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MIC CHIGAN STATEU

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This is to certify that the

thesis entitled

ELASTIC AND THERMAL PROPERTIES OF PARTICULATE
COMPOSITES WITH INHOMOGENEOUS INTERPHASES

presented by

Wei Wang

has been accepted towards fulfillment
of the requirements for

MaStef degree in Mechanics

5 Major pr'gfessor

 

Date 1/1121!

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

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TO AVOID FINES Mum on or him dd. duo.

DATE DUE DATE DUE DATE DUE

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU I: An Nflrmatlvo Amen/Equal Opportunny lawman
Wanna-9.1

 

 

 

ELASTIC AND THERMAL PROPERTIES OF PARTICULATE
COMPOSITES WITH INHOMOGENEOUS INTERPHASES

by

Wei Wang

A THESIS

Submitted To
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Materials Science and Mechanics

1994

ABSTRACT

ELASTIC AND THERMAL PROPERTIES OF PARTICULATE
COMPOSITES WITH INHOMOGENEOUS INTERPHASES

by

Wei Wang

A particulate composite that consists of three phases: inclusion, interphase and
matrix, is considered. A mathematical procedure that is based on the composite spheres
assemblage method (CSA) and the generalized self-consistent method (GSC) is used to
evaluate the effective moduli and the thermal expansion coefficients of the composite. The
inclusion and matrix are assumed to be homogeneous and isotropic, and the interphase is
isotropic and inhomogeneous. The perfect bonding conditions are assumed at both the
inclusion—interphase and interphase-matrix interfaces. The power, linear and cubic func-
tions are chosen to simulate the variation in the Young’s modulus, Poisson’s ratio and the
thermal expansion coefficient in the interphase. In the analysis, the CSA method is
employed to calculate the effective bulk modulus and thermal expansion coefficient of the
composite, and the GSC method to calculate the effective shear modulus. The solution is
obtained either in a closed form or in a form of infinite series.

The numerical results are presented, which illustrate the effects of interphase. It is
shown that inhomogeneity of the interphase in particle reinforced composite does have an

influence the on effective properties of composite.

DEDICATION

To my wife, Yu Kui, my sister, my brother in law and my parents

iii

ACKNOWLEDGMENTS

I would like to express my gratitude to Dr. Iwona Jasiuk, my advisor, for her gui-
ance, encouragement, and support throughout the course of this research work. I thank her

for the excellent advice that I received in my research field.

I also would like to thank Dr. Nicholas Altiero and Dr. Dahsin Liu for their great

help when I met difficulties.

Grateful acknowledgment is extended to the Department of Materials Science and

Mechanics as well as all the faculty for the support and direction they gave me.

I would also like to acknowledge the financial support of the Research for Excel-

lence Fund from State of Michigan.

Finally, I am very grateful to those who have made my work become reality.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
NOMENCLATURE ........................................................................................................... xi
1. INTRODUCTION ........................................................................................................... 1
2. PROBLEM FORMULATION AND SOLUTION METHOD ........................................ 7
2.1. EFFECTIVE BULK MODULUS ............................................................................. 7
2.2. EFFECTIVE SHEAR MODULUS ........................................................................ 15
2.3. EFFECTIVE THERMAL EXPANSION COEFFICIENT ..................................... 24
3. NUMERICAL ANALYSIS ........................................................................................... 28
3.1. BULK MODULUS ANALYSIS ............................................................................ 28
3.2. SHEAR MODULUS ANALYSIS .......................................................................... 50
3.3. THERMAL EXPANSION COEFFICIENT ANALYSIS ....................................... 62
4. CONCLUSION .............................................................................................................. 75 ’
5. APPENDICES ............................................................................................................... 76
APPENDIX A ................................................................................................................ 78
APPENDIX B ................................................................................................................ 81
APPENDIX C ................................................................................................................ 83
APPENDIX D ................................................................................................................ 90
APPENDIX E ................................................................................................................ 93
APPENDIX F ................................................................................................................ 97
APPENDIX G .............................................................................................................. 101
APPENDIX H .............................................................................................................. 104

APPENDIX I .............................................................................................................. 107
BIBLIOGRAPHY ............................................................................................................ 1 10

vi

LIST OF TABLES

Table ............................................................................................................................... page

2.1 Direction cosines between Cartesian coordinates and spherical coordinates ................ 8

vii

LIST OF FIGURES

Figure ............................................................................................................................ page
1.1 The single composite sphere .......................................................................................... 4
1.2 The composite sphere assemblage model ...................................................................... 5
1.3 The generalized self-consistent model ........................................................................... 6
2.1 Relation between Cartesian coordinates and spherical coordinates .............................. 7
3.1 Variation of Young’s modulus in the interphase ..................................................... 28,29
3.2 Cubic variation simulating power variation ................................................................. 33
3.3 Effective bulk modulus vs. volume fraction f for no interphase case .......................... 36

3.4 Effective bulk modulus vs. f for a inhomogeneous interphase stiffer than

matrix; interphase has power variation in Young’s modulus and a constant

Poisson’s ratio .............................................................................................................. 37
3.5 Effective bulk modulus vs. f for a homogeneous interphase stiffer than matrix;

interphase has a constant Young’s modulus and a constant Poisson’s ratio ................ 38
3.6 Comparison between Fig. 3.3 and Fig. 3.4 .................................................................. 39
3.7 Effective bulk modulus vs. f for an inhomogeneous interphase softer than

matrix; interphase has power variation in Young’s modulus and a constant

Poisson’s ratio .............................................................................................................. 40
3.8 Effective bulk modulus vs. f for a homogeneous interphase softer than matrix;

interphase has a constant Young’s modulus and a constant Poisson’s ratio ................ 41
3.9 Comparison between Fig. 3.7 and Fig. 3.8 .................................................................. 42

3.10 Bulk modulus vs. f for inhomogeneous interphase with interchange in proper-

viii

ties ............................................................................................................................. 43
3.11 Effective bulk modulus vs. f for a homogeneous interphase with interchange

in the properties ........................................................................................................ 44
3.12 Comparison between Fig. 3.10 and Fig. 3.11 ............................................................ 45
3.13 Effective bulk modulus vs. f; the power variation of Young’s modulus in

interphase is simulated by cubic variation ................................................................ 46
3.14 Comparison of effective bulk moduli from different variations of Young’s

modulus in interphase ............................................................................................... 47
3.15 Bulk moduli of composite with different variations of Poisson’s ratio and a

uniform variation of Young’s modulus in the interphase .......................................... 48
3.16 Bulk moduli of composite with different variations of Poisson’s ratio and a

linear variation of Young’s modulus in the interphase .............................................. 49
3.17 Effective shear modulus vs. f for no interphase case ................................................. 52
3.18 Effective shear modulus vs. f for an inhomogeneous interphase that is stiffer

than matrix; interphase has a power variation Young’s modulus and a con-

stant Poisson’s ratio .................................................................................................. 53
3.19 Effective shear modulus vs. f for a homogeneous interphase that is stiffer than

matrix; interphase has constant a Young’s modulus and a constant Poisson’s

ratio ........................................................................................................................... 54
3.20 Comparison between Fig. 3.18 and Fig. 3.19 ............................................................ 55
3.21 Effective shear modulus vs. f for an inhomogeneous interphase softer than

matrix; interphase has power variation in Young’s modulus and a constant

Poisson’s ratio ........................................................................................................... 56
3.22 Effective shear modulus vs. f for a homogeneous interphase softer than

matrix; interphase has a constant Young’s modulus and a constant Poisson’s

ratio ........................................................................................................................... 57

3.23 Comparison between Fig. 3.21 and Fig. 3.22 ............................................................ 58

ix

3.24 Effective shear modulus vs. f for an inhomogeneous interphase with the inter-
change of properties .................................................................................................. 59

3.25 Effective shear modulus vs. f for a homogeneous interphase with the inter-

change of the properties ............................................................................................ 60
3.26 Comparison between Fig. 3.24 and Fig. 3.25 ............................................................ 61
3.27 Variation of thermal expansion coefficient in the interphase ..................................... 62
3.28 Effective thermal expansion coefficient vs. f for no interphase case ......................... 65

3.29 Effective thermal expansion coefficient vs. f for an inhomogeneous interphase

with power variation in the thermal expansion coefficient ....................................... 66
3.30 Effective thermal expansion coefficient vs. f for an inhomogeneous interphase

with the constant thermal expansion coefficient ....................................................... 67
3.31 Comparison between Fig. 3.29 and Fig. 3.30 ............................................................ 68
3.32 Effective thermal expansion coefficient vs. f for a homogeneous interphase

having a power variation in the thermal expansion coefficient ................................ 69
3.33 Effective thermal expansion coefficient vs. f for a homogeneous interphase

having a constant thermal expansion coefficient ...................................................... 70
3.34 Comparison between Fig. 3.32 and Fig. 3.33 ............................................................ 71
3.35 Comparison of effective thermal expansion coefficient for the case of differ-

ent variations of thermal expansion coefficients in the interphase ........................... 72
3.36 Comparison of effective thermal expansion coefficient for the case of differ-

ent variations of Young’s modulus in the interphase ................................................ 73
3.37 Comparison of effective thermal expansion coefficient for the case of differ-

ent variations of Poisson’s ratio in the interphase ..................................................... 74

NOMENCLATURE

RVE ..................................................................................... Representative volume element
CSA ............................................................................ Composite sphere assemblage model
GSC ................................................................................. Generalized self-consistent model
r, 0, (p .................................................................................................. Spherical coordinates
x, y, z ................................................................................................... Cartesian coordinates
u,, ue, u.p .................................................................. displacements in spherical coordinates
ux, uy, uZ .................................................................. displacements in Cartesian coordinates
p, l, m, e ...... superscripts which refer to inclusion(particle), interphase(layer), matrix and

equivalent homogeneous phase

C i, .............................................................. unknown constants in bulk moduli calculations
D; ............................................................. unknown constants in shear moduli calculations
C in .................................................... unknown constants in thermal expansion coefficients
1' .................................................................................... superscript which refers to p, l, m, e
n ..................................................................... subscripts which refer to numbers 1, 2, 3 ......
t .................................................... subscript which refers to thennal expansion coefficients
t ......................................................................................................... thickness of interphase
a, b, c ..................................................................... radii of inclusion, interphase and matrix
f .................................................................................................................... volume fraction
PB ...................................................................................................................... perfect bond
E ................................................................................................................ Young’s modulus

xi

v ..................................................................................................................... Poisson’s ratio

G .................................................................................................................... shear modulus
K ...................................................................................................................... bulk modulus
7. ..................................................................................................................... Lame constant
0‘", 099, 0w ................................................................................................. normal stresses
1: r0’ 19¢, 1: (pr ..................................................................................................... shear stresses
8", 809, amp .................................................................................................... normal strains
7,9, 79¢, 7W ........................................................................................................ shear strains
W ...................................................................................................................... strain energy
P" .................................................................................... constants in interphase simulation
Q" ................................................................................... constants in interphase simulation
R ..................................................................................... constants in interphase simulation
S ...................................................................................... constants in interphase simulation

xii

1. INTRODUCTION

In a particle reinforced composite, the particle usually bears the major part of the
load. However, the important factor that will influence the reinforcement mechanism is the
interface. The interface, serving as a load transfer between the matrix and inclusions, is
known to have an important effect on the composite response [see Hughes (1991), Kerans
et a1. (1989), Wright (1990) and Jayararnan (1993), for example]

Many composites are made up of inclusions which are perfectly bonded to a sur-
rounding matrix. This model has been studied extensively [for reviews, see Christensen
(1979), Hashin (1983), Hine et a1. (1993), for example]. However, in many composites,
there is an interface layer between inclusions and matrix. This interface layer exists as a
natural consequence of composites processing or is intentionally introduced into the com-
posite to improve the properties of composite. It is a three dimensional region between the
inclusion and matrix and is usually called an interphase. The interphase can be very thin as
is the interphase between tungsten wire reinforcement and steel, or relatively thick as in
vapor deposited boron fiber in epoxy [Bogan and Hinder (1993)]. Although its volume
fraction is small as compared with the inclusions and matrix volume fractions, its exist-
ence is important and it strongly affects both the local field and the overall properties of
composites [Theocaris (1986)]. The structure of the interphase is very complicated [see
Hull (1981), Drzal (1983), Chawla (1987), Date et a1 (1993), for example]. For simplicity,
in the theoretical investigations many researchers modeled the interphase as a homoge-
neous region. These studies include those of Agarwal (1974), Mikata and Taya (1986),
Pagano (1988), Jasiuk and Tong (1989), Benveniste et al. (1989), Chen et al. (1990),
Pagano and Tandon (1990), Tong and Jasiuk (1990), Maurer (1990), Sullivan and Hashin

(1990), Qiu and Weng (1991), Dasgupa and Bhandarkar (1992), Bostroin et a1 (1992),
Theocaris and Demakos (1992), Herve and Zaoui (1993), Gregory (1993), and many oth-
ers.

The advantage of the homogeneous interphase model is that it gives relatively
simple mathematical analysis. However, it may not give a good representation of the com—
plex state at the interface that may be due to a chemical reaction or diffusion, for example.
In order to account for this complex microstructure, recently, a number of researchers
studied the inhomogeneous interphase. These publications include those of Theocaris et
a1. (1985), Theocaris (1986), Sideridis (1988), Papanicolaou et a1. (1989), Sottos et a1.
(1989), J ayaraman et a1. (1991), Jayaraman and Reifsnider (l992a,b), Jasiuk and Kouider
(1993), Bogan and Hinder (1993), Nassehi et a1. (1993), and other.

Most of the above papers discussed the fiber reinforced composite. In this thesis,
we evaluate the effective bulk modulus, shear modulus and thermal expansion coefficient
of particle reinforced composite. We assume that the particles are spherical in shape and
that there is an inhomogeneous interphase between each particle and the matrix. We fol-
low the classical elasticity methods to solve these problems and use some simplifying
assumptions

1. The composite is modeled as a representative volume element (RVE), Fig. 1.2
and Fig. 1.3, (see, Hashin ( 1983)].

2. The three phases: particle, interphase and matrix are linearly elastic.

3. The particle and matrix are homogeneous and isotropic.

4. The interphase is isotropic and inhomogeneous with the Young’s modulus, Pois-
son’s ratio and thermal expansion coefficient having a radial variation.

5. There is perfect bonding between the phases.

6. The temperature changes uniformly in all the phases.

To determine the effective elastic moduli of a multiphase composite, we follow Hashin

(1983) and subject the composite to homogeneous boundary conditions

o
ui(s) = 8‘.ij (1.1)

where 83. are constant strain tensors. Then by the average strain theorem

5.. = 59. (1.2)
where the overbars denote volume average over composite. According the linearity of the

equations of elastrcrty, the average stresses oil. and strarns 81.]. must obey followrng rela-
tion

—— _ — _ 0
or} ‘ ijklekz ‘ ijktekt (1'3)

where ijk, is defined as the effective elastic stiffness of a composite.

In our analysis, we evaluate the effective bulk modulus and thermal expansion
coefficient by using the composite spheres assemblage model (CSA)[Hashin and Rosen
(1964)]. In the CSA model we have three concentric spheres having radii a, b and c (Fig.
1.1), which correspond to the radius of particle, the outer radius of the interphase, and the
outer radius of matrix, respectively. The microstructure of composite is represented by a
collection of such various sizes of composite spheres which completely fill the space (Fig.
1.2). The ratios of radii a/b and a/c are taken to be constants for each composite sphere.
The major advantage of this model is that we can evaluate the effective bulk modulus by
using the single composite sphere (Fig. 1.1), and its result represents the effective bulk
modulus of composite spheres assemblage model (CSA) [Christensen (1979, p. 50)].

We evaluate the shear modulus by using the generalized self -consistent model
(GSC) [Christensen and Lo (1979)]. In the GSC method, a single composite sphere (Fig.
1.1) is embedded in the infinitely extended medium of yet unknown effective properties.

This geometric model is shown in Fig. 1.3. In both methods the volume fraction f of parti-
cles is defined by f = a3/c3. In this thesis, the superscripts p, l, m, e denote the inclusion
(particle), interphase (layer), matrix, and effective medium, respectively.

This thesis is an extension of previous theoretical works which dealt with the sub-

' f inhomogeneous interphases and which addressed the

[Theocaris

    

    
 
 

m.
m

N

. .
\Qs

\

\\\\

\

\ \

has
V\

\\\\\\\

           
    
                     

  

so.

\s ss
ss . s; \' s w
‘ \\\\ \ .

Fig. 1.2

Composite spheres assemblage model

 

s

s
Vs

 

Fig. 1.3

Generalized self-consistent model

2. FORMULATION AND SOLUTION METHOD

2.1. EFFECTIVE BULK MODULUS
We use composite spheres assemblage model (CSA) to evaluate the bulk modulus of par-
ticulate reinforced composite. We subject a single composite sphere to the homogeneous

surface displacements, which in Cartesian coordinate system are given by

._ 0
ux—e x
0
u = .
._ 0
“Z _ 8222
where
0 _ 0 _ O _ 0
an — a” — Eu - e" (2.2)

[see Hashin (1983)]. It is convenient for us to use the spherical coordinate system (see

Fig. 2.1) which is related to the Cartesian system as given in Table 2.1,

N f

 

V~<

 

 

Fig. 2.1
Relation between Cartesian coordinates
and spherical coordinates

 

 

 

 

 

 

 

 

 

x y z

sincpcosO sintpsinO costp

costpcose costpsinO —sin(p

-sin6 c059 0
Table2.1

Direction cosines between Cartesian and spherical coordinates

and the Cartesian coordinates can be expressed by using spherical coordinates as
x = rsintpcosO
y = rsintpsinO (2.3)
2 = rcosrp

Using Table l, we express the displacements in the spherical coordinates as

u, = uxsrntpcose + uysrntpsrnO + uzcostp

u

q, = uxcosocose + uycososinO - uzsintp (2.4)

“9 = -uxsrn9 + uycose

Thus, when we substitute eqns. (2.3) into eqns. (2.1) and then put them into eqn. (2.4), we

get
u, = e‘r’rr
uq, = 0 (2.5)
“a = 0

Due to the spherically symmetric nature of the problem, the displacements in the bulk

modulus calculations are in the form

u, = u,(r)
u.p = 0 (2.6)
"9 = 0

that is, the displacement in r direction depends on r only and the displacement components

in (p and 0 direction are zero. Next, we use this displacement formulation approach to

determine the unknown displacement u (r)

The govemrng equations that we use to calculate the displacements are given
below:

1) Strain-displacement Relations

 

=3u,
r
e = 1 3u9+ u mu
99 rsintp50+ r r ‘P
13 u
=7§6¢+ 7
1 an due no (2.7)
7’9: r—_sin(p39r +37 7
1311, cottpu 1 auq,
'7an ' — “e + was
auq, u¢+13u
Y‘1” = 57' - — +115:
2) Constitutive Equations (Stress-strain Relations)
on = 26 (err+ fie) 1,9 = Gay,0
v "oo = 07%
0'99 — 20 (269 + T3258) {W = 67‘” (2.8)
(”2260ap ‘P+-1—_-2v e) e=€rr+809+etptp

3) Equilibrium Equations (with no body forces)

30,, 1 31,9 31w
5— sin<p+— 7-"(26 sintp— Geesintp— O'q, ¢sinrp+1 (”comp-+39,6 +-a—¢ sintp =
a: 9 1 are ac
’ - _ - ‘P- 99 2.9
‘37 srncp+ r (3tresrntp+3-(—p srn¢+3§ +219¢costpj - 0 ( )
at 3: Bo
re . . Ocp W . _
8—; srnrp+;(3t¢rsrntp+a—e + (ow-696) costp+5(—p srntp) 0

where G rs shear modulus and v is Poisson’s ratio. In the isotropic material they have the
following relation

10

_ E
G _ W (2.10)

where E is the Young’s modulus.

Substituting eqns. (2.6) into eqns. (2.7) we get the expressions for strains in terms
of displacements. Substituting the strain expressions into eqns. (2.8) we get the stresses in
terms of displacements, and then substituting them into equilibrium eqns. (2.9), we get the

equilibrium equation which can be used to find displacement u, as follows

2
_ _ "r 3E 8E 28E 3_a_E
Er2 (v 1) (2v 1) (1+v)-a——2 +r(2E+§—ra- 2rv5—- W a +2 N 37

,1

av 8v
+4EVra— —2Ev2 r3— -4Ev- 2Ev2 +4Ev3 )g— l:'+(--4Ev+2Ev —4Ev3 (2'11)
8v
+ZVr—g—E— 2v2r g—E- 4V3 rg—E+4Ev2r r-a——-2Er+2Er%-: )u =0

In eqn. (2.11), Young’s modulus E and Poisson’s ratio v are taken as the function of
radius r, that is E = E (r) and v = v (r) . The procedure to obtain eqn. (2.11) is given in
the appendix A.

Since we assume that the matrix and inclusion are homogeneous and isotropic, the
Young’s Modulus and Poisson’s ratio are constant in these phases. If we let E and v to be
constant in eqn. (2.11) and simplify these equations, we get the following governing equa-

tion for displacement u, in the matrix and inclusion.

232u, au,
r ‘57 +2r; -2“, = O (2.12)
r

Eqn. (2.12) is a Euler equation. The general solution to this equation is

C1r3 + C2
u, = ___r2 (2.13)

where C1 and C2 are constants. Adopting eqn. (2.13) to the inclusion phase and matrix, it

gives us

11

g1 (2.14)
U? = C’lnr+ -—2-
r

where C1, C’l" , C’z" are constants. Superscripts p and m denote the inclusion (particle) and

matrix. The coefficient 0; in the particle phase is taken as zero in order to avoid the singu-

larity at the origin, and the corresponding stresses can be determined by eqns. (2.7) and

(2.8) as

 

(2.15)

 

 

"-1-2v'"_1+v'"
For the inhomogeneous isotropic interphase, eqn. (2.11) is the general governing
equation to determine the displacements in the interphase. Eqn. (2.11) can be expressed in
a more general form as

2

 

 

u, Bu,
8—13 +f(r)-a-; +g(r)u, = 0 (2-16)
where
A
fit) = 2 (r) (2.17)
Er (v— 1) (2v- 1) (1+v)
B
80) = 2 (r) (2.18)
Er (v—l) (2v—l) (1+v)
and

8E 35 2ar: aE

A(r) =r(2E+3—r- 2rva—- rv 5—+2r rv3—

av av
+4EVr-a—; - 215»er — 4Ev - 2Ev2 + 4Ev3)

12

av
B (r) = 4Ev + 2Ev3 — 4Ev3+4EVr57

av
—2Ev3ra—r - 4Ev - 2Ev2 + 4Ev3

are functions of the radial coordinate r only.
In general, the solution to the above equation cannot be found in a closed form, but
under certain conditions of coefficients )7 r) and g( r), the solution can be obtained in a form

of infinite series in terms of r. If f( r) and g( r) both have Taylor series expansion at r0, then

a convergent series solution of eqn. (2.16) exists of the form u, = 2 an (r - r0)". For
n = 0

more details about series solution, see Appendix B and refer to publications of Johnson
and Johnson (1982), Jayaraman et a1. (1991), Jayaraman and Reifsnider (l992a,b), Jasiuk
and Kouider (1993), Lutz and Ferrari (1993).

