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Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy
Accepted Manuscript Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy Dahai Zhang, Qingguo Fei, Dong Jiang, Yanbin Li PII: DOI: Reference:
S0263-8223(17)31652-5 https://doi.org/10.1016/j.compstruct.2018.02.082 COST 9432
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
25 May 2017 29 December 2017 26 February 2018
Please cite this article as: Zhang, D., Fei, Q., Jiang, D., Li, Y., Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy, Composite Structures (2018), doi: https://doi.org/10.1016/ j.compstruct.2018.02.082
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Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy
Dahai Zhang 1, 2, Qingguo Fei 1, 2,*, Dong Jiang 1, 3, Yanbin Li 1, 2 1
Institute of Aerospace Machinery and Dynamics, Southeast University, Nanjing 211189, China.
2
School of Mechanical Engineering, Southeast University, Nanjing 211189, China.
3
College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China.
* Corresponding author. Email address: [email protected] 1
Abstract The out-of-plane crushing resistance of fractal-like honeycombs with self-similar hierarchy are investigated numerically and analytically. The hierarchical honeycombs are developed via replacing each three-edge vertex of a regular hexagonal honeycomb by a smaller hexagon. Hierarchical honeycombs with higher orders are constructed by repeating this process. Theoretical solutions are derived based on the Super Folding Element (SFE) theory to predict the mean crushing forces (MCFs) of the hierarchical honeycombs. Numerical simulations are also conducted in conjunction with the theoretical solutions. Crushing resistances of three groups of hierarchical honeycombs with different relative densities are explored subsequently. Good agreement between results obtained using theoretical and numerical methods indicates that the theoretical predictions are reliable. The results show that hierarchy can significantly improve the crushing resistance of honeycombs. It is found that the MCFs of first to fourth order hierarchical honeycombs are improved by 71%, 114%, 201% and 309% compared with the regular honeycomb under the same relative density, respectively. It reveals that the amplification of the MCF from one generation to the next approaches a constant ( n 1.26 ) when the hierarchical order is sufficiently large. The potential maximum achievable amplifications of the MCF for hierarchical honeycombs of different relative densities are also discussed. Keywords: Honeycombs; Self-similar hierarchy; Out-of-plane; Crushing resistance.
2
1.
Introduction Biological materials always perform exceptionally in mechanical providing protection
and support for plant and animal bodies [1]. This is believed due to that nature has developed optimized approaches of achieving high structural efficiency. Among nature’s strategies in pursuing better mechanical efficiency, functional adaptation of the structures at all levels of hierarchy is the most common one. There are many natural materials displaying this hierarchical design, such as bones [2], teeth [3], woods [4] and tendons [5]. The fundamental concept of hierarchical design is to create complex cellular microstructures by placing material where most needed [2]. The mechanical efficiency of the resulting materials can be significantly improved. In recent years, the hierarchy has been widely introduced into design of artificial materials and structures, for example, lattice trusses [6, 7], composites [8, 9] and thin-walled tubes [10, 11]. As one type of typical biological materials, honeycombs have been widely used in many kinds of industries as protection or energy absorption components [12]. Numerous works have been proposed on exploring both in-plane and out-of-plane crushing performance of honeycombs by means of experimental, theoretical and numerical methods [13-17]. To understand the potential advantages of hierarchy in pursuing better crashworthiness, the hierarchy was also introduced into design of honeycombs by many researchers recently [18-38]. Accordingly, hierarchical honeycombs with various configurations have been constructed. One hierarchical development approach is to replace the solid walls of the honeycomb by cellular microstructures. The cellular microstructures can be various, such as sandwich structures [18, 19], hexagonal micro-cells [20, 21], triangular micro-cells [23-25] and negative Poisson’s ratio sub-structures [26]. Exhaustive studies were conducted subsequently to explore the effect of hierarchy on mechanical properties of honeycombs by Fan et al. [18], Taylor et al. [19, 21], Chen et al. [20], Vigliotti et al.[22], Sun et al. [23, 26], 3
Qiao et al. [24] and Chen et al.[25]. They proved that hierarchical honeycombs had super mechanical properties and damage tolerance compared with the equal mass solid-wall ones. Another hierarchical arrangement for honeycombs is to replace the three-edge vertexes of the honeycombs by cellular microstructures [27-38]. The cellular microstructures can also be various, among which self-similar hierarchical honeycombs drew the most attention. Ajdari et al. [27], Haghpanah et al. [28], Oftadeh et al. [29, 30] and Mousanezhad et al. [31-33] constructed a sort of vertex-based hierarchical honeycombs by replacing each three-edge vertex of a regular hexagonal honeycomb with a smaller hexagon. And by iterating this process, hierarchical honeycombs of higher orders were obtained. Ajdari et al. [27] investigated the mechanical behaviors of a set of 2D vertex-based hierarchical honeycombs. It was found that the stiffness of the hierarchical honeycombs of first and second orders can be up to 2.0 and 3.5 times respectively compared with the regular honeycomb. Haghpanah et al. [28] found that with increasing of the hierarchical order, stiffness could be further improved, 5.3 and 7.1 times respectively for third and fourth orders. Oftadeh et al. [30] explored the amplification of the effective elastic modulus from one generation to the next of a class of fractal-like hierarchical honeycombs. The amplification coefficient was about 1.2 when the hierarchical order was sufficiently high. Mousanezhad et al. [31, 32] developed a new class of 2D hierarchical honeycombs with negative Poisson’s ratio. Auxeticity was led by contracting in transverse directions under uniaxial compressive loadings. Also, a few non-self-similar vertex-based hierarchical honeycombs were developed and explored by Mousanezhad et al. [33], Chen et al.[34] and Li et al. [35]. The aforementioned works focused mainly on addressing the in-plane mechanical properties of the hierarchical honeycombs. However, it is well known that honeycombs offer much more promising energy absorption capacity in the out-of-plane direction [13]. Therefore, studies on exploring the out-of-plane crashing behaviors of hierarchical 4
honeycombs were also proposed recently [36-38]. Chen et al. [36] studied the out-of-plane mechanical behaviors of a set of non-self-similar hierarchical honeycombs using numerical and experimental methods. Zhang et al. [37] and Sun et al. [38] investigated the out-of-plane crashworthiness of the self-similar hierarchical honeycombs of first and second orders. They found that hierarchy greatly improved the crashworthiness and energy absorption capacity of the honeycomb. Specifically, the maximum specific energy absorption (SEA) of the first order and second order were improved about 80% and 185%, respectively. They aimed to address the effect of hierarchical microstructures size and only the hierarchical honeycombs up to second order were explored. Studies on the out-of-plane crushing behaviors of hierarchical honeycombs with higher orders were not found so far to the authors’ best knowledge. Therefore, more works are significantly necessary to conduct for better understanding the effect of hierarchy on out-of-plane crashworthiness of hierarchical honeycombs, especially for higher orders. This paper aimed to investigate the out-of-plane crushing resistance of fractal-like hierarchical honeycombs with a wider range of orders using numerical and analytical methods. In Section 2, the self-similar vertex-based hierarchical honeycombs were developed. In Section 3, theoretical solutions for the mean crushing force (MCF) of the hierarchical honeycombs were derived based on the Super Folding Element (SFE) theory. Numerical modeling was presented in Section 4. In Section 5, the simulation results on three groups of hierarchical honeycombs were presented and compared with the theoretical results. In Section 6, discussions were presented to profoundly reveal the effect of hierarchical orders on amplification of the MCF of hierarchical honeycombs. 2.
Configurations of the hierarchical honeycombs Based on Oftadeh et al.’ work [30], fractal-like hierarchical honeycombs investigated in
this paper are constructed by replacing each three-edge vertex of a regular hexagonal 5
honeycomb by a smaller self-similar network. Hierarchical honeycombs with higher orders can be developed by iterating this process. Fig. 1 shows a most common hexagonal honeycomb with expanding angle of 120 as well as a Y-shaped unit cell extracted from the honeycomb. On behalf of the whole honeycomb, the Y-shaped unit cell is widely used in exploring the out-of-plane mechanical properties of the honeycomb for simplicity. Here, the regular honeycomb is defined as zeroth order. The unit cells of hierarchical honeycombs of zeroth to fourth orders are illustrated in Fig. 2. Fig. 3 shows a typical process of development of the i th order hierarchical honeycomb through hierarchy of the i 1 th order hierarchical honeycomb. All the fractal-like honeycombs explored in this paper are constructed by repeating this process. A set of hierarchical structural organization parameter i are defined to determine the configurations of hierarchical honeycombs, where i is defined by ratio of the newly introduced to the previous hexagonal edge lengths
i li / li 1
(1)
where li and li 1 are respectively the edge lengths of the newly introduced and the previous hexagonal edges. It is noted that i varies from 1 to n. l 0 is the edge length of the Y-shaped unit cell (zeroth order), which is half of the edge length of the regular honeycomb, as shown in Fig. 1. Accordingly, the edge length of the hierarchical honeycomb of i th order can be calculated from i li j l0 j 1
(2)
As shown in Fig. 3, in order to avoid overlapping of the newly introduced edges with the preexisting edges, some geometric constrains should be imposed and meet the following requirement. For the nth hierarchical order, 6
0 li li 1 n li l 0 i 1
(3)
Substituting Eqs. (1) and (2) into Eq. (3), the geometric constrains can be rewritten in terms of the structural organization parameter i
0 i 1 n i j 1 i 1 j 1
(4)
For simplicity, the wall thickness of a given hierarchical honeycomb is prescribed to be uniform. Relative density of the nth order hierarchical honeycomb, , compared to the material density, s , can be calculated from [30]
n 1 t l 3i 1 li n2 0 s 3 i 1 l0
(5)
where t n is the wall thickness of the nth order hierarchical honeycomb. Substituting Eqs. (1) and (2) into Eq. (5), one can have
n i t 1 1 3i 1 j n l 3 i 1 j 1 0
(6)
Adjust t n for maintaining the same relative density with different i , i.e., 0 , where 0 =
1 t0 is the relative density of the zeroth order hierarchical honeycomb compared 3 l0
to the materials density. Wall thickness of the nth order hierarchical honeycomb is n i tn t0 1 3i 1 j j 1 i 1
(7)
As stated above, previous works have been carried out by Zhang et al. [37] and Sun et al. [38] to explore the effect of i on mechanical properties of the hierarchical honeycombs up to second order. In this paper, we seek to understand the effect of hierarchy order on the 7
crushing resistance of hierarchical honeycombs with higher orders. Therefore, hierarchical honeycombs with same i but different orders are developed for simplicity. Oftadeh et al. [30] investigated how the optimal structural organization evolved as the relative density changed. They suggested i 1 / 2 on seeking the maximum amplification of the effective elastic modulus, and which is also applied in this paper. Substituting i 1 / 2 into Eq. (6), one can have n
1 3 tn 3 2 l0
(8)
Also, the wall thickness and wall length of the hierarchical honeycombs of nth order are respectively n
2 t n t0 3
(9)
n
1 l n l0 2
3.
