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Ballistic limit evaluation for impact of cylindrical projectiles on honeycomb panels
ARTICLE IN PRESS Thin-Walled Structures 48 (2010) 55–61
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Ballistic limit evaluation for impact of cylindrical projectiles on honeycomb panels Gholam Hossein Liaghat a, Ali Alavi Nia b,, Hamid Reza Daghyani c, Mojtaba Sadighi c a
Mechanical Engineering Department, Tarbiat Modarres University, Tehran, Iran Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran c Mechanical Engineering Department, Amir Kabir University, Tehran, Iran b
a r t i c l e in fo
abstract
Article history: Received 31 July 2008 Received in revised form 4 May 2009 Accepted 17 July 2009 Available online 21 August 2009
In this paper the perforation of honeycomb panels under impact of cylindrical projectiles is studied and the ballistic limit velocity is determined by the energy method. The results show that the increase of panel thickness, compression and shear yield strengths and cell wall thickness increase the ballistic limit velocity, while an increase in the size of the cell and projectile mass decrease it. These results are in good agreement with available experimental data. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Impact Crushing Ballistic limit Honeycomb
1. Introduction Honeycomb is a thin walled structure constructed from open cells with very thin walls. The cells are usually hexagonal; however they can have any other geometries. Honeycombs are extensively used as energy absorbers, because of their individual properties. Sandwich panels with honeycomb core are used in transportation and aerospace industries because of their high stiffness and specific strength. Honeycombs are impacted in different situations by projectiles, and the impact damage varies from indentation of sandwich skins to complete perforation of the panel. Therefore, the study of structural behaviour of honeycombs is a high demand of advanced industries. The first study of honeycomb crush was done by McFarland [1] who proposed a semi empirical model to predict the crushing strength of cellular structures with hexagonal cells. This model was then developed by other researchers considering the bending and extensional deformations. Wierzbicki [2] introduced an angle element to predict the crushing load of honeycombs under quasi static axial loading. Abramowicz and Wierzbicki [3] modified this model for axisymmetric and asymmetric deformation modes. The honeycombs response to quasi static and impact loads were experimentally studied by other researchers [4–6]. Perforation of sandwich panels with honeycomb core by projectiles were studied analytically by Hoo Fatt and Park [7] using a semi Corresponding author.
E-mail address: [email protected] (A.A. Nia). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.07.008
empirical model for the core response. The crushing behaviour of honeycomb panels under quasi static axial loading was later studied by Liaghat et al. [8] modifying the Wierzbicki model. The dynamic crushing of honeycomb panels impacted by cylindrical blunt projectiles was also investigated evaluating the minimum velocity for densification of panels [9]. Alavi Nia et al. [10] studied the ballistic resistance of aluminium honeycomb panels against the cylindrical steel projectiles experimentally. In the present study, an analytical model is introduced to predict the ballistic limit of metallic honeycombs.
2. Fundamental assumptions Since the perforation mechanism in honeycomb is very complicated; a structural model is developed based on the following assumptions:
the honeycomb behaviour is rigid, perfectly plastic, the projectile is a rigid, blunt cylinder, the projectile diameter is greater than the cell size, the cell walls after impact are first torn, then densification occurs followed by the shear and exit of the plug, and the tearing of cell walls occurs on a circle with a diameter of 1.5 times the projectile diameter and continues to a depth of 0.7 panel’s thickness (because the damaged zone diameter is equal to 1.5 times of the projectile diameter [5]).
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3. Energy dissipation during perforation In addition to internal energy consumed in folding (Ei), there are other mechanisms which participate in energy dissipation as follows:
the necessary energy for cell walls tearing (Et), plug shearing energy (Esp), plug kinetic energy (EKp), (the plug moves ahead of the projectile with the same velocity as the projectile).
and for one hexagonal cell (Fig. 2) which is composed of four such units the dissipated energy in complete crushing will be: b H2 ð5Þ Ei ¼ 4E ¼ M0 134:4H þ 75:40C þ 76:48 b h
Defining Ai as: A1 ¼ 134:4 A2 ¼ 75:40
ð6Þ
A3 ¼ 76:48
Eq. (5) will be written as follows: b H2 Ei ¼ M0 A1 H þ A2 C þ A3 b h
These energies are determined by the following methods.
ð7Þ
3.1. Evaluation of Ei Wierzbicki used the angle element (Fig. 1) for analysis of crushing of aluminium honeycomb panels [2]. Based on his analysis plastic energy which is dissipated in deformation fields has three components:
E1 which is due to extension of a toroidal surface (with larger
If the area of deformed region after impact is considered to be equal to nA times the cell area, the total energy consumed in folding process is given by: b H2 ð8Þ Eit ¼ nA M0 A1 H þ A2 C þ A3 b h where nA is defined as follows:
and smaller radii a and b, respectively) and is equal to Hb E1 ¼ 33:6M0 ðEq: ð5:1Þ in ½2Þ h
nA ¼ ð1Þ
M0, H and h are fully plastic moment of section, folding half wavelength and cell wall thickness, respectively. E2 which is due to movement of horizontal hinge lines and is equal to: E2 ¼ 6pCM0 ðEq: ð5:2Þ in ½2Þ
ð2Þ
where C is the cell edge.
