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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.
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A simple auxetic tubular structure with tuneable mechanical properties
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2016 Smart Mater. Struct. 25 065012
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A simple auxetic tubular structure with
tuneable mechanical properties
Xin Ren
, Jianhu Shen
, Arash Ghaedizadeh
, Hongqi Tian
Yi Min Xie
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476,
Melbourne 3001, Australia
Key Laboratory of Trafc Safety on Track, School of Trafc & Transportation Engineering, Central South
University, Changsha 410075, Hunan Province, Peoples Republic of China
XIE Archi-Structure Design (Shanghai)Co., Ltd, 1436 Jungong Road, Yangpu District, Shanghai 200433,
Peoples Republic of China
Received 5 January 2016, revised 31 March 2016
Accepted for publication 19 April 2016
Published 13 May 2016
Auxetic materials and structures are increasingly used in various elds 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, Poissons ratio, tubular structure, tube, metallic metamaterial, plastic
deformation, buckling
(Some gures may appear in colour only in the online journal)
1. Introduction
Materials and structures with negative Poissons 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 auxeticsby 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 [69], fracture
toughness [10], vibration control [11]and negative com-
pliance [1214]. 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
eld, such as angioplasty stents [1618], annuloplasty rings
[19]and oesophageal stents [20,21]. For the patterns of all
these medical devices, the cellular conguration 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 signicant inu-
ence on the nite-sized 3D tubular structures.
Recently, Mohsenizadeh et al [8]concluded that auxetic
foam-lled square tube is superior to empty and conventional
foam-lled square tubes in terms of crashworthiness indicators
through experimental method, and Hou et al [9]obtained the
optimal Poissons ratio of the lled material for three foam-lled
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 Poissons ratio and
Youngs 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 [2426], 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 rst step of the design framework is to generate a
buckling-induced auxetic tubular structure. Similar to the
geometry conguration of the Bertoldis work [24], a planar
sheet is shown in gure 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 gure 1(b). It should be noted that the
shell model of the tubular structure is very similar to the
nite-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-
sons 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 xed to avoid rigid rotation. Shell
elements were used for buckling analysis.
The commercial nite 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.
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 specic, the conguration of the
anticipated buckling modes should contain geometry similar
to the alternating ellipsoidal pattern in our previous and oth-
ers work [24,2633]. The results of the rst two buckling
modes are presented in gure 2(eigenvalues of the 1st and
2nd modes are 8.731 43×10
and 1.186 56×10
respectively). Apparently, the rst buckling mode meets the
requirement of selecting desirable modes, while the second
buckling mode is unqualied 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 gure 3,the
corresponding deformation scale factor (DSF)(0.014 65 in this
case)is dened as PSF=100%. Other values of PSF are
dened 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 rst 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
conguration 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 gure 4were fabricated
using 3D printing (Shapeways, New York)technique with raw
brass as their base material. The specic 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
9.0%, wall thickness=4.04 mm; PSF=20%: overall mass=
165.56 g, relative density=848.8 kg m
wall thickness=4.03 mm).
Figure 2. The rst two buckling modes of the initial tubular
structure: (a)conguration of the rst buckling mode; (b)
conguration of the second buckling mode.
Figure 3. The geometries of the rst buckling mode at different values of PSF from 0% to 100%.
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
was employed using a
Shimazu machine. A camera was used to record the defor-
mation procedure to measure the evolution of the Poissons
ratio of the tubular structures.
As can be seen in gure 4, the centres of two rotation part
of six layers were marked with small points. The experimental
value of Poissons ratio was calculated using image proces-
sing method from six layers of the tubular structures, as
shown in gure 5. The equation of calculating Poissons ratio
of one layer of the tubular structure is shown in formula (1)
and the equation of calculating overall Poissons ratio of the
tubular structure is shown in formula (2)
=- =-D
zZ i25, 1
() ()
vv i
425, 2
¯() ()
where Δd=dD,Z
is the diameter of the tube before deformation, dis the real-
time diameter of the tube during deformation, Z
is the height
of the ith layer before deformation, z
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 Bertoldisnding 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 gure 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 gure 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 veried 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 Poissons ratio using an imaging processing method from two perspective views: (a)top view; (b)front view.
