Asymptotic Homogenization

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Alexander L. Kalamkarov1

Fellow ASMEDepartment of Mechanical Engineering,

Dalhousie University,P.O. Box 1000,

Halifax, NS, B3J 2X4, Canadae-mail: [email protected]

Igor V. AndrianovInstitute of General Mechanics,

Rheinisch-Westfälische Technische Hochschule(Technical University of Aachen),

Templergraben 64,Aachen D-52062, Germany

Vladyslav V. Danishevs’kyyPrydniprov’ska State Academy of Civil

Engineering and Architecture,Chernishevs’kogo 24a,

Dnipropetrovsk 49600, Ukraine

Asymptotic Homogenization ofComposite Materials andStructuresThe present paper provides details on the new trends in application of asymptotic ho-mogenization techniques to the analysis of composite materials and thin-walled compos-ite structures and their effective properties. The problems under consideration are impor-tant from both fundamental and applied points of view. We review a state-of-the-art inasymptotic homogenization of composites by presenting the variety of existing methods,by pointing out their advantages and shortcomings, and by discussing their applications.In addition to the review of existing results, some new original approaches are alsointroduced. In particular, we analyze a possibility of analytical solution of the unit cellproblems obtained as a result of the homogenization procedure. Asymptotic homogeniza-tion of 3D thin-walled composite reinforced structures is considered, and the generalhomogenization model for a composite shell is introduced. In particular, analytical for-mulas for the effective stiffness moduli of wafer-reinforced shell and sandwich compositeshell with a honeycomb filler are presented. We also consider random composites; use oftwo-point Padé approximants and asymptotically equivalent functions; correlation be-tween conductivity and elastic properties of composites; and strength, damage, andboundary effects in composites. This article is based on a review of 205 references.DOI: 10.1115/1.3090830

Keywords: composite materials, thin-walled composite reinforced structures, asymptotichomogenization, unit cell problems, effective properties

IntroductionThe rapidly increasing popularity of composite materials and

tructures in recent years has been seen through their incorpora-ion in the mechanical and civil engineering, aerospace, automo-ive and marine applications, as well as in biomedical and sportroducts. Success in practical application of composites largelyepends on a possibility to predict their mechanical properties andehavior through the development of the appropriate mechanicalodels. The micromechanical modeling of composite structures,

owever, can be rather complicated as a result of the distributionnd orientation of the multiple inclusions and reinforcementsithin the matrix, and their mechanical interactions on a local

microlevel. Therefore, it is important to establish such micro-echanical models that are neither too complicated to be devel-

ped and applied nor too simple to reflect the real mechanicalroperties and behavior of the composite materials and structures.

The micromechanical analysis of composites has been the sub-ect of investigation for many years. According to Willis 1, theumerous methods in mechanics of composites can be classifiednto four broad categories: asymptotic, self-consistent, variational,nd modeling methods. There are no rigorous boundaries betweenhese categories.

The self-consistent methods and the general “one-particle”chemes for approximate evaluation of the effective propertiesave been reviewed in Refs. 2,3. Our present review deals withhe asymptotic approaches that are capable of analyzing the com-osite materials and structures with constituents with high contrastn their material properties. For many problems that we will dis-uss below, other analytical or numerical approaches are not asffective as the asymptotic homogenization.

First, we will deal with the regular composites. The coefficientsf the corresponding equations modeling mechanical behavior of

1Corresponding author.
Published online March 31, 2009. Transmitted by Victor Birman.
pplied Mechanics Reviews Copyright © 20

om: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02

the composite solid are rapidly varying periodic functions in spa-tial coordinates. Accordingly, the resulting boundary-value prob-lems BVPs are very complex. A look at numerical methods,applied directly to an original boundary-value problem for a com-posite solid, shows that they are not always convenient and aresometimes even inappropriate in their standard form. Therefore, itis important to develop analytical methods based on rigorousmathematical techniques. At present, asymptotic techniques areapplied in many cases in micromechanics of composites. Variousasymptotic approaches to the analysis of composite materials haveapparently reached their conclusion within the framework of themathematical theory of asymptotic homogenization. Indeed, theproof of the possibility of homogenizing a composite material of aregular structure, i.e., of examining an equivalent homogeneoussolid instead of the original inhomogeneous composite solid, isone of the principal results of this theory. Theory of homogeniza-tion has also indicated a method of transition from the originalproblem which contains in its formulation a small parameter re-lated to the small dimensions of the constituents of the compositeto a problem for a homogeneous solid. The effective properties ofthis equivalent homogeneous material are determined through thesolution of so-called local problems formulated on the unit cell ofthe composite material. These solutions also enable calculation oflocal stresses and strains in the composite material. The indicatedresults are fundamentals of the mathematical theory of homogeni-zation. In the present paper we will review the basics of theasymptotic homogenization and the analytical solutions of the unitcell problems for laminated, fiber-reinforced and particulate com-posites. Afterward, we will generalize the obtained results for therandom composites. We will also analyze thin-walled compositestructures, damage in composite materials and boundary effects,as well as the approximate links between the conductivity andelastic problems for the composite materials.

Following this Introduction the rest of the paper is organized asfollows: the asymptotic homogenization technique is presented in
Sec. 2. Section 3 deals with the unit cell problems. In Sec. 4 we
MAY 2009, Vol. 62 / 030802-109 by ASME

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ill discuss the use of two-point Padé approximants TPPAs.ection 5 deals with the use of asymptotically equivalent func-

ions AEFs. That is followed by the review of techniques appliedo random composites in Sec. 6. Section 7 studies the existinginks between the conductivity and elastic problems for composite

aterials. Asymptotic homogenization of three-dimensional thin-alled composite reinforced structures is discussed in Sec. 8. Theeneral homogenization model for a composite shell is presented.symptotic homogenization techniques in the study of strength

nd damage and the boundary effects in composite materials areeviewed in Sec. 9. Conclusions and some generalizations in these of asymptotic homogenization are presented in Sec. 10.

Asymptotic Homogenization MethodFor the past 25 years homogenization methods have proven to

e powerful techniques for the study of heterogeneous media.ome of these classical tools today include multiple-scale expan-ions 4–8, G- and -convergences 9,10, and energy methods11,12.

An approach based on Fourier analysis has been proposed inefs. 13,14. This method works in the following way. First,riginal operator is transformed into an equivalent operator in theourier space. The standard Fourier series is used to expand theoefficients of the operator and a Fourier transform is used toecompose the integrals. Next, the Fourier transforms of the inte-rals are expanded using a suitable two-scale expansion, and theomogenized problem is finally derived by merely neglectingigh-order terms in the above expansions when moving to theimit as the period tends to zero.

The method of orientational averaging was proposed in Ref.15. It is based on the following assumptions: A characteristicolume repeated throughout the bulk of the composite is isolatedrom the composite medium. The properties of the composite as ahole are assumed to be the same as those of this characteristicolume. In the case of ideally straight fibers the set of fibers isepresented in the form of the array of unidirectional reinforcedylinders. We should also mention papers on homogenization us-ng wavelet approximations 16 and nonsmooth transformations17.

In this section we describe a variant of homogenization ap-roach that will be used further. For simplicity, we will start with2D heat conduction problem. However, these results will remain

orrect for other kinds of transport coefficients such as electricalonductivity, diffusion, magnetic permeability, etc. Due to theell-known longitudinal shear–transverse conduction analogy, seeef. 18, the elastic antiplane shear deformation can also bevaluated in a similar mathematical way. This will be followed byhe summary on the asymptotic homogenization applied to thelasticity problem for a 3D composite solid. Analogoussymptotic homogenization technique has been developed for aumber of more complicated nonlinear models, see Refs. 5,11.

Let us consider a transverse transport process through the peri-dic composite structure when the fibers are arranged in a periodicquare lattice, see Fig. 1.

The characteristic size l of inhomogeneities is assumed to beuch smaller than the global size L of the whole structure: lL.ssuming the perfect bonding conditions on the interface be-

ween the constituents, the governing BVP can be written as fol-ows:

ka 2ua

x12 +

2ua

x22 = − fa in a, um = uf ,

kmum

n= kf uf

non 2.1

ere and in the sequel variables indexed by m correspond to theatrix and those indexed by f correspond to the fibers; index a

akes both of these references: a=m or a= f . Generally, BVP 2.1

30802-2 / Vol. 62, MAY 2009


allows a number of different physical interpretations, but here it isdiscussed with a reference to the heat conduction. Then, in theabove expressions, ka are the heat conductivities of the constitu-ents, ua is a temperature distribution, fa is a density of heatsources, and /n is a derivative in the normal direction to theinterface . Let us now consider the governing BVP 2.1 usingthe asymptotic homogenization method 4–6,19–23. We will de-fine a natural dimensionless small parameter = l /L, 1, char-acterizing the rate of heterogeneity of the composite structure.

In order to separate micro- and macroscale components of thesolution we introduce the so-called slow x and fast y coordi-nates

xs = xs, ys = xs−1, s = 1,2 2.2

and we express the temperature field in the form of an asymptoticexpansion

ua = u0x + u1ax,y + 2u2

ax,y + ¯ 2.3

where x=x1e1+x2e2 and y=y1e1+y2e2, e1 and e2 are the Cartesianunit vectors. The first term u0x of expansion 2.3 represents thehomogeneous part of the solution; it changes slowly within thewhole domain of the material and does not depend on fast coor-dinates. All the further terms ui

ax ,y, i=1,2 ,3 , . . ., describe localvariation in the temperature field on the scale of heterogeneities.In the perfectly regular case the microperiodicity of the mediuminduces the same periodicity for ui

ax ,y with respect to fast vari-ables

ukax,y = uk

ax,y + Lp 2.4

where Lp=−1lp, lp= p1l1+ p2l2, and ps=0, 1, 2, . . ., l1 and l2are the fundamental translation vectors of the square lattice.

