspect that the sheet should be
buckled here as well: for each point in the region buckled II, one can find a
point for the same $W/t$ and a smaller length $L/t$ in the effectively buckled
region buckled I. Formulated the other way around: from a sheet effectively
buckled in region buckled (I), the sheet should remain buckled if it was
longer (larger $L$ for same $W$). We think that the absence of buckling in
region buckled (II) is a flaw of our model that oversimplifies the curvature
decay profiles and how $L$ affects the scaling of the energy terms in equation
(10).

Figure 5: Buckling criterion $\epsilon<\epsilon_{c}$ (eq. (15)). In the region
buckled I, the buckling criterion is satisfied. Elsewhere, the sheet should be
smooth except in region buckled II (see text for details). The solid line is
the scaling law with exponent 2/3 and a numerical factor $a=3.235$.

This model captures the thick-sheet regime and the fact that buckling does not
exist below a threshold value $W_{c}/t$ identified by a dashed line in the
figure. The value of the prefactor $\alpha=-0.43$ sets the value of the lower
bound for the transition $W_{c}/t=56$ and the value of the prefactor $a$.

An additional red solid line is added for the transition regime in the thin-
sheet limit $W/t=a\times(L/t)^{2/3}$ with $a=3.235$. This regime consistently
matches with the region buckled I for $W/t$ and $L/t$ sufficiently large. This
regime can be found from the model if the term for the longitudinal stretching
is neglected (see main article for the scaling argument with $L^{4}\gg
W^{4}$). This also justifies that for thick sheets, both numerical and
experimental data points should shift towards smaller values of $W/t$.

##  Appendix B Experiments with paper

The self-contact experiment was performed with acetate sheets (see figure 1 of
the main article) and standard paper sheets of thickness $t=0.125$ mm. Figure
6 presents the state of a rectangular sheet curved by self-contact for acetate
sheets (squares and crosses (x)) and paper (circles and crosses (+)).

Figure 6: State of rectangular sheets of dimensions $L,W,t$ with contact of
the mid-length points. The plot represents experiments with acetate sheets and
paper. The scaling law $W/t\sim(L/t)^{2/3}$ is represented with two different
numerical prefactors.

Paper sheets buckle more easily than acetate sheets: in the region between the
two scaling relations $W/t=3.15(L/t)^{2/3}$ and $W/t=3.7(L/t)^{2/3}$ in figure
6, paper sheets are buckled but acetate sheets open smoothly. This difference
can be rationalized by introducing anisotropy and two different Young’s moduli
in the equation (10) for the energy

$\tilde{\mathcal{E}}\sim\tilde{\mathcal{E}}_{b,\perp}+\tilde{\mathcal{E}}_{b,\parallel}+\tilde{\mathcal{E}}_{s,\parallel}$
(16)

in which

$\tilde{\mathcal{E}}_{b,\perp}\sim LWt\times
E_{\perp}t^{2}({c_{0}}^{2}+{c_{e}}^{2})$ (17)

is the transverse bending energy,

$\tilde{\mathcal{E}}_{b,\parallel}\sim LWt\times
E_{\parallel}t^{2}\frac{W^{4}}{L^{4}}(c_{0}-c_{e})^{2}$ (18)

is the longitudinal bending energy and

$\tilde{\mathcal{E}}_{s,\parallel}\sim LWt\times
E_{\parallel}\frac{W^{8}}{L^{4}}(c_{0}-c_{e})^{4}$ (19)

is the longitudinal stretching energy $E_{\parallel}$ and $E_{\perp}$ are two
Young’s moduli with $E_{\perp}<E_{\parallel}$ in which $\parallel$ refers to
the direction parallel to the paper fibers and $\perp$ the direction
perpendicular (see Ref. [32] Matsumoto et al.). Here, the paper sheets are cut
such as the fibers are aligned with $L$ which means that the smaller modulus
$E_{\perp}$ appears in the transverse bending energy term. Figure 7 represents
the phase diagram obtained with $E_{\perp}/E_{\parallel}=0.71$. This value is
chosen to match the thin-sheet scaling regime with prefactor 3.7 found in the
experiments.

Figure 7: State diagram predicted by the toy model with anisotropy (equation
(16), $E_{\perp}/E_{\parallel}=0.71$). The dashed line and the solid red line
(with factor 3.235) are the same as in figure 5 for the isotropic model
$E_{\parallel}=E_{\perp}$. The solid blue line is the scaling law with a
prefactor 3.7 in good agreement with the experiments performed with paper
sheets.

