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Dynamic indentation of auxetic and non-auxetic honeycombs under large deformation
Composite Structures 207 (2019) 323–330
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Dynamic indentation of auxetic and non-auxetic honeycombs under large deformation
T
L.L. Hu , M.Zh. Zhou, H. Deng ⁎
Department of Applied Mechanics & Engineering, School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510275, PR China
ARTICLE INFO
ABSTRACT
Keywords: Re-entrant honeycomb Negative Poisson’s ratio Indentation resistance Large deformation Theoretical analysis
It is generally acknowledged that the indentation resistance or hardness of auxetic materials is higher than that of their conventional counterparts under elastic deformation. However, this property of the auxetic material may not always be superior to that of the non-auxetic materials when the deformation is relatively large with plasticity considered. In this study, we come up with an index to quantitatively depict the indentation resistance of the hexagonal honeycombs under large deformation. The indentation resistance of both the auxetic and nonauxetic hexagonal honeycombs is compared and discussed. Results show that in the premise of honeycombs possessing the same relative density, the indentation resistance of auxetic hexagonal honeycombs is not always higher than that of the non-auxetic honeycombs. This phenomenon is verified by the numerical simulations. Further analysis shows that there is a critical value of the absolute value of Poisson’s ratio, which is determined by the cell-wall length ratio, to estimate the higher indentation resistance between the auxetic and non-auxetic hexagonal honeycombs. The influence of indentation velocity is also analyzed based on numerical simulations. This present work is supposed to shed light on the design and evaluation of the indentation resistance for both auxetic and conventional honeycombs.
1. Introduction Auxetic cellular materials, such as honeycombs and foams with reentrant structures [1,2], have attracted many researchers’ attention over the past decades due to their peculiar behavior and the resulting enhanced properties [3]. Literatures reported that auxetic cellular materials have some improved properties compared to their conventional counterparts, including energy-absorbing ability [4], buckling resistance [5,6], shear resistance [7], indentation resistance (hardness) [4,8,9] and so on. Many potential engineering applications benefitting from these enhanced properties have been reported, such as protective sporting equipment [10,11], cushions [12] and energy absorbers [10,13,14]. As for the improvement of indentation resistance in auxetic materials, it is commonly believed that this property is due to the flow of material towards the site of the indenter, causing the densification of auxetic material under the indenter [16,17], as shown in Fig. 1. Extensive works have been carried out experimentally [4,10,18–22], numerically [9,23] and theoretically [8,20–22] to study the indentation resistance of auxetic foams and other homogeneous materials with a negative Poisson’s ratio. Alderson et al. [19] examined the elastic in-
⁎
dentation resistance of an anisotropic UHMWPE foam, which showed a hardness up to twice as hard as the conventional material at low loads in its hardest direction. A follow-up research [21] showed that for enhanced indentation resistance to be found in auxetic UHMWPE foams, auxetic behavior and structure integrity are required. Chan and Evans [20] analyzed the indentation resistance of a polymeric foam with auxetic behavior under elastic deformation and showing a better indentation resilience for auxetic foam compared with conventional foam. Based on the classical elasticity theory, the indentation resistance (or hardness) of an isotropic material is proportional to (1 2) m with m depending on the type of the indenter [15,19,24] and the Poisson’s 0.5. ratio of isotropic homogeneous materials is limited to 1 This can partially explain the increase in hardness when the Poisson’s ratio of auxetic foams decreases from 0 to -1. A detailed theoretical analysis was carried by Argatov et al. [8] to study the indentation compliances for isotropic homogeneous materials with negative Poisson’s ratio based on elastic theories. The studies mentioned above mainly focused on the elastic macrohomogeneous behavior of auxetic materials (mostly foams) under indentation and the minimum of Poisson’s ratio is limited to -1. However, according to Gibson and Ashby [25], the Poisson’s ratio of a cellular
Corresponding author. E-mail address: [email protected] (L.L. Hu).
https://doi.org/10.1016/j.compstruct.2018.09.066 Received 15 June 2018; Received in revised form 10 August 2018; Accepted 19 September 2018 Available online 21 September 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved.
Composite Structures 207 (2019) 323–330
L.L. Hu et al.
Nomenclature 0
h b t l d F y
Mp Y
dW
angle between the bevel edge and the horizontal edge of the cell length of the cell’s horizontal edge out-of-plane thickness of the honeycomb cell-wall thickness of the honeycomb length of the cell’s bevel edge instant change of the cell-wall angle force strain of the cell along the loading direction value of the cell-wall angle at a certain instant during the deformation 1 full plastic bending moment of the cell walls, Mp = 4 Y bt 2 yield stress of the cell-wall material
dH L F xr
s
0
instant work done by the external force acting on a typical cell true stress of a typical cell average stress of the honeycomb (index of the indentation resistance) instant change of the cell’s height,dH = 2l ·[sin sin( d )] instant wide of a typical cell average force residual displacement (the displacement at the end of the indentation process where the force decreases to zero) Poisson’s ratio of the hexagonal honeycomb density of the cell-wall material density of the honeycomb the critical value of the magnitude of Poisson’s ratio
material, like the re-entrant hexagonal honeycomb, can be much smaller than −1. And in the structural point of view, honeycomb materials with various microstructures have inhomogeneous properties [26–28] and the behavior of auxetic materials under large deformation needs to be taken into consideration. Moreover, when the cellular material is used in practical engineering especially as part of an energyabsorbing system, the plasticity of the material is the key property that affects the performance [29,30]. Some recent studies [23,31] on the indentation of auxetic composites did consider the structure of the material or the composite system, but still only the elastic properties were studied. So the plastic behavior of auxetic cellular materials under indentation needs further investigation. In this present paper, an analytical formula of the indentation resistance of both re-entrant and conventional hexagonal honeycomb under large deformation is obtained. The relationship between the indentation resistance of the honeycomb and the cells’ geometrical parameters as well as the mechanical properties of the base material is revealed. The theoretical predictions are then verified by numerical simulations. Finally, the influence of indentation velocity on the honeycombs’ indentation resistance and the relationship between the
Fig. 2. Configuration of re-entrant hexagonal honeycomb under indentation.
