NASATechnical
Memorandum
106344
................
_r
?
Theimoelastic Response of Metal Matrix Composites With Large,Diameter Fibers Subjected to Therma! Gradien!s
Jacob Aboudi ...................
and MarekJerzy
Pindera
University of Virginia Charlottesville, Virginia and Steven M. Arnold ....... Lewis Research center Cleveland, Ohio
October 1993

.............
.............................................
(NASATMI06344) RESPONSE
__i
I%IASA
OF
N94I4823
THERMOELASTIC METAL
WITH LARGEDIAMETER SUBJECTED TO THERMAL
MATRIX
COMPOSITES
FIBERS GRADIENTS G3/24
Unclas
i,
0189400
;
WITH
THERMOELASTIC LARGEDIAMETER
RESPONSE FIBERS
OF METAL SUBJECTED
MATRIX COMPOSITES TO THERMAL GRADIENTS
by Jacob
Aboudi
1 and
MarekJerzy
Pindera
University of Virginia Charlottesville, Virginia Steven National
M. Arnold
Aeronautics and Space Administration Lewis Research Center Cleveland,
Ohio
ABSTRACT
A new micromechanical composites utilize
subjected
classical
field quantities,
theory
to thermal
homogenization in the present
the macrostructure
is presented
gradients. schemes
approach
approach
In contrast in the course
the actual
of the composite.
homogenization
for the response
Examples
in predicting
to existing
microstructural
details
of thinwalled
approach number produce mating
socalled
of decoupling
stress
The results
and macromechanical
estimates
is the usefulness of thermal
matrix
illustrate
of the present
with
that the classical
in the presence
field quantities,
gradients
composites
and temperature
of the microscopic
with
of the classical
include composites with or nonuniformly spaced,
analyses
of the composite,
for macroscopic
the local fluctuations
in the presence details
composites.
finite dimensions
Also demonstrated
distributions
microstructural
fibers,
conservative
and overestimating
of the composite.
gradient
micromechanical
of largediameter excessively
functionally
coupled
limitations
metal
that
and macroscopic
are explicitly
that illustrate
matrix
theories
microscopic
largediameter fibers when subjected to thermal gradients. These examples a finite number of fibers in the thickness direction that may be uniformly thus admitting
metal
micromechanical
of calculating
are offered
the response
of heterogeneous
while
quantities approach
of a finite
gradient
both underesti
in different
in generating
by appropriately
may
regions favorable
tailoring
the
internal
of the composite.
NOMENCLATURE
ui(S),
Ti(S)

displacement
and traction
co, _o

average values of strains and homogeneous boundary conditions
E/j, a;j

average
values
JVisiting Professor, Faculty of Engineering,
of strains
components
stresses
and stresses
TeIAviv University, RamatAviv
on the surface in
a
composite
in a representative
69978, Israel
S of a composite subjected
volume
element
to

strains
Cqkl

elements
p,q,r

indices
used to identify
the cell (p,q,r)
a,_,y

indices
used to identify
the subcell
d_a ), h fs, I T

dimensions

volume

local
k!a_q)

coefficients
of heat conductivity
T(al3v)

temperature
field in the subcell

temperature
at the center

coefficients
in the temperature

components
of the heat flux vector

average
_(a) X1
_ffs) _(_,) ,X2
q_a_T)
L(_av)
l (Lrh,n)
,X3
and stresses
at the point Xk in a composite
of the effective
stiffness
of the subceU
of the subcell
subcell
tensor
(al3Y)
(al3T) in the pth
coordinates
values
of
of the material
of the subcell
the
subceU
heat
interfacial

displacement

x 1 displacement

coefficients
components

associated
the subcell
displacement
coefficients
associated
the subcell
displacement
coefficients expansion
associated of the subceU
heat
with the linear
(ct[3T)
(a[_t) component
q_at_,)
when
of l,m,n
fluxes
(a[3"/)
at the center
displacement
coefficients
flux
the subcell
for other values
in the subcell
component
associated
of the subcell
within
in the subcell
surface
of subcell
(al3T)
(a[3T)
expansion

