Alexander L. Kalamkarov1
Fellow ASME
Department of Mechanical Engineering,
Dalhousie University,
P.O. Box 1000,
Halifax, NS, B3J 2X4, Canada
e-mail: [email protected]
Igor V. Andrianov
Institute of General Mechanics,
Rheinisch-Westfälische Technische Hochschule
(Technical University of Aachen),
Templergraben 64,
Aachen D-52062, Germany
Vladyslav V. Danishevs’kyy
Prydniprov’ska State Academy of Civil
Engineering and Architecture,
Chernishevs’kogo 24a,
Dnipropetrovsk 49600, Ukraine
Asymptotic Homogenization of
Composite Materials and
Structures
The present paper provides details on the new trends in application of asymptotic homogenization techniques to the analysis of composite materials and thin-walled composite structures and their effective properties. The problems under consideration are important from both fundamental and applied points of view. We review a state-of-the-art in
asymptotic 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 also
introduced. In particular, we analyze a possibility of analytical solution of the unit cell
problems obtained as a result of the homogenization procedure. Asymptotic homogenization of 3D thin-walled composite reinforced structures is considered, and the general
homogenization model for a composite shell is introduced. In particular, analytical formulas for the effective stiffness moduli of wafer-reinforced shell and sandwich composite
shell with a honeycomb filler are presented. We also consider random composites; use of
two-point Padé approximants and asymptotically equivalent functions; correlation between conductivity and elastic properties of composites; and strength, damage, and
boundary effects in composites. This article is based on a review of 205 references.
关DOI: 10.1115/1.3090830兴
Keywords: composite materials, thin-walled composite reinforced structures, asymptotic
homogenization, unit cell problems, effective properties
1
Introduction
The rapidly increasing popularity of composite materials and
structures in recent years has been seen through their incorporation in the mechanical and civil engineering, aerospace, automotive and marine applications, as well as in biomedical and sport
products. Success in practical application of composites largely
depends on a possibility to predict their mechanical properties and
behavior through the development of the appropriate mechanical
models. The micromechanical modeling of composite structures,
however, can be rather complicated as a result of the distribution
and orientation of the multiple inclusions and reinforcements
within the matrix, and their mechanical interactions on a local
共micro-兲 level. Therefore, it is important to establish such micromechanical models that are neither too complicated to be developed and applied nor too simple to reflect the real mechanical
properties and behavior of the composite materials and structures.
The micromechanical analysis of composites has been the subject of investigation for many years. According to Willis 关1兴, the
numerous methods in mechanics of composites can be classified
into four broad categories: asymptotic, self-consistent, variational,
and modeling methods. There are no rigorous boundaries between
these categories.
The self-consistent methods and the general “one-particle”
schemes for approximate evaluation of the effective properties
have been reviewed in Refs. 关2,3兴. Our present review deals with
the asymptotic approaches that are capable of analyzing the composite materials and structures with constituents with high contrast
in their material properties. For many problems that we will discuss below, other analytical or numerical approaches are not as
effective as the asymptotic homogenization.
First, we will deal with the regular composites. The coefficients
of the corresponding equations modeling mechanical behavior of
1
Corresponding author.
Published online March 31, 2009. Transmitted by Victor Birman.
Applied Mechanics Reviews
the composite solid are rapidly varying periodic functions in spatial coordinates. Accordingly, the resulting boundary-value problems 共BVPs兲 are very complex. A look at numerical methods,
applied directly to an original boundary-value problem for a composite solid, shows that they are not always convenient and are
sometimes even inappropriate in their standard form. Therefore, it
is important to develop analytical methods based on rigorous
mathematical techniques. At present, asymptotic techniques are
applied in many cases in micromechanics of composites. Various
asymptotic approaches to the analysis of composite materials have
apparently reached their conclusion within the framework of the
mathematical theory of asymptotic homogenization. Indeed, the
proof of the possibility of homogenizing a composite material of a
regular structure, i.e., of examining an equivalent homogeneous
solid instead of the original inhomogeneous composite solid, is
one of the principal results of this theory. Theory of homogenization has also indicated a method of transition from the original
problem 共which contains in its formulation a small parameter related to the small dimensions of the constituents of the composite兲
to a problem for a homogeneous solid. The effective properties of
this equivalent homogeneous material are determined through the
solution of so-called local problems formulated on the unit cell of
the composite material. These solutions also enable calculation of
local stresses and strains in the composite material. The indicated
results are fundamentals of the mathematical theory of homogenization. In the present paper we will review the basics of the
asymptotic homogenization and the analytical solutions of the unit
cell problems for laminated, fiber-reinforced and particulate composites. Afterward, we will generalize the obtained results for the
random composites. We will also analyze thin-walled composite
structures, damage in composite materials and boundary effects,
as well as the approximate links between the conductivity and
elastic problems for the composite materials.
Following this Introduction the rest of the paper is organized as
follows: the asymptotic homogenization technique is presented in
Sec. 2. Section 3 deals with the unit cell problems. In Sec. 4 we
Copyright © 2009 by ASME
MAY 2009, Vol. 62 / 030802-1
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
will discuss the use of two-point Padé approximants 共TPPAs兲.
Section 5 deals with the use of asymptotically equivalent functions 共AEFs兲. That is followed by the review of techniques applied
to random composites in Sec. 6. Section 7 studies the existing
links between the conductivity and elastic problems for composite
materials. Asymptotic homogenization of three-dimensional thinwalled composite reinforced structures is discussed in Sec. 8. The
general homogenization model for a composite shell is presented.
Asymptotic homogenization techniques in the study of strength
and damage and the boundary effects in composite materials are
reviewed in Sec. 9. Conclusions and some generalizations in the
use of asymptotic homogenization are presented in Sec. 10.
2
Asymptotic Homogenization Method
For the past 25 years homogenization methods have proven to
be powerful techniques for the study of heterogeneous media.
Some of these classical tools today include multiple-scale expansions 关4–8兴, G- and ⌫-convergences 关9,10兴, and energy methods
关11,12兴.
An approach based on Fourier analysis has been proposed in
Refs. 关13,14兴. This method works in the following way. First,
original operator is transformed into an equivalent operator in the
Fourier space. The standard Fourier series is used to expand the
coefficients of the operator and a Fourier transform is used to
decompose the integrals. Next, the Fourier transforms of the integrals are expanded using a suitable two-scale expansion, and the
homogenized problem is finally derived by merely neglecting
high-order terms in the above expansions when moving to the
limit as the period tends to zero.
The method of orientational averaging was proposed in Ref.
关15兴. It is based on the following assumptions: A characteristic
volume 共repeated throughout the bulk of the composite兲 is isolated
from the composite medium. The properties of the composite as a
whole are assumed to be the same as those of this characteristic
volume. In the case of ideally straight fibers the set of fibers is
represented in the form of the array of unidirectional reinforced
cylinders. We should also mention papers on homogenization using wavelet approximations 关16兴 and nonsmooth transformations
关17兴.
In this section we describe a variant of homogenization approach that will be used further. For simplicity, we will start with
a 2D heat conduction problem. However, these results will remain
correct for other kinds of transport coefficients such as electrical
conductivity, diffusion, magnetic permeability, etc. Due to the
well-known longitudinal shear–transverse conduction analogy, see
Ref. 关18兴, the elastic antiplane shear deformation can also be
evaluated in a similar mathematical way. This will be followed by
the summary on the asymptotic homogenization applied to the
elasticity problem for a 3D composite solid. Analogous
asymptotic homogenization technique has been developed for a
number of more complicated nonlinear models, see Refs. 关5,11兴.
Let us consider a transverse transport process through the periodic composite structure when the fibers are arranged in a periodic
square lattice, see Fig. 1.
The characteristic size l of inhomogeneities is assumed to be
much smaller than the global size L of the whole structure: l Ⰶ L.
Assuming the perfect bonding conditions on the interface ⳵⍀ between the constituents, the governing BVP can be written as follows:
ka
冉
冊
⳵ 2u a ⳵ 2u a
+ 2 = − fa
⳵ x21
⳵ x2
km
⳵ um
⳵uf
= kf
⳵n
⳵n
in ⍀a,
on ⳵ ⍀
allows a number of different physical interpretations, but here it is
discussed with a reference to the heat conduction. Then, in the
above expressions, ka are the heat conductivities of the constituents, ua is a temperature distribution, f a is a density of heat
sources, and ⳵ / ⳵n is a derivative in the normal direction to the
interface ⳵⍀. Let us now consider the governing BVP 共2.1兲 using
the asymptotic homogenization method 关4–6,19–23兴. We will define a natural dimensionless small parameter ␧ = l / L, ␧ Ⰶ 1, characterizing the rate of heterogeneity of the composite structure.
In order to separate micro- and macroscale components of the
solution we introduce the so-called slow 共x兲 and fast 共y兲 coordinates
x s = x s,
y s = xs␧−1,
共2.2兲
s = 1,2
and we express the temperature field in the form of an asymptotic
expansion
ua = u0共x兲 + ␧ua1共x,y兲 + ␧2ua2共x,y兲 + ¯
共2.3兲
where x = x1e1 + x2e2 and y = y 1e1 + y 2e2, e1 and e2 are the Cartesian
unit vectors. The first term u0共x兲 of expansion 共2.3兲 represents the
homogeneous part of the solution; it changes slowly within the
whole domain of the material and does not depend on fast coordinates. All the further terms uai 共x , y兲, i = 1 , 2 , 3 , . . ., describe local
variation in the temperature field on the scale of heterogeneities.
In the perfectly regular case the microperiodicity of the medium
induces the same periodicity for uai 共x , y兲 with respect to fast variables
uak 共x,y兲 = uak 共x,y + L p兲
共2.4兲
where L p = ␧−1l p, l p = p1l1 + p2l2, and ps = 0 , ⫾ 1 , ⫾ 2 , . . ., l1 and l2
are the fundamental translation vectors of the square lattice.
The spatial derivatives are defined as follows:
⳵
⳵
⳵
=
+ ␧−1
⳵ xs ⳵ xs
⳵ ys
共2.5兲
Substituting expressions 共2.2兲, 共2.3兲, and 共2.5兲 into the governing
BVP 共2.1兲 and splitting it with respect to equal powers of ␧ one
comes to a recurrent sequence of problems:
um = u f ,
共2.1兲
Here and in the sequel variables indexed by m correspond to the
matrix and those indexed by f correspond to the fibers; index a
takes both of these references: a = m or a = f. Generally, BVP 共2.1兲
030802-2 / Vol. 62, MAY 2009
Fig. 1 Composite material with hexagonal array of cylindrical
fibers
⳵2ua1 ⳵2ua1
+
=0
⳵ y 21 ⳵ y 22
冏冋
km
in ⍀,
f
关um
1 = u1兴兩⳵⍀ ,
⳵ um1
⳵ u1f
⳵ u0
− kf
= 共k f − km兲
⳵m
⳵m
⳵n
册冏
⳵⍀
共2.6兲
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
ka
冉
⳵2ua1
⳵2ua1
⳵2ua2 ⳵2ua2
⳵ 2u 0 ⳵ 2u 0
+2
+ 2 + 2
2 +
2 +2
⳵ x1 ⳵ y 1
⳵ x2 ⳵ y 2 ⳵ y 1 ⳵ y 2
⳵ x1
⳵ x2
in ⍀
= − fa
冏冋
f
关um
2 = u2兴兩⳵⍀,
k
⳵ um2
m
⳵m
−k
⳵ u2f
f
⳵m
=k
⳵ u1f
f
⳵n
−k
⳵ um1
m
⳵n
册冏
冊
具kij典
where
具kij典 = 关共1 − c兲km + ck f 兴␦ij +
⳵⍀
共2.7兲
and so on.
Here ⳵ / ⳵m is a derivative in the normal direction to the interface ⳵⍀ in the fast coordinates y 1, y 2.
The BVP 共2.6兲 allows evaluation of the higher-order component
uai 共x , y兲 of the temperature field; owing to the periodicity condition 共2.4兲 it can be considered within only one periodically repeated unit cell. It follows from the BVP 共2.6兲 that variables x and
y can be separated in u1共x , y兲 by assuming
u1共x,y兲 =
⳵ u0共x兲
⳵ u0共x兲
U1共y兲 +
U2共y兲
⳵ xl
⳵ x2
共2.8兲
where U1共y兲 and U2共y兲 are local functions for which problem
共2.6兲 yields the following unit cell problems:
⳵2U1共y兲 ⳵2U1共y兲
+
= 0,
⳵ y 21
⳵ y 22
f
Um
1 共y兲 = U1共y兲,
km
⳵2U2共y兲 ⳵2U2共y兲
+
=0
⳵ y 21
⳵ y 22
⳵ Um1 共y兲
⳵ U1f 共y兲
− kf
= 共k f − km兲m1,
⳵m
⳵m
km
in ⍀
on ⳵ ⍀
f
Um
2 共y兲 = U2共y兲
⳵ Um2 共y兲
⳵ U2f 共y兲
− kf
⳵m
⳵m
on ⳵ ⍀
= 共k f − km兲m2
共2.9兲
where m1 and m2 are components of a unit normal to the interface
⳵⍀ in coordinates y 1, y 2.
In order to determine the effective heat conductivity, the BVP
共2.7兲 should be considered. Let us apply to Eq. 共2.7兲 the following
homogenization operator over the unit cell volume ⍀0:
冋冕冕
共·兲dy 2dy 3 +
⍀m
0
冕冕
册
共·兲dy 2dy 3 L−2
⍀in
0
Terms containing ua2 will be eliminated by means of the Green
theorem and taking into account the boundary conditions 共2.7兲 and
the periodicity condition 共2.4兲, which yields
冉
冊
关共1 − c兲km + ck f 兴
+
冊 冕冕 冉
冕冕 冉
⳵ 2u 0 ⳵ 2u 0
km
+
+
L2
⳵ x21
⳵ x22
⳵2um1
kf
dy 1dy 2 + 2
⳵ x2 ⳵ y 2
L
⍀in
0
⍀m
0
⳵2um1
⳵ x1 ⳵ y 1
冊
⳵2u1f
⳵2u1f
+
dy 1dy 2
⳵ x1 ⳵ y 1 ⳵ x2 ⳵ y 2
= − 关共1 − c兲f m + cf f 兴
共2.10兲
where c is the fiber volume fraction.
Let us note a difference in the right-hand side of Eq. 共2.10兲
when k f → 0 and k f = 0. Assume that f f = f m = f. Then for any k f
⫽ 0 we get an expression −f in the right-hand side of Eq. 共2.10兲.
But for k f = 0 we get there a different expression −f共1 − c兲. That
represents an explanation to the following “paradox” pointed out
in Refs. 关24,25兴:
lim lim u共x1,x2,k f ,␧兲 ⫽ lim u共x1,x2,0,␧兲
k f →0 ␧→0
␧→0
The homogenized heat conduction equation can be obtained by
substituting expression 共2.8兲 for uគ 1共x , y兲 into Eq. 共2.10兲, which
yields
Applied Mechanics Reviews
⳵ u20共x兲
= − 具f典
⳵ xi ⳵ x j
+
kf
L2
冕冕
␦il
⍀in
0
km
L2
冕冕
⍀m
0
共2.11兲
␦il
⳵ Umj
dy 1dy 2
⳵ yl
⳵ U jf
dy 1dy 2
⳵ yl
共2.12兲
where 具f典 = 共1 − c兲f m + cf f is the effective density of heat sources,
␦ij is Kronecker’s delta, indices i , j , l = 1 , 2, and the summation
over the repeated indices is implied.
Note that in general the homogenized material will be anisotropic, and 具kij典 in Eq. 共2.11兲 is a tensor of effective coefficients of
heat conductivity. Tensor 具kij典 is defined by expression 共2.12兲, and
it can be readily calculated as soon as the unit cell problems 共2.9兲
are solved and the local functions U1共y兲 and U2共y兲 are found. Unit
cell problems 共2.9兲 can be solved analytically or numerically. The
approximate methods of their analytical solution will be presented
in Sec. 3.
If a periodic heterogeneous medium is made of constituents
with moderately different properties, the homogenized equations
preserve a local character of the original equations. The coefficients of the homogenized equations can be explicitly expressed
in terms of the solutions of the unit cell problems. However, when
a heterogeneous medium consists of materials with highly different properties, the homogenized constitutive relation may reveal a
nonlocal structure. Theory for this case was developed by Allaire
关26兴 and Zhikov 关27兴. This made it possible to analyze the highergradient effects in the overall behavior of heterogeneous media
关28–32兴.
Asymptotic homogenization procedure strongly depends on the
following three parameters: the natural small dimensionless parameter ␧ characterizing the rate of heterogeneity of the composite
structure, on the ratio of material properties of matrix and inclusion ␧1 = km / k f , and on the volume fraction of inclusions c. The
above obtained results are formally valid for ␧1 ⬃ 1 and c ⬃ 1. For
␧1 Ⰶ 1 and c ⬃ 1 the effective heat conductivity 具k典 in the first
approximation does not depend on km; for c Ⰶ 1 and ␧1 ⬃ 1 k f must
be omitted in 具k典, etc. But numerical error in calculating effective
conductivity 具k典, obtained by assuming ␧1 ⬃ 1, c ⬃ 1, is not essential, and for simple isotropic inclusions 共spheres, ellipsoids, cylinders, and parallelepipeds兲 in isotropic matrix it can be used in the
first approximation for any kind of differences between properties
of the constituent materials and their volume fractions. For the
higher-order homogenization approach this conclusion can be
wrong.