indentation resistance and the absolute value of Poisson’s ratio for both auxetic and non-auxetic hexagonal honeycombs are studied. 2. Theoretical analysis A typical configuration of the re-entrant hexagonal honeycomb under indentation is shown in Fig. 2. The cell-wall material of the honeycomb is assumed to be rigid-perfectly plastic, which is generally accepted when analyzing metallic or polymeric honeycombs. A typical cell is chosen for theoretical analysis, like the analysis on the traditional hexagonal honeycombs [27,28] and the circular cell honeycombs [26,32]. The configuration of the deformed cell is sketched by the red dashed line in Fig. 2. The angle between the bevel edge and the horizontal edge of the cell is named as the cell-wall angle 0 . h and l are the lengths of the cell’s horizontal edge and the bevel edge, respectively.
Fig. 1. The deformation profile of non-auxetic and auxetic materials under indentation [15]
324
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2.1. Indentation resistance
honeycombs’ relative density,
s
, and the cell size, as shown in Eq. (6),
where is the density of the honeycomb and cell-wall material.
Unlike the deformation of the re-entrant honeycombs under flat plate crushing [33], the deformation of each individual cell along the transverse direction of the entire model is different under indentation. Since the indentation resistance of the honeycomb is mostly contributed by the cells right underneath the indenter, the following theoretical analysis considers only the deformation pattern of the cells underneath the indenter. Based on the collapse process of the honeycombs’ cells observed in numerical simulations (the details will be described in Section 3), the cells under the indenter deform symmetrically like the cases in flat plate crushing condition [33]. Moreover, the cell walls can be assumed no bend or fold during compression, and the length of the cell walls is postulated to remain unchanged during the cell’s collapse process. The work done by the indentation force is dissipated by the plastic hinges occurring in the corners of the re-entrant hexagonal cells. There are 8 plastic hinges within one typical cell to dissipate the energy produced by the external force, as shown in Fig. 2. At a certain instant during the deformation with the cell-wall angle being equal to , the instant work done by the external force can be obtained
= s
t l
2sin
( 0
h l
+2
(
h l
)
cos
0
)
s
is the density of the
. (6)
By substituting Eq. (6) into Eq. (5), the average stress of auxetic hexagonal honeycombs and non-auxetic hexagonal honeycombs can be rewritten as functions of the honeycombs’ relative density instead of the cell-wall thickness, as shown in Eq. (7) 2
4 Y
( + 2 ( ) s
h l
=
1 0
d
y
h l 2
cos 0
) sin
(h / l ) 2
2
0
·arctan
1
h +1 l h 1 l
tan 20 ,
0
(0°, 90°)
=
2
2 Y
( + 2 ( ) s
h l
h l 2
2
cos 0
) sin
(h / l ) 2
1
0
2arctan
h +1 l h 1 l
tan 20
,
0
(90°, 180°)
(1)
(7)
where Mp = is the fully plastic bending moment of the cell walls and Y , b and t are the yield stress of the cell-wall material, out-of-plane thickness of the honeycomb and cell-wall thickness of the honeycomb, respectively. d is the instant change of the cell-wall angle. Then the force, F , can be derived
The average stress shown in Eqs. (5) and (7) can be used as an index to depict the indentation resistance of the hexagonal honeycombs. The higher the index, the higher the indentation resistance is. Based on Eqs. (5) and (7), the relationship between the index of the indentation resistance and the initial cell-wall angle can then be given for honeycombs with the same cell-wall thickness or the same relative density, as shown in Fig. 4. To get a better comparison result, we use a normalized value based on the 45° model. The parameters used to plot the figures are adopted from the numerical simulations in Section 3. For both cases, the index increases with the increase of initial cell-wall angle for auxetic honeycombs while decreases for non-auxetic honeycombs. As it shows in Fig. 4(a), for honeycombs with the same cell-wall thickness, the index of the honeycombs with NPR effect is always higher than that of the honeycombs without NPR effect. It indicates that hexagonal honeycombs with negative Poisson’s ratio manifest stronger indentation resistance than the traditional non-auxetic hexagonal honeycombs in the premise that they have the same cell-wall thickness. But for honeycombs possessing the same relative density, as shown in Fig. 4(b), the index of the auxetic hexagonal honeycombs does not always exceed that of the non-auxetic hexagonal honeycombs. The relative magnitude of the index depends on the initial cell-wall angle of the hexagonal
dW = 8Mp d , 1 bt 2 4 Y
F = lim
dW
0 dH
d
=
Y bt
2
l
· lim d
0 sin
d sin(
d )
=
2 Y bt , lcos
(2)
where dH = 2l·[sin sin( d )] is the instant change of the cell’s height. The instant wide of a typical cell with cell-wall angle being equal to is (3)
L = 2h 2lcos . Thus the true stress of a typical cell can be obtained 2 F Y (t / l ) = h Lb 2cos l cos
=
(
)
. (4)
The true stress-strain curves of typical cells can then be obtained based on Eq. (4) and geometric relationships, as shown in Fig. 3. Fig. 3 shows the evolution of the bearing force under the indenter with consideration of both the cells’ large deformation and the flow of material towards the indenter caused by the negative Poisson’s ratio effect (NPR effect). It can depict the indentation resistance of the honeycomb to a certain extent, as the bearing force of the cell varies with different initial cell-wall angle during collapse. To thoroughly demonstrate the indentation resistance of the whole deformation process of a typical cell, average stress is deduced by calculating the integration of the true stress (Eq. (4)) over strain (area between the curve and the strain-axis in Fig. 3) for both auxetic hexagonal honeycombs ( 0 (0°, 90°) ) and non-auxetic hexagonal honeycombs ( 0 (90°, 180°) )
sin 0
=
1 0
d
y
=
2 Y (t / l ) (h / l ) 2
1
h +1 l h 1 l
·arctan
2 Y (t / l ) 2sin 0 (h / l)2
1
tan 20 ,
2arctan
h +1 l h 1 l
0
(0°, 90°)
tan 20
,
0
(90°, 180°) (5)
Fig. 3. True stress-strain curves of typical cells with various initial cell-wall angles.