integrals
in the subcell
(a[3y)
heat fluxes

unit cell
(al]q)
l = m = n = O; higherorder

of a composite
of the subcell
(a[3 T)
terms
in the secondorder
expansion
terms
in the firstorder
expansion
of
terms
in the firstorder
expansion
of
ut a_'_)
with the linear u_ _th') with the linear u_ cxt3_') with
the quadratic
displacement
2
ut a_t)
term xt a)z in the secondorder
coefficients
associated
expansion
of the subcell
coefficients
,
with
x_ )2 in the
secondorder
term
X_"Y)2in the
secondorder
ut _')
the quadratic
displacement
ut al3't)
local strain
components
in the subcell
((x[_T)
local
components
in the subcell
(a_y)
stress
of the stiffness
elements
of
the
(products
of the stiffness
average
values
tensor
thermal
of the material
tensor tensor
of
the
of the
in the subcell
material
and the thermal
subcell
l = m = n = 0; higherorder
@Z)o,o)
term
of the subcell
elements
_j(l,m,n )
the quadratic
displacement
associated
expansion
S(ai_.t)
with
stress
in
(o_[_y)
the
subcell
expansion
coefficients)
components
stress components
for other
(otl3T)
_!_13f) values
when
of l,m,n
surface
integrals
of the subcell
interracial
stresses
_(a) = +__d_)/2 _t_ I_') at xl
surface
integrals
of the subcell
interfacial
stresses
t_
surface
integrals
of the subcell
interracial
stresses
_[_'t)
(_) _) at x2
= +hfJ2
at x3 (_') = +I_,/2,
1.0 INTRODUCTION The
past
thirty
site materials. aerospace fied
The
years
have
applications
structural
into different
categories
of the matrix
intermetallic
matrix
based
phase
on the
type
cation
into
oriented)
shortfiber,
material
sporting
For example,
used
or continuous,
in the development
and
Historically,
on the geometry
and ceramic
oriented
growth
(CMC)
composites
oriented
materials
and distribution matrix
to contain
accessories
composite
polymeric
matrix
or random
recreational
composites,
(PMC), are
have
metal four
phase.
thereby and
to advanced been
classi
of the reinforcement
the reinforcement or random,
and use of compo
classes The
providing
unidirectional
matrix
phase (MMC),
of compo
reinforcement further
classifi
(continuous
and
composites.
Typically, buted
based
of matrix
can be finitelength
from
components.
phase.
(IMC),
tremendous
range
and engine
and the type
sites
seen
the reinforcement
in a statistically
or macroscopically
is macroscopically
micromechanical
phase
approaches
homogeneous have
been
in the various uniform
classes fashion
with properties developed
3
during
of composite such
that
materials
is distri
the
resulting
twophase
that do not vary
spatially.
Numerous
the :past thirty
year s, as discussed
by
Aboudi such
(1991),
to calculate
composites
given
micromechanical with
properties
that properties
stress
and strain
niques
material
which
applicability tures
imposes with
1969).
inclusion
fine such
tively
largediameter
and fibers
to potential
coupling
between
1 for situations
following
a new concept
certain
This
idea
who
have
class
of composites.
variable
involving
at each point
within
boundary
conditions
to such
of stress
to the overall
to the thickness
be admitted.
in composites
of a single
ply,
of
mierostmc
composite,
and
Composites smalldiameter
containing
rela
such as B/A1 or SiC/Ti,
approach
based
and remains
and the global and
with
of local
the range
fine
that can
is suspect,
gradients,
limits
of the
microscopic
treatment
principle
with very
reinforced
at
on the assumption
as the clearly
deformation
dimensions
gradients
Alternatively,
the effective
and
is based
to composites
for instance.
thermal
(RVE)
by evaluating
states
tech
response.
on the concept
to be established This
will be discussed
of due
is illustrated
in more
detail
in
in the
section.
Recently, to achieve
approach
the microstructure
either
analyses
composites
homogenization
specified
decoupling
and global
of the traditional
conditions,
analyses
unidirectional
with respect
of the
to in the literature
of deformation
carbon,
and reliability
Such
of local
with respect
include
homogeneous
referred
averages
micromechanical
element
only on the local a priori.
decoupling
on the severity
and the classical
based
effective
volume
wellestablished
volume
to apply
The
elastic
tractions.
the local and global
sometimes
dimensions)
as graphite
the applicability
representative
homogenization
microstructures
fibers
Figure
The
boundary
the various
to be known
of a principle
constraints
an RVE
point
of the classical
(i.e.,
surface
to decoupling
are assumed
(Malvern,
or prescribed
ancl the ability
at a given
of the applicability
homogeneous
analyses.
The of the
by a set of effective
that relate
of
phases.
microstmcture
structural
parameters
socalled
of a definable
amounts
characterized
complicated
properties
of the individual
the heterogeneous
continuum in more
or macroscopic)
properties
to replace
in applying
continuum
properties
action
under
effective
and
as the constitutive
assumption
This
that point
it possible
displacements
the heterogeneous
called
distribution
can be used
components
is the existence
(often
homogeneous
are defined
central
an element.
makes
an equivalent
of surface
average
geometry,
subsequently
elastic
The
the
analysis
composite
in terms
the
has
required
been
coined
spacings
properties,
sizes
traditional
methods
involving response
pursued the
term The
and
of the internal
characteristics
to given
vigorously
by Japanese
functionally
gradient
idea
between
tailoring
involves
individual
shapes. of changing