The anisotropy of paper has an impact on the buckling threshold and
consistently on the post-buckling morphology. Figure 8 illustrates two
morphologies obtained with a paper sheet of dimension $L=21$ cm, $W=6$ cm for
the paper cut with (a) the fibers in the longitudinal direction (same
orientation as in figure 6) and (b) the fibers in the transverse direction.
From a visual comparison, the sheet (a) with the fibers parallel to $L$ is a
higher state of crumpling than (b). This is confirmed by the number of d-cones
(4 for (a) and 2 for (b)) and recess angle of the longitudinal centerline
$v=0$.

Figure 8: Morphology of a paper sheet with mid-length points contact for a
sheet with (a) the paper fibers in the longitudinal direction and (b) the
fibers in the transverse direction. An small adhesive strip is used to
maintain the paper in self-contact.

##  Appendix C Spring network model

Figure 9 (a) represents a unit cell made of 8 vertices arranged in a cubic
structure. Two types of linear springs are used:

  * •

springs of length 1 with a spring constant $k_{1}$ for the first neighbours
(solid lines)

  * •

springs of length $\sqrt{2}$ with a spring constant $k_{2}$ for the second-
nearest neighbours (dashed lines)

###  C.1 Structure

Figure 9: (a) Schematics of unit cell with $2\times 2\times 2$ vertices. The
two colors indicates the upper and lower layers. The first neighbours are
connected by linear springs $k_{1}$ with length 1 and the second neighbours on
faces are connected by springs $k_{2}$ of length $\sqrt{2}$. (b)
Representation of a sheet with $9\times 5\times 2$. The mid-length vertices
are highlighted in green.

Figure 9 (b) represents a small sheet made of $9\times 5\times 2$ vertices.
The structure is at rest without additional constraints and the elastic energy
is nil (all springs have a length 1 or $\sqrt{2}$). The vertices at mid-length
are highlighted on the upper face. To force self-contact and bending, a
supplementary spring linking the two mid-length points is added.

###  C.2 Spring constants

The value of the springs constants $k_{1}$ and $k_{2}$ is set to have the same
force-displacement relation for the stretching along the sides and the
diagonals of the cubic cells. Figure 10 presents the displacements of two
points at a distance $r$ for an applied force $F$ and $-F$ in the cases where
the two points are on the longitudinal direction $\vec{x}$ and on the diagonal
direction $\vec{x}+\vec{y}$.

Figure 10: Displacement $\delta r$ of a pair of points separated by a distance
$r$ for a stretching force of magnitude $F$. The size of the sheet is
$81\times 81\times 2$ vertices and $r$ is small enough ($r\leq 10$) to avoid
finite size effect of the sheet. The displacement is computed for the force
$F$ either in a longitudinal direction (squares, $F$ parallel to springs
$k_{1}$) or in a diagonal direction (diamonds, $F$ parallel to springs
$k_{2}$). The inserted images represents the elastic energy for the $15\times
15$ vertices in the middle of the sheet.

The superimposition of the simulation results for the force applied in the
longitudinal and the diagonal directions is obtained for
$k_{1}/k_{2}=\sqrt{2}$.

###  C.3 Mechanical thickness

The thickness of the spring-network is calibrated by the simulation of
stretching and bending tests. Figure 11 represents a sheet made of $81\times
31\times 2$ vertices submitted to (a) an in-plane stretching force $F_{S}$ and
(b) to an out-of-plane bending force $F_{B}$.

Figure 11: Sheet made of $81\times 31\times 2$ vertices submitted to (a) a
stretching deformation and (b) a bending deformation. The forces are such as
$F_{S}/F_{B}=L^{2}$ ($L=80$) and they are distributed on the width of the
strip (only one force vector is represented for clarity). The displacements
for the extension (a) and the deflection (b) are used to calibrate the
effective thickness of the sheet according to eq. 23

For an elastic plate of dimension $L\times W\times t$, the stretching force-
displacement relation is given by:

$F_{S}=E\frac{Wt}{L}\delta_{S}$ (20)

with $\delta_{S}$ the displacement parallel to the applied force $F_{S}$ and
$t$ the thickness.

