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A simple auxetic tubular structure with tuneable mechanical properties
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2016 Smart Mater. Struct. 25 065012
(http://iopscience.iop.org/0964-1726/25/6/065012)
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A simple auxetic tubular structure with
tuneable mechanical properties
Xin Ren
1,2
, Jianhu Shen
1
, Arash Ghaedizadeh
1
, Hongqi Tian
2
and
Yi Min Xie
1,3
1
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476,
Melbourne 3001, Australia
2
Key Laboratory of Traffic Safety on Track, School of Traffic & Transportation Engineering, Central South
University, Changsha 410075, Hunan Province, People’s Republic of China
3
XIE Archi-Structure Design (Shanghai)Co., Ltd, 1436 Jungong Road, Yangpu District, Shanghai 200433,
People’s Republic of China
E-mail: mike.xie@rmit.edu.au
Received 5 January 2016, revised 31 March 2016
Accepted for publication 19 April 2016
Published 13 May 2016
Abstract
Auxetic materials and structures are increasingly used in various fields because of their unusual
properties. Auxetic tubular structures have been fabricated and studied due to their potential to
be adopted as oesophageal stents where only tensile auxetic performance is required. However,
studies on compressive mechanical properties of auxetic tubular structures are limited in the
current literature. In this paper, we developed a simple tubular structure which exhibits auxetic
behaviour in both compression and tension. This was achieved by extending a design concept
recently proposed by the authors for generating 3D metallic auxetic metamaterials. Both
compressive and tensile mechanical properties of the auxetic tubular structure were investigated.
It was found that the methodology for generating 3D auxetic metamaterials could be effectively
used to create auxetic tubular structures as well. By properly adjusting certain parameters, the
mechanical properties of the designed auxetic tubular structure could be easily tuned.
Keywords: auxetic, Poisson’s ratio, tubular structure, tube, metallic metamaterial, plastic
deformation, buckling
(Some figures may appear in colour only in the online journal)
1. Introduction
Materials and structures with negative Poisson’s ratio (NPR)
exhibit counter-intuitive behaviour, i.e. under uniaxial compres-
sion (tension), these materials and structures contract (expand)
transversely. They are also named as ‘auxetics’by Evans [1].
Because of the uncommon feature which is equipped by
auxetics, these auxetic materials and structures are superior to
conventional materials and structures in terms of indentation
resistance [2,3], shear resistance [4], synclastic behaviour [5],
enhanced resilience [5], energy absorption [6–9], fracture
toughness [10], vibration control [11]and negative com-
pliance [12–14]. Recently, Grima et al [15]found that a
regular conventional sheet of rubber-like material can be
converted to an auxetic metamaterial by using non-symmetric
quasi-random cuts.
As one branch of auxetics, auxetic tubular structure or
auxetic stent has attracted much research effort towards
exploring its applications as foldable devices in the medical
field, such as angioplasty stents [16–18], annuloplasty rings
[19]and oesophageal stents [20,21]. For the patterns of all
these medical devices, the cellular configuration is prede-
signed. The conventional straight forward method is to roll
the 2D auxetic sheets to tubes. However, the cellular tubes
designed in this method may not have auxetic behaviour
under large compressive strain. Besides, most of the existing
auxetic stents only demonstrate auxetic behaviour in tension.
To the best knowledge of authors, no systematic research has
been carried out on auxetic tubular structures under large
compressive strain which is required when stents scaffold are
inserted into the blood vessels. Gatt et al [22]proposed a
three-dimensional tubular system with a typical planar two-
Smart Materials and Structures
Smart Mater. Struct. 25 (2016)065012 (9pp)doi:10.1088/0964-1726/25/6/065012
0964-1726/16/065012+09$33.00 © 2016 IOP Publishing Ltd Printed in the UK1
dimensional system constructed from rotating rigid units.
They mentioned that the edge effect had a significant influ-
ence on the finite-sized 3D tubular structures.
Recently, Mohsenizadeh et al [8]concluded that auxetic
foam-filled square tube is superior to empty and conventional
foam-filled square tubes in terms of crashworthiness indicators
through experimental method, and Hou et al [9]obtained the
optimal Poisson’s ratio of the filled material for three foam-filled
tubes in terms of energy absorption though FE method. How-
ever, all the tubes they used were conventional one, invest-
igation on auxetic tubes are rare, particularly in compression.
Grima et al [23]investigated the effect of Poisson’s ratio and
Young’s modulus of 2D honeycomb structures on the formation
of the tubular structures and indicated that the semi re-entrant
honeycombs had a natural tendency to form cylindrical tubes.
