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1972

A comparison of analytical and experimental approaches in A comparison of analytical and experimental approaches in

determining the thermoelastic stresses around a cylindrical determining the thermoelastic stresses around a cylindrical

inclusion of elliptical cross-section inclusion of elliptical cross-section

Kenneth Byron Oster

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Recommended Citation Recommended Citation Oster, Kenneth Byron, "A comparison of analytical and experimental approaches in determining the thermoelastic stresses around a cylindrical inclusion of elliptical cross-section" (1972). Masters Theses. 5326. https://scholarsmine.mst.edu/masters_theses/5326

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A COMPARISON OF ANALYTICAL AND EXPERIMENTAL APPROACHES

IN DETERMINING THE THERMOELASTIC STRESSES AROUND A

CYLINDRICAL INCLUSION OF ELLIPTIC CROSS-SECTION

BY

KENNETH BYRON OSTER, 1935-

A THESIS

Presented to the Faculty of the Graduate School of the

UNIVERSITY OF MISSOURI-ROLLA

In Partial Fulfillment of the Requirements for the Degree

MASTER OF SCIENCE IN ENGINEERING MECHANICS

1972

Approved by

Thesis T2740 66 pages c.l

ABSTRACT

Three approaches are used to determine the principal stress

differences in an infinite elastic medium containing a cylindrical

inclusion of elliptic cross-section due to a uniform temperature

change. First, the two-dimensional equations of thermoelasticity

are solved resulting in equations for the stress components anywhere

in the medium material. Second, boundary displacements based on

stress-strain relations are used as input displacements in a finite

element program of the medium. Third, these same boundary displace­

ments are used as input displacements on a photoelastic model

simulating the medium.

The problem is treated as one of plane strain in the first

approach and one of generalized plane stress in the latter two

approaches. The latter two approaches are equivalent to a second

boundary-value problem in the theory of elasticity.

ii

Comparison is made of the resulting principal stress differences

using the above three approaches.

iii

ACKNOWLEDGEMENT

The author would like to express his appreciation to

Dr. Peter G. Hansen, Chairman of the Engineering Mechanics Department,

University of Missouri-Rolla, for the valuable advise and assistance

given during the inception and completion of this thesis.

iv

TABLE OF CONTENTS

Page

A.B s TRACT ••••••••••••••••••••••••••••••••••••••••••••• I •••••••••.•••• ii

ACKNOWLEDGEMENT •••••••••••••••••••••••••••••••••••••••••••••••••••• iii

LIST OF ILLUSTRATIONS •••••••••••••••••••••••••••••••••••••••••••••• v

LIST OF TABLES ••••••••••••••••••••••••••••••••••••••••••••••••••••• vii

LIST OF SY~OLS •••••••••••••••••••••••.•••••••••••••••••••••••••••• viii

I. INTRO DUCT ION ••••••••••••••••••••••••••••••••••••••••••••••

A. Review of Literature ••.................••.............

B. Theoretical and Experimental Considerations .......... .

c. Use of Elliptic Coordinates ...•.....•.•.....•...••....

II. THEORY OF ELASTICITY •••..••.•......• , , . , . , •• , . , •.• , , . , ..•.

A. Stresses in Medillm .................................... .

B. Boundary Displacements.

III. FINITE ELEJ.vlENT METHOD •••••••••••••••••••••••••••••••••••••

IV. EXPERIMENTAL ANALYSIS •••••••••••••••••••••••••••••••••••••

v. CO~ ARISON OF RESULTS •••••••••••••••••••••••••••••••••••••

VI. CONCLUSIONS •••••••••••••••••••••••••••••••••••••••••••••••

1

1

2

6

11

11

19

25

30

42

54

BIBLIOGRAPHY. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • 56

VITA............................................................... 57

LIST OF ILLUSTRATIONS

Figure

1.

2.

3.

Quartz Inclusions in Porcelain After Firing .....••...•.•.....•

Diagram of Coordinate Sys tern Used .•••.•..........•••••••....•.

Loci of Principal Stress Differences in Medium Based on

v

Page

4

7

Theory of Elasticity. . • • . • . . . . . • • . • . . . . . • • . • . • . . . . . . . . . • • • • . . • 17

4. Loci of Principal Stress Differences at Interface Near Major

Axis Based on Theory of Elasticity............................ 18

5. Sign Convention of Rectangular and Elliptic Displacement

Components.................................................... 23

6.

7.

8.

9.

Finite Element Grid Used ..•••.•..•...•....•....•...•..••......

Finite Element Grid Near Inclusion ........••.........••..•..••

Finite Element Grid Near Small Radius ..•....................••

Calibration Specimen ......................................... .

10. Loading Method to Determine the Modulus of Elasticity of the

Polyurett1ane ................................................. .

11. Photoelastic Fringes Around Inclusion Using a Light Field •....

12. Photoelastic Fringes Around End of Inclusion Using a Light

Field ........................................................ .

13. Photoelastic Fringes Around Inclusion Using a Dark Field ..... .

14. Photoelastic Fringes in Quadrant Containing Centroids of

Finite Elements Used for Comparison (Light Field) ............ .

15. Principal Stress Difference in Medium Along Major Axis .•...•..

16. Principal Stress Difference at Centroid of Elements Adjacent

27

28

29

31

33

36

37

38

39

43

to Interface (1-23).................................... . . . . . . . 44

17. Principal Stress Difference at Centroid of Elements 70-92 ••... 45

vi

Figure Page

18. Principal Stress Difference at Centroid of Elements 173-184 •.•. 46

19. Principal Stress Difference at Centroid of Elements 197-208 •••. 47

20. Principal Stress Difference at Centroid of Elements 22 7-232 •.•• 47

21. Principal Stress Difference at Centroid of Elements 233-238 •••. 48

22. Principal Stress Difference at Centroid of Elements 239-244 •••• 48

23. Isoclinic Patterns and Stress Trajectories Based on

Photoelastic Me thad . ........................................... 50

24. Isoclinic Patterns and Stress Trajectories Based on Theory

of Elasticity.................................................. 51

25. Traction Stresses at Interface Between Inclusion and Medium

Based on Theory of Elasticity ..•.•..••.••..••••.••••.•.•.•.•••• 53

vii

LIST OF TABLES

Page

I. Properties of Materials Used •.•••..•...•..•....•.•...•.•.•...• 34

II. Principal Stress Differences at Centroids of Elements

Located on Pho toe las tic Model •.•...•.•...••.••••.•..•..•••.•.• 40

III. Principal Stress Differences at Locations Along the Major

Axes of Photoelastic Model.................................... 41

a

b

c

E

f

G

J

k

K

N

p

q

t

T

u

u~

u n

v

x,y

viii

LIST OF SY~1BOLS

Major semiaxis of an ellipse

Minor semiaxis of an ellipse

Half distance between foci of ellipse

Modulus of elasticity

Material fringe value

Modulus of elasticity in shear

Stretch ratio of the transformation from cartesian to

elliptic coordinates

Constant = a"T

Physical constant

Fringe order

Maximum principal stress

Minimum principal stress

Model thickness

Temperature above uniform initial temperature

Component of displacement in the x-direction

Component of displacement in the ~-direction

Component of displacement in the n-direction

Component of displacement in the y-direction

Cartesian coordinates

Coefficient of thermal expansion, angle between the principal

stress direction at a point and the ~-axis through the same

point

tanh E;, = b /a 0 0 0

E" (a."-a' )T

E z

s 8

v

~,n

0 z

0~

0 n

T~n

¢

X

Unit normal strain in the z-direction

Function of the complex quantity ~+in

Angle between principal stress axis at a point and the

x-axis

Poisson's ratio

Elliptic coordinates

Unit tensile stress in the z-direction

Unit tensile stress in the ~-direction

Unit tensile stress in the n-direction

Unit shear stress

Inclination of the curve n;constant to the x-axis

Airy's stress function

ix

The quantities associated with the inclusion are identified by

primes while those associated with the medium are identified by double

primes.

