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3 1293 00892 9782

This is to certify that the

dissertation entitled

Topics in Plane Elasticity : A Semi-Infinite
Bimaterial Strip Under Thermal Loading and
A Circular Inclusion in a Half-Plane and
An Infinite Strip Under Thermo-Mechanical Loadings

presented by
Mikyoung Lee

has been accepted towards fulfillment
of the requirements for

 

Ph . D . degree in Mechanics

 

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TOPICS IN PLANE ELASTICITY’: A.SEHI-INFINITB
BIHAIERIAL.STRIP‘UIDER.IHEIHAL.LOADINC AND
A.CIRCULAR IRCUUSION Il.A.HALF-PLAHE AND

AN INFINITE STRIP UNDER,IHEBIDHHECHANIGAL.LOADINCS

BY

Mikyoung Lee

A.DISSEEIKTIOH

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Metallurgy, Mechanics, and Materials Science

1991

Topics in Plane Elasticity : a Sell-Infinite
Material Strip under Ibernal Loading and
a Circular Inclusion in a.fla1f-P1ane and
an Infinite Strip under Iberia-Mechanical Landings
By
Mikyoung Lee

Three problems in plane elasticity are solved exactly in this
dissertation. First, the elastic fields due to a circular inclusion
embedded in a half-plane subjected either to a uniform uniaxial loading
at infinity parallel to the edge boundary, or a uniform non-shear type
eigenstrain loading are investigated. The second case is when the
inclusion is embedded in an infinite strip. Both the inclusion and the
matrix are assumed linear elastic and isotropic. In the above two
cases, the interface between the inclusion and the surrounding material
is perfectly bonded, or allows pure sliding (shear tractions are
specified to vanish). The Papkovich-Neuber displacement potentials in a
form of infinite series are used in the analysis. In these problems the
joint effect of the free surface (or free surfaces) and interface
conditions ( perfectly bonded or pure sliding) is studied.

The third problem is of a semi-infinite bimaterial strip that undergoes
a constant temperature change. Following Bogy (1968), the Airy stress
function method, Mellin transform, and conformal mapping are used to
investigate the asymptotic behavior of stresses at the interface near

the edge of two strips.

ACKNOWLEDGMENTS

I would like to thank those who have helped in this work. This
includes Professor Iwona Jasiuk, who supported me as a friend and
as an academic advisor; my guidance committee members: Professor
Nicholas Altiero, Professor Robert Soutas-Little, and Professor
David Yen, who gave me a chance to complete this work. They are
sincerely appreciated.

And Professor E. Tsuchida from Saitama University, Professor T.
Mura from Northwestern University and Dr. J. Lau from Hewlett-
Packard share my special thanks for their inspiration.

Finally, I want to thank my dad for his encouragement and
unlimited support.

Financial support of this work by the National Science
Foundation Grant No. MSM-88lO920 and the State of Michigan through

its 1989-91 Research Excellence Fund is gratefully acknowleged.

TABLE OF CONTENTS

LIST OF FIGURES v
NOMENCLATURE ix
CHAPTER 1 INTRODUCTION ...................................... 1
CHAPTER 2 A CIRCULAR INCLUSION IN A HALE-PLANE .............. 6
2.1 INTRODUCTION ...................................... 6
2.2 METHOD OF SOLUTION ................................ 7
2.3 RESULTS AND DISCUSSION ............................ 20
CHAPTER 3 A CIRCULAR INCLUSION IN AN INFINITE STRIP ......... 32
3.1 INTRODUCTION ...................................... 32
3.2 METHOD OF SOLUTION ................................ 32
3.3 RESULTS AND DISCUSSION ............................ 45
CHAPTER 4 A SEMI-INFINITE BIMATERIAL STRIP .................. 61
4.1 INTRODUCTION ...................................... 61
4.2 METHOD OF SOLUTION ................................ 62
4.3 RESULTS AND DISCUSSION ............................ 76
CHAPTER 5 CONCLUSIONS ....................................... 80
REFERENCES ................................................... 81
APPENDIX ..................................................... 90

iv

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

LIST OF FIGURES

Circular inclusion in a half-plane.

0 vs. l/a for a uniaxial loading a - p when P = 100
Y? yy 0

for perfect bonding (dashed lines) and pure sliding

(solid lines) for a half-plane case.

*
0 vs. l/a for an eigenstrain loading a - e - 6
xx yy 22

when P - 100 for perfect bonding (dashed lines) and

pure sliding (solid lines) for a half-plane case.

a and 0 along x axis for eigenstain loading
xx YY

e * - e *, e *= 2 e * when r = 100 and a = 0.8.
xx zz yy xx

Solid lines represent the case when an inclusion is
located near the free surface, while the dashed lines
correspond to the case, when the inclusion is embedded

in the infinite medium.

Jump in the tangential displacement [us] along the

inclusion-matrix interface for eigenstain loading when
P - 100 and a - 0.5 for half-plane. (pure sliding case)

Shear stress 0 vs. 0 for eigenstrain loading when

r0

P - 100 and a - 0.5 for a half-plane case.

V

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

(perfect bonding case)

Displacement ux along the free surface (x - -1) for

a uniaxial loading for a half-plane case when a - 0.5.
Circular inclusion in an infinite strip.

0 vs. l/a for a uniaxial loading a - p when P = 100
YY YY °

for perfect bonding (dashed lines) and pure sliding

(solid lines) for an infinite strip case.

* * *
0 vs. l/a for an eigenstrain loading 6 - e - e
YY xx YY 22

when P - 100 for perfect bonding (dashed lines) and

pure sliding (solid lines) for an infinite strip case.

axx and 0 along x axis for eigenstain loading

6 * - e *, e *— 2 Ex: when P - 100 and a — 0.8.

YY

Solid lines represent the case when an inclusion is
located near the free surface, point lines represent
the case when an inclusion is embedded in the strip,
while dashed lines correspond to the case, when the

inclusion is embedded in the infinite medium.

Jump in the tangential displacement [uo] along the

inclusion-matrix interface for eigenstrain loading when

P - 100 and a - 0.5 for an infinite strip case.
vi

Figure

Figure

Figure

Figure

Figure

Figure

13

14

15

16

17

18

(pure sliding case)

Shear stress are vs. 0 for eigenstrain loading when

P - 100 and a - 0.5 for an infinite strip.

(perfect bonding case)

Displacement ux along the free surface (x - -1) for

a uniaxial loading for an infinite strip case when

P - 100.

Displacement ux along the free surface (x - -1) for

a uniaxial loading for an infinite strip case when

P - 0.01.

Displacement ux along the free surface (x - -1) for

* * *
e - e - e for an infinite strip case when P -
xx yy 22

Displacement ux along the free surface (x - -l) for

* * *
e - e - 2 e for an infinite strip case
xx 22 yy

when P - 100.

Displacement ux along the free surface (x - -1) for

* * * ‘ , ,
e '- e - 2 exx for an infinite strip case

yy 22
when P - 100.

vii

100.

Figure

Figure

Figure

Figure

Figure

19

20

21

22

23

Displacement ux along the free surface (x - -l) for

* * * , ,
e - c - £22 for an infinite strip case

xx YY
when P - 0.01.

Displacement ux along the free surface (x = -l) for

* *
e - e - 2 c * for an infinite strip case
xx 22 yy

when P - 0.01.

Displacement ux along the free surface (x - -1) for

* * *
e - e - 2 e for an infinite strip case
yy 22 xx

when P - 0.01.
Two quarter planes and two semi-infinite strips.

Stresses at the interface of a bimaterial semi-infinite

strip.

viii

a -
E -
G -
h -
K -
Po '
ui -
a -
¢0’ ¢1 -
¢ -
eij -
*
Eij -
V -
aij -
l" —

NOMENCLATURE

radius of inclusion
Young's modulus

shear modulus

the width of the strip
bulk modulus

uniaxial loading
displacement in i-direction

coefficient of thermal expansion

Papkovich- Neuber displacement potentials

Airy stress function

elastic strain tensor

inelastic strain tensor

Poisson's ratio

stress tensor

the ratio of shear modulus (C/G)

ix

CHAPTER 1

Three problems in plane elasticity are solved exactly in this
dissertation. First, the problem of a circular inclusion embedded in a
half-plane subjected either to a uniform uniaxial loading at infinity
parallel to the edge boundary, or a uniform non-shear type eigenstrain
loading is investigated. The details are included in Chapter 2. The
second case is when the inclusion is embedded in an infinite strip. This
solution is discussed in Chapter 3. Both the inclusion and the matrix
are assumed linear elastic and isotropic. In the above two cases, the
interface between the inclusion and the surrounding material is
perfectly bonded, or allows pure sliding (shear tractions are specified
to vanish). The Papkovich-Neuber displacement potentials in a form of
infinite series are used in the analysis. In the above problems the
joint effect of free surface (or free surfaces) and interface conditions
(perfectly bonded or pure sliding) is studied. The third problem is of
a semi-infinite bimaterial strip that undergoes a constant temperature
change. The solution is given in Chapter 4. Following Bogy (1968), the
Airy stress function method, Mellin transform, and conformal mapping are
used to investigate the asymptotic behavior of stresses at the interface
near the edge of.two strips.

The presence of inhomogeneities (inclusions) in an elastic material
has been the subject of extensive investigations due to the importance
of stress concentrations in the design. When inclusions exist in a
material, the stress is intensified in their vicinity. The degree of

this stress disturbance depends on the mismatch of the elastic
1

2
constants, the shape, the size, the location of the inclusions, the
boundary conditions at the inclusion-matrix interface, the loading, and

other factors.

As a literature background most of the papers dealing with
inclusions (see Mura, 1987), along with Eshelby's celebrated paper
(Eshelby, 1957), involve the cases when the inclusion is embedded in an
infinitely extended material and is perfectly bonded.

The two dimensional problem of a semi-infinite plate with a hole
subjected to a uniaxial loading was solved by Jeffery (1920). Jeffery's
paper, however, contains an error and the solution does not satisfy the
traction free boundary conditions at the straight edge. The above
solution was corrected by Mindlin (1948). This problem has been
reexamined recently by Callias and Markenskoff (1989), who analyzed the
singularity of the hoop stress at a hole, as the hole approaches the
free boundary. The problem of a circular inclusion in a half-plane
undergoing an expansion was solved by Richardson (1969). The inclusion
was assumed to be of the same material as the matrix. The problem of
a semi-infinitely extended material with a perfectly bonded circular
inclusion was addressed by Saleme (1958). Coincidentally, Saleme's
solution also contains an error, since it does not include the rigid
body displacement of the inclusion, as pointed out by Shioya (1967).
Most of the above authors used bipolar coordinates and Airy stress
functions in a form of infinite series in their analysis.

The problem of an infinite strip with a hole under a uniaxial
tension loading was solved by Rowland (1929). In his paper a solution
was sought by the successive approximation method. Later, Howland and
Stevenson (1933) considered a more general problem which includes

unsymmetrical cases. Wang (1946) studied perforated shear webs. He

3
considered an infinitely long plate of a finite width with a plain
circular hole and rectangular plate with a plain or reinforced Circular
hole. The problem of determining the stresses in a perforated strip
loaded in tension through a close-fitting rigid pin filling the hole was
solved by Theocaris (1954). In his paper, there are some limitations
like that the inclusion, which represents a pin, is rigid and the ratio
of the diameter of the hole and the breadth of the strip does not exceed
0.5. The analytical solution of an infinite strip having an
unsymmetically located perforating hole was investigated by Ling (1956).
This can be considered as a complete solution because it is valid in the
entire strip, unlike the previous solution by Rowland and Stevenson
(1933) for a symmetrically perforated strip. Tamate (1956) studied the
effect of a circular hole in an infinite strip subjected to a pure twist
by using the perturbation method.

