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A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance
Accepted Manuscript A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance Huan Wang, Zixing Lu, Zhenyu Yang, Xiang Li PII: DOI: Reference:
S0263-8223(18)31753-7 https://doi.org/10.1016/j.compstruct.2018.10.024 COST 10274
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
14 May 2018 3 September 2018 8 October 2018
Please cite this article as: Wang, H., Lu, Z., Yang, Z., Li, X., A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.10.024
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A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance Huan Wang, Zixing Lu, Zhenyu Yang*, Xiang Li Institute of Solid Mechanics, Beihang University, Beijing 100083, P.R. China Abstract Recently, auxetic honeycombs have attracted considerable attention for their excellent mechanical properties, especially the potential applications for energy absorption. In this paper, a novel re-entrant auxetic honeycomb is proposed and named as re-entrant star-shaped honeycomb (RSH), with the in-plane impact responses explored theoretically and numerically. Three types of microstructural deformation modes are observed
under
different
impact
velocities,
including
low-velocity mode,
medium-velocity mode and high-velocity mode. Moreover, a deformation map is summarized to illustrate the effects of the impact velocity and the relative density on the deformation modes. Two typical plateau stress regions are detected under low-velocity impact loading, and the second plateau stress is almost twice higher than the first one. The transverse contraction of the RSH mainly occurs at the first plateau region, which is different for the classical re-entrant honeycomb (RH). In addition, the absorbed energy of the RSH decreases slightly and then increases with the impact velocity. The results of the finite element simulations suggest that the RSH shows more excellent impact resistance, compared with the RH and SSH with same cell wall thickness. Furthermore, the effects of the cell wall thickness and the impact velocity on the crushing strength of the RSH are discussed, and the theoretical results are in good agreement with the finite element simulations.
Keywords Auxetic honeycomb; Deformation mode; Plateau stress; Impact velocity; Energy absorption *
Author to whom correspondence should be addressed. Electronic mail: [email protected]
Nomenclature Young’s modulus of matrix material yielding stress of matrix material Poisson’s ratio of matrix material length of the horizontal cell wall length of the inclined cell wall depth of honeycomb in out-of-plane direction cell wall thickness of honeycomb length of a RSH unit cell cell wall angle defined in Fig. 1 maximum bending moment of inclined strut of the RSH in Fig. 12a maximum bending moment of inclined strut in Fig. 12b plastic bending moment of inclined strut of the unit cell of RSH in Fig. 12a plastic bending moment of the double-thickness inclined strut in Fig. 12b plastic bending moment of a single inclined strut in Fig. 12b critical impact velocity between low-velocity and medium-velocity mode critical impact velocity between medium-velocity and low-velocity mode ratio of the strain range of the first plateau region to that of the whole plateau region
density of matrix material density of honeycomb relative density of honeycomb impact velocity absorbed energy of honeycomb absorbed energy per unit volume of honeycomb volume of honeycomb low-velocity crushing strength high-velocity crushing strength contact stress between the top rigid plate and the RSH contact stress between the bottom rigid plate and the RSH first plateau stress of the RSH under low-velocity mode second plateau stress of the RSH under low-velocity mode time instant at the beginning of a collapse period time instant at the end of a collapse period height of the representative block at height of the representative block at variation of the momentum from to
1. Introduction Auxetic material, also known as material with negative Poisson’s ratio, expands when stretched and contracts when compressed [1]. Auxetic materials exist in nature, such as cancellous bone [2], cow teat skin [3], aquatic salamander skin [4], -cristobalite [5], and some face-centered cubic (FCC) crystals [6]. First man-made auxetic material was produced by transforming conventional low-density open-cell foam into a re-entrant structure [7]. For the past few decades, auxetic materials have received extensive attention all over the world because of the excellent mechanical properties, including enhancements of shear resistance, indentation resistance, high fracture toughness, and synclastic curvature, etc. [8-13], which promise the auxetic material to be an excellent energy absorber [14]. These potential applications recently triggered many interests in investigating the dynamic properties of the auxetic material with experiments, theoretical analysis and
finite element simulations. Zhang et al. [15] systematically discussed the in-plane crushing behaviors of auxetic re-entrant honeycombs (RHs) with different cell wall aspect ratios and cell-wall angles, and concluded that the plateau stresses increase quadratically with the increase of impact velocity. A further comparison between the RHs and conventional hexagon honeycombs was performed by Liu et al. [14] and Hou et al. [16], and they confirmed that the RHs can absorb more energy than conventional hexagon honeycombs under the same impact strain range. Jin et al. [17] and Chang et al. [18] used RHs as sandwich structure cores to improve the blast resistance and dynamic response. Meanwhile, some researchers focused on some other typical auxetic honeycombs, such as double arrowhead honeycombs (DAHs). Qiao et al. [19, 20] investigated the in-plane impact responses of DAHs theoretically and numerically, and confirmed that the energy absorption capacities of functionally graded DAHs could be improved only under high velocity impact. To gain a better jounce bumper with excellent viscoelasticity characteristic to dissipate crushing energy, Wang et al. [21] adopted the double arrowhead structure as the filler and found that the load-displacement curve is smoother, which has a beneficial effect on the vibration performance of entire vehicle. Many researchers have done a great deal of researches on the dynamic crushing behaviors of typical auxetic materials, and obtained some interesting results which are helpful for the engineering application. As we all know, there are two main mechanisms of negative Poisson’s ratio, the re-entrant mechanism and the rolling-up mechanism [22]. How to design auxetic materials with better mechanical properties is an interesting issue that researchers are paying close attention to. Some researchers added straight ribs or sinusoidal-shaped ribs into each cell of the typical re-entrant structure to improve its in-plane mechanical properties and energy absorption capacities [23-28]. Harkati et al. [29] developed a refined analytical model of a multi-reentrant honeycomb taking into account the effect of bending and shearing. Theocaris et al. [30] and Meng et al. [31] analyzed the equivalent mechanical behaviors of star-shaped honeycombs (SSHs). Ma et al. [32], being inspired by the star-shaped structure, proposed a novel re-entrant square honeycomb, and studied the influence of geometric features on the
in-plane dynamic crushing behaviors by finite element method. Lu et al. [33] proposed two novel auxetic 3D cross chiral structures with higher Young’s modulus. Amer Alomarah et al. [34] firstly introduced a new re-entrant chiral auxetic structure which combines the microstructure features of the RHs and the anti-tetrachiral honeycomb. They found the new auxetic structure shows more excellent tensile properties and auxetic features compared with the RHs. With the development of 3D printing technology, it gives us a chance to manufacture these micro-cellular structures easily. Some researchers [35, 36] have shown experimental study about these structures produced by 3D printing. In this paper, a novel re-entrant auxetic structure is proposed to obtain superior energy absorption capacity. The crushing behavior of this new auxetic honeycomb under the in-plane impact loading is systematically studied and compared with the RH and SSH. Both the theoretical strength for the low-velocity and high-velocity crushing are deduced and compared with the results of finite element simulations. In addition, the effects of the cell wall thickness and the impact velocity on the crushing strength of the RSH are discussed in detail. 2. Re-entrant star-shaped honeycombs 2.1 Geometric models RH (Fig. 1a) and SSH (Fig. 1b) are the two typical auxetic honeycombs based on the re-entrant mechanism. Combining the microstructure features of these two typical auxetic honeycombs, a new auxetic honeycomb (Fig. 1c) is obtained and therefore named as re-entrant star-shaped honeycomb (RSH). Fig. 1 also shows the unit cells of the RH, SSH and RSH. The geometries of these structures can be determined by the length of the horizontal cell wall , the length of the inclined cell wall , the angle and the cell wall thicknesses .
Fig. 1. Combining the re-entrant honeycomb (a) and the star-shaped honeycomb (b), a new re-entrant star-shaped honeycomb (c) can be obtained.
For an auxetic honeycomb, the deformation mode and energy absorption performance are highly related to the two important factors: relative density and impact velocity. Based on the previous theoretical analysis [37], the relative densities of the RH, SSH and RSH can be calculated, respectively, as
RH
* 1 4 l h SSH t s l 1 2 2l sin 4 h
SSH
RSH where
,
respectively. respectively.
is
(1)
(2)
2
* h l 2t l cos 8 RSH 1t s 2 l h l 2sin 2cos cos
and
(3)
are the relative densities of the RH, SSH and RSH,
,
and
are the densities of the RH, SSH and RSH,
is the density of the bulk material. In this paper, the length of the
horizontal cell wall angle
* RH 1 t h l 2 s 2 l h l sin
is
, the length of inclined cell wall
is
, and the
for RH, SSH and RSH. When the cell wall thickness of the RSH is set
to be 0.1200, 0.2435, 0.3706 and 0.5022 mm, the corresponding relative densities of the RSH
are obtained as 0.05, 0.1, 0.15 and 0.2, respectively. Fig. 2 gives the
variation of relative densities of the RH, SSH and RSH with different cell wall thicknesses and different angles. Relative densities of the RH and SSH increase
linearly with the increase of cell wall thickness, while that of the RSH has a nearly linear increase. In addition, the relative density of the RH has a lower rate of increase with the increase of cell wall thickness than that of the SSH and RSH (Fig. 2a). Relative densities of all these three auxetic honeycombs increase nonlinearly with the increase of angle
(Fig. 2b).