The spatial derivatives are defined as follows:

xs=

xs+ −1

ys2.5

Substituting expressions 2.2, 2.3, and 2.5 into the governingBVP 2.1 and splitting it with respect to equal powers of onecomes to a recurrent sequence of problems:

2u1a

y12 +

2u1a

y22 = 0 in , u1

m = u1f ,

kmu1m

m− kf u1

f

m= kf − km

u0

n 2.6

Fig. 1 Composite material with hexagonal array of cylindricalfibers

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ka 2u0

x12 +

2u0

x22 + 2

2u1a

x1 y1+ 2

2u1a

x2 y2+

2u2a

y12 +

2u2a

y22

= − fa in

u2m = u2

f , kmu2m

m− kf u2

f

m= kf u1

f

n− kmu1

m

n

2.7

nd so on.Here /m is a derivative in the normal direction to the inter-

ace in the fast coordinates y1, y2.The BVP 2.6 allows evaluation of the higher-order component

iax ,y of the temperature field; owing to the periodicity condi-ion 2.4 it can be considered within only one periodically re-eated unit cell. It follows from the BVP 2.6 that variables x andcan be separated in u1x ,y by assuming

u1x,y =u0x

xlU1y +

u0xx2

U2y 2.8

here U1y and U2y are local functions for which problem2.6 yields the following unit cell problems:

2U1yy1

2 +2U1y

y22 = 0,

2U2yy1

2 +2U2y

y22 = 0 in

U1my = U1

f y, U2my = U2

f y on

kmU1my

m− kf U1

f ym

= kf − kmm1, kmU2my

m− kf U2

f ym

= kf − kmm2 on 2.9

here m1 and m2 are components of a unit normal to the interface in coordinates y1, y2.

In order to determine the effective heat conductivity, the BVP2.7 should be considered. Let us apply to Eq. 2.7 the followingomogenization operator over the unit cell volume 0:

0

m·dy2dy3 +

0in

·dy2dy3L−2

erms containing u2a will be eliminated by means of the Green

heorem and taking into account the boundary conditions 2.7 andhe periodicity condition 2.4, which yields

1 − ckm + ckf 2u0

x12 +

2u0

x22 +

km

L2 0

m 2u1

m

x1 y1

+2u1

m

x2 y2dy1dy2 +

kf

L2 0

in 2u1

f

x1 y1+

2u1f

x2 y2dy1dy2

= − 1 − cfm + cf f 2.10

here c is the fiber volume fraction.Let us note a difference in the right-hand side of Eq. 2.10

hen kf →0 and kf =0. Assume that f f = fm= f . Then for any kf

0 we get an expression −f in the right-hand side of Eq. 2.10.ut for kf =0 we get there a different expression −f1−c. That

epresents an explanation to the following “paradox” pointed outn Refs. 24,25:

limkf→0

lim→0

ux1,x2,kf, lim→0

ux1,x2,0,

The homogenized heat conduction equation can be obtained byubstituting expression 2.8 for u1x ,y into Eq. 2.10, which
ields
pplied Mechanics Reviews


kij u0

2xxi xj

= − f 2.11

where

kij = 1 − ckm + ckfij +km

L2 0

mil

Ujm

yldy1dy2

+kf

L2 0

inil

Ujf

yldy1dy2 2.12

where f = 1−cfm+cf f is the effective density of heat sources,ij is Kronecker’s delta, indices i , j , l=1,2, and the summationover the repeated indices is implied.

Note that in general the homogenized material will be aniso-tropic, and kij in Eq. 2.11 is a tensor of effective coefficients ofheat conductivity. Tensor kij is defined by expression 2.12, andit can be readily calculated as soon as the unit cell problems 2.9are solved and the local functions U1y and U2y are found. Unitcell problems 2.9 can be solved analytically or numerically. Theapproximate methods of their analytical solution will be presentedin Sec. 3.

If a periodic heterogeneous medium is made of constituentswith moderately different properties, the homogenized equationspreserve a local character of the original equations. The coeffi-cients of the homogenized equations can be explicitly expressedin terms of the solutions of the unit cell problems. However, whena heterogeneous medium consists of materials with highly differ-ent properties, the homogenized constitutive relation may reveal anonlocal structure. Theory for this case was developed by Allaire26 and Zhikov 27. This made it possible to analyze the higher-gradient effects in the overall behavior of heterogeneous media28–32.

Asymptotic homogenization procedure strongly depends on thefollowing three parameters: the natural small dimensionless pa-rameter characterizing the rate of heterogeneity of the compositestructure, on the ratio of material properties of matrix and inclu-sion 1=km /kf, and on the volume fraction of inclusions c. Theabove obtained results are formally valid for 11 and c1. For11 and c1 the effective heat conductivity k in the firstapproximation does not depend on km; for c1 and 11 kf mustbe omitted in k , etc. But numerical error in calculating effectiveconductivity k , obtained by assuming 11, c1, is not essen-tial, and for simple isotropic inclusions spheres, ellipsoids, cylin-ders, and parallelepipeds in isotropic matrix it can be used in thefirst approximation for any kind of differences between propertiesof the constituent materials and their volume fractions. For thehigher-order homogenization approach this conclusion can bewrong.

Let us now consider asymptotic homogenization of an elasticityproblem for a 3D periodic composite material occupying region with a boundary S, see Fig. 2.

We assume that the region is made up by the periodic rep-etition of the unit cell Y in the form of a parallelepiped withdimensions Yi, i=1,2 ,3. The elastic deformation of this compos-ite solid is described by the following BVP:

ij

xj= f i in , ux = 0 on S 2.13

ij = cijklekl

, eij =

1

2 ui

xj+

uj

xi 2.14

where cijkl is a tensor of elastic coefficients. The coefficients cijklare assumed to be periodic functions with a unit cell Y. Here andin the sequel all Latin indices assume values of 1, 2, and 3, and
repeated indices are summed.
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The introduction of the fast variables yi=xi /, i=1,2 ,3,, simi-ar to Eq. 2.2, into Eqs. 2.13 and 2.14 and the rule of differ-ntiation 2.5 leads to the following BVP:

ij

xj+

1

ij

yj= f i in , ux,y = 0 on S 2.15

ij x,y = cijkly

uk

xlx,y 2.16

The next step is to expand the displacements and, as a result,he stresses into the asymptotic expansions in powers of the smallarameter , similar to expansion 2.3,

ux,y = u0x,y + u1x,y + 2u2x,y + ¯ 2.17

ij x,y = ij

0x,y + ij1x,y + 2ij

2x,y + ¯ 2.18

here all above functions are periodic in y with the unit cell Y.ubstituting Eqs. 2.17 and 2.18 into Eqs. 2.15 and 2.16,hile considering at the same time the periodicity of ui in y,

eveals that u0 is independent of the fast variable y, see Ref. 5or details. Subsequently, equating terms with similar powers of esults in the following set of equations:

ij0x,yyj

= 0 2.19

ij1x,yyj

+ij

0x,yxj

= f i 2.20

here

ij0 = cijkl uk

0

xl+

uk1

yl 2.21

ij1 = cijkl uk

1

xl+

uk2

yl 2.22

ubstitution of Eq. 2.21 into Eq. 2.19 yields

yjcijkl

uk1x,yyl

=cijkly

yj

uk0xxl

2.23

ue to the separation of variables in the right-hand side of Eq.2.23 the solution of Eq. 2.23 can be written as follows, similaro Eq. 2.8:

un1x,y =

uk0xxl

Nnkly 2.24

here Nnkly n ,k , l=1,2 ,3 are periodic functions with a unit

x2

(b)

y1

x1

Matrix

Y

Ω y3

Reinforcement

y2

x3

(a)

Reinforcement

ε

Fig. 2 „a… 3D periodic composite and „b… unit cell Y

ell Y satisfying the following equation:

30802-4 / Vol. 62, MAY 2009


yjcijmny

Nmkly

yn = −

cijkl

yj2.25

It is observed that Eq. 2.25 depends only on the fast variable yand it is entirely formulated within the unit cell Y. Thus, theproblem 2.25 is appropriately called an elastic unit cell problem.Note that instead of boundary conditions, this problem has a con-dition of a periodic continuation of functions Nm

kly.If inclusions are perfectly bonded to matrix on the interfaces of

the composite material, then the functions Nmkly together with the

expressions cijkl+cijmnyNmkly /ynnj

c, i=1,2 ,3, should becontinuous on the interfaces. Here, nj

c are the components of theunit normal to the interface.

The next important step in the homogenization process isachieved by substituting Eq. 2.24 into Eq. 2.21, and the result-ing expression into Eq. 2.22. The result is then integrated overthe domain Y of the unit cell with volume Y remembering totreat x as a parameter as far as integration with respect to y isconcerned. After canceling out terms that vanish due to the peri-odicity, we obtain the homogenized global problem

Cijkl

2uk0x

xj xl= f i in , u0x = 0 on S 2.26

where the following notation is introduced:

Cijkl =1

YY

cijkly + cijmnyNm

kl

yndv 2.27

Similarly, substituting Eq. 2.24 into Eq. 2.21 and then integrat-ing the resulting expression over the domain of the unit cell Yyields

ij0 =

1

YY

ij0ydv = Cijkl

uk0

xl2.28

Equations 2.26 and 2.28 represent the homogenized elastic-

ity BVP. The coefficients Cijkl given by Eq. 2.27 are the effectiveelastic coefficients of the homogenized material. They are readilydetermined as soon as the unit cell problem 2.25 is solved andthe functions Nm

kly are found. It is observed that these effectivecoefficients are free from the complications that characterize theoriginal rapidly varying elastic coefficients cijkly. They are uni-versal for a composite material under study and can be used tosolve a wide variety of boundary-value problems associated withthe given composite material.

