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Pergamon Pms pk
AVERAGE
ELASTICPLASTIC BEHAVIOR COMPOSITE MATERIALS
OF
S. C. LEN, C. C. YANG and T. MURA Department of Civil Engineering. Northwestern University, Evanston. IL 60208. U.S.A.
and T.
IWARUMA
Department of Civil Engineering. Pohoku University. Aobaku. Sendai.980. Japan AbstractA new approach to estimate the overall behavior of an inhomogeneous body is applied to investigate the average elasticplastic state of composite materialsin small deformation. The method is based on the concept of the average field of an infinite body with inhomogeneities. which is replaced by an equivalent body with homogeneous inclusions having appropriate eigenstrains that are composed of the actual plastic misfit strains and the properly determined eigenstrains by the quivalent replacement of the inf~omogeneities.A simple example is given for a pure shear state of the elasticperfectly plastic composite materials with spherical inhomogeneiti~. The overall hardening rate. quantitative estimate of ductility improvement. and unloading behavior are discussed.
I, INTRODUCIION to undcrstitnd the overall mechanical behavior ofcomposite materials has been one of the major subjbwts in many cngin~cring fields for many years. When the volume fraction of inhomogcneitics is small, Eshclby’s method evaluates such average behavior quite reasonably and explicitly including the shape etfect of dispersoids (Eshelby. 1957). However, when the intcrdction between inclusions becomes significant as the volume fraction increases, the socalled selfconsistent method has been introduced by KrGner (1958). Budi3nsky (1965) and Hill (1965). This method can apply even for crystalline materials as a limiting case when the matrix portion of composites vanishes. Hutchinson (1964, 1970) has utilized this method to estimate the elasticplastic incremental relations of crystalline materials and composites. fwakuma and NematNasser f 1984) and NematNasser and lwakuma (1985) have extended the same method for finite deformations to predict ductile insktbility of strain localization. The method had been generalized and modified by Willisand Acton (1976) and Willis (1977). However, theselfconsistent method fails to give a satisfactory estimate when the disprrsoids are either voids or perfectly rigid, and yields unacceptable results when the volume fraction exceeds a certain value. Recently several researches have been conducted to evaluate the average elastic moduli of composite materials. As far as the average field is concerned, the fo~ulation becomes very simple by the MoriTanaka theory (Mori and Tanaka, 1973). Using the modification of the MoriTanaka method, Bcnvcniste (1987) and Mori and Wakashima (1989) have estimated the average elastic moduli of composite materials. Here the aspect of a back stress is introduced to take into account the interaction of inhomogeneities and matrix materials. Moreover the results become consistently identical in both cases where either the farfield stress or farfield displacement is prescribed. In other words, the estimation of material properties can be carried out independently of the farfield boundary conditions. This is strongly desirable from a physical point of view. The prediction by this method seems to give reasonable results even when the volume fraction becomes large. When the shape of all inhomogeneities with the same elasticity is spherical, it can be shown that the overalt elastic moduli coincide with either one of the bounds calculated through a variational method by Hashin and Shtrikman (1963). Attempting
18.59
s c.
lY64J
LI> YI ul
Here we apply this method to evaluate average elasticplastic materials
within
perfectlyplastic
the framework
of small deformations.
relations
For simplicity.
of composite
a simple elastic
material is considered for the matrix and inhomo~enrities.
One of the main
objectives is to calculate the average hardening coefficient of such twophase composites.
2. DETERbflS.4TION
Consider a sufficiently distributed
OF APPROXIMATE
ELASTIC
PLASTIC
BEHAVIOR
large body D which contains identically
and oriented inhomogeneities.
shaped and randomI>
R = R, +RI +R, +. . . . the total volume fraction
of which is denoted by_/: The elastic moduli of the matrix ,U ( = DR)
and inhomogeneities
are (? and c*. respectively. These materials are elastic perfectlyplastic
with the yield shear
stresses r:! for the matrix and r:! for the inhomogeneity. The following
method based on that of Mori and Tanaka. yields the same results for
the cases when either the surface traction or the surface displacement is prescribed on CD. As a matter of fact. this method has a great advantage because no boundary condition is necessary to determine the overall state. However. for the sake of clear comprehension. WC‘ here consider the case when the surface traction (fi6”) body whose outward
unit
normal
is applied on the boundary of fhc
vector is denoted by 2. The
case when the surface
displacement is prcscribcd is given in the Appendix. The inhomogeneities arc distributed randomly and arc of the same shape. For the sake of simplicity. WC consider only one rcprcscntativc
volume
I: which contains only one inhomogcncity.
rcproscntativc volume he similar If no inhomogcncity C(C’
C”)
gcncitics
and provitlcs
exists.