The quantities associated with the interface between the inclu­

sion and the medium are identified with a zero subscript.

1

I. INTRODUCTION

Stress concentrations in elastic bodies due to material discon­

tinuities as a result of inclusions have been the concern of many

investigators. Of special interest is the stress concentration caused

by an inclusion having an elliptic cross-sectional geometry. This

type of inclusion cross-section simulates the two-dimensional Griffith

crack used in fracture mechanics as discussed by Tetelman [1]*. It

also approximates the general shape of inclusions found in most

engineering materials such as metals, ceramics, etc. Adequate means

for determining the stress conditions that will result in material

failure for single and multiple inclusions when subjected to different

loading and temperature conditions is the concern oftllls investigator.

A. Review of Literature

Investigators performing studies to determine the stresses

around inclusions having an elliptic cross-section have been primarily

concentrating on problems where the medium is subjected to a uniform

tension force, compression force or shear couple acting at infinity

or at a great distance from an inclusion. Donnell [2] derived

theoretical equations by the Airy stress function technique for the

stresses in a plate containing an elliptical discontinuity with

stiffness k times that of the plate for tensile loads and for shear

loads applied at the edge of the plate.

Hardiman [3] used a method which consists of finding the complex

potentials inside and outside the inclusion which: (1) give the

appropriate stresses at infinity or in the inclusion, and (2) give

* denotes references in the bibliography

2

continuity of displacement and mean stresses across the interface.

Complex constants were determined for simple tension at any orienta­

tion, all-round tension, and simple shear, all applied at infinity.

Also, complex constants were determined for simple tension and all­

round tension in the inclusion. Symm [4] derived expressions for the

stresses at points both inside and outside of an elliptic inclusion

in an elastic plate from complex potentials of Hardiman for load

cases of simple tension, all-round tension and simple shear.

Chen [5] made a two-dimensional study of an elliptic elastic

inclusion embedded in an anisotropic medium where the medium is sub­

jected to a uniform stress or couple at infinity.

Mindlin and Cooper [6] considered the thermoelastic stresses

due to a uniform temperature change of an elastic medium containing

a cylindrical inclusion of elliptic cross-section. They determined

by the Airy stress function technique the components of stress at

the interface between the inclusion and medium. Equations for the

location along the boundary of the stationary values of p, q, and

p - q were derived. Also, equations for the maximum traction across

the interface, the homogeneous stress throughout the inclusion and the

axial stress in the inclusion and along the interface were determined.

B. Theoretical and Experimental Considerations

It is noted that studies have not been performed on problems

of elliptical inclusions using other analytical methods such as the

finite element or experimental methods such as the use of photoelasticity.

The study here attempts to compare the results that would be obtained

from both a finite element and a photoelasticity approach with the

3

results obtained from the theory of elasticity. The problem considered

here is one of uniform heating, or cooling, of a body composed of

an infinite elastic medium in which is embedded an infinitely long cylinder

of elliptic cross-section which has not only a different coefficient of

thermal expansion but also has different constants of elasticity from

the medium material. Therefore, the purpose of this study is to

derive the thermoelastic stresses (after Mindlin and Cooper) in the

medium around a cylindrical inclusion of elliptic cross-section due

to a uniform temperature change and to compare the results with

those obtained from a finite element program and a photoelasticity

study.

A practical example as to the results of thermoelastic stresses

around inclusions approaching an elliptic cross-section is shown

in Figure 1. It shows quartz inclusions in porcelain after firing.

Due to the high thermal stress around the inclusions, microcracks

can be noted in the glassy matrix material. An adequate means by

which the location of the critical stresses initiating these micro­

cracks can be found for any elliptic cross-section and combination

of material properties is the concern of this investigator.

Since the difference of principal stresses is proportional

to one of the simplest measurements that can be made with the

photoelastic technique, it is used as a means of comparing the three

approaches. To make the comparison independent of the change in

temperature and in the difference in the coefficient of thermal ex­

pansion between the medium and inclusion material, the difference of

4

Figure 1. Quartz Inclusions tn Forcelain After Firing

5

principal stress is divided by the absolute value of 6 [6:::E 11 (a''-a')T).

The resulting quotient is dimensionless. The parameter o is a function

of the temperature change, the difference in coefficients of thermal

expansion and the modulus of elasticity of the surrounding medium.

The elasticity problem is treated as one of plane strain in

elliptical coordinates which reduces to the determination of a~, an'

and T~n as functions of~ and n only. Boresi [7, page 137) has shown

that the governing differential equation for the stress function

for plane strain and generalized plane stress is the same where body

forces are absent. Therefore, the use of generalized plane stress

conditions in the other two approaches is possible for comparison

purposes.

The finite element method and the photoelastic technique used

are equivalent to a second boundary-value problem in elasticity.

The stresses in the interior of the elastic medium which is in equilib­

rium are determined from the displacements of the interface between

the medium and inclusion. These displacements are determined from

displacement equations for the boundary as derived from stress-strain

relations from the theory of elasticity. A check as to the validity

of these displacement equations are made by assuming an inclusion

infinitely rigid as compared to the medium material which has a

coefficient of thermal expansion equal to zero. This results in

the free expansion of the inclusion in the ~.n plane due to a uniform

change in temperature.

The geometry and elastic properties of the inclusion and medium

are kept constant in comparing the three approaches. The ratio of

the minor and major semiaxes of the boundary, b /a , is assumed to 0 0

be 0.20 where b =0.20 in. and a =1.0 in. This initial geometry 0 0

of the boundary is based on the approximate elliptical shape of

an actual quartz inclusion as reported by Hansen [8] and the limita-

tions of the available tooling for producing the photoelastic model.

For this study the inclusion material is assumed to be aluminum

while the surrounding medium is polyurethane. Average material pro-

6

perties are assumed except for the modulus of elasticity of the medium

material which is found by calibration.

