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MECH 503 Introduction to Mechanics of Defects in Solids

Chapter 3 Theory of Inclusions and Inhomogeneities

Reference book: T. Mura, Micromechanics of defects in Solids.

Micromechanics is a subject that encompasses mechanics related to microstructures

of materials. The methods employed here is a continuum theory of elasticity yet itsapplications cover a broad area relating to the mechanical behavior of materials:

plasticity, fracture and fatigue, constitutive equations, composite materials,

ploycrystals, etc. These subjects are treated here by means of a powerful and unified

method that is called the eigenstrain method. In this chapter, we will briefly

introduce the general theory of eigenstrain method in section 3.1; in section 3.2

typical problems and solutions relating to inclusions and inhomogeneities are

analyzed by this method in detail; in section 3.3 applications of the solution of the

inclusions and inhomogeneities in classical nucleation of phase transformation and

overall moduli of the composites are demonstrated.

3.1 Eigenstrain Method and Solution Technique

3.1.1 Concept of EigenstrainEigenstrain is a generic name given to such nonelastic strains as thermal

expansion, phase transformation, initial strains, plastic strains and misfit strains. Then

eigenstress is a generic name given to self-equilibrated internal stresses caused by

one or several of these eigenstrains in bodies which are free from any other external

force and surface constraint. The eigenstress fields are created by the incompatibility

of the eigenstrains. For example, when a part of material (Fig1.1) has itstemperature raised by T , thermal stress ij is induced in the material by the

constraint from the part that surrounds . The thermal expansion T , where is thelinear thermal expansion coefficient constitutes the thermal expansion strain,

(1.1)Tijij =

*

D

Fig1.1 Inclusion

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The thermal expansion strain is the strain caused when can be expandedfreely with the removal of the constraint from the surrounding part. The actual strain

is then the sum of the thermal and elastic strains. The elastic strain is related to the

thermal stress by Hookes law. The thermal expansion strain (1.1) is a typical

example of an eigenstrain.

When an eigenstrain is prescribed in a finite subdomain in a

homogenous material (see figure 1.1) and it is zero in the matrix , then iscalled an inclusion. If a subdomain

*ij

D D in a material has elastic moduli different

from those of the matrix, then is called an inhomogeneity. Applied stresses will be

disturbed by the existence of the inhomogeneity. This disturbed stress field can be

simulated by an eigenstress field by considering a fictitious eigenstrain in in a

homogeneous material.

D

*

ij

In the following section the solution technique to solve the field equations of

eigenstrain problems by using elasticity theory will be reviewed. The technique aims

at finding displacements , strainiu ij , and stress ij at an arbitrary point

when a free body is subjected to a given distribution of eigenstrain .

),,( 321 xxxx

D*

ij

3.1.2 Solution Technique for Eigenstrain ProblemsI. Hookes law

For infinitesimal deformations considered here, the total strain ij is regarded as

the sum of elastic strain and eigenstrain ,ije*

ij

(1.2)*ijijij e +=

The total strain must be compatible, so

(ijjiij uu ,,

2

1+= ) (1.3)

where jiji xuu = /, . The elastic strain is related to stress ij by Hookes law:

( )*

klklijklklijklijCeC ==

(1.4)or

( )*, kllkijklij uC = (1.5)

where are elastic moduli and the summation convention for repeated indices is

employed. Since is symmetric

ijklC

ijklC ijlkijkl CC = , we have lkijlkklijkl uCuC ,, = . In the

domain where , (1.5) becomes0* =ij

lkijklklijklij uCC ,== (1.6)

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II. Equilibrium conditions

When eigenstresses are calculated, material domain D is assumed to be free

from any external force and any surface constraint. If these conditions for the free

body are not satisfied, the stress field can be constructed from the superposition of theeigenstress of the free body and the solution of a proper boundary value problem.

The equations of equilibrium are

( )3,2,10, == ijij (1.7)

The boundary conditions for free external surface forces are

0=jij n (1.8)

where is the exterior unit normal vector on the boundary ofD . By substituting

(1.5) into (1.7) and (1.8), we have

jn

*

,, jklijklljkijkl CuC = (1.9)

and

jklijkljlkijkl nCnuC*

, = (1.10)

It can be seen that the contribution of to the equations of equilibriums is similar to

that of a body force since the equations of equilibrium under body force with zero

are

*ij

iX

*

ij iljkijkl XuC =, . Similarly, behaves like a surface force on the

boundary. Thus, it can be said that elastic displacement field caused by in a free

body is equivalent to that caused by body force and surface force .

In subsequent sections, D in most cases is considered as an infinitely extended body

(infinite body), and condition (1.8) is replaced by the condition

jkjijkl nC*

*

ij

*

, jklijklC jklijkl nC*

( ) xx 0 forij .

