UNCORRECTED PROOF - Clemson University

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UNCORRECTED PROOF - Clemson University

1

J Mater Sci

DOI 10.1007/s10853-006-0864-3

2 Lattice Boltzmann method based computation

3 of the permeability of the orthogonal plain-weave

4 fabric preforms

5 M. Grujicic K.M. Chittajallu Shawn Walsh

6 Received: 9 July 2005 / Accepted: 7 November 2005 / Published online: j

7 Ó Springer Science+Business Media, LLC 2006

8

9 Abstract Changes in the permeability tensor of fabric

10 preforms caused by various modes of fabric distortion

11 and fabric-layers shifting and compacting is one of the

12 key factors controlling resin flow during the infiltration

13 stage of the common polymer-matrix composite liquid-

14 molding processes. While direct measurements of the

15 fabric permeability tensor generally yield the most

16 reliable results, a large number of fabric architectures

17 used and numerous deformation and layers rear-

18 rangement modes necessitates the development and

19 the use of computational models for prediction of the

20 preform permeability tensor. The Lattice Boltzmann

21 method is used in the present work to study the effect

22 of the mold walls, the compaction pressure, the fabric-

23 tows shearing and the fabric-layers shifting on the

24 permeability tensor of preforms based on orthogonal

25 balanced plain-weave fabrics. The model predictions

26 are compared with their respective experimental

27 counterparts available in the literature and a reason-

28 ably good agreement is found between the corre-

29 sponding sets of results.

30

31 Nomenclature

33 34 f Particle distribution function

36 35 ff Fiber volume fraction

M. Grujicic (&) Æ K.M. Chittajallu

Department of Mechanical Engineering,

Clemson University, 241 Engineering Innovation Building,

Clemson, SC 29634-0921, USA

e-mail: mica.grujicic@ces.clemson.edu

h Fabric thickness (m)

37 38

/ Relative dimensionless shift of the adjacent fabric 39 40

layers

41

K Permeability tensor of the fabric (m 42 43

2 )

L In-plane quarter cell dimension (m)

44 45

p Pressure (Pa)

46 47

a Shear angle (deg.)

48 49

rf Fiber radius (m)

50 51

s Relative shift of the adjacent fabric layers (m) 52 53

e Particle velocity component

54 55

q Fluid point density (particles/lattice point) 56 57

W Collision operator

58 59

t Time (s)

60 61

s Time relaxation parameter

62 63

ti Velocity component weighting factor

64 65

u Fluid nodal velocity (lattice parameters/time 66 67

increment)

68

m Fluid kinematic viscosity (m 69 70

2 /s)

x Nodal position vector

71 72

73

Subscripts

74

75

76

bot Quantity associated with the bottom surface of 78 79

the fabric

80

top Quantity associated with the top surface of the 81 82

fabric

83

84

85

86

Superscripts

87

B Quantity associated with the bottom channel 89 90

eq Equilibrium quantity

91 92

F Quantity associated with the fabric

93 94

T Quantity associated with the top channel

95 96

UNCORRECTED PROOF

S. Walsh

Army Research Laboratory—WMRD AMSRL-WM-MD,

Proving Ground, Aberdeen, MD 21005-5069, USA

Journal : 10853 Dispatch : 11-9-2006 Pages : 12

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123


100 97 Introduction

98

99

101 Over the last two decades, major advances in the resin-

102 injection technologies have enabled the processing of

103 high-performance polymer-matrix composites expand

104 from their aerospace roots to diverse military and civil

105 applications. At the same time, the processing science

106 has become an integral part of the composite-manu-

107 facturing technology so that empiricism and

108 semi-empiricism have given way to a greater use of

109 computer modeling and simulations of the fabrication

110 processes. Among the modern polymer-matrix com-

111 posite manufacturing techniques, liquid-molding pro-

112 cesses such as Resin Transfer Molding (RTM),

113 Vacuum Assisted Resin Transfer Molding (VARTM)

114 and Structural Reaction Injection Molding (SRIM)

115 have a prominent place. A detailed review of the major

116 liquid molding processes can be found in the recent

117 work of Lee [1]. One common feature to all these

118 composite fabrication processes is the use of low-

119 pressure infiltration of the porous fabric preforms with

120 a viscous fluid (resin). Conformation of the fabric

121 preforms to the ridges and recesses in the mold and the

122 applied pressure can induce significant distortions and

123 deformations in the fabric as well as give rise to shifting

124 of the individual fabric layers and, in turn, cause sig-

125 nificant change in local permeability of the preform.

126 Since the infiltrating fluid follows the path of least

127 resistance, local changes in the fabric permeability can

128 have a great influence on the mold filling process

129 affecting the filling time, the filling completeness and

130 the formation of pores and dry spots. Hence, a better

131 understanding of the changes in fabric permeability

132 caused by various local distortions, shearing and

133 shifting of its tows is critical for proper design of the

134 liquid molding fabrication processes.