C” dcnoks
the ovcrnll
Ihc overall
Jisturhancc
Let the shape of the
to that of the irlhotliogclicity.
in local
slrain
uniform liclds
ci,,, ~IKI ri,, tlcnolc the avoragc distiirhcrl
of both
slrcss
rospcclivcly. Then. since the overall stress Ii&i
5” Jistributcs
plastic
strain.
uniformly. The
Ihc malrilt
cxiskncc
u hcrc
;intf inhoriiof”ioitics.
liclds of the matrix
5” =
of inhonloLet
and inhonlogcncily,
is prcscribcd. we must have
( I ../‘)ri,, +,/‘ci,, = 0.
(1)
Mori and Tanaka ( 1973) have shown that the average disturbed strain tield due to the existence of an inhomogcncity
vanishus outside of the inhomogeneity,
shapes of the representative volume V and inhomogeneity
providctl that the
R,, are taken as similar.
Hcncc
the average disturbed strain field due to the inhomogeneity exists only inside of the inclusion in the representative volume. Howcvcr, the overall strain disturbance is no longer Lcro cvcn outs&
of the inclusion
Thcrcforc,
bccuusc the surface displacement
= c:(~“+tyc/;,),
f?‘+ri,, where
is free in the very far Ii&i.
the average lieId in the matrix can be written as
C’;1 is the avcragc
(2)
plastic strain in the matrix. and c,, is the overall uniform disturbance.
In other words, CL,is to take into account the interaction between neighboring rcprcsentativc volumes. Then,
in the inhomogeneity
5” + CT0=
R,.
c*(2”+c,,+(,T)fT:;).
(3)
whcrc ;’ expresses the disturbed strain field due to the cxistcncc of the inhomogencity and ) indicates the avcragc over R, ; misfit by plasticity inside the rcprcscntativc volume ; ( Z,P,is the avcragc plastic strain in Rx. Since all inhomogcncitics arc of the same shape with the same material properties. the average over R, is identical with that over R. Substitution
whcrc
of eqn (2) into eqn (3) yields
1861
Estimating elastic moduli of’ composites
AZ’ = t; 2;.
(5)
From eqn (4). the local disturbed strain field 7 due to plasticity is caused only by the misfit of the plastic strain AZ’. It is natural because the superposition of the uniform plastic strain (ZT,) over the entire body does not disturb the entire stress field. Applying the equivalent inclusion method. we have the equivalency condition in one inhomogeneity as
do+& = c*(c
‘(~“+,j,)+(~)A~p)
(ha)
= ~{~‘(~“+~,~,)+(~)(A~P+(~*))~,
(6b)
where C* denotes the eigenstrain. Then the local disturbance ;I is given by (see Mum. 1982)
;;,(.2) = 
c,,~,IA&~,(~)+&:,(F))
i(G,~,,,(.~p)+G,,,,,(.~F,)
d:.
(7)
When the inhomogeneity has the ellipsoidal shape. integration of eqn (7) over f2Kleads to (7) = s’(AC’+(C*)).
(8)
where s’bccomcs constant in terms of dimensions of inhomogencity and Poisson’s ratio of the matrix, and is called Eshelby’s tensor (Eshelby. 1957). Substitution of eqn (8) into eqn (6b) yields c?,~= ri,,, + C?(.% I’)(A?‘+ (Z”)).
(9)
whcrc ris the identity tensor. From cqns (I) and (9). wc have _ (?!I = /‘c’(.? ?)(A?+
(Z*)).
(10)
f’utting cqn (IO) back into cqn (9), we obtain dl, = (I /)c”(.c
lj(Ai’+
(F*)).