The measurement of the displacement orthogonal to the stretching direction
$\delta_{S}^{*}$ defines the Poisson ratio
$\nu=\delta_{S}^{*}/\delta_{S}=0.29$. The bending force-displacement relation
reads

$F_{B}=\frac{3EI}{L^{3}}\delta_{B}\quad\mathrm{with}\quad
I=\frac{EW{t}^{3}}{12(1-\nu^{2})}\,.$ (21)

Combining eqs. (20) and (21), one can get the ratio

$\frac{F_{B}}{F_{S}}=\frac{1}{4(1-\nu^{2})}\left(\frac{t}{L}\right)^{2}\frac{\delta_{B}}{\delta_{S}}$
(22)

One obtains the effective thickness of the two-layer sheet,

$t=\sqrt{4(1-\nu^{2})L^{2}\frac{F_{B}}{F_{S}}\frac{\delta_{S}}{\delta_{B}}}$
(23)

The simulation presented in figure 11 is performed with
${F_{B}}/{F_{S}}=(1/L)^{2}$, with $L=80$ for the 80 intervals between the 81
vertices on the longitudinal direction. The amplitudes obtained in the
simulation are $\delta_{S}=1.0897$ and $\delta_{B}=1.7224$. According to eq.
(23), the thickness of the spring model is

$t=1.52$ (24)

###  C.4 Dynamics of the vertices and equilibrium

The dynamics of the vertices is numerically solved by iteration of the forces
on each vertex. The equation of motion for each vertex is:

$m\frac{d^{2}\vec{r}_{i}}{dt^{2}}=-\Gamma\frac{d\vec{r}_{i}}{dt}-\Sigma_{j}\vec{F_{ij}}$
(25)

with $\vec{r}_{i}$ the location of the vertex $i$, $m=1$ the mass of the
vertex, $\Gamma$ a damping parameter and $\Sigma_{j}$ the force from the
springs $k_{1}$ and $k_{2}$ connecting the neighbours vertices $j$ to the
vertex $i$. The damping coefficient $\Gamma$ is in the range $10^{-4}$ to
$0.5$, depending on the network size.

Figure 12: (a) Short-length sheet with $11\times 471\times 2$ vertices with
mid-length points contact. (b) Sheet obtained by longitudinal replication of a
transverse profile $1\times 471\times 2$ from (a). The sheet has a total
$1701\times 471\times 2$ vertices.

A simple first-order integration method is used to solve the equation of
motion (25). This method is known to produce inaccurate time resolution of
differential equations. However, the main objective of this simulation is to
identify the final state of the sheet. To minimize the number of iterative
steps to converge towards equilibrium, a large integration step is taken with
$\Delta t=1$ for spring constants $k_{1}=0.3568$ and $k_{2}=0.2523$,
satisfying the condition $k_{1}=\sqrt{2}k_{2}$. A very small random noise is
added to the position (of the order of $10^{-6}$ as compared to 1 the typical
distance between two vertices) in order to avoid symmetry breaking degeneracy
issues.

The sheet profiles are first computed for short-length sheets. Figure 12 (a)
presents the stationary configuration of a sheet made of $11\times 471\times
2$ vertices after multiple iteration of the equation of motion for each
vertices ($2\>10^{4}$ iteration steps). In the limit of small length, the
sheet configuration matches with Euler’s elastica model for slender structures
(pure bending limit). Figure 12 (b) presents the initial state of the sheet’s
configuration with $1701\times 471\times 2$ obtained by replication of the
transverse profile in figure 12 (a). The simulation of the equation of motion
is performed by maintaining a contact between the two mid-length points marked
by a circle in figure 12 (b).

Figure 13 presents the configuration of a sheet of dimensions $1401\times
401\times 2$ vertices during the simulation after a number of iteration steps
$\Delta t$.

Figure 13: Sequence showing the evolution of a sheet of $1401\times 401\times
2$ vertices during the simulation of the spring network model. Time step
$\Delta t/1000=1,26,51,76$.

In principle, the simulations should be run until equilibrium is reached.
Because simulations are time consuming, they are stopped when buckling and
stress focusing appear: the goal of the simulation is to identify the
existence of buckling and not to follow in detail the post-buckling
equilibrium shape. In figure 3 (c) of the main article, the simulation points
for the smooth opening (circles) are obtained for sheets at mechanical
equilibrium. For the data points corresponding to buckled sheets (squares and
star), equilibrium is not reached for some of the largest sheets. It does not
affect the validity of the threshold since the minimal stretching keeps on
decreasing during stress focusing.

###  C.5 Dynamics and threshold

Figure 14 presents the evolution of the minimal strain
$\epsilon_{uu}\times(W/t)$ for 3 sheet sizes as a function of the simulation
time step. The 3 sheets have 471 vertices in the transverse direction and
1701, 1751 and 1801 vertices in the longitudinal direction. The data for 1751
vertices is shifted horizontally for clarity.