Such
grading
inclusions,
offers
the compliance
4
(cf. Yamanouchi to describe
as by using
a number
of composite
of the composite
parameters
the properties
as well
an approach
input
researchers materials
spatially
microstructure
this
taken
root.
et al.,
1990)
newly
emerging
of the material
by using
inclusions
different
of advantages
smactural
has
elements
with over
the
by varying
more the
lamination Grading
sequence or tailoring
ponent
allows
the final level
or dropping the
design
severe
aerospace
engines
tendency
to circuit
the thickness
tailoring
the microstructure
reduced,
if not eliminated,
lead
These
strategies
for the
bal
analysis)
axis
configurations
for
the
cannot
couples
2. The
fibers
can admit
to the current
functionally
in the x2x3
composite
is reinforced
(Figure
2a), or both
axis
reinforcing
gradient fibers
different
microstructural
(FG) can
by periodic (Figure
direction,
be either
thermoelastic configurations
2b).
properties. whose
science,
be
state,
couple
implicitly suffer
approach
is
the glowith
in the presence plate
with a fingradient,
in the direction
of the x2
of the xl
hereafter
axis,
adjacent Further,
the model
to the applied
with
to a temperature
between
or finitelength.
the
theoretical
of composites
is a composite
In the direction
currently
from
of the material
of fibers
and
assume
analytical
behavior
hetero
approach, schemes
a new
and subjected
spacing
the
by the foregoing
reinforcement)
herein
of these
computational
As implied
of local
Consequently,
response
can
microstructures
of appropriate
which
arrays
of
judiciously
moment
tailored
is those
to analyze
the fiber
continuous
bending
micromechanical
largediameter
plane
by
materials
microsmacture
considered
is the
irrespective
homogenization
As a result,
need
graded,
the problem
to infinity
the functionally The
confidence.
the heterogeneous
to respond
classical
of the principle
with
and
analysis.
that
gra
or airfoil)
However,
that explicitly
traditional
materials,
validity
be used
H extending
section,
liner
with
by the lack
of the
and
of warping.
composites
the global
applications aircraft
be present
the thermal
materials
for
of the temperature
will
the severity
graded with
graded
In particular,
vary.
tailored
and
in order
or the x3
called
functionally
gradients.
see Figure
to the material
in advanced
as a combustor
of processing,
of applicability
(e.g.
ite thickness
into
candidates
is Considered.
from
areas
ideal
tendency
are handicapped
material
microstructures
of thermal
considerations
process
consequence
material,
be derived
in the following
explicitly
This
material
of functionally
limits
upon
and (that
tailored
instance. com
material
structures
(such
decreasing
however,
of the
of an RVE
shortcomings
component
in the
are
a direct
direction.
that may
activities
the
in analyzing
presented
design
for
or a structural
and structural
of possible
thermal
For instance,
consequently
response
on
from
of a heterogeneous
benefits
elaborated
existence
boards.
activities,
discussion
used
geometry,
material
structural
the number
or heterogeneous
mierostrueture
further
the entire
ranging
outofplane
to increased
materials.
brief
of a composite
microstructures
of a structural
in the
potential
tailored
gradients,
a homogeneous
geneous
increasing
with
whether
have
thereby
thermal
to bend
The
crosssectional
both the material
This brings
materials
involving
across
integrate
product.
sense,
the
applications.
Composite
dient
to reduce
microstructure
to truly
and final
in the purest
specific
internal
the designer
design
plies
each
admits
thermal
arrays
may
array
of
a variety
of
gradient
can
be
investigated(including unidirectional and bidirectional arrayswith uniform spacings
in the FG
direction,
as well
graded
materials
consisting
functionally ously
changing
tions,
metallicrich
tures whereas
properties
those
In addition,
subjected
regions
are placed
regions
exposed
schemes
to thermal
ness direction.
gradients,
Hence,
direction
gradients
can therefore
This inplane
force
directional
is the first
and
SiC
moment
fibers,
number
of uniformly
results
are normalized
effective tion value
and
problem
quantities
of interest Finally,
spacing
advantages moment
2.0
uniformly
compared
examples
in the FG of using
resultants
sites
_eralizafions,
in the
illustrating direction
effective
obtained
the effect
graded
tempera
of the various
largediameter
fibers
in the thick
are required
in the presence
section
thickness
in the
of thermal
fiber
volume
uni
of the
fraction.
These
by f'trst generating
using
a suitable
in the
thermal
is considered
the corresponding quadratically spaced
the
homogenizaboundary
a bidirectionally
directions
with regard
with
as a function
obtained
to the uniformly
by presenting
reinforced
gradient,
Similarly,
using
composites
to lower
row of fibers
properties
of linearly,
are compared
OF THE
CLASSICAL
micromechanical
use
of simple
differential
ing techniques
with those
finite
of fibers
composite.
applica
approach.
predictions rows
continu
In such
(or plies)
for a fixed
in both inplane
produce
reinforced
and
the various
homogenized
confi
and cubically
varying
configuration,
to reducing
and
inplane
force
the and
are discussed.
various
include
these
fibers
functionally
APPLICABILITY The
approach
homogeneous
spaced
with
in the FG direction
employing
with
fibers
temperature
of the individual
of an equivalent
with
fiber
properties
exposed