E3 which is due to movement of inclined hinge lines and is equal to: H2 ðEq: ð5:3Þ in ½2Þ E3 ¼ 19:12M0 b
deformed region area ðp=4Þð1:5 DÞ2 ¼ Ac unit cell area
where D is the projectile diameter and Ac is the area of one unit cell (Fig. 2) which is equal to: pffiffiffi ð9Þ Ac ¼ 3S2 S is the cell size (Fig. 2). Honeycomb panel surface consists of rectangles and each rectangle enclosing one unit cell. Eq. (9) gives the area of this rectangle. Therefore: nA ¼ 1:02ðD=SÞ2
ð10Þ
ð3Þ
Therefore, the total dissipated energy for one unit composed of two angle elements (Fig. 1) is equal to sum of these three components, thus: b H2 ð4Þ E ¼ M0 33:6H þ 6pC þ 19:12 b h
3.2. Evaluation of Et Considering Es, Aw and nw as the fracture energy per unit area of honeycomb material, the section area of a torn wall and the number of torn walls, respectively, we will have: Et ¼ nw Aw Es
Fig. 1. Typical folding element composed of two angle elements.
ð11Þ
Fig. 2. Area of one unit cell.
ARTICLE IN PRESS G.H. Liaghat et al. / Thin-Walled Structures 48 (2010) 55–61
where nw is equal to [11]: D nw ¼ 7:5 S
ð12Þ
Derivation of Eq. (12) is described further in appendix. Since the effective crushing distance by densification is about 0.7 times the panel thickness (t) [12], then Aw is: Aw ¼ 0:7th
ð13Þ
the shear stress achieves the shear strength of honeycomb material, ty. The resulting plug is approximately a cylinder which its diameter is equal to 1.5D. The sheared area of plug is equal to nwhtp (where tp is the plug length and is equal to 0.3t). Therefore, dissipated energy due to plug shear is approximately equal to the product of average shear force ð12ty nw htp Þ and the distance travelled (tp). Consequently it can be written as: Esp ¼
1 ty nw htp2 2
ð15Þ
Substituting Eq. (12) into Eq. (15), it yields:
Consequently, Eq. (11) is given by: Et ¼ 5:25ðDth=SÞEs
57
ð14Þ
Esp ¼ 0:3375ðty Dht 2 =SÞ
ð16Þ
3.3. Evaluation of Esp
3.4. Evaluation of EKp
Target- projectile system and exiting plug from honeycomb are shown in Fig. 3. Fig. 4 shows this system in different stages of perforation process. The shearing failure of cell walls occurs when
Considering V¯ as the average velocity of plug during motion and neglecting friction dissipation during perforation, we may write: EKp ¼
1 2 mp V 2
ð17Þ
where mp is the plug mass and is equal to: mp ¼ r Vp ¼ r
p 4
ð1:5DÞ2 t ¼ 1:77ðrD2 tÞ
ð18Þ
r is honeycomb density and Vp is plug volume. If the residual velocity of the projectile is assumed to be zero, then the ballistic limit velocity, V50, is equal to initial velocity of the projectile. The reduction of the velocity of the projectile from the initial velocity, V, to the complete stop is considered to be linear. Therefore, V¯ is the average velocity during motion. The movement path of plug is the sum of the plug and the projectile lengths, and the distance from impact position till complete exit of projectile from target, x, is the sum of the panel thickness and the projectile length, (l), (Fig. 4). Therefore: x¼lþt
ð19Þ
The initial velocity of the plug (V1p) is considered the same as the projectile velocity, (V). From Fig. 4 one may write: V1p l þ tp l þ tp ¼ ) V1p ¼ V V lþt lþt
ð20Þ
Fig. 3. Target–projectile system and exiting plug.
Since the final velocity of the plug is zero, the average velocity of the plug, V¯p, is obtained from: 1 V l þ tp ð21Þ V p ¼ V1p ¼ 2 2 lþt
Substituting Eqs. (18) and (21) in Eq. (17) yields: 2 0:7 EKp ¼ 0:22rtD2 1 V2 1 þ l=t
ð22Þ
4. Energy balance The energy balance is written as follows: Ep ¼ Eit þ Et þ Esp þ EKp
ð23Þ
the projectile’s initial kinetic energy is equal to: Fig. 4. Target–projectile system geometry in different stages of perforation process.
Ep ¼
1 mV 2 2
ð24Þ
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where m is the projectile mass. Substituting Eqs. (8), (14), (16), (22) and (24) in Eq. (23) results in: 1 b H2 n þ 5:25ðDth=SÞEs mV 2 ¼ M0 nA A1 H þ A 2 C þ A 3 b 2 h 2 0:7 þ0:3375ðty Dht 2 =SÞ þ 0:22rtD2 1 V2 ð25Þ 1 þ ðl=tÞ where n is the folds number during crushing. Eq. (25) can be written as: V2 b H2 nA5 þ A6 þ A7 fm A4 g ¼ A1 H þ A2 C þ A3 ð26Þ M0 b h where parameters A4–A7 be given by: l þ 0:3t A4 ¼ 0:44rD2 t lþt
ðDth=SÞEs M0
A7 ¼ 0:675
ðty Dht 2 =SÞ M0
ð27Þ
1 m A4
Eq. (26) is written as: V2 b H2 nA5 þ A6 þ A7 ¼ a A1 H þ A2 C þ A3 M0 b h
@f b H2 lt A5 2 ¼ 0 ¼ 0 ) a A1 H þ A2 C þ A3 b @n h n
ð38Þ
Solving Eqs. (35)–(38) simultaneously, one can obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A2 A3 Ch2 b¼ A21 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A22 2 H¼ C h A1 A3 t 2
ð39Þ
ð40Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 A1 A3 A22 C 2 h
ð41Þ
Using Eq. (6), these relations may be written as: ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 b ¼ 0:683 Ch2 ¼ 0:569 Sh2 H ¼ 0:821
Defining a as:
a¼
ð37Þ
n¼
A5 ¼ 2:04ðD=SÞ2 A6 ¼ 10:5
@f t ¼ 0 ) 2H ¼ 0 @l n
ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 C 2 h ¼ 0:569 S2 h ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 C 2 h ¼ 0:879t= S2 h
ð28Þ
n ¼ 0:609t=
ð29Þ
Substituting Eqs. (42) and (43) in Eq. (29) results in: ( )0:5 rffiffiffi pffiffiffiffiffiffiffiffiffiffi 3 C þ A6 þ A7 V50 ¼ M0 a 137:78tA5 h
ð42Þ ð43Þ ð44Þ