To simulate the inhomogeneity in the elastic constants of the interphase, we use
three alternate variations:

case 1. Power variation

QI

E’ = 11>1 ((51) v' = constant (2.19)
case 2. Linear variation
5’ = [3(5) +Q2 v’ = constant (2.20)

case 3. Cubic variation
15" = P (5)3+Q (I)2+R(5) +5 v’ — P'(5)3+Q'(5)2+R'(5) +S' (221)
3 a 3 a a - a a a '

where, P, Q, R, S are constants

Substituting eqn. (2. 19) into eqn. (2.11), the governing equation is reduced to

azu Bu 2le
2 r r _ =

13

This is the Euler equation. The solution is given by

 

 

“Ir .—.- C‘llr‘”-I-Clzr"2 (2°23)
where
lOv’-—2
s1: -1 1+Q+ 9+Q3+——————Q (2.24)
2 v’-1
’-2
s2 = -1 1+Q— 9+Q2+M (2.25)
2 v’—1

C11 and C12 are constants. The superscript 1 represents the interphase (layer).
Substituting eqn. (2.23) into eqn. (2.7) and eqn. (2.8), we obtain the expression for
stress 0'".

For the cases 2 [eqn. (2.20)] and 3 [eqn. (2.21)], we follow the same procedure.
First, we get the series solution for the displacement in the interphase and then substitute it
into eqns. (2.7) and (2.8) to get the stresses. Thus, we know the expressions for displace-

ments and the stresses in the inclusion, matrix and interphase. These expressions include

five unknown constants C’l’ , C'l", C3", C11 and Ca. In order to evaluate these constants, we

use the boundary conditions. If we assume perfect bonding between the phases, we have at

r = a
0P 2 O”
" 1' r (2.26)
u: = u.
and at r = b
1
o = o’"
'1' " (2.27)
it, = u',"
In addition, on the outer surface we have
_ 0
ur((:) — enc (2.28)

as given in eqn. (2.5).

14

The bulk modulus is usually defined as [Mase and Mase (1992), Frederick and
Chang (1972)]

32..

H

6..
K = 1?. = __‘_'_ (2.29)
are

by which we infer that the bulk modulus K relates the pressure p to the volume change
given by the cubical dilatation 811° According to summation convention, we have
on. = o"+ Ow + 099 and a“. = e"+ 8W + 800. In our case, from eqns. (2.7) and (2.8),

we find on = OW = 699’ e" = em, = 909' So, eqn. (2.29) can be written as

K ._._. " . (2.30)

 

Combining eqns. (1.3) and (2.30), the bulk modulus of the composite [Hashin
(1983), Jasiuk and Kouider (1993)], Can be expressed as

 

(26'1”(l+v"‘) 4C3)G"'
5,, _ O’,"(C) __ 1—2v’” :3—

o ' o o
8" 3877' 3817'

K‘ = (2.31)

 

 

 

M

where 6'" is the average stress in the composite sphere, which is equal to the stress in the

matrix on the outer boundary a; (c) by using CSA model. The superscripts e and m rep-

resent the effective composite and the matrix, respectively.
It is worthwhile to mention that, if when the GSC model (Fig. 1.3) is used to eval-
uate the effective bulk modulus, it gives the identical result to that of above CSA model

[Christensen (1979)]. but it is more complicated to use.

15

2.2. EFFECTIVE SHEAR MODULUS

In this section, we evaluate the effective shear modulus of particulate composite.
The CSA model gives us a simple mathematical formalism in the bulk modulus evalua-
tion, but we cannot use it to evaluate the shear modulus uniquely. The reason is as fol-
lows. In the case where the simple shear type displacements are prescribed on the surface
of the composite sphere, the resulting boundary stresses are not those corresponding to a
state of simple shear stress. Correspondingly, when the simple shear stresses are pre-
scribed on the boundary, the resulting surface deformation state is not that of a simple
shear deformation. Accordingly, CSA model can only give the bounds on the shear modu-
lus. Therefore, as an alternate method, we choose the generalized self-consistent model
(GSC) to evaluate the effective shear modulus. In this method a single composite sphere
(Fig. 1.1) is embedded in the infinitely extended medium of yet unknown effective proper-
ties of the composite.

Again we first use the Cartesian coordinate system first, and we subject this com-

posite system to the remote displacement boundary conditions

“x = egyy
_ o
u), — exyx (232)
u2 = 0

Substituting eqn. (2.3) into eqn. (2.32) and then substituting them into eqn. (2.4),
we get
u, = egyr(sin(p) 2sin29

_ 0 '
“e .. exyrsrntpCOSZO (2.33)

u:

0 . .
q, exyrsm2tpsrn20

NIH

Guided by the eqn. (2.33), which represents the deformation of a homogeneous

medium, a general solution for the heterogeneous problem is in the form

16

u, = U, (r) (sintp) 2sin20
“e = US (r) sintpcosZO (2.34)
u(p = U,p (r) sin2tpsin20
Following the same procedures as in the bulk modulus calculations, we substitute

eqn. (2.34) into eqns. (2.7), (2.8) and the equilibrium eqns. (2.9). Consequently, each
equation can be separated into the sum of two parts

H (r) sinztp + J(r) = o (2.35)
where H (r) and J (r) can be expressed as

H (r) = h (E, v, ur, ue, 11¢)

. (2.36)
J(r) =1 (E, v, u,, rte, u¢)
We equate to zero the coefficients of sinztp and the term independent of (p, that is
H (r) = O
(2.37)
J (r) = 0

The three equilibrium equations give six equations, but only three of them are indepen-

dent. These three independent goveming equations are given as follows

 

 

at}, aqu, 30,,
where
E

I“ ' [rm-1)]

- zg—Evzr — £in + 35 — 35w.» 4E - 413v2 + 253—er
L = r Br Br Br Br
2 (1+v)(-1+2v)r2

-gEVr—2Ev—2E+E?—vr—§§r
L = r Br Br
3 (l+v)r

 

17

 

 

 

 

 

—12Ev2—Eg—vr —a—§vr +12E— ZaEv2r+aEr+ZEa—vvr
L = r ar ar ar ar
4 (l+v)(— 1+2v)
BZU, av, 30,,
L537 —L637_ +L7U,+L8—a—r -3L9Uq, = 0 (2.39)
where
_ '(v-l)E
L5 — - —1+2v]
’ C(r)
__(1+v)(—1+2v) r
-g§v2r—SE+13Ev+2Eg—Vr+4Egvv2 +2Ev2— 435v 3r+2§fvr-16Ev3
L =
7 (1+v)(-1+2v)2 r2
L8 = [(—13-t-E2v)r]
av av
- g§v3r+2g§ Vr— 8Ev3 -2%;Ev v2+4Eg;v2r+zEv2+7Ev+2Ea—;r—3E
L =
9 (1+v)(-1+2v)2r2
av
C(r)= ---%;r-t-4Ev-I-2Ev2 —4Ea—r —Vr+2-g—EVr
8E 3v
__ 2 _ _ 3__ 3 _ 2 _
+gr(E)v r Zarv 4Ev +2E8rv r 2E
and

(yo—2%) = 0 (2.40)

A computer program given in Appendix C was written by using MAPLE to simplify the
above algebraic manipulations. Next, we solve the differential equation system (2.38),

(2.39) and (2.40) to determine U ,, U9 and Uq,

For the homogeneous matrix and inclusion

E = constant = constant (2.41)

18

Then, eqns. (2.38), (2.39) and (2.40) give us

Bu, 2 q,
—r§— -4(1—v)U,—(1—2v)r 3—
’ r (2.42)

auop
—2(1—2v)r3—r +12(1—v) U¢=0

2
-(1-v)r23—U'-2(1-v)r3£]’+ (5 —8v) U
(Br Br ’

(2.43)
BUq,
+3537 —3 (3-4v) (1,, = 0
U9— 2U“, = 0 (2.44)

They are the same as in Christensen and Lo (1979). The method to solve the above differ-

ential system is to introduce an operator

r"3—:U=d(d—l 1)...(d-n+1)U (2.45)

So, we change the differential system to the general linear algebraic system. We can solve
the algebraic system easily, and the final solution of the differential system depends on the
roots of the algebraic system; again this technique to solve above differential system is
standard [for example, see Wyie and Barret (1982) p. 209)]. We will discuss it later in
more details in eqns. (2.55)—(2.57). Finally, the displacements in the homogeneous

medium are given as

 

6v +(5- 4v) 1 . .
u, = (Dlr— 1_ 2v D2 r 33+3D— +-(l ——)2v D4?) (srntp)zsrn20 (2.46)
7—4
“0 = (Dlr-—(—1——§—Y—)-D2r3-ZD3-17+20412)sin(p00820 (2.47)
" V r r
1 (7—4v) 3 1 1 . .
u = — (D r-———D r —2D — +2D —) srn2 srn20 (2.48)

where D1, D2, D3 and D4 are constants.

Guided by above equations, we can get displacements for different phases.

19

in the inclusion, O < r < a:

 

 

up = Dpr— 6v” Df’r3 (sintp)3sin20
(7 ’4VP) 3) .
up = r— —-———Dpr srn c0520 2.49
e ( 1 1_ 2vP 2 (P ( )
l (7 '4Vp) 3 . .
P = _ — .—
u.p 2 [D‘l’r 1_ 2vP D’z’r )srnthsrn29
1n the matrix b < r < c
6v’" 5 4vP
u'," = ( Tr— 0'2" r3+ 3D’3"- 4+-(————)-D"' 4—1-2)(sintp)2 sin20
1- 2V“ (1- 2 VP) 421‘
(7 '4Vp) 3 1 1 J .
um = mr - ————Dmr - ZDm— + 2Dm— srn cos20 2.50)
0 ( 1 1_ va 2 3 r4 4 r2 ‘9 (
1 (7 “4VP) 3 l 1 j . .
u'" = - "'r— ———D’"r —2D’"— + 2D’"— srn2 srn20
cP 2( 1 1_ va 2 3 r4 4 r2 (P

and in the equivalent homogeneous phase, c < r < co:

+-(5 4V‘)D 1)
D‘r + 3D§;— —— sin 2sin20
"i=2 (1 — v) D“ 2 ( (p)
= (D‘r— ZD‘— l +ZDZ— r33) sintpcosZO (2.51)
ufp= 9WD -2D;-1— +2D§— )sin2tpsin20

Some terms in the above equations are missing in order to avoid the unbounded

solution. The coefficient D? is specified by the displacement boundary conditions on the
remote outer boundary given in eqn. (2.33), that is D: = 82),

o n p p
The expressrons for the corresponding stresses off, 6‘39, 6:”, 1,9, 10¢, 15,, off,

m e e e
03,0,30’ 036' Tnii’ Tetp’ Tm ’ 0'", 0300’ 0'

8 e . . .
, (pr 99’ Ire, 13¢ and Tvr can be obtained by substituting

the above displacements (2.49), (2.50) and (2.51) into eqns. (2.7) and (2.8).

Next, we consider the inhomogeneous interphase region. Eqns. (2.38), (2.39) and

20

(2.40) are the general governing equations to determine the displacements in the inter-
phase. If we assume that the Young’ modulus and Poisson’s ratio have a radial variation,
the closed form solution usually cannot be obtained easily for this differential system

unless we consider a special case. In order to solve these equations we consider the power

variation case, that is E,“ = P (r/ a) Q and v1 = constant.
Then, eqns. (2.38) - (2.40) are reduced to
320

at], 2 q,
45— + (2Qv’-4—Q+4v’)U,+r (-1+2v’)-——2
’ 3’ (2.52)

Buq,
+r(Q+2) (—1+2v’)3—r + (12—12v—2Qv+Q)u¢=0

2 ,aZU, ,au, , ,
r (-1+v) +r(Q+2)(—1+v) +(—8v+5—2QV)U
W 57 ' (2.53)
3 all" 9 12’ 6 ' U —0
U9— 2Uq, = 0 (2.54)

If we introduce the operator eqn. (2.45), the differential system of eqns. (2.52) and

(2.53) changes to
(-d+4v’+2Qv’-4—Q) U,
2 2 (2.55)
+ (2v’d +2v’d—d -d— Qd+2dQv’—2Qv‘+Q+ 12— 12v) 0,, = o
(- d2 ~d+v’d2+v’d+dQv’ -dQ— 8v’+5 - ZQV’) U, (2 56)
+(3d+12v’—9+6Qv’)U¢=O '
Eliminating Uq, from eqn. (2.55) and eqn. (2.56), we finally get
(—1+v’)d4+ (-2Q+2v’+2Qv’-2)d3
2 I 2 l 2
+ v- -13v— v— +13d
(Q Q Q Q ) (2.57)

+(15Q-17Qv’—14v’+ Q2— 3szl+ 14) d
+4Q-4Qv’+24v’-4Q2 1= o

Eqn. (2.57) is a characteristic equation to determine U ,, which will give us four roots d1,

21

d2, d3 and d4. The calculation steps are given in the Appendix D.

For the our specified problem, the roots of characteristic equation usually follow
two cases:

case 1. all roots are real and unequal.

case 2. there are two different complex conjugate roots.

For the case 1, the displacements in the interphase are given as

“'2 1 d3

u’, = (12% +D’2r +D3r +05%“) sintpzsin29
uf, = 2 (blrd‘ + bzrd’ + [23rd3 + b4rd‘) sincpcos29 (2.58)
uzp = (blrd1 + bzrd’ + b3rd3 + b4rd‘) sin2cpsin29

For the case 2, if the complex roots are

dl=m1+ml

d2=m1—m1

d3 = m2 + inz (2'59)
(14 = m2 - inz
the displacements are given as
u', = [D’lcos (nllnr) r’"‘ + Dlzsin (nllnr) 2"“ + D; cos (nzlnr) r’"2 (2 60)
+Dl4sin (nzlnr) rmz] sintpzsin29 '
“b = 2[b1cos (nllnr) r'"1 + bzsin (nllnr) 1"“ + b3 cos (nzlnr) rm2 (2 61)
+b4sin(n21nr)r’"2]sin(pc0529
142, = [121 cos (nllnr) r’"' + bzsin (nllnr) r’"' + b3cos (nzlnr) r’”2 (2 62)

+b4sin (nzlnr) rmz] sin(p2sin29
The constants b1, b2, b3, (24 in eqns. (2.61)-(2.62) can be expressed in terms of unknown

constants DI , DI , D’ , D3. This is shown in the Appendix B. After we find ul,, us, ufp ,

we can substitute them into eqns. (2.7) and (2.8) to obtain the expressions for 0"", age,

I
‘P‘P’

l

1 z
0 1,9, 19¢ and 1,”.

22

Till now, we have the unknown coefficients D‘l’, D5, D11, DI , DI , DI , D’l", D’z",
3" . 2". §, DZ-
Although the perfect bond conditions produce eighteen equations at r = a, r = b

and r = c, only twelve equations are independent. These are

u, — u, u,p — u,p 0'" — 0'" Ire — trO at r - b. (2.63)
m _ e m _ e m _ e _ e _ .
u, — Ur 11¢ — 11¢ 0'" — 0'" ‘Cre — Ire at r — C.

Next we apply the Eshelby formula, derived by Eshelby (1956), to evaluate the
strain energy stored in the configuration of Fig. 1.3 [Christensen and Lo (1979)]. Accord-

ing to this formula, we have the elastic strain energy W under the applied displacement

conditions at infinity

Wamp— = w0 +— :1 (0,312,149 -o9 n uf)ds (2.64)

where S is the surface of the composite sphere defined by r=c, 63. and u? are stresses

and displacements in the composite in the absence of inclusion, and of]. and uf are stresses

and displacement disturbances in the effective medium. mep is the elastic strain energy

stored in the model of composite given in Fig. 1.3, W0 is the strain energy in the sphere

having the effective properties of composite. Since the outer equivalent region has the

properties of composite, which we want to evaluate, we can conclude that

Wcomp = W0 (2.65)
Combining eqn (2.64) and eqn (2.65), we obtain

I(Of.jn u 0—o'f.)jnju‘.’)dS= 0 (2.66)

For our problem, eqn. (2.66) yields

23

o o o
J(Ggru,+12¢ufp+1°9ur Girur-Tiouo’Tee 149) (IS = O (2.67)

where
d5 = czsintpdedcp (2.68)
Since this composite is subjected to the displacement boundary conditions at infinity, the
stresses and displacements in the composite without inclusion are given as
u‘,’ = Dir(sintp) 2511129

113 = Dirsintpcos29

ug= éD‘; rsin2<psin29
(2.69)
0° = zc‘D‘ir (sintp) 2$11129
13¢=ZGeD1rsintpc0320
1:29: G‘Dirsinthsin26

where D: = 82y. The displacements u:,u‘ u9 are given in eqn. (2. 51). By using eqns.

“w,

(2.7) and (2.8), we can find air, 1"

“p and 1:9. Now substituting all of these stresses and

displacements into eqn. (2.67), we obtain the condition that D: = 0. We find the effective
shear modulus by solving the twelve equations (2.63) for the twelve unknowns first and
then setting D: = O. The expression D: = 0 involves the effective shear modulus G‘,
which we solve for. Appendix F illustrates the procedure how to determine 0‘. In this
equation, the effective Poisson’s ratio ve does not appear and thus we can solve for the

unknown 0‘ explicitly.

24

2.3 EFFECTIVE THERMAL EXPANSION COEFFICIENTS

In this chapter, we evaluate the effective thermal expansion coefficient of particu-
late composite. Many papers were published on this subject and on thermal stresses [Levin
(1967), Rose and Hashin (1970), Takao and Taya (1985), Jasiuk et a1. (1988), Tong and
Jasiuk (1990), Benveniste and Dvorak (1990), Adams (1991), Lee et a1. (1991), Siboni
and Benveniste (1991), Li et a1. (1992), Sutcu (1992), Kouider and Jasiuk (1993), and oth-
ers]. But most of these publications either consider the case of no interphase or assume
that the interphase is homogeneous. Some papers discussed the inhomogeneous interphase
situation, but they didn’t predict the thermal expansion coefficient of particulate composite
with the inhomogeneous interphase. Our present study will demonstrate the effect of inho-
mogeneous interphase on thermal expansion coefficient of composite. In the analysis, we
use composite spheres assemblage model (CSA) to evaluate the thermal expansion coeffi-
cient.

Considering the Cartesian coordinates, we let a composite sphere shown in Fig.
1.1 be subjected to a uniform temperature change. Then, the corresponding volumetric
average strain is given by

EU = OtikATSkj (2.70)

where over bar represents the volumetric average, AT is the temperature change, 0t”. are

the unknown thermal expansion coefficients and 517 is Kronecker delta defined as

89=0ifi¢j

,1. 1111:)

and by using the strain displacement relations, we find the displacements in the Cartesian

spherical coordinate systems as

x xx 7‘ rr
“y = 3% “e = 0 (2.71)
u = '8' z u =

25

where a” = e = ezz = e"

yy
Eqn. (2.71) is similar to eqn. (2.1) and eqn. (2.5), and the analysis is similar to the
one for the effective bulk modulus. Again, since the loading is radial, the displacements

are of the form

14, = u,(r)
u9 = O (2.72)
uq, = 0

Here we have the same strain-stress relations and equilibrium equation as eqn.
(2.7) and eqn. (2.9), but the constitutive equations are different. Note for thermo-elastic
problem, the general constitutive equations are

oi]. = Cijk,(ek,—aknAT8nl) (2.73)

where i,j = r, 9, (p

Therefore, for our problem eqn. (2.8) can be modified as

V
C" = 20 (8"- aAT+ We)

 

v
099 = 26 (296 - ctAT+1_ 2ve)
v
TrO = 07w
199 = 0799
Tcpr = GYQ)!‘

where e = €rr+899+€¢¢-3CXAT

Again the same approach is followed as in the bulk modulus evaluation and the

governing equation that can be used to determine u r is given as

26

32“ BE 65 233 3a_E
Er2(v—1)(2v-1)(1+v)—22+r(2E+a_ r- 2rv-a—r-r rv a +2rv a—r
+4Evra—v - zEv2r3—V — 4Ev - 251:2 + 411:);3 )—'—" + (4Ev + 2Ev2 - 4Ev3
Br Br 3r (2.75)
+2vr§§ awry—411335 +4EvrM-2Er-1-2Er )u,
Br Br r87“ (Jr
8E 30% aaE 3v _
+r2 (1+v)2 (Zava— +2Ev-a—-— “a—r —2Ea—a—r-Eg—;)T—O

where E = E (r), v = v (r) and 0t = 0t (r). For the homogeneous media: the inclusion
and the matrix, eqn. (2.75) was reduced to eqn. (2.12). The displacements in the inclusion
and the matrix, are given in eqn. (2.14). Substituting eqn. (2.14) into eqn. (2.57), the

stresses are

 

 

62 _ EPCf, aPEPAT
" 1-2vP l-2vP

EpCfl 2E"'Cf5 a’"E"'T

(2.76)

m—

" 1—2v’" (l+v"')r3— 1-2v’"

 

For the inhomogeneous interphase case, we choose power, linear and cubic functions to

simulate El, 011 and v’. A special case is given as

Q N
E’ = mg) 01’ = 114(2) v’ = constant (2.77)

If we substitute (2.77) into eqn. (2.75) and simplify it, we get

1 r N
r(1+v)(Q+N)ATM(Z)

232“
-2)u2+ _1+v = o (2.78)

I
r2 22—2 +r(Q+2)3—+(12vQ

V1

 

 

The closed form solution can be obtained for the displacement ul, from eqn. (2.78)
(see Appendix G) as

u’, = Ci]: (r, v’, Q, N) + cfizp (r, v’, Q, N) +11 (r, v', Q, N, M, AT, 0) (2.79)

where §, [3 and n are functions in terms of r, v’, Q, N, M, AT, a, which are given in

27

Appendix F. For other cases, we can use series solution method to determine 14'.
After determining the displacements in the interphase, we substitute them into

eqns. (2.7) and (2.74), and the corresponding stresses can be found. The next step is to

evaluate the five unknowns C51 , C11, sz, Cf; and C3. Similarly as in the bulk modulus

analysis, we assume that perfect bonding conditions at the particle-interphase and inter-
phase-matrix interfaces are applied to our problem which give us eqns. (2.26) and (2.27).
Since our CSA model is only subjected to a uniform temperature change, the outer surface

is traction free, so we have the additional equation given as
0",", (c) = 0 (2.80)

Combining eqns. (2.26), (2.27) and (2.80), we can evaluate the five unknowns.

The definition of thermal expansion coefficient is given in eqn. (2.70) that is
éik

22 -.- fiakj (2.81)

G

Since only 5" exists in the composite, see eqn. (2.71), the thermal expansion coefficient

of composite can be expressed as

m1

_ ff
01,, - T (2.82)

D

where E" is average strain in radial direction. From eqn. (2.71) and using the CSA model,

the definition of the average strain can be expressed as

 

 

u u'" c
5.. = J = '( ) (2.83)
‘J r c
So eqn. (2.82) is written as
"5
t
2 “Tm Cf“? 284
‘ CAT " AT ( ’ )

where the superscript e denotes effective composite.