(10)
Theoretical solutions The R-I shock theory developed by S.R. Reid et al. [39] has been widely used to predict
crashworthiness of honeycombs under dynamic loading. Based on their theory, the dynamic MCF of a honeycomb can be expressed as
Ppld Ppls
v2 S D
(11)
where Ppls and Ppld are the quasi-static and dynamic MCFs, respectively; v is the crushing velocity; D and S are respectively the densification strain and the cross-section area of the honeycomb perpendicular to the crushing direction. It is seen that only the quasi-static term, i.e. Ppls , is determined by the specific configuration of the honeycomb, while the inertia
8
v2 term, i.e. S , is mainly determined by the impact velocity. Hence, the main task is to D predict the quasi-static term. To theoretically predict the quasi-static MCF of the fractal-like hierarchical honeycombs, the Super Folding Element (SFE) method developed by Wierzbicki and Abramowicz [40-42] is utilized. The basic assumption of the theory is that the length of the local buckling wave 2H for each fold remains constant during the formation of each buckle or fold. Also, the wall thickness of the thin-walled structures is assumed to be the same. Fig. 4 shows the simplified folding mechanism of each fold. Chen and Wierzbicki [43] further simplified the kinematically admissible model consisting of trapezoidal, toroidal and cylindrical surfaces with moving hinge lines by basic folding element. They gave the expression of the MCF for a multi-cell section thin-walled structure P
2 0t NA 3
(12)
Where 0 denotes the flow stress of the material; t is the wall thickness; N is the number of contributing flanges (for example, N=3 and N=15 respectively for the unit cells of hierarchical honeycombs of zeroth order and first order, as shown in Fig. 2; A is the area of the cross-section of all cell walls. A basic folding element consists of some extensional and compressional elements and three stationary hinge lines, the mean crushing resistance can thus be obtained. The MCF can be derived by considering the energy equilibrium of the system. The total energy is conservative during the process of progressive folding for the honeycomb, which can be expressed as
Wext U dissipation
(13)
where Wext and U dissipation denote the external work done by compression and the energy dissipated by plastic deformation, respectively. Considering the external work done by 9
compression is dissipated by plastic deformation in bending and membrane, the equilibrium equation can be rewritten by integrating Eq. (13) for one wavelength
P 2 H U bending U membrane
(14)
where U bending and U membrane denote the bending energy and membrane energy dissipated in the generation of one fold, respectively. In reality, a folding element can never be compressed completely flattened. Therefore, a correction coefficient is introduced to consider the effective compression distance. The value of can be obtained by dividing the effective compression distance by the wavelength 2 H , as
/ 2H
(15)
Wierzbicki and Abramowicz [40, 41] found that the effective crushing distance is about 70-75%, i.e., 0.70 0.75 . Here, 0.70 [44] is adopted for simplicity. 3.1. The bending energy Fig. 5 shows the illustration of the basic folding mechanism of one contributing flange. Three horizontal stationary hinge lines are developed on the flange after deformation. The bending energy U bending can be calculated by summing up the energy dissipated at each stationary hinge line, as 3
U bending M 0 i Lc
(16)
i 1
1 where M 0 0 t 2 is the fully plastic bending moment of the flange per width; i and 4
Lc denote the rotation angles at the hinge lines and the total length of all flanges,
respectively. A widely used assumption is that the flanges are completely flattened, which means i at the three hinge lines are respectively / 2 , and / 2 , as shown in Fig. 5(c). Accordingly, the bending energy can be drawn as
10
U bending 2M 0 Lc
(17)
3.2. The membrane energy As shown in Fig. 6, three membrane elements are developed after deformation for each flange, one in extension and two in compression. The membrane energy dissipation for one contributing flange M m can be evaluated by integrating the extensional and compressional area, which is M m 2M 0 H 2 / t
(18)
It is assumed by Chen and Wierzbicki [43] that the membrane elements developed near the corner line were the same for different types of angle elements. Hence, the membrane energy dissipated can be easily calculated by summing up all the contributing flanges U membrane NM m
(19)
However, Zhang and Zhang [45] found that generation of the membrane elements was significantly sensitive to the type and central angle of the angle element. They suggested dividing the structure into a number of constitutive structural elements and deriving the membrane energy dissipation of each single constitutive structural element respectively. Then, the membrane energy dissipated over the collapse of a fold can be obtained by summing up each single constitutive structural element. Fig. 7 shows the division of constitutive elements in a typical hierarchical honeycomb, the third order hierarchical honeycomb specifically. Two types of constitutive structural elements are identified accordingly: corner element and Y-shaped element, as shown in Fig. 8. Actually, only the two types of angle elements are identified for all of the hierarchical honeycombs. 3.2.1. Corner element The membrane energy dissipation of the corner element has been widely analyzed by Chen and Wierzbicki [43], and Zhang and Zhang [45, 46]. As shown in Fig. 8(a), each corner 11
element consists of two contributing flanges. The membrane energy dissipation of the corner element can be obtained by summing up the two flanges, as
M mcorner 2M m 4M 0 H 2 / t
(20)
Zhang and Zhang [45] modified Eq. (20) based on experimental results by considering the effect of central angle and width of the element, as
M
corner m
4M 0 H 2 tan( / 2) ( ) 0.6 t 0.164( B / t ) (tan( / 2) 0.06 / tan( / 2))
(21)
where is the central angle of the corner element; B is the width of the element. 3.2.2. Y-shaped element Compared with the corner element, folding mechanism of the Y-shaped element is much more complicated. An inextensional deformation mode is observed for the Y-shaped element in numerical simulations, as shown in Fig. 9. The membrane energy of the Y-shaped
M mY shaped can be calculated by summing up the membrane energy of each panel and that dissipated by thickening of the panels as [47] M mY shaped ( ) M mA M mB M mC M mthickning 2M 0 H 2 (4 tan( / 4) 2 sin( / 2) 3 sin ) t
(22)
A B C thickning where M m , M m , M m and M m denote the membrane energy dissipated in the
folding of panel A, B, C and by thickening of the panels, respectively; is the central angle of the Y-shaped element. Considering the effect of dimensions of the element including the width and the thickness, Eq. (22) can be further modified [45] as
M
Y shaped m
2 M 0 H 2 5.41(4 tan( / 4) 2 sin( / 2) 3 sin ) ( ) t ( B / t ) 0.6
(23)
3.2.3. Other types of element As stated above, only corner element and Y-shaped element are identified for the hierarchical honeycombs investigated in this paper. Hence, only the membrane energy 12
dissipation for these two types of elements are derived and other types of elements are not specifically derived. However, more types of constitutive elements may be identified in hierarchical honeycombs with other configurations. For example, T-shaped element, crisscross element and 4-panel angle element, etc. Despite the difference of various constitutive elements, the membrane energy dissipation for a specific constitutive element,
M mEi , can be calculated by integrating the extensional and compressional area of the element during crushing
M mEi 0tds
(24)
s
Divide the structure into a number of constitutive elements and derive the membrane energy dissipation of each single element. Then, the whole membrane energy dissipated for any kind of honeycomb over the collapse of a fold can be obtained by summing up all the energy dissipation for each constitutive element, as U membrane N Ei M mEi
(25)
where N Ei is the number of the specific constative element with membrane energy Ei dissipation of M m .
3.3. Mean crushing force The equivalent elements of the constitutive elements in the hierarchical honeycombs explored in this paper are primarily corner element of 120o and Y-shaped element of
120o , as shown in Fig. 7. Substituting the specific angles into Eqs. (21) and (23), the membrane energies of these elements can be obtained, as
M
corner m
23.91M 0 H 2 1 (120 ) t ( B / t ) 0.6 o
M mY shaped (120o )
71.83M 0 H 2 1 t ( B / t ) 0.6 13
(26)
(27)
The whole membrane energy dissipation can be expressed as
U membrane N c M mcorner NY M mY shaped
(28)
where N c and N Y are respectively the number of corner elements and Y-shaped elements in a specific hierarchical honeycomb. Substituting Eqs. (17) and (28) into Eq. (14), one can have
P 2H 2M 0 Lc N c M mcorner NY M mY shaped
(29)
The wavelength H can be determined by the stationary condition of the MCF P 0 H
(30)
2Lc t ( B / t ) 0.6
(31)
Therefore
H
where 23.91N c 71.83NY , substituting Eq. (31) into Eq. (29), the MCF for a hierarchical honeycomb is obtained as
P
1
M0
2Lc t ( B / t ) 0.6
(32)
1 Eq. (32) could be rewritten by substituting M 0 0 t 2 as well as taking the effective 4
crushing distance coefficient 0.70 into account P 0.3571 0 t 1.5
2Lc ( B / t ) 0.6
(33)
For hierarchical honeycombs with other configurations, Eq. (29) could be revised into a general formulation by replacing Eq. (28) with Eq. (25). By doing this, Eq. (29) can be revised as
P 2H 2 M 0 Lc N Ei M mEi
14
(34)
Via substituting specific values of N Ei and M mEi into Eq. (34) as well as the solution of H using Eq. (30) subsequently, the MCFs of fractal-like honeycombs with other configurations could be addressed accordingly. 4.