Let us now consider asymptotic homogenization of an elasticity
problem 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 repetition of the unit cell Y in the form of a parallelepiped with
dimensions ␧Y i, i = 1 , 2 , 3. The elastic deformation of this composite solid is described by the following BVP:
⳵␴ij␧
= fi
⳵xj
in ⍀,
␧
␴ij␧ = cijklekl
,
u␧共x兲 = 0
eij␧ =
冉
on S
1 ⳵ ui␧ ⳵ u␧j
+
2 ⳵ x j ⳵ xi
冊
共2.13兲
共2.14兲
where cijkl is a tensor of elastic coefficients. The coefficients cijkl
are assumed to be periodic functions with a unit cell Y. Here and
in the sequel all Latin indices assume values of 1, 2, and 3, and
repeated indices are summed.
MAY 2009, Vol. 62 / 030802-3
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
冉
x3
Ω
y3
Reinforcement
Y
y2
x2
y1
Reinforcement
ε
x1
(a)
Fig. 2
(b)
„a… 3D periodic composite and „b… unit cell Y
The introduction of the fast variables y i = xi / ␧, i = 1 , 2 , 3,, similar to Eq. 共2.2兲, into Eqs. 共2.13兲 and 共2.14兲 and the rule of differentiation 共2.5兲 leads to the following BVP:
⳵␴ij␧ 1 ⳵␴ij␧
+
= fi
⳵xj ␧ ⳵ yj
in ⍀,
u␧共x,y兲 = 0
on S
␴ij␧ 共x,y兲 = cijkl共y兲
⳵ xl
共x,y兲
共2.15兲
共2.16兲
The next step is to expand the displacements and, as a result,
the stresses into the asymptotic expansions in powers of the small
parameter ␧, similar to expansion 共2.3兲,
u␧共x,y兲 = u共0兲共x,y兲 + ␧u共1兲共x,y兲 + ␧2u共2兲共x,y兲 + ¯ 共2.17兲
␴ij␧ 共x,y兲 = ␴ij共0兲共x,y兲 + ␧␴ij共1兲共x,y兲 + ␧2␴ij共2兲共x,y兲 + ¯ 共2.18兲
where all above functions are periodic in y with the unit cell Y.
Substituting Eqs. 共2.17兲 and 共2.18兲 into Eqs. 共2.15兲 and 共2.16兲,
while considering at the same time the periodicity of u共i兲 in y,
reveals that u共0兲 is independent of the fast variable y, see Ref. 关5兴
for details. Subsequently, equating terms with similar powers of ␧
results in the following set of equations:
⳵␴ij共0兲共x,y兲
=0
⳵yj
共2.19兲
⳵␴ij共1兲共x,y兲 ⳵␴ij共0兲共x,y兲
+
= fi
⳵yj
⳵xj
共2.20兲
where
␴ij共0兲 = cijkl
␴ij共1兲 = cijkl
冉
冉
⳵ u共k0兲 ⳵ u共k1兲
+
⳵ xl
⳵ yl
⳵ u共k1兲 ⳵ u共k2兲
+
⳵ xl
⳵ yl
冊
冊
共2.21兲
共2.22兲
Substitution of Eq. 共2.21兲 into Eq. 共2.19兲 yields
冉
冊
⳵ u共k1兲共x,y兲
⳵
⳵ cijkl共y兲 ⳵ u共k0兲共x兲
cijkl
=
⳵ yl
⳵ xl
⳵yj
⳵yj
共2.23兲
Due to the separation of variables in the right-hand side of Eq.
共2.23兲 the solution of Eq. 共2.23兲 can be written as follows, similar
to Eq. 共2.8兲:
un共1兲共x,y兲 =
⳵ u共k0兲共x兲 kl
Nn 共y兲
⳵ xl
共2.24兲
where Nkl
n 共y兲 共n , k , l = 1 , 2 , 3兲 are periodic functions with a unit
cell Y satisfying the following equation:
030802-4 / Vol. 62, MAY 2009
共2.25兲
It is observed that Eq. 共2.25兲 depends only on the fast variable y
and it is entirely formulated within the unit cell Y. Thus, the
problem 共2.25兲 is appropriately called an elastic unit cell problem.
Note that instead of boundary conditions, this problem has a conkl
共y兲.
dition of a periodic continuation of functions Nm
If inclusions are perfectly bonded to matrix on the interfaces of
kl
the composite material, then the functions Nm
共y兲 together with the
共c兲
kl
expressions 关共cijkl + cijmn共y兲 ⳵ Nm共y兲 / ⳵y n兲n j 兴, i = 1 , 2 , 3, should be
共c兲
continuous on the interfaces. Here, n j are the components of the
unit normal to the interface.
The next important step in the homogenization process is
achieved by substituting Eq. 共2.24兲 into Eq. 共2.21兲, and the resulting expression into Eq. 共2.22兲. The result is then integrated over
the domain Y of the unit cell 共with volume 兩Y兩兲 remembering to
treat x as a parameter as far as integration with respect to y is
concerned. After canceling out terms that vanish due to the periodicity, we obtain the homogenized global problem
C̃ijkl
⳵ u␧k
冊
⳵ Nmkl共y兲
⳵
⳵ cijkl
cijmn共y兲
=−
⳵yj
⳵ yn
⳵yj
Matrix
⳵2u共k0兲共x兲
= fi
⳵ x j ⳵ xl
in ⍀,
u共0兲共x兲 = 0
where the following notation is introduced:
C̃ijkl =
1
兩Y兩
冕冉
cijkl共y兲 + cijmn共y兲
Y
on S
冊
⳵ Nmkl
dv
⳵ yn
共2.26兲
共2.27兲
Similarly, substituting Eq. 共2.24兲 into Eq. 共2.21兲 and then integrating the resulting expression over the domain of the unit cell Y
yields
具␴ij共0兲典 =
1
兩Y兩
冕
Y
␴ij共0兲共y兲dv = C̃ijkl
⳵ u共k0兲
⳵ xl
共2.28兲
Equations 共2.26兲 and 共2.28兲 represent the homogenized elasticity BVP. The coefficients C̃ijkl given by Eq. 共2.27兲 are the effective
elastic coefficients of the homogenized material. They are readily
determined as soon as the unit cell problem 共2.25兲 is solved and
kl
the functions Nm
共y兲 are found. It is observed that these effective
coefficients are free from the complications that characterize the
original rapidly varying elastic coefficients cijkl共y兲. They are universal for a composite material under study and can be used to
solve a wide variety of boundary-value problems associated with
the given composite material.
It should be noted that the solution of the global problem 共2.26兲
for the equivalent homogenized material will not be satisfactory in
the vicinity of the boundary of the solid S, i.e., at the distances of
order of ␧. From the standpoint of homogenization theory a
boundary-layer problem should be considered. Boundary effects
will be discussed in Sec. 9.
3
Unit Cell Problems
As we have seen in Sec. 2, the derivation of the homogenized
equations for the periodic composites includes solution of the unit
cell problems, i.e., problems 共2.9兲 and 共2.25兲. In some particular
cases these problems can be solved analytically producing exact
solutions, for example, for laminated composites and gridreinforced structures, see Refs. 关5,33–35兴. The explicit formulas
for effective moduli are very useful, especially for the design and
optimization of composite materials and structures 关35,36兴. But in
general case, the unit cell problems cannot be solved analytically,
and therefore the numerical methods should be used. In some
cases, the approximate analytical solutions of the unit cell problems can be found, and the explicit formulas for the effective
coefficients can be obtained due to the presence of additional
small parameters within the unit cell, not to be confused with the
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
small parameter of inhomogeneity. In particular, use of the parameter ␧1 = km / k f will be demonstrated in Sec. 4. As a rule, the problems in micromechanics of composites are multiscale. Consequently, it is very difficult to solve them analytically or
numerically. But, at the same time, that opens wide opportunities
for application of the asymptotic methods.
For a small volume fraction of inclusions, c Ⰶ cmax, one can use
the three-phase model 关37–40兴. It has been proposed by Bruggeman 关41,42兴 and is based on the following assumption: The periodically heterogeneous composite structure is approximately replaced by a three-phase medium consisting of a single inclusion, a
matrix layer, and an infinite effective medium with certain homogenized mechanical properties. Asymptotic justification of the
three-phase composite model is given in Ref. 关37兴.
For laminated composite materials unit cell problems 共2.9兲 and
共2.25兲 are one dimensional, and they can be solved analytically.
Using this analytical solution, the effective properties of laminated
composites can be obtained in the explicit analytical form from
Eqs. 共2.12兲 and 共2.27兲, see Refs. 关5,35兴. In the more complicated
case of generally anisotropic constituent materials the explicit formulas for effective elastic, actuation, thermal conductivity, and
hygroscopic absorption properties of laminated smart composites
are derived by Kalamkarov and Georgiades 关43兴. In particular, the
following explicit formula for the effective elastic coefficients of a
laminated composite in the case of generally anisotropic constituent materials is derived in Ref. 关43兴:
−1
C̃ijkl = 具Cijkl典 − 具Cijm3Cm3q3
Cq3kl典
(a)
(b)
Fig. 3 „a… Fiber-reinforced composite with fiber volume fraction close to maximum and „b… asymptotic model
−1
−1
−1
+ 具Cijm3Cm3q3
典具C−1
q3p3典 具C p3n3Cn3kl典
where the angular brackets denote a rule of mixture, and as earlier
indicated all Latin indices assume values of 1, 2, and 3, and repeated indices are summed.
For fiber-reinforced periodic composites the unit cell problem
共2.25兲 becomes two dimensional, and it can be solved analytically
for some simple geometries, or numerically, see Refs. 关5,44–47兴.
In particular, the numerical results for the effective elastic moduli
of the incompressible porous material are obtained in Ref. 关48兴.
It is important to obtain the approximate solution of the unit
cell problem valid for all values of material parameters and volume fraction of constituents. For that purpose various interpolation procedures can be applied. In this section we will introduce
an asymptotic technique based on a modification of a boundary
shape perturbation approach 关49兴. Some other techniques developed in Refs. 关50–52兴 will be discussed in Secs. 4 and 5.
For small volume fraction of inclusions the solution can be
represented in the form of series of the Weierstrass elliptic functions 关53–59兴 or their 3D generalization 关60兴.
For large inclusions c ⬇ cmax one can use lubrication approximations 关38,61兴. In this approach the unit cell problem with
curved boundaries of inclusion is replaced by a much simpler
problem for a strip 共in 2D case兲, see Fig. 3, or a layer 共in 3D case兲.
In 2D case the following equation can be used instead of Eq.
共2.6兲:
⳵2um1
=0
⳵ y 21
f
um
1 = u 1,
in strip − 0.5⌬ ⱕ y 1 ⱕ 0.5⌬,
km
⳵ um1
⳵ u1f
⳵ u0
− kf
= 共k f − km兲
⳵m
⳵m
⳵n
f
um
1 = u1兩r=A,
冏
km
共3.1兲
⳵ um1
⳵ u1f
⳵ u0
− kf
= 共k f − km兲
⳵r
⳵r
⳵n
冏
共3.2兲
r=A
It is shown in Ref. 关4兴, Chap. 6, Sec. 3, see also Refs. 关5,62兴,
that for axially symmetric domains the periodicity continuation
condition 共2.4兲 can be replaced by zero boundary conditions at the
center and at the outer boundary ⳵⍀0 of the unit cell:
u1f = 0兩r=0
共3.3兲
um
1 = 0兩r=Ro
共3.4兲
It should be noted that such a replacement is justified for the first
approximation of the asymptotic approach, but it may be wrong
for the higher approximations.
In Eq. 共3.4兲 the square shape of ⳵⍀0 can be defined as
Ω0m
Ω0
r
y3
θ
f
− ⬁ ⱕ y2 ⱕ ⬁
R0
for y 1 = ⫾ 0.5⌬
and the same problem by replacing y 1 and y 2. Here ⌬ is a minimal
distance between inclusions.
Let us now describe the modification of a boundary shape perturbation approach. We introduce the polar coordinates r
= 冑y 21 + y 22, ␪ = arctan共y 2 / y 1兲 in the plane of the unit cell plane, see
Fig. 4. Then Eq. 共2.6兲 can be written as follows:
Applied Mechanics Reviews
⳵2ua1 1 ⳵ ua1 1 ⳵2ua1
+
+
=0
⳵ r2 r ⳵ r r2 ⳵ ␪2
y2
∂Ω
∂Ω0
A
L/2
Fig. 4 Unit cell of the regular square lattice of cylindrical
fibers
MAY 2009, Vol. 62 / 030802-5
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
R共␰兲 =
R0
cos共␰兲
共3.5兲
where R0 = L / 2 is the radius of the inscribed circle, ␰ = ␪ − ␪0 can
be considered as a small parameter 共␰ ⬍ 1兲, ␪ = ␪0 − ␲ / 4 ¯ ␪0
+ ␲ / 4, and ␪0 = ␲n / 2, n = 0 , 1 , 2 , . . ..
Solution of the unit cell problems 共3.1兲–共3.4兲 is represented in
the form of asymptotic expansion in powers of ␰. This expansion
should be invariant if ␰ is replaced by −␰; thus it should contain
only even powers of ␰:
ua1 = ua1,0 + ␰2ua1,2 + ␰4ua1,4 ¯
共3.6兲
The boundary condition 共3.4兲 is formulated at r = R共␰兲. Consequently, if we now substitute expansion 共3.6兲 directly into condition 共3.4兲, then parameter ␰ will be present in the arguments of the
functions um
1,j , j = 0 , 1 , 2 , . . ., and splitting the input problem with
respect to ␰ will not be possible. In order to eliminate ␰ from the
arguments of um
1,j the boundary condition 共3.4兲 should be transferred from the original contour r = R共␰兲 to the inscribed circle r
= R0 by means of the Taylor expansion
m
2
um
1 兩r=R0/cos共␰兲 = u1,0兩r=R0 + ␰
+ ␰4
冏冉
R0
2
冏 冏
5R0 ⳵ um
1,0
24 ⳵ r
⳵r
+
r=R0
R20 ⳵2um
1,0
2
8 ⳵r
⳵ u0
,
⳵n
u1f = C3r
C p = C p,0 + ␰2C p,2 + ␰4C p,4 + O共␰6兲,
冊冏
r=R0
+ ¯ 共3.7兲
⳵ u0
⳵n
共3.8兲
共3.9兲
p = 1,2,3
The coefficients of expansion 共3.9兲 are as follows:
C1,2 = −
␭␹
,
1 − ␭␹
␭␹
,
共1 − ␭␹兲2
C1,4 = −
C2,0 = −
␭
A 2,
1 − ␭␹
C2,2 =
1 ␭␹共1 − 3␭␹兲
,
2 共1 − ␭␹兲3
C3,4 =
C3,0 = −
␭ 2␹
A 2,
共1 − ␭␹兲2
C2,4 =
␭共1 − ␹兲
1 − ␭␹
C3,2 =
␭␹共␭ − 1兲
共1 − ␭␹兲2
1 ␭2␹共1 − 3␭␹兲 2
A ,
2 共1 − ␭␹兲3
1 ␭␹共␭ − 1兲共1 − 3␭␹兲
2
共1 − ␭␹兲3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.74
0.76
0.77
0.78
Expansion 共3.9兲 PA 共3.12兲 Formulas 共3.13兲 Perrins et al. 关64兴
1.210
1.470
1.811
2.306
3.270
7.106
73.92
—
—
—
—
1.247
1.544
1.918
2.417
3.145
4.386
7.409
10.91
15.29
20.18
35.01
1.223
1.506
1.879
2.395
3.172
4.517
7.769
11.46
15.99
21.04
36.60
冉
C p,0 + C p,2 − C p,0
Cp =
共3.10兲
1 − 3␭␹ 2
␰4C p,4
=
␰
␰2C p,2 2共1 − ␭␹兲
共3.11兲
Ranges of the variables in expression 共3.11兲 are as follows: −1
ⱕ ␭ ⱕ 1, 0 ⱕ ␹ ⱕ 1, and 0 ⱕ ␰2 ⱕ 共␲ / 4兲2. It can be easily seen that
expansion 共3.9兲 diverges in the case of perfectly conductive nearly
touching fibers when ␭ → 1 and ␹ → 1. In order to eliminate this
singularity, the Padé approximants 共PAs兲 can be applied 关63兴. In
the case under consideration the Padé approximants to expansion
共3.9兲 are as follows:
1.222
1.500
1.860
2.351
3.080
4.342
7.433
11.01
15.44
20.43
35.93
冊
C p,4 2
␰
C p,2
C p,4 2
␰
C p,2
共3.12兲
As a second possibility to avoid divergence at ␭ → 1 and ␹ → 1,
the following approximate estimation of the overall sum of expansion 共3.9兲 can be proposed. The first term C p,0 共zero-order approximation兲 represents a solution of the unit cell problems
共3.1兲–共3.4兲 when the outer boundary ⳵⍀0 of the unit cell is replaced by a circle of radius R0. At this step Eq. 共3.1兲 as well as the
boundary conditions 共3.2兲 and 共3.3兲 are strictly satisfied, but there
exists a discrepancy in the boundary condition 共3.4兲. All the next
terms of the expansion tend to reproduce the original square shape
of ⳵⍀0 in order to satisfy the boundary condition 共3.4兲 more accurately. On the other hand, the original shape of ⳵⍀0 can be
restored exactly in the zero-order approximation if R0 in the expression for ␹ = A2 / R20 is substituted by R共␰兲 defined in Eq. 共3.5兲.