III. General expressions of elastic fields for given eigenstrain distributions

The case where a given material is infinitely extended is of particular interest for

the mathematical simplicity of the solution as well as for its practical importance that

we will discuss in 3.2. When the solution is applied to inclusions problems, it can be

assumed with sufficient accuracy that the materials are infinitely extended since the

size of the inclusions is relatively much small compared to the size of the macroscopic

material samples.

Since the linear theory of elasticity allows for the superposition of solutions, we can

expand in the Fourier series form,( )x*ij

, (1.11)( ) ( ) ( )xx = iijij exp

**

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and for each single wave ( ) ( )x iij exp*

, we suppose that the solution of (1.9) may

be expressed in the same form:

( ) ( ) ( )xx = iuu ii exp~

(1.12)

Substituting (1.11) and (1.12) into (1.9), we have

jklijkljlkijkl iCuC *= (1.13)

Expression (1.13) stands for three equations for determining the three unknown iu for

given*

ij . Using the notation

( ) jlijklik CK = , jklijkli iCX *= (1.14)

the expression (1.13) can be written as:

( )ikik XuK = (1.15)

then iu is obtained as

( ) ( ) ( ) DNXu ijji /= , (1.16)

where are cofactors of the matrix( )ijN K , and ( )D is the determinant of K .Substituting (1.16) into (1.12), we have

( ) ( ) ( ) ( ) ( )xx = iDNiCu ijlmnijkli exp~ 1* (1.17)

Then the displacement field can be easily obtained as superpositions of the solution

for single waves, namely,

( ) ( ) ( ) ( ) ( )xx =

iDNCiu ijlmnjlmni exp1*

(1.18)

Similarly, if is given by the Fourier integral form,*ij

( ) ( ) ( )

= xx diijij exp

** , (1.19)

where

( ) ( ) ( )

= xxx diijij exp)2(

*3*

, (1.20)

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we have

( ) ( ) ( ) ( ) ( ) xx diDNCiu ijlmnjlmni

= exp1* (1.21)

When Greens functions ( )'xx ijG are defined as

( ) ( ) ( ) ( ) ( ){ } xxxx '' diDNG ijij =

exp213 (1.22)

(1.21) can be written as

( ) ( ) ( ) ''' xxxxx dGCu lijmnjlmni

= ,

* (1.23)

Sometimes Greens function are called the fundamental solution because ( )'xx ijG isthe displacement component in the direction at point x when a unit body force in

the direction is applied at point in the infinitely extended material.

ix

jx'

x

With the displacement field in hand, the strain and stress field can be obtained

directly,

( ) ( ) ( ) ( ){ } '''' xxxxxxx dGGC lijkljikmnklmnij +=

,,

*

2

1 (1.24)

and

( ) ( ) ( ) ( ))( *,* xxxxxx ''' klqlkpmnpqmnijklij dGCC +=

. (1.25)

If we can obtain the Greens functions, the displacement field, strain field and stress

field can be calculated by equations (1.23), (1.24) and (1.25), Greens functions have

been obtained explicitly only for isotropic and transversely isotropic materials.

Therefore, for practical calculations of other materials the expression (1.21) is much

more convenient.

IV. Static Greens functions

Here we just list the result of explicit expression of the static Greens function of

isotropic materials for further usage:

( ) (

+

+= 2/

22

8

1xxx

xG jiijijij

x ) (1.26)

where ( ) 2/1iixxx == x , and are lame constants. Substituting (1.26) into (1.23),

(1.24) and (1.25), the closed solutions of the eigenstrain problem of the isotropicmaterial are obtained.

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3.2 Isotropic ellipsoidal inclusion and ellipsoidal inhomogeneities

In this part, explicit formulae are derived for elastic fields caused by

ellipsoidal inclusions following the method discussed in 3.1. As to ellipsoidal

inhomogeneities, we emphasize the equivalent inclusion method, since it provides aconsistent method whereby the results of ellipsoidal inclusion are used.

3.2.1 Isotropic ellipsoidal inclusions

An ellipsoidal inclusion is considered in an isotropic infinite body. Eigenstrainsgiven in the ellipsoidal domain are assumed to uniform (constant). The solution of the

problem has been given out by Eshelby (1957) who is a pioneer in the field of

micromechanics. Expressions for the solution are different for interior points(points

inside the inclusion) and exterior points(points outside the inclusion). Eshelbys most

valuable result is that the strain and stress fields are uniform for the interior points.

