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x __ 2 thesisentitled _ ‘_
FAILURE CRITERION OF A MULTI-LAYERED

GLASS—FIBER-REINFORCED-EPOXY COMPOSITE
MATERIAL AND THE APPLICABILITY OF THE

STREE FIELD THEORY
presented by

Ching-Fong Chen

has been accepted towards fulfillment
of the requirements for

M . S .
degree in Metallurgy

 

 

 

 

Datem c

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

 

 

MSU

LIBRARIES
‘—

 

 

RETURNING MATERIALS:

Place in book drop to
remove this checkout from
your record. FINES will
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stamped below.

 

 

 

 

 

 

FAILURE CRITERION OF A MULTI-LAYERED GLASS—FIBER-
REINFORCED-EPOXY COMPOSITE MATERIALS
AND
THE APPLICABILITY OF THE STRESS FIELD THEORY

By

Ching—Fong Chen

A THESIS

submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Metallurgy, Mechanics and Materials Science

1983

6 were

ABSTRACT
FAILURE CRITERION OF A MULTI-LAYERED GLASS-FIBER-
REINFORCED-EPOXY COMPOSITE MATERIAL

AND .
THE APPLICABILITY OF THE STRESS FIELD THEORY

By

Ching-Fong Chen

Mechanical properties of a composite were experimentally investigated
as a function of the orientation. These results can be explained by a
superposition of failure criteria of the fiber failure mode and the matrix
failure mode. The nature of the failure mode as a function of fiber
orientation, 6 , was examined by using optical and scanning electron
microscopy. Some preliminary investigations were also conducted to deter—
mine the elastic stress-strain distributions. For that purpose, a laser
interferometric strain gage (I.S.G.) measuring system was used. The
adaptation of this technique to a poorly reflecting composite material
poses some experimental difficulties. These difficulties were overcome by
a successful bonding of a metallic layer on the surface of the sample.
Preliminary results of the 1.3.0. investigation indicate a free-edge effect

of the stress-strain distribution in accordance with existing theories.

ACKNOWLEDGMENTS

This research is particially supported by the Ford Motor Company.
I would like to thank.Mr. Subhasish Sircar and Mr. Brian B. Schultz
for their help in accomplishing some of the experimental work. Dr.
Masaharu Kato is also appreciated for the data analysis. I would like
to thank Dr. John F. Martin for his guidance throughout the work, es-
pecially for giving me the chance to use the I.S.G. equipment. Special
thanks goes to Dr. Kalinath Mukherjee. He offered me the.chance to do
this topic and gave me inspiration and guidance constantly. Without
his help this thesis could not have been finished. He is an excellent

teacher, a friend and the advisor.

ii

TABLE OF CONTENTS

LIST OF FIGURE

I. INTRODUCTION

II. EXPERIMENTAL METHODS

(a)
(b)
(e)
(d)
(e)
(f)
(g)

Material and Sample Preparation

Tensile Test

Scanning Electron Microscopy

Computer Analysis and Calculation

Fundamentals of the Interferometric Strain Gage (I.S.G.)
Application of the I.S.G. Technique for an Epoxy Composite

Cyclic Stress—Strain Test

III. RESULTS AND DISCUSSION

(a)
(b)
(c)
(d)

Static Tensile Properties
Failure Criterion
Validity of I.S.G.Techique for Composites

Stress Distribution Across the Top Surface

IV.‘ CONCLUSIONS

APPENDIX:

Tensor Transformation

LIST OF REFERENCES

iv

13
13
20
25
27
27
31
41
44
46
47

49

LIST OF FIGURES

Figure
1. Unidirectional Layer of the Composite
2. Multidirectional Layer of the Composite
3. Three Dimensional Schematic of the Laminate
4. Geometry of Tensile Sample
5. Geometry of Cyclic Loading Sample
6. Locations of the Indentations for the I.S.G. on a Test Sample
7. Interferometric Strain Gage
8. Fundamental of I.S.G.
9. Indentations on Sample (_S E M Micrograph )
10. Interference Pattern
ll. Schematic Diagram for the Laser Interferometric Strain
Measuring System
12. The Indentation in the Region that Contains Both Fibers
and Epoxy
13. Indentations in the Region that Contains Epoxy Only
14. Geometry of the Indentation
15. Schematic Stress—Strain Behavior of a Linear Elastic Material
16. Schematic Stress-Strain Behavior of a Viscoelastic Material
17. Tensile Stress-Strain Curve of the Composite for
Different Orientations
l8. Schematic Presentation 6f the Three Regions 6f the Tensile