Gibson and Ashby [25], built the relationship between the 325
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Fig. 4. Dependence of honeycombs’ relative magnitude of average stress on initial cell-wall angle for (a) honeycombs with the same cell-wall thickness and (b) honeycombs with the same relative density.
honeycomb, which means that honeycombs with NPR effect do not always have higher indentation resistance than those without NPR effect. This conclusion is counterintuitive and other auxetic structures should be investigated in the future to re-evaluate the feature of “high indentation resistance” under large deformation.
to prevent the honeycomb from out-of-plane buckling. The honeycomb models and rigid rod were meshed using a 4-node shell element (Shell 163 in LS-DYNA) for a balance between the computational efficiency and accuracy. Belytschko-Tsay formulation is used for this element. The element size of the models is selectively refined such that the elements near the rigid rod have an element size of 1 mm and the size of the elements increases up to 5 mm with the distance departing from the loading area. The convergence study showed that this meshing scheme can provide both accurate results and time efficiency. An automatic single surface contact is applied on the honeycombs, so the program automatically determines which surfaces within the model may come into contact. An automatic surface-to-surface contact is defined between the rigid rod and honeycomb specimen. Contact frictions are not considered for both contact definitions.
3. Numerical verifications 3.1. Numerical model The numerical models of re-entrant hexagonal honeycombs are built using ANSYS/LS-DYNA. Based on symmetry considerations, we consider only half of the in-plane domain in the numerical models to save computational time, as shown in Fig. 2. The size of the entire model is about 875 mm × 640 mm with more than 1000 cells included. The lengths of the typical cell’s horizontal edge h and the bevel edge l are h = 21 mm and l = 10 mm, respectively. The out-of-plane thickness of the honeycomb b is set as 1 mm. By changing the cell-wall angle, four types of re-entrant honeycombs are obtained with 0 being equal to 30°, 45°, 60° and 75°, respectively. To compare the mechanical properties of auxetic honeycombs with non-auxetic honeycombs, a set of hexagonal honeycombs with 0 being equal to 120°, 135° and 150° is also considered in the numerical simulation. The indenter used in the simulation is a rigid rod with a radius of 126 mm. Density is a key factor in affecting the mechanical properties of cellular materials, thus different cellular materials are usually compared with their relative density being the same. In numerical simulations, the cell-wall thickness t is set as 0.5 mm, 0.8 mm, 1.13 mm, 1.44 mm, 1.83 mm, 1.61 mm and 1.2 mm for the honeycombs with 0 = 30°, 45°, 60°, 75°, 120°, 135° and 150°, respectively, to ensure = 0.167. For the honeycombs possessing the same relative density s honeycombs possess the same cell-wall thickness, the cell-wall thickness is set to be 0.8 mm. The cell-wall material of the honeycomb is assumed to be isotropic elastic-perfectly plastic with Young’s modulus 70 Gpa and yield stress 3 y = 255 MPa. The density of the cell-wall material is 2700 kg/m . The honeycomb block is fixed on the bottom side and subjected to crush by the indenter on the top, as shown in Fig. 2. An initial velocity is exerted on the indenter to crush the honeycomb and three levels of crushing velocity are adopted in the numerical simulation: 10 m/s, 30 m/s and 80 m/s. The material of the indenter is set to be rigid material with Young’s modulus 210 GPa. And the density of the indenter varies with different crushing velocities to ensure an identical initial crushing energy of 58.2 J. Symmetric constraints are set on the left boundaries of the models. The out-of-plane displacement of all the nodes is also constrained so as
3.2. Deformation mode The deformation evolution of the honeycombs with various cell-wall angles is exhibited in Fig. 5 with the impact velocity of 10 m/s and 80 m/s. The “disp” in Fig. 5 is the displacement of the indenter. High impact velocity and large cell-wall angle will lead to high irregularity of the deformed cells, as shown in Fig. 5. For the 30 ° model with the impact velocity being equal to 10 m/s, the cells under the indenter deform uniformly and the deformation transits smoothly from the contact area to the free area. With the increase of impact velocity, the cells under the indenter tend to deform layer-by-layer while the cells away from the indenter barely influenced. And more cells are stretched between contact area and free area due to a relatively large shear load. Similar behavior can also be found in the 45° and 60 ° cases. The increase of cellwall angle also affects the deformation. Compared to the 30 ° model, the 45° and 60 ° model exhibit more noticeable deformation bands during the indentation process and more cells are stretched to an irregular shape. 3.3. Force-displacement curves The indentation force in the numerical simulations is obtained directly from the contact force between the indenter and the honeycomb. And the displacement is obtained by tracking the movement of the contact point between the honeycomb and the lowest point of the indenter. Under low-velocity indentation (10 m/s), the force-displacement curves of the honeycombs with the same cell-wall thickness and the same relative density are shown in Fig. 6. The diminution of the displacement after the force reaching the maximum value indicates the rebound of the indenter attributed to the recovery of the honeycombs’ elastic deformation. 326
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Fig. 5. Deformation evolution of re-entrant honeycombs with various cell-wall angle under different crushing velocities.
