- Email: [email protected]
In–plane dynamic crushing behavior and energy absorption of honeycombs with a novel type of multi-cells
Thin-Walled Structures 117 (2017) 199–210
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
In–plane dynamic crushing behavior and energy absorption of honeycombs with a novel type of multi-cells
MARK
⁎
Dahai Zhanga,b,d, Qingguo Feic, , Peiwei Zhanga,d a
Department of Engineering Mechanics, Southeast University, Nanjing 210096, China Faculty of Science, Engineering and Technology, Swinburne University of Technology, John Street, Hawthorn, VIC 3122, Australia c School of Mechanical Engineering, Southeast University, Nanjing 210096, China d Key Laboratory of Lightweight and Reliability Technology for Engineering Vehicle, College of Hunan Province, Changsha 410114, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Honeycombs Multi-cell Deformation mode Energy absorption In-plane
In order to pursue better crashworthiness and higher energy absorption efficiency, a new type of multi-cell honeycomb (quadri-arc) was designed and followed by a series of numerical studies on in-plane dynamic crushing behavior and energy absorption property under different impact loading. Meanwhile, simulations of a regular circular single-cell honeycomb were also conducted as comparisons. Three distinct deformation modes were identified from the observation of deformation profiles: quasi-static, transition and dynamic, respectively. Simulations indicate that deformation modes are not only influenced by impact velocity but also sensitive to relative density of the honeycomb, based on which a deformation map was summarized. Furthermore, the plateau stress and energy absorption of the quadri-arc honeycomb as well as the circular honeycomb were explored and compared, during which effects of impact velocity and relative density of the honeycomb were discussed in detail. The results show that a much higher plateau stress and better energy absorption efficiency for the quadri-arc honeycomb are predicted compared to the circular honeycomb, especially in the quasi-static case. The investigation suggests that design of the quadri-arc multi-cell will enhance the crashworthiness and energy absorption capacity of honeycombs.
1. Introduction As one typical type of cellular material, honeycomb structures have been widely employed in many engineering applications, including personal protective equipment, automotive industry, transportation and aeronautics engineering due to their desirable mechanical properties such as outstanding lightweight, high strength, heat insulation and excellent performance in energy absorption [1]. With increase of safety requirements, optimization and improvement of honeycomb structures for pursuing better crashworthiness and higher energy absorption efficiency have become advanced research hotpots recently [1–3]. Generally, the stress-strain curve of a honeycomb structure subjected to in-plane impact loading contains three distinct regimes: a linear elastic regime, a plateau regime and a densification regime. Among the three regimes, energy capacity of the structure mainly relies on the plateau regime, which is dominated by elastic buckling, plastic buckling and plastic collapse of the unit cells [4]. Therefore, geometry sensitivity is a very important characteristic of honeycombs, the mechanical properties of a honeycomb structure are not only determined by the matrix material, but also extremely be sensitive to cell ⁎
topology properties [5–9]. By changing the geometrical parameters of the cells such as cell type, cell size, and ratio of wall thickness-to-length and so on, the honeycombs may show a different mechanical property and energy absorption capacity [5]. As is well known, honeycomb cellular structure is one classic biomimetic material, which was first developed originating from the nature honeycomb in a nest. So honeycomb structure with hexagonal cells have drawn the most attention of researchers. However, hexagonal honeycombs are not suitable for all applications, many other types of simple cells have been introduced and designed for specific purpose, including circular, triangular and square cells. Mechanical characteristics and energy absorption capacity of honeycomb structures with different simple types of single cells have been fully researched theoretically, experimentally and numerically [4,5,8–16]. Gibson and Ashby [4] systematically studied the fundamental mechanical properties of honeycomb structures with hexagonal cells as well as triangular and square cells based on micro-structure model and theoretically derived the equation to predict the plateau stress of honeycombs. Ruan et al. [9] numerically investigated the in-plane dynamic behavior of hexagonal honeycomb structures, they observed three deformation modes and discussed the
Corresponding author. E-mail address: [email protected] (Q. Fei).
http://dx.doi.org/10.1016/j.tws.2017.03.028 Received 29 September 2016; Received in revised form 15 March 2017; Accepted 23 March 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Nomenclature
RB ,RC lB tB ,tC κB ,κC ρB , ρC ρB , ρC , ρ ρs V0 Es μs σys
V Relative impact velocity Vy Yield velocity c0 Elastic wave speed in the matrix material εY Yield strain of the matrix marital ε Nominal compressive strain δ Vertical displacement VC1, VC 2 Critical velocity σ (ε ) Nominal compressive stress VB Wave speed εcr Nominal strain at initial stress peak εd Locking strain λ ,C , C1, C2, C3 Constants Plateau stress σpd ,σpq
Constitutive cell Radius of honeycomb Cell length of the quadri-arc honeycomb Cell thickness Cell thickness-to-length ratio Density of the honeycomb Relative density of the honeycomb Density of matrix material Impact velocity Young's modulus of matrix material Poisson's ratio of matrix material Yield stress of matrix material
[21–23] have also been introduced and studied. Generally, multi-cell honeycomb structures show considerably desirable mechanical properties and energy absorption capacity compared with the single-cell honeycombs. Fig. 1 shows a Kagome honeycomb structure and a diamond honeycomb structure, their micro-structures are magnified and given as well as the constitutive cells. It is noticed from the figure that a multicell honeycomb is essentially constructed by cross connection, combination or superimposition of simple single-cells. For Kagome honeycomb, the single-cells are triangular cells while for diamond honeycomb, the single-cells are triangular or hexagonal cells. In this paper, a new type of multi-cell honeycomb as shown in Fig. 2 is introduced and followed by a serious of numerical analyses on dynamic behavior and energy absorption properties under in-plane impact loadings. The new multi-cell honeycomb is constructed from circular cells and named as quadri-arc multi-cell honeycomb based on the geometry of its unit cells. This paper is organized as follow. In Section 2, details of the microstructure of quadri-arc honeycomb is described. Finite element (FE) model of the honeycomb is established and validated. During which, a FE model of regular circular honeycomb is established as comparison. In Section 3, deformation modes of the quadri-arc honeycomb are investigated and compared with the circular honeycomb. Moreover, an overall deformation map of the quadri-arc honeycomb is summarized and the critical velocities of different deformation modes are proposed based on analysis of deformation profiles of honeycombs with different relative densities under different impact velocities. In Section 4, the dynamic impact response of the quadri-arc honeycomb is investigated by analyzing the reaction stress-strain curve of the impact plate, during which effects of impact velocity and relative density are discussed. Meanwhile, the plateau stress of the quadri-arc honeycomb is compared with the circular honeycomb. In Section 5, energy absorption property of the quadri-arc honeycomb is further studied and compared with the regular circular honeycomb.
influences of cell wall thickness and impact velocity on localized deformation and plateau stress of the honeycomb. Papka et al. [10,11] conducted experimental and numerical studies of uniaxial and biaxial crushing of a honeycomb with circular cells, they presented a full scale numerical simulation method which can successfully predict the major material parameters of interest and compared well with the experimental results. An analytical model was established by Hu et al. [12,13] to deduce the crushing strength of the hexagonal honeycombs and was in good agreement with the numerical simulation results. Besides, a lot of works were presented to discuss the crushing behavior of honeycombs with different cell micro-topologies. Wang et al. [14,15] investigated the in-plane mechanical properties of periodic honeycomb structures with seven different single cell types and derived initial yield surfaces of periodic metal honeycombs under a combined in-planes stress state. The empirical equation for honeycomb structures filled with equilateral triangular or quadratic cells and regular or staggered micro-arrangement at high impact velocities were formulated in terms of impact velocity, cell geometric and topology parameters by Liu et al. [16]. Studies mentioned above mainly focused on the honeycombs with simple single-cells. However, honeycomb structures with multi-cells were found to be highly efficient energy absorption structures and received more and more research interests recently [17–23]. A multicell honeycomb structure is usually constituted by a number of simple cells with various angles and by different connection factors. Kagome honeycomb as shown in Fig. 1(a) is one typical multi-cell honeycomb constructed by a combination of hexagon and triangle cells. A lot of works have been presented [17–20] to research the mechanical characteristics of Kagome honeycombs under impact loading and it was found that Kagome honeycombs are better choice under the targets of energy absorption capacity compared with honeycombs with simple single-cells. Besides, other types of multi-cell honeycomb structures such as diamond cell honeycombs [19,20] and chiral cell honeycombs
Fig. 1. Two typical multi-cell honeycombs and their constitutive cells.
200
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 2. Quadri-arc multi-cell honeycomb and a unit cell.