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 Poissons ratio
described in gure 5and formulas (1)and (2), the value of
Poissons 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 dene the metallic tubular structure
with PSF=0% is non-auxetic. The experimental values of
Poissons ratio as a function of displacement for the specimen
with PSF=20% are shown in gure 7.
So far, the experimental result has conrmed 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 inuence 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 conguration of the FE model is shown in
gure 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 Youngs 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 Poissons ratio as a function of
displacement for the model with PSF=20%.
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 forcedisplacement
curves from the simulation with that from experiments. The
deformation process of the tubular structure with PSF=20%
was shown in gure 8, which is nearly identical to the
deformation process from experiments shown in gure 6(b).
The FE model was further validated by comparing the
forcedisplacement curves as shown in gure 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
dened 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 difcult to simulate using FE models.
4.3. Comparison of rubber and brass tubular structures
According to the previous ndings 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 Yongs modulus of 1 MPa was
chosen in FE simulations. The deformation patterns for the
same geometry with two different material models were
shown in gure 10.
The FE results illustrate that the base material has a
signicant 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 nding is similar
to the nding we have observed for the test of 3D auxetic
4.4. Effect of PSF on auxetic behaviour
The magnitude of the PSF not only determines the geometric
conguration 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 gure 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 forcedisplacement curves of the
tubular structure with PSF=20%, between experiment and FE
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 gures 11(a)and (b),
when the tubular structure has a PSF of 60%, the trends of the
curves of Poissons ratio-displacement and forcedisplace-
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 signicant 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 gure 12, where Es is the elastic modulus and Ep is
the strain hardening modulus.
We can see that the differences in Poissons 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 densication strain
increases signicantly 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 Poissons 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.
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 signicant 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 eld 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].
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).
[1]Evans K E, Nkansah M A, Hutchinson I J and Rogers S C 1991
Molecular network design Nature 353 124
[2]Argatov I I, Guinovart-Díaz R and Sabina F J 2012 On local
indentation and impact compliance of isotropic auxetic
materials from the continuum mechanics viewpoint Int. J.
Eng. Sci. 54 4257
[3]Coenen V L and Alderson K L 2011 Mechanisms of failure in
the static indentation resistance of auxetic carbon bre
laminates Phys. Status Solidi b248 6672
[4]Choi J B and Lakes R S 1992 Nonlinear properties of polymer
cellular materials with a negative Poissons ratio J. Mater.
Sci. 27 467884
[5]Lakes R S 1987 Foam structures with a negative Poissons
ratio Science 235 103840
[6]Evans K E 1991 Auxetic polymers: a new range of materials
Endeavour 15 1704
[7]Alderson A and Alderson K L 2007 Auxetic materials Proc.
Inst. Mech. Eng. G221 56575
[8]Mohsenizadeh S, Alipour R, Shokri Rad M,
Farokhi Nejad A and Ahmad Z 2015 Crashworthiness
assessment of auxetic foam-lled tube under quasi-static
axial loading Mater. Des. 88 25868
Figure 12. Effect of plastic strain hardening on mechanical properties of designed tubular structures: (a)Poissons ratio as a function of
displacement for models with PSF=20% and base material with different values of Ep/Es; (b)force as a function of displacement for
models with PSF=20% and base material with different values of Ep/Es.
Smart Mater. Struct. 25 (2016)065012 X Ren et al
[9]Hou S, Liu T, Zhang Z, Han X and Li Q 2015 How does negative
Poissons ratio of foam ller affect crashworthiness? Mater.
Des. 82 24759
[10]Choi J B and Lakes R S 1996 Fracture toughness of re-entrant
foam materials with a negative Poissons ratio: experiment
and analysis Int. J. Fract. 80 7383
[11]Chen C P and Lakes R S 1996 Micromechanical analysis of
dynamic behavior of conventional and negative Poissons
ratio foams J. Eng. Mater. Technol. 118 2858
[12]Grima J N, Caruana-Gauci R, Wojciechowski K W and
Evans K E 2013 Smart hexagonal truss systems exhibiting
negative compressibility through constrained angle
stretching Smart Mater. Struct. 22 084015
[13]Pozniak A A, Kaminski H, Kedziora P, Maruszewski B,
Strek T and Wojciechowski K W 2010 Anomalous
deformation of constrained auxetic square Rev. Adv. Mater.