3. NUMERICAL ANALYSIS

3.1 BULK MODULUS ANALYSIS

In this thesis, we consider that the inclusion and matrix are homogeneous and iso-
tropic, and we assume an idealized model of interphase, in which the interphase is inho-
mogeneous but isotropic and has Young’s modulus and Poisson’s ratio varying spatially in

a radial coordinate, eqns (2.19)-(2.21). Variations of E are illustrated in Fig. 3.1a,b

 

 

 

 

 

 

 

 

 

 

 

EA r 3 r 2 r
(Ff-Pug) +93%) ”‘2’”
Q1
E" E‘ = mg)
‘3
2.8 ,8 El = P2(g)+Q2
2 E
e \\ e»
8 E t '
>" £3 4—»
g 8 .><
.... J: E
e E
B
4—4—> .5
4 ’2 >
d L h
2.
radius p
Fig.3.1a

Schematic variation of Young’s modulus in the interphase

In our calculation, we keep the values of E” , E'" constant and change the thickness

t of the interphase. When r=a, from the relation in Fig. 3.1, we have E1 = E” , then we can
get P1 = E” from E’ = P1 (r/a)Q‘. When r = b,we have E1 = E'", and finally we get

Ql = [In (Em/EpH/[ln (b/a)] P1 = E" (3.1)

Analogous to the power variation case, for linear variation, we have

28

29

  

3 2
E’ = 103(2) +9.15) mg) +5

 

 

  

E’ = P2(-:2)+Q2

 

 

 

 

 

Fig. 3.1b
Variation of Young’s modulus in the composite’s interphase

30

_bEp—Ema _a(Em"Ep)
Qz'W P2“—‘1:T— (32)

and for cubic variation case

223 = 2a3b (1?" - 19")

2b4 - 3123a - 3b2a2 + 70312 — 3a4

_ 3a3 (bEp—bgn—ali'"+aEp)

Q3 _ 2b4 - 3123a - 3b2a2 + 7a3b - 3a4
_ 6a4 (Em - B")
_ 2b4 - 3b3a - 3b2a2 + 7a3b — 3a4
_ 2b4li‘”-3530122",—390230-1-6a3bE"’+a3bEm—3a4Efl
_ 2b4 - 3b3a - 3b202 + 7a3b - 3a“

 

 

(3.3)

 

R

S

 

where we assume that when r=a, ad-r-E' (r) = 0; r=a, $15 (r) = 0. We have similar pro-

cedure for v1. It should be emphasized that we can change the parameters P3, Q3, R, S,

P', Q', R' and S' in eqn. (2.21) to obtain different variations of E1 and v1 in the inter-
phase. For example, if we let P3 = 0, Q3 = 0 and R at 0, S 99 0, we obtain linear varia-
tion in Young’ modulus. When P3 = 0 and Q3 #0, R #0, S ,4 0, we get the quadratic
variation of Young’modulus in the interphase.

Also, when Ql = 0, P1 at 0, eqn. (2.19) will give us a homogeneous interphase sit-
uation, and when Q1 = 0, P1 = E” , eqn. (2.19) yields no interphase situation. Now, in the
numerical examples presented in this thesis, we first consider the following elastic proper-
ties:

P =
V 0.3 VI = 0.3
GP = ZSGPA vm = 0.4 (3.4)

Q
E1=P(g) G’”=1.0GPA

and other elastic properties can be obtained directly from the following relationships

31

 

 

G(3}.+ZG) G=E/(2(1+v))
E:
1+0 K=3/(E(1-2v))
v=>./(2(7.+G)) [2:230 (3.5)
2_ Ev 3
‘ (1+v)(1-2v) E=2G(1+v)

Now, we start the numerical analysis by considering different variations in the
interphase. First, we consider the inhomogeneous interphase case where the Young’s mod-
ulus has power variation. Substituting eqn. (3.4) into eqn. (3.1), we can determine the con-
stant P. If we choose the different thicknesses t as follows

t= 0.010 b = 1.10
t = 0.050 b = 1.050
t = 0.020 b = 1.020
t = 0.0080 b = 1.0080
t = 0.0050 b = 1.0050
t = 0 b = 0

(3.6)

and substitute these t’s into eqn. (3.1), we can calculate the values of constant Q. We sub-
stitute 0, b, c, Q as well as eqn. (3.4) into eqns. (2.26), (2.27) and (2.28), solve for the

unknown coefficients, and substitute them into eqn. (2.31). We can obtain the effective

bulk modulus in terms of volume fraction f = (03/c3) and the elastic properties of inclu-

sion, interphase and matrix. We plot it for different values of thickness 1‘, (see Fig. 3.4). In
order to compare the power variation result with the homogeneous interphase assumption,

we introduce the following relationship

b r Q2
(15(5) dr

5’, = b_a (3.7)

 

Eqn. (3.7) gives the same “area” average of Young’s modulus as the power variation in

interphase. Following the above steps and setting Q1 = 0, P1 = Ell we obtain the homo-

geneous interphase case (Fig. 3.5). Finally, we take P1 = E'", Q1 = 0 and plot the no

interphase case as Fig. 3.3, which agrees with Christensen and Lo’s (1979) result as

32

expected. Observing Fig. 3.4 and Fig. 3.5, we find how the bulk modulus depends on the
thickness 1‘, and we observe that the thicker is the interphase, the higher is the bulk modu-
lus, and the effect of interphase is higher at the larger’volume fraction of particles. We
choose t = 0.10 and t = 0.050 cases from Fig3.4 and compare with the same thickness
situation in Fig. 3.5, and then compare with perfect bond situation. This comparison is
given in Fig. 3.6. We are also considering the following two situations for the bulk modu-

lus case. One is when the interphase is softer than the matrix and has a power variation in

Ii" and v1 is constant, and the second is when the properties of the particles and the matrix
are interchanged. Then the corresponding homogeneous interphase assumptions are con-
sidered too, the results of comparison are similar to the above analysis. These results are
displayed in Fig. 3.7 - Fig. 3.12. We find that the power variation interphase model gives
the lower prediction for the effective bulk modulus than the homogeneous interphase
model and the higher prediction than the perfect bonding case. This agrees with Jasiuk and
Kouider (1993). In this case, the homogeneous interphase model overestimates the effec-
tive bulk modulus.

Before we study the linear and cubic variations in Young’s modulus in the inter-
phase, we should discuss the series solution and test its convergence. Here, we choose one
case from eqn. (3.6), with b = 1.050 for our analysis. When we fix the relation
b = 1.050, from eqn. (3.1), we find that the Young’s modulus in the power variation has
the relation .

15’ = 65 (g).64 (3.8)
Now, we use the cubic variation to simulate this power variation as shown in Fig. 3.2

So, the approximate expression can be achieved as

3 2
15’ = —541314.1362(—;.) +1697798.147(£) -1775237.386(§) +618818.3794(3.9)

by using the curve fitting method. Substituting eqn. (3.9) and eqn. (3.4) into eqn. (2.16),

 

33

    
   

 

 

 

 

 

 

60 «F
3 2
2 5‘ = — 541314.1362 (2’2) +1697798.147(-2E)
SO
-1775237.386(g) + 618818.3794'
40»
30 .1_
20 «-
10 «-
1i01 1i02 1i03 1i04 1105
fig3 2 r/a
cubic variation simulating power variation
we obtain
8211, au, 0 3 10
'a—rj'l’fU'); +8(")“r" (' )
where
7'3 r2 r
-2706570.679—3 + 6791192590—2 - 5325712126; + 1237636750
f(r) = a3 02 (3.11)
r [- 54131413652 + 169779814712 - 17752373862 + 618818.379)
a 0
r3 r2 2,
-309322.364—3 - 485085.186—2 + 2028842729 H — 1237636750
0 a 0
g (r) = 3 2 (3.12)
r2 (— 541314.136’—3 + 169779814712- — 17752373862 + 618818.379)
0 0

f (r) and g (r) both have the Taylor series expansion as

f(r) = anh-ro)" (3.13)
n=0

34

g(r) = Zgn(r—r9)n (3.14)

n=0
If r9 = 0 is a regular singular point, then according to Fuchs’ theorem, we can

find a convergent series solution as
u = 2 a,r""" (3.15)

But r0 = 0 is located at the center of inclusion and not in the interphase. One can also

show that the other regular singular points lie outside the interphase region a < r < b, so

we choose ordinary point r0 in the interphase. It is also more convenient to choose
0 5 19 S. b, since this will require a smaller radius of convergence and thus a number of

terms in the series will be smaller than for the case when r9 is outside this interval. The

series solution is expressed as follows

u. = Z 4.0-6)" (216)
n=0

Eqns. (3.16), (3.14) and (3.13) are substituted into the differential eqn. (3.10) and coeffi-

cients of same powers of (r — r9) are set equal to zero to determine 0,,. Since the center
point in the interphase gives a faster covergence than the boundary point, we take
r0 = (0+b) / 2 and this is an ordinary point in our series solution. In this case,
r9 = 1.0250. The MAPLE computer program was written to solve this problem. Finally,
we get solution of eqn. (3.10) in the form

u, = c“, 2 b,,(r-1.02511)"+c‘2 2 any-1.02511)" (3.17)
n=0 n=0

where bu and on are known, and C11 and C12 are the unknown constants. After we have

the expression given in eqn. (3.17), the effective bulk modulus can be evaluated easily fol-

lowing the previous steps.

35

The bulk moduli for the interphase properties given in Fig. 3.2 are shown in Fig.
3.13. These two curves almost overlap with each other, and the small difference comes
mainly from the small difference between the cubic polynomial fit and the power varia-
tion. We must mention that when the series solution includes over sixty terms, the curve 1
in Fig. 3.13 does not change when the number of terms is increased. This implies that sixty
terms series solution will give convergent solution for this particular case. We have also
tested the convergence of our series by using the ratio test that is

0 (r-r )"+l
lim "+1 0 <1 (3.18)
n—>oo

 

 

 

n
0 ,2 (r - r9)
We found that our series solutions satisfy this condition

Repeating the above procedures, we can derive bulk moduli for cases given in

eqns. (2.20) and (2.21). First, we assume that v’ is constant and take different variations

of E," . The comparison of linear variation of E, cubic variation of El, uniform interphase
and no interphase case are given in Fig. 3.14. We observe that the homogeneous inter-

phase yields the highest modulus. Again, the tendency of this results is the same as in Jas-
iuk and Kouider (1993). Then, we let E‘ be constant and compare different variations of
V, (Fig. 3.15). We also let El be the linear variation and compare the different variations

of v’ (3.16). We find that when 15’ is fixed, power variation of v’ gives highest value in

effective bulk modulus and uniform case gives lowest result. This tendency is opposite to

the one for varying E1.
Observing Fig. 3.14, we may infer that the smoother variation of Young’s modulus

in the interphase gives the lower estimation of the effective bulk modulus, while the

rougher variation of v1 gives the lower estimation of the effective bulk modulus.

 

36

BULK MODULUS
(no interphase)

 

54 . 16 .
50"

74‘

g .9.

1t

8

g t=0(no interphase)

g Km =4. 667(GPA )
Kp=54.167(GPA)

E _ #03

a, 3" ope-mam)
G'"=1(GPA)

E 10:04

E

 

 

 

Fig. 3.3 Effective bulk modulus vs. volume fraction f for the no interphase case.

37

   

 

 

 

 

 

 

 

BULK MODULUS
25.. (inhomogeneous interphase)
’4
Q
. E: El _ p2(:) '
i i .
20" .4 g
5’ '3.
>- .§
2 1. €80 c
D. 2. t=0.0050 >1“
9. 3. c=0.008a
k 4. t=0.028
Vt 5. t=0.050
b 6. t-0.10
§ 15" Km=4.667(GPA)
g 119:0.3
a =25(GPA)
“2 =0.3
Gm=I(GPA)
E v’"=0.4 r a,
a El = P1(-)
U a
H
5,,
1.5.5?
I l l l l l l
T I T I I I j
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t'
Fig. 3.4 Effective bulk modulus of the composite with an inhomogeneous interphase.

38

 

  

 

 

 

 

 

 

 

BULK MODULUS
(homogeneous interphase)
35-- ‘ EA
b
Ep 5‘ =IP1(r/0)Q‘dr/(b-0)
a 32‘ ‘2
'3 .
E 3 La
30" 00 .9
a
+1.;
5 r
2 252-
g 1. taro
2: 2. 6:0. 005a
3. t=0.00Ba
g 4. t~0.020
g 5. t=0.058
q 60 t=0.18
20--
% K"'=4.667(GPA)
E Kp=54.167(GPA)
1P=0.3
“2 912200111)
2% fi’éuam)
- =0.4
N 15- 2 2/
:1 I Q '
.9 E = jP,(r/a) 'dr/(b—a)
a
(P1 and Q, are from Fig. 3.3)
10-
4.665;.
I I L I l I I
I I I I I I I
0 0.1 0.2 0.3 0.4 0 5 0.6 0.

f
Fig. 3.5 Effective bulk modulus of the composite with a homogenous interphase.

39

BULK MODULUS
(power-homogeneous)

30""

 

 

1. t=0(no interphase)
_\ 2. t=0.05a(power variation)
a 25.. 3. t=0.05a(homogeneoua interphase) ,.
u 4. t=0.01a(power variation) :'
9: 5. t=0.01a(homogeneous interphase)
92 K'"=4.667(GPA) :'
b KP=54.167(GPA) 2.:
1% 19:03 .2
g =25(GPA)
22' =03
I»: Gm=1(GPA)
é v”'=0.4
in
§
2
H 15-
h
n
102- ..............

4.65-

k l I l I I I

l I I T I T I

0 0.1 0 2 0 3 0.4 0 5 0 6 0.7
f

Fig. 3.6 Comparison between the inhomogeneous and homogeneous interphase cases.

 

 

 

 

 

 

 

 

BULK MODULUS
(inhomogeneous interphase)
16" 5‘
Q.
. E” . 5‘ = 101(2)
% 3
3: i .2
1r ’3’ '9' 1
5‘5, ?~ ‘
1. tag ‘42
fl 2. €80.0058 Dr
'1 3. t=0.008a
{‘5 12- 4. t=0.02a
" 5. t=0.05a
k 6'. t=0.18
In
E K'"=4.667(GPA)
Q vP=0.3
g =25(GPA)
1m- =0.3 5
E G’"=1(GPA)
m v""=0.4 r
g E = PI(;)
an
S ._
.// 6'
er ’/
4.651
I I I I L I I
I I f T I I I
0 0.1 0.2 03 150.4 05 06 07

Fig. 3.7 Effective bulk modulus vs. f for an inhomogeneous interphase softer than matrix;
Interphase has a power variation in Young’s modulus and a constant Poisson’s ratio.

41

 

 

 

 

 

 

 

BULK MODULUS
(homogeneous interphase)
1e~- EA 2
b
Ep 5‘ = IP1(r/a)Q'dr/(b-a) 3
.3 g , a
E i 1’4"
2.“ .3. _._.
4—75
1. no "
—~ 2. t=0.005a
E _ 3. t=0.008a
2 1‘“ 4. t:0.02a
& 5. t=0.05a
m 6. t=0.1a
E K"'=4.667(GPA)
q KP=54.167(GPA)
g vP=0.3
.4 mi =25(GPA)
g v=dL3
v'”=0.4
g .
g E’ = [P1 (r/a)Q'dr/ (b —a)
E a
a 81’ (P1 and Q1 are from Fig. 3.7)
6'
.,,//
6+
4.661
I I I l I_ I I
I I I r T T I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t
Fig. 3.8 Effective bulk modulus vs. f for a homogeneous interphase softer than matrix;
Interphase has a constant Young’s modulus and a constant Poisson’s ratio.

42

BULK MODULUS ,1
(power-homogeneous) ;

(soft interphase)

 

 

16--
141--
1. t=0(no interphase)
2. t=0. 05a (homogeneous variation)
I; L 3. t=0.05a (power interphase)
g 12 4. t=0.1a (homogeneous variation)
c 5. t=0.1a(power interphase)
K"'=4.667(GPA)
{g KP=54.167(GPA)
g has ,3
9 =25 ( GPA )
g m. =0.3
G'"=I(GPA)
E v"'=0.4
0:
g
E .
6"- ......................
4.661 .....
I I L I I I I
F I I I I I I
0 0.1 0 2 0.3 0.4 0 5 0.6 o 7

Fig.3.9 The comparison between Fig. 3.7 and Fig. 3.8.

43

 

 

 

 

 
   

 

 

 

 

5‘ 161 BULK MODULUS
(inhomogeneous interphase)
interchange of properties

504*
Q:
,
EA 1:4 = 191(5)
E'"
_ .. ~° .
\ ’5 g S

.. =~ .

33 .2 -

{D

i; H

V)

b 30..

g

E 1. 12:0

2. t=0.005a

“ 3. t=0.008a

E 4. t=0.028

H 5. t=0.058

Q “P 6. t=0.1a

E K'"=4.667(GPA)

vP=0.4
=1(GPA)
=0.3
G'"=25(GPA)
1_ 5 '
10-- E _ P’(a)
I I I I I I I
I I I I I I I
0 0.1 0.2 0.3 f 0.4 0.5 0.6 0.7

Fig. 3.10 Effective bulk modulus vs. f for an inhomogeneous interphase with
interchange of the properties.

BULKHMDDULUS

 

 

   
  

 

 

 

 

 

54.161
(homogeneous interphase)
interchange of properties
50'
b
E’ = IP1(r/a)Q'dr/(b-a)
EA 0
E
3 E” I
.- 3 ."
‘°‘ 2 i 1" i
3 g i
A 3 .a
g 4‘15?
3 *—>
14
to
b
30*
§
B 1. t=0
K 2. t=0.005a
g 3. t=0.00Ba
m 4. t=0.028
5. t=0.05a
E 6. t=0.1a
£320.. K'"=4.667(GPA)
E yr=o.4
=1(GPA)
=0.3
G’"=25(GPA)
v’"=0.3
b
13': [P1 (r/a)Q'dr/ (b-a) \
10' a ,
(PI and Q1 are from Fig. 3.10)
I I I I I I I
I I I I I I I

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

f
Fig. 3.11 Effective bulk modulus vs. f for a homogeneous interphase with interchange
of the properties.

45

 

 

54.151 sum: MODULUS
(power-homogeneous)
(interchange of properties)
5m-
1. t=0(no interphase)
2. t=0.05a (homogeneous interphase)
3. t=0.05a (power variation)
4. t=O.1a (homogeneous interphase)
5. t=0.1a (power variation)
4 g. KP=4.667(GPA)
K”=54.167(GPA)
v'"=0.3
"'=25(GPA)
—~ =0.3
E GP=1(GPA)
E vP=0.4
3.;
I0
g...
§
I4
E
0)
g
20-
E
............. 1
10*-
4
I I I I I I I
I I I I I I I
0 0 1 0 2 0 3 0.4 0 5 0 6 0 7
f

Fig. 3.12 The comparison between Fig. 3.10 and Fig. 3.11.

.BUEJKJMCHNILIHE
(cubic-power variations)

20.3-

20.20

20.1.

19.9‘

19 e 8'”

19.70

 

19.6-

 

 

0.74 01742 01744 0:746 0174: 0275
t

 

 

 

 

2 2%
£5.
10-
1:.
U:
E “T 1. t=0.05(cubic simulation)
q “- 2. t=0.05a (power variation)
g K'"=4.667(GPA)
.4 12? KP=54.167(GPA)
§ =0.3
_ =25(GPA)
g 1° $0.3
H G’"=1(GPA)
8 9" v’”=0.4
E".
h. 5"
u:
"661 i t : e : : :
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

f'
Fig. 3.13 Effective bulk modulus vs. f; the power variation of Young’s modulus in
interphase is simulated by cubic variation.

47

    

 

 

 

 

EFFECTIVE BULK MODULUS x‘ (GPA)

 

 

BULK MODULUS
2r-
E
22- A
.g 23‘
g.
E g
1s~ .
~1———£———u~
pr
1&-
1r-
1. t=0(no interphase)
2. t=0.05a(cubic variation)
3. t=0.05a(linear variation)
4. t=0.05a(uni£orm interphase)
1’ ' K'"=4.667(GPA)
EW=51L6NGTM)
tsp-=0.3
fiflfiumGEA)
10' =03
GWEJWGBM)
v”'=0.4
/'//
8‘— /,.//
en-
4137
I I I I I I L
1 I l I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

f
Fig. 3.14 Comparison of effective bulk moduli from different variations of Young’s
moduli in the interphase

EFFECTIVE BULK MODULUS x‘ (GPA)

48

 

 

24.7.1 V'" = 0.4
VP = 0.3
24.64. EP = 65 (GPA)
E’" = 2.8(GPA)
24.5.. b = 1.050
24.44?
24 .3... 1. cubic variation of v1
2. linear variation of v
3. uniform variation of v
24.2..

 

 

0.745 0:746 01747 f 0I74e 0:749

g

VA

251*-

”J.

 

20-

inclusion(particlc)

 

15"

 

 

 

 

 

 

 

 

 

10-

J

 

j
1'

f

0 0.1 0.2 0.3 0.4 0.5 0.

0

     
 
 
   
 

Z75

   
   

 

J

  

0.7

Fig. 3.15 Bulk moduli of composite with different variations of Poisson’s ration

and uniform variation of Young’s modulus in the interphase.

49

 

    
   

v’" = 0.4 3
23.5 VP = 0.3

E” = 65 (GPA)

15'" = 2.8(GPA)
23 «b

b = 1.050

  
    
 

22. s- 1. cubic variation of v’

2. linear variation of v
3. uniform variation of v

22.

 

A L J I l
v r f

0.73 0.73m.734o.7360.7380.74 0.7420.7440.7460.7480.75
\ t /
EA v4

 

 

 

      
 

N
‘F

.0
'2’

H N
r 9.?
Young's modulus _

inclusionanicle) 1!;

 

Poisson's ratio

0.:
1‘

 

 

3

H
*

 

 

 

 

 

 

 

 

EFFECTIVE BULK MODULUS x‘ (GPA)

 

 

124
10
0-
@-
0i1 0‘.2 0‘.3 tofi 0i5 0i6 0i?-

Fig. 3.16 Bulk moduli of composite with different variations of Poisson’s ratio and
linear variation of Young’s modulus in the interphase.

50

3.2. SHEAR MODULUS ANALYSIS

In this section, we conduct the numerical analysis of the effective shear modulus.
We assume that the interphase between the matrix and inclusion is isotropic and inhomo-
geneous, and that the Young’s modulus of interphase has the elastic properties changing
with the radial distance from the inclusion boundary, and the Poisson’s ratio is constant.
We illustrate our analytical results by considering a particulate composite, with the proper-
ties K"11 = 4.667GPA, Kp = 54.167GPA, G“[1 = 1.0GPA, Gp = 25.0GPA,
v’" = 0.4, v? = 0.3 and v' = 0.3, which has an interphase region with spatially varying

properties given in eqn. (2.19). In our example we assume that E." (a) = E" and

E’ (b) = E'" (Fig. 3.1a.b); constants P1 and Q1 are found from these two consu'aints

[eqn. (3.1)]. For comparison we also consider a homogeneous interphase case such that

E’ = fip(r/a)9dr/(b-a) which can be obtained from eqn. (2.19) by setting Q1 = o
and P1 = E’. We can also obtain the perfect bonding case by setting Q1 = O and

P1 = F". We substitute the properties of the composite [eqn.(3.4)] into eqn. (2.57), and

solve this fourth order equation. We get two complex conjugate roots as in eqn. (2.59). So
we take the expressions (2.60), (2.61) and (2.62) as our displacements results. By using

eqns. (2.63) and the remote boundary condition (2.33), we can obtain a set of linear equa-

tions to be solved for the unknown coefficients. Then we set the coefficient D: = O and

we can obtain a quadratic equation for Gc . We neglect the negative root of the quadratic
equation and finally, we achieve the relation
a3
0‘ = 17(7) (3.19)
c
where F represents function in terms of volume fraction and depends on the properties of
inclusion, interphase and matrix, and superscripts e symbolizes the effective composite

region. We plot this power variation case in Fig. 3.18. The results that shows the same ten-

51

dency as the bulk modulus results. We see that the thickness of interphase has an impor-
tant influence on the effective shear modulus. The effect of interphase is more pronounced
at the higher volume fractions. The homogeneous interphase case is given in Fig. 3.19 and
the no interphase case is shown in Fig. 3.17. Similarly as in the bulk modulus analysis, we
compare the power variation case and homogeneous interphase model case in Fig. 3.18.
Again the perfect bonding condition yields lowest results, the power variation gives the
intermediate values and the homogeneous interphase model yields the highest shear mod-
ulus. This agrees with the effective bulk modulus results when the same comparison is
made. Comparing to Fig. 3.7 - Fig. 3.11 in the bulk modulus study, the analogous conse-
quences are obtained for softer interphase case (Fig. 3.21 - Fig. 3.23) and case when there
is an interchange of properties between the matrix and inclusion (Fig. 3.24 - Fig. 3.26). As
in the bulk modulus analysis, we would like to have calculated other variations of inter-
phase [eqn.(2.20), eqn.(2.21)]. However since these involve much complicated mathemat-
ical manipulations, we will leave it for the future study.