The problem of a sliding inclusion in an infinite material was
addressed in two'dimensions by Tsuchida et a1. (1986) and Kouris et a1.
(1988), and in three dimensions by Mura and Furuhashi (1984), Mura et
a1. (1985), and Jasiuk et a1. (1987). The problem of a sliding hemi-
spherical or hemi-spheroidal inclusion at a free surface was solved by
Kouris and Mura (1989), Kouris et a1. (1989), and Tsuchida et a1.
(1990).

The half-plane problem in three dimensions for eigenstrain loading
was solved by Mindlin and Cheng (1950) and Aderogba (1976) for a
spherical inclusion, and by Sea and Mura (1979) for an ellipsoidal
inclusion. The problem of an inhomogeneity near the free surface
subjected to all-around tension at infinity was solved by Tsutsui and
Saito (1973) for a spherical inclusion and Tsuchida and Mura (1983) for

a spheroidal inclusion.

4

The three dimensional problems of an infinite strip with a prolate
spheroidal cavity subjected to an axisymmetric pressure and a uniaxial
tension were investigated by Tsuchida and Yaegashi (1981a, 1981b). The
solution of this problem is based upon Boussinesq's stress function
approach.

In this dissertation we consider a problem of a sliding circular
inclusion embedded in a half-plane or a semi-infinite strip. We are
interested in the joint effect of the free surface and sliding boundary

condition. This problem has not been addressed before.

When two different materials are bonded together and subjected to a
temperature change, the thermal stress will arise due to the different
thermal expansion coefficients.

A number of studies on bimaterial strips subjected to temperature
change are reported in literature. The first solution, based on beam
theory, was given by Timoshenko (1925). The problem of edge effects was
first addressed by Aleck (1949). The more recent solutions include
approximate solutions (Hess, 1969; Chen and Nelson, 1979; Chen et a1.,
1982; Blech and Kantor, 1984; Suhir, 1986, 1989; Gerstle and Chambers,
1987) and elasticity solution of Kuo (1989), to mention some. The finite
element solutions of this problem are discussed by Lau (1989).
Additional references are given in Eischen et a1. (1990). The solutions
to the related problems include the bimaterial subjected to mechanical
loading (Bogy, 1968; Hess, 1969). Also, the singularity of the
interfacial stresses at the edge is discussed by Bogy (1970), and
Blanchard and Ghoniem (1989) for the case of bonded dissimilar quarter
planes, and by Hein and Erdogan (1971), and Dempsey and Sinclair (1981)
for the case of the bimaterial wedges. For the above mentioned

geometries the singularity is shown to depend on the elastic constants.

5
In this dissertation we consider a problem of a bimaterial semi-infinite
strip. This problem was studied before as mentioned above.- But the
existing results are approximate, using either strength of materials
approach or finite element method. Here we solve this problem exactly

by using theory of elasticity. We find that stresses are Singular at

the interface at the edge of the strip.

CHAPTER 2 : A Circular Inclusion in.A.Ha1f-P1sne

In this part of dissertation, the plane elasticity problem of a
circular inclusion embedded in a half-plane is considered. The solution
will apply to two physical cases of a disk-like inclusion embedded in a
semi-infinite plate (plane stress), or a cylindrical inclusion embedded
in a half-space (plane strain). The inclusion has different elastic
constants from those of the matrix. The inclusion may represent the
cross-section of a fiber located near the free surface of a composite
material. The loading is either a uniaxial stress at infinity parallel
to the edge boundary, or a uniform non-shear type eigenstrain
(transformation strain) given by cij, such that i - j, where i,j -
x,y,z. Both the inclusion and the surrounding matrix are assumed to be
linear elastic and isotropic. The interface between the inclusion and
the matrix allows sliding: the shear tractions are specified to vanish
and no separation takes place in the normal direction. The perfect
bonding case is ineluded for comparison. In this study the joint effect

of a traction free edge boundary and sliding is investigated.

METHOD OF SOEUTION

Consider a semi-infinite material having a circular inclusion
subjected to a uniaxial loading or uniform eigenstrains. Let the origin
of coordinates be at the center of the circular inclusion and the x—axis

be directed down into the interior of the semi-infinite body. It is

7

convenient to use a polar coordinate system, which is related to the

Cartesian system by x - r cos0 and y - r sino. Without loss of

generality, the distance between the center of inclusion and the point

on the free boundary closest to the inclusion is taken as unity.

Therefore, the free surface (the straight edge of the plate or the plane

surface of the semi-infinite body) is given by x - -l. The radius of

inclusion is taken as r - a.
The following boundary conditions are used:

1) the traction free condition on the surface x - -1:

( ) ( ) - 0

axx x--l - axy x--1

ii) either the pure sliding conditions at the interface of the

inclusion:

) - (

0
r-a rr

( )

0
rr r-a ’

- (arfl) - 0

(are) r-a

r-a

(Ur) - (E )

r-a r r-a

(2.1)

(2.2)

or the perfect bonding conditions at the interface of the inclusion:

) - (

0’
r-a rr

( )

0
rr r-a

)

(are r-a - (ar0)r-a

(Er) r-a

(u )

r r-a

(2.3)

(“0)r-a - (Eo)r-a

iii) either a uniaxial stress applied at infinity:
0 - 0, a - 0, a - po ‘ (2.4)
or the vanishing tractions for eigenstrain case:

a - 0, i,j - x,y. (2.5)
In the preceding expressions the quantities defined in the
inclusion are denoted by a superior bar.
The Papkovich-Neuber displacement potentials are used to solve the
above boundary value problems. The displacement potentials, ¢O and d1,
that account for the disturbance due to the inhomogeneity and the effect

of free surface, are given as follows:

i) for the matrix (r > a), ¢0 and Ol-are

 

 

 

cos m0

(1) i ¢0 - C0 ( -Ao log r +m§1 Am )

1 rm

¢ _ C 2 B cos m0 , (2.6)

L 1 0 m-O m rm

r -Ax
(II) ¢O - CO I” ¢1(A) e cos Ay d)

0

9-
H
I

 

CO Jo A ¢2(A) e-Ax cos Ay dA (2.7)
0

9

ii) and for the inclusion (r < a), ¢O and ¢l are

(111) ‘0 - Co E A r“ cos n0 (2.8)

_ — n
d1 - CO 2 En r cos n0
where

p for a uniaxial loading
0

C - '
* *
O 266 for an eigenstrain loading (6* - 6* , 6* or e )
xx yy 22

where G is the shear modulus of the matrix and e

* *
xx’ eyy and 622 are the

eigenstrains. An, Bm’ Anand in are the unknown constants and ¢1(A) and
¢2(A) are the unknown functions of A, which are determined from the
boundary conditions (2.1)-(2.5). Note that (I) is expressed by a suffix
m and (III) by a suffix n. For reference, the relations between the
displacement potentials ¢0 and $1 and the stresses and displacements are
given in the Appendix.

When a uniaxial loading is applied, the displacement potentials are

given as

'6

(IV) «to - 8_o <~+I><y2- x2> (2.9)
4

 

The stresses and displacements obtained from (IV) are given as

10

axx - 0

xy - O

ayy - po

20 ux - - 3%: p0 x

26 uy - fgi po y
where

3 - 4v for plane strain

x - %—i—5 for plane stress

(2.10)

where u is Poisson's ratio. The stresses and displacements, given in

equation (2.10), are expressed in polar coordinates as

a - p i (1 - cos 20)

2
1
Ora - po 5 sin 20
a - p 1 (1 + cos 20)
90 O -

r
26 ur - po Z {(5-1) - 2cos 20}

2C uo - pO 3 sin 20.

N

(2.11)

ll
' When the inclusion is subjected to an eigenstrain loading, the
displacements inside the inhomogeneity are the sum of displacements of

the inclusion if there would be no constraint from the matrix

ux - (e + n 622) X
* * *
uy - (cyy + q fizz) y (2.12)

where

{ u for plane strain
’7-

0 for plane stress

and the elastic displacements obtained from (III). The displacements,
given in (2.12), are expressed in polar coordinates as

* * * *
+ (exx + eyy) + (exx - eyy) cos 20 }

* g *
ur - 2 { 2" 6zz

* I * *
u - 2 ( cyy - Exx) sin 20. (2.13)

The corresponding stresses are zero.

Note, that the stress components derived from (I) and (II) vanish
at infinity. Therefore, for the eigenstrain loading the boundary
condition (2.5) is automatically satisfied. For the case of uniaxial
loading, do and d1, given by (IV), yield the uniaxial tension p0.
Therefore, the boundary condition (2.4) is also automatically satisfied.
To satisfy the traction free boundary condition (2.1) at the free

surface, (I) is considered as follows:

12

(I)-1 ¢0 - - C0 A0 log r
¢ _ C A cos m0
0 0 1 m -;m--
(I)-2 ¢ _ C g B cos m0
1 0 m~0 m -;m—- '

Using one of the following mathematical relations

cos mo _ (-1)m

 

m
r

m
r

Am-1 eAx cos Ay dA
(m-l)! 0

(x < 0)

sin mo _ (-1)""1 I” Am-I eAx sin Ay 81
O

(m-l)!

(I)-2 is expressed in Cartesian coordinates as

(I)-2*

 

-l)m m-l Ax
¢ - C E ( A IQ A e cos Ay d)
0 0 m-l (m-l)! m 0

 

 

m
:-1:)! Bm I: Am-1 eAx cos Ay dA.
m-

As for (I)-l,

l 2 2
¢0 - -C0 A0 log r - - C0 A0 5 log (x + y )

and the stresses are obtained directly from equation (2.17)

cos 20

r2

 

as

(2.14)

(2.15)

(x < 0)

(2.16)

(2.17)

13

 

r2
and using (2.15) they are expressed as
axx - C0 A0 Jo A eAx cos Ay dA
0
(x < 0) (2.19)

Ax
axy - - C0 A0 J0 A e sin Ay dA.
0

Using equations (2.7), (2.16) and (2.19), the boundary conditions (2.1)

are written as

(axx) IQ A2 A n+1 A g (-l)m m-l 6A
__ - [ ¢ (A) e + ( ___ -A) ¢ (A) e + A ,____ A e
co x--1 0 1 2 2 m-l m (m-l)!

m
-mgl (2:: +A) Am-2 e.A B (-1) + A0 A.1 e.A ] cos Ay dA - 0 (2.20)

2 m (m-l)!

m
(-1) m-le-A

 

(axy) 2 A n-1 A
l-EA [361(k) e +(_§_~\)¢2(A) e -m§1A

“‘ A
Co x - m (m-l)!
m
+m§1 (5:1 +A> *m'z 6'2 Bm('1) - A0 A'le'A ] sin Ay ax — o. (2.21)
2 _—

(m-l)!

Setting the quantities in the brackets in equations (2.20)-(2.21) to
zero, two equations for the two unknowns ¢1(A) and ¢2(A) are obtained.