Figure 14: Evolution of the minimal strain (longitudinal compression) during 3
simulations with 471 vertices for the transverse direction and 1701, 1751 and
1800 vertices for the longitudinal direction. The dashed line is the threshold
value -1.16. A video file at [movie URL] for a representation of the strain
map and the sheet profile during a simulation of a sheet with 1801x471x2
vertices.  shows the evolution of the sheet with $L=1800$. The abrupt
transition corresponds to the transition between 4 d-cones and 2 d-cones.

For the shortest sheet $1701\times 471\times 2$, the normalized strain stays
above the threshold marked by a dashed line. For 1751 and 1801, the strain is
decreasing with the iteration number (opening of the sheet represented in
figure 13). The color and symbol change indicates a sign change for the
curvature of the sheet transverse profiles at $u=\delta/2$ (see criteria in
figure 3 (c) of the main article and figure 3 (b) for the definition of
$\delta$).

The simulation with 1801 vertices explores a longer simulation time and a
second and more abrupt transition is observed after $675\times 10^{3}$ time
steps and the minimal strain almost double. This second transition is
associated with the traveling and merging of the stress focusing points
(figure 3 (d) with the separation $\delta$). The video at [movie URL] for a
representation of the strain map and the sheet profile during a simulation of
a sheet with 1801x471x2 vertices.  shows the evolution of the sheet profile
$1751\times 471\times 2$ and the two transitions (first transition around
$300\times 10^{3}$ and second, more abrupt transition at $675\times 10^{3}$).

Figure 15: Representation of the simulation results to obtain the parameter
region $(L/t,W/t)$ with stress focusing. The experimental points are not
presented to avoid confusion. The dashed line is the same as in figure 2 of
the main article.

Although it is close, the curvature inversion does not exactly match with the
threshold $-1.24$. One should expect that the compression threshold depends on
the local curvature of the sheet. Since the location of the minimal strain is
not self-similar (see figure 3 (d) of the main article with $\delta(W)$), one
should expect that the appropriate normalization of $\epsilon_{uu}$ is not
exactly $W/t$ but a slightly modified function that accounts for the parameter
$\delta$, at the second order. Besides, the simulation includes added noise,
which could explain fluctuating dynamics, notably if an instability is about
to be reached (see the plot for 1851 vertices with the curvature sign that
fluctuates around $50-100\times 10^{3}$ times steps).

The simulation $1701\times 471\times 2$ was performed from an initial
configuration taken from the sheet $1751\times 471\times 2$ by removing 25
vertices on both ends, just before the transition. This is why the minimum
strain increases for the simulation $1701\times 471\times 2$. This strategy
was done to be sure that the smooth opening is stable for the configuration
$1701\times 471\times 2$.

###  C.6 State diagram

The simulation of the spring network model for different sheets sizes is used
to establish the buckling region in parameters ($L/t$, $W/t$). Figure 15 is
the same figure that figure 3 in the main article excepted that the simulation
data are presented. The experimental data for the acetate sheets are removed
from the plot (those data points are distributed around the dashed line).