Fig2.1 An ellipsoidal inclusion with

1x

2x

3x

From (1.23) we have

( ) ( ) ',* dxGCu lijmnjlmni ='xxx (2.1)

where (see Fig 2.1) is given by

(2.2)1///2

3

2

3

2

2

2

2

2

1

2

1 ++ axaxax

Substituting (1.26) into (2.1) and after lengthy calculation the displacement field can

be obtained. An attractive conclusion of this isotropic ellipsoidal inclusion is that the

strain (and therefore the stress) is uniform inside the inclusion. And the strain inside

the inclusion can be written as

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(2.3)*

klijklij S =

and the Eshelbys tensor has the following properties:

ijlkjiklijkl SSS == (2.4)

The stress inside the inclusion can be written as

( )** klmnklmnijklijijklij SCeC == (2.5)

Here we just list explicit expressions of the Eshelbys tensor for sphere shapes of

inclusions.

Sphere ( aaaa ===321

)

)1(15

57333322221111

v

vSSS

=== ,

)1(15

15332222111133331122331122

v

vSSSSSS

====== (2.6)

)1(15

54313123231212

v

vSSS

===

and the stress inside the sphere inclusion are

( ) ( )*

33

*

22

*

1111115

152

)1(15

152

115

16

v

v

v

v

v

+

+

= (2.7.1)

( )*

1212115

572

v

v

=

(2.7.2)

All other stress components are obtained by the cyclic permutation of (1,2,3). As to

points outside the inclusion, the stress and strain are nonuniform and quite different

from those inside it.

I. Energy of inclusions

First we consider the case when the body D is free from any external force and

surface constraint, but eigenstrains are prescribed in*ij . The elastic strain energy

is

dDeWD

ijij= 21*

(2.8)

where and

*

ijijije = ( )ijjiij uu ,,21

+= . Integrating by parts we obtain

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0,, == dDudSnudDuD

ijij

S

jiij

D

jiij (2.9)

since 0=jij n (free surface condition) on S and 0, =jij (equilibrium condition) in

and is the exterior unit vector normal to S. Therefore, we haveD jn

dDWD

ijij=**

2

1 (2.10)

If is an ellipsoidal inclusion (of volume V) and is uniform, *ij ij in is also

uniform. Then (2.8) is written as

**

2

1ijijVW = (2.11)

ij has been obtained by (2.5) for isotropic materials. When the inclusion is of

spherical inclusion, (2.11) can be written as

( )( ) ( ){

}))(57(

)15(4145

8

2*

31

2*

23

2*

12

*

11

*

33

*

33

*

22

*

22

*

11

2*

33

2*

22

2*

11

3*

+++

++++++

=

v

vv

aW

(2.11a)

Consider now the case when the body D , containing inclusion , is additionallysubjected to external surface traction F (i.e., the surface force is prescribed on S).

The displacement field is the sum of u and , where is the displacement if act

alone in the absence of eigenstrains, and is due to eigenstrains prescribed in the

inclusions. Then, the elastic strain energy is

i

i

0

iu0

iu iF

iu

( )( )dDuuWD

ijjijiijij ++=*

,

0

,

0*

2

1 , (2.12)

where . Since0

,

0

lkijklij uC= 0, =jij in andD 0=jij n on S, the integration by parts

gives

( ) =+D

jijiij dDuu 0,0

, (2.13)

It is also seen that

( ) =D

ijjiij dDu 0*

,

0 (2.14)

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since .( ) 0and)( 0,0,0,*,0,*,0 ==== dDuuuuuCuD

jiijjiijkllkijjilkijklijjiij

The elastic strain energy, therefore, becomes

= dDdDuW ijijD

jiij

*0

,

0*

2

1

2

1 (2.15)

It is interesting to note that the elastic strain energy is the sum of the two energies

caused respectively by and . What about the energy for displacement boundary

conditions?

iF*

ij

II. Interaction energy of inclusion

The total potential energy of a body subjected to a surface traction andeigenstrains in

iF*

ij is defined by

( ) +=S

iii dSuuFWW0*

(2.16)

where on S, and is defined by(2.12). If in , W becomesijij Fn =0

*W 0* =ij D

=S

ii

D

jiij dSuFdDuW00

,

0

0

2

1 (2.17)

If on S, W becomes0=iF

( )

== dDdDuW ijijD

ijjiij

**

,12

1

2

1

(2.18)

10 WWWW = . (2.19)

Then we have

dDdSuFW ijijS

ii

== *0 (2.20)

If is an ellipsoidal inclusion and is uniform, *ij

*0

ijijVW == . (2.21)

Under a constant temperature condition, the elastic strain energy of a body isthe Helmholtz free energy of the body (W* in eq. (2.15)). The total potential energy

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(2.16) is the sum of the elastic strain energy of the body and the potential energy of an

external force and corresponds to the Gibbs free energy of the body. One should

mention that (2.18) could also be considered to be the Gibbs free energy of the body

when an external force is absent and an inclusion is a source of the stress field. Eq.