in the APPLICATIONS
imposed
by the continuum
subsequently
composite
guration.
fibers
surface
to be valid
using the present
the
gradients.
of a single
schemes
in a composite
by
that
thermal
of composites
of how many
to be addressed
produced
thermoelastic
scheme
answered
phases
can include
to test the applicability
ply consists
question
resultants
spaced
each
configurations
fiber
are ceramicrich.
it possible
homogenization
be finally
of the
the response
i.e., when
ceramic
temperatures
makes
predicting
for classical
question
to elevated
These
severe
in the vicinity
the fundamental
thickness
and
involving
formulation
when
arrays).
of metallic
for applications
the present
homogenization
as multiphase
or variable
and
Reuss
matrix
given
by Aboudi
(1991).
these
wellestablished
or numerical
As stated
the
existence
6
effective
various
section, of
arrays
and
has
ability
gen
boundor fibers
recently
assumption the
their
models,
of inclusions
approaches
RVE
of compoand
cylinder
the central an
properties schemes
concentric
of periodic
of these
in the preceding
SCHEMES
selfconsistent
method,
analyses
A discussion
is
to calculate
hypotheses,
the MoriTanaka
phase.
techniques
used
and Voigt
schemes,
approximate
surrounding
approaches
HOMOGENIZATION
been
in applying to
apply
homogeneousboundaryconditionsto suchanelement.Thesehomogeneous can be specified
or in terms
where
in terms
of prescribed
coordinates
marion.
of surface
surface
n i is the unit outward
Cartesian
and
either
normal
tively.
displacements
(1)
Ti(S)
(2)
the prescribed
the constants
of the following
V is the volume
placements
enclosed
ui are continuous;
ous medium;
and
Cijkl are defined
body
forces
given
index
by equations
1!1
+ ujni)dS
l 1 _t_(rixj
+ rjxi)dS
S. The above
relations
= _
by the surface
vanish.
and repeated
x i are the
implies
averaged
(1) and
sumstrains
(2), respec
relations
ij(Xk) dV
the tractions
S of the composite,
e ° and G ° are the volume
conditions
_ij=__!Gij(Xk)dV=
where
surface
c ° are constants,
boundary
Eij =
= o° nj
on the boundary
e ° and
medium
This is a consequence
u (S) =E°xj
vector
of the surface,
under
conditions
tractions
For an inhomogeneous
stresses
boundary
_(uinj
Ti are continuous
Under
the above
(3)
(4)
hold provided
at all interfaces
conditions,
that: the dis
of the heterogene
the effective
elastic
moduli
as
5ij = cijn n
In practice, boundary from
conditions
the behavior
to the bounding the
entire
defines
the average
and stresses
are calculated
for an RVE
of the compositeatlarge. surface
composite, the
strains
composite's
of the RVE, its average
that result whose
which
are the same can
properties.
from
behavior
of homogeneous is indistinguishable
the homogeneous
boundary
as the boundary
conditions
be calculated. To
the application
macroscopic
By applying
behavior
macroscopic
(5)
qualify
This as
average
an RVE,
conditions applied
behavior, the
volume
to
in turn, of the
elementusedto calculateaveragecompositebehaviormust meettwo criteria. First, it must be sufficiently small with respectto the dimensionsof the compositeatlargein order to be considereda material point in the equivalenthomogeneouscontinuum (i.e. h _ H, see Figure 1). Second,
it must
be sufficiently
so that to the first order conditions
in which
arrays,
the repeating
conditions
boundary
boundary
Clearly,
bounding
a typical
tions.
RVE
of the RVE
As a result,
even
permitting uum.
the definition
In contrast,
of a single defined)
the
ply, the variation invalidates
These
local
phenomena
in
homogenization
individual
together
the individual composite ure
phases
with
1). This,
effects
equivalent
in turn,
may
such as localized
of the composite
field
as graphite
unit
on which
cell
or
alter
local
from
conductivity
for instance.
deformation
gradients
within
will
not vary
significantly,
thereby
homogeneous
contin
with respect
the RVE
may
is
give
rise in
about
the applicability
traditional
thermal
The size of the RVE
questions
the
produce
characteristics
will play
to unexpected
conductivities
to identical
obviously
is based.
thermal
gradients
gradient
that it can be
properties
neglected
may
to the thickness
(assuming
of effective
the thermal
subjected
a very
condi
different
arrangement
occupying
boundary
RVE
which
In such
of homogeneous
fibers
the
fibers.
near the
in the equivalent
For instance,
different
while
to compo
effects
the concept
coupling
or carbon
boundarylayer
within
within
is limited
of fibers
type
the RVE
of interest
properties
and the temperature
fiber
boundary
of the
approaches
largediameter
effective
"hot spots"
of the
of periodic
that the homogeneous
number
at a point
directional
the
case
1)
boundary
(5) independent
in the
inhomogeneous
within
schemes.
are quite
such
of either
quantities
with their
which
1963).
one to disregard
property
localglobal
micromechanical phases
quantifies
assumptions
the
in equation
on the deformation
large
of highly
of the quantities
of the
rooted
diameters
with relatively
the basic
variations
conditions
application
of a material
in composites
(Hill,
allowing
upon
field
applied
a sufficiently
in the presence
the compositeatlarge,
properties
of the aforementioned
composite,
h, see Figure
on the type of loading.
small
contains
(i.e. d _
by both sets of homogeneous
elastic
symmetry
very
induced
phase
as the RVE provided
depending
with
of the entire
surfaces
are
of applicability
by fibers
volume
conditions
by either