C. Use of Elliptic Coordinates

In problems dealing with curvilinear boundaries such as the

elliptic boundary it is advisable to use a coordinate system other

than the Cartesian coordinate system. The elliptic coordinate

system was used here since one of its variables, ~' can be made con-

stant over the inclusion boundary. As shown in Figure 2, the curves

~=constant are a set of confocal ellipses whose foci are (±c,O).

The curves n=constant are a set of hyperbolas that are both confocal

with these ellipses and orthogonal to them. In terms of rectangular

coordinates x,y the elliptical coordinates ~,n are defined by the

following transformation:

x + iy = c cosh(~+in) = c cosh ~ (1)

Using the relations between hyperbolic functions along with

their relation to circular functions, it can be shown that Equation (1)

reduces to

y ·1']=90°

I -sl I I W>':WAs(sO';';ts';'J';'s&<;f:J I I I ---- X

1']=252° 1J=28SO 1']=270°

Figure 2. Diagram of Coordinate System Used

"'--.1

8

X+ iy c(cosh ~ cos n + i sinh ~ sin n) (2)

Equating real and imaginary parts,

X = C cosh ~ COS n ( 3)

y = c sinh ~ sin n (4)

resulting in the rectangular coordinates in terms of elliptic

coordinates.

Consider the elliptic boundary at the interface between medium

and inclusion as ~ having semimajor and semiminor axes a and b , 0 0 0

respectively. It can be shown that the coordinates along any ellipti-

cal curve are given by

X = a COS n (5)

y = b sin n (6)

Substituting these expressions for x and y along the boundary into

Equations (3) and (4), it can be shown that

tanh ~ = b /a = y 0 0 0

which is constant for any given values of a and b.

(7)

The transformation of the type x + iy = f(~+in) is shown by

Coker and Filon [9, page 152] to have the following relationship

between J, the stretch ratio, and ¢, the inclination of the curve

n=constant to the x-axis.

Jei¢ = d(x+iy)/d(s+in) ( 8)

Therefore, in elliptic coordinates

Jei¢ = c sinh(~+in)

= c(sinh s cos n + i cosh ~ sin n) (9)

From the exponential definition of circular functions it can be

shown that ei¢ = cos ¢ + i sin ¢. Therefore, Equation (9) can be

reduced to real and imaginary components,

J cos ¢ = c sinh s cos n

J sin ¢ = c cosh s sin n

(10)

(11)

9

To obtain the stretch ratio J in terms of s and n, Equations (10)

and (11) are each squared then added together to eliminate the ¢ terms.

In terms of double angles the stretch ratio can then be found from

(12)

Also, by dividing Equation (11) by Equation (10), an expression for

tan ¢ can be obtained which is also in terms of s and n.

tan ¢ = coth s tan n (13)

Relationships between the elliptical coordinate s and any set

of semimajor and semiminor axes a,b can be realized from Equation (7).

tanh ~ = b/a (14)

10

sinh .; = bj~}-b2 (15)

cosh .; = a/ p (16)

sinh 2.; 2 2 = 2ab/(a -b ) (17)

cosh 2.; (a2 +b 2) I (a 2 -b2) (18)

-2.; (a-b)/ (a+b) e = (19)

Equations (14) through (19) are used in preparing simple computer

programs for calculating the hyperbolic and exponential functions

necessary for determining the stresses in the medium material and the

boundary displacements without having to use special computer sub-

routines.

11

II. THEORY OF ELASTICITY

A. Stresses in Medium

The stresses in curvilinear coordinates can be determined as

shown by Coker and Filon [9, page 163] from the following equations,

(20)

(21)

T i;n (22)

where J is the stretch ratio and X is the Airy stress function. The

sum of the normal stresses cri; and on results in

(23)

The compatibility equation for plane strain and generalized plane

stress is shown by Boresi 2

[7, page 136] to be V (o~+on)=O when no

body forces are present. Therefore, from Equation (23) v4x=O.

The stress function is biharmonic as required by the compatibility

condition of the plane theory of elasticity.

The Airy stress functions, X' and x", used by Mindlin and Cooper

[6] for the inclusion and medium, respectively, are biharmonic

functions in elliptic coordinates, leading to single-valued displace-

ments.

x' A cosh 2~ cos 2n + B(cosh 2!; + cos 2n) (24)

12

x" -2~ -2~ De cos 2n + F~ + H(e + cos 2n) (25)

where A, B, D, F, H are constants whose values are found from the

boundary conditions. The following assumptions are made to establish

these boundary conditions:

1. Plane strain conditions apply.

2. Continuous displacement and traction across the interface

between the two materials is present.

3. Zero stress conditions in the medium are present at points

located far from the boundary of the inclusion.

The first assumption of plane strain requires on z=constant,

that the sum of the forces on the medium and inclusion in the z-direc-

tion must equal to zero. Therefore,

f 2'1T f ~0 a 'J 2d~dn J 21T ~1 a "J 2d~dn (26) + lim fr 0

0 0 z 0 so z 1;-+oo

1

Where Jo2n fot;o J2drdn ~ is the cross-sectional area of the inclusion

and lim J02 '1T

~ -roo 1

1;1 2 f~ J d~dn is the area of the surrounding medium.

0

Two

restrictions as to the application of a solution based on this con-

clition of plane strain must be realized. First, the calculated

stresses are not applicable within a distance (a) from the intersections

of the cylindrical surface 1;=~0 with the two planes z=constant. Also,

if the composite body is bounded by two traction-free planes at

z•constant, the solution would not be applicable for E'>>E".

13

The second assumption of continuity of traction and displacement

requires

0 ' t; =0" T '= ~ , t;n T 11 on t; = t;0 t;n

U ' = U " U ' = U " U ' = U " on <=" = s:: t; t; ' n n ' z z s "'o

The third assumption of zero stress at a distance far from the

inclusion requires

lim(ot;", on", cr 2 ", Tt;n") = 0

~-+co

The condition u ' = u " requires s z z z

from the condition lim 0 " = o, k = a"T. z

E;-+oo

' = s " = k. z

Therefore,

Equations for the constants in the above stress functions which

satisfy these boundary conditions are derived by Mindlin and Cooper

[6] and are listed below.

