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AVERAGE STRESS IN MATRIX AND AVERAGE ELASTIC ENERGY OF MATERIALS

WITH MISFITTING INCLUSIONS*

T. MORIt and K.

TANAKA7

Hrtving noted an important role of image stress in work hardening of dispersion hardened materials,‘*.3)

the present paper discusses 8 method of calculating the average internal stress in the matrix of a material

containing inclusions with transformation strain. It is shown that the average stress in the matrix is uni-

form throughout the material snd independent of the position of the domain where the average treatment

is carried out. It is also shown that the 8ctual stress in the matrix is the average stress plus the locally

fluctuating stress, the average of which vanishes in the matrix. Average elastic energy is also considered

by taking into account the effects of the interaction among ths inclusions and of the presence of the free

boundary.

COSTRAIXTE MOYENNE DANS LA MATRICE ET ENERGIE ELASTIQUE MOYENNE DES

XATERIAUZi CONTEKATU’T DES INCLUSIONS IMPARFA~S

Ayant remarque que la force image joue un &le important dans la consolidation des mat&iaux

durcls par dispersion, les auteurs proposent ici une m6thode de calcul de la contrainte inteme moyenne

dam la metrice d’un mat&iau conbnant des inclusions pr&entant des d6formations dues $ une trans-

formation, et montrent, que 18 contrainte moyenne dans la matrice est uniforme 8. travera le matbriau

et indbpendsnte de 18 position de la zone dans laquelle le traitement moyen est effect&. 11s montrent

aussi que ia contra&e r&&e dans 18 matrice est &8le $, 18 somme de 18 contrainte moyenne et de la

contfsinte locale variable dont 18 moyenne pour toute 18 metrice tend vers z&o. L’&ergie &stique

moyenne est Bgalement calculb en tenant compte des effets d’interaction entre les inclusions et de 18

presence

u

joint libre.

DIE JlITTLERE SPANNUNG IN DER MATRIX UND DIE MITTLERE ELASTISCHE

ENERGIE VON MATERIALIEN MIT EINSCHLUSSEN

rv’schdem die gro&. Bedeutung der Bildkraft fiir die Verfestigu~ von dispersions geh~e~n

Materi81ien’s+SL beront wurde, diskutiert die vorliegende Arbeit eine Method8 z u r Beschreibung der mitt-

leren inneren Spannung in der Matrix eines Materials, das Einschliisse mit UmW8ndlUngSVen3p8nnUngen

enthitlt, Es wird gezeigt, d8B die mittlere Spannung in der Matrix im ganzen Material gleichfiirmig und

unabhitngig van der Lage des Bemichs ist; fiir den die Behandlung durchgefiihrt wurde. AuRerdem wird

gezeigt, d8B die aktuelle Spennung in der Matrix gleich der mittleren Spannung plus einer lokat

fluktuierenden Spannung ist, deren Mittelwert iiher die gesamte Matrix verschwindet. Die mittlere

eIastische Energie wird ebenf8lls diskutiert un t e r ~~c~~ehti~ng der W~hseIwjrkung‘~~ekte

znischen den Einschliissen und der Gegenwart der freien Oberfliiche.

1. INTRODUCTION

-4s

Brown pointed out in describing the present

situation,(f) some papers have recently appeared which

discuss work-hardening of dispersion hardened mate-

rial~.(*-~) All these papers deal with internal stress

which developed as a result of plastic deformation

occurring only in the matrix. References (2) and (4)

pursue energy consideration and agree with each other

wit’h respect to the hardening rate, while references

(3) and (5) discuss the role of int,ernal stress in the

matrix in work hardeni~~~. Although Brown presented

a comprehensive vipn- of what was discussed in these

papers,

(1) we would like nonetheless to report our

own understanding. for Brown’s explanation and

derivation of some stresses and strains were, in details,

not completely acceptable from our point of view and

we feel that an alternative treatment is possible.

Therefore, the results of some of our calculations

of the internal stress and of elastic energy will be

presented here. Since the internal stress developed

* Received July 22, 1972: revised September 8, 1972.

t Department of Metallurgic81 Engineering, Tokyo Institute

of Technology, Ookayama, Meguro, Tokyo, Japan.

$ Now at: National Research Institute for Metals, Nakame-

guro, Meguro, Tokyo, Japan.

-.

by uniform plastic deformation occurring only in the

matrix can be duplica~d by giving uniform tranafor-

mation strain to inclusions alone, internal stress will

be charscterized in the following sections by inclusions

with uniform transformation strain. Thus, some ofthe

results will be directly applicable to a material with

misfitting precipitates.