Summarizing the effective bulk modulus and shear modulus results we find that
the homogeneous variation of Young’s modulus in interphase overestimates the effective
elastic moduli in our study. This agrees with the observation in Jasiuk and Kouider (1993),
while homogeneous variation of Poisson’s ratio in interphase underestimates the effective

bulk modulus.

52

SHEAR MODULUS
(no interphase)

 

20'”

15v 1:20 (120 interphase)

fl

:5 GP=25(GPA)

- 6'" =1 (GPA)

'b has
v"'=0.4

 

 

 

 

Fig. 3.17 Effective shear modulus vs. f for the no interphase case

53

 

 

 

 

 

 

 

SHEAR.HODULUS
(inhomogeneous interphase)
E
in» A
g
‘5»
10-- >8:
1. t=0
*’ 2. t=0.005a
g 4. t-0.023
V- 5. t=0.0Sa ’
t, 6'. t-0.1a
GP=25(GPA) ‘
6" G’"=1(GPA)
vp=0.3
v7=0.4
v=0.3 r I
E’ = -
, (a)
4" /‘
2“ ,,
1
J l I l l l 1
fl 1 l l I I l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

f
Fig. 3.18 Effective shear modulus vs. f for an inhomogeneous interphase stiffer than
matrix; interphase has a power variation Young’s modulus and a constant Poisson’s ratio.

 

 

 

 

 

 

 

19* SHEAR MODULUS
(homogeneous interphase)
E
16" A b
E)" E’ = ]P,(r/a)Qldr/(b—a)
a 3 l / a
. .. 8 5'"
14- ;. ..
g g ‘r
.a a E
<—‘—>
12* >'
1. t=0
2. 1:80.0058
3. t=0.008a
«10" 4. £30.028
a 5. t=0.0Sa
9, 6. £230.18
'20 GP=25(GPA)
G’"=1(GPA)
9" v”=0.3
v"'=0.4
v’=o.3
b
6.. E': IP1(r/a)Q‘dr/(b—a) //
a
(P I and Q, are from Fig. 3.16)
4-11-
2.1-
1
I I I I I l I
I I I I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 3.19 Effective shear modulus vs. f for a homogeneous interphase stiffer than matrix;
Interphase has a constant Young’s modulus and a constant Poisson’s ratio.

55

 

 

15" SHEAR MODULUS
(power-inhomogeneous)
16--
14%
12» "
1. t=0(no interphase)
2. t=0.05a(power variation)
3. t=0.05a(homogeneous interphase) '
«10" 4. t-0.1a(power variation)
5 5. tan-0.1a (homogeneous interphase)
{5' GP=25(GPA) '
G'"=I(GPA)
W vP=0.3
v’"=0.4
v’=o.3 1
6‘-
4-0-
2-- .................
1 I I l I l I I
I I I I I I I
0 0 1 0.2 0.3 0.4 0.5 . 0.

Fig. 3.20 Comparison between Fig. 3.18 and Fig. 3.19.

56

 

 

 

 

 

 

 

 

SHRAR.MODULUS
7“ (inhomogeneous interphase)
EA
Q
. E” E’ = 11>l (5)
a 3 E” a
9» g g r_.
"° '5 1
a—gi
5,-- ¢——-> r
1. tsO >
2. t=0.00Sa
3. t=0.008a
4. t=0.02a /
5. t=0.05a /
-\ 6. t=0.1a
E 4+
o K'"=4.667(GPA)
is has
3P=25(GPA)
=0.3
G'"=1(GPA)
3.. (=0.4 r
P1:
2.-
1-
i i : i i i i
0 0 1 0 2 0.3 0.4 0 5 o 6 0.7

f

Fig. 3.21 Effective shear modulus vs. f for an inhomogeneous interphase softer than

matrix; Interphase has a power variation in Young’s modulus and a constant Poisson’s
lflUO.

57

 

  

 

 

 

 

 

 

 

SHEAR.MODULUS
(homogeneous interphase)
7-:-
E
‘ b
51’ El = JP1(r/a)Q‘dr/(b-a)
% .3 a
6.. E i
L". .5 . ,
a 3
g 3 fl‘-
2...? i
H
d—L—h r
5~- 1. £80 >
2. t=0.0058
3. t=0.008a
4. t-0.02a
5. t=0.058
6. t-0.1a
E4. _ K’"=4.667(GPA)
g KP=54.167(GPA)
‘5 v”=0.3
=25(GPA)
=0.3
G'"=1(GPA)
v’"=0.4
3-- b
E’ = JPl(r/a)Q'dr/(b—a)
a
2--
1-1
} i i i i i i
0 0.1 0.2 0 3 0.4 0 5 0.6 0.7

Fig. 3.22 Effective shear modulus vs. f for a homogeneous interphase softer than matrix;
interphase has a constant Young’s modulus and a constant Poisson’s ratio.

G‘ (GPA)

58

SHEAR MODULUS
(power-homogeneous)

(soft interphase)

 

 

1. t=0. 05a (homogeneous variation)
4“ 2. t=0.05a (power interphase)
3 . t=0 . 1a (homogeneous variation)
4 . t=0.1a (power interphase)
K'"=4.667(GPA)
KP-—-54.167(GPA)
vp=0.3
=25(GPA)
v=0.3
G'"=1(GPA)
3" v"'=0.4
2m-
1.:
I I I I I I
I I I I I I
o 0 1 0 2 0.3 0.4 0 s .

Fig. 3.23 The comparison between Fig. 3.21 and Fig. 3.22

59

 

 

 

 

 

o‘ (GPA)

 

 

 

2% SHEAR MODULUS
(inhomogeneous interphase)
interchange of properties
20¢ EA I} = P1( )
\ 3.3%?—
5 8 g
>‘ 5
15" E .5
‘49
d—L—N
>r
1. t=0
2. t=0.005a
3. t=0.008a
10._ 4. t=0.02a
5. t=0.05a
6. t=0.18
K'"=4.667(GPA)
vP=0.4 \
=I(GPA) \‘
=0.3 \
G’"=25(GPA) \
S~~ v’"=0.3
Q \.
z r I g
E = P197)
1 i i i i i i
0 0.1 O 2 O 3 0.4 0 S 0 6 0 7

Fig. 3. 24 Effective shear modulus vs. f for an inhomogeneous interphase with the
interchange of properties.

 
 

 

 

 

 

 
  
  

 

 

 

25 SHEAR MODULUS
(homogeneous interphase)
interchange of properties
b
15‘ = IPl(r/a)Q‘dr/(b—a)
5‘ a
3"“ E
3.153
.2 g
3 .3.
>‘ a
g .5
15" A—flgJ
2
E5.
‘5 1. tBO
2. ts0.005a
3. t=0.008a
1o-- 4. t=0.02a
5. t=0.05a
6. t=0.18
K'"=4.667(GPA)
vP=o.4
=I(GPA)
=0.3
G’"=25(GPA)
5“ v"'=0.3
b
E’ = [P1 (r/a)Q'dr/ (b—a)
a
I I I 4 I I I
T I T I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
f

Fig. 3.25 Effective shear modulus vs. f for a homogeneous interphase with the
interchange of the properties

61

 

 

 

 

 

62

3.3. THERMAL EXPANSION COEFFICIENT ANALYSIS
We assume that the thermal expansion coefficients in the inclusion, matrix and

interphase are follows

or? = 5 x 10'6/0C
-6
a'" = 66 X 10 /0C (3.20)

erN
0"- (a)

where M, N are constants. We can sketch eqn.(3.20) in Fig. 3.27.

 

 

 

 

 

 

 

 

 

 

 

a A a’"
N
I r
on = M —
( a)
E
Q)
'8
8
§ on?
I:
'2
g t .x
e- a
q—p
o g E
'8 ”g
[5 E'
3
G
a ""
a a
b
Le +
C
« F
> r
radius

Fig. 3.27 Variation of a thermal expansion coefficient in the interphase.

 

where
M = or” (3.21)
N = (3.22)
b
I" (a)

We substitute eqn. (3.20) and (3.4) into the MAPLE computer program (Appendix H),

which gives thermal expansion coefficients for the inhomogeneous interphase with a
power variation at. The results are given in Fig. 3.29.
I N . 1 b N .
When we replace CL = M(r/a) With or = [IaM(r/a) dr] / (b -a) 1n eqn.

(3.20), we obtain Fig. 3.30. The comparison of the results between Fig. 3.29 and Fig. 3.30

are given in Fig. 3.31. Then we assume that the interphase is homogeneous, and we
re 1 El - P Q‘ El - ' ' ' I

p ace — 1 (r/ a) by — 34 (GPA) in eqn. (3.4). For power variation (1 and
constant (1.1 cases are illustrated in Fig. 3.32 and Fig. 3.33 respectively. We also compare
above two cases in Fig. 3.34, and then, we set b = a and calculate no interphase case in

Fig. 3.28. Finally, we assume that El has a power variation and v1 is constant, compare

those cases of different variations of (I! in Fig. 3.35. By studying Fig. 3.28 - Fig. 3.33, we
find that again the effect of interphase is pronounced and it is higher for the larger volume
fraction of particles and for the thicker interphase, and no interphase assumption produces

highest values of effective thermal expansion coefficient. The comparison shows that there
is not a big difference among different variations of at. But, we can see that these results
have the same tendency as elastic moduli study with El having different variations. The
tendency is that the smoother variation of or! gives the lower estimation of the effective

thermal expansion coefficient. In other words, the uniform variation of at overestimate the

thermal expansion coefficient. We may conclude that the influence of the thickness of

64

interphase is greater than the different variations of thermal expansion coefficient in inter-
phase, which have the same volumetric average value.
We also check the relationship between the thermal expansion coefficient and bulk

modulus that is given in Christensen and Lo (1979) for the two phase model.

 

1 1
(OED-am)(—-—)
ot" = ot’"+ 1 K: Km (3.23)
F-F

We find that our result for no interphase cases is identical to the eqn. (3.23) [Appendix I].
So we prove that eqn. (3.23) is valid in CSA model. This infers that the similar relation
may be obtained to our interphase problem. Here, we introduce two numerical conse-
quences to illustrate the possibility that we may obtain the similar relation. The first conse-
quence is the comparison of effective thermal expansion coefficient from different
variations of Young’s modulus in the interphase (Fig. 3.36). We compare this result (Fig.

3.36) to the similar result (Fig. 3.14) in bulk modulus analysis, we find that they have the

same tendency which is that uniform variation of El overestimates both K‘ and oc‘, and

the smoother variation of El gives lower estimations. The second consequence is the
Comparison of effective thermal expansion coefficient from different variations of Pois-
son’s ratio in the interphase (Fig. 3.37). We also compare Fig3.37 with Fig. 3.15 in bulk
modulus study. We note that the same tendency is achieved for both effective bulk modu-

lus and thermal expansion coefficient. When the Poisson’s ratio in the interphase has uni-
form variation, we obtain highest value in K‘ and of, and the sharper variation of v1 gives

the lower estimation of K‘ and or“. These two numerical consequences strongly support
our conjecture that there may be a relation similar to eqn. (3.23) in our interphase problem.

But for the mathematical proof, again we leave for the future study.

65

{RHERmUUD.EXTUUWSICHV«COEHHRICIIHflP
(perfect bond)

Gel-051*

50-05‘

 

 

 

 

 

4e-05‘
G
E
’8
30-05“
v”=0.3
v"'=0.4
GP=25(GPA)
Gm=1(GPA)
20-05‘“
or’" = 66 x 104/”C
01" = 5x 10"/°C
1e-054h
Se-Os .m
1 1 1 1 1
0 0.2 0.4 0.6 0.8 1
f

Fig. 3.28 Effective thermal expansion coefficient vs. f for the no interphase case.

or" (loo)

THERMAL EXPANSION COEFFICIENT
(inhomogeneous interphase

with power variation at)

660-06

60-051"

Se-OS‘

4e-OS“

1. £30

2. t=0.0058 g

3"09' 3. t80.0088 \
4. t-0.02a ‘
5

, GP=25(GPA)
2"” G'"=I(GPA)

or” = 66x 10"/°C
of = 5x 104/"C

E'=E'"(‘;’)Q

U

1e-05'1 N
01' = a” (g)

1 r

I j

0 0.2 0.
f

Fig. 3.29 Effective thermal expansion coefficient vs. f for an inhomogeneous
interphase with a power variation in the thermal expansion coefficient

 

I
I

4 0.6

 

 

67

66 e-O 6 THERMAL EXPANSION COEFFICIENT
(inhomogeneous interphase

with constant 011 )

I

60-054

50-054

I

40-05-

0t‘ (l0C)

t=0
t=0.0058
t=0.0088
t=0.028
t=0.05a
t:0.1a

vP=0.3
v’"=0.4 \
\

=0.3
2.—os- GP=25(GPA)
G’”=I(GPA)
a” = 66x10“/°C
of = 5x10“/°c

_ E’ = E'"(a/r)Q’
ot’ = jjot”(a/r)9'dr/(b—a)

I

3e-05-

fiynunu

10-05-

 

l I I
I I I
0 0.3 0.4 0.6

f
Fig. 3.30 Effective thermal expansion coefficient vs. f for an inhomogeneous
interphase with the constant thermal expansion coefficient.

. d_.

68

{RHIHUIALrIDKRAJHSIOQIICOEHHKICHJHMT

    

66 —0
. (inhomogeneous interphase)
(power variation Cit-constant 01’)
66-051-
5e-OS'”
40-05"
8
in
‘8
30-05“-
1. t=0(no interphase)
2. t=0.05a (constant or)
3. t=0.05a (power variation 0t!)
4 t=0.1a (constant )
“'05" 5. t=0.1a(power variation Otl)
vP=0.3
v”'=0.4
v’=o.3
GP=25(GPA)
G’"=I(GPA)
10-05“ a” = 5x10"/"C
ot'" = 66x10“/°C
I I I
I T I I

 

 

0 0 . 2 0 .4 0 . 6 0 . a
f
Fig. 3.31 Comparison between Fig. 3.29 and Fig. 3.30.

69

 

 

66e-06 THERMAL EXPANSION COEFFICIENT
(homogeneous interphase
with power variation at)
60-05-1-
5e-05--
4o-os.t
G
%" 1 t-O
'8 2. tsO. 005a
3. t=0.00Ba
4. t=0.02a
3o-05.L 5. t=0.05a
6. t=0.1a ,\
v”=0.3 \\
v;"=0.4
v=a3
GP=25(GPA)
G’"=I(GPA)
a.-os.. a’" = 66x10“/°c
a" = 5x104/°c
E’ = 34 (GPA)
N
a
0t, = am (7)
10-05..
I l l I
I r I I
0 0.2 0.4 0.6 0.9

f

Fig.3.32 Effective thermal expansion coefficient vs. f for a homogeneous interphase
having a power variation in the thermal expansion coefficient.

70

 

 

66 06 flflHEmUfllL IUKRAAHEIOMI(flQETTEDCIIMflP
" (homogeneous interphase
with constant or!)
60-05"
50-05" ‘\
4e-OS“
G
I:
”a 1. t=0
2. t=0.0058
3. t=0.008a
” 4. t=0.028
3"“ 5. t=0.05a
6. t=0.18
vp=0.3
v"'=0.4 \
v’=0.3 \
Gp=25(GPA) \
G’"=IGPA)
20-05“
or" = 66x10’6/“C
01" = 5 x10“/°c
E’ = 34(GPA)
a' = fiM(a/r)"’dr/ (b —a)
10-05"
1 1 r r
I 1 I 1
0 0.2 0.4 0,5 0.8

f
Fig. 3.33 Effective thermal expansion coefficient vs. f for a homogeneous interphase
having a constant thermal expansion coefficient

71

 

6 THERMAL EXPANSION COEFFICIM
6 "06 (homogeneous interphase)
(power variation Oil-constant at)
60-05"
50-05"
“-054-
G
i:
‘8
30-051”
1. t-0(no interphase)
2. ts0.05a(constant (ll)
3. t=0.05a (power variation Otl)
4. t=0.1a(constant or’)
5. t=0.1a (power variation Otl)
“'05" vP=0.3
v”'=0.4
v’=o.3
GP=25(GPA)
G'"=1(GPA)
01’ = 5x10"/°C
1"05" a” = 66x10‘6/0C
I I I
I I I

 

0 o . 2 0 .4 0 .
f
Fig. 3.34 Comparison between Fig. 3.32 and Fig. 3.33

72

£EHIHUMAIHlEXIUUMSIIHV CKMIFTHDCIIHWT

VA

     
 

60-05-

 

 

 

 

 

 

 

 

 

 

 
  
 
 
 
 

Se-05«
I
i 40-05~
‘8
3,4,5“ 1. uniform variation of 01'
2. linear variation of 01
3. cubic variation of 01
20-05-b
0 0 . 2 0 f
K t=0. 05a (power variation E‘) )
vP=0.3
v’"=0.4

1.211980-051- “—03

6'" =25( GPA)
Gm=1( GPA)

or" = 5 x 104/"C
or” = 66 x 104/"c

1.201980-05-

1.191980-05<-

1.181980-05--

1.171980-05dr

1.161980-05--

1.151980-05‘-

 

1.14198e-05 :41 1 1 1
0.79 0.792 0.794 t 0.796 0.798 0.0

\ J

Fig. 3.35 Comparison of effective thermal expansion coefficient for the case of
different variations of thermal expansion coefficients in the interphase.

 

 

 

 

73

{RHIHUIAIrlEXIUUMSIIHV CKHQEFURZIEEEP

    

 

 

 

EA
60-051-
.3
5.-05. g
E"
>-
40-054-
1. uniform variation of E1

01‘ (I00

 

 

 

 

 

 
 
  
 
 
 

 

 

 

 

3 -05~-
‘ 2. linear variation of El
3. cubic variation of El
20-054-
le-OSu : ‘
o 0.2 /.
K tsO. 05a (power variation 01') \
1.130-05 4- ”(4:03
v’"=0.4
1.120-OSd- V£413
Gp=25(GPA)
1.110—05 '1' Gm=1(GPA)
1.1e-05 -_ 01” = 5x10"/°C
_ -6 o
1.09.45" 01’" - 66x10 / C
1.080-05-r
1.07._05,.
1.060-05d-
1,05.—05-.
0.79 0:792 0:794 f 0:796 0:798 0.0

\

Fig. 3.36 Comparison of effective thermal expansion coefficient for the case of
different variations of Young’s modulus in the interphase.

J

74

amuuuuan EQHUUWEDON'CKHflEFICHJflWP

 
 
  

60-05"

Poisson’s ratio

50-051h

 

 

 

 

 

 

 

 

 

 
 
 
  
 
 

 

 

G - ..
it 4e 05
‘8 > r
3 -05--
‘ 1. cubic variation of v:
2. linear variation of v
3. uniform variation of v
20—05"
le-O 1 1
o 0.2 0.4
r
( taO. 05a (power variation 01’)
_ , vP=0.3
1.160 05 v"'=0.4
51.0.3
Gp=25(GPA)
1.1a-05+ G'"=1(GPA)
ot" = 5x10"/°C
or" = 66x10“/°C
1.12¢-05«
1.100-05«
1.08a-05h
0.79 0:792 0:794 : 0:796 0:798 6.8

 

 

k J

Fig. 3.37 Comparison of effective thermal expansion coefficient for the case of
different variations of Poisson’s ratio in the interphase.

 

4.CONCLUSION

We obtained the effective bulk modulus, shear modulus and thermal expansion
coefficient of particle reinforced composites with inhomogeneous interphases. We
explored the effects of interphases. both inhomogeneous and homogeneous, on the overall
elastic pr0perties of composites as well as thermal properties and showed that the study of
interphase in composite materials is important. The elastic and thermal properties of inter-
phase strongly influence the overall elastic and thermal properties of composite. Control-
ling the properties of interphase in order to improve the characteristics of composite is an
important topic in composite manufacturing.

In this thesis we consider an idealized interfacial model, in which the interphase is
inhomogeneous but isotrOpic. A more realistic model is the one that accounts for the
local anisotropy and randomness of the constitutive law of interphase. This is being stud-
ied by Ostoja-Starzewski and Jasiuk (1992), Jasiuk and Ostoja-Starzewski (1993) and
Ostoja-Starzewski et a1. (1994).

7C

APPENDICES

5.APPENDICES

NOMENCLATURE

u[r]=u [ r] .......................................................................................................................... u ,u
[theta]=u [9] ...................................................................................................................... “e
u[phi]=u [0] ....................................................................................................................... u,p
U[r]=U [,1 ......................................................................................................................... U ,
U[theta]= U [0] ................................................................................................................. U9
U[phi]= U [11] .................................................................................................................... Uq,
epsilon[rr]=e ["1 ............................................................................................................... a"
epsilon[theta,theta]=e [9’ 0] .............................................................................................. 899
epsilon[phi,phi]=e [ db 0] .................................................................................................. 8W
epsilon[r,theta]=e [ r, 9] ..................................................................................................... ere
epsilon[theta,phi]=£ [9‘ 0] ............................................................................................... 86¢
epsilon[r,phi]=e [r’ 4’] ....................................................................................................... em
sigma[rr]=0' [ rrl ............................................................................................................... a"
sigma[theta,theta]=o [9’ 9] .............................................................................................. 0'99
sigma[phi,phi]=0‘ [ 0. 0] ................................................................................................... 611) (p
tau[r,theta]=t [a 9] ........................................................................................................... '1: r0
tau[theta,phi]=1: [9, 4,] ...................................................................................................... 19¢
tau[r,phi]=t Ir. 0] .............................................................................................................. 1: r0
b[1],b[2],b[3],b[4]=b[1] ,bm ’bl3] ’bl4] ........................................................ bl, b2, b3, b4
D[1],D[2],D[3],D[4]=D[ I] ,D 121 ’D13] ,0 [4] ............................................. D’l, 05,191,, Df,

76

 

77

m[1],m[2],m[3],m[4]=mm,m[21,m[3],m[4] ............................................. ml, m2, m3, m4
n[1],n[2],n[3],n[4]=n [1] , n [2] , n [3] , n [4] ................................................. n1, n2, n3, n4

P,Q .............................................................................................................................. P1, Q1
nu(p),nu(l),nu(m) ................................................................................................... v”, v’, v’"
G(p),G(m) ................................................................................................................. 0°, 0'"
D[pl],D[p2] ............................................................................................................... 01,01;
D[ll],D[12],D[l3],D[l4] .............................................................................. D11, 1);, D’ , of,
D[m1],D[m2],D[m3],D[m4] .................................................................... D'l", Dif, D3",fo
D[el],D[e3],D[e4] ............................................................................................. D; , 0;, D;
E(p),E(m) ................................................................................................................... E", E'"
alpha(p),alpha(m) ....................................................................................................... 01p, 01'"
C[ptl] ............................................................................................................................... Cfl
C[lt1],C[lt2] .............................................................................................................. (3:1,sz

C[mt1],C[mt2] ......................................................................................................... Cf}, Cf;

78

APPENDIX A

This computer program generates the eqn. (2.11) in bulk
modulus the study.