Solving these equations, ¢1(A) and ¢2(A) are found as

l4

 

o (A) - A 1'1 c'2A (n-ZA) + E [ Am(-1)m Am'l c'2*(s-2A)
1 0 m-l _(;:1)T—
m 2 2
_ B (-1) An-2 e-2A (n - 1 - 4A )1 (2.22)
“ (m-l)! 2

111
P20) - -2A0 A’1 of” - “121 [2 A11('1) Am-1 8-21
(m-l)!
m
- _%m£%;l_ (n+2A) Am‘z c'2*]. (2.23)
m- 1

In order to satisfy the boundary conditions (2.2) or (2.3) at the

inclusion's interface, the following relation is used

~Ax n (Ar)n
e cos Ay - “20 (-1) cos n0. (2.24)
n!

 

(II) is rewritten by using equation (2.24) as follows

*
(II)

dO - CO n20 an rn cos n0
¢1 - Co n20 fin rn cos n0 (2'25)

(_1)n An
a - d1(A) ox (2.26)
o .

and

where

m m
“n ' Ao “n1 + m§1 “n2 Am +m21 “n3 Bm

5n ' AO pnl + mgl ”:2 An +m21 5:3 Bm (2°27)
n-l
a - (s - 2A) (-1)n A c'2A dA

n!

O

m+n-l
am - I“ (A - 2A) (-1)m+n A c'2A dA
n2 -———————'

2 2 m+n-2
m _ (x -1 -4A ) (_1)m+n+1 A e-2A dA
0 2 (m-l)! n!

n
finl - [a (-2) (-1)n _2_ e’ZA dA (2.28)
o

n!

m+n

fl:2 _ JG (-2) (_1)m+n A e-2A dA
0 (m-l)! n!

m+n-1
Em3 - Jo (x + 2A) (-1)n+m * c'2A dA.
o

(m-l)! n!

16

Using Euler's Integral of second kind

fo'axxbdx-M (a,b>0)
o

and the relation
Jo %— e'sZ cos {a d5 - - % log (a2 + 22)
0

equations (2.28) can be simplified as

a
n1 n n

an2

 

where

nH-n

 

mm
m _ (-1) e,,, Am+n c1 _ (-1> (m+n)!
7n m! n! m! n! 2m+n+l
0

(2.29)

(2.30)

(2.31)

(2.32)

17

and

log 2 n - 0
en _ o (2.33)
73-1 n 2 1
n

Next, the stresses and displacements are derived from the displacement
potentials given by (I), (II)* and (III), and the equations (2.11) are
used for a uniaxial loading and equations (2.13) for eigenstrain loading
cases. Substituting these results in the sliding boundary conditions
(2.2), the following four equations are obtained:

1) the continuity of normal tractions (a ) - (3
rr r-a rr

)

r-a'

 

n20 [8A1 An + SBl Bn-l + 532 Bn+1 + 3(11 an + 861 fln-l + 882 Bn+l
A P
— - - - - - 0 o (l-cos 20)
' sA1 An ' s81 Bn-l ' S32 Bn+1 ] °°s “9 -— ' ——
a2 C0 2

(2.34)

ii) the condition of continuity of normal displacements

(ur)r-a - (ur)r-a:

n20 [{kAl An + kBl Bn-l + kBZ Bn+1 + ka1 an + kfil fin-l + kfl2 fin+l}

1 _ _. _. _.
‘ f ‘kAI An + k81 Bn-l + k82 Bn+1 } 1 °°S no '

fig - fig 2 ( (l-x) - 2 cos 2o ) +
a C 4
0
20 a * * * * *
E- 5 { 2n £22 + (cxx + eyy) + (cxx - eyy) cos 20} (2.35)

0

18

iii) the continuity of tangential tractions (ar0)r-a - (ar0)r-a:

B

n21 [tAl An + t81 Bn-1 + t82 n+1 + talan + t81 fin-l + t82 fin+1

1

tA1 An - tBl Bn-l - th sin 20 (2.36)

Bn+1] sin n9 - - __

Co

l\>

iv) and the condition of vanishing shear tractions given by either

(ar0)r-a- 0 :

n21 [ tA1 An + tBl Bn-l + th Bn+l] sin n0 - 0 (2.37)

or (ar0)r-a
The problem of the perfectly bonded inclusion can be solved by using the
first three equations, (2.34)-(2.36), and the last one, (2.37), is

replaced by the condition of continuity of tangential displacements

(u0)r-a- (“0)r-a :
n21 [‘ 1A1 An + 131 Bn-1 + 132 Bn+1 + 1o1 “n + lfil fin-l + 1fl2
5 } - 1 I 1 A + 1 E + 1 E }] sin n0 - - p0 a sin 20
n+1 F— A1 n Bl n-l B2 n+1 -— 2
0
+353 ( .* - «2* )sin 20 (2.3a)
CO 2 yy xx

where F - C / G, and G and C are the shear moduli of the matrix and

inclusion, respectively, and

Al

a1

A1

a1

A1

Sal

Al

al

Note that the terms involving barred quantities kAl’ kBl’ 1A1""

obtained directly from kal’

to the rigid body motion of the matrix.

-n

n+1
a

-n

n+1
a

('n) a- 9

n(n+1)

 

n+2
a

n-2

- n(n-l)a ,

n(n+l)

 

n+2
a

- (-n)(n-1)an'2.

19

 

 

 

 

 

 

_ ~(n-l+n)
kBl "____"’
28n-l
(n-l-Ic)an-1
kfil 2 '
_ -(n-1-n)
131 ‘_—-'——— ’
28n-l
1 _ (s-n+1)a“‘1
51 2 '
s _ (n-l)(n+2)
Bl n ’
2a
5 (n-l)(n-l-s) an.2
51 2 ’
n(n-l)
t81 n '
2a
t _ (l-n)(n-l-n) an'2
51 2 '

kfll’ 1&1....

82

82

B2

82

B2

82

82

82

~(n+l+x)

28n+1

(n+l-x)an+1

 

-(n+1+n)

(2.39)

2an+l

-(A:+n+1)an+l

2

 

(2.40)

(n+l)(n+1+n)

 

28n+2

(n+1)(n-2)sn

 

2
(2.41)

(n+l)(n+l+n)

 

2an+2
-(1+n)n an
2
(2.42)

can be

by replacing n with E.

The displacement containing the coefficients a1 and 60 corresponds

Therefore, a

and 60 are taken

20

as zero. The displacements containing A1 and 30 represent the rigid
body motion of the inclusion in x-direction. Since the terms containing
A1 and DO give the same rigid body displacement of the inclusion, one of
the two constants can be chosen to be zero. In this paper, 30 - 0 and
A1 fl 0 (B0 # 0, A1 - 0 will give the same result).

Equating the coefficients of sin no and cos no in the left and
right sides of either equations (2.34)-(2.37) for the sliding case, or
(2.34)-(2.36) and (2.38) for the perfect bonding case, an infinite set
of algebraic equations for the unknown constants An, Bn’ An and in is
obtained. Each of these four sets of unknowns is associated with an
integer n, which varies from zero or one to infinity. In the numerical
calculations the infinite series are truncated at n - N. For the
examples included in this paper N ranges from 2 to 33. N is chosen in
such a way that the boundary conditions are satisfied to 3 or more
significant figures, which means that the error involved is 1.0 x 10-3%
or less. The acCuracy is improved by increasing a number of terms in
the series. The solution holds for the whole range of a, 0 < a < 1. As
a 4 1, only the number of terms needs to be increased. When a perfectly
bonded or sliding inclusion is embedded in the infinite matrix, i.e. the
inclusion is far away from the free surface, the solution is given by
finite series involving n-0 and n-2. After An’ Bn, Kn and in are

evaluated the stresses and displacements are known everywhere in the

inclusion and the matrix.

RESULTS AND DISCUSSION

In the numerical examples considered in this Chapter the plane

strain case is assumed. This case is of interest in the light of

21
applications to composite materials reinforced with unidirectional
fibers. For simplicity v - 3 - 0.3 is taken.

Figs. 2-3 illustrate the stress ayy at points M, N, N', P, and P'
(see Fig. l) for perfect bonding (dashed lines) and pure sliding (solid
lines) cases for different positions of the inclusion relative to the
free surface. The radius "a" is varied from 0.2 to 0.99, while the
center of the inclusion remains at a unit distance from the free surface
(Fig. l). The loading conditions include the remote uniaxial stress 0yy
- po (Fig. 2) and the eigenstrain loading e:x - €;y - (:2 (Fig. 3).

The inclusion is assumed to be stiffer than the matrix such that the
ratio P - C/G - 100 (note that the higher the F, the more pronounced the
effect of interface). It can be observed that the effect of free
surface contributes significantly to the stress disturbance when the
inclusion is embedded less than one diameter away from the free surface
(which corresponds to l/a < 3).

When the uniaxial loading is applied (Fig. 2) and the inhomogeneity
is perfectly bonded, the maximum stress ayy occurs in the inclusion at
the point N', which is closest to the free surface. The stress in the
matrix near the inclusion is almost zero since the inclusion, which is
stiffer than the matrix, carries the load. It is also observed that the
stress a at the free surface is less than the applied load when the
inclusion is close to the free surface.

Quite a different behavior is observed when the sliding takes place
at the interface. The highest stress occurs in the matrix near the free
surface when the inclusion is close to the free boundary. It is
interesting to observe that the hoop stress in the sliding inclusion at
points N' and P' and the vicinity is compressive, while the perfect

bonding solution yields tensile stresses in these regions. This

compressive streSs in the sliding inclusion is due to the fact that the

22
sliding interface does not transfer shear tractions and due to the
curvature of the inclusion. Therefore, the matrix, which is allowed to
flow more freely around the inclusion causes the local compressive
stress in the inclusion.

When the inclusion is soft (F << 1), the presence of the free
surface gives rise to a very high stress concentration in the matrix at
point N. For this case the contribution of interface is negligible.
This is expected, because in a limiting case when the inclusion becomes
a hole (F - 0), both perfect bonding and sliding cases reduce to the
same result. In this limit case the present solution (for plane stress
case) coincides with Mindlin's solution (Mindlin, 1948). When inclusion
is away from the free surface, the solution agrees with the result for
the inclusion embedded in the infinitely extended material
(Muskhelishvili, 1953; Eshelby, 1957), as expected.

When discussing the solution of a sliding inclusion, which is
subjected to either a uniaxial remote loading or an eigenstrain loading
(non-dilatational), a comment needs to be made about the tensile normal
tractions that may arise at the inclusion-matrix interface. These
tensile tractions imply a tendency to debond in the normal direction,
but the debonding is not allowed by the sliding boundary conditions. In
order to avoid the possibility of debonding, the case of remote loading
may be considered in conjunction with the eigenstrain loading, when the
eigenstrains are chosen in a way that the total radial stress is
compressive all along the interface as suggested by Muskhelishvili
(1953). This case can be obtained by a superposition of these two
solutions. However, it is more informative to separate these two cases
when giving the numerical results. The more realistic boundary
conditions for this problem would allow for a local debonding. This

would give rise to a mixed boundary value problem. The analytical

23
solutions to a problem, when a sliding inclusion is subjected to a
uniaxial tensile loading at infinity and debonding is allowed are given
by Stippes et a1. (1962), Noble and Hussain (1969), Margetson and
Morland (1970), and Rear et a1. (1973), among others. In this
dissertation, the debonding in the normal direction is not allowed, for
simplicity.

