Global Journal of Engineering Sciences (GJES)
A
3D Star-Shaped zero Poisson’s Ratio Honeycomb with Cubic Symmetry
Authored by Zhang Wenzhi
Abstract
A
three-dimensional (3D) star-shaped zero Poisson’s ratio (ZPR) honeycomb with
cubic symmetry is proposed in this paper based on a twodimensional (2D)
star-shaped ZPR honeycomb. Based on the Timoshenko beam theory and the energy
method, the analytical formulas for the equivalent Young’s modulus and the
specific stiffness of the new honeycomb are obtained, which are verified by the
numerical analysis and the experiments. The results show that the proposed new
honeycomb has the ZPR effect in all three principal directions with the same
equivalent Young’s modulus. With this excellent property, the novel honeycomb
is of good application prospect in some important fields, such as
anti-collision facilities and wing morphing. In addition, the influence of the
geometric parameters on this new honeycomb’s elastic performance is further
studied. The results show that the new honeycomb’s elastic performance can be
significantly improved by increasing either the thin struts’ slant angle or the
ratio of width to height of the thin struts. This study provides a new idea for
the design of 3D ZPR honeycombs.
Keywords: Zero Poisson’s ratio (ZPR);
Equivalent parameters; Timoshenko beam theory; Three dimensional (3D);
Star-shaped honeycomb
Introduction
Metamaterial
is a kind of artificial composite material or structure with extraordinary
physical properties [1]. As one type of Metamaterial, the zero Poisson’s ratio
(ZPR) metamaterial and the negative Poisson’s ratio (NPR) metamaterial are of
abnormal Poisson’s ratio effect [2-3]. Due to their unique properties in energy
absorption and coaxial curvature, these materials are playing an increasingly
important role in some important fields such as aerospace, transportation and
biomedicine.
Research
on the NPR honeycombs is becoming abundant [4-5], while research on the ZPR
honeycombs is relatively lagging. Most research on the ZPR honeycombs focuses
on the 2D honeycombs, such as the semi re-entrant (SRE) honeycomb [6-9],
accordion honeycomb [9-11], star-shaped ZPR honeycomb [12], reversed semi
re-entrant (RSRE) honeycomb [13] and mixed cruciform honeycomb [14-16]. Some
other research focuses on forming novel 2D ZPR honeycombs by combining multiple
structures [17], adding structural thin struts [18] or changing the shape of
structural thin struts [19-23].
Compared
with 2D honeycombs, 3D honeycombs have the potential to perform the same
equivalent elastic properties in all three principal directions, which has been
demonstrated by some NPR honeycombs (e.g., [24-29]). However, there are not
many honeycombs with ZPR effect in all three principal directions, and it is
even rarer for 3D ZPR honeycombs to exhibit the same equivalent Young’s modulus
in all three principal directions. Wang et al. [30] extended the 2D SRE
honeycomb, RSRE honeycomb and their hybrid cellular to 3D, but only the 3D RSRE
honeycomb showed ZPR effect in three principal directions. Chen et al. [31]
designed a 3D honeycomb which shows NPR effect in two principal directions and
ZPR effect in another principal direction. Yang and Ma [32-33] proposed two
kinds of 2D U-shaped honeycombs, which were expanded into a series of 3D ZPR
honeycombs, but the mechanical properties are mainly reflected in one principal
direction.
This
paper proposes a 3D star shaped ZPR honeycomb with the same equivalent elastic
properties in three principal directions. The equivalent elastic properties of
this novel honeycomb are analyzed in the elastic range, and the relation
between its geometric parameters and equivalent elastic properties is discussed.
The
unit cell of the proposed novel 3D star-shaped ZPR honeycomb is a regular
hexahedron surrounded by six 2D star-shaped ZPR honeycombs of the same size. As
shown in Figure 1, the length of the strut is defined as l , the section of the strut is t × t, the slant angle between the struts is
2θ, and the equivalent length of the unit cell is 2L
= 2l sinθ This work is to study the star-shaped honeycomb of the
concave octagon, so we define.45O< θ<90O.