It should be noted that the solution of the global problem 2.26for the equivalent homogenized material will not be satisfactory inthe vicinity of the boundary of the solid S, i.e., at the distances oforder of . From the standpoint of homogenization theory aboundary-layer problem should be considered. Boundary effectswill be discussed in Sec. 9.

3 Unit Cell ProblemsAs we have seen in Sec. 2, the derivation of the homogenized

equations for the periodic composites includes solution of the unitcell problems, i.e., problems 2.9 and 2.25. In some particularcases these problems can be solved analytically producing exactsolutions, for example, for laminated composites and grid-reinforced structures, see Refs. 5,33–35. The explicit formulasfor effective moduli are very useful, especially for the design andoptimization of composite materials and structures 35,36. But ingeneral case, the unit cell problems cannot be solved analytically,and therefore the numerical methods should be used. In somecases, the approximate analytical solutions of the unit cell prob-lems can be found, and the explicit formulas for the effectivecoefficients can be obtained due to the presence of additional
small parameters within the unit cell, not to be confused with the


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mall parameter of inhomogeneity. In particular, use of the param-ter 1=km /kf will be demonstrated in Sec. 4. As a rule, the prob-ems in micromechanics of composites are multiscale. Conse-uently, it is very difficult to solve them analytically orumerically. But, at the same time, that opens wide opportunitiesor application of the asymptotic methods.

For a small volume fraction of inclusions, ccmax, one can usehe three-phase model 37–40. It has been proposed by Brugge-an 41,42 and is based on the following assumption: The peri-

dically heterogeneous composite structure is approximately re-laced by a three-phase medium consisting of a single inclusion, aatrix layer, and an infinite effective medium with certain homog-

nized mechanical properties. Asymptotic justification of thehree-phase composite model is given in Ref. 37.

For laminated composite materials unit cell problems 2.9 and2.25 are one dimensional, and they can be solved analytically.sing this analytical solution, the effective properties of laminated

omposites can be obtained in the explicit analytical form fromqs. 2.12 and 2.27, see Refs. 5,35. In the more complicatedase of generally anisotropic constituent materials the explicit for-ulas for effective elastic, actuation, thermal conductivity, and

ygroscopic absorption properties of laminated smart compositesre derived by Kalamkarov and Georgiades 43. In particular, theollowing explicit formula for the effective elastic coefficients of aaminated composite in the case of generally anisotropic constitu-nt materials is derived in Ref. 43:

Cijkl = Cijkl − Cijm3Cm3q3−1 Cq3kl

+ Cijm3Cm3q3−1 Cq3p3

−1 −1Cp3n3−1 Cn3kl

here the angular brackets denote a rule of mixture, and as earlierndicated all Latin indices assume values of 1, 2, and 3, and re-eated indices are summed.

For fiber-reinforced periodic composites the unit cell problem2.25 becomes two dimensional, and it can be solved analyticallyor some simple geometries, or numerically, see Refs. 5,44–47.n particular, the numerical results for the effective elastic modulif the incompressible porous material are obtained in Ref. 48.

It is important to obtain the approximate solution of the unitell problem valid for all values of material parameters and vol-me fraction of constituents. For that purpose various interpola-ion procedures can be applied. In this section we will introducen asymptotic technique based on a modification of a boundaryhape perturbation approach 49. Some other techniques devel-ped in Refs. 50–52 will be discussed in Secs. 4 and 5.

For small volume fraction of inclusions the solution can beepresented in the form of series of the Weierstrass elliptic func-ions 53–59 or their 3D generalization 60.

For large inclusions ccmax one can use lubrication approxi-ations 38,61. In this approach the unit cell problem with

urved boundaries of inclusion is replaced by a much simplerroblem for a strip in 2D case, see Fig. 3, or a layer in 3D case.

In 2D case the following equation can be used instead of Eq.2.6:

2u1m

y12 = 0 in strip − 0.5 y1 0.5, − y2

u1m = u1

f , kmu1m

m− kf u1

f

m= kf − km

u0

nfor y1 = 0.5

nd the same problem by replacing y1 and y2. Here is a minimalistance between inclusions.

Let us now describe the modification of a boundary shape per-urbation approach. We introduce the polar coordinates ry1

2+y22, =arctany2 /y1 in the plane of the unit cell plane, see

ig. 4. Then Eq. 2.6 can be written as follows:



2u1a

r2 +1

r

u1a

r+

1

r2

2u1a

2 = 0 3.1

u1m = u1

f r=A, kmu1m

r− kf u1

f

r= kf − km

u0

n

r=A

3.2

It is shown in Ref. 4, Chap. 6, Sec. 3, see also Refs. 5,62,that for axially symmetric domains the periodicity continuationcondition 2.4 can be replaced by zero boundary conditions at thecenter and at the outer boundary 0 of the unit cell:

u1f = 0r=0 3.3

u1m = 0r=Ro

3.4

It should be noted that such a replacement is justified for the firstapproximation of the asymptotic approach, but it may be wrongfor the higher approximations.

In Eq. 3.4 the square shape of 0 can be defined as

(a)

(b)

Fig. 3 „a… Fiber-reinforced composite with fiber volume frac-tion close to maximum and „b… asymptotic model

y2

y3r

θΩ0

m

A

R0

L/2

Ω0f

∂Ω

∂Ω0

Fig. 4 Unit cell of the regular square lattice of cylindrical
fibers
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R =R0

cos3.5

here R0=L /2 is the radius of the inscribed circle, =−0 cane considered as a small parameter 1, =0− /4¯0 /4, and 0=n /2, n=0,1 ,2 , . . ..Solution of the unit cell problems 3.1–3.4 is represented in

he form of asymptotic expansion in powers of . This expansionhould be invariant if is replaced by −; thus it should containnly even powers of :

u1a = u1,0

a + 2u1,2a + 4u1,4

a¯ 3.6

he boundary condition 3.4 is formulated at r=R. Conse-uently, if we now substitute expansion 3.6 directly into condi-ion 3.4, then parameter will be present in the arguments of theunctions u1,j

m , j=0,1 ,2 , . . ., and splitting the input problem withespect to will not be possible. In order to eliminate from therguments of u1,j

m the boundary condition 3.4 should be trans-erred from the original contour r=R to the inscribed circle rR0 by means of the Taylor expansion

u1mr=R0/cos = u1,0

m r=R0+ 2R0

2 u1,0

m

r

r=R0

+ 45R0

24

u1,0m

r+

R02

8

2u1,0m

r2 r=R0

+ ¯ 3.7

ventually, we obtain

u1m = C1r + C2r−1

u0

n, u1

f = C3ru0

n3.8

here

Cp = Cp,0 + 2Cp,2 + 4Cp,4 + O6, p = 1,2,3 3.9

he coefficients of expansion 3.9 are as follows:

C1,0 =

1 − , C2,0 = −

1 − A2, C3,0 = −

1 − 1 −

C1,2 = −

1 − 2 , C2,2 =2

1 − 2A2, C3,2 = − 11 − 2

C1,4 = −1

2

1 − 31 − 3 , C2,4 =

1

2

21 − 31 − 3 A2,

C3,4 =1

2

− 11 − 31 − 3 3.10

ere = kf −km / kf +km, =A2 /R02=c /cmax, and A=a /, where

is a radius of fiber in slow variables.Let us now examine convergence of expansion 3.9. The ratios

f the third-to-second constitutive terms are the same for any p:

4Cp,4

2Cp,2=

1 − 3

21 − 2 3.11

anges of the variables in expression 3.11 are as follows: −11, 01, and 02 /42. It can be easily seen that

xpansion 3.9 diverges in the case of perfectly conductive nearlyouching fibers when →1 and →1. In order to eliminate thisingularity, the Padé approximants PAs can be applied 63. Inhe case under consideration the Padé approximants to expansion
3.9 are as follows:
30802-6 / Vol. 62, MAY 2009


Cp =

Cp,0 + Cp,2 − Cp,0Cp,4

Cp,22

1 −Cp,4

Cp,22

3.12

As a second possibility to avoid divergence at →1 and →1,the following approximate estimation of the overall sum of expan-sion 3.9 can be proposed. The first term Cp,0 zero-order ap-proximation represents a solution of the unit cell problems3.1–3.4 when the outer boundary 0 of the unit cell is re-placed by a circle of radius R0. At this step Eq. 3.1 as well as theboundary conditions 3.2 and 3.3 are strictly satisfied, but thereexists a discrepancy in the boundary condition 3.4. All the nextterms of the expansion tend to reproduce the original square shapeof 0 in order to satisfy the boundary condition 3.4 more ac-curately. On the other hand, the original shape of 0 can berestored exactly in the zero-order approximation if R0 in the ex-pression for =A2 /R0

2 is substituted by R defined in Eq. 3.5.In this case the boundary conditions 3.2–3.4 are exactly satis-fied, and the solution converges for all values of and . Thus weobtain

C1 = cos2

1 − cos2, C2 = −

1 − cos2A2,

C3 = −1 − cos21 − cos2

3.13

The obtained solution satisfies Eq. 3.1 only approximately,but further comparison with the known numerical results showsthat the error of this approximation is not successive.