Fig. 2. Variation of relative densities of the RH, SSH and RSH. (a) Variation of relative densities with cell wall thicknesses (h=10 mm, l=5 mm,
=30 ); (b) variation
of relative densities with angle θ (h=10 mm, l=5 mm, t=0.2435 mm). 2.2 Finite element models The dynamic crushing behaviors and energy absorption properties of the new RSH are systematically studied by finite element method with the commercial software package ANSYS/LS-DYNA explicit code, and the dynamical performances of RH are also simulated for a comparison. The diagrammatic sketches of finite element models of the RSH and the RH are shown in Fig. 3. The matrix material of honeycomb is aluminum alloy AA6060 T4 and modeled as rate independent elastic-perfectly plastic with Young’s modulus MPa, and Poisson’s ratio
=68.2 GPa, density
=2700 kg/m3, yield stress
=80
=0.3 [38]. The local plasticity can be descripted by the
constitutive relationship of the matrix material. As shown in Fig. 3, the auxetic honeycomb models for numerical simulation are positioned between two rigid plates, with the bottom plate fixed and the upper plate impacting the honeycomb along the y direction with a certain constant velocity. The
impact velocity
varies from 1 to 110 m/s in order to analyze the effect of velocity on
deformation modes and energy absorption capabilities. For comparison, the geometrical parameters of RSH and RH are set to be same, including the length of the horizontal cell wall h, the length of the inclined cell wall l and the angle wall thickness
. The cell
varies with the change of the relative density. A 4-node element, type
Shell163, is used to mesh honeycomb cell walls. Considering the convergence of the calculation and mesh sensitivity, five integration points through the thickness are adopted in the following simulations, with element size of 0.5 mm. In order to improve the computation efficiency, the depth of all the honeycombs in out-of-plane direction is the same as the element size. The out-of-plane displacement of each node in each finite model is constrained to ensure the plane strain situation, with all the nodes of honeycombs free to move in the in-plane directions (x and y direction in Fig. 3). To avoid probable penetration, automatic single surface contact is defined on all the surfaces of honeycomb cell walls for self-contacting, and automatic surface-to-surface contact is defined between the contact faces of honeycombs and the two rigid plates. The emphasis of this paper is focused on the enhancement of the in-plane impact resistance of the honeycomb with novel structures. The possible presence of the cell defects is neglected for the simulations in this paper.
Fig. 3. Diagrammatic sketches for auxetic honeycombs under in-plane impact loading. (a) RH; (b) RSH.
3. In-plane crushing behaviors of RSH under impact loading In this section, three types of deformation modes for the RSH under the y-directional impact are studied and compared with the RH. Both the theoretical crushing strengths for low-velocity impact and high-velocity impact are developed and compared with the results of the finite element simulations. 3.1 Finite element analysis 3.1.1 Low-velocity mode Fig. 4 shows the low-velocity modes of RSH and RH under the impact velocity of 1 m/s along the y direction. As we can see, the RH and RSH shrink horizontally when compressed along the y direction, and an obvious “X”-shaped band can be observed for both the RH and RSH topologies in the initial impact stage (Fig. 4b). The “X”-shaped band of the RSH continue to propagate with the time until the top half of the “X”-shaped band intersects
with
its
bottom
half
(Fig.
4c-f).
Then,
a
parallelogram-shaped core forms in the RSH and begins to collapse (Fig. 4g and h). The shapes of unit cells of both RH and RSH change into parallelogram shapes during the impact processes. The cell walls of the RH begin to collapse after few unit cells change into parallelogram shapes (Fig. 4e-h). However, almost all the unit cells of the RSH maintain parallelogram shapes at the same time (Fig. 4g), and only after that the layer-by-layer collapse begins (Fig. 4h). Fig. 5 shows more details of the local deformation processes of the RSH. The number of inclined struts per inclined edge of the parallelogram shape in the RSH is very different from that in the RH (Fig. 5d). Each inclined edge of parallelogram shape in the RSH has two struts, but only one in the RH, which results in the increasing of the stress level and the production of the second plateau region in the strain-stress curve of the RSH (Fig. 6).
(a)
(b)
(e)
(c)
(f)
(d)
(g)
(h)
Fig. 4. Deformation processes of the RSH and RH under low-velocity impact ( =1 m/s) along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). The red dashed lines
are added in the figures to highlight the localized deformation regions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 5. A typical localized deformation processes of the RSH picked from Fig. 4. (a) Stress free state; (b)-(c) initial deformation; (d) the formation of parallelogram-shaped core; (e)-(g) the parallelogram-shaped core is collapsed layer by layer; (h) densification.
Based on the analysis of Fig. 4 and Fig. 5, we can divide the low-velocity collapse process of the RSH into five distinct phases (Fig. 6). The whole RSH structure enters the elastic state firstly (state I). After reaching the initial peak stress, the plastic hinges form at both ends of the two inclined opposite struts (BC and EF) and one end of the four inclined struts (BG, BH, EI and EJ) of some unit cells in the RSH (Fig. 5b). The rest of inclined struts (AB and ED) of these unit cells keep unchanged. So that an “X”-shaped band is observed. As the local deformation with “X”-shaped band grows up and propagates, the normal contact stress between the RSH and the top rigid plate tends to be stable and the first plateau stress region occurs (state II). The state III is a transitional region during which the “X”-shaped band disappears and the parallelogram-shaped core is forming, meanwhile the stress in Fig. 6 increases. After that, more plastic hinges form at both ends of the four inclined opposite edges (A B , B G , E J and E D in Fig. 5d) of the parallelogram structure. The layer-by-layer collapse process of the parallelogram-shaped core follows. As a result a higher stress level, the second plateau stress region, is formed (state IV). Eventually, each adjacent row of cells collapse and contact, so that the stress increases sharply and the compressive densification is detected (state V).
Fig. 6. The strain-stress curves of the RSH and RH obtained by finite element simulations with the same conditions in Fig. 4, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 4. The five
deformation states of the RSH are also appended. 3.1.2 Medium-velocity mode Fig. 7 shows the deformation processes of the RSH under the medium-velocity impact with velocity of 25 m/s along the y direction. With the increase of the impact velocity, the inertia effect begins to emerge. Several rows of cells near the impact end are slightly crushed, and localized collapses from the top layer step-by-step are observed for the RSH in the initial impact stage (Fig. 7b-d). With the increase of impact displacement, an incomplete “X”-shaped band can be observed for the RSH (Fig. 7e-g). Unlike the low-velocity mode, most unit cells of the RSH do not change into the parallelogram shapes (Fig. 5d) at the same time, so that there is not a parallelogram-shaped core formed.
(a)
(e)
(b)
(f)
(c)
(d)
(g)
(h)
Fig. 7.Deformation processes of the RSH and RH under the impact velocity of 25 m/s along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). The red dashed lines
are added in the figures to highlight the localized deformation regions.
According to the deformation process of the RSH (Fig. 7), the medium-velocity mode can be divided into four distinct phases (Fig. 8). Firstly, the elastic deformation of the entire RSH structure is detected (state I). When the contact stress between the top rigid plate and the RSH reaches the initial stress peak, the plastic hinges begin to form at both ends of the struts of the re-entrant structures in the RSH. Plastic hinges
also form at the middle position of several rows of cells near the impact end. So we can see the localized collapse from the top layer step-by-step in Fig. 7b-d, which results in the stress undulation in state II. The state III is a plateau stress region during which the incomplete “X”-shaped band forms and then disappears gradually. Due to the exist of the localized collapse at the top layer and the incomplete “X”-shaped band, some unit cells of the RSH change into various parallelograms with various geometrical parameters, and some collapse directly without changing into parallelograms. Eventually, each adjacent row of cells contact and the compressive densification occurs (state IV).
Fig. 8. The strain-stress curves of the RSH and RH obtained by finite element simulations with same conditions in Fig. 7, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 7. The four deformation states of the RSH are also appended. 3.1.3 High-velocity mode When the RSH is subjected to a relatively high impact velocity, saying in Fig. 9, the cell walls near the impact end are crushed first. With the increase of compression strain, the “I” localized deformed band forms and propagates forward row-by-row to the fixed end until the RSH is totally crushed. The RH also shows the “I”-mode during high impact velocity crushing [15]. Remarkably, there is no similar
“X”-shaped band and parallelogram-shaped core observed. Due to the high impact velocity, the cell walls at the impact end do not get much time to shrink transversely, so that the RSH at the high-velocity mode do not display obvious dynamic negative Poisson’s ratio.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 9. Deformation processes of the RSH under high velocity impact ( =100 m/s) along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ).
According to the deformation processes in Fig. 9, the high-velocity mode can be divided into three distinct phases (Fig. 10). The deformation of the RSH is elastic and uniform throughout the structure at the beginning (state I). Beyond the initial peak stress, the plastic hinges begin to form at the ends and the middle positions of all the horizontal and inclined struts near the impact end. And then the “I”-shaped band forms at the upper edge of the RSH and propagates forward layer-by-layer to the fixed end, which results the repeatable stress undulation in state II (Fig. 10). Eventually, each adjacent row of cells contact and the compressive densification occurs (state III). In order to describe the collapse process accurately, a
cells block is taken
from the finite element model in Fig. 9 and exhibited in Fig. 11. A repeatable collapse process can be observed in Fig. 11b-e, which indicates that the deformation configuration of the two top half cells of the block in Fig. 11b are the same as that of the two bottom half cells of the block in Fig. 11e. Also the same states can be found between Fig. 11c and Fig. 11f, Fig. 11d and Fig. 11g. Furthermore, the collapse process of the RSH is regular and repeatable, which give us a chance to formulate the
high-velocity crushing strength
.
Fig. 10. The strain-stress curves of the RSH and RH obtained by finite element simulations with same conditions in Fig. 9, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 9. The three deformation states of the RSH are also appended.
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
Fig. 11. Close-up view of the typical dynamic collapse processes of the RSH obtained by the finite element simulation. (a) The original configuration of the block; (b)-(e) the collapse processes of the two upper unit cells; (e)-(h) the collapse processes of the two lower unit cells.
3.2 Theoretical analysis 3.2.1 Low-velocity mode When the bending moments of its cell walls exceed the plastic limit, plastic collapse will take place in the honeycomb, which results in the plateau region of the strain-stress curve. According to the literature [37], the upper limit of the plateau stress can be deduced by the relationship between the working done by external force on the entire unit cell and the plastic dissipation at all hinges of a unit cell, and the lower limit can be given by the relationship between the maximum bending moment of a single strut in unit cells and the plastic bending moment of this single strut.
