Curved Kirigami SILICOMB cellular structures with zero Poissons ratio for large deformations and morphing
Y.J.Chena,b, F. Scarpab,*,C. Remillatb,c, I.R. Farrowb, Y.J.Liud, J.S.Lenga,*
aP.O. Box 3011, Centre for Composite Materials and Structures, No. 2 YiKuang
Street, Science Park of Harbin Institute of Technology (HIT), Harbin, 150080, China bAdvanced Composites Centre for Innovation and Science, University of Bristol,
Bristol BS8 1TR, UK cAerospace Engineering, University of Bristol, BS8 1TR, UK dP.O. Box 301, Department of Aerospace Science and Mechanics, No. 92 West dazhi
Street, Harbin Institute of Technology (HIT), Harbin, P.R. China. *Corresponding authors:
Tel.: +44(0) 1173315306; fax: +44(0) 1179272771.
Email address: [email protected] (F. Scarpa)
Tel.:+86(0) 451-86402328; fax: +86(0) 451-86402328.
Email address: [email protected] (J.S. Leng) Abstract: The work describes the design, manufacturing and parametric modeling of
a curved cellular structure (SILICOMB) with zero Poissons ratio produced using
Kirigami techniques from PEEK films. The large deformation behavior of the cellular
structure is evaluated using full-scale finite element (FE) methods and experimental
tests performed on the cellular samples. Good agreement is observed between the
mechanical behavior predicted by the FE modeling and the 3-point bending
compression tests. Finite Element simulations have been then used to perform a
parametric analysis of the stiffness against the geometry of the cellular structures,
showing a high degree of tailoring that these cellular structures could offer in terms of
minimum relative density and maximum stiffness. The experimental results show also
high levels of strain-dependent loss factors and low residual deformations after cyclic
large deformation loading.
Keywords: cellular structures, zero Poissons ratio, finite element (FE), loss factor, stiffness, morphing.
INTRODUCTION
During the last decades, cellular and lattice structures have attracted significant
attention from researchers within the mechanics of solids community due to their
significant lightweight and out-of-plane stiffness properties(Bitzer, 1997). Cellular
solids have been widely developed and used as sandwich cores in a variety of
engineering applications, ranging from marine to aerospace and automotive
lightweight constructions(Alderson et al., 2010a; Alderson et al., 2010b). The
conventional hexagonal honeycomb is a typical example of a cellular core
configuration, with unit cells made of ribs with equal length and an internal cell angle
of 30o (Gibson and Ashby, 1997). Over-expanded centresymmetric honeycombs with
special orthotropic properties can also be developed when varying the cell wall aspect
ratio and internal cell angle, always with positive values (Bezazi et al., 2005; Gibson
and Ashby, 1997). Hexagonal honeycombs with convex configuration (i.e., positive
internal cell angle leading to positive Poissons ratio - PPR) exhibit anticlastic or
saddle-shape curvatures when subject to out-of-plane bending(Evans, 1991; Lakes,
1987; Masters and Evans, 1996). On the opposite, if the in-plane Poissons ratios of
the cellular structures are negative (i.e.,auxetic, or NPR), the curvature is synclastic,
with a resulting dome-shaped configuration(Evans and Alderson, 2000a; Lakes, 1987;
Miller et al., 2010). In comparison with conventional cellular structures, the auxetic
cellular configurations feature increased in-plane shear modulus and enhanced
indentation resistance (Evans, 1991; Evans and Alderson, 2000b; Lakes, 1987; Milton,
1992). Auxetic cellular structures have also been used to prototype morphing wings
(Bettini et al., 2010; Bornengo et al., 2005; Martin et al., 2008; Spadoni et al., 2006),
radomes(Scarpa et al., 2003), adaptive and deployable structures(Hassan et al., 2008).
Both types of honeycombs can be used as the core of sandwich composites, but make
difficult to be used in the morphing cylindrical applications (Grima et al., 2010). In
that sense, cellular structures with zero Poissons ratio(ZPR) do not produce
anticlastic and synclastic curvature behaviors, but can be used to manufacture cores
for tubular structures(Grima et al., 2010), such as air intakes.