Let us check the obtained solution in the case of perfectly con-ductive fibers kf /km=, the case that usually leads to main com-putational difficulties. Table 1 displays numerical results for theeffective conductivity k evaluated on the basis of expansion3.9 and improved expressions 3.12 and 3.13. The obtainedsolutions are also compared with the data from Perrins et al. 64.It should be pointed out that the method given in Ref. 64 is notapplicable in the limiting case of perfectly conductive nearlytouching fibers kf /km= and c→cmax=0.7853, . . . when rapidoscillation of the temperature field occurs on the microlevel. Bothof the present solutions 3.12 and 3.13 predict this case cor-rectly with a considerably small discrepancy between them.

Finally, we assume the solution of the unit cell problem in theform 3.8 and 3.13. Comparison of the obtained results with thedata from Perrins et al. 64 for different values of conductivity of

Table 1 Effective conductivity Šk‹ /km of the regular square lat-tice of perfectly conductive cylindrical fibers „kf /km=…

c

The present solutions; Cp are determined by

Perrins et al. 64Expansion 3.9 PA 3.12 Formulas 3.13

0.1 1.210 1.247 1.223 1.2220.2 1.470 1.544 1.506 1.5000.3 1.811 1.918 1.879 1.8600.4 2.306 2.417 2.395 2.3510.5 3.270 3.145 3.172 3.0800.6 7.106 4.386 4.517 4.3420.7 73.92 7.409 7.769 7.4330.74 — 10.91 11.46 11.010.76 — 15.29 15.99 15.440.77 — 20.18 21.04 20.430.78 — 35.01 36.60 35.93

fibers is presented in Fig. 5.



→f

a

4

cfeap

FsR

Fad

A

Downloaded Fr

The behavior of the effective conductivity at kf /km→ and ccmax can be verified by comparison with the asymptotic formula

rom O’Brien 65 shown in Fig. 6.The case kf /km= should be checked separately, and the results

t this limit are shown in Table 2.

Two-Point Padé ApproximantsBergman 66 showed that for the two-component isotropic

omposites the effective conductivity k is a Stieltjes function of

1=km /kf. This fact was used with one-point Padé approximantsor evaluating bounds of the effective parameters, see Refs.18,67. On the other hand, it is possible to obtain asymptoticxpansion for k as a function of 1 for 11 and 11. It givespossibility to use TPPAs generated by two different power ex-

ansions of Stieltjes function 68–78.

ig. 5 Effective conductivity Šk‹ in the perfectly regular case:olid curves—the present solution „3.13…; circles—data fromef. †64‡

100

k/km

120

kf/k

m

c = 0.784

c = 0.785

101

102

103

104

105

100

80

60

40

20

1

140

ig. 6 Asymptotic behavior of Šk‹ in the perfectly regular caset kf /km\, c\cmax: solid curves—the present solution;ashed curves—the asymptotic formula from Ref. †65‡

Table 2 Effective conductivity Šk‹ /km in the case kf /km=

Volumefraction c

k /km, formulafrom Ref. 65

k /km,present solution

0.784 74.41 73.340.785 139.5 138.40.7853 281.0 279.3



The notion of TPPA has been defined in Ref. 63. Let us as-sume that

fz = i=0

aizi when z → 0

i=0

biz−i when z → 4.1

The TPPA is represented by the rational functionk=0

m akzk /k=0

n bkzk, where k+1 k=0,1 , . . . ,n+m+1 coefficients

of a Taylor expansion, if z→0, and m+n+1−k coefficients of aLaurent series, if z→, coincide with the corresponding coeffi-cients of the series 4.1.

Tokarzewski 69,70 and Tokarzewski et al. 72–74 investi-gated the TPPA for a nonequal, finite number of terms of twopower expansions of the Stieltjes functions at zero and at infinity.Under some assumptions they proved that the diagonal TPPAsform sequences of lower and upper bounds uniformly convergingto the Stieltjes function.

The general situation when the TPPA corresponding to an arbi-trary number of terms of power expansions at zero and infinity hasbeen studied in the real domain by Tokarzewski and Telega75,76. They extended the fundamental inequalities derived forthe PA to the general TPPA. They proved the following theoremthat is very useful for practical applications.

The TPPAs for the Stieltjes function, represented by the powerexpansions at zero, Rzn=1

cnzn, and at infinity, Rzn=0

C−nz−n, obey the following inequalities for k=1,2 , . . .,2Mk=1,2 , . . . ,2M +1:

− 1kM/Mk − 1kM + 1/M + 1k − 1kRz

− 1k−1M/M − 1k − 1kM + 1/Mk − 1k−1Rz4.2

where Rz stands for the limit as M tends to infinity of M /Mk,M +1 /Mk, z is real and positive, and M /Nk

= j=0M jz

j / j=0N jz

j. Here k denotes the given number of coeffi-cients of power expansions at infinity matched by the TPPA rep-resented by M /Nk.

The inequalities 4.2 have the consequence that M /Mk andM +1 /Mk form upper and lower bounds for Rz obtainableusing only the given number of coefficients and that the use ofadditional coefficients improves the bounds.

The above theorem has been successfully used for the study ofthe effective heat conductivity for a periodic square array of cyl-inders of conductivity kf =h embedded in a matrix of conductivitykm=1. As an input for calculating TPPA the coefficients of theexpansions of ex in powers of h−1 for h−11 and in powersof 1 / h−1 for h→ have been used. The sequences of TPPAuniformly converging to the effective conductivity are shown inFigs. 7 and 8. The best bounds obtained by the TPPAs, namely,18 /181 and 18 /182, are presented in Fig. 8. In Figs. 7 and 8the asymptotic solution obtained by McPhedran et al. 79 isdrawn for comparison.

It follows that the TPPAs allow us to evaluate the effectivemoduli for a range of parameters much wider than the PA 18,67.For example, for =0.78539 the TPPA approach leads to veryrestrictive bounds, whereas the PA method fails see Fig. 7.

5 Asymptotically Equivalent FunctionsThe asymptotic formulas for the effective conductivity k for

ccmax and ccmax do not provide complete representation ofthe effective conductivity for any arbitrary c. Therefore, it is nec-essary to obtain the values of functions k for the intermediate
values of c. TPPA cannot be used in this case because the
MAY 2009, Vol. 62 / 030802-7


aftafp

a

Tn

wftmb

2

0

Downloaded Fr

symptotic expressions for k for ccmax contain the logarithmicunctions. The solution of this problem can be found by applyinghe asymptotically equivalent functions 80 or the quasifractionalpproximations in the other terminology. Let us assume that theunction fz in the limit z→ is described by a nonrational ex-ression Fz:

fz = Fz for z → 5.1a

nd

fz = i=0

cizi for z → 0 5.1b

hen the AEFs should also contain similar nonrational compo-ents. In general, the AEFs can be produced from Eqs. 5.1a and5.1b as follows:

fz i=0

m

izzii=0

n

izzi 5.2

here i and i are considered not as constants but as someunctions of z. Functions iz and iz are chosen in such a wayhat i the expansion of the AEFs 5.2 in powers of z for z→0

atches the perturbation expansion 5.1b and ii the asymptoticehavior of the AEFs 5.2 for z→ coincides with the expres-

Fig. 7 The sequences of †M /M‡0, †Mformly converging to the effectivesquare array of cylinders. Curves †M„a……. The bounds †18/18‡1 and †18/18

Fig. 8 The TPPA upper and lower ba square array of densely packed=0.785 the bounds coincide. For =0tive. For higher volume fractions lower and upper bounds rapidly incr

Table 3 The constan

a1 a2

SC array 1.305 0.231BCC array 0.129 0.413FCC array 0.0753 0.697

30802-8 / Vol. 62, MAY 2009


sion Fz Eq. 5.1a. Normally, such approach leads to satisfac-tory results, see Refs. 81–90.

Now let us consider an application of the method of AEFs forcalculation of the effective heat conductivity of the infinite regulararray of perfectly conducting spheres embedded in a matrix withunit conductivity. Sangani and Acrivos 91 obtained the follow-ing expansion for the effective conductivity k :

k = 1 − 3c− 1 + c + a1c10/31 + a2c11/3

1 − a3c7/3 + a4c14/3 + a5c6

+ a6c22/3 + Oc25/3−1

5.3

Here we consider three types of space arrangement of spheres,namely, the simple cubic SC, body centered cubic BCC, andface centered cubic FCC arrays. Constants ai for these arrays aregiven in Table 3.

In the case of perfectly conducting large spheres c→cmax,where cmax is the maximum volume fraction for a sphere theproblem can be solved by means of a reasonable physical assump-tion that the heat flux occurs entirely in the region where spheresare in a near contact. Thus, the effective conductivity is deter-mined in the asymptotic form for the flux between two spheres,which is logarithmically singular in the width of a gap, justifyingassumption 92

‡1, and †M /M‡2, M=2,4,6,12,18 uni-nductivity e„h… „h=2 /1… of the‡2 are indistinguishable „solid line—are very restrictive.

ds on the effective conductivity forighly conducting cylinders. For 53, 0.78539 bounds are very restric-0.78539816 the difference betweenes.

a1 , . . . ,a6, in Eq. „5.3…

a4 a5 a6

5 0.0723 0.153 0.01054 0.257 0.0113 0.005621 0.0420 0.0231 9.1410−7

/Mco

/M‡

ounh

.78

eas

ts

a3

0.400.760.74



wtncTa

tc

Ha

e−=

ocnmtmtf

p

SBF

Fu

A

Downloaded Fr

k = − M1 ln − M2 + O−1 5.4

here =1− c /cmax1/3 is the dimensionless width of a gap be-ween the neighboring spheres, →0; M1=0.5cmaxp, p is theumber of contact points at the surface of a sphere; and M2 is aonstant, dependent on the type of space arrangement of spheres.he values of M1, M2, and cmax for the three types of cubic arraysre given in Table 4.

