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A simple auxetic tubular structure with tuneable mechanical properties



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).
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Supplementary resource (1)

... As an alternative to the 2D beam/shell-dominated auxetic microstructures, auxetic microstructure generated by arraying periodic orthogonal elongated holes (typically with an elliptical shape) onto a sheet structure was proved to be beneficial to achieve targeted auxetic performance [54,[56][57][58][59]. The auxeticity of this kind of structure is attributed to the local buckling instability in the connected region among the orthogonal elongated holes. ...
... By introducing imperfection to alter the initial geometry with a simple building cell, Shen et al. [64] posted a design exhibiting initial auxeticity. To quantitatively determine the magnitude of the buckling mode to alter the initial geometry, Ren and his coworkers [65] suggested a pattern scale factor (PSF) method and proposed several 3D bulk [58,65] and 2D tubular [17,57] auxetic meta-materials. In compression, the elliptical holes are transformed into peanut-shaped holes. ...
... The dual gradient in unit-cell geometry is key to the auxeticity found in these two natural cellular materials [11]. Moreover, 3D auxetic tubular structures have been generated by adjusting the unit-cell geometry [13], whose auxeticity is induced by the elastic instability of the cell ligament under compression. Thus, we predict that the combined pore size and wall thickness gradients in cellular structure can be used to trigger the elastic instability-induced auxetic behavior, and these auxetic cellular materials with dual-gradient structures provide reference to designing auxetic metamaterials. ...
... Further, the proposed auxetic mechanism based on the dual-gradient architecture can be extended to cellular structures with common convex polygons or patterns, where the constituent unit cells are not restricted to a specific unit geometry, such as a re-entrant unit, making these auxetic metamaterial fabrication processes relatively simple and easier to scale up. Additionally, due to the synergy of pore size gradient and wall thickness gradient, the buckling behavior in dual-gradient auxetic metamaterials occurs gradually with respect to the dimension rather than rapidly buckling through the entire structures in the early stage of deformation, as previously reported in uniform or single gradient structures [7,8,13], thus facilitating the maintenance of integral strength. The generality of this proposed design strategy can be tested by applying it to a class of 3D printed gradient cellular structures with different unit topologies and verifying the existence of auxeticity in several other plant parenchyma tissues with dual-gradient porosity. ...
... Ren et al. developed metallic Table 1 The value of the geometric ratios. auxetic tubular structures based on the latest methodology for generating 3D auxetic metamaterials [42]. The tubular structure exhibited auxetic behavior in compression and tension, making it an excellent stent candidate. ...
... Auxetic tubes that had the auxetic cell space in the walls removed were also researched, in addition to the non-auxetic tube filled with auxetic foam. Due to their potential to adapt as stents, Ren et al. [27] created an auxetic tube that displays a negative Poisson's ratio and auxetic behavior in both compression and tension. Analytical and computational techniques were used by Scarpa et al. [28] to investigate the mechanical characteristics of tubular constructions made up of centrally symmetric cells. ...
... A novel auxetic composite was generated by filling re-entrant honeycombs with foam, and the NPR-enhanced effect of the auxetic structure on energy absorption was investigated [21]. The mechanical behavior of an auxetic tubular structure was investigated in both compression and tension, and its properties were found to be adjustable by varying the magnitude of the pattern scale factor [22]. Auxetic chiral lattice composites were also discovered to possess superior stiffness and energy absorption capability [23]. ...
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The human head is most vulnerable to injury during activities such as road traffic and sports. To mitigate the risk of traumatic brain injury (TBI), helmets serve as an important protective device. This study proposes a hedgehog biomimetic helmet with auxetic lattice liners in the shape of a hemisphere. The helmeted head impact configuration is built based on a high bio-fidelity head-neck finite element model incorporated into our novel helmet model. Biomechanical responses including acceleration, intracranial pressure, and von Mises strain of head are extracted from the simulation model to assess TBI risks. The results indicate that the helmet featuring auxetic lattice liners outperforms those without liners or with other liner designs, offering superior protection. Compared to the threshold, the novel helmet design was found to reduce the head injury criterion value by 72.65%. Additionally, parametric studies of lattice’s bar radius for uniform and graded auxetic lattice liners are discussed. Finally, this study also carries out the optimization design of lattice strut radius and height, resulting in a lightweight auxetic lattice liner with superior protective performance. The outcomes of this study extend the application of auxetic materials and provide guidance for designing helmet liners that better mitigate TBI.