In this case the boundary conditions 共3.2兲–共3.4兲 are exactly satisfied, and the solution converges for all values of ␭ and ␹. Thus we
obtain
C1 =
␭␹ cos2共␰兲
,
1 − ␭␹ cos2共␰兲
C3 = −
Here ␭ = 共k f − km兲 / 共k f + km兲, ␹ = A2 / R20 = c / cmax, and A = a / ␧, where
a is a radius of fiber in slow variables.
Let us now examine convergence of expansion 共3.9兲. The ratios
of the third-to-second constitutive terms are the same for any p:
030802-6 / Vol. 62, MAY 2009
c
1−
where
C1,0 =
The present solutions; C p are determined by
⳵ um1,0
Eventually, we obtain
−1
um
1 = 共C1r + C2r 兲
Table 1 Effective conductivity Šk‹ / km of the regular square lattice of perfectly conductive cylindrical fibers „kf / km = ⴥ…
C2 = −
␭
A2 ,
1 − ␭␹ cos2共␰兲
␭关1 − ␹ cos2共␰兲兴
1 − ␭␹ cos2共␰兲
共3.13兲
The obtained solution satisfies Eq. 共3.1兲 only approximately,
but further comparison with the known numerical results shows
that the error of this approximation is not successive.
Let us check the obtained solution in the case of perfectly conductive fibers k f / km = ⬁, the case that usually leads to main computational difficulties. Table 1 displays numerical results for the
effective conductivity 具k典 evaluated on the basis of expansion
共3.9兲 and improved expressions 共3.12兲 and 共3.13兲. The obtained
solutions are also compared with the data from Perrins et al. 关64兴.
It should be pointed out that the method given in Ref. 关64兴 is not
applicable in the limiting case of perfectly conductive nearly
touching fibers 共k f / km = ⬁ and c → cmax = 0.7853, . . .兲 when rapid
oscillation of the temperature field occurs on the microlevel. Both
of the present solutions 共3.12兲 and 共3.13兲 predict this case correctly with a considerably small discrepancy between them.
Finally, we assume the solution of the unit cell problem in the
form 共3.8兲 and 共3.13兲. Comparison of the obtained results with the
data from Perrins et al. 关64兴 for different values of conductivity of
fibers is presented in Fig. 5.
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
The notion of TPPA has been defined in Ref. 关63兴. Let us assume that
f共z兲 =
Fig. 5 Effective conductivity Šk‹ in the perfectly regular case:
solid curves—the present solution „3.13…; circles—data from
Ref. †64‡
The behavior of the effective conductivity at k f / km → ⬁ and c
→ cmax can be verified by comparison with the asymptotic formula
from O’Brien 关65兴 shown in Fig. 6.
The case k f / km = ⬁ should be checked separately, and the results
at this limit are shown in Table 2.
4
Two-Point Padé Approximants
Bergman 关66兴 showed that for the two-component isotropic
composites the effective conductivity 具k典 is a Stieltjes function of
␧1 = km / k f . This fact was used with one-point Padé approximants
for evaluating bounds of the effective parameters, see Refs.
关18,67兴. On the other hand, it is possible to obtain asymptotic
expansion for 具k典 as a function of ␧1 for ␧1 Ⰶ 1 and ␧1 Ⰷ 1. It gives
a possibility to use TPPAs generated by two different power expansions of Stieltjes function 关68–78兴.
140 k /k
120
c = 0.785
80
60
c = 0.784
40
20
1
k f/km
100 101
兺az
i
i
when z → 0
i=0
⬁
兺bz
i
i=0
−i
when z → ⬁
冧
共4.1兲
The TPPA is represented by the rational function
m
n
兺k=0
akzk / 兺k=0
bkzk, where k + 1 共k = 0 , 1 , . . . , n + m + 1兲 coefficients
of a Taylor expansion, if z → 0, and m + n + 1 − k coefficients of a
Laurent series, if z → ⬁, coincide with the corresponding coefficients of the series 共4.1兲.
Tokarzewski 关69,70兴 and Tokarzewski et al. 关72–74兴 investigated the TPPA for a nonequal, finite number of terms of two
power expansions of the Stieltjes functions at zero and at infinity.
Under some assumptions they proved that the diagonal TPPAs
form sequences of lower and upper bounds uniformly converging
to the Stieltjes function.
The general situation when the TPPA corresponding to an arbitrary number of terms of power expansions at zero and infinity has
been studied in the real domain by Tokarzewski and Telega
关75,76兴. They extended the fundamental inequalities derived for
the PA to the general TPPA. They proved the following theorem
that is very useful for practical applications.
The TPPAs for the Stieltjes function, represented by the power
⬁
expansions at zero, R共z兲 ⬵ 兺n=1
cnzn, and at infinity, R共z兲
⬁
−n
⬵ 兺n=0C−nz , obey the following inequalities for k = 1 , 2 , . . .,
2M共k = 1 , 2 , . . . , 2M + 1兲:
共− 1兲k关M/M兴k ⬍ 共− 1兲k关共M + 1兲/共M + 1兲兴k ⬍ 共− 1兲kR共z兲
共共− 1兲k−1关M/共M − 1兲兴k ⬍ 共− 1兲k关共M + 1兲/M兴k ⬍ 共− 1兲k−1R共z兲兲
共4.2兲
m
100
冦
⬁
102 103
104 105
Fig. 6 Asymptotic behavior of Šk‹ in the perfectly regular case
at kf / km \ ⴥ, c \ cmax: solid curves—the present solution;
dashed curves—the asymptotic formula from Ref. †65‡
Table 2 Effective conductivity Šk‹ / km in the case kf / km = ⴥ
Volume
fraction c
具k典 / km, formula
from Ref. 关65兴
具k典 / km,
present solution
0.784
0.785
0.7853
74.41
139.5
281.0
73.34
138.4
279.3
Applied Mechanics Reviews
where R共z兲 stands for the limit as M tends to infinity of 关M / M兴k,
关共M + 1兲 / M兴k, z is real and positive, and 关M / N兴k
N
j
j
= 兺M
j=0␣ jz / 兺 j=0␤ jz . Here k denotes the given number of coefficients of power expansions at infinity matched by the TPPA represented by 关M / N兴k.
The inequalities 共4.2兲 have the consequence that 关M / M兴k and
关共M + 1兲 / M兴k form upper and lower bounds for R共z兲 obtainable
using only the given number of coefficients and that the use of
additional coefficients improves the bounds.
The above theorem has been successfully used for the study of
the effective heat conductivity for a periodic square array of cylinders of conductivity k f = h embedded in a matrix of conductivity
km = 1. As an input for calculating TPPA the coefficients of the
expansions of ␭e共x兲 in powers of h − 1 for h − 1 Ⰶ 1 and in powers
of 1 / 共h − 1兲 for h → ⬁ have been used. The sequences of TPPA
uniformly converging to the effective conductivity are shown in
Figs. 7 and 8. The best bounds obtained by the TPPAs, namely,
关18/ 18兴1 and 关18/ 18兴2, are presented in Fig. 8. In Figs. 7 and 8
the asymptotic solution obtained by McPhedran et al. 关79兴 is
drawn for comparison.
It follows that the TPPAs allow us to evaluate the effective
moduli for a range of parameters much wider than the PA 关18,67兴.
For example, for ␸ = 0.78539 the TPPA approach leads to very
restrictive bounds, whereas the PA method fails 共see Fig. 7兲.
5
Asymptotically Equivalent Functions
The asymptotic formulas for the effective conductivity 具k典 for
c Ⰶ cmax and c ⬇ cmax do not provide complete representation of
the effective conductivity for any arbitrary c. Therefore, it is necessary to obtain the values of functions 具k典 for the intermediate
values of c. TPPA cannot be used in this case because the
MAY 2009, Vol. 62 / 030802-7
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 7 The sequences of †M / M‡0, †M / M‡1, and †M / M‡2, M = 2 , 4 , 6 , 12, 18 uniformly converging to the effective conductivity ␭e„h… „h = ␭2 / ␭1… of the
square array of cylinders. Curves †M / M‡2 are indistinguishable „solid line—
„a……. The bounds †18/ 18‡1 and †18/ 18‡2 are very restrictive.
sion F共z兲 共Eq. 共5.1a兲. Normally, such approach leads to satisfactory results, see Refs. 关81–90兴.
Now let us consider an application of the method of AEFs for
calculation of the effective heat conductivity of the infinite regular
array of perfectly conducting spheres embedded in a matrix with
unit conductivity. Sangani and Acrivos 关91兴 obtained the following expansion for the effective conductivity 具k典:
asymptotic expressions for 具k典 for c ⬇ cmax contain the logarithmic
functions. The solution of this problem can be found by applying
the asymptotically equivalent functions 关80兴 or the quasifractional
approximations in the other terminology. Let us assume that the
function f共z兲 in the limit z → ⬁ is described by a nonrational expression F共z兲:
for z → ⬁
f共z兲 = F共z兲
共5.1a兲
冉
具k典 = 1 − 3c − 1 + c + a1c10/3
and
⬁
f共z兲 =
兺cz
i
for z → 0
i
共5.1b兲
+ a6c22/3 + O共c25/3兲
i=0
Then the AEFs should also contain similar nonrational components. In general, the AEFs can be produced from Eqs. 共5.1a兲 and
共5.1b兲 as follows:
兺
␣i共z兲zi
冒兺
i=0
␤i共z兲zi
−1
共5.3兲
Here we consider three types of space arrangement of spheres,
namely, the simple cubic 共SC兲, body centered cubic 共BCC兲, and
face centered cubic 共FCC兲 arrays. Constants ai for these arrays are
given in Table 3.
In the case of perfectly conducting large spheres 共c → cmax,
where cmax is the maximum volume fraction for a sphere兲 the
problem can be solved by means of a reasonable physical assumption that the heat flux occurs entirely in the region where spheres
are in a near contact. Thus, the effective conductivity is determined in the asymptotic form for the flux between two spheres,
which is logarithmically singular in the width of a gap, justifying
assumption 关92兴
n
m
f共z兲 ⬇
冊
1 + a2c11/3
+ a4c14/3 + a5c6
1 − a3c7/3
共5.2兲
i=0
where ␣i and ␤i are considered not as constants but as some
functions of z. Functions ␣i共z兲 and ␤i共z兲 are chosen in such a way
that 共i兲 the expansion of the AEFs 共5.2兲 in powers of z for z → 0
matches the perturbation expansion 共5.1b兲 and 共ii兲 the asymptotic
behavior of the AEFs 共5.2兲 for z → ⬁ coincides with the expres-
Fig. 8 The TPPA upper and lower bounds on the effective conductivity for
a square array of densely packed highly conducting cylinders. For ␸
= 0.785 the bounds coincide. For ␸ = 0.7853, 0.78539 bounds are very restrictive. For higher volume fractions ␸ ⱖ 0.78539816 the difference between
lower and upper bounds rapidly increases.
Table 3 The constants a1 , . . . , a6, in Eq. „5.3…
SC array
BCC array
FCC array
030802-8 / Vol. 62, MAY 2009
a1
a2
a3
a4
a5
a6
1.305
0.129
0.0753
0.231
⫺0.413
0.697
0.405
0.764
⫺0.741
0.0723
0.257
0.0420
0.153
0.0113
0.0231
0.0105
0.00562
9.14⫻ 10−7
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
‹k›/km
Table 4 The constants M1, M2, and cmax
SC array
BCC array
FCC array
M1
M2
cmax
␲/2
0.7
2.4
7.1
冑3␲ / 8
冑2␲ / 6
冑3␲ / 2
冑2␲
␲/6
具k典 = − M 1 ln ␹ − M 2 + O共␹−1兲
共5.4兲
where ␹ = 1 − 共c / cmax兲1/3 is the dimensionless width of a gap between the neighboring spheres, ␹ → 0; M 1 = 0.5cmaxp, p is the
number of contact points at the surface of a sphere; and M 2 is a
constant, dependent on the type of space arrangement of spheres.
The values of M 1, M 2, and cmax for the three types of cubic arrays
are given in Table 4.
On the basis of “limiting” solutions 共5.3兲 and 共5.4兲 we develop
the AEFs valid for all values of the volume fraction of inclusions
c 苸 关0 , cmax兴:
具k典 = 共P1共c兲 + P2c共m+1兲/3 + P3 ln ␹兲/Q共c兲
共5.5兲
Here rational functions P1共c兲 and Q共c兲 and constants P2 and P3
are determined as follows:
m
Q共c兲 = 1 − c − a1c10/3,
P1共c兲 =
兺␣c
i
i/3
i=0
P2 = 0
+
P2 = − 共P1共cmax兲
for n = 1,
共m+1兲/3
Q共cmax兲M 2兲/cmax
14
12
10
8
6
4
2
for n = 2
The AEF 共5.5兲 takes into account m leading terms of expansion
共5.3兲 and n leading terms of expansion 共5.4兲. Coefficients ␣i are
␣0 = 1,
␣3 = 2 − Q共cmax兲M 1 / 共3cmax兲,
␣10 = −a1
equal
to
j/3
− Q共cmax兲M 1 / 共10c10/3
and
␣ j = −Q共cmax兲M 1 / 共jcmax
兲,
j
max 兲,
= 1 , 2 , . . . , m − 1 , m, j ⫽ 3 , 10.
Increment of m and n leads to the growth of the accuracy of the
obtained solution 共5.5兲. Let us illustrate this dependence in the
case of SC array. We calculated 具k典 for different values of m and
n. In Fig. 9 our analytical results are compared with experimental
measurements from Meredith and Tobias 关93,94兴 共black dots兲. Details on these data can be found in Ref. 关95兴. Finally, we restrict
m = 19 and n = 2 for all types of arrays, as they provide a satisfactory agreement with numerical data and a rather simple analytical
form of the AEF 共5.5兲.
Numerical results for the BCC and the FCC arrays are displayed in Figs. 10 and 11, respectively. For BBC array the obk /km
12
0.1 0.2 0.3 0.4 0.5 0.6 c
0
Fig. 10 Effective conductivity Šk‹ / km of the BCC array versus
volume fraction of inclusions c
tained AEF 共5.5兲 is compared with the experimental results from
McKenzie and McPhedran 关96兴 and McKenzie et al. 关97兴. For
FCC array the experimental data are not available; therefore we
are comparing with the numerical results obtained in Ref. 关97兴
using the Rayleigh method. The agreement between the analytical
solution 共5.5兲 and the numerical results is quite satisfactory.
6
Random Composites
Two-phase composites with random microstructure were analyzed by Drugan and Willis 关98兴 and Drugan 关99兴. They employed
the Hashin–Shtrikman variational principle. A numerical implementation of this work was carried out by Segurado and Llorca
关100兴.
Percolation effects are very important for the analysis of composite materials 关101,102兴. On the other hand, in many cases composite materials can be studied without taking into account the
percolation effects. For example, foam concrete usually has the
chaotic distribution of pores with no clusters. The structure of
dispersed composites manufactured by a cold drawing or uniform
pressure is nearly regular with no clusters. In these cases a concept of shaking geometry 关103,104兴 can be very useful.
It should be noted that many of commonly used bounds, for
example, the Hashin–Shtrikman variational bounds, demonstrate
divergence and become almost out of practical use for the densely
packed 共but not percolated兲 composites with the highly different
properties of the constituent materials. Therefore, in order to obtain a reasonable estimation of the effective properties, the improved bounding models should be developed, such that they do
not allow appearance of cluster chains, even if a volume fraction
of one of the constituents is beyond the percolation limit. Such
improved bounds for the effective transport properties of a random composite material with cylindrical fibers are proposed in
Ref. 关105兴.
Let us now consider a nonregular fiber-reinforced composite.
Center of each fiber can randomly deviate within a circle of diameter d, whereas these circles themselves form a regular square
lattice of a period l, see Fig. 12.
10
m=5, n=1
8
‹k›/km
m=10, n=2
6
m=19, n=2
4
2
m=5, n=2
0
0.1
0.2
0.3
0.4
0.5 C
Fig. 9 Effective conductivity Šk‹ / km of the SC array versus volume fraction of inclusions c
Applied Mechanics Reviews
17.5
15
12.5
10
7.5
5
2.5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 c
Fig. 11 Effective conductivity Šk‹ / km of the FCC array versus
volume fraction of inclusions c
MAY 2009, Vol. 62 / 030802-9
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
matrix Ωm
interface ∂Ω
fibre Ωf
l3
x3
l2
e3
e2
x2
L
l
Fig. 12 General view of the shaking-geometry composite
material
Fig. 13 Deviation of the fibers about the regular square lattice
Such kind of a microstructure is usually referred to as a
shaking-geometry composite 关103,104兴. From a practical point of
view it may correspond to a random “shaking” of the fibers about
the periodic lattice caused by some fabrication or technological
reasons. The deviation parameter ␦ = d / l describes the rate of nonregularity of the structure; its maximum value ␦max is determined
by the case when neighboring fibers are nearly touching each
other. Values of ␦ cannot be higher than ␦max since that will mean
a penetration of the neighboring fibers. A simple geometrical calculation yields ␦max = 1 − 冑c / cmax. Let us also assume that k f ⬎ km.