(2.17) is the Gibbs free energy of the body when only an external force is a source of

the stress field. Therefore, the interaction term (2.19) is an extra part of the Gibbsfree energy of the body, produced by the coexistence of the two sources of the stress

field. It is often called the interaction Gibbs free energy between and .*ij iF

3.2.2 Ellipsoidal inhomogeneities

When the elastic moduli of an ellipsoidal subdomain of a material differ from

those of the remainder (matrix), the subdomain is called an ellipsoidal

inhomogeneity. Voids, cracks and precipitates are examples of the inhomogeneity. A

material containing inhomogeneities is free from any stress field unless a load is

applied. On the other hand, a material containing inclusions is subjected to an internalstress (eigenstress) field, even if it is free from all external tractions. If an

inhomogeneity contains an eigenstrain, it is called an inhomogeneous inclusions.

Most of the precipitates in alloys and martensites in phase transformation are

inhomogeneous inclusions. Eigenstrains inside these inhomogeneous inclusions are

misfit and phase transformation strains.

Eshelby first pointed out that the stress disturbance in an applied stress due to

the presence of an inhomogeity can be simulated by an eigenstress caused by an

inclusion when the eigenstrain is chosen properly. This equivalency will be called the

equivalent inclusion method.

I. Equivalent inclusion method for inhomogeneity

Consider an infinitely extended material with the elastic moduli ,

containing an ellipsoidal domain

ijklC

, with the elastic moduli . is called an

ellipsoidal inhomogeneity. We investigate the

*

ijklC

disturbance in an applied stress caused

by the presence of this inhomogeneity. Let us denote the applied stress at infinity by

and the corresponding strain by0

ij ( )0,0,2

1ijji uu + . The stress disturbance and the

displacement disturbance are denoted by and , respectively. The total stress

(actual stress) is , and the total displacement is . Stress components

ij iu

ijij +

0

ii uu +

0

ij are self-equilibrium; that is,

0, =jij (2.22)

and 0=ij at infinity. When a finite body is considered,

0=jij n (2.23)

on the boundary. Hookes law is written as

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( )( ) +=+

+=+

-Din

in

,

0

,

0

,

0

,

*0

lklkijklijij

lklkijklijij

uuC

uuC

(2.24)

The equivalent inclusion method is used to simulate the stress disturbance

using the eigenstress resulting from an inclusion which occupies the space .Consider an infinitely extended homogeneous material with the elastic moduli

everywhere, containing domain

ijklC

with eigenstrain . has been introduced here

arbitrarily in order to simulate the inhomogeneity problem by use of inclusion method.

Such an eigenstrain is called an equivalent eigenstrain. When this homogeneous

materials is subjected to the applied strain

*

ij*

ij

( )0,0,02

1ijjiij uu += at infinity, the resulting

total stress, distortion, and elastic distortion, respectively, are , ,

and in . Then, Hookes law yields

ijij +0

jiji uu ,0

, +*

,

0

, ijjiji uu +

( )( ) +=+

+=+

-Din

in

,

0

,

0

*

,

0

,

0

lklkijklijij

kllklkijklijij

uuC

uuC

(2.25)

where . The necessary and sufficient condition for the0

,

0

lkijklij uC= equivalency of

the stress and strain in the above two problems of inhomogeneity (eq. (2.24)) and

inclusion (eq. (2.25)) is

( ) +=+ in)( *,0,,0,* kllklkijkllklkijkl uuCuuC (2.26a)

or

( +=+ in)( *00* klklklijklklklijkl CC (2.26b)

As mentioned in the preceding sections, kl in the above equation can be obtained as

known function of when the eigenstrain problem in the homogeneous materials

solved. Thus Eq. (2.26) determines for a given , in such a manner that

equivalency holds. After obtaining , the stress can be found from (2.24)

or (2.25).

*

kl*

kl0

kl

*

kl ijij +0

If0

ij is a uniform stress,*

kl is also uniform in . Then, from (2.3) we have

*

klijklij S = (2.27).

Substituting this into (2.26b) leads to

( ))(**0*0*

klmnklmnklijklmnklmnklijkl SuCSC +=+ (2.28)

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from which the six unknown components of , are determined (note that is

proportional to or ).

*

kl*

kl0

kl0

kl

II. Inhomogeneous inclusion

Sometimes the inhomogeneity may involve its own eigenstrain. An example is

the formation of martensite blades in quenched carbon steel and precipitations in

alloys. Let a material D , containing an ellipsoidal inhomogeneous inclusion , be

under a stress field . is the applied stress if the material is homogeneous,

i.e. having no inclusions.

ijij +0

0

ij

ij is the sum of the two stress disturbances, one caused by

the inhomogeneity, and the other, by the eigenstress associated with eigenstrain in