2. Numerical models
ρC = ρC / ρs =
1 ⎛ tC ⎞ 1 π ⎜ ⎟ = πκC 2 ⎝ RC ⎠ 2
(2)
2.1. Finite element modeling where ρC is the authentic density of the circular honeycomb. For better comparison, parameters including mass, volume and matrix material of both models are all set to be the same. By substituting Eqs. (1) and (2) into ρB = ρC , the relationship between RB and RC can be derived as RB = 4RC . To systematically explore the in-plane crushing behavior and energy absorption of the quadri-arc honeycomb, FE models of the quadri-arc and regular circular honeycombs are established. The commercial software package ABAQUS is employed and the step type is DynamicExplicit. Here, the quadri-arc honeycomb is established with cell size RB = 20 mm, composed of 5 cells in X1 direction and 6 cells in X2 direction while cell size for the circular honeycomb is RC = 5 mm with 10 cells in X1 direction and 12 cells in X2 direction. Other parameters of the two honeycombs are all the same including cell wall thickness tB = tC = 0.2 mm, height in out-plane direction h = 2 mm. A rigid flat plane with constant-velocity V0 is modeled to impact the honeycomb and the honeycomb is taken to be clamped at its bottom surface, periodic boundary conditions are applied in each side of the structure to prevent out-of-plane buckling and eliminate boundary effects on simulation results. As shown in Fig. 4, both honeycombs are meshed using hourglass controlled, four nodes, reduced integration shell element (S4R) with
As illustrated in Fig. 2, the micro-structure of the novel multi-cell honeycomb is quadri-arc, which is constructed from circular cells. The configuration of the quadri-arc cell is defined by the following parameters: RB denotes the radius of the constitutive circular cells, tB denotes the thickness of unit cells, lB denotes the length of unit cells and κB = tB / lB denotes thickness-to-length ratio. It can be easily derived that lB = RB . Hence, the relative density of the quadri-arc honeycomb is
⎛t ⎞ ρB = ρB / ρs = 2π ⎜ B ⎟ = 2πκB ⎝ RB ⎠
(1)
where ρB is the authentic density of the quadri-arc honeycomb; ρs is the density of the matrix material. In order to better understand the impact behavior of the quadri-arc honeycomb (Fig. 3(a)), a model of single-cell honeycomb with regular circular cells (also are the constitutive cells of the quadri-arc honeycomb) is also established and simulated as a comparison, as shown in Fig. 3(b). Characteristic parameters of the circular honeycomb are radius of the circular cells RC , cell wall thickness tC and thickness-tolength ratio κC = tC / RC , the relative density of circular honeycomb could be obtained as
Fig. 3. Geometric configurations and boundary conditions of the honeycombs.
201
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 4. 3D view of the numerical models.
five integration points across the thickness [9]. A general contact is adopted to simulate the self-contact of cells, while a surface-to-surface contact is adopted to simulate the contact between the honeycomb and the rigid flat plane. The friction coefficient is set as 0.3. A strain-rate independent idealistic elastoplastic model [24] is employed here and the property of matrix material is listed in Table 1. A convergence test on element size is conducted firstly for both honeycombs. As shown in Fig. 5, the simulated mean crushing forces gradually stabilizes and converges when the element size reduces to 2.0 mm. Therefore, 1.0 mm is adopted in both models conservatively and all results given below are convergent.
c0 =
Es / ρs
(4)
εY = σys / Es
(5)
where c0 is the speed of the longitudinal elastic stress wave; εY is the yield strain of matrix material. Here, a dimensionless normalized parameter V = V0 / Vy is introduced to recognize the effect of impact velocity and address the inertial effect. Figs. 7–9 show deformation patterns of the quadri-arc honeycomb as well as the circular honeycomb under three different impact velocities at different compressive strains. Symbol ε denotes the compressive nominal strain along the impact direction, which is calculated from
2.2. Validation analysis
(6)
ε = δ / L1 To validate the accuracy of the numerical models, a validation analysis is conducted in this section. There are many analogous studies on the circular honeycombs available in literature, e.g. [25]. In this study, simulation of a circular honeycomb with 10 × 10 unit cells, cell diameter of 7.9 mm and wall thickness of 0.3 mm under quasi-static inplane crushing is carried out [25]. The modeling approach is the same as that mentioned in Section 2.1 and the simulation results are compared with the literatures, as shown in Fig. 6. It is seen that both force-displacement curves and deformation patterns are in good agreement. Therefore, the numerical models are considered reliable.
where δ is vertical displacement and L1 is the initial length of the honeycomb along impact direction. As shown in Fig. 7, global deformation occurs when the compressive nominal strain is small (ε = 13.4%) for both quadri-arc and regular circular honeycombs in the quasi-static case (V = 0.4 ), which produces an X-shaped deformation mode. With increase of compression (ε = 41.3%), deformation modes of these two kinds of honeycombs show significant difference. Specifically, a rectangular localized band is developed in the middle of the quadri-arc honeycomb while the circular honeycomb continues to deform as an X-shaped mode. At last (ε = 74.1%), both honeycombs are completely crushed, as shown in Fig. 7(c). At a higher impact velocity of V = 4.0 , the inertia influence begins to emerge and global deformation mode is replaced by local deformation mode for both quadri-arc and circular honeycombs (see Fig. 8). A number of cells near the impact edge of the honeycomb are slightly crushed and localized layer-by-layer collapses are observed for the quadri-arc honeycomb. The compactness degree of the collapse presents a trend of layer-by-layer decreasing from the impact edge to the
3. Dynamic crushing behavior In this section, deformation modes of the quadri-arc multi-cell honeycomb under various in-plane impact velocities are studied and compared with the circular honeycomb. 3.1. Deformation modes It always shows different deformation modes when the impact velocity changed for honeycomb structures [26]. The yield velocity Vy is a very important indicator widely employed to address the inertial effect [27–29]. Zou et al. [29] indicates that yield velocity is the lowest velocity that governs a change in material response for cellular materials. The yield velocity is defined as a material's property
Vy = c0 εY
Table 1 Material property of the matrix material.
(3) 202
Matrix material
Young's modulus Es
Yield stress σys
Poisson's ratio μs
Density ρs
Aluminum (Al)
69 GPa
76 MPa
0.33
2700 kg/m3
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
2×3 and 4×6 cells for the quadri-arc and circular honeycombs, respectively. The calculated Poisson's ratios of different cases are listed in Table 2. It can be seen that the Poisson's ratio of the quadri-arc honeycomb is close to zero and not sensitive to the impact velocity. On the other hand, the Poisson's ratio of the circular honeycomb is much larger and decreases with the increase of impact velocity. Fig. 10 gives the deformations of a unit cell extracted from the central area of the quadri-arc honeycomb as well as the corresponding cells of the circular honeycomb under different impact velocities. As shown in Figs. 7–9, the deformation of unit cells of the quadri-arc honeycomb are similar for different impact velocities, a typical example (V = 4.0 ) is drawn in Fig. 10(a). As shown in Fig. 10(a), no significant change is observed in the horizontal direction with crush proceed, which leads to an almost zero Poisson's ratio for the quadri-arc honeycomb. While for circular honeycomb, the unit cells are crushed smoothly and fully stretched perpendicularly to the impact direction, which leads to a large Poisson's ratio, as shown in Fig. 10(b). However, in a higher impact velocity case (see Fig. 10(c)), the unit cells are crushed irregularly with a smaller stretch perpendicularly to the impact direction, which leads to a smaller Poisson's ratio compared to the low impact velocity case. As stated above, The relative density and wall thickness of both honeycombs are set to be the same, which leads to a difference in cell size. The number of cell rows in the crushing direction are 6 and 12 for the quadri-arc and circular honeycombs, respectively. Scale effects are observed in localization of the honeycombs, see Figs. 7–9. Take the case of V = 4.0 for example, although localized layer-by-layer collapses are observed with crushing proceed, relatively homogeneous collapse is observed in each row. As shown in Fig. 8(a), one row is crushed for the quadri-arc honeycomb while two rows are crushed for the circular honeycomb in the corresponding time. Almost uniform collapse is observed for the one row crushed in the quadri-arc honeycomb, while significant difference is found for the two rows crushed in the circular honeycomb. Specifically, denser collapse is observed for the row closer
Fig. 5. Convergence analyses for mean crushing force under different element size.
clamped edge. Afterward, the honeycomb is completely crushed layer by layer. For the circular honeycomb, a V-shaped deformation mode is observed and remains until the honeycomb is totally crushed. In Fig. 9, a much higher impact velocity (V = 12.0 ) is conducted and it is found that no obvious local deformation (X-shaped or V-shaped) appears for both honeycombs through the whole crushing process. Under high impact velocity, the transverse inhomogeneity is ignored because the inertia effect plays a dominant role in deformation of the honeycombs. An I-shaped deformation mode perpendicular to the impact direction is observed at the impact edge and propagates forward layer by layer to fixed edge with proceeding of the impact for both quadri-arc honeycomb and circular honeycomb, which is just a manner of wave propagation. Besides, an interesting phenomenon is observed that the Poisson's ratios of the two honeycombs are significantly different. Sub-structures ABCD and A′B′C′D′ are extracted from the center of the honeycombs to calculate the Poisson's ratio, as shown in Fig. 7. The sub-structure has
Fig. 6. Comparison of results between FE simulation and literature [25] for a circular honeycomb: (a) Crushing force-displacement curves; (b) Deformation patterns.
203
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 7. Deformation modes of honeycombs under quasi-static impact (V = 0.4 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%.
and crushed to address the size effect on contact forces. Fig. 11. shows the contact forces of the quadri-arc honeycombs with different cells but same relative density. The contact forces of the two honeycombs are very close, which indicates that cell size has no significant effect on
to the impact surface. Similar observation is also made in the other cases. It indicates that cell size has an effect on the localization process in some degree. Moreover, a quadri-arc honeycomb with 10×12 cells were modeled
Fig. 8. Deformation modes of honeycombs under middle-velocity impact (V = 4.0 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%.
204
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 9. Deformation modes of honeycombs under high-velocity impact (V = 12.0 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%. Table 2 Poisson's ratios of the honeycombs under different impact velocities.
V = 0.4 V = 4.0 V = 12.0
Quadri-arc honeycomb
Circular honeycomb
0.006 0.018 0.008
0.424 0.451 0.323
Fig. 11. Effect of cell size on contact force of the quadri-arc honeycomb.
ized as follows. Three types of modes are found when the impact velocity varies and can be divided into quasi-static mode, transition mode and dynamic mode (see Fig. 12). A quasi-static mode appears with low impact velocity, which can be characterized by an X-shaped band followed by a rectangular localized band developing with increasing of compression. A transition mode appears for a higher impact velocity, in which layer-by-layer localized collapses is observed and densification degree of the collapse presents a trend of decreasing layer-by-layer from the impact edge to the clamped edge. The third type is a dynamic mode, which appears when the impact velocity is sufficiently high. An I-shaped mode as a manner of wave propagation is observed in this mode. A deformation map of the quadri-arc honeycomb with different relative densities subjected to different impact velocity is provided in Fig. 12. It can be found that deformation mode of the quadri-arc honeycomb is determined not only by the impact velocity, but also the relative density. The critical velocity at which deformation modes change from one to another increases linearly with increasing of
Fig. 10. Deformation of the typical unit cells: a) Quadri-arc honeycomb (V = 4.0 ); b) Circular honeycomb (V = 4.0 ); c) Circular honeycomb (V = 12.0 ).
contact force. Same phenomenon was also reported in literatures [4,5,28] that the crashworthiness of a honeycomb is determined by the relative density rather than cell size. 3.2. Deformation modes map Different deformation modes have been observed for the quadri-arc honeycomb under different impact velocities, which can be summar205
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 12. Deformation modes map of the quadri-arc honeycomb.