ZPR implies also that the solid does not deform laterally when subjected to
mechanical loading (tensile or compressive) (Lira et al., 2011). In the field of
morphing skins, a candidate solution proposed by some authors is sandwich cellular
cores with flexible face-sheets (Bubert et al., 2010; Olympio and Gandhi, 2007,
2010a). For one-dimensional morphing applications, the confinement of the lateral
deformation belonging to cellular structures (with PPR and NPR) will result in a
substantial increase in the effective modulus along the morphing direction(Olympio
and Gandhi, 2007, 2010b). To solve this problem, Olympio (Olympio and Gandhi,
2010a, b)and Bubert (Bubert et al., 2010) have proposed the use of zero Poissons
ratio cellular structures for cores to be used in 1D morphing skins. Another
application of ZPR cellular structures has been previously identified by Engelmayr et
al.(Engelmayr et al., 2008), who demonstrated the feasibility of using accordion-like
honeycomb microstructures as scaffolds with controllable stiffness and anisotropy for
myocardial tissue engineering. Grima (Grima et al., 2010) has recently proposed a
new zero Poissons ratio cellular structure based on a semi re-entrant honeycomb
configuration. Another contribution to the field of zero Poissons ratio cellular
structures is the SILICOMB configuration developed by Lira, Scarpa and other
authors(Lira et al., 2009; Lira et al., 2011), which is inspired by the amorphous silica
lattice geometry and can be considered as a generalization of the accordion
honeycomb(Olympio and Gandhi, 2007). Much of the previous work cited has
focused on the mechanical behavior of planar cellular configurations. However, less
significant amount of work has been dedicated to tubular or curved cellular structures.
Jeong and Ruzzene(Jeong and Ruzzene, 2004) have investigated the wave
propagation behavior in cylindrical grid-like structures with hexagonal and re-entrant
shape leading to NPR effect. Scarpa et al. have investigated the variation of axial
stiffness, radial Poissons ratio and buckling behavior of tubular configurations with
re-entrant unit cell shape(Scarpa et al., 2008). However, to the best of the authors
knowledge, no specific work has been done on curved cellular configurations with
non-tubular layout. Moreover, no similar work has been done on cellular structures
with zero Poissons ratio behavior.
In this work, the stiffness coefficient related to various curved SILICOMB
configurations is investigated for the first time with FE simulation and experimental
methods. A full-scale curved SILICOMB topology is simulated using FE analysis,
with boundary conditions representing experimental compression tests. The curved
cellular honeycomb samples with zero Poissons ratio have been manufactured using
Kirigami techniques applied to thermoplastics (PEEK). Kirigami is a manufacturing
method combining Origami (ply-folding procedures) together with cut patterns, and is
particular suited to produce composite cellular structures with complex tessellations
and geometry, such as a morphing wingbox concept developed by some of the
Authors(Saito et al., 2011). Curved SILICOMB PEEK cellular structures are currently
evaluated as possible cores in sandwich reflectors, due to the low thermal capacity,
high-temperature performance and good mechanical behavior of thermoplastics PEEK.
The experimental results have been used to benchmark the FE model, and perform a
parametric analysis of the stiffness of the curved SILICOMB configurations versus
the
tests
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ETRY AND
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ellular confi
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ellular struc
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ILICOMB
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under
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ernal
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e cell
ng to
0 the
been
simulated using the commercial FE analysis package (ANSYS, version 11.0, Ansys
Inc.). Full-scale FE models have been prepared using 4-node element SHELL181
with three translation degrees of freedom (DOF) along the x, y and z directions, and
three rotational DOFs around the x, y and z axes. This element is also well suited for
linear, but also large rotation, and/or large strain non-linear applications. The
simulations were performed on a full-scale volume model of 3.54 cells with an
average mesh size of / 5L (Figure 2), representative of the samples manufactured
using the Kirigami technique. Based on this meshing size, a total number of 8960
elements were obtained. The deformation of the curved cellular sample subjected to
central loading was simulated initially using a linear FE approach. To investigate from
a numerical point of view of the energy absorption capability and damping properties
further, quasi-static non-linear analysis (with Newton-Raphson algorithms) and
non-linear transient analysis (with Full method) were used to describe the
force-displacement relation under the loading and uploading cycles. Contact pairs
were created between the peak nodes of inside strips and the surface of the adjacent
strips to prevent element penetration during the simulations (Figure 3). The CONTA
175 element was used to model the contact elements (the peak nodes of the strips),
while the TARGE 170 elements were representing the surfaces of the adjacent strips.
