In-plane crushing and energy absorption performance of multi-layer regularly arranged circular honeycombs

Composite Structures 96 (2013) 726–735

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In-plane crushing and energy absorption performance of multi-layer regularly arranged circular honeycombs Deqiang Sun a,⇑, Weihong Zhang b,⇑, Yucong Zhao a, Guozhi Li a, Yueqing Xing a, Guifen Gong a a b

Shaanxi University of Science and Technology, 710021 Xi’an, Shaanxi, China Engineering Simulation and Aerospace Computing (ESAC), School of Mechanical Engineering, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’an, Shaanxi, China

a r t i c l e

i n f o

Article history: Available online 26 October 2012 Keywords: Multi-layer regularly arranged circular honeycombs Deformation mode Dynamic plateau stress Energy absorption model Densification strain

a b s t r a c t The dynamic deformation modes, plateau stresses and energy absorption performance of multi-layer regularly arranged circular honeycombs are investigated numerically under the in-plane crushing loadings at the impact velocities 3–250 m/s, with aim to disclose the influences of configuration parameters and impact velocity on them. The numerical results are presented in the forms of diagrams, curves and tables. With increasing impact velocities, the double ‘‘V’’-shaped (quasi-static homogeneous mode), ‘‘V’’-shaped (transition mode) and ‘‘I’’-shaped (dynamic mode) collapse bands are observed in turn. A simplified energy absorption model is put forward and employed to describe the energy absorption performance. It is shown that the optimal energy absorption per unit volume is related to dynamic plateau stress and dynamic densification strain and that the optimal energy absorption efficiency is the reciprocal of dynamic densification strain. From physical analysis and discussion, the empirical formulas of critical velocity of mode transition, plateau stress and dynamic densification strain are given based on the numerical results. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.

Introduction As important two dimensional (2D) lightweight structures with good energy absorption performance, the multi-layer regularly arranged circular honeycombs (MRACHs) always act as cushioning materials and have many applications in the fields of aviation, packaging, military affairs, construction, vehicle manufacturing, and so on. Much theoretical, finite element (FE) and experimental work has been carried out for the circular honeycombs. By using the theoretical analysis, Chung and Waas [1] derived the equivalent elastic mechanics of circular and elliptic honeycombs and analyzed the sensitivities of honeycomb stiffness to three types of initial imperfections. Papka and Kyriakides [2] firstly presented the load–displacement response of polycarbonate circular honeycombs under the in-plane uniaxial compression experimentally and numerically. Chung and Waas [3] also presented the crushing response of circular polycarbonate honeycombs under in-plane uniaxial loading through a combination of experiment and numerical simulation, and discussed the reasons for the orthotropic response of the honeycombs. The response of circular honeycombs to in-plane biaxial compression has also been reported in some literature. Papka and Kyriakides [4,5] developed a testing facility that ⇑ Corresponding authors. Tel./fax: +86 29 86168316. E-mail addresses: [email protected] (D.Q. Sun), [email protected] (W.H. Zhang).

allowed the polycarbonate circular honeycombs to be compressed simultaneously and independently in two orthogonal directions. They recorded the stress–strain responses in both directions as well as the full-field view of the deformation by using a video camera. The effect of the biaxiality ratio on the energy absorption capacity of circular polycarbonate honeycombs was comprehensively discussed. Chung and Waas [6–8] investigated the compressive responses of circular polycarbonate honeycombs under in-plane static and dynamic biaxial loadings where the specimens were compressed in one direction with an initial load in the perpendicular direction. Hu et al. [9] conducted in-plane uniaxial and equi-biaxial compression tests on the circular polycarbonate honeycombs. A special test rig was designed to carry out the inplane equi-biaxial compression tests in a conventional universal testing machine. The deformation characteristics of circular polycarbonate honeycombs under uniaxial compression were quantitatively discussed by tracking the variations of the cell parameters. The initiation of deformation inhomogeneity and its evolution in circular polycarbonate honeycombs are characterized. Karagiozova and Yu [10] studied the post-collapse characteristics and strain localization of ductile circular honeycombs under in-plane uniaxial and biaxial loadings. For some investigations mentioned above, several low loading rates were set for the dynamic loadings and in fact the specimens were still subject to the quasi-static loadings. The circular cell rows of honeycombs in previous investigations were arranged in the