Fig. 12. (a) The initial configuration of the unit cell at the beginning of the first plateau region (state II); (b) the initial configuration at the beginning of the second plateau region (state IV), as well as the final configuration at the end of state III.
During the collapse process of the RSH, two plateau regions occur and the second plateau stress level is almost twice higher than the first one. Fig. 12 shows the initial configurations of the RSH unit cell at the beginning of the first and second plateau regions with the plastic hinges highlighted by red circles. Eight plastic hinges form in a unit cell at the beginning of both first and second plateau regions. The angle between the strut AB and AF at the first plateau region (state II) remains unchanged (Fig. 12a) until the unit cell of the RSH changes into a parallelogram shape (Fig. 12b).
The same situation also occurs in the angle between the strut CD and DE (Fig. 12). In addition, the configuration in Fig. 12b is exactly the initial configuration at the beginning of the second plateau region (state IV). When the four plastic hinges at both ends of the strut EF, and one end of the strut BG and EI (Fig. 12a) rotate a degree of
, the relationship between the working done
by external force on the entire unit cell and the plastic work at the joints can be given as 4M p1 ybL0l sin
where
is the plastic bending moment of the inclined strut,
angle of the hinge,
(4) is the rotational
is the depth of the RSH in out-of-plane direction, and
original length of a unit cell of RSH,
is the
. One has
for a rectangular beam. From Eq. (4), the upper limit of the first plateau stress
can be obtained as
t p1 l
2
ys h l 2sin 2cos sin
(5)
The relationship between the maximum bending moment of a single strut and its plastic bending moment
can be expressed as
M p1 M max1 where the first plateau stress
, and
(6) . From Eq. (6), the lower limit of
can be written as
t p1 l
2
ys h l 2sin 2cos sin
(7)
From Eqs. (5) and (7), it should be noted that the upper and lower limit of the first plateau stress of the RSH are same. With the same methods above, the upper and lower limit of the second plateau stress
can be obtained. The initial configuration
at the beginning of the second plateau region (state IV) has two struts per inclined
edge (Fig. 12b). This inclined edge can be considered as one strut whose thickness is equal to twice that of an original strut. So that the two opposite inclined edges would produce four plastic hinges at both ends. Thus, the upper limit of the second plateau stress
can be deduced from 2M p2 ybL0l sin
where
(8)
is the plastic bending moment of the equivalent strut, one has . From Eq. (8), the upper limit of the second plateau stress
can be
expressed as 2 2 ys t p2 l h l 2sin 2cos sin
(9)
The lower limit can be given by the relationship between the maximum bending moment of a single inclined strut inclined strut
and the plastic bending moment of a single
as
M p3 M max2 where
,
(10)
. Nominal strain and nominal stress are
adopted in this paper. And the length of unit cell after deformation is set to be its original length. Thus, one has stress
. The lower limit of the second plateau
can be expressed as
t p2 l
2
2 ys
h l 2sin 2cos sin
(11)
From Eqs. (9) and (11), we can see that the upper and lower limit of the second plateau stress are equal. In case of the honeycombs with only one plateau region in low-velocity mode, like the regular hexagonal honeycomb and the RH, it is easy to ensure their low-velocity crushing strengths, which are exactly their plateau stresses. However, for the RSH studied in this paper, it has two different plateau regions and the second one is almost two times higher than the first one. To formulate the low-velocity crushing strength of
the RSH, a proportional coefficient
is introduced and defined as the ratio of the
strain range of the first plateau region to that of the whole plateau region. Thus, the low-velocity crushing strength is expressed as
0 p1 1 p2 Considering both the upper and lower limit of
(12) and
are same, we can
2 ys 2 t 0 l h l 2sin 2cos sin
(13)
simplify Eq. (12) as
According to the strain-stress curve of the RSH in Fig. 6, it can be found that the strain range corresponding to the first plateau region is almost from 5% to 30%, and that corresponding to the second one is almost from 50% to 75%. Considering the effect of the stress of the transitional region on the low-velocity crushing strength, is set to be 0.6 for the specific structure in this paper. 3.2.2 High-velocity mode Based on the close-up view of the high-velocity collapse process in Fig. 11, a representative block consisting of two upper and lower adjacent unit cells 1 and 2 (Fig. 13a) is employed to reveal the deformation mechanism. The initial ( final (
) and the
) structure feature of the block in a collapse period are shown in Fig. 13b
and c, respectively, where
is the crushing time. The deformation states in Fig. 13b
and c correspond to that of cells in the finite element models in Fig. 11b and e, respectively.
and
in Fig. 13 are the crushing stress at the top and the bottom of
the representative block, respectively. It is worth noting that
equals
in the
low-velocity mode in section 3.2.1, but they are not equal for the situation under the high impact velocity. Actually,
also represents the contact stress between the top
rigid plate and the RSH, which is the high-velocity crushing strength we want to formulate here. The unit cell 2 in Fig. 13 is almost in a static state, so the contact stress between the bottom rigid plate and the RSH, the low-velocity crushing strength
, can be approximately equal to
. Hou et al. [16] and Hu et al. [39] adopted the
same assumption to analyze the high-velocity crushing strength of the RH and the hexagonal honeycomb, respectively. This assumption can be confirmed further by Fig. 14, in which the black line represents
in Eq. (13).
Fig. 13. (a) A representative block; (b) the initial configuration of the block at the beginning of a collapse period period
; (c) the final configuration at the end of a collapse
.
Fig. 14. The contact stress between the bottom rigid plate and the RSH under various impact velocities.
At the beginning of a collapse period (Fig. 13b), the strut AF just joined the
densified region and the other cell walls of the upper unit cell deform slightly. More significantly, the upper unit cell in Fig. 13b and the lower unit cell in Fig. 13c have same configuration, which means that the momentum of cell walls of the upper unit cell at
equals that of cell walls of the deformed lower unit cell at
, i.e.,
p1T0 p2Tf
where
and
(14)
are the momentum of the upper unit cell at
lower unit cell at
and that of the
. More specifically, one has
T0 T0 T0 T0 T0 Tf Tf Tf Tf Tf pAF pCD ; pAB pCK ; pEF pDN ; pBG pKO ; pBH pKP ;
(15)
T0 T0 T0 T0 T0 Tf Tf Tf Tf Tf pEI pNQ ; pEJ pNR ; pBC pKL ; pCD pLM ; pDE pMN ;
where
is the momentum.
The cell walls of the lower unit cell almost hold still at the beginning of a collapse period (Fig. 13b). Thus, the momentum of the lower unit cell at
,
, can be given
as p2T0 0
(16)
T0 T0 T0 T0 T0 T0 T0 T0 T0 pKC pKO pKP pKL pLM pMN pNR pNQ pND 0
(17)
To be more concrete, one has
In addition, the cell walls of the upper unit cell at
are collapsed and joined the
densified region. Therefore, all the cell walls of that has the same constant velocity as the densified region, and their momentums can be given as Tf Tf Tf Tf Tf Tf Tf pAB pCB pDE pEF pBG pBH pEITf pEJ stblv
(18)
Tf pAF stbhv
(19)
Therefore, the momentum of the upper unit cell at
,
, is
Tf Tf Tf Tf Tf Tf Tf Tf Tf p1Tf pAF pAB pCB pDE pEF pGB pHB pEI pEJ
Thus, the variation of the momentum from
to
,
p p1Tf p2Tf p1T0 p2T0
Combining Eqs. (14)-(20), Eq. (21) is simplified as
(20)
, is (21)
p 8l h stbv
(22)
Based on the conservation of momentum, one has
bL0
1 2 dt p
Tf
T0
where
for the RSH. From Eq. (23), the high-velocity
crushing strength,
, is expressed as
d As shown in Fig. 13, and
(23)
1 Tf T0
and
Tf
T0
1dt 0
p bL0 Tf T0
(24)
are the height of the representative block at
, respectively. Hence, the cycle time of the high-velocity collapse process is
Tf T0
H 0 H f 2l cos 5t v v
By substituting Eqs. (22) and (25) into Eq. (24),
(25)
can be simplified as
ht 8 s l l d 0 v2 t h 2 cos 5 2sin 2 cos l l
(26)
3.3 Effects of impact velocity and cell wall thickness Fig. 15a shows plateau stresses of the RSH obtained by finite element method under low-velocity mode, medium-velocity mode and high-velocity mode, and also gives theoretical results, including the low-velocity crushing strength high-velocity crushing strength
and the
. The local details of the plateau stresses under
low-velocity mode are enlarged in Fig. 15b. As we can see, the results of the finite element simulations are consistent with the theoretical results. The cell wall thickness and the impact velocity are the two key factors to determine the plateau stress. As the impact velocity increases under low-velocity mode, the inertial effect leads to the increase of the first plateau stress, and the second one decreases slightly because the number of unit cells which turn into parallelogram shapes at the same time drop. The plateau stress under high-velocity mode is in direct proportion to the square of the impact velocity, while that increases a little for the medium-velocity mode.
Fig. 15. (a) Variation of plateau stress for the RSH with different cell wall thicknesses under low-velocity mode, medium-velocity mode and high-velocity mode; (b) close-up view of the plateau stresses under low-velocity mode.
