Micromechanics and structural response of functionally graded...

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1

2 Micromechanics and structural response of functionally graded,3 particulate-matrix, fiber-reinforced composites

4 Guy M. Genin a, Victor Birman b,*

5 aDepartment of Mechanical, Aerospace, and Structural Engineering, Washington University, St. Louis, MO 63130, USA6 bMissouri University of Science and Technology (formerly, University of Missouri-Rolla), Engineering Education Center,7 One University Boulevard, St. Louis, MO 63121, USA

8

1 0a r t i c l e i n f o

11 Article history:12 Received 8 February 200813 Received in revised form 11 July 200814 Available online xxxx15

16 Keywords:17 Optimization of laminated composites18 Blast loading19 Micromechanics20 Composite materials21 Ashby method for materials selection22

2 3a b s t r a c t

24Reinforcement of fibrous composites by stiff particles embedded in the matrix offers the25potential for simple, economical functional grading, enhanced response to mechanical26loads, and improved functioning at high temperatures. Here, we consider laminated plates27made of such a material, with spherical reinforcement tailored by layer. The moduli for this28material lie within relatively narrow bounds. Two separate moduli estimates are consid-29ered: a ‘‘two-step” approach in which fibers are embedded in a homogenized particulate30matrix, and the Kanaun–Jeulin (Kanaun, S.K., Jeulin, D., 2001. Elastic properties of hybrid31composites by the effective field approach. Journal of the Mechanics and Physics of Solids3249, 2339–2367) approach, which we re-derive in a simple way using the Benveniste (1988)33method. Optimal tailoring of a plate is explored, and functional grading is shown to34improve the performance of the structures considered. In the example of a square, simply35supported, cross-ply laminated panel subjected to uniform transverse pressure, a modest36functional grading offers significant improvement in performance. A second example sug-37gests superior blast resistance of the panel achieved at the expense of only a small increase38in weight.39! 2008 Elsevier Ltd. All rights reserved.

40

41

42 1. Introduction

43 Much of Liviu Librescu’s research career was devoted to the science of developing improved structural systems through44 novel structural and material reinforcement schemes (Librescu, 1976; Librescu and Song, 2006). In this spirit, we contribute45 an article to this special edition of the International Journal of Solids and Structures devoted to his memory that focuses on46 optimizing a lightweight, dynamically loaded composite panel through careful selection of material properties and a novel47 reinforcing scheme.48 Hybrid composite materials consisting of an isotropic polymer matrix reinforced by both particles and unidirectional fi-49 bers offer the potential for simple functional grading to tailor mechanical response and reduce stress concentrations around50 attachments and discontinuities. Local particle reinforcement can increase stiffness and strength at key locations at the51 expense of a relatively small increase in weight. Moreover, polymeric composite structures subjected to thermal loading52 exhibit matrix deterioration at high temperatures, but a matrix reinforced with ceramic particles offers the potential to53 increase structural endurance and load-carrying capacity at high temperatures (Birman and Byrd, 2007).54 Effective application of hybrid particulate-matrix fiber-reinforced composites involves analysis of their stiffness in55 structural systems with either uniform or variable property distributions. For example, these structures may represent

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0020-7683/$ - see front matter ! 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2008.08.010

* Corresponding author. Tel.: +1 314 516 5431; fax: +1 314 516 5434.E-mail addresses: [email protected] (G.M. Genin), [email protected] (V. Birman).

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56 thin-walled shells or plates with in-surface variable stiffness. A particularly simple and attractive grading scheme involves57 embedding particles only in the outer layers of a laminate, achieving maximal increases in bending stiffness with a minimum58 of additional weight. Tailoring the volume fraction of particles, especially on the outer laminae of a composite, is far simpler59 than the alternative of varying fiber volume fraction or fiber orientations within a lamina.60 A broad range of homogenization methods exists to predict the properties of composite materials containing fibrous61 or particulate inclusions (e.g., Tucker and Liang, 1999; Torquato, 2001; Kakavas and Kontoni, 2006). The accuracy of62 these methods has been the subject of numerous investigations. Hu and Weng (2000) compare micromechanical models,63 including the double-inclusion method (Hori and Nemat-Nasser, 1993), the models of PonteCastaneda and Willis (1995),64 the model of Kuster and Toksöz (1974), and the Mori–Tanaka model (1973), with each model showing ranges of accu-65 racy. The Mori–Tanaka approach, applied in Benveniste’s (1987) generalized form in this paper, is considered accurate66 for predicting the elastic moduli of particulate-reinforced matrices only for volume fractions of inclusions below 40%67 (Kwon and Dharan, 1995; Sun et al., 2007). On the other hand, Noor and Shah (1993) showed that the Mori–Tanaka68 method provides an accurate prediction of the properties of fiber-reinforced composites even at a high volume fraction69 of fibers. In the following analysis, we study rigorous bounds in conjunction with estimates to assess the degree to70 which estimates can be trusted.71 General approaches to the characterization of a hybrid composite consisting of three different phases have been proposed72 by Yin et al. (2004) and Kanaun and Jeulin (2001). The Kanaun–Jeulin approach is based upon an effective field method with73 the assumption that the strain field within individual inclusions can differ for each population. We apply this approach in74 this paper, and re-derive it through a generalization of the modification of the Mori–Tanaka method suggested by Benveniste75 (1987); for the stiffness tensor of a three-phase composite, the methods yield identical predications. Many other authors76 have presented additional techniques for composite materials with multiple classes of reinforcement. Luo and Weng77 (1989) considered a three-phase concentric cylinder model consisting of a fiber, a coating (‘‘intermediate matrix”), and a ma-78 trix, and applied the resulting Eshelby (1957, 1959) tensors through a Mori–Tanaka formulation. The resulting estimates79 coincided with the lower Hill–Hashin bounds, or fell within them. Benveniste et al. (1989) and Dasgupta and Bhandarkar80 (1992) considered modifications of Benveniste’s method to model composites with coated inclusions.81 Bounds on material properties (e.g., moduli of elasticity, shear moduli, bulk modulus, and Poisson’s ratios) are calculated82 in this article using techniques summarized in Torquato (2001). Bounds such as those of Weng (1992) and the three-point83 bounding technique (e.g. Milton and Phan-Thien, 1982), which include descriptions of reinforcement geometry, are far tigh-84 ter than those of Voigt and Reuss (Hill, 1952), Hashin (1962), or Hashin and Shtrikman (1962, 1963).85 This paper begins with a re-derivation of the Kanaun–Jeulin estimate for the stiffness tensor of a three-phase composite,86 based on an extrapolation of the Benveniste approach (1987) for the case of multiple types of inclusions (fibers and parti-87 cles). Subsequently, bounds and additional estimates are developed in a two-step fashion: bounds and estimates are first88 employed for the response of a particulate matrix, and these are subsequently employed to establish bounds and estimates89 for the response of a particulate-reinforced matrix additionally reinforced by unidirectional fibers. Finally, the advantages of90 embedding a small amount of particles in fiber-reinforced materials are illustrated through consideration of a simply sup-91 ported cross-ply composite plate subjected to static, instantaneous, or blast pressure loadings. In all of these examples, add-92 ing glass particles to selected glass/epoxy laminae within a laminated plate resulted in significant reductions of maximum93 deformations achieved with very modest additional weight.