(11)
On the other hand. the cquivalcncy condition, cqn (6). with cqn (8) results in Ac”+(p*)
= [c(s(_ct’)(s/‘(J:_~)i]
‘{(c’__c*)c
 ‘#j+c*AC_‘}
(12)
Therefore the average stress ticlds in the matrix and inhomogcncity are obtained from eqns (IO) ( 12) iIs follows : 0
CT

+(T.,,=cJ
“/c’(.~i)[z’(cc’*)Isr(s1‘,)]
5”+a’l* = P+(l
‘((&f*)c
‘#~+CftC\~P),
r)c(J’7)[C(ct_c’*)(s_r(J’1’,>]’ x r(c'C*)c''a"+C"*Ac'PI.
(13)
In order to calculate the average strain tield, it is convenient to express its elastic part. From cqns (2) and (4). we can write 6, =
P’(P+d,,,).
c;,= c ‘(o’” + (J,,,) +
(f)
 Afp,
(14)
where E;, and ?9 are the elastic part of the average strain fields in the matrix and inhomogeneity, respectively. Let CY denote the overall average strain. and Zp the average plastic strain defined by
(15) Then the overall average elastic strain can be given by
zrf_y = ( I ,/‘)a:, +f’a;,. Considering
eqns (8) and (11). and substituting
the overall stressstrain
eqn (1t) into eqn (16). we finally obtain
relation as
= [~_(~~*)~~~(~I’))]
f_FP
(16)
‘[;&I
_/‘)(~~*)s’)~‘(p +,/‘(I
3. AN
EXAMPLE:
As an example.
SPHERICAL
INHOMOGENEITY
let the shape of inhomogencities
a spherical inhomogeneity
can be decomposed
whcrc the 7’ and I” components
j‘,(?~*,(~~)AKp].
IN AN ISOTROPIC
(17)
BODY
bc spherical. The Eshelby tensor for
into two parts as
correspond
to volumetric
I’r,kl = IA,,&,.
7: = ii’,
and shcnr deformation,
rcspcc
tivcly. and
in which “;,, indicates the Kroncckcr
tlclta. I lcnccforth
wc cxprcss such a tensor as
whcrc
(I’)) and v is Poisson’s ratio of the matrix material. Let IL and i. dcnotc the Lame constants of the matrix matcriul, and IL* and >.* be the corresponding
constants of the inhomogcncity.
c:= (3h. 2/L),
Then
cf* = (3h.f.
where ti is the bulk modulus dclined by (i.+2/~/3).
3/L*),
(30)
Substitution
of eqns (19) and (20) into
eqn (13) results in
a’“fS”
=
(zI)(KK”) __._..~._
(.p+(l_/)
[
_,
(/I_ __
l)(/L_IL*) _ _ _ ..,. _ n
1
+ ( I _/‘I
3(r~_
_. (T
I)tiK*
.4
where
7(/1 I)iL/L* . ~. ~mmi 
1
Afp,
(21)
1863
Estimating elastic moduli of composites
A E K (zf(z
I);@K*).
B = vuww
(22)
1);tpp’).
Similarly eqn (17) becomes
K(1 fp
j)(K
=
h.*b P(1 lmP’)B
~KA
’
2lrB
+_I(1 f)
1c7 0
(z 1):h.*)
,(P IIflr’)
[
&p* 1
(23)
3.2. Pure shear Overall shearing behavior can be easily estimated as the simplest case, because only one component of stress tensors is necessary. Let the body considered be subjected to pure shearing in the ,y,.x2plane. Then the yield condition
in particular
becomes as simple as
(244. (24b) in the matrix and inhomogeneity need only the ?’ components
respectively as an average. From eqns (21) and (23) we
to obtain
(2Sa)
Stuge I: both mareriuls we rlusric. When both materials are in elasticity, no plastic strain exists and eqn (2%)
a?? =
yields
1(I 4(/I.!(/I2p
where the following
I(IJ)(l,,,)b
nondimensional
I,] _ ElZ=2C’
_/(Iml { ‘
I (I /)(I
E
m)/? I I”
(26)
parameter is introduced :
(27)
Equation
(26) exactly coincides with the result by Mori and Wakashima
(1989).
Stage 2a: i/the matrix yiefdrJ7r.u. At the initial yielding. the yeild condition eqn (24a) is satisfied, and therefore putting AC:* = 0 into eqn (ZSa), we obtain the overall stress at the first yielding (a??), as
(28) Since the matrix material keeps satisfying the yield condition even after this initial yielding, its local stress level remains the same as the yield stress. Therefore from eqn (25a) :
S. C. Lls rf al.
IYM
Eliminating
Af:yI from eqns (25,) and (79). we obtain
2!((F~P)Iz
m+(l = Wj$ _(,
fi)(l f)(l _nr)(B_f(B_
m) ,)1]~~~
AS the plastic strain exists only in the matrix.
(I f‘,(l _____
A&‘p: = (E:,),~.
m) M
Therefore
I ;‘_
(30)
from eqns (15)
and (29) :
(3’) From eqns (30) and (3 I), the overall stressstrain
relation can be obtained as
(32)
Lvhich shows ;I bilinear relation together with eqn (26). The cocflicicnt in the first term ot the righthand side of cqn (32) represents the avcrngc ovcrall hardening rate after the first yicltiing. Similarly state insiik
elimination
of AC:, from eqns (25b) and (39) Icads to the local stress
an inhoniogcncity
as
(‘T”+6,)),_
=
$2 _ ’ y
Jr”
/
(33)
,$(I
.SI(IJ~C’ 3 : I/WI tlrc*itrlrc~r~rc!c/c,trc,;t~, yic*llls. As the applied stress incrcnscs. the stress st;ttc inside inhomogcncitics
incrcascs to achieve its yield condition.
Then
substitution
of ccln
(33) into ecln (24b) results in
and the maximum stress (c:),),, can be obtained as
(CT:)?),, = fr  (I /‘)r :’ +j.r:. Since L:‘: bccomcs arbitrary
(34)
at this stress level, the overall strain cannot bc dctermincd
uniquely, and thus this state can be called ultimate. Namely the ultimate state is achieved when the ovcrail stress reaches the average value of the yield stresses of the matrix and inhomogcncity S~uqr
3
?,.. : i/‘ I/W irrltorrrr?gcneir~
yiddr
becomes plastic first, the yield condition
first.
On the contrary,
if the inhomogeneity
eqn (24b) is satisfied first. Then substituting
eqn
(25b) into eqn (J&b), we obtain the initial yield overall stress as
(cf
),
?
= I
C._ ._ ,
Then the calculations and discussions cxprcssions :
_
(1 J’NP ‘)(I ml n1
similar
T:‘,
(35)
1
to those in Stage 2a lead to the following
Eslimaling elastic moduliof composites 2p(CEP),
* = a72 +
f(l
m)
1865
5F
m
(36)
Y
and
[email protected]:=
/ (1
f[l (I ew(Bm(l/0(1f>
uyl_
/))(I
f)
I)11 *“Y
.
(37)
From eqns (36) and (37). the overall hardening behavior can be obtained as
(1 ml
I)
/
OYz = 2P(l/9)(lf)+++ The ultimate maximum
7:.
(I/?)(If)+f
state is then achieved at the same stress level obtained
(38) at Stage 3; i.e. the
stress coincides with the average yield stress as given in eqn (34).
4. DISCUSSION
In the preceding section, the simplest example is solved to show the feasibility of the present method. Although
the obtained results show the average behavior, a clear phenom
enological tendency can bc obtained explicitly and quantitatively. Comparison
of eqn (28) with eqn (35) leads to the following
relation : the matrix will
yield first if
rf:
111
I=,?>
I P(l
5r
othcrwisc the inhomogcncity
will become.plastic
(39)
ml) first. In the following
several numerical
cxamplcs. Poisson’s ratio of the matrix material is kept equal to 0.3 unless otherwise stated. Figure I shows the cases when the inhomogeneity
is chosen to be stiffer and stronger
than the matrix : i.e. m = 100 and I = 3.0 or I = 1.2. Addition
l
ixhomogeneity
+
matrix yielding
fg/.’ y’,
0
of such particles will increase
yielding
,...__.__. _ k1.2
(
m=lOO
v=o.3
1
2 2cIE, J 7:
Fig. I. Overall
elasticplastic
behavior of composite materials shear. Y = 0.3. n! = 100.
with spherical inclusions in pure
I866
m
0.1 . 0.2 v=o.3 0.01
not only the initial
0
0.2
0.4
0.6
stitl’ncss but also the ultimate
0.8
1.0
strength.
f
Furthermore
the initial
yield
point can hc also cnhan~cti if’ the material paramctcrs ;lrc chosen so that the matrix yields Kirst, i1s indicatctl by the solid lines in the ligurc. Figure 2 summariLcs the overall initial elastic moduii in terms of the modulus ratio m itntl volume t’raction J
In the C;ISC ol’ spherical inhomogcncitics.
prcdictcd moduli coincide with the lower bound (SW tlashin MI > I, ilnd that thoy arc iclcntical with
it can bc shown that the
and Shtrikman,
1963) when
the upper hound when ttl < I. Figure
3 shows
h=O.Ol
O.Ol0
0.2
0.4
0.6
0.8
1.0
f Fig. 3.