Sci. 23 16974
[14]Xie Y M, Yang X, Shen J, Yan X, Ghaedizadeh A, Rong J,
Huang X and Zhou S 2014 Designing orthotropic materials
for negative or zero compressibility Int. J. Solids Struct. 51
[15]Grima J N, Mizzi L, Azzopardi K M and Gatt R 2016 Auxetic
perforated mechanical metamaterials with randomly oriented
cuts Adv. Mater. 28 3859
[16]Kuribayashi K, Tsuchiya K, You Z, Tomus D, Umemoto M,
Ito T and Sasaki M 2006 Self-deployable origami stent
grafts as a biomedical application of Ni-rich TiNi shape
memory alloy foil Mater. Sci. Eng. A419 1317
[17]Kuribayashi K and You Z 2006 Deployable stent US Patent
[18]Ley T J, Kveen G L, Ehr T G J, Brown B J and Friesen D L
2002 Stent congurations US Patent 6416538B1
[19]Burriesci G and Bergamasco G 2011 Annuloplasty prosthesis
with an auxetic structure US Patent 8034103B2
[20]Ali M N and Rehman I U 2011 An Auxetic structure
congured as oesophageal stent with potential to be used for
palliative treatment of oesophageal cancer; development and
in vitro mechanical analysis J. Mater. Sci., Mater. Med. 22
[21]Ali M, Buseld J C and Rehman I 2014 Auxetic oesophageal
stents: structure and mechanical properties J. Mater. Sci.,
Mater. Med. 25 52753
[22]Gatt R, Caruana-Gauci R, Attard D, Casha A R, Wolak W,
Dudek K, Mizzi L and Grima J N 2014 On the properties of
real nite-sized planar and tubular stent-like auxetic
structures Phys. Status Solidi b251 3217
[23]Grima J N, Oliveri L, Attard D, Ellul B, Gatt R, Cicala G and
Recca G 2010 Hexagonal honeycombs with zero Poissons
ratios and enhanced stiffness Adv. Eng. Mater. 12 85562
[24]Bertoldi K, Reis P M, Willshaw S and Mullin T 2010 Negative
Poissons ratio behavior induced by an elastic instability
Adv. Mater. 22 3616
[25]Bertoldi K, Boyce M C, Deschanel S, Prange S M and
Mullin T 2008 Mechanics of deformation-triggered pattern
transformations and superelastic behavior in periodic
elastomeric structures J. Mech. Phys. Solids 56 264268
[26]Mullin T, Willshaw S and Box F 2013 Pattern switching
in soft cellular solids under compression Soft Matter 9
[27]Ren X, Shen J, Ghaedizadeh A, Tian H Q and Xie Y M 2015
Experiments and parametric studies on 3D metallic auxetic
metamaterials with tuneable mechanical properties Smart
Mater. Struct. 24 095016
[28]Taylor M, Francesconi L, Gerendás M, Shanian A,
Carson C and Bertoldi K 2014 Low porosity metallic
periodic structures with negative Poissons ratio Adv. Mater.