Numerical simulations
4.1. Finite element modeling In order to predict the buckling behavior of the fractal-like hierarchical honeycombs with various sources of nonlinearity, out-of-plane crushing processes are simulated using the explicit finite element (FE) code ABAQUS/Explicit. ABAQUS/Explicit is a general purpose FE program with dynamic and nonlinear analysis capabilities and has been widely employed to manage complex problems and simulate highly nonlinear problems. It performs dynamic analysis using a Lagrangian formulation and integrating the equations of motion in time explicitly by means of central differences. Using a large number of small time increment makes the explicit solution advantageous because each increment is relatively quick to calculate and no convergence iterations are needed. As shown in Fig. 10, the FE model consists of three parts, two rigid walls and a honeycomb sandwiched between the rigid walls. The rigid walls are positioned just 1mm above or under the honeycomb for computing efficiency. A constant velocity of 0.5m/s towards the honeycomb is assigned to the top rigid wall at its reference node as motion of the top rigid wall is governed by its reference node. The bottom rigid wall is fully fixed. The hourglass controlled, four nodes, reduced integration shell element S4R with a large-strain formulation and five integration points across the wall thickness is used to mesh the honeycomb. The nonlinear contacts in the FE models are simulated using the built-in contact models in ABAQUS/Explicit. Specifically, the contacts between the rigid walls and the honeycomb are simulated using SURFACE-TO-SURFACE contact model. Besides, to avoid possible 15
self-interpenetration, the physical contacts between the surfaces of the honeycomb are simulated using GENERAL CONTACT ALGORITHM contact model. The contact algorithm enforces contact constrains using a penalty contact method in ABAQUS/Explicit. The friction coefficient is set as 0.2. Symmetric boundary conditions are imposed on the non-intersecting edges of the unit cell. Nonlinear constitutive model is employed to simulate the basic material of the cell walls, which is assumed to be elastic, perfectly plastic [16] with Young’s modulus of E 69 GPa , the yield stress of 0 76 MPa , Poisson’s ratio of 0.3 and density of
s 2700 kg/m 3 . 4.2. Validation of FE models A convergence test is firstly conducted to determine the optimum mesh size for the numerical models. Simulated MCF and computing time for the Y-shaped unit cell of a regular honeycomb (zeroth order) with l0 1.833mm and t0 0.15mm are plotted in Fig. 11. As shown in Fig. 11, the MCF gradually stabilizes and converges when the element size reduces to 0.4 mm while the computing time increases significantly with decreasing of the element size. For both the accuracy of results and computing efficiency consideration, 0.2 mm (9 elements on each cell wall) is conservatively chosen for this case. It is noted that the wall length of the hierarchical honeycomb decreases sharply with hierarchical order n increases, see Eq. (10). Therefore, the mesh size should be adjusted for different hierarchical orders. To validate the accuracy of the FE modeling method, simulations of zeroth order hierarchical honeycombs are conducted and compared with experimental results. The zeroth order hierarchical honeycomb, i.e., the regular honeycomb, is common in engineering applications, thus there are numerous experimental data available in literature. In this study, a
3 3 and a 5 5 unit cell of the zeroth order hierarchical honeycombs with l0 6mm and t0 0.75mm are simulated [48]. The modelling approach is the same as mentioned above, 16
and the force-strain curves and deformation patterns are plotted in Fig. 12. It is seen that both the numerical force-strain curves and the deformation patterns are in good agreement with the experimental results [48]. Moreover, the accuracy of the cell model is validated by comparing the numerical results with that of the whole model as well as theoretical solutions conducted by Gibson and Ashby [13]. Based on their work, crush strength of the regular honeycomb in out-of-plane direction is t l
5/ 3
pl 5.6 0
(35)
where l is the wall thickness of the honeycomb, which is double of the edge length of the Y-shaped unit cell. Fig. 13 shows the crushing stress-strain curves and deformation patterns of the Y-shaped unit cell model and a whole model ( 6 7 unit cell) with l0 1.833mm and t0 0.15mm . The crushing strengths obtained from simulation and Eq. (35) are listed in Table
1. The result obtained from the current theoretical solutions, i.e., Eq. (33) is also listed. It is seen that both the collapse mode and crushing response are in good agreement by using unit cell model and whole model. And also, both the simulation and the current theoretical results compare very well with the prior theoretical result (Eq. (35)). Hence, the numerical modeling approach used in this paper are reliable. 5.
Results Note with increasing of the hierarchical order n, micro-cells of the hierarchical
honeycomb increase greatly. It is a big challenge for the simulation computing, even use the cell model. Therefore, only the hierarchical honeycombs up to fourth order are simulated in this paper. Three sets of hierarchical honeycombs with different relative densities are developed. Each set of hierarchical honeycombs are developed based on the same Y-shaped 17
unit cell extracted from a regular honeycomb (zeroth order). The three unit cells of regular honeycombs (zeroth order) have the same edge length of l0 112mm , the same initial height of H 0 200mm and different wall thicknesses of t0 1.0mm , 1.5mm and 2.0mm , respectively. The relative density of the three sets of hierarchical honeycombs are respectively 0.0052 , 0.0077 and 0.0103 . The configurations of the three sets of hierarchical honeycombs are listed in Table 2. Numbers of the corner and Y-shaped elements for each hierarchical honeycomb are also given. 5.1. Deformation pattern Fig. 14 shows the deformation pattern of the set of hierarchical honeycombs with relative density of 0.0052 at the same crushing displacement of h =160mm. It is seen that all the honeycombs are crushed layer-by-layer. The length of the local buckling wave for each fold of each hierarchical honeycomb remains constant during the formation of buckling. Also, the folding pattern of each corner element or Y-shaped element is similar for all hierarchal honeycombs. A considerable difference is that with the increase of hierarchical order n, the wavelength is greatly shortened and the number of complete folds increases significantly. Similar observations are also drawn in the other two sets of hierarchical honeycombs. It is concluded that the deformations of the hierarchical honeycombs meet with the assumption of SFE theory perfectly. 5.2. Crushing response Fig. 15 shows the crushing responses of the set of hierarchical honeycombs with relative density of 0.0052 . As shown in Fig. 15(a), with hierarchical order increasing, the contact force increases significantly. Also, the effective crushing distance becomes shorter. The initial peak forces (PFs) and MCFs are shown in Fig. 15(b), it should be noted that the MCFs presented here are the average forces in the plateau region for the densification region 18
makes no significant sense to energy absorption. It is seen that the MCF is improved greatly with increasing of hierarchical order. Even better, the PF keeps almost a constant for the considered cases. The other two sets of hierarchical honeycombs perform a similar trend but with different magnitudes. More specifically, the MCFs are improved 71%, 114%, 201% and 309% while the PFs increase only 8.7%, 10%, 9.6% and 8.9% for the first to fourth order hierarchical honeycombs with relative density of 0.0052 , respectively. The results reveal that hierarchy can greatly improve the crashworthiness of honeycombs. Substituting the specific number of continuous elements into Eq. (33), the MCFs of the hierarchical honeycombs of different orders can be obtained. Take the second order hierarchical honeycomb for example, where N c 12 , N Y 9 , the MCF is derived as
P 174.9 0 l20.2 t21.8
(36)
The predicted MCFs for all of the hierarchical honeycombs are listed in Table 3. It is seen that the theoretical solutions compare very well with the numerical results for the lower orders (zeroth to second orders), but slightly smaller than the numerical results for the larger orders (third and fourth orders). Specifically, the maximum error is -11.9%, which is observed in the fourth order hierarchical honeycomb with relative density of 0.0052 . This is due to the effective crushing distance coefficient is taken as 0.70 for all of the orders. However, the value slightly decreases with hierarchy order and wall thickness. So, it is overestimated for the third and fourth order hierarchical honeycombs. In another word, the theoretical predictions are slightly conservative for higher orders. Modify the theoretical predictions by adopting the effective crushing distance obtained from simulations. Better agreement is reached between the numerical results and theoretical predictions, as shown in Table 3. However, the error is still very large for the first-order hierarchical honeycomb with relative density of 0.0103 after modification, which is +11.7%. This is because unstable global buckling takes place only for this case (as shown in Fig. 16), which makes it 19
considerably different from the other cases and not meeting with the basic assumptions of the current theoretical solutions. The MCF becomes much less for this case due to the unstable global buckling. In addition to this special case, the maximum error between the numerical results and theoretical predictions is -7.9% in the case of fourth order hierarchical honeycomb with relative density of 0.0052 . Considering the complexity of the problem and assumptions adopted, the results indicate that the theoretical solutions give a good prediction for the MCFs of hierarchical honeycombs. 6.
Discussions As stated in Section 5, the MCF increases with the hierarchical order significantly for
the considered cases. To seek the general rule of the amplification of MCF, an nth order hierarchical honeycomb is considered here. For the nth order hierarchical honeycomb, numbers of the corner elements N c and Y-shaped elements N Y can be calculated as Nc
3 n (3 1) 2
N Y 3n
(37) (38)
Substituting Eqs (37) and (38) into Eq. (33), MCF of the hierarchical honeycomb of nth order can be derived as P 0.3571 0 t 3 / 2
2Lc (107.70 3n 35.87) ( B / t ) 0.6
(39)
Eq. (39) can be further simplified for the nth ( n 1 ) hierarchy honeycomb by n 1 substituting Lc 3 ln , B 0.5 l n
P 6.6 3n1 (3n1 1) 0 ln0.2 tn1.8
(40)
Note Eq. (40) is suitable only when n 1 due to B l0 when n 0 . Therefore, Eq. (40) should be modified for n 0 , as 20
Pn0 13.2 0 l00.2 t01.8
(41)
Substituting Eqs. (9) and (10), Eq. (40) can be rewritten in terms of wall length and wall thickness of the zeroth order hierarchical honeycomb (regular honeycomb), as P 6.6 3n1 (3n1 1) 0.42 n 0 l00.2 t01.8
(42)
P 19.8 1.26n 0 l00.2 t01.8
(43)
With n increasing
Moreover, define the amplification of MCF from one generation to the next as
i Pi / Pi 1
(44)
It is seen from Eq. (43) that when n is sufficiently large, the amplification of the MCF from one generation to the next approaches a constant of n 1.26 . The amplification of MCF of the nth order ( n 1 ) hierarchical honeycomb compared with the regular honeycomb (zeroth order) can also be obtained as n
An i Pn / P0 0.5 3n 1 (3n 1 1) 0.42 n
(45)
i 0
With n increasing
An 1.5 1.26n
(46)
Fig. 17 shows the amplification of the MCF of hierarchical honeycombs up to tenth order. It is seen that the MCF increases exponentially with increasing of hierarchical order. This feature is validated by numerical simulations conducted in the previous section, as shown in Fig. 17. The results indicated that the prediction of the amplification of MCF for hierarchical honeycombs is reliable. However, the MCF of the hierarchical honeycomb cannot increases unlimitedly. Combine Eqs. (9) and (10), the ratio of wall thickness to wall length for the nth hierarchical honeycomb is obtained as 21
n
n
tn 4 t0 ln 3 l0
(47)
Rewritten Eq. (47) in terms of the relative density of the honeycomb, one can have
4 n 3
n
3
(48)
It is seen from Eq. (48) that n also increases exponentially with hierarchical order. More specifically, the increase of n is greater than that of An . When n is too large, the assumptions of the current theoretical solutions would be invalid. Therefore, in order to get a hierarchical honeycomb with a high order, the relative density of the honeycomb should be sufficiently small. Fig. 18 shows the design spaces for n 1 / 15 and n 1 / 20 , specifically. To meet the limiting condition requirement, the relative density and hierarchical order can only be selected from the shaded area. Considering the constrain of n 1 / 15 , maximum achievable amplification of MCF of hierarchical honeycombs for different relative densities are shown in Fig. 19. 7.