135 In simple terms, permeability can be defined as a

136 (tensorial) quantity which relates the local average

137 superficial velocity vector of the fluid with the associ-

138 ated pressure gradient. In polymer-matrix composite

139 liquid-molding processes such as the RTM and the

140 VARTM, the porous medium consists of woven- or

141 weaved-fabric preforms placed in the mold and the

142 fluid flow of interest involves preform infiltration with

143 resin. Complete infiltration of the preform with resin is

144 critical for obtaining high-integrity and high-quality

145 composite structures. The knowledge of the preform

146 permeability and its changes due to fabric bending,

147 shearing, compression, shifting, etc. is crucial in the

148 design of a composite fabrication process (e.g., in the

149 design of the tool plate, or for placement of the resin

150 injection ports).

123

J Mater Sci

While direct experimental measurements of the

fabric permeability is generally considered as most

accurate and most reliable, it suffers from a number of

limitations such as: (a) experimental set-ups can be

elaborate and the measurements very time consuming;

(b) experimental measurements have to be done for

each new architecture and/or deformation state of the

fiber preform; and (c) experiments involve measurements

of the (macro) preform length-scale flow

parameters (e.g., the flow length or the shape and the

size of a flow front) are used to infer a (meso) fabric/

fiber length-scale parameter (the permeability tensor).

In more complex preform architectures, this procedure

is a formidable task and can be accompanied with

substantial experimental and data reduction errors.

Consequently, development of the computational

models for prediction of the preform permeability

tensor to complement experimental measurements has

become a standard practice.

For the computational modeling approach to be

successful in predicting permeability of the fabric

preforms, it must include, in a correct way, both the

actual architecture of the fabric and the basic physics of

the flow through it. A schematic of the relatively simple

orthogonal plain-weave one-layer fabric architecture

is shown in Fig. 1. As seen in Fig. 1, the fabric

consists of orthogonal (warp and weft) yarns, which are

woven together to form an interconnected network.

Each yarn, on the other hand, represents a long con-

UNCORRECTED PROOF

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Fig. 1 A schematic of (a) the top view and (b) the edge view of a

one-layer orthogonal plain-weave fabric preform

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J Mater Sci

180 tinuous length of interlocked fibers, normally grouped

181 into bundles (threads) twisted together to form the

182 final yarn. In addition, the fabric involves a network of

183 empty pores and channels. When a fabric like the one

184 shown in Fig. 1 is being infiltrated, the resin flows

185 mainly through the pores and the channels. However,

186 since the fabric tows are porous (pores and channel on

187 a finer length scale exist between the fibers in tows),

188 the resin also flows within the yarn. Thus, when pre-

189 dicting the effective permeability of a fabric, the

190 computational model should account for both compo-

191 nents of the resin flow.

192 Prediction of the permeability of porous medium

193 has been the subject of intense research for at least last

194 two decades. Due to space limitations in this paper, it is

195 not possible to discuss all the models proposed over

196 this period of time. Nevertheless, one can attempt to

197 classify the models. One such classification involves the

198 following main types of models for permeability pre-

199 diction in the porous media: (a) the phenomenological

200 models based on the use of well-established physical

201 concepts such as the capillary flow (e.g., [2, 3]) or the

202 lubrication flow (e.g., [4]). These models generally

203 perform well within isotropic porous media with a

204 simple architecture; (b) the numerical models which

205 are based on numerical solutions of the governing

206 differential equations. These models generally attempt

207 to realistically represent the architecture of the fiber

208 preform but, due to limitations in the computer speed

209 and the memory size, are ultimately forced to the

210 introduction of a number of major simplifications (e.g.,

211 [5, 6]); and (c) the models which are based on a balance

212 between the fabric-architecture and the flow physics

213 simplifications, enabling physically based predictions of

214 the preform permeability within reasonably realistic

215 fabric architectures [7].

216 Over the last decade, the Lattice Boltzmann method

217 has become a respectable alternative to the classical

218 Navier–Stokes equations based computations of vari-

219 ous fluid flow phenomena. In particular, the lattice

220 Boltzmann method has been found to offer computa-

221 tional advantage in the analyses of fluid flow in media

222 with a complex geometry [8, 9], in multi-length-scale

223 porous media [10], moving boundary multi-phase flow

224 [11]. Since, in general, fabric preforms act as a porous

225 media with complex multi-length-scale architectures,

226 and the dynamics of their filling during the resin

227 injection stage of the RTM and VARTM processes,

228 the Lattice Boltzmann method has been used in the

229 present work to compute the permeability components

230 of fabric preforms under different modes of deforma-

231 tion. The permeability components are next assembled

232 into a permeability tensor.