26 236570
[29]Overvelde J T B and Bertoldi K 2014 Relating pore shape to
the nonlinear response of periodic elastomeric structures
J. Mech. Phys. Solids 64 35166
[30]Overvelde J T B, Shan S and Bertoldi K 2012 Compaction
through buckling in 2D periodic, soft and porous structures:
effect of pore shape Adv. Mater. 24 233742
[31]Wang P, Shim J and Bertoldi K 2013 Effects of geometric and
material nonlinearities on tunable band gaps and low-
frequency directionality of phononic crystals Phys. Rev. B
88 014304
[32]Willshaw S and Mullin T 2012 Pattern switching in two and
three-dimensional soft solids Soft Matter 8174750
[33]Shen J, Zhou S, Huang X and Xie Y M 2014 Simple cubic
three-dimensional auxetic metamaterials Phys. Status Solidi
b251 151522
[34]Hanssen A G, Hopperstad O S, Langseth M and Ilstad H 2002
Validation of constitutive models applicable to aluminium
foams Int. J. Mech. Sci. 44 359406
[35]Pozniak A A and Wojciechowski K W 2014 Poissons ratio of
rectangular anti-chiral structures with size dispersion of
circular nodes Phys. Status Solidi b251 36774
Smart Mater. Struct. 25 (2016)065012 X Ren et al

Supplementary resource (1)

... Therefore, preliminary applications of auxetic structures have extended to many fields, including medical equipment, intelligent materials, civil engineering [27][28][29]. So far, various auxetic structures have been systematically investigated, e.g., re-entrant structures [30], rigid rotation structures [31,32], chiral structures [33,34], bucklinginduced structures [35][36][37][38], etc. In particular, as a special part of auxetic structures, auxetic tubular structures require further research with the intention of capitalizing on the excellent auxetic behaviour for multi-functional applications [39][40][41]. ...
... Bhullar et al. [47] adopted rotating rigid square units to design and manufacture auxetic metal and polymer stents, using advanced manufacturing technology. Ren et al. [36] designed buckling-induced metallic tubular structures whose mechanical properties can be tuned by adjusting the value of the pattern scale factor. Limited investigations listed above showed that auxetic tubular structures are able to produce obvious radial shrinkage under axial compressive load, which is beneficial to improve mechanical properties. ...
... Auxetic tubular structures are designed through the coordinate transformation approach that convolves 2D auxetic structures made of periodically arranged elliptic unit cells into 3D auxetic tubes, and extending along with radius directions, as shown in Fig. 1. Based on our previous works of auxetic tubes [36], three groups of SSTs were designed and the geometrical configurations of them are shown in Fig. 2. The three groups of SSTs included an auxetic SST (ASST) with alternatingly ellipse holes, a non-auxetic SST (NSST) with circular holes and a traditional nonauxetic SST (TSST) without holes. As shown in Fig. 3(a), the main parameters of the unit cell contain the vertical and horizontal length L, the thickness t of the ribs, the long-axis a and the shortaxis b of the ellipse holes. ...
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Concrete-filled stainless steel tube (CFSST) members take advantage of the high strength and the outstanding corrosion resistance to act as an important role in civil engineering structures. However, the steel tube could not provide the perfect confinement effect for the core concrete during the initial elastic compression stage because Poisson’s ratio of the concrete is smaller than that of the stainless steel tube (SST). In this paper, a novel concrete-filled auxetic stainless steel tube (CFASST) composite structure was designed and manufactured to actively restrain the concrete, making the best use of the desirable deformation characteristics of auxetic tubular structures. The axial compressive performance of these CFASST members and their control factors were investigated experimentally and numerically. Test results were discussed in detail which included failure modes, load versus displacement curves and strain analysis. Finally, parametric analyses were conducted to further study the effects of different parameters (Poisson’s ratio, thickness of the stainless tube) on the CFSST composite structure under axial compression. It was found that CFASST composite structures possess an unusual deformation mode and an improved confinement effect.
... Auxetic foam materials exhibit an opposite deformation behavior to conventional materials under axial load. Specifically, auxetic foam materials contract (expand) laterally under uniaxial compression (tension) [1][2][3]. Along with the uncommon behavior, auxetic foam material has wide application in many fields [4][5][6][7] due to its superior shear resistance [8][9][10], indentation resistance [11], fracture resistance [12], energy absorption [13][14][15] and vibration isolation [16]. ...
... Dhanaseker et al. [32] simulated the mechanical properties of auxetic layers embedded masonry by two numerical methods. In another work of Dhanaseker et al. 3 [33], the mechanical properties of masonry walls under different impact velocities were analyzed which indicated that the auxetic fiber layer could significantly reduce the in-plane stress of masonry walls. ...