As it shows in Fig. 6(a), for honeycombs with the same cell-wall thickness, the distinction of force-displacement curves among different cell-wall angles is not significant for both auxetic and non-auxetic honeycombs. But for honeycombs with the same relative density, as shown in Fig. 6(b), the peak value of indentation force increases and the
indentation depth decreases with the increase of cell-wall angle for auxetic honeycombs. The results for non-auxetic honeycombs are contrary to that of auxetic honeycombs. These results are consistent with the theoretical curves we obtained in Section 2, as shown in Fig. 4. To numerically verify the theoretical results, the average force during the 327
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Fig. 6. Dependence of honeycombs’ indentation force-displacement curves on cell-wall angle for (a) honeycombs with the same cell-wall thickness and (b) honeycombs with the same relative density.
are normalized based on the 45° model. The normalized values of the FEM results are also added in Fig. 4 for comparison, indicating a good agreement between the theoretical predictions and numerical simulations. 4. Discussions 4.1. Velocity effect of indentation The force-displacement curves of honeycombs with a cell-wall angle of 30° under different indentation velocities is shown in Fig. 7. It is shown that when the indentation velocity is relatively low (10 m/s and 30 m/s), the indentation force increase monotonically with the increase of indentation depth. However, under high-velocity indentation, the force reaches the peak value at an early stage of the indentation process and decreases monotonically till the rebound of the indenter. The large fluctuation of the curve under high crushing velocity is due to the propagation and reflection of stress wave. Similar phenomenon can be found in many other researches [27,34,35] dealing with high velocity crushing response of honeycombs. To compare the indentation force of the honeycombs with different cell-wall angles under various indentation velocities in a more direct way, the average forces of the honeycombs with the same relative density are calculated based on Eq. (8), as shown in Fig. 8. The results show that the average value of the indentation force increases with the increase of indentation velocity and the cell-wall angle, indicating a high indentation resistance for honeycombs with large cell-wall angle under high-velocity indentation.
Fig. 7. Dependence of honeycombs’ indentation force-displacement curves on indentation velocity with a cell-wall angle of 30°.
4.2. Auxetic V.S. non-auxetic honeycombs In the above section, we conclude that the indentation resistance of the auxetic and non-auxetic honeycombs depends on the geometry of the cell’s structure, so we can get honeycombs with various indentation resistance by choosing different cell-wall angles of the honeycomb. However, the change of honeycombs’ cell-wall angle can affect the Poisson’s ratio of the honeycomb. Thus, in the engineering practice, the indentation resistance of the honeycombs is also of interest when the absolute values of the honeycombs’ Poisson’s ratio are the same. Gibson and Ashby [25] built the relationship between the hexagonal honeycombs’ Poisson’s ratio and the geometrical factors of the honeycomb, as shown in Eq. (9).
Fig. 8. Dependence of honeycombs’ average force on cell-wall angle and indentation velocity (same relative density).
whole indentation process is calculated
F =
Fdx xr
,
(8)
=
where Fdx is the integration of the force in the numerical simulation over the displacement of the indenter and x r is the residual displacement (the displacement at the end of the indentation process where the force decreases to zero). Then the average forces calculated by Eq. (8)
(
sin2 h l
cos
0
0
) cos
0
(9)
Combining Eqs. (5), (7) and (9), the relationship between the honeycombs’ index of indentation resistance (average stress) and the absolute value of Poisson’s ratio can be given, as shown in Fig. 9. It is 328
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Fig. 9. Dependence of honeycombs’ average stress on the absolute value of Poisson’s ratio for honeycombs with (a) the same cell-wall thickness and (b) the same relative density.
same relative density, the indentation resistance of auxetic hexagonal honeycombs is not always higher than that of the non-auxetic honeycombs, which depends on the cell-wall angle. As we keep the magnitude of the Poisson’s ratio of both the auxetic and non-auxetic honeycombs identical, it is interesting that there is a critical value of Poisson’s ratio, 0 . Only when the absolute value of Poisson’s ratio exceeds 0 will the indentation resistance of auxetic hexagonal honeycombs surpass that of non-auxetic honeycombs. And this critical value of Poisson’s ratio is determined by the cell-wall length ratio, h , of the hexagonal honeyl comb. The conclusions are counterintuitive and other auxetic structures should also be investigated in the future to re-evaluate their indentation resistance under large deformation. Besides, numerical simulations reveal the influence of indentation velocity and cell-wall angle on the indentation resistance of the honeycombs. It is shown that the indentation resistance increases with the increase of the indentation velocity and the cell-wall angle. This present work is supposed to shed light on the design and the evaluation of the indentation resistance for both auxetic and non-auxetic honeycombs. Fig. 10. Dependence of the critical value of Poisson’s ratio on the honeycombs’ cell-wall length ratio.
Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China under Grant Nos. 11472314 and 11772363 and the support from the Science and Technology Program of Guangzhou, China under Grant No. 201803030037.
obvious that for honeycombs with the same cell-wall thickness, when the Poisson’s ratios are the same, the index of indentation resistance of the auxetic honeycombs is always higher than that of the non-auxetic honeycombs. This conclusion is consistent with the result shown in Fig. 4(a). But for the honeycombs with the same density, the index of the auxetic honeycombs is higher than that of the non-auxetic honeycombs only when the magnitude of Poisson’s ratio exceeds a critical value, 0 . Based on Eqs. (7) and (9), 0 is a function of the cell-wall length ratio, h , and can be determined once the cell-wall length ratio is l
given. The dependence of Fig. 10.