. By Hookes law,

p

ij

( )( ) +=++=+

-Dinin

,

0

,

0

,

0

,

*0

lklkijklijij

p

kllklkijklijij

uuCuuC

(2.29)

where0

,

0

lkijklij uC= in ,D

00, =jij in ,D

on surface , (2.30)ijij Fn =0 S

in ,0, =jij D

on surface ,

0=jij n S

The inhomogeneous inclusion is simulated by an inclusion in the homogeneous

material with eigenstrainp

ij plus equivalent eigenstrain ,*

ij

( )( ) +=+

+=+

-Din

in

,

0

,

0

*

,

0

,

0

lklkijklijij

kl

p

kllklkijklijij

uuC

uuC

(2.31)

Eigenstrain is a fictitious one, introduced for this simulation. The equivalency

between (2.31) and (2.29) holds when

*

ij

( ) +=+ in)( *00* klpklklklijklpklklklijkl CC (2.32)

If is a uniform stress field and is a given uniform eigenstrain in , (2.32) is

satisfied by taking as the solution of the inclusion problem with a uniform

eigenstrain . Instead of (2.27), we now have

0

ijp

ij

ij

*

ij

p

ij +

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( ) ***,, )(2

1mnklmn

p

mnmnklmnkllkkl SSuu =+=+= (2.33)

where***

ij

p

ijmn += (2.34)

Substituting (2.33) into (2.32), we have the equation to determine ,**mn

( ))( ****0**0*0 klmnklmnklijklpklmnklmnklijklijij SCSC +=+=+ (2.35)

from which all components of are determined.**

ij In the absence of the applied stress,

this becomes

( ))( ******* klmnklmnijklpklmnklmnijklij SCSC == (2.36)

from which are determined.**

ij

III. Energies of inhomogeneities

First we consider the case of a body containingD inhomogeneous inclusions and free

from any external force or surface constraint. A given eigenstrain in is denoted by

. The elastic strain energy is the same as (2.8)p

ij

dDeWD

ijij= 21*

(2.37)

where andpijijije = ( ijjiij uu ,,

2

1+= ) . Integrating by parts we obtain from (2.37)

dDW pijij

= 2

1*(2.38)

if is an ellipsoid and is uniform, (2.38) becomespij

p

ijijVW 2

1* = (2.39)

where V is the volume of and ij is given by (2.36).

Next, consider the case when a body D , containing an inhomogeneous inclusion

with eigenstrain , is subjected to an external surface force . Thenp

ij iF

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( )( )dDuuWD

p

ijjijiijij ++= ,0

,

0*

2

1(2.40)

where and are the displacement and stress when acts in the absence of , so0iu0

ij iF

0

,

0

lkijklij uC= , (2.41))( ,0

,

*0 p

kllklkijklijij uuC +=+

The expression for may be further simplified because of the following relations.*W

( ) =+D

jijiij dDuu 0,0

, (2.42)

and

*0

,

0

,

*0

,

0

,

**

,

0

,

,

0

,,

0

)(

)()(

ijjijiij

ijlkijklkllk

ijij

p

ijjilkijkl

p

ijjilkijkl

p

ijjiij

u

uCu

uuC

uuCu

+=

+=

+=

=

(2.43)

where is a fictitious eigenstrain introduced in the equivalent inclusion method .*

ij

Thus we have

+= dDdDdDuW pijijijijD

jiij 2

1

2

1

2

1 *00,

0*(2.44)

The second term above can be considered as interaction term between inhomogeneity

and external stress. Discussions on different cases.

IV. Interaction energy of inhomogeneity

When a body contains an inhomogeneity and is subjected to on surface, the

total potential energy of a body (the

iF

Gibbs free energy) is

( )( ) ( )

( )

++=

+++=

S

iiiijij

D

jiij

S

iii

D

jijiijij

dSuuFdDdDu

dSuuFdDuuW

0*00

,

0

0

,

0

,

0

2

1

2

1

2

1

(2.45)

If the material is homogeneous, W becomes

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=S

ii

D

jiij dSuFdDuW00

,

0

02

1

(2.46)

The interaction energy between and the inhomogeneity is defined byiF

dDWWW ijij

== *002

1 (2.47)

If is uniform and is an ellipsoidal inhomogeneity, becomes uniform and0

ij *

ij

VW ijij*0

2

1= (2.48)

W is ofparticular interest in fracture mechanics (note: here depends on ).*ij

0ij

3.3 Application of the theory of inclusions and inhomogeneitiesIn this part, we demonstrate how the above theory and solutions are applied to

estimate macroscopic mechanical properties of composites and to determine the

morphology of the precipitates and martensites in phase transformation.

3.3.1 Estimate macroscopic mechanical properties of composites

We begin with the somewhat general properties of the internal stress of inclusionsnot explicitly mentioned in the preceding chapters.