Fig. 13. History of dynamic responses of the quadri-arc honeycomb.
4.1. Effect of impact velocity
relative density of the honeycomb, which means that increasing of the relative density delays the onset of the deformation mode. Two empirical equation for the two critical velocities are proposed respectively as
VC1 = 0.57 + 11.37ρB
(7)
VC 2 = 2.0 + 71.62ρB
(8)
Fig. 13 illustrates the nominal stress-strain curves at both front and bottom surfaces for quadri-arc honeycomb under three different impact velocities, during which the nominal stress σ (ε ) is calculated from σ (ε ) = F / A, where F is the reaction force of the impact plate and A is the cross section area of the structure normal to the longitudinal direction. As shown in Fig. 13(a), a typical stress-strain curve including three distinct dynamic response stages: a linear elastic stage initially, a long collapse plateau stage in the middle and a compressive densification stage at last is observed for the quadri-arc honeycomb. An important observation in Fig. 13(a) is that the plateau stress at the front surface increases with impact velocity. The phenomenon is a common trend, which has been reported and discussed widely in literatures [9,12,29–31]. However, impact velocity has much less effect on the
4. Dynamic impact response In this section, Dynamic impact response of the quadri-arc honeycomb is investigated by analyzing the reaction stress-strain curve of the impact plate. 206
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 14. Difference of the nominal stresses at the front and bottom surfaces for the case of V = 0.4 .
Fig. 15. Nominal stresses on the bottom surface at different impact velocities.
response at the bottom surface, as shown in Fig. 13(b). This is may be due to the stresses at the bottom surface are the results of compaction waves travelling from the front surface are far lower in intensity than the compaction wave [31]. Note the deformation of the honeycomb walls is mainly bending dominated, the wave of global uniaxial compaction can be called a bending wave [32,33]. Fig. 14 shows the nominal stress-strain curves at both front and bottom surfaces of the quasi-static case (V = 0.4 ). As shown in Fig. 14, there is a gap Δt between the times when stress firstly reaches the maximums at the front surface and bottom surface respectively. Moreover, the nominal stress curves at the bottom surfaces of different impact velocities are plotted in Fig. 15. It should be noted that the zero of the horizontal axis is the time where stress reaches the maximum at the front surface for each case. It can be seen that the time gaps are basically the same for the three cases although the crushing processes are significantly different. The wave speed can be calculated hereby, which is about VB = L1/ Δt = 307.7 m/s. Generally, there are initial peaks in the stress-strain curves. However, the plateau stage is the main interest discussed here, which is the most important part of energy absorption capacity for a honeycomb structure. Furthermore, the values of the peaks are uncertain and heavily influenced by the cut-off frequency of a wave filter in the FE calculation [9], so the initial peaks are not discussed in this paper. The plateau stress is defined as the average nominal stress between the initial peak and the compressive stress corresponding to the locking strain
Fig. 16. Comparisons of the dynamic responses for quadri-arc honeycomb and circular honeycomb under different impact velocities: a) V = 0.4 ; b) V = 4.0 ; c) V = 12.0 .
εd = 1 − Δρ
where εcr is the nominal strain at the initial stress peak; εd is the locking strain; λ is a constant which is determined by the type of the honeycomb cells and the impact velocity and ideally equal to 1, but is usually larger than 1 for the honeycomb cannot wholly crushed under impact loading. As shown in Fig. 15(a), significant influence of impact velocity on stress-strain curves is observed. The quasi-static stress-strain curve is much more stable and can be obviously divided into three regimes while a higher curve with strong fluctuation is found for the high impact velocity. As stated above, when the impact velocity is sufficiently high, a propagation of plane plastic wave liked deformation mode is observed. A significant influence caused by the inertia effect makes the dynamic stress-strain curve big difference from the quasistatic situation. A shock wave theory was suggested for analysis of the
ε
σp =
∫ε d σ (ε) dε cr
εd − εcr
(10)
(9) 207
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 17. Comparison of plateau stresses for the quadri-arc and circular honeycombs under different impact velocities.
Fig. 20. Effect of impact velocity on energy absorbed by the quadri-arc honeycomb.
dynamic plateau stress and a widely used theoretical model was derived by Reid [29,30] using the R-I shock theory
Table 3 Plateau stresses for the quadri-arc and circular honeycombs. Relative velocity
Circular honeycomb (Mpa )
Quadri-arc honeycomb (Mpa )
Difference
V = 0.4 V = 4.0 V = 12.0
0.1035 0.1919 0.9235
0.1509 0.2567 0.9785
+45.8% +33.8% +5.96%
σpd = σpq + CV0 2
(11)
where σpd and σpq are dynamic and quasi-static plateau stress, respectively; C is a coefficient of the inertia effect. It can be seen that the dynamic plateau stress consists of two terms: a constant quasi-static term determined by the micro-structure of honeycomb and a changing dynamic term determined by inertia effect. So the stress-strain curve of the honeycomb rises rapidly with increase of impact velocity.
4.2. Comparison with regular circular honeycomb For comparison, the simulated corresponding uniaxial stress-strain curves of the quadri-arc multi-cell honeycomb as well as the circular honeycomb under different impact velocities are plotted in Fig. 16. It could be seen from Fig. 16 that the quadri-arc honeycomb shows an obviously higher stress-strain curves than the circular honeycomb in quasi-static case (V = 0.4 ) and middle-velocity case (V = 4.0 ), but no apparent difference in the high velocity case (V = 12.0 ). Generally, higher plateau stress signifies better impact resistance capacity. From this standpoint, quadri-arc honeycomb performs much better on impact resistance compared with circular honeycomb under quasi-static and middle-velocity. It also should be noted that the quadri-arc honeycomb has an obvious lagging densification region in the quasi-static case, which makes it more perfect. Moreover, comparison of the plateau stresses of two kinds of honeycombs under different impact velocities are illustrated in Fig. 17 and specific data are given in Table 3. As shown in the diagrams, much larger plateau stresses are observed for the quadriarc honeycomb under quasi-static and middle-velocity impact. Specifically, 45.8% larger for quasi-static impact and 33.8% larger for middlevelocity impact. For high impact velocity, the difference is much smaller, only 5.96% higher plateau stress is obtained for the quadriarc honeycomb. As stated in Eq. (11), the dynamic plateau stress of a honeycomb consists of two terms: a quasi-static term and an inertia term. The quasi-static term is a constant determined by the microstructure of the honeycomb while the inertia term is determined by inertia effect and varying along the impact velocity. Therefore, difference of micro-structures of honeycombs cause a big difference in quasi-static term and a small difference in inertia term. Since inertia term is negligible in low-velocity cases and becomes the dominating part when the impact velocity is sufficient high, quadri-arc honeycomb is a better choice with its advantage degrading with increase of impact velocity.
Fig. 18. Plateau stresses of the quadri-arc honeycomb versus relative velocities of different cell thickness-to-length ratios.
Fig. 19. Plateau stresses of quadri-arc honeycomb versus cell thickness-to-length ratio under different impact velocities.
208
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 22. Comparison of energy absorbed by the quadri-arc and circular honeycombs.
Table 4 Energy absorbed by the quadri-arc and circular honeycombs under different velocities. Relative velocity
Circular honeycomb (kJ /kg )
Quadri-arc honeycomb (kJ /kg )
Difference
V = 0.4 V = 4.0 V = 12.0
0.3493 0.5854 2.153
0.5459 0.8211 2.294
+56.3% +40.3% +6.55%
Fig. 23. SEA of the quadri-arc honeycomb versus cell thickness-to-length ratio under different impact velocities.
σpd = σpq + (C1 κB2 + C2 κB + C3) V0 2 Fig. 21. Energy absorption history for quadri-arc honeycomb and circular honeycomb under different impact velocities: a) V = 0.4 ; b) V = 4.0 ; c) V = 12.0 .