Fig. 3 illustrates ayy at chosen points along the x axis for
eigenstrain loading e:x - 6;y - 6:2 for both perfect bonding and pure
sliding. The highest stress occurs in the matrix at the free surface
for the sliding case. Also, the free surface gives rise to higher
stresses in the matrix and in the inclusion near the free surface for
both interfacial conditions. Fig. 4 gives stress oxx and oyy along the
x axis when the eigenstrain loading e:x - 622- l, eyy - 2 exx is
applied, F - 100 and a - 0.8 for the perfect bonding case. The effect
of free surface is illustrated by comparing the stress field of the
inclusion located near the free surface (solid lines) and of the
inclusion embedded in the infinite matrix (dashed lines). Note that the
stresses are uniform when a perfectly bonded inclusion is embedded in
the infinite medium as expected from Eshelby's solution (Eshelby, 1957),
but they cease to be uniform when the inclusion is located near the free

surface. Fig. 5 illustrates the jump in the tangential displacement

[no] along the interface of the sliding inclusion for different

* * * * * * *
eigenstrain loadings e - e , e - 2e and e - e /2, while 5 -
yy xx yy xx yy xx 22
*
e , and P - 100, when the inclusion is located a distance of one

XX

radius away from the free surface (a - 0.5). Note that the free surface
influences the degree of sliding. It is interesting to observe that the
sliding occurs for the dilatational eigenstrain loading for the half-
space problem, while in the infinite space the sliding cylindrical

inclusion would experience no sliding due to this symmetric eigenstrain

24
loading. Fig. 6 shows the shear stress along the interface of the
perfectly bonded inclusion for the above eigenstrain loadings. Fig. 7
gives the displacement ux along the free surface for 0 s y s l for both
perfect bonding (dashed lines) and sliding (solid lines) cases, when a
remote uniaxial loading is applied. The inclusion is embedded a
distance of one radius away from the free boundary, which implies a -
0.5. Note that the displacement is Significantly affected by P. The
influence of interface can also be observed. The determination of the
displacements at the free surface can be of interest to experimentalists
since the differential strains can be measured by the experimental
techniques such as stereoscopy and interference microscopy (Cox et al.,
1988). These experimental and theoretical results can be used as a
nondestructive technique to predict the nature of the defects inside a
material (Mura, 1985).

In this chapter, Papkovich-Neuber displacement potentials in an
infinite integral form and infinite series in polar coordinates are
used. One of the advantages of using displacement potentials for this
problem is that the formulation is similar in both two and three
dimensions (Ishiwata, 1986; Tsuchida and Mura, 1983). The other one is
that we can get displacements directly, while Airy's stress function
method demands integration of strain or stress to determine the

displacements.

25

 

 

 

 

 

Fig. 1 Circular inclusion in a half-plane.

26

 

2.00

1 75_ I I‘ - 100

1.50- 1"

~ -. ~ .
~ - ".127: ;-:_—...—.__- -_'=.-=.= ==== =-‘= = 3

‘

/Po

1.25-

77

l .00-

 

 

0.75- /
0.50-
0.25-
- 0.00-
-0.25-

-0.50-

 

 

 

Stress Concentration.Factor a

-0.75-

-I.00

 

 

 

l | l l l l I l
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

l/a
Fig. 2 ayy vs. l/a for a uniaxial loading oyy - po when P - 100 for
perfect bonding case (dashed lines) and pure sliding

(solid lines) for a half-plane case.

27

 

3.5
* * *-
ii 3.0- N M ‘n " ‘yy ‘ $22
*1» 1‘ - 100
w .
N 2.5- \ ,
\ H \ .\

77
N

C)
l
2

 

 

 

Stress Concentration Factor 0"

 

 

 

 

_200 I 4| l l r I 1 I
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
l/a
Fig. 3 0 vs. 1/a for an eigenstrain loading ex: - 5y; - 62: when

P - 100 for perfect bonding (dashed lines) and pure sliding

(solid lines) for a half-plane case.

*’
33!

Stress ,/ 2G6

Fig. 4

28

 

 

 

 

perfect bonding

 

 

 

 

distance x

along x axis for eigenstain loading

a and 0

xx

c * - e *, c *- 2 e * when r - 100 and a - 0.8.
xx 22 yy xx

Solid lines represent the case when an inclusion is
located near the free surface, while the dashed lines

correspond to the case, when the inclusion is embedded

in the infinite medium.

29

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toss MCHUUOH Cumumcomwo Mom oommuoucu xauuma-con5HO:H

one wcoao mesa uaoaoouHamHn HoHuCOMcou on» Ca case n .wwm

c cams

on. cm. 9.1. cm. $9 an ac o¢ ON 0

p — p b P — n — P ”Foal

 

 

IN — .6!
load...
.6ch NNU
. s. em
.36..
a co. ow

 

 

NNU I NNU .
R .s .s unod
u 956.2...» 939 .

. . . . .. - cod

 

 

 

30

.omco Ocofia-mHo£ c you Amoco mauncon uoomuonv

00H I L Cons waunmoa :HouuchwHo you s .m> muo mmouum umonm o .wwm

c meflw

00.. cm. 0.: amp 2.: oo oo o... ON a

h — F - p _ c n p p s n . .VcOlu

 

 

Indl
INdI
I «6| KN
U
an UN
ad
to

 

NW
Ul- U

A a. lad
wfifltfion uoomuom

b p n b lP L h n ”no

 

 

 

31

0.,

3.. I xv .093 ocoaaéaon a now wawncoa

Howxcac: a new commuam ovum one macaw x: uaoaoouaanwo n .mwm

% 005mg
0.0 0H0 J0 N0 0.0

 

_ 0.0

III 00H I .H IN.O

 

 

 

‘
‘II““
I! II“
’1’ “‘
IOHI‘U“U- I ’H'U'l'l"""""'ll'"".'"'."""""I.
I H I h
I
I 0
III I? O
’I.’
I] N

 

 

 

CHAPTER 3 : A.Circu1ar Inclusion in an Infinite Strip

In this part of dissertation, the plane elasticity problem of a
circular inclusion embedded in an infinite strip is considered. The
method of solution is similar to the previous problem (the case of
inclusion in a half-plane discussed in Chapter 2). For completeness we
include the details in this chapter. The solution will apply to two
physical cases of a disk-like inclusion embedded in a thin strip (plane
stress), or a cylindrical inclusion embedded in a very thick strip

(plane strain).

METHOD OF SOLUTION

A circular inclusion in an infinite strip under either a uniaxial
loading at infinity parallel to the edge boundaries, or a uniform non-
shear type eigenstrain (transformation strain) is considered.

Let the origin of coordinates be at the center of the circular
inclusion and the x-axis be directed down as shown in Fig. 8. The
inclusion is assumed to be located an equal distance between two
surfaces and the distance between the center of inclusion and the points
on the free boundaries closest to the inclusion is taken as unity.
Therefore, the free surfaces (the edges of the strip) are given by
x - :1. The radius of inclusion is taken as r - a.

The following boundary conditions are used:

i) the traction free condition on the surfaces x - -1 and x - 1:

32

33

(axx)x-il - (axy)x-i1 - 0

ii) either the pure sliding conditions at the interface of the

inclusion:

) - <‘ )

0
r-a rr

(

0
rr r-a

(ar0)r-a - (5r0)r-a - 0

(Ur) - (E >

r-a r r-a

(3.1)

(3.2)

or the perfect bonding conditions at the interface of the inclusion:

( ) (arr)

a -
rr r-a r-a

(ar0)r-a - (3r0)r-a

(Er)r-a

(u)

r r-a

(u0)r-a - (Ea)r-a

iii) either a uniaxial stress applied at infinity:

0 - 0, a - O, a - p
xx xy yy 0

or the vanishing tractions for eigenstrain case:

aij " 0, i,j - x)Y°

(3.3)

(3.4)

(3.5)

34
In the preceding expressions the quantities defined in the

inclusion are denoted by a superior bar.

The Papkovich-Neuber displacement potentials are given as follows:

i) for the matrix (r > a), ¢0 and d1 as

 

 

cos 2m0
(I) ‘0 - cO ( -Ao log r +m§1 Am ""EET")
1 r
¢ _ C 2 B cos (2m+1)0 (3.6)
L 1 O m-O m 2m+1
r
(II) F ¢0 - CO I” ¢1(A) cosh Ax cos Ay dA
4 O
¢1 - C0 IQ A ¢2(A) sinh Ax cos Ay dA (3.7)
‘ O

 

ii) and for the inclusion (r < a), do and ¢1 as

(III) ¢o - c0 “20 Kn r2“ cos 2n0 (3.8)

- 2n+l

d1 - CO n20 Bn r cos (2n+1)0

where

p for a uniaxial loading
0

Co - * * * * *

265 for an eigenstrain loading (6 - e , e or e )
xx yy

where G is the shear modulus of the matrix and 6* , 5* and e* are the
xx yy 22
eigenstrains. Am, B Kn and in are the unknown constants and ¢1(A) and

p2(A) are the unknown functions of A, which are determined from the

35
boundary conditions (3.1)-(3.5). Note that (I) is expressed by a suffix
m and (III) by a suffix n.

When a uniaxial loading is applied, the displacement potentials are

 

given as
(IV) I 450 - 22 (n+1)(y2- x2) (3.9)
. ¢1 ' ‘ $9 "
where
3 - by for plane strain
K - %—i—5 for plane stress (3.10)

where u is Poisson's ratio. The stresses and displacements obtained

from (3.9) are expressed in polar coordinates as

l

arr - po _ (1 - cos 20)
2
a - p 1 sin 20
r0 0 ’
a - p 1 (l + cos 20) (3 ll)
00 o E '

r
26 ur - po Z {(5-1) - 2cos 20)

r
26 uo - po _ sin 20

N

36
When the inclusion is subjected to an eigenstrain loading, the
displacements inside the inhomogeneity are the sum of displacements in

the inclusion if there would be no constraint from the matrix

* £ * * * ‘k *
ur - 2 { 2n £22 + (exx + eyy) + (exx - eyy) cos 20 }
u: - g ( e;y - e:x) sin 29. (3.12)

where

v for plane strain
0 _ (3.13)

O for plane stress

and the displacements obtained from III. The stresses in the
inhomogeneity are obtained from III.

Note, that the stress components derived from (I) and (II) vanish
at infinity. Therefore, for the eigenstrain loading the boundary
condition (3.5) is automatically satisfied. For the case of uniaxial
loading, ¢o and ¢1, given by (IV), yield the uniaxial tension p0.
Therefore, the boundary condition (3.4) is also automatically satisfied.
To satisfy the traction free boundary condition (3.1) at the free

surfaces, (I) is considered as follows:

(I)-1 ¢o - - C0 A0 log r

fl cos 2m0
‘0 ' Co m-l Am r2m (3.14)
(I)'2 cos (2m+1)0

 

¢o ' Co g—o Bm r(2m+1)

37

Using two of the following mathematical relations

 

 

m
cos m0 _ (-l) Am-l eAx cos Ay dA
m (m-l)! 0

(x < O)

 

m-l
sin m0 _ ('1) Am-l eAx sin Ay dA.