Inspired by a planar auxetic metamaterial induced by
elastic instability [24–26], we selected an available void
fraction of 0.69 (larger than 0.34 in [24]), which demonstrated
buckling-induced auxetic behaviour, to generate a tubular
structure with normal circular holes. During FE simulations,
we found that when the base material of the designed tubular
structure is rubber, the tubular structure did show auxetic
behaviour, as expected. However, this auxetic behaviour
disappeared when the base material was replaced with brass.
In this paper we implement the latest methodology of
generating buckling-induced auxetic metamaterials proposed in
our previous paper [27], to generate a simple auxetic tubular
structure which could be easily tuned by one parameter named
as the pattern scale factor (PSF). An in-depth investigation on
the designed auxetic tubular structure is carried out both
experimentally and numerically. Although the focus point is on
the compressive auxetic properties of the designed structure,
tensile auxetic performance is also investigated. A series of
parametric studies on the designed auxetic tubular structure are
executed by using the experimentally validated FE models.
2. Designing auxetic tubular structures
Similar to our previous work [27], the methodology of gen-
erating auxetic tubular structures can be summarised as four
steps. Firstly, designing buckling-induced auxetic tubular
structure; secondly, carrying on buckling analysis of the
initial tubular structure with the linear elastic base material;
thirdly, identifying the desirable buckling mode; lastly,
altering the initial tubular structure using the desirable
buckling mode.
2.1. Designing buckling-induced auxetic tubular structure
The first step of the design framework is to generate a
buckling-induced auxetic tubular structure. Similar to the
geometry configuration of the Bertoldi’s work [24], a planar
sheet is shown in figure 1(a)with a void fraction of 0.69.
Under planar constraint, it demonstrates a NPR behaviour
induced by elastic instability. Using coordinate transforma-
tion method, the planar sheet can be transferred into a tubular
structure as shown in figure 1(b). It should be noted that the
shell model of the tubular structure is very similar to the
finite-sized 3D tubular structure designed by Gatt et al [22].
2.2. Buckling analysis of the initial tubular structure with linear
elastic base material
The second step is to perform buckling analysis to obtain
desirable buckling modes with auxetic performance under
uniaxial compression. The modulus of 110 GPa and the Pois-
son’s ratio of 0.38 were used in the simulation. Lanczos was
chosen as the eigensolver in buckling analysis. In this work, the
out-of-plane rotational degree of freedom on top and bottom
nodes of the FE model was constrained. The degree of freedom
of bottom nodes along compressive direction was also con-
strained. The nodal movement on the top nodes was allowed to
move in the loading direction. It should be noted that one of the
nodes in the bottom was fixed to avoid rigid rotation. Shell
elements were used for buckling analysis.
The commercial finite element software package ABA-
QUS (Simulia, Providence, RI)was implemented for running
buckling analysis. ABAQUS/Standard was adopted for linear
buckling analysis using a Lanczos eigensolver. The number
of eigenvalues requested was set as 10. The tubular structure
was built using shell elements (ABAQUS element type SA)
Figure 1. Geometries of the planar sheet with auxetic behaviour induced by elastic instability (L=147 mm, H=98 mm, void
fraction=0.69):(a)the planar sheet with periodic circular holes; (b)the corresponding tubular structure.
2
Smart Mater. Struct. 25 (2016)065012 X Ren et al
with a shell thickness of 6 mm. Symmetrical mesh seed was
distributed to the FE model which sustained the uniaxial
compressive force.
2.3. Identifying the desirable buckling mode
The desirable buckling mode was selected based on similar
patterns observed from previous research on planar auxetic
structures. To be more specific, the configuration of the
anticipated buckling modes should contain geometry similar
to the alternating ellipsoidal pattern in our previous and oth-
er’s work [24,26–33]. The results of the first two buckling
modes are presented in figure 2(eigenvalues of the 1st and
2nd modes are 8.731 43×10
−4
and 1.186 56×10
−4
respectively). Apparently, the first buckling mode meets the
requirement of selecting desirable modes, while the second
buckling mode is unqualified because the holes are irregular.
2.4. Quantifying the desirable buckling mode and form the
auxetic tubular structure
The method of quantifying the desirable buckling mode is
similar to our previous work [27]. The PSF was employed to
quantify the adjustment on the initial tubular structure using the
desirable buckling mode. In the present work, when the edge of
the elliptical void is just closed, as shown in figure 3,the
corresponding deformation scale factor (DSF)(0.014 65 in this
case)is defined as PSF=100%. Other values of PSF are
defined accordingly, such as 50% with a DSF of 0.007 325, and
the 0% is the initial tubular structure without any adjustment.