94 2. Re-derivation of the Kanaun–Jeulin estimate using a Benveniste-type approach

95 Consider a material where two different types of isotropic inclusions are distributed within an isotropic matrix. The prop-96 erties of the matrix are identified in the subsequent solution with the subscript i = 1, while two types of the inclusions are97 denoted by i = 2 and i = 3. We take phase 2 to be spherical particles and phase 3 to be aligned fibers. We derive the stiffness98 tensor of such a material through generalization of Benveniste’s (1987) solution for a particulate composite with a single99 type of inclusions.

100 The approach is based on the following assumptions:

101 1. All material phases are isotropic and linearly elastic.102 2. The perturbed strain in the matrix due to the presence of inclusions is not affected by the interaction of the two types of103 inclusions. In other words, each type of inclusion i = 2 and i = 3 affect the strains in the matrix, but the perturbed matrix104 strain due to the interaction between these inclusions is assumed to be of second order.105 3. Phase 3 is represented by aligned fibers of circular cross section; i.e., the composite material is a lamina with embedded106 particles. This assumption is needed to utilize the Eshelby tensor for cylindrical inclusions. In general, the derivation107 shown below is independent of the orientation and shape of fibers and particles as long as the corresponding Eshelby108 tensor is known.109

110 The average stress !!ro" and average strain !!eo" tensors for the material under consideration are related through the effec-111 tive stiffness tensor L:

!ro # L!eo; !1"113113

2 G.M. Genin, V. Birman / International Journal of Solids and Structures xxx (2008) xxx–xxx




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114 where boldface symbols represent tensors and bars denote a volumetric average over the entire composite.115 The effective stiffness tensor for a matrix with two different types of embedded inclusions can be written as a general-116 ization of the expression proposed by Hill (1963)117

L # L1 $ f2!L2 % L1"A2 $ f3!L3 % L1"A3; !2"119119

120 where Li is the stiffness tensor of the ith phase, f2 and f3 are volume fractions of the corresponding types of inclusions, and the121 tensors of concentration factors A2 and A3 represent the relationships between the tensors of average strains in the corre-122 sponding inclusions !!ei" and the mean remote strain tensor !!eo":123

!e2 # A2!eo; !e3 # A3!eo !3"125125

126 Note that according to the second assumption, this approach does not explicitly account for the interaction of different types127 of inclusions. Accordingly, it is applicable only if at least one type of inclusion has a relatively small volume fraction.128 Expanding the Mori and Tanaka (1973) ideas, the tensors of average strain in the matrix and in the phases are represented129 as130

!e1 # !eo $ ~e2 $ ~e3;!e2 # !eo $ ~e2 $ ~e3 $ e02;!e3 # !eo $ ~e2 $ ~e3 $ e03;

!4"132132

133 where ~ei are tensors of perturbations superimposed on the average strain in the matrix as a result of the presence of the cor-134 responding inclusions, and e0i are tensors of average perturbed strain in the inclusions relative to the tensor of average strain135 in the matrix.136 The tensors of average stresses in the inclusions can now be expressed in terms of the stiffness of the matrix:137

L2!!e0 $ ~e2 $ ~e3 $ e02" # L1!!e0 $ ~e2 $ ~e3 $ e02 % e&2";L3!!e0 $ ~e2 $ ~e3 $ e03" # L1!!e0 $ ~e2 $ ~e3 $ e03 % e&3";

!5"139139

140 where e&i are tensors of average correlation strain in the corresponding type of inclusions. These tensors are related to the141 tensors of perturbations in the inclusions by

e&2 # S%12 e02; e&3 # S%1

3 e03; !6"143143

144 where Si are the fourth-order Eshelby tensors, presented in Appendix A for the cases of spherical inclusions and for infinitely145 long cylindrical inclusions (fibers).146 From (4) and (5), the tensors of perturbation strains can be expressed in terms of the tensors of average strain in the cor-147 responding inclusions as148

e0i # SiL%11 !L1 % Li"!ei !i # 2;3": !7"150150

151 As also directly follows from (4):152

e0i # !ei % !e1: !8"154154

155 The subsequent transformation requires expression of the tensors of average strain in each type of inclusion in terms of156 the tensor of the average strain in the matrix; i.e., the linear transformations Ti such that157

!ei # Ti!e1: !9"159159

160 This is easily accomplished using (7) and (8)

Ti # 'I$ SiL%11 !Li % L1"(%1; !10"162162

163 where I is a fourth-order unit tensor.164 The tensors of concentration factors are now determined by expressing the tensor of the applied strain, which also rep-165 resents the average strain in an equivalent homogeneous material, in terms of strains in the constituent phases through the166 rule of mixtures:167

!eo # f1!e1 $ f2!e2 $ f3!e3: !11"169169

170 Using (3) and (9) in (11) yields171

Ai # Ti!f1I$ f2T2 $ f3T3"%1: !12"173173

174 Using the concentration tensors (12) in the tensor of effective stiffness (2) yields an estimate coinciding with that derived175 by Kanaun and Jeulin (2001) using the effective field method176

L # L1 $X3

i#2

fi!Li % L1"Ti!f1I$ f2T2 $ f3T3"%1: !13"178178

G.M. Genin, V. Birman / International Journal of Solids and Structures xxx (2008) xxx–xxx 3




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179 Computations evaluating (13) can be performed simply using the tensor decomposition presented byWalpole (1984) and180 summarized recently by Sevostianov and Kachanov (2007). Note that for the case of a single type of inclusions this result181 converges to the formula derived by Benveniste (1987).

182 3. ‘‘Two-step approach to the bounding and estimation of the elastic response of a particulate-matrix, fiber-reinforced183 composite material

184 Bounds and estimates are combined here to evaluate the effective stiffness tensor of an isotropic matrix material rein-185 forced by a random dispersion of identical spherical particles. Subsequently, these bounds and estimates are used to repre-186 sent the matrix of a fiber-reinforced composite material, and additional bounds and estimates are applied to model this187 fiber-reinforced composite. An underlying assumption is that the spherical particles are small compared to the fiber radii.188 In the example that will be studied, /1 and /2 refer to the volume fractions of epoxy and (relatively stiff) identical spher-189 ical particles, respectively, within the particulate-reinforced matrix; these relate to the overall volume fractions of epoxy (f1)190 and spherical particles (f2) as /2 = f2/(f1 + f2), with /1 = 1 % /2. The dense packing limit for identical spherical particles is that191 /2 cannot exceed approximately 0.63 (Torquato, 2001).

