Rev.
(2014)
92-98 J.A. Baimova, S.V. Dmitriev, A.V. Korznikov, and R.R. Mulyukov
92 Adv. Mater. Sci. 39
E.A.
Korznikova,
MECHANICAL BEHAVIOR OF CRUMPLED SHEET
MATERIALS SUBJECTED TO UNIAXIAL COMPRESSION
E.A. Korznikova1, J.A. Baimova1, S.V. Dmitriev1,2, A.V. Korznikov1,3,
and R.R. Mulyukov1,4
1
Institute for Metals Superplasticity Problems, Russian Academy of Sciences, Khalturina 39, Ufa 450001, Russia
2
St. Petersburg State Polytechnical University, Polytechnicheskaya 29, St. Petersburg 195251, Russia
3
National Research Tomsk State University, Lenina 36, Tomsk 634050, Russia
4
Bashkir State University, Zaki Validi 32, Ufa 450074, Russia
Received: July 24, 2014
Abstract. Crumpled sheet materials appear at different scales ranging from crumpled graphene
composed of nanoscopic structural units to crumpled paper balls and crumpled foils. There exist
a number of studies analyzing fractal geometry of crumpled materials, while their mechanical
properties are not well addressed in the literature. In the present study the uniaxial compression
of crumpled graphene is simulated with the help of molecular dynamics and the results are
contrasted to the experimentally measured stress-density curves for differently packed crumpled
paper balls. The distinctions in the mechanical response of the two studied materials are discussed based on the differences in their structure.
1. INTRODUCTION
Crumpled sheet materials represent a new class of
structures attracting a great interest because of their
unique mechanical and physical properties. Such
materials seem to be very promising for applications in the fields where extremely large specific
surface area is required [1], which is in its turn governed by the topology of crumpled configuration [2].
Examples of applications for these materials range
from virus capsids and polymerized membranes to
folded engineering materials and geological formations [3]. Hence, understanding of crumpling mechanisms and factors governing the mechanical behavior of these materials is crucial for various research
areas.
Investigation of crumpled graphene and some
other three-dimensional carbon structures is one of
recent trends in modern materials science due to
some promising results and perspectives revealed
[1,4]. One of the new findings is that crumpling elimi-
nates layer stacking followed by transition to graphite
and can be obtained on a commercial scale at a
low price [1]. Together with combination of prominent properties of flat graphene [5-9] it opens great
opportunities for graphene-based bulk
nanomaterials. A drastic increase of graphene specific capacitance caused by crumpling was shown
in [10]. Authors have mentioned that using
uncrumpled graphene sheets as binder for the
crumpled particles eliminates the need for commonly
used inactive binder materials and can further improve device performances. Crumpling of graphene
can be performed in controllable [11] or uncontrollable [1,5] manner. Adjustable crumpling of graphene
on polymer laminates can be used for artificialmuscle actuators development [11].
Stress concentrates in some points during crumpling, exceeds the yield limit of the material and
leads to irreversible plastic deformations concentrated along narrow creases and point singularities
Corresponding author: E.A. Korznikova, e-mail: [email protected]
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Mechanical behavior of crumpled sheet materials subjected to uniaxial compression
[12,13]. Permanent changes in the folds grid complicate the study of the process by means of simulation. Rahul Narain et al. [13] proposed an adaptive remeshing concept in frame of finite elements
simulation which consists in using the crushed mesh
in the crumpled regions and retaining coarse mesh
in the flat areas. The other powerful tool for investigation of various two dimensional structures including graphene and its properties is molecular dynamics simulation [14-19]. On the other hand, physical
modeling can be considered as simple and reliable
approach to study the properties of crumpled materials [3,20-22].
Crumpled paper [3,20], aluminum foil [22,23], and
graphene [1] are the most common materials for
studying. Paper sheet was a subject of investigation for the longest time among all above mentioned
matters. It was also proved that it can be used as a
physical model of crumpled graphene materials [5].
Fold-length distribution for crumpled paper sheets
is well-described by a log-normal distribution [22]
and crumpling patterns exhibit the fractal properties characterized by two fractal dimensions which
are independent on the sheet size and ball diameter [24]. In [3] it was revealed that mechanical behavior of randomly folded sheets in one-dimensional
stress state is governed by the shape dependence
of crumpling network entropy.
Experiments with constrained folded rods [25]