Moreover, the non-linear and transient methods were controlled with the STABILIZE
command. To activate the command, an energy-dissipation ratio of 0.2 obtained from
the experimental test results was used in both non-linear FE methods. According to
the data sheet provided by the supplier, the PEEK material shows an isotropic
mechanical behavior ( 1.8cE GPa= , =0.4c ), which were used as the basic
mechanical constants in the FE model analysis.
T
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quivalent st
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ample of cont
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To better un
tiffness coe
oneycomb
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pplied displ
the radius
tion (x dire
also constr
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nderstand th
efficient ( K
sample is
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ween the node
lacement pe
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ection) (Fig
rained alon
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e versus di
ar region i
he stiffness
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assumed a
ICOMB conf
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erpendicula
while the tw
gure 4). Th
g the two
he outer su
splacement
s defined a
s coefficient
roduced as
as a homog
figuration
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ar to the mi
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he nodes at
other y an
urface could
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as the stiff
nt of the cu
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iddle
were
t the
nd z
d be
n be
fness
urved
ence.
uboid
structure with the length 8 cos ( )H , the width 2b and the height eH . Based on the
homogeneous cuboid structure, the equivalent stiffness coefficient ( eqK ) can be
expressed as:
= ceqe
E AKH
(1)
where the symbols cE , A and eH stand for the elastic modulus of the core
material, the cross-sectional area and effective height of the cuboid structure,
respectively. For simplicity, the cross-section A can be formulated as:
=16 cos ( )A Hb (2)
Substituting Eq. (2) into Eq. (1), the expression of equivalent stiffness coefficient
( eqK ) becomes:
16 cos ( )= ceqe
bHEKH
(3)
The relative density of the curved sample is then given by the volume ratio of the core
material ( cV ) within a representative unit cell and the unit cell ( rV ):
= cc r
VV
(4)
where and c represent respectively the density of the curved sample and the core
material. The volumes of the core material and the unit cell can be therefore
calculated as:
2 2= 4* * +( - )* /cos( ) *c o iV H b R R t (5)
2 2=( - )* * *cos ( )r o iV R R H (6)
where is the central angle of the representative unit cell, oR and iR are the outer
and inner radius of the curved sample (Figure 1(a)). Substituting Eq. (5) and Eq. (6)
into Eq. (4) and considering the non-dimensional parameters expressions, the relative
den
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mbination
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o the final cu
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Figu
le 1 Geometrometry l length L /(ml length H /(m
gle /(Degregle /(Degre
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was reached
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mm) mm) ee) ee)
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her semi-cir
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btained after
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10 10 20 -20
um thermof
was subseque
5(b)). In a
rcular moul
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cotch-Weld
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arameters of
matic diagram
MB samples mGeometryThickness Depth of wInner radiuCentral an
forming ma
ently perfor
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ld (Figure 5
eparated rec
ure samples
2216 B/A
room temp
f the curved
m of curved S
manufacturedy
of wall t /(mwall b /(mm)us iR /(mm)
ngle of unit ce
achine (Form
rmed at low
way, circul
5(c)). The th
ctangle and
by bonding
Gray).The
perature (23
d sample are
ILICOMB sa
mm) )
ell /(Degre
mech FM1)
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lar strips w
third step of
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ample
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were
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trips
-part
lular
o 72
Table
ue 25 5 88 97
Experimental testing
The stiffness coefficient of the curved SILICOMB cellular sample was obtained
from cyclic compression tests performed using a materials testing machine (Instron
