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Article published in
Engineering Structures, Vol. 266 (2022) 114656
https://doi.org/10.1016/j.engstruct.2022.114656
In-plane Impact Behavior of 3D-printed Auxetic Stainless Honeycombs
Yiyi Zhou1, Yunfan Li2, Dan Jiang3, Yu Chen4, Yi Min Xie5, Liang-Jiu Jia2*1
1. School of Civil Engineering and Architecture, Changzhou Institute of Technology, No. 666 Liaohe
Road, Changzhou 213000, China.
2. Department of Disaster Mitigation for Structures, College of Civil Engineering, Tongji University,
No. 1239, Siping Road, Shanghai 200092, China.
3. College of Civil Engineering, Nanjing Tech University, No. 30 Puzhu South Road, Nanjing 211800,
China.
4. College of Civil Engineering, Fuzhou University, Fuzhou 350116, China.
5. Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne
3001, Australia.
ABSTRACT
In order to fully comprehend dynamic responses of metallic auxetic honeycombs,
drop hammer impact tests and corresponding finite element (FE) analyses were
conducted. The three-dimensional (3D) printing technology was adopted to
manufacture stainless steel specimens with varying cell side length-to-thickness ratios
and cell configurations. The specimens were crushed under three distinct energy inputs,
and the deforming process and relevant mechanical parameters, e.g., the Poisson's ratio,
failure mode, plateau stress and energy absorption capacity were studied. The FE results
* Corresponding author. Tel.: +8618321909263
Email address: lj_jia@tongji.edu.cn (L.-J. Jia).
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can generally compare well with the experimental ones and facilitate clarifying the
impact mechanisms of the experimental results. Improvements of the auxetic specimens
in energy absorption were verified through comparison with that of the corresponding
convex honeycomb. The results reveal that the specimen with an auxetic honeycomb
has higher plateau stress and specific energy absorption with less deformation when the
geometric size is the same. A small side length-to-thickness ratio and input impact
energy can lead to greatly improved energy absorption efficiency.
Keywords: Re-entrant, Auxetic, Honeycomb, Impact behaviors, Energy absorption,
3D-printing, Stainless steel.
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1. Introduction
Negative Poisson's ratio (NPR) materials and structures, often referred to as
“auxetics” by Evens [1] exhibit unique transverse shrinkage (expansion) properties
under uniaxial compression (tension). They exhibit unique mechanical characteristics,
including high shear bearing capacity, fracture resistance, energy absorption, and
resistance to collapse [2-5]. Numerous cellular auxetic materials with various structural
configurations have been proposed during the last two decades, including re-entrant
structures [6], chiral structures [7], rotating polygonal structures [8], perforated sheet
structures [9,10], and others [11-13].
The honeycomb structure is a well-known example of a periodic porous structure.
It consists of a tiny cross-sectional area with a thin-walled plate, leading to a relatively
low equivalent density. The primary deformation mode of the honeycomb structure is
crushing deformation when subjected to various in-plane impact loads, with the impact
energy absorbed through massive deformation. Due to the above particular properties,
honeycomb structures offer a wide range of potential applications, e.g., anti-collision
and buffer structures [14-16].
The concept of a re-entrant hexagon honeycomb was initially introduced in the
1980s [17] and has piqued the curiosity of a large number of researchers. Earlier
research concentrated on quasi-static and elastic models with relatively low
deformation capacity. Masters and Evans [18] investigated effects of buckling and
stress on collapse deformation of honeycombs, and developed a theoretical model for
estimating the elastic properties, and showed that the thickness and span ratio was a
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critical parameter defining the mechanical properties. Grima et al. [19] investigated two
types of elastomer and optimized Masters's theoretical formula. Yang et al. [20]
proposed a micropolar elasticity theory to investigate the link between the Poisson's
ratio of a re-entrant honeycomb and the geometric parameters of the cell. Wan et al. [21]
discovered that the NPR of re-entrant honeycomb is no longer constant under
substantial deformation, but dramatically varies with strain. Levy [22] investigated
expansion characteristics of concave corners in the direction perpendicular to the
external tension and the mechanical behavior of re-entrant honeycombs subjected to
large deflection. Lee et al. [23] and Scarpa et al. [24] demonstrated that the Poisson's
ratio and Young's modulus of a re-entrant hexagonal honeycomb are dependent on the
geometric characteristics of the elements. Wu [25] converted the influencing variables
of honeycomb core deformation due to expansion, shear, and deflection to equivalent
elastic parameters in order to maximize applicability of the theoretical model to
evaluating the in-plane bearing capacity. Cheng et al. [26] designed re-entrant unit cells
with different variable stiffness factors, and realized the tunability of stiffness from the
aspect of tuning the densification strain.
In terms of dynamic performance and energy absorption, Qi et al. [27] investigated
the impact and close-in blast response of sandwich panels with re-entrant honeycomb
cores, both experimentally and numerically. Hu et al. [28] developed a dynamic model
to predict the plateau stress of re-entrant honeycombs under quasi-static, low-speed and
high-speed impact loading. Zhang et al. [29] conducted finite element (FE) simulation
of re-entrant honeycombs through Abaqus and presented the deformation modes of
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structures under dynamic loading. Yang [30] numerically investigated the structural
reaction of a re-entrant honeycomb under rigid wall extrusion. Logakannan et al. [31]
investigated dynamic crushing of three-dimensional re-entrant honeycombs. It was
revealed that as the impact velocity varied, not only did the deformation mode vary, but
the specific energy absorption of the structure rose dramatically when the velocity
exceeded a threshold value. Acanfora et al. [32, 33] studied the lattice core structure
with different configurations and demonstrated that configurations with lattice cores
are able to maximize the energy absorption capacity, while preserving thickness and
mass reduction. Huang [34] presented a honeycomb structure with significant
deformation capacity to function as an anti-lock braking system. Stiffness of the
structure under in-plane compression and the failure mode were presented through
theoretical analyses.
Impact properties of metallic honeycomb structures have also been investigated in
recent years. Ruan et al. [35] investigated in-plane crushing behaviors of aluminum
honeycombs in two orientations and developed an empirical formula for evaluating the
plateau stress. Yasuki et al. [36] created a deformable barrier out of aluminum
honeycombs and tested it in a vehicle collision simulation. Xiao et al. [37] investigated
dynamic responses of an aluminum honeycomb sandwich beam under local impact
loading. Deformation of the re-entrant hexagonal honeycomb core was classified into
two parts: local compression deformation and bending tensile deformation. Li et al. [38]
investigated energy absorption capacity of two re-entrant hexagonal structures with
NPR fabricated of brass. The results reveal that when the compression speed increases,
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so does their energy absorption capacities. Xi et al. [39] studied rotatable re-entrant
honeycombs made of 3D-printed stainless steel and hypothesized that adjusting the
rotation of cellular cells could improve energy absorption performance. Harris et al. [40]
created a stainless steel origami honeycomb. It was found that combining an origami
mechanism with a standard honeycomb could achieve the NPR effect and energy
absorption in three directions. Han et al. [41] studied a 3D printed square auxetic tubular
lattice structures, and the results show that the structures have stronger energy
absorption capacity under axial and lateral loads.
Steel, as the most extensively used material, offers favorable mechanical
performance at a reasonable cost. Utilization of the 3D-printing technology to create
steel honeycomb structures is urgently needed and has a wide range of applications.