The organization of the paper is as follows: A brief

overview of the fabric architectures and the Lattice

Boltzmann method is presented in ‘‘Computional

procedure’’ section. The application of the Lattice

Boltzmann method to revealing the role of various

fabric distortion and layers-compaction and shifting

phenomena is presented and discussed in ‘‘Results and

discussion’’ section. The main conclusions resulted

from the present work are summarized in ‘‘Conclusions’’

section.

Computational procedure

Fabric architecture

Single-layer fabrics

In this work, only (un-sheared and sheared) balanced

orthogonal plain-weave fabric is considered. Due to

the in-plane periodicity, the fabric architecture can be

represented using a unit cell. The entire orthogonal

plain-weave fabric can then be obtained by repeating

the unit cell in the in-plane (x- and y-) directions. A

schematic of one quarter of a plain-weave unit cell with

the appropriate denotation for the system dimensions

are shown in Fig. 2. In a typical plain-weave fabric, the

fabric thickness (h) to the quarter cell in-plane

dimension (L) ratio, h/L, is small (0.01–0.1), while the

tow cross-section is nearly elliptical in shape with a

large (width-to-height) aspect ratio (five or larger). The

geometry of the tows within the cell can be described

using various mathematical expressions (e.g., [12]) for

the top, ztop(x,y), and the bottom, zbot(x,y), surfaces of

the fabric, respectively. In the present work, the fol-

UNCORRECTED PROOF

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Fig. 2 Schematic of one quarter of the unit cell for a one-layer

balanced orthogonal plain-weave fabric

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263 lowing sinusoidal functions originally proposed by Ito

264 and Chou [13] are used:

266

ztopðx; yÞ

¼ h

2

zbotðx; yÞ

¼ h

2

sin 2p

L

2p

x þ sin y ð1Þ

L

2p 2p

sin x þ sin y ð2Þ

L L

268 269 Since the fiber tows have typically a near elliptical

270 cross-section, Eqs. (1) and (2) only approximate the

271 actual tow cross-section shape. Nevertheless, they are

272 used in the present work since they greatly simplify

273 permeability calculations in the distorted fabric and are

274 generally considered as a good approximation for the

275 actual tow cross-section shape.

276 A simple examination of Fig. 2 shows that within a

277 single-layer orthogonal plain-weave fabric unit cell,

278 one can identify three distinct domains:

The top channel, Region T : ztop\ z\ h

2

280 The fabric, Region F : zbot\ z\ztop

282

The bottom channel, Region B :

h

2

\ z\zbot

284 285 Regions T and B contain only the resin, while region

286 F can generally contain both the fiber tows and the

287 resin. The resin flow through a unit cell is analyzed in

288 the present work by considering only the flow within

289 the top and the bottom channels. In our previous work

290 [7], it was shown that the resin flow within the fabric

291 has a minor contribution to the overall permeability of

292 the perform.

293 Multi-layer fabrics

294 In typical RTM and VARTM processes, the preforms

295 may contain several fabric layers. In such multi-layer

296 preforms, nesting and compaction can generally have a

297 significant effect and must be included when predict-

298 ing preform permeability. Numerous experiments

299 (e.g., [14, 15]) confirmed that permeability varies with

300 the number of layers. A schematic of two types of two-

301 layer plain-weave fabric preforms analyzed in the

302 present work is given in Fig. 3(a) and (b). The two

303 types are generally referred to as ‘‘in-phase’’ and ‘‘out-

304 of-phase’’ fabric architectures or laminates. The

305 mathematical expressions for the top and the bottom

306 surfaces in each preform layer can be readily defined

307 using Eqs. (1) and (2).

123

Sheared fabrics

As pointed out earlier, when the fabric preform is

forced to conform to the ridges and recesses of a mold,

it may locally undergo shear deformation. Such

deformation can significantly affect local permeability

of the preform. As shown in Fig. 4, when a balanced

square-cell plain-weave fabric is sheared, the unit cell

size increases and to make the calculations of preform

permeability manageable, the shear angle a =tan –1 (m/n)

is generally allowed to take only the values corresponding

to relatively small integers m and n.