The value of plateau stress can directly reflect the energy absorption ability of honeycomb, which means high plateau stress usually corresponds to stronger energy absorption ability. The absorbed energies of the RSH with same conditions in Fig. 15 are illustrated in Fig. 16. Note that the absorbed energy of the RSH with the minimum impact velocity is not the least. Specifically, about 8 m/s, 10 m/s and 12 m/s are the three impact velocities corresponding to the least absorbed energy of the RSH for t=0.1200 mm, t=0.2345 mm, and t=0.3706 mm, respectively. Interestingly, 8 m/s, 10 m/s and 12 m/s are exactly the first critical impact velocity (Eq. (30)) between the low-velocity mode and the medium-velocity mode, which indicates the RSH with this first critical impact velocity absorbs the lowest energy. Under the first critical impact velocity, few unit cells of the RSH simultaneously change into the new parallelogram structure with two struts per inclined edge and the second plateau region in strain-stress curve is disappearing. We already know that the second plateau stress is almost twice higher than the first one. The disappearance of the second plateau region inevitably results in the decrease of absorbed energy when the impact velocity is low and the inertial effect is not obvious. When almost all the unit cells change into the new parallelogram structure firstly and then collapse, the RSH can absorb more impact energy than the situation of most unit cells collapsing directly under a
relatively low velocity impact. In other words, the increase of the impact velocity has less contribution to the absorbed energy than the new structure under low-velocity mode. As the impact velocity increases, the inertial effect prevails. Thus the absorbed energy rises when the impact velocity is faster than the first critical impact velocity.
Fig. 16. Variation of absorbed energy for the RSH with different cell wall thicknesses under various impact velocities. 4. Discussions 4.1 The main failure mechanism of RSH The dynamic crushing behaviors of honeycombs are studied by finite element method with the commercial software package ANSYS/LS-DYNA in this paper, and the matrix material of honeycomb is modeled as elastic-perfectly plastic. The failure mechanism of honeycomb cell walls can be either elastic buckling or plastic collapse, depending on its slenderness ratio [40]. When the axial load applied to cell wall exceeds the critical force
, elastic buckling would occur firstly. According to the
results of Timoshenko and Gere [41], one has
Pcrit
n 2 2 Es I l2
(27)
where n is the end constraint factor that depends on the degree of the constraint to rotation at the ends of cell wall. For the cell wall of honeycomb loaded by axial force,
the constraint on the wall by neighboring joints lies between these limits, that is 0.5 n 2 [37]. For the hexagonal honeycomb, Gibson and Ashby [37] used the value n=0.69 because the joints can rotate. The axial load applied to cell wall EF (Fig. 12a) equals
. When
, the failure mechanism of honeycomb cell wall would be elastic buckling. The elastic buckling stress of RSH,
, can be given as
n2 2 Es t e l 6cos h l 2sin 2cos 3
The first plateau stress
(28)
is already given in section 3.2.1 (Eqs. (5) and (7)),
which is the minimum plastic collapse stress of RSH. When e p1 , the elastic buckling of RSH would occurs firstly before it enters the yielding stage. Thus, the critical cell wall slenderness ratio of RSH can be determined as
6 cos t 2 2ys l n Es sin
(29)
When the lower limit value of n is taken, that is n=0.5, and t/l is 0.004941. However, the minimum value of t/l in this paper is 0.024 when the relative density of RSH is 0.05, which is much larger than 0.004941. Therefore, plastic collapse is the main failure mechanism in our numerical models, which is also further confirmed by the finite element simulations in section 3.1. It is can be observed that plastic hinges form at the ends and middle position of cell walls of RSH in Fig. 4, 7 and 9. 4.2 Deformation modes map Three types of deformation modes for the RSH are detected under different impact velocities in section 3, including low-velocity mode, medium-velocity mode and high-velocity mode. Deformation modes are sensitive to the variation of impact velocity and relative density. Therefore, a deformation map of the RSH is summarized in Fig. 17, where the black and red line are the critical transitions between two adjacent modes for the RSH, and the orange and blue line are that for the RH [14].
When impact velocity is below the black line in Fig. 17, the deformation mode of RSH is considered as the low-velocity mode. When impact velocity is over the red line in Fig. 17, the deformation mode of RSH is the high-velocity mode. When impact velocity is between the black line and the red line in Fig. 17, the deformation mode of RSH is distinguished as the medium-velocity mode. According to this map, the RSH would require higher impact velocity for entering the high-velocity mode than the RH, which indicates that the RSH has better impact resistance than the RH. The critical impact velocity depends linearly on the relative density, which means the critical impact velocity rises with the relative density increasing. The empirical formulas of the two critical impact velocities of the RSH are achieved, respectively, as
where
vc1 4.75 43RSH
(30)
vc2 30 130RSH
(31)
is the first critical impact velocity between the low-velocity mode and the
medium-velocity mode,
is the second critical impact velocity between the
medium-velocity mode and the high-velocity mode.
Fig. 17. Deformation modes map of the RSH.
4.3 Comparisons with the RH and SSH The comparisons of energy absorption capacities for the RH, SSH and RSH under different impact loadings are investigated. Due to the different overall lengths and heights between the three honeycombs, the absorbed energy per unit volume adopted, which is defined as
, where
is
is the volume of honeycombs. Fig.
18 shows the comparisons of energy absorption capacities between the RH, SSH and RSH under different impact velocities and Table 1 presents the specific data. The impact velocities of 1 m/s, 25 m/s and 100 m/s exactly correspond to the three deformation modes of the RSH. It is easy to see that the RSH can absorb more impact energy than the RH with same cell wall thickness under all the three impact velocities. Specifically, the absorbed energy of the RSH is 64.26% larger than that of the RH under the impact velocity of 1 m/s and 49.85% larger under the impact velocity of 100 m/s. The reason that the RSH under low-velocity impact can absorb much more energy can be attributed to that the unit cells of the RSH turn into parallelogram shapes with two struts per inclined edge. This new parallelogram-shaped core results in the second plateau region in the strain-stress curves of the RSH. However, the RSH under high-velocity impact does not have enough time to change the shape of unit cells and therefore collapses directly. In addition, the absorbed energy of the RSH under the impact velocities of 1 m/s and 25 m/s is close to that of the SSH. The absorbed energy of the SSH increases rapidly after the compressive strain reaching 0.6 under the impact velocities of 1 m/s and the compressive strain reaching 0.7 under the impact velocities of 25 m/s. This rapid increase indicates that the densification of the SSH is much too early.
Fig. 18. Comparisons of energy absorption capacities for the RH, SSH and RSH under
different impact velocities (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ).
Table 1 Absorbed energy by the RH and RSH under different velocities (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). (kJ/m3)
Velocity (m/s)
(kJ/m3)
1
117.40
71.48
+64.24%
25
153.85
102.48
+50.13%
100
1207.62
810.32
+49.03%
4.4 Auxetic performances The variation of the dynamic Poisson’s ratio and the corresponding transverse strain of the RH, SSH and RSH under different y-directional impact velocities are shown in Fig. 19a-c, respectively. Dividing the average shrinkage mass of the transverse lengths of the nodes between the right and left edge of the finite element model by the original transverse length is the transverse strain here. And the dynamic Poisson’s ratio is defined as the negative ratio of the transverse strain and the longitudinal strain. Auxetic honeycombs contract literally when compressed longitudinally. Thus the transverse strain and the longitudinal strain here represent compressive strain, and only the absolute values of them are picked. As shown in Fig. 19, the properties of auxetic performances for RH, SSH and RSH are rather obvious, especially in the low-velocity mode ( =1 m/s) and the medium-velocity mode ( =25 m/s). The transverse contraction of the RSH under the low-velocity mode occurs mainly at the first plateau region, while that of the RH and SSH happens at the entire impact process (Fig. 19a).
Fig. 19. Variation of the dynamic Poisson’s ratio of the RH, SSH and RSH under different impact velocities along the y direction (h=10 mm, l=5 mm, t=0.2435 mm, =30 ). (a)
=1 m/s; (b)
= 25 m/s; (c)
=100 m/s.
5. Conclusions Combining two classical auxetic structures, RH and SSH, a novel auxetic structure named as RSH is proposed with the in-plane crushing behaviors of this new honeycomb studied systematically. Three types of deformation modes for the RSH are detected under different impact velocities, including low-velocity mode, medium-velocity mode and high-velocity mode. An obvious “X”-shaped band can be observed in the initial impact stage under low-velocity crushing, then most unit cells turn into parallelogram shapes with two struts per inclined edge at the same time which leads to the second plateau region. The second plateau stress is almost twice higher than the first one. The localized collapse
forms firstly at several rows of cells near the impact end, after that an incomplete “X”-shaped band can be observed under medium-velocity impact loading. The “I”-shaped band forms at the upper edge of the RSH and propagates forward layer-by-layer to the fixed end under high-velocity mode. The auxetic performance for the RSH is rather obvious, especially under the low-velocity and the medium-velocity mode. And the transverse contraction of the RSH mainly occurs at the first plateau region. The results of finite element simulation show that the RSH can absorb more impact energy than the RH and SSH with same cell wall thickness. Specifically, the absorbed energy of the RSH is 64.26% larger than that of the RH under the impact velocity of impact velocity of
and 49.85% larger under the
. Unlike the RH in which the absorbed energy increases
with the increase of the impact velocity, the absorbed energy of the RSH decreases slightly and then increases with the increase of the impact velocity. The velocity with the least absorbed energy of the RSH is exactly the first critical impact velocity between the low-velocity and the medium-velocity mode. In addition, the RSH would require higher impact velocity for entering the high-velocity mode than the RH, which indicates that the RSH has better impact resistance. Furthermore, theoretical models of the low-velocity and high-velocity crushing strength are developed. The upper and the lower limit of the two plateau stresses are formulated based on the theory of plastic dissipation under low-velocity impact. The high-velocity crushing strength is given based on the conservation of momentum. Both the theoretical low-velocity and high-velocity crushing strengths are coincident with the finite element method results. Acknowledgement The authors thank the support from the National Natural Science Foundation of China (Grant No. 11672013, 11672014 and 11472025) and Defense Industrial Technology
Development
Program
(Grant
No.
JCKY2016601B001
and
JCKY2016205C001) and the Fundamental Research Funds for the Central Universities.