> u[r](r,theta,phi):=u[r](r);
um(r, 9, 0) I= “M(T)

 

> u[theta](r,theta,phi):=0;
u[e](r, 0, 11)) := O

 

> u[phi](r,theta,phi):=0;

u[“(n 9, 0) := 0

 

> epsilon[rr]:=diff(u[r](r,theta,phi),r);

81171:: 5: u[r](r)

 

 

> epsilon[theta,theta]:=1/(r*sin(phi))*diff(u[theta](r,theta,phi),theta)+u[r](r,theta,p
hi)/r+(cot(phi)/r)*u[phi](r,theta,phi);
“I r]( r)

elm].— r

 

> epsilon[phi,phi]:=diff(u[phi](r,theta,phi),phi)/r+u[r](r,theta,phi)/r;
u (r)
._ _[_’1_

810.111" ,

 

> gamma[r, theta]:=1/(r*sin(phi))*diff(u[r](r,theta,phi),theta)+diff(u[theta](r,theta,
phi),r)-u[theta](r,theta,phi)/r;

Ytr.91:=0

 

> gamma[theta,phi]:=difl(u[theta](r,theta,phi),phi)/r-cot(phi)*u[theta](r,theta,phi)/
r+1/(r*sin(phi))*diff(u[phi](r,theta,phi),theta);

719.9130

 

> gamma[r,phi]:=difl(u[phi](r,theta,phi),r)-u[phi](r,theta,phi)/r+diff(u[r](r,theta,phi)
.phi)/r;

71M] :0

 

> e:=epsilon[rr]+epsilon[theta,theta]+epsilon[phi,phi];

21 um“)
e:= —um(r) +2—

8r r

 

79

> sigma[rr]:=E(r)/(1+nu(r))*(epsilon[rr]+(nu(r)/(1-2*nu(r)))*(e));

a um(r)
J v(r) [[5 um(r)]+ 2 -—r-—]
+

a
._ Em (514mm 1-2v(r)

 

o .—
["1 1+v(r)

 

> sigma[theta,theta]:=E(r)/(1+nu(r))*(epsilon[theta,theta]+(nu(r)/(1-2*nu(r)))*(e)):

a u[rl(r)
u[rl(r) V(r) Eulrfir) +2 r
+

._ Em r 1- 2v(r)
[9'91"- 1+v(r)

 

O'

 

> sigma[phi,phi]:=E(r)/(1+nu(r))*(epsilon[phi,phi]+(nu(r)/(1-2*nu(r)))*(e));

a 2u[r](r)
“M(r) v”) Bram“) + r

 

 

 

 

 

E( r) +
0’ ._ r l - 2 V(r)
1M!" 1+v(r)
> tau[r,theta]:=E(r)/2/(1+nu(r))*gamma[r,theta];
11 n 01 := O
> tau[theta,phi]:=E(r)/2/(1+nu(r))*gamma[theta.Phi];
1‘ 0, 0] := O
> taulr,phi]:=E(r)/2/(1+nu(r))*gamma[r,phi];
1:” 01 := 0

 

> equi1:=diff(sigma[rr],r)*sin(phi)+1/r*(2*sigma[rr]*sin(phi)-sigma[theta,theta]*si
n(phi)-sigma[phi,phi]*sin(phi)+tau[r,phi]*cos(phi)+diff(tau[r,theta],theta)+diff(ta
Ulnphi].phi)*sin(phi))=0;

(
313 ) 3 )‘ v(r)%1
8r (r Bram” +1-2V(r)

1+v(r)

 

equiI :=

 

\

Rm (9- (r)]+ ”(”7“ [3% >)
- arum 1'2V(") ar r +E(r) [ill (fl)
(1+v(r))2 13’: ‘”

 

 

 

80

a a
—v(r) %1 v(r)%1 —v(r)
Br Br

+ +2

l-2V(r) (l-2v(r))2

a ' W N
- (r)
82 Br “”1 u[r](r)
v(r) [[ar? um(r)]+2 r -2 r2

1-2v(r) /(l+v(r)) srn(¢)+

/ )
H > (_a_ (>]+ “”701 '( )
r arul’] r l-2v(r) srn ¢

1+v(r)
“mm v(r)%1 .
E(r)[ r +1-2v(r)Jsrn(¢)
1+v(r)

 

 

+

 

 

 

 

2

 

-2

 

r=0

“M(r)

a
%1’=[Eulrl(r)]+2—r_

 

> eqn1:=collect(equi1,[diff(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor):
> eqn2:=normal(eqn1*(-1+2*nu(r))/sin(phi)):

> eqn3:=co|lect(eqn2,[difl(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor):

> eqn4:=normal(eqn3*(-1+2*nu(r))*(1+nu(r))’\2*r’\2)z

> eqn5:=collect(eqn4,[diff(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor);

372 [’1
[a J [a J 2 (a J 3
-2 —E(r) rv(r)- —E(r) rv(r) +2 —E(r) rv(r)
Br Br Br

+ 4 E(r) [§v(r)]rv(r) - 2 E(r) [3 v(r)]rV(r)2 - 4 E(r) v(r)
r 3r

eqn5 := E(r)r2(-1+v(r))(-l+ 2 v(r))(l +v(r)) [flu (r)]+r[[%E(r)]r

\
-2E(r)v(r)2+4E(r)v(r)3+2E(r) in (r) + 2 59-130) rv(r)
8r [’1 ) 3r

 

[a ] 2 [a ] 3 a J 2
-2 -E(r) rv(r) -4 —E(r) rv(r) +4E(r) —v(r) rv(r)
Br Br Br

+ 2E(r) [gr-v(r)]r-1-4 E(r) v(r) +2 E(r) v(r)2 - 4E(r) v(r)3 - 2 E(r))

um(r)=0

 

81

APPENDIX B

Here, we discuss the condition of existence of convergent series solution. The first
concept we shall need is that of an analytic function: A function f(r) is said to be analytic
at a point r = r0 if and only if it has a Taylor series at r = r0 which represents the func-

tion in some neighborhood of r = r0. In particular, polynomial functions are analytic

everywhere, and rational functions are analytic at all points where they are defined, i.e., at

all points except the zeros in denominators. Using the concept of an analytic function, we
can classify the point r0 about which we seek a solution of eqn. (2.16), here we just con-
sider the general form of (2. 16) and neglect the physical meaning of g (r) and f (r) . If the
coefficient functions f (r) and g (r) in eqn. (2.16) are both analytic at r = r0, the r0 is

said to be an ordinary point of the equation.

r0 but if the

If at least one of the functions f (r) and g (r) is not analytic at r

functions defined by the products (r - r0) f (r) and (r — r0) 23 (r) are analytic at
r = r0, then r0 is said to be a regular singular point of the equation.

If at least one of the products (r - r0) f ( r) and (r — r0) 2g (r) is not analytic at
r = r0 then r0 is said to be an irregular singular point of equation.

If r0 is an ordinary point of (2.16), then a convergent series solution exists of the

form u, = 2 an (r — r0) ", if r0 is regular singular point, then convergent series solution
n = 0
exists of the form

u, = Zan(r—r0)"+k (B.1)
n=0

If r0 is an irregular point, there are in general no solutions with expansion consisting

solely of powers of r — re. The method is known as the Frobenius Method and the condi-

82

tions of
existence of the series solution are called the Fuchs’ theorem.

Even though the conditions of Fuchs’ theorem are satisfied, it may not be possible
to obtain two in dependent solutions.

Let us suppose now that r0 is a regular singular point of eqn. (2.16). This means

that both (r - r0) f (r) and (r — r0) 23 (r) are analytic at r0 and can be therefore written

in the form

(r—r0)f(r) =p0+p1(r—r0) +p2(r-ro)2+... (B2)

(r—r0)g(r)=q0+q1(r-r0)+q2(r—ro)2+... .
By substituting eqn. (3.1) and eqn. (B.2) into eqn. (2.16), we obtain the equation
k2 + (p0 — 1) k+ q0 = O. This quadratic equation is known as the indicial equation of
eqn. (2.16). If the roots of this indicial equation differ by an integer, this process yields

only one solution [Johnson and Johnson (1982), p. 47). However, a second independent

solution can be found by assuming u, = 0 (r) u,1 (r) , where u,l is the first series solu-
tion, and then determining o (r) so that the product 0 (r) u,1 (r) will satisfy the given

differential equation. By using MAPLE (1992) symbolic manipulation program, we can

solve eqn (2.16) in the series solution using Fuchs’ method.

83

APPENDIX C

This computer program generates the eqns. (2.38), (2.39) and
(2.40) in the shear modulus study.

> u[r](r,theta,phi):=U[r](r)*(sin(phi))"2*sin(2*theta);
um(r, 0, o) := 0mm sin(¢)2 sin(2 e)

 

> u[theta](r,theta,phi):=U[theta](r)*sin(phi)*cos(2*theta);
(r, 9, ¢) := U (r) sin(tp) cos(2 9)

“161 191

 

> u[phi](r,theta,phi):=U[phi](r)*sin(2*phi)*sin(2*theta);
um(r, 9, 11)):= Um(r) sin(2 (p) sin(2 9)

 

> epsilon[rr]:=diff(u[r](r,theta,phi),r);

3
em ;= [5 Um(r)]sin(¢)2 sin(2 0)

 

> epsilon[theta,theta]:=1/(r*sin(phi))*diff(u[theta](r,theta,phi),theta)+u[r](r,theta,p
hi)/r+(cot(phi)/r)*u[phi](r,theta,phi);

U (r) sin(2 9) U (r)sin(¢)2sin(2e)
[91 [fl
8 := -2 +

cot(¢) Um(r) sin(2 0) sin(2 9)
+

r

 

> epsilon[phi,phi]:=difl(u[phi](r,theta,phi),phi)/r+u[r](r,theta,phi)/r;
Um(r)cos(2¢)sin(2e) U[r](r)sin(¢)zsin(29)
2 +

810.0]; r r

 

 

> gamma[r, theta] :=1/(r*sin(phi))*diff(u[r](r,theta,phi),theta)+diff(u[theta](r,theta,
phi),r)-u[theta](r,theta,phi)/r;

 

sin(tp) U[rl(r)cos(26) a
Y[r,0]:=2 r +[$U[91(r))sin(¢)cos(2 9)
U[e](r) sin(¢)cos(26)

 

r

 

> gamma[theta,phi]:=diff(u[theta](r,theta,phi),phi)/r-cot(phi)*u[theta](r,theta,phi)/
r+1/(r*sin(phi))*diff(u[phi](r,theta,phi),theta);

Um(r) cos(¢) cos(2 9) cot(¢) U
719.01: '

[91(r)sin(¢)cos(2 9)

r r

Um(r) sin(2 (p) cos(2 9)

r sin(¢)

> gamma[r,phi]:=diff(u[phi](r,theta,phi),r)-u[phi](r,theta,phi)/r+diff(u[r](r,theta,phi)
.phi)/r;

 

+2

 

Um(r) sin(2 0) sin(2 9)

 

a
7W] := [3r- U[¢](r)]sin(2 o) sin(2 6) -

U[rl(r) sin(¢) sin(2 9) cos(¢)

r

 

+2
r

 

> e:=epsilon[rr]+epsiIon[theta,theta]+epsilon[phi,phi];

U191") sin(2 9) 0mm sin(¢)2 sin(2 0)
+2

 

 

e := [a U (r)]sin((1>)2 sin(2 9) - 2
3r ['1

r r

cot(¢) Um(r) sin(2 o) sin(2 0) Um(r) cos(2 1p) sin(2 0)
+ +2
r r

> sigma[rr]:=E(r)/(1+nu(r))*(epsilon[rr]+(nu(r)/(1-2*nu(r)))*(e));

 

 

 

a o 2 e a ' 2 .
OI ]:=E(r)[[—U (r)]srn(¢) s1n(29)+v(r) [(—U (r)]sm(¢) sm(29)
rr 8" [’1 ar [’1

U (r) sin(2 0) Um(r) sin(¢)2 sin(2 e)

9
[1 +2
r r

cot(¢) U[“(r) sin(2 41) sin(2 0) Um(r) cos(2 11>) sin(2 9) 1/
+ +2 (1‘2V(r)

r r
)1/(1+v(r))

> sigma[theta,theta]:=E(r)/(1+nu(r))*(epsiIon[theta,theta]+(nu(r)/(1-2*nu(r)))*(e));
11mm sin(2 9) U[rl(r)sin(¢)2sin(29)
010,91:=E(r) -2 +

 

 

-2

 

 

 

 

r r

cot(¢)Um(r)sin(2¢)sin(2e) a 2
+ +v(r) EUMU) sin(¢) sin(20)

r

Um(r) sin(2 9) 0mm sin(¢)2 sin(2 e)
.4.

2
r r

 

 

-2

85

 

 

cot(¢) Ul¢l(r) sin(2 0) sin(2 9) Um(r) cos(2 0) sin(2 9)}
+ +2 (1-2V(r)

r r
)]/(1+V(r))

> sigma[phi,phi]:=E(r)/(1+nu(r))*(epsilon[phi,phi]+(nu(r)/(1-2*nu(r)))*(e));
; U[ (r) cos(2¢)sin(29) U (r) sin(111)28in(2 9) [
01 m
+ + v(r)

010.01’=E(r)[2 r r

Um(r) sin(2 e) Um(r) sin(¢)2 sin(2 a)
+2

 

(9- U (r)]sin(¢)2 sin(2 0) - 2
3r "1

cot(¢) Um(r) sin(2 o) sin(2 9) Um(r) cos(2 o) sin(2 9) 1/
(1 - 2 v(r)

r

 

+ +2

r r
)1/(1+V(r))

> tau[r,theta]:=E(r)/2/(1+nu(r))*gamma[r,theta];
1 sin(¢)Um(r) cos(20) a
tlr.9]:=-2-E(r) 2 r + $U[9](r) sin(¢)cos(29)
Um(r) sin(o) cos(2 0)}
- (1 + v(r))

r

 

 

> tau[theta,phi]:;E(r)/2/(1+nu(r))*gamma[theta,phi];

 

 

1 Um(r) cos(qr) cos(2 9) cot(¢) U[e](r) sin((p) cos(2 9)
110.61 :=§E(r) - r
Um(r)sin(2¢)cos(26)]/
+2 _ (1 +v(r))
rsrn(¢)

 

> tau[r,phi]:=E(r)/2/(1+nu(r))*gamma[r,phi];

 

 

l, a . . Uw(r) sin(2q>)sin(29)
tlr.¢]:=-2-E(r) 5‘11“”) srn(2¢)sm(29)- r
U[r](r) sin(tp) sin(2 6) cos(¢)]/
+2 (1 +v(r))
r

 

Since computer confused U[theta] and U[phi] with U(theta) and
U(phi), when it take derivative with them. We replace U[theta] and
U[phi] by U[t] and U[p].

 

86

 

> U[phi]:=U[p]:

> U[theta]:=U[t]:

> equit :=diff(sigma[rr],r)*sin(phi)+1/r*(2*sigma[rr]*sin(phi)-sigma[theta,theta]*si
n(phi)-sigma[phi,phi]*sin(phi)+tau[r,phi]*cos(phi)+diff(tau[r,theta],theta)+diff(ta
u[r,phi],phi)*sin(phi))=0:

> eqn1 :=simplify(equi1,trig):

> eqn2:=expand(eqn1/(-1/2*sin(2*theta)*sin(phi))):

> eqn3:=subs(cos(phi)"2=1-sin(phi)"2,eqn2):

> eqn4:=expand(eqn3):

> eqn5:=sort(eqn4,[sin(phi)],plex):

> eqn6:=op(1,eqn5):

> eqn7:=coeff(eqn6,sinmhi)"2):

> eqn8:=collect(eqn7,[diff(U[r](r),r,r),diff(U[r](r),r),U[r](r),diff(U[phi](r),r),U[phi](r)],
factor)=0:

> eqn9:=normal(eqn8*(1 +nufi))*(—1+2*nu(r))/2):

> eqn10:=collect(eqn9,[diff(U[r](r),r,r),diff(U[r](r),r),Ur(r),diff(U[phi](r),r),U[phi](r)],
factor):

> eqn1 1 :=normal(eqn10/(-1+2*nu(r))):

> eqn122=collect(eqn1 1 ,[diff(U[r](r),r,r),diff(U[r](r),r),U[r](r),diff(U[phi](r),r),U[phi](
r)],factor);

2

E(r) (v(r) - 1) [5; (IMO?) (( a

a 2
-1 +2v(r) arE(r)]r-2E(r) [arv(r)]v(r) r

 

eqn12:=-
3 a 2 2 a
+4E(r)v(r) - EEO) v(r) r-2E(r)v(r) -2 SEO) v(r)r

+ 2 [82r- E(r)]v(r)3 r + 4 E(r) [g v(r)]v(r) r - 4 E(r) v(r) + 2 E(r)]

a 2 a 3
{—U (r)]/(r (-l + 2 v(r)) (1 +v(r))) + [5 E(r) +4[— E(r))v(r) r
Br ['1 8r

8 2 a 3 2
+2[§;E(r)]v(r) r-2[EE(r)]v(r)r-1-l6 E(r) v(r) -2E(r) v(r)

a a
— 13 E(r) v(r) - 4 E(r) [5 v(r)]vm2 r - 2 E(r) [5; v(r)]r]Ul,,(r)/(

a
E —U
(-l+2v(r))2(l+v(r))r2)-3 (”(1% mm]
r(-1+2v(r))

 

- 3 {-7 E(r) v(r)

- 2 E(r) v(r)2 + 3 E(r) - 2 (327: E(r)]v(r) r - 2 E(r) [%v(r)]r + 8 E(r) V(7)3

87

a 3 a 2 i 2
+4[arE(r)JV(r) r-4E(r)[arv(r))v(r) r+2(arE(r)]V(r) TJUIPIU)

/((-1+ 2 v(r))2 (1 +v(r)) r2) =0
This is one of six equations from equilibrium equations. We choose
it as our governing eqn. (2.39).
> eqn13:=expand(eqn6-eqn7*sin(phi)"2):
> eqn14:=col|ect(eqn13,[diff(U[phi](r),r),diff(U[theta](r),r),U[phi](r),U[theta](r)],fac
tor)=0:
> eqn15:=normal(eqn14*(1+nu(r))*(-1+2*nu(r))/2):
> eqn16:=collect(eqn15,[diff(U[phi](r),r),diff(U[theta](r),r),U[phi](r),U[theta](r)],fac
tor);

a a
E(r) [EUIPIUJ E(r) [Eff/[‘10)] 2
eqn16z=2 - +2 -7 E(r)v(r)-2E(r) v(r)

r I“

f ax

\
+ 3 E(r) - 2 (3 E(r)]v(r) r - 2 E(r) [2 v(r) H» 8 E(r) v(r)3
Br Br )

 

+4-a—E 3413 -a-() 2+2313( )2U()
8r (r) v(r) r- (r) arvr v(r) r 8r r) v(r r lplr

/((1+ v(r))(-1+ 2 v(r)) r2) - {-7 E(r) v(r) - 2 E(r) v(r)2 + 3 E(r)

- 2 [_a_ E(r))vm r - 2 E(r) [9- v(r)]r + 8 E(r) v(r)3 + 4 (3 E(r))v(r)3 r
8r Br 8r

3 2 a 2
- 4 E(r)[; v(r)JV(r) H 2 [5 E(r))VU) ’]U[11(’)/((1 + v(r))

(-1+ 2 v(r)) r2) = O
This is one of six equations from equilibrium equations.
> equi2:=diff(tau[r,theta],r)*sin(phi)+1/r*(3*tau[r,theta]*sin(phi)+diff(tau[theta,phi]
,phi)*sin(phi)+diff(sigma[theta,theta],theta)+2*tau[theta,phi]*cosmhi))=0:
> eqn21:=simplify(equi2,trLg):
> eqn22:=expand(eqn21/(cos(2*theta)/2)):
> eqn23:=subs(cos(phi)"2=1-sin(phi)"2,eqn22):
> eqn24z=expargeqn23)z

 

> eqn25:=sort(eqn24,[sin(phi)],plex):

> eqn26:=op(1,eqn25):

> eqn27:=coeff(eqn26,sin(phi)"2):

> eqn28:=collect(eqn27,[diff(U[r](r),r),U[r](r),diff(U[theta](r),r,r),diff(U[theta](r),r),
U[theta](r),U[phi](r)],factor)=0:

88

> eqj129:=normal(eqn28*(1+nu(r))/2):
> eqn210:=co||ect(eqn29,[diff(U[r](r),r),U[r](r),diff(U[theta](r),r,r),diff(U[theta](r),r)
,U[theta](r),U[phi](r)],factor);

3
101170.10) 1 .2 1
r(-l+2v(r)) +213”) [SPUUIMJ‘LE

(
[2 E(r)]v(r) r + 2 E(r) v(r) + [3 E(r)]r + 2 E(r) - E(r) [31 v(r))r]
( Br Br Br

(EU ) ((1+v( )))-1-
(ar In” ' r ’2

 

eqn210 := -

 

(
{-a- E(r))wr) r + 2 E(r) v(r) + (3 E(r))r+ 2 E(r) - E(r) (P- v(r))r)
( Br Br Br

(4 v(r)-5)E(r) U[p](r)

Um(r)/((l+v(r))r2)-2 (-1+2v(r))r2 +Um(r)(-4E(r)

 

+ 4 E(r) v(r)2 + E(r) {-a- v(r)]r- 2 E(r) [2- v(r)]v(r) r + [2- E(r))v(r) r
Br Br 8r

+2 —E(r) v(r) r- —E(r) r ((l+v(r))(-1+2v(r))r2)=0
Br Br

This is one of six equations from equilibrium equations.
> eqn211 :=expand(eqn26-eqn27*sinjphi)"2)=0:
> eqn212:=factor(eqn21 1);
(0mm - 2 U,,,<r))<v(r)-1)E(r) _
(1 +v(r)) (-1 + 2 v(r)) :2 '
This is one of six equations from equilibrium equations.
> equi3:=diff(tau[r,phi],r)*sin(phi)+1/r*(3*tau[r,phi]*sin(phi)+diff(tau[theta,phi],the
ta)+(sigma[phi,phi]-sigma[theta,theta])*cos(phi)+diff(sigma[phi,phi],phi)*sin(ph
i))=0:
> eqn31:=simplity(equi3,trig):
> eqn32:=expand(eqn31/(sin(2*theta)*cos(phi))):
> eqn33:=subs(cos(phi)"2=1-sin(phi)"2,eqn32):

 

eqn212 := -8

 

 

> eqn34:=expand(eqn33):

 

> eqn35:=sort(eqn34,[sinmhi)],plex):

 

> eqn36:=op(1,eqn35):
> eqn37:=coeff(eqn36,sin(phi)"2):

 

> eqn38:=collect(eqn37,[diff(U[r](r),r),U[r](r),diff(U[phi](r),r,r),diff(U[phi](r),r),U[ph
i](r)],factor): -

 

> eqn39:=normal(eqn38*(1+nu(r))):

89

> eqn310:=collect(eqn39,[diff(U[r](r),r),U[r](r),diff(U[phi](r),r,r),diff(U[phi](r),r),U[p
hi](r)],factor)=0;

E(r) 3U (r)
(3’ M ]+U [4m )+4E( )v( )2+E( )[3v(r)]r
r(-1+2V(r)) "'(r) ' r r r r 3’

- 2 E(r) [%v(r))v(r) r+ [% E(r))v(r) r+ 2 (% 13(79)er r

a 82
- (I): E(r)]r)/((l +v(r)) (-1 + 2v(r)) r2) + E(r) [5’3 U[p](r)]+

f
(313(r))v(r) r+ 2 E(r) v(r) + (3130))” 2 E(r) - E(r) [311(0)]
Br Br Br

K

(

—aa U1 l(r))/(r(1+v(r)))—[12E(r)v(r)2-12E(r)+E(r)[—a v(r))r
K r P 8r

3a 9—12) 29-12) (12
+8r r) v(r)r- Br (r r+ 8r r Vr r

a
- 2 E(r) (;v(r))v(r) r]U[p](r)/((l +v(r)) (-1 +2 v(r)) r2) =0

 

eqn310 := -

 

This is one of six equations from equilibrium. We choose it as our
governing equation (2.38).