Note that the opposite case can be treated in the same mathematical way using the well-known duality relation 关106兴 具k共k f , km兲典
= 具k共km , k f 兲典−1.
Kozlov 关107兴 showed that a regular lattice possesses the extreme effective properties among the corresponding shakinggeometry random structures. Originally this result was proved for
the case of the dilute composites. Berlyand and Mityushev
关103,104兴 generalized Kozlov’s result 关107兴 to the nondilute cases.
Therefore, a solution for the perfectly regular lattice can be considered as a lower bound on the effective transport coefficient.
Below we will see that it almost coincides with the corresponding
Hashin–Shtrikman lower bound 关108–110兴.
On the other hand, the upper bound can be obtained by replacing the input nonregular assembly of fibers of radius a by the
regular lattice of fibers of radius a + d / 2. Such estimation is also
known as a security-spheres approach. It has been originally proposed by Keller et al. 关111兴 and was further extended by Rubenfeld and Keller 关112兴 and Torquato and Rubinshtein 关113兴. For
details and references see also Ref. 关114兴. In the case of highcontrast composites this bound appears to be essentially better
than Hashin–Shtrikman’s bound.
Improved bounds on the effective conductivity 具k典 of the random shaking-geometry composites, see Fig. 13, are deduced directly from the solution obtained by means of the improved
method of boundary shape perturbation, discussed in Sec. 3. Following the analytical results 关103,104兴, solution for the perfectly
regular lattice is assumed as the lower bound. The upper bound is
obtained by the security-spheres approach. In both cases we assume that there are no clusters of fibers in the composite under
study. However, we should note that for a purely random distribution of cylindrical fibers the percolation threshold is reached at
the volume fraction of inclusions c p ⬇ 0.41, see Ref. 关101兴.
Let us introduce lower K1 and upper K2 bounds for 具k典 such that
K1 ⱕ 具k典 ⱕ K2, and let us denote the conductivity of the perfectly
regular material as a function K0 of the fiber volume fraction c,
i.e., 具k典 兩␦=0 = K0共c兲. Then, the lower bound K1 is given by the
solution in the perfectly regular case at ␦ = 0, see Refs.
关103,104,107兴:
030802-10 / Vol. 62, MAY 2009
K1 = K0共c兲
共6.1兲
In order to obtain the upper bound K2 we replace the original
nonregular assembly of fibers of radius a by a regular lattice of
fibers of radius a + d / 2. This estimation yields
K2 = K0共c + 2␦冑ccmax + ␦2cmax兲
共6.2兲
For comparison we also provide Hashin–Shtrikman’s variational
bounds 关108–110兴, see also Ref. 关38兴:
K1 = km +
c
1/共k f − km兲 + 共1 − c兲/共2km兲
共6.3兲
1−c
1/共km − k f 兲 + c/共2k f 兲
共6.4兲
K2 = k f +
Numerical examples are shown in Fig. 14. We can observe that
the lower bound 共6.1兲 almost coincides with the Hashin–
Shtrikman bound 共6.3兲. At the same time, the situation with the
upper bound is different. For the low-contrast case 共k f / km → 1兲 the
Hashin–Shtrikman bound 共6.4兲 is better. But for the high-contrast
case 共k f / km → ⬁兲 the improved bound 共6.2兲 provides essentially
better results, while the Hashin–Shtrikman bound 共6.4兲 becomes
almost useless. A simple practical recommendation is that from
two upper bounds 共6.2兲 and 共6.4兲 the lowest one should be chosen.
7 Correlation Between Conductivity and Elastic Properties of Composites
It is of interest to establish certain links 共strict or approximate兲
between the solutions of transport and elasticity problems for
composite materials. Such cross-property relations become very
useful if one of them can be more easily calculated or measured
experimentally. As soon as the effective properties reflect certain
morphological information about the composite medium, one
might expect that extracting useful knowledge about one property
would allow determining the other properties. Unfortunately, exact results can be obtained very rarely, see Refs. 关18,115–117兴.
Explicit cross-property relations for the anisotropic two-phase
composite materials have been obtained by Sevostianov and
Kachanov 关118,119兴 and Sevostianov et al. 关120,121兴. Correlations between elastic moduli and thermal and electric conductivities of the anisotropic composite materials are found in Refs.
关122–124兴.
Manevitch et al. 关125兴, see also Refs. 关19,21,126,127兴, developed an approach that reduces the original 2D elasticity problem
to a form resembling the transport problem. Moreover, in some
particular cases it allows to establish direct analogy between the
effective elastic and transport properties.
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 14 Bounds on Šk‹ in the nonregular case. Solid curves—the present solution: the lower bound „6.1… at ␦ = 0 and the upper bound „6.2… for different
values of ␦. Dashed curves—the Hashin–Shtrikman bounds „6.3… and „6.4…. „a…
Dilute composite: c = 0.2 and „b… densely packed composite: c = 0.7.
Let us consider 2D composite material with square inclusions
shown in Fig. 15. We assume that the matrix and inclusions are
made of orthotropic materials. Governing equations of the plane
elasticity problem can be written as follows 共index a represents
both m for matrix and f for fibers兲:
a
=0
Ba1uaxx + 共Ba3 + Ba12兲uayx + Ba3vxy
共7.1兲
a
a
+ Ba3uxy
=0
Ba2vayy + 共Ba3 + Ba12兲vxy
共7.2兲
a
␤vayy + ␧共1 + ␣兲vaxx + ␧uxy
=0
m
m
m
Here ␣ = Bm
12 / B3 and ␤ = B2 / B1 .
Asymptotic splitting in the first approximation yields two independent equations 关125–127兴:
We assume that
m m m
f
f
f
f
兵Bm
1 ,B2 ,B3 ,B12其 = ␭兵B1,B2,B3,B12其
共7.3兲
We also assume the perfect bonding conditions on the interfaces
x = ⫾ a and y = ⫾ a:
um = u f
共7.4兲
vm = v f
共7.5兲
Sm = S f
共7.6兲
for
x = ⫾ a,
f
Tm
1 = T1
共7.7兲
for
y = ⫾ a,
f
Tm
2 = T2
共7.8兲
Ta2 = Ba2vay + Ba12uax , and Sa = Ba3共uay + vax 兲.
Here
m
Let us introduce parameter ␧2 = Bm
3 / B1 . Further we will treat ␧2
m
m
m
as a small parameter and assume that Bm
1 ⬃ B2 and B12 ⬃ B3 . Then
Ta1 = Ba1uax + Ba12vay ,
Eqs. 共7.1兲 and 共7.2兲 can be rewritten as follows:
a
=0
uaxx + ␧共1 + ␣兲uayy + ␧vxy
uaxx + ␧共1 + ␣兲uayy = 0
共7.9兲
␤vayy + ␧共1 + ␣兲vaxx = 0
共7.10兲
Equation 共7.9兲 must be solved with conditions 共7.4兲, 共7.6兲, and
共7.7兲, and Eq. 共7.10兲, with conditions 共7.5兲, 共7.6兲, and 共7.8兲. Using
m
m
m
smallness of the parameters Bm
12 / B1 , and B12 / B2 , these conditions
can be rewritten as follows:
for Eq. 共7.9兲,
um = u f
for x = ⫾ a
f
um
x = ␭ux
and y = ⫾ a
for x = ⫾ a
f
f
共umy + vm
x 兲 = ␭共u y + vx兲
for y = ⫾ a
共7.11兲
共7.12兲
共7.13兲
for Eq. 共7.10兲,
vm = v f
for x = ⫾ a
f
vm
y = ␭v y
and
y= ⫾a
for y = ⫾ a
f
f
共umy + vm
x 兲 = ␭共u y + vx兲
for x = ⫾ a
共7.14兲
共7.15兲
共7.16兲
Conditions 共7.13兲 and 共7.16兲 connect boundary-value problems
for ua and va. It was proposed in Ref. 关126兴 to replace conditions
共7.13兲 and 共7.16兲 by the following ones:
Fig. 15 Composite material with square inclusions
Applied Mechanics Reviews
for y = ⫾ a,
umy = ␭uyf
for x = ⫾ a,
f
vm
x = ␭vx
That allows replacing the elasticity BVP by the two transport
BVPs with some approximation. Error of this approximation depends on the elastic energy of deformation. The contact conditions
for shear forces Sa are not fully satisfied, but contribution of Sa
into the energy of deformation depends on coefficients Ba3, 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
cross-sectional shapes, for example, circle or elliptical fibers.
MAY 2009, Vol. 62 / 030802-11
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
8 Asymptotic Homogenization of Thin-Walled Composite Reinforced Structures
In the numerous engineering applications the composite materials used are in the form of thin-walled structural members like
shells and plates. Their stiffness and strength combined with the
reduced weight and associated material savings offer very impressive possibilities. It is very common that the reinforcing elements
such as embedded fibers or surface ribs form a regular array with
a period much smaller than the characteristic dimensions of the
whole composite structure. Consequently, the asymptotic homogenization analysis becomes applicable.
The asymptotic homogenized model for plates with periodic
inhomogeneities in tangential directions has been developed for
the first time by Duvaut 关128,129兴. In these works asymptotic
homogenization procedure was applied directly to a 2D plate
problem. Later, Andrianov et al. 关130兴 applied homogenization
method to analyze statical and dynamical problems for the ribbed
shells.
Evidently, the asymptotic homogenization method cannot be
applied directly to the cases of 3D thin composite layers if their
small thickness 共in the direction of which there is no periodicity兲
is comparable with the small dimensions of the periodicity cell 共in
the two tangential directions兲. To deal with the 3D problem for a
thin composite layer, a modified asymptotic homogenization approach was proposed by Caillerie 关131,132兴 in the heat conduction
studies. It consists of applying a two-scale asymptotic formalism
directly to the 3D problem for a thin inhomogeneous layer with
the following modification. Two sets of “rapid” coordinates are
introduced. Two tangential coordinates are associated with the
rapid periodic variation in the composite properties. The third one
is in the transverse direction and is associated with the small
thickness of the layer, and it takes into account that there is no
periodicity in this transverse direction. There are two small parameters, one a measure of periodic variation in two tangential
directions and the other is a measure of a small thickness. Generally, these two parameters may or may not be of the same order of
magnitude. But commonly in practical applications they are small
values of the same order. Kohn and Vogelius 关133–135兴 adopted
this approach in their study of a pure bending of a thin, linearly
elastic homogeneous plate with wavy surfaces.
The generalization of this approach to the most comprehensive
case of a thin 3D composite layer with wavy surfaces 共that model
the surface reinforcements兲 was offered by Kalamkarov 关5,33,34兴,
see also Ref. 关35兴. In these works the general asymptotic homogenization model for composite shell was developed by applying
the modified two-scale asymptotic technique directly to 3D elastic
and thermoelastic problems for a thin curvilinear composite layer
with wavy surfaces. The homogenization models were also developed in the cases of the nonlinear problems for composite shells,
see Refs. 关136,137兴. The developed homogenization models for
composite shell were applied for the design and optimization of
composite and reinforced shells 关35,36兴. Most recently, this technique was adopted in modeling of smart composite shells and
plates in Refs. 关138–142兴. The general homogenization model for
composite shell has found numerous applications in the analysis
of various practically important composite structures. Georgiades
et al. 关143兴 and Challagulla et al. 关144–146兴 studied gridreinforced and network thin composite generally orthotropic
shells as well as the 3D network reinforced composite structures.
Saha et al. 关147,148兴 analyzed the sandwich composite shells and,
in particular, the honeycomb sandwich composite shells made of
generally orthotropic materials. Asymptotic homogenization was
also applied to calculate the effective properties of the carbon
nanotubes by Kalamkarov et al. 关149,150兴.
030802-12 / Vol. 62, MAY 2009
Fig. 16 „a… Curvilinear thin 3D reinforced composite layer and
„b… unit cell Ω␦
Let us now summarize the above introduced general homogenization model for composite shell, see Refs. 关5,35兴 for details. Consider a general thin 3D composite layer of a periodic structure
with the unit cell ⍀␦ shown in Fig. 16. In this figure, ␣1, ␣2, and
␥ are the orthogonal curvilinear coordinates, such that the coordinate lines ␣1 and ␣2 coincide with the main curvature lines of the
midsurface of the carrier layer and coordinate line ␥ is normal to
its midsurface 共␥ = 0兲.
Thickness of the layer and the dimensions of the unit cell of the
composite material 共which define the scale of the composite material inhomogeneity兲 are assumed to be small as compared with
the dimensions of the structure in whole. These small dimensions
of the periodicity cell are characterized by a small parameter ␦.
The unit cell ⍀␦, see Fig. 16共b兲, is defined by the following
relations:
−
␦h1
2
⬍ ␣1 ⬍
␦h1
2
,
␥⫾ = ⫾
−
␦
2
␦h2
2
⬍ ␣2 ⬍
⫾ ␦F⫾
冉
␦h2
2
␥− ⬍ ␥ ⬍ ␥+ ,
,
␣1 ␣2
,
␦h1 ␦h2
冊
Here, ␦ is the thickness of the layer, and ␦h1 and ␦h2 are the
longitudinal dimensions of the periodicity cell ⍀␦. Functions F⫾
define the geometry of the upper 共S+兲 and lower 共S−兲 reinforcing
elements, for example, the ribs or stiffeners, see Figs. 16 and 17.
If there are no reinforcing elements, then F+ = F− = 0, and the composite layer has a uniform thickness of order of ␦, like it is, for
example, in the case shown in Fig. 18.
The periodic inhomogeneity of the composite material is modeled by the assumption that the elastic coefficients cijkl 共␣1 , ␣2 , ␥兲
are periodic functions in variables ␣1 and ␣2 with a unit cell ⍀␦.
The elasticity problem for the above 3D thin composite layer is
formulated as follows:
⳵␴ij
− fi = 0
⳵␣j
␴ij = cijkl共␣1, ␣2, ␥兲ekl,
(a)
Fig. 17
ekl =
冉
1 ⳵ uk ⳵ ul
+
2 ⳵ ␣l ⳵ ␣k
冊
(b)
„a… Wafer-reinforced shell and „b… unit cell
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
␮
ⴱ␭␮
⳵ bi3
1 ⳵ biⴱ␭
␤
+
=0
h␤ ⳵␰␤
⳵z
1 ⫾ ⴱ␭␮
ⴱ␭␮
n b + n⫾
3 bi3 = 0
h ␤ ␤ i␤
at z = z⫾
共8.6b兲
Fig. 18 Sandwich composite shell with a honeycomb filler
␴ijn j⫾ = pi⫾
共8.1兲
Here f i, p⫾
i , and uk represent body forces, surface tractions, and
displacement field, respectively, and n j⫾ is the unit normal to the
upper and lower wavy surfaces ␥⫾ = S⫾共␣1 , ␣2兲.
We introduce the following fast variables, ␰ = 共␰1 , ␰2兲, and z:
␰1 =
␣ 1A 1
,
␦h1
␰2 =
␣ 2A 2
,
␦h2
z=
␥
␦
where A1共␣兲 and A2共␣兲 are the coefficients of the first quadratic
form of the midsurface of a carrier layer 共␥ = 0兲.
The displacements and stresses are expressed in the form of the
following two-scale asymptotic expansions:
ui共␣, ␰,z兲 = ui共0兲共␣兲 + ␦ui共1兲共␣, ␰,z兲 + ␦2ui共2兲共␣, ␰,z兲 + ¯
␴ij共␣, ␰,z兲 = ␴ij共0兲共␣, ␰,z兲 + ␦␴ij共1兲共␣, ␰,z兲 + ␦2␴ij共2兲共␣, ␰,z兲 + ¯
共8.2兲
As a result of asymptotic homogenization procedure, see Refs.
关5,35兴 for details, the following relations for the displacements
and stresses are derived:
u 1 = v 1共 ␣ 兲 − ␦
u 2 = v 2共 ␣ 兲 − ␦
z ⳵ w共␣兲
+ ␦U1␮␯e␮␯ + ␦2V1␮␯␶␮␯ + O共␦3兲
A1 ⳵ ␣1
z ⳵ w共␣兲
+ ␦U2␮␯e␮␯ + ␦2V2␮␯␶␮␯ + O共␦3兲u3 = w共␣兲
A2 ⳵ ␣2
+ ␦U3␮␯e␮␯ + ␦2V3␮␯␶␮␯ + O共␦3兲
共8.4兲
Here and in the sequel Latin indices assume values of 1, 2, and 3;
Greek indices assume values of 1 and 2; and repeated indices are
summed; the midsurface strains are denoted as follows: e11 = e1
and e22 = e2 共elongations兲, e12 = e21 = ␻ / 2 共shear兲, ␶11 = k1 and ␶22
= k2 共bending兲, and ␶12 = ␶21 = ␶ 共twisting兲.