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S0263-8223(18)31753-7 https://doi.org/10.1016/j.compstruct.2018.10.024 COST 10274
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
14 May 2018 3 September 2018 8 October 2018
Please cite this article as: Wang, H., Lu, Z., Yang, Z., Li, X., A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.10.024
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A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance Huan Wang, Zixing Lu, Zhenyu Yang*, Xiang Li Institute of Solid Mechanics, Beihang University, Beijing 100083, P.R. China Abstract Recently, auxetic honeycombs have attracted considerable attention for their excellent mechanical properties, especially the potential applications for energy absorption. In this paper, a novel re-entrant auxetic honeycomb is proposed and named as re-entrant star-shaped honeycomb (RSH), with the in-plane impact responses explored theoretically and numerically. Three types of microstructural deformation modes are observed
under
different
impact
velocities,
including
low-velocity mode,
medium-velocity mode and high-velocity mode. Moreover, a deformation map is summarized to illustrate the effects of the impact velocity and the relative density on the deformation modes. Two typical plateau stress regions are detected under low-velocity impact loading, and the second plateau stress is almost twice higher than the first one. The transverse contraction of the RSH mainly occurs at the first plateau region, which is different for the classical re-entrant honeycomb (RH). In addition, the absorbed energy of the RSH decreases slightly and then increases with the impact velocity. The results of the finite element simulations suggest that the RSH shows more excellent impact resistance, compared with the RH and SSH with same cell wall thickness. Furthermore, the effects of the cell wall thickness and the impact velocity on the crushing strength of the RSH are discussed, and the theoretical results are in good agreement with the finite element simulations.
Keywords Auxetic honeycomb; Deformation mode; Plateau stress; Impact velocity; Energy absorption *
Author to whom correspondence should be addressed. Electronic mail: [email protected]
Nomenclature Young’s modulus of matrix material yielding stress of matrix material Poisson’s ratio of matrix material length of the horizontal cell wall length of the inclined cell wall depth of honeycomb in out-of-plane direction cell wall thickness of honeycomb length of a RSH unit cell cell wall angle defined in Fig. 1 maximum bending moment of inclined strut of the RSH in Fig. 12a maximum bending moment of inclined strut in Fig. 12b plastic bending moment of inclined strut of the unit cell of RSH in Fig. 12a plastic bending moment of the double-thickness inclined strut in Fig. 12b plastic bending moment of a single inclined strut in Fig. 12b critical impact velocity between low-velocity and medium-velocity mode critical impact velocity between medium-velocity and low-velocity mode ratio of the strain range of the first plateau region to that of the whole plateau region
density of matrix material density of honeycomb relative density of honeycomb impact velocity absorbed energy of honeycomb absorbed energy per unit volume of honeycomb volume of honeycomb low-velocity crushing strength high-velocity crushing strength contact stress between the top rigid plate and the RSH contact stress between the bottom rigid plate and the RSH first plateau stress of the RSH under low-velocity mode second plateau stress of the RSH under low-velocity mode time instant at the beginning of a collapse period time instant at the end of a collapse period height of the representative block at height of the representative block at variation of the momentum from to
1. Introduction Auxetic material, also known as material with negative Poisson’s ratio, expands when stretched and contracts when compressed [1]. Auxetic materials exist in nature, such as cancellous bone [2], cow teat skin [3], aquatic salamander skin [4], -cristobalite [5], and some face-centered cubic (FCC) crystals [6]. First man-made auxetic material was produced by transforming conventional low-density open-cell foam into a re-entrant structure [7]. For the past few decades, auxetic materials have received extensive attention all over the world because of the excellent mechanical properties, including enhancements of shear resistance, indentation resistance, high fracture toughness, and synclastic curvature, etc. [8-13], which promise the auxetic material to be an excellent energy absorber [14]. These potential applications recently triggered many interests in investigating the dynamic properties of the auxetic material with experiments, theoretical analysis and
finite element simulations. Zhang et al. [15] systematically discussed the in-plane crushing behaviors of auxetic re-entrant honeycombs (RHs) with different cell wall aspect ratios and cell-wall angles, and concluded that the plateau stresses increase quadratically with the increase of impact velocity. A further comparison between the RHs and conventional hexagon honeycombs was performed by Liu et al. [14] and Hou et al. [16], and they confirmed that the RHs can absorb more energy than conventional hexagon honeycombs under the same impact strain range. Jin et al. [17] and Chang et al. [18] used RHs as sandwich structure cores to improve the blast resistance and dynamic response. Meanwhile, some researchers focused on some other typical auxetic honeycombs, such as double arrowhead honeycombs (DAHs). Qiao et al. [19, 20] investigated the in-plane impact responses of DAHs theoretically and numerically, and confirmed that the energy absorption capacities of functionally graded DAHs could be improved only under high velocity impact. To gain a better jounce bumper with excellent viscoelasticity characteristic to dissipate crushing energy, Wang et al. [21] adopted the double arrowhead structure as the filler and found that the load-displacement curve is smoother, which has a beneficial effect on the vibration performance of entire vehicle. Many researchers have done a great deal of researches on the dynamic crushing behaviors of typical auxetic materials, and obtained some interesting results which are helpful for the engineering application. As we all know, there are two main mechanisms of negative Poisson’s ratio, the re-entrant mechanism and the rolling-up mechanism [22]. How to design auxetic materials with better mechanical properties is an interesting issue that researchers are paying close attention to. Some researchers added straight ribs or sinusoidal-shaped ribs into each cell of the typical re-entrant structure to improve its in-plane mechanical properties and energy absorption capacities [23-28]. Harkati et al. [29] developed a refined analytical model of a multi-reentrant honeycomb taking into account the effect of bending and shearing. Theocaris et al. [30] and Meng et al. [31] analyzed the equivalent mechanical behaviors of star-shaped honeycombs (SSHs). Ma et al. [32], being inspired by the star-shaped structure, proposed a novel re-entrant square honeycomb, and studied the influence of geometric features on the
in-plane dynamic crushing behaviors by finite element method. Lu et al. [33] proposed two novel auxetic 3D cross chiral structures with higher Young’s modulus. Amer Alomarah et al. [34] firstly introduced a new re-entrant chiral auxetic structure which combines the microstructure features of the RHs and the anti-tetrachiral honeycomb. They found the new auxetic structure shows more excellent tensile properties and auxetic features compared with the RHs. With the development of 3D printing technology, it gives us a chance to manufacture these micro-cellular structures easily. Some researchers [35, 36] have shown experimental study about these structures produced by 3D printing. In this paper, a novel re-entrant auxetic structure is proposed to obtain superior energy absorption capacity. The crushing behavior of this new auxetic honeycomb under the in-plane impact loading is systematically studied and compared with the RH and SSH. Both the theoretical strength for the low-velocity and high-velocity crushing are deduced and compared with the results of finite element simulations. In addition, the effects of the cell wall thickness and the impact velocity on the crushing strength of the RSH are discussed in detail. 2. Re-entrant star-shaped honeycombs 2.1 Geometric models RH (Fig. 1a) and SSH (Fig. 1b) are the two typical auxetic honeycombs based on the re-entrant mechanism. Combining the microstructure features of these two typical auxetic honeycombs, a new auxetic honeycomb (Fig. 1c) is obtained and therefore named as re-entrant star-shaped honeycomb (RSH). Fig. 1 also shows the unit cells of the RH, SSH and RSH. The geometries of these structures can be determined by the length of the horizontal cell wall , the length of the inclined cell wall , the angle and the cell wall thicknesses .
Fig. 1. Combining the re-entrant honeycomb (a) and the star-shaped honeycomb (b), a new re-entrant star-shaped honeycomb (c) can be obtained.
For an auxetic honeycomb, the deformation mode and energy absorption performance are highly related to the two important factors: relative density and impact velocity. Based on the previous theoretical analysis [37], the relative densities of the RH, SSH and RSH can be calculated, respectively, as
RH
* 1 4 l h SSH t s l 1 2 2l sin 4 h
SSH
RSH where
,
respectively. respectively.
is
(1)
(2)
2
* h l 2t l cos 8 RSH 1t s 2 l h l 2sin 2cos cos
and
(3)
are the relative densities of the RH, SSH and RSH,
,
and
are the densities of the RH, SSH and RSH,
is the density of the bulk material. In this paper, the length of the
horizontal cell wall angle
* RH 1 t h l 2 s 2 l h l sin
is
, the length of inclined cell wall
is
, and the
for RH, SSH and RSH. When the cell wall thickness of the RSH is set
to be 0.1200, 0.2435, 0.3706 and 0.5022 mm, the corresponding relative densities of the RSH
are obtained as 0.05, 0.1, 0.15 and 0.2, respectively. Fig. 2 gives the
variation of relative densities of the RH, SSH and RSH with different cell wall thicknesses and different angles. Relative densities of the RH and SSH increase
linearly with the increase of cell wall thickness, while that of the RSH has a nearly linear increase. In addition, the relative density of the RH has a lower rate of increase with the increase of cell wall thickness than that of the SSH and RSH (Fig. 2a). Relative densities of all these three auxetic honeycombs increase nonlinearly with the increase of angle
(Fig. 2b).