0
Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2018.09.066.
on the cell-wall length ratio, h , is shown in
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Dynamic indentation of auxetic and non-auxetic honeycombs under large deformation
T
L.L. Hu , M.Zh. Zhou, H. Deng ⁎
Department of Applied Mechanics & Engineering, School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510275, PR China
ARTICLE INFO
ABSTRACT
Keywords: Re-entrant honeycomb Negative Poisson’s ratio Indentation resistance Large deformation Theoretical analysis
It is generally acknowledged that the indentation resistance or hardness of auxetic materials is higher than that of their conventional counterparts under elastic deformation. However, this property of the auxetic material may not always be superior to that of the non-auxetic materials when the deformation is relatively large with plasticity considered. In this study, we come up with an index to quantitatively depict the indentation resistance of the hexagonal honeycombs under large deformation. The indentation resistance of both the auxetic and nonauxetic hexagonal honeycombs is compared and discussed. Results show that in the premise of honeycombs possessing the same relative density, the indentation resistance of auxetic hexagonal honeycombs is not always higher than that of the non-auxetic honeycombs. This phenomenon is verified by the numerical simulations. Further analysis shows that there is a critical value of the absolute value of Poisson’s ratio, which is determined by the cell-wall length ratio, to estimate the higher indentation resistance between the auxetic and non-auxetic hexagonal honeycombs. The influence of indentation velocity is also analyzed based on numerical simulations. This present work is supposed to shed light on the design and evaluation of the indentation resistance for both auxetic and conventional honeycombs.
1. Introduction Auxetic cellular materials, such as honeycombs and foams with reentrant structures [1,2], have attracted many researchers’ attention over the past decades due to their peculiar behavior and the resulting enhanced properties [3]. Literatures reported that auxetic cellular materials have some improved properties compared to their conventional counterparts, including energy-absorbing ability [4], buckling resistance [5,6], shear resistance [7], indentation resistance (hardness) [4,8,9] and so on. Many potential engineering applications benefitting from these enhanced properties have been reported, such as protective sporting equipment [10,11], cushions [12] and energy absorbers [10,13,14]. As for the improvement of indentation resistance in auxetic materials, it is commonly believed that this property is due to the flow of material towards the site of the indenter, causing the densification of auxetic material under the indenter [16,17], as shown in Fig. 1. Extensive works have been carried out experimentally [4,10,18–22], numerically [9,23] and theoretically [8,20–22] to study the indentation resistance of auxetic foams and other homogeneous materials with a negative Poisson’s ratio. Alderson et al. [19] examined the elastic in-
⁎
dentation resistance of an anisotropic UHMWPE foam, which showed a hardness up to twice as hard as the conventional material at low loads in its hardest direction. A follow-up research [21] showed that for enhanced indentation resistance to be found in auxetic UHMWPE foams, auxetic behavior and structure integrity are required. Chan and Evans [20] analyzed the indentation resistance of a polymeric foam with auxetic behavior under elastic deformation and showing a better indentation resilience for auxetic foam compared with conventional foam. Based on the classical elasticity theory, the indentation resistance (or hardness) of an isotropic material is proportional to (1 2) m with m depending on the type of the indenter [15,19,24] and the Poisson’s 0.5. ratio of isotropic homogeneous materials is limited to 1 This can partially explain the increase in hardness when the Poisson’s ratio of auxetic foams decreases from 0 to -1. A detailed theoretical analysis was carried by Argatov et al. [8] to study the indentation compliances for isotropic homogeneous materials with negative Poisson’s ratio based on elastic theories. The studies mentioned above mainly focused on the elastic macrohomogeneous behavior of auxetic materials (mostly foams) under indentation and the minimum of Poisson’s ratio is limited to -1. However, according to Gibson and Ashby [25], the Poisson’s ratio of a cellular
Corresponding author. E-mail address: [email protected] (L.L. Hu).
https://doi.org/10.1016/j.compstruct.2018.09.066 Received 15 June 2018; Received in revised form 10 August 2018; Accepted 19 September 2018 Available online 21 September 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature 0
h b t l d F y
Mp Y
dW
angle between the bevel edge and the horizontal edge of the cell length of the cell’s horizontal edge out-of-plane thickness of the honeycomb cell-wall thickness of the honeycomb length of the cell’s bevel edge instant change of the cell-wall angle force strain of the cell along the loading direction value of the cell-wall angle at a certain instant during the deformation 1 full plastic bending moment of the cell walls, Mp = 4 Y bt 2 yield stress of the cell-wall material
dH L F xr
s
0
instant work done by the external force acting on a typical cell true stress of a typical cell average stress of the honeycomb (index of the indentation resistance) instant change of the cell’s height,dH = 2l ·[sin sin( d )] instant wide of a typical cell average force residual displacement (the displacement at the end of the indentation process where the force decreases to zero) Poisson’s ratio of the hexagonal honeycomb density of the cell-wall material density of the honeycomb the critical value of the magnitude of Poisson’s ratio
material, like the re-entrant hexagonal honeycomb, can be much smaller than −1. And in the structural point of view, honeycomb materials with various microstructures have inhomogeneous properties [26–28] and the behavior of auxetic materials under large deformation needs to be taken into consideration. Moreover, when the cellular material is used in practical engineering especially as part of an energyabsorbing system, the plasticity of the material is the key property that affects the performance [29,30]. Some recent studies [23,31] on the indentation of auxetic composites did consider the structure of the material or the composite system, but still only the elastic properties were studied. So the plastic behavior of auxetic cellular materials under indentation needs further investigation. In this present paper, an analytical formula of the indentation resistance of both re-entrant and conventional hexagonal honeycomb under large deformation is obtained. The relationship between the indentation resistance of the honeycomb and the cells’ geometrical parameters as well as the mechanical properties of the base material is revealed. The theoretical predictions are then verified by numerical simulations. Finally, the influence of indentation velocity on the honeycombs’ indentation resistance and the relationship between the
Fig. 2. Configuration of re-entrant hexagonal honeycomb under indentation.
indentation resistance and the absolute value of Poisson’s ratio for both auxetic and non-auxetic hexagonal honeycombs are studied. 2. Theoretical analysis A typical configuration of the re-entrant hexagonal honeycomb under indentation is shown in Fig. 2. The cell-wall material of the honeycomb is assumed to be rigid-perfectly plastic, which is generally accepted when analyzing metallic or polymeric honeycombs. A typical cell is chosen for theoretical analysis, like the analysis on the traditional hexagonal honeycombs [27,28] and the circular cell honeycombs [26,32]. The configuration of the deformed cell is sketched by the red dashed line in Fig. 2. The angle between the bevel edge and the horizontal edge of the cell is named as the cell-wall angle 0 . h and l are the lengths of the cell’s horizontal edge and the bevel edge, respectively.
