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A finite element analysis of a 3D auxetic textile structure for composite reinforcement
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2013 Smart Mater. Struct. 22 084005
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IOP PUBLISHING SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 22 (2013) 084005 (8pp) doi:10.1088/0964-1726/22/8/084005
A finite element analysis of a 3D auxetic
textile structure for composite
reinforcement
Zhaoyang Ge1,2, Hong Hu2and Yanping Liu2
1College of Textiles, Donghua University, Shanghai, People’s Republic of China
2Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon,
Hong Kong
E-mail: tchuhong@polyu.edu.hk
Received 25 January 2013, in final form 29 April 2013
Published 22 July 2013
Online at stacks.iop.org/SMS/22/084005
Abstract
This paper reports the finite element analysis of an innovative 3D auxetic textile structure consisting of three
yarn systems (weft, warp and stitch yarns). Different from conventional 3D textile structures, the proposed
structure exhibits an auxetic behaviour under compression and can be used as a reinforcement to manufacture
auxetic composites. The geometry of the structure is first described. Then a 3D finite element model is
established using ANSYS software and validated by the experimental results. The deformation process of the
structure at different compression strains is demonstrated, and the validated finite element model is finally used
to simulate the auxetic behaviour of the structure with different structural parameters and yarn properties. The
results show that the auxetic behaviour of the proposed structure increases with increasing compression strain,
and all the structural parameters and yarn properties have significant effects on the auxetic behaviour of the
structure. It is expected that the study could provide a better understanding of 3D auxetic textile structures and
could promote their application in auxetic composites.
(Some figures may appear in colour only in the online journal)
Nomenclature
2l1Spacing of two adjacent warp yarns in the same
layer
l2Spacing of two neighbouring weft yarns
r1Initial radius of the warp yarn
r2Initial radius of the weft yarn
θInclination angles of the straight line of the weft
yarn
vPoisson’s ratio of the 3D structure
E11 Longitudinal Young’s modulus of yarns
E22,E33 Transverse Young’s moduli of yarns
G23 Transverse shear modulus of yarns
G12,G13 Longitudinal shear moduli of yarns
v12,v13 ,v23 Poisson’s ratios of yarns in the longitudinal and
transverse directions
k1Ratio of r1and r2
k2Ratio of l1and r2
k3Ratio of l2and r2
k4Ratio of the longitudinal Young’s moduli of the
warp and weft yarns
1. Introduction
Auxetic materials are those having negative Poisson’s
ratio (NPR) [1,2]. Unlike conventional materials, an
auxetic material may contract when compressed along
a perpendicular direction, which results in a unique
feature that the material can concentrate itself under
the compressive load to better resist the load [3,4].
This special feature, in combination with other improved
properties, such as enhanced fracture toughness and shear
modulus, has made auxetic materials very attractive for many
potential applications, such as automotive [5], aerospace and
defence [6], sports equipment, etc.
Among many man-made auxetic materials, auxetic
composites occupy an important place. According to Alderson
10964-1726/13/084005+08$33.00 c
2013 IOP Publishing Ltd Printed in the UK & the USA
Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
et al [7], two approaches could be used to fabricate
auxetic composites. The first one is to fabricate laminated
composites by using suitable stacking sequences based
on non-auxetic constituent materials (reinforcement and
matrix) [8], because the auxetic behaviour in a laminated
composite can be achieved through the shear–extension or
extension–shear coupling behaviour of individual lamina
(single layer unidirectional fibre reinforced composite). By
selecting a suitable orientation of the reinforced fibres [9],
auxetic effects could be obtained in in-plane [10,11] or
out-of-plane [12,13] in the composites. The study in [8] also
showed that highly anisotropic behaviour of the individual
lamina material is one of the requirements for obtaining
auxetic behaviour in a laminated composite. This makes
carbon fibre/epoxy resin a more suitable choice than either
Kevlar/epoxy resin or glass fibre/epoxy resin. The second
approach is to include auxetic elements in a composite, or
to use auxetic constituents, for example, an auxetic matrix,
auxetic reinforcement or both [8]. Wei et al theoretically
demonstrated that auxeticity of composites could be obtained
when the auxetic inclusion volume fraction exceeds a critical
value and the ratio of Young’s modulus between the inclusion
and matrix is within a definite interval [14]. Hou et al
presented a numerical analysis of a composite structure
with in-plane isotropic negative Poisson’s ratio by randomly
including re-entrant equilateral triangles [15]. The concept
they proposed could be extended to 3D composite structures
with isotropic negative Poisson’s ratio. Wang et al showed
that composites with inclusions of negative bulk modulus
could lead to a negative Poisson’s ratio and an extreme
damping, which meant that damping tended to infinity and
the composite was globally stable without constraints [16].