On the basis of “limiting” solutions 5.3 and 5.4 we develophe AEFs valid for all values of the volume fraction of inclusions 0,cmax:

k = P1c + P2cm+1/3 + P3 ln /Qc 5.5

ere rational functions P1c and Qc and constants P2 and P3re determined as follows:

Qc = 1 − c − a1c10/3, P1c = i=0

m

ici/3

P2 = 0 for n = 1, P2 = − P1cmax

+ QcmaxM2/cmaxm+1/3 for n = 2

The AEF 5.5 takes into account m leading terms of expansion5.3 and n leading terms of expansion 5.4. Coefficients i arequal to 0=1, 3=2−QcmaxM1 / 3cmax, 10=−a1

QcmaxM1 / 10cmax10/3, and j =−QcmaxM1 / jcmax

j/3 , j1,2 , . . . ,m−1,m, j3,10.Increment of m and n leads to the growth of the accuracy of the

btained solution 5.5. Let us illustrate this dependence in thease of SC array. We calculated k for different values of m and. In Fig. 9 our analytical results are compared with experimentaleasurements from Meredith and Tobias 93,94 black dots. De-

ails on these data can be found in Ref. 95. Finally, we restrict=19 and n=2 for all types of arrays, as they provide a satisfac-

ory agreement with numerical data and a rather simple analyticalorm of the AEF 5.5.

Numerical results for the BCC and the FCC arrays are dis-layed in Figs. 10 and 11, respectively. For BBC array the ob-

Table 4 The constants M1, M2, and cmax

M1 M2 cmax

C array /2 0.7 /6CC array 3 /2 2.4 3 /8CC array 2 7.1 2 /6

k /km

0 0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

12

C

m=5, n=1

m=5, n=2

m=10, n=2

m=19, n=2

ig. 9 Effective conductivity Šk‹ /km of the SC array versus vol-
me fraction of inclusions c


tained AEF 5.5 is compared with the experimental results fromMcKenzie and McPhedran 96 and McKenzie et al. 97. ForFCC array the experimental data are not available; therefore weare comparing with the numerical results obtained in Ref. 97using the Rayleigh method. The agreement between the analyticalsolution 5.5 and the numerical results is quite satisfactory.

6 Random CompositesTwo-phase composites with random microstructure were ana-

lyzed by Drugan and Willis 98 and Drugan 99. They employedthe Hashin–Shtrikman variational principle. A numerical imple-mentation of this work was carried out by Segurado and Llorca100.

Percolation effects are very important for the analysis of com-posite materials 101,102. On the other hand, in many cases com-posite materials can be studied without taking into account thepercolation effects. For example, foam concrete usually has thechaotic distribution of pores with no clusters. The structure ofdispersed composites manufactured by a cold drawing or uniformpressure is nearly regular with no clusters. In these cases a con-cept of shaking geometry 103,104 can be very useful.

It should be noted that many of commonly used bounds, forexample, the Hashin–Shtrikman variational bounds, demonstratedivergence and become almost out of practical use for the denselypacked but not percolated composites with the highly differentproperties of the constituent materials. Therefore, in order to ob-tain a reasonable estimation of the effective properties, the im-proved bounding models should be developed, such that they donot allow appearance of cluster chains, even if a volume fractionof one of the constituents is beyond the percolation limit. Suchimproved bounds for the effective transport properties of a ran-dom composite material with cylindrical fibers are proposed inRef. 105.

Let us now consider a nonregular fiber-reinforced composite.Center of each fiber can randomly deviate within a circle of di-ameter d, whereas these circles themselves form a regular squarelattice of a period l, see Fig. 12.

0 0.1 0.2 0.3 0.4 0.5 0.6

2

4

6

8

10

12

14

c

‹k›/km

Fig. 10 Effective conductivity Šk‹ /km of the BCC array versusvolume fraction of inclusions c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

2.5

5

7.5

10

12.5

15

17.5

c

‹k›/km

Fig. 11 Effective conductivity Šk‹ /km of the FCC array versus
volume fraction of inclusions c
MAY 2009, Vol. 62 / 030802-9


svtrrboacNc=

tgtTsBH

irkpfdct

drmlrossbt

Kris

Fm

0

Downloaded Fr

Such kind of a microstructure is usually referred to as ahaking-geometry composite 103,104. From a practical point ofiew it may correspond to a random “shaking” of the fibers abouthe periodic lattice caused by some fabrication or technologicaleasons. The deviation parameter =d / l describes the rate of non-egularity of the structure; its maximum value max is determinedy the case when neighboring fibers are nearly touching eachther. Values of cannot be higher than max since that will meanpenetration of the neighboring fibers. A simple geometrical cal-

ulation yields max=1−c /cmax. Let us also assume that kf km.ote that the opposite case can be treated in the same mathemati-

al way using the well-known duality relation 106 kkf ,km kkm ,kf −1.Kozlov 107 showed that a regular lattice possesses the ex-

reme effective properties among the corresponding shaking-eometry random structures. Originally this result was proved forhe case of the dilute composites. Berlyand and Mityushev103,104 generalized Kozlov’s result 107 to the nondilute cases.herefore, a solution for the perfectly regular lattice can be con-idered as a lower bound on the effective transport coefficient.elow we will see that it almost coincides with the correspondingashin–Shtrikman lower bound 108–110.On the other hand, the upper bound can be obtained by replac-

ng the input nonregular assembly of fibers of radius a by theegular lattice of fibers of radius a+d /2. Such estimation is alsonown as a security-spheres approach. It has been originally pro-osed by Keller et al. 111 and was further extended by Ruben-eld and Keller 112 and Torquato and Rubinshtein 113. Foretails and references see also Ref. 114. In the case of high-ontrast composites this bound appears to be essentially betterhan Hashin–Shtrikman’s bound.

Improved bounds on the effective conductivity k of the ran-om shaking-geometry composites, see Fig. 13, are deduced di-ectly from the solution obtained by means of the improvedethod of boundary shape perturbation, discussed in Sec. 3. Fol-

owing the analytical results 103,104, solution for the perfectlyegular lattice is assumed as the lower bound. The upper bound isbtained by the security-spheres approach. In both cases we as-ume that there are no clusters of fibers in the composite undertudy. However, we should note that for a purely random distri-ution of cylindrical fibers the percolation threshold is reached athe volume fraction of inclusions cp0.41, see Ref. 101.

Let us introduce lower K1 and upper K2 bounds for k such that

1 k K2, and let us denote the conductivity of the perfectlyegular material as a function K0 of the fiber volume fraction c,.e., k =0=K0c. Then, the lower bound K1 is given by theolution in the perfectly regular case at =0, see Refs.

Ll

matrix Ωm

fibre Ωf

interface ∂Ω

x2

x3

e2

e3 l2

l3

ig. 12 General view of the shaking-geometry compositeaterial

103,104,107:

30802-10 / Vol. 62, MAY 2009


K1 = K0c 6.1

In order to obtain the upper bound K2 we replace the originalnonregular assembly of fibers of radius a by a regular lattice offibers of radius a+d /2. This estimation yields

K2 = K0c + 2ccmax + 2cmax 6.2

For comparison we also provide Hashin–Shtrikman’s variationalbounds 108–110, see also Ref. 38:

K1 = km +c

1/kf − km + 1 − c/2km6.3

K2 = kf +1 − c

1/km − kf + c/2kf6.4

Numerical examples are shown in Fig. 14. We can observe thatthe lower bound 6.1 almost coincides with the Hashin–Shtrikman bound 6.3. At the same time, the situation with theupper bound is different. For the low-contrast case kf /km→1 theHashin–Shtrikman bound 6.4 is better. But for the high-contrastcase kf /km→ the improved bound 6.2 provides essentiallybetter results, while the Hashin–Shtrikman bound 6.4 becomesalmost useless. A simple practical recommendation is that fromtwo upper bounds 6.2 and 6.4 the lowest one should be chosen.

7 Correlation Between Conductivity and Elastic Prop-erties of Composites

It is of interest to establish certain links strict or approximatebetween the solutions of transport and elasticity problems forcomposite materials. Such cross-property relations become veryuseful if one of them can be more easily calculated or measuredexperimentally. As soon as the effective properties reflect certainmorphological information about the composite medium, onemight expect that extracting useful knowledge about one propertywould allow determining the other properties. Unfortunately, ex-act results can be obtained very rarely, see Refs. 18,115–117.

Explicit cross-property relations for the anisotropic two-phasecomposite materials have been obtained by Sevostianov andKachanov 118,119 and Sevostianov et al. 120,121. Correla-tions between elastic moduli and thermal and electric conductivi-ties of the anisotropic composite materials are found in Refs.122–124.

Manevitch et al. 125, see also Refs. 19,21,126,127, devel-oped an approach that reduces the original 2D elasticity problemto a form resembling the transport problem. Moreover, in someparticular cases it allows to establish direct analogy between the

Fig. 13 Deviation of the fibers about the regular square lattice

effective elastic and transport properties.



smeb

W

Wx

aE

ly

A

Downloaded Fr

Let us consider 2D composite material with square inclusionshown in Fig. 15. We assume that the matrix and inclusions areade of orthotropic materials. Governing equations of the plane

lasticity problem can be written as follows index a representsoth m for matrix and f for fibers:

B1auxx

a + B3a + B12

a uyxa + B3

avxya = 0 7.1

B2avyy

a + B3a + B12

a vxya + B3

auxya = 0 7.2

e assume that

B1m,B2

m,B3m,B12

m = B1f ,B2

f ,B3f ,B12

f 7.3

e also assume the perfect bonding conditions on the interfaces= a and y= a:

um = uf 7.4

vm = v f 7.5

Sm = Sf 7.6

for x = a, T1m = T1

f 7.7

for y = a, T2m = T2

f 7.8

Here T1a=B1

auxa+B12

a vya, T2

a=B2avy

a+B12a ux

a, and Sa=B3auy

a+vxa.