Fig. 1. The deformation profile of non-auxetic and auxetic materials under indentation [15]
324
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2.1. Indentation resistance
honeycombs’ relative density,
s
, and the cell size, as shown in Eq. (6),
where is the density of the honeycomb and cell-wall material.
Unlike the deformation of the re-entrant honeycombs under flat plate crushing [33], the deformation of each individual cell along the transverse direction of the entire model is different under indentation. Since the indentation resistance of the honeycomb is mostly contributed by the cells right underneath the indenter, the following theoretical analysis considers only the deformation pattern of the cells underneath the indenter. Based on the collapse process of the honeycombs’ cells observed in numerical simulations (the details will be described in Section 3), the cells under the indenter deform symmetrically like the cases in flat plate crushing condition [33]. Moreover, the cell walls can be assumed no bend or fold during compression, and the length of the cell walls is postulated to remain unchanged during the cell’s collapse process. The work done by the indentation force is dissipated by the plastic hinges occurring in the corners of the re-entrant hexagonal cells. There are 8 plastic hinges within one typical cell to dissipate the energy produced by the external force, as shown in Fig. 2. At a certain instant during the deformation with the cell-wall angle being equal to , the instant work done by the external force can be obtained
= s
t l
2sin
( 0
h l
+2
(
h l
)
cos
0
)
s
is the density of the
. (6)
By substituting Eq. (6) into Eq. (5), the average stress of auxetic hexagonal honeycombs and non-auxetic hexagonal honeycombs can be rewritten as functions of the honeycombs’ relative density instead of the cell-wall thickness, as shown in Eq. (7) 2
4 Y
( + 2 ( ) s
h l
=
1 0
d
y
h l 2
cos 0
) sin
(h / l ) 2
2
0
·arctan
1
h +1 l h 1 l
tan 20 ,
0
(0°, 90°)
=
2
2 Y
( + 2 ( ) s
h l
h l 2
2
cos 0
) sin
(h / l ) 2
1
0
2arctan
h +1 l h 1 l
tan 20
,
0
(90°, 180°)
(1)
(7)
where Mp = is the fully plastic bending moment of the cell walls and Y , b and t are the yield stress of the cell-wall material, out-of-plane thickness of the honeycomb and cell-wall thickness of the honeycomb, respectively. d is the instant change of the cell-wall angle. Then the force, F , can be derived
The average stress shown in Eqs. (5) and (7) can be used as an index to depict the indentation resistance of the hexagonal honeycombs. The higher the index, the higher the indentation resistance is. Based on Eqs. (5) and (7), the relationship between the index of the indentation resistance and the initial cell-wall angle can then be given for honeycombs with the same cell-wall thickness or the same relative density, as shown in Fig. 4. To get a better comparison result, we use a normalized value based on the 45° model. The parameters used to plot the figures are adopted from the numerical simulations in Section 3. For both cases, the index increases with the increase of initial cell-wall angle for auxetic honeycombs while decreases for non-auxetic honeycombs. As it shows in Fig. 4(a), for honeycombs with the same cell-wall thickness, the index of the honeycombs with NPR effect is always higher than that of the honeycombs without NPR effect. It indicates that hexagonal honeycombs with negative Poisson’s ratio manifest stronger indentation resistance than the traditional non-auxetic hexagonal honeycombs in the premise that they have the same cell-wall thickness. But for honeycombs possessing the same relative density, as shown in Fig. 4(b), the index of the auxetic hexagonal honeycombs does not always exceed that of the non-auxetic hexagonal honeycombs. The relative magnitude of the index depends on the initial cell-wall angle of the hexagonal
dW = 8Mp d , 1 bt 2 4 Y
F = lim
dW
0 dH
d
=
Y bt
2
l
· lim d
0 sin
d sin(
d )
=
2 Y bt , lcos
(2)
where dH = 2l·[sin sin( d )] is the instant change of the cell’s height. The instant wide of a typical cell with cell-wall angle being equal to is (3)
L = 2h 2lcos . Thus the true stress of a typical cell can be obtained 2 F Y (t / l ) = h Lb 2cos l cos
=
(
)
. (4)
The true stress-strain curves of typical cells can then be obtained based on Eq. (4) and geometric relationships, as shown in Fig. 3. Fig. 3 shows the evolution of the bearing force under the indenter with consideration of both the cells’ large deformation and the flow of material towards the indenter caused by the negative Poisson’s ratio effect (NPR effect). It can depict the indentation resistance of the honeycomb to a certain extent, as the bearing force of the cell varies with different initial cell-wall angle during collapse. To thoroughly demonstrate the indentation resistance of the whole deformation process of a typical cell, average stress is deduced by calculating the integration of the true stress (Eq. (4)) over strain (area between the curve and the strain-axis in Fig. 3) for both auxetic hexagonal honeycombs ( 0 (0°, 90°) ) and non-auxetic hexagonal honeycombs ( 0 (90°, 180°) )
sin 0
=
1 0
d
y
=
2 Y (t / l ) (h / l ) 2
1
h +1 l h 1 l
·arctan
2 Y (t / l ) 2sin 0 (h / l)2
1
tan 20 ,
2arctan
h +1 l h 1 l
0
(0°, 90°)
tan 20
,
0
(90°, 180°) (5)
Fig. 3. True stress-strain curves of typical cells with various initial cell-wall angles.