It was also reported that honeycomb structures with negative
Poisson’s ratio [17] and auxetic fibre networks [18] could be
used as embedding structures for making auxetic composite
structural materials. In recent years, the application of auxetic
fibres and yarns in fabricating composites has attracted a great
deal of attention [19–21]. Alderson et al [20] showed that
the use of auxetic fibres as reinforcement could considerably
increase the interface strength between the matrix and fibre
because the diameter of the auxetic fibres becomes bigger
when it is pulled out from the matrix.
3D textile structures including woven, knitted and braided
structures have been widely used as composite reinforce-
ments [22]. Compared to conventional 2D laminated compos-
ites, 3D textile structural composites have superior mechan-
ical performance, such as improved structural integrity, en-
hanced fracture toughness and through-the-thickness strength
against delamination. They have been widely used in aircraft,
marine craft, automotives, civil infrastructure and medical
prostheses [23] and so on. However, most of the 3D textile
structures for composite reinforcement found in the literature
are non-auxetic structures. Although a 2D plain woven fabric
made of auxetic helical yarns was used to manufacture
auxetic composites [21], 3D auxetic textile structures made of
non-auxetic yarns for composite reinforcements are still very
limited.
This paper reports a finite element (FE) analysis of a
novel 3D auxetic textile structure especially developed for
Figure 1. 3D auxetic textile structure: (a) unloaded state; (b) under
compression.
composite reinforcement. Both the 2D geometrical model
and 3D FE model of the structure under compression in the
through-the-thickness direction are presented and compared
with experimental results. The effects of structural parameters
and yarn properties on the auxetic behaviour of the structure
are predicted based on the validated FE model. It is expected
that the study could provide a better understanding of 3D
auxetic textile structures and promote their application in
auxetic composites.
2. 3D auxetic textile structure and its geometry
As shown in figure 1, the 3D auxetic textile structure
developed for composite reinforcement consists of three yarn
systems, i.e., weft yarns, warp yarns and stitch yarns. It was
fabricated on a prototype which had been specially developed
at the Institute of Textiles and Clothing of The Hong Kong
Polytechnic University [24], as shown in figure 2. From
figure 1(a), it can be seen that both the weft and warp yarns
are straight at the initial state. They are alternately placed
along two orthogonal directions of the fabric plane, and are
bound together by the stitch yarns through the fabric thickness
direction. The stitch yarns used were elastic yarns to provide
a better stability of the structure. Since the warp yarns in
each layer are arranged one in and one out, void spaces are
created among the warp yarns. In addition, the positions of the
warp yarns in two adjacent layers are shifted by a half-yarn
spacing. As shown in figure 1(b), this arrangement of the
warp yarns can result in the contraction of the structure in
the weft direction (xdirection) due to crimping of the weft
yarns when compressed in the fabric thickness direction (y
direction). Different from the warp yarns, the weft yarns
are fully arranged. This arrangement makes the warp yarns
still keep a straight state under compression. As a result,
the extension of the fabric structure in the warp direction
(zdirection) is negligible. Therefore, the structure will have
a negative Poisson’s ratio in the weft direction and a zero
Poisson’s ratio in the warp direction under compression.
The cross-section of the structure is shown in figure 3,
in which the auxetic behaviour of the structure in the weft
direction under compression can be better demonstrated. As
shown in figure 3(a), a unit-cell of the structure is outlined
by broken lines. It should be pointed out that the void
spaces of the warp yarns may result in the reduction of the
stability of the structure. Since the structure is developed
for composite reinforcement, the compressible matrix will
be filled in the void spaces of the structure during the
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Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
Figure 2. Prototype for fabricating the 3D auxetic textile structure.
Figure 3. Cross-section of the 3D auxetic structure: (a) unloaded
state; (b) under compression.
composite manufacturing process. Therefore, the stability of
the composite structure can be assured.
According to the previous 2D geometrical model [25],
as shown in figure 4, the Poisson’s ratio of the structure in
the weft direction is given by equation (1), based on the
assumptions in which both the radii and lengths of the weft
and warp yarns are kept constant during the compression of
the structure in the thickness direction.