We define the average stress of body D as

==D

kjik

D

ijij dDxVdDV ,)/1()/1( (3.1)

Integrating by parts and noting 0, =jij in D, the above equation is always equal to

=

SkjikijdSnxV )/1(

(3.2)

One can see from (3.2) that the average stress caused by inclusions is zero because of

the absence of boundary traction. If the remote applied stress on the boundary S is

uniform and equals to , than (3.2) can be reduced to0ij

000 )/1()/1(ij

S

kjik

S

kjikij dSnxVdSnxV === (3.3)

Similarly, we define average strain of body D as

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+==D

ijji

D

ijij dDuuVdDV )(2

1)/1()/1( ,, (3.4)

Using divergence theorem

+=S

ijjiij dSnunuV )(2

1)/1( (3.5)

If the remote applied displacement on the boundary is uniform which implies

mimi xu0= (3.6)

Substituting (3.6) in to (3.5), we get

0

ijij = (3.7)

Then the effective moduli of a composite is defined as

klijklij C = (3.8)

or inversely

klijklij s = (3.9)

whereijklC is the average elastic moduli of the composite and ijkls is the average

compliance of the composite.

3.3.2 Bound of the effective moduli of compositesThe evaluation of average elastic moduli (the shear modulus and bulk modulus) of

composite materials or polycrystals is one of the classical problems in

micromechanics. The pionner work on this subject have been done by Voigt and

Reuss. The Voigt approximation gives upper bounds and the Reuss approximation

gives lower bounds of the average elastic moduli.

First, let us see the Voigt approximation. Consider a composite material which

consists of randomly distributed isotropic inhomogeneities with various elastic moduli

, , , and a1ijklC

2

ijklCn

ijklC isotropic matrix with the shear modulus (see figure

3.2). The average or effective modulus

0

ijklC

ijklC of the composites materials will be

investigated here. The volume fractions of the inhomogeneities are denoted by , ,

, and that of the remainder(matrix) by . The shapes of the inhomogeneities are

arbitrary.

1c 2c

nc 0c

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Let us assume that the uniform displacement condition mimi xu0

= is prescribed on

the boundary, then the average elastic strain of the composite material is from

equation (3.7). The average stress of the composite material is

0

ij

0

0

kl

n

r

r

ijklrij Cc =

= (3.18)

Fig.3.2 Composite material containing inhomogeneities

D

n

ijklC

0

ijklC

n

where we assume that all points in the composite has the same strain that equals to the

applied uniform strain . Substituting (3.18) in to (3.8) and eliminate from the

equation, we get the average shear modulus and the average bulk modulus as

0

ij0

ij

=

=n

r

rrc0

. (3.21)

==

n

rrrKcK 0 (3.22)

Second, let us see the Reuss approximation. If we apply uniform stress on the

boundary, then the average stress of the composite material is (equation (3.3)). If

we assume that all the elements of the composite material are subjected to a uniform

stress equal to . Then the average strain of the composite is expressed by ((3.4))

0

ij

0

ij

0

ij

=

==n

r

kl

r

ijklr

D

ijij scdDV

0

0)/1( . (3.23)

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where is the compliance tensor of r-th inhomogeneity. Substituting (3.23) into

(3.9) and eliminating from the equation, we get:

r

ijkls

0

ij

1

0/

=

=

n

r

rrc (3.24)

1

0

/

=

=

n

r

rr KcK (3.25)

It is easy to shown that the Voigt approximation and Reuss approximation are the

upper and lower bounds of the true average elastic moduli with the help of the

minimal potential energy and minimal compliance energy (Hill, 1952).

3.3.3 Calculation of the effective moduli of composites--- Eshelbys methodLet us consider again the composite material shown in Fig 3.1. We evaluate the

average shear strain when a uniform shear stress is given as =012 on the

boundary. When the relation between and is obtained, the average shear modulus

is determined by (3.9). The stress in the composite material varies from place and

place. Here it is denoted by , where120

12 + 12 is the disturbance due to

inhomogeneities. Eshelbys equivalent inclusion method in 3.2 is employed. The

composite material is simulated by a homogeneous material having the same elastic

moduli as those of the matrix and containing inclusion r with eigenstrain , where

is the r-th inclusion with the same volume and location as the r-th

inhomogeity, The equivalency equation is

*

12

.,...,2,1 nr= r

( )( )12012

*

1212

0

12012

0

12

2

2

+=

+=+

r

in r (3.26)

where

,2,2 01200

12

*

12121212 == S (3.27)

and is Eshelbys tensor;ijklS ( ) ( )001212 115/54 vvS = for spherical inclusions. The

equivalent eigenstrain in is obtained from (3.26) as*12 r

( ) rr g/0

120

*

12 = , (3.27)where

( )012120 2 += rr Sg . (3.28)