Gibson and Ashby [25], built the relationship between the 325
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Fig. 4. Dependence of honeycombs’ relative magnitude of average stress on initial cell-wall angle for (a) honeycombs with the same cell-wall thickness and (b) honeycombs with the same relative density.
honeycomb, which means that honeycombs with NPR effect do not always have higher indentation resistance than those without NPR effect. This conclusion is counterintuitive and other auxetic structures should be investigated in the future to re-evaluate the feature of “high indentation resistance” under large deformation.
to prevent the honeycomb from out-of-plane buckling. The honeycomb models and rigid rod were meshed using a 4-node shell element (Shell 163 in LS-DYNA) for a balance between the computational efficiency and accuracy. Belytschko-Tsay formulation is used for this element. The element size of the models is selectively refined such that the elements near the rigid rod have an element size of 1 mm and the size of the elements increases up to 5 mm with the distance departing from the loading area. The convergence study showed that this meshing scheme can provide both accurate results and time efficiency. An automatic single surface contact is applied on the honeycombs, so the program automatically determines which surfaces within the model may come into contact. An automatic surface-to-surface contact is defined between the rigid rod and honeycomb specimen. Contact frictions are not considered for both contact definitions.
3. Numerical verifications 3.1. Numerical model The numerical models of re-entrant hexagonal honeycombs are built using ANSYS/LS-DYNA. Based on symmetry considerations, we consider only half of the in-plane domain in the numerical models to save computational time, as shown in Fig. 2. The size of the entire model is about 875 mm × 640 mm with more than 1000 cells included. The lengths of the typical cell’s horizontal edge h and the bevel edge l are h = 21 mm and l = 10 mm, respectively. The out-of-plane thickness of the honeycomb b is set as 1 mm. By changing the cell-wall angle, four types of re-entrant honeycombs are obtained with 0 being equal to 30°, 45°, 60° and 75°, respectively. To compare the mechanical properties of auxetic honeycombs with non-auxetic honeycombs, a set of hexagonal honeycombs with 0 being equal to 120°, 135° and 150° is also considered in the numerical simulation. The indenter used in the simulation is a rigid rod with a radius of 126 mm. Density is a key factor in affecting the mechanical properties of cellular materials, thus different cellular materials are usually compared with their relative density being the same. In numerical simulations, the cell-wall thickness t is set as 0.5 mm, 0.8 mm, 1.13 mm, 1.44 mm, 1.83 mm, 1.61 mm and 1.2 mm for the honeycombs with 0 = 30°, 45°, 60°, 75°, 120°, 135° and 150°, respectively, to ensure = 0.167. For the honeycombs possessing the same relative density s honeycombs possess the same cell-wall thickness, the cell-wall thickness is set to be 0.8 mm. The cell-wall material of the honeycomb is assumed to be isotropic elastic-perfectly plastic with Young’s modulus 70 Gpa and yield stress 3 y = 255 MPa. The density of the cell-wall material is 2700 kg/m . The honeycomb block is fixed on the bottom side and subjected to crush by the indenter on the top, as shown in Fig. 2. An initial velocity is exerted on the indenter to crush the honeycomb and three levels of crushing velocity are adopted in the numerical simulation: 10 m/s, 30 m/s and 80 m/s. The material of the indenter is set to be rigid material with Young’s modulus 210 GPa. And the density of the indenter varies with different crushing velocities to ensure an identical initial crushing energy of 58.2 J. Symmetric constraints are set on the left boundaries of the models. The out-of-plane displacement of all the nodes is also constrained so as
3.2. Deformation mode The deformation evolution of the honeycombs with various cell-wall angles is exhibited in Fig. 5 with the impact velocity of 10 m/s and 80 m/s. The “disp” in Fig. 5 is the displacement of the indenter. High impact velocity and large cell-wall angle will lead to high irregularity of the deformed cells, as shown in Fig. 5. For the 30 ° model with the impact velocity being equal to 10 m/s, the cells under the indenter deform uniformly and the deformation transits smoothly from the contact area to the free area. With the increase of impact velocity, the cells under the indenter tend to deform layer-by-layer while the cells away from the indenter barely influenced. And more cells are stretched between contact area and free area due to a relatively large shear load. Similar behavior can also be found in the 45° and 60 ° cases. The increase of cellwall angle also affects the deformation. Compared to the 30 ° model, the 45° and 60 ° model exhibit more noticeable deformation bands during the indentation process and more cells are stretched to an irregular shape. 3.3. Force-displacement curves The indentation force in the numerical simulations is obtained directly from the contact force between the indenter and the honeycomb. And the displacement is obtained by tracking the movement of the contact point between the honeycomb and the lowest point of the indenter. Under low-velocity indentation (10 m/s), the force-displacement curves of the honeycombs with the same cell-wall thickness and the same relative density are shown in Fig. 6. The diminution of the displacement after the force reaching the maximum value indicates the rebound of the indenter attributed to the recovery of the honeycombs’ elastic deformation. 326
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Fig. 5. Deformation evolution of re-entrant honeycombs with various cell-wall angle under different crushing velocities.