There are still several limitations in previous studies. To begin, elastic models under
quasi-static conditions and perfectly plastic metal models under dynamic conditions are
of primary interests. Traditional mechanical models for 3D-printed stainless steel lack
validation. Second, the majority of metal honeycomb structures are composed primarily
of aluminum alloy and brass, and there is still a scarcity of research on dynamic
performance of 3D-printed steel honeycomb structures. The side length-to-thickness
ratio of an auxetic honeycomb plays a dominant effect on its impact performance.
Nonetheless, this parameter has seldom been studied. Meanwhile, the 3D-printing
method provides a convenient approach for manufacturing auxetic configurations. A
comprehensive study on the impact performance of the 3D-printed stainless steel
auxetic structures and corresponding critical parameters is of great importance for both
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research and engineering applications.
This study focuses on the impact behaviors of 3D-printed auxetic and convex
stainless steel honeycombs. First, the manufacturing process of 3D-printed stainless
steel honeycomb structures is introduced, which is a fresh attempt at intelligent
construction. Second, high deformation capacity of the steel honeycombs is primarily
owing to plastic deformation of the periodical structure, leading to a complex deforming
process. This study investigates the crushing mode, energy dissipation capacity of
auxetic and convex specimens with varying side length-to-thickness ratio under
different input impact energies. To give a quantitative discussion on the impact
performance of the auxetic specimens, several critical indices were investigated,
including nominal stress and strain, plateau stress, energy absorption capacity, specific
energy absorption and the Poisson's ratio. It has been found in this study that the cell
side length-to-thickness ratio and input energy have significant effects on the impact
performance of the auxetic specimens.
2. Specimen design and manufacturing
2.1 Specimen design
Six re-entrant and convex hexagonal honeycomb specimens were designed in this
study. There are three re-entrant hexagonal specimens with a wall thickness of 1.0 mm:
A1.0-cr, A1.0-med and A1.0-max, where the input impact energy is 1500 J, 4500 J and
13500 J, respectively. A0.6-cr and A0.6-max are two re-entrant hexagonal specimens
with a wall thickness of 0.6 mm. H1.0-cr is a convex hexagonal specimen with a wall
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thickness of 1.0 mm under the impact energy of 1500 J. Fig. 1 shows the designed
dimensions of each specimen. Preliminary numerical analysis results indicate that the
impact energy of 1500 J corresponds to the plateau stage for the A1.0-cr and the
compaction stage for A0.6-cr. This energy was respectively amplified by three and nine
times to achieve medium- and high-speed impact.
The designed configuration of the tested specimens takes into account the
following factors: to begin, the wall thicknesses of the specimens were set at 1.0 mm
and 0.6 mm, respectively, since 0.6 mm is the smallest thickness that can currently be
obtained by 3D-printing of stainless steel. Because the side lengths of both regular and
re-entrant hexagonal units were fixed at 10 mm, the side length-to-thickness ratios of
inclined side of the specimens were 10 and 50/3, respectively. Each re-entrant hexagon
unit has a horizontal side length of 20 mm and an included angle of 60 degrees between
the horizontal and the inclined sides. Each specimen has five horizontal units and five
layers along the vertical direction. As a result, the length, width and height of all the
specimens are the same, i.e., 140×86.6×50 mm. The main geometrical and testing
parameters are introduced in Table 1.
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Fig. 1 Configuration of specimens (unit: mm).
Table 1 Main geometrical and testing parameters.
2.2 Specimen manufacturing
To date, the stainless steel honeycomb structure manufactured using the 3D-
printing technology is still rare, and the manufacturing process has not been detailed in
the literature [39, 40, 42, 43]. The manufacturing process of the specimens in this study
can be primarily divided into six phases.
(1) Setting up equipment and materials: the specimens were printed using the
Selective Laser Melting (SLM) technique in a manner of layer by layer printing, with
an EOSINT M280 metal 3D printer shown in Fig. 2(a). The dimension of working
No. Specimen Mass (g)
Impact
energy
(J)
Theoretical
height (m)
Actual
height
(m)
Deviation
Hammer
mass
(kg)
1 A 1.0-cr 795 1500 0.515 0.508 -1.44%
297
2 A 0.6-cr 495 1500 0.515 0.517 0.31%
3 H 1.0-cr 575 1500 0.515 0.512 -0.66%
4 A 1.0-med 795 4500 1.546 1.546 -0.01%
5 A 0.6-max 495 13500 4.638 4.646 0.17%
6 A 1.0-max 795 13500 4.638 4.637 -0.03%
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platform of the 3D printer is 250×250×300 mm, and the printing accuracy is 0.02 mm.
For the 3D-printing, stainless steel 316L metal powder with particle sizes ranging from
5 to 13 microns was employed, and the mechanical property of the powder is given in
Table 2.
(2) 3D-printing: a stainless steel base plate was inserted into the printer cavity,
which was then filled with printing ingredients. The material was melted at a high
temperature when the printing sitting head moved to the specified place. After printing
one layer, the printer continued the process for the following layer until printing of the
specimen was finished as shown in Fig. 2(b).
(3) Model surface powder cleaning: after printing, remaining powder adheres to
surfaces of the specimens. To avoid powder condensation on the external and internal
surfaces of honeycomb specimens, air flow and vibration shown in Fig. 2(c) were used
for powder treatment.
(4) Heat treatment: to remove residual stress, the specimens were heated in a
closed furnace as illustrated in Fig. 2(d) at 650 °C for twenty hours.
(5) Hot isostatic pressing: the specimens were placed in a closed container shown
in Fig. 2(e), and isostatic pressure and high temperature were applied simultaneously.
According to the literature [44], hot isostatic pressing is able to reduce the porosity to
below 1%, and significantly reduces the yield strength while increasing the elongation.
Although the specimens were layered printed, the coupon test and impact test results
did not show any delamination fracture.
(6) High pressure water cutting: using a high-pressure water pistol, the treated
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specimens were sliced from the stainless steel base plate.
Table 2 Mechanical property of the powder.
Material 3D printer
Yield stress
(MPa)
Tensile
strength (MPa)
Elongation (%)
316L EOS
≥
400
≥
600
≥
30
(a) EOSINT M280
3D printer (b) Melting printing (c) Powder cleaning (d) Heat treatment (e) Hot isostatic
pressing machine
Fig. 2 Illustration of manufacturing process (a) EOSINT M280 3D printer, (b) Melting printing,
(c) Powder cleaning, (d) Heat treatment and (e) Hot isostatic pressing.
3. Impact test
The drop hammer impact testing machine was employed to perform impact tests
on the specimens, and the test setup is shown in Fig. 3(a). The drop hammer impact
testing machine has an input energy ranging from 1000 J to 23000 J. During the test,
the force can be obtained from the load cell of the hammer, and the displacement can
be obtained through a high-speed camera with a frequency of 2000 frames per second.
The specimens were marked with little dots to record the deformation, and determine
the values of the Poisson’s ratio. Five points at the top surface of each specimen, i.e.,
D1 to D5, were selected. As for transverse strain, eight points at four different rows, i.e.,
P1 to P8, were selected as shown in Fig. 3(b).