demonstrated that the lowest level of energy was
observed in case of spiral state (i.e. completely
stacked) but due to self avoidance the system is
topologically frustrated and only local minimal energy state can be reached. Increasing confinement
strength leads to more ordered pattern and to the
decrease of energy difference between the current
disordered state and the ground level. Concentration of elastic energy around fold lines and its weak
dependence on the length of the fold in a crumpled
sheet was shown in [26,27]. Total energy of two
folds is much greater than the energy of a single
fold that is as long as the combined length of the
two folds [26]. More recently it was discovered that
no matter how tightly a ball of paper is crumpled,
2/3 of the space inside the ball remains empty because of packing constraints [28].
Effect of hydrostatic pressure on the mechanical and structural properties of bulk carbon
nanomaterials was considered in [14,15]. These
materials show distinct relations between the pressure and volume strain. Molecular dynamics simulations realized in [29] have shown that similarly to
the results reported in paper [24] crumpled graphene
possess a fractal scale invariance with a fractal di-
(a)
93
(b)
Fig. 1. (a) Photograph of a hand-made paper ball
as the initial building unit for experimentally investigated crumpled sheet material. (b) The schematic
description of the compression device, where F is
the load (force) applied, h heigth, a width of the
orhogonal cell, and b the length of the orhogonal
cell. A front view cross section presents an example
of simple cubic arrangement of paper balls.
mension of D (),r ( hYZ
TYZ
dT aRc
RS]
Ve
the latter of crumpled paper D 3 ,.r
In this work mechanical behavior of the materials composed of crumpled paper balls is studied
experimentally and crumpled graphene is studied
by means of molecular dynamics simulations.
Crumpled paper sheet for experimental investigation is chosen due to its low ductility [5] and numerous studies performed on this material
[3,20,21,26,29]. Another novel feature of this work
is the study of the effect of packing type on deformation behavior of crumpled paper balls that was
not analyzed earlier despite a plenty of studies performed.
2. EXPERIMENTAL DETAILS
Paper sheets of standard size (297x210 mm) with
density 80 g/m2 and ~0.1 mm thickness are folded
by hands into approximately spherical balls (Fig.
1a). The structural units obtained are placed into a
rigid orthogonal box with the cross-section area
S = :S = 435S530 mm (Fig. 1 b) in the three different ways: random (R), simple cubic (SC), and face
centered cubic (FCC) packing. Initial height h of the
box vary for different packing types from h = 368.8
to 398 mm. Amount of crumpled paper balls placed
in the box is 445 for R packing, 441 for SC, and 476
for FCC packing with the initial density 0.0248;
0.0245, and 0.0248 g/cm3, respectively. The degree
of compression ( h) is recorded as the function of
the applied compressive load F. Compressive stress
is L = F/S. Loading up to the maximal value of the
force took about 20 minutes.
94
E.A. Korznikova, J.A. Baimova, S.V. Dmitriev, A.V. Korznikov, and R.R. Mulyukov
Fig. 2. (a) The building unit of crumpled structure composed of 6 bended graphene flakes. (b) Initial structure of bulk crumpled graphene.
3. SIMULATION SETUP
Crumpled graphene nanostructure is simulated using the large-scale atomic/molecular massively parallel simulator (LAMMPS) package [30] with the
adaptive intermolecular reactive empirical bond order (AIREBO) potential [31] describing carbon-carbon interactions. The AIREBO potential has been
successfully used for solving different problems such
as stability range of graphene with the defects [32],
thermoconductivity of hybrid graphene [16], properties of discrete breathers in hydrogenated graphene
[33]. Additionally, the van-der-Waals forces acting
between structural units are modeled using the
Lennard-Jones potential with the equilibrium distance
) mR Ue
YV ZZ R]
ae
Ve
Z
R]
V Vc
Xj W ( VK
for each carbon-carbon pair not connected by the
valent bonds. The Nose-Hoover thermostat is used
to control the system temperature around 300K [34].
Fig. 2a shows the single unit of simulated
crumpled material, which consists of 6 graphene
flakes of different shape and orientation. Eight such
building units having different orientation in space,
are placed without overlapping in the vertices of a
cube to produce a computational cell for crumpled
material shown in Fig. 2b. This system contains
N = 10960 atoms, and the periodic boundary conditions are applied to the simulation cell. A time step
of 0.2 fs is used to integrate the equations of atomic
motion.
The uniaxial compression is applied to the computational cell in three directions (i) 
=
,
=
=
xx
yy
zz
0, (ii) 
=