Because the result of the first buckling mode obtained
from buckling analysis was uniform. Unlike our previous
work [27]where a representative volume element (RVE)was
employed, the adjustment was directly applied to the whole
configuration of the initial tubular structure in this study.
3. Experiment
3.1. Fabrication of metallic tubular structure for experiments
The specimens of the tubular structure in figure 4were fabricated
using 3D printing (Shapeways, New York)technique with raw
brass as their base material. The specific manufacturing proce-
dure of the metallic tubular structure was same as our previous
work [27]. The material properties of the printed raw brass
material were measured through standard tensile tests which
were completed in the previous work [27].(PSF=0%: overall
mass=164.10 g, relative density=841.2 kg m
−3
,masserror=
9.0%, wall thickness=4.04 mm; PSF=20%: overall mass=
165.56 g, relative density=848.8 kg m
−3
,masserror=12.3%,
wall thickness=4.03 mm).
Figure 2. The first two buckling modes of the initial tubular
structure: (a)configuration of the first buckling mode; (b)
configuration of the second buckling mode.
Figure 3. The geometries of the first buckling mode at different values of PSF from 0% to 100%.
3
Smart Mater. Struct. 25 (2016)065012 X Ren et al
3.2. Uniaxial compression tests on tubular structures
The mechanical performance of the printed tubular structures
was tested using standard quasistatic uniaxial compression
tests, and the strain rate of 10
−3
s
−1
was employed using a
Shimazu machine. A camera was used to record the defor-
mation procedure to measure the evolution of the Poisson’s
ratio of the tubular structures.
As can be seen in figure 4, the centres of two rotation part
of six layers were marked with small points. The experimental
value of Poisson’s ratio was calculated using image proces-
sing method from six layers of the tubular structures, as
shown in figure 5. The equation of calculating Poisson’s ratio
of one layer of the tubular structure is shown in formula (1)
and the equation of calculating overall Poisson’s ratio of the
tubular structure is shown in formula (2)
e
e
=- =-D
D
vdD
zZ i25, 1
i
xy
zii
() ()
å
=´vv i
1
425, 2
i
2
5
¯() ()
where Δd=d−D,Z
i
=Z
i−1
−Z
i+1
,Δz=z
i−1
−z
i+1
.D
is the diameter of the tube before deformation, dis the real-
time diameter of the tube during deformation, Z
i
is the height
of the ith layer before deformation, z
i
is the height of the ith
layer during deformation.
3.3. The comparison of auxetic behaviour of the two tubular
structures from experiments
According to the Bertoldi’sfinding regarding NPR behaviour
induced by elastic instability [18], here we utilised a similar
geometry mentioned in their work to generate a tubular
structure as shown in figure 4(a). In our previous study [27],
we found that the loss of auxetic behaviour in metallic auxetic
metamaterials. To verify the loss of auxetic behaviour will
also occur towards tubular structures, here we made a com-
parison experimentally.
The experimental deformation processes for two tubular
metallic samples of PSF=0% and PSF=20% are illu-
strated in figure 6. The initial metallic tubular structure with
PSF=0% is non-auxetic, and the altered metallic tubular
structure with PSF=20% illustrates auxetic behaviour. This
result further verified that the phenomenon of the loss of
auxetic behaviour not only occurs for 3D metamaterials, but
also exists for tubular structures. In addition, using the latest
Figure 4. Two test samples for the tubular structure (scale bar:
10 mm):(a)initial tubular structure (PSF=0%);(b)altered tubular
structure (PSF=20%).
Figure 5. Calculating Poisson’s ratio using an imaging processing method from two perspective views: (a)top view; (b)front view.
4
Smart Mater. Struct. 25 (2016)065012 X Ren et al
methodology of generating 3D metallic metamaterials, the
loss of auxetic behaviour towards tubular structure could be
easily obtained again.
According to the method of calculating Poisson’s ratio
described in figure 5and formulas (1)and (2), the value of
Poisson’s ratio for the tubular structure with PSF=0% could
not be calculated properly because the marked points in the
same layer were not in same horizontal level. The ideal
auxetic deformation pattern is that the diameter of the tubular
structures in different height could change evenly under
uniaxial deformation. However, from the deformation pattern
shown in 6(a), we can define the metallic tubular structure
with PSF=0% is non-auxetic. The experimental values of
Poisson’s ratio as a function of displacement for the specimen
with PSF=20% are shown in figure 7.