to the inclusion
energy
unit cell is interpreted
the range
composites,
strain
the effective
conditions,
reinforced
small
making
are replaced
periodic
sites
the elastic
is the same,
manner
large with respect
of the
gradients
in
in the homogeneous
boundary and
conditions
produce
in relation
an important
(Fig
unexpected
to the thickness role
in the above
scenario. The copic coarse
preceding
approach
based
or spatially
decoupling
the
discussion
on the concept
variable local
raises
of an RVE
microstructure.
response
from
the
in the presence
In light global
of the traditional
of large
of this discussion,
response
by
calculating
thermal the
micros
gradients
current pointwise
and
practice effective
of
thermoelasticpropertiesof functionally gradedmaterialswithout regard to whether the actual microstructureadmitsthe presenceof an RVE, and subsequentlyusing thesepropertiesin the global analysis of the heterogeneous material,remainsto be justified. These issueswere discussedqualitatively as early as 1974by Pagano(1974) with regard to mechanicalloading of macroscopicallyhomogeneouscomposites.No further work in this areaappearsto have been publishedin the open literaturesincethen.In orderto resolvetheseissues,a model is required thatexplicitly couplesthe microstructuraland macrostructuralanalyses.The model presentedin the following sectionis a stepin this directionfor applicationsinvolving compositeswith uniformly or nonuniformly spaced,largediameterfibers subjectedto throughthethicknessthermal gradients. 3.0 ANALYTICAL MODEL The heterogeneous ing block nated
composite
or repeating
by the triplet
shown
unit cell
given
in Figure
(0_13_/).Each
index
tx, 13, _t takes
tive position
of the given
subcell
the unit cell
along
and x3 axes,
since
these
direction, given
are
number.
the x2
the periodic
remains
x 3, the corresponding (p,q,r)
and
an infinite
tions. set
The
forp
indices
= 1, 2 ..... range
material
reinforced
through
homogenization
applied response
f'trst step,
in the x2x3
the
of the entire
subcell
taken
plane.
of fibers
of the
cell, requiting
approach
boundaryvalue
the temperature
Rather,
plate.
to note
whose
Thus
the response
couples problem
distribution
desig
9
given
configuration
a
x2 and
is designated
by the tri
or FG direction,
in the x2
media
and x3
as well
as bi
unit cell
in the
an entire
of local
action
This is what is meant
be
obtained
column
of
cannot
be
coupled
to the
by the statement
with the global is solved
direc
by a different
can
to be explicitly
composite
within
two directions,
comprises
in the foregoing
or the FG
the cell
properties
details
of
identifies
that the repeating
principle
the rela
dimensions
can be represented
of each cell
in the heterogeneous
The
in the thickness
the RVE
the microstructural outlined
cell
effective
the
indicate
p which
of multiphase
of cells in the FG direction.
explicitly
subcells
of the subcells
of the composite
the unit cell
RVE
below.
of eight
the x 1 axis
dimensions
a given
considerations
the
along
For the other
Thus
It is important
to be
thickness
column
within
allowing
as explained
to an individual
The thermal
each
build
for the
with a running index
M is the number
configurations.
is not
spanning
that the present
the dimensions
unit cell to unit cell. The
M, where
the basic
1 or 2 which
h2, and I l, I2, are fixed
whereas
constant
using
x2 and x3 axis, respectively.
are designated
parameters,
framework
cells
on the values
q and r are introduced.
occupying
present
such
3. This unit cell consists
the xl, hi,
2 can be constructed
of q and r due to the periodicity
of thermoelastic
directionally
from
the FG direction
We note thatp
plet
along
directions,
d_ p), d_ °), can vary
cell along
in Figure
analysis.
in two steps.
is determined
In the
by solving
the heatequationundersteadystateconditionsin eachsubregionor cell of the composite.Since the compositeis periodic in thex2x3 plane, it is sufficient to determine the distribution of temperatures
in a single
continuity
and
row
compatibility
is indeed
indistinguishable
tribution
in the
stresses
of cells
conditions from
entire
thermal
problem,
are satisfied.
occupying
generated
subject
These
only,
the
composite,
and
row of cells is considered
that the given
Given
internal
dis
strains
in each
conditions.
nature
and
subregion
As in the case
due to periodic
cell
the temperature
displacements,
equations
boundary
that appropriate
ensure
plane.
the equilibrium
continuity
provided
conditions
cells in the x2x3
by solving
to appropriate
only a single
the FG dimensions
the adjacent
volume
are subsequently
the composite
spanning
of
of the
of the composite
in the x 2x 3 plane. The
analytical
by the first author sites,
referred of cells
tion
the
modeled
(Paley
method
to consmact
the entire
by three
the repeating and strains
in each
given
cell
in turn,
with the
sense
respect
which
midpoints.
to determine
makes
condition
to include
center
solution
continuity
10
cell,
to the given
phases
and
subcell
sur
geometry
of
for the stresses state of strain
subcell
of tractions
in terms
and
center
and
field a linear
between
adjacent of a
cells,
displacement
boundaryvalue
geometric
that gra
problem,
of the composite.
into an arbitrary additional
dis
and diplace
displacements
of the
or
The coeffi
in adjacent
properties
unit cell is subdivided multiple
block
and the unknown
displacements
or effective
character
midpoint.
in the expansion,
center
is