## References

  * Rayleigh (1888) Lord Rayleigh, “On the bending and vibration of thin elastic shells, especially of cylindrical form,” Proceedings of the Royal Society of London 45, 105–123 (1888).
  * Witten (2007) Thomas A Witten, “Stress focusing in elastic sheets,” Reviews of Modern Physics 79, 643 (2007).
  * Cerda and Mahadevan (1998) Enrique Cerda and L Mahadevan, “Conical surfaces and crescent singularities in crumpled sheets,” Physical Review Letters 80, 2358 (1998).
  * Chaïeb  _et al._ (1998) Sahraoui Chaïeb, Francisco Melo, and Jean-Christophe Géminard, “Experimental study of developable cones,” Physical review letters 80, 2354 (1998).
  * Cerda  _et al._ (1999) Enrique Cerda, Sahraoui Chaieb, Francisco Melo, and L Mahadevan, “Conical dislocations in crumpling,” Nature 401, 46–49 (1999).
  * Ben Amar and Pomeau (1997) M Ben Amar and Y Pomeau, “Crumpled paper,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 453, 729–755 (1997).
  * Lobkovsky  _et al._ (1995) Alex Lobkovsky, Sharon Gentges, Hao Li, David Morse, and Thomas A Witten, “Scaling properties of stretching ridges in a crumpled elastic sheet,” Science 270, 1482–1485 (1995).
  * Sultan and Boudaoud (2006) Eric Sultan and Arezki Boudaoud, “Statistics of crumpled paper,” Physical review letters 96, 136103 (2006).
  * Andresen  _et al._ (2007) Christian André Andresen, Alex Hansen, and Jean Schmittbuhl, “Ridge network in crumpled paper,” Physical review E 76, 026108 (2007).
  * Balankin  _et al._ (2010) Alexander S Balankin, Didier Samayoa Ochoa, Israel Andrés Miguel, Julián Patiño Ortiz, and Miguel Ángel Martínez Cruz, “Fractal topology of hand-crumpled paper,” Physical Review E 81, 061126 (2010).
  * Cambou and Menon (2011) Anne Dominique Cambou and Narayanan Menon, “Three-dimensional structure of a sheet crumpled into a ball,” Proceedings of the National Academy of Sciences 108, 14741–14745 (2011).
  * Croll  _et al._ (2019) Andrew B Croll, Timothy Twohig, and Theresa Elder, “The compressive strength of crumpled matter,” Nature communications 10, 1–8 (2019).
  * Gottesman  _et al._ (2015) Omer Gottesman, Efi Efrati, and Shmuel M Rubinstein, “Furrows in the wake of propagating d-cones,” Nature communications 6, 1–7 (2015).
  * Chopin and Kudrolli (2016) Julien Chopin and Arshad Kudrolli, “Disclinations, e-cones, and their interactions in extensible sheets,” Soft Matter 12, 4457–4462 (2016).
  * Rogers and Huang (2009) John A Rogers and Yonggang Huang, “A curvy, stretchy future for electronics,” Proceedings of the National Academy of Sciences 106, 10875–10876 (2009).
  * Luo  _et al._ (2011) Jiayan Luo, Hee Dong Jang, Tao Sun, Li Xiao, Zhen He, Alexandros P Katsoulidis, Mercouri G Kanatzidis, J Murray Gibson, and Jiaxing Huang, “Compression and aggregation-resistant particles of crumpled soft sheets,” ACS nano 5, 8943–8949 (2011).
  * Mao  _et al._ (2012) Shun Mao, Zhenhai Wen, Haejune Kim, Ganhua Lu, Patrick Hurley, and Junhong Chen, “A general approach to one-pot fabrication of crumpled graphene-based nanohybrids for energy applications,” ACS nano 6, 7505–7513 (2012).
  * Ma  _et al._ (2012) Xiaofei Ma, Michael R Zachariah, and Christopher D Zangmeister, “Crumpled nanopaper from graphene oxide,” Nano letters 12, 486–489 (2012).
  * Yan  _et al._ (2013) Wei Yan, Wen-Yu He, Zhao-Dong Chu, Mengxi Liu, Lan Meng, Rui-Fen Dou, Yanfeng Zhang, Zhongfan Liu, Jia-Cai Nie, and Lin He, “Strain and curvature induced evolution of electronic band structures in twisted graphene bilayer,” Nature communications 4, 1–7 (2013).
  * Reis (2015) Pedro M Reis, “A perspective on the revival of structural (in) stability with novel opportunities for function: from buckliphobia to buckliphilia,” Journal of Applied Mechanics 82 (2015).
  * Holmes (2019) Douglas P Holmes, “Elasticity and stability of shape changing structures,” Current opinion in colloid & interface science (2019).
  * Schroll  _et al._ (2011) Robert D Schroll, Eleni Katifori, and Benny Davidovitch, “Elastic building blocks for confined sheets,” Physical review letters 106, 074301 (2011).
  * Fuentealba  _et al._ (2015) JF Fuentealba, O Albarrán, E Hamm, and E Cerda, “Transition from isometric to stretching ridges in thin elastic films,” Physical Review E 91, 032407 (2015).