5. Energy absorption efficiency
4.3. Plateau stress map
In this section, energy absorption capacity of the quadri-arc honeycomb subjected to in-plane impact loadings is investigated. A specific energy absorbed indicator SEA is adopted [31] to evaluate the energy absorption capabilities of honeycomb and sandwich materials, which is defined as
Effect of the relative density and impact velocity on plateau stress is further studied. Figs. 18 and 19 illustrate the plateau stresses of quadriarc honeycomb of different relative densities under different impact velocities. As shown in Fig. 18, the plateau stress is in a quadratic relation with impact velocity, which can be explained by Eq. (11). Besides, it can be seen from Fig. 19 that the plateau stress is also in a quadratic relation with cell thickness-to-length ratio (relative density) under high dynamic impact velocity. Hence, constant C in Eq. (11) can be formulated as
C = C1 κB2 + C2 κB + C3
(13)
εd
∫ε σ (ε) dε W cr SEA = = M ρs ρ
(14)
where W is the total energy absorbed by honeycomb and M is the total mass of honeycomb; ρ is the relative density of honeycomb. Based on Eq. (14), the absorbed energy per unit mass for the quadriarc honeycomb under different impact velocities versus the nominal strain normal to the longitudinal direction are depicted in Fig. 20. It is seen that impact velocity has great influence on energy absorbed,
(12)
where C1, C2, C3 are constants, so Eq. (11) can be rewritten as 209
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Acknowledgements
energy absorbed is significantly increased along with increase of impact velocity. Moreover, energy absorbed by the circular honeycomb are also calculated and illustrated in Fig. 21 as comparisons. Under quasistatic or middle-velocity, quadri-arc honeycomb shows better efficiency than the regular circular honeycomb. But when the impact velocity is sufficiently high, no obvious advantage is observed for the quadri-arc honeycomb. Compassion of the energy absorbed ultimately by the two honeycombs under different impact velocities are illustrated in Fig. 22 and specific data are given in Table 4. It can be seen that much larger absorbed energy are observed for the quadri-arc honeycomb under quasi-static and middle-velocity impact. Specifically, 56.3% larger for the quasi-static impact and 40.3% larger for the middle-velocity impact. For the high impact velocity, the difference is little with only 6.55% more energy absorbed by the quadri-arc honeycomb. As stated in Eq. (14), energy absorbed is essentially the integral along compressive strain of plateau stress. Therefore, big difference of the plateau stress for quasi-static or middle-velocity case results in big difference of energy absorbed while small difference of the plateau stress for highvelocity case results in little difference of energy absorbed. In a word, quadri-arc honeycomb is a much better choice for energy absorption efficiency compared with regular circle honeycomb. But its advantage will lose out with increase of impact velocity. Effect of the relative density and impact velocity on SEA is further studied. Fig. 23 shows SEA of the quadri-arc honeycombs with different relative densities at different impact velocities. It is seen that SEA of the quadri-arc honeycombs increases with both impact velocity and relative density nonlinearly. The trend of SEA is similar with that of the plateau stress, as shown in Fig. 19. It is also observed from Fig. 23 that SEA for the impact velocity V = 0.2 and V = 1.0 are almost the same. It indicates that there is no significant inertial effect when the impact velocity is smaller than the yield velocity. Only when the impact velocity is larger than the yield velocity, i.e., V > 1.0 , significant inertial effect appears.
This work was supported by a research grant by the National Natural Science Foundation of China (No. 11572086), a research grant by Jiangsu Natural Science Foundation of China (No. BK2012318) and a research grant by Key Laboratory of Lightweight and Reliability Technology for Engineering Vehicle, College of Hunan Province (No. 2016kfjj08). References [1] Q. Zhang, X. Yang, P. Li, et al., Bioinspired engineering of honeycomb structure–Using nature to inspire human innovation, Progress. Mater. Sci. 74 (2015) 332–400. [2] J. Fazilati, M. Alisadeghi, Multiobjective crashworthiness optimization of multilayer honeycomb energy absorber panels under axial impact, Thin-Walled Struct. 107 (2016) 197–206. [3] J. Qiao, C. Chen, In-plane crushing of a hierarchical honeycomb, Int. J. Solids Struct. 85 (2016) 57–66. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, 1999. [5] Y. Liu, X.C. Zhang, The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs, Int. J. Impact Eng. 36 (1) (2009) 98–109. [6] R. Oftadeh, B. Haghpanah, D. Vella, et al., Optimal fractal-like hierarchical honeycombs, Phys. Rev. Lett. 113 (10) (2014) 104301. [7] R. Oftadeh, B. Haghpanah, J. Papadopoulos, et al., Mechanics of anisotropic hierarchical honeycombs, Int. J. Mech. Sci. 30 (81) (2014) 126–136. [8] S.A. Galehdari, M. Kadkhodayan, S. Hadidi-Moud, Analytical, experimental and numerical study of a graded honeycomb structure under in-plane impact load with low velocity, Int. J. Crashworthiness 20 (4) (2015) 387–400. [9] D. Ruan, G. Lu, B. Wang, et al., In-plane dynamic crushing of honeycombs—a finite element study, Int. J. Impact Eng. 28 (2) (2003) 161–182. [10] S.D. Papka, S. Kyriakides, Biaxial crushing of honeycombs—part 1: experiments, Int. J. Solids Struct. 36 (29) (1999) 4367–4396. [11] S.D. Papka, S. Kyriakides, In-plane biaxial crushing of honeycombs—part II: analysis, Int. J. Solids Struct. 36 (29) (1999) 4397–4423. [12] L.L. Hu, T.X. Yu, Dynamic crushing strength of hexagonal honeycombs, Int. J. Impact Eng. 37 (5) (2010) 467–474. [13] L. Hu, F. You, T. Yu, Effect of cell-wall angle on the in-plane crushing behavior of hexagonal honeycombs, Mater. Des. 46 (2013) 511–523. [14] A.J. Wang, D.L. McDowell, In-plane stiffness and yield strength of periodic metal honeycombs, J. Eng. Mater. Technol. 126 (2) (2004) 137–156. [15] A.J. Wang, D.L. McDowell, Yield surfaces of various periodic metal honeycombs at intermediate relative density, Int. J. Plast. 21 (2) (2005) 285–320. [16] Y. Liu, X.C. Zhang, The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs, Int. J. Impact Eng. 36 (1) (2009) 98–109. [17] X. Zhang, H. Zhang, Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs, Compos. Struct. 96 (2013) 143–152. [18] Z. Zhang, S. Liu, Z. Tang, Crashworthiness investigation of kagome honeycomb sandwich cylindrical column under axial crushing loads, Thin-Walled Struct. 48 (1) (2010) 9–18. [19] Z. Zhang, S. Liu, Z. Tang, Comparisons of honeycomb sandwich and foam-filled cylindrical columns under axial crushing loads, Thin-Walled Struct. 49 (9) (2011) 1071–1079. [20] H. Fan, F. Jin, D. Fang, Uniaxial local buckling strength of periodic lattice composites, Mater. Des. 30 (10) (2009) 4136–4145. [21] F. Scarpa, S. Blain, T. Lew, et al., Elastic buckling of hexagonal chiral cell honeycombs, Compos. Part A: Appl. Sci. Manuf. 38 (2) (2007) 280–289. [22] A. Klepka, W.J. Staszewski, D. Di Maio, et al., Impact damage detection in composite chiral sandwich panels using nonlinear vibro-acoustic modulations, Smart Mater. Struct. 22 (8) (2013) 084011. [23] A. Airoldi, P. Bettini, M. Zazzarini, et al., Failure and energy absorption of plastic and composite chiral honeycombs, Struct. Shock Impact Xii. (2012) 101–114. [24] D. Zhang, D. Jiang, Q. Fei, et al., Experimental and numerical investigation on indentation and energy absorption of a honeycomb sandwich panel under lowvelocity impact, Finite Elem. Anal. Des. 117 (2016) 21–30. [25] Y. Wang, P. Xue, J.P. Wang, Comparing study of energy-absorbing behavior for honeycomb structures, InKey Eng. Mater. 462 (2011) 13–17.. Trans Tech Publications. [26] Z. Wang, Z. Lu, S. Yao, et al., Deformation mode evolutional mechanism of honeycomb structure when undergoing a shallow inclined load, Compos. Struct. 147 (2016) 211–219. [27] S.R. Reid, C. Peng, Dynamic uniaxial crushing of wood, Int. J. Impact Eng. 19 (5) (1997) 531–570. [28] G. Lu, T.X. Yu Energy absorption of structures and materials, Elsevier. [29] Z. Zou, S.R. Reid, P.J. Tan, S. Li, J.J. Harrigan, Dynamic crushing of honeycombs and features of shock fronts, Int. J. Impact Eng. 36 (1) (2009) 165–176. [30] J.J. Harrigan, S.R. Reid, C. Peng, Inertia effects in impact energy absorbing materials and structures, Int. J. Impact Eng. 22 (9) (1999) 955–979. [31] X.C. Zhang, L.Q. An, H.M. Ding, Dynamic crushing behavior and energy absorption of honeycombs with density gradient, J. Sandw. Struct. Mater. 16 (2) (2014) 125–147. [32] A. Hönig, W.J. Stronge, In-plane dynamic crushing of honeycomb. Part I: crush band initiation and wave trapping, Int. J. Mech. Sci. 44 (8) (2002) 1665–1696. [33] A. Hönig, W.J. Stronge, In-plane dynamic crushing of honeycomb. Part II: application to impact, Int. J. Mech. Sci. 44 (8) (2002) 1697–1714.