ð45Þ
The fully plastic moment M0 is equal to: 5. Calculation of ballistic limit
M0 ¼
s0 h2
ð46Þ
4
Assuming: 2H ¼
t n
ð30Þ
and using Lagrange multipliers, the function f is considered as follows: f ¼ z lg
Considering a constant flow stress, s0 and substituting Eqs. (46) and (29) in Eq. (45) yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 58:50t s0 D2 ðh=SÞ5=3 þ 10:5ðDthEs =SÞ V50 ¼ m A4
ð31Þ
þ0:675ðty Dht 2 =SÞ
o0:5
ð47Þ
or: b H2 t f ¼ a A1 H þ A2 C þ A3 nA5 þ A6 þ A7 l 2H b h n
ð32Þ
where l is the Lagrange multiplier and g and z are defined as follows, respectively: g ¼ 2H
z¼
t n
V2 M0
ð33Þ
ð34Þ
To minimise f, the following conditions must be satisfied: @f b H ð35Þ ¼ 0 ) a A1 þ 2A3 nA5 2l ¼ 0 @H h b @f H H2 ¼ 0 ) a A1 A3 2 nA5 ¼ 0 @b h b
By concentrating on the contribution of each term of Eq. (47) to ballistic limit values, it is seen that the effect of A4 on ballistic limit is not important, because the ratio A4/m for test numbers 1 and 2 (reported in [5]), are equal to 0.002 and 0.001, respectively. Thus neglecting A4 only causes 0.2% variation in the ballistic limit velocity. Note that A4 in Eq. (47) indicates to the plug mass which in turn shows the small effect of plug kinetic energy on ballistic limit. Therefore, with enough accuracy, one may use the following equation instead of Eq. (47): 1 n V50 ¼ pffiffiffiffiffi 58:50t s0 D2 ðh=SÞ5=3 þ 10:5ðDthEs =SÞ m o0:5 ð48Þ þ0:675ðty Dht 2 =SÞ and mass of the projectile is:
ð36Þ
m¼
p 4
D2 lrp
ð49Þ
ARTICLE IN PRESS G.H. Liaghat et al. / Thin-Walled Structures 48 (2010) 55–61
where rp is density of the projectile. Combining Eqs. (48) and (49) results in: ( )0:5 1 5=3 2 V50 ¼ þ 13:37ðthEs =DSÞ þ 0:86ðty ht =DSÞ ½74:48t s0 ðh=SÞ rp l ð50Þ
6. Comparison of theoretical and experimental results To compare the analytical results of the present study with the available experimental data the studies of Goldsmith and Louie [5] and Alavi Nia et al. [10] are used. Goldsmith and Louie [5] measured the ballistic limit of honeycomb panels impacted with cylindrical and spherical projectiles. The specifications for two types of test specimens with cylindrical projectiles, are listed in Table 1. Both specimens are 5052-aluminum with yield and ultimate strengths and panel thickness of 255 and 290 MPa and 19.05 mm, respectively. The shear strength of material is 165 MPa [13], and the fracture energy per unit area of it is 10,000 J/square m [14]. The predicted values for ballistic limit with the measured one are compared in Table 2. Alavi Nia et al. [10] obtained experimentally the ballistic limit velocities of four types of 5052-aluminum honeycomb panels
59
impacted with cylindrical projectiles which their mass, diameter and length are 12 g, 8.7 and 26.1 mm, respectively. The specifications of these panels are listed in Table 3 and their mechanical properties are as indicated in previous paragraph. The calculated and the measured ballistic limit velocities are listed in Table 4. The results show that the present model can predict the ballistic limit velocity of aluminium honeycomb panels with an acceptable accuracy. The reason for differences between the predicted and measured values are explained as:
Using simplified assumptions especially assumption 5 in
Section 2, which is based on experimental observations and vary from one test to another. Approximation in calculation of plastic deformation energy, because the folding mechanism in honeycomb is very complicated and somewhat irregular whereas it is assumed here that the folds occur in symmetric mode. Neglecting energy dissipation for deletion of bonds between honeycomb cell walls. Neglecting the rate effects.
In Fig. 5 the relative contribution of energy dissipation in folding, shearing, tearing and kinetic energy of plug for two cases of Table 1 is shown. It is clear from this figure that folding and
Table 1 Target and projectile data for tests [5]. Test number
Honeycomb specification
Honeycomb density (kg/ m3)
Projectile diameter (mm)
Projectile length (mm)
Projectile mass (gr)
Cell size (mm)
Cell wall thickness (mm)
1 2
3.175-0.0254-72.1 6.35-0.0254-36.8
72.1 36.8
6.32 6.32
19.05 19.05
4.66 4.66
3.175 6.35
0.0254 0.0254
Table 2 Comparison of experimental [5] and analytical (Eq. (50)) ballistic limit velocities. Test number
Ballistic limit velocity (m/s)
1 2
Difference percentage
Analytical (Eq. (50))
Experimental [5]
35. 20 21. 81
32.6 23.6
8 7.6
Table 3 Honeycomb panels specifications. Honeycomb specification
Cell size (mm)
Cell walls thickness (mm)
Density (kg/m3)
Panel thickness (mm)
3.175-0.05-129.6 4.76-0.05-91.2 4.76-0.076-129.6 4.76-0.038-70.4
3.175 4.76 4.76 4.76
0.05 0.05 0.076 0.038
129.6 91.2 129.6 70.4
19.05 25.4 12.7 9.52
Table 4 Comparison between calculated and experimental V50 [10]. Honeycomb specification
3.175-0.05-129.6 4.76-0.05-91.2 4.76-0.076-129.6 4.76-0.038-70.4
Projectile
Projectile shape
D (mm)
L/D
8.7
3
Cylindrical (blunt nose)
Ballistic limit (m/s)
DV50 (%)
Experimental
Calculated [1]
47.56 40.83 36.5 19
50.7 44.25 40.1 19.8
+6.6 +8.3 +9.8 +4.2
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Fig. 5. Comparison of contribution of different mechanisms in energy dissipation.