Fig. 2. Variation of relative densities of the RH, SSH and RSH. (a) Variation of relative densities with cell wall thicknesses (h=10 mm, l=5 mm,
=30 ); (b) variation
of relative densities with angle θ (h=10 mm, l=5 mm, t=0.2435 mm). 2.2 Finite element models The dynamic crushing behaviors and energy absorption properties of the new RSH are systematically studied by finite element method with the commercial software package ANSYS/LS-DYNA explicit code, and the dynamical performances of RH are also simulated for a comparison. The diagrammatic sketches of finite element models of the RSH and the RH are shown in Fig. 3. The matrix material of honeycomb is aluminum alloy AA6060 T4 and modeled as rate independent elastic-perfectly plastic with Young’s modulus MPa, and Poisson’s ratio
=68.2 GPa, density
=2700 kg/m3, yield stress
=80
=0.3 [38]. The local plasticity can be descripted by the
constitutive relationship of the matrix material. As shown in Fig. 3, the auxetic honeycomb models for numerical simulation are positioned between two rigid plates, with the bottom plate fixed and the upper plate impacting the honeycomb along the y direction with a certain constant velocity. The
impact velocity
varies from 1 to 110 m/s in order to analyze the effect of velocity on
deformation modes and energy absorption capabilities. For comparison, the geometrical parameters of RSH and RH are set to be same, including the length of the horizontal cell wall h, the length of the inclined cell wall l and the angle wall thickness
. The cell
varies with the change of the relative density. A 4-node element, type
Shell163, is used to mesh honeycomb cell walls. Considering the convergence of the calculation and mesh sensitivity, five integration points through the thickness are adopted in the following simulations, with element size of 0.5 mm. In order to improve the computation efficiency, the depth of all the honeycombs in out-of-plane direction is the same as the element size. The out-of-plane displacement of each node in each finite model is constrained to ensure the plane strain situation, with all the nodes of honeycombs free to move in the in-plane directions (x and y direction in Fig. 3). To avoid probable penetration, automatic single surface contact is defined on all the surfaces of honeycomb cell walls for self-contacting, and automatic surface-to-surface contact is defined between the contact faces of honeycombs and the two rigid plates. The emphasis of this paper is focused on the enhancement of the in-plane impact resistance of the honeycomb with novel structures. The possible presence of the cell defects is neglected for the simulations in this paper.
Fig. 3. Diagrammatic sketches for auxetic honeycombs under in-plane impact loading. (a) RH; (b) RSH.
3. In-plane crushing behaviors of RSH under impact loading In this section, three types of deformation modes for the RSH under the y-directional impact are studied and compared with the RH. Both the theoretical crushing strengths for low-velocity impact and high-velocity impact are developed and compared with the results of the finite element simulations. 3.1 Finite element analysis 3.1.1 Low-velocity mode Fig. 4 shows the low-velocity modes of RSH and RH under the impact velocity of 1 m/s along the y direction. As we can see, the RH and RSH shrink horizontally when compressed along the y direction, and an obvious “X”-shaped band can be observed for both the RH and RSH topologies in the initial impact stage (Fig. 4b). The “X”-shaped band of the RSH continue to propagate with the time until the top half of the “X”-shaped band intersects
with
its
bottom
half
(Fig.
4c-f).
Then,
a
parallelogram-shaped core forms in the RSH and begins to collapse (Fig. 4g and h). The shapes of unit cells of both RH and RSH change into parallelogram shapes during the impact processes. The cell walls of the RH begin to collapse after few unit cells change into parallelogram shapes (Fig. 4e-h). However, almost all the unit cells of the RSH maintain parallelogram shapes at the same time (Fig. 4g), and only after that the layer-by-layer collapse begins (Fig. 4h). Fig. 5 shows more details of the local deformation processes of the RSH. The number of inclined struts per inclined edge of the parallelogram shape in the RSH is very different from that in the RH (Fig. 5d). Each inclined edge of parallelogram shape in the RSH has two struts, but only one in the RH, which results in the increasing of the stress level and the production of the second plateau region in the strain-stress curve of the RSH (Fig. 6).
(a)
(b)
(e)
(c)
(f)
(d)
(g)
(h)
Fig. 4. Deformation processes of the RSH and RH under low-velocity impact ( =1 m/s) along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). The red dashed lines
are added in the figures to highlight the localized deformation regions.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 5. A typical localized deformation processes of the RSH picked from Fig. 4. (a) Stress free state; (b)-(c) initial deformation; (d) the formation of parallelogram-shaped core; (e)-(g) the parallelogram-shaped core is collapsed layer by layer; (h) densification.
Based on the analysis of Fig. 4 and Fig. 5, we can divide the low-velocity collapse process of the RSH into five distinct phases (Fig. 6). The whole RSH structure enters the elastic state firstly (state I). After reaching the initial peak stress, the plastic hinges form at both ends of the two inclined opposite struts (BC and EF) and one end of the four inclined struts (BG, BH, EI and EJ) of some unit cells in the RSH (Fig. 5b). The rest of inclined struts (AB and ED) of these unit cells keep unchanged. So that an “X”-shaped band is observed. As the local deformation with “X”-shaped band grows up and propagates, the normal contact stress between the RSH and the top rigid plate tends to be stable and the first plateau stress region occurs (state II). The state III is a transitional region during which the “X”-shaped band disappears and the parallelogram-shaped core is forming, meanwhile the stress in Fig. 6 increases. After that, more plastic hinges form at both ends of the four inclined opposite edges (A B , B G , E J and E D in Fig. 5d) of the parallelogram structure. The layer-by-layer collapse process of the parallelogram-shaped core follows. As a result a higher stress level, the second plateau stress region, is formed (state IV). Eventually, each adjacent row of cells collapse and contact, so that the stress increases sharply and the compressive densification is detected (state V).
Fig. 6. The strain-stress curves of the RSH and RH obtained by finite element simulations with the same conditions in Fig. 4, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 4. The five
deformation states of the RSH are also appended. 3.1.2 Medium-velocity mode Fig. 7 shows the deformation processes of the RSH under the medium-velocity impact with velocity of 25 m/s along the y direction. With the increase of the impact velocity, the inertia effect begins to emerge. Several rows of cells near the impact end are slightly crushed, and localized collapses from the top layer step-by-step are observed for the RSH in the initial impact stage (Fig. 7b-d). With the increase of impact displacement, an incomplete “X”-shaped band can be observed for the RSH (Fig. 7e-g). Unlike the low-velocity mode, most unit cells of the RSH do not change into the parallelogram shapes (Fig. 5d) at the same time, so that there is not a parallelogram-shaped core formed.
(a)
(e)
(b)
(f)
(c)
(d)
(g)
(h)
Fig. 7.Deformation processes of the RSH and RH under the impact velocity of 25 m/s along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). The red dashed lines
are added in the figures to highlight the localized deformation regions.
According to the deformation process of the RSH (Fig. 7), the medium-velocity mode can be divided into four distinct phases (Fig. 8). Firstly, the elastic deformation of the entire RSH structure is detected (state I). When the contact stress between the top rigid plate and the RSH reaches the initial stress peak, the plastic hinges begin to form at both ends of the struts of the re-entrant structures in the RSH. Plastic hinges
also form at the middle position of several rows of cells near the impact end. So we can see the localized collapse from the top layer step-by-step in Fig. 7b-d, which results in the stress undulation in state II. The state III is a plateau stress region during which the incomplete “X”-shaped band forms and then disappears gradually. Due to the exist of the localized collapse at the top layer and the incomplete “X”-shaped band, some unit cells of the RSH change into various parallelograms with various geometrical parameters, and some collapse directly without changing into parallelograms. Eventually, each adjacent row of cells contact and the compressive densification occurs (state IV).
Fig. 8. The strain-stress curves of the RSH and RH obtained by finite element simulations with same conditions in Fig. 7, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 7. The four deformation states of the RSH are also appended. 3.1.3 High-velocity mode When the RSH is subjected to a relatively high impact velocity, saying in Fig. 9, the cell walls near the impact end are crushed first. With the increase of compression strain, the “I” localized deformed band forms and propagates forward row-by-row to the fixed end until the RSH is totally crushed. The RH also shows the “I”-mode during high impact velocity crushing [15]. Remarkably, there is no similar
“X”-shaped band and parallelogram-shaped core observed. Due to the high impact velocity, the cell walls at the impact end do not get much time to shrink transversely, so that the RSH at the high-velocity mode do not display obvious dynamic negative Poisson’s ratio.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 9. Deformation processes of the RSH under high velocity impact ( =100 m/s) along the y direction (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ).
According to the deformation processes in Fig. 9, the high-velocity mode can be divided into three distinct phases (Fig. 10). The deformation of the RSH is elastic and uniform throughout the structure at the beginning (state I). Beyond the initial peak stress, the plastic hinges begin to form at the ends and the middle positions of all the horizontal and inclined struts near the impact end. And then the “I”-shaped band forms at the upper edge of the RSH and propagates forward layer-by-layer to the fixed end, which results the repeatable stress undulation in state II (Fig. 10). Eventually, each adjacent row of cells contact and the compressive densification occurs (state III). In order to describe the collapse process accurately, a
cells block is taken
from the finite element model in Fig. 9 and exhibited in Fig. 11. A repeatable collapse process can be observed in Fig. 11b-e, which indicates that the deformation configuration of the two top half cells of the block in Fig. 11b are the same as that of the two bottom half cells of the block in Fig. 11e. Also the same states can be found between Fig. 11c and Fig. 11f, Fig. 11d and Fig. 11g. Furthermore, the collapse process of the RSH is regular and repeatable, which give us a chance to formulate the
high-velocity crushing strength
.
Fig. 10. The strain-stress curves of the RSH and RH obtained by finite element simulations with same conditions in Fig. 9, where the seven circular points represent the corresponding deformation configurations of the RSH in Fig. 9. The three deformation states of the RSH are also appended.
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
Fig. 11. Close-up view of the typical dynamic collapse processes of the RSH obtained by the finite element simulation. (a) The original configuration of the block; (b)-(e) the collapse processes of the two upper unit cells; (e)-(h) the collapse processes of the two lower unit cells.
3.2 Theoretical analysis 3.2.1 Low-velocity mode When the bending moments of its cell walls exceed the plastic limit, plastic collapse will take place in the honeycomb, which results in the plateau region of the strain-stress curve. According to the literature [37], the upper limit of the plateau stress can be deduced by the relationship between the working done by external force on the entire unit cell and the plastic dissipation at all hinges of a unit cell, and the lower limit can be given by the relationship between the maximum bending moment of a single strut in unit cells and the plastic bending moment of this single strut.