Analysis of the Equivalent Elastic Properties
Because of the cubic symmetry
of its cellular units, the novel honeycomb proposed in this study has the same
equivalent elastic properties in three principal directions, which are not
impacted when the scale of honeycombs is a
× b × c,a ≠ b ≠ c. In this paper, 1/8 of the unit cell is selected as
a simplified structure for analysis, and only the situation under uniaxial load
in the Y direction is discussed (Figure 2 (a)). Assuming that the longitudinal
displacement
Equivalent Poisson’s ratio
When the simplified structure
is subjected to the load in the Y direction, deformation only occur with strut
AB and strut AC, and the vector of the lateral displacement at point B and that
at point C both point to the interior of the unit cell, which has nothing to do
with the equivalent transverse strain. Therefore, the equivalent transverse
strain is
εx =εz =
0(1)
and the equivalent Poisson’s
ratio is obtained to be
νyx =νyz =
0(2)
Namely, the proposed structure
is zero Poisson’s ratio honeycomb.
Equivalent young’s modulus
In the next we derive the
equivalent Young’s modulus based on Timoshenko beam theory. From the principle
of virtual work
Numerical Analysis
In order to verify the
analytical results above, the unit cell shown in Figure 1(c) is numerically
analyzed using the Timoshenko beam unit B31, and the size is divided by 0.1.
The finite element (FE) results are shown in Figure 6 with the boundary
condition shown in Figure 3. In addition, the node displacements of planar HJKL
and planar DFJK were coupled by constraint equations for uniform expansion of
cell units.
Experiment
In order to verify the
theoretical results and the finite element results, three sets of samples are
prepared by the 3D printer of the SLA principle (Figure 4(a)). The geometrical
parameters of the samples are listed in Table 1 and the scale of honeycombs in
each set is 6× 6× 6 . The elastic modulus of the base material is 2286MPa and
the Poisson’s ratio of the base material is 0.397 . A compression test is
performed on the samples with quasi-static load with the speed of 0.2mm min by
a universal material testing machine. Micrometers are set around the sample to
obtain the deformation in another two principal directions, as shown in Figure
3(b).
Results and Discussion
Figure 5 shows that when the
uniaxial compression is loaded in the Y direction, transverse displacements of
the same value simultaneously occur with the central nodes in the X direction
and the Z direction, while no transverse displacements occur with the corner
nodes. The FE results are in consistency with our discussion in section 3.1.
This result verifies that the structure designed in this paper is zero
Poisson’s ratio honeycomb and reveals the formation mechanism of zero Poisson’s
ratio effect.
It can be seen from Table 2
that the analytical results are in good agreement with the FE results, namely vyx=vyz=0,
and that there is a slight deviation in the experimental results. The reason
for this deviation may be i. There are manufacturing errors in the 3D printed
test specimens; ii. There exist certain zones where thin struts overlap and
interlace in the manufactured honeycomb; iii. The influence of boundary
conditions exists when the number of unit cells is small.
Figure
6 shows the influence of geometric parameters on normalized Young’s modulus EY/E0 and equivalent Poisson’s ratio vyxor(vyz). The results show that the FE results are in
good agreement with the curves of the analytical formula, and the normalized
Young’s modulus EY/E0increases with the increase of θ or t/l, and the growth rate first slows down and then increases
rapidly. The results also show that the changes of θ and t / l have no effect on the performance of zero Poisson’s ratio.
Figure 7 shows the
influence of the novel honeycomb’s geometric parameters in comparison with the
FE results and the analytical solutions. The results show that the specific
stiffness β increases with the increase of the slant
angle θ , and the growth rate first slows down and then
increases rapidly, while β also increases with the increase of
the width to height ratio t / l of the thin strut, and the
growth rate first increases rapidly and then tends to slow down. Figure 8 shows
the relationship between the normalized Young’s modulus and the relative
density obtained from the analytical solution. It can be seen from Figure 8
that the normalized Young’s modulus is approximately equal to the square of the
relative density. That is, bending deformation is the dominant deformation
mechanism of the new three-dimensional zero Poisson’s ratio honeycomb.
Conclusion
In this paper, a novel
three-dimensional (3D) star-shaped zero Poisson’s ratio (ZPR) honeycomb with
cubic symmetry is developed based on a two-dimensional (2D) star-shaped ZPR
honeycomb. An analytical model is developed to investigate the Young’s modulus
of the proposed new honeycomb, which is verified by the finite element (FE)
results and the experimental results. The influence of the geometric parameters
(the inclination angle θ and
the widthto- height ratio t / l of
the thin struts) on the honeycomb’s elastic performance is studied in detail.
The results show that the
proposed honeycomb has ZPR effect in three principal directions with the same
equivalent Young’s modulus. Moreover, the increase of the width-to-height ratio t / l of this new honeycomb’s struts
improves the stiffness of the material without having any impact on the ZPR
effect. Therefore, the novel honeycomb is of good application prospect in
various fields, such as anti-collision facilities and wing morphing
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