2. AVERAGE INTERNAL STRESS IN MATRIX

For simplicity, the case where elastic constants

Ciinl are uniform throughout a specimen V, will be

considered. Suppose the specimen has N inclusions

which are, on the macroscopic scale, uniformly dis-

tributed in the matrix. When an inclusion V acquires

uniform transformation strain EijT, total strain eij’ is

introduced into the specimen. E

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ACTA NETALLURGICA, VOL. 21, 1973

inside the inclusion. cij

m is given by

Epj* =

E,,

’

+

4j, li(xa

~‘)1/2}

Dtx’)>

(1)

where Gki is Green’s function for an infinite homogen-

eous body with elastic constants Cijkl.@) It can be

shown t,hat when aU inclusions acquire identical

transformation strain pijT, the specimen as a whole

undergoes a shape change and the average strain,

(Q~~)~~, that descibes the specimen shape change is

given by

(EijF)gTb = fe* jTt

(2)

where f is a volume fraction of the inclusions. From

this, let us assume the following : (1) the average of

total strains, in domain V,, from all the inclusions,

is also equal t,o

fc i i

f

VR

is

a

representative domain

for the specimen.

This implies that V has a sufficient

number of inclusions, say, Y inclusions,

MV/V, = f,

and

VR

is not in a neighborhood close to the boundary

1V,,l.

Assumption (I) is realistic. As an example, let

us consider the lattice constants of martensite. The

macroscopically determined ltverage lattice constants

of a martensite single crystal (if it were present) are

believed to be equal to those measured by a narrow

beam X-ray technique if interstitial atoms are uni-

formly distributed. Assumption (1) was justified by

Eshelby when &ij

T = &aij (bij is the Kronecker

delta).‘?)

Let us consider the case where

V,

has a shape

similar to that of the inclusions. Then total strain

cijF is expressed as

M

P M

s

-j-

2 e,,iyx. Pf,

(3)

I=1

where .zijm(x, xp) is the quant,ity defined by equation

(1) when the P-th inclusion is at xp, ~~~~‘~(5,p) is t.he

image strain due to the P-th inclusion, and

VR

is

assumed to contrtin inclusions 1 to i@. Apparently,

the second and the third sums are eIast,ic and do

not fluctuate much in VR.

Therefore, it, is meaningful

to define the average of the second and the third

sums in equation (3) in

V,

as

t,he average elastic strain

(%j)Y,3

As shown in a previous paper,(*) when a single inclusion

V with uniform transformation strain is within an

infinite homogeneous body, volume integrals of total

stmin and stress vanish if the integration is carried

out, in the region

V’ - I’,

where V’ is a domain

surrounding

V

and is of shape similar to that, of I’.

Applying this conclusion, the average of the first sum

of equation (3) in V, is calculated as

i l f

where XijKlare Eshelby’s tensors and are equal to the

integrals in equation (1) when x is within I’ in equa-

tion (l).(6) Assumption (1) and equat.ions (3) through

(5) give

( ij)yn = -f K jmnemnT - ejjT)-

(6)

Thus, the average elastic strain defined above is

independent of the posit,ion and the size of I’,.

The average internal &ress, (~i~~o)~-~,efined by

(Oi$O>va ~i$&&-R’ (7)

is also independent of the position and the size of

V,,

insofar as

VR

can be regarded as a representative

volume. (aijo) v,

is the average in V, of the sum of

the image stresses of all the inclusions and the stresses

of the inchtsions outside

V,

when they are in an

infinite body. However, it should be noted that. this

sum itself is nearly constant in V,.

Next, let us consider the following sum,

where ai j

m x, zp)

is internal stress due to the P-th

inclusion at xp within

V,

when it, is in an infinite

body. This sum, ai,* (1, M) cannot be assumed nearly

constant in V,;

instead, it fluctuates and t,he wave

length of the fluctuation is apparent.ly of the order of

the inter-inclusion spacing. cijm (1,

X)

is called 1ocaIIy

fluctuating stress, to which nearby inclusions obviously

contribute predominantly. However, the average of

this locally fluctuating st,ress in the matrix of Vn can

be shown to be zero. Instead of averaging equation

(8) directly, let us take the following approach.

Consider a fixed point in V,. First imagine V,

without inclusions. Next introduce an inclusion into

V,

such that the fixed point is always outside the

inclusions and record (T$~~~V. Here, uilm is the stress

due to the inclusion when it is in an infinite body snd

SV is a small volume element around the fixed point.