,

=

=0,
and
(iii)

=

,

=

=
0
yy
xx
zz
zz
xx
yy
with the strain rate of 3 (ad-1. In the unloading
process, is reduced gradually with the same strain
rate. The degree of anisotropy of the structure is
estimated by comparing the longitudinal ( L) and
transverse ( T) components of the resulting normal
stress. For example, for compression with 
=
,
xx

=

=
0,
and
are
defined
as
and
(
+
yy
zz
L
T
xx
yy
)/2, respectively. It is found that, in spite of relazz
tively small size of the computational cell, the longitudinal and transverse stress components measured in the three tests differ within 5%. In the following, the results for the uniaxial compression are
presented as the average for the three tests with
different compression axes.
To study the effect of random orientation of the
structural units on the mechanical response of
crumpled graphene, several random realizations of
the computational cell are checked out. It is found
that the curves coincide well which indicates that
the random misorientation of building units has no
significant effect on the properties of these structures.
4. RESULTS AND DISCUSSION
Firstly, the results of the compression tests for the
materials composed of paper balls are presented.
Stress-density curves for compression of the arrays
of paper balls are shown in Fig. 3a for the three
packing types. All the dependences within the studied range of density are nearly linear with considerable force increase upon compaction. Slope of the
curve slowly grows with the increase of the initial
density of the material. R and SC packing show
similar behavior in contrast to FCC packing, which
exhibits a larger strengthening. At the strain =
l/l0=0.26 ( ~0.0325 g/cm3) FCC packing shows
25% larger stress than R and SC packing. Higher
rigidity of the FCC packing can be explained by the
Mechanical behavior of crumpled sheet materials subjected to uniaxial compression
95
(a)
(a)
(b)
(b)
(c)
Fig. 3. He
c
Vddsde
c
RZ Tfc
g
Vd Se
RZVUUfc
ZXT
pression of paper ball structure including (a) loading for three cases of different packing types of the
crumpled paper and (b) loading-unloading cycle for
the case of SC packing. (c) Photograph of a crumpled
paper ball cross section after compression.
more homogeneous structure with smaller voids
between paper balls in comparison to R and SC
packing.
Fig. 3b presents the L( ) dependence for the
case of SC packing including loading up to =
0.0475 g/cm3 followed by unloading. The loading part
of this curve can be approximated by the two linear
functions with different slopes. A larger slope is observed for stress L>3.5 MPa. The increase of the
slope of L( ) can be explained by the changes in
deformation mechanisms. After achieving a certain
value of density corresponding to complete compression of all the initially existing ridges and folds,
Fig. 4. Stress as the function of density at T = 300K
for simulated crumpled graphene subjected to
uniaxial compression (a) and unloading after deformation 
=0.5 (b).
xx
the formation of new ridges and folds is required for
further deformation, which is the reason for force
growing with the number of folding events. The comparison of initial loose structure of a crumpled paper ball having relatively small amount of ridges and
folds (Fig. 1a) with the one after compression (Fig.
3c) favors this assumption. At the first stage of unloading down to a half of the maximum value of
stress no change of material density is observed.
Density of the structure after complete unloading
decreases down to 0.039 g/cm3, which is about 83%
of the density after loading and 1.5 times higher
than that of initial material density.
Secondly, the numerical results for the uniaxial
compression of the crumpled graphene simulated
at 300K are described. The stress as the function
of density for crumpled graphene is presented in
Fig. 4a. The L( ) and T( ) curves calculated for
uniaxial compression are shown by open circles and
black squares, respectively. It can be seen that the
curves demonstrate a nonlinear behavior and the
material is compression-resistant which is similar
to that shown in [28]. The slope of the stress-density curves starts to increase rapidly with increase
in density after the density of graphite ( g =