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In this study, three levels of input impact energy were employed. The impact
energy for the first level is 1500 J, and the loaded specimens include A1.0-cr, A0.6-cr,
and H1.0-cr. The hammer head weighs 297 kg, and theoretically the initial height of the
hammer head is 0.52 m. The impact energy of the second level is 4500 J, and the loaded
specimen was A1.0-med, with a theoretical initial height of 1.55 m. The impact energy
of the third level is 13500 J, and the loaded specimens include A1.0-max and A0.6-max,
with a theoretical initial height of 4.64 m. As seen in Table 1, there are minor deviations
in the actual initial heights.
Fig. 3 Test setup (a) Layout, (b) Specimen and target points.
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A
1.0
-
cr
ε
=0.2
ε
=
0.379
A0.6
-
cr
ε=0.2
ε=0.4
ε=0.6
ε=0.71
H1.0
-
cr
ε
=0.2
ε
=0.4
ε
=0.463
A 1.0
-
med
ε
=0.2
ε
=0.4
ε
=0.6
ε
=0.626
A 0.6
-
max
ε
=0.2
ε
=0.4
ε
=0.6
ε
=0.8
ε
=0.93
A1.0
-
max
ε
=0.2
ε
=0.4
ε
=0.6
ε
=0.791
Fig. 4 Crushing processes of specimens.
4 Experimental results
4.1 Deformation mode
The failure process of the specimens was recorded using the photos taken by the
high-speed camera. The failure process was illustrated in Fig. 4 with a nominal strain
increment of 0.2 still the maximum deformation, where the nominal strain is defined as
the relative deformation between the top and bottom edges of the specimen divided by
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the initial height of the specimen. The following phenomena were noticed during the
crushing process.
For the impact energy of 1500 J, the crushed configuration of the specimen showed
that Specimens A1.0-cr and H1.0-cr did not achieve the compaction stage. The
maximum nominal strains of A1.0-cr and H1.0-cr were 0.38 and 0.46, respectively. The
maximum nominal strain of Specimen A0.6-cr was 0.71, which was entirely crushed
after the testing. For the impact energy of 4500 J and 13500 J, all of the specimens were
entirely crushed and achieved the compaction stage.
During the crushing process of the re-entrant specimen A0.6-max, damage
occurred immediately at the top area adjacent to the hammer head, which is different
from the other specimens. All other specimens began to deteriorate in the middle section,
and the cells showed oblique shear failure. It can be also found that the crushing
deformation of Specimens A1.0-cr and A1.0-max were not symmetric, which can be
due to initial imperfections, e.g., inevitable eccentricities of the testing machine,
manufacturing deviations or unflatness of the loading platform. When the re-entrant
specimens were completely crushed, the deformation mode of each element was
essentially the same, bending to the inner side and yielding a substantial NPR. However,
the distortion of regular cells initiated in the middle of the specimens, and the damaged
cells formed an X-shaped configuration as shown in Fig. 4.
4.2 Nominal stress and strain
During crushing of the specimen, the transient force on the hammer head was
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recorded by the load cell. The nominal stress was calculated as the measured transient
force divided by the external cross sectional area of the specimen. The crushing
processes of the specimens were recorded by high-speed photography, and this enabled
measurements of projectile displacement in order to compute the nominal strain.
Through the analysis of nominal stress and strain, the elastic limit stress, compaction
stress and corresponding strain values were obtained. The plateau stress [45, 46] is
calculated by Eq. (1):
𝜎=
∫𝜎(𝜀)d𝜀
(1)
where σ is the plateau stress, σ(ε) is the nominal stress varying with the nominal
strain, 𝜀 is the nominal strain corresponding to the elastic limit stress and 𝜀 is the
maximum strain value before the material is compacted, and according to the literature
[46], 𝜀 is calculated by Eq. (2):
[
() ∫𝜎(𝜀)
d𝜀]|= 0 (2)
In the test，Specimens A0.6-cr, A1.0-med, A0.6-max and A1.0-max were pressed
to complete crushing. The nominal stress-strain curves of the four specimens can be
classified into three distinct stages, i.e., elastic stage, plateau stage and compaction
stage, showing the typical characteristics of energy absorbing honeycomb structures.
Table 3 provides full nominal stress and strain data.
In order to study their impact responses, the hammer mass, M, and an initial
velocity, v0, was applied to the specimen, with the initial kinetic energy Ke as Mv2 / 2.
The normalized initial kinetic energy 𝐾 can be expressed as:
𝐾=
area (3)
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where 𝐴 area is the cross-sectional area of the profile of the honeycombs. 𝜎 is the
effective yield stress, which is formulated by:
𝜎 =
(4)
Relative density of the honeycombs can be expressed by:
Δ𝜌 = 𝜌∗/𝜌 (5)
where 𝜌∗ is the apparent density of cellular material, and 𝜌 is the density of applied
base material.
Table 3 Compaction stress and strain of test results.
Specimen Stress Strain Impact
energy (J) 𝐾
Ultimate
elastic stress
(MPa)
Plateau
stress
(MPa)
Compaction
stress (MPa)
Ultimate
elastic
strain
Compaction
strain
Final
strain
A 1.0-cr 7.035 NA NA 0.034 NA 0.376 1500 0.347
A 0.6-cr 2.422 1.511 2.479 0.046 0.575 0.709 1500 0.963
H 1.0-cr 4.623 NA NA 0.036 NA 0.460 1500 1.234
A 1.0-med 8.765 4.940 4.508 0.073 0.457 0.622 4500 1.040
A 0.6-max 4.209 2.339 5.074 0.062 0.673 0.923 13500 8.671
A 1.0-max 13.839 9.167 24.795 0.049 0.682 0.785 13500 3.120
Note: 𝐾
is the impact response.
Table 4 Energy absorption of experimental results.
Specimen Total energy absorption
Volumetric energy absorption
(J/cm³)
Specific energy absorption
(J/g)
Energy
absorption
in elastic
stage (J)
Energy
absorption
in plateau
stage (J)
Total
Energy
absorption
(J)
Impact
energy
(J)
Energy
absorption
ratio
In
elastic
stage
In
Plateau
stage
In
Total
In
Elastic
stage
In
Plateau
stage
In
Total
A 1.0-cr 41.898 NA 843.902 1500 56.26% 0.069 NA 1.392 0.053 NA 1.062
A 0.6-cr 32.274 484.448 896.324 1500 59.75% 0.053 0.799 1.479 0.065 0.979 1.811
H 1.0-cr 56.568 NA 826.064 1500 55.07% 0.093 NA 1.363 0.098 NA 1.437
A 1.0-med 196.965 1132.449 2301.620 4500 51.15% 0.325 1.868 3.797 0.248 1.424 2.895
A 0.6-max 53.409 873.393 4976.351 13500 36.86% 0.088 1.441 8.209 0.108 1.764 10.053
A 1.0-max 118.782 3571.776 7683.115 13500 56.91% 0.196 5.892 12.674 0.149 4.493 9.664
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①
①
② ②
A1.0-cr A0.6-cr
①
①
② ②
H1.0-cr A1.0-med
①
①
② ②
A0.6-max A1.0-max
Fig. 5 Nominal stress-strain curves and energy absorption efficiency curves of (a) A1.0-cr, (b)
A0.6-cr, (c) H1.0-cr, (d) A1.0-med, (e) A0.6-max and (f) A1.0-max.