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Auxetic foam materials contract (expand) laterally under uniaxial compressive (tensile) load. Due to superior characteristics of auxetic foam, e.g., shear resistance and in-plane indentation resistance, studies of auxetic foam composites have been increasing in recent years. In this paper, a novel cement-based auxetic foam composite is designed, fabricated and experimentally investigated. The influence of foam hole density, mass fraction and age on the flexural and compressive strength of the composite is analyzed. The failure modes and crack development of the specimen are examined. It is found that the flexural and compressive strength of composite are improved at the curing age of 7 and 14 days, and reduced at the curing age of 28 days with the incorporation of auxetic foam. And the flexural compression ratio of the composite is greater than that of matrix material. The integrity of the specimen is preserved during the compression failure process of cement-based auxetic foam composites. It is indicated that the incorporation of auxetic foam improves the toughness and deformation behavior of composites.
... Auxetic structures have demonstrated good potentials to be used as auxetic nails [52], tubular [53] and energy absorption structures [54,12,55,56,57]. Moreover, recent studies show that the auxetic structures can play an important role for electrical-skin (e-skin) applications due to their extraordinary performance in controlling the electronic responses and conforming to human skins [58,59,60]. ...
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Taking advantage of the powerful design potentials of isogeometric analysis, an integrated shape and size optimization framework for designing tetra chiral and anti-chiral auxetics is proposed to be capable of obtaining excellent designs without complicated implementation efforts. The framework utilizes a non-uniform rational basis spline (NURBS) based parametrization method that describes the chiral and anti-chiral structures with a small number of size and shape parameters. With this effective framework, systematic design studies considering both plane strain and stress conditions are performed to provide bounding graphs for the best achievable auxeticities under different stiffness requirements. Designs with tunable effective properties are also provided to demonstrate the capabilities of the proposed framework. The potential for electronic-skin applications is illustrated.
... Tubular, pipelike structures are among the mechanical metamaterials particularly sought in engineering applications. For this reason, numerous proposals can be found in the literature, one group of which concerns medical applications [35][36][37][38][39][40][41]. ...
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Auxetic tubular structures are widely known structures, characterized by a negative Poisson’s ratio upon stretching and deformation in the axial and transverse directions, which have numerous application possibilities. In this paper, tubular structures were realized by rolling up planar auxetic structures and using rigid square frames as unit cells. Planar and tubular structures were built from square frames that were 3D printed with plastic or laser-cut from metal. The changes in linear dimensions of the studied structures were based on a hinge mechanism, the functioning of which was experimentally verified on different solutions leading to square unit cells. To connect the square frames of the structure, an innovative solution was used in the form of rotation axes on their surface at a preset distance from the edge of the square frame. The geometric parameter thus introduced was used to determine the relative change in the size of the structure when stretched (i.e., when moving from the closed to the open position).
... After five years, the artificial auxetic material was firstly fabricated using foams composed of concave ligaments in 1987 [5]. After that, an increasing number of auxetics were reported, i.e., re-entrant hexagon [6][7][8], arrow structure [9,10], perforated plate [11][12][13] and chiral structures [14,15]. Auxetics was endowed with many superior performances due to the special deformation including dent resistance [16], synclastic surface [17] and energy absorption performance [18,19]. ...
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Auxetic metamaterials tend to be fabricated using additive manufacturing and laser cutting due to their special porous microstructures. Although the auxetic metamaterials have many applications currently, the high cost and the low efficiency of available manufacturing are adverse to expend their application range. In this work, the auxetic chiral honeycomb is assembled crosswise using the slotted wave plate. Effects of wave radius, plate thickness, slot percentage, and base material on the Poisson’s ratio and mechanical performance are explored experimentally and numerically. The results show that assembled auxetic chiral honeycomb (AACH) exhibits lower peak force and high plateau stress than the conventional assembled one. With the increase of wave radius and plate thickness, the energy absorption (EA) and specific energy absorption (SEA) would increase. As for the different material combinations, when the base materials in vertical and horizontal wave plates adopt stainless steel and aluminum, respectively, the AACH would exhibit desirable EA, SEA, and auxetic behavior. These findings provide a new approach to the manufacture of auxetics at a low cost, which is beneficial for potential applications.