If the introduction of the inclusion is repeated many

times in a random manner, Caij”6V becomes propor-

tional to the integral, Sv,_ v aiim6 V, where the center

of the inclusion is conversely fixed at the opposite

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AND

TAKAKA:

MATERIALS WITH MIISFITTISG INCLUSTOSS

673

position of the above fixed point from the center of

Ti,.

Because of the statement following equation (4),

this integral vanishes. Since this is t,rue for every

inclusion, the average of ~iirrn rom ali the actually

present, inclusions in VR

at a point within the matrix in

V, becomes zero. That is,

@ij”(ll M)), = 0.

(9)

The total average stress, (CT&~, in the matrix within

1B is the sum of {G,~O}~~and (~,~“(l, M)jnl. From

equations (6), (7) and (9),

with

Csijr*

= %?JL&nnT - 4%

(11)

where oiila? is stress within a single inclusion when it is

in an infinitely extended body. {aJIM is independent

of the size and position of

V and

of the shape of the

specimen.

In summary, the internal stress at any point in the

matrix is the uniform average stress (aJM plus the

locally fluctuating stress from nearby inclusions.

This locally fluctuating stress is averaged to be zero

in the matrix. It is important to note that (a&X is

the average, within the matrix, of the sums of the

st’resses of the inclusions when they are present in an

infinite body (Cbijm) and of the image stresses of all

the inclusions (X(rijim). The relative contributions of

Cuijg and of Ceijim are not generally determined.

However, as shown in the appendix, the contribution

from each term can be calculated, if the specimen is of

ellipsoidal shape. Especially when the specimen is

similar in shape to the inclusions, (u~~);,~ s solely due

to LT,,‘“.

Bs Brown mentioned,“) when it moves in the mat-

rix as a straight line, a dislocation feds, as a whole,

only t,he average stress-(oij)4,f, since the average of the

locally fluctuating stress in the matrix is zero. In

such a case. an extra applied stress equal to - (u,~)~~

ia needed to move a dislocation from the

case

where

there is no internal stress. If &ijT s regarded as pro-

duced by plastic deforma~,ion, t,he above hardening is

work hardening which Brown and Stobbs identified

as due to the image stre ) and to which energy

balance consideration was applied.f2) The equivalency

of both treat,ment>shas been discussed by Hart.t5)

rom

the present understanding of internal st.ress,

Hart’s treatment of work hardening of dispersion

hardened materiaW5) is identified as that which

emphasizes and estimates the role of the locally

fluctuating stress. As t’reated by Hart,

a

dislocation

feels obstruct,ion from t,he locally fluctuating stress

when it is flexible.

Brown and St,obbs also considered

t,he role of the locally fluctuating stress in work

hardening in a somewhat di~erent manner.(3~ It is

important to note that, hardening due to this factsor

operates as an energy dissipation mechanism.“*5)

Because of its fluctuating and position-dependent

character, estimation of hardening due to the locally

~uctuating stress involves certain approximations and

seems to depend on the choice of particular situations

to

be

considered; thus, it would not seem to be as

uniquely and simply performable as estimation of

hardening due to the average stress in the matrix,

taij)M*

3. ~NTE~CTIO~ AMONG INCLUSIONS AND

ENERGY CONSIDERATION

That the average internal stress in the matrix is

equal to -fa..I””

naturally indicates that t’he actual

stress within cn inclusion is, on the average, equal to

aijlm - foijIco.

Thus, elastic energy per unit. volume

of the specimen, EeEft,s given by

E,1” = -f(l - f)oii%ilT/2.

(12)

In Reference (2), the term (1 - f) was omitt,ed on the

assumption that the interaction among inclusions can

be ignored. However, equation (12) is a correct

expression which takes into account interaction among

inclusions together with the effect of the presence of

the free boundary of the specimen. If f is small,

(1 - f) in equation (12) can be, for all practical

purposes, replaced by unity. However, whenf is large,

ignoring the effect,s of the interaction among inclusions

and the presence of the free boundary, as was done

previously,@) may lead to a significant error.

As already mentioned, Hart clarified,t5) in the case

of n-ork hardening of dispersion hardened materials,

that a. hardening rate derived from consideration of

energy balancet2) is essentially equivalent to that

derived from consideration of the average internal

stress in the matrix.cxv3’ If equat,ion (12) is used for

an elastic energy calculation,

exact

agreement of the

hardening rates oalculat,ed by the two approaches is

verified.

ACKNOWLEDGEMENTS

We

appreciate discussion with Professor T. Mura of

Northwestern Universit,y and the encouragement, he

gave us. Dr. L. M. Brown not only supplied us with a

copy of his paper before publication but also corn.

mented on our &udy, and we greatly appreciat,e his

kindness.