Fig. 12. (a) The initial configuration of the unit cell at the beginning of the first plateau region (state II); (b) the initial configuration at the beginning of the second plateau region (state IV), as well as the final configuration at the end of state III.
During the collapse process of the RSH, two plateau regions occur and the second plateau stress level is almost twice higher than the first one. Fig. 12 shows the initial configurations of the RSH unit cell at the beginning of the first and second plateau regions with the plastic hinges highlighted by red circles. Eight plastic hinges form in a unit cell at the beginning of both first and second plateau regions. The angle between the strut AB and AF at the first plateau region (state II) remains unchanged (Fig. 12a) until the unit cell of the RSH changes into a parallelogram shape (Fig. 12b).
The same situation also occurs in the angle between the strut CD and DE (Fig. 12). In addition, the configuration in Fig. 12b is exactly the initial configuration at the beginning of the second plateau region (state IV). When the four plastic hinges at both ends of the strut EF, and one end of the strut BG and EI (Fig. 12a) rotate a degree of
, the relationship between the working done
by external force on the entire unit cell and the plastic work at the joints can be given as 4M p1 ybL0l sin
where
is the plastic bending moment of the inclined strut,
angle of the hinge,
(4) is the rotational
is the depth of the RSH in out-of-plane direction, and
original length of a unit cell of RSH,
is the
. One has
for a rectangular beam. From Eq. (4), the upper limit of the first plateau stress
can be obtained as
t p1 l
2
ys h l 2sin 2cos sin
(5)
The relationship between the maximum bending moment of a single strut and its plastic bending moment
can be expressed as
M p1 M max1 where the first plateau stress
, and
(6) . From Eq. (6), the lower limit of
can be written as
t p1 l
2
ys h l 2sin 2cos sin
(7)
From Eqs. (5) and (7), it should be noted that the upper and lower limit of the first plateau stress of the RSH are same. With the same methods above, the upper and lower limit of the second plateau stress
can be obtained. The initial configuration
at the beginning of the second plateau region (state IV) has two struts per inclined
edge (Fig. 12b). This inclined edge can be considered as one strut whose thickness is equal to twice that of an original strut. So that the two opposite inclined edges would produce four plastic hinges at both ends. Thus, the upper limit of the second plateau stress
can be deduced from 2M p2 ybL0l sin
where
(8)
is the plastic bending moment of the equivalent strut, one has . From Eq. (8), the upper limit of the second plateau stress
can be
expressed as 2 2 ys t p2 l h l 2sin 2cos sin
(9)
The lower limit can be given by the relationship between the maximum bending moment of a single inclined strut inclined strut
and the plastic bending moment of a single
as
M p3 M max2 where
,
(10)
. Nominal strain and nominal stress are
adopted in this paper. And the length of unit cell after deformation is set to be its original length. Thus, one has stress
. The lower limit of the second plateau
can be expressed as
t p2 l
2
2 ys
h l 2sin 2cos sin
(11)
From Eqs. (9) and (11), we can see that the upper and lower limit of the second plateau stress are equal. In case of the honeycombs with only one plateau region in low-velocity mode, like the regular hexagonal honeycomb and the RH, it is easy to ensure their low-velocity crushing strengths, which are exactly their plateau stresses. However, for the RSH studied in this paper, it has two different plateau regions and the second one is almost two times higher than the first one. To formulate the low-velocity crushing strength of
the RSH, a proportional coefficient
is introduced and defined as the ratio of the
strain range of the first plateau region to that of the whole plateau region. Thus, the low-velocity crushing strength is expressed as
0 p1 1 p2 Considering both the upper and lower limit of
(12) and
are same, we can
2 ys 2 t 0 l h l 2sin 2cos sin
(13)
simplify Eq. (12) as
According to the strain-stress curve of the RSH in Fig. 6, it can be found that the strain range corresponding to the first plateau region is almost from 5% to 30%, and that corresponding to the second one is almost from 50% to 75%. Considering the effect of the stress of the transitional region on the low-velocity crushing strength, is set to be 0.6 for the specific structure in this paper. 3.2.2 High-velocity mode Based on the close-up view of the high-velocity collapse process in Fig. 11, a representative block consisting of two upper and lower adjacent unit cells 1 and 2 (Fig. 13a) is employed to reveal the deformation mechanism. The initial ( final (
) and the
) structure feature of the block in a collapse period are shown in Fig. 13b
and c, respectively, where
is the crushing time. The deformation states in Fig. 13b
and c correspond to that of cells in the finite element models in Fig. 11b and e, respectively.
and
in Fig. 13 are the crushing stress at the top and the bottom of
the representative block, respectively. It is worth noting that
equals
in the
low-velocity mode in section 3.2.1, but they are not equal for the situation under the high impact velocity. Actually,
also represents the contact stress between the top
rigid plate and the RSH, which is the high-velocity crushing strength we want to formulate here. The unit cell 2 in Fig. 13 is almost in a static state, so the contact stress between the bottom rigid plate and the RSH, the low-velocity crushing strength
, can be approximately equal to
. Hou et al. [16] and Hu et al. [39] adopted the
same assumption to analyze the high-velocity crushing strength of the RH and the hexagonal honeycomb, respectively. This assumption can be confirmed further by Fig. 14, in which the black line represents
in Eq. (13).
Fig. 13. (a) A representative block; (b) the initial configuration of the block at the beginning of a collapse period period
; (c) the final configuration at the end of a collapse
.
Fig. 14. The contact stress between the bottom rigid plate and the RSH under various impact velocities.
At the beginning of a collapse period (Fig. 13b), the strut AF just joined the
densified region and the other cell walls of the upper unit cell deform slightly. More significantly, the upper unit cell in Fig. 13b and the lower unit cell in Fig. 13c have same configuration, which means that the momentum of cell walls of the upper unit cell at
equals that of cell walls of the deformed lower unit cell at
, i.e.,
p1T0 p2Tf
where
and
(14)
are the momentum of the upper unit cell at
lower unit cell at
and that of the
. More specifically, one has
T0 T0 T0 T0 T0 Tf Tf Tf Tf Tf pAF pCD ; pAB pCK ; pEF pDN ; pBG pKO ; pBH pKP ;
(15)
T0 T0 T0 T0 T0 Tf Tf Tf Tf Tf pEI pNQ ; pEJ pNR ; pBC pKL ; pCD pLM ; pDE pMN ;
where
is the momentum.
The cell walls of the lower unit cell almost hold still at the beginning of a collapse period (Fig. 13b). Thus, the momentum of the lower unit cell at
,
, can be given
as p2T0 0
(16)
T0 T0 T0 T0 T0 T0 T0 T0 T0 pKC pKO pKP pKL pLM pMN pNR pNQ pND 0
(17)
To be more concrete, one has
In addition, the cell walls of the upper unit cell at
are collapsed and joined the
densified region. Therefore, all the cell walls of that has the same constant velocity as the densified region, and their momentums can be given as Tf Tf Tf Tf Tf Tf Tf pAB pCB pDE pEF pBG pBH pEITf pEJ stblv
(18)
Tf pAF stbhv
(19)
Therefore, the momentum of the upper unit cell at
,
, is
Tf Tf Tf Tf Tf Tf Tf Tf Tf p1Tf pAF pAB pCB pDE pEF pGB pHB pEI pEJ
Thus, the variation of the momentum from
to
,
p p1Tf p2Tf p1T0 p2T0
Combining Eqs. (14)-(20), Eq. (21) is simplified as
(20)
, is (21)
p 8l h stbv
(22)
Based on the conservation of momentum, one has
bL0
1 2 dt p
Tf
T0
where
for the RSH. From Eq. (23), the high-velocity
crushing strength,
, is expressed as
d As shown in Fig. 13, and
(23)
1 Tf T0
and
Tf
T0
1dt 0
p bL0 Tf T0
(24)
are the height of the representative block at
, respectively. Hence, the cycle time of the high-velocity collapse process is
Tf T0
H 0 H f 2l cos 5t v v
By substituting Eqs. (22) and (25) into Eq. (24),
(25)
can be simplified as
ht 8 s l l d 0 v2 t h 2 cos 5 2sin 2 cos l l
(26)
3.3 Effects of impact velocity and cell wall thickness Fig. 15a shows plateau stresses of the RSH obtained by finite element method under low-velocity mode, medium-velocity mode and high-velocity mode, and also gives theoretical results, including the low-velocity crushing strength high-velocity crushing strength
and the
. The local details of the plateau stresses under
low-velocity mode are enlarged in Fig. 15b. As we can see, the results of the finite element simulations are consistent with the theoretical results. The cell wall thickness and the impact velocity are the two key factors to determine the plateau stress. As the impact velocity increases under low-velocity mode, the inertial effect leads to the increase of the first plateau stress, and the second one decreases slightly because the number of unit cells which turn into parallelogram shapes at the same time drop. The plateau stress under high-velocity mode is in direct proportion to the square of the impact velocity, while that increases a little for the medium-velocity mode.
Fig. 15. (a) Variation of plateau stress for the RSH with different cell wall thicknesses under low-velocity mode, medium-velocity mode and high-velocity mode; (b) close-up view of the plateau stresses under low-velocity mode.