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
Nominal stress (MPa)
Nominal strain
A1.0-cr Nominal stress
1
2
0.0
0.2
0.4
0.6
0.8
A1.0-cr Energy absorption capacity
Energy absorption capacity
0.0 0.2 0.4 0.6 0.8
0
5
10
15
Nominal stress (MPa)
Nominal strain
A0.6-cr Nominal stress
0.0
0.2
0.4
0.6
0.8
Energy absorption capacity
A0.6-cr Energy absorption capacity
1
2
0.0 0.2 0.4 0.6 0.8
0
5
10
15
Nominal stress (MPa)
Nominal strain
H1.0-cr Nominal stress
1
2
0.0
0.2
0.4
0.6
0.8
H1.0-cr Energy absorption capacity
Energy absorption capacity
0.0 0.2 0.4 0.6 0.8
0
5
10
15
20
25
30
Nominal stress (MPa)
Nominal strain
1
2
A1.0-med Nominal stress
0.0
0.2
0.4
0.6
A1.0-med Energy absorption capacity
Energy absorption capacity
0.0 0.2 0.4 0.6 0.8 1.0
0
25
50
75
100
Nominal stress (MPa)
Nominal strain
A0.6-max Nominal stress
0.0
0.2
0.4
0.6
Energy absorption capacity
A0.6-max Energy absorption capacity
1
2
0.0 0.2 0.4 0.6 0.8 1.0
0
25
50
75
100
A1.0-max Nominal stress
Nominal stress (MPa)
Nominal strain
0.0
0.2
0.4
A1.0-max Energy absorption capacity
Energy absorption capacity
12
18
(a) (b)
(c)
Fig. 6 Energy absorption versus nominal strain curves (a) Total energy absorption, (b) Volumetric
energy absorption and (c) Specific energy absorption.
4.3 Energy absorption
The energy absorption capacity is of great importance for the auxetic structures
under the impact loading, where energy absorption efficiency is an important index.
According to the literature [39, 40], it can be calculated by Eq. (6):
β(ε)=∫()
() (6)
where β(ε) represents the energy absorption capacity and is defined as the ratio of the
energy absorbed under the given nominal strain to the corresponding nominal stress.
The nominal stress-strain curve and energy absorption efficiency curve are combined
0.0 0.2 0.4 0.6 0.8 1.0
0
1000
2000
3000
4000
5000
6000
Total energy absorption (J)
Nominal strain
A1.0-cr Total energy absorption
A0.6-cr Total energy absorption
H1.0-cr Total energy absorption
A1.0-med Total energy absorption
A0.6-max Total energy absorption
A1.0-max Total energy absorption
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
12
Volumetric energy absorption (J/cm
3
)
Nominal strain
A1.0-cr Volumetric energy absorption
A0.6-cr Volumetric energy absorption
H1.0-cr Volumetric energy absorption
A1.0-med Volumetric energy absorption
A0.6-max Volumetric energy absorption
A1.0-max Volumetric energy absorption
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
Specific energy absorption(J/g)
Nominal strain
A1.0-cr Specific energy absorption
A0.6-cr Specific energy absorption
H1.0-cr Specific energy absorption
A1.0-med Specific energy absorption
A0.6-max Specific energy absorption
A1.0-max Specific energy absorption
19
in Fig. 5. The right side of each figure presents the instantaneous deformed
configuration of the specimen when the nominal stress reaches the elastic limit stress
and compaction stress. Points ① and ② in Fig. 5 present instantaneous deformed
configurations of the specimen when the nominal stress reaches the elastic limit stress
and compaction stress, respectively.
The energy absorption efficiency of all re-entrant hexagonal specimens has a
process of initial increasing and then falling, and shows better energy absorption
capacity in the plateau stage. The energy absorption efficiency curves and nominal
stress-strain curves show an apparent antisymmetric trend, that is, when the nominal
stress increases, the energy absorption efficiency decreases, and vice versa, as shown
in Fig. 5.
The total energy absorbion 𝐸 in the impact process was calculated by the
transient force and the measured displacement as shown in Fig. 6(a). The energy
absorbed in different deformation stages were obtained by the elastic limit stress,
compaction stress and corresponding strain values given in Table 4. In order to
accurately describe energy absorption of the specimen, specific energy absorption,
𝐸 [31, 39], in Fig. 6(b), defined as energy absorption per unit mass, is computed by
the following formula:
𝐸 =
(7)
where Q is the mass of the specimen. The volumetric energy absorption 𝐸(𝜀) [39],
defined as energy absorption per unit volume is calculated in Eq. (8):
𝐸(ε) = ∫σ(ε)dε
(8)
20
From the energy absorption results of each specimen given in Fig. 6(c) and Table
4, since the volumes of all specimens are equal, the trend of volumetric energy
absorption 𝐸(ε) is consistent with the varying tendency of the total absorbed energy
𝐸. The specimen with the highest energy absorption ratio is A0.6-cr, which absorbs
59.75% of the input energy, and the specimen with the lowest one is A0.6-max, which
absorbed 36.86% of the input energy. Most of the specimens absorb 51.15% to 59.75%
of the input impact energy. Deformation in the elastic stage is relatively limited, and
the absorbed energy is minimal, accounting for 3.60% to 8.50% of the total absorbed
energy. The energy absorbed at the plateau stage accounts for around 50% of the total
absorbed energy, whereas the energy absorbed within the compaction stage accounts
for approximately 40%. When the side length-to-thickness ratio of the auxetic specimen
decreases, the plateau stress increases dramatically. As the input impact energy
increases, the plateau stress also increases.
5. Finite element analysis
The impact properties of the re-entrant and convex hexagon specimens were
numerically investigated in this study employing the ABAQUS explicit module.
Models were meshed with S4R elements. A convergence study has been conducted,
indicating a proper mesh size of 2 mm. In the simulation, the specimen is placed on the
base plate, and a fixed constraint is applied to the base plate. The base material in the
tests is stainless steel 316L, with the following material parameters: elastic modulus E
of 200 GPa, density of 7.81 kg/m3, and the Poisson's ratio of 0.30. The Johnson-Cook
(JC) model was employed to simulate the dynamic elastoplastic behaviors of the
specimens under impact loading. A 3D-printed tensile coupon with the same post-
printing processing was manufactured and tested to obtain the critical parameters of the
21
Johnson-Cook model given in Table 5. In addition, the strain rate effect was considered
in the FE analyses, and it is related with parameter C in the Johnson-Cook model, and
a value of 0.1 was employed according to the literature [47].
Since it was found that the experimental deformation of some specimens was
asymmetric, which can be caused by initial defects, a finite element model with initial
defects is established. The first three modes of the corresponding numerical model were
chosen, and the size of the initial imperfection is taken as one-tenth of the cell wall
thickness.
Five integration points are defined along the thickness direction to address the
needs for convergence and accuracy. The top surface of the honeycomb model and the
bottom surface of the hammer head were set as face-to-face contact in the analyses,
while the other portions were set as the self-contact algorithm. The frictional coefficient
along the tangential direction was set as 0.30. The FE models are illustrated in Fig. 7.
The external sizes of the honeycomb model are 140 mm, 86.60 mm, and 50 mm in the
x, y, and z orientations, respectively. The thicknesses of the models were set to 0.6 mm
and 1 mm, respectively.
The model was situated between the hammer head and the loading platform, and
both the hammer head and the loading platform were defined as rigid bodies. The
impact action of the hammer head was simulated by a predetermined field velocity of
the hammer hitting on the top surface of the specimen. The initial speeds of the hammer
were respectively defined as 3.18 m/s, 5.51 m/s, and 9.54 m/s, corresponding to the
input impact energy of 1500 J, 4500 J, and 13500 J.