> eqn31 1:=expand(eqn36-eqn37*sin(ph0A2):

> eqn312:=factor(eqn31 1)=O:

> eqn313:=eqn312*rA2*(1+nu(r))/2/E(r);

eqn313 := Um(r) - 2 U[p](r) = O

This is one of six equations from equilibrium equations. We choose
it as our govemiann. (2.40).

90

APPENDIX D

This computer program generates the eqns. (2.52), (2.53), (2.55),
(2.56) and (2.57) in the shear modulus study.

> gov1:= -1/r*E(r)/(-1+2*nu(r))*diff((U[r])(r),r)+(U[r])(r)*(-4*E(r)+4*E(r)*nu(r)’\2+E
(r)*diff(nu(r),r)*r-2*E(r)*diff(nu(r),r)*nu(r)*r+diff(E(r),r)*nu(r)*r+2*diff(E(r),r)*nu(r
)"2*r-difl(E(r),r)*r)/(1+nu(r))/(-1+2*nu(r))/r’\2+E(r)*diff(diff((U[phi])(r),r),r)+(diff(
E(r),r)*nu(r)*r+2*E(r)*nu(r)+diff(E(r),r)*r+2*E(r)-E(r)*diff(nu(r),r)*r)/r/(1+nu(r))‘d
iff((U[phi])(r),r)-(12*E(r)*nu(r)’\2-12*E(r)+E(r)*difl(nu(r),r)*r+diff(E(r),r)*nu(r)*r-d
iff(E(r),r)*r+2*diff(E(r),r)*nu(r)’\2*r—2*E(r)*diff(nu(r),r)*nu(r)*r)/(1+nu(r))/(-1+2*n
u(r))/r’\2*(U[phi1)(r) = 0:

:gov2:= (U[theta])(r)-2*(U[phi])(r) = 0:

> gov3:= -E(r)*(nu(r)-1)/(-1+2*nu(r))*diff(diff((U[r])(r),r),r)-1/r*(diff(E(r),r)*r-2*E(r)*
diff(nu(r),r)*nu(r)’\2*r+4*E(r)*nu(r)’\3-diff(E(r),r)*nu(r)’\2*r-2*E(r)*nu(r)A2-2*difl(
E(r),r)*nu(r)*r+2*diff(E(r),r)*nu(r)’\3*r+4*E(r)*diff(nu(r),r)*nu(r)*r-4*E(r)*nu(r)+2
*E(r))/(-1+2*nu(r))'\2/(1+nu(r))‘difl((U[r])(r),r)+(5*E(r)+4’diff(E(r),r)*nu(r)’\3*r+2
*diff(E(r),r)*nu(r)’\2*r-2'diff(E(r),r)*nu(r)*r+16*E(r)*nu(r)A3-2*E(r)*nu(r)’\2-13*E(
r)*nu(r)-4*E(r)*diff(nu(r),r)*nu(r)A2*r-2*E(r)*diff(nu(r),r)‘r)/(-1+2*nu(r))’\2/(1+nu(
r))/r’\2*(U[r])(r)-3*E(r)/r/(-1+2*nu(r))*diff((U[phi])(r),r)-3*(-7*E(r)*nu(r)-2*E(r)*nu
(r)A2+3*E(r)-2*diff(E(r),r)*nu(r)*r-2*E(r)*diff(nu(r),r)*r+8*E(r)*nu(r)’\3+4*difl(E(r
),r)*nu(r)’\3*r-4*E(r)*diff(nu(r),r)*nu(r)’\2*r+2*diff(E(r),r)*nu(r)’\2*r)/(-1+2‘nu(r))’\
2/(1+nu(r))/r’\2*(Uu3hi])(r) = 0:

> E(r):=P*(r/a)"Q:

> nu(r):=nu:

m“ :
_>_gov3:

> eqn1:=normal(gov1):
> eqn2:=normal(qov3):
> eqn3:=sort(eqn1,[diff(U[r](r),r),U[r](r),diff(U[phi](r),r,r),diff(U[phi](r),r),U[phi](r)])

a ‘1
eqn.? := - [1(5 Um(r)j - 2 Qv Um(r) - 4v Um(r) + Q Um(r) + 4 Um(r)

 

32 1 a?-
+r2[5;3U”](r)J-2r2V(5’3Um(r)]-4rv%l~2er%l+rQ%1

+ 2r%l + 12v U[¢](r) +2 Qv U[M(r) - 12 U[¢](r)- Q U[“(r))l’['-;--r/(r2

(-1+2v))=0

a
%1:=; U[¢](r)

91

This equation is equivalent to eqn. (2.52)
> eqn4:=sort(eqn2,[diff(U[r](r),r,r),diff(U[r](r),r),Ur(r),diff(U[phi](r),r),U[phi](r)]);

r 32 82 3
NM :=- [:TP(PV (I): Um(r))- r2 (57 Um(r))- r Q[E Um(r))

a a .2 2
+2rv EUMU) +er EUMU) -2r arUm(r) +3r arUW(r)

- 9 Umm + 6 Qv Um(r)+12v Umm - 2 Qv Um(r) +5 Um(r)

- 8 v Um(r)]/(rz (-1 + 2v)) =0

This equation is equivalent to equation (2.53)
> eqn5:=-d*U[r]+(4*nu+2*Q*nu-4-Q)*U[r]+(2*nu-1)*d*(d-1 )*U[phi]+(-Q+4*nu+2*
Q*nu-2)*d*U[phi]+(-2*Q*nu+Q+12-12'nu)'U[phi]=0:
> eqn6:=(-1+nu)*d*(d-1)*U[r]+(2*nu-2+Q*nu-Q)*d*U[r]+(-8*nu+5-2*Q*nu)*U[r]+
3*d*Ujphi]+(12*nu-9+6*Q*nu)*U[phi]=O:
> eqn7:=collect(eqn5,[U[r],U[phi]],distributed,factor);
eqn7:=(-d+4v+2 Qv-4-Q) Um

+(-d2-d+2vd2+2vd—dQ+2dQv-2QV+Q+12-12v)Um=O

This equation is equivalent to eqn. (2.55)
> eqn8:=collect(eqn6,[U[r],U[phi]].distributed,factor);

eqn8:=(vd2+vd-dZ-d+dQv-dQ-8v+5-2Qv) Um
+(3d+6Qv+12v-9)Um=0
This equation is equivalent to eqn. (2.56)
> eqn9:=eqn7*(12*nu-9+6*Q*nu+3*d):
> eqn1 0:=eqn8*(-2*Q*nu+Q+12-12*nu+2*nu*d’\2+2*nu*d-d’\2-d-d*Q+2*d*Q*nu)

 

> eqn11:=eqn10-eqn9:

> eqn12:=simplify(eqn11):

> eqn13:=collect(eqn12,[U[r]],distributed,factor):

> eqn14:=normal(eqn13/(2‘nu-1)/U[r]):

> eqn15:=collect(eqn14,[d]);

eqn15:=(v— l)d‘+(2Qv-2Q-2+2v)d3+(Q2v-Qv+13-Q2-13v-Q)d2

+(-17Qv+15Q-3Q2v+14-14v+Q2)d-4Qv+4Q+24v-24
- 4 Q2 v = 0
This equation is equivalent to eqn. (2.57)

> eqn16:=solve(eqn15,{d});

92

 

 

=--%2+-

4 2 Jv-1

eqn16.={d1 ILsz Q2 4Q+8Qv 29+29v+2%1}

 

d

 

{ -_%121./QZV- Q2 4Q+8Qv 29+29v+2%1}
4% JV- 1 ’
{d_ 1%2:1,[Q2v- Qz-4Q+8Qv 29+29v 2%1}

_ 47 ,

 

 

Jv- 1
_%12 _1_2-JQ2v Q2 4Q+8Qv 29+29v 2%1}
4% JV 1

 

 

d

In.
%1. = 25Q2v2- 22Q2v+Q2+100Qv -144Qv+44Q+100v2- 200v+100)

2Qv-2Q-2+2v
v-l

Four roots of eqn. (2.57).
Note: Poisson’s ratio in this appendix represents Poisson’s ratio in
interphase.

 

%2 :=

93

APPENDIX E

This computer program determines unknown constants of eqns
(2.58), (2.61) and (2.62) in the shear modulus study.

> U[r](r):=D[1]*r’\d[1]+D[2]*rAd[2]+D[3]*rAd[3]+D[4]*r"d[4];

d d d d
°_.- m 12) m 141
Um(r). D“ r +Dl21r +Dl3lr «+0le

1

 

> U[phi](r):=b[1]*r’\d[1]+b[2]*r"d[2]+b[3]*r"d[3]+b[4]*r"d[4];

d d d d
'= [U [3) [3] (4]
U[¢](r). blllr +b[2]r +bmr +bl4lr

 

> eqn1:=(rA2*nu*diff(diff((U[r])(r),r),r)-r’\2‘diff(difl((U[r])(r),r),r)+r*Q*nu*diff((U[r])(r
),r)+2*r*nu*diff((U[r])(r),r)-2*r*diff((U[r])(r),r)-r*Q*diff((U[r])(r),r)+3*r‘diff((U[phi])
(r).r)+6*Q*nU*(U[phi])(r)+12*nU*(U[phi])(r)-9*(U[phi])(r)-8*nU*(U[rl)(r)+5*(U[r])(
r)-2*Q*nu*(U[r])(r))=O:

> eqn2:=collect(eqn1 ,[r],distributed,factor):

> eqn3:=sort(eqn2,[r’\d[1],r’\d[2],r"d[3Lr"d[4]],plex):

> eqn4:=op(1,eqn3):

> eqn5:=coeff(eqn4,r/\d[1])=O:

> eqn6:=coeff(eqn4,r’\d[2])=0:

> eqn7:=coeff(eqn4,r"d[3])=0:

> eqn8:=coeff(eqn4,r’\d[¢fl)=0:

> b[1]:=solve(eqn5,b[1]);

2
bm"'(50111+QVD111d111'Dmdm'ZQVDm+VD111d111'Dmd111

2
+vadm -8va-QDmdm)/(6Qv+3dm-9+12v)

 

> b[2]:=solve(eqn6,b[2]);

2
b —-(-2Qva+5Dm+QvD -8va-D d -D d

[2] [2] [2] [2]

2
+VD121d121' QDmdm+vadm )/(6Qv-9+3dm+12v)

121‘ 121d121

 

> b[3]:=solve(eqn7,b[3]);

2
b ”("SVD131+5D[31+QVDmdtal'QDl3ld131+VDI31dl3l +VDl3ldl3l

2
'2QVD131'Dlald131 'D131d131)/(6QV+12V+3d131'9)

131‘

 

> b[4]:=solve(eqn8,b[4]):

'2 2
b141"'(5D141'D141d141 +VD141d141 +VD141d141'D141d141'8VD141

+QVD141d141'2QVDH]'QDt4ld141)/('9+6QV+3d141+12v)

 

When the roots of the characteristic functions are complex,

94

> d[1]:=m[1]+l*n[1];d[2]:=m[1]-l*n[1];d[3]:=m[2]+l*n[2];d[4]:=m[2]-l*n[2];

dmz=mm+lnm
d121:=m111'1"111
dmz=mm+1nm
d14lz=m121'1"121

 

> eqn1:=(rA2*nu*diff(diff((U[r])(r),r),r)-r’\2*diff(diff((U[r])(r),r),r)+r*Q*nu*diff((U[r])(r
),r)+2*r*nu*diff((U[r])(r),r)-2*r*diff((U[r])(r),r)-r*Q*diff((U[r])(r),r)+3*r*diff((U[phi])
(r).r)+6*0*nU*(U[Phi])(r)+12"“U*(U[p])(r)-9*(U[phil)(r)-8*nU*(U[r])(r)+5*(U[r])(r)-
2*Q*nu*(U[r])Q))=O:

> U[r](r):=D[1]*cos(n[1]*In(r))*r’\m[1]+D[2]*sin(n[1]*ln(r))*r’\m[1]+D[3]*cos(n[2]*ln
(r))*rAm[2]+D[4]*sin(n[2]*ln(r))*r’\m[2];

U[r](r) := Dm cos(nm 1n(r)) r'"m + Dm sin( 1n(r)) r'"m

"[11

+Dl3lcos(n[2]ln(r)) r I11+Dm sin(nm 1n(r))r m

 

> U[phi](r):=b[1]*cos(n[1]*ln(r))*r’\m[1]+b[2]*sin(n[1]*In(r))*r’\m[1]+b[3]*cos(n[2]*l
n(r))*r"m[2]+b[4]*sin(n[2]*|n(r))*r’\m[2];

U[¢](r) := billcos(nlll111(r))'m‘”+ bm sin(nm 1110.)) r'"‘”

+ bl3l cos(nm 1n(r)) r'"m + bl4l sin( ln(r)) rmm

"[21

 

> eqn2:=collect(eqn1,[r],distributed,factor):

> eqn3:=sort(eqn2,[r/\mj1],r’\m[2]]):

> eqn4:=op(1,eqn3):

> eqn5:=coeff(eqn4,r/\m[1])=O:

> eqn6:=subs({cos(n[1]*ln(r))=x,sin(n[1]*ln(r))=y},eqn5):
> eqn7z=colIect(eqn6,[x,y],factor):

> eqn8:=op(1,eqn7):

> eqn81:=coeff(eqn8,x)=0:

> eqn82:=coeff(eqn8,y)=0:

> eqn83:=solve({eqn81 ,eqn82},{b[1],b[21}):

> eqn9:=coeff(eqn4,r’\m[2]):

> eqn10:=subs({cos(n[2]*ln(rn=u,sin(n[2]*ln(r))=v},eqn9):

 

> eqn1 1 :=collect(eqn10,[u,vj,factor):
> eqn1 11:=coeff(eqn1 1,u)=0:

 

> eqn1 12:=coeff(eqn1 1,v)=0:
> eqn1 13:=solve({eqn1 1 1,eqn112},{b[3],b[4]}):

> assign({eqn83,eqn1 13});
> bl1li=bl1k

95

1 2 2 2
2’111"'3('4Q2v D111'Q0111m111'15D111'Q0111'2111+6D121'2111m111

2
-n”Dmmm -1-4van[l -[26VD "(Hm l-m3QvD

-7Qvamm- 11VD111m:11'2VDmm[112[216QvDu:

ll

2 3 2
'Dmnm m111+2Q0111m111+VD111m111 +822111’2'111'2222111'22111

2 3 2
+24VD111'4D111'2111'2D121'2111'D121'2111'Dlllml”

2
11 111221114"12111QVD1m111+"111VD121m111

2 3
+2Qv2 22111222111 “22212122111 +SVD12122111+16QVD111

2 2
+2Qv Dmmm-I-4Qv D

2
+van mm+3QvD

2
[2]n[121m1_2Q VD“ l“m l-nllZszDn]

2 2
+2Qv Dmnm-ZQV Dmn

1]

-Qvam[ +2sz Dmn“

[I] ll

2 2
+2Qvamm +3QD[2 1n111)/(

9+4Q2v2-6mm+n 2-1-m2 +4vam- 12Qv)

 

[ll [1]
> b[2]:=b[2];

b ---l-(150 D
"'3 1212222111 121"

2
[2] +7QVD121m111'22113QVD111

[111]

2
'6VD111"111"2111'3Q0121m111'2D121m111 "801212221120111'211

2
-24va+4Dmnm +oDmnmmm-4vanmz-16Qle2l

2
+3QD111'211112+SVD111"111+2VD1m1112+IIVD121m111+l6QV D121
2

2 - 3 3
+4Q V2 22121290121221” 22111221112221”2 “222111an 2222121222!”

2 2 2 2
'ZQV D[2]m[1]+2Qv D[1]n[1]+2Qv D[2]n[1]+VD[lln[l]m[ll
2

111
3QvD

2

3 2
"VD121m111'mmVthlnm+2m11192VD121+QD121m

-2Q2van“ 1+QVD1211222112'4QVD 2

111""2111 12122111

-Dmn 22+4QvD mm+2sz2mDrzl1[]-2QZV2D[2]m

[ll [1] nil]

2 2 2 2
-2Qv 0121222111 )/(9+4Q2 v -6mm+nm]2 2222111 +4vam-12Qv

[ll

 

96

> b[3]:=b[3];

1( 2 2 2
b[31"'3 'Q0131m121 '"1210141m121 'D131'2121 m121+6D141"121m121'15D13

3 2 2
+5vanm+vanm +3Qvanm -11vamm-16Qv 0131

3 2 2
+vam +3 QDl3lm121+4VD131n121 22222213122212] -3QvD[4 n

[2] [2]

2 2
2222131222121'QD13122121+16QVD131+2QDI41MIZIQVDHm“
2
'4Q2V0131+8D131m121+24VDI312134D3222‘“ 2D14 "[21
2
-7Qvamm-6vanmmm- 121412212 '[D 3112mm +'2121‘222141222121

2 2
+VDlunlzlm1u'QVD131m121'ZQZVD124122121'2Q2VD1312"

2 2 2 2- 2
+2Qv22212D1411+2QVD13112m1+2QV 22131222121 2Qv D131"

[2]

121
2 2 W

+2QV2121221312" +2Qv2 221412212 l-1-4Qv 22141229122212]

9+4Q2v2-12Qv-6mm+n

> b[4]:=b[4];

1

2 2
22141” 3 (3 QDIBIn121+ 15 DUI-422221412212] +2vamm 26221312212] "212]

~3QDI4 Wm W-t-llvD Wm 1-2013122121224222)” 1-16Qva

3
+Dl4lml2] "D131"121m1212 +m121D141"1212+QDI41121212+V0131"121

2
+2m121Q2VDI41+QVD141m1212'VD1-11m1213+VD131221212"121

2
-2Qv2Dmm +2Q2v2Dmnm-2QZVD

2
1212+ "2121 2 4 QV "2121)

 

2 2
+2Qv 014122 [3122121

2 2 2
'3QVD141"121 'm121VD141"121“2m1219VD131'2121’2m12192V 0141

2
2'4le] Qv 2213122121 +QDl4lml

2
+16Qv D[:]+5vD[3ln[2]+7 QVDmmm'6VD[31"121m[21

-2Qv2Dmm

[2] [2]

2
2122+2Qv 221312212 1+4Q2v D141

2
'2221412'222121 8Dmmm+4Dmnm-3Qvanm

2 2 2
22213122121 3)/(9+4sz -12Qv-6mm+nm +2212] +4vam)

Note: Poisson’s ratio in this appendix represents Poisson’s ratio in
interphase.

[2]

97

APPENDIX F

This computer program determines the shear modulus of the
composite with a power variation Young’s modulus in the
interphase.

> for a from 10"(-6) by 0.01 to 0.9 do

> nu(p):=3/10;

> G(D)==25;

> nu(l):=3/10;

> P:=65;

> G(m):=1;

> b := (?/?)*a;(lnput thickness value of interphase.)

> c:=1;

> nu(m) :=4J10;

> O :=(In(E(m)/E(p))/(ln(b/a));

> m[1]:=?;(lnput one real component of eqn(2.59).)

> m[2]:=?;(lnput another real component of eqn(2.59).)
> n[1]:=?;(lnput one imaginary component of eqn(2.59).)
> n[2]:=?;(lnput another imaginary component of eqn(2.59).)
> C:=3/10;

> b[1] :=-1/3*(-D[1]*n[1]’\2*m[1]+8*nu(l)*D[1]*n[1]/\2-24*Q*nu(l)’\2*D[1]-15*D[1]-
D[2]*n[1]A3-D[1]*m[1}’\3+2*D[1]*m[1]A2-4*D[1]*n[1]A2+8*D[1]*m[1]-2*D[2]*n[1]
-32*nu(l)A2*D[1]+44‘nu(l)*D[1]+4*nu(I)A2*D[1]*m[1]A2+nu(l)*D[2]*n[1]’\3-Q*D[
1]"m[1]’\2+4"nu(l)’\2"D[2]"n[1]-6"nu(|)"D[1]*m[1]/\2+3"Q"D[2]"n[1]-Q"D[1]*n[1]A
2-n[1]*D[2]*m[1]A2+3*Q*D[1]*m[1]+16*Q*nu(l)*D[1]+nu(l)*D[1]*m[1]A3-4*nu(l)
A2"D[1]"’n[1]’\2+6"D[2]“'n[1]"m[1]+n[1]"’nu(l)"D[2]*m[1]/\.’2+8"nu(l)’\2"D[2]"n[1]"'
m[1]-Q*nu(l)*D[1]*m[1]A2-4’Q’\2*nu(l)’\2*D[1]+4*nu(l)’\2*D[1]*m[1]+nu(l)*D[2]*
n[1]-15*nu(l)*D[1]*m[1]+nu(l)*D[1]*n[1]A2*m[1]+6*Q*nu(|)’\2*D[1]*m[1]-2*Q’\2*
nu(l)*D[2]*n[1]-2*Q*nu(l)’\2*D[1]*n[1]A2+4*Q*nu(l)A2*D[2]*n[1]*m[1]-2*Q’\2*nu(
I)*D[1]*m[1]+2*Q"2*nu(l)’\2*D[2]*n[1]+2*Q*nu(l)/\2*D[1]*m[1]’\2+2*Q’\2*nu(I)'\2
*D[1]*m[1]-4*n[1]*Q‘nu(l)*D[2]*m[1]+6*Q*nu(l)’\2*D[2]*n[1]+3*Q*nu(l)*D[1]*n[1
]A2-11*Q*nu(l)*D[1]*m[1]-7*Q*nu(l)*D[2]*n[1]-14*nu(l)*D[2]*n[1]*m[1])/(8*nu(l)*
m[1]+16*Q*nu(l)A2+n[1]I‘2-12'Q‘nu(l)+9+16*nu(l)’\2+m[1 ]A2+4*Q*nu(l)*m[1]+
4*Q’\2*nu(l)A2-6*m[1]-24*nu(l));

> b[2] :=1/3*(Q*D[2]*m[1]A2+15‘D[2]-D[1]*n[1]A3+D[2]*m[1N-8*D[2]*m[1]-2*D[
2]*m[1]’\2-2*D[1]*n[1]+4*D[2]*n[1]A2+32*nu(I)A2*D[2]-44*nu(l)*D[2]-D[1]*n[1]*
m[1]’\2+Q*D[2]*n[1]A2+m[1]*D[2]*n[1]’\2+3*Q*D[1]*n[1]+6*D[1]*n[1]*m[1]-3*Q*
D[2]*m[1]+nu(l)*D[1]*n[1]+6*nu(l)*D[2]*m[1]A2+nu(l)*D[1]*n[1]A3-nu(l)*D[2]*m[
1]A3+4*nu(I)/‘2*D[1]*n[1]-4*nu(l)’\2*D[2]*m[1]A2-4*nu(l)'\2*D[2]*m[1]+4*Q’\2*nu
(l)’\2*D[2]+24*Q*nu(I)’\2*D[2]-8*nu(l)*D[2]*n[11A2+15*nu(l)*D[2]*m[1]-16*Q*nu(
I)*D[2]+4*nu(l)’\2*D[2]*n[1]A2+Q*nu(l)*D[2]*m[1]A2+nu(l)*D[1]*n[1]*m[1]’\2-m[1
]*nu(l)*D[2]*n[1]A2-2*Q"2*nu(l)*D[1]*n[1]-3*Q*nu(l)*D[2]*n[1]A2+4*Q*nu(l)’\2*D
[1]*n[1]*m[1]+2*Q*nu(l)’\2*D[2]*n[1]A2-4*Q*nu(l)*D[1]*n[1]*m[1]—2*Q’\2*nu(l)’\2
*D[2]*m[1]+2*Q’\2*nu(|)’\2*D[1]*n[1]-2*Q*nu(l)’\2*D[2]*m[1]A2+8*m[1]’nu(l)’\2*