As it shows in Fig. 6(a), for honeycombs with the same cell-wall thickness, the distinction of force-displacement curves among different cell-wall angles is not significant for both auxetic and non-auxetic honeycombs. But for honeycombs with the same relative density, as shown in Fig. 6(b), the peak value of indentation force increases and the
indentation depth decreases with the increase of cell-wall angle for auxetic honeycombs. The results for non-auxetic honeycombs are contrary to that of auxetic honeycombs. These results are consistent with the theoretical curves we obtained in Section 2, as shown in Fig. 4. To numerically verify the theoretical results, the average force during the 327
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Fig. 6. Dependence of honeycombs’ indentation force-displacement curves on cell-wall angle for (a) honeycombs with the same cell-wall thickness and (b) honeycombs with the same relative density.
are normalized based on the 45° model. The normalized values of the FEM results are also added in Fig. 4 for comparison, indicating a good agreement between the theoretical predictions and numerical simulations. 4. Discussions 4.1. Velocity effect of indentation The force-displacement curves of honeycombs with a cell-wall angle of 30° under different indentation velocities is shown in Fig. 7. It is shown that when the indentation velocity is relatively low (10 m/s and 30 m/s), the indentation force increase monotonically with the increase of indentation depth. However, under high-velocity indentation, the force reaches the peak value at an early stage of the indentation process and decreases monotonically till the rebound of the indenter. The large fluctuation of the curve under high crushing velocity is due to the propagation and reflection of stress wave. Similar phenomenon can be found in many other researches [27,34,35] dealing with high velocity crushing response of honeycombs. To compare the indentation force of the honeycombs with different cell-wall angles under various indentation velocities in a more direct way, the average forces of the honeycombs with the same relative density are calculated based on Eq. (8), as shown in Fig. 8. The results show that the average value of the indentation force increases with the increase of indentation velocity and the cell-wall angle, indicating a high indentation resistance for honeycombs with large cell-wall angle under high-velocity indentation.
Fig. 7. Dependence of honeycombs’ indentation force-displacement curves on indentation velocity with a cell-wall angle of 30°.
4.2. Auxetic V.S. non-auxetic honeycombs In the above section, we conclude that the indentation resistance of the auxetic and non-auxetic honeycombs depends on the geometry of the cell’s structure, so we can get honeycombs with various indentation resistance by choosing different cell-wall angles of the honeycomb. However, the change of honeycombs’ cell-wall angle can affect the Poisson’s ratio of the honeycomb. Thus, in the engineering practice, the indentation resistance of the honeycombs is also of interest when the absolute values of the honeycombs’ Poisson’s ratio are the same. Gibson and Ashby [25] built the relationship between the hexagonal honeycombs’ Poisson’s ratio and the geometrical factors of the honeycomb, as shown in Eq. (9).
Fig. 8. Dependence of honeycombs’ average force on cell-wall angle and indentation velocity (same relative density).
whole indentation process is calculated
F =
Fdx xr
,
(8)
=
where Fdx is the integration of the force in the numerical simulation over the displacement of the indenter and x r is the residual displacement (the displacement at the end of the indentation process where the force decreases to zero). Then the average forces calculated by Eq. (8)
(
sin2 h l
cos
0
0
) cos
0
(9)
Combining Eqs. (5), (7) and (9), the relationship between the honeycombs’ index of indentation resistance (average stress) and the absolute value of Poisson’s ratio can be given, as shown in Fig. 9. It is 328
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Fig. 9. Dependence of honeycombs’ average stress on the absolute value of Poisson’s ratio for honeycombs with (a) the same cell-wall thickness and (b) the same relative density.
same relative density, the indentation resistance of auxetic hexagonal honeycombs is not always higher than that of the non-auxetic honeycombs, which depends on the cell-wall angle. As we keep the magnitude of the Poisson’s ratio of both the auxetic and non-auxetic honeycombs identical, it is interesting that there is a critical value of Poisson’s ratio, 0 . Only when the absolute value of Poisson’s ratio exceeds 0 will the indentation resistance of auxetic hexagonal honeycombs surpass that of non-auxetic honeycombs. And this critical value of Poisson’s ratio is determined by the cell-wall length ratio, h , of the hexagonal honeyl comb. The conclusions are counterintuitive and other auxetic structures should also be investigated in the future to re-evaluate their indentation resistance under large deformation. Besides, numerical simulations reveal the influence of indentation velocity and cell-wall angle on the indentation resistance of the honeycombs. It is shown that the indentation resistance increases with the increase of the indentation velocity and the cell-wall angle. This present work is supposed to shed light on the design and the evaluation of the indentation resistance for both auxetic and non-auxetic honeycombs. Fig. 10. Dependence of the critical value of Poisson’s ratio on the honeycombs’ cell-wall length ratio.
Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China under Grant Nos. 11472314 and 11772363 and the support from the Science and Technology Program of Guangzhou, China under Grant No. 201803030037.
obvious that for honeycombs with the same cell-wall thickness, when the Poisson’s ratios are the same, the index of indentation resistance of the auxetic honeycombs is always higher than that of the non-auxetic honeycombs. This conclusion is consistent with the result shown in Fig. 4(a). But for the honeycombs with the same density, the index of the auxetic honeycombs is higher than that of the non-auxetic honeycombs only when the magnitude of Poisson’s ratio exceeds a critical value, 0 . Based on Eqs. (7) and (9), 0 is a function of the cell-wall length ratio, h , and can be determined once the cell-wall length ratio is l
given. The dependence of Fig. 10.
0
Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2018.09.066.
on the cell-wall length ratio, h , is shown in
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l
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5. Conclusion In the present paper, an index of indentation resistance is brought up and is deduced for both re-entrant and conventional hexagonal honeycomb, which is a function of the honeycombs’ geometrical factors and the relative density, as shown in Eqs. (5) and (7). Numerical simulations are conducted to confirm the validity of the analytical predictions based on the index. Results show that for honeycombs with the same cell-wall thickness, the indentation resistance of auxetic hexagonal honeycombs is always higher than that of the non-auxetic hexagonal honeycombs. But in the premise of honeycombs possessing the 329
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