3343, load cell: 1KN) with a constant displacement rate of 2 mm/min in controlled
displacement mode (Figure 6). A flat steel plate was placed on the top of the base
plate of the machine to provide sufficient contact area for the support of both bases of
the samples. Therefore, the bases of the curved sample can slide horizontally on the
flat steel plate while the top of the curved sample was always in contact with the
upper plate of the machine. However, the slight horizontal sliding of the curved
sample has no obvious effect on the compression deformation and the external
loading along the vertical direction. It can also be found that no obvious horizontal
sliding is occurring during the experimental tests. In order to increase the contact area
between the bases and the flat steel plate, both bases of the curved sample were cut to
keep horizontal contact with the flat steel plate (Figure 6). Therefore, the actual
effective height (25mm) is lower than the theoretical value and is obtained by
measurement. The force and displacement values were recorded by a BlueHill control
and acquisition software. The cyclic tests were conducted at room temperature and
under controlled humidity conditions. The experimental result curves for force versus
strain are shown in Figure 7, with each test containing four different maximum strains
(40%, 60%, 80% and 100%). The strain ( ) has been defined as the ratio between the
compression deformation and the effective height ( eH ). For increasing compression
values, the force-strain curve shows a linear increase for strains lower than 0.4,
followed by a non-linear increase trend until the maximum strain is reached. The
non-linear increase is mainly attributed to the contact between the inner strips.
Com
F
Test
than
#1)
mparison b
Figure 8 sho
t #3) and th
n 0.4. A goo
and the FE
Figure 6
Figure 7 Ex
between the
ows the com
he FE pred
od agreeme
E simulation
6 Curved SILI
xperimental c
RESULTS
e experimen
mparison be
dictions (Lin
ent has been
n (nonlinea
ICOMB samp
cyclic compre
AND DISC
nts and FE
etween expe
near, Nonli
n observed
ar and transi
ple experime
ession force v
CUSSIONS
E model
erimental re
near and T
between the
ient method
ntal setup
vs. strain curv
S
esults (Test
Transient) fo
e experimen
ds) when th
ves
#1, Test #2
or strains lo
ntal result (
he displacem
2 and
ower
(Test
ment
was lower than 6.5 mm. The nonlinear and transient FE simulations tended to provide
a stiffer response beyond this displacement value because of the increased stiffness
contact existing between the top nodes belonging to the internal and adjacent strips
(see Figure 3). However, these pairs of nodes were not always in contact during the
experiment since they were not perfectly aligned due to the manufacturing
imperfections, and therefore the expected significant increase in force was not
effectively taking place. The numerical loss factors (see the following paragraph for
the definition of loss factor used) were also lower than the experimental ones. Against
an experimental value of 0.2, the nonlinear quasi-static simulation provided a loss
factor of 0.14 and the transient dynamic one of 0.17. The stabilization algorithm used
during the ANSYS simulations involved the use of an energy approach to calculate
the virtual equivalent viscous damping necessary for the stability, and its definition is
different from the hysteretic damping calculated through the experimental curves.
Moreover, a linear reduction of the damping during the load step was necessary to
guarantee the numerical convergence of the solution, and this aspect also contributed
to the effective decrease of the numerical loss factor. Although no complete contact
between the top nodes of the samples was observed all the time, some contact friction
was nevertheless occurring, an aspect not represented by the numerical simulations.
All these factors concurred to produce hysteresis curves showing some discrepancies
with the experimental ones (Figure 8). However, the elastic part of the force-strain
slope has been captured well by different simulations, even using the linear elastic FE
model. The numerical results showed in general a slight softer response (1.5%)
compared to the experimental result from Test #2. The stiffness decrease was also
observed against the results from the experimental Test #3 (4.8%); however, a slightly
higher stiff response (6.3%) was observed against the results from Test #1. These
results provide sufficient evidence to use the elastic FE model to analyze and discuss
the parametric dependence between the curved SILICOMB stiffness and the geometry
parameters of the unit cell in the following sections.