Table 6 shows the coefficient of variation (CoV) of the critical mechanical
22
parameters between the numerical and experimental results. The FE and experimental
results generally compare well with each other, as shown in Fig. 8 and Table 6. The
nominal stress of most results is quite close to the plateau stress, and the final strain of
FE results is slightly larger than the experimental ones. For Specimen H1.0-cr, the
ultimate deformation strain of the experimental result was well predicted by the FE
results, and the plateau stress of the FE results is a bit higher. In the discussion section,
the failure mechanisms of the FE and experimental models are compared in depth.
Table 5 Parameters of Johnson-Cook model.
A (MPa) B (MPa) N C
400 1047 0.7043 0.1
Table 6 Comparison of main mechanical parameters between experimental and numerical results.
Specimen
Ultimate elastic
stress (MPa)
,
,
Plateau stress
(MPa)
𝜎
,
𝜎
,
Total energy
absorption (J)
𝐸
,
𝐸
,
Test FE Test FE Test FE
A 1.0-cr 7.035 6.020 0.856 843.902 1446.654 1.714
A 0.6-cr 2.422 1.824 0.753 1.631 1.511 0.926 896.324 1476.001 1.647
H 1.0-cr 4.623 5.180 1.120 826.064 1378.424 1.669
A 1.0-med 8.765 7.574 0.864 5.730 4.940 0.862 2301.620 4433.040 1.926
A 0.6-max 4.209 8.421 2.001 2.360 2.339 0.991 4976.351 13054.003 2.623
A 1.0-max 13.839 10.576 0.764 9.270 9.167 0.989 7683.115 13263.948 1.726
CoV 0.453 0.065 0.199
Notes: 𝜎, is the ultimate elastic stress of test, 𝜎, is the ultimate elastic stress of FE. 𝜎, is the plateau
stress of test, 𝜎, is the plateau stress of FE. 𝐸, is the total energy absorption of test, 𝐸, is the total
energy absorption of FE.
23
(a) (b)
(c) (d)
Fig. 7 Finite element models and mesh diagrams (a) Re-entrant cell model, (b) Regular cell model,
(c) Re-entrant cell mesh and (d) Regular cell mesh.
(a) A1.0-cr (b) A0.6-cr
(c) H1.0-cr (d) A1.0-med
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
Nominal stress (MPa)
Nominal strain
abaqus A1.0-cr
test A1.0-cr
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
Nominal stress (MPa)
Nominal strain
abaqus A0.6-cr
test A0.6-cr
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
Nominal stress (MPa)
Nominal strain
abaqus H1.0-cr
test H1.0-cr
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
Nominal stress (MPa)
Nominal strain
abaqus A1.0-med
test A1.0-med
24
(e)A0.6-max (f) A1.0-max
Fig. 8 Nominal stress and strain curves of experimental and FE results with (a) A1.0-cr, (b) A0.6-
cr, (c) H1.0-cr, (d) A1.0-med, (e) A0.6-max and (f) A1.0-max.
6. Discussion
6.1 Cell configuration
The most conventional geometric form of honeycomb is the regular hexagonal one,
which is broadly employed as anti-collision and energy absorption structures, such as
vehicle and high-speed train anti-collision beams [36, 48, 49]. The application of a re-
entrant hexagonal cell honeycomb is relatively uncommon, and mechanical property
data has become critical to its arrangement. One essential purpose of this study is to
evaluate differences in impact characteristics between re-entrant and auxetic
honeycomb structures.
Comparing the experimental results of Specimens A1.0-cr and H1.0-cr, both
specimens achieved the plateau stage while did not reach the compaction stage under
the impact energy of 1500 J. The area enveloped by the nominal stress-strain curve
represents the performance of energy absorption as shown in Fig. 9(a). The elastic limit
stress of Specimen A1.0-cr was 7.04 MPa, and that of Specimen H1.0-cr was 4.62 MPa.
In the plateau stage, the nominal stress of Specimen A1.0-cr was always greater than
0.0 0.2 0.4 0.6 0.8 1.0
0
25
50
Nominal stress (MPa)
Nominal strain
abaqus A0.6-max
test A0.6-max
0.0 0.2 0.4 0.6 0.8 1.0
0
25
50
Nominal stress (MPa)
Nominal strain
abaqus A1.0-max
test A1.0-max
25
that of Specimen H1.0-cr, indicating the auxetic specimen has a higher plateau stress.
The total absorbed energy of the re-entrant specimen A1.0-cr was 840.63 J, and the
maximum nominal strain was 0.38. The maximum nominal strain of the convex
specimen H1.0-cr was 0.46, and the total absorbed energy was 826.06 J. Intuitively,
Specimen A1.0-cr absorbed more total energy while exhibiting less deformation and a
higher nominal stress.
Due to the large elastoplastic deformation shown in Fig. 4, the side edges of
Specimen H1.0-cr cells were crushed and its width expanded, resulting in relatively
poor cushioning effectiveness. For Specimen A1.0-cr, the concave angle tended to close
and the folding edges were significantly bent, leading to NPR effect of the specimen.
The material tended to shrink internally and the density also increased. The auxetic
effect led to larger nominal stress and thus less deformation for the same impact energy
compared with the convex one.
As illustrated in Fig. 9(b), the following energy absorption behavior is identified
up to the plateau stage. The slope of volumetric energy absorption change curve of the
auxetic specimen A1.0-cr is greater than that of the convex specimen H1.0-cr. Specimen
A1.0-cr has a higher volumetric energy absorption under the same displacement. The
mass of Specimen A1.0-cr is a bit greater than that of Specimen H1.0-cr, while the slope
of the specific energy absorption curve of the former is still greater. Therefore, the
auxetic specimens have higher specific energy absorption compared with
corresponding convex ones under the same displacement while subjected to the same
impact energy, indicating better energy dissipation capacity of the re-entrant specimens
26
when the mass is the main concern.
FE analyses results of the convex specimen under the impact energy of 4500 J and
13500 J were supplemented to further compare the impact responses of the auxetic and
convex specimens. The FE model now includes two virtual specimens: H1.0-med and
H1.0-max, which simulates convex specimens with a thickness of 1.0 mm under input
impact energies of 4500 J and 13500 J, respectively.
By comparing the FE results of Specimens A1.0-med and H1.0-med under the
impact energy of 4500 J, and also those of Specimens A1.0-max and H1.0-max under
the impact energy of 13500 J, all the specimens reach the compaction stage, and the
overall trend of the mechanical properties of the specimens is consistent. The following
conclusions can be obtained: (1) The elastic limit stress and compaction stress of the
auxetic specimens are greater than those of the convex ones, as illustrated in Fig. 9(c)
and (e). When the auxetic specimens were completely crushed, they deformed less than
the corresponding convex ones. The plateau stresses of the auxetic specimens are higher,
and the nominal stresses increase considerably near the compaction stage, allowing
them to absorb more energy at this moment. (2) As shown in Fig. 9(d) and (f), because
the auxetic specimens absorb more energy before compaction, they exhibit a larger
slope of volumetric energy absorption curve than those of the corresponding convex
specimens under the input impact energy of 1500 J. Their specific energy absorption
curve also develop faster at the strain of compaction point and demonstrate greater
energy absorption performance per unit mass.
27
(a) (b)
(c) (d)
(e) (f)
Fig. 9 Comparison of mechanical properties of auxetic and convex specimen under different input
impact energy.