6. Conclusions A new multi-cell honeycomb constructed from circular single-cells is proposed and named as quadri-arc multi-cell honeycomb in this paper. Finite element method is employed to study the dynamic behavior and energy absorption properties of this new kind of honeycomb under inplane impact loadings. Simulations of a regular circular cell honeycomb are also conducted as comparison in order to better understand the mechanical property of the quadri-arc honeycomb. Three different types of deformation modes are observed for the quadri-arc honeycomb under different impact velocities, including quasi-static mode, transition mode and dynamic mode, respectively. An X-shaped localized band followed by a rectangular localized band developed with increase of compression at quasi-static impact loading. A transition mode is present when the impact velocity is moderate. Layer-by-layer localized collapse is observed, whose densification degree decreases generally layer-by-layer along the impact direction. When impact velocity is sufficiently high, an I-shaped mode as a manner of wave propagation is observed. Furthermore, dynamic impact responses and energy absorbed of the quadri-arc honeycomb are studied and compared with the regular circular honeycomb. Results show that the quadri-arc multi-cells could significantly improve the plateau stress and energy absorption efficiency, especially under quasi-static and middle-velocity impacts. 45.8% higher plateau stress and 56.3% more absorbed energy are obtained in the quasi-static case, specifically. However, the advantage is losing out with increase of impact velocity for initial effect becomes dominate in the high-velocity cases. One can conclude that the quadriarc honeycomb is a better choice compared with the regular circular honeycomb. 210
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
In–plane dynamic crushing behavior and energy absorption of honeycombs with a novel type of multi-cells
MARK
⁎
Dahai Zhanga,b,d, Qingguo Feic, , Peiwei Zhanga,d a
Department of Engineering Mechanics, Southeast University, Nanjing 210096, China Faculty of Science, Engineering and Technology, Swinburne University of Technology, John Street, Hawthorn, VIC 3122, Australia c School of Mechanical Engineering, Southeast University, Nanjing 210096, China d Key Laboratory of Lightweight and Reliability Technology for Engineering Vehicle, College of Hunan Province, Changsha 410114, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Honeycombs Multi-cell Deformation mode Energy absorption In-plane
In order to pursue better crashworthiness and higher energy absorption efficiency, a new type of multi-cell honeycomb (quadri-arc) was designed and followed by a series of numerical studies on in-plane dynamic crushing behavior and energy absorption property under different impact loading. Meanwhile, simulations of a regular circular single-cell honeycomb were also conducted as comparisons. Three distinct deformation modes were identified from the observation of deformation profiles: quasi-static, transition and dynamic, respectively. Simulations indicate that deformation modes are not only influenced by impact velocity but also sensitive to relative density of the honeycomb, based on which a deformation map was summarized. Furthermore, the plateau stress and energy absorption of the quadri-arc honeycomb as well as the circular honeycomb were explored and compared, during which effects of impact velocity and relative density of the honeycomb were discussed in detail. The results show that a much higher plateau stress and better energy absorption efficiency for the quadri-arc honeycomb are predicted compared to the circular honeycomb, especially in the quasi-static case. The investigation suggests that design of the quadri-arc multi-cell will enhance the crashworthiness and energy absorption capacity of honeycombs.
1. Introduction As one typical type of cellular material, honeycomb structures have been widely employed in many engineering applications, including personal protective equipment, automotive industry, transportation and aeronautics engineering due to their desirable mechanical properties such as outstanding lightweight, high strength, heat insulation and excellent performance in energy absorption [1]. With increase of safety requirements, optimization and improvement of honeycomb structures for pursuing better crashworthiness and higher energy absorption efficiency have become advanced research hotpots recently [1–3]. Generally, the stress-strain curve of a honeycomb structure subjected to in-plane impact loading contains three distinct regimes: a linear elastic regime, a plateau regime and a densification regime. Among the three regimes, energy capacity of the structure mainly relies on the plateau regime, which is dominated by elastic buckling, plastic buckling and plastic collapse of the unit cells [4]. Therefore, geometry sensitivity is a very important characteristic of honeycombs, the mechanical properties of a honeycomb structure are not only determined by the matrix material, but also extremely be sensitive to cell ⁎
topology properties [5–9]. By changing the geometrical parameters of the cells such as cell type, cell size, and ratio of wall thickness-to-length and so on, the honeycombs may show a different mechanical property and energy absorption capacity [5]. As is well known, honeycomb cellular structure is one classic biomimetic material, which was first developed originating from the nature honeycomb in a nest. So honeycomb structure with hexagonal cells have drawn the most attention of researchers. However, hexagonal honeycombs are not suitable for all applications, many other types of simple cells have been introduced and designed for specific purpose, including circular, triangular and square cells. Mechanical characteristics and energy absorption capacity of honeycomb structures with different simple types of single cells have been fully researched theoretically, experimentally and numerically [4,5,8–16]. Gibson and Ashby [4] systematically studied the fundamental mechanical properties of honeycomb structures with hexagonal cells as well as triangular and square cells based on micro-structure model and theoretically derived the equation to predict the plateau stress of honeycombs. Ruan et al. [9] numerically investigated the in-plane dynamic behavior of hexagonal honeycomb structures, they observed three deformation modes and discussed the
Corresponding author. E-mail address: [email protected] (Q. Fei).
http://dx.doi.org/10.1016/j.tws.2017.03.028 Received 29 September 2016; Received in revised form 15 March 2017; Accepted 23 March 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Nomenclature
RB ,RC lB tB ,tC κB ,κC ρB , ρC ρB , ρC , ρ ρs V0 Es μs σys
V Relative impact velocity Vy Yield velocity c0 Elastic wave speed in the matrix material εY Yield strain of the matrix marital ε Nominal compressive strain δ Vertical displacement VC1, VC 2 Critical velocity σ (ε ) Nominal compressive stress VB Wave speed εcr Nominal strain at initial stress peak εd Locking strain λ ,C , C1, C2, C3 Constants Plateau stress σpd ,σpq
Constitutive cell Radius of honeycomb Cell length of the quadri-arc honeycomb Cell thickness Cell thickness-to-length ratio Density of the honeycomb Relative density of the honeycomb Density of matrix material Impact velocity Young's modulus of matrix material Poisson's ratio of matrix material Yield stress of matrix material
[21–23] have also been introduced and studied. Generally, multi-cell honeycomb structures show considerably desirable mechanical properties and energy absorption capacity compared with the single-cell honeycombs. Fig. 1 shows a Kagome honeycomb structure and a diamond honeycomb structure, their micro-structures are magnified and given as well as the constitutive cells. It is noticed from the figure that a multicell honeycomb is essentially constructed by cross connection, combination or superimposition of simple single-cells. For Kagome honeycomb, the single-cells are triangular cells while for diamond honeycomb, the single-cells are triangular or hexagonal cells. In this paper, a new type of multi-cell honeycomb as shown in Fig. 2 is introduced and followed by a serious of numerical analyses on dynamic behavior and energy absorption properties under in-plane impact loadings. The new multi-cell honeycomb is constructed from circular cells and named as quadri-arc multi-cell honeycomb based on the geometry of its unit cells. This paper is organized as follow. In Section 2, details of the microstructure of quadri-arc honeycomb is described. Finite element (FE) model of the honeycomb is established and validated. During which, a FE model of regular circular honeycomb is established as comparison. In Section 3, deformation modes of the quadri-arc honeycomb are investigated and compared with the circular honeycomb. Moreover, an overall deformation map of the quadri-arc honeycomb is summarized and the critical velocities of different deformation modes are proposed based on analysis of deformation profiles of honeycombs with different relative densities under different impact velocities. In Section 4, the dynamic impact response of the quadri-arc honeycomb is investigated by analyzing the reaction stress-strain curve of the impact plate, during which effects of impact velocity and relative density are discussed. Meanwhile, the plateau stress of the quadri-arc honeycomb is compared with the circular honeycomb. In Section 5, energy absorption property of the quadri-arc honeycomb is further studied and compared with the regular circular honeycomb.
influences of cell wall thickness and impact velocity on localized deformation and plateau stress of the honeycomb. Papka et al. [10,11] conducted experimental and numerical studies of uniaxial and biaxial crushing of a honeycomb with circular cells, they presented a full scale numerical simulation method which can successfully predict the major material parameters of interest and compared well with the experimental results. An analytical model was established by Hu et al. [12,13] to deduce the crushing strength of the hexagonal honeycombs and was in good agreement with the numerical simulation results. Besides, a lot of works were presented to discuss the crushing behavior of honeycombs with different cell micro-topologies. Wang et al. [14,15] investigated the in-plane mechanical properties of periodic honeycomb structures with seven different single cell types and derived initial yield surfaces of periodic metal honeycombs under a combined in-planes stress state. The empirical equation for honeycomb structures filled with equilateral triangular or quadratic cells and regular or staggered micro-arrangement at high impact velocities were formulated in terms of impact velocity, cell geometric and topology parameters by Liu et al. [16]. Studies mentioned above mainly focused on the honeycombs with simple single-cells. However, honeycomb structures with multi-cells were found to be highly efficient energy absorption structures and received more and more research interests recently [17–23]. A multicell honeycomb structure is usually constituted by a number of simple cells with various angles and by different connection factors. Kagome honeycomb as shown in Fig. 1(a) is one typical multi-cell honeycomb constructed by a combination of hexagon and triangle cells. A lot of works have been presented [17–20] to research the mechanical characteristics of Kagome honeycombs under impact loading and it was found that Kagome honeycombs are better choice under the targets of energy absorption capacity compared with honeycombs with simple single-cells. Besides, other types of multi-cell honeycomb structures such as diamond cell honeycombs [19,20] and chiral cell honeycombs
Fig. 1. Two typical multi-cell honeycombs and their constitutive cells.
200
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 2. Quadri-arc multi-cell honeycomb and a unit cell.
2. Numerical models
ρC = ρC / ρs =
1 ⎛ tC ⎞ 1 π ⎜ ⎟ = πκC 2 ⎝ RC ⎠ 2
(2)
2.1. Finite element modeling where ρC is the authentic density of the circular honeycomb. For better comparison, parameters including mass, volume and matrix material of both models are all set to be the same. By substituting Eqs. (1) and (2) into ρB = ρC , the relationship between RB and RC can be derived as RB = 4RC . To systematically explore the in-plane crushing behavior and energy absorption of the quadri-arc honeycomb, FE models of the quadri-arc and regular circular honeycombs are established. The commercial software package ABAQUS is employed and the step type is DynamicExplicit. Here, the quadri-arc honeycomb is established with cell size RB = 20 mm, composed of 5 cells in X1 direction and 6 cells in X2 direction while cell size for the circular honeycomb is RC = 5 mm with 10 cells in X1 direction and 12 cells in X2 direction. Other parameters of the two honeycombs are all the same including cell wall thickness tB = tC = 0.2 mm, height in out-plane direction h = 2 mm. A rigid flat plane with constant-velocity V0 is modeled to impact the honeycomb and the honeycomb is taken to be clamped at its bottom surface, periodic boundary conditions are applied in each side of the structure to prevent out-of-plane buckling and eliminate boundary effects on simulation results. As shown in Fig. 4, both honeycombs are meshed using hourglass controlled, four nodes, reduced integration shell element (S4R) with
As illustrated in Fig. 2, the micro-structure of the novel multi-cell honeycomb is quadri-arc, which is constructed from circular cells. The configuration of the quadri-arc cell is defined by the following parameters: RB denotes the radius of the constitutive circular cells, tB denotes the thickness of unit cells, lB denotes the length of unit cells and κB = tB / lB denotes thickness-to-length ratio. It can be easily derived that lB = RB . Hence, the relative density of the quadri-arc honeycomb is
⎛t ⎞ ρB = ρB / ρs = 2π ⎜ B ⎟ = 2πκB ⎝ RB ⎠
(1)
where ρB is the authentic density of the quadri-arc honeycomb; ρs is the density of the matrix material. In order to better understand the impact behavior of the quadri-arc honeycomb (Fig. 3(a)), a model of single-cell honeycomb with regular circular cells (also are the constitutive cells of the quadri-arc honeycomb) is also established and simulated as a comparison, as shown in Fig. 3(b). Characteristic parameters of the circular honeycomb are radius of the circular cells RC , cell wall thickness tC and thickness-tolength ratio κC = tC / RC , the relative density of circular honeycomb could be obtained as
Fig. 3. Geometric configurations and boundary conditions of the honeycombs.