... Some of these studies, particularly ones on negative Poisson's ratios, examine auxetics through model structures and macromodels [8][9][10], an approach which was found to be highly useful in elucidating the properties of such unusual systems. Significant advances on the subject of negative compressibility have also been made [7,17,[46][47][48][49][50][51][52][53], as well as on potential applications (e.g., smart textiles [54] biomedical devices [55,56], etc.). ...
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Boron arsenate, BAsO4, is a β-cristobalite-like crystal which has been reported to exhibit the rather unusual property of negative linear compressibility behaviour at elevated pressures, that is expanding rather than shrinking in a linear dimension when subjected to pressure. This work proposes a ‘geometry—deformation mechanism’-based mathematical model to aid the discernment of the manner how this anomalous pressure behaviour is achieved. The model makes use of data obtained from DFT simulations over an extended range of pressures, including extreme pressure conditions, and rigorously explains the macroscopic properties of this material in terms of the nanoscale deformations. More specifically, through this model, it was possible to decipher the different contributions to the deformation mechanism and compressibility properties of BAsO4. Moreover, for the first time, it was shown that a rule related to the sum of angles of tetrahedrally coordinated atoms is so robust that it applies at the extreme pressures studied here.
... A majority of the deformations are localized at the hinges, resulting in contrasting behavior of the material from its constituents. Ren et al. [33] designed and fabricated a simple 3D auxetic structure with tuneable mechanical properties exhibiting symmetric negative Poisson's ratio under both compression and tensile loading. However, the structure proposed by them experiences stress concentrations at the ends of the voids at relatively low loading conditions leading to low load bearing capacity. ...
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Mechanical metamaterials are generally two-dimensional periodic structures or three-dimensional cellular structures which exhibit mechanical properties beyond the ordinary. Due to size and boundary effects, three-dimensional mechanical metamaterials typically display anisotropic changes even if they are isotropic in construction and composition. However, in this study, the comprehensive design and fabrication of a three-dimensional axisymmetric auxetic structure that exhibits uniform and axisymmetric transverse deformation under longitudinal compression loading are proposed. Extending the concept of two-dimensional periodically perforated auxetic sheet structures to the third dimension, the design of the metamaterial was generated by revolving a two-dimensional parabolic curve along the axis of rotation and subsequently perforating the structure periodically with elliptical voids varying in size longitudinally along the curvature of the structure in order to promote the exhibition of isotropic negative Poisson’s ratio. Furthermore, this study elucidates the significance of the perforations by comparing the metamaterial structure to a so-called plain structure.
... As 2D metamaterials cannot meet all needs, three-dimensional (3D) metamaterials with better performance are of interest [34][35][36][37][38][39][40][41]. For example, NPR tubular structures [17,42,43], tension-torsion coupling structures [44,45], double-arrow energy-absorbing structures [46,47], torsional structures [44,48,49], and 3D hexagonal reentrant structures [50][51][52][53] stand out. Of the 3D re-entrant structures, Li et al. [54] proposed a new 3D NPR concave lattice based on a 2D NPR structure summarized by predecessors. ...
Full-text available
Mechanical metamaterials are of interest to researchers because of their unique mechanical properties, including a negative Poisson structure. Here, we study a three-dimensional (3D) negative-Poisson-ratio (NPR) metal metamaterial lattice structure by adding a star structure to the traditional 3D concave structure, thus designing three different angles with a modified NPR structure and control structure. We further study the mechanical properties via finite element numerical simulations and show that the stability and stiffness of the modified structures are improved relative to the control structure; the stability decreases with increasing star body angle. The star angle has the best relative energy absorption effect at 70.9°. The experimental model is made by selective laser melting (SLM) technology (3D printing), and the compression experiment verification used an MTS universal compressor. The experimental results are consistent with the changing trend in finite element simulation.
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In the past decades, auxetic metamaterials have attracted extensive attention due to their desirable properties. In order to further improve the energy absorption properties of auxetic metamaterials, this work investigates the design of lightweight auxetic metamaterials based on the elliptic perforated plate. The material in the large deformation region is retained and the material in the small deformation region is removed as much as possible. The experimental and numerical results indicate that the mechanical properties of the novel metamaterials are almost unchanged, but the specific energy absorption is substantially enhanced owing to the reduced material overall and the distribution of the optimized materials to the most needed area. In this paper, the influence of the amount and mode of the material removal on the mechanical properties of the auxetic metamaterial is explored through the numerical method. The proposed auxetic metamaterials with lightweight characteristics have potential applications in many fields, e.g., civil engineering, aerospace and vehicle engineering.