Conclusions Numerical and analytical investigation on determining the crashworthiness of
fractal-like honeycombs with vertex-based self-similar hierarchy were proposed. By analyzing three groups of hierarchical honeycombs with various relative densities, the effect of the hierarchy on the mean crushing force (MCF) of the hierarchical honeycombs was discussed and quantified. The main conclusions are summarized as follows: 1)
Theoretical models to predict the MCFs of the hierarchical honeycombs were established based on the Super Folding Element (SFE) theory; Finite element (FE) models were also established and validated by prior theory and experimental results.
2)
Numerical simulations on hierarchical honeycombs up to fourth order of three different 22
relative densities were conducted. Results obtained using the current theoretical solutions were in good agreement with the numerical results, especially after modifying the effective crush distance. 3)
It is proved that hierarchy could significantly improve the crashworthiness of the honeycombs. According to the simulations, the MCF was improved 71%, 114%, 201% and 309% respectively for the first to fourth order while the initial peak force (PF) keeps almost constant.
4)
The amplification of the MCF from one generation to the next was theoretically predicted, which is a constant ( n 1.26 ) when the hierarchical order is sufficiently larger. Moreover, the amplification of the MCF of a given hierarchical honeycomb was also given and which is proved to be determined only by the hierarchical order.
5)
It was validated that the relative density should be sufficiently small for pursuing a high order hierarchical honeycomb. The potential maximum achievable amplifications of the MCFs for hierarchical honeycombs of some specific relative densities were also discussed.
Acknowledgements The authors would like to acknowledge the research grants from the National Natural Science Foundation of China (11572086, 11602112).
23
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26
Figure captions Fig. 1. Schematic diagram of the unit cell of a regular honeycomb (i.e., zeroth order). Fig. 2. Unit cells of the hierarchical honeycombs of zeroth to fourth orders. Fig. 3. Structural hierarchy of unit cell of the i 1 th order hierarchical honeycomb. Fig. 4. Simplified folding mechanism of each fold [40, 41]. Fig. 5. The global geometry of the basic folding mechanism: (a) Before deformation; (b) After deformation; (c) Rotation angles. Fig. 6. Illustration of the membrane elements. Fig. 7. Division of constituent elements in the third order hierarchical honeycomb. Fig. 8. Scheme of the constituent elements: (a) Corner element; (b) Y-shaped element. Fig. 9. Illustration of the generation of membrane elements for the Y-shaped element: (a) Observation from simulation; (b) Sketch map. Fig. 10. Schematic diagram of the FE model for the out-of-plane crushing. Fig. 11. Effect of element size on simulation results and computing times. Fig. 12. Validation of the modeling method: (a) Force-strain curves; (b) Deformation patterns. Fig. 13. Comparison of simulation results between the cell model and whole model: (a) Stress-strain curves; (b) Deformation patterns. Fig. 14. Isometric view of the deformation patterns of the hierarchical honeycombs ( 0.0052 ) of zeroth to fourth orders at the crushing displacement of h =160mm. Fig. 15. Crushing response of the set of hierarchical honeycombs with a relative density of
0.0052 : (a) Force-displacement curves; (b) PFs and MCFs. Fig. 16. Deformation patterns of the first-order hierarchical honeycomb with the relative density of 0.0103 at the crushing displacement of h =70mm. Fig. 17. Amplification of the MCF of hierarchical honeycomb for different orders. 27
Fig. 18. Design spaces of the hierarchical honeycombs when n 1 / 15 and n 1 / 20 . Fig. 19. Maximum achievable amplification of the MCF of hierarchical honeycombs for different relative density in the constraining of n 1 / 15 .
Table captions Table 1. Comparisons of crushing strength between simulation and theoretical results. Table 2. Configurations of the hierarchical honeycombs of zeroth to fourth orders for three different relative densities. Table 3. Comparison of the numerical simulations and theoretical predictions on MCFs.
28
Y
X
Fig. 1. Schematic diagram of the unit cell of a regular honeycomb (i.e., zeroth order).
29
0th order
1st order
2nd order
3rd order
4th order
Fig. 2. Unit cells of the hierarchical honeycombs of zeroth to fourth orders.
30
li li-1
ti i th order Hierarchy
ti-1
Fig. 3. Structural hierarchy of unit cell of the i 1 th order hierarchical honeycomb.
31
Horizontal hinge line
Plastic flow area
Fig. 4. Simplified folding mechanism of each fold [40, 41].
32
B Ψ
π/2
π
Ψ
π/2 2H
Ψ Ψ (a)
(b)
(c)
Fig. 5. The global geometry of the basic folding mechanism: (a) Before deformation; (b) After deformation; (c) Rotation angles.
33
B 45°
2H
Corner line
Fig. 6. Illustration of the membrane elements.
34
①
②
Fig. 7. Division of constituent elements in the third order hierarchical honeycomb.
35
B B
β
α
(a)
(b)
Fig. 8. Scheme of the constituent elements: (a) Corner element; (b) Y-shaped element.
36
H Panel B
90°-β/2
β
Panel A
Panel C
(a)
(b)
Fig. 9. Illustration of the generation of membrane elements for the Y-shaped element: (a) Observation from simulation; (b) Sketch map.
37
x
V
y
Top rigid wall
Unit cell
y
x Fully fixed bottom rigid wall
x y
Symmetric boundary condition in local x direction
Fig. 10. Schematic diagram of the FE model for the out-of-plane crushing.
38
Fig. 11. Effect of element size on simulation results and computing times.
39
(a)
(b)
Fig. 12. Validation of the modeling method: (a) Force-strain curves; (b) Deformation patterns.
40
(a)
(b)
Fig. 13. Comparison of simulation results between the cell model and whole model: (a) Stress-strain curves; (b) Deformation patterns.
41
0th order
1st order
2nd order
3rd order
4th order
Fig. 14. Isometric view of the deformation patterns of the hierarchical honeycombs ( 0.0052 ) of zeroth to fourth orders at the crushing displacement of h =160mm.
42
(a)
(b)
Fig. 15. Crushing response of the set of hierarchical honeycombs with a relative density of
0.0052 : (a) Force-displacement curves; (b) PFs and MCFs.
43
Fig. 16. Deformation patterns of the first-order hierarchical honeycomb with the relative density of 0.0103 at the crushing displacement of h =70mm.
44
Fig. 17. Amplification of the MCF of hierarchical honeycomb for different orders.
45
n 1 / 15
n 1 / 20
0.02 0.015
0.01 0.005
Fig. 18. Design spaces of the hierarchical honeycombs when n 1 / 15 and n 1 / 20 .
46
0.0029 0.0039 0.0051 0.0069 0.0091 0.0162 0.0289
0.0122 0.0217
Fig. 19. Maximum achievable amplification of the MCF of hierarchical honeycombs for different relative density in the constraining of n 1 / 15 .
47
Table 1. Comparisons of crushing strength between simulation and theoretical results. Simulation results
Cruhsing strength (MPa)
Whole-model
Cell-model
2.09
1.99
48
Theoretical results Wierzbicki et al. Current solution Eq. (35) Eq. (33) 2.07
2.12
Table 2. Configurations of the hierarchical honeycombs of zeroth to fourth orders for three different relative densities. Hierarchy Relative order density 0th
1st
2nd
3rd
4
th
Wall thickness Wall length Ratio of wall ti (mm) li (mm) thickness to length i
0.0052
1.0
0.0077
1.5
0.0103
2.0
0.01786
0.0052
0.6667
0.01190
0.0077
1.0
0.0103
1.3333
0.02391
0.0052
0.4444
0.01587
0.0077
0.6667
0.0103
0.8889
0.03175
0.0052
0.2963
0.02116
0.0077
0.4444
0.0103
0.5926
0.04233
0.0052
0.1975
0.02821
0.0077
0.2963
0.0103
0.3951
Nc
NY
0
1
3
3
12
9
39
27
120
81
0.00893 112
0.01339
56
0.01786
28
0.02391
14
0.03175
7
0.04233 0.05644
49
Table 3. Comparison of the numerical simulations and theoretical predictions on MCFs. Hierarchy Relative order density
th
0
st
1
2nd
3rd
4th
FEM Theoretical predication Modified theoretical prediction P (kN)
P (kN)
Error
P (kN)
Error
0.0052
2.67
2.58
-3.4%
0.72
2.51
-5.9%
0.0077
5.78
5.40
-7.3%
0.71
5.33
-7.7%
0.0103
9.03
8.94
-1.0%
0.71
8.81
-2.4%
0.0052
4.57
4.59
+0.4%
0.71
4.53
-1.0%
0.0077
9.69
9.53
-1.7%
0.70
9.53
-1.7%
0.0103
14.23
15.90
+11.7%
0.70
15.90
+11.7%
0.0052
5.71
6.01
+5.3%
0.71
5.92
+3.7%
0.0077
13.46
12.48
-7.3%
0.69
12.67
-5.9%
0.0103
21.73
20.91
-3.8%
0.68
21.53
-0.9%
0.0052
8.04
7.67
-4.6%
0.68
7.90
-1.7%
0.0077
17.78
15.91
-10.5%
0.66
16.87
-5.1%
0.0103
29.15
26.71
-8.4%
0.66
28.32
-2.7%
0.0052
11.05
9.73
-11.9%
0.67
10.17
-7.9%
0.0077
22.57
20.18
-10.6%
0.66
21.40
-5.2%
0.0103
35.91
33.87
-5.4%
0.65
36.48
+1.6%
50
S0263-8223(17)31652-5 https://doi.org/10.1016/j.compstruct.2018.02.082 COST 9432
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
25 May 2017 29 December 2017 26 February 2018
Please cite this article as: Zhang, D., Fei, Q., Jiang, D., Li, Y., Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy, Composite Structures (2018), doi: https://doi.org/10.1016/ j.compstruct.2018.02.082
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Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy
Dahai Zhang 1, 2, Qingguo Fei 1, 2,*, Dong Jiang 1, 3, Yanbin Li 1, 2 1
Institute of Aerospace Machinery and Dynamics, Southeast University, Nanjing 211189, China.
2
School of Mechanical Engineering, Southeast University, Nanjing 211189, China.
3
College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China.