192 3.1. Bounds and estimates of the mechanical response of the particle-reinforced matrix

193 The first step of the ‘‘two-step” procedure is to estimate and bound the effective bulk modulus, Kpm, and the effective194 shear modulus, Gpm, of the combination of isotropic spherical particles and isotropic matrix. The matrix has bulk modulus195 K1 and shear modulus G1, and the spherical particles have bulk modulus K2 and shear modulus G2.196 Beran and Molyneux (1966) and McCoy (1970) obtained three-point bounds on Kpm and Gpm, for two-phase composites.197 These involve two parameters, f2 and g2 (additionally, g1 = 1 % g2, f1 = 1 % f2) that characterize the shape and distribution of198 the two phases (e.g. Torquato, 1991) and must be evaluated numerically. This procedure is computationally expensive, and199 only a limited number of microstructures have been characterized. Values range between 0.15/2 < f2 < /2 and 0.5/2 < g2 < /2

200 (Torquato, 1991). For randomly spaced spherical particles, f2 ) 0.211/2 and g2 ) 0.483/2 (Torquato, 2001). The Milton and201 Phan-Thien improvement on the McCoy (1970) three-point bounds Gpm is202

hGi% /1/2!G2 % G1"2

heGi$ N6 Gpm 6 hGi% /1/2!G2 % G1"2

h~Gi$H; !14"

204204

205 where206

N #128K $ 99

G

D E

f$ 45 1

G

D E

g

30 1G

D E

f

6K % 1

G

D E

f$ 6 1

G

D E

g2K $ 21

G

D E

f

; !15"208208

H #3hGigh6K $ 7Gif % 5hGi2f6h2K % Gif $ 30hGig

; !16"210210

211 in which h i denotes a weighted average (for example, hGi = G1/1 + G2/2, hGif = G1f1 + G2f2, hGig = G1g1 + G2g2), and a tilde212 represents a reverse-weighted average, e.g. heGi # G2/1 $ G1/2.213 Milton’s (1981) form of the three-point Beran and Molyneux (1966) bounds on the bulk modulus of an isotropic two-214 phase composite is215

hKi% /1/2!K2 % K1"2

heK i$ 2!d%1"d hG%1i%1

f

6 Kpm 6 hKi% /1/2!K2 % K1"2

h~Ki$ 2!d%1"d hGif

; !17"217217

218 where d = 3.219 The ‘‘three-point” estimates lying between these bounds incorporate detailed information on the geometry and distribu-220 tion of the spherical particles (Torquato, 2001):221

/2K21

Ke1# 1% !d$ 2"!d% 1"G1K21l21

d!K1 $ 2G1"/1f2 !18"

223223

224 and225

/2l21

le1# 1% 2G1j21l21

d!K1 $ 2G1"/1f2 %

f2!d2 % 4"G1!2K1 $ 3G1" $ g2!dK1 $ !d% 2"G1"2

2d!K1 $ 2G1"2l2

21/1; !19"227227

228 where

d # 3; j21 # K2 % K1

K2 % AG1; je1 # Ke % K1

Ke % AG1; l21 # G2 % G1

G2 % BG1; le1 # Ge % G1

Ge % BG1; A # 2!d% 1"

d; and

B # dK1=21 $ !d$ 1"!d% 2"G1=dK1 $ 2G1

:230230





guy

Inserted Text

"

guy

Sticky Note

All of the "K's" in this equation that have multiple subscripts (K21 and Ke1 on the left, and K21 on the right) should be lowercase "kappa"

guy

Cross-Out

guy

Sticky Note

The subscript "1" should be deleted.

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231 3.2. Bounds on and estimates of the mechanical response of a fiber- and particle-reinforced lamina

232 We now study the mechanics of a lamina containing aligned fibers of volume fraction f3 embedded in an isotropic matrix233 of volume fraction f1 + f2. The matrix mechanical properties are those of the homogenized particle-reinforced matrix dis-234 cussed in Section 3.1; that is, the matrix is an isotropic continuumwith bulk modulus Kpm and shear modulus Gpm. The fibers235 are isotropic with bulk modulus Kfib and shear modulus Gfib. The dense packing limit for fibers requires f3 < 0.83 (Torquato,236 2001). In general, five moduli are needed to describe the linear elastic response of a transversely isotropic material. However,237 since both the homogenized matrix and the fibers are taken to be isotropic, and since the laminate is transversely isotropic,238 only three of the effective moduli are independent (Hill, 1964). In the following, we present bounds and estimates for: (1) the239 transverse shear modulus, GT, describing resistance to shearing in a plane perpendicular to the fiber axes; (2) the transverse240 bulk modulus, KT, defined in such a way that the elastic modulus ET for stretching perpendicular to the direction of the fibers241 is ET # 9KTGT=!3KT $ GT"; and (3) the longitudinal–transverse shear modulus GLT describing resistance to shearing in a plane242 containing the fiber axes. The remaining two moduli enumerated below are dependent upon these first three: (1) the lon-243 gitudinal elastic modulus EL describing resistance to stretching parallel to the fibers, and (2) the longitudinal-transverse Pois-244 son’s ratio mLT that dictates the ratio between tensile strain resulting from uniaxial stretching parallel to the fibers and the245 associated compressive strain perpendicular to the fibers.

246 3.2.1. Transverse shear modulus247 The Silnutzer three-point lower bound on GT

e is given by Eqs. (14) and (15) (Silnutzer, 1972, reported in Torquato, 2001)248 with /1 replaced with (f1 + f2), /2 replaced with f3, g2 = 0.276f3, f2 # 0:691f 3 $ 0:0428f 23 , and h i replaced with h if, where, for249 example, hGif * (f1 + f2)Gpm + f3Gfib.250 The Gibiansky and Torquato (1995) upper bound for GT

e is tighter than the Silnutzer bound. Making the above substitu-251 tions, the Gibiansky–Torquato upper bound is obtained from Eq. (14) using the following definition for H:

H%1 #

1G%1 $ K%1

max

* +%1

g

$ 1Kmax

; h 6 % 1Kmax

2hK%1if $ hG%1ig %H $ Z' (2

h~G%1ig $ 2h~K%1if; % 1

Kmax6 h 6 1

Gfib

2 1G%1

fib $ K%1

* +%1

g

$ 1Gfib

; h P 1Gfib

;

8>>>>>>>>>>><

>>>>>>>>>>>:

!20"

253253

254 where Kmax is the greater of Kpm and Kfib, Gfib > Gpm, h # !h~G%1ig % 2!H=Z"h~K%1if"=!1$ !H=Z"", H #!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g1g2!G

%1fib % G%1

pm"2

q, and

255 Z #!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!f1f2!K%1

fib % K%1pm"

2q

.256 An estimate for GT

e within these bounds is provided by Eqs. (18) and (19), with d=2.