The following notation is used in Eq. 共8.4兲:
bijlm =
⳵ Ulm
⳵ Ulm
1
n
n
cijn␤
+ cijn3
+ cijlm
⳵␰␤
h␤
⳵z
共8.5a兲
bijⴱlm =
⳵ Vlm
⳵ Vlm
1
n
n
cijn␤
+ cijn3
+ zcijlm
⳵␰␤
h␤
⳵z
共8.5b兲
lm
The functions Ulm
n 共␰1 , ␰2 , z兲 and Vn 共␰1 , ␰2 , z兲 in Eqs. 共8.3兲,
共8.5a兲, and 共8.5b兲 are solutions of the unit cell problems. Note that
all the above functions are periodic in variables ␰1 and ␰2 with
periods A1 and A2, respectively. The above mentioned unit cell
problems are formulated as follows:
␭␮
1 ⳵ bi␭␤␮ ⳵ bi3
+
=0
h␤ ⳵␰␤
⳵z
Applied Mechanics Reviews
ⴱ␭␮
␭␮
典e␭␮ + ␦2具b␣␤
典 ␶ ␭␮
N␣␤ = ␦具b␣␤
共8.3兲
␴ij = bij␮␯e␮␯ + ␦bijⴱ␮␯␶␮␯
1 ⫾ ␭␮
␭␮
n b + n⫾
3 bi3 = 0
h ␤ ␤ i␤
where n+i and n−i are components of the normal unit vector to the
upper 共z = z+兲 and lower 共z = z−兲 surfaces of the unit cell, respectively, defined in the coordinate system ␰1 , ␰2 , z.
If inclusions are perfectly bonded to matrix on the interfaces of
lm
the composite material, then the functions Ulm
n and Vn together
共c兲 ␭␮
共c兲 ␭␮
with
the
expressions
关共1 / h␤兲n␤ bi␤ + n3 bi3 兴
and
共c兲 ␭␮
*␭␮兴 should be continuous on the interfaces.
关共1 / h␤兲n␤ b*i␤ + n共c兲
b
3 i3
共c兲
Here ni are the components of the unit normal to the interface.
It should be noted that, unlike the unit cell problems of “classical” homogenization models, e.g., Eqs. 共2.9兲 and 共2.25兲, those
set by Eqs. 共8.6a兲 and 共8.6b兲 depend on the boundary conditions
z = z⫾ rather than on periodicity in the z direction.
lm
After local functions Ulm
n 共␰1 , ␰2 , z兲 and Vn 共␰1 , ␰2 , z兲 are found
from the unit cell problems given by Eqs. 共8.5a兲, 共8.6a兲, 共8.5b兲,
ⴱlm
and 共8.6b兲, the functions blm
ij 共␰1 , ␰2 , z兲 and bij 共␰1 , ␰2 , z兲 given by
Eqs. 共8.5a兲 and 共8.5b兲 can be calculated. These local functions
define stress ␴ij, as it is seen from Eq. 共8.4兲. They also define the
effective stiffness moduli of the homogenized shell. Indeed, constitutive relations of the equivalent anisotropic homogeneous
shell—which is between the stress resultants N11, N22 共normal兲,
and N12 共shear兲 and moment resultants M 11, M 22 共bending兲, and
M 12 共twisting兲 on one hand, and the midsurface strains e11 = e1,
e22 = e2 共elongations兲, e12 = e21 = ␻ / 2 共shear兲, ␶11 = k1, ␶22 = k2
共bending兲, and ␶12 = ␶21 = ␶ 共twisting兲 on the other—can be represented as follows 关5,35兴:
at z = z⫾
共8.6a兲
ⴱ␭␮ 3
␭␮
M ␣␤ = ␦2具zb␣␤
典e␭␮ + ␦具zb␣␤
典 ␶ ␭␮
共8.7兲
The angular brackets in Eq. 共8.7兲 denote averaging by the integration over the volume of the 3D unit cell:
具f共␰1, ␰2,z兲典 =
冕
f共␰1, ␰2,z兲d␰1d␰2dz
⍀
The coefficients in constitutive relations 共Eq. 共8.7兲兲 具b␣␤␭␮典,
具b ␣␤␭␮典 = 具zb␭␮␣␤典, and 具zbⴱ␣␤␭␮典 are the effective stiffness
moduli of the homogenized shell. The midsurface strains
e␭␲共␣1 , ␣2兲 and ␶␭␮共␣1 , ␣2兲 can be determined by solving a global
boundary-value problem for the homogenized anisotropic shell
with the constitutive relations 共8.7兲, see Refs. 关5,35兴 for details. It
should be noted, as can be observed from Eq. 共8.7兲, that there is a
following one-to-one correspondence between the effective stiffness moduli and the extensional, 关A兴, coupling, 关B兴, and bending,
关D兴, stiffnesses familiar from the classical composite laminate
theory, see, e.g., Ref. 关151兴:
ⴱ
MAY 2009, Vol. 62 / 030802-13
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
冤
␦具b11
␦具b22
␦具b12
11典
11典
11典
22
22
␦具b11典 ␦具b22典 ␦具b12
22典
12
12
␦具b11典 ␦具b22典 ␦具b12
A B
12典
= 2 ⴱ11
2 ⴱ12
B D
␦ 具b11 典 ␦2具bⴱ22
典
␦
具b
11
11 典
2 ⴱ22
2 ⴱ22
2 ⴱ12
␦ 具b11 典 ␦ 具b22 典 ␦ 具b22 典
2 ⴱ12
2 ⴱ12
␦2具bⴱ12
11 典 ␦ 具b22 典 ␦ 具b12 典
冋 册
The unit cell problems given by Eqs. 共8.5a兲, 共8.6a兲, 共8.5b兲, and
共8.6b兲 have been solved analytically for a number of structures of
a practical interest, and the explicit analytical formulas for the
effective stiffness moduli have been obtained for the following
types of composite and reinforced shells and plates: angle-ply
fiber-reinforced shells and grid-reinforced and network shells
关5,33,35,143–146兴; rib- and waferlike reinforced shells
关5,34,35,140,142,152兴; sandwich composite shells, in particular,
the honeycomb sandwich composite shells made of generally
orthotropic materials 关5,35,141,147,148兴; and carbon nanotubes
关149,150兴.
As the examples of these results, we will present here the analytical results for the effective stiffness moduli of a waferreinforced shell shown in Fig. 17 and a sandwich composite shell
with a honeycomb filler shown in Fig. 18.
The nonzero effective stiffness moduli of the wafer-reinforced
shell shown in Fig. 17 are obtained as follows, see Refs.
关5,34,35,152兴 for details:
具b11
11典
=
具b22
22典 =
E1共3兲
E2共3兲
=
共3兲
G12
,
具zb11
11典
具zbⴱ12
12 典 =
共3兲
G12
=
具bⴱ11
11 典
E2共3兲
共3兲 共3兲
12共1 − ␯12
␯21 兲
12
+
共1兲
G12
12
冉
96H4
␲ 5A 1h 1
冑
共1兲
G12
⬁
兺
共1兲
G23
n=1
⬁
兺
共2兲
G13
n=1
冉冑
关1 − 共− 1兲n兴
tanh
n5
F1共w兲 =
Ht1
,
h1
J1共w兲 =
F2共w兲 =
Ht2
,
h2
+ E2共1兲J1共w兲
共3兲 共3兲
␯21
E1
共3兲 共3兲
12共1 − ␯12
␯21 兲
冊 冉
H 3t 1
− K1 +
12
h1
2E0t0
1−
␯20
030802-14 / Vol. 62, MAY 2009
J2共w兲 =
S2共w兲 =
共H2 + H兲t2
2h2
共4H3 + 6H2 + 3H兲t2
12h2
+
冑3 EHt
4
a
共1兲
G23
n ␲ A 1t 1
共1兲
2H
G12
冊
1−
␯20
+
冑3 EHt
12 a
冉
冊
冉
冊
*
具zbⴱ11
22 典 = 具zb11 典 =
冑3 EH3t
2t30
␯ 0E 0 H 2t 0
+ Ht20 +
+
2
3
144 a
1 − ␯0 2
冉
冊
冤
2t30
3+␯
H 2t 0
E0
EH3t
+ Ht20 +
+
3
2共1 + ␯0兲 2
12共1 + ␯兲a 4冑3
−
共8.9兲
2 ␯ 0E 0t 0
冑3 EHt
E 0t 0
+
共1 + ␯0兲 12 a
冑3 EH3t
2t30
E 0 H 2t 0
+ Ht20 +
+
2
3
48 a
1 − ␯0 2
具zbⴱ12
12 典 =
冊
具b12
12典 =
,
ⴱ22
具zbⴱ11
11 典 = 具zb22 典 =
,
H 3t 2
− K2
h2
冉冑
关1 − 共− 1兲n兴
tanh
n5
冊
共8.11兲
22
+ E1共2兲J2共w兲 ,
共2兲
G12
共H2 + H兲t1
,
2h1
S1共w兲 =
共4H3 + 6H2 + 3H兲t1
,
12h1
E1共2兲S2共w兲 ,
=
共2兲
G13
n ␲ A 2t 2
共2兲
2H
G12
Here the superscripts indicate the elements of the unit cells ⍀1,
⍀2, and ⍀3, see Fig. 17共b兲; A1 and A2 are the coefficients of the
first quadratic form of the midsurface of a carrier layer; and
共w兲
共w兲 共w兲
共w兲 共w兲
F共w兲
1 , F2 , S1 , S2 , and J1 , J2 are defined as follows:
22
具b11
22典 = 具b11典 =
where
K1 =
共2兲
G12
共8.8兲
共8.10兲
共3兲 共3兲
1 − ␯12
␯21
共3兲 共3兲
12共1 − ␯12
␯21 兲
ⴱ22
具zbⴱ11
22 典 = 具zb11 典 =
冑
冥
␦2具zb12
11典
␦2具zb12
22典
2
␦ 具zb12
12典
␦3具zbⴱ12
11 典
3
␦ 具zbⴱ12
22 典
␦3具zbⴱ12
12 典
共3兲 共3兲
␯21
E1
E1共3兲
具zbⴱ22
22 典 =
96H4
K2 = 5
␲ A 2h 2
22
具b11
11典 = 具b22典 =
共1兲 共w兲
ⴱ22
具zb22
22典 = 具b22 典 = E2 S1 ,
具zbⴱ11
11 典 =
␦2具zb22
11典
␦2具zb22
22典
2
␦ 具zb12
22典
␦3具zbⴱ22
11 典
3
␦ 具zbⴱ22
22 典
␦3具zbⴱ12
22 典
The nonzero effective stiffness moduli of the sandwich composite shell with a honeycomb filler shown in Fig. 18 are obtained
as follows, see Refs. 关5,35兴 for details:
+ E2共1兲F1共w兲 ,
共3兲 共3兲
1 − ␯12
␯21
22
具b11
22典 = 具b11典 =
具b12
12典
+
共3兲 共3兲
1 − ␯12
␯21
E1共2兲F2共w兲 ,
␦2具zb11
11典
␦2具zb22
11典
2
␦ 具zb12
11典
␦3具zbⴱ11
11 典
3
␦ 具zbⴱ22
11 典
␦3具zbⴱ12
11 典
128H
⬁
兺
共冑3␲5At兲 n=1
冉
tanh
␲共2n − 1兲At
2H
共2n − 1兲5
冊
冥
共8.12兲
In Eq. 共8.12兲, the first terms define the contribution from the top
and bottom carrier layers of the sandwich shell, while the latter
terms represent the contribution from the honeycomb filler. E0 and
␯0 are the properties of the material of the carrier layers, and E
and ␯ of the honeycomb foil material. We have confined our attention here by the case of equal coefficients of the first quadratic
form of the midsurface of the shell, i.e., A1 = A2 = A.
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
9 Boundary Effects, Strength, and Damage in Composite Materials
While asymptotic homogenization leads to a much simpler
problem for an equivalent homogeneous material with certain effective properties, the construction of a solution in the vicinity of
the boundary of the original composite solid remains beyond capabilities of the classical homogenization, see, e.g., Refs.
关5,153,154兴. In order to determine stresses and strains near the
boundary, a boundary-layer problem should be considered in extension to the asymptotic homogenization. A boundary-layer
method in asymptotic homogenization was developed by
Kalamkarov 关5兴, Sec. 7, and used to solve a problem of a transversal crack in a periodic composite, and by Andrianov et al.
关130兴 in the theory of ribbed plates and shells. This approach was
further developed by Kalamkarov and Georgiades 关139兴 in
asymptotic homogenization of smart periodic composites. The exponential decay of boundary layers was proved in Ref. 关6兴 for the
problems with a simple geometry.
New generalized integral transforms for the analytical solution
of the boundary-value problems for composite materials have
been developed by Kalamkarov 关5兴, Appendix B, and Kalamkarov
et al. 关155兴.
The properties of boundary layers in periodic homogenization
in rectangular domains, which are either fixed or have an oscillating boundary, are investigated in Ref. 关156兴. Such boundary layers
are highly oscillating near the boundary and decay exponentially
fast in the interior to a nonzero limit that the authors called a
boundary-layer tail. It is shown that these boundary-layer tails can
be incorporated into the homogenized equation by adding dispersive terms and a Fourier boundary condition.
Although finding the explicit analytical solutions of boundarylayer problems in the theory of homogenization still remains an
open problem, the effective numerical procedures have been proposed in Refs. 关157,158兴.
Asymptotic homogenization approach can be effectively used
not only to calculate the effective properties of composites but
also to analyze their strength and damage. That follows from a
very important advantage of the asymptotic homogenization that,
in addition to the effective properties, it allows to determine with
a high accuracy the local stresses and strains defined by a microstructure of a composite material. A number of publications are
related to the formulation of the failure criteria based on the
asymptotic homogenization, see Refs. 关159–162兴.
Until recently, in the study of strength of composite materials
most typical was a phenomenological approach based on the failure criterion for the equivalent homogeneous anisotropic material,
see, e.g., Ref. 关38兴. It is of interest to develop such strength criteria for the composite materials that will take into account the
phenomenological failure criteria for each individual constituent
material. To achieve that, a concept of stress and strain concentration functionals for the composite materials was proposed in Ref.
关163兴, which allows expressing stresses and strains in the constituent materials in terms of the stresses and strains in the equivalent
homogenized material. Both the effective properties and the local
characteristics are taken into account in this approach. Particularly
important results could be produced in this way if the analytical
expressions for the stress and strain concentration functionals
could be obtained. That is a reason why this approach was used in
Refs. 关159,161兴 only for laminated composites, for which the unit
cell problems become one dimensional and thus solvable in explicit analytical expressions. It is possible to extend the concept of
stress and strain concentration functionals to 2D and 3D cases by
applying the methods introduced in Secs. 3–5 of the present paper.
According to the approach 关159,160兴, the general failure criteria
in stresses 共or strains兲 of the constituent materials are written first.
Then, the stress 共or strain兲 concentration tensors are substituted
into these failure criteria. And finally, the resulting expressions are
homogenized. In the opinion of the authors of the present paper,
the last procedure requires more detailed substantiation since as a
Applied Mechanics Reviews
result of averaging the local stress pikes will be cut off. More
substantiated criterion is offered in Ref. 关162兴, where it is suggested to find such a limiting value, for which the failure begins at
least at one point of any constituent of the composite.
Luo and Takezono 关164兴 used the asymptotic homogenization
method to obtain the effective mechanical properties of the fiberreinforced ceramic matrix composites and to derive the homogenized damage elastic concentration factor for the unidirectional
and cross-ply laminated composites. They introduced the internal
variables to describe the evolution of the damage state under
uniaxial loading and as a subsequence the degradation of the material stiffness.
Let us note that the application of homogenization approach for
the damage analysis assumes, as a rule, the uniform distribution of
sources of failure, for example, the uniform distribution of cracks
in 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 strength
of the composites.
It is suggested in Refs. 关165–169兴 to develop the damage progression models entirely on the basis of asymptotic homogenization, without any complementary phenomenological assumptions.
The authors mentioned that they remain within the applicability of
the asymptotic homogenization, which is limited to the early
stages of failure. For a correct description of the advanced stages
of failure one has to supplement these models with a phenomenological counterpart since the homogenization is not applicable
anymore. In particular, the classical continuum formulation is
used in Ref. 关165兴, but an internal length parameter is introduced
in the damage progression model, as a consequence of the microscopic balance of energy and a Griffith-type microcrack propagation criterion.
Asymptotic homogenization techniques in combination with the
phenomenological assumptions related to the damage in composites are developed in Refs. 关170–174兴.
A continuum-scale analysis to account for the damage produced
by evolving internal boundaries and employing methods of fracture mechanics on a smaller scale is offered in Refs. 关175–177兴.
The assumption of statistical homogeneity on a smaller scale
yields the macroscale damage-dependent model by employing a
homogenization principle. In this case, the physical details on the
smaller scale are not lost. The considered approximation is a particular case of the micromechanical damage approach that treats
each microphase as a statistically homogeneous medium
关178–185兴. Local damage variables are introduced to represent the
state of damage in each phase and the effective material properties
are defined thereafter. The overall damage model is subsequently
obtained by means of homogenization.
The problem of fatigue life prediction is studied in Ref. 关186兴
using homogenization with two temporal coordinates. In this approach the original boundary-value problem is decomposed into
coupled microchronological 共fast time-scale兲 and macrochronological 共slow time-scale兲 problems. The life prediction methodology was validated numerically against the direct cycle-by-cycle
simulations.
The simultaneous microscopic and macroscopic analyses at
each loading step are proposed in Refs. 关187,188兴. Such approach
leads to a very high volume of computations but it gives a possibility to take into account the evolution of damage and the effect
of loading history.