Let us introduce parameter 2=B3m /B1

m. Further we will treat 2s a small parameter and assume that B1

mB2m and B12

m B3m. Then

qs. 7.1 and 7.2 can be rewritten as follows:

uxxa + 1 + uyy

a + vxya = 0

Fig. 14 Bounds on Šk‹ in the nonreglution: the lower bound „6.1… at =0values of . Dashed curves—the HashDilute composite: c=0.2 and „b… dense

Fig. 15 Composite material with square inclusions



vyya + 1 + vxx

a + uxya = 0

Here =B12m /B3

m and =B2m /B1

m.Asymptotic splitting in the first approximation yields two inde-

pendent equations 125–127:

uxxa + 1 + uyy

a = 0 7.9

vyya + 1 + vxx

a = 0 7.10

Equation 7.9 must be solved with conditions 7.4, 7.6, and7.7, and Eq. 7.10, with conditions 7.5, 7.6, and 7.8. Usingsmallness of the parameters B12

m /B1m, and B12

m /B2m, these conditions

can be rewritten as follows:for Eq. 7.9,

um = uf for x = a and y = a 7.11

uxm = ux

f for x = a 7.12

uym + vx

m = uyf + vx

f for y = a 7.13

for Eq. 7.10,

vm = v f for x = a and y = a 7.14

vym = vy

f for y = a 7.15

uym + vx

m = uyf + vx

f for x = a 7.16

Conditions 7.13 and 7.16 connect boundary-value problemsfor ua and va. It was proposed in Ref. 126 to replace conditions7.13 and 7.16 by the following ones:

for y = a, uym = uy

f

for x = a, vxm = vx

f

That allows replacing the elasticity BVP by the two transportBVPs with some approximation. Error of this approximation de-pends on the elastic energy of deformation. The contact conditionsfor shear forces Sa are not fully satisfied, but contribution of Sa

into the energy of deformation depends on coefficients B3a, and it

is small in comparison with the contribution of the other terms.Similar approach can be used in the case of fibers of some other

r case. Solid curves—the present so-the upper bound „6.2… for different

Shtrikman bounds „6.3… and „6.4…. „a…packed composite: c=0.7.

ulaandin–

cross-sectional shapes, for example, circle or elliptical fibers.

MAY 2009, Vol. 62 / 030802-11


8p

rsrssawe

ithpms

asittpsdtiritprdamvte

ctsetawosccnpcoersSigan

0

Downloaded Fr

Asymptotic Homogenization of Thin-Walled Com-osite Reinforced Structures

In the numerous engineering applications the composite mate-ials used are in the form of thin-walled structural members likehells and plates. Their stiffness and strength combined with theeduced weight and associated material savings offer very impres-ive possibilities. It is very common that the reinforcing elementsuch as embedded fibers or surface ribs form a regular array withperiod much smaller than the characteristic dimensions of thehole composite structure. Consequently, the asymptotic homog-

nization analysis becomes applicable.The asymptotic homogenized model for plates with periodic

nhomogeneities in tangential directions has been developed forhe first time by Duvaut 128,129. In these works asymptoticomogenization procedure was applied directly to a 2D plateroblem. Later, Andrianov et al. 130 applied homogenizationethod to analyze statical and dynamical problems for the ribbed

hells.Evidently, the asymptotic homogenization method cannot be

pplied directly to the cases of 3D thin composite layers if theirmall thickness in the direction of which there is no periodicitys comparable with the small dimensions of the periodicity cell inhe two tangential directions. To deal with the 3D problem for ahin composite layer, a modified asymptotic homogenization ap-roach was proposed by Caillerie 131,132 in the heat conductiontudies. It consists of applying a two-scale asymptotic formalismirectly to the 3D problem for a thin inhomogeneous layer withhe following modification. Two sets of “rapid” coordinates arentroduced. Two tangential coordinates are associated with theapid periodic variation in the composite properties. The third ones in the transverse direction and is associated with the smallhickness of the layer, and it takes into account that there is noeriodicity in this transverse direction. There are two small pa-ameters, one a measure of periodic variation in two tangentialirections and the other is a measure of a small thickness. Gener-lly, these two parameters may or may not be of the same order ofagnitude. But commonly in practical applications they are small

alues of the same order. Kohn and Vogelius 133–135 adoptedhis approach in their study of a pure bending of a thin, linearlylastic homogeneous plate with wavy surfaces.

The generalization of this approach to the most comprehensivease of a thin 3D composite layer with wavy surfaces that modelhe surface reinforcements was offered by Kalamkarov 5,33,34,ee also Ref. 35. In these works the general asymptotic homog-nization model for composite shell was developed by applyinghe modified two-scale asymptotic technique directly to 3D elasticnd thermoelastic problems for a thin curvilinear composite layerith wavy surfaces. The homogenization models were also devel-ped in the cases of the nonlinear problems for composite shells,ee Refs. 136,137. The developed homogenization models foromposite shell were applied for the design and optimization ofomposite and reinforced shells 35,36. Most recently, this tech-ique was adopted in modeling of smart composite shells andlates in Refs. 138–142. The general homogenization model foromposite shell has found numerous applications in the analysisf various practically important composite structures. Georgiadest al. 143 and Challagulla et al. 144–146 studied grid-einforced and network thin composite generally orthotropichells as well as the 3D network reinforced composite structures.aha et al. 147,148 analyzed the sandwich composite shells and,

n particular, the honeycomb sandwich composite shells made ofenerally orthotropic materials. Asymptotic homogenization waslso applied to calculate the effective properties of the carbon
anotubes by Kalamkarov et al. 149,150.
30802-12 / Vol. 62, MAY 2009


Let us now summarize the above introduced general homogeni-zation model for composite shell, see Refs. 5,35 for details. Con-sider a general thin 3D composite layer of a periodic structurewith the unit cell shown in Fig. 16. In this figure, 1, 2, and are the orthogonal curvilinear coordinates, such that the coordi-nate lines 1 and 2 coincide with the main curvature lines of themidsurface of the carrier layer and coordinate line is normal toits midsurface =0.

Thickness of the layer and the dimensions of the unit cell of thecomposite material which define the scale of the composite ma-terial inhomogeneity are assumed to be small as compared withthe dimensions of the structure in whole. These small dimensionsof the periodicity cell are characterized by a small parameter .

The unit cell , see Fig. 16b, is defined by the followingrelations:

−h1

2 1

h1

2, −

h2

2 2

h2

2, − +,

=

2 F 1

h1,

2

h2

Here, is the thickness of the layer, and h1 and h2 are thelongitudinal dimensions of the periodicity cell . Functions F

define the geometry of the upper S+ and lower S− reinforcingelements, for example, the ribs or stiffeners, see Figs. 16 and 17.If there are no reinforcing elements, then F+=F−=0, and the com-posite layer has a uniform thickness of order of , like it is, forexample, in the case shown in Fig. 18.

The periodic inhomogeneity of the composite material is mod-eled by the assumption that the elastic coefficients cijkl 1 ,2 ,are periodic functions in variables 1 and 2 with a unit cell .

The elasticity problem for the above 3D thin composite layer isformulated as follows:

ij

j− f i = 0

ij = cijkl1,2,ekl, ekl =1

2 uk

l+

ul

k

Fig. 16 „a… Curvilinear thin 3D reinforced composite layer and„b… unit cell Ω

(a) (b)

Fig. 17 „a… Wafer-reinforced shell and „b… unit cell



Hdu

wf

f

a

HGsa=

app

A

Downloaded Fr

ijnj = pi

8.1

ere f i, pi, and uk represent body forces, surface tractions, and

isplacement field, respectively, and nj is the unit normal to the

pper and lower wavy surfaces =S1 ,2.We introduce the following fast variables, = 1 ,2, and z:

1 =1A1

h1, 2 =

2A2

h2, z =

here A1 and A2 are the coefficients of the first quadraticorm of the midsurface of a carrier layer =0.

The displacements and stresses are expressed in the form of theollowing two-scale asymptotic expansions:

ui,,z = ui0 + ui

1,,z + 2ui2,,z + ¯

ij,,z = ij0,,z + ij

1,,z + 2ij2,,z + ¯

8.2

As a result of asymptotic homogenization procedure, see Refs.5,35 for details, the following relations for the displacementsnd stresses are derived:

u1 = v1 − z

A1

w1

+ U1e + 2V1

+ O3

u2 = v2 − z

A2

w2

+ U2e + 2V2

+ O3u3 = w

+ U3e + 2V3

+ O3 8.3

ij = bije + bij

8.4

ere and in the sequel Latin indices assume values of 1, 2, and 3;reek indices assume values of 1 and 2; and repeated indices are

ummed; the midsurface strains are denoted as follows: e11=e1nd e22=e2 elongations, e12=e21= /2 shear, 11=k1 and 22k2 bending, and 12=21= twisting.The following notation is used in Eq. 8.4:

bijlm =

1

h

cijn

Unlm

+ cijn3Un

lm

z+ cijlm 8.5a

bijlm =

1

h

cijn

Vnlm

+ cijn3Vn

lm

z+ zcijlm 8.5b

The functions Unlm1 ,2 ,z and Vn

lm1 ,2 ,z in Eqs. 8.3,8.5a, and 8.5b are solutions of the unit cell problems. Note thatll the above functions are periodic in variables 1 and 2 witheriods A1 and A2, respectively. The above mentioned unit cellroblems are formulated as follows:

1

h

bi

+bi3

z= 0

1n

bi + n3

bi3 = 0 at z = z 8.6a

Fig. 18 Sandwich composite shell with a honeycomb filler

h



1

h

bi

+bi3

z= 0

1

h

nbi

+ n3bi3

= 0 at z = z 8.6b

where ni+ and ni

− are components of the normal unit vector to theupper z=z+ and lower z=z− surfaces of the unit cell, respec-tively, defined in the coordinate system 1 ,2 ,z.