Figure 8 Experimental result and FE prediction curves for force vs. displacement
Energy absorption and loss factor
Figure 9 shows the experimental results for total energy absorbed per unit volume
( w ) versus maximum strain ( MAX =40%, 60%, 80% and 100%). The total energy
absorbed per unit volume has been defined as:
= /w W V (8)
where W and V represent the total energy absorbed and total volume of the curved
SILICOMB samples respectively. The three tests showed similar trends, with an
average value ( 3/J m ) related to the total energy absorbed per unit volume of 26.73
3.36 for the first cycle ( MAX =40%), 57.414.67 for the second cycle ( MAX =60%),
104.178.11 for the third cycle ( MAX =80%) and 196.6910.51 for the fourth cycle
( MAX =100%). The total energy absorbed per unit volume showed a non-linear
increase for incremental maximum strains. The experimental results for loss factor ( )
versus maximum strain ( MAX ) are shown in Figure 10(a). The loss factor is calculated
by the following expression:
= WW
(9)
where W is the total work done on the sample. All the experimental results showed
similar trends, with an average value related to the loss factor equal to 0.2000.015
for the first cycle ( MAX =40%), 0.1800.007 for the second cycle ( MAX =60%),
0.1700.002 for the third cycle ( MAX =80%) and 0.1850.007 for the fourth cycle
( MAX =100%). For increasing maximum strains, the value of the loss factor first
tended to decrease, then increased up to a minimum value occurring around MAX
=80%. To explain the variation trend of loss factor further, Figure 10(b) shows one
hysteretic cycle curve of the experimental tests for compression force versus strain
(Test #2). It can be ascribed the non-linear behavior of force-strain curve, which
results from the contact between the inner strips.
Figure 9 Experimental results for total energy absorbed per unit volume w vs. maximum strain MAX
Par
REL
S
SIL
vert
that
With
With
Figu
( )
5o t
decr
Sim
0.85
resp
Figure 10 E
rametric an
LATION BE
Similarly to
LICOMB co
tex touching
t:
hin the red
hin the blue
ure 11 show
).In the pres
to 85o. For
reases from
milarly, the u
5 and 2 to
pectively, on
Experimental
nalysis
ETWEEN AN
planar cent
onfiguration
g each othe
dotted line:
e dotted line
ws the value
sent study, t
r increasing
m 22.86 to 2
upper bound
o 0.18) wh
n the oppos
l results curvecompre
NGLES ( A
tresymmetri
ns have limi
er. By inspe
2cos (cos(
es of ( /H
he range of
g angles,
2, while the
d for dec
hen the ang
ite, the low
es: (a) loss faession force v
AND ) AN
ic re-entran
iting cell w
ecting Figu
( ))
))
/ L ) versus
f the interna
, the corre
e lower boun
creases from
gle varyi
wer bound v
actor vs. mvs. strain
ND ASPEC
nt configura
all aspect r
ure 1(b), it
angle ( ) f
al cell angle
esponding u
nd increase
m 20.80 to 1
ing between
alue of i
maximum strai
T RATIO (
tions (Scarp
atios to avo
is possible
for different
es and
upper boun
es from 0.18
1.82 (16.23
n 25o, 45o
ncreases fro
ain MAX ; (b)
)
pa et al., 20
oid the oppo
to demons
(
(
t internal an
is varied f
nd value of
8 to 2 for
to 1.42, 9.7
o, 65o and
om 0.85 to
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osite
trate
(10)
(11)
ngles
from
f
=5o .
70 to
85o,
9.70
(1.42 to 16.23, 1.82 to 20.80 and 2 to 22.86) for the same set of values. In this study,
the value of is limited between an upper bound of 5.50 and a lower bound of 0.73
for internal cell angles=20o and =-20o .