10
Conclusions and Generalizations
Asymptotic homogenization is a mathematically rigorous powerful tool for analyzing composite materials and structures. The
proof of the possibility of homogenizing a composite material of a
regular structure, i.e., of examining an equivalent homogeneous
solid instead of the original inhomogeneous composite solid, is
one of the principal results of this theory. Method of asymptotic
homogenization has also indicated a procedure of transition from
the original problem 共which contains in its formulation a small
MAY 2009, Vol. 62 / 030802-15
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
parameter related to the small dimensions of the constituents of
the composite兲 to a problem for a homogeneous solid. The effective properties of this equivalent homogeneous material are determined through the solution of the unit cell problems. Important
advantage of the asymptotic homogenization is that, in addition to
the effective properties, it allows to determine with a high accuracy the local stress and strain distributions defined by the microstructure of composite materials.
The present paper reviews the state-of-the-art in asymptotic homogenization of composite materials and thin-walled composite
structures. Using 204 references we have presented a variety of
existing methods, pointed out their advantages and shortcomings,
and discussed their applications. In addition to the review of existing results, some new original approaches have also been offered. In particular, we discussed possible methods of analytical
solution of the unit cell problems obtained as a result of the
asymptotic homogenization. The asymptotic homogenization of
3D thin-walled composite reinforced structures is considered, and
the general homogenization model for a composite shell is introduced. In particular, the analytical formulas for the effective stiffness moduli of wafer-reinforced shell and sandwich composite
shell with a honeycomb filler are presented. We also discussed
random composites; use of two-point Padé approximants and asymptotically equivalent functions; the correlation between conductivity and elastic properties of composites; and strength, damage, and boundary effects in composites.
In conclusion, we would like to refer to some generalizations in
the application of the asymptotic homogenization. Generalization
that accounts for nonlinearity of transport problems for fiber composites is proposed in Refs. 关68,71,189兴. The unit cell problems
are formulated as the minimization problems, and some bounds
for the effective properties are extended to the nonlinear problems
and calculated using the two-point Padé approximants.
Many of the above discussed results can be generalized for the
inclusions with cross sections slightly different from the canonical
by means of the boundary shape perturbation technique 关49兴 as
well as for the quasiperiodic composites 关190兴. Generalizations on
account of anisotropy of the constituent materials are developed in
Refs. 关191,192兴.
We would also like to refer to the application of the asymptotic
homogenization in the analysis of stressed composite materials
and structures 关193,194兴, in the study of a threshold phenomenon
关195–197兴, in the investigation of the analytical properties of the
effective parameters 关198–203兴, and to a new approach based on
the integral equations 关204兴.
The fundamental aspects of homogenization, including nonlinear homogenization, nonconvex and stochastic problems, as well
as several applications in micromechanics, thin films, smart materials, and structural and topology optimization, are presented in
Ref. 关205兴.
Research in asymptotic homogenization of composites is actively continuing. And it is certain that it will bring many more
results of both fundamental and practical significance.
Acknowledgment
This work was supported by the Natural Sciences and Engineering Research Council of Canada 共NSERC兲 共for A.L.K.兲, by the
German Research Foundation Grant No. WE736/25-1 共for I.V.A.兲,
and by the Alexander von Humboldt Foundation, Institutional
Academic Co-Operation Program Grant No. 3.4-Fokoop-UKR/
1070297 共for V.V.D.兲.
References
关1兴 Willis, J. R., 1983, “The Overall Elastic Response of Composite Materials,”
Trans. ASME, J. Appl. Mech., 50, pp. 1202–1209.
关2兴 Buryachenko, V. A., 2001, “Multiparticle Effective Field and Related Methods
in Micromechanics of Composite Materials,” Appl. Mech. Rev., 54, pp. 1–47.
关3兴 Markov, K. Z., 2000, “Elementary Micromechanics of Heterogeneous Media,”
Heterogeneous Media: Micromechanics Modeling Methods and Simulations,
K. Z. Markov and L. Preziozi, eds., Birkhäuser, Boston, pp. 1–162.
030802-16 / Vol. 62, MAY 2009
关4兴 Bakhvalov, N., and Panasenko, G., 1989, Averaging Processes in Periodic
Media: Mathematical Problems in Mechanics of Composite Materials, Kluwer, Dordrecht.
关5兴 Kalamkarov, A. L., 1992, Composite and Reinforced Elements of Construction, Wiley, Chichester, NY.
关6兴 Bensoussan, A., Lions, J.-L., and Papanicolaou, G., 1978, Asymptotic Methods
in Periodic Structures, North-Holland, Amsterdam.
关7兴 Panasenko, G. P., 2005, Multi-Scale Modeling for Structures and Composites,
Springer, Berlin.
关8兴 Awrejcewicz, J., Andrianov, I. V., and Manevitch, L. I., 1998, Asymptotic
Approaches in Nonlinear Dynamics: New Trends and Applications, Springer,
Berlin.
关9兴 Dal Maso, G., 1993, An Introduction to 2 – 4 at. %-Convergence, Birkhäuser,
Boston.
关10兴 Oleynik, O. A., Shamaev, A. S., and Yosifian, G. A., 1992, Mathematical
Problems in Elasticity and Homogenization, North-Holland, Amsterdam.
关11兴 V. Berdichevsky, V. Jikov, and G. Papanicolaou, eds., 1999, Homogenization,
World Scientific, Singapore.
关12兴 Tartar, L., 1990, “H-Measure, A New Approach for Studying Homogenization,
Oscillation and Concentration Effects in Partial Differential Equations,” Proc.
- R. Soc. Edinburgh, Sect. A: Math., 115, pp. 193–230.
关13兴 Allaire, G., and Conca, C., 1998, “Boundary Layers in the Homogenization of
a Spectral Problem in Fluid-Solid Structures,” SIAM J. Math. Anal., 29共2兲,
pp. 343–379.
关14兴 Conca, C., and Lund, F., 1999, “Fourier Homogenization Method and the
Propagation of Acoustic Waves Through a Periodic Vortex Array,” SIAM J.
Appl. Math., 59共5兲, pp. 1573–1581.
关15兴 Lagzdins, A. Zh., Tamuzh, V. P., Teters, G. A., and Kregers, A. F., 1992,
Orientational Averaging in Mechanics of Solids, Longman Scientific and Technical, Essex, UK/Wiley, New York.
关16兴 Dorobantu, M., and Enquist, B., 1998, “Wavelet-Based Numerical Homogenization,” SIAM J. Math. Anal., 35共2兲, pp. 540–559.
关17兴 Pilipchuk, V. N., and Starushenko, G. A., 1997, “A Version of Non-Smooth
Transformations of Variables for One-Dimensional Elastic Systems of Periodic
Structures,” J. Appl. Math. Mech., 61共2兲, pp. 265–274.
关18兴 Milton, G. W., 2002, The Theory of Composites, Cambridge University Press,
Cambridge.
关19兴 Andrianov, I. V., Awrejcewicz, J., and Manevitch, L. I., 2004, Asymptotical
Mechanics of Thin-Walled Structures: A Handbook, Springer, Berlin.
关20兴 Berlin, A. A., Wolfson, S. A., Oshmyan, V. G., and Enikolopyan, N. S., 1990,
Principles for Polymer Composites Design, Chemistry, Moscow, in Russian.
关21兴 Manevitch, L. I., Andrianov, I. V., and Oshmyan, V. G., 2002, Mechanics of
Periodically Heterogeneous Structures, Springer, Berlin.
关22兴 Pobedrya, B. Ye., 1984, Mechanics of Composite Materials,Moscow State
University Press, Moscow, in Russian.
关23兴 Sanchez-Palencia, E., 1980, Non-Homogeneous Media and Vibration Theory,
Springer, Berlin.
关24兴 Bakhvalov, N. S., and Eglit, M. E., 1995, “The Limiting Behavior of Periodic
Media With Soft Media Inclusions,” Comput. Math. Math. Phys., 35共6兲, pp.
719–730.
关25兴 Bakhvalov, N. S., and Eglit, M. E., 2001, Homogenization of Some Multiparametric Problems 共Analysis and Application, ISAAC 2001兲, Kluwer, Dordrecht,
pp. 31–49.
关26兴 Allaire, G., 1992, “Homogenization and Two-Scale Convergence,” SIAM J.
Math. Anal., 23, pp. 1482–1518.
关27兴 Zhikov, V. V., 2000, “On an Extension of the Method of Two-Scale Convergence and Its Applications,” Mat. Sb., 191共7兲, pp. 973–1014.
关28兴 Bouchitté, G., and Bellieud, M., 2002, “Homogenization of a Soft Elastic
Material Reinforced by Fibers,” Asymptotic Anal., 32, pp. 153–183.
关29兴 Camar-Eddine, M., and Milton, G. W., 2004, “Non-Local Interactions in the
Homogenization Closure if Thermoelastic Functional,” Asymptotic Anal., 41,
pp. 259–276.
关30兴 Cherednichenko, K. D., 2006, “Two-Scale Asymptotics for Non-Local Effects
in Composites With Highly Anisotropic Fibres,” Asymptotic Anal., 49, pp.
39–59.
关31兴 Cherednichenko, K. D., Smyshlyaev, V. P., and Zhikov, V. V., 2006, “NonLocal Homogenized Limits for Composite Media With Highly Anisotropic
Periodic Fibres,” Proc. - R. Soc. Edinburgh, Sect. A: Math., 136, pp. 87–114.
关32兴 Smyshlyaev, V. P., and Cherednichenko, K. D., 2000, “On Derivation of Strain
Gradient Effects in the Overall Behaviour of Periodic Heterogeneous Media,”
J. Mech. Phys. Solids, 48, pp. 1325–1357.
关33兴 Kalamkarov, A. L., 1987, “On the Determination of Effective Characteristics
of Cellular Plates and Shells of Periodic Structure,” Mech. Solids, 22, pp.
175–179.
关34兴 Kalamkarov, A. L., 1989, “The Thermoelasticity Problem for Structurally
Nonuniform Shells of Regular Structure,” J. Appl. Mech. Tech. Phys., 30, pp.
981–988.
关35兴 Kalamkarov, A. L., and Kolpakov, A. G., 1997, Analysis, Design and Optimization of Composite Structures, Wiley, Chichester, NY.
关36兴 Kalamkarov, A. L., and Kolpakov, A. G., 1996, “On the Analysis and Design
of Fiber Reinforced Composite Shells,” Trans. ASME, J. Appl. Mech., 63共4兲,
pp. 939–945.
关37兴 Andrianov, I. V., Danishevs’kyy, V. V., and Kalamkarov, A. L., 2007,
“Asymptotic Justification of the Three-Phase Composite Model,” Compos.
Struct., 77共3兲, pp. 395–404.
关38兴 Christensen, R. M., 2005, Mechanics of Composite Materials, Dover, New
York.
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
关39兴 Christensen, R. M., and Lo, K. H., 1979, “Solutions for Effective Shear Properties in Three Phase Sphere and Cylinder Models,” J. Mech. Phys. Solids, 27,
pp. 315–330.
关40兴 Kalamkarov, A. L., and Liu, H. Q., 1998, “A New Model for the Multiphase
Fiber-Matrix Composite Materials,” Composites, Part B, 29共5兲, pp. 643–653.
关41兴 Bruggeman, D. A. G., 1935, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen, I. Dielecktrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys., 416共7兲,
pp. 636–664.
关42兴 Bruggeman, D. A. G., 1935, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen, I. Dielecktrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys., 416共8兲,
pp. 665–679.
关43兴 Kalamkarov, A. L., and Georgiades, A. V., 2002, “Modeling of Smart Composites on Account of Actuation, Thermal Conductivity and Hygroscopic Absorption,” Composites, Part B, 33共2兲, pp. 141–152.
关44兴 Jiang, C. P., Xu, Y. L., Cheung, Y. K., and Lo, S. H., 2004, “A Rigorous
Method for Doubly Periodic Cylindrical Inclusions Under Longitudinal Shear
and Its Applications,” Mech. Mater., 36, pp. 225–237.
关45兴 Rodríguez-Ramos, R., Sabina, F. J., Guinovart-Díaz, R., and Bravo-Castillero,
J., 2001, “Closed-Form Expressions for the Effective Coefficients of FibreReinforced Composite With Transversely Isotropic Constituents. I. Elastic and
Square Symmetry,” Mech. Mater., 33, pp. 223–235.
关46兴 Sabina, F. J., Bravo-Castillero, J., Rodríguez-Ramos, R., Guinovart-Díaz, R.,
and Valdiviezo-Mijangos, O. C., 2002, “Overal Behavior of Two-Dimensional
Periodic Composites,” Int. J. Solids Struct., 39, pp. 483–497.
关47兴 Guinovart-Díaz, R., Rodríguez-Ramos, R., Bravo-Castillero, J., Sabina, F. J.,
Otero-Hernández, J. A., and Maugin, G. A., 2005, “A Recursive Asymptotic
Homogenization Scheme for Multi-Fibrous Elastic Composites,” Mech.
Mater., 37, pp. 1119–1131.
关48兴 Bakhvalov, N. S., Bogachev, K. Yu., and Eglit, M. E., 1996, “Numerical
Calculation of Effective Elastic Moduli for Incompressible Porous Material,”
Mech. Compos. Mater., 32共5兲, pp. 579–587.
关49兴 Guz’, A. N., and Nemish, Yu. N., 1987, “Perturbation of Boundary Shape in
Continuum Mechanics,” Sov. Appl. Mech., 23共9兲, pp. 799–822.
关50兴 Andrianov, I. V., Bolshakov, V. I., Danishevs’kyy, V. V., and Weichert, D.,
2007, “Asymptotic Simulation of Imperfect Bonding in Periodic FibreReinforced Composite Materials Under Axial Shear,” Int. J. Mech. Sci., 49共3兲,
pp. 1344–1354.
关51兴 Andrianov, I. V., Danishevs’kyy, V. V., and Kalamkarov, A. L., 2008, “Micromechanical Analysis of Fiber-Reinforced Composites on Account of Influence
of Fiber Coatings,” Composites, Part B, 39共5兲, pp. 874–881.
关52兴 Andrianov, I. V., Danishevs’kyy, V. V., Guillet, A., and Pareige, Ph., 2005,
“Effective Properties and Micro-Mechanical Response of Filamentary Composite Wires Under Longitudinal Shear,” Eur. J. Mech. A/Solids, 24, pp. 195–
206.
关53兴 Fil’shtinskii, L. A., and Bardzokas, D., 1995, “Averaging the Electrical Properties of Fiber-Reinforced Metal Composites,” Mech. Compos. Mater., 31共4兲,
pp. 390–396.
关54兴 Fil’shtinskii, L. A., and Bardzokas, D., 1997, “Averaging the Electrical Properties of Fiber-Reinforced Metal Composites With Hollow Fibers,” Mech.
Compos. Mater., 33共3兲, pp. 269–274.
关55兴 Fil’shtinskii, L. A., and Shramko, Yu. V., 1998, “Averaging the Physical Properties of Fibrous Piezocomposites,” Mech. Compos. Mater., 34共1兲, pp. 87–93.
关56兴 Grigolyuk, E. I., and Fil’shtinskii, L. A., 1990, Perforated Plates and Shells,
Nauka, Moscow, in Russian.
关57兴 Grigolyuk, E. I., and Fil’shtinskii, L. A., 1992, Periodic Piecewise Homogeneous Elastic Structures, Nauka, Moscow, in Russian.
关58兴 Grigolyuk, E. I., and Fil’shtinskii, L. A., 1994, Regular Piecewise Homogeneous Structures With Defects, Fizmatlit, Moscow, in Russian.
关59兴 Mol’kov, V. A., and Pobedrya, B. Ye., 1985, “Effective Characteristics of a
Unidirectional Fiber Composite With a Periodic Structure,” Mech. Solids,
20共2兲, pp. 117–127.
关60兴 Berdichevsky, V. L., 1983, Variational Principles of the Continuum Mechanics, Nauka, Moscow, in Russian.
关61兴 Frankel, N. A., and Acrivos, A., 1967, “On the Viscosity of a Concentrated
Suspension of Solid Spheres,” Chem. Eng. Sci., 22, pp. 847–853.
关62兴 Lukkassen, D., Meidell, A., and Vigdergauz, Sh., 2003, “On the Elastic Deformation of Symmetric Periodic Structures,” Q. J. Mech. Appl. Math., 56共3兲,
pp. 441–454.
关63兴 Baker, G. A., and Graves-Morris, P., 1996, Padé Approximants, Cambridge
University Press, Cambridge.
关64兴 Perrins, W. T., McKenzie, D. R., and McPhedran, R. C., 1979, “Transport
Properties of Regular Arrays of Cylinders,” Proc. R. Soc. London, Ser. A, 369,
pp. 207–225.
关65兴 O’Brien, R. W., 1977, “Properties of Suspensions of Interacting Particles,”
Ph.D. thesis, University of Cambridge, Cambridge.
关66兴 Bergman, D. J., 1978, “The Dielectric Constant of a Composite Material—A
Problem in Classical Physics,” Phys. Rep., Phys. Lett., 43共9兲, pp. 377–407.
关67兴 May, S., Tokarzewski, S., Zachara, A., and Cichocki, B., 1994, “Continued
Fraction Representation for the Effective Thermal Conductivity Coefficient of
a Periodic Two Component Composite,” Int. J. Heat Mass Transfer, 37, pp.
2165–2173.
关68兴 Telega, J. J., Tokarzewski, S., and Galka, A., 2000, “Effective Conductivity of
Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants,” Acta Appl. Math., 61, pp. 295–315.