2 2 2 (2 7) A = c o(l-y )K/4y = A c /4 0

2 2 B c 2/4 (28) B = [ny-(l+y) ]A/(1-y ) = 0

2 D c 2/4 (29) D = -2yA/ (1-y) = 0

2 2 2 2 F c2/4 (30) F = 2[n(l+y )-2(l+y) ]yA/(1-y ) 0

2 H c2/4 (31) H = nyA/(1-y ) 0

where

14

(2m'+l)(m"+l) 2(2m"+l)-l K = ----------~--~~~~~~--~-------------n [n (m' -1) -4 (m '+ 1)+2 (m"+l) ]-2 (m"+l-n) y -\ l+y) 2

(32)

m' = v'/(1-v') (33)

m" = v"/(1-v") (34)

n = (1-G"/G')/(1-v") (35)

8 = E" (a" -a' ) T (36)

Substitution of the expressions for J and X" into the stress

Equations (20), (21), and (22) results in the general equations for

the stress components anywhere in the medium material based on a

uniform temperature change.

a " ~

F sinh 2~ 0

2(cosh 2~ - cos 2n) 2 _ D e-2~[sinh 2~ cos 2n + sinh2 2~ _ l]

0 (cosh 2~ - cos 2n) 2

-H [ sinh 2~ cosh 2~ _ l] 0 (cosh 2~ - cos 2n) 2

(37)

-F sinh 2~ 2 a" = o + D e-2~[sinh 2~ cos 2n +sinh 2~ _ l] n 2(cosh 2~ - cos 2n) 2 0 (cosh 2~ - cos 2n) 2

T~n II

-2~ +H [e (cosh 2~ - 2cos

0 (cosh 2~ - cos

F sin 2n 0

= 2(cosh 2~ - cos 2n) 2

H cosh 2~ 0

sin 2n

(cosh 2~ - cos

2 2n)+cos 2n] (38) 2n) 2

D sin -2~ 2n)+l] 2n[e (cosh 2~ - cos 0

(cosh 2~ - cos 2n) 2

(39)

15

where F 2 2 2

= 2oK[n(l+y )-2(l+y) ]/(1-y) (40) 0

2 2 D = -26(1-y )K/(1-y) (41)

0

H = noK ( 42) 0

Using the relationships between the elliptical coordinates ~,n and

a,b, Equations (5), (6), (14)--(19) in conjunction with the stress

Equations (37), (38), and (39), a numerical determination of the

stress components at any point in the medium can be performed.

The principal stresses (p,q) at any point in the medium material

based on 0~, 0 , and T, can be determined from the usual equations n sn

since the elliptic coordinate system is orthogonal at every point.

p,q (43)

In this study a comparison was made of the stress components

along the interface between the inclusion and medium as calculated

from Equations (37), (38), and (39) with those obtained from the

equations derived by Mindlin and Cooper [6] for along the boundary.

Identical results were obtained. The equations for the stress com-

ponents along the interface derived by Mindlin and Cooper are given

as follows:

where

0 " ~

B +A (j_) 0 0

0 " ::: H (1-2\fi)-D (<P+\flcos 2n) n o o

T " = A 1f! sin 2n ~n o

(44)

(~ ~ ) 0

(45)

(46)

16

2 2 ~

1-y -(l+x )cos 2n 2 2

l+y -(1-y )cos 2n (47)

~ 2 = 2 2

l+y -(1-y )cos 2n (48)

The loci of constant principal stress differences in the medium

based on the equations derived from the theory of elasticity are

shown in Figures 3 and 4. Only one quadrant is shown since the stress

pattern is symmetrical with respect to the x andy axes.

So that the differences of principal stress determined from the

theory of elasticity could be compared readily with the finite element

method, a special computer program was written. This program calcu-

lates the theoretical principal stress differences at the centroids

of the finite elements established in the finite element program.

The location of each element centroid i~ elliptic coordinates is found

by first determining the a and b values corresponding to the rectangular

coordinates of the centroid and c, the half distance between foci

of the confocal ellipses.

The magnitude of a and b are found from the following equations

which are derived from the general equation of the ellipse in rectangu­

lar coordinates (x2/a2 + y 2/h 2 = 1) and the relationship between a, b,

and c (a2 = b2 + c 2).

a =

2 2 2 4 4 4 2 2 2 2 2 2 1/2 [c +x +y +(c +x +y -2c x +2c y +2x y ) ]1/2

2 (49)

2 2 2 4 4 4 2 2 2 2 2 2 1/2 b = [-c +x +y +(c +x +y -2c x +2c y +2x y ) ]1/2 (SO)

2

-. c:: ·--., Col c:: 0 -en ·-Q I

>-

y

.8

.6

1.0

.4

1.5

.2

INCLUSION

00 .2 .4

MEDIUM

.6 .8 1.0

X- Distance (in.)

• Location of maximum principal stress ( 1]=26.0°)

1.2 See Fiv.4

1.4 1.6 1.8 X

Figure 3. Loci of Principal Stress Differences in Medium Based on Theory of Elasticity

,_. "

.I 0

-c -(I)

~ I

X .06

E 0 '-

LL.

~ .04 c 0 -(I) ·-0

02

MEDIUM

INCLUSION

3.40 -~--..

• Location of minimum principal stress (7]=3~

0 I I I I I I I I I I I .,., I. c ,, I I ,, I ' I X .90 .92 .94 .96 .98 1.00 1.02 1.04 1.06

Distance From Y-Axis (in.l

Figure 4. Loci of Principal Stress Differences at Interface Near Major Axis Based on Theory of Elasticity

1-' 00

19

The a and b values are used to determine the functions of ~, Equations

(14) through (19), necessary to calculate the stresses at the element

centroid. The n-value corresponding to the element centroid is de-

termined from Equation (5) or (6). The stress components at the

element centroids are then found from Equations (37), (38), and (39).

The angle 8 which the principal stress at any point in the medium

makes with the x-axis can be found from the following equation:

e = ¢ - a (51)

where ¢ = angle between the ~-axis at the point and the x-axis

= arctan (coth ~ tan n)

a = angle between the principal stress direction at a point

the ~-axis through that point 2T~

- 1/2 arctan [cr -d ] ~ n

This angle is used to obtain the stress trajectories based on the

theory of elasticity for comparison with those obtained from photo-

elasticity.

B. Boundary Displacements

The general displacement equations for U~ and Un in the directions

~ increasing and n increasing, respectively, are derived by Mindlin

and Cooper [6].

4GJU~ = 2(1-V)Ju~ - 2 ~ + c2(EaT - 2VGk)sinh 2~

4GJUn = 2(1-v)Jun - 2 ~- c2(EaT - 2vGk)sin 2n

(52)

(53)

20

where -i¢ i¢ 2 u~ + iun = e f Je (V x+iR)ds (54)

2 and R is the conjugate of V X· The constant k is the strain in the

z-direction and based on the assumed boundary conditions must be

equal to a"T. Equations (52) and (53) are the usual forms for the

displacements in curvilinear coordinates [9, page 164] with the

addition of the terms containing (EaT-2vGk), which account for the

thermal dilatation and strain in the z-direction. ,.