257 3.2.2. Transverse bulk modulus258 The Silnutzer three-point lower bound on KT

e is given by Eq. (17) with, as above, /1 replaced with (f1 + f2), /2 replaced with259 f3, g2 = 0.276f3, f2 # 0:691f 3 $ 0:0428f 23 , and h i replaced with hif (Torquato, 2001).260 A tighter upper bound is the Gibiansky–Torquato upper bound (Gibiansky and Torquato, 1995; Torquato, 2001):

KTe 6 hKif %

!f1 $ f2"f3!K f % Kpm"2hGifheK if $

GfGpm$Kf

Kf$heGif

: !21"262262

263 An estimate for KTe is obtained by Eqs. (18) and (19), with d = 2.

264 3.2.3. Longitudinal–transverse shear modulus265 The shear modulus GLT

e must lie within the Hashin and Rosen (1964) bounds

hGif $ Gfib

heGif $ Gpm

Gpm 6 GLTe 6 hGif $ Gpm

heGif $ Gfib

Gfib: !22"267267

268 The Halpin and Tsai (1967) semi-empirical equations provide estimates within these bounds for values of the parameter n in269 the range 0 6 n 6 25:

GLTe + 1$ nCHTf3

1% CHTf3Gpm; !23"

271271

272 where CHT # '!Gfib=Gpm" % 1(='!Gfib=Gpm" $ n(. The value n = 2 has been shown to provide a good estimate for laminae contain-273 ing regularly spaced, aligned fibers, and will be adopted in the following (Jones, 1975)Q3 .





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274 3.2.4. Longitudinal elastic modulus275 For a transversely isotropic two-phase lamina, EL

e is dictated by the bounds or estimates of the aforementioned properties276 (Hill, 1964; Torquato, 2001):

ELe # hEif $

4!mpm % mf "2

1kfib

% 1kpm

" #2

1k

$ %

f% 1KT

e $ GTe=3

!; !24"

278278

279 where m is Poisson’s ratio of the isotropic particulate-reinforced matrix (pm) or fibers (f), and ki = Ki + Gi/3 for each phase i.

280 3.2.5. Lateral/transverse Poisson’s ratio281 The effective Poisson’s ratio mLTe for contraction in the transverse plane associated with stretching parallel to the fibers is282 similarly dictated by the other material constants of the laminate (Hill, 1964; Torquato, 2001):

mLTe # hmif $!mpm % mf"

1kpm

% 1kfib

" # 1k

$ %

f% 1KT

e $ GTe=3

!: !25"

284284

285 4. Framework to assess tailoring of fiber-reinforced laminates with spherical particles

286 We apply the Ashby (2005) method to determine through material selection charts the grading of glass spheres within287 the matrix of a fiberglass laminate that optimizes the mechanical response to static, instantaneous, and blast loading with288 a minimum of additional weight. Desirable features to enhance the response to an applied surface pressure are reductions in289 deflection and stress, and increases in structural and material strength. We focus on choosing the volume fraction of spheres290 in each lamina of a symmetric, cross-ply laminated plate that optimizes weight and stiffness in response to these loading291 conditions. We first assess the benefits of tailoring in a square, statically- or instantaneously loaded plate by considering292 the effects of all possible spatial distributions of particles in laminates containing prescribed volume fractions of fibers.293 An objective function indicates which of these tailorings improve upon the stiffness to weight ratio of these laminates.294 We then assess the efficacy of tailoring in a specific blast-loaded panel. We restrict our attention to cases in which the num-295 ber of laminae N in the plate or sandwich panel is even, and all laminae are of the same thickness hk. Both representative296 examples illustrate the advantages of adding a relatively small amount of spherical particles to the matrix of fiber-reinforced297 composites.298 We consider a square, simply supported cross-ply plate of side dimension a, thickness h, and density q. The mass m of299 such a panel is300

m #XN

k#1

qkhka2 # a2hN

XN

k#1

qk * a2h!q; !26"302302

303 in which qk is the density of the kth lamina in the laminate, given as a volume-weighted average of the densities qmatrix,304 qspheres, and qfibers of the matrix, spheres, and fibers in the lamina:305

qk # f !k"1 qmatrix $ f !k"2 qspheres $ f !k"3 qfibers; !27"307307

308 and !q is the mean density of the plate. For the S-glass fiberglass considered in the following examples, qmatrix = 1500 kg/m3

309 and qspheres = qfibers = 2500 kg/m3.

310 4.1. Static and instantaneous loading

311 The exact solution for the static peak deflection w(x,y, t) due to an applied uniform pressure po on one face of an un-312 damped anisotropic plate is given by Lekhnitskii (1968), and the solution for an instantaneously applied dynamic loading313 by Soedel (2004). As described in Appendix B, we base design computations on the peak deflections resulting from a one-314 term approximation to these solutions. The approximation is obtained in terms of x and y coordinates in the plane of the315 plate, originating from one corner of the plate, by assuming a displacement field of the form w(x,y, t) = b(t)W11(x,y), in which316 t is time and W11(x,y) = sin(p x/a)sin(py/a) represents the first mode of deformation. For a slender, isotropic plate, the accu-317 racy of this approximation when compared to the exact solution is within 2.5%; for plates of the anisotropy range studied in318 the following examples, the accuracy is within 6%. The peak deflections of the plate subjected to a static or instantaneous319 load occur at the center of the plate and can be written (Appendix B)320

wmaxstatic +

16poa4

p6

& '!D11 $ 2!D12 $ 2D66" $ D22"%1

( )and wmax

instantaneous # 2wmaxstatic; !28"

322322

323 respectively. The composite plate stiffnesses Dij of a symmetric laminate containing an even number N laminae, each of324 thickness hk = h/N, are325





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DPROOF

Dij #Z %h=2

%h=2Qijz2dz # 2

XN=2

k#1

Q !k"ij h3

k112

$ k% 12

& '2 !

# 2hN

& '3XN=2

k#1

Q !k"ij k2 % k$ 1

3

& ': !29"

327327

328 The reduced stiffnesses can be written as follows for the kth lamina (e.g. Jones, 1999):329

Q11 Q12 0Q12 Q22 00 0 Q66

2

64

3

75

!k"

#

ZkEL $ !1% Zk"ET

1% !mLT"2!ET=EL"mLTET

1% !mLT"2!ET=EL"0

mLTET

1% !mLT"2!ET=EL"ZkE

T $ !1% Zk"EL

1% !mLT"2!ET=EL"0

0 0 GLT

2

66666664

3

77777775

!k"

; !30"

331331

332 where Zk = 1 for 0" laminae oriented with the fibers parallel to the 1-direction and Zk = 0 for 90" laminae oriented with fibers333 perpendicular to the 1-direction.334 Eq. (28) can be solved for the panel thickness, h:

h3 + bmax8poa4N

3

wmax

!XN=2

k#1

!Q !k"11 $ 2!Q !k"

12 $ 2Q !k"66 " $ Q !k"