  * Boudaoud  _et al._ (2000) Arezki Boudaoud, Pedro Patrício, Yves Couder, and Martine Ben Amar, “Dynamics of singularities in a constrained elastic plate,” Nature 407, 718–720 (2000).
  * Roman and Pocheau (2012) Benoît Roman and Alain Pocheau, “Stress defocusing in anisotropic compaction of thin sheets,” Physical review letters 108, 074301 (2012).
  * Cerda  _et al._ (2004) Enrique Cerda, Lakshminarayanan Mahadevan, and José Miguel Pasini, “The elements of draping,” Proceedings of the National Academy of Sciences 101, 1806–1810 (2004).
  * Lobkovsky and Witten (1997) Alexander E Lobkovsky and TA Witten, “Properties of ridges in elastic membranes,” Physical Review E 55, 1577 (1997).
  * Cerda and Mahadevan (2003) Enrique Cerda and Lakshminarayanan Mahadevan, “Geometry and physics of wrinkling,” Physical review letters 90, 074302 (2003).
  * Vandeparre  _et al._ (2011) Hugues Vandeparre, Miguel Piñeirua, Fabian Brau, Benoit Roman, José Bico, Cyprien Gay, Wenzhong Bao, Chun Ning Lau, Pedro M Reis, and Pascal Damman, “Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains,” Physical Review Letters 106, 224301 (2011).
  * Barois  _et al._ (2014) Thomas Barois, Loïc Tadrist, Catherine Quilliet, and Yoël Forterre, “How a curved elastic strip opens,” Physical review letters 113, 214301 (2014).
  * Pini  _et al._ (2016) Valerio Pini, JJ Ruz, Priscila M Kosaka, O Malvar, Montserrat Calleja, and J Tamayo, “How two-dimensional bending can extraordinarily stiffen thin sheets,” Scientific reports 6, 1–6 (2016).
  * Matsumoto  _et al._ (2018) Daichi Matsumoto, Tomohiko G Sano, and Hirofumi Wada, “Pinching an open cylindrical shell: Extended deformation and its persistence,” EPL (Europhysics Letters) 123, 14001 (2018).
  * Taffetani  _et al._ (2019) Matteo Taffetani, Finn Box, Arthur Neveu, and Dominic Vella, “Limitations of curvature-induced rigidity: How a curved strip buckles under gravity,” EPL (Europhysics Letters) 127, 14001 (2019).
  * Audoly and Pomeau (2010) Basile Audoly and Yves Pomeau, “Elasticity and geometry: from hair curls to the non-linear response of shells,” (2010).
  * Kramer and Witten (1997) Eric M Kramer and Thomas A Witten, “Stress condensation in crushed elastic manifolds,” Physical Review Letters 78, 1303 (1997).
  * Seung and Nelson (1988) Hyunjune Sebastian Seung and David R Nelson, “Defects in flexible membranes with crystalline order,” Physical Review A 38, 1005 (1988).
  * Curtin and Scher (1990) WA Curtin and H Scher, “Mechanics modeling using a spring network,” Journal of Materials Research 5, 554–562 (1990).
  * Ostoja-Starzewski (2002) Martin Ostoja-Starzewski, “Lattice models in micromechanics,” Appl. Mech. Rev. 55, 35–60 (2002).
  * Omori  _et al._ (2011) T Omori, T Ishikawa, D Barthès-Biesel, A-V Salsac, J Walter, Y Imai, and T Yamaguchi, “Comparison between spring network models and continuum constitutive laws: Application to the large deformation of a capsule in shear flow,” Physical Review E 83, 041918 (2011).
  * DiDonna (2002) Brian Anthony DiDonna, “Scaling of the buckling transition of ridges in thin sheets,” Physical Review E 66, 016601 (2002).
  * (41) See Supplemental Material at [movie URL] for a representation of the strain map and the sheet profile during a simulation of a sheet with 1801x471x2 vertices., .
  * Hutchinson (2010) John W Hutchinson, “Knockdown factors for buckling of cylindrical and spherical shells subject to reduced biaxial membrane stress,” International Journal of Solids and Structures 47, 1443–1448 (2010).
  * Alava and Niskanen (2006) Mikko Alava and Kaarlo Niskanen, “The physics of paper,” Reports on progress in physics 69, 669 (2006).
  * Peterson  _et al._ (1968) JP Peterson, P Seide, and VI Weingarten, “Buckling of thin-walled circular cylinders,” (1968).
  * Singer  _et al._ (2002) J Singer, J Arbocz, and T Weller, “Vol. 2, shells, built-up structures, composites and additional topics,” Buckling Experiments: Experimental methods in buckling of thin-walled structures (2002).
  * Gerasimidis  _et al._ (2018) S Gerasimidis, E Virot, JW Hutchinson, and SM Rubinstein, “On establishing buckling knockdowns for imperfection-sensitive shell structures,” Journal of Applied Mechanics 85 (2018).
  * Virot  _et al._ (2017) Emmanuel Virot, Tobias Kreilos, Tobias M Schneider, and Shmuel M Rubinstein, “Stability landscape of shell buckling,” Physical review letters 119, 224101 (2017).