Ballistic impact of GLARE™ fiber–metal laminates

Composite Structures 61 (2003) 73–88 www.elsevier.com/locate/compstruct

Ballistic impact of GLAREe fiber–metal laminates Michelle S. Hoo Fatt a

a,*

, Chunfu Lin a, Duane M. Revilock Jr. b, Dale A. Hopkins

b

Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA b NASA Glenn Research Center, 21000 Brookpark Road, Cleveland, OH 44135, USA

Abstract Analytical solutions to predict the ballistic limit and energy absorption of fully clamped GLARE panels subjected to ballistic impact by a blunt cylinder were derived. The analytical solutions were based on test results from NASA Glenn. The ballistic limit was found through an iterative process such that the initial kinetic energy of the projectile would equal the total energy dissipated by panel deformation, delamination/debonding and fracture. The transient deformation of the panel as shear waves propagate from the point of impact was obtained from an equivalent mass–spring system, whereby the inertia and stiffness depend on the shear wave speed and time. Predictions of the ballistic limit from the resulting non-linear differential equation were within 13% of the test data. The deformation energy due to bending and membrane accounted for most of the total energy absorbed (84–92%), with the thinner panels absorbing a higher percentage of deformation energy than the thicker panels. Energy dissipated in delamination represented 2–9% of the total absorbed energy, with the thinner panels absorbing a lower percentage of delamination energy than the thicker panels. About 7% of the total energy was attributed to tensile fracture energy of the glass/epoxy and aluminum. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: GLARE; Aluminum–glass/epoxy laminate; Global deformation; Delamination; Tensile fracture; Energy partition

1. Introduction This paper is concerned with the development of an analytical model to predict the energy absorption and ballistic resistance of GLARE 5, a fiber–metal laminate consisting of alternating layers of 2024-T3 aluminum alloy and S2-glass/epoxy laminates. The objective of this paper is to use ballistic impact tests results from NASA Glenn to develop an analytical model for ballistic perforation of GLARE 5. The analytical solutions will provide insights into damage mechanisms and energy absorption of this particular fiber–metal laminate and can be used as a design tool for fiber–metal laminates in general. It has been known for some time that a fiber–metal laminate with combined metal and composite, can enhance energy absorption and increase the ballistic limit of either the metal or composite from which it is made from [1–5]. Vlot and Krull [3] attributed the increase in energy absorption of GLARE to the significant increase of tensile strength of glass fibers at very high-strain rate *

Corresponding author. Tel.: +1-330-972-6308; fax: +1-330-9726027. E-mail address: [email protected] (M.S. Hoo Fatt).

since such an increase would not be as significant in bare aluminum. In Ref. [3], Vlot used a non-linear elastic impact model to calculate the transient deflection and impact force under low-velocity conditions but did not address delamination and fracture of GLARE that would occur under a ballistic impact situation. In this paper, we consider ballistic impact of GLARE 5 panels. In addition to finding the ballistic limit of GLARE, we will calculate the energy absorbed by various mechanisms during ballistic perforation of GLARE. Partitioning the energy during impact perforation allows one to compare the contribution of various energy absorbing mechanisms and provides valuable information for improving the ballistic performance of the fiber–metal laminates. While the main source of energy absorption in a thin aluminum plate is plastic deformation, it is often delamination in a laminated composite [6,7]. Except for those with toughened resins, many composite laminates are brittle when compared to ductile metals and have relatively low-interlaminar shear strength, which causes them to delaminate easily upon impact. The delamination process is very favorable in ballistic impact since it allows the panel to deform in a more efficient membrane state. Delamination also reduces the panel bending

0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00036-9

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M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

Nomenclature a span of the panel Adeb debonding area Adel delamination area Aij membrane stiffness cs shear wave speed Dij bending stiffness E YoungÕs modulus Ep plastic modulus Eav average panel stiffness Ebm bending and membrane energy Edeb debonding energy Edel delamination energy Eij laminate stiffness Ep petaling energy et energy density for tensile failure Et tensile fracture energy Etot total energy F impact load Gav average shear stiffness of GLARE Gij laminate shear stiffness GIIC Mode II interlaminar fracture toughness h laminate thickness hAl aluminum ply thickness hG=E S2-glass/epoxy ply thickness ILSS interlaminar shear strength Kb bending stiffness Km membrane stiffness L length of indenter or projectile me equivalent mass of panel M0 projectile mass Mx0 , My0 fully plastic bending moments Mxy0 fully plastic twisting moment N the total number of layers nt the number of layers undergoing tensile fracture

stiffness and allows higher transverse deformations to emanate away from the projectile. Thus a more likely mode of failure for a laminated panel is one involving large global panel deformation and tensile fracture rather than a very localized deformation and/or transverse shear fracture. During high-velocity impact of a plate, three-dimensional stress waves propagate through the thickness and laterally from the point of impact. If the contact duration is much higher than the transition time for throughthickness waves, the response is dominated by twodimensional, transverse shear waves. The transverse shear waves propagate laterally until they become bending and/or membrane waves. These waves may never reach the boundary of the panel before projectile perforation. This is in contrast to low-velocity impact whereby

np Nx0 , Ny0 Nxy0 P P0 r R Rdeb Rdel R0 t U V0 Vi Vri V50 w W x, y z D e ecr eAl eG=E mij m P qav q rf r0 n

the number of layers undergoing petaling fully plastic membrane forces fully plastic in-plane shear force static load load causing plastic flow in aluminum radial coordinate projectile radius debonding radius delamination radius equivalent projectile radius time strain energy projectile initial velocity velocity at the end of phase i reduced velocity at phase i ballistic limit transverse plate deflection work done by external force in-plane coordinates through-thickness coordinate central transverse deflection strain fracture strain fracture strain for aluminum fracture strain for S2 glass/epoxy PoissonÕs ratio of laminate PoissonÕs ratio of panel total potential energy average density of GLARE density tensile fracture strength yield strength deformation zone

transverse shear waves have reached and reflected off the boundary of the panel many times before panel perforation. Olsson [8] identified the above-mentioned highand low-velocity impact scenarios in terms of the panel response dominated by dilatational waves, flexural waves and quasi-static loading. Since all of the damaged GLARE panels from NASA Glenn underwent extensive panel deformation before projectile perforation, we assume that the impact response is dominated by two-dimensional transverse wave propagation, i.e., we neglect the effect of stress waves propagating through-thickness. In particular, we propose to use an equivalent mass–spring system with time varying inertia and stiffness to calculate the impact response of the panel. Requiring dynamic equilibrium and conservation of energy will allow us to calculate the

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

ballistic limit of the GLARE. High-strain rate material properties will be approximated from data found in the open literature. Finally, we will compare analytical predictions of the ballistic limit and energy absorption with NASA test results.