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
Nominal stress (MPa)
Nominal strain
A1.0-cr Nominal stress
H1.0-cr Nominal stress
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
Specific energy absorption (J/g)
Nominal strain
A1.0-cr Specific energy absorption
H1.0-cr Specific energy absorption
0
1
2
A1.0-cr Volumetric energy absorption
H1.0-cr Volumetric energy absorption
Volumetric energy absorption (J/cm
3
)
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
Nominal stress (MPa)
Nominal strain
A1.0-med Nominal stress
H1.0-med Nominal stress
0.0 0.2 0.4 0.6 0.8 1.0
0
4
8
Specific energy absorption (J/g)
Nominal strain
H1.0-med Specific energy absorption
A1.0-med Volumetric energy absorption
0
4
8
H1.0-med Volumetric energy absorption
A1.0-med Specific energy absorption
Volumetric energy absorption (J/cm
3
)
0.0 0.2 0.4 0.6 0.8 1.0
0
25
50
Nominal stress (MPa)
Nominal strain
A1.0-max Nominal stress
H1.0-max Nominal stress
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
Specific energy absorption (J/g)
Nominal strain
H1.0-max Specific energy absorption
A1.0-max Volumetric energy absorption
0
5
10
H1.0-max Volumetric energy absorption
A1.0-max Specific energy absorption
Volumetric energy absorption (J/cm
3
)
28
(a) (b)
A1.0-cr A0.6-cr H1.0-cr A1.0-med A0.6-max A1.0-max
(c)
Fig. 10 Poisson's ratio versus nominal vertical strain curves (a) Selected point position and (b)
Curves (c) Deformation diagram of specimen when the Poisson's ratio reaches the peak value.
(a) (b)
Fig. 11 Nominal stress and specific energy absorption versus nominal strain curves with (a) A1.0-
cr and A0.6-cr and (b) A1.0-max and A0.6-max.
6.2 Poisson’s ratio
As a typical auxetic structure, the Poisson's ratio of specimens is an important
parameter, and it can be evaluated using the following formula:
𝑣 = −
(9)
0.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Poisson's ratio
Vertical strain
A1.0-cr
A0.6-cr
H1.0-cr
A1.0-med
A0.6-max
A1.0-max
-0.811
-0.565
-0.510-0.473
-0.434
0.668
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
Nominal stress (MPa)
Nominal strain
A1.0-cr Nominal stress
A0.6-cr Nominal stress
0.0
0.5
1.0
1.5
2.0
A1.0-cr Specific energy absorption
A0.6-cr Specific energy absorption
Specific energy absorption (J/g)
0.0 0.2 0.4 0.6 0.8 1.0
0
30
60
Nominal stress (MPa)
Nominal strain
A0.6-max Nominal stress
A1.0-max Nominal stress
0
3
6
9
12
A0.6-max Specific energy absorption
A1.0-max Specific energy absorption
Specific energy absorption (J/g)
29
where 𝜀 is the transverse nominal strain and 𝜀 is the nominal vertical stain. The
varying process of the Poisson's ratio was calculated by recording the position data of
key points of the specimens. The method to obtain the transverse strain is illustrated in
Fig. 10(a), eight endpoints at four different rows, i.e., P1 to P8, were selected. The
transverse strain of each row was obtained by dividing the relative deformation of each
row by the initial distance of the two endpoints in the same row. The obtained transverse
strain values of the four rows were finally averaged as the transverse strain of the
specimen. Then the Poisson's ratio can be obtained readily.
According to the Poisson's ratio versus nominal vertical strain curves in Fig. 10(b),
all re-entrant specimens show good auxetic performance, with the minimum Poisson's
ratio ranging from -0.43 to -0.81. The Poisson’s ratio is always located in the negative
range for the auxetic specimens. During the compaction stage, the Poisson's ratio of all
specimens declines rapidly and then gradually increases. Taking the performance of re-
entrant specimens with a thickness of 1.0 mm under different impact energies, when
the impact energy is small, the Poisson's ratio is lower, and the strain corresponding to
the lowest Poisson's ratio is smaller. When the input energy is large, the Poisson's ratio
of the specimen is larger, i.e., the auxetic effect is less apparent, and the strain is larger
when the specimen reaches its lowest Poisson's ratio. The convex specimen presents an
entirely different trend, that is, the Poisson's ratio increases rapidly and then decreases
gradually in the process of compression, and its Poisson’s ratio is always located in the
positive range.
30
6.3 Side length-to-thickness ratio
The specimens in this study have two thicknesses of 1.0 and 0.6, with side length-
to-thickness ratios of 10 and 50/3 on the inclined side, respectively. From the
experimental results in this study, the specimens with different side length-to-thickness
ratios exhibited distinguished performance. For the input impact energy of 1500 J,
Specimen A1.0-cr achieved the plateau stage, and the ultimate nominal strain was 0.38,
and the total absorbed energy was 840.63 J. Specimen A0.6-cr achieved the compaction
stage, and the ultimate nominal strain was 0.71, and the total absorbed energy was
894.32 J. The specific energy absorption of Specimen A0.6-cr is 1.81 J/g at the final
stain, and that of Specimen A1.0-cr is 1.06 J/g. In this situation, the input impact energy
caused Specimen A0.6-cr to reach the compaction stage while Specimen A1.0-cr
remains in the plateau stage, the specimen with a large side length-to-thickness ratio
A0.6-cr can fully utilize the energy absorption capacity in all three stages and can
achieve a better energy absorption behavior, as shown in Fig. 11(a).
When the input impact energy is 13500 J, the highest nominal strain of Specimen
A1.0-max was 0.79, and the total absorbed energy was 7683.12 J. The total absorbed
energy was 4572.24 J, with Specimen A0.6-max having a maximum nominal strain of
0.93. Both specimens reached the compaction stage. The specific energy absorption of
Specimen with a large side length-to-thickness ratio A0.6-max is 10.05 J/g in total
compaction, and the value of Specimen with a small side length-to-thickness ratio A1.0-
max is 9.66 J/g.
When the full deformation is obtained, the specific energy absorption of the two
31
specimens is approximately the same, as shown in Fig. 11(b). The maximum stress of
the specimen, on the other hand, should be limited to be employed as a buffer structure,
since if the stress is too high when the specimen is fully compacted, it may destroy the
device that must be protected by the buffer structure. To control the stress in this
scenario, a maximum strain at the compaction stress, can be employed. The specific
energy absorption of the specimen with a small side length-to-thickness ratio A1.0-max
at the strain of compaction stress is 4.49, the value is around three times than the value
1.76 for Specimen A0.6-max, illustrating its advantages.
As a result, the smaller the side length-to-thickness ratio of the auxetic specimens
while crushing failure occurs, the greater the specific energy absorption value, and the
greater the change in the slope of the curve. Therefore, the specific energy absorption
of the auxetic specimen with a smaller side length-to-thickness ratio is higher.
When the specimen is built as a buffer structure, the energy absorption in the
plateau stage is considered the most important energy absorption index because the
deformation interval is longer, the stress is lower, and the energy interval is more stable.
Dividing the energy absorption in the plateau stage by the mass of the specimen, one
can obtain the specific energy absorption within the plateau stage. Its relationship with
the side length-to-thickness ratio is a crucial parameter for measuring energy absorption
capacities of the specimens.