Compacted fabrics

UNCORRECTED PROOF

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J Mater Sci

Fig. 3 x – z section of a quarter of the unit cell for: (a) an

in-phase; and (b) an out-of-phase two-layer orthogonal plainweave

fabric

When the fabric is subjected to compression during

mold closing in the RTM process or during evacuation

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J Mater Sci

Fig. 4 Effect of fabric shearing on the size of the quarter unit

cell (denoted using heavy dashed lines) in balanced plain-weave

fabric architectures: (a) un-sheared fabric; (b) fabric sheared by

an angle a = tan –1 (1/3) in the x-direction

322 of the vacuum bag in the VARTM process, it under-

323 goes a number of changes such as: the cross-section of

324 the fiber tow flattens, the pores and gaps between the

325 fibers inside tows as well as between individual tows

326 are reduced, the tows undergo elastic deformation,

327 inter-layers shifting (nesting), etc. A typical compres-

328 sion-pressure versus preform thickness curve for a

329 woven fabric is depicted in Fig. 5 [16]. The curve

330 shown in Fig. 5 has three distinct parts: two linear and

331 one non-linear. In the low-pressure linear and the non-

332 linear portions of the pressure versus thickness curve,

333 preform compaction is dominated by a reduction of the

334 pore and the gap sizes between the fibers in tows. In

335 the high-pressure linear region of the pressure versus

336 thickness curve, on the other hand, preform compac-

337 tion involves mainly tow bending and nesting as well as

338 fiber compression. Typical liquid molding processes

339 such as RTM or VARTM involve pressures which

340 correspond to the high-pressure linear pressures versus

341 thickness region. Hence, the effect of fabric compac-

Fig. 5 A typical compaction-pressure versus preform-thickness

curve for a plain-weave fabric architecture

tion on permeability of the fabric preform associated

with the high-pressure linear compaction regime is

investigated in this section.

To quantify the effect of preform compaction (in the

high-pressure linear region) on permeability of the

balanced orthogonal plain-weave fabric, the beambending

based micro-mechanical model developed in a

series of papers by Chen and Chou [17–19] is utilized in

the present work. The model of Chen and Chou

[17–19] is based on a number of well-justified

assumptions such as: (a) the fabric is considered to

extend indefinitely in the x – y plane and, hence, can

be represented using the unit cells such as the one

shown in Fig. 2; (b) tows in the fabric are treated as a

transversely isotropic solid material; (c) the fabric is

subjected to the compaction pressure only in the

through-the-thickness direction, and can freely adjust

its shape in the x – y plane. This assumption is less

justifiable in the regions of the fabric which are in

direct contact with the mold where friction can impose

some lateral constraint to the fabric. No attempt was

made in the present work to quantify this effect; (d)

since the compaction analyzed corresponds to the highpressure

linear region, no voids or gaps are assumed to

exist between the fibers in tows or between the tows;

(e) during fabric compaction, the cross-section area of

the tows is assumed to remain unchanged but the shape

of the cross-section undergoes a change; and (f) as

compaction proceeds, the deformation of the tows

leads to an increase in the effective volume fraction of

UNCORRECTED PROOF

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372 the fibers in the fabric and, in the limit of complete

373 compaction of the tows, the volume fraction of the

374 fibers in the fabric becomes equal to that in the indi-

375 vidual tows.

376 In order to derive a relationship between the

377 reduction in the fabric thickness, the effective volume

378 fraction of the fibers and various distributions and

379 magnitudes of the applied compaction pressure, Chen

380 and Chou [17–19] applied a simple procedure from the

381 solid-mechanics beam theory. Toward that end, the

382 one-quarter unit cell shown in Fig. 2 is first simplified

383 by replacing the two warp and the two weft tows with

384 four beams. Next based on the symmetry of the sim-

385 plified model, it is shown that the problem can be

386 further simplified using a single beam and the appro-

387 priate distribution of the applied and contacting pres-

388 sures, Fig. 6. The model of Chen and Chou [17–19] is

389 utilized in the present work to compute the effect of

390 the compaction pressure on the channel heights

391 (h T (x,y) and h B (x,y)) and on the fabric thickness,

392 h F (x,y ) fields. These fields are used to quantify the

393 effect of fabric compaction on the effective perme-

394 ability of a one-layer orthogonal plain-weave fabric.

395 Nested fabrics

396 As mentioned earlier, shifting of fabric layers followed

397 by their more compact packing (the phenomenon

398 generally referred to as layers ‘‘nesting’’) can have a

399 major effect on the effective fiber density in the pre-

form and, hence, on permeability of the preform.

Nesting of the fabric layers can particularly take place

under high applied pressures which are sufficient to

overcome inter-tow friction. The thickness reduction in

balanced orthogonal plain-weave fabrics whose geometry

is represented by Eqs. (1) and (2), due to layers

nesting has been analyzed by Ito and Chou [13] who

derived the following relation for the fabric thickness

reduction caused by nesting:

Dh nesting ¼

8

2 cos / x

2

2 sin

h

/ j xj

>< 2

2 cos / x

2

2 sin / >:

j xj

2

cos / y

2 ; / j xj

cos / y p

;

2 2

sin / y

2 ; / j xj

sin / y p

;

2 2

p

2 ; / y

UNCORRECTED PROOF

Fig. 6 A schematic of the pressure distribution on a curved

beam used in the calculation of permeability of un-sheared onelayer

orthogonal plain-weave fabrics

123

/ x

p

2

j j p; / y

p

2 ;p

2

/ y

j/ xj

p; p

2

p

2

p

/ y

p

ð3Þ

where / x ¼ 2p

L sx and / y ¼ 2p

L sy are dimensionless while

sx and sy are the dimensional relative shifts of the

adjacent layers in the x- and y-directions, respectively.

Two non-nesting cases associated with zero nesting,

reduction in the fabric thickness can be identified: (a)

/x = /y = 0 which corresponds to the iso-phase lami-

nate case and (b) /x = /y = ± p/2 corresponding to

the out-of-phase laminate case.

The relations given in Eq. (3) are used in the present

work to examine the effect of layers nesting on fabric

permeability. While, in general, fabric compaction

during the high-pressure linear compaction stage can

involve both elastic distortions (tow bending) and

layers nesting, the two modes of fabric compaction are

generally considered as decoupled and can be considered

separately.

The Lattice Boltzmann method

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J Mater Sci

As stated earlier, the computation of the fabric-preform

permeability tensor is carried out in the present work

using the Lattice Boltzmann method. Within this

method, the (continuum) physical space is replaced with

a large number (typically 10 6 –10 7 ) of discrete points

(nodes). When such nodes do not reside within a solid

(e.g., the points located outside the mold walls or reinforcing

fibers), they are assumed to be populated with

‘‘fluid’’ particles. Particles residing on a node are

allowed to have only one of a finite number of velocities

(typically the pores with a direction toward the nearest

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438 and the next nearest neighbor nodes). For example, in

439 the D3Q19 lattice used in the present work, Fig. 7, any

440 particle residing at node denoted as 0 can have either a

441 zero velocity (be stationary) or have a velocity with a

442 direction toward the nearest neighbor nodes (nodes 1, 3,

443 5, 7, 9, 18) and toward the next nearest neighbor nodes

444 (nodes 2, 4, 6, 8, 10–17). The magnitude of the velocity is

445 the same for all particles and it is equal to a ratio of the

446 lattice constant and the time increment. Within the

447 Lattice Boltzmann method, then, one computes the

448 temporal and spatial evolutions of a distribution func-

449 tions for such particles. Macroscopic fluid quantities

450 such as the point density q(x,t), and the velocity u(x,t)

451 (where x is the node position vector and t the time), can

452 then be computed as simple moment sums of these

453 particle distribution functions as:

455 and

J Mater Sci

qðx; tÞ

¼ Xn

uðx; tÞ

¼

i¼1

P n

i¼1

fiðx; tÞ

ð4Þ

fiðx; tÞei

qðx; tÞ

457 where fi(x,t) is the particle distribution function, i is the

458 velocity component, n is the total number of velocity

459 components per node (1 + the number of nearest

460 neighbors (6) + the number of next nearest neighbors

461 (12) = 19, in the case of the D3Q19 lattice).

462 The evolution of the particle distribution function

463 fi(x,t) is governed by the Lattice Boltzmann equation as:

6

5

X

12

13

7

17

4

16 18

Z

0

ð5Þ

fiðxþei; t þ 1Þ

¼ fiðx; tÞþXiðx;


ð6Þ

where the time increment is set explicitly to 1.

The collision operator, Wi(x,t) on the right hand side

of Eq. (6), represents the rate of change of the particle

distribution function due to particle collisions at the

nodes. Following Bhatnager et al. [20], the collision

operator is greatly simplified (the so-called BGK simplification)

through the use of a single time relaxation

parameter s as:

Xiðx; tÞ

¼ 1

s fiðx; tÞ

f eq

i ðx; tÞ

ð7Þ

where fi eq (x,t) is the equilibrium particle distribution

function at the node with a position vector, x, at time, t,

while s controls the rate at which the particle distribution

function approaches its equilibrium value.

The equilibrium particle distribution is represented

by relations:

f eq

0

and

f eq

i

qðxÞ ðx; tÞ

¼

3

ðx; tÞ

¼ tiqðxÞ UNCORRECTED PROOF

Fig. 7 The D3Q19 lattice consisting of 19 particle velocity

vectors including the zero-magnitude velocity vector for stationary

particles denoted as ‘‘O’’

9

8

11

3

1

15

10

14

Y

2

1

3

jux; ð tÞj2

2

1 þ 3ei ux; ð tÞþ

3

2 3eiei : ux; ð tÞux;

ð tÞ

jux; ð tÞj

2

ð8Þ

for i 6¼ 0; ð9Þ

and for the D3Q19 lattice, ti = 1/6 for the velocities

along the Cartesian axes (the ones with the direction

toward the nearest neighbor nodes) and ti = 1/36 for

the remaining velocities. Chen et al. [21] showed that

the formulation of the Lattice Boltzmann method

presented above yields a velocity field which is a

solution to the Navier–Stokes equation with the kinematic

viscosity equal to:

s 0:5

m ¼

3

ð10Þ

It should be noted that the equilibrium particlevelocity

distribution function defined by Eqs. (8) and

(9) are derived under the condition that the imposed

(average) density/velocity are consistent with the

ones obtained using Eqs. (4) and (5).