98

D[1]*n[1]+2*m[1]*Q’\2*nu(l)*D[2]+6*Q*nu(l)’\2*D[1 ]*n[1]-6*Q*nu(l)/\2*D[2]*m[1]
-7'Q*nu(l)*D[1]*n[1]-14*nu(l)*D[1]*n[1]*m[11+1 1*Q'nu(l)*D[2]*m[1])/(8"’nu(l)*m
[1]+16*Q'nu(l)’\2+n[11A2-12*Q'nu(l)+9+16*nu(l)"2+m[1]A2+4*Q*nu(l)*m[1]+4*
Q"2*nu(l)A2-6*m[1]-24*nu(l));

> b[3] :=-1/3*(-D[4]*n[21AS-D[3]*m[21"3+2*D[3]*m[2]"2-2"D[4]*n[2]-4*D[3]*n[2}’\2
+8*D[3]*m[2]-32*nu(|)/\2*D[3]+44*nu(l)*D[3]-1 5*D[3]-Q*D[3]*n[2]’\2-D[3]*n[2]’\
2*m[2]-Q*D[3]*m[2}’\2-n[2]*D[4]*m[2]’\2+6*D[4]*n[2]*m[2]+3*Q*D[3]*m[2]+3*Q
*D[4]*n[2]-15*nu(|)*D[3]*m[2]+8*nu(l)*D[3]*n[2]/\2+nu(I)*D[4]*n[2]+nu(l)*D[4]*n
[2]“3+nu(l)*D[3]*m[2]’\3+4*nu(I)A2*D[4]*n[2]-6*nu(l)*D[3]*m[2]“2+4*nu(l)’\2*D[
3]*m[2]’\2-4*Q’\2*nu(l)’\2*D[3]-24*Q*nu(l)’\2*D[3]+16*Q*nu(l)*D[3]-4*nu(l)A2*D[
3]*n[2]’\2+4*nu(l)’\2*D[3]*m[2]+nu(l)*D[3]*n[2}’\2*m[2]+n[2]*nu(I)*D[4]*m[2]’\2-
Q'nu(l)*D[3]*m[2]’\2+3*Q*nu(l)*D[3]*n[2]’\2-4*n[2]*Q*nu(I)*D[4]*m[2]—2*Q"2*n
u(l)*D[4]*n[2]-2*Q*nu(I)’\2*D[3]*n[2]/\2+2*Q*nu(i)’\2*D[3]*m[2]’\2-2*Q"2*nu(l)*
D[3]*m[2]+4*Q*nu(I)A2*D[4]*n[2]*m[2]+2*Q"2*nu(l)’\2*D[3]*m[2]+2*Q’\2*nu(l)’\
2*D[4]*n[2]+6*Q*nu(|)A2*D[3]*m[2]+6*Q*nu(I)A2*D[4]"n[2]+8*nu(l)’\2*D[4]*n[2]*
m[2]-14'nu(l)*D[4]*n[2]*m[2]—1 1*Q'nu(I)*D[3]*m[2]-7’Q*nu(|)*D[4]*n[2])/(n[2]’\2
+m[2}’\2+16*nu(|)’\2+4*Q*nu(I)*m[2]-24*nu(l)—6*m[2]-12*Q‘nu(l)+8*nu(l)*m[2]+
9+4*Q’\2*nu(l)’\2+1 6*Q*nu(l)"2);

> b[4] :=1/3*(3*Q*D[3]"n[2]-D[3]*n[2]’\3+D[4]*m[2]“3-8*D[4]*m[2]-2*D[4]*m[2]"2~
2*D[3]*n[2]+4*D[4]*n[2]/\2-44‘nu(i)*D[4]+32*nu(l)’\2* D[4]+1 5* D[4]+Q*D[4]*m[2
1A2-D[3]*n[2]*m[2]’\2+m[2]*D[4]*n[2]"2+Q*D[4]*n[2]’\2-3*Q*D[4]*m[2]+6*D[3]*n
[2]*m[2]-16'Q*nu(l)*D[4]-4*nu(I)’\2*D[4]*m[21"2+nu(l)*D[3]*n[2]-nu(|)*D[4]*m[2]
"3+nu(I)*D[3]*n[2]’\3-8*nu(I)*D[4]*n[2]’\2+6*nu(I)*D[4]*m[21"2+4*Q’\2*nu(l)’\2*
D[4]+24*Q*nu(l)’\2*D[4]-4*nu(I)’\2’D[4]*m[2]+4*nu(I)A2*D[4]*n[2]’\2+4'nu(|)’\2*
D[3]*n[2]+1 5*nu(l)*D[4]*m[2]+nu(l)*D[3]*n[2]*m[21"2+Q*nu(l)*D[4]*m[2]’\2-m[2
]*nu(I)*D[4]*n[2]’\2+6*Q*nu(l)’\2*D[3]*n[2]-6*Q*nu(I)A2*D[4]*m[2]-2*Q*nu(l)’\2*
D[4]*m[2}'\2-2*Q"2*nu(l)*D[3]*n[2]+2*Q’\2*nu(I)’\2*D[3]*n[2]-3*Q*nu(|)*D[4]*n[
2P2+2*Q*nu(l)A2*D[4]*n[2]"2-4*m[2]*Q*nu(|)*D[3]*n[2]+2'm[2]*Q’\2*nu(l)*D[4]
+4*m[2]*Q‘nu(l)’\2*D[3]*n[2]-2*m[2]*Q"2*nu(I)A2*D[4]+8*m[2]*nu(I)’\2*D[3]*n[2
]-7*Q*nu(l)*D[3]*n[2]+1 1 *Q'nu(l)*D[4]*m[2]-14*nu(l)*D[3]*n[2]*m[2])/(n[2]’\2+m
[21"2+1 6*nu(I)A2+4*Q*nu(l)*m[2]-24*nu(l)-6*m[2]-12*Q'nu(l)+8*nu(l)*m[2]+9+
4*Q"2*nu(l)’\2+1 6*Q*nu(l)’\2);

 

> eqn1 :=D[p1]*a-6'nu(p)*D[p2]*aA3/(1-2*nu(p)) = D[l1]*cos(n[1]*|n(a))*a"m[1]+
D[I2]*sin(n[1]*ln(a))*a/\m[1]+D[l3]*cos(n[2]*ln(a))*a’\m[2]+D[l4]*sin(n[2]*ln(a))*
aAml2];

 

> eqn2 :=1/2*D[p1]*a-(7-4‘nu(p))*D[p2]*a’\3/(2-4*nu(p)) = b1*cos(n[1]"'ln(a))"aA
m[1]+b2*sin(n[1]*In(a))*a’\m[1]+b3‘cos(n[2]*ln(a))*a’\m[2]+b4*sin(n[2]*ln(a))*a
"mIZJ;

 

> eqn3:=2*G(p)*D[p1]-6*nu(p)*D[p2]*aA2*G(p)/(-1+2*nu(p)) = -P*aAm[1]*(-nu(l)*
cos(n[1]*ln(a))"m[1]+cos(n[1]*ln(a))*m[1]+nu(l)*sin(n[1]*ln(a))*n[1]-sin(n[1]*ln(
a))*n[1]+2‘nu(l)*cos(n[1]*ln(a)))/(1+nu(l))/(-1+2*nu(l))/a*D[l1]+P‘a"m[1]*(nu(l)*
sin(n[1]*ln(a))*m[1]-2*nu(l)*sin(n[1]*In(a))-cos(n[1]*ln(a))*n[1]+nu(l)*cos(n[1]*l
n(a))*n[1]-sin(n[1]*ln(a))*m[1])/(1+nu(l))/(-1+2*nu(l))/a*D[I2]-P*a’\m[2]*(cos(n[2
]*In(a))*m[2]-nu(I)*cos(n[2]*ln(a))*m[2]+nu(l)*sin(n[2]*ln(a))*n[2]-sin(n[2]*ln(a))
*n[2]+2*nu(l)*cos(n[2]*ln(a)))/(1+nu(l))/(-1+2*nu(l))/a*D[|3]+P*a’\m[2]*(nu(l)*sin

99

(n[2]*ln(a))*m[2]+nu(l)*cos(n[2]*ln(a))*n[2]-sin(n[2]*ln(a))*m[2]-2*nu(l)*sin(n[2]
*ln(a))-cos(n[2]*In(a))*n[2])/(1+nu(l))/(-1+2*nu(|))/a*D[l4]+6*nu(l)*(b1*cos(n[1]*
ln(a))*a’\m[1]+b2‘sin(n[1]*ln(a))*a’\m[1]+b3‘cos(n[2]*ln(a))*a’\m[2]+b4*sin(n[2]
‘ln(a)[aAm[2])*P/(1+nu(l))/(-1+2*nu(l))/a;

 

> eqn4:=2*G(p)*D[p1]+2*G(p)*a"2*(2*nu(p)+7)/(-1+2*nu(p))*D[p2] = P/(1+nu(l))
/a*D[l1]*cos(n[1]*ln(a))*a’\m[1]+P/(1+nu(l))/a*D[|2]*sin(n[1]*In(a))*a’\m[1]+P/(1
+nu(|))/a*D[l3]*cos(n[2]*|n(a))*a’\m [2]+P/(1 +nu(l))/a*D[l4]*sin (n[2]*ln(a))*a/\m [2
]+P*(-b2*sin(n[1]*ln(a))*a'\m[1]-b4*sin(n[2]*ln(a))*a’\m[2]+b2*cos(n[1]*ln(a))*n[
1]*a/\m[1]+b2*sin(n[1]*ln(a))*a’\m[1]*m[1]-b1*sin(n[1]*ln(a))*n[1]*a’\m[1]+b1*c
os(n[1]*ln(a))*a/\m[1 ]*m[1]+b4*cos(n[2]*in(a))*n[2]*a’\m[2]-b1*cos(n[1]*ln(a))*a
Am[1]-b3"sin(n[2]"|n(a))"’n[2]"a’\m[2]+b3"cos(n[2]"ln(a))"a’\m[2]"’m[2]+b4"sin(n[
2]*ln(a))*a’\m[2]*m[2]-b3*cos(n[2]*ln(a))*a’\m[2])/(1 +nu(l))la;

> eqns :=D[l1]*cos(n[1]*ln(b))*b’\m[1]+D[l2]*sin(n[1]*ln(b))*b’\m[1]+D[l3]*cos(n[2
]*ln(b))*b’\m[2]+D[l4]*sin(n[2]*ln(b))*W[2] = D[m1]*b-6‘nu(m)*D[m2]*b’\3/(1-
2*nu(m))+3* D[m31/b’\4+(5-4*nu(m))*D[m 4]/(1 -2*nu(m))/b"2;

 

> ean :=b1*cos(n[1]*ln(b))*b’\m[1]+b2'sin(n[1]*ln(b))*b’\m[1]+b3‘cos(n[2]*ln(b))
*b’\m[2]+b4*sin(n[2]*ln(b))*b’\m[2] = 1/2*D[m1]*b-1/2"(7-4*nu(m))*D[m2]*b’\3/(
1-2*nu@))-D[m3]/b"4+D[m 4]/b’\2;

> eqn7 :=-P*b’\Q*b’\m[1]*(2'nu(l)*cos(n[1]*In(b))+nu(l)*sin(n[1]*ln(b))*n[1]+cos(n
[1]*ln(b))*m[1]-nu(l)*cos(n[1]*ln(b))*m[1]-sin(n[1]*ln(b))*n[1])/(a"Q)/(1+nu(l))/(-
1+2*nu(l))/b*D[I1]+P*b’\Q*b’\m[1]*(-cos(n[1]*ln(b))*n[1]+nu(l)*cos(n[1]*In(b))*n
[1]-sin(n[1]*ln(b))*m[1]+nu(l)*sin(n[1]*ln(b))*m[1]-2*nu(l)*sin(n[1]*ln(b)))/(a"Q)/
(1+nu(l))/(-1+2*nu(l))/b*D[l2]-P*b"Q*b’\m[2]*(2*nu(I)*cos(n[2]*in(b))+nu(l)*sin(
n[2]*ln(b))*n[2]-nu(l)*cos(n[2]*ln(b))*m[2]+cos(n[2]*ln(b))*m[2]-sin(n[2]*ln(b))*n
[2])/(a"Q)/(1+nu(l))/(-1+2*nu(l))/b*D[l3]+P*b"Q*b’\m[2]*(-sin(n[2]*ln(b))*m[2]+n
u(l)*sin(n[2]*ln(b))*m[2]-cos(n[2]*ln(b))*n[2]+nu(l)*cos(n[2]*ln(b))*n[2]-2*nu(l)*s
in(n[2]*ln(b)))/(a"Q)/(1+nu(l))/(-1+2*nu(l))/b*D[I4]+6*nu(l)*(b1*cos(n[1]*ln(b))*b
“m[1]+b2‘sin(n[1]*In(b))*b"m[1]+b3'cos(n[2]*In(b))*b’\m[2]+b4*sin(n[2]*ln(b))*
b’\m[2])*P*b"Q/(1+nu(l))/(-1+2*nu(l))/(a’\Q)/b = 2*G(m)*D[m1]-6*nu(m)*b"2*G
(m)/(-1+2*nu(m))*D[m2]-24*G(m)*D[m3]/b"5-4/b’\3*(nu(m)-5)*G(m)/(-1+2‘nu(
m))*D[m4]:

> ean :=P*b"O/(a"Q)/(1 +nu(l))/b*D[l1]*cos(n[1]*ln(b))*b'\m[1]+P*b"Q/(a"Q)/(1+
nu(l))/b*D[|2]*sin(n[1]*ln(b))*b’\m[1]+P’b"Q/(a"Q)/(1+nu(I))/b*D[l3]*cos(n[2]*ln
(b))*b’\m[2]+P*b"Q/(a’\Q)/(1+nu(|))/b*D[I4]*sin(n[2]*ln(b))*b’\m[2]-P*b’\Q*(b2*s
in(n[1]*ln(b))*b’\m[1]+b4’sin(n[2]*ln(b))*b’\m[2]-b2*cos(n[1]*In(b))*n[1]*b’\m[1]-
b2‘sin(n[1]*ln(b))*b’\m[1]*m[1]+b1*sin(n[1]*In(b))*n[1]*b’\m[1]-b1*cos(n[1]*|n(b
))*b’\m[1]*m[1]-b4*cos(n[2]*ln(b))*n[2]*b’\m[2]+b1*cos(n[1]*ln(b))*b’\m[1]+b3*s
in(n[2]*in(b))*n[2]*b’\m[2]-b3*cos(n[2]*In(b))*b’\m[2]*m[2]-b4*sin(n[2]*ln(b))*b’\
m[2]*m[2]+b3*cos(n[2]*ln(b))*b’\m[2])/(a’\Q)/(1+nu(l))lb = 2*G(m)*D[m1]+2*G(
m)*b’\2*(2*nu(m)+7)/(-1+2*nu(m))*D[m2]+16*G(m)*D[m3]/b"5-4*G(m)*(1+nu(
m))/b’\3/(-1+2*nu(m))*D[m4];

 

 

> eqn9 :=D[m1]*c+6*nu(m)*c’\3/(-1 +2*nu(m))*D[m2]+3*D[m3VcM+(-5+4*nu(m))
/(-1 +2*nu(m))/c’\2*D[m 4] = D[e1 ]*c+3*D[e3]/c’\4+(-5+4*nu(e))/(-1+2*nu(e))/c"
2*D[e4];

> eqn10 := 1/2*D[m1]*c-1/2*(-7+4*nu(m))‘c’\3/(-1+2*nu(m))*D[m2]-D[m3]/c’\4+

100

D[m4]/c’\2 = 1/2*D[e1]*c-D[eS]/c’\4+D[e4]/c"2;

> eqn11 := G(m)*D[m1]-3*nu(m)*c’\2*G(m)/(-1+2*nu(m))*D[m2]-12*G(m)/c’\5*D[
m3]-2/c’\3*(nu(m)-5)*G(m)/(-1+2*nu(m))'D[m4] = G(e)*D[e1]-12*G(e)*D[93]/c
A5-2/c"3*(nu(e)-5)*G (e)/(-1 +2*nu(e))*D[e4];

> eqn12 :=2*G(m)*D[m1]+2*G(m)*c’\2*(2*nu(m)+7)/(-1+2*nu(m))*D[m2]+16*G(
m)/c"5*D[m3]-4*G(m)/c’\3*(1+nu(m))/(-1+2*nu(m))*D[m4] = 2*G(e)*D[e1]+16*
G(e)*D[eB]/c’\5-4*G(e)/c"3*(1 +nu(e))/(-1 +2*nu(e))*D[e4];

> x:=solve({eqn1,eqn2,eqn3,eqn4,eqn5,eqn6,eqn7,eqn8,eqn9,eqn10,eqn1 1,eq
n12},{D[p1],D['p2],D[I1],D[l2],D[l3],D[l4],D[m1],D[m2],D[m3],D[m4],D[e3],D[e4]
D'

> assign(x);

> G1 :=D[e4];

> G2:=G1=O;

> n:=solve(GZ,G(e));

> print(n);

> readlib(unassign):

> unassign(’D[p1],D[p2],D[l1].D[l2],D[l3].D[I4],D[m1].D[m2],D[m3].D[m4],D[63].
D[e4l');

> 0d:

> end

101

APPENDIX G

This computer program generates the eqns. (2.75), (2.78) and
(2.79) in thermal expansion coefficient study.

> u[r](r,theta,phi):=u[r](r):

> u[theta](r,theta,phi):=0:

> u[phi](r,theta,phi):=0:

> epsilon[rr1:=difl(u[r](r,theta,phi),r):

> epsilonltheta,theta]:=1/(r*sin(phi))*diff(u[theta](r,theta,phi),theta)+u[r](r,theta,p
hi)/r+(cot(phi)/r)*u[phi](r,theta,phi):

> epsilon[phi,phi]:=diff(u[phi](r,theta.phi),th/r+u[r](r,theta,phi)/r:

> gamma[r, theta]:=1/(r*sin(phi))*dift(u[r](r,theta,phi),theta)+diff(u[theta](r,theta,
phi),r)-u[theta](r,theta,phi)/r:

> gamma[theta,phi]:=diff(u[theta](r,theta,phi),phi)/r-cot(phi)*u[theta](r,theta,phi)/
r+1/(r’sin(phi))*diff(u[phi](r,theta,phi),theta):

> gamma[r,phi]:=ditf(u[phi](r,theta,phi),r)-u[phi](r,theta,phi)/r+diif(u[r](r,theta,phi)
,phi)/r:

> e:=epsilon[rr]+epsiIon[theta,theta]+epsilon[phi,phi]:

> sigma[rr]:=E(r)/(1 +nu(r))*(epsiIon[rr]-alpha(r)*T-1-(nu(r)/(1 -2*nu(r)))*(e-3*alpha(
0’0):

> sigma[theta,theta]:=E(r)/(1+nu(r))*(epsiIon[theta,theta]-alpha(r)*T+(nu(r)/(1 -2*
nu(r)))*(e-3*alpha(r)*T)):

> sigma[phi,phi]:=E(r)/(1+nu(r))*(epsilon[phi,phi]-a|pha(r)*T+(nu(r)/(1-2*nu(r)))*(
e-3*alpha(r)*T)):

> tau[r,theta]:=E(r)/2/(1+nu(Q)*gamma[r,theta]:

> tau[theta,phi]:=E(Q/2/(1 +nu(r))‘1amma[theta,phi]:

> tau[r,phi]:=E(r)/2/(1 +nu(r))‘gamma[r, phi]:

> equi1 :=diff(sigma[rr],r)*sin(phi)+1/r*(2'sigma[rr]*sin(phi)-sigma[theta,theta]*si
n(phi)-sigma[phi,phi]*sin(phi)+tau[r.phi]*cos(phi)+diff(tau[r,theta],theta)+diff(ta
u[r,phi],phi)*sin(phi))=0:

> eqn1:=collect(equi1 ,[diff(u[r](r),r,r),diff(u[rj(r),r),u[r](r)],factor):

> eqn2:=normal(eqn1j—1+2*nu(r))/sin(phi)):

> eqn3:=collect(eqn2,[diff(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor);

2

1-1 +v(r))E1r1[5;u[,,(r)] a a
1+v(r) +[[5;E(r)]r-2[57E(r)]rv(r)

- —E(r) rv(r) +2 —E(r) rv(r) +4E(r) -v(r) rv(r)
Br Br

 

eqn.? :=

3r

- 2 E(r) [%v(r))rv(r)2 - 4E(r) v(r) - 2 E(r) v(r)2 +4 E(r)v(r)3 + 2 E(r))

102

a ( 2) a
a—umU) r(-1+2v(r))(1+v(r)) -2 - EEO.) rv(r)

1’ i 2 1’ i ’ 1’ J 2
+ —E(r) rv(r) +2 —E(r) rv(r) -2E(r) —v(r) rv(r)

Br Br Br
-2E(r)v(r)-E(r)v(r)2+2E(r)v(r)3-E(r)(%v(r)]r+E(r)]um(r)/(
(-1+2v(r))(1+v(r))2r2)+[2 [Bar E(r)]a(r)v(r)+2E(r)v(r)[aar 010)]

-[%E(r)]a(r)-2E(r) 01(r) [gr-v(r))— E(r)[aar a(r))]T/(- l+2v(r))=

O

This equation is equivalent to eqn. (2.75), and Poisson’s ratio in this
equation represents Poisson’s ratio in interphase.

> E(r):=P*(r/a)AQ:

> alpha(r):=M*(r/a)’\N:

> nu(r):=nu:

> eqn4:=eqn3:

> eqn5:=collect(eqn4,[diff(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor):
> eqn6:=normal(eqn5/(P*(r/a)"Q)):

> eqn7:=colIect(eqn64difl(u[r1(r),r,r),diff(u[r](r),r),u[r](r)1,factor):
> eqn8:=normal(eqn7*(1+nu)*r’\2/(-1+nu)):

> eqn9:=collect(eqn8.[diff(u[r](r),r,r),diff(u[r](r),r),u[r](r)],factor);

 

 

a2 a (QV+V-l)u[,](r)
eqn9:= [_11 (r)]r2+(Q+2)r[—u[r ](r)]-2

 

8F "'11 -1+v
TM(L]~r(Q+N)(1+v)
+ a =0
-1+v

This equation is equivalent to eqn. (2.78)
> x:=dsolve(egn9,u[r](r)):

 

> assign(x);
> Ur1:=u[r](r):

 

> u[r](r):=collect(Ur1,LC1,_02],factor);
I!” jl—+;Q I-1+v +951
u (r) — -4

.1” ](sz- 3N- Q-Qv-QN+QNv+3vN-NZ)

m[1+vQ+/1+v+%1]
_CI/(%3 %2)-4 Jl+v

103

(NZv-3N-Q-Qv-QN+QNv+3vN-NZ)_Cz/(%3 %2)
TMH”+‘)a('”)(Q+N)(1 +v)
%3%2

 

+4

 

%1:=JQ2v+10Qv+9v-9-2Q-Q2
%2:= /-1+VQ+3 l-l+v+%1+2N -1+v
%3:=-/-1+v Q-3 /-1+v +%1-2N -1+v

This equation is equivalent to eqn. (2.79), and Poisson’s ratio in this
equation represents Poisson’s ratio in interphase.