关69兴 Tokarzewski, S., 1996, “Two-Point Padé Approximants for the Expansions of
Applied Mechanics Reviews
Stieltjes Functions in Real Domain,” J. Comput. Appl. Math., 67, pp. 59–72.
关70兴 Tokarzewski, S., 1996, “N-Point Padé Approximants to Real-Valued Stieltjes
Series With Non-Zero Radii of Convergence,” J. Comput. Appl. Math., 75, pp.
259–280.
关71兴 Tokarzewski, S., and Andrianov, I. V., 2001, “Effective Coefficients for Real
Non-Linear and Fictitious Linear Temperature-Dependent Periodic Composites,” Int. J. Non-Linear Mech., 36共1兲, pp. 187–195.
关72兴 Tokarzewski, S., Blawzdziewicz, J., and Andrianov, I. V., 1994, “Two-Point
Padé Approximants for Formal Stieltjes Series,” Numer. Algorithms, 8, pp.
313–328.
关73兴 Tokarzewski, S., Blawzdziewicz, J., and Andrianov, I. V., 1994, “Effective
Conductivity for Densely Packed Highly Conducting Cylinders,” Appl. Phys.
A, 59, pp. 601–604.
关74兴 Tokarzewski, S., Galka, A., and Andrianov, I. V., 1997, “Bounds on the Effective Transport Coefficients of Two-Phase Media From Discrete Theoretical
and Experimental Data,” Comp. Assist. Mech. Eng. Sc., 4, pp. 229–241.
关75兴 Tokarzewski, S., and Telega, J. J., 1996, “S-Continued Fraction to Complex
Transport Coefficients of Two-Phase Composites,” Comp. Assist. Mech. Eng.
Sc., 3, pp. 109–119.
关76兴 Tokarzewski, S., and Telega, J. J., 1996, “Two-Point Padé Approximants to
Stieltjes Series Representations of Bulk Moduli of Regular Composites,”
Comp. Assist. Mech. Eng. Sc., 3, pp. 121–132.
关77兴 Tokarzewski, S., and Telega, J. J., 1997, “S-Continued Fraction Method for the
Investigation of a Complex Dielectric Constant of Two-Phase Composite,”
Acta Appl. Math., 49, pp. 55–83.
关78兴 Tokarzewski, S., Telega, J. J., Pindor, M., and Gilewicz, J., 2003, “A Note on
Total Bounds on Complex Transport Moduli of Parametric Two-Phase Media,”
ZAMP, 54, pp. 713–726.
关79兴 McPhedran, R. C., Poladian, L., and Milton, G. W., 1988, “Asymptotic Studies
of Closely Spaced Highly Conducting Cylinders,” Proc. R. Soc. London, Ser.
A, 415, pp. 185–196.
关80兴 Slepyan, L. I., and Yakovlev, Yu. S., 1980, Integral Transforms in the Nonstationary Problems of Mechanics, Sudostroyenie, Leningrad, in Russian.
关81兴 Andrianov, I. V., Danishevs’kyy, V. V., and Tokarzewski, S., 1996, “Two-Point
Quasifractional Approximants for Effective Conductivity of a Simple Cubic
Lattice of Spheres,” Int. J. Heat Mass Transfer, 39共11兲, pp. 2349–2352.
关82兴 Andrianov, I. V., Danishevs’kyy, V. V., and Tokarzewski, S., 2000, “Quasifractional Approximants for Effective Conductivity of a Regular Arrays of
Spheres,” Acta Appl. Math., 61, pp. 29–35.
关83兴 Andrianov, I. V., Starushenko, G. A., and Danishevs’kyy, V. V., 1999,
“Asymptotic Determination of the Thermal Conductivity of a Pile Field,” Soil
Mech. Found. Eng. 共Engl. Transl.兲, 36共1兲, pp. 31–36.
关84兴 Andrianov, I. V., Schevchenko, V. V., and Kholod, E. G., 1995, “Asymptotic
Methods in the Statics and Dynamics of Perforated Plates and Shells With
Periodic Structure,” Technische Mechanik, 15共2兲, pp. 141–157.
关85兴 Andrianov, I. V., Starushenko, G. A., and Tokarzewski, S., 1998, “Homogenization Procedure and Padé Approximants in the Theory of Composite Materials With Parallelepiped Inclusions,” Int. J. Heat Mass Transfer, 41共1兲, pp.
175–181.
关86兴 Andrianov, I. V., Starushenko, G. A., and Tokarzewski, S., 1999, “Torsion of a
Non-Homogeneous Bar With Periodic Parallelepiped Inclusions,” Technische
Mechanik, 19共1兲, pp. 5–12.
关87兴 Andrianov, I. V., Starushenko, G. A., Danishevs’kyy, V. V., and Tokarzewski,
S., 1999, “Homogenization Procedure and Padé Approximants for Effective
Heat Conductivity of Composite Materials With Cylindrical Inclusions Having
Square Cross-Sections,” Proc. R. Soc. London, Ser. A, 455, pp. 3401–3413.
关88兴 Tokarzewski, S., Andrianov, I. V., and Danishevs’kyy, V. V., 1997, “The Investigation of a Complex Conductivity of Regular Arrays of Spheres,” Theoretical Foundations of Civil Engineering: Warsaw University of Technology,
5, pp. 414–425.
关89兴 Tokarzewski, S., Andrianov, I. V., and Danishevs’kyy, V. V., 2001, “Parametric Complex Bounds on Effective Transport Coefficients of Two-Phase Media,” Theoretical Foundations of Civil Engineering: Warsaw University of
Technology, 9, pp. 431–440.
关90兴 Tokarzewski, S., Andrianov, I. V., Danishevs’kyy, V. V., and Starushenko, G.
A., 2001, “Analytical Continuation of Asymptotic Expansions of Effective
Transport Coefficients by Padé Approximants,” Nonlinear Anal. Theory, Methods Appl., 47, pp. 2283–2292.
关91兴 Sangani, A. S., and Acrivos, A., 1983, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London, Ser. A, 386, pp. 263–275.
关92兴 McPhedran, R. C., and Milton, G. W., 1981, “Bounds and Exact Theories for
the Transport Properties of Inhomogeneous Media,” Appl. Phys. 共Berlin兲, 26,
pp. 207–220.
关93兴 Meredith, R. E., and Tobias, C. W., 1960, “Resistance to Potential Flow
Through a Cubical Array of Spheres,” J. Appl. Phys., 31, pp. 1270–1273.
关94兴 Meredith, R. E., and Tobias, C. W., 1962, “Conduction in Heterogeneous Systems,” Adv. Electrochem. Electrochem. Eng., 2, pp. 15–47.
关95兴 McPhedran, R. C., and McKenzie, D. R., 1978, “The Conductivity of Lattices
of Spheres. 1. The Simple Cubic Lattice,” Proc. R. Soc. London, Ser. A, 359,
pp. 45–63.
关96兴 McKenzie, D. R., and McPhedran, R. C., 1977, “Exact Modelling of Cubic
Lattice Permittivity and Conductivity,” Nature 共London兲, 265, pp. 128–129.
关97兴 McKenzie, D. R., McPhedran, R. C., and Derrick, G. H., 1978, “The Conductivity of Lattices of Spheres. II. The Body-Centred and Face-Centred Lattices,”
Proc. R. Soc. London, Ser. A, 362, pp. 211–232.
关98兴 Drugan, W. J., and Willis, J. R., 1996, “A Micromechanics-Based Nonlocal
Constitutive Equation and Estimates of Representative Volume Element Size
MAY 2009, Vol. 62 / 030802-17
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
for Elastic Composites,” J. Mech. Phys. Solids, 44共4兲, pp. 497–524.
关99兴 Drugan, W. J., 2003, “Two Exact Micromechanics-Based Nonlocal Constitutive Equations for Random Linear Elastic Composite Materials,” J. Mech.
Phys. Solids, 51共9兲, pp. 1745–72.
关100兴 Segurado, J., and Llorca, J., 2002, “A Numerical Approximation to the Elastic Properties of Sphere-Reinforced Composites,” J. Mech. Phys. Solids,
50共10兲, pp. 2107–21.
关101兴 Molchanov, S. A., 1991, “Ideas in the Theory of Random Media,” Acta Appl.
Math., 22, pp. 139–282.
关102兴 Buryachenko, V. A., and Kushch, V. I., 2006, “Effective Transverse Elastic
Moduli of Composites at Non-Dilute Concentration of a Random Field of
Aligned Fibers,” ZAMP, 57共3兲, pp. 491–505.
关103兴 Berlyand, L., and Mityushev, V., 2001, “Generalized Clausius-Mossotti Formula for Random Composite With Circular Fibres,” J. Stat. Phys., 102共1/2兲,
pp. 115–145.
关104兴 Berlyand, L., and Mityushev, V., 2005, “Increase and Decrease of the Effective Conductivity of Two Phase Composites Due to Polydispersity,” J. Stat.
Phys., 118共3–4兲, pp. 481–509.
关105兴 Andrianov, I. V., Danishevs’kyy, V. V., and Weichert, D., 2008, “Simple
Estimation on Effective Transport Properties of a Random Composite Material With Cylindrical Fibres,” ZAMP, 59共5兲, pp. 889–903.
关106兴 Helsing, J., Milton, G. W., and Movchan, A. B., 1997, “Duality Relations,
Correspondences and Numerical Results for Planar Elastic Composites,” J.
Mech. Phys. Solids, 45共4兲, pp. 565–590.
关107兴 Kozlov, G. M., 1989, “Geometrical Aspects of Averaging,” Russ. Math. Surveys, 44共2兲, pp. 91–144.
关108兴 Hashin, Z., and Shtrikman, S., 1963, “A Variational Approach to the Theory
of Elastic Behavior of Multiphase Materials,” J. Mech. Phys. Solids, 11, pp.
127–40.
关109兴 Hashin, Z., and Shtrikman, S., 1962, “A Variational Approach to the Theory
of the Effective Magnetic Permeability of Multiphase Material,” J. Appl.
Phys., 33, pp. 3125–3131.
关110兴 Hashin, Z., 1983, “Analysis of Composite Materials–A Survey,” ASME J.
Appl. Mech., 50, pp. 481–505.
关111兴 Keller, J. B., Rubenfeld, L. A., and Molyneux, J. E., 1967, “Extremum Principles for Slow Viscous Flows With Applications to Suspensions,” J. Fluid
Mech., 30, pp. 97–125.
关112兴 Rubenfeld, L. A., and Keller, J. B., 1969, “Bounds on the Elastic Moduli of
Composite Media,” SIAM J. Appl. Math., 17, pp. 495–510.
关113兴 Torquato, S., and Rubinshtein, J., 1991, “Improved Bounds on the Effective
Conductivity of High-Contrast Suspensions,” J. Appl. Phys., 69共10兲, pp.
7118–7125.
关114兴 Torquato, S., 2002, Random Heterogeneous Materials: Microstructure and
Macroscopic Properties, Springer, New York.
关115兴 Benveniste, Y., 1995, “Correspondence Relations Among Equivalent Classes
of Heterogeneous Piezoelectric Solids Under Anti-Plane Mechanical and InPlane Electrical Fields,” J. Mech. Phys. Solids, 43, pp. 553–571.
关116兴 Gibiansky, L. V., and Torquato, S., 1995, “Rigorous Links Between the Effective Conductivity and Elastic Moduli of Fibre-Reinforced Composite Materials,” Philos. Trans. R. Soc. London, Ser. B, 343, pp. 243–278.
关117兴 Gibiansky, L. V., and Torquato, S., 1996, “Connection Between the Conductivity and Elastic Moduli of Isotropic Composites,” Proc. R. Soc. London,
Ser. A, 452, pp. 253–283.
关118兴 Sevostianov, I., and Kachanov, M., 2002, “Explicit Cross-Property Correlations for Anisotropic Two-Phase Composite Materials,” J. Mech. Phys. Solids, 50, pp. 253–282.
关119兴 Sevostianov, I., and Kachanov, M., 2003, “Correlations Between Elastic
Moduli and Thermal Conductivities of Anisotropic Short Fiber Reinforced
Thermoplastics: Theory and Experimental Verification,” Mater. Sci. Eng., A,
360, pp. 339–344.
关120兴 Sevostianov, I., Verijenko, V. E., and Kachanov, M., 2002, “Cross-Property
Correlations for Short Fiber Reinforced Composites With Damage and Their
Experimental Verification,” Composites, Part B, 33, pp. 205–213.
关121兴 Sevostianov, I., Sabina, F. J., Bravo-Castillero, J., Guinovart-Diaz, R., and
Rodriguez-Ramos, R., 2006, “Cross-Property Connections for FiberReinforced Composites With Transversely-Isotropic Constituents,” Int. J.
Fract., 142, pp. 299–306.
关122兴 Sevostianov, I., 2002, “Correlation Between Mechanical and Conductive
Properties of Porous/Microcracked Metals,” Theor Appl. Mech., 28–29, pp.
289–324.
关123兴 Sevostianov, I., 2003, “Explicit Relations Between Elastic and Conductive
Properties of a Material Containing Annular Cracks,” Philos. Trans. R. Soc.
London, Ser. A, 361, pp. 987–999.
关124兴 Sevostianov, I., and Kachanov, M., 2004, “Connection Between Elastic and
Conductive Properties of Microstructures With Hertzian Contacts,” Proc. R.
Soc. London, Ser. A, 460, pp. 1529–1534.
关125兴 Manevitch, L. I., Pavlenko, A. V., and Koblik, S. G., 1979, Asymptotic Methods in the Theory of Elasticity of Orthotropic Bodies, Viszha Shkola,
Donetzk, Kiev, in Russian.
关126兴 Andrianov, I. V., Danishevs’kyy, V. V., and Weichert, D., 2002, “Asymptotic
Determination of Effective Elastic Properties of Composite Materials With
Fibrous Square-Shaped Inclusions,” Eur. J. Mech. A/Solids, 21, pp. 1019–
1036.
关127兴 Andrianov, I. V., Danishevs’kyy, V. V., and Weichert, D., 2008, “Analytical
Study of the Load Transfer in Fibre-Reinforced 2D Composite Materials,”
Int. J. Solids Struct., 45, pp. 1217–1243.
关128兴 Duvaut, G., 1976, “Analyse fonctionnelle et méchanique des milieux contin-
030802-18 / Vol. 62, MAY 2009
关129兴
关130兴
关131兴
关132兴
关133兴
关134兴
关135兴
关136兴
关137兴
关138兴
关139兴
关140兴
关141兴
关142兴
关143兴
关144兴
关145兴
关146兴
关147兴
关148兴
关149兴
关150兴
关151兴
关152兴
关153兴
关154兴
关155兴
关156兴
关157兴
关158兴
ues,” Proceedings of the 14th IUTAM Congress, North-Holland, Amsterdam,
pp. 119–132.
Duvaut, G., 1977, Comportement macroscopique d’une plaque perforée périodiquement 共Lecture Notes in Mathematics Vol. 594兲, Springer, Berlin, pp.
131–145.
Andrianov, I. V., Lesnichaya, V. A., and Manevitch, L. I., 1985, Homogenization Methods in the Statics and Dynamics of Ribbed Shells, Nauka, Moscow, in Russian.
Caillerie, D., 1981, “Equations de la diffusion stationnaire dans un domaine
comportant une distribution périodique d’inclusions aplaties de grande conductivité,” C. R. Acad. Sci., Ser. I: Math., 292共1兲, pp. 115–118.
Caillerie, D., 1984, “Thin Elastic and Periodic Plates,” Math. Methods Appl.
Sci., 6, pp. 159–191.
Kohn, R. V., and Vogelius, M., 1984, “A New Model for Thin Plates With
Rapidly Varying Thickness,” Int. J. Solids Struct., 20, pp. 333–350.
Kohn, R. V., and Vogelius, M., 1985, “A New Model for Thin Plates With
Rapidly Varying Thickness, II: A Convergence Proof,” Q. J. Mech. Appl.
Math., 43, pp. 1–22.
Kohn, R. V., and Vogelius, M., 1986, “A New Model for thin Plates With
Rapidly Varying Thickness, III: Comparison of Different Scalings,” Q. J.
Mech. Appl. Math., 44, pp. 35–48.
Kalamkarov, A. L., 1988, “Geometrically Nonlinear Problem of a thin Composite Layer With Undulating Surfaces of Periodic Structure,” Mech. Solids,
23共5兲, pp. 40–45.
Kalamkarov, A. L., Duvaut, G., and Léné, F., 2002, “A New Asymptotic
Model of Flexible Composite Shells of a Regular Structure,” Int. J. Eng. Sci.,
40, pp. 333–343.
Kalamkarov, A. L., and Kolpakov, A. G., 2001, “A New Asymptotic Model
for a Composite Piezoelastic Plate,” Int. J. Solids Struct., 38共34–35兲, pp.
6027–6044.
Kalamkarov, A. L., and Georgiades, A. V., 2002, “Micromechanical Modeling of Smart Composite Structures,” Smart Mater. Struct., 11, pp. 423–434.
Kalamkarov, A. L., and Georgiades, A. V., 2004, “Asymptotic Homogenization Models for Smart Composite Plates With Rapidly Varying Thickness:
Part I–Theory,” Int. J. Multiscale Comp. Eng., 2共1兲, pp. 133–148.