Stress-Strain Curve

iv

Page

10
ll
12
14
15
16
17

19

22

23
24
28
28

29

30‘

Figure Page
19. Comparison of Fracture Surface for Various Orientations: 32
20. SEM Micrograph of a 00 Sample 33
21. SEM Micrograph of a 900 Sample 35
22. SEM Micrograph of a 900 Sample 36
23. Positive Rotation of Principal Material Axes from 37

Arbitrary XAY Axes

24. n = 1.20 40
25. n = 1.16 ' 40
26. Static Strength vs. Angle 42
27. Extensometer Strain vs. Stress 43

28. Distribution of Ox Along Surface of the Sample 45

I. INTRODUCTION

An operationally simple strength tensor theory for fiber reinforced
composites has been proposed by Tsai and wu(1). This theory has in-

creased the ability of curve fitting based on the prediction

Fioi + Fijoioj = 1 (l)
where F1 and Fij are strength tensors, and i,j = 1, 2,..., 6.
Hashim and Rotem (2) used a simple failure criterion to predict the
deformation of an unidirectional fiber reinforced laminae. This approach
is based on the two different failure modes. One is the fiber—failure
mode and the other is the matrix-failure mode.

For the former mode,

2
CA cos 6 ( )
criterion was used,
where,
0A = 011is the stress in the fiber direction.

O is the applied stress.

6 is the angle between applied stress and the fiber direction.

For the latter mode,

0 T
(73) 4" (Pt 1 ‘3’
T

criterion was used,

where,

T = 012

a; is the failure stress in transverse loading alone.

TS is the failure stress in shear loading alone.

The critical angle 6c, up to which equation (2) holds, can

also be calculated from this theory,

tan 6 = -—- (4)

Rotem (3) extended the application of this theory to the angle plies
laminae with various reinforcement angles under different strain
fields. Rotem (4) also proposed a general fatigue failure criterion

for multidirectional laminate. The static failure is expressed by:

o: = KXx Us (5)
l
p 2 2 -' 2 ‘
s K K
«wen
o r

where,

s . .
0' is the failure stress.

 

 

 

2
KXX = _%_. 1 -sec 29 + ( u+sec29)tag 26
n + tan 26
K — 1 ( H+SeCZG )tan26
KY ‘ - 2 2
n +tan 26

The composite material that is studied in this program is neither a
unidirectional laminate nor a multidirectional laminate. This ma-
terial consists of alternate unidirectional and random reinforced
layers. Hence in the present case, a simplified approached is adopted.
This approach is based upon the strength theory of Tsai and wu, and
failure criterion of Hashin and Rotem. In stead of assuming the
qua dratic form for the matrix failure mode, an exponent n shall be

used for this special material, i.e.

T n
2 12 _
(S) + ( s>-1 (7)
“2 T12

 

 

where a: , 0:2 are material constants. For the fiber-failure mode,
the experimental result shows that the angle is very small and the
strength does not chang very much within this range. Hence, we shall

assume that:

where do is the failure stress of the 00 sample.

Recently attention has been given to the problem of stress dis—
tributions in laminated plates. It is well known that laminated plate
theory (LPT) takes into account the stresses in the plane of the
laminate, ox, 0y and Txy; namely, a plane stress state is assumed.
Puppo and Evensen (5) proposed a model in which a set of anisotropic
layers are separated by isotropic adhesive layers. They assumed that
each anisotropic layer carries only inplane loads and exists in a state
of generalized plane stress, while the isotropic layers carry the
interlaminar shear stress.

After these, Pipes and Pagano (6) obtained the solution for the
exact equations of elasticity on the free edge region. For this
solution a symmetric laminate under uniform axial stress was assumed.
The Moire technique was utilized by Pipes (7) to examine the surface
displacement of the symmetric angle-ply laminate under axial extension.
Pagano (8) proposed a new theory to define the complete stress field

with an arbitary composite laminate. The results were compared with

the numerical results by Wang and Crossman (9).

In order to develop a general characterization of the various
failure modes, it is necessary to have a practical means of laminate
stress analysis. The finite element method has been widely used by
some authors (10, 11). This is accomplished through the assumption
of a simplified displacement field according to the classical lamina-
tion theory. Hence, its application is limited to the determination
of the force distribution rather than the stress distribution. There—
fore, experimentation is the primary method so far to find the stress
distribution in order to determine the failure loading and mode of
fracture.

An Interferometric Strain Gage (I.S.G.) is a very powerful tech-
nique for measuring the local stress-strain behavior (12, 13, 14, 15).
Unfortunately, its application is mostly limited to the metallic ma—
terials, because the necessary condition for using the I.S.G. is to
have a very smooth reflecting surface. The composite materials are
not very good reflecting materials. Thus, in order to apply this
technique to composite materials, it is necessary to metallize the
surface. There are several ways to metallize an epoxy composite
surface, e.g., vapor deposition, coating, etc.. A relatively successful
metallizing technique was developed for the present case. Using such
a technique the surface stress distribution of our laminate is experi-
mentally determined and these results are compared with the stress

distributions predicted by Wang and Crossman (9), and Pagano (8).