shearing mechanisms dissipate more than 97% of the projectile energy; whereas tearing dissipates only about 2% of that and the contribution of plug kinetic energy is not considerable and can be neglected. Furthermore, increasing the cell size results in decrease of the folding role and increasing the shearing role in energy dissipation; because Eit is approximately proportional to S5/3 whereas Esp is proportional to S1.
Fig. 6. Geometric presentation of intersection of damaged zone with honeycomb panel cell-walls.
Table 5 Evaluation of k for Eq. (51).
7. Conclusions Number
This paper introduces an analytical method to determine the ballistic limit velocity of metallic honeycombs impacted by cylindrical projectiles. This method is based on the assumption that the total kinetic energy of the projectile is dissipated in folding and crushing of honeycomb, tearing of cell walls, forming and shearing of the plug and its exit from the target. The ballistic limit velocity determined by Eq. (50) shows the following results:
An increase in panel strength in compression and shear and its
fracture energy per unit area increase the ballistic limit velocity, which is a rational result. Increase in panel thickness causes an increase in number of folds and this in turn increases crushing energy and the ballistic limit. Increase in wall thickness, means that the fully plastic moment is increased, therefore, the plastic deformation energy and consequently the ballistic limit velocity will increase. Enlarging the cell size results in decreasing the number of deformed cell walls and the necessary energy for plastic deformation; therefore, it decreases the ballistic limit velocity. Increasing the projectile mass, decreases the necessary velocity for a specified amount of energy needed for perforation; therefore, decreases the ballistic limit velocity. Comparing three terms of Eq. (50) shows that for type one specimen of Table 1 as an example, folding, plug shearing and tearing of cell walls consume about 64%, 39% and 2% of the projectile’s kinetic energy, respectively. This means that the folding and tearing mechanisms have the most and the least effects among the failure mechanisms of perforation of honeycomb respectively.
Appendix A graphical method is used for determination of number of cell walls which are torn due to impact of the projectile. Fig. 6 shows a honeycomb panel with hexagonal cells. In this figure the circles show the damaged zones which their diameter are 1.5 times the
(D/S)
Position of circle centre A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 4 6 8 2 3 4 6 8 2 3 4 6 8 2 3 4 6 8
B
C
nw
K
12 14 21 30 32 8 16 18 32 36 10 14 18 32 34 12 12 20 30 40
6 4.7 5.3 5 5.3 4 5.3 4.5 5.3 3.5 5 4.7 4.5 5.3 5.5 6 4 5 5 5
D
* * * * * * * * * * * * * * * * * * * *
projectile diameter. By drawing circles with diameters equal to 2S, 3S, 4S, 6S and 8S, the number of torn cell walls are determined. In Fig. 6 only circles with a diameter equal to 2S are shown. Circles A, B, C and D show damage zones for impact of projectiles which their centres are at the corner, centre and middles of two edges of honeycomb cell, respectively. It is assumed that the number of torn walls nw to be proportional to the ratio of (1.5D/S), then: nw ¼ kð1:5D=SÞ
ð51Þ
The number of nw from Fig. 6 for five different ratios of (1.5D/S) are counted and the results are listed in Table 5. It is clear from this table that the average value of k for 20 cases is equal to 4.995 (ffi5). Therefore, Eq. (51) yields:
nw ¼ 7:5ðD=SÞ
ð52Þ
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References [1] McFarland RK. Hexagonal cell structures under post-buckling axial load. AIAA Journal 1963;1(6):1380–5. [2] Wierzbicki T. Crushing analysis of metal honeycombs. International Journal of Impact Engineering 1983;1(2):157–74. [3] Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet metal columns. ASME Journal of Applied Mechanics 1989;56:113–20. [4] Wu E, Jiang W. Axial crush of metallic honeycombs. International Journal of Impact Engineering 1997;19(5/6):439–56. [5] Goldsmith W, Louie DL. Axial perforation of aluminum honeycombs by projectiles. International Journal of Solid and Structures 1995;32(8/ 9):1017–46. [6] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. International Journal of Impact Engineering 1992;12(2):241–62. [7] Hoo Fatt MS, Park KS. Perforation of honeycomb sandwich plates by projectiles. Composites, Part A: Applied Science and Manufacturing 2000;31:889–99.