Kalamkarov, A. L., Saha, G., and Georgiades, A. V., 2007, “General Micromechanical Modeling of Smart Composite Shells With Application to Smart
Honeycomb Sandwich Structures,” Compos. Struct., 79共1兲, pp. 18–33.
Kalamkarov, A. L., Georgiades, A. V., Challagulla, K., and Saha, G., 2006,
“Micromechanics of Smart Composite Plates With Periodically Embedded
Actuators and Rapidly Varying Thickness,” Journal of Thermoplastic Composite Materials, 19共3兲, pp. 251–276.
Georgiades, A. V., Kalamkarov, A. L., and Challagulla, K., 2006,
“Asymptotic Homogenization Model for Generally Orthotropic Reinforcing
Networks in Smart Composite Plates,” Smart Mater. Struct., 15, pp. 1197–
1210.
Challagulla, K., Georgiades, A. V., and Kalamkarov, A. L., 2007,
“Asymptotic Homogenization Modelling of Thin Composite Network Structures,” Compos. Struct., 79共3兲, pp. 432–444.
Challagulla, K., Georgiades, A. V., and Kalamkarov, A. L., 2007,
“Asymptotic Homogenization Model for Three-Dimensional Network Reinforced Composite Structures,” J. Mech. Mater. Struct., 2共4兲, pp. 613–632.
Challagulla, K., Georgiades, A. V., Saha, G., and Kalamkarov, A. L., 2008,
“Micromechanical Analysis of Grid-Reinforced Thin Composite Generally
Orthotropic Shells,” Composites, Part B, 39共4兲, pp. 627–644.
Saha, G., Kalamkarov, A. L., and Georgiades, A. V., 2007, “Asymptotic
Homogenization Modeling and Analysis of Effective Properties of Smart
Composite Reinforced and Sandwich Shells,” Int. J. Mech. Sci., 49共2兲, pp.
138–150.
Saha, G., Kalamkarov, A. L., and Georgiades, A. V., 2007, “Effective Elastic
Characteristics of Honeycomb Sandwich Composite Shells Made of Generally Orthotropic Materials,” Composites, Part A, 38共6兲, pp. 1533–1546.
Kalamkarov, A. L., Veedu, V., and Ghasemi-Nejhad, M. N., 2005, “Mechanical Properties Modeling of Carbon Single-Walled Nanotubes: An Asymptotic
Homogenization Method,” J. Comput. Theor. Nanosci., 2共1兲, pp. 124–131.
Kalamkarov, A. L., Georgiades, A. V., Rokkam, S., Veedu, V., and GhasemiNejhad, M. N., 2006, “Analytical and Numerical Techniques to Predict Carbon Nanotubes Properties,” Int. J. Solids Struct., 43共22–23兲, pp. 6832–6854.
Reddy, J. N., 1997, Mechanics of Laminated Composite Plates: Theory and
Analysis, CRC, Boca Raton, FL.
Georgiades, A. V., and Kalamkarov, A. L., 2004, “Asymptotic Homogenization Models for Smart Composite Plates With Rapidly Varying Thickness:
Part II–Applications,” Int. J. Multiscale Comp. Eng., 2共1兲, pp. 149–172.
Guz’, A. N., and Kokhanenko, Yu. V., 1995, “Edge Effects in Composites,”
Int. Appl. Mech., 31共3兲, pp. 165–181.
Dumontet, H., 1986, “Study of a Boundary Layer Problem in Elastic Composite Materials,” Model. Math. Anal. Numer., 20, pp. 265–286.
Kalamkarov, A. L., Kudryavtsev, B. A., and Bardzokas, D., 1991, “New
Generalized Integral Transforms in Axisymmetric Boundary-Value Problems
in the Mechanics of Composites,” Mech. Compos. Mater., 27共6兲, pp. 648–
657.
Allaire, G., and Amar, M., 1999, “Boundary Layer Tails in Periodic Homogenization,” COCV, 4, pp. 209–243.
Mishuris, G., and Öchsner, A., 2005, “Edge Effects Connected With Thin
Interfaces in Composite Materials,” Compos. Struct., 68, pp. 409–417.
Versieux, H. M., and Sarkis, M., 2006, “Numerical Boundary Corrector for
Elliptic Equations With Rapidly Oscillating Periodic Coefficients,” Commun.
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Numer. Methods Eng., 22, pp. 577–589.
关159兴 Gorbachev, V. I., and Pobedrya, B. Ye., 1985, “On Some Fracture Criteria for
Composite Materials,” Izv. Akad. Nauk Arm. SSR, Ser. Mekh., 38, pp. 30–
37.
关160兴 Pobedrya, B. Ye., 1988, “About Failure Criteria of StructurallyNonhomogeneous Materials,” Plastichnost i. Razrushenie Tverdih Tel,
Nauka, Moscow, pp. 170–175, in Russian.
关161兴 Pobedrya, B. Ye., and Gorbachev, V. I., 1988, “Strength Criteria for Layered
and Fibrous Composites,” Problemi Mashinostroyeniya i Avtomatizazii, Vol.
21, Mashinostroyenie, Moscow, pp. 65–68.
关162兴 Muravleva, L. V., 2007, “On Some Composite Failure Criteria Based on the
Averaging Method,” Mech. Solids, 42共3兲, pp. 471–480.
关163兴 Pobedrya, B. Ye., and Gorbachev, V. I., 1984, “Stress and Strain Concentration in Composite Materials,” Mech. Compos. Mater., 20共2兲, pp. 141–148.
关164兴 Luo, D., and Takezono, S., 2003, “The Numerical Simulation of the Mechanical Behavior of CFCC With Matrix Anisotropic Damage by Homogenization Method,” Mater. Sci. Res. Int., 9共1兲, pp. 61–68.
关165兴 Dascalu, C., Bilbie, G., and Agiasofitou, E. K., 2008, “Damage and Size
Effects in Elastic Solids: A Homogenization Approach,” Int. J. Solids Struct.,
45, pp. 409–430.
关166兴 Devries, F., 1991, “A Fatigue Damage Model by Fibres Breaks in Unidirectional Composites,” Composite Material Technology, Proceedings of the 14th
Annual Energy-Sources Technology Conference and Exhibition, Houston,
TX, ASME, New York, pp. 137–144.
关167兴 Devries, F., Dumontet, H., Duvaut, G., and Léné, F., 1989, “Homogenization
and Damage for Composite Structures,” Int. J. Numer. Methods Eng., 27, pp.
285–298.
关168兴 Leguillon, D., and Sanchez-Palencia, E., 1982, “On the Behaviour of a
Cracked Elastic Body With or Without Friction,” J. Mec. Theor. Appl., 1共2兲,
pp. 195–209.
关169兴 Terada, K., and Kikuchi, N., 2001, “A Class of General Algorithms for Multiscale Analysis of Heterogeneous Media,” Comput. Methods Appl. Mech.
Eng., 190, pp. 5427–5464.
关170兴 Ghosh, S., Bai, J., and Raghavan, P., 2007, “Concurrent Multi-Level Model
for Damage Evolution in Microstructurally Debonding Composites,” Mech.
Mater., 39共3兲, pp. 241–266.
关171兴 Ghosh, S., Lee, K., and Raghavan, P., 2001, “A Multi-Level Computational
Model for Multi-Scale Damage Analysis in Composite and Porous Materials,” Int. J. Solids Struct., 38, pp. 2335–2385.
关172兴 Ghosh, S., Ling, Y., Majumdar, B., and Kim, R., 2000, “Interfacial Debonding Analysis in Multiple Fiber Reinforced Composites,” Mech. Mater., 32,
pp. 561–591.
关173兴 Léné, F., 1986, “Damage Constitutive Relations for Composite Materials,”
Eng. Fract. Mech., 25共5–6兲, pp. 713–728.
关174兴 Raghavan, P., and Ghosh, S., 2005, “A Continuum Damage Mechanics
Model for Unidirectional Composites Undergoing Interfacial Debonding,”
Mech. Mater., 37, pp. 955–979.
关175兴 Allen, D. H., 2001, “Homogenization Principles and Their Application to
Continuum Damage Mechanics,” Compos. Sci. Technol., 61, pp. 2223–2230.
关176兴 Allen, D. H., Jones, R. H., and Boyd, J. G., 1994, “Micromechanical Analysis
of a Continuous Fiber Metal Matrix Composite Including the Effects of Matrix Viscoplasticity and Evolving Damage,” J. Mech. Phys. Solids, 42共3兲, pp.
505–529.
关177兴 Costanzo, F., Boyd, J. G., and Allen, D. H., 1996, “Micromechanics and
Homogenization of Inelastic Composite Materials With Growing Cracks,” J.
Mech. Phys. Solids, 44共3兲, pp. 333–370.
关178兴 Chaboche, J. L., Kruch, S., and Pottier, T., 1998, “Micromechanics Versus
Macromechanics: A Combined Approach for Metal Matrix Composite Constitutive Modeling,” Eur. J. Mech. A/Solids, 17, pp. 885–908.
关179兴 Fish, J., Yu, Q., and Shek, K., 1999, “Computational Damage Mechanics for
Composite Materials Based on Mathematical Homogenization,” Int. J. Numer. Methods Eng., 45, pp. 1657–1679.
关180兴 Jain, J. R., and Ghosh, S., 2008, “Homogenization Based 3D Continuum
Damage Mechanics Model for Composites Undergoing Microstructural Debonding,” ASME J. Appl. Mech., 75共3兲, p. 031011.
关181兴 Kruch, S., Chaboche, J. L., and Pottier, T., 1996, “Two-Scale Viscoplastic
and Damage Analysis of Metal Matrix Composite,” Damage and Interfacial
Debonding in Composites, G. Z. Voyiadjis and D. H. Allen, eds., Elsevier,
Amsterdam, pp. 45–56.
关182兴 Matzenmiller, A., Lubliner, J., and Taylor, R. L., 1995, “A Constitutive
Model for Anisotropic Damage in Fiber-Composites,” Mech. Mater., 20, pp.
Applied Mechanics Reviews
125–152.
关183兴 Voyiadjis, G. Z., and Kattan, P. I., 1993, “Micromechanical Characterization
of Damage-Plasticity in Metal Matrix Composites,” Damage in Composite
Materials, G. Z. Voyiadjis, ed., Elsevier, Amsterdam, pp. 67–102.
关184兴 Voyiadjis, G. Z., and Park, T. A., 1992, “A Plasticity-Damage Theory for
Large Deformation of Solids–I: Theoretical Foundation,” Int. J. Eng. Sci.,
30共9兲, pp. 1089–1106.
关185兴 Voyiadjis, G. Z., and Park, T. A., 1996, “Elasto-Plastic Stress and Strain
Concentration Tensors for Damage Fibrous Composites,” Damage and Interfacial Debonding in Composites, G. Z. Voyiadjis and D. H. Allen, eds.,
Elsevier, Amsterdam, pp. 81–106.
关186兴 Oskay, C., and Fish, J., 2004, “Fatigue Life Prediction Using 2-Scale Temporal Asymptotic Homogenization,” Int. J. Numer. Methods Eng., 61共3兲, pp.
329–359.
关187兴 Kouznetsova, V. G., Brekelmans, W. A. M., and Baaijens, F. P. T., 2001, “An
Approach to Micro-Macro Modeling of Heterogeneous Materials,” Comput.
Mech., 27, pp. 37–48.
关188兴 Choi, J., and Tamma, K. K., 2001, “Woven Fabric Composites. Part 1: Prediction of Homogenized Elastic Properties and Micromechanical Damage
Analysis,” Int. J. Numer. Methods Eng., 50, pp. 2285–2298.
关189兴 Gałka, A., Telega, J. J., and Tokarzewski, S., 1997, “Nonlinear Transport
Equation and Macroscopic Properties of Microheterogeneous Media,” Arch.
Mech., 49共2兲, pp. 293–319.
关190兴 Andrianov, I. V., Awrejcewicz, J., and Diskovsky, A. A., 2006, “Homogenization of Quasi-Periodic Structures,” ASME J. Vibr. Acoust., 128共4兲, pp.
532–534.
关191兴 Kalamkarov, A. L., Hassan, E., Georgiades, A. V., and Savi, M. A., 2009,
“Asymptotic Homogenization Model for 3D Grid-Reinforced Composite
Structures With Generally Orthotropic Reinforcements,” Compos. Struct.,
89共2兲, pp. 186–196.
关192兴 Saha, G., and Kalamkarov, A. L., 2009, “Micromechanical Thermoelastic
Model for Sandwich Composite Shells Made of Generally Orthotropic Materials,” Journal of Sandwich Structures and Materials, 11共1兲, pp. 27–56.
关193兴 Kolpakov, A. G., 2004, Stressed Composite Structures: Homogenized Models
for Thin-Walled Nonhomogeneous Structures With Initial Stresses, Springer,
Berlin.
关194兴 Kolpakov, A. G., 2007, Composite Materials and Structural Elements With
Initial Stresses, Siberian Branch of Russian Academy of Sciences, Novosibirsk, in Russian.
关195兴 Pernin, J. N., 1999, “Diffusion in Composite Solid: Threshold Phenomenon
and Homogenization,” Int. J. Eng. Sci., 37, pp. 1597–1610.
关196兴 Pernin, J. N., and Jacquet, E., 2001, “Elasticity in Highly Heterogeneous
Composite Medium: Threshold Phenomenon and Homogenization,” Int. J.
Eng. Sci., 39, pp. 755–798.
关197兴 Pernin, J. N., and Jacquet, E., 2001, “Elasticity and Viscoelasticity in Highly
Heterogeneous Composite Medium: Threshold Phenomenon and Homogenization,” Int. J. Eng. Sci., 39, pp. 1655–1689.
关198兴 Balagurov, B. Ya., 2001, “Effective Electrical Characteristics of a TwoDimensional Three-Component Doubly-Periodic System With Circular Inclusions,” J. Exp. Theor. Phys., 92共1兲, pp. 123–134.
关199兴 Balagurov, B. Ya., 2001, “Partial Moments of Electric Field Strength in the
Problem of Conduction of Binary Composites,” J. Exp. Theor. Phys., 93共4兲,
pp. 824–832.
关200兴 Balagurov, B. Ya., and Kashin, V. A., 2000, “Conductivity of a TwoDimensional System With a Periodic Distribution of Circular Inclusions,” J.
Exp. Theor. Phys., 90共5兲, pp. 850–860.
关201兴 Balagurov, B. Ya., and Kashin, V. A., 2005, “Analytic Properties of the
Effective Dielectric Constant of a Two-Dimensional Rayleigh Model,” J.
Exp. Theor. Phys., 100共4兲, pp. 731–741.
关202兴 Mityushev, V. V., Pesetskaya, E. V., and Rogosin, S. V., 2007, Analytical
Methods for Heat Conduction in Composites and Porous Media, WileyVCH, Amsterdam.
关203兴 Mityushev, V., Obnosov, Yu., Pesetskaya, E., and Rogosin, S., 2008, “Analytical Methods for Heat Conduction in Composites,” Mathematical Modelling and Analysis, 13共1兲, pp. 67–78.
关204兴 Parnell, W. J., and Abrahams, I. D., 2008, “A New Integral Equation Approach to Elastodynamic Homogenization,” Proc. R. Soc. London, Ser. A,
464, pp. 1461–1482.
关205兴 2007, Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials, P. Ponte Castañeda, J. J. Telega, and B.
Gambin, eds., Springer, Berlin.
MAY 2009, Vol. 62 / 030802-19
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Alexander L. Kalamkarov obtained his Masters (1975) and Ph.D. (1979) degrees from the Moscow
Lomonosov State University (USSR) and the Doctor of Sciences degree (1990) from the Academy of Sciences of the USSR. Since 1993 he is a Professor at the Department of Mechanical Engineering at the
Dalhousie University in Halifax, Nova Scotia, Canada. His academic career spans more than 30 years in
research 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 250
papers in the refereed journals and conference proceedings, and he also holds two patents in the area of
smart materials. He has reported his research results at numerous international conferences and has
presented six invited keynote lectures. Dr. Kalamkarov is a Member of several editorial and advisory boards
in the area of composite materials and smart structures. He is a Fellow of the ASME and a Fellow of the
CSME.
Igor V. Andrianov obtained his Masters of Applied Mechanics degree (1971) and Ph.D. degree in Structural Mechanics (1975) from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor of
Sciences 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 Dnepropetrovsk
Civil Engineering Institute. Currently he is a Research Scientist at the Rheinisch-Westfälische Technische
Hochschule (Technical University of Aachen, Germany). Dr. Andrianov is the author or co-author of 11
books and over 250 papers in refereed journals and conference proceedings. He has presented papers at
numerous international conferences and has supervised 21 Ph.D. theses. His research interests belong to
mechanics of solids, nonlinear dynamics, and asymptotic methods.
Vladyslav V. Danishevs’kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciences
degree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering and
Architecture, Dnipropetrovsk, Ukraine. He is an Associate Professor at this State Academy. He has authored
1 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 among
young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research
Fellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of the
Alexander von Humboldt Foundation (2007). He has conducted research at the Institute of General Mechanics in the Technical University of Aachen, Germany (2001–2002 and 2006). He was a NATO Research
Officer at the University of Rouen, France (2003–2004). His research interests belong to the mechanics of
heterogeneous materials and structures, asymptotic methods, and nonlinear dynamics.
030802-20 / Vol. 62, MAY 2009
Transactions of the ASME
Downloaded 01 Apr 2009 to 137.226.184.67. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm