Average elastic-plastic behavior of composite materials

Average elastic-plastic behavior of composite materials

ollZo-768392 f5oD+ .?a Pergamon Pms pk AVERAGE ELASTIC-PLASTIC BEHAVIOR COMPOSITE MATERIALS OF S. C. LEN, C. C. YANG and T. MURA Department of ...

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ollZo-768392

f5oD+

.?a

Pergamon Pms pk

AVERAGE

ELASTIC-PLASTIC 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. Aoba-ku. Sendai.980. Japan Abstract-A new approach to estimate the overall behavior of an inhomogeneous body is applied to investigate the average elastic-plastic 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 elastic-perfectly 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 so-called self-consistent 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 elastic-plastic incremental relations of crystalline materials and composites. fwakuma and Nemat-Nasser 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, theself-consistent 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 Mori-Tanaka theory (Mori and Tanaka, 1973). Using the modification of the Mori-Tanaka 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 far-field displacement is prescribed. In other words, the estimation of material properties can be carried out independently of the far-field 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 elastic-plastic materials

within

perfectly-plastic

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 two-phase 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 ( = D-R)

and inhomogeneities

are (? and c*. respectively. These materials are elastic -perfectly-plastic

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:(~“+ty-c/;,),

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’-(c-c’*)Is-r(s-1‘,)]

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 stress-strain

eqn (1-t) 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: = i-i’,

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)(K--K”) __._..~._

(.p+(l_/)

[

_,

(/I-_ __

l)(/L-_IL*) _ _ _ ..,. _ n

1

+ ( I _/‘I

3(r~_

_. (T

I)tiK*

.4

where

7(/1-- I)iL/L* . ~. ~-mm-i- ---

1

Afp,

(21)

1863

Estimating elastic moduli of composites

A E K- (z-f(z-

I);@--K*).

B = vuww-

(22)

1);tp-p’).

Similarly eqn (17) becomes

K-(1 f-p

-j-)(K-

=

h.*b P--(1 -l-m-P’)B

~KA



2lrB

+_I-(1 -f)

1c7 -0

(z- 1):-h.*)

,(P- IIf-lr’)

[

&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,-.x2-plane. 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-(I-J)(l-,,,)b

non-dimensional

I,] _ -ElZ=2C’

_/-(I-ml { ‘-

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 stress-strain

relation can be obtained as

(32)

Lvhich shows ;I bilinear relation together with eqn (26). The cocflicicnt in the first term ot the right-hand 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(C-EP),

* = a72 +

f(l

-m)

1865

5F

m

(36)

Y

and

[email protected]:=

/ (1

f[l -(I -ew(Bm(l-/0(1-f>

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)(l-f)+++ The ultimate maximum

7:.

(I-/?)(I--f)+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

ix-homogeneity

+

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

elastic-plastic

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

elastic-plastic

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 of-hardening

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 8-l

(414

when the matrix yields first, and

D=

when the inhomogeneity

,+//I+[) 1-B

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(i--r,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 Cu-SiOl 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 step-by-step

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 loading-unloading

by the amount

achieved

where

of composite

materials

elastic-perfectly-plastic

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

N-here X- is the bulk modulus

f (1 -x-l

I-

li-=

I-(I-/)(I-k)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 Levy-Mises-type 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 Levy-Mises-type in the same manner

of the plastic deformation

behavior of loading--unloading 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. DAAL03-X8-K-0019.

REFERENCES

Estimating Ncmat-Nasser,

S. and Iwakuma.

X55-65. Tanaka, K. and Mori,

T. (1985).

Elastic-plastic

T. (1970). The hardming

l&f. 931-941. Willis. J. R. (1977). Bounds and self-consistent

Mech. Phw. SolicO 25 . f&‘O’ Willis.

1871

elastic moduli of composites composites at tinite strams.

of crystals by non-deforming

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, 163-177.

APPENDIX:

CASE WHEN

SURFACE

DISPLACEMENT

IS PRESCRIBED

Consider that the surface displacement or the far-field strain E-O 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”+Zv-d~), &, = <*(P”

in

+Zo -$).

Comparison of eqns (Al) and (A?) leads to the following written in the following form :

respectively. Then these stresses can

(V-f&). in

(A?)

52,.

(A31

expression. and the equivalency condition can be also

citr= CT*I(~“+~,,-F::)+(~~l-d,,)-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).

([email protected]

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 = {e-rc[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?‘)(7-3)At’.

or inverscfy. p

-

f’

= IC-(C-C*):S-I(S-T,;t-‘[tC-(I

-./xC--C*t3;rT-‘ci+~(f

--~)fC-l?)(3-r)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 d-6,

=

manipulation

er ui after substitution

of eqns (A6).

(A9)

and (AIO)

into

a-/e(3-~[~-(~-~“)(3-/(3-~~]-‘((e-C*)~-’a+e*A9’).

rc?~B~ = $+(I -/,~(3-~[~-(~-~*)(3-/(3-~):]-‘~(c-~g)~-’a+~*A~p~. which coincide with eqn (13).

(Al 11