The displacement components U~ and Un along the boundary of

the inclusion are realized by substituting into Equations (52), (53),

and (54) the expression for the Airy's stress function of the inclu-

sion (X') and evaluating the resulting equations at ~ = ~ • From 0

Equation (23)

CY'+o' ~ n

(55)

Substituting into Equation (55) the expression for X', Equation (24),

h 1 . . f n2 ' . n2X' -- 8B/ c2. t e resu t1ng express1on or v X 1s v Since B = B c2/4 0

from Equation (28), V2x' = 2B • 0

The condition that R is the conjugate

of v2x• requires they must satisfy the Cauchy-Riemann's equations,

and

Therefore, R=C, a constant, and

V2x' + iR = 2B + iC 0

(56)

(57)

Multiplying by the expression for Jei<P, Equation (9), and integrating,

'..h 2 f Je1~(V X 1 +iR)d~ = 2B c(2 cosh ~ cos n - cosh ~ - cos n)

0

-2Cc sinh ~ sin n + i[c(4B sinh ~ sin n 0

+2C cosh~ cos n- c cos n- c sinh~)] (58)

-i¢ The expression for e of Equation (54) is found by dividing

Equation (12) by Equation (9), resulting in

(59)

The product of Equation (59) and (58) results in the expression

for u 1 + iu 1 in Equation (54). Equating real and imaginary parts, ~ n

U I

~

and U I

n

= c [4B sinh ~ cosh ~(cos2n - sin2n) ( . h2c + . 2 )1/2 o S1n <,. S1n n

2 -2B sinh ~ cosh ~ cos n - 2B sinh ~ cos n

0 0

-2C sin n cos n(sinh2~ + cosh2~)

2 +C cosh ~ sin n cos n + C cosh ~ sin n] (60)

= c [4B sin n cos n(sinh2~ + cosh2~) ( . h2c + . 2 )1/2 o S1n <,. S1n n

-2B cosh2~ sin n - 2B cosh ~ sin n cos n 0 0

-c sinh ~ cosh ~ cos n] (61)

21

The constant C can be found by evaluating the total displacement

of the inclusion in the ~-direction at~= 0 (see Figure 2). The

22

displacement in the ~-direction at s = 0 is zero. The component of

displacement in the ~-direction is found from Equation (52).

1 ~ 2 U~' = 4G'J[2(1-V')Ju~' - a~ + c (E'a'T-2V'G'k)sinh 2s)

1-v' 1 = (~)u~' - 2G'J(2A sinh 2~ cos 2n + 2B sin 2s)

cz + 4G'J(E'a'T-2V'G'k)sinh 2s (62)

At ~ = o, u~' = o,

0 = 1-v' c Czcr-)[sin n (-2C sin n cos n + c sin n cos n + c sin n)]

Therefore, C = 0

Setting C 0 in Equation (60) and (61) , the expressions for

u~' and un' in terms of double angles are

B c u~' = 0 [2/2 sinh 2~ cos 2n -sinh 2~(cos 2n+l) 112

(cosh 2~ - cos 2n) 112

- (cosh 2~ - 1) 112 (cos zn + 1)] (63)

B c and u '

0 [ z/2 cosh 2~ sin 2n = 2n)l/2 n (cosh 2~ - cos

- (cosh 2~ + 1)(1- cos Zn)l/Z- (cosh 2~ + 1) 112 sin Zn]

(64)

Substituting the above two equations into the total displacement

Equations (52) and (53) results in the following displacement equations

for the inclusion in terms of elliptic coordinates.

23

B c l-v' o U~' = (~) I [212 sinh 2~ cos 2n

s (cosh 2~- cos 2n) 1 2

- sinh 2~(cos 2n + 1) 112 - (cosh 2~ - 1) 1/ 2 (cos 2n + 1)]

U I

n

c2sinh 2~ [A 4G'J 0 cos 2n + B0 - E'a'T + 2v'G'a"T]

l-v' = (Z'G') B c

0

(cosh 2~ - cos [2/2 cosh 2~ sin 2n

2n)l/2

(65)

- (cosh 2~ + 1)(1- cos 2n) 112 - (cosh 2~ +!)sin 2n]

2 . 2 c s~n n + 4G'J [A0 cosh 2~ + B0 - E'a'T + 2v'G'a"T] (66)

Displacements u and v in Cartesian coordinates can be realized from

the following figure, which illustrates the relationship between

displacements in these coordinate systems.

I

y

-- ------

Ue- // 7]=Const

I I

I

cp ......... ..... _ +---,.__.... ______ x _ .....

....... --------(=const

-, \ \

Figure 5. Sign Convention of Rectangular and Elliptic Displacement Components

Us = u cos ¢ + v sin ¢

U = -u sin ¢ + v cos ¢ n

24

(6 7)

( 68)

Solving for the cos¢ and sin¢ in Equations (10) and (11), respectively,

and substituting into Equations (67) and (68) results in equations for

Us and u n

in tenns of u, v, s, and n. Solving for u and v,

u ' sinh s cos n - u ' cosh s sin n J [ s n ] u=-

sinh2i; 2 h2s . 2 c cos n + cos s~n n

(69)

and

us ' cosh i; sin n + U I sinh i; cos n J n

v ::::- [ ] c sinh2 i;

2 h2s . 2 cos n + COS Sl.n n (70)

Using the rectangular displacement Equations (69) and (70) in

conjunction with the elliptic displacement Equations (65) and (66),

a determination of the x and y components of the boundary displace-

ments (t,; = s ) can be made. These displacements were used in this 0

study as the input displacements for both the finite element method

and the photoelasticity study.

The validity of these displacement equations were verified by

assuming an infinitely rigid inclusion in a medium material having

a low modulus of elasticity and subjected to a uniform temperature

change. The displacement of the boundary based on these material

properties was easily determined from the free expansion of the

inclusion and Poisson's effect due to the constant load in the

z-direction. Identical displacement values were obtained.

I II. FINITE ELEHENT METHOD

The use of the finite element method in structural analysis

evolved from the development of digital computers. A comprehensive

presentation of the process in using the finite element method is

given by Zienkiewicz [10]. He presents the process of minimizing

the total potential energy of the system with respect to nodal

displacements, the process generally employed. This process is

the basis of the program used in this study.

The finite element procedure and program by Wilson [11] was

used in this study as the means of determining the differences of

principal stresses around an inclusion of elliptic cross-section due

to a constant temperature change. Wilson's program provides a

procedure for determining stresses and displacements in axisymmetric

solids as well as for plane stress conditions. A constant strain

triangular element is used with quadrilateral elements being divided

into four triangular elements. The stress in the quadrilateral is

considered to be the average of the values obtained for the four

triangular elements.

25

Since the plane area considered in this study is symmetric with

respect to the major and minor axes of the ellipse, the program was

setup as a plane stress problem using only one quadrant to be separated

into finite elements. To approach the condition of an infinite

medium the size of the plane containing the inclusion was taken

as lOO"xlOO" making one quadrant 50"x50".

26

Of the two possible elements that can be used in the program,

the quadrilateral element was used to a greater extent since it is

equivalent to four triangular elements. The triangular elements were

used only to increase the size of the finite elements further away

from the elliptical boundary. The nodal point locations were selected

to lie at the intersection of selected ellipses (~=constant) and

hyperbolas (n=constant). Due to the higher variation in strain

near the interface of the inclusion, the size of the elements were

made smaller in this area. A better representation of the strain

conditions was realized around the inclusion.

A total of 310 elements interconnected by 307 nodal points were

used to provide a grid for the medium. Figures 6, 7, and 8 illustrate

the finite element grid used within 5" of the major and minor axes

of the inclusion. These 263 elements consist of the elements located

in one quadrant of the lO"xlO" transparent material used in the

photoelasticity study.