 

cos m0 _ l Am-l e-Ax cosAy dA
m (m-l)! O

 

 

sin m0 _ l Am-l e-Ax sin Ay dA
m (m-l)! O

(I)-2 is expressed in Cartesian coordinates as

2m
2 (-l) A A2m-1 eAx

¢ - c
0 “'1 (2m-15'!J 0

2m
('1) I 2m Ax
¢ - C B A e cos Ay dA.

cos Ay dA

(1)-2*

 

 

L

A will be replaced by -A if x > 0.

As for (I)-1,
l 2 2
¢0 - -C0 A0 log r - - C0 A0 5 log (x + y )

and the stresses are obtained directly from equation (3.17) as

cos 20

t2

 

(x > 0)

(3.15)

(x < 0)

(3.16)

(3.17)

38

 

sin 20
xy - C0 A0 2 (3.18)
r
and using (3.15) they are expressed as
axx - C0 A0 I” A eAx cos Ay dA
0
(x < O)
axy - C0 A0 Im A eAx sin Ay dA.
0
(3.19)
axx - C0 A0 Jm A e-Ax cos Ay dA
O
(x > 0)

-Ax .
oxy - Co A0 J0 A e Sln Ay dA.
0

Using equations (3.7), (3.16) and (3.19), the boundary conditions (3.1)

are written as

- I” A2 [ ¢1(A) cosh A - ¢2(A)(i%; coshA-A sinhA)
0

2m
+ £1 A (’1) Azm'l e" - 20 B 1 (2 +3) 3““ e'A
m' m (2m-1)! m m (2m)! 2

 

-1

+ A A e'* 1 cos Ay dA - o (3.20)

0

a
i (":Z)x--l - I: A2 [ ¢1(A) sinhA - 2%: ¢2(A) sinhA + ¢2(A)A cosh A

Co
2m 2m-l
_ E A (-1) A2m-le-A + g B (-1) ("-1+A) A2m-le-A

”'1 ” (2m-1)! ”'0 In(2m): 2

 

39

- A 3'1

o e" 1 sin Ay dA - o (3.21)

Setting the quantities in the brackets in equations (3.20)-(3.21) to
zero, two equations for the two unknowns ¢1(A) and ¢2(A) are obtained.

Solving these equations, ¢1(A) and ¢2(A) are found as

2A A 2m-1 2A

 

 

 

 

 

 

 

¢1(A) _ A0 (5 +e -2A) A'1 + mgl m A (n + e -2A)
sinh 2A+2A (2m-1)! sinh 2A+2A
m-O (2m)! (sinh 2A + 2A) 2 '
-1 A 2m-1
¢2(A) _ A0 2A , + mil m 2A
sinh 2A+2A (2m-1)! sinh 2A+2A
B -2A 2m-l

m-O (2m)! sinh 2A+2A

In order to satisfy the boundary conditions (3.2) or (3 3) at the

inclusion's interface, the following relations are used

2n .
(Ar)
cosh Ax cos Ay - E cos 2n0 (3.24)
n-O “5;?—

§ (Ar)2n+1

_ cos(2n+l)0
“ 0 (2n+1)!

sinh Ax cos Ay -

(II) is rewritten by using equations (3.24) as follows

40

¢O - Co n20 an r2n cos 2n0

 

 

*
(II) ¢1 - co “:0 an r2n+1 cos(2n+1)a
where
A2“
an - ¢1(A) dA
0 2n!
A2n+l
Bu - A 11:20) __ dA
0 (2n+l)!
and
m m
an - A0 anl + mgl an2 Am +m20 an3 Bm
m m
fin - A0 finl + mil fin2 Am +mEO fin3 Bm
where
; _ 2n-1
anl - A (n + e 2‘ -21) X 1 dA
0 (2n)! sinh2A + 2A

2A
m

a _ (x + e - 2A) A2n+2m-l 1
n2

(2n)! (2m-1)! sinh2A + 2A

dA

 

 

£?""§

(K2_1 -432) A2n+2m-1 1

2 (2n)!(2m)! sinh2A+2A

dA

 

an3

a
£?""h

(3.25)

(3.26)

(3.27)

41

dA

 

fl - A2n+1 2
“1 0 (2n+l)! sinh2A+2A

A2n+2m+1

 

m
Bn2 - Jo
0

2A
fin3

(n -e' + 2A)

£7“"'h

Using these relations

k
Ik - A dA
0 sinh2A+2A

and

 

k -2A
Mk _ A e dA
0 sinh2A+2A

(2n+l)!(2m-l)!

2
sinh2A+2A

dA

2m+2n+l
A 1 dA

 

(2m)!(2n+1)! sinh2A+2A

equations (3.28) can be simplified as

m 1

“n1 ' l” I2m+2n-1 + M2m+2n-1

 

(2n)!(2m-l)!

2

m l n -l

[

an2

1
(2n)!

 

“no ' 2n-1

(2n)!(2m-1)! 2

[x I + M

- 2

- 2

2m + 2n-1 12m+2n+11

- 2 I

2n-1 2n]

12m+2n1

(3.28)

(3.29)

(3.30)

(3.31)

42
1
3:1 ' [2

I ]
(2n+1)(2m-1)! 2m+2n+1

1
(2m)!(2n+1)!

 

5:2 I" I2m+2n+1 ' M2m+2n+1+ 2 12m+2n+21

- ___l___ [2

I ]
(2n+l)! 2n+1

finO

Next, the stresses and displacements are derived from the displacement
potentials given by (I), (II)* and (III), and the equations (3.11) are
used for uniaxial loading and equations (3.12) for eigenstrain loading
cases. Substituting these results in the sliding boundary conditions

(3.2), the following four equations are obtained:

1) the continuity of normal tractions (a ) - (0
rr r-a rr

)

r-a'

g A + s B

n-O [8A1 n Bl a + s

+ s Bn + Sal n 51 fin-l + $52 fin

n-l B2

A p
0 o (l-cos 20)
’ SA1 An

- sBl Bn-l ~ 8B2 Bn ] cos n0 - - __ - 2

 

N
O

(3.32)
ii) the condition of continuity of normal displacements

(ur> - (E )

r-a r r-a'

n20 [{kAl An + knl Bn-1 + ksz Bn + ka1 an + kfil 5n-1 + kaz 5n}

1 _ _ _ _ _ ._
- f {kAl An + kBl Bn-l + kB2 Bn } ] cos n0 -

43

fig - fig 3 ( (l-x) - 2 cos 20 ) +
a C 4
0
2G a * * * * *
E- 5 l 20 £22 + (exx + eyy) + (exx - eyy) cos 20) (3.33)
0

iii) the continuity of tangential tractions (ar0)r-a - (ar0)r-a:

n21 [tAl An + tBl Bn-l + tBZ Bn + talan + tfil fln-l + tflZ fin

_ P
B ] sin n0 - - __
n

Co

H

- tA1 An - tB1 Bn-l - t82 sin 20 (3.34)

M

iv) and the condition of vanishing shear tractions given by either

(ar0)r-a- 0 :

n21 [ CA1 An + ‘31 an-1 + :32 an] sin n0 - o (3.35)

) - O.

or a
( r0 r-a

The problem of perfectly bonded inclusion can be solved by using the
first three equations, (3.34)-(3.36), and the last one, (3.37), is

replaced by the condition of continuity of tangential displacements

(u0)r-a- (“0)r-a :

E B + l a

n-l [‘ 1A1 An + 131 Bn-1 + 132 n a1 n + 151 5n-1 + 152 5n}

44

- 1 { I A + I B + I B }] sin n0 - - p0 a sin 20
- Al n Bl n-l B2 n -— -
P CO 2

+ 2f: ( c* - e* ) sin 29 (3.36)
CO 2 yy xx

where F - 5 / G, and G and E are the shear moduli of the matrix and

inclusion, respectively, and

 

 

 

 

 

 

 

 

 

 

 

-2n ~(2n-l+n) -(2n+l+n)
I‘M" 2+1’ 1‘1”" “—‘2—1" REV—7.11— ’
a n 2a n- 2a n
(2n-1-Ic)a2n-1 (2n+l-Ic)a2n+1
k - 2na2n'1 k - k -
a1 ’ 31 2 ’ B2 2
(3.37)
-2n ~(2n-1-n) -(2n+l+x)
lAl' ’ 1131' —— ' 1132 '———'
2n+l 2n-l n+1
a 2a 2a
1 _ (-2n) a2n-1 1 _ (n-2n+1)s2“'1 1 _ -(x+2n+1)a2n+1
a1 ’ £1 ’ 62
2 2
(3.38)
2n(2n+l) (2n-1)(2n+2) (2n+1)(2n+1+n)
s -_9 S - 9 s - 9
A1 2n+2 Bl 2n 32 n +2
a 2a 2a
2n-2 2n
3 _ 2n(2n-1)a2n-2, s _ (2n-l)(2n-1-x) a ’ s _ (2n+l)(2n-2)a
al fll 52
2 2
(3.39)
t _ 2n(2n+1) t _ 2n(2n-1) t _ (2n+l)(2n+1+n)
A1 —’ B1 —’ 82

a2n+2 232“ 2a 2n+2

45

 

 

2n-2 2n
t _ (-2n)(2n-1)a2n-2, t _ (1-2n)(2n-1-u) a ’ t _ -(1+2n)2n a
a1 fil 2 flz 2
(3.40)
Note that the terms involving barred quantities k k I ... can be

Al’ 31' 'Al’

obtained directly from k by replacing x with E.

al’ kp1' 1a1"°'

Note that the equations (3.32)-(3.36) are same as the corresponding
equations in Chapter 2 but equations (3.37)-(3.40) differ from the ones
in Chapter 2. Equating the coefficients of sin no and cos no in the
left and right sides of either equations (3.34)-(3.37) for the sliding
case, or (3.34)-(3.36) and (3.38) for the perfect bonding case, an

, K

infinite set of algebraic equations for the unknown constants An’ Bn n

and Bu is obtained.

RESUEIS AND DISCUSSION

In this chapter the plane strain case is considered for numerical
examples. u and 5 are chosen to be 0.3. Figs. 9-10 show the stresses
ayy at points M, N and N' (or P and P') for both perfectly bonded case
(dashed line) and pure sliding (solid line) case as the radius "a" of
the inclusion is varied from 0.2 to 0.9. Comparison with the half-plane
case given in Chapter 2 is included. As expected, the tendency of
stress distribution is quite similar for both cases. For details,
compare Figs. 2-3 with Figs. 9-10.

Fig. 11 shows that the stresses axx’ ayy along the x axis for
eigenstrain loading ex:- Ez: , ey;- 2 6x: when P - 100 and a)- 0.8.

Solid lines represent the case when an inclusion is located near the

free surface, the point lines are for the case when the inclusion is in

46
the infinite strip, while the dashed lines correspond to the case, when
the inclusion is embedded in the infinite medium. Note that the
stresses in the half plane fall between the solutions for the infinite
plane case and strip case as expected.