ν= − 2(r1+r2)(sin θ−θcos θ) +l1(cos θ−1)
2(r1+r2)(cos θ+θsin θ−1)−l1sin θ
×2(r1+r2)
l1
.(1)
3. Finite element analysis
3.1. 3D FE model
The major limitation of the geometrical model is that the
mechanical properties of yarns could not be included in
the analysis. Therefore, it could not provide an accurate
prediction of the auxetic behaviour of the structure. In order
to get more precise prediction of the auxetic behaviour of the
structure under compression, a 3D finite element analysis was
conducted in this work.
According to the observation from the fabricated fabric
sample shown in figure 5, the following assumptions were first
made.
Figure 4. 2D geometrical model of the structure under
compression.
(1) All the unit-cells have the same geometrical shape and
size. They have the same deformation under compression
of the structure. Therefore, a unit-cell is sufficient to
represent the deformation behaviour of the auxetic textile
structure.
(2) The stitch yarn is totally flexible and cannot withstand
the compression loads. Therefore, its effect on the
deformation behaviour of the structure under compression
loads in the fabric thickness direction could be omitted
from the FE analysis.
(3) Both the weft and warp yarns are totally straight at the
initial state when no compression load is applied to the
structure.
(4) No slippage takes place at the initial contacting points of
the weft and warp yarns during the compression process.
(5) All the yarns are considered as transversely isotropic
elastic materials having linear elastic properties and their
moduli are kept constant during the deformations of the
structure.
On the basis of the above assumptions, a 3D FE model
was first set up. As the compression process of the structure
is a large deformation process, and involves interactions of
several yarn objects, a solid element type was selected in order
to improve convergence during the calculations.
The 3D FE model of a complete unit-cell of the fabric
structure is shown in figure 6. It was built according to the
geometrical shape of the structure and meshed with solid
element SOLID186 of ANSYS via a mapped mesh method.
SOLID186 could be used as a high order 3D 20-nodes solid
element which exhibits quadratic displacement behaviour.
It has the properties of large deflection and large strain
capabilities.
The boundary conditions applied to the FE model under
compression in the vertical direction (ydirection) are shown
in figure 7. Due to the symmetrical feature of the structure,
two vertical planes X0–X00and Z0–Z00, which were defined
as the central plane of the unit-cell perpendicular to the xaxis
(figure 7(a)) and the central plane perpendicular to the zaxis
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Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
Figure 5. 3D auxetic fabric sample.
Figure 6. 3D FE model of the structure at the initial state.
(figure 7(b)), were set up to be fixed during the compression
process. Meanwhile, the vertical boundary planes A–A0and
B–B0perpendicular to the xaxis were set up to be freely
moved in the xdirection (figure 7(a)), and the vertical
boundary planes C–C0and D–D0perpendicular to the zaxis
were set up to be freely moved in the zdirection (figure 7(b)).
To make them remain as vertical planes under compression,
a set of coupled degrees of freedom were applied to these
boundary planes by using the ‘CP’ command. To simulate
the compression test, the bottom plane of the FE model in
the ydirection was fixed and a displacement in the negative y
direction was applied to the top plane of the model.
The interactions of yarns are also defined. The contact
pairs, defined as the surface to surface contacts, are applied
for the contacts between the weft and warp yarns. The
3D contact elements ‘CONTA174’ and ‘TARGE170’ are
alternately placed on the contact surfaces of the solid elements
from bottom to top. The behaviour of the contact elements is
always bonded.
Figure 7. Boundary conditions of the FE model: (a) front view;
(b) lateral view.
Transversely isotropic elastic properties were defined for
the solid elements. The longitudinal Young’s moduli E11 for
the warp and weft yarns were obtained from the experimental
results. From the nature of transversely isotropic materials,
the following relationships exist: E22 =E33,G12 =G13, and
G23 =0.5E22/(1+v23 ). The transverse Young’s moduli E22
were assumed to be equal to E11/15 based on the braided
yarns used. Data for G12 were assumed to be G23, since the
braided yarns used in this study had similar shear behaviour
in both longitudinal and transverse directions. The Poisson’s
ratio in both the transverse and longitudinal directions of
the yarns was 0.2 [26]. The specified geometrical parameters
and yarn material properties for the FE analysis are listed in
table 1.
3.2. Validation by experiment
In order to verify the validity of the above FE model, the
auxetic fabric sample shown in figure 5was compressed
at a speed of 2 mm min−1on an Instron 5566 machine
equipped with two circular compression platens of 150 mm in
diameter and a 10 kN load cell. The same method as proposed
in [25] was used to measure the lateral size change of the
structure and to obtain the Poisson’s ratio values at different
compression strains.