* Corresponding author. Email address: [email protected] 1
Abstract The out-of-plane crushing resistance of fractal-like honeycombs with self-similar hierarchy are investigated numerically and analytically. The hierarchical honeycombs are developed via replacing each three-edge vertex of a regular hexagonal honeycomb by a smaller hexagon. Hierarchical honeycombs with higher orders are constructed by repeating this process. Theoretical solutions are derived based on the Super Folding Element (SFE) theory to predict the mean crushing forces (MCFs) of the hierarchical honeycombs. Numerical simulations are also conducted in conjunction with the theoretical solutions. Crushing resistances of three groups of hierarchical honeycombs with different relative densities are explored subsequently. Good agreement between results obtained using theoretical and numerical methods indicates that the theoretical predictions are reliable. The results show that hierarchy can significantly improve the crushing resistance of honeycombs. It is found that the MCFs of first to fourth order hierarchical honeycombs are improved by 71%, 114%, 201% and 309% compared with the regular honeycomb under the same relative density, respectively. It reveals that the amplification of the MCF from one generation to the next approaches a constant ( n 1.26 ) when the hierarchical order is sufficiently large. The potential maximum achievable amplifications of the MCF for hierarchical honeycombs of different relative densities are also discussed. Keywords: Honeycombs; Self-similar hierarchy; Out-of-plane; Crushing resistance.
2
1.
Introduction Biological materials always perform exceptionally in mechanical providing protection
and support for plant and animal bodies [1]. This is believed due to that nature has developed optimized approaches of achieving high structural efficiency. Among nature’s strategies in pursuing better mechanical efficiency, functional adaptation of the structures at all levels of hierarchy is the most common one. There are many natural materials displaying this hierarchical design, such as bones [2], teeth [3], woods [4] and tendons [5]. The fundamental concept of hierarchical design is to create complex cellular microstructures by placing material where most needed [2]. The mechanical efficiency of the resulting materials can be significantly improved. In recent years, the hierarchy has been widely introduced into design of artificial materials and structures, for example, lattice trusses [6, 7], composites [8, 9] and thin-walled tubes [10, 11]. As one type of typical biological materials, honeycombs have been widely used in many kinds of industries as protection or energy absorption components [12]. Numerous works have been proposed on exploring both in-plane and out-of-plane crushing performance of honeycombs by means of experimental, theoretical and numerical methods [13-17]. To understand the potential advantages of hierarchy in pursuing better crashworthiness, the hierarchy was also introduced into design of honeycombs by many researchers recently [18-38]. Accordingly, hierarchical honeycombs with various configurations have been constructed. One hierarchical development approach is to replace the solid walls of the honeycomb by cellular microstructures. The cellular microstructures can be various, such as sandwich structures [18, 19], hexagonal micro-cells [20, 21], triangular micro-cells [23-25] and negative Poisson’s ratio sub-structures [26]. Exhaustive studies were conducted subsequently to explore the effect of hierarchy on mechanical properties of honeycombs by Fan et al. [18], Taylor et al. [19, 21], Chen et al. [20], Vigliotti et al.[22], Sun et al. [23, 26], 3
Qiao et al. [24] and Chen et al.[25]. They proved that hierarchical honeycombs had super mechanical properties and damage tolerance compared with the equal mass solid-wall ones. Another hierarchical arrangement for honeycombs is to replace the three-edge vertexes of the honeycombs by cellular microstructures [27-38]. The cellular microstructures can also be various, among which self-similar hierarchical honeycombs drew the most attention. Ajdari et al. [27], Haghpanah et al. [28], Oftadeh et al. [29, 30] and Mousanezhad et al. [31-33] constructed a sort of vertex-based hierarchical honeycombs by replacing each three-edge vertex of a regular hexagonal honeycomb with a smaller hexagon. And by iterating this process, hierarchical honeycombs of higher orders were obtained. Ajdari et al. [27] investigated the mechanical behaviors of a set of 2D vertex-based hierarchical honeycombs. It was found that the stiffness of the hierarchical honeycombs of first and second orders can be up to 2.0 and 3.5 times respectively compared with the regular honeycomb. Haghpanah et al. [28] found that with increasing of the hierarchical order, stiffness could be further improved, 5.3 and 7.1 times respectively for third and fourth orders. Oftadeh et al. [30] explored the amplification of the effective elastic modulus from one generation to the next of a class of fractal-like hierarchical honeycombs. The amplification coefficient was about 1.2 when the hierarchical order was sufficiently high. Mousanezhad et al. [31, 32] developed a new class of 2D hierarchical honeycombs with negative Poisson’s ratio. Auxeticity was led by contracting in transverse directions under uniaxial compressive loadings. Also, a few non-self-similar vertex-based hierarchical honeycombs were developed and explored by Mousanezhad et al. [33], Chen et al.[34] and Li et al. [35]. The aforementioned works focused mainly on addressing the in-plane mechanical properties of the hierarchical honeycombs. However, it is well known that honeycombs offer much more promising energy absorption capacity in the out-of-plane direction [13]. Therefore, studies on exploring the out-of-plane crashing behaviors of hierarchical 4
honeycombs were also proposed recently [36-38]. Chen et al. [36] studied the out-of-plane mechanical behaviors of a set of non-self-similar hierarchical honeycombs using numerical and experimental methods. Zhang et al. [37] and Sun et al. [38] investigated the out-of-plane crashworthiness of the self-similar hierarchical honeycombs of first and second orders. They found that hierarchy greatly improved the crashworthiness and energy absorption capacity of the honeycomb. Specifically, the maximum specific energy absorption (SEA) of the first order and second order were improved about 80% and 185%, respectively. They aimed to address the effect of hierarchical microstructures size and only the hierarchical honeycombs up to second order were explored. Studies on the out-of-plane crushing behaviors of hierarchical honeycombs with higher orders were not found so far to the authors’ best knowledge. Therefore, more works are significantly necessary to conduct for better understanding the effect of hierarchy on out-of-plane crashworthiness of hierarchical honeycombs, especially for higher orders. This paper aimed to investigate the out-of-plane crushing resistance of fractal-like hierarchical honeycombs with a wider range of orders using numerical and analytical methods. In Section 2, the self-similar vertex-based hierarchical honeycombs were developed. In Section 3, theoretical solutions for the mean crushing force (MCF) of the hierarchical honeycombs were derived based on the Super Folding Element (SFE) theory. Numerical modeling was presented in Section 4. In Section 5, the simulation results on three groups of hierarchical honeycombs were presented and compared with the theoretical results. In Section 6, discussions were presented to profoundly reveal the effect of hierarchical orders on amplification of the MCF of hierarchical honeycombs. 2.
Configurations of the hierarchical honeycombs Based on Oftadeh et al.’ work [30], fractal-like hierarchical honeycombs investigated in
this paper are constructed by replacing each three-edge vertex of a regular hexagonal 5
honeycomb by a smaller self-similar network. Hierarchical honeycombs with higher orders can be developed by iterating this process. Fig. 1 shows a most common hexagonal honeycomb with expanding angle of 120 as well as a Y-shaped unit cell extracted from the honeycomb. On behalf of the whole honeycomb, the Y-shaped unit cell is widely used in exploring the out-of-plane mechanical properties of the honeycomb for simplicity. Here, the regular honeycomb is defined as zeroth order. The unit cells of hierarchical honeycombs of zeroth to fourth orders are illustrated in Fig. 2. Fig. 3 shows a typical process of development of the i th order hierarchical honeycomb through hierarchy of the i 1 th order hierarchical honeycomb. All the fractal-like honeycombs explored in this paper are constructed by repeating this process. A set of hierarchical structural organization parameter i are defined to determine the configurations of hierarchical honeycombs, where i is defined by ratio of the newly introduced to the previous hexagonal edge lengths
i li / li 1
(1)
where li and li 1 are respectively the edge lengths of the newly introduced and the previous hexagonal edges. It is noted that i varies from 1 to n. l 0 is the edge length of the Y-shaped unit cell (zeroth order), which is half of the edge length of the regular honeycomb, as shown in Fig. 1. Accordingly, the edge length of the hierarchical honeycomb of i th order can be calculated from i li j l0 j 1
(2)
As shown in Fig. 3, in order to avoid overlapping of the newly introduced edges with the preexisting edges, some geometric constrains should be imposed and meet the following requirement. For the nth hierarchical order, 6
0 li li 1 n li l 0 i 1
(3)
Substituting Eqs. (1) and (2) into Eq. (3), the geometric constrains can be rewritten in terms of the structural organization parameter i
0 i 1 n i j 1 i 1 j 1
(4)
For simplicity, the wall thickness of a given hierarchical honeycomb is prescribed to be uniform. Relative density of the nth order hierarchical honeycomb, , compared to the material density, s , can be calculated from [30]
n 1 t l 3i 1 li n2 0 s 3 i 1 l0
(5)
where t n is the wall thickness of the nth order hierarchical honeycomb. Substituting Eqs. (1) and (2) into Eq. (5), one can have
n i t 1 1 3i 1 j n l 3 i 1 j 1 0
(6)
Adjust t n for maintaining the same relative density with different i , i.e., 0 , where 0 =
1 t0 is the relative density of the zeroth order hierarchical honeycomb compared 3 l0
to the materials density. Wall thickness of the nth order hierarchical honeycomb is n i tn t0 1 3i 1 j j 1 i 1
(7)
As stated above, previous works have been carried out by Zhang et al. [37] and Sun et al. [38] to explore the effect of i on mechanical properties of the hierarchical honeycombs up to second order. In this paper, we seek to understand the effect of hierarchy order on the 7
crushing resistance of hierarchical honeycombs with higher orders. Therefore, hierarchical honeycombs with same i but different orders are developed for simplicity. Oftadeh et al. [30] investigated how the optimal structural organization evolved as the relative density changed. They suggested i 1 / 2 on seeking the maximum amplification of the effective elastic modulus, and which is also applied in this paper. Substituting i 1 / 2 into Eq. (6), one can have n
1 3 tn 3 2 l0
(8)
Also, the wall thickness and wall length of the hierarchical honeycombs of nth order are respectively n
2 t n t0 3
(9)
n
1 l n l0 2
3.