201
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 4. 3D view of the numerical models.
five integration points across the thickness [9]. A general contact is adopted to simulate the self-contact of cells, while a surface-to-surface contact is adopted to simulate the contact between the honeycomb and the rigid flat plane. The friction coefficient is set as 0.3. A strain-rate independent idealistic elastoplastic model [24] is employed here and the property of matrix material is listed in Table 1. A convergence test on element size is conducted firstly for both honeycombs. As shown in Fig. 5, the simulated mean crushing forces gradually stabilizes and converges when the element size reduces to 2.0 mm. Therefore, 1.0 mm is adopted in both models conservatively and all results given below are convergent.
c0 =
Es / ρs
(4)
εY = σys / Es
(5)
where c0 is the speed of the longitudinal elastic stress wave; εY is the yield strain of matrix material. Here, a dimensionless normalized parameter V = V0 / Vy is introduced to recognize the effect of impact velocity and address the inertial effect. Figs. 7–9 show deformation patterns of the quadri-arc honeycomb as well as the circular honeycomb under three different impact velocities at different compressive strains. Symbol ε denotes the compressive nominal strain along the impact direction, which is calculated from
2.2. Validation analysis
(6)
ε = δ / L1 To validate the accuracy of the numerical models, a validation analysis is conducted in this section. There are many analogous studies on the circular honeycombs available in literature, e.g. [25]. In this study, simulation of a circular honeycomb with 10 × 10 unit cells, cell diameter of 7.9 mm and wall thickness of 0.3 mm under quasi-static inplane crushing is carried out [25]. The modeling approach is the same as that mentioned in Section 2.1 and the simulation results are compared with the literatures, as shown in Fig. 6. It is seen that both force-displacement curves and deformation patterns are in good agreement. Therefore, the numerical models are considered reliable.
where δ is vertical displacement and L1 is the initial length of the honeycomb along impact direction. As shown in Fig. 7, global deformation occurs when the compressive nominal strain is small (ε = 13.4%) for both quadri-arc and regular circular honeycombs in the quasi-static case (V = 0.4 ), which produces an X-shaped deformation mode. With increase of compression (ε = 41.3%), deformation modes of these two kinds of honeycombs show significant difference. Specifically, a rectangular localized band is developed in the middle of the quadri-arc honeycomb while the circular honeycomb continues to deform as an X-shaped mode. At last (ε = 74.1%), both honeycombs are completely crushed, as shown in Fig. 7(c). At a higher impact velocity of V = 4.0 , the inertia influence begins to emerge and global deformation mode is replaced by local deformation mode for both quadri-arc and circular honeycombs (see Fig. 8). A number of cells near the impact edge of the honeycomb are slightly crushed and localized layer-by-layer collapses are observed for the quadri-arc honeycomb. The compactness degree of the collapse presents a trend of layer-by-layer decreasing from the impact edge to the
3. Dynamic crushing behavior In this section, deformation modes of the quadri-arc multi-cell honeycomb under various in-plane impact velocities are studied and compared with the circular honeycomb. 3.1. Deformation modes It always shows different deformation modes when the impact velocity changed for honeycomb structures [26]. The yield velocity Vy is a very important indicator widely employed to address the inertial effect [27–29]. Zou et al. [29] indicates that yield velocity is the lowest velocity that governs a change in material response for cellular materials. The yield velocity is defined as a material's property
Vy = c0 εY
Table 1 Material property of the matrix material.
(3) 202
Matrix material
Young's modulus Es
Yield stress σys
Poisson's ratio μs
Density ρs
Aluminum (Al)
69 GPa
76 MPa
0.33
2700 kg/m3
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
2×3 and 4×6 cells for the quadri-arc and circular honeycombs, respectively. The calculated Poisson's ratios of different cases are listed in Table 2. It can be seen that the Poisson's ratio of the quadri-arc honeycomb is close to zero and not sensitive to the impact velocity. On the other hand, the Poisson's ratio of the circular honeycomb is much larger and decreases with the increase of impact velocity. Fig. 10 gives the deformations of a unit cell extracted from the central area of the quadri-arc honeycomb as well as the corresponding cells of the circular honeycomb under different impact velocities. As shown in Figs. 7–9, the deformation of unit cells of the quadri-arc honeycomb are similar for different impact velocities, a typical example (V = 4.0 ) is drawn in Fig. 10(a). As shown in Fig. 10(a), no significant change is observed in the horizontal direction with crush proceed, which leads to an almost zero Poisson's ratio for the quadri-arc honeycomb. While for circular honeycomb, the unit cells are crushed smoothly and fully stretched perpendicularly to the impact direction, which leads to a large Poisson's ratio, as shown in Fig. 10(b). However, in a higher impact velocity case (see Fig. 10(c)), the unit cells are crushed irregularly with a smaller stretch perpendicularly to the impact direction, which leads to a smaller Poisson's ratio compared to the low impact velocity case. As stated above, The relative density and wall thickness of both honeycombs are set to be the same, which leads to a difference in cell size. The number of cell rows in the crushing direction are 6 and 12 for the quadri-arc and circular honeycombs, respectively. Scale effects are observed in localization of the honeycombs, see Figs. 7–9. Take the case of V = 4.0 for example, although localized layer-by-layer collapses are observed with crushing proceed, relatively homogeneous collapse is observed in each row. As shown in Fig. 8(a), one row is crushed for the quadri-arc honeycomb while two rows are crushed for the circular honeycomb in the corresponding time. Almost uniform collapse is observed for the one row crushed in the quadri-arc honeycomb, while significant difference is found for the two rows crushed in the circular honeycomb. Specifically, denser collapse is observed for the row closer
Fig. 5. Convergence analyses for mean crushing force under different element size.
clamped edge. Afterward, the honeycomb is completely crushed layer by layer. For the circular honeycomb, a V-shaped deformation mode is observed and remains until the honeycomb is totally crushed. In Fig. 9, a much higher impact velocity (V = 12.0 ) is conducted and it is found that no obvious local deformation (X-shaped or V-shaped) appears for both honeycombs through the whole crushing process. Under high impact velocity, the transverse inhomogeneity is ignored because the inertia effect plays a dominant role in deformation of the honeycombs. An I-shaped deformation mode perpendicular to the impact direction is observed at the impact edge and propagates forward layer by layer to fixed edge with proceeding of the impact for both quadri-arc honeycomb and circular honeycomb, which is just a manner of wave propagation. Besides, an interesting phenomenon is observed that the Poisson's ratios of the two honeycombs are significantly different. Sub-structures ABCD and A′B′C′D′ are extracted from the center of the honeycombs to calculate the Poisson's ratio, as shown in Fig. 7. The sub-structure has
Fig. 6. Comparison of results between FE simulation and literature [25] for a circular honeycomb: (a) Crushing force-displacement curves; (b) Deformation patterns.
203
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 7. Deformation modes of honeycombs under quasi-static impact (V = 0.4 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%.
and crushed to address the size effect on contact forces. Fig. 11. shows the contact forces of the quadri-arc honeycombs with different cells but same relative density. The contact forces of the two honeycombs are very close, which indicates that cell size has no significant effect on
to the impact surface. Similar observation is also made in the other cases. It indicates that cell size has an effect on the localization process in some degree. Moreover, a quadri-arc honeycomb with 10×12 cells were modeled
Fig. 8. Deformation modes of honeycombs under middle-velocity impact (V = 4.0 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%.
204
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 9. Deformation modes of honeycombs under high-velocity impact (V = 12.0 ) at different compressive strains: a) ε = 13.4%; b) ε = 41.3%; c) ε = 74.1%. Table 2 Poisson's ratios of the honeycombs under different impact velocities.
V = 0.4 V = 4.0 V = 12.0
Quadri-arc honeycomb
Circular honeycomb
0.006 0.018 0.008
0.424 0.451 0.323
Fig. 11. Effect of cell size on contact force of the quadri-arc honeycomb.
ized as follows. Three types of modes are found when the impact velocity varies and can be divided into quasi-static mode, transition mode and dynamic mode (see Fig. 12). A quasi-static mode appears with low impact velocity, which can be characterized by an X-shaped band followed by a rectangular localized band developing with increasing of compression. A transition mode appears for a higher impact velocity, in which layer-by-layer localized collapses is observed and densification degree of the collapse presents a trend of decreasing layer-by-layer from the impact edge to the clamped edge. The third type is a dynamic mode, which appears when the impact velocity is sufficiently high. An I-shaped mode as a manner of wave propagation is observed in this mode. A deformation map of the quadri-arc honeycomb with different relative densities subjected to different impact velocity is provided in Fig. 12. It can be found that deformation mode of the quadri-arc honeycomb is determined not only by the impact velocity, but also the relative density. The critical velocity at which deformation modes change from one to another increases linearly with increasing of
Fig. 10. Deformation of the typical unit cells: a) Quadri-arc honeycomb (V = 4.0 ); b) Circular honeycomb (V = 4.0 ); c) Circular honeycomb (V = 12.0 ).
contact force. Same phenomenon was also reported in literatures [4,5,28] that the crashworthiness of a honeycomb is determined by the relative density rather than cell size. 3.2. Deformation modes map Different deformation modes have been observed for the quadri-arc honeycomb under different impact velocities, which can be summar205
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 12. Deformation modes map of the quadri-arc honeycomb.
