Micrometeoroid impact-induced damage of GLAss fiber REinforced aluminum fiber-metal laminates

Journal Pre-proof Micrometeoroid impact-induced damage of GLAss fiber REinforced aluminum fibermetal laminates Md.Zahid Hasan PII:

S0094-5765(19)31367-0

DOI:

https://doi.org/10.1016/j.actaastro.2019.10.039

Reference:

AA 7735

To appear in:

Acta Astronautica

Received Date: 21 August 2019 Revised Date:

9 October 2019

Accepted Date: 22 October 2019

Please cite this article as: M.Z. Hasan, Micrometeoroid impact-induced damage of GLAss fiber REinforced aluminum fiber-metal laminates, Acta Astronautica, https://doi.org/10.1016/ j.actaastro.2019.10.039. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

1

Micrometeoroid impact-induced damage of GLAss fiber REinforced

2

aluminum fiber-metal laminates

3 4 5 6 7

Given name: Md.Zahid; Family name: Hasan Designing Plastics and Composite Materials Department, Montan University Otto Gloeckel Street 2, 8700 Leoben, Austria * E-mail ID: [email protected]

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Abstract

9

Through several decades of development, engineers have made the GLAss fiber REinforced

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aluminum material mature for aviation structures, e.g., the fuselage of Airbus A380. A request

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like ‘GLARE + impact’ in a web search engine gives hundreds if not thousands of scientific

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articles address the high impact energy dissipation by GLAss fiber REinforced aluminum.

13

GLAss fiber REinforced aluminum has a good prospect in the field of spacecraft protection

14

against micrometeoroids and orbital debris. However, it is hard to comprehend a rational

15

reason why a thorough search of the relevant literature yielded only a couple of articles

16

concern with the ballistic impact of GLAss fiber REinforced aluminum. Handful of studies

17

interrogated the GLAss fiber REinforced aluminum damage using analytical, numerical and

18

experimental methods. No physical model, yet, has been proposed and validated to capture

19

the GLAss fiber REinforced aluminum damage upon collision with micrometeoroids in the

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low Earth orbit environment. This study, therefore, introduces a new numerical model, based

21

on the smoothed particle hydrodynamics and finite element methods, able to integrate the

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exorbitant strain rate of GLAss fiber REinforced aluminum constituents and approximate the

23

cataclysmic amount of energy dissipates in the shockwave-induced collapse of GLAss fiber

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REinforced aluminum. The model assumed the S2-glass/FM94-epoxy composite to be

25

orthotropic elastic prior to the onset of damage. Following the damage initiation, the energy-

26

based orthotropic softening governed the damage accumulation of composite blocks. Using

27

the model, an impact of a 2 mm 2024-T3 aluminum sphere on the GLAss fiber REinforced 1

28

aluminum 5-6/5-0.4 target predicted pronounced petalling of the front face aluminum layer,

29

spallation of the rear face aluminum layer, and buckling of the inner aluminum layers. By

30

contrast, the composite blocks conserved the imparted energy through membrane stretching

31

before being pierced. An experimental campaign, with the aid of a two-stage light-gas gun

32

facility, was pursued to interrogate the model accuracy. It was found that the model predicted

33

many of the experimental observations with a high degree of fidelity.

34 35

Keywords: micrometeoroid, hypervelocity impact, petalling, membrane stretching, debris cloud. Nomenclature Al ALE CFL EEM EOS FEM FMLs FRE FTIS GF/EP GF/PP GLARE HEL HVI

Aluminum Arbitrary-Lagrangian-Eulerian Courant-Friedrichs-Lewy Energy content of Eroded GLARE Mass Equation Of State Finite Element Method Fiber Metal Laminates FRiction sliding Energy Forward Time Integration Scheme Glass Fiber reinforced Epoxy Glass Fiber reinforced PolyPropylene GLAss fiber REinforced aluminum Hugoniot Elastic Limit Hypervelocity Impact

IDE IEG IEP ISS KEG KEP SPH SS MMOD

Interface Debonding Energy Internal Energy of GLARE Internal Energy of Projectile International Space Station Kinetric Energy of GLARE Kinetric Energy of Projectile Smoothed Particle Hydrodynamics Stainless Steel Micrometeoroids and Orbital Debris

36

1. Introduction

37 38 39

1.1 Impact shields

40

rapidly expand the chain process of formation of secondary fragments, which is known as the

41

cascade effect (the Kessler syndrome) [2]. Some experts believe, a cascading effect has

42

already started, at least at the altitudes of 900-1000 km [2]. Meanwhile, the quest to explore

43

the deep space has made the space-launch window shorter, adding more flotsam into the key

Space debris has cluttered the low Earth orbit [1, 2]. Mutual collisions of space debris may

2

44

orbits. Space debris imposes an immense threat to the survival of reconnaissance satellites and

45

manned spacecraft maneuvering the low Earth and geosynchronous orbits. Only the size of

46

the debris, however, does not decide the threat threshold. Millimeter-sized micrometeoroids,

47

even, could pierce the protection shield and spacecraft bulkhead, attributed to their high

48

velocity reaching 11 km/s and beyond [3]. The material of a protection shield continues to be

49

an open problem that affects the design and payload of a spacecraft.

50

Fiber-reinforced composite materials have been used in the primary shielding system of

51

manned spacecraft, for example, the International Space Station (ISS) [4]. Dissimilar

52

protection screens have been employed for different modules of ISS based on the protection

53

requirements [5]. One of the newest concepts suggests multi-layer low-weight cladding

54

shields oriented in a dumbbell shape to screen a critical space vehicle in two directions [5].

55

Note that the hypervelocity impact-induced damage of a composite protection screens appears

56

in dissimilar modes, to name a few, transverse micro-cracking, punch shear, delamination,

57

fiber breakage, and spallation [6-11]. The harsh space environment makes the inflicted

58

damage grow at a distressing rate. A GLARE protection shield, by contrast, favors the

59

structural integrity. Because, GLARE reinforces the crack bridging mechanism through

60

multiple load paths, attributed to its alternate metal/composite stacking sequence [12].

61

Furthermore, GLARE combines the synergistic advantages of high energy dissipation by

62

isotropic monolithic thin aluminum (Al 2024-T3) sheets and strength of orthotropic S2-

63

glass/FM94-epoxy (GF/EP) composite [13-15]. A detailed interrogation of GLARE damage

64

modes, yet, is imperative. By obtaining adequate information on the damage modes, a better

65

protection shield may be devised to defeat the micrometeoroids and orbital debris (MMOD).

66 67

3

68 69 70

1.2 Existing numerical models

71

18, 19, 20, 21], countable numerical investigations have been published in the open literature

72

to partially replace the experimental characterization of fiber-metal laminates (FMLs).

73

Sitnikova et al. [22] employed: a three dimensional (3D) progressive damage model for

74

woven glass-fiber reinforced polypropylene (GF/PP) composite, Johnson-Cook plasticity

75

model for Al-alloy sheets, and cohesive zones between these mating layers. The model could

76

reproduce blast failure modes, in good agreement with the experiments enlisted in Ref. [23].

77

Incorporating instantaneous failure of ceased elements, even though, underestimated the load-

78

carrying ability of GF/PP.

Concomitant with many experimental studies on the impact behavior of GLARE [16, 17,

79

Next, Guan et al. [24] further extended the 3D progressive damage model using strain-rate

80

dependent plasticity for the Al-alloy of Al-PP/PP 0°/90° 2/1 and 5/4. The model predicted a

81

higher permanent displacement of 2/1 specimens compared to that of the 5/4 grades for

82

impact velocities up to 150 m/s. The 5/4 grades suffered significant tensile fractures. Rough

83

contacts associated the mating layers, allowing no debonding or delamination. By

84

comparison, the FEM model of Karagiozova et al. [25] accommodated cohesive zones at the

85

Al-GF/PP interfaces to reconstruct debonding. The model confirmed the dependency of

86

laminate transverse velocity on the through-thickness properties of GF/PP, although, ignored

87

the damage of GF/PP continua.

88

Instead of a discrete cohesive zone approach, Yaghoubi et al. [26] opted for surface-to-

89

surface contacts between Al-layers and GF/EP composite blocks. Their model could predict

90

the ballistic limit of GLARE 5-3/2 beams and reconstruct plastic hinging and thinning of the

91

outer Al-skins. An erosion scheme when met the strain-based failure criteria removed the torn

92

fibers from the computational domain. In a complementary study, Fan et al. [27] proposed a

93

numerical model using the framework outlined by Guan et al. [24]. At low velocity impacts, 4

94

the model exhibited fiber fracture of GF/EP and plastic deformation of Al around the

95

perforation zone of FMLs 2/1 and 4/3, in accordance with the experiments.