Theoretical displacement components (u,v) obtained from

Equations (69) and (70) for along the elliptic boundary were used

as input displacements for the nodal points along the interface.

The v-displacements for nodal points lying along the x-axis as well

as the u-displacements for nodal points lying along the y-axis were

set equal to zero to simulate the boundary conditions along these

axes.

The resulting principal stress differences from Wilson's

computer program are used for comparison with the theoretical and

experimental approaches as shown in Figures 15 through 21.

27

y

259

256

Figure 6. Finite Element Grid Used

y

208 I 207

196 I 195

184

~~~--~~1.1 I l I I I f;l---69- 47 _ _ _ _ I I 1 46- 2 41--+---f------&...--..L... 23-1

INCLUSION

" I I ) 1\ '!1::111~1 ...... I X

SCALE : 111 = 0.2011

Figure 7. Finite Element Grid Near Inclusion

N 00

7

INCLUSION

SCALE: I"= 0.02 11

Figure 8. Finite Element Grid Near Small Radius

94

70 I 93

N \0

IV. EXPERIMENTAL ANALYSIS

The photoelasticity method was chosen for the experimental

approach because it is a whole-field method and the differences of

the principal stresses from this method provides an easy means of

comparison with analytical approaches.

30

The material used for the medium was a polyurethane plastic

produced by Photolastic~ Inc., Malvern, Pennsylvania, known commercial­

ly as PSM-4, [12]. Due to its high sensitivity to stress (f=3-5 psi/

fringe/in.) and low stiffness (E•lOOO psi), a preliminary study

showed it provided an adequate material for this study. The material

was obtained in sheet form, lO"xlO", with a thickness of 1/4".

An elliptical hole was made in the center of the plastic sheet

with a high speed router. A template having the same overall dimen-­

sions as the plastic sheet and with the desired elliptical hole

machined at the center was used as guide. Dry ice (solid carbon

dioxide) was used as a refrigerant to produce the necessary increase

in stiffness in the polyurethane for machining. Adjustment of the

template relative to the polyurethane sheet was found necessary due

to the thermal contraction of the polyurethane during freezing.

The inclusion material (plug) was made from a 1/2" aluminum

plate. The elliptical geometry necessary to produce the needed

medium displacement upon insertion was scribed on one side of the

plate. Machining of the boundary was performed with a vertical mill

for rough cutting and a sanding wheel for finishing. Periodic checks

of dimensions transverse to the major axis and along the major axis

31

were made during the sanding operation to insure correct dimensions

of the plug.

Calibration of the photoelastic material was performed using

a stepped tensile specimen taken from the polyurethane sheet used

in this study. The tensile specimen contained three widths as

shown in Figure 9. The stepped specimen provided more data than

a uniform width specimen resulting in a better average for the

material fringe value f and the modulus of elasticity E. Due to

the difficulty of obtaining an accurate value for Poisson's ratio

v in the calibration, it was assumed to be 0.46 based on similar

materials.

- 1n. - ln. 0 49. 022.

0 '

' ~ 0 "- 0.341n.

Figure 9. Calibration Specimen

32

The material fringe value for the polyurethane material was

found using a circular polariscope, i.e., a transmission polariscope

with quarter-wave plates added. The circularly polarized light

produced by the quarter-wave plates resulted in nondirectional sen­

sitivity, therefore, eliminating the isoclinics on the model specimen.

A light field was used by a parallel-crossed arrangement of the

polariscope elements, i.e., polarizing axes of polarizer and analyzer

parallel, fast axis of the first quarter-wave plate aligned with the

slow axis of the second quarter-wave plate. Loads were recorded

which produced 1/2-fringe orders at each of the three sections of

the calibration specimen.

The material fringe value f is a measure of the stress difference

required per fringe order formed for a unit thickness and can be found

from the following equation.

f = (p-q)t/N

where p,q normal stresses along the principal axes.

t thickness of the calibration specimen.

N = indicated fringe order.

( 71)

The material fringe value for the polyurethane material was found

to be 0.91 psi/fringe/inch.

The modulus of elasticity of the model material was determined

by placing the calibration specimen in a horizontal position and

loading it by means of a pulley system as shown in Figure 10. This

33

was found necessary due to the difficulty of measuring axial deforma-

tions when the specimen was placed in a vertical position. Axial

deformation was measured between marks scribed on the face of each

of the three sections of the calibration specimen. From these

deformations due to known axial loads and the dimensions of each of

the three cross-sections of the specimen, the average value for the

modulus of elasticity E could be easily determined from Hooke's

Law. The value of the modulus of elasticity for the polyurethane

material was found to be 450 psi. All three material constants

are listed in Table I.

calibration specimen

weight

Figure 10. Loading Method to Determine the Modulus of Elasticity of the Polyurethane

34

Table I. Properties of Materials Used

Modulus of Poisson's Fringe Material Elasticity Value

E Ratio

f

Polyurethane 450 psi 0.46 0.91 psi/

(PSM-4) fringe/in.

Aluminum 10x106 psi 0.33 --

Since both the material fringe value and the modulus of elasticity

for the polyurethane from the calibration test were lower in value

than the values given by [12], there resulted a greater number of

fringes on the model than was indicated in the preliminary study.

Figures 11 through 14 show the resulting photoelastic fringe

patterns after the aluminum plug was inserted into the elliptical

hole of the model material and placed in line with the circular

polariscope. The model material ,.;as supported by means of clamps

along the top edge. Fringes produced by the weight of the model

material as well as the plug were made negligible by placing the major

axis of the ellipse in the vertical direction. The weight of the

plastic and aluminum produces less than 1/10 of a fringe around the

inclusion.

Figure 11 shows the 1/2-order fringes as a result of a light

field. Figure 12 is a closeup of the fringes around the end of the

ellipse of Figure 11. The two visible dots in line with the major

35

axis are 0.25 and 0.50 inches from the end of the ellipse. The shadows

near the boundary of the ellipse, which are more pronounced adjacent to

the end of the ellipse, are due to model surface reflections of the

polarized light due to a slight dimpling of the model material in these

areas. Note that some indication as to the order of the fringes in these

shaded areas is apparent.

Figure 13 shows the full order fringes around the inclusion as

a result of a dark field. The field used in any of the photos can

be realized by noting whether the two holes near the center of the

plug can be seen or not. If they can be seen, a light field was

used, and the dark lines represent 1/2-order fringes. If the holes

cannot be seen, a dark field was used, and the dark lines represent

full order fringes.