Overall hardening
rale of composite materials with spherical inclusions in pure shear. when yielding of the matrix occurs first Y = 0.3.
Estimating
elastic mduli
of
1867
composites
t
inhomogeneity yielding
*
matrix yielding
0.6 0.6
v =0.3,
m=O.Ol
40
30
20
10
0
 
tll3  Ml60
2~LEK Fig. 4. Overall
elasticplastic
behavior of composite materials shear. Y = 0.3, m = 0.0 I.
changes of the hardening rate after the first yielding; eqn (32). in the case when material
with spherical inclusions
in pure
i.e. the coefficient of the first term of
parameters are chosen so that the matrix yields first.
The lines in the figure indicate the contours of the constant rate with respect to the elastic modulus ratio WI and the volume fraction/:
The smaller the modulus ratio becomes, or the
smaller the volume fraction is. the smaller this rate ofhardening
becomes.
As is clear from cqn (38). this hardening rate is independent of m in the case when the inhomogcncity
yields first. The tcndcncyjs
volume fraction
opposite to those in Fig. 3. and the larger the
is, the smaller the hardening
rate bccomcs, as can bc observed from the
dashed lines in Fig. I. On the other hand, ifsoftcr
and wcakcr materials arc chosen for inhomogcncities,
the
overall material shows a more ductile property. as shown in Fig. 4 : i.e. M = 0.01 and I = l/3 or l/60. The ultimate strength at which both phases bccomc plastic is reduced by addition of such inhomogencitics.
However,
the softer the inhomogeneity
if the matrix material
is chosen so that it yields first,
is, the larger becomes the total deformation
state. as indicated by the solid lines in the figure. On the contrary,
at the ultimate
if the inhomogeneity
yields first, the overall material yields at a relatively lower level of the applied stress, and no improvement
of ductility is observed.
In order to estimate this improvement
of ductility quantitatively,
we define a measure
of ductility D by
(40)
the denominator
of which expresses the yield strain of the matrix alone, and D becomes
unity when no inhomogeneity
exists. Using eqns (32), (34) and (38), we can express this
ductility measure by
D=~+Pu/wo .111 8l
(414
when the matrix yields first, and
D=
when the inhomogeneity
,+//I+[) 1B
yields first. In the latter case, ductility
(4lW is independent of the ratio
s. c.
I X6X
LIN
t!t ul
of the elastic moduli m. as is clear from eqn (41 b). The relations obtained from eqn (41) are shown in Fig. 5 as the contour lines in terms of the yield stress ratio I and elastic modulus ratio RI. From eqns (34) and (39). it can be shown that the strength ofcomposites increases by addition of inhomogeneities if I P I. Therefore. the softer and stronger inhomogeneity increases both the strength and ductility of composite materials. As for the hardening rate of composite materials. Tanaka and Mori (1970) have derived a simple formula as
(473)
while the present method leads to eqn (32). which reads h=
fhr( I  p, 2/l
(42b)
i_ll(ir,l)qjf,
The discrepancy lies only in the last term of the den~~minator of eqn (42b). and it can be negligibly small whenf’is very small. In fact. since eqn (47a) does not sufliciently take into account the interaction effects, it applies only whcn_/‘is small and yields unacceptable results whenf’= I. In comparison with experimental data, accuracy has been examined (see Mura, 1982). An example is a CuSiOl single crystal, where the SiOr particle is the spherical inhomogcncity ; 1~= 46.1 GN II‘, I[* = 31.3 GN m .‘, v = 0.33 andf‘= 0.0052 ; eqn (42a) pivcs tr/& = 2. IX x IO ‘, while eqn (42h) yields h/Z/c = 2. I9 x IO ‘. The volume fraction is so small that the dilrercncc is small. Since cqn (42a) accounts for approximately 65% of
t
I0 7 5
3 2
1 0.7 0.: 0.2 0.:
0.;1
.O(11
.Ol
.l
1
10
100
1000
m Fig. 5. &gxc of ductility improvement of composite nwtorials with spherical inclusions in pure &rtr, Y = O.j.j’= 0.1. In the region indicated by B. the yielding WXU~~ first tn the inclusion?., while the matrix yields first in A.