The value of plateau stress can directly reflect the energy absorption ability of honeycomb, which means high plateau stress usually corresponds to stronger energy absorption ability. The absorbed energies of the RSH with same conditions in Fig. 15 are illustrated in Fig. 16. Note that the absorbed energy of the RSH with the minimum impact velocity is not the least. Specifically, about 8 m/s, 10 m/s and 12 m/s are the three impact velocities corresponding to the least absorbed energy of the RSH for t=0.1200 mm, t=0.2345 mm, and t=0.3706 mm, respectively. Interestingly, 8 m/s, 10 m/s and 12 m/s are exactly the first critical impact velocity (Eq. (30)) between the low-velocity mode and the medium-velocity mode, which indicates the RSH with this first critical impact velocity absorbs the lowest energy. Under the first critical impact velocity, few unit cells of the RSH simultaneously change into the new parallelogram structure with two struts per inclined edge and the second plateau region in strain-stress curve is disappearing. We already know that the second plateau stress is almost twice higher than the first one. The disappearance of the second plateau region inevitably results in the decrease of absorbed energy when the impact velocity is low and the inertial effect is not obvious. When almost all the unit cells change into the new parallelogram structure firstly and then collapse, the RSH can absorb more impact energy than the situation of most unit cells collapsing directly under a
relatively low velocity impact. In other words, the increase of the impact velocity has less contribution to the absorbed energy than the new structure under low-velocity mode. As the impact velocity increases, the inertial effect prevails. Thus the absorbed energy rises when the impact velocity is faster than the first critical impact velocity.
Fig. 16. Variation of absorbed energy for the RSH with different cell wall thicknesses under various impact velocities. 4. Discussions 4.1 The main failure mechanism of RSH The dynamic crushing behaviors of honeycombs are studied by finite element method with the commercial software package ANSYS/LS-DYNA in this paper, and the matrix material of honeycomb is modeled as elastic-perfectly plastic. The failure mechanism of honeycomb cell walls can be either elastic buckling or plastic collapse, depending on its slenderness ratio [40]. When the axial load applied to cell wall exceeds the critical force
, elastic buckling would occur firstly. According to the
results of Timoshenko and Gere [41], one has
Pcrit
n 2 2 Es I l2
(27)
where n is the end constraint factor that depends on the degree of the constraint to rotation at the ends of cell wall. For the cell wall of honeycomb loaded by axial force,
the constraint on the wall by neighboring joints lies between these limits, that is 0.5 n 2 [37]. For the hexagonal honeycomb, Gibson and Ashby [37] used the value n=0.69 because the joints can rotate. The axial load applied to cell wall EF (Fig. 12a) equals
. When
, the failure mechanism of honeycomb cell wall would be elastic buckling. The elastic buckling stress of RSH,
, can be given as
n2 2 Es t e l 6cos h l 2sin 2cos 3
The first plateau stress
(28)
is already given in section 3.2.1 (Eqs. (5) and (7)),
which is the minimum plastic collapse stress of RSH. When e p1 , the elastic buckling of RSH would occurs firstly before it enters the yielding stage. Thus, the critical cell wall slenderness ratio of RSH can be determined as
6 cos t 2 2ys l n Es sin
(29)
When the lower limit value of n is taken, that is n=0.5, and t/l is 0.004941. However, the minimum value of t/l in this paper is 0.024 when the relative density of RSH is 0.05, which is much larger than 0.004941. Therefore, plastic collapse is the main failure mechanism in our numerical models, which is also further confirmed by the finite element simulations in section 3.1. It is can be observed that plastic hinges form at the ends and middle position of cell walls of RSH in Fig. 4, 7 and 9. 4.2 Deformation modes map Three types of deformation modes for the RSH are detected under different impact velocities in section 3, including low-velocity mode, medium-velocity mode and high-velocity mode. Deformation modes are sensitive to the variation of impact velocity and relative density. Therefore, a deformation map of the RSH is summarized in Fig. 17, where the black and red line are the critical transitions between two adjacent modes for the RSH, and the orange and blue line are that for the RH [14].
When impact velocity is below the black line in Fig. 17, the deformation mode of RSH is considered as the low-velocity mode. When impact velocity is over the red line in Fig. 17, the deformation mode of RSH is the high-velocity mode. When impact velocity is between the black line and the red line in Fig. 17, the deformation mode of RSH is distinguished as the medium-velocity mode. According to this map, the RSH would require higher impact velocity for entering the high-velocity mode than the RH, which indicates that the RSH has better impact resistance than the RH. The critical impact velocity depends linearly on the relative density, which means the critical impact velocity rises with the relative density increasing. The empirical formulas of the two critical impact velocities of the RSH are achieved, respectively, as
where
vc1 4.75 43RSH
(30)
vc2 30 130RSH
(31)
is the first critical impact velocity between the low-velocity mode and the
medium-velocity mode,
is the second critical impact velocity between the
medium-velocity mode and the high-velocity mode.
Fig. 17. Deformation modes map of the RSH.
4.3 Comparisons with the RH and SSH The comparisons of energy absorption capacities for the RH, SSH and RSH under different impact loadings are investigated. Due to the different overall lengths and heights between the three honeycombs, the absorbed energy per unit volume adopted, which is defined as
, where
is
is the volume of honeycombs. Fig.
18 shows the comparisons of energy absorption capacities between the RH, SSH and RSH under different impact velocities and Table 1 presents the specific data. The impact velocities of 1 m/s, 25 m/s and 100 m/s exactly correspond to the three deformation modes of the RSH. It is easy to see that the RSH can absorb more impact energy than the RH with same cell wall thickness under all the three impact velocities. Specifically, the absorbed energy of the RSH is 64.26% larger than that of the RH under the impact velocity of 1 m/s and 49.85% larger under the impact velocity of 100 m/s. The reason that the RSH under low-velocity impact can absorb much more energy can be attributed to that the unit cells of the RSH turn into parallelogram shapes with two struts per inclined edge. This new parallelogram-shaped core results in the second plateau region in the strain-stress curves of the RSH. However, the RSH under high-velocity impact does not have enough time to change the shape of unit cells and therefore collapses directly. In addition, the absorbed energy of the RSH under the impact velocities of 1 m/s and 25 m/s is close to that of the SSH. The absorbed energy of the SSH increases rapidly after the compressive strain reaching 0.6 under the impact velocities of 1 m/s and the compressive strain reaching 0.7 under the impact velocities of 25 m/s. This rapid increase indicates that the densification of the SSH is much too early.
Fig. 18. Comparisons of energy absorption capacities for the RH, SSH and RSH under
different impact velocities (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ).
Table 1 Absorbed energy by the RH and RSH under different velocities (h=10 mm, l=5 mm, t=0.2435 mm,
=30 ). (kJ/m3)
Velocity (m/s)
(kJ/m3)
1
117.40
71.48
+64.24%
25
153.85
102.48
+50.13%
100
1207.62
810.32
+49.03%
4.4 Auxetic performances The variation of the dynamic Poisson’s ratio and the corresponding transverse strain of the RH, SSH and RSH under different y-directional impact velocities are shown in Fig. 19a-c, respectively. Dividing the average shrinkage mass of the transverse lengths of the nodes between the right and left edge of the finite element model by the original transverse length is the transverse strain here. And the dynamic Poisson’s ratio is defined as the negative ratio of the transverse strain and the longitudinal strain. Auxetic honeycombs contract literally when compressed longitudinally. Thus the transverse strain and the longitudinal strain here represent compressive strain, and only the absolute values of them are picked. As shown in Fig. 19, the properties of auxetic performances for RH, SSH and RSH are rather obvious, especially in the low-velocity mode ( =1 m/s) and the medium-velocity mode ( =25 m/s). The transverse contraction of the RSH under the low-velocity mode occurs mainly at the first plateau region, while that of the RH and SSH happens at the entire impact process (Fig. 19a).
Fig. 19. Variation of the dynamic Poisson’s ratio of the RH, SSH and RSH under different impact velocities along the y direction (h=10 mm, l=5 mm, t=0.2435 mm, =30 ). (a)
=1 m/s; (b)
= 25 m/s; (c)
=100 m/s.
5. Conclusions Combining two classical auxetic structures, RH and SSH, a novel auxetic structure named as RSH is proposed with the in-plane crushing behaviors of this new honeycomb studied systematically. Three types of deformation modes for the RSH are detected under different impact velocities, including low-velocity mode, medium-velocity mode and high-velocity mode. An obvious “X”-shaped band can be observed in the initial impact stage under low-velocity crushing, then most unit cells turn into parallelogram shapes with two struts per inclined edge at the same time which leads to the second plateau region. The second plateau stress is almost twice higher than the first one. The localized collapse
forms firstly at several rows of cells near the impact end, after that an incomplete “X”-shaped band can be observed under medium-velocity impact loading. The “I”-shaped band forms at the upper edge of the RSH and propagates forward layer-by-layer to the fixed end under high-velocity mode. The auxetic performance for the RSH is rather obvious, especially under the low-velocity and the medium-velocity mode. And the transverse contraction of the RSH mainly occurs at the first plateau region. The results of finite element simulation show that the RSH can absorb more impact energy than the RH and SSH with same cell wall thickness. Specifically, the absorbed energy of the RSH is 64.26% larger than that of the RH under the impact velocity of impact velocity of
and 49.85% larger under the
. Unlike the RH in which the absorbed energy increases
with the increase of the impact velocity, the absorbed energy of the RSH decreases slightly and then increases with the increase of the impact velocity. The velocity with the least absorbed energy of the RSH is exactly the first critical impact velocity between the low-velocity and the medium-velocity mode. In addition, the RSH would require higher impact velocity for entering the high-velocity mode than the RH, which indicates that the RSH has better impact resistance. Furthermore, theoretical models of the low-velocity and high-velocity crushing strength are developed. The upper and the lower limit of the two plateau stresses are formulated based on the theory of plastic dissipation under low-velocity impact. The high-velocity crushing strength is given based on the conservation of momentum. Both the theoretical low-velocity and high-velocity crushing strengths are coincident with the finite element method results. Acknowledgement The authors thank the support from the National Natural Science Foundation of China (Grant No. 11672013, 11672014 and 11472025) and Defense Industrial Technology
Development
Program
(Grant
No.
JCKY2016601B001
and
JCKY2016205C001) and the Fundamental Research Funds for the Central Universities.
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