A comprehensive investigation on the extremely large post-buckling deformation of perforated cylindrical shells is conducted using experiments and verified with an analytical shell model and nonlinear finite element simulations. A “waisted” post-buckling configuration, which is characterized by uniform shrinking in the middle section of the perforated cylindrical shell, is identified. The waisted behavior is attributed to the triggering of a pattern transformation under compressive load that shows special hyperelastic metamaterial characteristics. The load-carrying capacity of waisted post-buckling suffers a sudden drop and then recovers when the holes are completely collapsed and closed. Plenty of design parameters can be utilized to enrich variations of the waisted post-buckling responses. The negative Poisson's ratio induced by pattern transformation plays a key role in forming the waisted post-buckling modes. This special hyperelastic metamaterial behavior can be easily achieved by fixing the boundaries and adjusting the geometric parameters of the shell. By comparing the characteristics of a porous cylindrical shell with those appeared for an equivalent porous panel, it is highlighted that pattern transformation can occur in a thinner porous cylindrical shell without lateral support. The waisted post-buckling modes of a perforated cylindrical shell are stable, and the shell is invulnerable to the progressively increasing applied loads. In comparison, an ordinary cylindrical shell may snap from one mode to another in the post-buckling process. Moreover, we find that some non-closed cylindrical panels can also buckle into the waisted-like modes. These findings can be applied to the construction of functional devices for soft robotics, actuators, and structural protection for facilities, etc.
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By introducing a periodic array of pores in an elastic matrix, instabilities with wavelengths that are of the order of the size of the microstructure can be triggered. Interestingly, these instabilities can be utilized to design a novel class of responsive materials. Possible applications include materials with unusual properties such as negative Poisson's ratio, phononic and photonic switches and colorful and reconfigurable displays. Although shape plays an important role in the design and performance of periodic materials, so far only the non-linear response of structures with circular and elliptical pores has been investigated and the effect of the pore shape on the structural response has not yet been explored. Here, we numerically explore the effect of pore shape on the non-linear response of a square array of pores in an elastomeric matrix. Our results show that pore shape can be used effectively to design material with desired properties and to control attractive features of soft porous systems, such as their stiffness, critical strain and negative Poisson's ratio.
Perforated systems with quasi-disordered arrays of slits are found to exhibit auxetic characteristics almost as much as their traditional ordered "rotating-squares" counterparts. This provides a highly robust methodology for constructing auxetics that may be used for various practical applications such as skin grafting, where a high degree of precision may not always be achievable.
Auxetic metamaterials are synthetic materials with microstructures engineered to achieve negative Poisson's ratios. Auxetic metamaterials are of great interest because of their unusual properties and various potential applications. However, most of the previous research has been focused on auxetic behaviour of elastomers under elastic deformation. Inspired by our recent finding of the loss of auxetic behaviour in metallic auxetic metamaterials, a systematic experimental and numerical investigation has been carried out to explore the mechanism behind this phenomenon. Using an improved methodology of generating buckling-induced auxetic metamaterials, several samples of metallic auxetic metamaterials have been fabricated using a 3D printing technique. The experiments on those samples have revealed the special features of auxetic behaviour for metallic auxetic metamaterials and proved the effectiveness of our structural modification. Parametric studies have been performed through experimentally validated finite element models to explore the auxetic performance of the designed metallic metamaterials. It is found that the auxetic performance can be tuned by the geometry of microstructures, and the strength and stiffness can be tuned by the plasticity of the base material while maintaining the auxetic performance.
