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spaced, thus admitting so-called functionally gradient composites. The results illustrate that the classical .... are ideal candidates ...... ___[Q_2(_?t,o)/h_. +Q_(_,,_ ...

NASATechnical

Memorandum

106344

................

_r

?

Thei-moelastic Response of Metal Matrix Composites With Large,Diameter Fibers Subjected to Therma! Gradien!s

Jacob Aboudi ...................

and Marek-Jerzy

Pindera

University of Virginia Charlottesville, Virginia and Steven M. Arnold ....... Lewis Research center Cleveland, Ohio

October- 1993

-

.............

.............................................

(NASA-TM-I06344) RESPONSE

__i

I%IASA

OF

N94-I4823

THERMOELASTIC METAL

WITH LARGE-DIAMETER SUBJECTED TO THERMAL

MATRIX

COMPOSITES

FIBERS GRADIENTS G3/24

Unclas

i,

0189400

;

WITH

THERMOELASTIC LARGE-DIAMETER

RESPONSE FIBERS

OF METAL SUBJECTED

MATRIX COMPOSITES TO THERMAL GRADIENTS

by Jacob

Aboudi

1 and

Marek-Jerzy

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 thin-walled

approach number produce mating

so-called

of decoupling

stress

The results

and macro-mechanical

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 large-diameter excessively

functionally

coupled

limitations

metal

that

and macroscopic

are explicitly

that illustrate

matrix

theories

microscopic

large-diameter 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

TeI-Aviv University, Ramat-Aviv

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 p-th

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; higher-order

--

of a composite

of the subcell

(a[3 T)

terms

in the second-order

expansion

terms

in the first-order

expansion

of

terms

in the first-order

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 second-order

coefficients

associated

expansion

of the subcell

coefficients

,

with

x_ )2 in the

second-order

term

X_"Y)2in the

second-order

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; higher-order

@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)

_!_13-f) 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)

short-fiber,

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 finite-length

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

two-phase

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

large-diameter

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 small-diameter

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

well-established

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

so-called

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 x2-x3

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 finite-length.

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

large-diameter

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

out-of-plane

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

cross-sectional

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 bi-directional arrayswith uniform spacings

in the FG

direction,

as well

graded

materials

consisting

functionally ously

changing

tions,

metallic-rich

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

large-diameter

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 bi-directionally

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 ceramic-rich.

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 multi-phase

or variable

and

Reuss

matrix

given

by Aboudi

(1991).

these

well-established

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

self-consistent

method,

analyses

A discussion

is

to calculate

hypotheses,

the Mori-Tanaka

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

E-ij =

= 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 composite-at-large. 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 composite-at-largein 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

boundary-layer

within

within

is limited

of fibers

type

the RVE

of interest

properties

and the temperature

fiber

boundary

of the

approaches

large-diameter

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

local-global

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 composite-at-large,

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,large-diameterfibers subjectedto through-the-thicknessthermal 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 x2-x3

the

of the entire

subcell

taken

plane.

of fibers

of the

cell, requiting

approach

boundary-value

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 multi-phase

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 heatequationundersteady-stateconditionsin eachsub-regionor cell of the composite.Since the compositeis periodic in thex2-x3 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

sub-region

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 x2-x3

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 2-x 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

mid-points.

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

boundary-value

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

mid-point.

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

closed-form

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 continously-reinforced,

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 doubly-periodic

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 higher-order 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 x2-x3 The

unknown

coefficients

problem

continuity

lines

and the solution

as those however,

the considered be treated nectivity

decoupled)

conditions

boundary-value

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 composite-at-large, 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 mid-point

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 second-order

the displacement expansion

field in the subeell (al_') of the p-th . _(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) ) /2-h_,2-_12 _

(40)

higher-order

of the higher-order

citly

in terms

of the

non-vanishing

are obtained

sense,

it is convenient

zeroth-order

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 ftrst-order

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 volume-averaged after lengthy

first-order

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 p-th

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_ ](p-l) 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 fh-st

- (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 1-direction.

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

Ti-A1 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) ¢