Computer program

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The computation of the fabric permeability tensor

has been carried out using H. W. Stockman’s Lattice

Boltzmann JB_PH code [22], a program for modeling

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502 multi-component flow and dispersion. Due to the use

503 of an optimization scheme and the BGK simplifica-

504 tion of the Lattice Boltzmann method, the code is

505 quite fast even on a single-CPU computer so that

506 complex solids geometries containing millions of

507 nodes can be modeled. Accuracy of the code has

508 been validated using four benchmark problems: (a)

509 permeability of a simple cubic array of spheres; (b)

510 Taylor-Aris dispersion between infinite plates; (c)

511 dispersion in a duct; and (d) dispersion in a simple

512 cubic array of spheres.

513 Results and discussion

514 Un-sheared single-layer plain-weave fabric

515 preforms

516 The Lattice Boltzmann method briefly overviewed in

517 ‘‘The Lattice Boltzmann method’’ section is used in

518 this section to analyze the velocity distributions

519 within the un-sheared single-layer balanced orthogo-

520 nal plain-weave quarter unit cell. It should be poin-

521 ted out that due to the symmetry of the unit cell with

522 respect to the z = 0 mid-plane, the velocity distribu-

523 tions within the top and the bottom resin channels

524 are expected to be identical and, hence, no transverse

525 flow of the resin through the fabric preform is

526 expected. Also, it should be noted that due to the

527 symmetry of the fabric geometry with respect to the

528 quarter unit-cell boundaries normal to the y-direc-

529 tion, a zero fluid-flow velocity is expected across

530 these boundaries.

531 The variations of the top- and bottom-channel

532 heights and of the fabric thickness in the x – y plane

533 within a quarter unit cell, used as input in the present

534 analysis, are shown in Fig. 8(a) and (b), respectively.

Fig. 8 (a) Resin channels

height and (b) fabric

thickness fields in an

un-sheared one-layer

orthogonal plain-weave fabric

preform

123

0.00005m

0.00015m

0.00025m

0.00035m

0.00045m

The variations of the x-component resin velocity within

the resin channels of a quarter unit cell over the x – y

planes corresponding to y = 0 and y = L/2 are for the

fixed imposed pressure gradient dP/dx = – Fxq =

1.0 · 10 7 Pa/m in the x-direction (Fx is the x-component

of the body force) are shown in Fig. 9(a) and (b).

The results displayed in Fig. 9(a) and (b) show that the

velocity distribution is symmetric about the z = 0 midplane.

In addition, a comparison of the results shown in

Fig. 9(a) and (b) with the ones shown in Fig. 8(a) and

(b) reveals that the magnitude of the x-component of

the resin velocity at a given x – y location correlates

inversely with the local height of the resin channel in

order to satisfy the continuity equation. It should be

noted that in order to improve clarity, the dimension of

the quarter unit cell in the z-direction is magnified ten

times in Fig. 9(a) and (b).

Effect of the number of layers in un-sheared

plain-weave fabric preforms

The Lattice Boltzmann method is used in the present

section to predict permeability of the balanced

un-sheared single- and multi-layer orthogonal plainweave

fabric architectures. In all the calculations carried

out in this section, as well as in the calculations

carried out in the previous section, the following unit

cell parameter and one-layer fabric thickness values

are used: L1 = L2 = L = 0.01 m and h = 0.001 m.

To determine the permeability tensor, the procedure

described in [22] is utilized. According to this procedure,

Kij component of the permeability tensor is

defined as:

Kij ¼ mUi

UNCORRECTED PROOF

Fj

0.0009m

Journal : 10853 Dispatch : 11-9-2006 Pages : 12

0.0007m

Article No. : 864

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0.0005m

0.0003m

0.0001m

J Mater Sci

ð11Þ

(a) (b)

535

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J Mater Sci

0.0017 0.0013

0.0009

0.0005

0.0001

Fabric Fabric

0.9e-7

2.7e-7

Fabric

1.8e-7

3.6e-7

4.5e-7

567 where Ui is the i-component of the superficial average

568 velocity and Fj the j-component of the body force. In

569 other words, components of the permeability tensor

570 are computed using the imposed pressure gradient

571 across the computational cell in a given direction and

572 the corresponding resulting average resin velocity. The

573 knowledge of spatial variation of the fluid pressure is

574 not required and is not calculated.

575 It should be noted that since the present analysis

576 considers solely the macro-scale fluid flow between the

(a)

(b)

Fig. 9 The distribution of the x-component of the resin velocity

in an un-sheared one-layer orthogonal plain-weave fabric

preform within (a) y = 0 and (b) y = L/2 planes

fabric layers and the mold walls and not the microscale

fluid flow between the fibers within the fabric, the

effects of capillarity/surface tension are deemed secondary.

To determine the effect of the number of fabric

layers on the effective permeability, the model developed

in the previous section is used for the cases of 1-,

2-, 3-, 5-, 10- and 20-layer in-phase orthogonal balanced

plain-weave fabric preforms in the absence of layer

nesting. The results of this calculation are presented in

Fig. 10. These results show that as the number of layers

increases, the permeability rises but at an ever

decreasing rate so that in fabric preforms with 10 or

more layers, the effect of the number of layers on

permeability becomes insignificant. This finding can be

easily rationalized by recognizing that as the number of

layers in the fabric increases, the effect of the bottom

and the top resin channels which are more restrictive to

the fluid flow (and thus reduce effective preform permeability)

decreases. Also shown in Fig. 10 are the

results of our recent calculations based on the use of a

lubrication flow model [7]. Since the two sets of the

results are in a very good agreement, the resin flow

through the fiber preform (neglected in the present

work and considered in [7]), appears to make a minor

contribution to the fabric preform permeability.

Effect of fabric shear on permeability

The effect of shear deformation (measured by the

magnitude of the shear angle h) on the effective permeability

of single-layer plain-weave fabric preforms is

displayed in Fig. 11. An example of the variation of the

Effective Preform Permeability, m 2

5E-10

0 3 6 9 12 15 18 21

Number of Fabric Layers in the Preform

UNCORRECTED PROOF

3E-09

2.5E-09

2E-09

1.5E-09

1E-09

Ref. [7]

Journal : 10853 Dispatch : 11-9-2006 Pages : 12

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Present Work

Fig. 10 The effect of the number of fabric layers on the effective

permeability of an un-sheared balanced plain-weave fabric

preform

123

577

578

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Preform Permeability Components, m 2

1E-09

8E-10

6E-10

4E-10

2E-10

Shear Angle, deg

Kxx [23]

Kyy [23]

608 top- and bottom-channel heights and of the fabric

609 thickness in the x – y plane within a quarter unit cell,

610 used as input in the present analysis, are shown in

611 Fig. 12(a) and (b), respectively. For comparison, the

612 experimental values of preform permeabilities ob-

613 tained in [23] are also shown in Fig. 11. While the

614 agreement between the corresponding computational

615 and the experimental values is only fair, the effect of

616 shear deformation on preform permeability appears to

617 be quite well predicted by the model. In addition, the

618 corresponding computed values of the in-plane off-

619 diagonal (K xy and K yx) elements of the effective per-

620 meability are very close as required by symmetry of the

621 orthogonal plain-weave fabric architecture.

622 When the fabric is sheared in the x-direction, as

623 shown in 4(b), weft tows are rotated but remain stress

624 free. Consequently, the dimension of the fabric-pre-

625 form unit cell in the y-direction is altered causing a

626 change in the effective fiber volume fraction in the unit

627 cell. This change in the fiber volume fraction can have

Kxx

Kyy

Kxy

Kyx

0

0 5 10 15 20 25 30

Fig. 11 The effect of shear on permeability of a single-layer

balanced orthogonal plain-weave fabric preform

Fig. 12 (a) Resin channels

height and (b) fabric

thickness fields in a one-layer

orthogonal plain-weave fabric

preform subjected to shear in

the x-direction by an angle of

a = tan –1 (1/3).

123

a significant effect on preform permeability at large

shear angles and, hence, must be taken into account.

The procedure described below is used to correct the

values of permeabilities in the sheared fabrics obtained

using the Lattice Boltzmann method.

To quantify the permeability correction described

above, the Kozeny–Carman relation (e.g., [24]) for

permeability of the porous media with a fibrous

architecture is used. According to this relation, permeability

of such media is given by:

K ¼ r2 3

f ð1 ffÞ

UNCORRECTED PROOF

0.00005m

0.00015m

0.00045m

0.00035m

0.00025m

cf 2

f

ð12Þ

where rf and ff are the fiber radius and the fiber volume

fraction, respectively, while c is a fibrous-medium

architecture-dependent constant. It should be noted

that Eq. (11) is based on the assumption that no shear

of the fibers accompanies fabric shear. Furthermore, it

is possible to more accurately account for the change in

the effective fiber volume fraction in the unit cell

through a direct simulation of the fabric shear process.

However, such an analysis was beyond the scope of the

present work.

When the fabric preform is sheared in the x-direction

by an angle h, the fiber volume fraction in fabric

tows changes as:

ff;h ¼

ff;o

sinð90 hÞ

ð13Þ

where the angle h is given in degrees and the subscripts

o and h are used to denote the value of a respective

quantity in the un-sheared fabric and in the fabric

sheared by an angle h, respectively.

To account for a shear-induced change in the fiber

volume fraction, the permeability values for sheared

0.0009m

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(a) (b)

0.0007m

0.0001m

0.0003m

0.0005m

J Mater Sci

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J Mater Sci

Effective Preform Permeability, m 2

8E-10

7.5E-10

7E-10

6.5E-10

6E-10

5.5E-10

5E-10

0 2 4 6 8 10 12 14 16

659 fabric preforms obtained using the Lattice Boltzmann

660 method are multiplied by the following correction

661 factor:

Kcorr ¼ f 2 3

o ð1 fhÞ

ð1 foÞ 3 f 2 h

This Work

Ref. [25]

Compaction Force, N

Fig. 13 Effect of compaction (represented by the magnitude of

the compaction force) on permeability of a one-layer un-sheared

orthogonal plain-weave fabric preform

ð14Þ

663 664 It should be noted that the effect of fabric shear on

665 permeability analyzed in the present work is fairly

666 simplified and does not include the role of potentially

667 important phenomena such as complex meso-scale tow

668 deformation modes during fabric shear.

669 Effect of preform compaction on permeability

670 The effect of the compaction force applied to the upper

671 and the lower (rigid and flat) molds on the effective

Fig. 14 (a) Resin channels

height and (b) fabric

thickness fields in an unsheared

one-layer orthogonal

plain-weave fabric preform

subject to a total compressive

force of 10.3 N via rigid, flat

upper and lower tool surfaces

0.00005m

0.00015m

0.00025m

0.00035m

permeability of a one-layer orthogonal plain-weave

fabric is displayed in Fig. 13. In these calculations, the

Young’s modulus is assigned a value of 22 GPa and a

sinusoidal distribution of the applied and the contacting

pressures is assumed [19]. Since fabric compaction

in the mold precedes resin infiltration, it is assumed

that the fibrous structure is supported only by the solid

material and not by the fluid. An example of the variation

of the top- and the bottom-channel heights and

of the fabric thickness in the x – y plane within a

quarter unit cell, used as input in the present analysis,

are shown in Fig. 14(a) and (b), respectively. For

comparison, the experimental results reported by

Sozer et al. [25] are also shown in Fig. 13. It is seen that

Dimensionless Shift in the y-Direction

1.5

0.5

1.0

0.76

0.49

0

0 0.5 1 1.5

Dimensionless Shift in the x-Direction

(a) (b)

UNCORRECTED PROOF

1

0.73

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0.44 0.37 0.30

0.0007m

0.55

0.0005m

0.0003m

0.0001m

0.38

Fig. 15 The effect of nesting on the ratio of fabric permeability

at the given values of layer shifts in the x- and the y-directions

and fabric permeability of an un-nested in-phase laminate fabric

123

672

673

674

675

676

677

678

679

680

681

682

683

684

685


686 a reasonably good agreement exists between both the

687 magnitude of the predicted preform permeability and

688 its change with the applied compaction force.

689 Effect of layer nesting

690 The effect of nesting (quantified by the magnitudes of

691 the dimensionless layer shifts in the x- and the y-

692 directions, /x and /y, respectively, in a two-layer

693 orthogonal plain-weave fabric is shown in Fig. 15. The

694 values displayed in Fig. 15 pertain to the ratio of fabric

695 permeability at the given values of /x and /y and fabric

696 permeability at / x = / y = 0. As expected, fabric nest-

697 ing gives rise to the reduction in fabric permeability.

698 Furthermore, for the case of a out-of-phase laminate

699 fabric (/ x = / y = ± p/2), fabric permeability is only

700 about 30% of its value in the in-phase laminate fabric.

701 This finding is in excellent agreement with its experi-

702 mental counterpart reported by Sozer et al. [25].

703 Conclusions

704 Based on the results obtained in the present work, the

705 following main conclusions can be drawn:

706 1. Effective permeability of the orthogonal plain-

707 weave fabric preforms can be determined compu-

708 tationally using the Lattice Boltzmann method for

709 the analysis of resin flow through tool-surface/

710 fabric-tow and tow/tow channels.

711 2. The computational approach presented in this

712 work enables assessment of the contribution that

713 various phenomena such as the mold walls, fabric

714 shearing, interlayer shifting and restacking as well

715 as fabric compaction due to the infiltration pres-

716 sure make to orthogonal plain-weave fabric-pre-

717 form permeability.

718 3. While no comprehensive set of experimental data

719 is available to fully test validity of the present

720 model, the agreement of the model predictions

721 with selected experimental results can be generally

722 qualified as reasonable.

123

Acknowledgements The material presented in this paper is

based on work supported by the U.S. Army Grant Number

DAAD19-01-1-0661. The authors are indebted to Drs. Walter

Roy, Fred Stanton, William DeRosset and Dennis Helfritch of

ARL for the support and a continuing interest in the present

work. The authors also acknowledge the support of the Office of

High Performance Computing Facilities at Clemson University.

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