22 " k2 % k$ 13

& ' !%1

* bmax8poa4N

3

wmax

!1U

& '; !31"

336336

337 where U can be interpreted as an effective stiffness, bmax = 1 for static loading of the plate, and bmax = 2 for instantaneous338 loading of the plate. Note that because of the symmetries of the problem considered, neither Q !k"

12 , nor Q !k"66 , nor the sum

339 !Q !k"11 $ Q !k"

22 " vary depending on whether a lamina is in the 0" or 90" orientation. Therefore, U is independent of the stacking340 chosen (e.g., a [0/90/90/0]s laminate will have the same value of U as a [90/0/90/0]s laminate.)341 The performance index for optimization can be identified by combining and substituting back into (26)342

m #2hN

a2XN=2

k#1

qk + a3!8bmaxa"1=3 po

wmax

& '1=3

N!qU%1=3: !32"344344

345 The goal is to find a panel of minimummassm that can provide a certain stiffness (po/wmax). While the number of laminae N346 is held constant, the thicknesses of the laminae are allowed to vary freely to reach this optimum. Under these constraints, the347 only free variables on the right hand side of (32) are the density and stiffness; the panel that provides the optimal stiffness348 per unit weight is that which maximizes the material performance index, M # U1=3=!q. As will be described in Section 5, the349 optimum degree of tailoring is found graphically through an Ashby-type material selection chart, a plot of !q against U for350 panels containing a prescribed volume fraction of fibers and all possible distributions of spherical particles. The Ashby351 ‘‘selection lines” are isoclines of the material performance index, M.

352 4.2. Blast loading

353 We consider the response of a specific plate to an explosive blast (Houlston et al., 1985), Gupta (1985), Gupta et al. (1987),354 Birman and Bert (1987), Librescu and Nosier (1990a), Librescu and Na (1998a,b)). Here, we modify the approximate solution355 (28) to predict the response of the panel to a blast overpressure, uniformly distributed over the surface of the panel but vary-356 ing with time according to the Friedlander equation (e.g., Birman and Bert, 1987)357

p!t" # po!1% !t=tp"" exp!%Kt=tp"; !33"359359

360 where po is the peak pressure, tp is a positive phase duration of the pulse and K is an empirical decay parameter. Reasonable361 values for the parameters are tp = 0.1 s and K = 2 (Librescu and Nosier, 1990b).362 The peak deflection can be estimated as (Appendix B)363

wmax!t" # wstaticmax

x21t

2p

K2 $x21t

2p

!W&!t"; !34"

365365

366 where367

W&!t" # p!t"po

% 2KK2 $x2

1t2p

!exp!%Kt=tp" % 1% 2K

K2 $x21t

2p

!cosx1t $ K%

K2 %x2t2pK2 $x2

1t2p

!sinx1tx1tp

!35"369369

370 and the fundamental frequency x1 is

x21 # p4

a4!qh 'D11 $ 2!D12 $ 2D66" $ D22(: !36"372372

373 As justified below (see the results associated with Fig. 7), the peak displacement for the example studied in Section 5374 could be approximated as375

wblastmax + wstatic

max 2% px1tp

!1$K"& '

# 16po

p2!qhx21

2% px1tp

!1$K"& '

: !37"377377





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378 Note that the first term in the parentheses in Eq. (37) is the same as Eq. (28) for an instantaneously applied load po; the sec-379 ond term reflects the effect of reaching the peak pressure po gradually over the duration of a blast loading.

380 5. Numerical results

381 Material constants of an isotropic epoxy matrix (E1 = 3.12 GPa and m1 = 0.38) containing spherical isotropic glass inclu-382 sions (E2 = 76.0 GPa and m2 = 0.25) were evaluated using the two estimates and compared to the Voigt and Reuss, Hashin–383 Shtrikman, and three-point bounds (Fig. 1). The three-point estimate lies within all bounds considered; the Benveniste esti-384 mate (shown above to be analogous to the Kanaun–Jeulin estimate for the case of a two-phase material) falls just below the385 three-point bounds and coincides with the Hashin–Shtrikman lower bound. Note that the associated predictions and esti-386 mates for the Poisson ratio were inverted: the Benveniste approach predicted slightly high values. While Benveniste predic-387 tions remain within the Hashin–Shtrikman bounds, they were slightly outside the three-point bounds, particularly at larger388 particle volume fractions, though the deviation remained quite small. Like the Mori–Tanaka theory, the Benveniste approach389 is acceptable only for relatively low particle volume fractions. As will be shown below, the Kanaun–Jeulin approach for a390 three-phase composite has these same ranges of applicability.391 For a particulate-matrix, fiber-reinforced material (glass fibers and particles, with identical moduli, and epoxy matrix),392 the bounds evaluated (Fig. 2) correspond to the upper and lower three-point bounds from Fig. 1, combined with the bounds393 presented in Section 3.2. The ‘‘two-step estimates” use the three point estimates for the particulate-reinforced matrix in con-394 junction with the estimates for fiber-reinforced composites presented in Section 3.2. The Kanaun–Jeulin estimate derives di-395 rectly from Section 2. All moduli in these graphs were normalized by those of the epoxy matrix.396 In a parameter study assessing the effects of varying volume fractions of spheres at prescribed volume fractions of fibers,397 the Kanaun–Jeulin estimate lies just outside of the three-point bounds, and is usually a reasonable approximation provided398 that the particle and fiber volume fractions are small. In all cases, the Kanaun–Jeulin estimate lies closest to the bounds for399 lowest volume fractions of inclusions (Fig. 2). The Kanaun–Jeulin estimate follows the lower Hashin–Shtrikman bounds (not400 shown) for the transverse elastic modulus (ET) and both shear moduli (GT and GLT), and lies just above the upper bound for401 the longitudinal elastic modulus (EL). The ‘‘two-step” estimate lies within the three-point bounds. Estimates for Poisson’s402 ratio obtained using both methods are close. When the two laminae are combined into a symmetric 0"/90" composite,403 the errors cancel and the Kanaun–Jeulin estimate provides estimates of some of the average reduced stiffness terms404 QT

ij # !2=h"PN=2

k#1Q!k"ij (Eq. (30)) that are far improved (Fig. 3). Plotted in Fig. 3 are the sum !QT

11 $ QT22" and the term QT

12, rel-405 evant to calculation of the bending stiffnesses Dij. Note that the third constant, QT

66, is equal to GLT.

Fig. 1. Estimates and bounds for the stiffening of an isotropic epoxy matrix by spherical glass inclusions.