Figure 11 Permissible ranges of values ( /H L ) vs. angle with different constant angles
RELATION BETWEEN STIFFNESS COEFFICIENT AND GEOMETRY
PARAMETERS
Figures 12 and 13 show the variation of the stiffness coefficient versus different
geometry parameters of the curved cellular configuration, which is represented by the
baseline 3.54 cells SILICOMB layout used for the manufactured samples. For
simplicity, the curvature radius ( ( ) / 2o iR R R= + ) has been used within all the
parametric analysis. The figures show results related to the variation of one or two
geometry parameters, while the other are kept constant. The baseline geometry
parameters are illustrated in Table 1. Figure 12(a) shows the FE prediction for the
non-dimensional stiffness coefficient ( / eqK K ) versus curvature radius ( /R L ) for
different wall lengths ( L ). For increasing /R L values, the non-dimensional stiffness
coefficient decreases when L is kept constant, on the opposite, / eqK K decreases for
increasing L values. The FE predictions related to the non-dimensional stiffness
coefficient versus the curvature radius ( /R L ) for different aspect ratios ( = /H L )
are shown in Figure 12(b). For increasing /R L , / eqK K decreases when is kept
constant. The magnitude of the non-dimensional stiffness decreases also for higher
values of cell wall aspect ratio.
The sensitivity of the non-dimensional stiffness versus the non-dimensional
curvature for different thickness ratios = /t L is shown in Figure 12(c). It is possible
to observe also in this case the pattern of a decreasing stiffness coefficient for
increasing /R L values, and a stiffening effect of the mechanical performance of the
cellular core for increasing values. The increase of the non-dimensional stiffness
for the curved cellular configuration for increasing thickness values is compatible
with the stiffening effect provided in planar and prismatic centresymmetric
honeycomb cells when the thickness of the walls is increased(Gibson and Ashby,
1997). Figure 12(d) shows how / eqK K varies versus the curvature for different depth
ratios = /b L being kept constant. Aside from the quasi-inverse dependence versus
/R L observed also for the other parametric analysis, the stiffness appears to increase
with the depth ratio. This is an interesting phenomenon, because in planar hexagonal
structures the through-the-thickness stiffness is dependent on the relative density only
(Gibson and Ashby, 1997), while the transverse shear in the plane (yz) tends to
approach a lower bound, the other (xz) is dependent on the unit cell geometry only
and not on the depth ratio(Scarpa and Tomlin, 2000). A similar behavior has been also
observed in SILICOMB planar honeycomb structures, although the dependence of the
transverse shear modulus yzG versus is far less marked than in centresymmetric
hexagonal honeycombs(Lira et al., 2011). The curved geometry of the cellular
stru
simu
coef
Figu
( /R
T
ang
non
beh
sim
con
R/L
ucture appea
ulation, th
fficient, eve
ure 12 FE pre
/ L ): (a) for dfor diffe
The depend
le for di
n-dimension
avior betwe
ilar behavi
figurations(
did contrib
ared to have
herefore pr
en more ma
ediction of th
different wall erent thicknes
ence of the
ifferent /R
nal curvature
een =-20o a
ior has be
(Lira et al.,
bute to the
e created equ
roviding an
rked for thi
he non-dimen
lengths ( L );ss ratios ( =
e non-dime
/ L values
e /R L has b
and =20o
en observe
2009). Th
increase of
uivalent ho
n increase
icker honeyc
sional stiffne
; (b) for diffe= /t L ); (d) fo
ensional stif
can be ob
been kept co
, with a m
ed for the
he use of de
f the stiffne
op stresses
to the n
comb sectio
ess coefficient
erent cell wallor different d
ffness coef
bserved in
onstant, /K
maximum va
Ex modulu
ecreasing n
ess. The re
during the 3
non-dimens
ons.