... The auxetic tube formed after the coordinate transformation of the perforated plate structure further expands the application range of the structure with the auxetic effect. On the other hand, Ren et al. [40] proposed for the first time a tubular structure that exhibits auxetic effects in both tension and compression states. Han et al. [41] designed a square-section tube in the form of a perforated plate. ...
... Their study shows that if the structure's geometric parameters are appropriately selected, the mechanical properties can be significantly improved. Hassanin et al. [16] investigated the penetration resistance of auxetic structures made of shape memory materials. ...
Humans have always sought the optimal use of materials around them and, in this field, inspired by nature, have succeeded in inventing various structures. As one example, lattice structures, which are lightweight, strong, and stiff, are used widely in various applications, including energy absorbers. Lattice structures with a negative Poisson's ratio have been developed as a new type of lattice structure. As a result of this feature, auxetic structures have unique properties like shear strength, penetration resistance, fracture toughness, crack resistance, and high energy absorbability. In this paper, the mechanical behavior of the auxetic panels made using the 3D metal printer method is investigated by experimental tests and finite element methods. Experiments are used to verify the accuracy of the numerical model. Using the DMLS method, samples were prepared from metal-based AlS10Mg Aluminum composition. The 3D printing method was used to fabricate samples. Afterwards, experimental tests were made and the mechanical properties of these materials were determined by tensile test and used in finite element simulations. Following the confirmation of the model's accuracy, the finite element simulation results are used to perform a parametric study and determine the appropriate geometry. The numerical analysis is conducted using ABAQUS software, which uses the nonlinear finite element method.
... Tabacu and Stanescu [33] examined auxetic, anti-tetra-chiral structures designed as tubes subjected to tensile quasi-static loads, establishing theoretical calculation models to estimate the reaction force under tensile loads. Ren et al. [34] developed a simple tubular structure exhibiting auxetic behavior in both compression and tension by extending a recently proposed design concept for 3D metallic auxetic metamaterials. Zhang et al. [35] developed a combined tubular structure with tunable stiffness, improving bearing capacity and stability through the length design of the central column. ...
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Tests were carried out to develop and manufacture various types of auxetic dowels using3D printing technology. These dowels were then used to connect L-type corner joint specimens forcase furniture, and their strength and stiffness were analyzed through experimental, theoretical, andnumerical means. In the scope of the study, eight different types of auxetic dowels including twoinclusion types, two inclusion sizes, and two dowel hole diameters, as well as a reference non-auxeticdowel, were designed. Accordingly, a total of 180 specimens that included 10 replications for eachgroup were tested; 90 were tested under tension and the remaining 90 were tested under compression.The results demonstrated that the assembly force required for the corner joints connected with auxeticdowels was significantly lower compared to non-auxetic dowels. Furthermore, the numerical andtheoretical analyses yielded similar outcomes in this study. Both analyses revealed that the dowelsused to connect the corner joints experienced substantial stresses during mounting and bending,ultimately leading to their failure. Upon concluding the test results, it was observed that the cornerjoints connected with dowels featuring rectangular inclusions exhibited superior performance whencompared to those with triangular inclusions. In light of these findings, it can be concluded thatfurther enhancements are necessary for auxetic dowels with rectangular inclusions before they canbe utilized as alternative fasteners for traditional dowels.