So far, the experimental result has confirmed that the
methodology of generating 3D metallic metamaterials can be
extended to design auxetic tubular structure as well. To the best
knowledge of authors, no one has conducted studies on auxetic
tubular structures about their compressive mechanical perfor-
mance. To illustrate the nonlinear effect on metallic auxetic
tubular structures, numerical investigation both on compressive
and tensile auxetic performance was executed on our designed
tubular structures. The influence of PSF and metal plasticity on
their mechanical properties was explored.
4. Finite element analysis
4.1. FE model for metallic tubular structure
The geometric configuration of the FE model is shown in
figure 5. ABAQUS/Explicit solver was employed in post-
buckling analysis for considering large deformation and self-
contacts [34]. Although the shell elements were used in the
stage of generating auxetic tubular structures, for obtaining
more accurate result in FE simulations, solid elements
(ABAQUS element type C3D8)were adopted for large
deformation analysis. Because the base material of the
experimental specimens was same as that of the printed
models in our previous work [27], the same bilinear elastic-
plastic material model with a Young’s modulus of 87 GPa and
a hardening modulus of 1.7 GPa were used in FE simulations.
Through comparing the FE models and the printed tub-
ular models, we found that the printed tubular models were
lighter than our initial design. The relative mass errors were
Figure 6. Experiments on two metallic tubular structures (scale bar: 10 mm):(a)global buckling of the metallic sample with PSF=0%; (b)
auxetic behaviour of the metallic sample with PSF=20%.
Figure 7. Experimental results of Poisson’s ratio as a function of
displacement for the model with PSF=20%.
5
Smart Mater. Struct. 25 (2016)065012 X Ren et al
9.0% and 12.3% for the two printed samples with PSF of 0%
and 20%, respectively. We found that the wall thickness of
the printed models was smaller than our initial design.
Adjustment of the FE models was made by decreasing the
thickness of the tubular structures when it is compared with
experimental results.
4.2. FE model validation
The mesh dependence analysis was performed, which is
similar to the work conducted by Pozniak et al [35]. A mesh
size with four layers of elements for the minimal link of the
FE models was adopted. The FE model was validated by
comparing the deformation process and force–displacement
curves from the simulation with that from experiments. The
deformation process of the tubular structure with PSF=20%
was shown in figure 8, which is nearly identical to the
deformation process from experiments shown in figure 6(b).
The FE model was further validated by comparing the
force–displacement curves as shown in figure 9. Although the
peak force of the experimental result is higher than that of FE
result, the overall trends of these two curves are similar.
Through checking the geometries of the printed sample after
the test, we found some broken parts on it, as shown in the
green dashed circles. However, the failure criterion was not
defined in the FE simulations. Therefore, we attributed the
difference between these two curves to the imperfection of the
printed specimen and the fracture of the minimal links in the
experiment, which are difficult to simulate using FE models.
4.3. Comparison of rubber and brass tubular structures
According to the previous findings of our work [27], the
initial auxetic behaviour of the buckling-induced metamater-
ial disappeared when the base material of rubber was replaced
by brass. Based on the work of Bertoldi et al [24], we selected
the similar geometry with a void fraction of 0.69, to generate
a tubular structure which should possess the auxetic beha-
viour induced by elastic instability. To verify the effect of
base material on the tubular structure with PSF=0%, we
conducted a comparison using FE method by changing mat-
erial model in ABAQUS setting. The linear elastic material
model for rubber with a Yong’s modulus of 1 MPa was
chosen in FE simulations. The deformation patterns for the
same geometry with two different material models were
shown in figure 10.
The FE results illustrate that the base material has a
significant effect on the auxetic performance of the buckling-
induced tubular structure. When the base material is replaced
from rubber to brass, the initial auxetic behaviour of buckling-
induced tubular structure will disappear. The finding is similar
to the finding we have observed for the test of 3D auxetic
metamaterials.
4.4. Effect of PSF on auxetic behaviour
The magnitude of the PSF not only determines the geometric
configuration of the designed tubular structure but also affects
auxetic performance of the metallic tubular structure. The
auxetic performance for different models with various values
of PSF was investigated using the validated FE models.