As an effective candidate for enhancing energy absorption, a range of foam materials have gained considerable popularity, in which the density, Young's modulus and plasticity of foam materials are considered critical to crashworthiness. Relatively speaking, less attention has been paid to the roles played by the Poisson's ratio of foam or cellular materials. More importantly, the interaction between different Poisson's ratios and thin-walled structures has been a critical yet under-studied issue. This paper aims to explore the effects of negative, zero and positive Poisson's ratio of auxetic foams, ranging from À1 to 0.5, on structural crashworthiness and seek optimal design for different foam-filled square, circular and conic tubes. In this study the specific energy absorption (SEA) and mean crushing force (MCF) are taken as the objective functions by using mathematical regression analysis. The sequential quadratic programming (SQP) and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) are employed for single and multiobjective design of foam-filled tubes with different Poisson's ratios, respectively. The optimal Poisson's ratio is obtained for these three different types of foam-filled tubes. By comparison we found that the crashworthiness of foam filled conic tube is the best, followed by circular and then squared tubes. The study provides new insights into material selection and design with a more favorable Poisson's ratio for crashworthiness.
There has been considerable interest in materials exhibiting negative or zero compressibility. Such materials are desirable for various applications. A number of models or mechanisms have been proposed to characterize the unusual phenomena of negative linear compressibility (NLC) and negative area compressibility (NAC) in natural or synthetic systems. In this paper we propose a general design technique for finding metamaterials with negative or zero compressibility by using a topology optimization approach. Based on the bi-directional evolutionary structural optimization (BESO) method, we establish a systematic computational procedure and present a series of designs of orthotropic materials with various magnitudes of negative compressibility, or with zero compressibility, in one or two directions. A physical prototype of one of such metamaterials is fabricated using a 3D printer and tested in the laboratory under either unidirectional loading or triaxial compression. The experimental results compare well with the numerical predictions. This research has demonstrated the feasibility of designing and fabricating metamaterials with negative or zero compressibility and paved the way towards their practical applications.
Elastic instability of soft cellular solids plays an increasingly important role in the creation of metamaterials with smart properties. Inspiration for much of this research comes from a planar metamaterial with negative Poisson's ratio behavior induced by elastic instability. Here we extend the concept of buckling induced pattern switch further to the design of a new series of three-dimensional metamaterials with negative Poisson's ratio over a large strain range. The highlight of this work is that our designs are based on very simple initial geometric shapes.Different deformation patterns of materials without and with auxetic behavior.
Auxetics, i.e. systems with a negative Poisson’s ratio, exhibit the unexpected property of becoming wider when stretched and narrower when compressed. This property arises from the manner in which the internal geometric units within the system deform when the system is submitted to a stress and may be explained in terms of ‘geometry–deformation mechanism’ based models. This work considers realistic finite implementa- tions of the well known rotating squares system in the form of (i) a finite planar structure and (ii) a tubular conformation, as one typically finds in stents. It shows that although the existing models of the Poisson’s ratios and moduli based on periodic systems may be appropriate to model systems where the geometry/deformation mechanism operate at the micro- or nano- (molecular) level where a system may be considered as a quasi infinite system, corrections to the model may need to be made when one considers finite structures with a small number of repeat units and suggests that for finite systems, especially for the 2D systems, the moduli as predicted by the periodic model may be significantly overestimating the moduli of the real system, even sometimes by as much as 200%. ?
Using Finite Element computer simulations, Poisson's ratio (PR) is determined for anti-chiral structures built on rectangular lattices with disorder introduced by stochastic distributions of circular node sizes. The investigated models are parameterized by the lattice anisotropy, the rib thickness, and the radii distribution of circular nodes. Three approaches are developed. The first approach, exact in the limit of infinitely large system and infinitely dense mesh, uses only planar elements (CPS3). Two other approaches are approximate and exploit one-dimensional elements utilizing the Timoshenko beam theory. It is shown that in the case of sufficiently large anisotropy of the studied structures PR can be highly negative, reaching any negative value, including those lower than inline image. Thin ribs and thin-walled circular nodes favor low values of PR. In the case of thick ribs and thick-walled circular nodes PR is higher. In both cases the dispersion of the values of circular nodes radii has a minor effect on the lowest values of PR. A comparison of the results obtained with three different approaches shows that the Timoshenko beam based approximations are valid only in the thin rib limit. The difference between them grows with increasing thickness.