96

1.3 Next quantum leap

97

The aforementioned numerical models offer upgrades, yet, have limitations, entail

98

assumptions and not been validated for HVI events. HVI features an extreme plastic

99

compression of materials, due to a rapid rise of pressure across the shockwave of high-

100

frequency bands. To investigate the hydrostatic material compression and shockwave-induced

101

bulk material failure, an experimental campaign is ideal. Technical challenges and expense of

102

experiments have ushered us in a new era of numerical models based on either an Eulerian or

103

a Lagrangian framework, however.

104

The Eulerian description redistributes materials in a fixed spatial grid. One would need to

105

iterate many derivatives to compute the velocity, pressure, density, temperature, etc., in the

106

fixed grid points, makes the computational burden paramount [28]. On the other side, at a

107

projectile/target interface, the highly distorted Lagrangian elements have the tendency to flip

108

back on itself, resulting in a negative mass and premature termination of an analysis [29]. In

109

pursuit of a stable and accurate numerical model, this study adopted the Arbitrary-

110

Lagrangian-Eulerian (ALE) framework of Autodyn-3D hydrocode, to reproduce the

111

perforation failure of a thick GLARE 5-6/5-0.4 laminate. The ALE framework accommodated

112

the single-phase non-linear equations of state, orthotropic constitutive relationships,

113

interactive individual material plane damage initiation criteria and energy-based damage

114

continuation criteria. The proposed model alleviated the deficiencies of stand-alone numerical

115

frameworks, reproduced the HVI-induced damage of GLARE, apportioned the energy

116

dissipated by different failure modes and reconstructed the detailed morphology of debris

5

117

clouds. Numerous failure modes were in good agreement with the experiments. The model,

118

nevertheless, had shortcomings will be elaborated where appropriate.

119

2. Modeling particulars

120 121 122

2.1 Geometry and material

123

in-plane dimension 100 mm × 100 mm and thickness 5 mm. The size of the fragment-

124

simulating hydrodynamic projectile represented micrometeoroids don’t leave a luminous trail

125

in the space environment [30]. A gap interaction method specified the frictionless contact

126

between the projectile and target placed 0.05 mm apart. The 11.64 mg projectile was launched

127

with an initial velocity at an incidence obliquity of 0° relative to the normal of the target. The

128

0° angle of incidence was chosen in all configurations to make sure the most transfer of

129

projectile momentum to the target. The GLARE 5-6/5-0.4 laminate comprised six Al 2024-T3

130

layers and five S2-glass/FM94-epoxy cross-ply 0°/90°/90°/0° composite-blocks. The analysis

131

addressed only the cross-ply GLARE configuration because of its superior impact resistance

132

to other stacking sequences [12, 31]. Each Al layer and GF/EP composite block was

133

respectively, 0.4 and 0.5 mm thick. A gap of 10 µm between the Al layer and GF/EP

134

composite block emulated the thickness of an inter-laminar interface assigned by surface-to-

135

surface contacts. The GLARE model had 10 pre-defined debonding interfaces. The composite

136

blocks were homogenized, thus, eliminated the need of specifying individual laminas in the

137

stacking sequence. To compute the stiffness matrix constants of GF/EP laminate, the fiber

138

direction and in-plane transverse to the fiber direction of the unidirectional GF/EP-lamina

139

were aligned with the global X- and Y- axis. The through-thickness material direction

140

followed the global Z-axis.

A 2 mm 2024-T3 Al-sphere impacted the front surface of the GLARE 5-6/5-0.4 target of

141

It is worth noticing that HVI experiments were conducted using 2 mm diameter spherical

142

stainless steel (SS) and Al projectiles launched at 5.5 km/s. The small diameter milligram 6

143

mass projectiles entirely disintegrated on impact and the SS projectiles caused exacerbated

144

damage to the outer Al skins of the GLARE 5-6/5-0.4 specimen. GLARE could efficiently

145

disperse the energy flux density of the SS projectile, despite the projectile density was higher.

146

Another thought provoking variable was the number of layers in the GLARE stack. Keeping

147

the experimental conditions identical, the GLARE 5-5/4-0.4 and 5-4/3-0.4 grades allowed

148

easy passage to the projectile leaving less damage in the impact zone. The thinner GLARE

149

configurations had proven not to be promising to defeat the micrometeoroids, because the

150

more irreversible work a GLARE laminate can transfer, the less will be the damage efficiency

151

of the orbital debris. The numerical analysis, therefore, reconstructed the HVI tests wherein

152

the 2024-T3 aluminum projectiles were dislodged against the thick GLARE 5-6/5-0.4

153

laminates.

154 155 156

2.2 Discretization

157

quarter model of the GLARE plate and projectile with proper symmetry and boundary

158

conditions was constructed. A mesh-resolved FE-model asked to change the number of

159

elements and mesh size. Refining the mesh smoothened the results. To ensure that the

160

predictions were further invariant with the mesh size, the model was finer discretized and

161

next, compared to the one exhibited in Fig. 1. The predictions of both models collapsed on the

162

same line in terms of shockwave pressure and kinetic energy of the target. The model,

163

subsequently, employed the mesh given in Fig. 1. A finer mesh size 0.1 mm × 0.1 mm

164

circumscribed up to 15 and 20 mm in the X- and Y-direction from the target center. The biased

165

and coarser mesh toward the target’s periphery reduced the computational zones and did not

166

affect the extent of GLARE damage, as the impact damage clustered around the target center.

167

Total 44 solid elements discretized the through-thickness direction: four elements across an

168

Al layer and a GF/EP composite block ensured a sufficient resolution.

Because of the symmetry of geometry, material properties and boundary conditions, only a

7

169

3D full-integrated constant stress hexahedral elements were assigned to 141,900 voxels of

170

each Al layer and GF/EP composite block. SPH particles were preferred for the projectile

171

discretization, as it is a meshless Lagrangian technique and does not entail the use of a

172

numerical grid to compute the spatial derivatives. The particle-based framework is free of

173

mesh tangling and distortion usually occur in large deformation of Lagrangian elements.

174

76,658 Lagrangian 3D Smoothed Particle Hydrodynamics (SPH) particles, of size 0.01 mm,

175

embodied the quarter projectile. The particle size maintained the computational accuracy at an

176

acceptable level.

177 178

Fig. 1. Finite element (FE) model of a GLARE 5-6/5-0.4 laminate.

179

3. Material models

180 181 182

3.1 Constitutive equations of GF/EP

183

to the findings in Refs. [32, 33]. The macroscopic orthotropic constitutive equations of GF/EP

The analysis presumed GF/EP-blocks to be linear elastic until the onset of failure, related

8

184

helped capturing the through-thickness deformation, for which the stress-strain relationship in

185

the incremental form is [34, 35]:

186

 ∆σ 11  C11 C12 ∆σ  C  22   21 C 22  ∆σ 33  C31 C32  = ∆ 0 τ 23    0  ∆τ 31   0 0    0  ∆τ 12   0 (1)

187 188 189

C13 C 23 C33

0 0 0

0 0 0

0 0

C 44 0

0 C55

0

0

0

  ∆ε 11   ∆ε    22    ∆ε 33    0  ∆γ 23  0   ∆γ 31    C 66   ∆γ 12  0 0 0

where i, j = 1, 2, 3 are the material directions; ∆σij and ∆εij are the stress and strain

190

increments; Cij is the stiffness coefficient. The global Z-axis followed the 11-direction, i.e.,

191

the through-thickness material direction. The X- and Y- axis were oriented respectively, in the

192

in-plane 22- and 33-direction (see Fig. 1) due to the prerequisites of the Autodyn hydrocode

193

[36] explained in section 5.1. The constitutive equations of GF/EP composite enlisted in

194

sections 3.1-3.5 complied with the mentioned coordinate system convention.

195

Strong shockwaves compress and distort composite materials near the HVI spot [8, 37, 38].