If inclusions are perfectly bonded to matrix on the interfaces ofthe composite material, then the functions Un

lm and Vnlm together

with the expressions 1 /hncbi

+n3cbi3

and

1 /hncb

i*

+n3cb

i3* should be continuous on the interfaces.

Here nic are the components of the unit normal to the interface.

It should be noted that, unlike the unit cell problems of “clas-sical” homogenization models, e.g., Eqs. 2.9 and 2.25, thoseset by Eqs. 8.6a and 8.6b depend on the boundary conditionsz=z rather than on periodicity in the z direction.

After local functions Unlm1 ,2 ,z and Vn

lm1 ,2 ,z are foundfrom the unit cell problems given by Eqs. 8.5a, 8.6a, 8.5b,and 8.6b, the functions bij

lm1 ,2 ,z and bijlm1 ,2 ,z given by

Eqs. 8.5a and 8.5b can be calculated. These local functionsdefine stress ij, as it is seen from Eq. 8.4. They also define theeffective stiffness moduli of the homogenized shell. Indeed, con-stitutive relations of the equivalent anisotropic homogeneousshell—which is between the stress resultants N11, N22 normal,and N12 shear and moment resultants M11, M22 bending, andM12 twisting on one hand, and the midsurface strains e11=e1,e22=e2 elongations, e12=e21= /2 shear, 11=k1, 22=k2bending, and 12=21= twisting on the other—can be repre-sented as follows 5,35:

N = b e + 2b

M = 2zb e + zb

3 8.7

The angular brackets in Eq. 8.7 denote averaging by the integra-tion over the volume of the 3D unit cell:

f1,2,z =

f1,2,zd1d2dz

The coefficients in constitutive relations Eq. 8.7 b ,

b

= zb , and zb

are the effective stiffness

moduli of the homogenized shell. The midsurface strainse1 ,2 and 1 ,2 can be determined by solving a globalboundary-value problem for the homogenized anisotropic shellwith the constitutive relations 8.7, see Refs. 5,35 for details. Itshould be noted, as can be observed from Eq. 8.7, that there is afollowing one-to-one correspondence between the effective stiff-ness moduli and the extensional, A, coupling, B, and bending,D, stiffnesses familiar from the classical composite laminate
theory, see, e.g., Ref. 151:
MAY 2009, Vol. 62 / 030802-13


aetfito

lrw

s

w

0

Downloaded Fr

A B

B D =

b1111 b11

22 b1112 2zb11

11 2zb1122 2zb11

12 b11

22 b2222 b22

12 2zb1122 2zb22

22 2zb2212

b1112 b22

12 b1212 2zb11

12 2zb2212 2zb12

12 2b11

11 2b1122 2b11

12 3zb1111 3zb11

22 3zb1112

2b1122 2b22

22 2b2212 3zb11

22 3zb2222 3zb22

12 2b11

12 2b2212 2b12

12 3zb1112 3zb22

12 3zb1212

8.8

The unit cell problems given by Eqs. 8.5a, 8.6a, 8.5b, and8.6b have been solved analytically for a number of structures of

practical interest, and the explicit analytical formulas for theffective stiffness moduli have been obtained for the followingypes of composite and reinforced shells and plates: angle-plyber-reinforced shells and grid-reinforced and network shells5,33,35,143–146; rib- and waferlike reinforced shells5,34,35,140,142,152; sandwich composite shells, in particular,he honeycomb sandwich composite shells made of generallyrthotropic materials 5,35,141,147,148; and carbon nanotubes149,150.

As the examples of these results, we will present here the ana-ytical results for the effective stiffness moduli of a wafer-einforced shell shown in Fig. 17 and a sandwich composite shellith a honeycomb filler shown in Fig. 18.The nonzero effective stiffness moduli of the wafer-reinforced

hell shown in Fig. 17 are obtained as follows, see Refs.5,34,35,152 for details:

b1111 =

E13

1 − 12321

3 + E12F2

w,

b2222 =

E23

1 − 12321

3 + E21F1

w,

b2211 = b11

22 =21

3E13

1 − 12321

3

b1212 = G12

3, zb1111 = b11

11 = E12S2

w,

zb2222 = b22

22 = E21S1

w,

zb1111 =

E13

121 − 12321

3+ E1

2J2w,

zb2222 =

E23

121 − 12321

3+ E2

1J1w

zb2211 = zb11

22 =21

3E13

121 − 12321

3,

zb1212 =

G123

12+

G121

12H3t1

h1− K1 +

G122

12H3t2

h2− K2 8.9

here

K1 =96H4

5A1h1G12

1

G231

1 − − 1n

n5 tanhG231

G121

nA1t1

2H

n=1

30802-14 / Vol. 62, MAY 2009


K2 =96H4

5A2h2G12

2

G132

n=1

1 − − 1n

n5 tanhG132

G122

nA2t2

2H

8.10

Here the superscripts indicate the elements of the unit cells 1,2, and 3, see Fig. 17b; A1 and A2 are the coefficients of thefirst quadratic form of the midsurface of a carrier layer; andF1

w ,F2w, S1

w ,S2w, and J1

w ,J2w are defined as follows:

F1w =

Ht1

h1, F2

w =Ht2

h2, S1

w =H2 + Ht1

2h1, S2

w =H2 + Ht2

2h2

J1w =

4H3 + 6H2 + 3Ht1

12h1, J2

w =4H3 + 6H2 + 3Ht2

12h2

8.11

The nonzero effective stiffness moduli of the sandwich com-posite shell with a honeycomb filler shown in Fig. 18 are obtainedas follows, see Refs. 5,35 for details:

b1111 = b22

22 =2E0t0

1 − 02 +

3

4

EHt

a, b12

12 =E0t0

1 + 0+

3

12

EHt

a

b2211 = b11

22 =20E0t0

1 − 02 +

3

12

EHt

a

zb1111 = zb22

22 =E0

1 − 02H2t0

2+ Ht0

2 +2t0

3

3 +

3

48

EH3t

a

zb2211 = zb

11*22 =

0E0

1 − 02H2t0

2+ Ht0

2 +2t0

3

3 +

3

144

EH3t

a

zb1212 =

E0

21 + 0H2t0

2+ Ht0

2 +2t0

3

3 +

EH3t

121 + a3 +

43

−128H

35Atn=1

tanh2n − 1At

2H

2n − 15 8.12

In Eq. 8.12, the first terms define the contribution from the topand bottom carrier layers of the sandwich shell, while the latterterms represent the contribution from the honeycomb filler. E0 and0 are the properties of the material of the carrier layers, and Eand of the honeycomb foil material. We have confined our at-tention here by the case of equal coefficients of the first quadratic
form of the midsurface of the shell, i.e., A1=A2=A.


9p

pftpbtmKvfapp

obe

iiafbbs

lop

naviasra

musrpmtehciecRcpsaAiTiht

A

Downloaded Fr

Boundary Effects, Strength, and Damage in Com-osite MaterialsWhile asymptotic homogenization leads to a much simpler

roblem for an equivalent homogeneous material with certain ef-ective properties, the construction of a solution in the vicinity ofhe boundary of the original composite solid remains beyond ca-abilities of the classical homogenization, see, e.g., Refs.5,153,154. In order to determine stresses and strains near theoundary, a boundary-layer problem should be considered in ex-ension to the asymptotic homogenization. A boundary-layer

ethod in asymptotic homogenization was developed byalamkarov 5, Sec. 7, and used to solve a problem of a trans-ersal crack in a periodic composite, and by Andrianov et al.130 in the theory of ribbed plates and shells. This approach wasurther developed by Kalamkarov and Georgiades 139 insymptotic homogenization of smart periodic composites. The ex-onential decay of boundary layers was proved in Ref. 6 for theroblems with a simple geometry.

New generalized integral transforms for the analytical solutionf the boundary-value problems for composite materials haveeen developed by Kalamkarov 5, Appendix B, and Kalamkarovt al. 155.

The properties of boundary layers in periodic homogenizationn rectangular domains, which are either fixed or have an oscillat-ng boundary, are investigated in Ref. 156. Such boundary layersre highly oscillating near the boundary and decay exponentiallyast in the interior to a nonzero limit that the authors called aoundary-layer tail. It is shown that these boundary-layer tails cane incorporated into the homogenized equation by adding disper-ive terms and a Fourier boundary condition.

Although finding the explicit analytical solutions of boundary-ayer problems in the theory of homogenization still remains anpen problem, the effective numerical procedures have been pro-osed in Refs. 157,158.

Asymptotic homogenization approach can be effectively usedot only to calculate the effective properties of composites butlso to analyze their strength and damage. That follows from aery important advantage of the asymptotic homogenization that,n addition to the effective properties, it allows to determine withhigh accuracy the local stresses and strains defined by a micro-

tructure of a composite material. A number of publications areelated to the formulation of the failure criteria based on thesymptotic homogenization, see Refs. 159–162.