Figure 14 shows the quadrant of the model containing the loca­

tions of the centroids of a selected number of the finite elements.

Also, increments of length are indicated along the major and minor

axes. The centroids and distances from the axes identified by black

dots were placed on the model material before insertion of the plug.

The magnitude of the principal stress differences at each of the

element centroids shown are listed in Table II. The principal stress

differences are based on the stress-optical relationship given by

Equation (71). Table III gives the corresponding values of the

principal stress differences for along the major and minor axes.

~isu~e, 11. !hotoe!aatie JJ:~M AX"ound Inclusion .. Ut~~~ • ~Itt Jt.e,ta

Table II. Principal Stress Differences at Centroids of Elements Located on Photoelastic Model

Element lit Element lit Element m No. No. No.

71 NV* 205 .28 245 .15

74 NV 207 .30 246 .15

77 NV 227 .65 247 .15

79 1.34 228 .54 248 .15

81 .85 229 .37 249 .11

83 .85 230 .22 250 .09

85 .69 231 .17 251 NEG

87 .69 232 .15 252 NEG

89 .72 233 .39 253 NEG

91 .56 234 .35 254 .07

173 NV 235 .24 255 .07

175 1.50 236 .20 256 NEG

177 .95 237 .11 257 NV

179 .54 238 .09 258 NEG

181 .48 239 .20 259 NEG

183 .46 240 .22 260 NV

197 1.43 241 .20 261 NV

199 .95 242 .13 262 NEG

201 . 56 243 .07 263 NV

203 .39 244 NEG**

* Not visible ** Less than 1/2 fringe

40

Table III. Principal Stress Differences at Locations Along the Major Axes of Photoelastic Model

Coordinates Coordinates

X y faT X y foT 1.25 0.0 1.50 o.o 0.50 .41

1.50 0.0 .68 0.0 1.00 .21

2.00 0.0 .27 0.0 2.00 NEG

2.50 0.0 .17 0.0 3.00 NEG

3.00 0.0 .10 0.0 4.00 NEG

4.00 o.o NEG*

* Less than 1/2 fringe

41

V. COMPARISON OF RESULTS

A graphical comparison of the principal stress differences

using the three approaches is illustrated in Figures 15 through 22.

The principal stress difference along the major axis is compared

in Figure 15 for the theoretical and photoelastic approaches. Good

correlation is indicated for the photoelastic readings taken.

Correlation at distances less than 0.25 inches from the tip of

the inclusion could not be verified due to the closeness of the

photoelastic fringes as shown in Figure 12.

Figure 16 compares the principal stress difference between

the theoretical and finite element approaches at the centroids of

the finite elements adjacent to the interface between the inclusion

and medium. Though some difference is noted in the magnitude of

the principal stress difference as a function of n, the overall

shape and location of stationary values (n=0°, 10.9°, 90°) are

identical.

Figures 17 through 22 compare the three approaches at confocal

ellipses to the inclusion based on the principal stress differences

at the centroids of finite elements. The parameters a and b are

42

given for each ellipse along with the finite elements whose centroids

are represented. Note that the correlation between the three approaches

improves the further away one moves from the inclusion boundary.

The explanation for the lack of correlation near the boundary

for the photoelastic results can be realized by examining the iso­

clinics (loci of points at which the directions of principal mean

4.0

3.0

p- q

181 2.0

1.0

• Photoelastic Results

Theoretica I Results

1.5 2.0 2.5 3.0 3.5 4.0 X

Distance From Y-Axis (in.)

Figure 15. Principal Stress Difference in Medium Along Major Axis

.1:'­w

4.5

4,0

3.5 Theoretical Results

3.0

p- q

lSI

2.5

2.0

1.5

Ot---~~~--~--~~~~~~--~--_. 0 10 20 30 40 50 60 70 80 90

'1 (Degrees)

Figure 16. Principal Stress Difference at Centroid of Elements Adjacent to Interface (1-23)

44

p- q

181

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Finite Element Method

•

a=I.029 in. b=0.316 in.

• Photoelastic Results

Theoretical Resu Its

• • • •

• 0 10 20 30 40 50 60 70 80 90

"1 (Degrees)

Figure 17. Principal Stress Difference at Centroid of Ele)llents 70-92

45

2.5

2.0

1.5

1.0

. 5

Theoret ica I Results

a=l.l2in. b= 0.54 in.

• Photoelastic Results

Finite Element Method

• • •

0 0 10 20 30 40 50 60 70 80 90 ., (Degrees)

Figure 18. Principal Stress Difference at Centroid of Elements 173-184

46

p-q 181

1.5

1.0

.5

0 0 10

a=l.25in. b= 0.77 in.

• Photoelostic Results

Finite Element Method

• •

20 30 40 50 60 70 "' (Degrees)

80 90

Figure 19. Principal Stress Difference at Centroid of Elements 197-208

a=l.49 in. 1.0 b=l.l2 in.

Theoretical

.5

• Photoelostic Results

Finite Element Method

00 10 20 30 40 50 60 70 "1 (Degrees)

80 90

Figure 20. Principal Stress Difference at Centroid of Elements 227-232

47

p- q

tal

p-q

w

1.0 a= 1.79 in. b= 1.49 in.

• Photoelastic .5 Results

0~--~--~~~--~--~--~~--~~~ 0 10 20 30 40 50 60 70

1.0

.5

80 90 "1 (Degrees)

Figure 21. Principal Stress Difference at Centroid of Elements 233-238

Finite Element Method

a =2.23 in. b = 2.00 in.

• Photoelastic Results

Theoretical Results

0+---~--_.--~~--~--~--~--~--~~~ 0 10 20 30 40 50 60 70 80 90

"1 (Degrees)

Figure 22. Principal Stress Difference at Centroid of Elements 239-244

48

49

stress are parallel to fixed directions) and corresponding stress tra­

jectories (lines tangent or perpendicular to the principal stress at

every point along the line) for both the theoretical and photoelastic

approaches. The isoclinics based on the theory of elasticity were

obtained by contour mapping of the angle 8 which the principal stress

at any point makes with the x-axis. Values of 8 are obtained from

Equation (51). Isoclinics were obtained in the photoelastic approach

by placing the photoelastic model containing the plug in the field

of a plane polariscope and noting the points of zero transmission

that moved when the axis of the polarizer and analyzer was rotated.

These points of zero transmission are associated with points at which

one of the principal stresses is parallel to the axis of polarization

of the polarizer. Figures 23 and 24 illustrate the isoclinics and

corresponding stress trajectories for the theoretical and photoelastic

approaches.

Due to the lack of traction between the photoelastic model

simulating the medium and the aluminum plug simulating the inclusion,

the isoclinics will touch the inclusion boundary at points whose

tangent to the x-axis are equal to the angle which the isoclinic

represents. The corresponding stress trajectories tend to be con­

focal and perpendicular to the elliptical boundary as shown in

Figure 23.

Based on the boundary condition of continuous traction across

the interface used in the derivation of the stress equations from

the theory of elasticity, the same configuration of isoclinics and

stress trajectories would not be expected. Figure 24 shows that the

STRESS TRAJECTORIES

90° 75° 60° 45°

ISOCLINICS

----------------------oo Figure 23. Isoclinic Patterns and Stress Trajectories

Based on Photoelastic Method

V1 0

STRESS TRAJECTORIES

goo 75°

ISOCLINICS

I I -..._ QO

Figure 24. Isoclinic Patterns and Stress Trajectories Based on Theory of Elasticity

VI ......

largest deviation between the two approaches is apparent mainly

near the interface. Note that the principal stress trajectories

are not confocal to the inclusion boundary when traction is assumed

between the two materials.