Fig. 13. History of dynamic responses of the quadri-arc honeycomb.
4.1. Effect of impact velocity
relative density of the honeycomb, which means that increasing of the relative density delays the onset of the deformation mode. Two empirical equation for the two critical velocities are proposed respectively as
VC1 = 0.57 + 11.37ρB
(7)
VC 2 = 2.0 + 71.62ρB
(8)
Fig. 13 illustrates the nominal stress-strain curves at both front and bottom surfaces for quadri-arc honeycomb under three different impact velocities, during which the nominal stress σ (ε ) is calculated from σ (ε ) = F / A, where F is the reaction force of the impact plate and A is the cross section area of the structure normal to the longitudinal direction. As shown in Fig. 13(a), a typical stress-strain curve including three distinct dynamic response stages: a linear elastic stage initially, a long collapse plateau stage in the middle and a compressive densification stage at last is observed for the quadri-arc honeycomb. An important observation in Fig. 13(a) is that the plateau stress at the front surface increases with impact velocity. The phenomenon is a common trend, which has been reported and discussed widely in literatures [9,12,29–31]. However, impact velocity has much less effect on the
4. Dynamic impact response In this section, Dynamic impact response of the quadri-arc honeycomb is investigated by analyzing the reaction stress-strain curve of the impact plate. 206
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 14. Difference of the nominal stresses at the front and bottom surfaces for the case of V = 0.4 .
Fig. 15. Nominal stresses on the bottom surface at different impact velocities.
response at the bottom surface, as shown in Fig. 13(b). This is may be due to the stresses at the bottom surface are the results of compaction waves travelling from the front surface are far lower in intensity than the compaction wave [31]. Note the deformation of the honeycomb walls is mainly bending dominated, the wave of global uniaxial compaction can be called a bending wave [32,33]. Fig. 14 shows the nominal stress-strain curves at both front and bottom surfaces of the quasi-static case (V = 0.4 ). As shown in Fig. 14, there is a gap Δt between the times when stress firstly reaches the maximums at the front surface and bottom surface respectively. Moreover, the nominal stress curves at the bottom surfaces of different impact velocities are plotted in Fig. 15. It should be noted that the zero of the horizontal axis is the time where stress reaches the maximum at the front surface for each case. It can be seen that the time gaps are basically the same for the three cases although the crushing processes are significantly different. The wave speed can be calculated hereby, which is about VB = L1/ Δt = 307.7 m/s. Generally, there are initial peaks in the stress-strain curves. However, the plateau stage is the main interest discussed here, which is the most important part of energy absorption capacity for a honeycomb structure. Furthermore, the values of the peaks are uncertain and heavily influenced by the cut-off frequency of a wave filter in the FE calculation [9], so the initial peaks are not discussed in this paper. The plateau stress is defined as the average nominal stress between the initial peak and the compressive stress corresponding to the locking strain
Fig. 16. Comparisons of the dynamic responses for quadri-arc honeycomb and circular honeycomb under different impact velocities: a) V = 0.4 ; b) V = 4.0 ; c) V = 12.0 .
εd = 1 − Δρ
where εcr is the nominal strain at the initial stress peak; εd is the locking strain; λ is a constant which is determined by the type of the honeycomb cells and the impact velocity and ideally equal to 1, but is usually larger than 1 for the honeycomb cannot wholly crushed under impact loading. As shown in Fig. 15(a), significant influence of impact velocity on stress-strain curves is observed. The quasi-static stress-strain curve is much more stable and can be obviously divided into three regimes while a higher curve with strong fluctuation is found for the high impact velocity. As stated above, when the impact velocity is sufficiently high, a propagation of plane plastic wave liked deformation mode is observed. A significant influence caused by the inertia effect makes the dynamic stress-strain curve big difference from the quasistatic situation. A shock wave theory was suggested for analysis of the
ε
σp =
∫ε d σ (ε) dε cr
εd − εcr
(10)
(9) 207
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 17. Comparison of plateau stresses for the quadri-arc and circular honeycombs under different impact velocities.
Fig. 20. Effect of impact velocity on energy absorbed by the quadri-arc honeycomb.
dynamic plateau stress and a widely used theoretical model was derived by Reid [29,30] using the R-I shock theory
Table 3 Plateau stresses for the quadri-arc and circular honeycombs. Relative velocity
Circular honeycomb (Mpa )
Quadri-arc honeycomb (Mpa )
Difference
V = 0.4 V = 4.0 V = 12.0
0.1035 0.1919 0.9235
0.1509 0.2567 0.9785
+45.8% +33.8% +5.96%
σpd = σpq + CV0 2
(11)
where σpd and σpq are dynamic and quasi-static plateau stress, respectively; C is a coefficient of the inertia effect. It can be seen that the dynamic plateau stress consists of two terms: a constant quasi-static term determined by the micro-structure of honeycomb and a changing dynamic term determined by inertia effect. So the stress-strain curve of the honeycomb rises rapidly with increase of impact velocity.
4.2. Comparison with regular circular honeycomb For comparison, the simulated corresponding uniaxial stress-strain curves of the quadri-arc multi-cell honeycomb as well as the circular honeycomb under different impact velocities are plotted in Fig. 16. It could be seen from Fig. 16 that the quadri-arc honeycomb shows an obviously higher stress-strain curves than the circular honeycomb in quasi-static case (V = 0.4 ) and middle-velocity case (V = 4.0 ), but no apparent difference in the high velocity case (V = 12.0 ). Generally, higher plateau stress signifies better impact resistance capacity. From this standpoint, quadri-arc honeycomb performs much better on impact resistance compared with circular honeycomb under quasi-static and middle-velocity. It also should be noted that the quadri-arc honeycomb has an obvious lagging densification region in the quasi-static case, which makes it more perfect. Moreover, comparison of the plateau stresses of two kinds of honeycombs under different impact velocities are illustrated in Fig. 17 and specific data are given in Table 3. As shown in the diagrams, much larger plateau stresses are observed for the quadriarc honeycomb under quasi-static and middle-velocity impact. Specifically, 45.8% larger for quasi-static impact and 33.8% larger for middlevelocity impact. For high impact velocity, the difference is much smaller, only 5.96% higher plateau stress is obtained for the quadriarc honeycomb. As stated in Eq. (11), the dynamic plateau stress of a honeycomb consists of two terms: a quasi-static term and an inertia term. The quasi-static term is a constant determined by the microstructure of the honeycomb while the inertia term is determined by inertia effect and varying along the impact velocity. Therefore, difference of micro-structures of honeycombs cause a big difference in quasi-static term and a small difference in inertia term. Since inertia term is negligible in low-velocity cases and becomes the dominating part when the impact velocity is sufficient high, quadri-arc honeycomb is a better choice with its advantage degrading with increase of impact velocity.
Fig. 18. Plateau stresses of the quadri-arc honeycomb versus relative velocities of different cell thickness-to-length ratios.
Fig. 19. Plateau stresses of quadri-arc honeycomb versus cell thickness-to-length ratio under different impact velocities.
208
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Fig. 22. Comparison of energy absorbed by the quadri-arc and circular honeycombs.
Table 4 Energy absorbed by the quadri-arc and circular honeycombs under different velocities. Relative velocity
Circular honeycomb (kJ /kg )
Quadri-arc honeycomb (kJ /kg )
Difference
V = 0.4 V = 4.0 V = 12.0
0.3493 0.5854 2.153
0.5459 0.8211 2.294
+56.3% +40.3% +6.55%
Fig. 23. SEA of the quadri-arc honeycomb versus cell thickness-to-length ratio under different impact velocities.
σpd = σpq + (C1 κB2 + C2 κB + C3) V0 2 Fig. 21. Energy absorption history for quadri-arc honeycomb and circular honeycomb under different impact velocities: a) V = 0.4 ; b) V = 4.0 ; c) V = 12.0 .