A comparison of the Poisson’s ratio–compression strain
curves obtained from the 2D geometrical model, the 3D
FE model and the experiment is shown in figure 8. It
can be seen that the curve from the FE model is very
close to the curve from the experiment. This implies that a
very good agreement is obtained between the FE analysis
and the experiment. Therefore, the proposed FE model is
validated by the experiment. However, the difference between
the geometrical analysis and the experiment is very large
although the variation trends of their curves are similar,
i.e., the auxetic behaviour of the structure increases with
increasing compression strain. This is normal, because the
yarn properties could not be included in the geometrical
analysis. Therefore, many factors such as the changes in
yarn cross-section and length were omitted in the geometrical
analysis. As a result, the geometrical analysis could not
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Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
Table 1. Geometrical parameters and yarn properties.
Radius
(mm)
Yarn spacing
(mm) E11 (MPa) E22 (MPa) G12 (MPa) G23 (MPa) v12 v23
Warp yarns 3.08 7.50 131.76 8.78 3.66 3.66 0.2 0.2
Weft yarns 1.01 7.50 144.96 9.66 4.03 4.03 0.2 0.2
Figure 8. Poisson’s ratio–compression strain curves obtained from
geometrical analysis, FE model and experiment.
provide an accurate prediction of the auxetic behaviour of the
structure.
4. Demonstration and simulation of auxetic
behaviour
The above validated FE model can be used to simulate the
auxetic behaviour of 3D textile structures made with different
structural parameters and yarn properties. First of all, the FE
model can provide a very clear demonstration on how a 3D
auxetic textile structure deforms under compression. Figure 9
shows the front view of the deformation process of the auxetic
structure calculated with the values listed in table 1at different
compression strains. It can be clearly seen that the structure
laterally contracts when compressed in the vertical direction
and the maximal auxetic effect is obtained when the weft
yarns in two adjacent layers come into contact each other.
Exceeding this critical state, if we continue to compress the
structure, the changes in the yarn cross-sections become the
main deformation mode of the structure, which may lead to a
positive Poisson’s ratio of the structure. In this case, the above
FE model is no longer valid due to changes in the boundary
conditions.
The parameters that can influence the auxetic behaviour
of the proposed 3D auxetic textile structure include the radii
and longitudinal Young’s moduli of the warp and weft yarns,
the spacing of two adjacent warp yarns in the same layer 2l1
and the spacing of two adjacent weft yarns in the same layer
l2. Keeping the same assumptions that yarns are transversely
isotropic elastic materials and E22 =E11/15, G12 =G23 , the
Figure 9. Front view of the deformation process of an auxetic
structure at different compression strains.
effects of the non-dimensional parameters k1,k2,k3, and k4as
listed in the nomenclature were analysed. In order to facilitate
the analysis, the radius of the weft yarn, set as 1 mm, and
its longitudinal Young’s modulus, set as 100 MPa, were kept
unchanged.
The effect of k1, which was defined as the radius ratio
of the warp and weft yarns, is shown in figure 10, in which
the Poisson’s ratio–compression strain curves of the structure
were plotted with different values of k1(k1=1–3) when other
non-dimensional parameters were kept unchanged (k2=7,
k3=7, k4=1). It should be noted that the variation scope
of the compression strain for each curve is from 0 to the
maximal value where the weft yarns in the two adjacent layers
come into contact. It can be seen that k1has a significant
effect on the auxetic behaviour of the structure and this effect
increases with increasing k1. This phenomenon is normal
because the void spaces between two adjacent layers of the
weft yarns become increased when k1increases. Therefore,
more void spaces are provided for the weft yarns to be more
easily crimped under compression of the structure. As high
crimping of the weft yarns will lead to a high contraction
of the structure, a high auxetic effect is obtained. Therefore,
the auxetic behaviour of the structure can be increased by
increasing k1, i.e., by increasing the difference in radii of the
warp and weft yarns.
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Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
Figure 10. Poisson’s ratio–compression strain curves for different
k1values.
Figure 11. Poisson’s ratio–compression strain curves for different
k2values.
The effect of k2, which was defined as the ratio of the
half spacing of two adjacent warp yarns in the same layer
and the radius of the weft yarns, is shown in figure 11, in
which the Poisson’s ratio–compression strain curves of the
structure were plotted with different values of k2(k2=7–10)
when other non-dimensional parameters were kept unchanged
(k1=3, k3=7, k4=1). It can be seen that k2can influence
the auxetic behaviour of the structure, but the effect is not
so significant as that of k1. It can be also seen that the
auxetic effect increases with decreasing k2. This is because
with decreasing k2, the spacing between the two adjacent warp
yarns in the same layer decreases. Therefore, the contraction
of the structure in the transverse direction with the same
compression strain will lead to a high transverse strain. As
a result, the auxetic effect of the structure increases with
decreasing k2. However, with decreasing k2, the void spaces
for the weft yarns to be freely deformed will be decreased,
which may limit the further contraction of the weft yarns in the
Figure 12. Poisson’s ratio–compression strain curves for different
k3values.
horizontal direction when compressed. Therefore, the auxetic
effect can be decreased when k2becomes too small.