t ( / eqK K ) vsl aspect ratiosepth ratios (
fficient vers
Figure 13
/ eqK present
alue occurr
us in plan
on-dimensio
sults from
3-point ben
sional stiff
s. curvature ra
s ( = /H L = /b L )
sus the inte
3(a). When
ted a symm
ring at =0
nar SILICO
onal curvat
the simulat
nding
fness
adius
); (c)
ernal
the
metric
o . A
OMB
tures
tions
rela
diffe
has
=0
SIL
whe
Fig
REL
DEN
D
ligh
dep
and
tend
stiff
obse
ated to the n
ferent curva
been oppos
0 , and th
LICOMB str
en is varie
gure 13 Non-d
LATION BE
NSITY
Density is
htweight san
endence of
different
ded to decr
fness was in
erved for th
non-dimens
ature radii (
site to the on
e stiffness
ructures do
ed between -
dimensional s
ra
ETWEEN T
an import
ndwich app
f / eqK K vers
curvature r
rease when
ncreased wh
he non-dime
sional stiffn
/R L ) are s
ne recorded
tended to
also exhib
-20o and 20
stiffness coef
adii ( /R L ): (
THE STIFF
tant design
plications.
sus the rela
radii /R L
L is kept
hen the /R
ensional sti
ness coeffic
shown in Fi
d for the dep
o decrease
it a minimu
0o(Lira et al.
fficient ( /K K(a) angle ( )
FNESS CO
n parameter
The carpet
ative density
. For incre
t constant, o
/ L value is
iffness coef
cient ( / eqK K
igure 13(b)
pendence ve
for increa
um for the
., 2009).
eqK ) vs. angle); (b) angle (
OEFFICIEN
r when se
t plot in F
y / c for
easing rela
on the oppo
s fixed. A s
fficient /K K
q ) versus th
. In this cas
ersus , with
asing R/L
in-plane Yo
e values for di
)
NT AND TH
electing co
igure 14(a)
r different w
ative densit
osite, the n
similar trend
eqK was var
he angle (
se, the beha
h a minimu
values. Pl
oungs mod
ifferent curva
THE RELAT
ore materia
) illustrates
wall length
ty / c , K
non-dimensi
d has also b
aried agains
) at
avior
um at
lanar
dulus
ature
TIVE
al in
s the
hs L
/ eqK K
ional
been
t the
relative density with different aspect ratios and different non-dimensional curvatures
(Figure 14(b)). For increasing / c , / eqK K decreases when is kept constant.
On the opposite, the non-dimensional stiffness did increase with the relative density
with a clear linear trend when /R L was kept constant.
Non-dimensional stiffness versus relative density and thickness ratios is shown
in Figure 14(c). For increasing / c , / eqK K values did exhibit a decrease when
was kept constant, while the opposite was true when /R L values were fixed. An
increase of the relative density by a factor of 2.3 led to a stiffness increase of 9.3
times for R/L=5.75. A different type of behavior has been observed in the carpet plot
illustrating the dependence of / eqK K versus / c for different depth ratios = /b L
and different /R L values (Figure 14(d)). For increasing / c values, / eqK K
showed a decrease when and /R L were kept constant. If / c was fixed, the
non-dimensional stiffness showed a significant increase for decreasing and /R L
values. As an example, when relative densities was 4.09 %, / eqK K increased by 42.3 %
for passing from 1.7 to 1.3, and for /R L passing from 7.75 to 5.75.
The non-dimensional stiffness tended to decrease for relative densities ranging
between 3.8% and 4.1%, and increasing internal cell angle values, while at
constant / c the stiffness showed an increase for lower /R L values (Figure 14(e)).