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[8] Liaghat GH, Sadighi M, Daghyani HR, Alavi Nia A. Crushing of metallic honeycombs under axial quasi static loads. Tehran University Journal 2003;37(1):145–56. [9] Liaghat GH, Daghyani HR, Sadighi M, Alavi Nia A. Dynamic crushing of honeycomb panels under impact of cylindrical projectiles. Amir Kabir University Journal 2003;14(53):68–79. [10] Alavi Nia A, Razavi SB, Majzoobi GH. Ballistic limit determination of aluminum honeycombs– experimental study. Materials Science and Engineering 2008;488:273–80. [11] Alavi Nia A. Dynamic Crushing of Honeycomb Panels Under Impact of Cylindrical Projectiles. Internal Report. Tarbiat Modarres University; 2001. [12] Abramowicz W. The effective crushing distance in axially compressed thinwalled metal columns. International Journal of Impact Engineering 1983;1(3):309–17. [13] Hatch JE. Aluminum: Properties and Physical Metallurgy. ASM 1984;358. [14] Atkins AG, Mai YW. Elastic and Plastic Fracture: Metals, Polymers, Ceramics, Composites, Biological Materials. Chichester, Newyork: Ellis Horwood, Halsted; 1985.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Ballistic limit evaluation for impact of cylindrical projectiles on honeycomb panels Gholam Hossein Liaghat a, Ali Alavi Nia b,, Hamid Reza Daghyani c, Mojtaba Sadighi c a
Mechanical Engineering Department, Tarbiat Modarres University, Tehran, Iran Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran c Mechanical Engineering Department, Amir Kabir University, Tehran, Iran b
a r t i c l e in fo
abstract
Article history: Received 31 July 2008 Received in revised form 4 May 2009 Accepted 17 July 2009 Available online 21 August 2009
In this paper the perforation of honeycomb panels under impact of cylindrical projectiles is studied and the ballistic limit velocity is determined by the energy method. The results show that the increase of panel thickness, compression and shear yield strengths and cell wall thickness increase the ballistic limit velocity, while an increase in the size of the cell and projectile mass decrease it. These results are in good agreement with available experimental data. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Impact Crushing Ballistic limit Honeycomb
1. Introduction Honeycomb is a thin walled structure constructed from open cells with very thin walls. The cells are usually hexagonal; however they can have any other geometries. Honeycombs are extensively used as energy absorbers, because of their individual properties. Sandwich panels with honeycomb core are used in transportation and aerospace industries because of their high stiffness and specific strength. Honeycombs are impacted in different situations by projectiles, and the impact damage varies from indentation of sandwich skins to complete perforation of the panel. Therefore, the study of structural behaviour of honeycombs is a high demand of advanced industries. The first study of honeycomb crush was done by McFarland [1] who proposed a semi empirical model to predict the crushing strength of cellular structures with hexagonal cells. This model was then developed by other researchers considering the bending and extensional deformations. Wierzbicki [2] introduced an angle element to predict the crushing load of honeycombs under quasi static axial loading. Abramowicz and Wierzbicki [3] modified this model for axisymmetric and asymmetric deformation modes. The honeycombs response to quasi static and impact loads were experimentally studied by other researchers [4–6]. Perforation of sandwich panels with honeycomb core by projectiles were studied analytically by Hoo Fatt and Park [7] using a semi Corresponding author.
E-mail address: [email protected] (A.A. Nia). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.07.008
empirical model for the core response. The crushing behaviour of honeycomb panels under quasi static axial loading was later studied by Liaghat et al. [8] modifying the Wierzbicki model. The dynamic crushing of honeycomb panels impacted by cylindrical blunt projectiles was also investigated evaluating the minimum velocity for densification of panels [9]. Alavi Nia et al. [10] studied the ballistic resistance of aluminium honeycomb panels against the cylindrical steel projectiles experimentally. In the present study, an analytical model is introduced to predict the ballistic limit of metallic honeycombs.
2. Fundamental assumptions Since the perforation mechanism in honeycomb is very complicated; a structural model is developed based on the following assumptions:
the honeycomb behaviour is rigid, perfectly plastic, the projectile is a rigid, blunt cylinder, the projectile diameter is greater than the cell size, the cell walls after impact are first torn, then densification occurs followed by the shear and exit of the plug, and the tearing of cell walls occurs on a circle with a diameter of 1.5 times the projectile diameter and continues to a depth of 0.7 panel’s thickness (because the damaged zone diameter is equal to 1.5 times of the projectile diameter [5]).
ARTICLE IN PRESS 56
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3. Energy dissipation during perforation In addition to internal energy consumed in folding (Ei), there are other mechanisms which participate in energy dissipation as follows:
the necessary energy for cell walls tearing (Et), plug shearing energy (Esp), plug kinetic energy (EKp), (the plug moves ahead of the projectile with the same velocity as the projectile).
and for one hexagonal cell (Fig. 2) which is composed of four such units the dissipated energy in complete crushing will be: b H2 ð5Þ Ei ¼ 4E ¼ M0 134:4H þ 75:40C þ 76:48 b h
Defining Ai as: A1 ¼ 134:4 A2 ¼ 75:40
ð6Þ
A3 ¼ 76:48
Eq. (5) will be written as follows: b H2 Ei ¼ M0 A1 H þ A2 C þ A3 b h
These energies are determined by the following methods.
ð7Þ
3.1. Evaluation of Ei Wierzbicki used the angle element (Fig. 1) for analysis of crushing of aluminium honeycomb panels [2]. Based on his analysis plastic energy which is dissipated in deformation fields has three components:
E1 which is due to extension of a toroidal surface (with larger
If the area of deformed region after impact is considered to be equal to nA times the cell area, the total energy consumed in folding process is given by: b H2 ð8Þ Eit ¼ nA M0 A1 H þ A2 C þ A3 b h where nA is defined as follows:
and smaller radii a and b, respectively) and is equal to Hb E1 ¼ 33:6M0 ðEq: ð5:1Þ in ½2Þ h
nA ¼ ð1Þ
M0, H and h are fully plastic moment of section, folding half wavelength and cell wall thickness, respectively. E2 which is due to movement of horizontal hinge lines and is equal to: E2 ¼ 6pCM0 ðEq: ð5:2Þ in ½2Þ
ð2Þ
where C is the cell edge.