2.09-2.33 g/cm3) is achieved. The effect of van-derWaals forces can possibly explain the strong non-
96
E.A. Korznikova, J.A. Baimova, S.V. Dmitriev, A.V. Korznikov, and R.R. Mulyukov
(a)
(b)
(c)
(d)
Fig. 5. Structural transformation of one building unit under uniaxial compression at the densities (a) 1.0, (b)
1.5, (c) 2.0, and (d) 2.8 g/cm3.
linear growth of L( ) and T( ) at densities higher
than g. Evolution of L under unloading after uniaxial
deformation 
= 0.5 ( ~1.76 g/cm3) is shown in red
xx
in Fig. 4b. The value of strain 
before unloading is
xx
chosen close to the strain = 0.4 reached for
crumpled paper compression. Considerable difference with unloading behavior of crumpled paper
material can be observed. A small rapid drop of
stress in the case of crumpled graphene is followed
by linear decrease of density down to complete
unloading. The slope of the unloading line is close
to the slope of the stress-density curve at the point
where unloading begins. The difference in the unloading curves can be attributed to outstanding elastic properties of crumpled graphene. Absence of vander-Waals forces effect can be explained by a relatively low density of the material ( < g).
Fig. 5 shows the structural transformation of one
graphene flake chosen near the center of the computational cell for uniaxial compression at T = 300K
at the densities (a) 1.0, (b) 1.5, (c) 2.0, and (d) 2.8
g/cm3. It can be seen that single structural unit of
bulk crumpled graphene can be easily deformed at
current loading scheme forming folds without sharp
ridges in contrast with the case of crumpled paper.
Different topography of two crumpled materials (e.g.
structure and amount of building units, number of
folds per unit, absence of sharp ridges which are
the main energy concentrators in case of crumpled
paper) also can be considered as the reason for
distinctions in the mechanical behavior of these two
materials under loading and unloading.
5. CONCLUSIONS
A comparative study of the evolution of mechanical
and structural characteristics of two materials composed of crumpled sheets, subjected to uniaxial
compression was performed by means of experimental study (material composed of crumpled paper balls) and molecular dynamics simulation
(crumpled graphene sheets). Three different paper
ball packing were tested, namely R, SC, and FCC
packing. The following conclusions are made as a
result of the present study.
1. FCC packing of paper balls shows higher strength
and higher rigidity at the same density (see Fig.
3a). This can be attributed to a more homogeneous
ctructure of the close-packed FCC lattice with
smaller voids between paper balls in comparison to
R and SC packing.
2. Stress-density curves for paper ball materials can
be approximated by a bilinear function within the
density range from 0.025 to 0.05 g/cm3. Crumpled
graphene demonstrates a highly nonlinear stressdensity curve whose slope starts to increase very
rapidly for > g, where g is the density of graphite.
3. The unloading curve for the paper ball material
differs very much from that of crumpled graphene
Mechanical behavior of crumpled sheet materials subjected to uniaxial compression
(see Figs. 3b and 4b). Unloading of the paper ball
material goes in two steps. First step is characterized by a sharp and large drop of stress at practically unchanged density and the second step is
charaterized by a nonlinear stress-density relation.
A small rapid drop of stress in the case of crumpled
graphene is followed by a linear decrease of stress
down to complete unloading. The slope of the unloading line is close to the slope of the stress-density curve at the point where unloading begins.
4. The differences in mechanical behavior of
crumpled paper and crumpled graphene materials
can be understood by comparing their structure and
type of bonding. Graphite, fullerites and crumpled
graphene aggregate into 3D solids by means of weak
van-der-Waals bonds, while the structural units are
formed by the carbon atoms connected with the
strong covalent sp2 bonds. All such materials can
be classified based on the types of chemical bonds
formed in them and on the number of the nearest
neighbors with which each atom forms covalent
bonds [35-37]. sp2-bonded carbon materials demonstrate extraordinary elastic properties. Paper