(10)
Theoretical solutions The R-I shock theory developed by S.R. Reid et al. [39] has been widely used to predict
crashworthiness of honeycombs under dynamic loading. Based on their theory, the dynamic MCF of a honeycomb can be expressed as
Ppld Ppls
v2 S D
(11)
where Ppls and Ppld are the quasi-static and dynamic MCFs, respectively; v is the crushing velocity; D and S are respectively the densification strain and the cross-section area of the honeycomb perpendicular to the crushing direction. It is seen that only the quasi-static term, i.e. Ppls , is determined by the specific configuration of the honeycomb, while the inertia
8
v2 term, i.e. S , is mainly determined by the impact velocity. Hence, the main task is to D predict the quasi-static term. To theoretically predict the quasi-static MCF of the fractal-like hierarchical honeycombs, the Super Folding Element (SFE) method developed by Wierzbicki and Abramowicz [40-42] is utilized. The basic assumption of the theory is that the length of the local buckling wave 2H for each fold remains constant during the formation of each buckle or fold. Also, the wall thickness of the thin-walled structures is assumed to be the same. Fig. 4 shows the simplified folding mechanism of each fold. Chen and Wierzbicki [43] further simplified the kinematically admissible model consisting of trapezoidal, toroidal and cylindrical surfaces with moving hinge lines by basic folding element. They gave the expression of the MCF for a multi-cell section thin-walled structure P
2 0t NA 3
(12)
Where 0 denotes the flow stress of the material; t is the wall thickness; N is the number of contributing flanges (for example, N=3 and N=15 respectively for the unit cells of hierarchical honeycombs of zeroth order and first order, as shown in Fig. 2; A is the area of the cross-section of all cell walls. A basic folding element consists of some extensional and compressional elements and three stationary hinge lines, the mean crushing resistance can thus be obtained. The MCF can be derived by considering the energy equilibrium of the system. The total energy is conservative during the process of progressive folding for the honeycomb, which can be expressed as
Wext U dissipation
(13)
where Wext and U dissipation denote the external work done by compression and the energy dissipated by plastic deformation, respectively. Considering the external work done by 9
compression is dissipated by plastic deformation in bending and membrane, the equilibrium equation can be rewritten by integrating Eq. (13) for one wavelength
P 2 H U bending U membrane
(14)
where U bending and U membrane denote the bending energy and membrane energy dissipated in the generation of one fold, respectively. In reality, a folding element can never be compressed completely flattened. Therefore, a correction coefficient is introduced to consider the effective compression distance. The value of can be obtained by dividing the effective compression distance by the wavelength 2 H , as
/ 2H
(15)
Wierzbicki and Abramowicz [40, 41] found that the effective crushing distance is about 70-75%, i.e., 0.70 0.75 . Here, 0.70 [44] is adopted for simplicity. 3.1. The bending energy Fig. 5 shows the illustration of the basic folding mechanism of one contributing flange. Three horizontal stationary hinge lines are developed on the flange after deformation. The bending energy U bending can be calculated by summing up the energy dissipated at each stationary hinge line, as 3
U bending M 0 i Lc
(16)
i 1
1 where M 0 0 t 2 is the fully plastic bending moment of the flange per width; i and 4
Lc denote the rotation angles at the hinge lines and the total length of all flanges,
respectively. A widely used assumption is that the flanges are completely flattened, which means i at the three hinge lines are respectively / 2 , and / 2 , as shown in Fig. 5(c). Accordingly, the bending energy can be drawn as
10
U bending 2M 0 Lc
(17)
3.2. The membrane energy As shown in Fig. 6, three membrane elements are developed after deformation for each flange, one in extension and two in compression. The membrane energy dissipation for one contributing flange M m can be evaluated by integrating the extensional and compressional area, which is M m 2M 0 H 2 / t
(18)
It is assumed by Chen and Wierzbicki [43] that the membrane elements developed near the corner line were the same for different types of angle elements. Hence, the membrane energy dissipated can be easily calculated by summing up all the contributing flanges U membrane NM m
(19)
However, Zhang and Zhang [45] found that generation of the membrane elements was significantly sensitive to the type and central angle of the angle element. They suggested dividing the structure into a number of constitutive structural elements and deriving the membrane energy dissipation of each single constitutive structural element respectively. Then, the membrane energy dissipated over the collapse of a fold can be obtained by summing up each single constitutive structural element. Fig. 7 shows the division of constitutive elements in a typical hierarchical honeycomb, the third order hierarchical honeycomb specifically. Two types of constitutive structural elements are identified accordingly: corner element and Y-shaped element, as shown in Fig. 8. Actually, only the two types of angle elements are identified for all of the hierarchical honeycombs. 3.2.1. Corner element The membrane energy dissipation of the corner element has been widely analyzed by Chen and Wierzbicki [43], and Zhang and Zhang [45, 46]. As shown in Fig. 8(a), each corner 11
element consists of two contributing flanges. The membrane energy dissipation of the corner element can be obtained by summing up the two flanges, as
M mcorner 2M m 4M 0 H 2 / t
(20)
Zhang and Zhang [45] modified Eq. (20) based on experimental results by considering the effect of central angle and width of the element, as
M
corner m
4M 0 H 2 tan( / 2) ( ) 0.6 t 0.164( B / t ) (tan( / 2) 0.06 / tan( / 2))
(21)
where is the central angle of the corner element; B is the width of the element. 3.2.2. Y-shaped element Compared with the corner element, folding mechanism of the Y-shaped element is much more complicated. An inextensional deformation mode is observed for the Y-shaped element in numerical simulations, as shown in Fig. 9. The membrane energy of the Y-shaped
M mY shaped can be calculated by summing up the membrane energy of each panel and that dissipated by thickening of the panels as [47] M mY shaped ( ) M mA M mB M mC M mthickning 2M 0 H 2 (4 tan( / 4) 2 sin( / 2) 3 sin ) t
(22)
A B C thickning where M m , M m , M m and M m denote the membrane energy dissipated in the
folding of panel A, B, C and by thickening of the panels, respectively; is the central angle of the Y-shaped element. Considering the effect of dimensions of the element including the width and the thickness, Eq. (22) can be further modified [45] as
M
Y shaped m
2 M 0 H 2 5.41(4 tan( / 4) 2 sin( / 2) 3 sin ) ( ) t ( B / t ) 0.6
(23)
3.2.3. Other types of element As stated above, only corner element and Y-shaped element are identified for the hierarchical honeycombs investigated in this paper. Hence, only the membrane energy 12
dissipation for these two types of elements are derived and other types of elements are not specifically derived. However, more types of constitutive elements may be identified in hierarchical honeycombs with other configurations. For example, T-shaped element, crisscross element and 4-panel angle element, etc. Despite the difference of various constitutive elements, the membrane energy dissipation for a specific constitutive element,
M mEi , can be calculated by integrating the extensional and compressional area of the element during crushing
M mEi 0tds
(24)
s
Divide the structure into a number of constitutive elements and derive the membrane energy dissipation of each single element. Then, the whole membrane energy dissipated for any kind of honeycomb over the collapse of a fold can be obtained by summing up all the energy dissipation for each constitutive element, as U membrane N Ei M mEi
(25)
where N Ei is the number of the specific constative element with membrane energy Ei dissipation of M m .
3.3. Mean crushing force The equivalent elements of the constitutive elements in the hierarchical honeycombs explored in this paper are primarily corner element of 120o and Y-shaped element of
120o , as shown in Fig. 7. Substituting the specific angles into Eqs. (21) and (23), the membrane energies of these elements can be obtained, as
M
corner m
23.91M 0 H 2 1 (120 ) t ( B / t ) 0.6 o
M mY shaped (120o )
71.83M 0 H 2 1 t ( B / t ) 0.6 13
(26)
(27)
The whole membrane energy dissipation can be expressed as
U membrane N c M mcorner NY M mY shaped
(28)
where N c and N Y are respectively the number of corner elements and Y-shaped elements in a specific hierarchical honeycomb. Substituting Eqs. (17) and (28) into Eq. (14), one can have
P 2H 2M 0 Lc N c M mcorner NY M mY shaped
(29)
The wavelength H can be determined by the stationary condition of the MCF P 0 H
(30)
2Lc t ( B / t ) 0.6
(31)
Therefore
H
where 23.91N c 71.83NY , substituting Eq. (31) into Eq. (29), the MCF for a hierarchical honeycomb is obtained as
P
1
M0
2Lc t ( B / t ) 0.6
(32)
1 Eq. (32) could be rewritten by substituting M 0 0 t 2 as well as taking the effective 4
crushing distance coefficient 0.70 into account P 0.3571 0 t 1.5
2Lc ( B / t ) 0.6
(33)
For hierarchical honeycombs with other configurations, Eq. (29) could be revised into a general formulation by replacing Eq. (28) with Eq. (25). By doing this, Eq. (29) can be revised as
P 2H 2 M 0 Lc N Ei M mEi
14
(34)
Via substituting specific values of N Ei and M mEi into Eq. (34) as well as the solution of H using Eq. (30) subsequently, the MCFs of fractal-like honeycombs with other configurations could be addressed accordingly. 4.