REFERENCES

1. L. M.

BROWN

Private communication to be publidwd in

Acta Met.).

0. K.

TAKAKA

and T. MORI, Acta Net. X8, 931 1970).

3. L. M.

BROWN

rend W. M.

STOBBS

Phil. Mw. eS 1185

1971).

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574

ACTA METALLURGICA, VOL. 21, 1973

4.

T. B&IRA,&tat.

onf. Me Pmpetiies of Materias at K.yoto,

Vol. 2. 1971).

5. E. W. HART Aeta Met. 80,275 1972).

6. J. D. ESEIELBY,

Proc. R. Sot. A %&

376 1957).

7. J. D. ESHELBY, J. app. Phys. 25,255 1954).

8. K. TANAKA and T. MORI, J. Bmicity. To be published.

9. Ii. TANAKA and T. MORI, Phil. Mag. 25,737 1971).

10. N. KINOSRITA and T. MURA, Whys. Status Solidi. 5, 759

1971).

APPENDIX

AVERAGE STRESS AND IMAGE STRESS IN A

UNIFORM ELLIPSOIDAL BODY CONTAINING

ELLIPSOIDAL INCLUSIONS WITH TRANS

FORMATION STRAIN

Suppose that an ellipsoidal body, VO, contains an

ellipsoidal inclusion I’ with transformation strain

~~~~~Elastic constants Cijkl are uniform throughout

the body. Let dijDo be stress, assuming the inclusion

to be in an infinite body, and let rrijim be image stress

to reflect the presence of the boundary

IV,,l

in the

actual body. The total internal stress in the actual

body, oij, is uijm + aijim.

Since for static equilibrium,

s

crij

dD =

I’0

r

(dijm + CT~;~)dD = 0,

ti r-0

.r

qjin dD =

- jjj-

dD -s,_pj =‘dD.

F-o

(Al)

As u,

m in

V

is uniform,‘@ the first term in equation

(Al) is expressed as

where ,,,(I’) are Eshelby’s tensors defined for the

shape of the inclusion. Reference (8) show that the

second term in equation (Al) is independent of the

size of the body

V,

and of the position of the inclusion

and is given by

_

-

4&nn(v)l~?nnTt

A31

where ~~~~~(Va) are Eshelby’s tensors appropriate

for the shape of the body I’,. Thus, the average, in

the matrix, of clijac of a single inclusion is written as

laii4).lI =

(aij”>~o-v

= {v/(vCl - V)}Cijkl

x [4c,,,CVo, - ~mAWmnT.

(W

Equations (A3) and (A4)

do not depend on the position

of the inclusion.

From equations (Al) and (A3),

s

UiiimdD = -VCijKI[Xktllnn(V~Vo)EmnT &klT].

(A5)

v

0

Consequently, the average of the image stress in the

body,

V,,

is calculated as

(air”“>r-a= -(vlvotc,j~,Es*,,,(v~cl,)e,,T - *?I*

(A6)

Equations (A5) and (A6) are independent of the shape

and the position of the inclusion. What is used in bhe

above derivation is uniformit’y of sbress and straiu

within an ellipsoidal inclusion with uniform trans-

formation strain when the inclusion is in an infinite

body. This uniformity is assured in an anisot,ropic

case.(r”j Thus, the above conclusions are valid when

the body is anisotropic.

Finally, the average stress in the matrix will be

considered when the body has many inclusions, the

number of which is N, all of which are identical in

shape and volume. When the body is similar in shape

to the inclusions, the result of equation (A4) becomes

zero and the average stress in the matrix becomes

identical to the sum of the average image stress

given by equation (A5). However, when the body has

a different shape, the average stress in the matrix is,

from equations (A4) and (Ati),

(ofj>Jf = -(Nv/v*)C;.jkl[Sklmn(VO)EmnT -

&klTl

+

(NV~~‘~)( I - V/VJ’Cii,,

x 1h n(VcJ - mAwmnT.

Here,

NV/ V0

is equal tof, the volume fraction of the

inclusion, and is finite. Since N is a large number,

V/V,

must be negligibly small compared to unity,

Consequently, the above value becomes

((3ij) izI = -~c~j*~[~*~~~(v)&~~T - &* lTI*

In terms of the stress, defined in equation

(I I),

which

is theinternal stress within a single inclusion in an

infinitely extended body, (ni,) is written as

Xaij)+lI = -f”iflm,

which is, of course, ident8ical to equation (10).