balls, on the other hand, deform with formation of
sharp ridges and there is no interaction between
the paper sheets analogous to the van-der-Waals
bonds which interact between sp2-bonded carbon
units.
Present study serves to some improvements in
the understanding in the area of crumpled materials
and opens perspectives for investigation in this direction, for example, studying the effect of various
deformational topology on physical properties within
one material.
ACKNOWLEGEMENTS
K.E.A. acknowledges financial support from the
Russian Foundation for Basic Research grant 1402-97029-p-povolzhie-a. S.V.D. is grateful for the financial support provided by the Russian Government Project 5-100-2020. J.A.B. gratefully thanks
Russian Science Foundation grant 14-13-00982.
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A.V. in conducting experiments is gratefully acknowledged. All the calculations were carried out on the
supercomputer of the Supercomputer Center of
Russian Academy of Sciences.
REFERENCES
[1] L. Zhang, F . Zhang, X. Yang, G. Long, Y. Wu
and T. Zhang // Sci. Rep. 3 (2013) 1408.
P
(Q
6H 7R]
R Z 9 HR RjR : EZVURAVo
R. C. Montes de Oca, A. Horta Rangel and M.
97
u BRc
e
nVk8c
fk%
%Phys. Rev. B 77 (2008)
125421.
[3] A.S. Balankin and O. Susarrey Huerta // Phys.
Rev. E 77 (2008) 051124.
[4] Z. Zhao, X. Wang, J. Qiu, J. Lin and D. Xu //
Rev. Adv. Mater. Sci. 36 (2014) 137.
P
+Q>6 Dg
Z
Ut %
%Rev. Adv. Mater. Sci. 34 (2013)
1.
P
,Q>6 Dg
Z
Ut %
%Rev. Adv. Mater. Sci. 30 (2012)
201.
P
-Q>6 Dg
Z
Ut R U6< HYVZVc R %
%J. Phys.
D: Appl. Phys. 46 (2013) 345305.
P
.Q>6 Dg
Z
Ut %
%Rev. Adv. Mater. Sci. 34 (2013)
12.
P
/Q
6H @ TY Vg>6 Dg
Z
Ut R U7C HV V g
// Rev. Adv. Mater. Sci. 37 (2014) 105.
[10] J. Luo, H. Jang and J. X. Huang // ACS Nano
7 (2013) 1464.
[11] J. Zang, S. Ryu, N. Pugno, Q. Wang, Q. Tu,
M.J. Buehler and X. Zhao // Nature Mat. 12
(2013) 321.
[12] M. Ben Amar and Y. Pomeau // Proc. Roy.
Soc. Lond. A 453 (1997) 729.
P)QG CRc
RZ I EW
RW
WR U ;Dt
7c
Z
V%
%ACM
Transactions on Graphics, Proceedings of
ACM SIGGRAPH 32 (2013) 1.
[14] J.A. Baimova, B. Liu, S.V. Dmitriev and
K. Zhou // Phys. Stat. Sol. (RRL) 8 (2014)
336.
[15] J.A. Baimova, R.T. Murzaev and S.V. Dmitriev
// Phys. Solid State 56(10) (2014) 1946.
[16] B. Liu, C. Reddy, J. Jiang, J. Baimova,
S. Dmitriev, A. Nazarov and K. Zhou // Appl.
Phys. Lett. 101 (2012) 211909.
[17] E.A. Korznikova and S.V. Dmitriev // Appl.
Phys. D. 47 (2014) 345307.
[18] E.A. Korznikova, Y.A. Baimova and S.V.
Dmitriev // Europhys. Lett. 102 (2013) 60004.
[19] E.A. Korznikova, Y.A. Baimova, S.V.
Dmitriev, R.R. Mulyukov and A.V. Savin //
JETP Letters 96 (2012) 222.
[20] S. Deboeuf, E. Katzav, A. Boudaoud,
D. Bonn and M. Adda-Bedia // Phys. Rev.
Lett. 110 (2013) 104301.
[21] G. A. Vliegenthart and G. Gompper // Nature
Mat. 5 (2006) 216.
[22] O. Bouaziz, J. P. Masse , S. Allain,
L. Orgeas and P. Latil // Mat. Sci. Eng. A 570
(2013) 1.
[23] A.S. Balankin, O.S. Huerta and V. Tapia //
Phys. Rev. E 88 (2013) 032402.
[24] A. S. Balankin, D. S. Ochoa, I. A. Miguel,
J. P. Ortiz and M.A.M. Cruz // Phys. Rev. E
81 (2010) 061126.
98
E.A. Korznikova, J.A. Baimova, S.V. Dmitriev, A.V. Korznikov, and R.R. Mulyukov
[25] E. Bayart, A. Boudaoud and M. Adda-Bedia //
Phys. Rev. E 89 (2014) 012407.
[26] A. E. Lobkovsky, S. Gentes, H. Li, D. Morse
and T. A. Witten // Science 270 (1995) 1482.
[27] J.A. Baimova, B. Liu and K. Zhou // Letters
on Materials 4(2) (2014) 96.
[28] K. Matan, R. B. Williams, T. A. Witten and
S. R. Nagel // Phys. Rev. Lett. 88 (2002)
076101.
[29] S.W. Cranford and M.J. Buehler // Phys. Rev.
B. 84 (2011) 205451.
[30] S. Plimpton // J. Comp. Phys. 117 (1995) 1.
[31] S. Stuart, A. Tutein and J. Harrison // J.
Chem. Phys. 112 (2000) 6472.
[32] J.A. Baimova, B. Liu, S.V. Dmitriev, K. Zhou
and A.A. Nazarov // EPL 103 (2013) 46001.
[33] B. Liu, J.A. Baimova, S.V. Dmitriev, X. Wang,
H. Zhu and K. Zhou // J. Phys. D Appl. Phys.
46 (2013) 305302.
[34] S. Nose //J Phys Soc Jpn. 70 (2001) 75.
[35] E.A. Belenkov and V. A. Greshnyakov // New
Carbon Materials 28 (2013) 273.
[36] E.A. Belenkov and V. A. Greshnyakov //
Phys. Solid State 55 (2013) 1754.
[37] A.N. Enyashin and A. L. Ivanovskii // Phys.
Status Solidi B 248 (2011) 1879.