Next, the behavior of displacements on the free surfaces is
investigated in Figs. 14-21. Figs. 14-15 illustrate the displacements
on the free surfaces for perfect bonding and pure sliding cases for
different positions of the inclusion relative to the free surface. The
radius "a" is varied as 0.1, 0.5, and 0.9, while the center of the
inclusion remains at the center of the strip. The loading condition is
the remote uniaxial loading ayy- p0. The soft inclusion (P - 0.01) has
a more pronounced effect on the displacements of the free surfaces.
That is simply because the softer inclusion can not sustain a uniaxial
loading and it deforms more than the stiffer inclusion. When the
inclusion is stiff (F - 100), the displacements for perfectly bonded
case are less pronounced than those for pure sliding case. It means
that pure sliding boundary condition actually helps the matrix to deform
more freely so that the displacements on the free surfaces are much
larger, comparatively.

The case of the eigenstrain loading (Figs. 16-21) has different
features. The stiffer inclusion has more effect on the displacements
along the free surfaces. The boundary condition does not contribute
much unless the size of inclusion is large enough such as a - 0.9. It
is interesting to observe that both boundary conditions have a very
similar influence on the deformation along the free surfaces under any

type of eigenstrain loading.

47

3.5.... 3.3.3.5 to 5 $934505" uoHaouHo

 

 

a .mE

 

lpo

Stress Concentration Factor 0

48

 

2.00
1 .75-
1 .50 -

1 .25-

73’

1.00-

P-IOO

0.75 " I,

0.50- I

0.25-

0.00-

-O.25- N'

-0.50-

-0.75-

 

 

 

 

 

-1.00

Fig. 9

1.0 2.0 3.0 4.0
1/ a

5.0

0 vs. l/a for a uniaxial loading ayy - po when P - 100 for

YY
for perfect bonding (dashed lines) and pure sliding

(solid lines) for an infinite strip case.

49

 

3.5

i?
fix}!

/ 2G6

YY

 

 

Stress Concentration Factor 0

 

 

 

 

 

1.0 2.0 3.0 4.0 5.0
1 / a * *
*
Fig. 10 0 vs. l/a for an eigenstrain loading 5 - e - 6 when
YY xx YY 22

F - 100 for perfect bonding (dashed lines) and pure sliding

(solid lines) for an infinite strip case.

50

 

 

 

 

 

 

 

 

4.0
_ a perfect bonding
300‘
2.0-
a: £5 *
6‘3
N 1.0-
‘\~ .
tn 0.0-
U}
G)
H «II
JJ .
V3 -1.o-
-2.0-
-300 0 l7 U r j I I ' T 1
-1.0 -0.6 -0.2 0.2 0.6 1.0 1.4
distance K
Fig. 11 axx and ayy along x axis for eigenstain loading
* * * *
(xx = 522. eyy= 2 exx . Solid lines represent the

case when an inclusion is located near the free surface
(a-0.8), point lines represent the case when an inclusion is
embedded in the strip, while dashed lines correspond to the

case, when the inclusion is embedded in the infinite medium.

51.

.omuo mfiuum ouacamca so you Amoco wcuofiam ousnv n.o I m can 00H I A

can? wsdvooa Gaouumcomao you ooomuouaa xwuuoa-:0unaao:«

on» wcoao _§s_ ucoaoooaanap Hoauaomcou onu :H mash NH .wam
e menm
.0“: Am do do .om .o

am: .0”: _

.0“: dw—

— —

 

CD
‘?

 

 

 

 

. wfiwvwfim oHDm
. _ . _ b _ p _ . — s _ r _ s _ . O _‘ .O

 

 

 

52

.owuum coasamca so you Ammoo weapaon uoouuoav

 

ooH I p cons wcapooa :chumsowfio you c .m> can mmouum umonm ma .mfim
e 0..”me
63-09 .o: .02. do? .8 .oo .0 .ON .0
. — . _ p _ . — . — . — . — p _ - noell

 

*

0

NH
0

*.

 

yawn—Eon uoomnom

. _

b

p

r

-

[F.Oul

 

 

53

AH- I xv.ooa I p cosh ammo mason ouaaamaw so you wcucooa

K

Houxowcs a now oomwusm ooum ofiu macaw : uaoaoocHAmHo «a .wam
moo—EEG
0; ad md .\..o md 0.0 To nd Nd to o.

 

sins :s

o

. x (((((( «Win45. o

 

 

 

 

 

\.\ IIIHIP.nV

\ \\\|

\\\ \\\

(x \\ m. Hm

‘\\ \ o
\
\
\
\\\\\
\\
\\\\ Illl..l..l.l||lll..ll.llll

.1‘ In‘ lli‘ 1000

 

0.0

 

°d / m moz

54

AH- I xvao.o I p cons omoo mason magnumca so you weavmoa

Housman: a you ooouuzm comm on» waoac x: unoaoooflnman ma .wam

m 353%:

0; one who no so. who one mo who so o.o.

 

 

 

 

 

 

 

 

 

H c r 0
xx
\‘ .w
msus ..
... z
. D
I m
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I x
III:
l. “mu
Iok.
3.ch . -.
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55

AH- I xv .ooa I p cons mason my“:

xx x
MN» I mm» I o How ooomuau ooum on» macaw : u
s s «

“wad no you

coaoooaemao 0H .wwm

m 8.53.5

0.? ad md Nd md md cud nd Nd —.o 0.0
_ _ _ _ _ _ _ _ _ .

 

dank

 

 

o “HRH“.II ilmw.wll
(@\L,.---

 

_'

 

 

 

 

*“3 mDZ/ x“ mDZ

56

h» an xx

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new ad- I xv oomwusm ovum on» muons x: uaoaoooaamwa NH .wwm
woos—Sm:—
04 md md Nd ad ad .vd nd Nd Fd od
_ _ _ F p _ _ _ p» mu.wuu

 

ecfiuh
mduownw
finsiw \

 

 

 

l
“

j

 

 

 

 

*“3 mflz/ ‘11 «oz

57‘

xx NN hx

omoo Qwuum ouficamca so you s u N I w I u

s s

X

you AH- I xv ooowuam ooum any wcoHu : uaofioooanuwa

.m mos—«Em—

oé 0.0 0.0 #0 0.0 0.0 #0 n0 Nd
_ _ _ c _ _ _ _

0H

F

.mE

_

.o 0.0 .

hol

 

81c
anssw

 

new”.--

 

\

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nII‘III-l

  

70.0I

10.0!

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find!

 

 

Ndl

1 _..Ol

 

 

0.0

 

*“3 woz/ ‘11 mm

AH- I xv .Ho.0 I 0 cos: nmuuo oumcmmcm so mom

an» xx» I.xxu uom ooomHSo ooum on» mcoao x5 usoaooomamma ma .wmm

.. .. .. n» 353me

0.? 0.0 0.0 Nd 0.0 0.0 .vd Md Nd Pd 0.0
_ t _ _ _ m _ _ .

_ ”OI

 

Sdnh

”*nnw "scam

 

58

 

 

 

 

 

 

 

 

*“3 mnz/ ‘11 moz

59

0.?

h» an

.H0.0 I h cogs omoo amuum oumcmmCm to mom u N I

VII .0

 

« s s
mom Am- I xv ooomuoo ooum on» wcoao x: uaoaoooanowa 0N .MHK
moo—EEG
0d 0d 5.0 0d 0.0 .vd nd Nd ..d 0.0
_ _ _ _ _ _ _ _ _ mwAUII

mcdnh
mdussw
muoniw .

  

 

 

 

 

 

 

 

*“3 "IDZ/ x“ wDZ

60

0.—

xx NM FA

40.0 I 0 sons omoo own—um oumammam so mom u N I u I u

.1 s 1..

 

 

 

mom 3.- I xv ooomudm ooum on”. wcomo x: ucoaoooHn—omo MN .0?“
m 3.335
0.0 0.0 5d 0d 0.0 .nd nd Nd Pd 0.0
_ _ _ _ _ _ — _ _ ON.O|
3.0”;
. a
flux m 19.0..

mdnownnw

 

 

..Edn

(mod:

 

00.0

 

*“3 mDZ/ x“ mDZ

CHAPTER 4 : A.Seli—Infinite Bi-sterial Strip

In the last part of this dissertation, the two dimensional
elasticity problem of perfectly bonded semi-infinite bimaterial strips
that are subjected to a constant temperature change is also considered.
Physically, this geometry may represent a bimaterial thermostat used in
electronic packaging. The strips are assumed to be linearly elastic and
isotropic, and have different elastic and thermal properties.

We follow the procedure used by Bogy (1968), Blanchard and Ghoniem
(1989) and Kuo (1989). We employ the Airy stress function approach
along with Mellin transform and apply the residue theorem to the
inversion integral in order to solve for the stress field at the
interface when the two bonded quarter planes are subjected to a constant
temperature change. Then this result is transformed by using the
function 2 - f(§) defining the Schwarz-Christoffel transformation. This
transformation maps the configuration of two quarter planes into the
desired geometry of two semi-infinite strips. In order to satisfy the
traction free boundary conditions we employ the iterative process used
by Kuo (1989). Then the asymptotic behavior of stresses near the edge at

the interface of two perfectly bonded strips is obtained.

61

62

METHOD OF SOUUTION

Consider the elasticity solution for the problem when the two
perfectly bonded semi-infinite strips are subjected to a constant
temperature change. In the notation used, the upper strip is denoted by
double prime, while the lower strip is described by single prime. The
problem considered is the plane elasticity problem. The solution is

formulated in terms of polar coordinates.

1. Solution for the two elastic bonded quarter planes

Initially, we solve the problem of two perfectly bonded quarter
planes (see Fig. 22). The method of solution for this part follows the
papers by Bogy (1968) and Blanchard and Ghoniem (1989). We use the Airy

stress function method. The governing equation is:
v4¢ - o (4.1)

The boundary conditions for the problem of two perfectly bonded quarter
planes are given as follows:

a) the traction free conditions:

I I a

000 - are - 0 at 0 - - 2
(4.2)

II ‘ I! l

000 - are 0 at 9 - + 2

b) the continuity conditions at the interface (0 - 0):

I II

”90 ' ”so

63

- arfl (4.3)

Next, we use Mellin transforms (Titchmarsh, 1948; Tranter, 1966) to
simplify the solution process. The transforms for ¢. aij and ui are

defined as:

3(S.9) - I” ¢(r,0) rs'ldr
0

A

s+l
aij(s,0) - I: aij(r,0)r dr (4.4)

ni(s,a) - I: ui(r,0)rsdr

The governing differential equation (4.1) is transformed to:

a“ 2‘ .
9—¢ + [(s+2)2 + $2] g_¢ + 52(s+2)2¢ -o (4.5)

d0“ d02

A

and the solution ¢ is given as:

180 -180 + Bei(s+2)0 - e-i(s+2)0 (4.6)

¢(s,0) - A e + B
The expressions for the transformed stresses and displacements in terms

of o are:

A
A

aro(s,0) - (5+1)d0

A A

aoo(s,0) - (s+1)¢ (4.7)

64

 

A A 2 A
_ __l__. , ms , m d. _ ..nfl;__
ur(8.0) 2y(s+1) {s(l+s 4 ) ¢(S.0) 4 d02 ¢(S.9)} 8p(s+1)
“ 1 2 as. 2 d. ‘
uo(s,0) - 2p(s+l)(s+2) {[s+s - 4 -2(s+l) ] d0 ¢(s,0)-
m d3. ‘
¢ (5.0))
4 d03
where
_Q_
1+” for plane stress
m - (4.8)
4(l-v) for plane strain,
and
T - n E a AT (4.9)

and p, E, a, and u are the shear modulus, Young's modulus, the

coefficient of thermal expansion, and Poisson's ratio, respectively, and

l for plane stress
n - (4.10)
l+u for plane strain
The quantities, given in (4.7), are substituted in the boundary
conditions (4.2)-(4.3). The traction free condition are + i ”a; - 0

at 0 - - g leads to

A' - % [B'(s+1) - B' sis”) (4.11)