Numerical simulations
4.1. Finite element modeling In order to predict the buckling behavior of the fractal-like hierarchical honeycombs with various sources of nonlinearity, out-of-plane crushing processes are simulated using the explicit finite element (FE) code ABAQUS/Explicit. ABAQUS/Explicit is a general purpose FE program with dynamic and nonlinear analysis capabilities and has been widely employed to manage complex problems and simulate highly nonlinear problems. It performs dynamic analysis using a Lagrangian formulation and integrating the equations of motion in time explicitly by means of central differences. Using a large number of small time increment makes the explicit solution advantageous because each increment is relatively quick to calculate and no convergence iterations are needed. As shown in Fig. 10, the FE model consists of three parts, two rigid walls and a honeycomb sandwiched between the rigid walls. The rigid walls are positioned just 1mm above or under the honeycomb for computing efficiency. A constant velocity of 0.5m/s towards the honeycomb is assigned to the top rigid wall at its reference node as motion of the top rigid wall is governed by its reference node. The bottom rigid wall is fully fixed. The hourglass controlled, four nodes, reduced integration shell element S4R with a large-strain formulation and five integration points across the wall thickness is used to mesh the honeycomb. The nonlinear contacts in the FE models are simulated using the built-in contact models in ABAQUS/Explicit. Specifically, the contacts between the rigid walls and the honeycomb are simulated using SURFACE-TO-SURFACE contact model. Besides, to avoid possible 15
self-interpenetration, the physical contacts between the surfaces of the honeycomb are simulated using GENERAL CONTACT ALGORITHM contact model. The contact algorithm enforces contact constrains using a penalty contact method in ABAQUS/Explicit. The friction coefficient is set as 0.2. Symmetric boundary conditions are imposed on the non-intersecting edges of the unit cell. Nonlinear constitutive model is employed to simulate the basic material of the cell walls, which is assumed to be elastic, perfectly plastic [16] with Young’s modulus of E 69 GPa , the yield stress of 0 76 MPa , Poisson’s ratio of 0.3 and density of
s 2700 kg/m 3 . 4.2. Validation of FE models A convergence test is firstly conducted to determine the optimum mesh size for the numerical models. Simulated MCF and computing time for the Y-shaped unit cell of a regular honeycomb (zeroth order) with l0 1.833mm and t0 0.15mm are plotted in Fig. 11. As shown in Fig. 11, the MCF gradually stabilizes and converges when the element size reduces to 0.4 mm while the computing time increases significantly with decreasing of the element size. For both the accuracy of results and computing efficiency consideration, 0.2 mm (9 elements on each cell wall) is conservatively chosen for this case. It is noted that the wall length of the hierarchical honeycomb decreases sharply with hierarchical order n increases, see Eq. (10). Therefore, the mesh size should be adjusted for different hierarchical orders. To validate the accuracy of the FE modeling method, simulations of zeroth order hierarchical honeycombs are conducted and compared with experimental results. The zeroth order hierarchical honeycomb, i.e., the regular honeycomb, is common in engineering applications, thus there are numerous experimental data available in literature. In this study, a
3 3 and a 5 5 unit cell of the zeroth order hierarchical honeycombs with l0 6mm and t0 0.75mm are simulated [48]. The modelling approach is the same as mentioned above, 16
and the force-strain curves and deformation patterns are plotted in Fig. 12. It is seen that both the numerical force-strain curves and the deformation patterns are in good agreement with the experimental results [48]. Moreover, the accuracy of the cell model is validated by comparing the numerical results with that of the whole model as well as theoretical solutions conducted by Gibson and Ashby [13]. Based on their work, crush strength of the regular honeycomb in out-of-plane direction is t l
5/ 3
pl 5.6 0
(35)
where l is the wall thickness of the honeycomb, which is double of the edge length of the Y-shaped unit cell. Fig. 13 shows the crushing stress-strain curves and deformation patterns of the Y-shaped unit cell model and a whole model ( 6 7 unit cell) with l0 1.833mm and t0 0.15mm . The crushing strengths obtained from simulation and Eq. (35) are listed in Table
1. The result obtained from the current theoretical solutions, i.e., Eq. (33) is also listed. It is seen that both the collapse mode and crushing response are in good agreement by using unit cell model and whole model. And also, both the simulation and the current theoretical results compare very well with the prior theoretical result (Eq. (35)). Hence, the numerical modeling approach used in this paper are reliable. 5.
Results Note with increasing of the hierarchical order n, micro-cells of the hierarchical
honeycomb increase greatly. It is a big challenge for the simulation computing, even use the cell model. Therefore, only the hierarchical honeycombs up to fourth order are simulated in this paper. Three sets of hierarchical honeycombs with different relative densities are developed. Each set of hierarchical honeycombs are developed based on the same Y-shaped 17
unit cell extracted from a regular honeycomb (zeroth order). The three unit cells of regular honeycombs (zeroth order) have the same edge length of l0 112mm , the same initial height of H 0 200mm and different wall thicknesses of t0 1.0mm , 1.5mm and 2.0mm , respectively. The relative density of the three sets of hierarchical honeycombs are respectively 0.0052 , 0.0077 and 0.0103 . The configurations of the three sets of hierarchical honeycombs are listed in Table 2. Numbers of the corner and Y-shaped elements for each hierarchical honeycomb are also given. 5.1. Deformation pattern Fig. 14 shows the deformation pattern of the set of hierarchical honeycombs with relative density of 0.0052 at the same crushing displacement of h =160mm. It is seen that all the honeycombs are crushed layer-by-layer. The length of the local buckling wave for each fold of each hierarchical honeycomb remains constant during the formation of buckling. Also, the folding pattern of each corner element or Y-shaped element is similar for all hierarchal honeycombs. A considerable difference is that with the increase of hierarchical order n, the wavelength is greatly shortened and the number of complete folds increases significantly. Similar observations are also drawn in the other two sets of hierarchical honeycombs. It is concluded that the deformations of the hierarchical honeycombs meet with the assumption of SFE theory perfectly. 5.2. Crushing response Fig. 15 shows the crushing responses of the set of hierarchical honeycombs with relative density of 0.0052 . As shown in Fig. 15(a), with hierarchical order increasing, the contact force increases significantly. Also, the effective crushing distance becomes shorter. The initial peak forces (PFs) and MCFs are shown in Fig. 15(b), it should be noted that the MCFs presented here are the average forces in the plateau region for the densification region 18
makes no significant sense to energy absorption. It is seen that the MCF is improved greatly with increasing of hierarchical order. Even better, the PF keeps almost a constant for the considered cases. The other two sets of hierarchical honeycombs perform a similar trend but with different magnitudes. More specifically, the MCFs are improved 71%, 114%, 201% and 309% while the PFs increase only 8.7%, 10%, 9.6% and 8.9% for the first to fourth order hierarchical honeycombs with relative density of 0.0052 , respectively. The results reveal that hierarchy can greatly improve the crashworthiness of honeycombs. Substituting the specific number of continuous elements into Eq. (33), the MCFs of the hierarchical honeycombs of different orders can be obtained. Take the second order hierarchical honeycomb for example, where N c 12 , N Y 9 , the MCF is derived as
P 174.9 0 l20.2 t21.8
(36)
The predicted MCFs for all of the hierarchical honeycombs are listed in Table 3. It is seen that the theoretical solutions compare very well with the numerical results for the lower orders (zeroth to second orders), but slightly smaller than the numerical results for the larger orders (third and fourth orders). Specifically, the maximum error is -11.9%, which is observed in the fourth order hierarchical honeycomb with relative density of 0.0052 . This is due to the effective crushing distance coefficient is taken as 0.70 for all of the orders. However, the value slightly decreases with hierarchy order and wall thickness. So, it is overestimated for the third and fourth order hierarchical honeycombs. In another word, the theoretical predictions are slightly conservative for higher orders. Modify the theoretical predictions by adopting the effective crushing distance obtained from simulations. Better agreement is reached between the numerical results and theoretical predictions, as shown in Table 3. However, the error is still very large for the first-order hierarchical honeycomb with relative density of 0.0103 after modification, which is +11.7%. This is because unstable global buckling takes place only for this case (as shown in Fig. 16), which makes it 19
considerably different from the other cases and not meeting with the basic assumptions of the current theoretical solutions. The MCF becomes much less for this case due to the unstable global buckling. In addition to this special case, the maximum error between the numerical results and theoretical predictions is -7.9% in the case of fourth order hierarchical honeycomb with relative density of 0.0052 . Considering the complexity of the problem and assumptions adopted, the results indicate that the theoretical solutions give a good prediction for the MCFs of hierarchical honeycombs. 6.
Discussions As stated in Section 5, the MCF increases with the hierarchical order significantly for
the considered cases. To seek the general rule of the amplification of MCF, an nth order hierarchical honeycomb is considered here. For the nth order hierarchical honeycomb, numbers of the corner elements N c and Y-shaped elements N Y can be calculated as Nc
3 n (3 1) 2
N Y 3n
(37) (38)
Substituting Eqs (37) and (38) into Eq. (33), MCF of the hierarchical honeycomb of nth order can be derived as P 0.3571 0 t 3 / 2
2Lc (107.70 3n 35.87) ( B / t ) 0.6
(39)
Eq. (39) can be further simplified for the nth ( n 1 ) hierarchy honeycomb by n 1 substituting Lc 3 ln , B 0.5 l n
P 6.6 3n1 (3n1 1) 0 ln0.2 tn1.8
(40)
Note Eq. (40) is suitable only when n 1 due to B l0 when n 0 . Therefore, Eq. (40) should be modified for n 0 , as 20
Pn0 13.2 0 l00.2 t01.8
(41)
Substituting Eqs. (9) and (10), Eq. (40) can be rewritten in terms of wall length and wall thickness of the zeroth order hierarchical honeycomb (regular honeycomb), as P 6.6 3n1 (3n1 1) 0.42 n 0 l00.2 t01.8
(42)
P 19.8 1.26n 0 l00.2 t01.8
(43)
With n increasing
Moreover, define the amplification of MCF from one generation to the next as
i Pi / Pi 1
(44)
It is seen from Eq. (43) that when n is sufficiently large, the amplification of the MCF from one generation to the next approaches a constant of n 1.26 . The amplification of MCF of the nth order ( n 1 ) hierarchical honeycomb compared with the regular honeycomb (zeroth order) can also be obtained as n
An i Pn / P0 0.5 3n 1 (3n 1 1) 0.42 n
(45)
i 0
With n increasing
An 1.5 1.26n
(46)
Fig. 17 shows the amplification of the MCF of hierarchical honeycombs up to tenth order. It is seen that the MCF increases exponentially with increasing of hierarchical order. This feature is validated by numerical simulations conducted in the previous section, as shown in Fig. 17. The results indicated that the prediction of the amplification of MCF for hierarchical honeycombs is reliable. However, the MCF of the hierarchical honeycomb cannot increases unlimitedly. Combine Eqs. (9) and (10), the ratio of wall thickness to wall length for the nth hierarchical honeycomb is obtained as 21
n
n
tn 4 t0 ln 3 l0
(47)
Rewritten Eq. (47) in terms of the relative density of the honeycomb, one can have
4 n 3
n
3
(48)
It is seen from Eq. (48) that n also increases exponentially with hierarchical order. More specifically, the increase of n is greater than that of An . When n is too large, the assumptions of the current theoretical solutions would be invalid. Therefore, in order to get a hierarchical honeycomb with a high order, the relative density of the honeycomb should be sufficiently small. Fig. 18 shows the design spaces for n 1 / 15 and n 1 / 20 , specifically. To meet the limiting condition requirement, the relative density and hierarchical order can only be selected from the shaded area. Considering the constrain of n 1 / 15 , maximum achievable amplification of MCF of hierarchical honeycombs for different relative densities are shown in Fig. 19. 7.
Conclusions Numerical and analytical investigation on determining the crashworthiness of
fractal-like honeycombs with vertex-based self-similar hierarchy were proposed. By analyzing three groups of hierarchical honeycombs with various relative densities, the effect of the hierarchy on the mean crushing force (MCF) of the hierarchical honeycombs was discussed and quantified. The main conclusions are summarized as follows: 1)
Theoretical models to predict the MCFs of the hierarchical honeycombs were established based on the Super Folding Element (SFE) theory; Finite element (FE) models were also established and validated by prior theory and experimental results.
2)
Numerical simulations on hierarchical honeycombs up to fourth order of three different 22
relative densities were conducted. Results obtained using the current theoretical solutions were in good agreement with the numerical results, especially after modifying the effective crush distance. 3)
It is proved that hierarchy could significantly improve the crashworthiness of the honeycombs. According to the simulations, the MCF was improved 71%, 114%, 201% and 309% respectively for the first to fourth order while the initial peak force (PF) keeps almost constant.