the displacement
at the subcell's
strains
(average)
solution
of the
on subcell
fiber
rectangular
homogeneous
is imposed subcell
the repeating
The
of a given
for homogenized
macroscopic
it possible
terms
composite
as a building
of a single
subcelis
The approximate
of cells,
of the
formula
The periodic
by approximating
by satisfying
individual
to the corresponding expressions
method
with the linear are obtained
fibrous
closedform
x_(co , _(1_), _(_) centered
between
a connectivity
is obtained
generalized
unit cell are thus representative
of cells.
macroscopically
of the displacement
associated centers
method
the
compo
In the original
phase.
consists
an approximate,
The solution
,. coorctmates
unit cell
developed
periodic
recently
1992).
in a matrix
of this repeating
some
most
unit cell that can be used
the name
given
and triply
unidirectional
embedded
The
one to obtain
in terms
necessary
is used
Hence
and
and Pindera,
a repeating
assemblage.
subcells
subcell
of the subcell
the generalized subcells
subcells.
in an average In addition,
dients
mawix
at the
of fibers
of the approach
of doubly
1991)
Aboudi
The properties
entire
or microvariables
response
(Aboudi,
1992;
one to identify
to the composite.
cells.
provides
array
of the
subcells
is a derivative
a continouslyreinforced,
composite.
in the local
placements ments
cells,
in the individual
of the
expansion
of cells
Aboudi,
unit cell allows
applied
cients
of
allows
properties
rounded
stress
and
problem
of the effective
method
as a doublyperiodic
of the assemblage
of the
for the above
in the treatment
to as the
method of
technique
In
number
of
detail
in
modelingthe repeatingunit cell. The procedurefor analyzinglocal stressand strainfields in this case,however, is the sameas that usedin the original methodof cells (i.e., by approximating local subcell displacementsusing linear expansionsin termsof local coordinatesin the individual subcells). Conversely,in the presentanalysisa higherorder theory is requiredin order to capture the local effects createdby the thermalgradient,the microstructureof the composite,and the finite dimensionin the FG direction. Accordingly,in the thermalproblemthe temperaturefield in each subceUof a repeatingunit cell is approximatedusing a quadratic expansion in local coordinatesalong the threecoordinatedirectionsassociatedwith the given subcell.In the solution for the local strainsandstresses,the displacementfield in the FG direction in eachsubcell is also approximatedusing a quadratic expansion in local coordinateswithin the subcell.The displacementfield in the x2 and x3 directions, however, is still approximated using a linear expansion
in local
in the x2x3 The
unknown
coefficients
problem
continuity
lines
and the solution
as those however,
the considered be treated nectivity
decoupled)
conditions
boundaryvalue
problem
classical
finite
dimension
from
the
of the composite's
outline
that summarizes
manipulations
in the individual in the Appendix necessary
solution
microstructure
conditions
in an average
of both material
method
treatments
in the FG
in the FG direction. approaches
sense
A fundamental
which
Accordingly,
direction
currently
of the sub
in solving
strains
the given
in this direction
of homogeneous
features
boundary
as well
set the present
employed
cannot the con
in terms
and the compositeatlarge, These
along
lies in the fact that
strains
homogenized
in both
by satisfying
effects
schemes.
for the homogenized
to define
of ceils.
and structural
homogenization
unit cell
coordinates
are obtained
This is due to the absence
the repeating
local
and stresses
and the previous
are not imposed
concepts.
gradient
the governing
boundary
expressions
micromechanical
of this new
strains
it is not possible
for both
the area of functionally
is presented
since
and quadratic
and generalized
the classical
gradients)
of the composite
classical
and
elements
provide
micromechanical
that hold
fields
the present
using
(which
for internal
tractions
contains
displacement
ment
character
with the linear
in the original
between
composite
(i.e.,
conditions
and
employed
cell midpoint
An
the periodic
associated
of displacements
difference,
using
to reflect
plane.
the thermal
similar
coordinates
as the
model
by researchers
apart
working
materials.
analytical
approach
equations subcells
for both the thermal
for the determination
will now be given.
so as not to obscure
to generate
the governing
11
of the temperature
A detailed
the basic equations.
and mechanical
concepts
derivation
and
of these
by the involved
problem displaceequations algebraic
in
3.1 Thermal Suppose
Analysis:
Formulation
that the composite
Let the composite the bottom
Problem
material
be subjected
surface
occupies
the region
to the temperature
(x I = H). Also,
0 =12:2 3.3.4
Boundary The
final
boundary surface
Conditions set of conditions
conditions must
equal
that the solution
at the top and bottom the normal
stress
for the displacement
surfaces.
tom surface
the condition
satisfy
are the
in the cell p = 1 at the top
_(I)= _Id_i) Xl 2 "
(l,q,r)
the temporal
stress
must
fit),
_t1113_')I
with f(t) describing
The normal
field
= f(t),
variation
that the surface
(37)
of this loading,
whereas
in the cell p = M at the bot
x 1 = H is rigidly
clamped
(say)
is imposed
(M,q,r)
ut213_') I
For other types
3.4
Mechanical Due
of boundary
Analysis:
to symmetry
cell is approximated
conditions,