> xi:=coeff(u[r](r),_C1);
m-mQ-me]
§:=-4 J-l+v (sz-3N-Q-Qv-QN+QNv+3vN-AP)/(
(-1-1+vQ-3/-1+v+%1-2N -1+v)
(-l+vQ+3/-1+v+%l+2N -1+v))

%1:=jQ2v+lOQv+9v-9-2Q-Q2

 

 

> beta:=coeff(u[r](r),_02):

r[_mj:1_:Q+j:-fi_v+%1]
e:=-4 J~1+v (sz-sN-Q-Qv-QN+QNv+3vN-Nl)/(

(-/-l+v Q-3./-1+v+%1-2N -1+v)
U-l+v Q+3 /-1+v+%1+2N -l+v))

%1:=JQ2v+10Qv+9v-9-2Q-Q2

 

 

> eta:=u[r](r)-alpha*_C1-beta*_02;
n :=4 TMr<~+l>a<-~>(Q +111) (1 +v)/(

(-/-1+v Q-3./-l+v+fQ2v+10Qv+9v-9-2Q-Q2-2N./-1+v)
(/-1+v Q+3 /-1+v+JQ2v+10Qv+9v-9-2Q-QZ+2N1-1+v))

 

 

 

104

APPENDIX H

This computer program evaluates the thermal expansion
coefficient of the composite with power variation Young’s
modulus and thermal expansion coeflicient in interphase.

 

> P:=65:

> Q:=(ln(E(rm/E(p))/(ln(b/a)):

> N:=:(In(alpha(m)/alpha(p))/(ln(b/a)):

> alpha(p):=5*10"(-6):

> E(p):=65: 2’

> nu(p):=3/10:

> nu(l):=3/10:

> E(m):=28/10:

> nu(m):=4/10:

> alpha(mpeenw-e): :22

> M:=5*10"(—6): '2

> b:=(?/?)*a:(lnput thickness value of interphase.)

> c:=1:

> eqn1 :=C[pt1]*a = 4*a"(-1/2*((-1+nu(l))’\(1/2)+(-1+nu(l))A(1/2)*Q-(10*Q‘nu(l)+
9*nu(l)+Q’\2*nu(l)-Q’\2-2*Q-9)’\(1/2))/(-1+nu(l))’\(1/2))*(-3*N+3*nu(l)*N-Q*N+Q
*N'nu(I)-NA2+N’\2*nu(l)-Q*nu(l)-Q)/(3*(-1+nu(l))A(1/2)+(-1+nu(l))“(1/2)*Q-(1O’
Q*nu(l)+9*nu(|)+Q"2*nu(I)-Q"2-2*Q-9)’\(1/2)+2*N*(-1+nu(l))’\(1/2))/(3*(-1+nu(l
))’\(1/2)+(-1+nu(l))’\(1/2)*Q+(10‘Q’nu(l)+9*nu(l)+Q’\2*nu(l)-Q"2-2*Q-9)"(1/2)+
2*N*(—1+nu(l))’\(1/2))*C[lt1]+4'aA(-1/2*((-1+nu(l))A(1/2)+(-1+nu(l))’\(1/2)*Q+(1O
*Q‘nu(l)+9*nu(I)+QA2*nu(l)-Q’\2-2*Q-9)’\(1/2))/(-1+nu(l))’\(1/2))*(-3*N+3*nu(l)*
N-Q*N+Q*N*nu(I)-N’\2+NI\2‘nu(l)-Q*nu(I)-Q)/(3*(-1+nu(l))A(1/2)+(-1+nu(l))A(1/
2)*Q-(10*Q*nu(l)+9*nu(I)+Q’\2*nu(l)-Q"2-2*Q-9)A(1/2)+2*N*(-1+nu(l))’\(1/2))/(
3*(-1+nu(l))’\(1/2)+(-1+nu(l))’\(1/2)*Q+(10'Q’nu(l)+9*nu(I)+Q’\2*nu(l)-Q’\2-2*Q
-9)’\(1/2)+2*N*(-1+nu(l))A(1/2))*C[It2]-4*T*M*a’\(N+1)*a'\(-N)*(1+nu(l))*(Q+N)/(
3*(-1+nu(l))’\(1/2)+(-1+nu(l))A(1/2)*Q-(10*Q*nu(|)+9*nu(l)+Q"2*nu(l)-Q"2-2*Q—
9)’\(1/2)+2*N*(-1+nu(l))’\(1/2))/(3*(-1+nu(l))“(1/2)+(-1+nu(l))A(1/2)*Q+(1O*Q*n
u(l)+9*nu(l)+Q"2*nu(l)-Q’\2-2*Q-9)’\(1/2)+2*N*(—1 +nu(l))’\(1/2)):

> eqn2 := -E(p)/(-1+2‘nu(p))*C[pt1]+T*aipha(p)*E(p)/(-1+2*nu(p)) = -2*P*a"(—1/
2*((-1+nu(l))A(1/2)+(-1+nu(l))A(1/2)*Q-(10*Q*nu(l)+9*nu(l)+Q’\2*nu(l)-Q’\2-2*Q
-9)’\(1/2))/(-1+nu(l))A(1/2))*(-3*N+3*nu(l)*N-Q*N+Q*N*nu(l)-NA2+NA2*nu(|)-Q"
nu(l)-Q)"(-(-1+nu(l))A(1/2)+5*(-1+nu(l))’\(1/2)*nu(l)-(-1+nu(l))A(1/2)*Q+(-1+nu(l)
)’\(1/2)*Q*nu(l)+(10*Q'nu(l)+9*nu(l)+Q’\2*nu(l)-Q’\2-2*Q-9)’\(1/2)-(10*Q*nu(l)+
9*nu(|)+Q"2*nu(l)-Q"2-2*Q-9)’\(1/2)*nu(l))/(1+nu(l))/(-1+2*nu(l))/(-1+nu(l))’\(1/
2)/a/(3*(-1+nu(l))A(1/2)+(-1+nu(l))’\(1/2)*Q-(10*Q*nu(l)+9*nu(|)+Q"2*nu(I)-Q"2
-2*Q-9)"(1/2)+2*N*(-1+nu(l))A(1/2))/(3*(-1+nu(l))A(1/2)+(-1+nu(l))“(1/2)*Q+( 1O
*Q*nu(l)+9*nu(l)+Q"2*nu(l)-Q"2-2*Q-9)’\(1/2)+2*N*(-1+nu(l))A(1/2))*C[lt1]-2*P
’a"(—1/2*((-1+nu(l))’\(1/2)+(-1+nu(l))A(1/2)*Q+(10*Q*nu(l)+9*nu(l)+Q"2*nu(|)-Q

 

 

 

 

 

105

A2-2'Q--9)"(1/2))/(-1~1-nu(l))’\(1/2))"(-:33"N+3*nu(|)*N-Q"N+Q"N"nu(l)-N’\2+N’\2"n
u(l)-Q*nu(I)-Q)*(5*(-1+nu(l))A(1/2)*nu(l)+(-1+nu(l)W1/2)*Q*nu(l)+(10*Q*nu(l)+
9*nu(I)+Q’\2*nu(I)-Q"2-2*Q-9)A(1/2)*nu(I)-(-1+nu(l))“(1/2)-(-1+nu(l))A(1/2)*Q-(
10‘Q’nu(l)+9*nu(l)+Q’\2*nu(l)-Q"2-2*Q-9)’\(1/2))/(1+nu(l))/(-1+2*nu(l))/(-1+nu(
l))’\(1/2)/a/(3*(-1+nu(l))A(1/2)+(-1+nu(l))“(1/2)*Q-(10*Q'nu(l)+9*nu(l)+Q"2*nu(|
)-Q"2-2*Q-9)"(1/2)+2*N*(-1 +nu(l))A(1/2))/(3*(-1+nu(l))“(1/2)+(-1+nu(l))“(1/2)‘
Q+(10*Q'nu(l)+9*nu(|)+Q/\2*nu(l)-Q’\2-2*Q-9)A(1/2)+2*N*(-1+nu(l))“(1/2))*C[lt
2]+4*(-3*a*N+aA(N+1 )*a’\(-N)*Q-a*Q+a’\(N+1 )*a’\(-N)*N’\2+a’\(N+1)*a"(—N)*N
+a’\(N+1)*a’\(-N)*Q*N-a/\(N+1)*aA(-N)*Q*N*nu(l)-a/‘(N+1)*aA(-N)*NA2*nu(I)+a
A(N+1)*a’\(-N)*Q*nu(l)+a’\(N+1)‘aA(-N)*nu(l)*N-a*Q*nu(l)+3*a*N*nu(l)-a*Q*N+
a'Q’N’nu(I)-a*N"2+a‘N’\2*nu(I))*1“M*P/(-1+2'nu(l))/(3*(-1+nu(l))“(1/2)+(-1+n
u(|))’\(1/2)*Q-(10’Q’nu(l)+9*nu(l)+Q’\2*nu(l)-Q"2-2*Q-9)"(1/2)+2*N*(-1+nu(l))“
(1/2))/(3*(-1+nu(l))’\(1/2)+(-1+nu(l))“(1/2)*Q+(10*Q*nu(l)+9*nu(l)+Q"2*nu(l)-Q
A2-2"Q-9)’\(1/2)+2"N"(-1 +nu(l))“(1/2))/a:

> eqns :=4*b’\(-1/2*((-1+nu(l))/\(1/2)+(-1+nu(l))“(1/2)*Q-(10'Q'nu(l)+9*nu(l)+Q"
2*nu(I)-Q"2-2*Q-9)’\(1/2))/(—1+nu(l))/\(1/2))*(-3*N+3*nu(l)*N-Q*N+Q*N*nu(I)-N
A2-1-N’\2"'nu(I)-Q"nu(l)-Q)/(3"’(-1+nu(l))/\(1/2)+(-1 +nu(l))A(1/2)*Q-(1O'Q‘nu(l)+9*
nu(I)+Q’\2*nu(l)-Q"2-2*Q-9)A(1/2)+2*N*(-1+nu(l))A(1/2))/(3*(-1+nu(l))“(1/2)+(-
1+nu(l))’\(1/2)*Q+(10*Q‘nu(i)+9*nu(l)+Q’\2*nu(l)-Q"2-2*Q-9)’\(1/2)+2*N*(-1+n
u(l))’\(1/2))*C[It1]+4*b"(-1/2*((-1+nu(l))"(1/2)+(-1+nu(l))“(1/2)*Q+(10*Q‘nu(l)+
9*nu(l)+Q"2*nu(I)—Q"2-2*Q-9)A(1/2))/(-1+nu(l))/\(1/2))*(-3*N+3*nu(l)*N-Q*N+Q
*N'nu(l)-N/\2+N’\2*nu(l)-Q*nu(I)-Q)/(3*(-1+nu(l))A(1/2)+(-1 +nu(l))/\(1/2)*Q-(10*
Q'nu(|)+9*nu(l)+Q’\2*nu(l)-Q’\2-2*Q-9)"(1l2)+2*N*(-1+nu(l))/\(1/2))/(3*(-1+nu(l
))’\(1/2)+(-1+nu(l))“(1/2)*Q+(10’Q‘nu(l)+9*nu(I)+Q’\2*nu(l)-Q’\2-2*Q-9)’\(1/2)+
2*N*(—1+nu(l))A(1/2))*C[lt2]-4*T"M*b’\(N+1)*a'\(-N)*(1+nu(l))*(Q+N)/(3*(-1+nu(l
))"(1/2)+(-1+nu(l))/\(1/2)*Q-(10*Q*nu(l)+9*nu(l)+Q’\2*nu(l)-Q’\2-2*Q-9)’\(1/2)+2
*N*(-1+nu(l))A(1/2))/(3*(-1+nu(l))A(1/2)+(-1+nu(l))“(1/2)*Q+(10*Q‘nu(l)+9*nu(l)
+Q’\2*nu(l)-Q"2-2*Q-9)’\(1/2)+2‘N*(-1 +nu(l))A(1/2)) = C[mt11*b+C[mt2]/b’\2:

> eqn4 :=-2*P*(b/a)"Q*b’\(-1/2*((-1+nu(l))’\(1/2)+(-1+nu(l))“(1/2)*Q-(10*Q‘nu(l)+
9*nu(l)+Q’\2*nu(I)-Q’\2-2*Q-9)"(1/2))/(-1+nu(l))/\(1/2))*(-3*N+3'nu(l)’N-Q*N+Q
*N’nu(l)-N’\2+N’\2*nu(l)-Q*nu(l)-Q)*(-(-1+nu(l))/\(1/2)+5*(-1+nu(l)W1/2)*nu(l)-(
-1+nu(l))A(1/2)*Q+(-1+nu(l))/‘(1/2)*Q*nu(l)+(1O’Q'nu(I)+9*nu(|)+Q’\2*nu(l)-Q’\2
-2*Q-9)’\(1/2)-(1 O‘Q’nu(l)+9*nu(l)+Q"2*nu(I)-Q"2-2*Q-9)’\(1/2)*nu(l))/(1+nu(l))
/(-1+2*nu(l))/(-1+nu(l))“(1/2)/b/(3*(-1+nu(l))A(1/2)+(-1+nu(l))A(1/2)*Q-(1O*Q*nu
(I)+9*nu(l)+Q’\2*nu(l)-Q"2-2*Q-9)"(1/2)+2*N*(-1+nu(l))’\(1/2))/(3"(-1+nu(l))’\(1/
2)+(-1+nu(l))A(1/2)*Q+(1O’Q'nu(l)+9*nu(l)+Q’\2*nu(I)-Q"2-2*Q-9)"(1/2)+2*N*(
-1+nu(l))A(1/2))*C[lt1]-2*P*(b/a)’\Q*b’\(-1/2*((-1 +nu(l))“(1/2)+(-1+nu(l))“(1/2)*Q
+(10*Q’nu(l)+9*nu(|)+QA2*nu(l)-Q"2-2*Q-9)’\(1/2))/(-1+nu(l))/\(1/2))*(-3*N+3*n
u(|)*N-Q*N+Q*N*nu(l)-N’\2+N’\2*nu(|)-Q*nu(l)-Q)*(5*(-1+nu(l))/\(1/2)*nu(l)+(-1
+nu(l))“(1/2)*Q*nu(l)+(10*Q’nu(|)+9*nu(i)+Q’\2*nu(l)-Q’\2-2*Q-9)"(1l2)*nu(l)-(-
1+nu(l))/\(1/2)-(-1+nu(l))A(1/2)*Q-(10*Q*nu(l)+9*nu(l)+Q"2*nu(I)-Q’\2-2*Q-9)"(
1/2))/(1+nu(l))/(-1+2‘nu(|))/(-1+nu(l))A(1/2)/b/(3*(-1+nu(l))A(1/2)+(-1+nu(l))/\(1/
2)*Q-(10*Q*nu(l)+9*nu(I)+Q/‘2*nu(I)-Q’\2-2*Q-9)"(1/2)+2‘N*(-1 +nu(l))“(1/2))/(
3*(-1+nu(l))’\(1/2)+(-1+nu(l))“(1/2)*Q+(10*Q*nu(l)+9*nu(l)+Q"2*nu(l)-Q"2-2*Q
-9)’\(1/2)+2*N*(-1+nu(l))/\(1/2))*C[lt2]+4*(-3*(b/a)’\N*b*N+b’\(N+1)*a’\(-N)*Q-(b
/a)"N*b*Q+b’\(N+1)*a’\(-N)*N’\2+b’\(N+1)‘aN-N)*N+b’\(N+1)*a’\(-N)*Q*N-b’\(N

106

+1 )*a"(-N)*Q*N*nu(l)-b"(N+1)*a"(-N)*N"2*nu(l)+b"(N+1 )*a"(-N)*Q*nu(l)+b"(N
+1 )‘a‘(—N)*nu(I)*N-(b/a)"N*b*Q*nu(l)+3*(b/a)"N*b*N*nu(I)-(b/a)"N*b*Q*N+(b/
a)"N*b*Q*N*nu(l)-(b/a)"N*b*N"2+(b/a)"N*b*N"2*nu(l))*T*M*P*(b/a)"Q/(-1 +2"
nu(l))/(3*(-1+nu(l))“(1/2)+(-1+nu(l))“(1/2)*Q-(10*Q*nu(l)+9*nu(l)+Q"2*nu(l)-Q"
2-2*Q-9)"(1/2)+2*N*(-1+nu(l))‘(1/2))/(3*(-1+nu(l))“(1/2)+(-1 +nu(l))"(1 /2)*Q+(1
0*Q’nu(I)+9*nu(l)+Q"2*nu(l)-Q"2-2*Q—9)"(1/2)+2*N*(-1+nu(l))‘(1/2))/b = -E(m
)/(-1+2*nu(m))*C[mt1]-2*E(m)/(1+nu(m))/b"3*C[mt2]+T*alpha(m)*E(m)/(-1+2*
nu(m)):

> eqn5 := -E(m)/(-1+2*nu(m))*m1-2*E(m)/(1+nu(m))/c"3*m2+T*
alpha(m)’EQn)/(—1+2*nu(m)) = 0:

> x:=solve({eqn1,eqn2,eqn3,eqn4,eqn5},{p1 ,l1,l2,m1 ,m2}):

> assign(x);

 

> alpha:=(m1+m2/c"3)/T:

> eq1:=evalf(alpha):

> eq2:=normal(eq1):

> eq3:=subs(a=f"(1/3),eq2):

107

APPENDIX I

This computer program proves that the relationship between
thermal expansion coefficient and bulk modulus for two phases
composite is valid in the CSA model for particle reinfored
composite.

For no interphase case:
> K[c]:=-(-4*K[m]*G[m]-3*K[p]*K[m]+4*f*G[m]*K[m]-4*f*G[m]*K[p])/(3'K[m]*f+4*
Glml-3*f*K[Pl+3*K[p]);

K ,=_ '4Km101m1’ 3 Kilelml +4wa Kw ' 4fGiml K1131

[Cl -
3 Kimlf+ 4 GM] 3me + 3 Kip]

(1.1)
> alpha[c]:= (3*f*E[p]*alpha[p]-3*f*E[p]*a|pha[p]*nu[m]-2*alpha[m]*f*E[m]+4*alp
ha[m]*t*E[m]*nu[p]+alpha[m]*E[p]+alpha[m]*E[p]*nu[m]+2*a|pha[m]*E[m]-4*al
pha[m]*E[m]"nu[p]-f*E[p]*alpha[m]-f*E[p]*alpha[m]*nu[m])/(2*f*E[p]-4*f*E[p]*n
u[m]-2*f*E[m]+4*f*E[m]*nu[p]+E[p]+E[p]*nu[m]+2*E[m]-4*E[m]*nu[pl);

o‘th:=(3fl'31110‘1131' 3"”511210‘1131"1»:1' 2 “1m1fE1 + 4 almlelml "1131 + “th Em

+ “133:1 Eipl vim] + 2 a1m1E1m1' 4 aimlElmlvlpi 'fE1p1a1m1'fE1p1 o‘1»:1"1ml)

/(2fE1p1' 4fElpl vim] ' 2fE1m1+ 4fElml v1131 + E1131+ Etp1v1m1+ 2 Elm

at]

l

'4Erm1V1p1)

(1.2)
From Christensen (1979), we have
> alpha[c]:=a|pha[m]+(alpha[p]-a|pha[m])/(1/K[p]-1/K[m])*(1/K[c]-1/K[m]);

111-111;; 1

 

(I I=CX +

 

 

(1.3)
Subbstite K[c][eqn. (1.1)] into above eqn. (1.3), we have
> alpha[c] := alpha[m]+(alpha[p]-alpha[m])/(1/K[p]-1/K[m])*(-1/(-4*K[m]*G[m]-3*
K[p]*K[m]+4*f*G[m]*K[m]-4*f*G[m]*K[p])*(3*K[m]*f+4*G[m]-3*f*K[p]+3*K[p])-1/
Kiml);

are] ;= O‘1»:1” (“1121' am)

108

 

 

[ 3K1111]linjr"""(;11'3fK11314'3K1m1 ]/[
’4K1m161ml 3K11: 1K1m1+4f01m1K1ml '1“me 1p] Kim]

1 1 J

Kip] Kim]

> a1 :=simplify(a|pha[c]);

a] 2w=-(-3a Kip 1K 3fK ‘1 Klm '4K fa”) [NIH-301 Ianf [P]

11 11 [pl [121 ml
+40‘1m1 KtmlfG1m1'4awK1nM011)/(

“(we +3KIP1KM 4fG +4fGllem)

[Ml [melll

(1.4)

Note, we have following relations
> nu:=1/2*(3*K-2*G)/(3*K+G); E:=9*G*K/(3*K+G);
1 3 K - 2 G

v:=—
23K+G

GK
3K+G

 

(1.5)
Substitue eqn. (1.5) into eqn. (1.2)

> nulplz=1/2*(3*Kipl-2*G[pl)/(3*Klpl+G[12]): Elpl:=9*G[p]*Kip1/(3*Kipl+G[pl):nui
m]:=1/2*(3*K[m]-2*G[m])/(3*K[m]+G[m]); E[m]:=9*G[m]*K[m]/(3*K[m]+G[m]);

v _13K1p1'261p1
[121'—
2 3Klpl+GlPl

 

E ._ 01p] K1131
1 1"
P 3 K1121 + Gm

 

13K1ml'201m]

v :=

[In]
2 3K! ml+Glml
Glleim}

E1m1 := K
3 1m 1+ Glml

> alpha[c]:= (3*f*E[p]*alpha[p]-3*f*E[p]*alpha[p]*nu[m]-2*alpha[m]*f*E[m]+4*a|p
ha[m]*f*E[m]*nu[p]+alpha[m]*E[p]+alpha[m]*E[p]*nu[m]+2*alpha[m]*E[m]-4*al
pha[m]*E[m]*nu[p]-f*E[p]*alpha[m]-f*E[p]*aIpha[m]*nu[m])/(2*i*E[p]-4*f*E[p]*n
u[m]-2*f*E[m]+4*i*E[m]*nu[p]+E[p]+E[p]*nu[m]+2*E[m]-4*E[m]*nu[pl):

109

> al1:=simplify(alpha[c]);

all := (3me 01m Kw + 4fK or

1131 [pl Glml ' 4 almlfGIm] K1m
+ 4 am G K

1m11m1'3fK or K )/(

[pl 1m1 1m1
4lepl Glml ' 4fG1m1K1ml + 3 Klpl K1m1 4' 4 Gim] Kiml)

(1.6)

eqn. (1.2) can be rearranged as eqn. (1.6)
Comparing eqn. (1.6) and eqn. (1.4)

> a1 := -(-3*alpha[m]*K[p]*K[m]-3*t*K[p]*alpha[p]*K[m]-4*K[p]*f*alpha[p]*G[m]+
3*alpha[m]*K[m]*f*K[p]+4*alpha[m]*K[m]*f*G[m]-4*alpha[m]*K[m]*G[m])/(4*K[
m]*G[m]+3*K[er[m]-4*1*G[m]*K[m]+4*f*G[m]*K[p]):

> a|1 := (3*i*K[p]*alpha[p]*K[m]+4*f*K[p]*alpha[p]*G[m]-4*aipha[m]*f*G[m]*K[m]
+3*alpha[m]*K[p]*K[m]+4*alpha[m]*G[m]*K[m]-3*t*K[p]*a|pha[m]*Klm])/(4*i*K[

p]*G[m]-4*f*G[m1*K[m]+3*K[p]*K[m]+4*G[m]*K[m]):
> eqn1:=a1-al1:

> eqn2:=simplify(eqn1);

eqn2 := 0

That implies eqn. (1.4) and eqn. (1.6) are identical. So, we prove the
relationship btween bulk modulus and thermal expansion coefficient
is valid in CCA model.

1+ 3 allellelm

]

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