5. Energy absorption efficiency
4.3. Plateau stress map
In this section, energy absorption capacity of the quadri-arc honeycomb subjected to in-plane impact loadings is investigated. A specific energy absorbed indicator SEA is adopted [31] to evaluate the energy absorption capabilities of honeycomb and sandwich materials, which is defined as
Effect of the relative density and impact velocity on plateau stress is further studied. Figs. 18 and 19 illustrate the plateau stresses of quadriarc honeycomb of different relative densities under different impact velocities. As shown in Fig. 18, the plateau stress is in a quadratic relation with impact velocity, which can be explained by Eq. (11). Besides, it can be seen from Fig. 19 that the plateau stress is also in a quadratic relation with cell thickness-to-length ratio (relative density) under high dynamic impact velocity. Hence, constant C in Eq. (11) can be formulated as
C = C1 κB2 + C2 κB + C3
(13)
εd
∫ε σ (ε) dε W cr SEA = = M ρs ρ
(14)
where W is the total energy absorbed by honeycomb and M is the total mass of honeycomb; ρ is the relative density of honeycomb. Based on Eq. (14), the absorbed energy per unit mass for the quadriarc honeycomb under different impact velocities versus the nominal strain normal to the longitudinal direction are depicted in Fig. 20. It is seen that impact velocity has great influence on energy absorbed,
(12)
where C1, C2, C3 are constants, so Eq. (11) can be rewritten as 209
Thin-Walled Structures 117 (2017) 199–210
D. Zhang et al.
Acknowledgements
energy absorbed is significantly increased along with increase of impact velocity. Moreover, energy absorbed by the circular honeycomb are also calculated and illustrated in Fig. 21 as comparisons. Under quasistatic or middle-velocity, quadri-arc honeycomb shows better efficiency than the regular circular honeycomb. But when the impact velocity is sufficiently high, no obvious advantage is observed for the quadri-arc honeycomb. Compassion of the energy absorbed ultimately by the two honeycombs under different impact velocities are illustrated in Fig. 22 and specific data are given in Table 4. It can be seen that much larger absorbed energy are observed for the quadri-arc honeycomb under quasi-static and middle-velocity impact. Specifically, 56.3% larger for the quasi-static impact and 40.3% larger for the middle-velocity impact. For the high impact velocity, the difference is little with only 6.55% more energy absorbed by the quadri-arc honeycomb. As stated in Eq. (14), energy absorbed is essentially the integral along compressive strain of plateau stress. Therefore, big difference of the plateau stress for quasi-static or middle-velocity case results in big difference of energy absorbed while small difference of the plateau stress for highvelocity case results in little difference of energy absorbed. In a word, quadri-arc honeycomb is a much better choice for energy absorption efficiency compared with regular circle honeycomb. But its advantage will lose out with increase of impact velocity. Effect of the relative density and impact velocity on SEA is further studied. Fig. 23 shows SEA of the quadri-arc honeycombs with different relative densities at different impact velocities. It is seen that SEA of the quadri-arc honeycombs increases with both impact velocity and relative density nonlinearly. The trend of SEA is similar with that of the plateau stress, as shown in Fig. 19. It is also observed from Fig. 23 that SEA for the impact velocity V = 0.2 and V = 1.0 are almost the same. It indicates that there is no significant inertial effect when the impact velocity is smaller than the yield velocity. Only when the impact velocity is larger than the yield velocity, i.e., V > 1.0 , significant inertial effect appears.
This work was supported by a research grant by the National Natural Science Foundation of China (No. 11572086), a research grant by Jiangsu Natural Science Foundation of China (No. BK2012318) and a research grant by Key Laboratory of Lightweight and Reliability Technology for Engineering Vehicle, College of Hunan Province (No. 2016kfjj08). References [1] Q. Zhang, X. Yang, P. Li, et al., Bioinspired engineering of honeycomb structure–Using nature to inspire human innovation, Progress. Mater. Sci. 74 (2015) 332–400. [2] J. Fazilati, M. Alisadeghi, Multiobjective crashworthiness optimization of multilayer honeycomb energy absorber panels under axial impact, Thin-Walled Struct. 107 (2016) 197–206. [3] J. Qiao, C. Chen, In-plane crushing of a hierarchical honeycomb, Int. J. Solids Struct. 85 (2016) 57–66. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, 1999. [5] Y. Liu, X.C. Zhang, The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs, Int. J. Impact Eng. 36 (1) (2009) 98–109. [6] R. Oftadeh, B. Haghpanah, D. Vella, et al., Optimal fractal-like hierarchical honeycombs, Phys. Rev. Lett. 113 (10) (2014) 104301. [7] R. Oftadeh, B. Haghpanah, J. Papadopoulos, et al., Mechanics of anisotropic hierarchical honeycombs, Int. J. Mech. Sci. 30 (81) (2014) 126–136. [8] S.A. Galehdari, M. Kadkhodayan, S. Hadidi-Moud, Analytical, experimental and numerical study of a graded honeycomb structure under in-plane impact load with low velocity, Int. J. Crashworthiness 20 (4) (2015) 387–400. [9] D. Ruan, G. Lu, B. Wang, et al., In-plane dynamic crushing of honeycombs—a finite element study, Int. J. Impact Eng. 28 (2) (2003) 161–182. [10] S.D. Papka, S. Kyriakides, Biaxial crushing of honeycombs—part 1: experiments, Int. J. Solids Struct. 36 (29) (1999) 4367–4396. [11] S.D. Papka, S. Kyriakides, In-plane biaxial crushing of honeycombs—part II: analysis, Int. J. Solids Struct. 36 (29) (1999) 4397–4423. [12] L.L. Hu, T.X. Yu, Dynamic crushing strength of hexagonal honeycombs, Int. J. Impact Eng. 37 (5) (2010) 467–474. [13] L. Hu, F. You, T. Yu, Effect of cell-wall angle on the in-plane crushing behavior of hexagonal honeycombs, Mater. Des. 46 (2013) 511–523. [14] A.J. Wang, D.L. McDowell, In-plane stiffness and yield strength of periodic metal honeycombs, J. Eng. Mater. Technol. 126 (2) (2004) 137–156. [15] A.J. Wang, D.L. McDowell, Yield surfaces of various periodic metal honeycombs at intermediate relative density, Int. J. Plast. 21 (2) (2005) 285–320. [16] Y. Liu, X.C. Zhang, The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs, Int. J. Impact Eng. 36 (1) (2009) 98–109. [17] X. Zhang, H. Zhang, Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs, Compos. Struct. 96 (2013) 143–152. [18] Z. Zhang, S. Liu, Z. Tang, Crashworthiness investigation of kagome honeycomb sandwich cylindrical column under axial crushing loads, Thin-Walled Struct. 48 (1) (2010) 9–18. [19] Z. Zhang, S. Liu, Z. Tang, Comparisons of honeycomb sandwich and foam-filled cylindrical columns under axial crushing loads, Thin-Walled Struct. 49 (9) (2011) 1071–1079. [20] H. Fan, F. Jin, D. Fang, Uniaxial local buckling strength of periodic lattice composites, Mater. Des. 30 (10) (2009) 4136–4145. [21] F. Scarpa, S. Blain, T. Lew, et al., Elastic buckling of hexagonal chiral cell honeycombs, Compos. Part A: Appl. Sci. Manuf. 38 (2) (2007) 280–289. [22] A. Klepka, W.J. Staszewski, D. Di Maio, et al., Impact damage detection in composite chiral sandwich panels using nonlinear vibro-acoustic modulations, Smart Mater. Struct. 22 (8) (2013) 084011. [23] A. Airoldi, P. Bettini, M. Zazzarini, et al., Failure and energy absorption of plastic and composite chiral honeycombs, Struct. Shock Impact Xii. (2012) 101–114. [24] D. Zhang, D. Jiang, Q. Fei, et al., Experimental and numerical investigation on indentation and energy absorption of a honeycomb sandwich panel under lowvelocity impact, Finite Elem. Anal. Des. 117 (2016) 21–30. [25] Y. Wang, P. Xue, J.P. Wang, Comparing study of energy-absorbing behavior for honeycomb structures, InKey Eng. Mater. 462 (2011) 13–17.. Trans Tech Publications. [26] Z. Wang, Z. Lu, S. Yao, et al., Deformation mode evolutional mechanism of honeycomb structure when undergoing a shallow inclined load, Compos. Struct. 147 (2016) 211–219. [27] S.R. Reid, C. Peng, Dynamic uniaxial crushing of wood, Int. J. Impact Eng. 19 (5) (1997) 531–570. [28] G. Lu, T.X. Yu Energy absorption of structures and materials, Elsevier. [29] Z. Zou, S.R. Reid, P.J. Tan, S. Li, J.J. Harrigan, Dynamic crushing of honeycombs and features of shock fronts, Int. J. Impact Eng. 36 (1) (2009) 165–176. [30] J.J. Harrigan, S.R. Reid, C. Peng, Inertia effects in impact energy absorbing materials and structures, Int. J. Impact Eng. 22 (9) (1999) 955–979. [31] X.C. Zhang, L.Q. An, H.M. Ding, Dynamic crushing behavior and energy absorption of honeycombs with density gradient, J. Sandw. Struct. Mater. 16 (2) (2014) 125–147. [32] A. Hönig, W.J. Stronge, In-plane dynamic crushing of honeycomb. Part I: crush band initiation and wave trapping, Int. J. Mech. Sci. 44 (8) (2002) 1665–1696. [33] A. Hönig, W.J. Stronge, In-plane dynamic crushing of honeycomb. Part II: application to impact, Int. J. Mech. Sci. 44 (8) (2002) 1697–1714.
6. Conclusions A new multi-cell honeycomb constructed from circular single-cells is proposed and named as quadri-arc multi-cell honeycomb in this paper. Finite element method is employed to study the dynamic behavior and energy absorption properties of this new kind of honeycomb under inplane impact loadings. Simulations of a regular circular cell honeycomb are also conducted as comparison in order to better understand the mechanical property of the quadri-arc honeycomb. Three different types of deformation modes are observed for the quadri-arc honeycomb under different impact velocities, including quasi-static mode, transition mode and dynamic mode, respectively. An X-shaped localized band followed by a rectangular localized band developed with increase of compression at quasi-static impact loading. A transition mode is present when the impact velocity is moderate. Layer-by-layer localized collapse is observed, whose densification degree decreases generally layer-by-layer along the impact direction. When impact velocity is sufficiently high, an I-shaped mode as a manner of wave propagation is observed. Furthermore, dynamic impact responses and energy absorbed of the quadri-arc honeycomb are studied and compared with the regular circular honeycomb. Results show that the quadri-arc multi-cells could significantly improve the plateau stress and energy absorption efficiency, especially under quasi-static and middle-velocity impacts. 45.8% higher plateau stress and 56.3% more absorbed energy are obtained in the quasi-static case, specifically. However, the advantage is losing out with increase of impact velocity for initial effect becomes dominate in the high-velocity cases. One can conclude that the quadriarc honeycomb is a better choice compared with the regular circular honeycomb. 210