IY69
Estimating elastic moduli of composites the experimentally
observed
hardening,
the present
method
also gives a lower
estimate
of
hardening. Behavior Since both
at unloading materials
can be also traced stepbystep
become
elastic
becomes the same as the initial the residual unloading
plastic starts
strains at
elasticity
a certain
occurs in the earlier
Hence the overall initial
behavior
yielding.
fied. A typical
behavior
positive
However,
stress
state
material
as CJ?, = 6,.
in loadingunloading
by the amount
achieved
where
of composite
materials
elasticperfectlyplastic
pressure component.
Therefore
this successive of {a,
(uY~),.). for the
eqn (34) is satis
is shown in Fig. 6.
materials letting
are concerned,
because plasticity h’ denote
the is not
the overall
Nhere X is the bulk modulus
f (1 xl
I
li=
I(I/)(Ik)a'
ratio defined
by
x_
E
h.’
(44)
K’ Since this bulk modulus
remains constant
tangent similar
at any stage ofdoformation,
bulk moduIu<.
11,changes with deformation C, K shows relations
bulk
from eqn (23).
c
as the instantaneous
if
(ay,),
hardening
when
of
For example.
effect of the kinematic
state is always
above. behavior
the accumulation point.
does not change even with the plastic deformation,
we obtain.
the overall
occurs at a?: = { (T,  ~(cJ?~)~). Therefore
the ultimate
affected by the hydrostatic modulus,
shear
stage than the virgin
As far as the LevyMisestype bulk modulus
to the procedure
unloading,
shifts the next yielding
shows the Bauschinger
although
similarly
after
of the composite.
due to yielding
c cU < (~7~)~. the next yielding yielding
immediately
as is given
k can bc considcrcd
On the other hand. the tangent
shear cocllicicnt
in cqns (26). (32) and (37). In the elastic
to those in Fib*. 2 for 11,//c, as is apparent
region.
from the comparison
with eqn (26). When the shape overall
of isotropic
instantaneous
Young’s
modulus
Since k remains ratio becomes A uniaxial
tangent
and Poisson
Corlstant
ratio can bc calculated
condition
the incompressibility can be also applied
shear. In this cast, the incompressibility
Fig. 6. Typical
in an isotropic
also shows an isotropy.
material
is spherical,
Therefore
the
the tangent
from eqns (26). (32). (37) and (43).
and 11,, 0 when both phases become plastic,
l/2. This rcllccts loading
inhomogcncitics
modulus
the tangent
of the LevyMisestype in the same manner
of the plastic deformation
behavior of loadingunloading of composite materials pure shear.
Poisson
plasticity. as that in pure
and the axisymmetry
with spherical inclusions
in
of
s. c.
18X
Li% ZI crl.
the loading condition leads to simplerelationsas (E,P):: = (E,P),~ =  1;2(&,P),,, (n = M,R). If the von Mises yield condition is employed. it can be easily shown that the initial yielding occurs at G 0, , = (CT?,), E J3(07,),. wherr_(upl), is given in eqn (28) or (35). Similarly the ultimate stress is calculated as (o’;),), = ,_/3i,.. The instantaneous tangent Young’s modulus E, is directly obtained from eqn (23). and the results are almost the same as those in Fig. 2 in the elastic range. The hardenins rates of Young’s modulus are obtained as
GW if the matrix yields first, and
(Jjb) if the inhomogcncity
bocomcs plastic first, whore
and rl and 11;trc dcfincd in ccln (22). Equation (45:;) shows the relations quite similar to those in Fig. 3, when HI = k. 5. CONC’l.USION
A simple eatcnsion of the method to cstimatc elastic motfuli of composites is proposed for an elastic plastic composite matcriaf. Obtained overall behavior shows a simple bilinear property in pure shear, and the B~lus~hin~~r elkt can be obscrvcd. A qu~lntitiitive evaluation of ductility improvement has also been carried out. Ac~kn0~clc~r~~~~~frtenr.v  WC
are graA’ul to Prof. T. Mori. Tokyo Institute olTcchnology. and L)r bl. t Iori, Tohoko Univursity, for their usct’ul suggestions and discussions. This rcsarch was supported by U.S. Army Research Oliicc Contract No. DAAL03X8K0019.