E3 which is due to movement of inclined hinge lines and is equal to: H2 ðEq: ð5:3Þ in ½2Þ E3 ¼ 19:12M0 b
deformed region area ðp=4Þð1:5 DÞ2 ¼ Ac unit cell area
where D is the projectile diameter and Ac is the area of one unit cell (Fig. 2) which is equal to: pffiffiffi ð9Þ Ac ¼ 3S2 S is the cell size (Fig. 2). Honeycomb panel surface consists of rectangles and each rectangle enclosing one unit cell. Eq. (9) gives the area of this rectangle. Therefore: nA ¼ 1:02ðD=SÞ2
ð10Þ
ð3Þ
Therefore, the total dissipated energy for one unit composed of two angle elements (Fig. 1) is equal to sum of these three components, thus: b H2 ð4Þ E ¼ M0 33:6H þ 6pC þ 19:12 b h
3.2. Evaluation of Et Considering Es, Aw and nw as the fracture energy per unit area of honeycomb material, the section area of a torn wall and the number of torn walls, respectively, we will have: Et ¼ nw Aw Es
Fig. 1. Typical folding element composed of two angle elements.
ð11Þ
Fig. 2. Area of one unit cell.
ARTICLE IN PRESS G.H. Liaghat et al. / Thin-Walled Structures 48 (2010) 55–61
where nw is equal to [11]: D nw ¼ 7:5 S
ð12Þ
Derivation of Eq. (12) is described further in appendix. Since the effective crushing distance by densification is about 0.7 times the panel thickness (t) [12], then Aw is: Aw ¼ 0:7th
ð13Þ
the shear stress achieves the shear strength of honeycomb material, ty. The resulting plug is approximately a cylinder which its diameter is equal to 1.5D. The sheared area of plug is equal to nwhtp (where tp is the plug length and is equal to 0.3t). Therefore, dissipated energy due to plug shear is approximately equal to the product of average shear force ð12ty nw htp Þ and the distance travelled (tp). Consequently it can be written as: Esp ¼
1 ty nw htp2 2
ð15Þ
Substituting Eq. (12) into Eq. (15), it yields:
Consequently, Eq. (11) is given by: Et ¼ 5:25ðDth=SÞEs
57
ð14Þ
Esp ¼ 0:3375ðty Dht 2 =SÞ
ð16Þ
3.3. Evaluation of Esp
3.4. Evaluation of EKp
Target- projectile system and exiting plug from honeycomb are shown in Fig. 3. Fig. 4 shows this system in different stages of perforation process. The shearing failure of cell walls occurs when
Considering V¯ as the average velocity of plug during motion and neglecting friction dissipation during perforation, we may write: EKp ¼
1 2 mp V 2
ð17Þ
where mp is the plug mass and is equal to: mp ¼ r Vp ¼ r
p 4
ð1:5DÞ2 t ¼ 1:77ðrD2 tÞ
ð18Þ
r is honeycomb density and Vp is plug volume. If the residual velocity of the projectile is assumed to be zero, then the ballistic limit velocity, V50, is equal to initial velocity of the projectile. The reduction of the velocity of the projectile from the initial velocity, V, to the complete stop is considered to be linear. Therefore, V¯ is the average velocity during motion. The movement path of plug is the sum of the plug and the projectile lengths, and the distance from impact position till complete exit of projectile from target, x, is the sum of the panel thickness and the projectile length, (l), (Fig. 4). Therefore: x¼lþt
ð19Þ
The initial velocity of the plug (V1p) is considered the same as the projectile velocity, (V). From Fig. 4 one may write: V1p l þ tp l þ tp ¼ ) V1p ¼ V V lþt lþt
ð20Þ
Fig. 3. Target–projectile system and exiting plug.
Since the final velocity of the plug is zero, the average velocity of the plug, V¯p, is obtained from: 1 V l þ tp ð21Þ V p ¼ V1p ¼ 2 2 lþt
Substituting Eqs. (18) and (21) in Eq. (17) yields: 2 0:7 EKp ¼ 0:22rtD2 1 V2 1 þ l=t
ð22Þ
4. Energy balance The energy balance is written as follows: Ep ¼ Eit þ Et þ Esp þ EKp
ð23Þ
the projectile’s initial kinetic energy is equal to: Fig. 4. Target–projectile system geometry in different stages of perforation process.