0263-8223/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.10.008

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stagger pattern. In cushioning applications, the MRACHs are always subject to the dynamic impact loadings with high impact velocities. Hereby it is very necessary and valuable to disclose the dynamic behaviors of MRACHs under the crushing loadings for the cushioning optimization. Even though the experimental method can be used to disclose the dynamic behaviors, it is impossible to prepare various specimens with all kinds of dimensions for experiments. Therefore the FE simulation method has become popular in the investigations on the dynamics of honeycombs, e.g. Ruan et al. [11] and Sun et al. [12,13] for regular hexagonal honeycombs, Zheng et al. [14], Li et al. [15] and Liu et al. [16] for irregular honeycombs, Ali et al. [17] for irregular hexagonal honeycombs, Liu and Zhang [18] and Qiu et al. [19,20] for triangular, square, kagome and rhombus honeycombs, etc. These FE simulations show that the configuration parameters and impact velocity affect the dynamic behaviors of 2D honeycombs. In this paper, the FE simulations of MRACHs under the in-plane (in direction x2, Fig. 1) crushing loadings are carried out by using the explicit FE product ANSYS/LS-DYNA at the impact velocities 3–250 m/s. The dynamic deformation modes, plateau stresses and energy absorption performance of MRACHs are presented in the forms of tables, diagrams and empirical formulas with the aim to disclose the influence laws of cell configuration parameters and impact velocity on the in-plane dynamic behavior of MRACHs. FE model The configuration of MRACHs is shown in Fig. 1. The radius and cell wall thickness of periodic representative cell are t and R, respectively. The depth of total MRACHs is b. The densities of MRACHs and cell wall base material are assumed as q⁄ is and qs,  , is respectively. Then the relative density of MRACHs, q

q ¼ q =qs ¼

pt

ð1Þ

2R

Fig. 2a is the FE model used for the in-plane crushing analysis of MRACHs, which is similar to the ones used in the FE investigations mentioned above. R is kept as 3 mm for all simulations. The commercial software ANSYS/LS-DYNA is employed here. The representative cell is periodically extended in the x1 and x2 directions to generate the entire specimen with 16 by 16 cells in directions x1 and x2, of which all cells are overlapped. The specimen is sandwiched between two rigid plates. The support plate, P2, is stationary, and the impact plate, P1, moves downwards in direction x2 at a constant impact velocity, m, until the specimen collapses absolutely. m is varied from 3 to 250 m/s. All nodes of specimen are

constrained from displacement in the out-of-plane direction (direction x3 in Fig. 1) to ensure the in-plane strain state of deformation. All cell walls of specimen are discretized with square Belytschko–Tsay Shell163 elements with five integration points and element edge length of 0.3 mm by the way of free meshing. Automatic single surface contact (ASSC) without friction is defined between the external surfaces of the specimen which may contact each other during crushing. Automatic surface-to-surface contact (ASTS) with the friction coefficient of 0.02 is defined between the specimen and the impact (or support) plate. The elastic linear strain-hardening (also known as bilinear, Fig. 2b) model is used to describe the stress–strain relation of cell wall material and the typical is aluminum with the following mechanical properties: Young’s modulus Es = 68.97 GPa, yield stress rys = 292 MPa, tangent modulus Etan = 689.7 MPa, Poisson’s ratio ms = 0.35 and density qs = 2700 kg/m3. Additionally, the behavior of cell wall material is treated as rate-independent. Some tentative simulations were carried out to determine the appropriate cell number and depth of specimens with consideration of the size sensibility on the dynamics of MRACHs. The size sensibility lies in not only cell number but also the depth of specimen. Firstly, with b kept as 8 mm, simulations of the MRACHs (R = 3 mm and t = 0.1 mm) were carried out with cell number of 4  4, 6  6, 8  8, 10  10, 12  12, 14  14, 16  16, 18  18 and 20  20 in directions x1 and x2 at the impact velocities of 20, 70, 125 and 200 m/s, respectively. The dynamic plateau stresses become stable when the cell number is more than or equal to 12  12. Then, more in-plane crushing simulations were carried out for the MRACHs (R = 3 mm and t = 0.1 mm) with 12  12 cells in directions x1 and x2 and b = 0.5, 3, 5, 8, 10 12 and 15 mm at the impact velocities of 20, 70, 125 and 200 m/s, respectively. The FE results show that the in-plane plateau stresses fluctuate less with the increase of b, and the proper depth of specimens should be larger than 8 mm. Therefore, in our simulations the depth and cell number of all specimens are respectively set to be 10 mm and 16  16 in directions x1 and x2, respectively. Deformation modes All force and energy response curves of specimens can be obtained for all simulations by using the commercial software LSPREPOSTD. The external work is converted into the kinetic energy, K, and internal energy, U, of specimen. The U includes the elastic strain energy and plastic dissipation, and the total energy absorption of specimen, E = U + K. As shown in Fig. 3, for the specimen with fixed cell radius and cell wall thickness, with increasing

R x2

t x3 b x1

(a)

(b)

(c)

Fig. 1. Configuration of MRACHs with 9 by 9 cells: (a) front view; (b) left view; (c) representative cell.

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Stress

P1 v

Etan

σ ys x2 x1 Es

P2 Strain

(a)

(b)

Fig. 2. FE model for in-plane crushing analysis of MRACHs: (a) front view; (b) stress–strain curve for bilinear strain-hardening cell wall base material.