4)
The amplification of the MCF from one generation to the next was theoretically predicted, which is a constant ( n 1.26 ) when the hierarchical order is sufficiently larger. Moreover, the amplification of the MCF of a given hierarchical honeycomb was also given and which is proved to be determined only by the hierarchical order.
5)
It was validated that the relative density should be sufficiently small for pursuing a high order hierarchical honeycomb. The potential maximum achievable amplifications of the MCFs for hierarchical honeycombs of some specific relative densities were also discussed.
Acknowledgements The authors would like to acknowledge the research grants from the National Natural Science Foundation of China (11572086, 11602112).
23
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honeycombs under uniaxial compression: experimental and numerical analyses. International Journal of Mechanical Sciences. 2016;111-112:125-33. [37] Zhang Y, Lu M, Wang CH, Sun G, Li G. Out-of-plane crashworthiness of bio-inspired self-similar regular hierarchical honeycombs. Composite Structures. 2016;144:1-13. [38] Sun G, Jiang H, Fang J, Li G, Li Q. Crashworthiness of vertex based hierarchical honeycombs in out-of-plane impact. Materials & Design. 2016;110:705-19. [39] Reid S R PC. Dynamic Uniaxial Crushing of Wood. International Journal of Impact Engineering. 1997;19(5):531-70. [40] Wierzbicki T, Abramowicz W. On the Crushing Mechanics of Thin-Walled Structures. Journal of Applied Mechanics. 1983;50:727-34. [41] Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet metal columns. Journal of Applied Mechanics. 1989;56:113-20. [42] Wierzbicki T. Crushig analysis of metal honeycombs. International Journal of Impact Engineering. 1983;1:157-74. [43] Chen W, Wierzbicki T. Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption. Thin-Walled Structures. 2001;39:287–306. [44] Tran T, Hou S, Han X, Chau M. Crushing analysis and numerical optimization of angle element structures under axial impact loading. Composite Structures. 2015;119:422-35. [45] Zhang X, Zhang H. Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs. Composite Structures. 2013;96:143-52. [46] Zhang X, Zhang H. Energy absorption of multi-cell stub columns under axial compression. Thin-Walled Structures. 2013;68:156-63. [47] Zhang X, Zhang H. Numerical and theoretical studies on energy absorption of three-panel angle elements. International Journal of Impact Engineering. 2012;46:23-40. [48] Zhang X, Zhang H, Wen Z. Experimental and numerical studies on the crush resistance of aluminum honeycombs with various cell configurations. International Journal of Impact Engineering. 2014;66:48-59.
26
Figure captions Fig. 1. Schematic diagram of the unit cell of a regular honeycomb (i.e., zeroth order). Fig. 2. Unit cells of the hierarchical honeycombs of zeroth to fourth orders. Fig. 3. Structural hierarchy of unit cell of the i 1 th order hierarchical honeycomb. Fig. 4. Simplified folding mechanism of each fold [40, 41]. Fig. 5. The global geometry of the basic folding mechanism: (a) Before deformation; (b) After deformation; (c) Rotation angles. Fig. 6. Illustration of the membrane elements. Fig. 7. Division of constituent elements in the third order hierarchical honeycomb. Fig. 8. Scheme of the constituent elements: (a) Corner element; (b) Y-shaped element. Fig. 9. Illustration of the generation of membrane elements for the Y-shaped element: (a) Observation from simulation; (b) Sketch map. Fig. 10. Schematic diagram of the FE model for the out-of-plane crushing. Fig. 11. Effect of element size on simulation results and computing times. Fig. 12. Validation of the modeling method: (a) Force-strain curves; (b) Deformation patterns. Fig. 13. Comparison of simulation results between the cell model and whole model: (a) Stress-strain curves; (b) Deformation patterns. Fig. 14. Isometric view of the deformation patterns of the hierarchical honeycombs ( 0.0052 ) of zeroth to fourth orders at the crushing displacement of h =160mm. Fig. 15. Crushing response of the set of hierarchical honeycombs with a relative density of
0.0052 : (a) Force-displacement curves; (b) PFs and MCFs. Fig. 16. Deformation patterns of the first-order hierarchical honeycomb with the relative density of 0.0103 at the crushing displacement of h =70mm. Fig. 17. Amplification of the MCF of hierarchical honeycomb for different orders. 27
Fig. 18. Design spaces of the hierarchical honeycombs when n 1 / 15 and n 1 / 20 . Fig. 19. Maximum achievable amplification of the MCF of hierarchical honeycombs for different relative density in the constraining of n 1 / 15 .
Table captions Table 1. Comparisons of crushing strength between simulation and theoretical results. Table 2. Configurations of the hierarchical honeycombs of zeroth to fourth orders for three different relative densities. Table 3. Comparison of the numerical simulations and theoretical predictions on MCFs.
28
Y
X
Fig. 1. Schematic diagram of the unit cell of a regular honeycomb (i.e., zeroth order).
29
0th order
1st order
2nd order
3rd order
4th order
Fig. 2. Unit cells of the hierarchical honeycombs of zeroth to fourth orders.
30
li li-1
ti i th order Hierarchy
ti-1
Fig. 3. Structural hierarchy of unit cell of the i 1 th order hierarchical honeycomb.
31
Horizontal hinge line
Plastic flow area
Fig. 4. Simplified folding mechanism of each fold [40, 41].
32
B Ψ
π/2
π
Ψ
π/2 2H
Ψ Ψ (a)
(b)
(c)
Fig. 5. The global geometry of the basic folding mechanism: (a) Before deformation; (b) After deformation; (c) Rotation angles.
33
B 45°
2H
Corner line
Fig. 6. Illustration of the membrane elements.
34
①
②
Fig. 7. Division of constituent elements in the third order hierarchical honeycomb.
35
B B
β
α
(a)
(b)
Fig. 8. Scheme of the constituent elements: (a) Corner element; (b) Y-shaped element.
36
H Panel B
90°-β/2
β
Panel A
Panel C
(a)
(b)
Fig. 9. Illustration of the generation of membrane elements for the Y-shaped element: (a) Observation from simulation; (b) Sketch map.
37
x
V
y
Top rigid wall
Unit cell
y
x Fully fixed bottom rigid wall
x y
Symmetric boundary condition in local x direction
Fig. 10. Schematic diagram of the FE model for the out-of-plane crushing.
38
Fig. 11. Effect of element size on simulation results and computing times.
39
(a)
(b)
Fig. 12. Validation of the modeling method: (a) Force-strain curves; (b) Deformation patterns.
40
(a)
(b)
Fig. 13. Comparison of simulation results between the cell model and whole model: (a) Stress-strain curves; (b) Deformation patterns.
41
0th order
1st order
2nd order
3rd order
4th order
Fig. 14. Isometric view of the deformation patterns of the hierarchical honeycombs ( 0.0052 ) of zeroth to fourth orders at the crushing displacement of h =160mm.
42
(a)
(b)
Fig. 15. Crushing response of the set of hierarchical honeycombs with a relative density of
0.0052 : (a) Force-displacement curves; (b) PFs and MCFs.
43
Fig. 16. Deformation patterns of the first-order hierarchical honeycomb with the relative density of 0.0103 at the crushing displacement of h =70mm.
44
Fig. 17. Amplification of the MCF of hierarchical honeycomb for different orders.
45
n 1 / 15
n 1 / 20
0.02 0.015
0.01 0.005
Fig. 18. Design spaces of the hierarchical honeycombs when n 1 / 15 and n 1 / 20 .
46
0.0029 0.0039 0.0051 0.0069 0.0091 0.0162 0.0289
0.0122 0.0217
Fig. 19. Maximum achievable amplification of the MCF of hierarchical honeycombs for different relative density in the constraining of n 1 / 15 .
47
Table 1. Comparisons of crushing strength between simulation and theoretical results. Simulation results
Cruhsing strength (MPa)
Whole-model
Cell-model
2.09
1.99
48
Theoretical results Wierzbicki et al. Current solution Eq. (35) Eq. (33) 2.07
2.12
Table 2. Configurations of the hierarchical honeycombs of zeroth to fourth orders for three different relative densities. Hierarchy Relative order density 0th
1st
2nd
3rd
4
th
Wall thickness Wall length Ratio of wall ti (mm) li (mm) thickness to length i
0.0052
1.0
0.0077
1.5
0.0103
2.0
0.01786
0.0052
0.6667
0.01190
0.0077
1.0
0.0103
1.3333
0.02391
0.0052
0.4444
0.01587
0.0077
0.6667
0.0103
0.8889
0.03175
0.0052
0.2963
0.02116
0.0077
0.4444
0.0103
0.5926
0.04233
0.0052
0.1975
0.02821
0.0077
0.2963
0.0103
0.3951
Nc
NY
0
1
3
3
12
9
39
27
120
81
0.00893 112
0.01339
56
0.01786
28
0.02391
14
0.03175
7
0.04233 0.05644
49
Table 3. Comparison of the numerical simulations and theoretical predictions on MCFs. Hierarchy Relative order density
th
0
st
1
2nd
3rd
4th
FEM Theoretical predication Modified theoretical prediction P (kN)
P (kN)
Error
P (kN)
Error
0.0052
2.67
2.58
-3.4%
0.72
2.51
-5.9%
0.0077
5.78
5.40
-7.3%
0.71
5.33
-7.7%
0.0103
9.03
8.94
-1.0%
0.71
8.81
-2.4%
0.0052
4.57
4.59
+0.4%
0.71
4.53
-1.0%
0.0077
9.69
9.53
-1.7%
0.70
9.53
-1.7%
0.0103
14.23
15.90
+11.7%
0.70
15.90
+11.7%
0.0052
5.71
6.01
+5.3%
0.71
5.92
+3.7%
0.0077
13.46
12.48
-7.3%
0.69
12.67
-5.9%
0.0103
21.73
20.91
-3.8%
0.68
21.53
-0.9%
0.0052
8.04
7.67
-4.6%
0.68
7.90
-1.7%
0.0077
17.78
15.91
-10.5%
0.66
16.87
-5.1%
0.0103
29.15
26.71
-8.4%
0.66
28.32
-2.7%
0.0052
11.05
9.73
-11.9%
0.67
10.17
-7.9%
0.0077
22.57
20.18
-10.6%
0.66
21.40
-5.2%
0.0103
35.91
33.87
-5.4%
0.65
36.48
+1.6%
50