Four FE models with thicknesses of 0.2, 0.4, 0.6, and 1.0 mm were developed for
assessment, denoted as A0.2-med, A0.4-med, A0.6-med, and A1.0-med, and simulating
an energy intake of 4500 J. The correlation between the specific energy absorption and
32
the side length-to-thickness ratio of four specimens at the plateau stage is shown in Fig.
12 and Table 7. The specific energy absorption at the plateau stage shows a nonlinear
increasing trend as the side length-to-thickness ratio decreases.
Table 7 Average specific energy absorption within plateau stage.
No.
Energy absorption within
plateau stage
(J)
Mass
(kg)
Average specific energy absorption
within plateau stage (J/kg)
A0.2-med 0.156 0.159 0.962453
A0.4-med 0.526 0.318 1.638616
A0.6-med 1.301 0.495 2.616828
A1.0-med 3.280 0.795 4.110528
Fig. 12 Average specific energy absorption within plateau stage versus side length-to-thickness
ratio curve.
Fig. 13 Nominal stress and specific energy absorption versus nominal strain curves of auxetic
specimens with a thickness of 1.0 mm.
10 20 30 40 50
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
specific energy absorption in plateau stage (J/kg)
Side length-to-thickness ratio
A1.0-med
A0.6-med
A0.4-med
A0.2-med
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
Nominal stress (MPa)
Nominal strain
A1.0-cr Nominal stress
A1.0-med Nominal stress
A1.0-max Nominal stress
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
Specific energy absorption (J/g)
Nominal strain
A1.0-cr Specific energy absorption
A1.0-med Specific energy absorption
A1.0-max Specific energy absorption
33
(a)
(b)
(c)
(d)
Fig. 14 Comparison of deformation mode between experimental and FE results (a) A0.6-max ε =
0.2, (b) A1.0-cr ε = 0.2, (c) A1.0-med ε = 0.2 and (d) H1.0-cr ε = 0.2.
34
6.4 Input impact energy
The input impact energy has a great effect on impact characteristics of the
specimens. When different input impact energies were applied to the same specimen,
as shown in Fig. 13, as the input impact energy increases, so do the plateau stress and
elastic limit stress of the specimen, as well as the specific energy absorption of the
auxetic specimen.
The deformation mode of the auxetic specimen is also affected by the input impact
energy. The experimental and FE results indicate that the auxetic specimen collapsed
from the first row that is closest to the hammer head when the input impact energy is
large and the side length-to-thickness ratio is large, such as A0.6-max, as seen in Fig.
14(a).
When the input impact energy is small and the side length-to-thickness ratio is low,
the deformation of the specimens can be classified as symmetrical or asymmetric. Some
specimens were eccentric under the impact loading, which is due to initial imperfections.
The FE model with an initial imperfection is considered in this study, and some
consistent results are found. As illustrated in Fig. 14(b), in asymmetric crushed models
such as A1.0-cr and A1.0-med, plastic deformation primarily concentrates near an
inclined line, and the deformation was created first from the cells in the middle section.
The deformation forms of other specimens are generally symmetrical, such as
A1.0-med and H1.0-cr, the cells buckle first at the middle part, then at both ends, and
starts to destroy from two diagonal directions at the same time, forming an X-shaped
crushing path. The corresponding results are also simulated by FE analyses, as shown
35
in Fig. 14(c) and Fig. 14(d).
7. Conclusion
Experiments and numerical analyses were conducted in this study to investigate
impact behaviors of 3D-printed stainless auxetic structures with re-entrant cell
honeycomb. The 3D-printed stainless structures have been demonstrated to be
mechanically strong and reliable, and have good potential to be employed in buffer
structures. The following conclusions can be drawn based on the experimental and
numerical results:
(1) The crushing processes of the auxetic and convex specimens can be divided
into three typical stages, i.e., elastic, plateau and compaction. In each stage, the nominal
stress of the auxetic specimen is greater than that of a convex one with the same
geometric sizes. Under the same input impact energy, the ultimate deformation of the
auxetic specimen is smaller, which absorbs more energy per unit volume and mass
compared with the corresponding convex one.
(2) The auxetic specimens in this study perform well in terms of the auxetic
performance, i.e., negative Poisson's ratio, with values of the Poisson’s ratio ranging
from -0.80 to -0.40. With an increasing input impact energy, the value of the minimum
Poisson's ratio increases, i.e., leading to a relatively poor auxetic effect. Meanwhile, the
strain corresponding to the minimum Poisson's ratio increases.
(3) The specific energy absorption of the auxetic specimen decreases with an
increasing side length-to-thickness ratio under the same input impact energy. The
36
specific energy absorption within the plateau stage increases nonlinearly as the side
length-to-thickness ratio decreases.
(4) The specific energy absorption of the auxetic and the convex specimens
increases with an increasing input impact energy. When the input impact energy is high
and the side length-to-thickness ratio of the specimen is large, the top row of cells close
to the hammer head is apt to be crushed first. When the input impact energy is low and
the side length-to-thickness ratio is small, the auxetic specimens can crush in an
asymmetric diagonal mode or a symmetric X-shaped mode. The experimental results
indicate that the auxetic specimens can crush in an asymmetric mode, and it is necessary
to consider initial imperfection during numerical simulations of the corresponding
specimens.
Acknowledgement
This study is supported by the National Natural Science Foundation of China (grant
number: 52178109) and Shanghai Municipal Science and Technology Major Project
(2021SHZDZX0100) and the Fundamental Research Funds for the Central Universities.
37
References
[1] Saxena KK, Das R, Calius EP. Three decades of auxetics research - materials with negative Poisson's ratio: a
review. Adv Eng Mater 2016;18:1847-70.
[2] Evans KE, Alderson A. Auxetic materials: Functional materials and structures from lateral thinking! Adv Mater
2000;12:617-28.
[3] Ren X, Das R, Tran P, Ngo TD, Xie YM. Auxetic metamaterials and structures: a review. Smart Mater Struct
2018;27.
[4] Choi JB, Lakes RS. Fracture toughness of re-entrant foam materials with a negative Poisson's ratio: Experiment
and analysis. Int J Fract 1996;80:73-83.
[5] Zhang Y, Ren X, Zhang XY, Huang TT, Sun L, Xie YM. A novel buckling-restrained brace with auxetic perforated
core: Experimental and numerical studies. Eng Struct 2021;249:113223.
[6] Masters IG, Evans KE. Models for the elastic deformation of honeycombs. Compos Struct 1996;35:403-22.
[7] Grima JN, Gatt R, Farrugia PS. On the properties of auxetic meta-tetrachiral structures. Phys Status Solidi B
2008;245:511-20.
[8] Grima JN, Farrugia PS, Gatt R, Attard D. On the auxetic properties of rotating rhombi and parallelograms: A
preliminary investigation. Phys Status Solidi B 2008;245:521-9.
[9] Taylor M, Francesconi L, Gerendas M, Shanian A, Carson C, Bertoldi K. Low porosity metallic periodic
structures with negative Poisson's ratio. Adv Mater 2014;26:2365-70.
[10] Zhang Y, Ren X, Han D, Cheng X, Jiang W, Zhang XG et al. Static and dynamic properties of a perforated
metallic auxetic metamaterial with tunable stiffness and energy absorption. Int J Impact Eng 2022;164:104193.
[11] Bertoldi K, Reis PM, Willshaw S, Mullin T. Negative Poisson's ratio behavior induced by an elastic instability.
Adv Mater 2010;22:361-6.