The following quantities in are easily calculatedr

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( ) rr gS /20

120121212 =

( )rr gS /)12(2

0

1200121212 = (3.29)

rg/0

12012

0

12 =+

.rr g/20

12012

0

12 =+

According to Tanaka-Moris theorem** (see the reference book), we have

012 =

dD

rD

(3.30)

and therefore ,=

=n

r

rrD

1

=

=n

r

rrr

D

gScdD

r 1

0

120121212 /))(12( . (3.31)

This does not , however, satisfy the condition for stress disturbance

012 = dDD

(3.32)

Therefore, we add the uniform stress

=

n

rrrr

gSc1

0

1201212

/))(12( (3.33)

to12 in (3.29). Consequently, the uniform strain

=

n

r

rrrgSc

1

0

0

1201212 2/))(12( (3.34)

is added to in(3.29). The average strain is120

12 +

( ) ( )( )==

+=+=n

r

rrr

n

r

rr

D

cgScgcdD1

00001212

1

12

0

12 /12/2

(3.35)

where 0 is the average strain in the matrix, and

00 / = . (3.36)

Insert (3.35) into (3.9), by using

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11

0 =+ =

n

r

rcc (3.37)

we have

( ) ({

++=

=

n

r

rrrSc

1

01212000 2/1/ )} . (3.38)

Similarly, the average bulk modulus is obtained as

( ) (

++= =

n

r

riijjrr KKSKKKcKK1

00003

1/1/ ) (3.39)

* (hw)

* More discussion here on Voight and Reuse.

3.3.4 Morphology of the precipitates and martensites in phase transformation

Precipitates and martensites in alloys are typical examples of inhomogeous

inclusions. The elastic strain energy caused by these particular inclusions will be

discussed. If the interfacial energy can be neglected, the morphology of precipitates

and martensites can be determined by the condition that the elastic strain energy

caused by the inclusion take a minimum value.

Consider an infinite isotropic matrix with shear modulus and Poissons ration ,

containing an ellipsoidal precipitate whose elastic constants are and . Barnett et

al calculated the elastic strain energy

v

* *v

per unit volume of the precipitate when

===

=

aaaaa

ij

p

ij

/0, 321

(3.40)

where is a constant misfit strain. Their results are

+

+

++=

1

*

*2* 11

112113/

v

v

KMMK

KVW

(3.41)

where

( ) ( ) )21/(1),21/(1 **** vvvv +=+= ,

( )

1,1)-()-1/(1

1,3/1

1,cos1)1/(1

3/2-22

12/322

>+=

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{ } ( ){ } 12*

2*

))1(2/)-1)/2(1-3(3M-v)2M(12v)-(1))1/((1

)12/)1(2/)13(3)45()1/(1(

+++

+=

v

vMMK

By minimizing (3.41), Barnett concluded that the minimum strain energy is achieved

by a flat ellipsoid when *

, and by a sphere when . >*

Tanaka and Moris theorem **

For simplicity, let us consider a homogeneous body containing an inclusion notnecessarily ellipsoidal (see Fig 3.1). At this stage of discussion, the shape of the

inclusion is not specified. Give to the inclusion a uniform eigenstrain . Let the

internal stress due

*

ij

1V

2V

Fig3.1 Inclusion with an arbitrary shape in an infinite domain ain

to be , implying that is calculated by assuming the body to be infinitely

extended. outside is safely given by

*

ijij

ij

ij

( ) ( )

=''

,* dxxxGCCx qlkpmnpqmnijklij (3.10)

Let us integrate over the domainij 12 VV as shown in Fig.3.1. are

ellipsoidal and

12 and VV

12 VV .

( ) ( )

( )

+

=

''

,

*

''

,

*

1

212

dxxxGCdxC

dxxxGCdxCdxx

qlkpmnpqmn

V

ijkl

qlkpmnpqmn

V

ijkl

VV

ij

(3.11)

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When we change the order of integration

( ) ( )

( )

1

212

'

,

*'

'

,

*'

+

=

V

qlkpmnpqmnijkl

V

qlkpmnpqmnijkl

VV

ij

dxxxGCCdx

dxxxGCCdxdxx

(3.12)

Since gives the total distortion at when the uniform

eigenstrain is defined in and we have assumed to be an ellipsoid, this distortion

is independent of and is determined by the shape of , as long as . This

condition is satisfied since . A similar remark applies to the second termin (3.12). Thus (3.12) is rewritten as

( ) 2

'

,

*

V

qlkpmnpqmn dxxxGC '

x

2V 2V'

x 2V 2'

Vx

2

'

Vx

( ) ( ) ( )[ ]*1

*

2

12

mnklmnmnklmnijkl

VV

ij VSVSCdxx =

(3.13)

where is the volume of the inclusion, and ( ) ( )12 and VSVS klmnklmn are Eshelbys

tensors. Equation (3.13) leads to interesting conclusions: The volume integral of

over is proportional to ; it is independent of the absolute position and size

of as long as ; it depends only on the shape of ; and itvanishes when are similar in shape and have the same orientation, that is ,

when and

ij

12 VV

12 and VV 12 VV 12 and VV

12 and VV

332211 /// bababa ==

( ) ( ) ( ) 1///:

,1///:

2

3

20

33

2

2

20

22

2

1

20

111

2

3

2

3

2

2

2

2

2

1

2

12

++

++

bxxbxxbxxV

axaxaxV(3.14)

When is an ellipsoidal inclusion we can have the limiting situation, . Then 1V

( ) ( ) ( )[ **22

mnklmnmnklmnijkl

V

ij SVSCdxx =

] , 2V . (3.15)

If are similar in shape, (3.15) vanishes,and2V

( ) 02

=

dxxV

ij , 2V (3.16)

Similarly, for the corresponding strain

( ) 02

=

dxxV

ij , 2V (3.17)

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3.4 Equilibrium shape of inclusions (microstructure patterns)

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The interaction among morphology, driving force, and evolution (light gray = martensite, dark gray =

austenite)

Multi-scale structure of the front zone and the material hierarchy

(light gray = martensite, dark gray = austenite)

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In many cases of microstructure evolution such as second phase precipitation

and phase transitions (see the figures above), we need to think about what is the final

equilibrium state (i.e., thermodynamic equilibrium) of the evolution, e.g., the volume

and shape of a second phase particle. Here we consider the simplest example where

the volume of the particle or inclusion is fixed and we are to determine the shape of

the inclusion that gives the lowest free energy. Basically the systems energy consists

of two terms: Surface energy and volume energy (see details in the paper by Johnson

and Cahn for 2D analysis (1984))

( ) ( ) += dVdAnshapeE ijijtot 21

where ( )n is the energy due to the interface (surface tension or interfacial energy)between the inclusion and the matrix and is assumed to be isotropic ( ( ) 0int =n , bothinclusion and matrix are elastic isotropic for simplicity). It is proportional to the area

of the interface (for a given volume). The second term is due to the elastic strain

energy and/or chemical free energy. Both terms depend on the shape of the inclusion

(for a given volume). In the following we only consider a single particle with

eigenstrain:

ijij 0* = .

For mathematical simplicity we assume that all possible shapes of the inclusion are

within the family of elliptical shapes characterized by their area and aspect ratio:

abA = , .ba/

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For convenience we change the variables a, b into u, v as:

( )2

bau

= ,

( )2

bav

+=

so that the area and the degree of non-circular in shape are represented as:

( 222

uvA = ) ,ba

ba

v

uU

+==

The total interfacial energy is

lE 0int = , (l is the circumference of ellipse)

( ) dxxa

xabal

a

=0 22

222

4,

, ( )22 /1 ab=

=

2,4

22

a

baaEl , hereEis an elliptical function. By Taylor expansion, the

interfacial energy can be re-expressed as

( )

+++= L420int

64

33

4

312, UUAUAE ..

Using the Eshelbys solution in this chapter we can write down the elastic strain

energy as function ofA and Uas

( )( )

( )( ) ( )( ) 2**2

*2

0211211

112,

U

UAUAEelastic

++++

+= ,

In the above, is the ratio of the shear modulus of the two phases, /*= 43= . Using Taylor expansion, the elastic strain energy becomes

( )( )( )

( )( )( ) ( )

( ) ( )

+

++

++

++

+

+= L4

2*2

222

**

*2

0

211

11

211

111

21

2, UU

AUAEelastic

.

The total energy of the system now has the form similar to that of Landau energy

expansion (see chapter 5)

( ) ( ) ( ) ( ) ( ) ( ) L442

20int

!4

1

2

1,,, UAFUAFAFUAEUAEUAE

elastictot++=+=

The minima of ( )UAEtot , will give the optimum Ufor a givenA. If we only use theexpansion up to 4th order, the values of U that give minima are determined by

( ) ( )

+==

242

6

10 UAFAFU

U

Etot (here is positive on physical ground) together

with the second derivative of the total energy. We have

4F

= 0if/6

0if0

242

2

FFF

F

U .

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Thus the question of equilibrium shape of inclusion has now been reduced to the sign

of , which is the result of inclusion size dependent competition between the

interfacial and bulk energy

)(2 AF

( )( )( )

( )( )2**2

002

211

112

2

3

++

+

=

AAAF .

By setting =0 we can obtain the critical size , which also gives a

nondimensional size parameter

)(2 AF cA

cAA/= . So, in this simple model, the parameterA

serves as the particle size and with increase in size there will be a transition from U=0

regime (i.e., circular inclusion) to the broken symmetry regime (U0) where the

inclusion exhibits elliptical shapes. It should be noted that the transition only occur in

the case of 1

503 chapter 3

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