2. Test results from NASA Glenn The Ballistic Impact Lab at NASA Glenn performed a series of ballistic impact tests on GLARE 5 and 2024T3 aluminum panels. The GLARE panels, experimental procedure and results from these tests are described below.

75

The configuration of GLARE and aluminum panels that were tested is described in Table 1. Panels of GLARE 5 having thickness of 0.1, 0.06 and 0.076 in. were purchased from Aviation Equipment Structures Inc., Costa Mesa CA. With the exception of the 0.076 in.-thick GLARE, all of the panels had aluminum and S2-glass/epoxy layers which were 0.02 in. thick. The 0.076 in.-thick GLARE had 0.02 in.-thick S2-glass/ epoxy layers but 0.012 in.-thick aluminum layers. Each S2-glass/epoxy layer had a layup of [0/90]s . The 2-layer systems shown in rows 4 and 5 consisted of two 0.100 and 0.076 in. panels, respectively, bonded together with epoxy. 2.2. Experimental procedure

2.1. Materials The GLARE laminates consist of thin high-strength aluminum alloy sheets bonded together in an autoclave with strong fiber adhesive prepregs. The prepregs are glass fibers in epoxy adhesive. The GLARE 5 grade is the result of an effort focused on optimizing the fiber metal laminate concept for the use in impact-prone structures. Specifically, GLARE 5 consists of alternating layers of 2024-T3 aluminum alloy and S2-glass/epoxy laminates. The panels were originally designed for use in the Boeing 777 impact resistant bulk cargo door and are being considered for jet engine containment cases.

The impact tests were conducted at room temperature using the gas gun shown in Fig. 1(a). The gas gun consisted of a pressure vessel with a pressure capacity of 10 MPa (1500 psi) and a volume of 2250 cc, connected via a high-speed solenoid valve to a stainless steel hollow barrel,
2 m long. The barrel had an outside diameter of 2.54 cm (1 in.) and an inside diameter of 1.28 cm (0.505 in.). Helium was used as the propellant. Ballistic impact tests were conducted on 17.8 cm (7 in.) square flat panels. The panels were clamped on all sides in a fixture with a 15.24 cm  15.24 cm (6 in.  6 in.) aperture as shown in Fig. 1(b). The projectiles were

Table 1 Description of GLARE and aluminum test panels Material

Al/glass–epoxy layup

Total panel thickness (in.)

Al thickness (in.)

GLARE-5 GLARE-5 GLARE-5 GLARE-5, 2-layer (0.100) GLARE-5, 2-layer (0.076) Aluminum 2024-T3 Aluminum 2024-T3 Aluminum 2024-T3 Aluminum 2024-T3

Al/GE/Al/GE/Al AL/GE/AL Al/GE/Al/GE/Al (Al/GE/Al/GE/Al)2 (Al/GE/Al/GE/Al)2 – – – –

0.100 0.060 0.076 0.208 0.160 0.020 0.063 0.125 0.250

0.020 0.020 0.012 0.020 0.012 0.020 0.063 0.125 0.250

Fig. 1. NASA GlennÕs 50 caliber impact lab setup: (a) 50 caliber gas gun and (b) test fixture (15.24 cm square aperture).

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M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88 240

Lasers

220 200 Ballistic Limit (m/s)

Specimen Projectile

Gun Barrel

180 160 140 120 100 80 60

2024 Al

GLARE

Bonded GLARE

40 20 0 0

Fig. 2. Schematic of experimental setup.

20

40

60

80

100

120

140

160

180

2

Areal Weight Density (N/m )

Fig. 3. Ballistic limit vs. areal weight density of 2024-T3 aluminum and GLARE 5 panels at room temperature.

flat-faced Ti–6Al–4V, AMS 4928 cylinders, 2.54 cm (1 in.) long with a 1.27 cm (0.5 in.) diameter and with a hardness of 36–37 HRC. A radius of 0.8 mm (0.0032 in.) was machined on the edge of the impacting face. The mass of the projectiles ranged between 14.05 and 14.20 g. As shown in Fig. 2, the velocity of the projectile prior to impact was measured as it exited the barrel by the interruption of two laser beams a known distance apart. The ballistic limit defined as the lowest velocity for complete penetration or perforation, was calculated by testing between 7 and 10 panels at each thickness of GLARE and between 7 and 14 panels of the 2024-T3 aluminum (see Table 2). The ballistic limit was achieved by impacting these panels at different velocities to narrow the range between perforation and no perforation. Generally, the velocity range between perforation and no perforation was relatively small, and there was little overlap (cases in which some projectiles did not perforate at higher velocities than others that did perforate). The highest speed without perforation is given in the last column of Table 2 to indicate the small scatter in the data. This means that in most cases there was little ambiguity in determining the ballistic limit. It is estimated that the error in the ballistic limit is within 6 m/s (3%) [9].

2.3. Results and discussion A plot of the ballistic limit as a function of areal weight density is given for the GLARE panels and 2024T3 aluminum in Fig. 3. The test results show that the ballistic limit for the 0.06 in.- and 0.076 in.-thick GLARE (solid triangles) was about 15% higher than 2024-T3 Al (solid diamonds). However, the ballistic limit of the 0.1 in.-thick GLARE was about the same as that for the aluminum. Thicker GLARE panels were made by bonding panels together (see two-layer bonded GLARE data or solid squares in Fig. 3). The ballistic limit for the GLARE was slightly higher than the aluminum for the two-layer bonded 0.1 in.-thick GLARE but not for the two-layer bonded 0.076 in.-thick GLARE, which had thinner layers of aluminum. The mixed set of results suggested that in some cases the GLARE could be used to raise the ballistic limit, while in other cases it cannot. One would need to develop simple analytical models that can be used to explain energy absorption of GLARE in order to explain the disparity in the results.

Table 2 GLARE impact test results Material

Panel thickness (mm)

Areal weight density (N/m2 )

No. of panels tested

Ballistic limit (lowest recorded perforation speed) (m/s)

Highest speed recorded without perforation (m/s)

GLARE 5 GLARE 5 GLARE 5 GLARE 5 2-ply (0.076 in.) GLARE 5 2-ply (0.100 in.) Aluminum 2024 Aluminum 2024 Aluminum 2024 Aluminum 2024

1.53 1.93 2.59 4.02 5.3 0.4 1.6 3.2 6.4

36.65 43.41 61.15 91.92 124.46 10.89 42.92 86.53 179.44

9 7 8 6 7 14 13 7 11

136 151 156 185 212 67 131 196 213

136 150 155 179 206 70 132 192 217

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

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Fig. 5. Geometry of GLARE panel impacted by blunt cylinder.

3. Problem formulation

Fig. 4. Cross-section showing projectile penetrating into two-layer bonded 0.100 in.-thick GLARE panel.

Fig. 4 shows the cross-sections of the two-layer bonded 0.1 in.-thick GLARE panel as the projectile penetrated through it. There is extensive delamination between plies in the glass/epoxy layers, but the bond between the glass/epoxy and aluminum seems to be intact. The distal layers undergo very large deflection and membrane stretching while the projectile has already perforated the frontal layers. Some of the plies in the glass/epoxy layers of the distal layers have already broken within the aluminum layers. Also note that the top layer of aluminum buckled since the adjacent glass/ epoxy layer would have fractured while the aluminum is deforming. An elastic unloading wave would have then induced compressive forces on the adjacent aluminum. The perforation energy of the GLARE is primarily attributed to fracture energy due to transverse shear and delamination and bending/membrane energy dissipation. However, one can also identify friction and buckling as possible sources of energy dissipation.

Consider a fully clamped GLARE panel with dimensions a  a  h, as shown in Fig. 5. It is subjected to projectile impact by a rigid, blunt cylinder of radius R, mass M0 and velocity V0 . The panel consists of N alternating layers of aluminum and glass/epoxy, where the outermost layers are aluminum and the aluminum and glass/epoxy layers are hAl and hG=E , respectively. The cross-section of the perforated GLARE panel suggests that the kinetic energy of the projectile can be dissipated in global deformation, including panel bending and membrane stretching; extensive delamination within the glass/epoxy plies or debonding between the aluminum and glass/epoxy layers; and tensile fracture of the glass/epoxy and aluminum. Petals indicate tensile fracture of the aluminum layers on the distal side of the panel. A simple energy balance is thus given by Etot ¼ Ebm þ Edel þ Edeb þ Et þ Ep

ð1Þ

where Ebm is the strain energy and plastic work in bending and membrane stretching; Edel is the delamination energy within glass/epoxy layers; Edeb is the energy dissipated in debonding aluminum and glass/epoxy layers; Et is the tensile fracture energy of the glass/epoxy; Ep is the fracture energy in petaling of aluminum layers. Closed-form expressions for each of these energy components will be developed in the following sections.

4. Global panel deformation Equivalent mass–spring systems are often used to find the dynamic response of panels subjected to low-velocity

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M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

Mo

Mo + me (ξ)

cs

ξ (t) = cst

∆

∆(t)

K (ξ )

Fig. 6. Wave propagation model using an equivalent mass–spring system.

motion using equivalent mass and spring would then have to be adjusted for these events. Our analytical model for high-velocity impact will incorporate both the wave propagation and stationary phases. To find the dynamic response of the panel in the wave propagation phase, we model the system as the equivalent simple mass and spring shown in Fig. 6, where D is the central deflection of the panel, and the inertia of the projectile and panel and spring stiffness depend on the extent of deformation n as shear waves emanate away from the point of impact. If the shear wave travels with a constant velocity cs , then n ¼ cs t. When the shear wave reaches the boundary, we set n ¼ a=2. In order to find functions for the inertia and stiffness, we first consider the static indentation response of GLARE. 4.1. Static response 4.1.1. Load–deflection Actual stress–strain curves from static tensile tests for 2024-T3 Al and unidirectional S2 glass/epoxy are shown together in Fig. 7. Specific (static) material properties for the S2 glass/epoxy and aluminum are taken from Refs. [12,13] and given in Tables 3 and 4, respectively. The

2000 1800 2024-T3 Al S2 Glass/Epoxy

1600 1400

stress, σ (MPa)

impact loading, i.e., when the load duration is long enough for transverse shear waves to reach and reflect off the boundary. Equivalent inertia for the projectile and plate and the equivalent spring stiffness are calculated by assuming a velocity field to evaluate the effective kinetic energy of the plate and using the static load–indentation response of the panel, respectively. The velocity and deformation fields occur over the entire span of the panel. The resulting equation of motion for the projectile and plate is solved to find the transient deflection of the projectile/plate at the point of impact. Equivalent mass–spring models have been successfully used to determine low-velocity impact response of plates [10] and even sandwich panels [11]. In some high-velocity impact situations, shear waves propagating from the point of impact may not have time to reach the boundary before the projectile perforates the plate. Equivalent inertia and stiffness parameters cannot be calculated over the entire span of the panel as they are in the low-velocity impact scenario. Such highvelocity impact problems involve wave propagation, whereby the inertia and stiffness depend on the extent of deformation, which varies with time. If the shear wave reaches the boundary before the projectile perforates the plate, the deformation zone becomes fixed and the equivalent mass and stiffness can be calculated for the fixed span of the panel. One can therefore envision two consecutive phases of motion for high-velocity impact: a wave propagation phase followed by a stationary wave phase. Careful observation of a section of a GLARE panel perforated at the ballistic limit suggests that there may have been time for the wave to reach the boundary because the damaged zone extended almost to the edge of the panel. However, this damaged profile corresponds only to final fracture of the aluminum layers. The panel may have spent a great deal of time in the wave propagation stage before total perforation. Furthermore, delamination and fracture of the glass/epoxy layers would have occurred prior to fracture of aluminum layers. Since delamination and fracture involve energy dissipation, they would change the kinetic energy or momentum of the projectile/plate. The equation of

1200 1000 800 600 400 200 0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

strain, ε

Fig. 7. Actual stress–strain curves for 2024-T3 Al and unidirectional S2 glass/epoxy.

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88 Table 3 Ply properties of unidirectional S2-glass/epoxy prepreg h ¼ 0:127 mm: ply thickness q ¼ 1980 kg/m3 : density E11 ¼ 52 GPa: longitudinal stiffness E22 ¼ 17 GPa: transverse stiffness E33 ¼ 17 GPa: through-thickness stiffness G12 ¼ 7 GPa: in-plane shear modulus G23 ¼ 7 GPa: out-of-plane shear modulus G13 ¼ 7 GPa: out-of-plane shear modulus m12 ¼ 0:25: PoissionÕs ratio m23 ¼ 0:32: PoissionÕs ratio m13 ¼ 0:25: PoissonÕs ratio s12 ¼ 75 MPa: in-plane shear strength s13 ¼ s23 ¼ 77 MPa: transverse shear strength rf ¼ 1779 MPa: tensile strength eG=E ¼ 0:0325: tensile failure strain et
0:5rf ef ¼ 28:9 MJ/m 3 : energy density to tensile failure

Table 4 Material properties of 2024-T3 aluminum q ¼ 2780 kg/m3 : density E ¼ 72:2 GPa: YoungÕs modulus Ep ¼ 1:3 GPa: plastic modulus r0 ¼ 350 MPa: yield strength rf ¼ 500 MPa: ultimate tensile strength eAl ¼ 0:18: tensile fracture strain (ductility)

aluminum is elastic–plastic with high ductility (18%), while the glass/epoxy is elastic and brittle (3.25% fracture strain). In order to obtain analytical expressions for the load–indentation response of the GLARE, we idealize the aluminum as rigid, linear strain hardening and the glass/epoxy as linear elastic (see Fig. 8). The yield strength and plastic modulus for the aluminum are denoted r0 and Ep . Because of the rigid plastic assumption, the panel will not deflect until the load reaches a finite value that causes plastic flow. Stress distributions corresponding to fully plastic bending and membrane states are shown separately in Fig. 9(a) and (b). Although the actual stress distribution at yield results from an interaction of bending moments and membrane forces, we will assume a limited interaction curve bounded by the fully plastic bending moment and membrane force, i.e., there is no interaction between bending and stretching. We will use the principle of minimum potential energy to calculate the load–deflection response of GLARE. Since the fully clamped panels undergo very large deflections, several times the panel thickness, before fracture, membrane forces would be induced and become progressively larger with increasing deflections. The total strain energy of the panel would consist of both bending and membrane energy. To express the total strain energy of the panel, we assume that deflections are large, strains are finite and that the in-plane deformations are negligible compared to the transverse

79

σG/E

E11

σA1 σo

Ep

ε Fig. 8. Idealized material behavior for 2024-T3 Al and unidirectional S2 glass/epoxy.

deflections. This allows us to write the strain energy in terms of transverse deflections only. For the symmetric GLARE laminate, the strain energy due to bending is given by  2   2  Z a=2 Z a=2  ow ow Ub ¼ 4 Mx0 þ My0 2 ox oy 2 R R  2  ow þ Mxy0 2 dx dy oxoy Z a=2 Z a=2 (  2 2 ow D11 þ2 ox2 R R  2 2  2  2  ow ow ow þ 2D12 þ D22 oy 2 ox2 oy 2  2 2 ) ow dx dy ð2Þ þ 4D66 oxoy where Mx0 , My0 and Mxy0 are the fully plastic bending and twisting moments and Dij is the bending stiffness. Expressions for Mx0 , My0 , and Mxy0 are derived by taking moments about the neutral surface of the plate. For instance, consider a typical GLARE panel with a total of N aluminum and glass–epoxy layers. The layup consists of top and bottom layers of aluminum so that there are ðN þ 1Þ=2 aluminum and ðN 1Þ=2 glass– epoxy layers. Also assume that the thickness of every aluminum and glass/epoxy layer are hAl and hG=E , respectively. Fig. 9(a) shows the stress distribution in the aluminum when the panel is in pure bending. From this, we calculate that the fully plastic bending moments pffiffiffi in the panel are Mx0 ¼ My0 ¼ M0 and Mxy0 ¼ M0 = 3, where 8 2 for N ¼ 3 > 0 hAl < 2r 1 2 r h þ 2r h ðh þ h Þ 0 Al Al G=E i M0 ¼ 4 0 hAl > :  1 þ 2 þ 3 þ    þ ðN 1Þ for N P 5 4

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M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88 2024-T3 Al S2 Glass/Epoxy

σo

σo

σo

σo

σo

σo

σo

N.A.

σo

σo

σo

σo

(a)

N.A.

(b)

Fig. 9. Stress distribution in aluminum: (a) fully plastic bending moment and (b) fully plastic membrane force.

This formula applies to standard GLARE panels and should be modified for the bonded GLARE panels since there will be two aluminum layers at the neutral surface instead of one. Assuming non-linear strain–displacement relations given by von Karman (see Ref. [14]) and neglecting inplane deformations, we obtain the following expression for the membrane strain energy of a symmetric GLARE panel: Z a=2Z a=2 ( "  2 2 # 1 ow Um ¼ 4 Nx0 2 ox2 R R "  2 #  ) 1 o2 w ow ow þ Ny0 þ Nxy0 2 dx dy 2 oy 2 ox oy (  4  4 Z Z 1 a=2 a=2 ow ow A11 þ A22 þ 2 R R ox oy  2  2 ) ow ow þ ð2A12 þ 4A66 Þ dx dy ð3Þ ox oy

deflection profile of the panel. Since we expect very large deformation or that the panel will be in a membrane state before fracture, we consider the following deformation profile, which closely resembles that of a stretched membrane:

where Nx0 ¼ Ny0 and Nxy0 are the fully plastic membrane and in-plane shear forces, Aij is the membrane stiffness. Note that the above expression is valid for analyses involving deformation of fully clamped plates but not for buckling of plates since the in-plane deformations are not negligible in this case. Expressions for Nx0 , Ny0 , and Nxy0 are derived by taking summation of membrane forces about the neutral surface. Thus using the corresponding stress diagrams for the typical GLARE panel shown in Fig. 9(b), one gets Nx0 ¼ Ny0 ¼ ðN 2þ1Þ r0 hAl and ffiffi r0 hAl . Again these expressions will be modNxy0 ¼ ðN2pþ1Þ 3 ified appropriately for bonded GLARE panels. The solution for the load–indentation response using the principle of minimum potential energy is very sensitive to the shape function that is used to describe the

and

wðx; yÞ ( ¼

D



D0 1

 2x 2 1 a



 2y 2 a

0 6 x 6 R; 0 6 y 6 R R 6 x 6 a=2; R 6 y 6 a=2 ð4Þ

2R 2 Þ. a

where D0 ¼ D=ð1 The above profile satisfies zero slope and deflection at boundaries, but non-zero slope at center. Substituting Eq. (4) into Eqs. (2) and (3) gives the following expressions for the bending and membrane energy 2

8ða 2RÞ 32ða 2RÞ ð2Mx0 þ 3Mxy0 ÞD þ 2 3a 45a4  ½9ðD11 þ D22 Þ þ 10D12 þ 80D66 D2

Ub ¼

Um ¼

2

2ða 2RÞ4 ða 2RÞ6 2 ð8N þ 15N ÞD þ x0 xy0 15a4 2205a8  ½1568ðA11 þ A22 Þ þ 2880ðA12 þ 2A66 Þ D4

ð5Þ

ð6Þ

The work done by the indentation load is W ¼ PD

ð7Þ

Minimizing P ¼ Ub þ Um W with respect to D gives P ¼ P0 þ ðKb þ Km1 ÞD þ Km2 D3 where 2

P0 ¼

8ða 2RÞ ð2Mx0 þ 3Mxy0 Þ 3a2

ð8Þ

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88 2

Kb ¼

81

2200

64ða 2RÞ ½9ðD11 þ D22 Þ þ 10D12 þ 80D66 45a2

2000 BONDED

1800

Km1

4ða 2RÞ4 ¼ ð8Nx0 þ 15Nxy0 Þ 15a6

Km2

4ða 2RÞ ¼ ½1568ðA11 þ A22 Þ þ 2880ðA12 þ 2A66 Þ 2205a8

1600 1400 P (N)

6

1200 1000

The first term of the right-hand side of Eq. (8) is the indentation load that would just cause plastic flow in the aluminum and is a consequence of our rigid-plastic assumption. The second linear term is partly due to bending and partly due to membrane forces (as a consequence of the rigid plastic assumption). The final term is due to the non-linear membrane effects. 4.1.2. Effect of debonding/delamination The load–indentation response of the 0.1 in.-thick GLARE panel is shown in Fig. 10. Since delamination within the glass/epoxy layers and debonding between the aluminum and glass/epoxy layers may alter the load– indentation response, we re-calculated the bending and membrane stiffness matrices for the panel assuming that each layer was unbonded and compared the load indentation response to the bonded case. The load– indentation curve for a completely debonded 0.1 in.-thick GLARE panel is also shown in Fig. 10. Notice that the bonded and unbonded responses are almost the same, i.e., the unbonded solution is only slightly lower than the bonded solution. This result seems odd at first since the fully bonded panel should be stiffer than an unbonded panel. However, this is only true when panel deflections are small (less than half of the panel thickness). When deflections are very large, membrane stretching resistance is much greater than bending resistance and since debonding does not alter the membrane stiffness of the panel, debonding has very little effect on the load–indentation response.

800 600 UNBONDED

400 200 0.0

0.5

1.0

1.5

2.0

2.5

∆ (mm)

Fig. 11. Load–indentation response of 0.1 in.-GLARE 5, considering only bending energy.

Fig. 11 shows the load–indentation response of the panel when deflections are small. The load–deflection curve was derived in Appendix A by using a cosine shape function that satisfies plate bending boundary conditions. Now one can see a significant difference in the limit load (intercept) and bending stiffness (slope) of the bonded and unbonded panels. 4.1.3. Effect of tensile fracture Tensile fracture would first occur when the tensile strain in the glass/epoxy reaches 3.25%. Since deflections are very large, we can ignore bending strains and assume that all of the glass/epoxy layers break at the same time when the membrane strain reaches 3.25%. After the glass/epoxy layers break, there is a sudden reduction in the panel stiffness. Both bending and membrane energy of the panel are calculated by setting the stiffness properties for the glass/epoxy equal to zero. Thus the bending and membrane stiffness are lower, but the form of the load–indentation relation remains the same. We thus distinguish two phases of response:

140

A. Before glass/epoxy breaks: 120

P (kN)

100

I P ¼ P0 þ ðKbI þ Km1 ÞD þ Km2 D3

Bonded

80

B. After glass/epoxy breaks:

60

II P ¼ P0 þ ðKbII þ Km1 ÞD þ Km2 D3

40

ð9Þ

ð10Þ

Unbonded

20

0 0

5

10

15

20

∆ (mm)

Fig. 10. Static load–indentation response for 0.1 in.-GLARE 5 panel.

The superscripts I and II denote quantities calculated before and after the glass/epoxy layers break, respectively. Note that there are no superscripts in P0 and Km1 because they are associated with plastic flow in the aluminum. The above expressions can be used to determine the load and energy at fracture once we determine

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M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

a critical deflection Dcr at which membrane strains approach the tensile fracture strain. The membrane strain in the panel is given by  2 1 ow ex ¼ ð11Þ 2 ox Thus an approximate strain at the edge of the projectile (x ¼ R, y ¼ 0) is given by e¼

8D2 ða 2RÞ

ð12Þ

2

The critical deflection that would cause tensile fracture can be given in terms of the fracture strain ecr rffiffiffiffiffi ecr Dcr ¼ ða 2RÞ ð13Þ 8 The above expression for Dcr can be substituted into Eqs. (9) or (10) to give the failure load. For instance, the failure load at which the glass/epoxy breaks is I D3G=E ð14Þ PG=E ¼ P0 þ ðKbI þ Km1 ÞDG=E þ Km2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where DG=E ¼ ða 2RÞ eG=E =8 is the critical displacement when the glass/epoxy breaks. The load at which the aluminum layers break is given by II D3Al ð15Þ PAl ¼ P0 þ ðKbII þ Km1 ÞDAl þ Km2 pffiffiffiffiffiffiffiffiffiffiffi where DAl ¼ ða 2RÞ eAl =8 is the critical displacement when the aluminum breaks. Furthermore, the bending and membrane energy absorbed before fracture is given by

Ebm ¼ P0 DG=E þ

ðKbI þ Km1 ÞD2G=E

2 þ P0 ðDAl DG=E Þ

þ þ

þ

I Km2 D4G=E

4

ðKbII þ Km1 ÞðD2Al D2G=E Þ 2 II Km2 ðD4Al D4G=E Þ 4

The response of the panel can then be found from the following equation of motion: € þ P0 ðtÞ þ ðKb ðnÞ þ Km1 ðnÞÞD ðM0 þ me ðnÞÞD þ Km2 ðnÞD3 ¼ 0

ð17Þ

where me ðnÞ ¼ phR2 qav þ

 6 4 R qav hn2 1 25 n

2

P0 ðnÞ ¼

8ðn RÞ ð2Mx0 þ 3Mxy0 Þ 3n2

Kb ðnÞ ¼

16ðn RÞ2 ½9ðD11 þ D22 Þ þ 10D12 þ 80D66 45n4

Km1 ðnÞ ¼

4ðn RÞ4 ð8Nx0 þ 15Nxy0 Þ 15n4

Km2 ðnÞ ¼

ðn RÞ ½1568ðA11 þ A22 Þ þ 2880ðA12 þ 2A66 Þ 2205n8

6

and n ¼ cs t. The solution for me is derived in Appendix B, while the expressions for the stiffness are obtained by setting a ¼ 2n in the static stiffness terms. If the shear wave reaches the boundaries, then n ¼ a=2 and P0 , Kb , Km1 and Km2 are then the same as those in Section 4.1.1. The shear wave reaches the panel boundary at tb ¼ a=ð2cs Þ. Since energy is dissipated during debonding/delamination and fracture, the above equation of motion must be modified. We assume that these events take place instantaneously when the contact force between the projectile and the panel equals that of the delamination load Pd and that there is a sudden loss of kinetic energy of the system at such time. The following section describes the energy loss due to delamination and tensile fracture of the glass/epoxy and aluminum.

5. Delamination ð16Þ The load at delamination/debonding Pd is approximated by

4.2. Impact response Pd2 ¼ The equivalent mass–spring system, shown with timevarying inertia and stiffness in Fig. 6, is used to model the dynamic response of the panel. The inertia and stiffness are expressed as time-varying functions by using the results from static indentation and setting the span of the panel equal to the distance traveled by the shear waves propagating from the point of impact, n. Shear wavespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi propagate with an average shear wave speed ffi cs ¼ Gav =qav , where Gav and qav are the average transverse shear modulus and density of the GLARE. When the distance traveled by the shear wave n is greater than 10 times the panel thickness, it becomes a bending wave.

8p2 Eav h3 GIIC 9ð1 m2 Þ

ð18Þ

where Eav is the average stiffness of the panel, PoissonÕs ratio is taken as m ¼ 0:3, and GIIC is the Mode II interlaminar shear fracture toughness of the glass/epoxy (delamination) or the bond between the aluminum and the glass/epoxy (debonding). Separate loads are calculated for delamination and debonding because values for the Mode II interlaminar shear fracture toughness are generally not the same. The delamination radius Rdel can be given in terms of the delamination load and the interlaminar shear stress of the glass/epoxy (ILSS)G=E

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

Rdel ¼

Pd 2phðILSSÞG=E

ð19Þ

Similarly, the debonding radius Rdeb is given in terms of the debonding load and the interlaminar shear stress between the glass/epoxy and the aluminun (ILSS)G=E–Al Rdeb ¼

Pd 2phðILSSÞG=E–Al

ð20Þ

83

8. Ballistic response of GLARE We distinguish two phases of motion that arise from different equations of motion before and after the glass/ epoxy layers break. Since the inertia and stiffness of system vary with time, the resulting differential equations must be solved numerically. Delamination causes a sudden loss of kinetic energy (or velocity) of the system. The following sections describe how the two phases of motion are affected by delamination and tensile fracture.

The energy due to delamination is given by G=E

Edel ¼ Adel GIIC

8.1. Phase I ð21Þ G=E

where Adel ¼ pR2del is the delamination area and GIIC is the Mode II interlaminar fracture toughness of the glass/ epoxy. Similarly, the debonding energy is G=E–Al

Edeb ¼ Adeb GIIC

ð22Þ G=E–Al

where Adeb ¼ pR2deb is the debonding area and GIIC is the Mode II interlaminar fracture toughness between glass/epoxy and aluminum.

The equation of motion before the glass/epoxy breaks is € þ P0 ðnÞ þ ðK I ðnÞ þ Km1 ðnÞÞD ðM0 þ me ðnÞÞD b I þ Km2 ðnÞD3 ¼ 0

I where KbI and Km2 are calculated from the Dij and Aij of all the glass/epoxy and aluminum layers. The initial conditions are

Dð0Þ ¼ 0; 6. Tensile fracture of glass/epoxy The tensile fracture energy of the glass/epoxy Et can be approximated by multiplying the fracture energy per unit volume (area under the stress–strain curve from a tensile test) by the volume of the hole left when the projectile passes through the laminate. Thus 2

Et ¼ nt et pR hG=E

ð23Þ

where nt is the number of glass/epoxy layers undergoing fracture and Et is the energy density to tensile fracture (toughness).

7. Petaling of aluminum As the projectile pierces through the aluminum, it bends back material around the periphery of the projectile. This phenomenon is petaling. We can approximate the plastic work in petaling Ep by assuming a hinge line equal to the circumference of the projectile and a bend angle of p=2. The total plastic work dissipated in bending of petals as the projectile passes through the entire panel is thus given by Ep ¼ n p

r0 p2 Rh2Al 4

ð24Þ

where np is the number of aluminum layers undergoing petaling.

ð25Þ

D_ ð0Þ ¼

M0 V0 ðM0 þ pR2 hqav Þ

The initial velocity of the panel is calculated from the resulting transfer of momentum at t ¼ 0. The contact force between the projectile and the panel is € F ðtÞ ¼ M0 D

ð26Þ

This contact force will eventually grow to cause debonding and delamination in the GLARE. Examination of the fractured GLARE panels showed that there was no evidence of debonding between the glass/epoxy and aluminum but significant delamination in the glass/epoxy. We therefore disregard debonding and consider only a sudden loss in kinetic energy of the system when delamination occurs. At t ¼ td , the contact force equals the delamination load given by Eq. (18) and there is a sudden loss of kinetic energy since energy is dissipated in Mode II interlaminar fracture. Denote velocity just before delamination as D_ ðtd Þ ¼ V1 and the reduced velocity right after delamination as Vr1 . Conservation of energy gives that 1 1 ðM0 þ me ÞV12 Edel ¼ ðM0 þ me ÞVr12 2 2

ð27Þ

where Edel is calculated from Eq. (21). The above equation is used to find a reduced velocity, which now becomes the initial velocity for the following sub-phase of motion. At t ¼ tG=E , the glass/epoxy breaks or D ¼ DG=E , where

84

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

eG=E ¼

8 2 > < 2DG=E2

ðn RÞ 2 > 8DG=E

:

ða 2RÞ

if n < a=2 2

ð28Þ

if n ¼ a=2

There are two cases in Eq. (28) because fracture of the glass/epoxy can occur before or after shear waves reach the boundary. There is another sudden drop in the velocity at this time since energy is dissipated in tensile fracture of the glass/epoxy. Denote the velocity just before the glass/epoxy breaks as D_ ðtG=E Þ ¼ V2 and the reduced velocity right after the glass/epoxy breaks as Vr2 . Another energy balance for finding Vr2 is as follows: 1 1 ðM0 þ me ÞV22 Et ¼ ðM0 þ me ÞVr22 2 2

ð29Þ

8.2. Phase II After the glass/epoxy breaks, a new equation of motion governs the system response: € þ P0 ðnÞ þ ðK II ðnÞ þ Km1 ðnÞÞD ðM0 þ me ðnÞÞD b II þ Km2 ðnÞD3 ¼ 0

ð30Þ

II where KbII and Km2 are the bending and membrane resistance that are calculated by setting material properties for the glass/epoxy equal to zero. The initial conditions for Eq. (30) are DðtG=E Þ ¼ DG=E , D_ ðtG=E Þ ¼ Vr2 . Motion continues until the aluminum breaks at t ¼ tAl or when D ¼ DAl and 8 2 < 2DAl 2 if n < a=2 ð31Þ eAl ¼ ðn RÞ 2 8D : Al 2 if n ¼ a=2 ða 2RÞ

Again fracture of the aluminum can occur before or after the wave reaches the boundary. The velocity of the system will be finite at this time, i.e., D_ ðtAl Þ ¼ V3 , since the kinetic energy of the system corresponding to this velocity should be equal to the petaling energy of the aluminum layers. Therefore, 1 Ep ¼ ðM0 þ me ÞV32 2

ð32Þ

sorbed by each mechanism is equal to the initial kinetic energy of projectile. Our preliminary calculations using static material properties for the glass/epoxy and aluminum in Tables 3 and 4, suggest that tensile strain rates range from 200 to 700 s 1 . Since both the glass/epoxy and aluminum are very rate-sensitive materials, we must introduce dynamic material properties for them. 9.1. High-strain rate material properties 9.1.1. S2 glass/epoxy The most extensive study on tensile high-strain rate behavior of S glass/epoxy is by Armenakas and Sciammarella [15]. Using an impact-loading machine propelled by explosives, Armenakas and Sciammarella found that the stiffness of unidirectional glass/epoxy laminates increases by about 50% when loaded to strain rates of about 500 s 1 . We therefore assume that Eij and Gij are 1.5 times that of the static values. We also assume that PoissonÕs ratios mij are the same as static values since rate effects in the longitudinal direction should be about the same in the transverse direction. Furthermore, we assume that as in E glass/epoxy [16], there would be a slight increase in the dynamic fracture strain to ef ¼ 4% and a 50% increase in the static energy density to tensile failure et ¼ 53:4 MJ/m 3 . Tsai et al. [17] recently performed dynamic delamination tests on S2 glass/epoxy and found that the dynamic Mode II fracture toughness for the S2 glass/epoxy is about the same as the static G=E Mode II fracture toughness GIIC ¼ 2 KJ/m2 . Likewise, the dynamic interlaminar shear strength for the S2 glass/ epoxy was assumed to be the same as the static interlaminar shear strength of E glass=epoxy ðILSSÞG=E ¼ 20 MPa [18] since the epoxy governs interlaminar properties in both the S2-glass/epoxy and E-glass/epoxy laminates. 9.1.2. 2024-T3 aluminum Under impact strain rates, the flow stress of the aluminum is relatively insensitive to strain rate, while its ductility decreases substantially. We assume that the dynamic flow stress of the aluminum is the same as the static flow stress, but the ductility of the aluminum was reduced from 18% to 5.7% [19]. 9.2. Transient response of 0.1 in.-thick GLARE

9. Results and discussion The equations of motion and required energy balance at the ballistic limit described in Section 8 are solved through an iterative scheme. An initial velocity must first be assumed in order to find deflections and velocity of the projectile and panel. The velocity of the system is then reduced instantaneously when delamination and fracture take place. The projectileÕs initial velocity is equal to the ballistic limit when the total energy ab-

The transient deflection and velocity of the 0.1 in.-thick GLARE panel are calculated and the results are shown in Fig. 12. The transient response of the strain at the edge of the projectile and the contact force between the projectile and panel are also shown in Figs. 13 and 14, respectively. Delamination occurred at td ¼ 0:003 ms and this led to a sudden drop in the velocity of about 3.41 m/s. It was found that the wave reached the boundary at tb ¼ 0:056 ms, which occurs during Phase I

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

160

0.02

0.04

td

0.06

0.08

0.10

Velocity

0.12

0.14

tG/E

tb

tAl

0.16 180 160 140

Velocity (m/s)

140 Deflection

120

120 100

100 I(a)

II

I(c)

I(b)

80

80

60

60

40

40

0.02

tG/E

td

tb

0.04

0.06

0.08

0.10

tAl 0.12

0.14

0 0.16

Time (ms)

Fig. 12. Transient variation of central deflection and velocity.

6

5

Strain (%)

4

3 I(a)

I(b)

I(c)

II

2

1

0 0.00

tb

td 0.02

0.04

0.06

t Al

t G/E 0.08 0.10 Time (ms)

0.12

0.14

wave reaches the boundary but this event is hardly noticeable in Fig. 12. The glass/epoxy breaks at tG=E ¼ 0:1 ms and there is another sudden loss of velocity (or kinetic energy) at this time. The contact force at this time shows a sudden load drop since the panel stiffness will be reduced when the glass/epoxy breaks. Finally, the aluminum breaks at about tAl ¼ 0:141 ms. 9.3. Energy partition

20

20 0 0.00

Deflection (10-1 mm)

0.00 180

85

The analytical model was used to predict similar response in the other GLARE panels and the ballistic limit was found to be within 13% of the test data, as shown in Fig. 15. Table 5 shows energy partition of the results. Most of the energy is absorbed in global panel deformation. Bending and membrane energy of the 0.06 in.-thick GLARE panel, the thinnest panel, accounts for 92% of the total energy, while they represent 84% of the total energy in the 2  0.1 in.-thick GLARE panel, thickest panel. This result emphasizes the use of thinner panels that would allow energy dissipation in the membrane stretching. Delamination energy accounts for 2% of the total energy absorbed in the 0.06 in.-thick GLARE but 9% of the total energy absorbed in the 2  0.1 in.-thick GLARE since there are more plies to delaminate in the thicker panels. Finally, the tensile fracture energy of the glass/epoxy and aluminum layers is only about 7% of the total energy absorbed during ballistic impact.

0.16

10. Concluding remarks

Fig. 13. Transient variation of strain at periphery of projectile.

28 26 24 22 Contact Force (kN)

20 18 16 14 12

I(a)

I(c)

I(b)

II

10 8 6 4 2 0 0.00

0.02

td

tb

0.04

0.06

t G/E 0.08 Time (ms)

0.10

t Al 0.12

0.14

0.16

Fig. 14. Transient variation of contact force.

motion. We distinguish Phases I (a), (b) and (c) in Fig. 12 as pre-delamination, post-delamination, post-propagation phases, respectively. The transient response of the strain and the contact force show a kink when the shear

An equivalent mass–spring system was used to find the dynamic response of GLARE subjected to impact by blunt titanium cylinders. Expressions for the inertia and stiffness were first derived by considering static indentation and then expressed as time-varying functions by setting the span of the panel equal to the distance traveled by shear waves propagating from the point of impact. Solving the resulting non-linear differential equation of motion and using energy conservation allowed us to calculate the ballistic limit and compare the energy absorbed by various mechanisms during ballistic impact of GLARE. The ballistic limit was found through an iterative process such that the initial kinetic energy of the projectile would balance the total energy dissipated by each mechanism. The analytical predictions of the ballistic limit were within 13% of the test data, while analytical predictions of the total energy dissipated at the ballistic were within 25% of test results. Our results also show that the deformation energy due to bending and membrane accounted for 84–92% of the total energy absorbed. Thinner panels absorbed a higher percentage of deformation energy than thicker panels since it was easier for

86

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88 300 V50 (m/s)-Experimental 275

V50 (m/s)-Analytical

250 225

V50 (m/s)

200 175 150 125 100 75 50 25 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Thickness (in)

Fig. 15. Variation of ballistic limit with panel thickness.

Table 5 Energy partition during ballistic perforation

V50 (m/s) (% diff) Ebm (J) Edel (J) Et (J) Ep (J) Etot (J) (% diff)

2  0:076 in.-GLARE 5

2  0:1 in.-GLARE 5

0.06 in.-GLARE 5

0.076 in.-GLARE 5

0.1 in.-GLARE 5

Analysis

Test

Analysis

Test

Analysis

Test

Analysis

Test

Analysis

Test

124.09 ()9.5%) 101.13 2.30 3.43 2.85 109.71 ()18.2%)

137.16

131.31 ()13.0%) 108.23 5.83 6.86 1.54 122.85 ()24.3%)

150.88

158.33 (+0.9%) 159.83 7.60 6.86 4.32 178.61 (+1.7%)

156.97

192.41 (+3.5%) 224.75 22.1 13.72 3.21 263.78 (+7.1%)

185.93

214.00 (+1.3%) 274.29 29.95 13.72 8.34 326.30 (+2.6%)

211.23

– – – – 134.04

– – – – 162.20

them to bend and stretch before fracture. The energy dissipated in delamination represented 2–9% of the total absorbed energy. The thinner panels absorbed a lower percentage of delamination energy than the thicker panels since they had fewer plies to delaminate. The remaining energy absorbed was about 7% and this was attributed tensile fracture energy of both glass/epoxy and aluminum. Since most of the energy is dissipated in panel deformation rather than delamination or fracture, one can improve the ballistic performance of the fiber–metal laminate by combining ductile metals with composites, whose energy absorption is enhanced by high-strain rates while they undergo large deformation. This is usually the case for most polymer-based composites since their stiffness and fracture toughness generally increase with strain rate, except at very high rates when the polymer is in the glassy state. The epoxy-based composite, which was used in the GLARE, resulted in a 15% increase in the ballistic limit for a given areal weight density of bare 2024 aluminum. However, Compston et al. [5] have shown that fiber–metal laminates with alternating 2024 aluminum and E-glass/polypropelene layers can increase the ballistic limit for a given areal

– – – – 175.55

– – – – 246.31

– – – – 317.90

weight density of the 2024 aluminum by almost 50%. Polypropelene is a thermoplastic, which not only has twice the fracture toughness of a thermoset such as epoxy, but a fracture toughness that increases significantly with strain rate.

Acknowledgements This work was supported by the Ohio Aerospace Institute under the 2000 OAI Core Collaborative Research program. The authors would like to thank the NASA Glenn Research Center, GE Aircraft Engines, Goodrich Aerospace and Honeywell for their participation in various aspects of this research.

Appendix A. Load–deflection response assuming bending deflection profile The following deflection is chosen to satisfy zero slope and deflection at boundaries, and slopes are zero at center of panel:

M.S. Hoo Fatt et al. / Composite Structures 61 (2003) 73–88

   D 2px 2py 1 þ cos wðx; yÞ ¼ 1 þ cos 4 a a

ðA:1Þ

p4 ½3D11 þ 3D22 þ 2D12 þ 4D66 D2 8a2 ðA:2Þ

and the membrane energy as  Um ¼

 3p2 p4 Nx0 þ 2Nxy0 D2 þ ½105ðA11 þ A22 Þ 16 8192a2 þ 50ðA12 þ 2A66 Þ D4

ðA:3Þ

The work done by the indentation load is  2 P a 2pR0 sin W ¼ 2 R0 þ D 4R0 2p a

ðA:4Þ

pffiffiffi where R0 ¼ pR=2 is an approximate radius that would allow us to integrate in rectangular coordinates. Thus the total potential energy is P ¼ Ub þ Um W

where D_ is the amplitude of the velocity profile and 2 D_ 0 ¼ D_ = 1 R . n

The above equation is a one-term approximation of a trigonometric series solution, which converges rapidly (within 5% of the exact solution). Using this deflection profile, one gets the bending energy as Ub ¼ 8Mxy0 D þ

87

The kinetic energy (KE) is then approximately  4 Z nZ n 1 x KE ¼ phR2 qav D_ 2 þ 2 qav hD_ 20 1 2 n R R  4 x  1 dx dy n

ðB:2Þ

where qav is the mass density of the facesheet. After integration of Eq. (B.2), one gets  6 1 2 R _2 2 2 2 _ KE ¼ phR qav D þ qav hn 1 D 2 25 n

ðB:3Þ

The kinetic energy using an effective mass me is also given as 1 KE ¼ me D_ 2 2

ðB:4Þ

Therefore, the effective facesheet mass is me ¼ phR2 qav þ

ðA:5Þ

 6 4 R qav hn2 1 25 n

ðB:5Þ

Minimizing P with respect to D gives P ¼ P0 þ ðKb þ Km1 ÞD þ Km2 D3

ðA:6Þ

where P0 ¼  Kb ¼

32R20 a 0 R0 þ 2p sin 2pR a



2 Mxy0

R20 p4

a 0 a2 R0 þ 2p sin 2pR a

2 ½3ðD11 þ D22 Þ þ 2ðD12 þ 2D66 Þ

R20 2 Km1 ¼  2 ð3p Nx0 þ 32Nxy0 Þ a 0 2 R0 þ 2p sin 2pR a Km2 ¼



R20 p4

a 0 512a2 R0 þ 2p sin 2pR a

2 ½105ðA11 þ A22 Þ

þ 50ðA12 þ 2A66 Þ

Appendix B. Effective mass of panel An effective mass of the panel can be approximated by assuming that velocity profile is similar to the deformation profile for the facesheet indentation: ( D_ ; for 0 6 x2 þ y 2 6 R2 2  2 w_ ðx; yÞ ¼ _  1 ny ; for R2 6 x2 þ y 2 6 n2 D0 1 nx ðB:1Þ

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