[12] Zhai XD, Gao JL, Liao HJ, Kirk CD, Balogun YA, Chen WN. Mechanical behaviors of auxetic polyurethane
foam at quasi-static, intermediate and high strain rates. Int J Impact Eng 2019;129:112-8.
[13] Shen H-S, Xiang Y. Effect of negative Poisson’s ratio on the postbuckling behavior of pressure-loaded FG-
GRMMC laminated cylindrical shells. Eng Struct 2021;243:112458.
[14] Gibson LJ, Ashby MF. Cellular solids: structure and properties. 2nd ed. New York: Cambridge University Press;
1997.
[15] Bitzer T. Honeycomb technology: materials, design, manufacturing, applications and testing. Dordrecht:
Springer Netherlands; 1997.
[16] Cutler J, Liber J. Understanding aircraft structures. 4th ed. Malden: Blackwell Pub; 2005.
[17] Frost HJ, Ashby MF. Deformation-mechanism maps. 1st ed. Oxford: Pergamon Press; 1982.
[18] Masters IG, Evans KE. Models for the elastic deformation of honeycombs. Compos Struct 1996;35:403-22.
[19] Grima JN, Ellul B, Attard D, Gatt R, Attard M. Composites with needle-like inclusions exhibiting negative
thermal expansion: A preliminary investigation. Compos Sci Technol 2010;70:2248-52.
[20] Yang DU, Lee S, Huang FY. Geometric effects on micropolar elastic honeycomb structure with negative
Poisson's ratio using the finite element method. Finite Elem Anal Des 2003;39:187-205.
[21] Wan H, Ohtaki H, Kotosaka S, Hu GM. A study of negative Poisson's ratios in auxetic honeycombs based on a
large deflection model. Eur J Mech A-Solid 2004;23:95-106.
[22] Levy O, Krylov S, Goldfarb I. Design considerations for negative Poisson ratio structures under large deflection
for MEMS applications. Smart Mater Struct 2006;15:1459-66.
[23] Lee JH, Choi JB, Choi K. Application of homogenization FEM analysis to regular and re-entrant honeycomb
structures. J Mater Sci 1996;31:4105-10.
[24] Scarpa F, Panayiotou P, Tomlinson G. Numerical and experimental uniaxial loading on in-plane auxetic
38
honeycombs. J Strain Anal Eng Des 2000;35:383-8.
[25] Wu YX, Wang F. Analysis for equivalent in-plane elastic parameters of the honeycomb core. Appl Mech Mater
2013;441:84-90.
[26] Cheng X, Zhang Y, Ren X, Han D, Jiang W, Zhang XG et al. Design and mechanical characteristics of auxetic
metamaterial with tunable stiffness. Int J Mech Sci 2022;223:107286.
[27] Qi C, Remennikov A, Pei LZ, Yang S, Yu ZH, Ngo TD. Impact and close-in blast response of auxetic
honeycomb-cored sandwich panels: Experimental tests and numerical simulations. Compos Struct 2017;180:161-78.
[28] Hu LL, Zhou MZ, Deng H. Dynamic crushing response of auxetic honeycombs under large deformation:
Theoretical analysis and numerical simulation. Thin-Walled Struct 2018;131:373-84.
[29] Zhang J, Lu G. Dynamic tensile behaviour of re-entrant honeycombs. Int J Impact Eng 2020;139.
[30] Yang S, Qi C, Guo DM, Wang D. Energy absorption of an re-entrant honeycombs with negative Poisson's ratio.
Appl Mech Mater 2012;148-149.
[31] Logakannan KP, Ramachandran V, Rengaswamy J, Ruan D. Dynamic performance of a 3D re-entrant structure.
Mech Mater 2020;148.
[32] Acanfora V, Saputo S, Russo A, Riccio A. A feasibility study on additive manufactured hybrid metal/composite
shock absorbers. Compos Struct 2021;268:113958.
[33] Acanfora V, Castaldo R, Riccio A. On the effects of core microstructure on energy absorbing capabilities of
sandwich panels intended for additive manufacturing. Materials 2022;15:1291.
[34] Huang J, Gong X, Zhang Q, Scarpa F, Liu Y, Leng J. In-plane mechanics of a novel zero Poisson's ratio
honeycomb core. Composites, Part B 2016;89:67-76.
[35] Ruan D, Lu G, Wang B, Yu TX. In-plane dynamic crushing of honeycombs—a finite element study. Int J Impact
Eng 2003;28:161-82.
[36] Yasuki T, Kojima S. Application of aluminium honeycomb model using shell elements to offset deformable
barrier model. Int J Crashworthiness 2009;14:449-56.
[37] Xiao DB, Chen XQ, Li Y, Wu WW, Fang DN. The structure response of sandwich beams with metallic auxetic
honeycomb cores under localized impulsive loading-experiments and finite element analysis. Mater Des 2019;176.
[38] Li D, Yin JH, Dong L, Lakes RS. Strong re-entrant cellular structures with negative Poisson's ratio. J Mater Sci
2018;53:3493-9.
[39] Xi HF, Xu JC, Cen SD, Huang SQ. Energy absorption characteristics of a novel asymmetric and rotatable re-
entrant honeycomb structure. Acta Mech Solida Sin 2021;34:550-60.
[40] Harris JA, McShane GJ. Impact response of metallic stacked origami cellular materials. Int J Impact Eng
2021;147.
[41] Han D, Ren X, Luo C, Zhang Y, Zhang XY, Zhang XG et al. Experimental and computational investigations of
novel 3D printed square tubular lattice metamaterials with negative Poisson’s ratio. Addit Manuf 2022;55:102789.
[42] Huang TT, Ren X, Zeng Y, Zhang Y, Luo C, Zhang XY et al. Based on auxetic foam: A novel type of seismic
metamaterial for Lamb waves. Eng Struct 2021;246:112976.
[43] Zhao X, Wei L, Wen D, Zhu G, Yu Q, Ma ZD. Bending response and energy absorption of sandwich beams
with novel auxetic honeycomb core. Eng Struct 2021;247:113204.
[44] Lavery N P, Cherry J, Mehmood S, Davies H, Girling B, Sackett E et al. Effects of hot isostatic pressing on the
elastic modulus and tensile properties of 316L parts made by powder bed laser fusion. Mater Sci Eng, A
2017;693:186-213.
[45] Li Z, Gao Q, Yang S, Wang L, Tang J. Comparative study of the in-plane uniaxial and biaxial crushing of
hexagonal, re-entrant, and mixed honeycombs. J Sandwich Struct Mater 2019;21:1991-2013.
[46] Fan X, Yin X, Tao Y, Chen M, Fang D. Mechanical behavior and energy absorption of graded honeycomb
39
materials under out-of-plane dynamic compression. Acta Mech Solida Sin 2015;36:114-22.
[47] Samantaray D, Mandal S, Borah U, Bhaduri AK, Sivaprasad PV. A thermo-viscoplastic constitutive model to
predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel. Mater Sci Eng, A
2009;526:1-6.
[48] Peng Y, Deng W, Xu P, Yao S. Study on the collision performance of a composite energy-absorbing structure
for subway vehicles. Thin-Walled Struct 2015;94:663-72.
[49] Hou S, Li Q, Long S, Yang X, Li W. Design optimization of regular hexagonal thin-walled columns with
crashworthiness criteria. Finite Elem Anal Des 2007;43:555-65.