0
_)
equation
(38) should
1 d_M)
be modified
(38)
accordingly.
Solution
considerations, by a secondorder
the displacement expansion
field in the subeell (al_') of the pth . _(a) _(I_), and _3 _') as in the local coordinates xl , x2
follows:
1..,_(a)2 utah>=wt_> +_:>,t _> + _t,xx 
ld_)2)Uta_,t
_t,x:
) + 1..,_(13)2
1,.2 ,,,toting,)  __,
1 ,.,_(7)2 ll_)Wta_v) + _t_x3  4
21
(39)
where
wt a_'),
wtaf_v),
which
problem.
quantifies
replaced
conditions
start
with
involve
S(aB.r)
,
,j (,,m,n)
of (l,m,n)
conditions
cells
are 56M unknown
thermal
problem.
and
of heat
Here,
mechanical
subsequently
fluxes
the
and
similar
heat
to those
and
employed of
equation
is
displacements Finally,
As in the
V_ al_),
determination
conduction
and temperature.
the governing
Ut _v),
The
of tractions
quantifies.
modify
subcell,
quantifies.
and the continuity
the continuity
stress
the equilibrium
thermal
equations
at
the boun
problem,
we
to accommodate
a 1 _ V_7)d! .
)12 h C2 zv 2 _(a) ) /2h_,2_12 _
(40)
higherorder
of the higherorder
citly
in terms
of the
nonvanishing
are obtained
sense,
it is convenient
zerothorder
equations
(40)
stresses which
These
coefficients
using
in the displacement
in a volumetric
)t
average
continuum.
unknown
integration
(xl
provides
stresses
equations
volume
equations
quantities:
field
coefficients
from
the appropriate
1 = m = n = 0, equation
following
be determined
equations,
of satisfying
the following
required
of the
of Equilibrium
In the course
values
center
cells p = 1 and M.
Equations
to define
at the
there
of the
replaces
the internal
the boundary
that
equilibrium
interfaces
dary
For
displacements
In this case,
parallels
by the three
the various
3.4.1
the
¢ taft'Y), Z_al_v), and V_ al_') must
in the thermal these
are
in the are needed
stress
Ut a13v).....
quantities
_t a13v).....
(31), (32) and (39)
and ftrstorder
stress
subcell,
to describe can
for
(40).
in terms
other
the governing
be evaluated
V_ °_133')by
in equation
components
whereas
expli
performing This yields
the the
of the unknown
field expansion:
(41)
(42) (43)
22
(44) 1
_)
2_alh')
gcl
(45)
hp.,
(46)
Satisfaction
of the equilibrium
the volumeaveraged after lengthy
firstorder
algebraic
equations
in the following
•.(a_,t) . in the different o,j(l.m.n)
stresses
manipulations
results
eight
subcells
(8) relations
among
(ot[]q,) of the pth
cell,
(see the Apendix):
(47)
where,
as in the case of equation
(20), the triplet
(a[3T) assumes
all permutations
of the integers
1
and 2.
3.4.2
Traction The
continuity
associated ensured
Continuity
Equations
of tractions
with the x 1 (FG) by the following
t 12St_(_!l.0)/h_
at the
direction,
subcell
interfaces,
equations
(33a)
as well
and (34a)
as between
imposed
individual
in an average
+ 12St_!o.1)/l_
" St_!,.o)/t3_ ](pl) Continuing,
if we
into equation
substitute
(A45),
equation
we obtain,
(A32)