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406 Functional grading for optimization of the stiffness to weight ratio was studied for a symmetric, 8-ply laminate consisting407 of epoxy, varying volume fractions f3 of fibers, and all possible volume fractions f2 of spherical reinforcing particles (ranging408 from f2 = 0 to the dense packing limit, f2 = 0.63(1 % f3); for example, for f3 = 0.5, the average density !q (Eq. (27) ranges from409 2000 kg/m3 (no particles) to 2315 kg/m3 (dense packing limit)). The set of all possible ‘‘stiffnesses”U for a laminate contain-410 ing 50% by volume fibers and varying volume fractions of spherical particles, tailored with all possible ply-by-ply spatial dis-411 tributions, is shown as a function of average density in Fig. 4 for four sets of mechanical properties: the upper and lower412 bounds from Fig. 2, the Kanaun–Jeulin estimate, and the two-step estimate. The shapes of the regions corresponding to each413 set of mechanical properties were nearly identical, other than an offset along the vertical axis. The scallops on the top of each414 region correspond to progressive addition of spherical reinforcement to the outermost plies: the leftmost scallop corre-415 sponds to a tailoring involving addition of spherical reinforcement to the outermost pair of plies, up to the maximum;416 the next scallop corresponds to addition of spherical reinforcement to the next pair of plies.417 Howmuch stiffening is too much? The selection lines are the isoclines of the performance indexM # U1=3=!q. The stiffness

418 U continues to increase as spherical particles are added to the inner two pairs of plies, but such configurations have a less419 optimal stiffness to mass ratio: the best solutions are those intersected by the highest optimization isoclines. Optimization420 isoclines (dashed lines, Fig. 4) appear straight because both axes of the plot are logarithmic. All !!q;U" points above the

Fig. 2. Estimates and bounds for the stiffening of a composite containing isotropic, aligned glass fibers of varying volume fractions f3 by spherical glassinclusions of volume fraction f2. f2 ranges from 0 to the dense packing limit, although not all models depicted are valid over this entire range. The upper andlower bounds are calculated using the upper and lower bounds from Fig. 1 in conjunction with the bounding procedures described in the text. Moduli arenormalized by those of epoxy.





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421 isocline that intersects the point corresponding to no reinforcement of spherical particles (lower left corner of the region,422 circled in Fig. 4) represent improvements in mechanical response through addition of spherical particles. The grading at423 the optimum (circled in Fig. 4) had no spherical particles in the inner four laminae, and the maximum possible volume frac-424 tion (f2 = 31.5%) in the outer four laminae. One of the worst possible gradings of spherical particles (also circled in Fig. 4) had425 the maximum possible volume fraction of spherical particles in the innermost pair of laminae, 70% of this volume fraction in426 the next pair, and no spherical particles in the outermost two pairs.427 Note that the region corresponding to the Kanaun–Jeulin estimate obscures the region corresponding to the lower bound428 except at the highest volume fractions of spherical reinforcement (highest average density); at these higher values, the solu-429 tion is least reliable, and the stiffness U can be seen to drop beneath the lower bound.430 When varying volume fractions of fibers are considered (Fig. 5), the domains of possible !!q;U" pairs for each volume frac-431 tion of fibers has approximately the same shape as those in Fig. 4, except that the region appears to rotate counterclockwise432 relative to the optimization isoclines for higher volume fractions of fibers. The result is that the optimal tailoring of the vol-433 ume fraction of spheres changes at the highest volume fraction fibers considered, so that adding the greatest possible volume434 of spheres to the outermost three pairs of laminae (rather than the outermost two pairs) becomes optimal. The details of the435 least optimal tailoring changes with fiber volume fraction, but in each case involve higher volume fractions of spheres close436 to the neutral axis of the laminate.437 To assess the role of modulus tailoring in blast-loading, a square, symmetric, 8-ply cross-ply laminate of thickness438 h = 0.008 m and in-plane dimension a = 0.5 m was considered. For such a laminate, the natural frequency x1 varied from439 approximately 250–800 Hz (Fig. 6). For this natural frequency range, and using the blast considered by Librescu and Nosier

Fig. 4. Log–log Ashby-type material selection chart, with selection lines (isoclines of the material performance index M # U1=3=!q". The shaded areascorrespond to all stiffnesses U and average densities !q attainable when tailoring a symmetric, 8-ply 0"/90" cross-ply laminate (volume fraction of glassfibers f3 = 0.5) using spherical glass inclusions (volume fraction f2 allowed to vary from lamina to lamina). Improved performance is achieved for laminatestailored to have higher values of M; these lie along higher isoclines, as shown.

Fig. 3. Estimates and bounds for the stiffening of a symmetric cross-ply composite containing aligned isotropic glass fibers of varying volume fractions f3 byspherical glass inclusions of volume fraction f2. Note that the third elastic constant for the in-plane mechanical response of the cross-ply, QT

66, is identical toGLT (Fig. 2). Moduli are normalized by the elastic modulus of epoxy.





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The line in this arrow was somehow made thicker in the journal's processing of this EPS file. Could you reduce the thickness of this line so that the arrow looks like an arrow? If you need a tiff image instead of an EPS file, please contact us.

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Our preference would be for this to read "from approximately 250Hz to 800Hz."

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440 (1990a,b) (tp = 0.1 s, K = 2), several oscillations occur in the panel during the positive pressure phase of the blast duration, tp441 (Fig. 7). The first and third terms in Eq. (35) are dominant over this time range. Using these two terms,W*(t) is approximately442 (dashed lines, Fig. 7)

W&!t" + p!t"po

% cosx1t # !1% !t=tp"" exp!%Kt=tp" % cosx1t !38"444444

445 and the peak displacement occurs near x1t = p so that

W&max + 1% p

x1tp

& 'exp %K p

x1tp

& '$ 1 + 2% p

x1tp!1$K": !39"

447447

448 This approximation to W&max is accurate to within a few percent (circles, Fig. 7). Substitution of this into Eq. (34) yielded the

449 approximation (37). For the specific panel studied here, nearly all additions of spherical tailoring improved the stiffness per450 unit mass ratio of the plate (Fig. 8).

Fig. 5. Log–log Ashby-type materials selection chart, with selection lines (isoclines of the material performance index M # U1=3=!q" to identify the optimaltailoring of symmetric 8-ply 0"/90" cross-ply laminates (volume fraction f3 of glass fibers) using spherical glass inclusions (volume fraction f2 allowed tovary from lamina to lamina). Improved performance is achieved for laminates tailored to have higher values ofM; these lie along higher isoclines, as shown.Certain distributions of spherical particles can be more effective in improving the stiffness-to-weight ratio than additional fibers.

Fig. 6. The fundamental frequencyx1 occurs over the range of 255–800 Hz for all possible composites and tailoring of a symmetric, 8-ply 0"/90" compositeof the dimensions shown.





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As in Fig. 4, this arrow became distorted in the journal's processing so that it no longer looks like an arrow. Could you fix this?