65

and the condition are + i 000 - 0 at 0 - + g gives

A" - i [B"(s+1) - E" c‘is“] (4.12)

It is convenient to rewrite the constants B', B", B' and B" in terms of

real and imaginary components:

8' - 2E + 2Fi B" - 20 + 2H1 (4.13)

3' - 23 - 2Fi E" - 26 - 2H1

Using the relations (4.13), the continuity conditions at 0 - 0:

A I A

i) uo(s,0) - u0"(s,0) gives

08 +(A-m')F + knG + k(-A+m")H - 0 (4.14)

where

I?

k - p’/u", E - 2 , n - - sin St, A - 2s + 2 + 2 c0526

"I "II

ii) ur (3,0) - ur (5.0) yields

 

' - - n - mI-rr _ @"T"
(1 + m ) E + 0F + k( 7 m ) G + k n H l6(s+l) (4.15)

where

7 - 2s + 2 - 2 coszé

"I "I "II AII

iii) 0 r9 + i099 - Ora + i 009 gives

-nE - AF - no + AH - o (4.16)

66

7E + 0F - 76 + 0H - 0 (4.17)

Combining equations (4.14)-(4.l7), the following system of four

simultaneous linear equations is obtained:

 

 

F' o "
1 n -7 n E 0 (4 18)
'I A 'I ”A F - I I_ n -
7+m n -k(1+m') kn G EI§(;E¥)I
n A-m' kn -k(A-m") H 0

The determinant of the above matrix is given by

2 2

3 n + (k - 1>2(1A - n2)2 - kg 1) + k2(k-l)(A-7)(1A - n2) (4.19)

"X" - k
where
k - km" - m', k - km" + m' (4.20)

The solution for the unknown constants E, F, G and H is

E " [7* k2 + "2 k3 +<k-1>Hn2-m1 [W] flifl

F - [n(vA-nz)(k-1) - 2k1nm"] [éifi;s+1;r"] “§“ (4.21)

 

H - [n(k-l)(n2-1*) ‘27"m ] [ i6(;+1; "J “in

67

G - [-q2k3 + 1Ak2 + (k-1>A(v2'7*)1[ 16(s+1; "] Hi“

Therefore, using equations (4.6), (4.7), (4.11)-(4.l3) and (4.21), the

transformed stresses at the interface (0 - 0) are given as:

or;(s,0) - - F 3(s+1)2 + E 4(s+1) sin sz - F 4(s+l)(cos sn + 1)

09;(s,0) - E 3(s+1)2 - E 4(s+1) (cos sx + 1) F - 4(s+1) sin St
(4.22)

ar;(s,0) - - H 8(s+l)2 - H 4(s+1) (cos sx + l) - G 4(s+1) sin sw

" II

000(s,0) - c 8(s+1)2 - c 4(s+1) (cos sx + 1) + a 4(s+1) sin sn

To recover the stresses in the real space we use Mellin's inversion

integral (Titchmarsh, 1948) defined as

+1” A
_ 1 -(s+2)
aij(r,9) 3;; lc_im aij(s,0)r ds (4.23)

We perform the contour integration of (4.23) by using Cauchy's residue
theorem (Marshen, 1973) and we obtain

-(s+2)]

aij(r,0) - ligl [(s-sl) aij (s,0)r (4.24)

where s1 is the root of the determinant "X" and it depends on the
combination of elastic constants.

Using (4.24), the stresses at the interface are given as

68
ar;(r,0)0_o - ar;(r,0)o_o - [-(2s + 3 + cos sn) (n(yA - "2)(k-1) -

2kynm") + [1Ak2 + n2k3 + A(k-l)(n2 - 1A)}sin sx]
s-s1

'(51+2) I I I I n n n n
[n m E a - k n m E a ]

 

g. "X" r
4 ds S-sl

000(r’0)0-0 - 000(r,0)9_0 - [(25 + l - cos sx){A(02 - 1A)(k - l) + 1Ak2

+nzk3} + {(k-1)(n3 - n1A) + 2k1nm"} sin SII]S_S

'(sl+2) I I I I n n n n
r [n m E a - k n m E a ] (4.25)

 

g. "X"
4 ds s-sl

2

' 2 2
arr(r’o)0-0 - [-(2s + 5 - cos sn) {1A k2 + n k3 + (k-l)(An - 1A )} +

sin sn((07A - n3)(k-1) - 2kvnm )]S_S
1

“(5 +2) I I I I n u n n
r [n m E a - k n m E a ]

 

T
s. “X" -
4 ds 3 s1

.. 2 2 2
arr(r,0)o_o - [-(25 + 5 - cos s«){- 0 k3 + 7Ak2 + (k-1)(An - 7A )} +

sin St {(k-1)(IAfl ' "3) + 27'7m'Hs-s
1

'(S+2) IIII IIIIIIII

d Afix" r 1 [n m E a - k n m E a ]
4 -— s-s
1

ds

 

Note that the stresses at the interface, given in (4.25), are of form:

-(sl+2)
- C r (4.26)

”ij(r'9)o-o ij
Recall that 51 is the root of the determinant (4.19), which lies in the
interval -2 < 51 < -1. It is shown by Bogy (1968), that when k2[4(k-1)-
k2] > 0, there exists one real root in that interval. This case will

give us the algebraic singularity of form r-A. This situation is of

69
interest in this paper. In general, the problems of this nature could
have stresses of forms r'A, r0 or 1n r. For the more complete
discussion on this topic see papers by Bogy (1968, 1970), Dundurs

(1969), and Blanchard and Ghoniem (1989).

2. Solution for the two elastic bonded semi-infinite strips

Next, we want to use the solution of two quarter planes to obtain
the desired solution for the two semi-infinite strips. To accomplish
this we use a conformal mapping (Nehari, 1952). The Schwarz-Christoffel
transformation for mapping the half-plane (Re(z)>0) into the interior of
semi-infinite strip (-h1<Re(§)<h2, Im(§)>0), as seen in Fig. 22, is

derived as

12
g - o —¢‘-— - o sin'1 is + p (4.27)
o /1_22

 

The constants c and B are determined by using

 

C - - hl’ Z - + i (4.28)
g - + h2, z - - i
Therefore,
h -h
n 1 2
z - - i sin { h1+h2 (g + 2 )} - w(§) (4.29)
10 i¢

where z - r e and g - p e
Next, we use the relations between the original and transformed

stresses (Muskhelishvili, 1953)

70

a +0¢¢-0 +099

a¢¢ - a + 2i ap¢ - (000 - a + 2i or
where
2 I
e210 _ 55 w 5;)
p 5 (C)

Using (4.29),

' 9w. :11_ _1I_
” (§) ' d; ' h1+h2 °°S‘n1+h2(‘ +

and

 

 

h -h
2 °°s‘ h :h (5 + ‘12_2 )’
210 §_ 1 2
e ' 2 (‘1) h -h
P COS{ J— (E + L2 )}

(4.30)

(4.31)‘

(4.32)

(4.33)

(4.34)

At the interface of the two semi-infinite strips, where ¢ - g , for the

case of h - h - h, equation (4.34) reduces to

1 2

(4.35)

71

Therefore, the relations (4 30) become
a + a - a + 000 (4.36)
- app + 210p¢ - 099 - a + 2iaro

Also, at the interface where ¢ - g

r - -i sin (fig 1) - - 12 sinh(¥ h) - sinh(¥ h) (4.37)

Therefore, the stresses at the interface of the two semi-infinite

elastic strips are:

0p;(p,§) - (n'm'E'a' - k n m E a )[7An(2km -k2 ) ' 0 2k3 315,51
-(sl+2)
or swim] (4.38)

__ flXfls_ 2h
4 ds 31

 

II I I I I II II II II 2
op¢(p,§) - (n m E o - k n m E o )[1An(2m +k 2) - n k3]5=51
' -(s1+2)
AT 12
a - "HS_ -8 sinh 211 (4.39)
ds 1

 

a¢¢(p, ,2) - (n m E a - k n"n"E"o")[2n2 1A(k- 1) - 7 22A (k- 1)
+ 72Ak2 + 702k3 - n 4(k-1) - 2702 (k3 + m )]
5351
T -(sl+2)
4 d_A"X"S 5-5 Sinh[§fi (4.40)
ds 1

II II II 2

- (n m E o' - k n m E o )[20 21A(k 1) - 7 22A (k 1) + 7 Ak

II E
a¢¢(p.2) 2

72
-(sl+2)

 

2 4 2 ' AT 12
- 77) k3 - ’7 (k-l) + 2717 111 13-5 i. "X" sinh 2h
1 4 ds s-s1

(4.41)

 

 

I E - I I I I - II II II II - - 2
app(p,2) (n m E a k n m E a )[( 23 5+cos sn) {1Ak2+n R3
2 2 3 "
+ (k-1)(An - 1A )} + sin sx {(07A - n )(k-l) - 2k1nm )]S_S
l
-(sl+2)
AT 12
d— "xlls-s sinh 2h (4.42)
ds 1 ‘
II E I I I I II II II II 2
app(p,2) - (n m E a - k n m E a )[(-25-5+cos SE) [-n k3 + 1Ak2
2 2 3 '
+ (k-1)(n A-VA )} + sin sx {(k-1)(vnk - n ) +21nm HS”s
1
-(sl+2)
AT . £2
4.d_ "X" Slnh[2h (4.43)
s-s
ds 1

Clearly, the shearing stresses in (4.38)-(4.39) and peeling stresses in
(4.40)-(4.41) satisfy continuity conditions at the interface.
So, the stresses at the interface of the two semi-infinite strips,

given in (4.38)-(4.43), are of form

-(sl+2)
E _ 12
oij(P.2) cij sinh[2h] (4.44)
When p 4 O,
- £2 ~ :2
51nh[2h] ~ 2h (4.45)

Therefore, the stresses are expressed as

73
2-(sl+2)

fl
Oij(P.2) ' Cij h (4-45)

Again, when -2 < 31 < -1 , the stresses are singular as p 4 0.
After the transformation of shape from two quarter planes to two

semi-infinite strips we need to check the tractions on the boundary (see

Fig. 22):
Zia
a) at v - O and -h S u s 0, e - -1. Therefore,
a" - o, a" - a" , a" - o (4.47)
uv uu yy vv

b) at v - O and O s u S h, e - -l

a - 0, o - a , a - 0 (4.48)

c) at u - -h and v > O

 

 

 

II II hv II II v2 II II h2
auv - - ayy h2 + v2 ' auu - ayy h2 + v2 ’ avv - ayy h2 + v2 (4'49)
d) at u - h and v > 0
I o 11" I v V2 p , h2
, a - (4.50)

a - a , a - a a -—-————-
uv yy h2 + v2 uu yy hZ + v2 vv yy h2 + v2
The stresses on the lateral surfaces, given in equations (4.49)-(4.50),
do not vanish, as specified by traction free boundary conditions.