... Analyzing Table 2, it is possible to conclude that the finite element methodology has been used by many authors all over the world, where the method's efficiency can be applied in the design of many different structures and perform many analyses such as energy absorption, kinking response, compression, deformation, and impact load. [20] Cylindrical Double V Compression ABAQUS Ren et al. [78] Circular Compression ABAQUS Yang et al. [79] Dimpled Uniaxial Compress ABAQUS Wu et al. [52] Hierarchical anti-tetrachiral In-plane ABAQUS Ruan et al. [47] Antichiral-Reentrant Compression -Wu et al. [53] Anti-tetrachiral In-plane ABAQUS Ren et al. [60] Performed Deformation ABAQUS Lei et al. [80] Digitally Reprogrammable Deformation ABAQUS Geng et al. [81] Chiral Mechanics Properties ABAQUS Farrell et al. [9] Deformed cell inspired Twist-deformation ABAQUS Gao et al. [82] Double Arrow Impact Loading LS-Ddyna Jiang et al. [83] Lattice Stress-Strain ABAQUS Hamzehei et al. [74] Anti-trichiral Compression ABAQUS Nejad et al. [84] Cellular-Reentrant Compression ABAQUS Zhang et al. [56] Peanut Shape Compression ANSYS Zhang et al. [85] Elliptical hole Compression ABAQUS Tabacu et al. [86] Anti-tetra chiral Reaction force LD-Dyna Zhang et al. [57] Elliptical hole Compression ABAQUS Gao et al. [55] Cylindrical Double V Energy absorption LS-Dyna Jiang et al. [87] Lattice Compression ABAQUS Zhang et al. [58] Asymmetrical re-entrant Compression ABAQUS Doudaran et al. [63] Anti-tetrachiral Energy absortion ABAQUS Doudaran et al. [63] Double-V Energy absortion ABAQUS Doudaran et al. [63] Re-entrant Energy absortion ABAQUS Han et al. [61] Elliptical hole Compression ABAQUS Ren et al. [33] Elliptical hole Energy absortion ABAQUS Novac et al. [88] Axisymmetric Chiral Compression Ls-dyna Solak et al. [89] Peanut Shape Mechanics Properties Workbench Zhang et al. [90] Anti-tetra chiral Effective Poisson's Ratio ABAQUS Effective Elastic Modulus ...
Auxetic materials and structures have been attracting attention due to their extraordinary mechanical properties that stand out due to their high capacity to absorb energy. Some types of auxetic tubular structures have been studied and designed in diverse engineering fields such as mechanical, aerospace and medical. This manuscript cites more than a hundred papers containing the definitions, designs, structural analyses and optimization, mechanical properties, and specific applications of tubular design of auxetic structures. It can be noted from the present paper that additive manufacturing has been one of the most common manufacturing techniques used in many cases to manufacture tubular structures, and numerical analysis was essential to analyzing the behavior of the structures. The purpose of this paper is to assist researchers and engineers in using the methodology step by step to develop and apply auxetic tubular structures.
Porous sandwich structure (foam) is a prerequisite in the field of construction, aerospace, and automotive sector, due to its energy-absorbing characteristics and sound-absorbing properties. The porous structure was previously difficult to manufacture, but is now feasible, thanks to additive manufacturing process or 3D printing technology. Complex shapes and structure can be produced by the novel production technique such as fused deposition modeling (FDM) known as three-dimensional printing. These studies emphasized the importance of porous structure, particularly foams, for the construction industry, automobile sector and aerospace industries for its high performance, light-weightiness, shock absorbing properties, etc. In this article, we convey the different processes as well as their main features which can be used to fabricate the foam structure having various topologies such as open cells and closed cells, with different fundamental designs and materials such as polymer, ceramic, and metals, with a particular emphasis on metallic and polymeric materials. In this regard, a comparison between the traditional foam manufacturing techniques and additively 3D printed foam structure was made, to study the various complexities associated with such method of fabrication and attempts to integrate such study have been divided into four categories: (1) architected porous structures, (2) conventional and 3D methods of polymeric and metallic foam construction, (3) foam lattice design and characterization, and lastly (4) foam properties and applications.
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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.