into equation
(A43)
(49)
directly,
and
equation
(A33)
respectively:
tst_g?l,o>/h, +St_&o_/h2](P_ =0 t stC_(d,)o,1)[ll
Finally
combining
equations
(A24)
and
+ StC_(_)o,1)
(A44),
and
]12
](p)
(50) (51)
=0
equations
(A25)
and
(A46),
yields,
respec
tively,
(P)
(p)
(52)
s)_(_:o,o) I =sh(_:o,o> I (p) (p) al
As indicated mental
previously,
unknown
coefficients
tions
(41) through
8.2.4
Displacement The
average
cz2)
(47) through
wt alh'), Ut a_'),
(53) are easily
Vt al_'t), Wt a_),
(53)
expressed
_t aId't), )_al_,),
in terms
of the funda
and V_ _')
using
equa
(46).
displacement basis
equations
)
at the
Continuity continuity interfaces.
Conditions conditions, This
i.e. equations
is accomplished
51
(35)
by fhst
 (36),
are now
substituting
imposed
equation
(39)
on an into
equation
(35a),
[ wt
+
then into equation
1
)
1137 )
+ T1 d2U_1_7 '
) ](p) =[
wt 21_') 
ld_)t_t
](p)
(54)
(35b)
1 h2V(a2.7) [ Wt °_17) + "_" 1 I
](p)
= [
wttt2y)
1 L.21z(a2y) + "4"n2,l
(p) h lX_,=lv) I
followed
1 .t2rr(21_') 2153')+ "_u2''1
by equation
](p)
(55)
(p) (56)
= heX_'2_> i
(35c)
[ Wt ctl_l) + 412W_
a_l)
1,21xz(ct_2) [ wt etl_2> + _'2" 1
](P)=
(p)
](p)
(57)
(p) (58)
and f'mally into equation
[ wt:l_, )
(36a)
1 d(f,
[email protected])+
1 d_P+l)2u_l[_,)
The other two displacement the present
case of normal
(59) provide ments between
24 relations
continuity
loading
applied
](p+l)=
relations
[ wt2157) + ld,_)_t2137)
the subcells and between neighboring
52
1 )2 U_267) ] q') (59) _d_
(36b) and (36c) are identically
in the x 1direction.
which must be imposed
+
to guarantee cells.
satisfied
for
Consequently,
equations
(54) 
the continuity
of the displace
Table
Material
E and thermal
1. Material
properties
of SCS6
SiC fiber
and titanium
E (GPa)
v
ct (10 6 m / m / °C)
SiC fiber
414.0
0.3
4.9
TiA1 matrix
100.0
0.3
9.6
v denote
the Young's
expansion,
modulus
and _: is the thermal
Table
2. Material
properties
and Poisson's
matrix.
_c(W / re°C)
400.0,
200.0,
40.0,
17.6
8.0
ratio,
respectively,
oc is the
SiC/Ti
composite
(vf = 0.40).
coefficient
conductivity.
of the SCS6
EA (GPa)
VA
226.0
0.30
Er
(GPa)
167.0
GA (GPa)
60.9
tXA ( 10 .6 m / m / °C)
tx T (10 6 m / m / °C)
_cA (W / m°C)
_:T (W / m°C)
6.15
7.90
164.80
16.20
Subindices
A and T denote
axial and transverse
quantifies,
53
respectively.
of
Table 3. Normalized
inplane
force and moment
resultants
for different
K:f / _:m ratios.
VI=3
50.0
0.7954
0.7895
0.9444
0.9209
25.0
0.8046
0.8000
0.9409
0.9177
5.0
0.8621
0.8631
0.9271
0.9019
2.2
0.9195
0.9158
0.9201
0.8924
50.0
0.9655
0.9684
0.9930
0.9873
25.0
0.9655
0.9684
0.9930
0.9873
5.0
0.9770
0.9789
0.9896
0.9842
2.2
0.9885
0.9895
0.9896
0.9842
VI = 20
$4
Replacement Scheme T2
T l H
RVE T1
Material continuum point
m i
A
A
Heterogeneous material with homogenized properties
Heterogeneous material with fine microstructure
d=O(_h)
"['_
>
i h=O((H) ¢