196

The nonlinear effect of shockwave beyond the Hugoniot elastic limit (HEL) of material

197

requires the deviatoric and volumetric strains to be demarcated in the form [39]:

198

199 200 201 202 203 204 205

 ∆σ 11  C11 ∆σ  C  22   21  ∆σ 33  C 31  =  ∆τ 23   0  ∆τ 31   0     ∆τ 12   0

C12

C13

0

0

C 22 C 32 0

C 23 C 33 0

0 0 C 44

0 0 0

0 0

0 0

0 0

C 55 0

 d 1 vol  ∆ε + ∆ε  0   11 3   1 d 0  ∆ε 22 + ∆ε vol  3  0  1   ∆ε 33d + ∆ε vol  0   3   ∆γ 23 0    ∆γ 31  C 66     ∆γ 12  

(2) where the volumetric strain increment (∆εvol) is defined as [29]:

∆ε vol = ∆ε11 + ∆ε 22 + ∆ε 33 (3)

9

206 207

and the deviatoric strain increment (∆εd) as [40]:

∆ε ijd = ∆ε ij − ∆ε vol

208 209 210 211

(4)

212

stress increments, yielding [40]:

213

∆ σ 11 =

214 215

(5)

216

∆ σ 22 =

217 218

(6)

219

∆ σ 33 =

220 221 222

(7)

223

(∆P) of volumetric dilatation [35]:

224

1 ∆P = − ( ∆ σ 11 + ∆ σ 22 + ∆ σ 33 ) 3

Expanding Eq. (2) and grouping the volumetric and deviatoric strains restructure the direct

1 d (C11 + C12 + C13 ) ∆ ε vol + C11 ∆ ε 11d + C12 ∆ ε 22 + C13 ∆ ε 33d 3

1 d (C 21 + C 22 + C 23 ) ∆ε vol + C 21 ∆ ε 11d + C 22 ∆ ε 22 + C 23 ∆ ε 33d 3

1 d (C 31 + C 32 + C 33 ) ∆ ε vol + C 31 ∆ ε 11d + C 32 ∆ ε 22 + C 33 ∆ ε 33d 3

A third of the trace of stress increment tensors results in the equivalent pressure increment

225 226 227

(8)

228

deviatoric strain would have generated volumetric stress, and volumetric strain would have

229

incited deviatoric stress [3]. Substituting Eqs. (5-7) in Eq. (8) coupled the volumetric and

230

deviatoric strains [4, 39]:

The orthotropic stiffness coefficients of GF/EP were not all equal. Consequently, the

∆P = −

231

232 233 234 235

1 [C11 + C 22 + C33 + 2(C12 + C 23 + C31 )]∆ε vol − 1 (C11 + C 21 + C31 )∆ε 11d 9 3

1 d − (C 21 + C 22 + C32 )∆ε 22 3 1 − (C31 + C 23 + C 33 )∆ε 33d 3 (9) where the effective bulk modulus (K') [3]:

10

1 [C11 + C 22 + C 33 + 2(C12 + C 23 + C 31 ) ] 9

236

K′ =

237 238 239

(10)

240

(9) accounts for a linear relationship between the pressure and volumetric strain. The rest

241

terms couple the pressure and deviatoric strain. The pressure and volumetric strain are non-

242

linearly related to each other under high strain rates [29].

243

relationship between the pressure and volumetric strain, therefore, replaced the linear

244

association.

245 246 247

3.2 Equation of state

248

volumetric thermodynamic response of GF/EP to shockwave pressure [3, 41, 42]:

249 250 251 252 253 254 255 256 257 258

determines the extent of compressibility of GF/EP. The first term of the right-hand side of Eq.

A polynomial non-linear

A polynomial formulation of Mie-Grüneisen equation of state (EOS) expressed the

P = K ′µ + A2 µ 2 + A3 µ 3 + ( B0 + B1 µ ) ρ 0 e (11) P = T1 µ + T2 µ 2 + B0 ρ 0 e (12)

when

when

(µ

(µ

>

<

0,

compression)

0,

expansion)

where volumetric strain, µ = ( ρ / ρ 0 ) − 1 . “0” in subscript is the state prior to the nucleation of shockwave; ρ0 is the material density

259

prior to the shockwave compression; P, ρ, and e are respectively, the hydrostatic pressure,

260

density, and specific internal energy following the shockwave compression; A2, A3, B0, B1, T1,

261

and T2 are the material constants. The impetus behind the use of a polynomial EOS comes

262

from the modeling flexibility it offers [43].

263 264 265

3.3 Damage nucleation in GF/EP

266

the Autodyn hydrocode. Because, it took into consideration the orthotropic nature of the

The numerical frame incorporated a modified form of Hashin's failure criteria available in

11

267

failure modes and the impact of progressive material degradation on the load carrying

268

capability. The criteria coupled the failure modes employing three failure surfaces [44]: (i)

269 270 271

2 11, f

e

272 273 274 275

e

276 277 278 279 280 281

σ =  11 σ  11, f

(ii)

2 22, f

e

2

2

  σ12  +    σ12, f

  σ13  +    σ13, f

2

  ≥ 1, in the 11-direction (σ11 ≥ 0) 

(13)

For the tensile fiber failure

σ =  22 σ  22, f

(iii)

2 33, f

For delamination

2

  σ12  +    σ12, f

2

  σ 23  +    σ 23, f

2

  ≥ 1 , in the 22-direction (σ22 ≥ 0) 

(14)

For the transverse matrix cracking

σ =  33 σ  33, f

2

2

  σ 23  +    σ 23, f

  σ13  +    σ13, f

2

  ≥ 1, in the 33-direction (σ33 ≥ 0) 

(15)

2 2 2 where e11, f , e22, f , and e33, f are the failure surfaces of the corresponding failure modes; f

282

designates the initial failure strength. The parabolic stress-based failure initiation criteria,

283

when reached the value of 1 or above at an element integration point, triggered the failure

284

modes there. The computation cycles checked and updated the failure status of element

285

integration points successively.

286 287 288

3.4 Damage growth in GF/EP

289

stresses updated the stress state of the damaged element. Introduced the non-linear strain-

290

softening in the 3D Hashin's failure criteria, Eq. (14) transformed to [4]:

Once a discrete element began to fail, a non-linear reduction of the respective failure

291 2

2 22, f

292

e

293 294

(16)

2

2

      σ 23 σ 22 σ12 = + + ≥1    σ (1 − D )   σ (1 − D )   σ (1 − D )  22  12  23   22, f  12, f  23, f

12

295

where Dij is the damage coefficient. An additional limit surface, defined in the material stress

296

space of each material plane, helped estimating the non-linear strain-softening of GF/EP. If

297

the stress state was computed to lie outside a limit surface, an iterative backward-Euler

298

procedure returned the stress point to the softening limit surface [44]. Doing so, an inelastic

299

crack strain ( εij ) accumulated in the material and was incorporated in the damage coefficient

300

[3, 34]:

301

 Lij Fij2  ε ijcr  Dij =   2G   F  ij   ij 

cr

i,

j

=

1,

2,

3

302 303 304

(17)

305

the fracture energy of each failure mode (see Table 1), and Lij specifies the characteristic

306

dimension of a finite cell cracks in the failure direction. Dij = 0 designated a pristine material

307

and was set to unity as soon as the material strength was entirely exhausted. The forward time

308

integration scheme (FTIS) updated the damage coefficient (Dij) in the incremental time steps,

309

e.g., for the progressive damage accumulation in the 22-direction [34]:

310 311 312 313 314 315 316

D22n+1 = D22n + ∆D22 + C ∗ ( D12 + D23 )

(18)

D12n +1 = D12n + ∆D12 + C ∗ ( D22 + D23 )

(19)

D23n+1 = D23n + ∆D23 + C ∗ ( D12 + D22 )

(20)

where C* is the coupling coefficient varies between 0 and 1. Given C* = 0, the damage

317

coefficients are uncoupled. This allows, for example, modeling the transverse shear failure of

318

matrix, when the fibers remain intact. In this study, C* = 0.2 predicted failures in close

319

proximity to the experiments.

320 321

3.5 Element stress and stiffness

where Fij stands for the initial failure stress in three normal and three shear directions, Gij is

13

322

After the onset of failure, the numerical scheme continuously updated the strength and

323

stiffness of damaged elements depending on the failure modes and current extent of material

324

degradation. The corresponding update of element elastic stiffness matrix conceded:

325

326

C11 (1 − D11 ) C12 (1 − Max( D11 , D22 )) C13 (1 − Max ( D11 , D33 )) 0  C (1 − Max ( D , D )) C22 (1 − D22 ) C23 (1 − Max ( D22 , D33 )) 0 11 22  21  C (1 − Max ( D11 , D33 )) C32 (1 − Max ( D22 , D33 )) C33 (1 − D33 ) 0 Cij =  31 0 0 0 α C44   0 0 0 0  0 0 0 0 

0 0 0 0

α C55 0

      0   α C66  0 0 0 0

(21)

327 328 329

The complete failure of material in the 22-direction enforced σ22 to zero and modified the

330

other directional stresses according to the loss of Poisson’s effect. Subsequent to the 22-

331

directional failure, the constitutive equation emerged as:

332

C11 (1 − D11 )  ∆σ 11    0   0     ∆σ 33  C31 (1 − Max( D11, D33 ))  = 0  ∆τ 23    ∆τ 31   0    0  ∆τ 12  

0 C13 (1 − Max( D11, D33 ))

0

0

0

0

0

0

0

C33 (1 − D33 )

0

0

0

0

α C44

0

0

0

0

α C55

0

0

0

0

 d 1 vol   ∆ε11 + 3 ∆ε  0   1 vol  d 0   ∆ε 22 + ∆ ε   3 0     ∆ε d + 1 ∆ε vol  33 0    3   0  ∆γ 23   α C66   ∆γ 31    ∆γ 12  

(22)

333 334 335

A similar expression confirmed the entire failure of material in the 33-direction, by

336

reducing the respective element stress and stiffness components to zero. The material damage

337

degraded the shear stiffness of GF/EP by a factor “α“. This study assigned a nominal value of

338

α = 0.2, consistent with the Refs. [3, 35]. An orthotropic post-failure option set the pressure of

339

a failed element to zero (bulk failure), given that two of the damage coefficients of a failure

340

surface had reached the value of 1, meant GF/EP suffered a breach in more than one direction.

341

If so, the element tensile stresses dropped to zero and the residual shear stiffness scaled down

342

the material shear strength. It is to notice that the model did not incorporate the delamination

14

343

mode of failure (see section 3.6), thus did not demand a modification of delamination-related

344

stress and stiffness. The material model of Al is briefly outlined in Appendix A.

345

346 347 348

3.6 Debonding criteria

349

when subjected to high impulsive loading [45]. The proposed GLARE model, to be pertinent,

350

accommodated only Al-GF/EP but no GF/EP-GF/EP interfaces. Surface-to-surface contacts

351

connected the mating Al layers and GF/EP composite blocks. A quadratic nominal stress-

352

based criterion [46]:

353

 σ n  σ s    +   ≥1 σ N  σ S 

Unless poorly manufactured, the composite-lamina interfaces of FMLs barely delaminate

a

b

a,

b

=

2

354 355 356

(23)

357

stresses in the normal and shear direction of an interface; σN and σS are the limiting nominal

358

stresses in the normal-only and shear-only mode of interface debonding. The accessible

359

experimental facilities did not allow to measure the Al-GF/EP interface strength at a high

360

strain-rate. Therefore, σN = 8.2 and σS

361

three-point bending experiments approximated the Al-GF/EP interface strength. The

362

numerical code invoked a surface interaction between the Al layers and GF/EP composite

363

blocks subsequent to the interface debonding.

364 365 366

4. Numerical implementation

367

since it could discriminate the material interfaces during the long duration penetration phase

368

[28]. Moreover, the ALE method allowed arbitrary adaptation of the element shape in

369

distorted impact zones. When failure criteria reached the limit threshold at all element

when met initiated the debonding of Al-GF/EP interfaces. In Eq. (23), σn and σs are the

=

46.6 MPa, respectively, from quasi-static peel and

The reference frame of numerical analysis was Arbitrary-Lagrangian-Eulerian (ALE),

15

370

integration points, ceased elements eroded into particles. The explicit integration scheme

371

preserved the inertia or nodal mass at all activated nodal degrees of freedom following the

372

element erosion.

373

Two different mesh resolutions: 165 (X) × 215 (Y) × 44 (Z) and 90 (X) × 115 (Y) × 22 (Z)

374

were evaluated to achieve a mesh-resolved solution. The energy profiles of the 165 (X) × 215

375

(Y) × 44 (Z) spatial grid, illustrated in Fig. 5, differ by 1.1 % from that of the 90 (X) × 115 (Y)

376

× 22 (Z) resolution. A finer mesh (315 (X) × 415 (Y) × 88 (Z)) had also been considered,

377

however, this dense mesh resolution clogged the data transfer between the hard drive and

378

RAM of cluster nodes. In order to avoid unexpected interruption of the simulation run, the

379

FE-analysis employed the mesh resolution 165 (X) × 215 (Y) × 44 (Z) (see Fig. 1).

380

The time-step size of computation had been adjusted based on the shockwave velocity and

381

element size. A time-step size between 0.0002 and 0.002 µs satisfied the convergence

382

criterion of Courant-Friedrichs-Lewy (CFL ≤ 1). The 0.0002 µs size was chosen at the phase

383

of shockwave onset and release of rarefaction wave to stabilize the solution. Following the

384

shockwave dissipation, the time step size was increased to 0.002 µs, whilst the kinetic energy

385

of the GLARE had taken over and the gradient of the material derivatives declined. The

386

iterations of many material derivatives at each time step demanded over 720 clock hours in a

387

20 CPU + 100 GB RAM cluster to reach the physical computation time of 130 µs. Making the

388

time step size half (0.0001 µs at the shock phase and 0.001 µs at the penetration phase) of the

389

implemented one (0.0002 µs at the shock phase and 0.002 µs at the penetration phase)

390

demonstrated a mere 0.8 % change of the energy profiles in Fig. 5. Courtesy to the grid

391

resolution and time integration scheme, the material derivatives accumulated an absolute error

392

of 0.019 upon ending the analysis at t = 130 µs, when the shock and release waves

393

disappeared from the computational domain, and the kinetic energy of the system approached

394

towards an asymtote. The accumulated error did not surpass the allowable relative

16

395

computational error considered 6 % being acceptable looking at the complexity of physical

396

phenomena involved into the shockwave-induced damage of GLARE. Besides, the

397

comparison of the numerical approximations against the experimental outcomes delineated

398

the fidelity of the computation (see Table. 5). To get to know more about error estimation,

399

please read Refs. [47, 48].

400

5. Material data

401 402 403 404 405

5.1 Material properties of GF/EP

406

material directions. While a Poisson’s ratio of less than 0.5 confirmed that the GF/EP is

407

compressible, the through-thickness bulk modulus signified the extent of compressibility. The

408

bulk modulus and density of material estimated the sound wave velocity, which made the

409

shockwave velocity a function of the particle velocity. If the material stress surpassed the

410

failure stress of a particular material direction, the directional damage of GF/EP started. Next,

411

damage accumulated in GF/EP following the enrichment of pre-defined energy at the crack

412

tips. A material characterization campaign was pursued in-house to derive the GF/EP

413

laminate properties. The main objective of this paper is to lend detailed insights in the

414

numerical model. The test methods of material characterization will be elaborated in a

415

companion study, therefore.

416 417

Table 1 Data set of a GF/EP 0°/90°/90°/0° composite block

Conjugate shockwave compression and material strength The Young’s and shear modulus determined the material stiffness of GF/EP in the six

Strength: Orthotropic Reference density [g/cm3] Young's modulus 11 [KPa] Young's modulus 22 [KPa] Young's modulus 33 [KPa] Poisson's ratio 12 Poisson's ratio 23 Poisson's ratio 31

Failure: Orthotropic softening 1.8 1.90e+007 3.083e+007 3.083e+007 0.4 0.114 0.4

Tensile failure stress 11 [KPa] Tensile failure stress 22 [KPa] Tensile failure stress 33 [KPa] Maximum shear stress 12 [KPa] Maximum shear stress 23 [KPa] Maximum shear stress 31 [KPa] Fracture energy 11 [J/m2]

6.90e+004 5.90e+005 5.90e+005 4.83e+004 8.69e+004 4.83e+004 83.375

17

Shear modulus 12 [KPa] Shear modulus 23 [KPa] Shear modulus 31 [KPa] Reference temperature [K] Specific heat [J/KgK] Volumetric response: Polynomial Bulk modulus A1 [KPa] Parameter A2 [KPa] Parameter B0 Parameter B1 Parameter T1 Parameter T2 Strength: Elastic Shear modulus [KPa]

3.89e+006 8.10e+006 3.89e+006 300 900 2.69e+007 2.69e+008 0 0 2.69e+007 0

Fracture energy 22 [J/m2] Fracture energy 33 [J/m2] Fracture energy 12 [J/m2] Fracture energy 23 [J/m2] Fracture energy 31 [J/m2] Damage coupling coefficient Erosion: Failure criteria satisfied

1e-006 1e-006 747 1e-006 1.378e+003 0.2

8.10e+006

418

* The through-thickness compressibility decides the volumetric strain of GF/EP upon HVI.

419

The volumetric strain is substantial in the short duration impulse phase and results in the bulk

420

failure of material at the impact site. It was, therefore, of high importance to align the local

421

11-coordinate direction with the material thickness (Z-axis), in line with the coordinate

422

system convention of the Autodyn hydrocode. Readers, would like to use the material data

423

and reproduce the results, are requested to strictly follow the coordinate orientation of GF/EP

424

composite explicated in Fig. 1 and Table 1.

425 426 427 428 429

5.2 Material properties of aluminum sheet and projectile

430

Al. Material constants, C1 specified the characteristic sound speed in 2024-T3 aluminum

431

alloy and S1 yielded the slope between the shockwave velocity and particle velocity.

432

Irreversible deformations were allowed for in the plastic model of Al using yield stress,

433

plastic flow rule and a predefined hardening law. The Al layers endured plastic hardening due

434

to the frequent loading and unloading, akin to the compression to and decompression from the

435

Hugoniot strain of Al (see Appendix A). Subsequent to the plastic hardening, if the material

436

strain exceeded the limiting failure strains in the three normal and three shear material

437

directions, the Al elements eroded.

Govern the bulk failure and plasticity The Mie-Grüneisen equation of state correlated the pressure with the volumetric strain of

18

438 439 440 441

Table 2 Data set of aluminum material model [17] Equation of state: Shock Reference density [g/cm3] Gruneisen coefficient Parameter C1 Parameter S1 Reference temperature [K] Specific heat [J/KgK] Strength: Steinberg Guinan Shear modulus [KPa] Yield stress [KPa] Maximum yield stress [KPa] Hardening constant Hardening exponent Melting temperature [K]

Failure: Material strain 2.785 2.0 5.328e+003 1.338 300 863 2.86e+007 2.6e+005 7.6e+005 310 0.185 1.22e+003

Tensile failure strain 11 Tensile failure strain 22 Tensile failure strain 33 Maximum shear strain 12 Maximum shear strain 23 Maximum shear strain 31 Post-failure option: Isotropic Erosion: Failure criteria satisfied

0.225 0.225 0.225 0.3181 0.3181 0.3181 For Al-sheet

442

6. Results and discussion

443 444 445

6.1 Transient behavior of GLARE

446

duration phase-I dilatational compression, and (ii) long duration phase-II penetration. Taking

447

the case VI = 7.11 km/s for instance, Fig. 2 delineates the two phases.

448 449 450 451

Short duration phase-I dilatational compression

452

velocity contour at t = 0.235 µs). At t = 0.77 µs, the distal layers away from the impact site

453

deflected downstream and the damage circumscribed the target center. A single elastic-plastic

454

shockwave extremely compressed the GLARE laminate throughout the thickness and inflicted

455

bulk failure to the frontal Al-6 layer. Besides, the Al-6 layer torn off.

456 457 458

Long duration phase-II penetration

459

traction-free rear face augmented the particle velocity of the incident wave without altering

The numerical analysis treated the HVI damage of GLARE in two distinct phases: (i) short

The dramatic increase of target velocity characterized the short duration phase-I (see the

After that, at t = 1.40 µs, the farthest Al-1 layer (rear face) started to debond since the

19

460

the wave direction. Petalling of the frontal Al-6 layer (front face) initiated. Transverse shear

461

deformation at the periphery of the impact zone started the debonding of the first three pre-

462

defined interfaces adjacent to the impact site. The front face Al-petals were distinguishable at

463

t = 3.32 µs, when dynamic bulging induced large plastic deformation to the rear part of

464

GLARE. Next, at t = 4.51 µs, the farthest Al-1 layer was pierced, attributed to the erosion of

465

Al elements suffered the failure strain. Following that, at t = 9.24 µs, the petalled area of the

466

front face enlarged and the pierced hole of the rear face widened. Thinning instability of the

467

Al-1 layer augmented the fragmentation process. At this instant, the debonded GF/EP-5-

468

block, next to the front face, flapped backward due to the release from volumetric

469

compression, which resulted in a momentum imbalance and consequently, torn off the petal

470

tips. As found at t = 21.05 µs, the transient kinetics of GLARE decelerated rapidly

471

accompanied by the lateral growth of petal cracks. All interfaces disintegrated around the HVI

472

spot.

473

In the later time steps, fractures of the front face and rear face propagated laterally,

474

discernible at t = 32.86 µs. The debonding was less confined to the HVI spot and emanated

475

towards the edge of GLARE. Effective plastic deformation pursued the same trend. The

476

volumetric strain of materials, however, was minute toward the periphery of the target. The

20

477 478

Fig. 2. Spatio-temporal gradient of absolute velocity of GLARE at VI = 7.11 km/s; to

479

facilitate the cognitive interpretation of GLARE failure modes, the imagery does not include

480

the fragmented projectile.

21

481

inner Al-layers buckled under compression and as an outcome, led to a near symmetric out-

482

of-plane deformation of GLARE related to the mid-thickness plane. The plate accumulated

483

most of the plastic damage at t = 44.61 µs and reached the permanent deformed state t = 130

484

µs subsequent to the recovery of the large transient deformations.

485

The permanent deformed state highlighted smaller pierced holes in GF/EP composite

486

blocks compared to the perforations of Al layers, since the debonded GF/EP-blocks stored

487

circa four times more energy through the favorable elastic stretching before being pierced. By

488

contrast, the Al layers in the rear half of GLARE endured pronounced plastic thinning and

489

bending deformation prior to perforation. Notice that a mismatch of the spatial time history of

490

damage accumulation is comprehendible for dissimilar impact velocities. The delineated

491

failure modes will be the common signatures of GLARE damage at the upper end of HVI

492

spectrum, however.

493 494 495

6.2 Failure of GF/EP composites

496

site experienced pronounced transverse compression, due to the shockwave-induced pressure

497

of 144 GPa at the HVI spot. The volumetric and deviatoric strains were coupled. The

498

transverse compression, consequently, incited orthotropic stresses in the material continuum.

499

Among the GF/EP composite blocks, the GF/EP-5 suffered the highest tensile stress in both

500

in-plane directions, σxx = σyy = +11.56 GPa at t = 0.472 µs (Fig. 3). Besides, an out-of-plane

501

compressive stress σzz = -20.03 GPa and shear stress τyz = 5.32 GPa were determined at the

502

periphery of impact zone. The in-plane principal stresses were two orders of magnitude higher

503

than the tensile failure strength (see Table 1). Moreover, shear fracture of elements was

504

visibly distinct at the edge of impact area. Tensile fiber failure and punch shear culminated in

505

the penetration of GF/EP-5 composite block next at t = 0.707 µs. The same stress components

506

led to the penetration of the distal GF/EP composite blocks; predicted, σxx = σyy = +8.60 GPa

At the phase of damage accumulation, the frontal GF/EP composite blocks at the impact

22

507

in the GF/EP-4 composite block at t =1.283 µs, together with σzz = -3.23 GPa and τyz = 3.81

508

GPa. As follows, at t = 2.144 µs, the projectile penetrated GF/EP-4 composite block ascribed

509

to the progressive orthotropic softening.

510 511

Fig. 3. Progressive degradation of the GF/EP-blocks; impact velocity 7.11 km/s; columns (a),

512

(b), (c), and (d) correspond to the time instances 0.472, 0.707, 1.283, and 2.144 µs,

513

respectively.

514

Once entirely penetrated, the stress components of GF/EP-4 composite block attenuated

515

below the failure stress at the periphery of the pierced hole. In no instance, the far-field

516

stresses surpassed the limiting failure stress. For clarity, the red color of the contour plots

517

locates the zones of high tensile stress (Fig. 3).

518

The distal GF/EP composite blocks endured lower impact-induced stresses. Because, the

519

reverberated shockwave disintegrated the projectile and dispersed the projectile- mass and

520

momentum prior to further penetration. Albeit the progressive fragmentation of the projectile,

23

521

the principal stress components of GF/EP composite blocks were beyond the limiting stress

522

threshold and consequently, allowed the complete perforation of GLARE at VI = 9 km/s. The

523

perforated holes formed a conical through-thickness tunnel. This suggests, the GF/EP-blocks

524

adjacent to the impact site endured more damage and let the fragmented projectile

525

downstream with a lower axial momentum. In other words, a strong shockwave formed close

526

to the front face, while the rear face received only widely distributed impact momentum and a

527

lower impact energy [51]. The tensile fiber failure mode of GF/EP assimilated the tensile

528

fiber breakage of the composite plies of GLARE specimens, exemplified in Fig. 9.

529 530 531

6.3 Debris cloud

532

comprise solid fragments and liquid particles [49, 50]. Given the HVI energy is high enough,

533

debris vaporizes and the expansion of the gas-vapor cloud results in the formation of a

534

diverging shockwave [51]. In this study, the Sesame multi-phase equation of state could

535

reproduce the phase change of material. The material enthalpy might have been of different

536

orders of magnitude based on the Sesame multi-phase formulation [52, 53]. For the sake of

537

simplicity, the debris cloud demonstrated in Fig. 4 includes only the solid material phase,

538

ascribed to the solid phase equations of state assigned to the target and projectile.

HVI of an Al projectile at VI = 6 km/s or beyond on an Al target dislodges debris clouds

539

The debris plume emanated when Al elements eroded upon reaching the failure strain in

540

three normal and three shear directions. The GF/EP-elements if failed at eight integration

541

points eroded as well. Using the erosion scheme, HVI of a 2 mm Al-sphere on a GLARE 5-

542

6/5-0.4 laminate evacuated debris clouds uprange and downrange in the perforation events

543

(Fig. 4). The uprange veil ejected in a tapered axisymmetric conical shape fringed by the

544

eroded particles of Al-layers. The GF/EP-debris and projectile remnants densely populated

545

the veil core. The debris distribution of uprange ejecta was very much alike for the impact

546

velocities.

24

547 548

549

550

551

552 25

553

Fig. 4. Evacuation of debris cloud uprange and downrange at t = 20 µs of computation; (a),

554

(b),

555

(c), and (d) correspond to the impact velocities 4.78, 7.11, 9, and 11 km/s, respectively; 1

556

symbolizes the detached spalls.

557

In comparison, the downrange ejecta was individualistic for each impact velocity. At VI =

558

7.11 km/s, the rear face dislodged no solid fragments, but particles. By contrast, at VI = 9

559

km/s, the rear face Al-layer spalled, corroborating the reflection of the compressive shock as a

560

tensile wave at the rear face. Couples of tiny GF/EP-fragments accompanied the spalls

561

downstream. At VI = 11 km/s, the launched Al-spalls were splintery since the plastic strain

562

energy was high enough to degrade and downsize them. Moreover, a large mass fraction of

563

defragmented projectile populated the downrange ejecta. Note that at higher impact velocities,

564

the downrange ejecta emerged in a narrower spreading angle about the centro-symmetry

565

plane: predicted 22.28° for VI = 11 km/s (Fig. 4d) compared to 33.81° for VI = 9 km/s (Fig.

566

4c). The uprange ejecta apprised an analogous trend of lateral dispersion with impact velocity,

567

though, formed a larger spreading angle due to the continuous lateral dispersion of the

568

disrupted mass (Figs. 4a through 4d). Alongside, higher impact velocities furnished more

569

damage to the GF/EP-blocks, visible by the dense packing of GF/EP-debris in the ejecta veils.

570 571 572

6.4 Energy partition

573

MpiVI2/2) transfers into the kinetic energy of the debris cloud, while another part of it transfers

574

into the reversible and irreversible work of the target. The projectile remnants in the uprange

575

and downrange ejecta budgets the restitutional energy (ER = MprVr2/2). The energy transferred

576

to GLARE (ET = EI - ER) divides into the GLARE internal and kinetic energy. The internal

577

energy mainly includes the elastic and plastic strain energy (IE) [54]. At a computational

578

cycle, the reference total energy (TE) was equal to the sum of the kinetic energy of GLARE

During the projectile/target interaction, part of the impact energy of the projectile (EI

=

26

579

(KEG) and projectile (KEP), the internal energy of GLARE (IEG) and projectile (IEP), the

580

friction sliding energy (FRE) between the projectile and GLARE, the interface debonding

581

energy (IDE) and the energy content of eroded GLARE mass (EEM) following impact (Eq.

582

(24)). The FRE was zero, due to the frictionless interaction between the projectile and

583

GLARE. The EEM encompassed the difference between TE and CTE in Fig. 5a. Since the

584

materials retained the solid phase, the energy balance in Eq. (24) did not apportion a phase

585

change energy.

586 587 588

TE = KEP + KEG + IEG + IEP + FRE + IDE + EEM (24)

Energy (J)

(a) Time history of energies 80

TE

60

CTE KEP

40

KEG

20

IEP IEG

0 0

30

60 Time (µs)

90

PDEP

120

PDEAl

589 590 Percentage internal energy of GLARE, %IEG

(b) Spatial distribution of internal energy

591 592

100 75

%IE by discrete thickness

50 Cumulative %IE 25 0 0

0.25

0.5

0.75

1

Normalized GLARE thickness (H/HG)

27

Percentage plastic damage energy by GLARE, %PDEG

(c) Spatial distribution of plastic energy 100 75

%PDE by discrete thickness

50

Cumulative %PDE 25 0 0

0.25

0.5

0.75

1

H/HG

593 594

Fig. 5. Spatial time history of different forms of energy for VI = 7.11 km/s; zero and one of

595

the normalized GLARE thickness (H/HG) symbolize the front face and rear face, respectively.

596

Fig. 5 outlines the impact energy partition for VI = 7.11 km/s. Fig. 5a shows a steep decline

597

of KEP, while KEG and IEG inclined upon impact. At t = 0.566 µs, KEG reached sharply the

598

maxima, approximated 29.71 J (46.68 % of the current total energy (CTE = 63.64 J)); IEG

599

inclined to 18.83 J (29.58 % of CTE), and KEP showed a sudden drop by 86.36 % from the

600

initial 78.41 J to 10.69 J. Next, KEG converted to the IEG; in relation, KEG reduced to 5.2 J

601

(10.94 % of CTE = 47.49 J) at t = 130 µs. At the same instant, IEG experienced an upsurge to

602

37.43 J (78.81 % of CTE) attributed to the KEG energy conversion; energy dissipation by

603

plastic strain and plastic hardening of the Al-layers (PDEAl) inclined to 14.6 J (39 % of IEG =

604

37.43 J), and KEP was 0.084 % of its initial KE (78.41 J). Element erosion reduced the total

605

system energy (TE), which approached towards an asymptote of circa 47.5 J (CTE) at t = 130

606

µs. At this instance, the debonding of Al-GF/EP interfaces dispensed 9.29 J of the projectile

607

impact energy.

608

Figs. 5b and 5c illustrate the individual and cumulative contribution of material layers to

609

the internal energy (%IE) and plastic damage energy (%PDE) of GLARE at t = 130 µs. As

28

610

seen, circa 56.54 % of the cumulative IE distributed non-uniformly in the normalized

611

thickness range 0 < H/HG < 0.55 (the front part of GLARE laminate), in which the GF/EP-

612

blocks alone contributed 38.21 % (see Fig. 5b). The rest 43.46 % of the cumulative IE was

613

near uniformly distributed in the thickness range 0.55 < H/HG < 1 (the rear part of GLARE

614

laminate).

615

The PDE traversed fluctuations across the thickness of GLARE (Fig. 5c). Approximately

616

49.76 % of the cumulative PDE accumulated in the thickness range 0 < H/HG < 0.55, while

617

the thickness range 0.55 < H/HG < 1 subsumed the remainder 50.24 %. Energy peaks of the

618

discrete %PDE curve at the front face and rear face stand for the energy dissipation by plastic

619

strain, plastic hardening and petalling. The triangles between these energy peaks refer to the

620

substantial energy budgeted by buckling and rupture of inner Al-layers. As aforementioned,

621

plastic strain and plastic hardening of Al-layers tapped about 14.6 J and rupture-engendered

622

Al-mass erosion evacuated the large 29.4 J.

623

By comparison, the GF/EP composite blocks conserved the internal energy primarily via

624

elastic deformation (constituted 18.22 J of the IEG = 37.43 J), despite a mere 1.56 J drained

625

in the mass erosion of GF/EP. The energy budget of mass erosion verifies larger perforated

626

holes of Al layers compared to that of GF/EP composite blocks. Looking at the energy

627

partition, it can be concluded that the increase of volume fraction of GF/EP composite blocks

628

will reinforce the elastic limit of GLARE, and a higher volume fraction of Al layers will

629

augment the plasticity-induced dissipation of impact energy.

630 631 632

6.5 Perforated and non-perforated deformation GLARE panel manufacture

633

Identical to the numerical model, the manufactured GLARE 5-6/5-0.4 laminates comprised

634

S2-glass fiber reinforced FM94-epoxy prepregs embedded between 2024-T3 Al-alloy sheets.

635

GF/EP ply and Al sheet were 0.125 and 0.4 mm thick. The fiber direction was aligned with

29

636

the rolling direction of Al sheets. A chromate coating on the Al sheet promoted its adhesion

637

with the GF/EP ply. The material layers were stacked in the required sequence manually and

638

cured in an autoclave for 3 h at a curing temperature of 120 °C under 6 bar pressure. Square

639

specimens with a dimension of circa 10 mm ×10 mm were cut using a diamond saw

640

afterwards. Ultrasonic C-scan of the specimens elucidated no manufacturing-induced flaws or

641

cutting-induced delaminations. Finally, four circular holes were drilled 1 cm away from the

642

specimen edge to mount the specimens on the hatch of the test chamber.

643 644

Fig. 6.– (a) Hand lay-up of the GLARE stack; (b) breather fabric and air-tight plastic foil

645

covering the GLARE laminates on the stainless steel bed of the autoclave prior to curing; red

646

arrows locate the position of GLARE laminates.

647 648 649

Test instruments

650

specimens at room temperature and 50 % relative humidity. Helium was used as the

651

propellant. Fig. 7 includes a pictorial image of the test set-up and Fig. 8 outlines the basic lay-

652

out of the gas gun. At the end of the target chamber, four bars on a hatch kept the target plate

653

fixed against the projectile. A witness plate, placed 60 mm downstream of the GLARE plate,

654

collected the debris cloud comprised fragmented projectile and GLARE materials. Two wide

655

beam lasers positioned perpendicular to the projectile trajectory measured the impact velocity

656

within ±0.01 km/s. A vibration sensor attached to the GLARE plate verified the impact

657

velocity reported by the laser beams. An identical sensor on the witness plate measured the

A two-stage light-gas gun was employed to conduct the HVI experiments on GLARE

30

658

impingement velocity of downrange debris cloud. High-speed videos could be captured

659

through two window openings in the target chamber. A high-speed camera of a million frame

660

rate, however, was not available during the test campaign. The explosive nature of HVI

661

destroyed the surface-adhered thermocouples a couple of times. To avoid further damage of

662

the test assets, the test agenda did not include measuring the temperature gradient of GLARE

663

specimens upon impact. Two GLARE specimens had been evaluated against the numerical

664

analysis of an identical impact velocity. In the simulation, the projectile impacted exactly the

665

target center. Only the test specimens impinged at or in the vicinity of the mid-zone were

666

chosen to deduce a reasonable comparison, therefore. Keeping the impact velocity, the

667

captured debris mass showed a maximum ±1.5 % deviation between the two experiments, and

668

the highest mismatch of the impingement velocity of the debris mass on the witness plate was

669

recorded ±1.8 %. Oscillations in the velocity signals from the laser beam and vibration sensor

670

had not been smoothened. Therefore, the standard deviation maxima of velocity was inferred

671

from the real time measurements. The allowable error maxima of the experimentally

672

determined values was set at 5 % considering the reproducibility of the experiments. The

673

author acknowledges that the randomicity of HVI experiments, thus, the deviation of

674

experimentally determined values, depends on the calibration of the test set-up. The mismatch

675

between the experiments and numerical analysis can be large, if the test conditions do not

676

resemble the simulative environment and in relation, the test specimens do not incur all the

677

damage modes predicted by the numerical analysis. Fine tuning of the gas-gun is imperative

678

to reproduce the tests and dislodge the projectile over the required flight trajectory to impinge

679

the target exactly at the middle. On top of that, a well-controlled manufacturing process is

680

required to circumvent the manufacturing-induced flaws and their impact on the disparity of

681

damage modes of test specimens.

31

682

683 684

Fig. 7. – (a) The two-stage light-gas gun; (b) mounting hatch; (c) a GLARE specimen

685

subjected to HVI; 1 and 2 stand for the mounting bar and witness plate; 3 identifies the

686

GLARE plate on the mounting bars; 4 indicates location of bored holes.

687 688 689 690 691 692 693

Fig. 8.– Schematic side-view of the two-stage light-gas gun.

32

694 695 696

Quantitative assessment: petals

697

GLARE model was reflected about the two symmetry planes to facilitate the cognitive

698

interpretation. The distinctive features of GLARE damage were the fracture and outward

699

petals of the front face around the central perforated hole. As the impact velocity increased

700

from 4.78 to 7.11 km/s, the front face petalled area reshaped from a near circular to an

701

elliptical contour (compare Figs. 9a and 9c). At VI = 4.78 km/s, the non-perforated rear face

702

sustained a crack at the tip of the circular cusp due to plastic yielding (Fig. 9b). At VI = 7.11

703

km/s, the rear face ruptured and generated a near rectangular petalled area, however (Fig. 9d).

704

The shape of the petalled areas was consistent with that of the experiments. At both impact

705

velocities, perforated holes of the GF/EP composite blocks had smaller openings than the

706

holes pierced in the Al-layers, in agreement with the experiment.

Fig. 9 compares the numerical predictions against the experimental outcomes. The quarter

707

By contrast, at VI = 4.78 km/s, the ten front face petals of the experiment were way more

708

than the four front face petals predicted by the analysis (Fig. 9a). If compared, at VI = 7.11

709

km/s, the front face petals were eight both in the experiment and simulation (Fig. 9c); on the

710

flip side, the predicted rear face petals were four compared to seven rear face petals of the

711

tested specimen (Fig. 9d).

33

712 713

Fig. 9. Predicted petals against experimental ones; (a) and (b) correspond respectively, to the

714

front face and rear face damage for VI = 4.78; (c) and (d) stand respectively, for the front face

715

and rear face petals for VI = 7.11 km/s; (e) and (f) show respectively, the front face and rear

716

face petals for VI = 9 km/s; (g) and (h) demonstrate respectively, the front face and rear face

717

petals for VI = 11 km/s; computation time 130 µs.

718 719

Table 3 Petals of experiments and simulations GLARE panel

Front face petals

Rear face petals

VI (km/s) Measurement

Prediction

Measurement

prediction

720

4.78 10 4 NP NP 7.11 8 8 7 4 GLARE 5-6/5-0.4 9 8 4 11 8 8 *NP = No perforation; the counted petals correspond to the pictorial images in Fig. 9; at VI =

721

4.78 and 7.11 km/s, the number of petals of tested specimens varied by ±2 ascribed to the

722

deviation of impact spot.

34

723

HVI at VI = 9 and 11 km/s added no extra front face petals compared to the event at VI =

724

7.11 km/s, but, redistributed the petals around a circular front face petalled area (Figs. 9e and

725

9g). It is anticipated that the front face became fracture-saturated beyond the ballistic limit

726

and dispensed the excess energy through thinning, buckling and transient vibration.

727

Conversely, the rear face petals were identical in number at VI = 7.11 and 9 km/s (compare

728

Figs. 9d and 9f). On the other hand, at VI = 11 km/s, the rear face petals multiplied to eight

729

and were four more than that at VI = 7.11 km/s (Figs. 9h vs 9d). Unlike the near rectangular

730

rear face petalled area at VI = 7.11 km/s, the HVI at VI = 11 km/s inscribed an elliptical

731

petalled area at the rear face.

732 733 734

Qualitative assessment: permanent deformation and debonding

735

experimental one at VI = 4.78 and 7.11 km/s. The model over-predicted debonding compared

736

to that of the experiments. Updating the strength and mode-mixity of model interfaces, based

737

on the hyperstrain-rate 104-106 s-1 [40, 42], will alleviate the deformation discrepancies.

Fig. 10 shows, the predicted out-of-plane deformation of GLARE was beyond the

738

739 740

Fig. 10. Predicted deformation contours against the experimental outcomes; (a) and (b) stand

741

for VI = 7.11 and 4.78 km/s, respectively.

35

742 743 744 745

Table 4 Measured and predicted failures in comparison GLARE panel

VI (km/s)

Front face failure, mm

Difference, %

Rear face failure, mm

((M-P)/M)*100

Difference % ((M-P)/M)*100

Measured

Predicted

Measured

Predicted

746

4.78 17.67 11 37.74 NP NP 7.11 14.40 16 -11.11 25.21 18.5 26.61 GLARE 5-6/5-0.4 9 24.3 20.2 11 37.7 22.6 *NP = No perforation; M = Measured; P = Predicted; the measured damage width

747

corresponds to the pictorial images in Fig. 9; the second tested specimen at VI = 4.78 km/s

748

suffered a 17 mm wide front face failure, while the surplus test at VI = 7.11 km/s narrowed the

749

front and rear face failure by circa -0.6 and -1.1 mm, respectively.

750 751 752 753 754 755

Quantitative assessment: Perforated holes For the quantitative assessment, diameter of the perforated holes in the front face and rear

756

face were compared (Table 4). The digital images, shown in Fig. 9, were post-processed to

757

take the scale measurements between the two closest points at the perforated holes.

758

Quantitative assessments revealed that the numerical model underestimated the front face

759

failure at VI = 4.78 km/s, despite the front face failure was in close agreement with the

760

experiment at VI = 7.11 km/s. This velocity inflicted a non-conservative rear face failure

761

against that of the experiment. The front and rear face succumbed to more damage with the

762

impact velocity, see for VI = 9 and 11 km/s.

763

Dissipated Energy

764

In experiments, the only viable way to approximate the dissipated energy by GLARE

765

damage is: subtract the debris cloud energy from the total impact energy. The energy of the

766

debris cloud is a function of the debris mass and debris velocity. The mass difference of the

767

target plate prior and posterior to the HVI test resulted in the erupted debris mass, while the

36

768

vibration sensor on the witness plate recorded the velocity of the debris cloud. Based on the

769

estimated debris mass and debris velocity, the experiments demonstrated that circa 64 and

770

73 % the projectile impact energy dissipated through the GLARE damage for VI = 4.78 and

771

7.11 km/s, respectively.

772 773

Table 5 The dissipated impact energy in tests and simulations GLARE panel

VI (km/s)

% impact energy dissipated Measured Predicted

Difference, % ((M-P)/M)*100

774

4.78 64 72 -12.5 7.11 73 82 -12.3 GLARE 5-6/5-0.4 9 87.6 11 91.4 *M = Measured; P = Predicted; the measurements of debris mass and debris velocity

775

encompassed a deviation of circa ±1.5 %; the dissipated impact energy of the test specimens

776

exhibited a ±3 % discrepancy, therefore.

777 778

Now, look at the numerical analysis: the GLARE damage dissipated circa 72 and 82 % of

779

the projectile impact energy for VI = 4.78 and 7.11 km/s, respectively. The validation

780

confirmed predicted values within appropriate tolerance levels of the experimental

781

estimations (see Table 5). As the VI increased to 9 and 11 km/s, the dissipated energy reached

782

respectively, 87.6 and 91.4 % of the projectile impact energy. The energy saturation of the

783

GLARE laminate above the shatter regime made the increase of energy dissipation

784

inconspicuous. The extent of irreversible work by the GLARE laminate implies that the

785

energy flux density to the spacecraft bulkhead will be less, which makes the GLARE laminate

786

a potential candidate for the protection shield.

787 788 789

7. Concluding remarks

790

the physical models have to handle the isotropic response of metal and orthotropic mechanics

791

of composites in one numerical framework. This study envisaged the analysis using an

Analyzing the hypervelocity impact of GLAss fiber REinforced aluminum laminates since

37

792

explicit dynamics finite element model of GLAss fiber REinforced aluminum in the state-of-

793

the-art Autodyn hydrocode. While captured the GLAss fiber REinforced aluminum failure

794

modes during the two penetration phases, the code conserved the energy of the computational

795

domain throughout the computation cycles.

796

In the short duration dilatation phase, the hypervelocity-induced shockwave compressed

797

the GLAss fiber REinforced aluminum laminate at an enormous strain-rate, reinforced by the

798

through-thickness compressibility of glass fiber reinforced epoxy composite blocks. At VI =

799

7.11 km/s, the aluminum layers suffered a strain-rate on average 2.25 times the strain-rate the

800

glass fiber reinforced epoxy composite blocks experienced in the impact zone. The disparity

801

between strain rates of frontal material layers was even greater: predicted ɛ̇zz = 87.61 and 10.3

802

µs-1 in the aluminum-6 layer and glass fiber reinforced epoxy-5 composite block, respectively.

803

The aluminum layers collapsed earlier, as a result. Next, the long duration penetration phase

804

provoked the propagation of debonding, splitting the GLAss fiber REinforced aluminum

805

laminate into sub-laminates. Debonding reduced the panel bending stiffness and allowed the

806

glass fiber reinforced epoxy composite blocks to deform in a more efficient membrane state.

807

If contrasted to the experiments, the model over-predicted inter-laminar interface debonding,

808

confirming weaker pre-defined interfaces compared against that of the manufactured

809

specimens. The interface strength of the model could be tuned to restrict the debonding. The

810

familiar phenomenon is, when the interfaces behave stronger, the materials fail a priori and

811

perturbs the computational convergence.

812

As seen, the detached aluminum-spalls degraded with the higher loading rate, indicated the

813

transition from adiabatic tension-shear to material erosion mode. The dissection of shockwave

814

at the GLAss fiber REinforced aluminum-interfaces attenuated the shockwave pressure away

815

from the impact site, attesting to use numerous material layers to promote the dispersion of

38

816

shock front. It is also evident, the heterogeneous fiber/matrix microstructure of glass fiber

817

reinforced epoxy-continua reinforces the pressure attenuation through wave reflection and

818

transmittance at the fiber/matrix interfaces. Composite heterogeneity, moreover, allows petals

819

to conform to the experimental ones. Because, the fiber direction favors the peeling of petals

820

and the transverse-fiber direction exerts friction to impede the rolling of petals. A

821

heterogeneous model is again man-hour and computer intense, and asks for a compromise

822

between the model accuracy and efficiency.

823

By and large, the model accuracy was appreciable. The computational burden, however,

824

seemed to be excessive. Because, the continuous material distribution in the geometrical and

825

material coordinate systems, and the non-linear yield surface affected the computational

826

efficiency. On the other side, the model benefitted from the introduction of progressive

827

softening of glass fiber reinforced epoxy and prevented the premature brittle failure of glass

828

fiber reinforced epoxy composite blocks. So on, the energy-based damage accumulation

829

criteria of glass fiber reinforced epoxy did not allow the iteration error to swamp the solution.

830

The proposed model, resultantly, was stable. Despite the mentioned missing features, the

831

model could reconstruct the failure modes and help analyze the physical phenomena of

832

damage evolution in GLAss fiber REinforced aluminum. Given the effort involved in

833

collecting the material data, developing the numerical model and reaching the computational

834

convergence, the exemplified predictions are credible as the first approximation of GLAss

835

fiber REinforced aluminum 5-6/5-0.4 response to micrometeoroid impacts. Perhaps, the

836

numerical model and enlisted material data enable predictive simulation campaigns to be

837

performed for highly complex GLAss fiber REinforced aluminum grades in the case when

838

test facilities are not accessible.

839

39

840 841 842

Appendix A. Material model of aluminum

843

dependent plasticity of Al-layers. The model assumed that at a strain-rate of 105 sec-1, the

844

yield stress reached the maximum limit and next, was independent of strain-rate. A shear

845

modulus, proportional to pressure and inversely proportional to temperature, made sure the

846

Bauschinger effect was included in the model. The shear modulus (G) and yield stress (Y)

847

read:

848

  G ′p  p  G ′   T G = G0 1 +   1 +  (T − 300 )  3  G0    G0  η 

The semi-empirical flow stress model of Steinberg-Guinan delineated the strain-rate

(A.1)

849 850

  Y p′  p  G ′   n Y = Y0 1 +   1 +  τ  (T − 300 )  (1 + βε ) 3  G0    Y0  η 

(A.2)

851 852 853 854

subject to Y0 [1 + βε ] ≤ Ymax n

where ɛ, β, n, T, η = v0/v signify the effective plastic strain, hardening constant, hardening

855

exponent, temperature, and compression, respectively. The subscripts p and T of primed

856

parameters stand for the derivatives at the reference pressure and temperature (T = 300 K, P =

857

0, ɛ = 0). Subscripts max and zero appoint the maximum value and reference state before the

858

nucleation of a shockwave. Al-elements failed when they reached the failure strain in three

859

normal and three shear directions.

860 861 862

Funding acknowledgements

863

commercial, or not-for-profit sectors.

864 865

Declarations of interest: none

This research did not receive any specific grant from funding agencies in the public,

40

866

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867

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Highlights  The penetration mechanism of GLAss fiber REinforced aluminum, subjected to hypervelocity impacts, was analyzed.  A shockwave of the high frequency band resulted in the bulk failure of outer aluminum skin at the impact site.  The pressure of shockwave attenuated away from the impact site due to the wave reflection and transmittance at the GLARE interfaces.  Composite blocks stored substantial internal energy through elastic stretching before being pierced.  For the impact velocity of 9 km/s, Al-spalls and GF/EP-splinters populated the downrange debris cloud.

Highlights • The penetration mechanism of GLAss fiber REinforced aluminum, subjected to hypervelocity impacts, was analyzed. • A shockwave of the high frequency band resulted in the bulk failure of outer aluminum skin at the impact site. • The pressure of shockwave attenuated away from the impact site due to the wave interference and transmittance at the GLARE interfaces. • GF/EP composite blocks stored substantial internal energy through elastic stretching before being pierced. • For the impact velocity of 9 km/s, Al-spalls and GF/EP-splinters populated the downrange debris cloud.