A different behavior has however been observed from the carpet plot representing
/ eqK K versus the relative density for different angles and different /R L values
(Figure 14(f)). Increases of the relative density showed very slight augmentation of
the non-dimensional stiffness when /R L was kept constant, on the opposite, / eqK K
tended to decrease significantly when was fixed. At constant values of / c ,
/K K
mag
Figu
/R Lt
REL
GEO
eqK tended
gnitudes for
ure 14 Non-d
L values: (athicknesses ra
LATION B
OMETRY PA
to provide
r higher a
dimensional s
a) and differenatios ; (d)
BETWEEN
PARAMETER
higher va
angles.
tiffness coeff
nt wall lengthand different
dif
THE EFF
RS
alues for d
ficient ( / eK Khs L ; (b) andt depth ratiosfferent angle
FECTIVE H
decreasing R
eq ) vs. relativ
d different as ; (e) and d
HEIGHT A
/R L , but f
ve density ( pect ratiosdifferent angl
AND THE
featured hi
/ c ) for diff ; (c) and diff
les ; (f) an
E UNIT C
gher
ferent
ferent nd
CELL
T
app
(Fig
curv
/R L
con
non
type
para
curv
non
sym
Hig
by a
sens
stru
Figu
The effectiv
lications, si
gure 4 (a)).
vature /R L
L values, t
stant, show
n-dimension
e of behav
ameterized
vature valu
n-dimension
mmetric dist
gh values of
a factor of
sitivity of
ucture.
ure 15 Relati
vs. /R L a
ve height H
ince it can b
The non-di
L is shown
the /eH R p
wing depend
nal height ve
vior was h
against th
ues /R L
nal height H
tribution ar
f /R L ten
4.7 when
the /eH R
on between t
at different L
eH is anoth
be used to d
imensional
in Figure 1
parameter t
dence close
ersus /R L
owever ob
he internal
(Figure 1
/eH R assum
ound =0o
nded to decr
/R L was
parameter
he effective h
L values; (b)
her parame
determine th
effective he
15(a) for dif
tended to lo
to ( / )R L
was found
bserved wh
cell angle
5(b)). Wh
med the sam
o in which
rease the no
increased f
against th
height and the
/eH R vs. a
ter to be c
he maximum
eight /eH R
fferent wall
ower its ma
1 . No signi
for differen
en the non
e and
hen /R L
me value at
/eH R assu
on-dimensio
from 5.75 to
he radius o
e unit cell geo
angle for
considered f
m compressi
R versus n
l lengths L .
agnitude wh
ficant sensi
nt values of
n-dimension
different n
was kept
=-20o and
umes its m
onal height
o 12.75, sh
f the curve
ometry param
different R
for enginee
ion deforma
non-dimensi
. For increa
hen L was
itivity abou
f L . A diffe
nal height
non-dimensi
constant,
d =20o , wi
maximum va
t at cons
howing a st
ved SILICO
meters: (a) eH/ L values
ering
ation
ional
asing
kept
t the
erent
was
ional
the
ith a
alue.
stant
rong
OMB
/e R
CONCLUSIONS
In this work, a novel curved cellular structure has been produced using Kirigami
techniques from PEEK films and investigated from a numerical and experimental
point of view. The focus of the analysis illustrated in this paper was the possible
relation between stiffness and geometry parameters of the curved cellular structure
and its unit representative cell. The fidelity of the FE full-scale models has been
validated by the experimental results. A parametric analysis has shown that it is
possible to obtain significant variations of the stiffness by intervening on the
geometry parameters of the curved sample. The analysis suggested also the possibility
of using curved SILICOMB configuration to design core structures with minimum
density and maximum stiffness coefficients. Experimental evidence also showed that
general strain dependence could be observed in the energy absorption capability and
loss factors of the curved samples. The large deformation capability with good shape
recovery shown by the curved PEEK SILICOMB samples coupled with the tailoring
of the mechanical properties suggest the potential use of these cellular structures at
zero Poissons ratio for shape morphing applications.
ACKNOWLEDGEMENTS
This research was performed under the European Commission project
NMP4-LA-2010-246067-, Acronym: M-RECT. The European Commission financial
support is strongly acknowledged. The authors would also like to thank the M-RECT
partners and the European Commission for making these investigations possible. The
Author Chen would also like to thank CSC (Chinese Scholarship Council) for funding
of his research work through the University of Bristol. Special thanks go also to the
University of Bristol and Harbin Institute of Technology.
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