The variation of the auxetic performance on the values of
PSF is shown in figure 11. We can see clearly that the auxetic
performance of the tubular structure can be adjusted by
changing the values of PSF. By increasing the value of PSF,
the effective strain range for the tubular structure under ten-
sion could be enlarged, while the effective strain range under
Figure 8. Deformation process of the FE model with PSF=20%.
Figure 9. Comparison of the force–displacement curves of the
tubular structure with PSF=20%, between experiment and FE
simulation.
6
Smart Mater. Struct. 25 (2016)065012 X Ren et al
compression will become smaller. Therefore, to obtain an
auxetic tubular structure which has a similar effective auxetic
strain both in compression and tension, a proper value of PSF
should be chosen. As can be seen in figures 11(a)and (b),
when the tubular structure has a PSF of 60%, the trends of the
curves of Poisson’s ratio-displacement and force–displace-
ment are nearly the same.
4.5. Effect of plastic strain hardening on auxetic performance
Because of the high ductility of raw brass (with an elongation
up to 0.3), it was chosen as the base material of the printed
tubular structures. Strain hardening, as a fundamental feature
of metal plasticity, has a significant effect on the mechanical
performance of cellular structures and materials. The effect of
plastic strain hardening on auxetic performance and load-
bearing capability of the designed tubular structures were
investigated by using the validated FE models. A bilinear
elastic-plastic material model was employed in FE models.
The FE results of the parametric study for strain hardening are
shown in figure 12, where Es is the elastic modulus and Ep is
the strain hardening modulus.
We can see that the differences in Poisson’s ratio for FE
models with different ratio of Ep/Es from 0.2 to 1.0 are very
small, especially when the ratio of Ep/Es is over 0.2. The
magnitude of the force at the point of densification strain
increases significantly from around 1 to 17 KN when the ratio
of Ep/Es enhances from 0 to 1.
Figure 10. Illustration of different deformation patterns of the tubular structure with PSF=0% with different base materials: (a)undeformed
shape; (b)FE model using elasto-plastic material model with isotropic linear strain hardening, for brass; (c)FE model using a linear elastic
material model for rubber.
Figure 11. Effect of PSF on mechanical performances both in compression and tension: (a)curves of Poisson’s ratio as a function of
displacement for metallic tubular structures with different PSF, (b)curves of force as a function of displacement for metallic tubular
structures with different PSF.
7
Smart Mater. Struct. 25 (2016)065012 X Ren et al
5. Concluding remarks
In this study, the latest methodology for generating 3D
auxetic metamaterials was successfully extended to the
development of metallic auxetic tubular structures. The per-
formance of a simple metallic auxetic tubular structure under
both compressive and tensile loading was investigated
experimentally and numerically. The effects of the PSF and
the plastic strain hardening on the auxetic performance and
other mechanical properties were examined using the vali-
dated FE models. From the obtained results, the following
conclusions could be drawn:
(a)A simple auxetic tubular structure has been designed,
fabricated and tested, which exhibits auxetic behaviour
in both compression and tension.
(b)The buckling-induced auxetic tubular structure would
lose its auxetic behaviour when the base material is
changed from an elastomer to a ductile metal.
(c)The latest methodology for generating 3D metallic
auxetic metamaterials has been further developed to
create a simple auxetic tubular structure, and the
effectiveness of the design approach has been validated
experimentally and numerically.
(d)The mechanical properties of the proposed tubular
structure can be tuned by adjusting the magnitude of the
PSF. When the PSF is set at a certain value (∼60% in
this study), it is possible to achieve a similar auxetic
performance under compression and tension.
The most significant feature of our designed auxetic
tubular structure is its tunability by simple control parameters,
i.e. the PSF and the plastic strain hardening ratio of base
material. The mechanical properties of the tubular structure
could be easily adjusted by these two parameters individually.
Most of the existing auxetic stents only exhibit auxetic
behaviour under tension. The designed simple tubular struc-
ture has auxetic behaviour in both compression and tension.
Therefore, our designed metallic tubular structure not only
has potential to be used in the medical field but could also be
employed in other structures (e.g. in armoured vehicles to
absorb impact loading). The same design approach could be
extended to the development of new composite materials and
structures with auxetic behaviour. Also, it will be a fasci-
nating future work to investigate the mechanical performance
of auxetic tubular structures composed of 2D auxetic meta-
material proposed by Grima et al [15].
Acknowledgments
This work was supported by the Australian Research
Council (DP140100213, DP160101400), the China Scholar-
ship Council (201306370057), the Major Program of the
National Natural Science Foundation of China (U1334208).
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