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451 6. Discussion

452 This article demonstrated how tailored distributions of particle reinforcement can improve the structural response of453 laminated plates, using fiberglass cross-ply panels as an illustration. A series of micromechanical models were considered454 to obtain the results in this paper. The Kanaun–Jeulin approach suffered from limitations identical to those of the Mori–Ta-455 naka estimate (Figs. 1 and 2). As expected from the Kanaun–Jeulin approach’s requirement that inclusion classes have min-456 imal mechanical interactions, the estimates were best for the lowest concentrations of inclusions (Fig. 2). Errors cancelled457 out for the case of a cross-ply laminate (Fig. 3). At moderate concentrations of inclusions, the Kanaun–Jeulin approach offers458 a simple and compact approach for estimating the time-dependent response of a plate (Fig. 6). However, this approach can459 provide estimates that fall outside of rigorous bounds on mechanical response (Fig. 2), and must therefore be used with care.460 For the example of a static or instantaneously applied pressure loading applied to a cross-ply panel, the optimization of461 the distribution of reinforcing particles proceeded via the Ashby approach, showing that a broad class of particle distribu-462 tions led to improved stiffness to weight ratios of the panels considered (Fig. 4). As expected, the optimal tailoring involved463 including spherical reinforcement in the outer plies, and excluding the reinforcement from the inner plies. Poorly-chosen464 tailoring of spherical reinforcement particles leads to a reduction in the stiffness to weight ratio of the panels. However,465 well-chosen spherical reinforcement, added to the outermost plies, was more effective than the uniform addition of fibers466 to all laminae (Fig. 5.)

Fig. 7. For the blast and plate considered, the peak displacement increased as the fundamental frequency, x1, increased. The time and magnitude of thispeak displacement was approximated to within a few percent by Eq. (37).

/

//

Fig. 8. For the specific symmetric 8-ply 0"/90" cross-ply laminate plate considered, nearly all increases in mass resulting from the addition of sphericalparticles leads to an improvement in the stiffness-to-weight ratio. The only exceptions are cases in which particles are added only to the innermost laminaein the composite. Both axes are linear.





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467 For blast loading, a closed form estimate was developed to predict the peak displacement of a panel in response to a Fried-468 lander pressure pulse. A blast-resistant panel can withstand only a limited deflection before losing strength; while other as-469 pects of blast resistance such as improved thermal resistivity from glass particles and the ability to tailor stiffness in the470 vicinity of stress concentrations are certainly important (e.g. Budiansky et al., 1993; Genin and Hutchinson, 1999), the focus471 here was exclusively on the improvement of stiffness to weight ratio. For the specific panel and blast chosen for detailed472 study, the peak displacement occurred at a well-defined moment shortly after the arrival of the blast ‘‘overpressure” wave.473 This time was a function of the fundamental frequency; the greater the fundamental frequency, the sooner after the blast and474 greater in magnitude was the peak displacement (Fig. 7). However, the magnitude of this peak decreased with both increas-475 ing stiffness and decreasing mass, and hence with increasing particle content in the laminate. The improvements from stiff-476 ness increases dominated over the deleterious effects of increased fundamental frequency, and the stiffness to weight ratio477 for the blast-loaded panel improved for nearly all additions of glass particles, no matter how poorly the tailoring was chosen478 (Fig. 8).

479 7. Uncited references

480 Vel and Batra (2003), Wang et al. (2000), Zhao et al. (1990), Zhao and Weng (1990b).Q1

481 Acknowledgements

482 This work was supported in part by the Army Research Office through contract W911NF-07-1-0290 (Program Manager:483 Dr. Bruce LaMattina), the National Institutes of Health through Grant HL079165, and the Johanna D. Bemis trust.

484 Appendix A. Eshelby tensors

485 Eshelby’s tensor for a spherical inclusion embedded within an isotropic matrix are the following (Eshelby, 1959;486 Torquato, 2001):

Sijkl #5m1 % 1

15!1% m1"dijdkl $

4% 5m115!1% m1"

!dikdjl$ dildjk"; !A1"488488

489 where m1 Poisson’s ratio for the matrix and dij is Kronecker’s delta. For needle-shaped inclusions, with the axis of the needles490 aligned in the 1-direction, the non-zero components are

S2222 # S3333 # 5% 4m18!1% m1"

;

S2211 # S3311 # m12!1% m1"

;

S2323 # 3% 4m18!1% m1"

;

S1212 # S1313 # 1=4:

!A2"

492492

493 Appendix B. Approximate response of a simply supported orthotropic plate to an arbitrary dynamic pressure loading

494 The Love equation for the displacement fieldw(x,y, t) of an undamped, square plate lying in the x–y plane that is subjected495 to a uniform, time-varying pressure p(t) is (e.g. Soedel, 2004)

Lfw!x; y; t"g$ !qhd2wdt2

!x; y; t" # p!t"; !B1"497497

498 where !q is the mean density of the plate, h is the thickness of the plate, and the operator L{w(x,y, t)} can be written as

Lfw!x; y; t"g # D11o4w!x; y; t"

ox4$ 2!D12 $ 2D66"

o4w!x; y; t"ox2oy2

$ D22o4w!x; y; t"

oy4: !B2"

500500

501 The exact solution to this problem can be written in terms of an infinite series using the standard Fourier decomposition502 (e.g. Soedel, 2004). We focus attention on an approximate solution incorporating only the first mode, W11(x,y):

w!x; y; t" + b1!t"W11!x; y" # b1!t" sinpxa

sinpya

; !B3"504504

505 where the modal participation factor b1(t) is found from the following convolution integral (Soedel, 2004):

b1!t" #1x1

Z t

0A1!s" sinx1!t % s"ds; !B4"

507507





guy

Sticky Note

the "jl" in the "deltajl" term should be a subscript (delta_jl). Our apologies for this error.

guy

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Our apologies for these oversights. Please delete these references.

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508 in which the fundamental frequencyx1 can be written as in Eq. (35), and the composite plate stiffnesses Dij are defined in Eq.509 (29). The amplitude A1(t) in the convolution integral, with units of acceleration, can be written

A1!t" #p!t"!qh

RX W11!x; y"dXRW2

11!x;y"dXX

# 16p!t"p2!qh ; !B5"

511511

512 where X represents the surface of the plate. Thus,

w!x; y; t" +16

p2!qhx1sin

pxa

sinpya

Z t

0p!s" sinx1!t % s"ds !B6"

514514

515 and516

wmax!t" # wa2;a2; t

" #+ 16p2!qhx1

Z t

0p!s" sinx1!t % s"ds: !B7"

518518

519 For the case of an instantaneously applied loading po at time t = 0, (B7) yields

wmax!t" +16po

p2!qhx21!1% cosx1t" # wstatic

max !1% cosx1t"; !B8"521521

522 where wstaticmax # 16po=p2qhx2

1 is the peak deflection that would result from the quasi-static application of a pressure po to the523 face of the plate.524 For a blast overpressure, uniformly distributed over the surface of the plate but varying with time according to the Fried-525 lander equation (Eq. (33)), Eq. (B7) yields

wmax!t" # wstaticmax

x21t

2p

K2 $x21t

2p

!W&!t"; !B9"