Until recently, in the study of strength of composite materialsost typical was a phenomenological approach based on the fail-

re criterion for the equivalent homogeneous anisotropic material,ee, e.g., Ref. 38. It is of interest to develop such strength crite-ia for the composite materials that will take into account thehenomenological failure criteria for each individual constituentaterial. To achieve that, a concept of stress and strain concentra-

ion functionals for the composite materials was proposed in Ref.163, which allows expressing stresses and strains in the constitu-nt materials in terms of the stresses and strains in the equivalentomogenized material. Both the effective properties and the localharacteristics are taken into account in this approach. Particularlymportant results could be produced in this way if the analyticalxpressions for the stress and strain concentration functionalsould be obtained. That is a reason why this approach was used inefs. 159,161 only for laminated composites, for which the unitell problems become one dimensional and thus solvable in ex-licit analytical expressions. It is possible to extend the concept oftress and strain concentration functionals to 2D and 3D cases bypplying the methods introduced in Secs. 3–5 of the present paper.ccording to the approach 159,160, the general failure criteria

n stresses or strains of the constituent materials are written first.hen, the stress or strain concentration tensors are substituted

nto these failure criteria. And finally, the resulting expressions areomogenized. In the opinion of the authors of the present paper,
he last procedure requires more detailed substantiation since as a


result of averaging the local stress pikes will be cut off. Moresubstantiated criterion is offered in Ref. 162, where it is sug-gested to find such a limiting value, for which the failure begins atleast at one point of any constituent of the composite.

Luo and Takezono 164 used the asymptotic homogenizationmethod to obtain the effective mechanical properties of the fiber-reinforced ceramic matrix composites and to derive the homog-enized damage elastic concentration factor for the unidirectionaland cross-ply laminated composites. They introduced the internalvariables to describe the evolution of the damage state underuniaxial loading and as a subsequence the degradation of the ma-terial stiffness.

Let us note that the application of homogenization approach forthe damage analysis assumes, as a rule, the uniform distribution ofsources of failure, for example, the uniform distribution of cracksin the matrix. It is clear that this assumption is far from the reality,but the obtained results can still be used to evaluate a true strengthof the composites.

It is suggested in Refs. 165–169 to develop the damage pro-gression models entirely on the basis of asymptotic homogeniza-tion, without any complementary phenomenological assumptions.The authors mentioned that they remain within the applicability ofthe asymptotic homogenization, which is limited to the earlystages of failure. For a correct description of the advanced stagesof failure one has to supplement these models with a phenomeno-logical counterpart since the homogenization is not applicableanymore. In particular, the classical continuum formulation isused in Ref. 165, but an internal length parameter is introducedin the damage progression model, as a consequence of the micro-scopic balance of energy and a Griffith-type microcrack propaga-tion criterion.

Asymptotic homogenization techniques in combination with thephenomenological assumptions related to the damage in compos-ites are developed in Refs. 170–174.

A continuum-scale analysis to account for the damage producedby evolving internal boundaries and employing methods of frac-ture mechanics on a smaller scale is offered in Refs. 175–177.The assumption of statistical homogeneity on a smaller scaleyields the macroscale damage-dependent model by employing ahomogenization principle. In this case, the physical details on thesmaller scale are not lost. The considered approximation is a par-ticular case of the micromechanical damage approach that treatseach microphase as a statistically homogeneous medium178–185. Local damage variables are introduced to represent thestate of damage in each phase and the effective material propertiesare defined thereafter. The overall damage model is subsequentlyobtained by means of homogenization.

The problem of fatigue life prediction is studied in Ref. 186using homogenization with two temporal coordinates. In this ap-proach the original boundary-value problem is decomposed intocoupled microchronological fast time-scale and macrochrono-logical slow time-scale problems. The life prediction methodol-ogy was validated numerically against the direct cycle-by-cyclesimulations.

The simultaneous microscopic and macroscopic analyses ateach loading step are proposed in Refs. 187,188. Such approachleads to a very high volume of computations but it gives a possi-bility to take into account the evolution of damage and the effectof loading history.

10 Conclusions and GeneralizationsAsymptotic homogenization is a mathematically rigorous pow-

erful tool for analyzing composite materials and structures. Theproof of the possibility of homogenizing a composite material of aregular structure, i.e., of examining an equivalent homogeneoussolid instead of the original inhomogeneous composite solid, isone of the principal results of this theory. Method of asymptotichomogenization has also indicated a procedure of transition from
the original problem which contains in its formulation a small
MAY 2009, Vol. 62 / 030802-15


pttmatrs

mseaifsa3tdnsryda

ttpafa

ibwaR

haet

eatR

tr

A

iGaA1

R

0

Downloaded Fr

arameter related to the small dimensions of the constituents ofhe composite to a problem for a homogeneous solid. The effec-ive properties of this equivalent homogeneous material are deter-

ined through the solution of the unit cell problems. Importantdvantage of the asymptotic homogenization is that, in addition tohe effective properties, it allows to determine with a high accu-acy the local stress and strain distributions defined by the micro-tructure of composite materials.

The present paper reviews the state-of-the-art in asymptotic ho-ogenization of composite materials and thin-walled composite

tructures. Using 204 references we have presented a variety ofxisting methods, pointed out their advantages and shortcomings,nd discussed their applications. In addition to the review of ex-sting results, some new original approaches have also been of-ered. In particular, we discussed possible methods of analyticalolution of the unit cell problems obtained as a result of thesymptotic homogenization. The asymptotic homogenization ofD thin-walled composite reinforced structures is considered, andhe general homogenization model for a composite shell is intro-uced. In particular, the analytical formulas for the effective stiff-ess moduli of wafer-reinforced shell and sandwich compositehell with a honeycomb filler are presented. We also discussedandom composites; use of two-point Padé approximants and as-mptotically equivalent functions; the correlation between con-uctivity and elastic properties of composites; and strength, dam-ge, and boundary effects in composites.

In conclusion, we would like to refer to some generalizations inhe application of the asymptotic homogenization. Generalizationhat accounts for nonlinearity of transport problems for fiber com-osites is proposed in Refs. 68,71,189. The unit cell problemsre formulated as the minimization problems, and some boundsor the effective properties are extended to the nonlinear problemsnd calculated using the two-point Padé approximants.

Many of the above discussed results can be generalized for thenclusions with cross sections slightly different from the canonicaly means of the boundary shape perturbation technique 49 asell as for the quasiperiodic composites 190. Generalizations on

ccount of anisotropy of the constituent materials are developed inefs. 191,192.We would also like to refer to the application of the asymptotic

omogenization in the analysis of stressed composite materialsnd structures 193,194, in the study of a threshold phenomenon195–197, in the investigation of the analytical properties of theffective parameters 198–203, and to a new approach based onhe integral equations 204.

The fundamental aspects of homogenization, including nonlin-ar homogenization, nonconvex and stochastic problems, as wells several applications in micromechanics, thin films, smart ma-erials, and structural and topology optimization, are presented inef. 205.Research in asymptotic homogenization of composites is ac-

ively continuing. And it is certain that it will bring many moreesults of both fundamental and practical significance.

cknowledgmentThis work was supported by the Natural Sciences and Engineer-

ng Research Council of Canada NSERC for A.L.K., by theerman Research Foundation Grant No. WE736/25-1 for I.V.A.,

nd by the Alexander von Humboldt Foundation, Institutionalcademic Co-Operation Program Grant No. 3.4-Fokoop-UKR/070297 for V.V.D..

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30802-16 / Vol. 62, MAY 2009


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A

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Alexander L. Kalamkarov obtained his Masters (1975) and Ph.D. (1979) degrees from the MoscowLomonosov State University (USSR) and the Doctor of Sciences degree (1990) from the Academy of Sci-ences of the USSR. Since 1993 he is a Professor at the Department of Mechanical Engineering at theDalhousie University in Halifax, Nova Scotia, Canada. His academic career spans more than 30 years inresearch and university teaching. His research interests belong to mechanics of solids, composite materials,and smart materials and structures. Dr. Kalamkarov has authored 3 research monographs and over 250papers in the refereed journals and conference proceedings, and he also holds two patents in the area ofsmart materials. He has reported his research results at numerous international conferences and haspresented six invited keynote lectures. Dr. Kalamkarov is a Member of several editorial and advisory boardsin the area of composite materials and smart structures. He is a Fellow of the ASME and a Fellow of theCSME.

Igor V. Andrianov obtained his Masters of Applied Mechanics degree (1971) and Ph.D. degree in Struc-tural Mechanics (1975) from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor ofSciences degree in Mechanics of Solids from the Moscow Institute of Electronic Engineering in 1990.During 1974–1977, he was a Research Scientist at the Dnepropetrovsk State University; during 1977–1990,an Associated Professor; and during 1990–1997, a Full Professor of Mathematics at the DnepropetrovskCivil Engineering Institute. Currently he is a Research Scientist at the Rheinisch-Westfälische TechnischeHochschule (Technical University of Aachen, Germany). Dr. Andrianov is the author or co-author of 11books and over 250 papers in refereed journals and conference proceedings. He has presented papers atnumerous international conferences and has supervised 21 Ph.D. theses. His research interests belong tomechanics of solids, nonlinear dynamics, and asymptotic methods.

Vladyslav V. Danishevs’kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciencesdegree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering andArchitecture, Dnipropetrovsk, Ukraine. He is an Associate Professor at this State Academy. He has authored1 monograph and over 50 refereed papers. Among his awards are the Soros Post-Graduate Student’s Award(1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement amongyoung scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO ResearchFellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of theAlexander von Humboldt Foundation (2007). He has conducted research at the Institute of General Me-chanics in the Technical University of Aachen, Germany (2001–2002 and 2006). He was a NATO ResearchOfficer at the University of Rouen, France (2003–2004). His research interests belong to the mechanics ofheterogeneous materials and structures, asymptotic methods, and nonlinear dynamics.

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