The effect of k3, which was defined as the ratio of the
spacing of two adjacent weft yarns in the same layer and
the radius of the weft yarn, is shown in figure 12, in which
the Poisson’s ratio–compression strain curves of the structure
were plotted with different values of k3(k3=5, 7, 9) when
other non-dimensional parameters were kept unchanged (k1=
3, k2=7, k4=1). It can be seen that the change of k3cannot
cause a fluctuation of Poisson’s ratio of the structure. This
means that the weft yarn has no obvious effect on the auxetic
behaviour of the structure. Since the arrangement of the weft
yarns in all the layers of the structure is the same, all the weft
yarns have the same bending shape during the compression
process, regardless of their spacing. Therefore, the auxetic
behaviour of the structure would remain nearly unchanged
when the value of k3changes.
The effect of k4, which was defined as the ratio of the
longitudinal Young’s moduli of the warp and weft yarns, is
shown in figure 13, in which the Poisson’s ratio–compression
strain curves of the structure were plotted with different values
of k4(k4=1/81, 1/9, 1, 9) when other non-dimensional
parameters were kept unchanged (k1=3, k2=7, k3=7).
It can be seen that k4has a significant effect on the auxetic
behaviour of the structure and this effect increases with
increasing k4.k4=1/81 is a special case for k1=3. In this
case, both the warp and weft yarns have the same bending
rigidity. As the two kinds of yarns have the same ability to
resist the deformation, the Poisson’s ratio of the 3D structure
becomes nearly zero. However, when k4is greater than 1/81,
the weft yarns become relatively softer than the warp yarns.
Since the auxetic behaviour comes from the deformation of
the weft yarns, the easy deformation of softer weft yarns can
lead to a high auxetic effect. Therefore, the auxetic effect of
the structure can be increased by increasing k4, i.e., increasing
the difference of the moduli of the warp and weft yarns.
An interesting phenomenon is observed when k4=9.
In this case, the auxetic behaviour of the structure decreases
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Smart Mater. Struct. 22 (2013) 084005 Z Ge et al
Figure 13. Poisson’s ratio–compression strain curves for different
k4values.
during the initial compression stage. This phenomenon
could be explained by analysing the deformation behaviour
of the weft yarns during the compression process. Under
compression, the weft yarns are subjected to both bending
and stretching deformations. While the bending deformation
produces an increased effect on the auxetic behaviour of the
structure, the stretching deformation produces a decreased
effect on the auxetic behaviour of the structure. Since the
weft yarns are more easily stretched at the initial deformation
due to straightening effect of braided yarns, a greater impact
on the auxetic behaviour is produced at the beginning of the
compression process. In particular, with increasing k4, the
difference in the longitudinal Young’s moduli of the warp and
weft yarns is increased, which makes the weft yarns more
easily stretched during the initial stage. As a result, an obvious
decreasing trend of the auxetic effect at the initial compression
is observed when k4=9.
5. Conclusions
The geometry of an innovative 3D auxetic textile structure
for composite reinforcement was described. Its FE model
was established using ANSYS software and validated
by the experimental results. The deformation process at
different compression strains was demonstrated and the
auxetic behaviour at different structural parameters and yarn
properties were simulated by using transversely isotropic
elastic materials in the FE model. The following conclusions
can be drawn from the analysis.
(1) The FE model can provide an accurate prediction
of the auxetic behaviour of the 3D auxetic textile
structure. Further work will consider the influence of
the structural parameters and yarn properties on the
mechanical behaviour of the structure under compression.
(2) The auxetic behaviour of the 3D auxetic textile structure
increases with increasing compression strain. It is
significantly affected by the structural parameters and
yarn properties. An increase of the ratios of the radii
and moduli between the warp and weft yarns and a
decrease of the spacing between two adjacent warp yarns
can increase the auxetic behaviour of the structure. The
spacing between two adjacent weft yarns in the same layer
has no obvious effect on the auxetic behaviour of the
structure.
Acknowledgment
The author would like to thank Research Grants Council
of Hong Kong Special Administrative Region Government
for funding support in the form of a GRF project (grant
no. 516510).
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