527527

528 where

W&!t" # p!t"po

% 2KK2 $x2

1t2p

!

exp!%Kt=tp" % 1% 2KK2 $x2

1t2p

!

cosx1t $ K%K2 %x2t2pK2 $x2

1t2p

!sinx1tx1tp

: !B10"530530

531 References

532 Ashby, M.F., 2005. Materials Selection in Mechanical Design, third ed. Elsevier, New York.533 Benveniste, Y., 1987. A new approach to the application of Mori–Tanaka theory in composite materials. Mechanics of Materials 6, 147–157.534 Benveniste, Y., Dvorak, G.J., Chen, T., 1989. Stress fields in composites with coated inclusions. Mechanics of Materials 7, 305–317.535 Beran, M., Molyneux, J., 1966. Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quarterly536 of Applied Mathematics 24, 107–118.537 Birman, V., Bert, C.W., 1987. Behavior of laminated plates subjected to conventional blast. International Journal of Impact Engineering 6, 145–155.538 Birman, V., Byrd, L.W., 2007. Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews 60, 195–216.539 Budiansky, B., Hutchinson, J.W., Evans, A.G., 1993. On neutral holes in tailored, layered sheets. Journal of Applied Mechanics 60, 1056–1057.540 Dasgupta, A., Bhandarkar, S.M., 1992. A generalized self-consistent Mori–Tanaka scheme for fiber-composites with multiple inter-phases. Mechanics of541 Materials 14, 67–82.542 Eshelby, J.D., 1957. The determination of the elastic field outside an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London,543 Series A 241, 376–396.544 Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proceedings of the Royal Society of London, Series A 252, 561–569.545 Genin, G.M., Hutchinson, J.W., 1999. Failures at attachment holes in brittle matrix laminates. Journal of Composite Materials 33 (17), 1600–1619.546 Gibiansky, L.V., Torquato, S., 1995. Geometrical-Parameter bounds on the effective moduli of composites. Journal of the Mechanics and Physics of Solids 43,547 1587–1613.548 Gupta, A.D., 1985. Dynamic analysis of a flat plate subjected to an explosive blast. Proceedings of the 1985 ASME International Computers in Engineering549 Conference 1, 491–496.550 Gupta, A.D., Gregory, F.H., Bitting, R.L., Bhattacharya, S., 1987. Dynamic response of an explosively loaded hinged rectangular plate. Computers and551 Structures 26, 339–344.552 Halpin, J.C., Tsai, S.W., 1967. Effects of environmental factors on composite materials. Air Force Technical Report AFML-TR-67-423, Wright Aeronautical553 Laboratories, Dayton, OH.554 Hashin, Z., 1962. The elastic moduli of heterogeneous materials. Journal of Applied Mechanics 29, 143–150.555 Hashin, Z., Rosen, B.W., 1964. The elastic moduli of fiber-reinforced materials. Journal of Applied Mechanics 31, 223–232.556 Hashin, Z., Shtrikman, S., 1962. A variational approach to the theory of the effective magnetic permeability of multiphase materials. Journal of Applied557 Physics 33, 3125–3131.558 Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of559 Solids 11, 127–140.560 Hill, R., 1952. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society of London, Series A 65, 349–354.561 Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372.562 Hill, R., 1964. Theory ofmechanical properties of fiber-strengthenedmaterials. I. Elastic behaviour. Journal of theMechanics and Physics of Solids 11, 357–372.563 Hori, M., Nemat-Nasser, S., 1993. Double-inclusion model and overall moduli of multi-phase composites. Mechanics of Materials 14, 189–206.564 Houlston, R., Slater, J.E., Pegg, N., Des Rochers, C.G., 1985. On the analysis of structural response of ship panels subjected to air blast loading. Computers and565 Structures 21, 273–289.566 Hu, G.K., Weng, G.J., 2000. The connections between the double-inclusion model and the Ponte Castaneda–Willis, Mori–Tanaka, and Kuster–Toksöz models.567 Mechanics of Materials 32, 495–503.





guy

Sticky Note

The "W_11^2(x,y) d omega" term should be the integrand rather than an upper limit of integration. That is, the denominator should be identical to the numerator, except that the "W_11" term is squared in the denominator.

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568 Jones, R.M., 1999. Mechanics of Composite Materials, second ed. Taylor & Francis, Oxford.569 Kakavas, P.A., Kontoni, D.-P., 2006. Numerical investigation of the stress field of particulate reinforced polymeric composites subject to tension.570 International Journal for Numerical Methods in Engineering 65, 1145–1164.571 Kanaun, S.K., Jeulin, D., 2001. Elastic properties of hybrid composites by the effective field approach. Journal of the Mechanics and Physics of Solids 49,572 2339–2367.573 Kuster, G.T., Toksöz, M.N., 1974. Velocity and attenuation of seismic waves in two-phase media: I. Theoretical formulation. Geophysics 39, 587–606.574 Kwon, P., Dharan, C.K.H., 1995. Effective moduli of high content particulate composites. Acta Metalurgica et Materialia 43, 1141–1147.575 Lekhnitskii, S.G., 1968. Anisotropic Plates. Gordon and Breach, New York.576 Librescu, L., 1976. Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff International, Leyden.577 Librescu, L., Na, S., 1998a. Dynamic response control of thin-walled beams to blast pulses using structural tailoring and piezoelectric actuation. Journal of578 Applied Mechanics 65, 497–504.579 Librescu, L., Na, S., 1998b. Dynamic response of cantilevered thin-walled beams to blast and sonic-boom loadings. Shock and Vibration 5, 23–33.580 Librescu, L., Nosier, A., 1990a. Response of Shear deformable elastic laminated composite flat panels to sonic boom and explosive blast loadings. AIAA581 Journal 28, 345–352.582 Librescu, L., Nosier, A., 1990. Dynamic response of anisotropic composite panels to time-dependent external excitations. 17th Congress of the International583 Council of the Aeronautical Sciences, ICAS-90-1.4R, pp. 2134–2144.584 Librescu, L., Song, O., 2006. Thin-walled Composite Beams: Theory and Application. Springer, Dordrecht, The Netherlands.585 Luo, H.A., Weng, G.J., 1989. On Eshelby’s S-tensor in a three-phase cylindrically concentric solid, and the elastic moduli of reinforced-reinforced composites.586 Mechanics of Materials 8, 77–88.587 McCoy, J.J., 1970. On the displacement field in an elastic medium with random variation of material properties. Recent Advances in Engineering Sciences,588 vol. 5. Gordon and Breach, New York.589 Milton, G.W., 1981. Bounds on the electromagnetic, elastic, and other properties of two-component composites. Physical Review Letters 46, 542–545.590 Milton, G.W., Phan-Thien, N., 1982. New bounds on effective elastic moduli of two-component materials. Proceedings of the Royal Society of London, Series591 A 380, 305–331.592 Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21, 571–574.593 Noor, A.K., Shah, R.S., 1993. 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