REFERENCES
Estimating NcmatNasser,
S. and Iwakuma.
X5565. Tanaka, K. and Mori,
T. (1985).
Elasticplastic
T. (1970). The hardming
l&f. 931941. Willis. J. R. (1977). Bounds and selfconsistent
Mech. Phw. SolicO 25 . f&‘O’ Willis.
1871
elastic moduli of composites composites at tinite strams.
of crystals by nondeforming
fnf. 1. sulj&
Srrucrures
particles and fibres. Acru h#eruff.
estimates for the overall properties of anisotropic
composites. J.
*.
J. R: and Acton. J. R. (1976). The overall elastic moduli of a dilute suspension
of spheres. Q. /. bfech.
Appl. hfuth. 29, 163177.
APPENDIX:
CASE WHEN
SURFACE
DISPLACEMENT
IS PRESCRIBED
Consider that the surface displacement or the farfield strain EO is prescribed instead of the surface traction treated in Section 2. Let E‘,, and 6 denote the disturbed strain fields of the matrix and inhomopeneity respectively. Then these disturbances must satisfy
(1 f)E, Cf& = iI
CA])
Let d,, and d,, denote the local stress fields in the matrix and inhomogeneity. be expressed as 8” = ~(2”+Zvd~), &, = <*(P”
in
+Zo $).
Comparison of eqns (Al) and (A?) leads to the following written in the following form :
respectively. Then these stresses can
(Vf&). in
(A?)
52,.
(A31
expression. and the equivalency condition can be also
citr= CT*I(~“+~,,F::)+(~~ld,,)ASr~ = c’((r’+s” From eqn [A+. (&c,,) and plastic deformation.
2:)+(~~,~V)(~~~fZ*):.
(A4b)
is considcrcd as a disturbed strain crm~po~tcnt due to the existence of inhom[~~cncity Thcrcfore. rcpflrcing (2) in cqtl (Xl hy (4, d,). WC bavc _ r,,Cu.=:
Eliminating
(A4a)
S(AEr+#Y).
(AS)
 E /‘.~(A2P+d*). %
(A61
4, from cqns (A 1) and (AS), wc obtain
Substitution as
ofcqn (AS) intocyn
(A4h) results in an Illtcrn;ltivecxp~ssion
Ljl) =
for the focal stress in the inhomogcncity
c~(z”+s, ir,)+(s~)(A/?+/?)).
(A71
Using eqns (AZ) and (A7). WCcan dclinc the over~lf stress by the svzragc stress as
= C(L”‘+~,,,i_:,)+/C(S~(~~r+E‘*), and substituting
eqn (A6) into it. we obtain ri = C(z”SP,)I’t(Ad’fHI).
(
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On the other hand. the equivalency condition cqn (A4) can bc rewritten
by substitution
ofeqns
(A.5) and (A6)
to obtain (AI’+Z*) Simple manipulation stress field :
6 = {erc[c(f
= [c(1
after substitution
/)(C’C*)Sj‘i(c‘~*)i”C~:+C*~~l.
(A9)
of cqn (At)) into cqn (AX) tcads to the foll~~win~ expression for the overa]]
/)ce
r)e[c(f
/)(c‘P)S]
_’ x (c t?‘)(73)At’.
or inverscfy. p

f’
= IC(CC*):SI(ST,;t‘[tC(I
./xCC*t3;rT‘ci+~(f
~)fCl?)(3r)A27. (AfO)
Equation
(AIO)
defines d for given S’,
but is cxactfy identical with eqn (17). which defines i for given do.
1872
S. C. Lw
Furthermore. several steps of cumbersome eqns (AZ) and (A7) result in d6,
=
manipulation
er ui after substitution
of eqns (A6).
(A9)
and (AIO)
into
a/e(3~[~(~~“)(3/(3~~]‘((eC*)~’a+e*A9’).
rc?~B~ = $+(I /,~(3~[~(~~*)(3/(3~):]‘~(c~g)~’a+~*A~p~. which coincide with eqn (13).
(Al 11