25 A K-u B U-u C E-u

20

C B

15 B

10

C

B C 5

BC B C A A 0 0 0.02

A 0.04

A 0.06

Energy absorption (J)

Energy absorption (J)

25

A 0.08

C B

15

C B

10

C

5 B 0 0

C

B

B

C

A

A

A

A 0.04

0.02

0.06

u (m)

u (m)

(a)

(b)

70

A 0.08

500 A K-u B U-u C E-u

60 50

C

40

C

30

C

20

C

10

B

C

A

B

B A

B

B A

A

A

0 0

0.02

0.04

0.06

0.08

u (m)

(c)

Energy absorption (J)

Energy absorption (J)

A K-u B U-u C E-u

20

A K-u B U-u C E-u

400

C

300 C 200

C C

100 B

0 0

C A

B

0.02

A

B

A

0.04

BA

0.06

B A

0.08

u (m)

(d)

Fig. 3. Energy absorption curves of specimen (R = 3 mm and t = 0.10 mm) under the increasing in-plane impact velocities: (a) m = 3 m/s; (b) m = 20 m/s; (c) m = 50 m/s; (d) m = 150 m/s.

impact velocities, the K and U increases significantly due to inertia effect. The energy absorption change in specimen causes the transition of deformation modes and enhancement of plateau stress with the increasing impact velocities. The enhancement of plateau stress will be discussed in plateau stress. Effect of the impact velocity Similar to the other honeycombs in the FE investigations mentioned above, it is found that the deformation of MRACHs under in-plane dynamic crushing can be catalogued into three modes, quasi-static homogeneous mode (QHM), transition mode (TM) and dynamic mode (DM), with the increasing impact velocities. For the QHM at low impact velocities, the typical deformation stages are shown in Fig. 4. At the initial stage, the deformation in specimen is linear and every cell deforms homogeneously with the transverse expansion (Fig. 4a). With further crushing, the ‘‘V’’-shaped collapse band firstly appears near the support plate and the deformation is not macroscopically homogeneous any more. Following the two ramifications of the collapse band, the

alternate parallel cells subsequently begin to collapse so that each of the slantwise cells nearest to the first collapse band is finally packaged by the surrounding four elliptic collapsing cells (Fig. 4c). Meanwhile the reverse ‘‘V’’-shaped collapse band is observed near the impact plate (Fig. 4c) that is evolved in the same propagation mode of the first ‘‘V’’-shaped collapse band. The packaged cells hold the quasi-circular shape for a long time and gradually collapse from two ends to the center of honeycombs until densification (Fig. 4d–f). The TM happening at the moderate impact velocities is shown in Fig. 5. The deformation is inhomogeneous and the first row cells near to the impact plate firstly collapse in the form of regularlyarranged orange segments (Fig. 5a). With further crushing, each row of cells deform in two different modes alternately near the collapse front, as shown in Fig. 5b–d. That is to say, for the second row, the odd columns of cells deform in the quasi-circular mode and the even columns of cells are crushed into the collapsing ellipses (Fig. 5b). However for the third row, the odd columns of cells are crushed into ellipses and the even columns of cells still keep quasi-circular (Fig. 5c). This causes that each quasi-circular cell is

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(a) ε = 0.076

(b) ε = 0.187

(c) ε = 0.285

(d) ε = 0.454

(e) ε = 0.644

(f) ε = 0.831

Fig. 4. Typical deformation stages of MRACHs for quasi-static homogeneous mode (R = 3 mm, t = 0.15 mm and m = 3 m/s).

(a) ε = 0.032

(b) ε = 0.128

(c) ε = 0.211

(d) ε = 0.387

(e) ε = 0.640

(f) ε = 0.819

Fig. 5. Typical deformation stages of MRACHs for transition mode (R = 3 mm, t = 0.15 mm and m = 50 m/s).

(a) ε = 0.026

(b) ε = 0.080

(c) ε = 0.256

(d) ε = 0.544

(e) ε = 0.765

(f) ε = 0.883

Fig. 6. Typical deformation stages of MRACHs for dynamic mode (R = 3 mm, t = 0.15 mm and m = 150 m/s).

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D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

vc2 150

DM TM QHM

v (m/s)

100

vc1

10

3 0.01

0.05

0.1

t/R Fig. 7. Deformation mode map of the MRACHs with different t/R ratios corresponding to different impact velocities.

surrounded by four quasi-elliptic collapsing cells, which is similar to the QHM. Moreover, the center cells deform earlier than those in both sides in the collapse front, which leads to the ‘‘V’’-shaped collapse bands, as shown in Figs. 5b–d. However after the stage (d), the cells on both sides in specimen near the support plate soon deform and touch the top collapse bands earlier than those in center (Fig. 5e). With the gradual collapse of quasi-circular cells, the specimen goes into densification (Fig. 5f). At the very high impact velocity, the DM occurs. The cells are crushed sequentially in the ‘‘I’’-shaped planar manner from the impact end, as shown in Fig. 6. The collapse band is oriented perpendicular to the loading direction and only a single row of cells are in the collapse process while the other cells behind them have already gone into densification. Critical velocity of mode transition

deformation and energy absorption. F increases quickly to the initial peak impact force with the corresponding initial displacement, u0, at the point C and then drops sharply to a low level during the yield phase. This is followed by a long collapse plateau with large displacement and a roughly constant force. Correspondingly, there is a long and nearly straight portion on the W–u curve before the curve turns upwards. The inflexion point of W–u curve, D, is the key point where densification starts and F begins to increase abruptly (Fig. 8)). Point D is called the optimal energy absorption point (OEAP) and the corresponding displacement, uD, is named as densification displacement. During densification phase, the energy absorption is obvious but the force becomes very large where the mechanics of specimen is similar to that of cell wall material. So, in the cushioning applications, most of external kinetic energy should be absorbed in the flat plateau stress phase before the OEAP. By fitting the W–u curve from u0 and uD, the corresponding external work W0 and WD can be determined. The dynamic plateau stress, rp, is defined as

rp ¼ ðW D  W 0 Þ=ðA0 ðuD  u0 ÞÞ ¼

1 A0 ðuD  u0 Þ

Z

uD

FðuÞdu

ð4Þ

u0

A0 is the cross-sectional area of specimen in impact direction. A one-dimensional shock wave model is proposed by Reid and Peng [21], in order to explain the inertia effect of wood under dynamic crushing. With the continuum assumption, the material is assumed to be at rest without deformation ahead the shockwave front; this region is under constant static yield strength. Behind the shock wave front, the material is fully compacted and has densification strain. The particle velocity in this region is approximated as m and the dynamic plateau stress is expressed as

rp ¼ r0 þ Av 2

ð5Þ

where r0 is the static plateau stress and Am2 is the dynamic enhancement caused by the inertia effect, which creates the shock wave. A is the relation parameter on impact velocity which is

 qs =eSD A¼q

ð6Þ

From our simulations, it is seen that the critical impact velocity, mc, at which the deformation mode switches from one type to another depends on the t/R ratio, as Ruan et al. [11] found for regular hexagonal honeycombs. Some simulations were carried out by changing the t/R ratio at different impact velocities. The corresponding deformation modes are marked in Fig. 7. The critical velocity of mode transition from QHM to TM, mc1, looks independent of the ratio of t/R. In our simulations mc1, is located in the range of 8–20 m/s. For most specimens, mc1  15 m/s. Similar to Ruan’ derivation [11], it is found that the critical velocity of mode transition from TM to DM, mc2 1 (t/R)1/2 for the MRACHs. Therefore the empirical equations for two critical velocities are

and eSD is the static densification strain. Eq. (5) is valid for hexagonal honeycombs [11,17] and irregular honeycombs [19]. Qiu et al. [19] classified the 2D cellular materials into two kinds, the bending-dominated and membrane-dominated structures. As expected by Qiu et al. [19], the quasi-static F–u curve possesses a relatively ‘‘flat-topped’’ plateau shown in Fig. 9, which implies that the MRACHs are the bending-dominated structures similar to type I structures. Qiu et al. [20] carried out lots of FE simulations for five kinds of regular honeycombs at the impact velocities 3.5–140 m/s. The plateau stresses can be well fitted by the following empirical expressions:

mc1  15 ðm=sÞ

ð2Þ

rp ¼ A1 rys q 2 þ ðC 2 q 2 þ C 1 q þ C 0 Þqs m2

ð7Þ

ð3Þ

rp ¼ A1 rys q 2 þ B1 q qs m2

ð8Þ

r9m ¼ A1 rys q 2 þ q qs m2 =ð1  B2 q Þ

ð9Þ

pffiffiffiffiffiffiffiffi

mc2  426 t=R ðm=sÞ Plateau stress

Fig. 8 shows the F–u and W–u response curves of specimen (R = 3 mm, t = 0.15 mm and m = 50 m/s). F is the impact force between impact plate and specimen; u is the displacement of impact plate; W is the external work which is obtained from the integration of corresponding F–u curve and approximately equal to the total energy absorption of specimen. The in-plane crushing loading causes the history with four different phases: linear elastic phase, yield phase, flat plateau stress phase and densification phase in turn for MRACHs (Fig. 8a). The linear elastic phase is characterized as a transient response with small

where A1, C2, C1, C0, B1 and B2 are the fitting coefficients depending on the configuration parameters of honeycomb cores. For MRACHs, in order to study the effects of t/R ratio on these fitting coefficients, eight specimens with different cell wall thicknesses are used. The FE simulations are carried out at the impact velocities 3, 20, 50, 70, 100, 125, 150, 175, 200 and 250 m/s, respectively. The corresponding dynamic plateau stresses are listed in the Table 1. It is found that Eq. (5) has good agreement with the FE results for the MRACHs at the impact velocities 3–250 m/s, as shown in Fig. 10. The values of A obtained by the least-square fitting are also listed in the last row.

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D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

II

2.0

F (KN)

Initial peak impact force

I C

1.5

A F-u curve I. Linear elastic phase II. Yield phase III. Flat plateau stress phase IV. Densification phase

A

1.0

100

A IV

III

A

A A

0.5

uD

0

0

0.04

u 0 0.02

80 D 60

WD

B B

40 B 20 0

0.08

0.06

B

B W-u

External work (J)

2.5

uD

B 0.02

0

0.04

u (m)

u (m)

(a)

(b)

0.06

0.08

Fig. 8. Load–displacement and external work curves of MRACHs under the in-plane crushing loading (R = 3 mm, t = 0.15 mm and m = 50 m/s): (a) F–u curve; (b) W–u curve.

stresses can still be best fitted by Eq. (10) for the MRACHs as shown in Fig. 11. The calculated A1 is 0.183 from the least-square fitting. In combination with Eq. (1). The static plateau stress is

5 A F-u

F (KN)

4

A

r0 ¼ 0:453rys

3 2

 2 t R

ð11Þ

A

1

Empirical expressions

A

A

A

0 0

0.02

0.04

0.06

0.08

u (m) Fig. 9. Typical quasi-static F–u curve of MRACHs (R = 3 mm, t = 0.15 mm and m = 3 m/s).

Static plateau stress At a very low impact velocity, e.g. 3 m/s, the kinetic energy of specimen K is less than 2% of the corresponding total energy E before the OEAP so that E is nearly equal to the internal energy U, as shown in Fig. 3a. Therefore this time the quasi-static plateau stress is about equal to the corresponding static one, as predicted by Wang and Fan [22]. The quasi-static plateau stresses are listed in fourth row of Table 1. From Qiu’s limit analysis [19] and FE simulations [20], it has been found that, for the regular triangular, square, kagome, rhombus and hexagonal honeycombs, r0 relies  2 proportionally which is corresponding to the left term of on q Eqs. (7)–(9) as

2

r0 ¼ A1 rys q

ð10Þ

 is varied by changing the t/R ratio with the For the MRACHs, q fixed R. The quasi-static plateau stress becomes larger with the increasing relative density. It is found that the quasi-static plateau

Qiu et al. [20] have introduced the dynamic enhancement stress due to the inertia effect can be fitted by the right terms of Eqs. (7)– (9), respectively. The number of specimen is assumed as n. Here we introduce the error, the absolute total squared error between the formula prediction and the FE results, which is Pn  i¼1 ðAðq; qs Þ  Ai Þ. The errors are 35.09, 487.69 and 56.52 corresponding to Eqs. (7)–(9) from the least-square fitting, respectively. The fitted curves are shown in Fig. 12 respectively. This shows that Eqs. (7) and (9) better fits the FE results for the MRACHs. From the least-square fitting, the calculated fitting coefficients are C2 = 0.815, C1 = 0.985, C0 = 0.0005 and B2 = 0.616, respectively. So the plateau stress of MRACHs can be calculated by the following two formulas:

rp ¼ 0:183rys q 2 þ ð0:815q 2 þ 0:985q  0:0005Þqs m2

ð12Þ

rp ¼ 0:183rys q 2 þ q qs m2 =ð1  0:616q Þ

ð13Þ

Energy absorption Energy absorption model The nominal engineering stress, r, and strain, e, of MRACHs are defined as respectively

Table 1 In-plane plateau stresses of aluminum MRACHs (R = 3 mm) under in-plane crushing loadings.

m (m/s)

t (mm) 0.03

0.05

0.07

0.1

0.15

0.2

0.25

0.3

3 20 50 70 100 125 150 175 200 250

0.0062 0.0155 0.0955 0.2030 0.4123 0.6464 0.8577 1.2338 1.6485 2.5685

0.0265 0.0371 0.1730 0.3545 0.7232 1.0673 1.5751 2.1609 2.7946 4.2319

0.0525 0.0681 0.2595 0.5297 1.0326 1.5332 2.3425 3.1001 3.9106 6.3112

0.1272 0.1376 0.3651 0.8319 1.5775 2.3252 3.3875 4.6896 5.8988 9.1069

0.3254 0.3051 0.8678 1.4222 2.2798 3.8812 5.2981 7.1740 9.1283 14.0669

0.5749 0.9366 1.5869 2.3215 3.8362 5.7083 7.7168 10.2251 12.9948 19.6175

0.9064 1.5185 2.2686 3.2164 5.1649 7.4964 9.7808 13.0103 16.5593 24.9070

1.3397 2.2168 3.1561 4.2250 6.4886 9.2226 12.2490 15.8827 20.0426 31.8000

A (kg/m3)

40.97

68.12

99.77

145.12

221.27

303.22

379.84

472.96

rp (MPa)

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D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

r ¼ F=A0 ; e ¼ u=H

35 t = 0.03 mm t = 0.05 mm t = 0.07 mm t = 0.10 mm t = 0.15 mm t = 0.20 mm t = 0.25 mm t = 0.30 mm

30 25

σp (m/s)

20

ð14Þ

where H is the initial length of specimen in impact direction. Similar to the F–u curve of MRACHs, the r–e curve has four phases with distinct characteristics, as shown in Fig. 13a. With the aim to evaluate the energy absorption performance of MRACHs conveniently, the whole r–e curve can be simplified as a piecewise function with four line segments as shown in Fig. 13b. The piecewise function is

8 > > > > > > <

r ¼ Ed e; 0 6 e < e0 rp 6 r 6 r0 ; e ¼ e0 r ¼ rp ; e0 < e 6 eD > > > r ¼ rp þ Es ðe  eD Þ; eD < e < 1 > > >

15 10 5

ð15Þ

:

0

0.01

0.05

The first phase is linear with the slope of in-plane dynamic Young’s modulus of MRACHs, Ed and the stress goes up to the initial peak value, rIP; the second is a vertical line where the stress drops from rIP to rp with a constant strain, e0; the third is a flat plateau oscillating about a fixed stress, rp, until the dynamic densification strain, eD; for the last phase the densified specimen have same mechanical properties of cell wall material and therefore the slope of fourth line segment is Es. Then,

-5 0

50

100

150

200

250

v (m/s) Fig. 10. rp–m curves of the MRACHs with different ratios of t/R (R = 3 mm).

Ed ¼ rIP =e0 ;

1.4 FE calculated results Fitting curve

1.2

e0 ¼ u0 =H; eD ¼ uD =H

ð16Þ

The energy absorption per unit volume, e, of specimen is defined as

1.0

e¼

Z

e

r de

ð17Þ

According to Eqs. (15)–(17), the simplified energy absorption model should be

0.8

8 e ¼ Ed e2 =2; 0 6 e < e0 > > > > 2 > > < e ¼ Ed e0 =2; e ¼ e0 e ¼ Ed e20 =2 þ rp ðe  e0 Þ; e0 < e 6 eD > > 2 > 2 > > e ¼ Ed e0 =2 þ rp ðe  e0 Þ þ Es ðe  eD Þ =2; > :

0.6 0.4 0.2

0

0

0.02

0.04

0.06

0.08

eD ¼ rp eD

Fig. 11. r0–t/R curves of the MRACHs.

5

5 FE results Fitting curve

2 1

3

2

0.04

0.06

0.08

0.1

0

3 2

1

1

0.02

FE results Fitting curve

4

A (102Kg/m3)

A (102Kg/m3)

4

3

00

ð19Þ

5 FE results Fitting curve

4

ð18Þ

eD < e < 1

It is found that the u0 is so small that e0  0 for all simulations. Therefore, the energy absorption per unit volume of MRACHs corresponding to the OEAP, eD, is

0.1

t/R

A (102Kg/m3)

σ0 (MPa)

0

0

0.02

0.04

0.06

0.08

0.1

0

0

0.02

0.04

0.06

t/R

t/R

t/R

(a)

(b)

(c)

Fig. 12. Fitted A–t/R curves of the MRACHs based on different equations: (a) Eq. 7; (b) Eq. 8; (c) Eq. 9.

0.08

0.1

733

D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

I

II

Stress σ (MPa)

A σ-ε curve I. Linear elastic phase II. Yield phase III. Flat plateau stress phase IV. Densification phase

8 6 4 2 0

0.1

0.2

0.3

0.4

0.5

Strain ε

0.6

0.7

σ IP

8 6

σp

4 2

εD

ε0 0

10

IV

III

10

Stress σ (MPa)

12

εD

ε0

0.8

0

0.9

0

0.1

0.2

0.3

0.4

(a)

0.5

Strain ε

0.6

0.7

0.8

0.9

(b)

Fig. 13. Stress–strain curves of MRACHs under the in-plane crushing loading (R = 3 mm, t = 0.25 mm and m = 50 m/s): (a) actual r–e curve; (b) simplified r–e curve.

Table 2 Dynamic densification strains of aluminum MRACHs (R = 3 mm) under in-plane crushing loadings.

m (m/s)

eD 3 20 30 50 70 100 125 150 175 200 250

mc2 (m/s) eDH kc for TM kv for TM (103) eDD

t (mm) 0.03

0.05

0.07

0.1

0.15

0.2

0.25

0.3

0.7799 0.8769 0.8927 0.9207 0.9404 0.9508 0.9558 0.9617 0.9676 0.9683 0.9775

0.7664 0.8579 0.8742 0.9047 0.9277 0.9354 0.9423 0.9459 0.9539 0.9598 0.9589

0.7470 0.8257 0.8501 0.8720 0.9143 0.9249 0.9299 0.9342 0.9399 0.9459 0.9489

0.7046 0.8108 0.8304 0.8606 0.8847 0.9149 0.9241 0.9250 0.9287 0.9328 0.9359

0.6679 0.7956 0.8046 0.8357 0.8649 0.8958 0.8981 0.9087 0.9130 0.9137 0.9169

0.5577 0.7706 0.7831 0.8171 0.8487 0.8750 0.8789 0.8809 0.8829 0.8898 0.8932

0.5380 0.7585 0.7711 0.7906 0.8207 0.8608 0.8740 0.8768 0.8793 0.8822 0.8915

0.5224 0.7843 0.7589 0.7840 0.8092 0.8583 0.8642 0.8707 0.8717 0.8754 0.8764

43 0.7799 0.8453 1.8692 0.9554

55 0.7664 0.8271 1.8801 0.9463

65 0.7470 0.8000 1.8492 0.9340

78 0.7046 0.7844 1.8676 0.9269

95 0.6679 0.7649 1.8561 0.9077

110 0.5577 0.7459 1.8111 0.8851

123 0.5380 0.7313 1.7493 0.8808

135 0.5224 0.7253 1.6445 0.8736

Energy absorption efficiency Here, we define a novel term, CD, named the dynamic cushioning coefficient as

C D ¼ rp =eD

ð20Þ

that characterizes the energy absorption efficiency of MRACHs. In cushioning applications, the external kinetic energy is hoped to be absorbed under a certain level of rp before the OEAP. A smaller CD implies that the MRACHs have better energy absorption efficiency. From the Eqs. (19) and (20), CD can be expressed as

C D  1=eD

ð21Þ

eD is the nominal strain of MRACHs corresponding to the OEAP called dynamic densification strain as mentioned above. Physically the dynamic cushioning coefficient is approximately equal to the reciprocal of dynamic densification strain. Therefore, in cushioning applications the larger dynamic densification strain means the better energy absorption efficiency. It is necessary to discuss the dynamic densification strain in view of evaluation of energy absorption performance of MRACHs. The corresponding dynamic densification strains of eight specimens used above are listed in Table 2 for different impact velocities. The critical velocities of mode transition from TM to DM, mc2, are listed in the fifteenth row of Table 2 for different t/R ratios. In our simulations, it is found that the ratio of t/R and impact velocity affect the dynamic densification strain of MRACHs

together under the in-plane dynamic impact loadings. In terms of  , determines the dythe t/R ratio, the nominal porosity, 1  kc q namic densification strain of MRACHs where kc is the correction coefficient. As mentioned above, different impact velocities lead to different deformation modes for the MRACHs with fixed configuration parameters. At the low impact velocity, the dynamic densification strain for the QHM, eDH, is approximately equal to the corresponding quasi-static one at the impact velocity of 3 m/s. For the DM, the ‘‘I’’-shaped planar deformation mode determines that the MRACHs with fixed configuration parameters have nearly identical dynamic densification strains at different high impact velocities. Therefore the corresponding average dynamic densification strain is viewed as the dynamic densification strain for the DM, eDD, listed in the last row of Table 2. So, eDH and eDD are only connected with the configuration parameters of MRACHs in the  ). k is the constant relation coefficient. As shown form of k (1  kc q in Fig. 14, eDH and eDD can be best fitted by respectively

eDH ¼ 0:813ð1  2:493q Þ

ð22Þ

eDD ¼ 0:959ð1  0:624q Þ

ð23Þ

From our simulation results listed in Table 2, it is seen that the dynamic densification strains for the TM, eDT, (listed in italics in Table 2 for different impact velocities) are linearly related to the impact velocities for the MRACHs with fixed configuration parameters. As shown in Fig. 15a, the eDT–m curves have same slope, kv for different t/R ratios. The scale coefficient between eDT

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D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

0.8

0.96 FE results Fitted curve

0.75

FE results Fitted curve

0.94

0.7 0.92

ε DD

ε DH

0.65 0.6

0.9

0.55 0.88

0.5 0.45

0.86 0

0.02

0.04

0.08

0.06

0.10

0.14

0.12

0.16

0

0.02

0.04

0.06

0.08

ρ

ρ

(a)

(b)

0.10

0.12

0.14

0.16

 curves of MRACHs under the in-plane crushing loading (R = 3 mm): (a) for the QHM; (b) for the DM. Fig. 14. eD–q

1.05

0.86

t = 0.03 mm t = 0.05 mm t = 0.07 mm t = 0.10 mm t = 0.15 mm t = 0.20 mm t = 0.25 mm t = 0.30 mm

1

0.95

0.82 0.8

kC

εDT

0.9

FE results Fitted curve

0.84

0.78

0.85 0.76 0.8 0.74 0.75

0.01

0.05

0.1

0.72

0.7 20

40

60

80

100

120

140

0.7

0

0.05

0.1

v (m/s)

ρ

(a)

(b)

0.15

2

 curves of the MRACHs with different t/R ratios (R = 3 mm). Fig. 15. (a): eDT–m curves; (b): kC–q

and m, kc, is proportionally connected with the nominal porosity,  . The calculated values of kv and kc are also listed in Table 2 1  kc q by using the least-square fitting. Therefore, the relation of eDT on m and t/R can be described as

eDT ¼ bð1 þ kv mÞð1  kc q Þ

ð24Þ

 ) as shown in Fig. 15b. By the least-square fitwhere kc = b (1  kc q ting, the empirical expression of eDT is

eDT ¼ 0:865ð1 þ 0:00182mÞð1  0:865q Þ

ð25Þ

Conclusions In this paper, by using the FE source codes ANSYS/LS-DYNA, the effects of t/R and impact velocity on the in-plane deformation modes, plateau stresses and energy absorption performance of MRACHs under in-plane crushing loadings are studied when t/R is from 0.01 to 0.1 and impact velocity is from 3 to 250 m/s. In view of size sensitivity of the FE model, some tentative FE simulations of MRACHs with different dimensions lead to the confirmation of the reliable FE model which has 16  16 cells in directions x1 and x2 and depth of 10 mm. Many FE simulations for 8 specimens of MRACHs with different configuration parameters are carried out at different impact velocities.

For the specimen of MRACHs, three different deformation modes, quasi-static homogeneous mode, transition mode and dynamic mode, are observed with the increasing impact velocities. For the QHM at low impact velocities, the ‘‘V’’-shaped collapse band firstly appears near the support plate and reverse ‘‘V’’-shaped collapse band is subsequently observed near the impact plate, which results in the double ‘‘V’’-shaped deformation pattern. For the TM at moderate impact velocities, the ‘‘V’’-shaped only appears near the impact plate. At the high impact velocity, a single row of cells are crushed sequentially in the ‘‘I’’-shaped planar manner from the impact end, behind which the cells have already been densified. The empirical expression of critical velocity of mode transition is given based on the numerical results. The numerical simulations show that in-plane plateau stresses are proportional to the square of impact velocities when all configuration parameters are kept constant under in-plane crushing loadings, which is consistent with the one-dimensional shock wave model proposed by Reid and Peng [21]. According to the three empirical expressions of plateau stress given by Qiu et al. [20], it is found that two of them can better fit the FE results. The corresponding two sets of empirical expressions are given based on our results. A simplified energy absorption model is put forward and employed to describe the energy absorption performance, which

D.Q. Sun et al. / Composite Structures 96 (2013) 726–735

shows that the dynamic densification strain determines the energy absorption efficiency. From physical analysis and discussion based on the FE numerical results, it is found that the t/R ratio and impact velocity affect the dynamic densification strain of MRACHs together under the in-plane dynamic impact loadings. The t/R ratio affects the nominal porosity. Different deformation modes lead to different laws of dynamic densification strain on the impact velocities. These are discussed in detail. The empirical expressions of dynamic densification strain are obtained by using least-square fitting. Acknowledgements This work is supported by National Natural Science Foundation of China (10925212, 90916027), 973 Program (2011CB610304), 111 Project (B07050), Natural Science Foundation of Shaanxi Province (2010JQ1011), Natural Foundation of Education Department of Shaanxi (11JK0534), and Initial Research Foundation of Shaanxi University of Science and Technology for Ph.D. (BJ12-15). References [1] Chung J, Waas AM. The in-plane elastic properties of circular cell and elliptical cell honeycombs. Acta Mech 1999;144(1–2):29–42. [2] Papka SD, Kyriakides S. In-plane crushing of a polycarbonate honeycomb. Int J Solid Struct 1998;35(3–4):239–67. [3] Chung J, Waas AM. Compressive response and failure of circular cell polycarbonate honeycombs under inplane uniaxial stresses. J Eng Mater Technol 1999;121(4):494–502. [4] Papka SD, Kyriakides S. Biaxial crushing of honeycombs—Part 1. Experiments. Int J Solid Struct 1999;36(29):4367–96. [5] Papka SD, Kyriakides S. In-plane biaxial crushing of honeycombs: Part II. Analysis. Int J Solid Struct 1999;36(29):4397–423. [6] Chung J, Waas AM. In-plane biaxial crush response of polycarbonate honeycombs. J Eng Mech 2001;127(2):180–93.

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