The magnitude of the shear stress along the interface based

on the theory of elasticity is illustrated in Figure 25. Points

on this curve were obtained from Equation (39) with ~=~ • 0

Comparison of the stress trajectories of Figure 24 based on

continuous traction along the boundary with the microcracks shown

in Figure 1 for the quartz inclusions indicates a similarity. Since

52

the microcracks in Figure 1 were produced due to a greater contraction

of the inclusion material the resulting stress trajectories would

be just reversed of those indicated in Figure 24. Note how closely

the stress trajectories compare with the microcracks. Also, as

indicated in Figure 4, the minimum principal stress based on the

condition investigated occurs along the boundary at ~=3.82°. This

corresponds closely with the point of initiation of the microcracks

of Figure 1. Since this point corresponds to the point of the maxi-

mum principal stress in the porcelain the initiation of a microcrack

would be expected here in the brittle material.

1!"1 181

2.2

2.0

I .8

1.6

1.4

1.2

1.0

.8

.6

.4

.2

0 0 10 20 30 40 50 60 70 80 90 "1 (Degrees)

Figure 25. Traction Stresses at Interface Between Inclusion and Medium Based on Theory of Elasticity

53

54

VI. CONCLUSIONS

Correlation of results of the finite element method with the

theory of elasticity was good, with greater correlation being realized

away from the interface. Refinement of the finite elements such as

making the elements smaller near the interface may improve this

correlation. The finite element method not only provides an easy way

of determining thermoelastic stresses around an inclusion of elliptic

cross-section but also provides a means of determining stress around

multiple inclusions and inclusions in a medium containing force systems

such as uniform tension, compression, or shear couple as studied by

other investigators mentioned previously. These force systems must

only satisfy the conditions of generalized plane strain when using

the method presented here.

Due to the lack of traction between the inclusion and medium

in the photoelastic study, correlation was not realized between the

theoretical and experimental approaches. This was found true

especially in the area around the inclusion where these traction

stresses were acting. The photoelastic approach used in this study

provides an adequate means of determining stresses around an inclu-

sion of elliptic cross-section where two conditions must be satisfied.

First, there must be negligible traction between inclusion and medium

material. Second, generalized plane stress conditions must be applicable.

As in the finite element method, other types of force systems can be

introduced along with multiple inclusions as long as the above two

conditions are met.

55

In using the photoelastic technique in determining the thermoelas-

tic stresses in the manner discussed in this study, the displacement

of the inclusion boundary due to the change in temperature must be

known. These displacements can be found from Equations (69) and (70)

at any point along the interface when ~=~ for a single inclusion 0

and approximated by the same equations when multiple inclusions are

being studied. The effect of the expansion or contraction of one

inclusion on another must be assumed negligible in the latter case.

When using the finite element method, one could eliminate the

need for the boundary displacements by dividing the inclusion or

inclusions as well as the medium into finite elements and setting

the problem up as one of plane strain. The input would then be

the overall temperature change. Plane strain can be realized in a

finite element program by use of the appropriate terms in the

elasticity matrix. Zienkiewicz [10, page 30] derives these terms

for the case of isotropic thermal expansion.

Based on the results of this study, further investigation

should be made into the analytical and experimental approaches that

can be used in a plastic analysis of an inclusion of elliptic

cross-section due to a uniform temperature change. Wilson's finite

element program [11], along with photoplasticity could be utilized

here to help further understand the phenomena of elastic and plastic

conditions in engineering materials.

BIBLIOGRAPHY

1. Tetelman, A.S. and McEvily, A.J. Jr., Fracture of Structural Materials, New York: John Wiley and Sons, Inc., 1967.

56

2. Donnell, L.H., "Stress Concentrations Due to Elliptical Dis­continuities in Plates Under Edge Forces," Theodore von Karman Annivers~ Volume; California Institute of Technology, pp. 293-309, 1941.

3. Hardiman, Jessie, "Elliptic Elastic Inclusion In an Infinite Elastic Plate," Quart. J. Mechs. Appl. Maths., Vol. 7, 2, pp. 226-230, 1954.

4. Symm, G. T., "Stresses In and Around Elliptic Elastic Inclusions," National Physical Lab., (NPL-MA-69), N69-18511, pp. 25, Oct. 1968.

5. Chen, W. T., ''On Elliptic Elastic Inclusion in Anisotropic Medium," ~uarterlJ[Journal of Mechanics and Applied Math., Vol. 20, Part 3, pp. 307-313, Aug. 1967.

6. Mindlin, Raymond D., and Cooper, Hilda L., "Thermoelastic Stress Around a Cylindrical Inclusion of Elliptic Cross-Section," Journal of A~lied Mechanics of the American Society of Mechanical Engineers, XVII, 3, pp. 265-268, 1950.

7. Boresi, Arthur P., Elasticity In Engineering Mechanics, New Jersey: Prentice-Hall, Inc.

8. Hansen, P.G., "Stress Systems Around Inclusions," Ph.D. Thesis, Washington University, Sever Institute of Technology, 1963.

9. Coker, E.G., and Filon, L.N.G., A Treatise on Photoelasticity, London, England: Cambridge University Press, 1957.

10. Zienkiewicz, O.C. and Cheung, Y.K., The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill Book Co., London, 196 7.

11. Wilson, E.L., ''A Digital Computer Program for the Finite Element Analysis of A~isymmetric Solids with Orthotropic Nonlinear Material Properties," Unpublished Report, Department of Civil Engineering, University of California, Berkeley, 1967.

12. Photolastic Bulletin P-1121, "Materials for Photoelastic Coatings and Photoelastic Models," Photolastic Inc., Malvern, PA.

57

VITA

Kenneth B. Oster was born in Kansas City, Missouri, July 11, 1935.

He received his elementary education at Ashland and Chapel Grade

Schools, Jackson County, and his secondary education at Raytown High

School, Raytown, Missouri, from which he graduated in June, 1953.

He attended Missouri Valley College, Marshall, Missouri, from

September, 1953 until June, 1956 and the University of Missouri­

Columbia, from June, 1956 until June, 1958. In June, 1958 he received

the degree of Bachelor of Science in Mathematics from M.V.C. and the

degree of Bachelor of Science in Civil Engineering from the University

of Missouri.

He held the position of Structures Engineer at North American

Aviation, Inc., Los Angeles, California, from June, 1958 until August,

1962. From August, 1962 until April, 1970 he held the position of

Structures Engineer and later Senior Structures Engineer at General

Dynamics Pomona Division, Pomona, California.

Since September, 1970, he has been a Graduate Assistant in the

Engineering Mechanics Department at the University of Missouri-Rolla,

and is presently working toward a Ph.D. in Civil Engineering at the

University of Missouri-Rolla.