Ep ¼
1 mV 2 2
ð24Þ
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where m is the projectile mass. Substituting Eqs. (8), (14), (16), (22) and (24) in Eq. (23) results in: 1 b H2 n þ 5:25ðDth=SÞEs mV 2 ¼ M0 nA A1 H þ A 2 C þ A 3 b 2 h 2 0:7 þ0:3375ðty Dht 2 =SÞ þ 0:22rtD2 1 V2 ð25Þ 1 þ ðl=tÞ where n is the folds number during crushing. Eq. (25) can be written as: V2 b H2 nA5 þ A6 þ A7 fm A4 g ¼ A1 H þ A2 C þ A3 ð26Þ M0 b h where parameters A4–A7 be given by: l þ 0:3t A4 ¼ 0:44rD2 t lþt
ðDth=SÞEs M0
A7 ¼ 0:675
ðty Dht 2 =SÞ M0
ð27Þ
1 m A4
Eq. (26) is written as: V2 b H2 nA5 þ A6 þ A7 ¼ a A1 H þ A2 C þ A3 M0 b h
@f b H2 lt A5 2 ¼ 0 ¼ 0 ) a A1 H þ A2 C þ A3 b @n h n
ð38Þ
Solving Eqs. (35)–(38) simultaneously, one can obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A2 A3 Ch2 b¼ A21 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A22 2 H¼ C h A1 A3 t 2
ð39Þ
ð40Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 A1 A3 A22 C 2 h
ð41Þ
Using Eq. (6), these relations may be written as: ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 b ¼ 0:683 Ch2 ¼ 0:569 Sh2 H ¼ 0:821
Defining a as:
a¼
ð37Þ
n¼
A5 ¼ 2:04ðD=SÞ2 A6 ¼ 10:5
@f t ¼ 0 ) 2H ¼ 0 @l n
ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 C 2 h ¼ 0:569 S2 h ffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffi 3 3 C 2 h ¼ 0:879t= S2 h
ð28Þ
n ¼ 0:609t=
ð29Þ
Substituting Eqs. (42) and (43) in Eq. (29) results in: ( )0:5 rffiffiffi pffiffiffiffiffiffiffiffiffiffi 3 C þ A6 þ A7 V50 ¼ M0 a 137:78tA5 h
ð42Þ ð43Þ ð44Þ
ð45Þ
The fully plastic moment M0 is equal to: 5. Calculation of ballistic limit
M0 ¼
s0 h2
ð46Þ
4
Assuming: 2H ¼
t n
ð30Þ
and using Lagrange multipliers, the function f is considered as follows: f ¼ z lg
Considering a constant flow stress, s0 and substituting Eqs. (46) and (29) in Eq. (45) yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 58:50t s0 D2 ðh=SÞ5=3 þ 10:5ðDthEs =SÞ V50 ¼ m A4
ð31Þ
þ0:675ðty Dht 2 =SÞ
o0:5
ð47Þ
or: b H2 t f ¼ a A1 H þ A2 C þ A3 nA5 þ A6 þ A7 l 2H b h n
ð32Þ
where l is the Lagrange multiplier and g and z are defined as follows, respectively: g ¼ 2H
z¼
t n
V2 M0
ð33Þ
ð34Þ
To minimise f, the following conditions must be satisfied: @f b H ð35Þ ¼ 0 ) a A1 þ 2A3 nA5 2l ¼ 0 @H h b @f H H2 ¼ 0 ) a A1 A3 2 nA5 ¼ 0 @b h b
By concentrating on the contribution of each term of Eq. (47) to ballistic limit values, it is seen that the effect of A4 on ballistic limit is not important, because the ratio A4/m for test numbers 1 and 2 (reported in [5]), are equal to 0.002 and 0.001, respectively. Thus neglecting A4 only causes 0.2% variation in the ballistic limit velocity. Note that A4 in Eq. (47) indicates to the plug mass which in turn shows the small effect of plug kinetic energy on ballistic limit. Therefore, with enough accuracy, one may use the following equation instead of Eq. (47): 1 n V50 ¼ pffiffiffiffiffi 58:50t s0 D2 ðh=SÞ5=3 þ 10:5ðDthEs =SÞ m o0:5 ð48Þ þ0:675ðty Dht 2 =SÞ and mass of the projectile is:
ð36Þ
m¼
p 4
D2 lrp
ð49Þ
ARTICLE IN PRESS G.H. Liaghat et al. / Thin-Walled Structures 48 (2010) 55–61
where rp is density of the projectile. Combining Eqs. (48) and (49) results in: ( )0:5 1 5=3 2 V50 ¼ þ 13:37ðthEs =DSÞ þ 0:86ðty ht =DSÞ ½74:48t s0 ðh=SÞ rp l ð50Þ
6. Comparison of theoretical and experimental results To compare the analytical results of the present study with the available experimental data the studies of Goldsmith and Louie [5] and Alavi Nia et al. [10] are used. Goldsmith and Louie [5] measured the ballistic limit of honeycomb panels impacted with cylindrical and spherical projectiles. The specifications for two types of test specimens with cylindrical projectiles, are listed in Table 1. Both specimens are 5052-aluminum with yield and ultimate strengths and panel thickness of 255 and 290 MPa and 19.05 mm, respectively. The shear strength of material is 165 MPa [13], and the fracture energy per unit area of it is 10,000 J/square m [14]. The predicted values for ballistic limit with the measured one are compared in Table 2. Alavi Nia et al. [10] obtained experimentally the ballistic limit velocities of four types of 5052-aluminum honeycomb panels
59
impacted with cylindrical projectiles which their mass, diameter and length are 12 g, 8.7 and 26.1 mm, respectively. The specifications of these panels are listed in Table 3 and their mechanical properties are as indicated in previous paragraph. The calculated and the measured ballistic limit velocities are listed in Table 4. The results show that the present model can predict the ballistic limit velocity of aluminium honeycomb panels with an acceptable accuracy. The reason for differences between the predicted and measured values are explained as:
Using simplified assumptions especially assumption 5 in
Section 2, which is based on experimental observations and vary from one test to another. Approximation in calculation of plastic deformation energy, because the folding mechanism in honeycomb is very complicated and somewhat irregular whereas it is assumed here that the folds occur in symmetric mode. Neglecting energy dissipation for deletion of bonds between honeycomb cell walls. Neglecting the rate effects.
