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: f.scarpa@bristol.ac.uk (F. Scarpa)

Tel.:+86(0) 451-86402328; fax: +86(0) 451-86402328.

Email address: lengjs@hit.edu.cn (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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s.

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parameter

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the accordi

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ellular confi

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MODEL

rs of the

hin the black

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figuration to

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and depth ra

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ellular struc

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ILICOMB

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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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ween the node

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ection) (Fig

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ar region i

he stiffness

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ICOMB conf

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gure 4). Th

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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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metry (a) and

MANUFACT

facturing

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mbination

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le 1 Geometrometry l length L /(ml length H /(m

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eparated rec

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ctangle and

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perature (23

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ILICOMB sa

mm) )

ell /(Degre

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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

003),

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.

REFERENCES European Project M-RECT. http://www.mrect.risk-technologies.com. Alderson, A., Alderson, K., Attard, D., Evans, K., Gatt, R., Grima, J., Miller, W., Ravirala, N.,

Smith, C., Zied, K., 2010a. Elastic constants of 3-, 4-and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading. Composites Science and Technology, 70: 1042-1048.

Alderson, A., Alderson, K., Chirima, G., Ravirala, N., Zied, K., 2010b. The in-plane linear elastic constants and out-of-plane bending of 3-coordinated ligament and cylinder-ligament honeycombs. Composites Science and Technology, 70: 1034-1041.

Bettini, P., Airoldi, A., Sala, G., Landro, L.D., Ruzzene, M., Spadoni, A., 2010. Composite chiral structures for morphing airfoils: Numerical analyses and development of a manufacturing process. Composites Part B: Engineering, 41: 133-147.

Bezazi, A., Scarpa, F., Remillat, C., 2005. A novel centresymmetric honeycomb composite structure. Composite Structures, 71: 356-364.

Bitzer, T., 1997. Honeycomb technology: materials, design, manufacturing, applications and testing. Chapman & Hall, London.

Bornengo, D., Scarpa, F., Remillat, C., 2005. Evaluation of hexagonal chiral structure for morphing airfoil concept. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 219: 185-192.

Bubert, E.A., Woods, B.K.S., Lee, K., Kothera, C.S., Wereley, N.M., 2010. Design and fabrication of a passive 1D morphing aircraft skin. Journal of intelligent material systems and structures, 21: 1699-1717.

Engelmayr, G.C., Cheng, M., Bettinger, C.J., Borenstein, J.T., Langer, R., Freed, L.E., 2008. Accordion-like honeycombs for tissue engineering of cardiac anisotropy. Nature materials, 7: 1003-1010.

Evans, K., 1991. The design of doubly curved sandwich panels with honeycomb cores. Composite Structures, 17: 95-111.

Evans, K., Alderson, K., 2000a. Auxetic materials: the positive side of being negative. Engineering Science and Education Journal, 9: 148-154.

Evans, K.E., Alderson, A., 2000b. Auxetic materials: Functional materials and structures from lateral thinking! Advanced materials, 12: 617-628.

Gibson, L.J., Ashby, M.F., 1997. Cellular solids: structure and properties, Second ed. Cambridge University Press.

Grima, J.N., Oliveri, L., Attard, D., Ellul, B., Gatt, R., Cicala, G., Recca, G., 2010. Hexagonal Honeycombs with Zero Poisson's Ratios and Enhanced Stiffness. Advanced Engineering Materials, 12: 855-862.

Hassan, M., Scarpa, F., Ruzzene, M., Mohammed, N., 2008. Smart shape memory alloy chiral honeycomb. Materials Science and Engineering: A, 481: 654-657.

Jeong, S.M., Ruzzene, M., 2004. Analysis of vibration and wave propagation in cylindrical grid-like structures. Shock and Vibration, 11: 311-332.

Lakes, R., 1987. Foam structures with a negative Poisson's ratio. Science, 235: 1038-1040. Lira, C., Scarpa, F., Olszewska, M., Celuch, M., 2009. The SILICOMB cellular structure:

Mechanical and dielectric properties. physica status solidi (b), 246: 2055-2062.

Lira, C., Scarpa, F., Tai, Y., Yates, J., 2011. Transverse shear modulus of SILICOMB cellular structures. Composites Science and Technology, 71: 1236-1241.

Martin, J., HeyderBruckner, J.J., Remillat, C., Scarpa, F., Potter, K., Ruzzene, M., 2008. The hexachiral prismatic wingbox concept. physica status solidi (b), 245: 570-577.

Masters, I., Evans, K., 1996. Models for the elastic deformation of honeycombs. Composite Structures, 35: 403-422.

Miller, W., Smith, C., Scarpa, F., Evans, K., 2010. Flatwise buckling optimization of hexachiral and tetrachiral honeycombs. Composites Science and Technology, 70: 1049-1056.

Milton, G.W., 1992. Composite materials with Poisson's ratios close to1. Journal of the Mechanics and Physics of Solids, 40: 1105-1137.

Olympio, K.R., Gandhi, F., 2010a. Flexible skins for morphing aircraft using cellular honeycomb cores. Journal of intelligent material systems and structures, 21: 1719-1735.

Olympio, K.R., Gandhi, F., 2010b. Zero Poissons ratio cellular honeycombs for flex skins undergoing one-dimensional morphing. Journal of intelligent material systems and structures, 21: 1737-1753.

Olympio, K.R., Gandhi, F., 2007. Zero-v cellular honeycomb flexible skins for one-dimensional wing morphing, Proc. of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii

Saito, K., Agnese, F., Scarpa, F., 2011. A Cellular Kirigami Morphing Wingbox Concept. Journal of intelligent material systems and structures, 22: 935-944.

Scarpa, F., Smith, C., Ruzzene, M., Wadee, M., 2008. Mechanical properties of auxetic tubular trusslike structures. physica status solidi (b), 245: 584-590.

Scarpa, F., Smith, F., Chambers, B., Burriesci, G., 2003. Mechanical and electromagnetic behaviour of auxetic honeycomb structures. Aeronautical Journal, 107: 175-183.

Scarpa, F., Tomlin, P., 2000. On the transverse shear modulus of negative Poissons ratio honeycomb structures. Fatigue & Fracture of Engineering Materials & Structures, 23: 717-720.

Spadoni, A., Ruzzene, M., Scarpa, F., 2006. Dynamic response of chiral truss-core assemblies. Journal of intelligent material systems and structures, 17: 941-952.