Therefore, we need to cancel these tractions by applying the additional

loading. Therefore, these additional stresses are, at u - -h,

 

 

 

 

-(s +2)
Lv 1
a" - - C (cosh— 2h ) hv
uv II h2 + v2
-(s +2)
I! 1 2
a" - C (cosh 2h ) v
uu II h2 + v2
and at u - h
-(s +2)
Lv 1
a, - c (cosh— 2h ) hv
uv I h2 + v2
-(s +2)
5! 1
a, - c (cosh 2h ) v2
uu I h2 + v2

 

 

where

CI - {-A[s coség + (s+4)cosi§i§lfi] - B[-s sinsJ
m'T - anTn
4 g. "XII...s

CII- {- -C[s coség + (s+4)cosi§i§l£] - D[-s sins"
3 gum:

and

2

A - 7) k2 + 0 k3 + (k-l) A (n2 - 1A)

(4.51)

(4.52)

2 '31

(4.53)

2 + (5+4) sin £§¢211 ])S_ '51

75

B - n(vA-n2)<k-1> - 2k1nm"
(4.54)

C - n(k-1)(n2-1A) -27nm'
2 2
D - -n k3 + 7Ak2 + (k-I)A(n -1A)

3. Iteration procedure for the boundary adjustment

For the remaining part of the solution we follow Kuo's (1989) paper
(sections 4 and 5). Initially, we solve the problem of two infinite
strips with the loading given in equations (4.51)-(4.52), but of the
opposite sign. Superposing this solution with the previous solution of
semi-infinite strips we could have normal tractions at the transverse
edge. These tractions need to be cancelled. So the additional problem
of the two quarter planes subjected to normal tractions needs to be
solved. In this step we follow solution of Bogy (1970). This loading
in turn gives rise to the additional lateral tractions which need to be
cancelled. This procedure will be repeated until it satisfies the
convergence criterion. The details of this iteration procedure are
given in Kuo's paper and are not repeated here.

For the numerical example considered in this paper, the iteration
procedure is done only once. The remaining tractions on the lateral
surface are very small and their contribution to the stresses along the

interface is negligible.

76

RESUETS AND DISCUSSION

In the numerical example we consider a thermostat having the following

properties:
Aluminum Molybdenum
u" - 0.345 u'- 0.293
E" - 70380 MPa E'- 325000 MPa
n -6 0 r -6 0
a - 23.6 x 10 / C a - 4.9 x 10 / C

We solve the case when the thickness is much smaller than the other
dimensions. Therefore, we have plane stress case. We take h1 - h2 - h.
The stress distribution at the interface of the semi-infinite bimetallic
molybdenum/aluminum strip is illustrated in Fig. 23. In the discussion
we assume that the temperature change AT > 0. Note that the axial
stress, a;p, in the aluminum strip is compressive and a;p in the
molybdenum strip is tensile, as expected, since the thermal expansion
coefficients a" > a'. The peeling stress a;¢ - aé¢ is compressive along
the interface. Kuo (1989) solves the same thermostat problem. However,
in his paper Figs. 4 and 5, which represent the axial stress, give the
results that do not agree with ours. Also, we observe that he solves
the problem of plane strain, although the plane stress case was supposed

to be considered. Note, that the elasticity solution yields singular

stresses at the edge at the interface of two strips. The singular term

-l/(s1+2)
is of form [p/h] . For the case of molybdenum/aluminum semi-

infinite strips 5 - -l.8875. The similar geometries such as two

1

dissimilar quarter planes or wedges also yield singular stresses at the

77
interface near the edge or corner and the form of singularity depends on
the combination of elastic constants, as shown by Bogy (1968, 1970),
Dundurs (1969), Hein and Erdogan (1971), Dempsey and Sinclair (1981),
and Blanchard and Ghoniem (1989), among others.

The problem of bimaterial thermostat has received considerable
attention in literature because of its application in the electronic
packaging industry. The different solution approaches to this problem
give different stresses at the interface. For example, the elasticity
solutions by Kuo (1989) and in this dissertation give the singular
stresses at the edge at the interface as expected from theory of
elasticity. The approximate methods based on strength of materials
approach and energy methods do not give singular stresses at the edge at
the interface. For example, Suhir (1989) imposes the condition of zero
shear stress at the edge along the interface as suggested by Razaqpar
(1987). There also exists a number of finite element solutions.

However many of these solutions are incorrect as discussed by Lau
(1989). The present result for the stresses at the interface at the
edge is in agreement with the finite element solution of Lau (1989). In
this dissertation we propose another solution for this very interesting
and challenging problem. Our solution is singular as expected by theory
of elasticity. In reality, however, the material cannot sustain
infinite stresses, but the proper interpretation of these results may
increase the understanding of the stress field. Also, our solution can
give guidance to the researchers using finite element method regarding

the nature of the elements to be employed.

78

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J

.muuun

mm

 

 

 

 

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IN ofimumwv
a... h; m; n; —.— ad 5.0 m.o nd —.0

- . F‘ — p. - PII._ . — . — p» — b — . éflfiull
IN.°I.

no.0

emu
1N6
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. lvd

 

 

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I

IV .0 3 / $831113

CHAPTER 5 : GOICUUSIONS

The plane elasticity problems of a semi-infinite bimaterial strip
subjected to a temperature change and a circular inclusion in a half-
plane and an infinite strip under thermo-mechanical loading have been
investigated analytically. We used Papkovich-Neuber displacement
potentials in an infinite series form and Airy stress functions,
respectively.

The numerical examples illustrate how the joint effect of free
surface, the mismatch of material constants, the geometry, the type of
loadings, and the boundary conditions effect the stresses and
displacements.

The study of the problem when an inclusion is embedded in a half
plane or an infinite strip shows that the stiffer the inclusion (or the
larger the P ) the more pronounced is the effect of interface. The
effect of free surface contributes significantly to the stress
disturbance when the inclusion embedded close to the free surface (or
surfaces). The stresses do actually increase locally due to the sliding
boundary condition. The solution of the semi-infinite bimaterial strip

gives the singular stresses at the interface at the edge of the strip.

80

Aderogba, K., 1976, "On Eigenstresses in a Semi-infinite Solid,”
Mathematical Proceedings of the Cambridge Philosophical Society, Vol.

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Aleck, B. J., 1949, ”Thermal Stresses in a Rectangular Plate Clamped
along the Edge,“ ASME Journal of Applied Mechanics, Vol. 16, pp. 118-

122.

Blanchard, J. P. and Ghoniem, N. M., 1989, "Relaxation of Thermal
Stress Singularities in Bonded Viscoelastic Quarter Planes," ASME

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Blech, J. J. and Kantor, Y., 1984, "An Edge Problem having no
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Bogy, D. B., 1970, "On the Problem of Edge-bonded Elastic Quarter
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Bogy, D. B., 1968, "Edge-bonded Dissimilar Orthogonal Elastic Wedges

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81

82
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83
Gerstle, F. P. and Chambers, R. 8., 1987, ”Analysis of End Stresses in
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Hess, M. S., 1969, "The End Problem for a Laminated Elastic Strip- II.
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Howland, R.C.J., and Stevenson, A.C., 1933, "Bi-Harmonic Analysis in a
Perforated Strip", Philosophical Transactions of the Royal Society of

London, Series A, Vol. 232, pp. 155-222.

Hussain, M. A., and Pu, S. L., 1971, "Slip Phenomenon for a Circular

Inclusion," ASME Journal of Applied Mechanics, Vol. 38, pp. 627-633.

Ishiwata, K., 1986, "Stress Concentration due to an Elliptic Hole or a
Griffith Crack in a Semi-infinite Elastic Plate under Tension,"

Master's Dissertation, Saitama University, Japan.

84
Jasiuk, 1., Tsuchida, E., and Mura, T., 1987, "The Sliding Inclusion
Under Shear," International Journal of Solids and Structures, Vol. 23,

pp. 1373-1385.

Jeffery, G. B., 1920, ”Plane Stress and Plane Strain in Bipolar
Coordinates," Transactions of Royal Society, London, Series A, Vol.

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Keer, L. M., Dundurs, J., and Kiattikomol, K., 1973, "Separation of a
Smooth Circular Inclusion from a Matrix," International Journal of

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Knight, R.C., 1934, "On the Stresses in a Perforated Strip," Quarterly

Journal of Mathermatics, Vol. 5, pp. 255-278.

Kouris, D., and Mura, T., 1989, "The Elastic Field of a Hemispherical
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the Mechanics and Physics of Solids, Vol. 37, pp. 365-379.

Kouris, D., Tsuchida, E., and Mura, T., 1988, "An Anomaly of Sliding

Inclusions," ASME Journal of Applied Mechanics, Vol. 53, pp. 724-726.

Kouris, D., Tsuchida, E., and Mura, T., 1989, "The Hemispheroidal
Inhomogeneity at the Free Surface of an Elastic Half Space," ASME

Journal of Applied Mechanics, Vol. 56, pp. 70-76.

Kuo, A.Y., 1989, "Thermal Stresses at the Edge of Bimetallic

Thermostat," ASME Journal of Applied Mechanics, Vol. 56, pp. 585-589.

85
Lau, J. M., 1989, "A Note on the Calculation of Thermal Stresses in
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Electronic Packaging, Vol. 111, pp. 313-320.

Margetson, J., and Morland, L. W., 1970, ”Separation of Smooth Circular
Inclusions from Elastic and Viscoelastic Plates Subjected to Uniaxial
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86
Mura, T., Jasiuk, I., and Tsuchida, E., 1985, "The Stress Field of a
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87
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88
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90

APPENDIX

The displacements and stresses are related to the Papkovich-Neuber

displacement potentials ¢0 and d1 as follows:

a) in Cartesian coordinates

 

 

a¢ 8¢
-—Q —-l-
ZGux 8x + x ax n ¢1
6d 06
ZGu - ‘—Q + x'—41
7 By By
2 2
0 ¢ 0 ¢ a¢
0 ___1 1 __1
axx ' 2 + x 2 ' 2 (”+1) ax
6x 3x
2 “ 2
3.10 111 1 5’31
”yy ' 2 + x 2 ' 2 (3") ax
6y 8y
2 2
,, _a¢0,xf’_"’1_1(x_1)i‘11
xy axay axay 2 6y
b) in polar coordinates
a¢ 6¢
ZGu - 5:9 + r coso 5:1 - x c030 ¢1
3¢ a¢
_ 1 ___ __1
ZGuo r 60 + cosfi 80 + n sino pl
6.216 :21. 1 811 J.
a - + r coso - (n+1) c030 + (3-n)
rr ar2 61.2 2 6r 2

(A1)

(A2)

£122 321
r 80

 

91

2 2
36 a 6 3 ¢ 3¢
_ %.——Q + 15 -——Q + 99§£'--l + % (x-l) c059 ‘—l + l (5+1)
r

 

a
00 fit 602 r 602 at 2
11:11.21
r 30
132% 1.310 £1 1 “1 1'
”r0 ‘ r arao ' r2 00 + °°So area + 2 ("'1’ 51“” 3?" ' 2 (”+1)
601
c ___
r 80
where
2 2
v p1 - v p2 - 0
and
2 62 02 02 1 a 1 62
V -—+—-—+— —+ —— .
8x2 fly2 at2 r 6r r2 802