In Fig. 5 the relative contribution of energy dissipation in folding, shearing, tearing and kinetic energy of plug for two cases of Table 1 is shown. It is clear from this figure that folding and
Table 1 Target and projectile data for tests [5]. Test number
Honeycomb specification
Honeycomb density (kg/ m3)
Projectile diameter (mm)
Projectile length (mm)
Projectile mass (gr)
Cell size (mm)
Cell wall thickness (mm)
1 2
3.175-0.0254-72.1 6.35-0.0254-36.8
72.1 36.8
6.32 6.32
19.05 19.05
4.66 4.66
3.175 6.35
0.0254 0.0254
Table 2 Comparison of experimental [5] and analytical (Eq. (50)) ballistic limit velocities. Test number
Ballistic limit velocity (m/s)
1 2
Difference percentage
Analytical (Eq. (50))
Experimental [5]
35. 20 21. 81
32.6 23.6
8 7.6
Table 3 Honeycomb panels specifications. Honeycomb specification
Cell size (mm)
Cell walls thickness (mm)
Density (kg/m3)
Panel thickness (mm)
3.175-0.05-129.6 4.76-0.05-91.2 4.76-0.076-129.6 4.76-0.038-70.4
3.175 4.76 4.76 4.76
0.05 0.05 0.076 0.038
129.6 91.2 129.6 70.4
19.05 25.4 12.7 9.52
Table 4 Comparison between calculated and experimental V50 [10]. Honeycomb specification
3.175-0.05-129.6 4.76-0.05-91.2 4.76-0.076-129.6 4.76-0.038-70.4
Projectile
Projectile shape
D (mm)
L/D
8.7
3
Cylindrical (blunt nose)
Ballistic limit (m/s)
DV50 (%)
Experimental
Calculated [1]
47.56 40.83 36.5 19
50.7 44.25 40.1 19.8
+6.6 +8.3 +9.8 +4.2
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Fig. 5. Comparison of contribution of different mechanisms in energy dissipation.
shearing mechanisms dissipate more than 97% of the projectile energy; whereas tearing dissipates only about 2% of that and the contribution of plug kinetic energy is not considerable and can be neglected. Furthermore, increasing the cell size results in decrease of the folding role and increasing the shearing role in energy dissipation; because Eit is approximately proportional to S5/3 whereas Esp is proportional to S1.
Fig. 6. Geometric presentation of intersection of damaged zone with honeycomb panel cell-walls.
Table 5 Evaluation of k for Eq. (51).
7. Conclusions Number
This paper introduces an analytical method to determine the ballistic limit velocity of metallic honeycombs impacted by cylindrical projectiles. This method is based on the assumption that the total kinetic energy of the projectile is dissipated in folding and crushing of honeycomb, tearing of cell walls, forming and shearing of the plug and its exit from the target. The ballistic limit velocity determined by Eq. (50) shows the following results:
An increase in panel strength in compression and shear and its
fracture energy per unit area increase the ballistic limit velocity, which is a rational result. Increase in panel thickness causes an increase in number of folds and this in turn increases crushing energy and the ballistic limit. Increase in wall thickness, means that the fully plastic moment is increased, therefore, the plastic deformation energy and consequently the ballistic limit velocity will increase. Enlarging the cell size results in decreasing the number of deformed cell walls and the necessary energy for plastic deformation; therefore, it decreases the ballistic limit velocity. Increasing the projectile mass, decreases the necessary velocity for a specified amount of energy needed for perforation; therefore, decreases the ballistic limit velocity. Comparing three terms of Eq. (50) shows that for type one specimen of Table 1 as an example, folding, plug shearing and tearing of cell walls consume about 64%, 39% and 2% of the projectile’s kinetic energy, respectively. This means that the folding and tearing mechanisms have the most and the least effects among the failure mechanisms of perforation of honeycomb respectively.
Appendix A graphical method is used for determination of number of cell walls which are torn due to impact of the projectile. Fig. 6 shows a honeycomb panel with hexagonal cells. In this figure the circles show the damaged zones which their diameter are 1.5 times the
(D/S)
Position of circle centre A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 4 6 8 2 3 4 6 8 2 3 4 6 8 2 3 4 6 8
B
C
nw
K
12 14 21 30 32 8 16 18 32 36 10 14 18 32 34 12 12 20 30 40
6 4.7 5.3 5 5.3 4 5.3 4.5 5.3 3.5 5 4.7 4.5 5.3 5.5 6 4 5 5 5
D
* * * * * * * * * * * * * * * * * * * *
projectile diameter. By drawing circles with diameters equal to 2S, 3S, 4S, 6S and 8S, the number of torn cell walls are determined. In Fig. 6 only circles with a diameter equal to 2S are shown. Circles A, B, C and D show damage zones for impact of projectiles which their centres are at the corner, centre and middles of two edges of honeycomb cell, respectively. It is assumed that the number of torn walls nw to be proportional to the ratio of (1.5D/S), then: nw ¼ kð1:5D=SÞ
ð51Þ
The number of nw from Fig. 6 for five different ratios of (1.5D/S) are counted and the results are listed in Table 5. It is clear from this table that the average value of k for 20 cases is equal to 4.995 (ffi5). Therefore, Eq. (51) yields:
nw ¼ 7:5ðD=SÞ
ð52Þ
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[8] Liaghat GH, Sadighi M, Daghyani HR, Alavi Nia A. Crushing of metallic honeycombs under axial quasi static loads. Tehran University Journal 2003;37(1):145–56. [9] Liaghat GH, Daghyani HR, Sadighi M, Alavi Nia A. Dynamic crushing of honeycomb panels under impact of cylindrical projectiles. Amir Kabir University Journal 2003;14(53):68–79. [10] Alavi Nia A, Razavi SB, Majzoobi GH. Ballistic limit determination of aluminum honeycombs– experimental study. Materials Science and Engineering 2008;488:273–80. [11] Alavi Nia A. Dynamic Crushing of Honeycomb Panels Under Impact of Cylindrical Projectiles. Internal Report. Tarbiat Modarres University; 2001. [12] Abramowicz W. The effective crushing distance in axially compressed thinwalled metal columns. International Journal of Impact Engineering 1983;1(3):309–17. [13] Hatch JE. Aluminum: Properties and Physical Metallurgy. ASM 1984;358. [14] Atkins AG, Mai YW. Elastic and Plastic Fracture: Metals, Polymers, Ceramics, Composites, Biological Materials. Chichester, Newyork: Ellis Horwood, Halsted; 1985.