II. EXPERIMENTAL METHODS

(a) Material and Sample Preparation:

In this material, a set of undirectional layers (Figure l) are
separated by adhesive layers containing mutidirectional random fibers
(Figure 2) to form a sandaich structure. The three dimensional sche-
matic structure (Figure 3) shows that all the unidirectional layers
are reinforced in the same direction and all the multidirectional
layers are in the random array. The angle 6 is defined as the angle
between the unidirectional reinforced direction and the applied load
ox. By using this angle a , the static tensile samples were made at
the following angles: 0°, 15°, 30°, 45°, 60°, and 90°. In order to
avoid those problems of slipping out and fracture at the grip, the
geometry of the static tensile sample is designed as shown in Figure
4.

The geometry of the cyclic loading sample is shown in Figure 5.
The cyclic loading sample was made in the 00 direction and the loca-
tions of the indentations are made along the top of the sample as
shown in Figure 6.

(b) Tensile Test:

All tensile samples were tested by using an Instron Floor Model
tensile testing machine.

(c) Scanning Electron Microscopy:

The fracture surface of the 00 and 900 samples were observed by

 

Figure l : Unidirectional Layer of the Composite

 

Figure 2: Multidirection Layer of the Composite

 

 

Figure 3: Three Dimensional Schematic
of the Laminate

10

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l

\ /
fl.787~|\

l-—'|.'|8'I ——+—— 1.568 —+—l.181——c

 

 

 

 

 

 

 

Figure 4 : Geometry of Tensile Sample
( Unit : inch )

11

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Figure 6 = Locations of the Indentations
for the I.S.G. on a Test

Sample

13

using the scanning electron microscope (HITACHI Model 415A). For
electron conduction in these nonmetallic materials, the samples were
metalized by vaccum vapor deposition. The coating material used was
copper. .

(d) Computer Analysis and Calculation:

Newton interpolation polynomial method was used for curve fitting
experimental data. The transition angle, 9c, between the two different
failure modes was also calculated by using the computer.

(e) Fundamentals of the Interferometric Strain Gage (I.S.G.):

The Interferometric Strain Gage (I.S.G.) is a noncontacting Laser
based measuring system which can measure strains over a very small
gage length (50—100 microns). Figure 7 shows the set-up of I.S.G..
Figure 8 illustrates the fundamental principles upon which the I.S.G.
is designed. A Vicker's hardness tester, with a pyramidal diamond,
was used to get the indentations which form a gage length on a sample
Figure 9. Incident monocromatic coherent laser light is reflected
by the indentations. These reflecting rays interfere to create inter-

ferrence fringes (Figure 10) when the following relation is satisfied:
d sina = nA (n = 0, i1, i2, ..... ) (9)
where: A = wavelength of the laser light.

n = order of reflection

d = space between two indentations

l4

 

Figure 7: Interferometric Strain Gage

 

15

 

 

 

 

 

 

 

Figure 8 = Fringe Generation Principles

l6

 

Figure 9 : Indentations on Sample
( SEM Micrograph)

I7"

 

gwsnothsw- _ '.

Oo.‘

 

Figure 10 : Interference Pattern

18

a = angle between incident and reflected laser rays.

The interference fringes consist of consecutive bright and dark
bands. Each.bright fringe is defined by an integer, n, from equation
(1). The I. S. G. measures strain by monitoring the movement of inter-
ference fringes.

For a fixed angle so, when tensile load is applied to the speciman,
the indentations move apart a distance 6d. From equation (9)

d sin do = nAo , since A is a constant for a given fringe, do must
decrease for tensile load, which means that the fringes will move toward

the laser beam by 6m, if a fixed angle so is chosen. The relationship

between fringe motion and displacement is defined as:

A
6d = ( Sin 0‘0 ) 6m (10)

where, 5m is the fraction of fringes.

 

A schematic diagram of the I. S. G. is shown in Figure 11. It has
two channels. Fringe motion of both channels can also be caused by a
rigid body motion which occurs in most loading. However, these effects
are cancelled out since one pattern moves toward the laser while the
other pattern moves away from it. Hence the relative displacement on

the specimen can be averaged out by the following equation:

A 5m + 6m
5d = -—---—- < 1 2 ) (ll)
2

 

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The equation of strain can be expressed by:

 

6d = 9. Gmrl' 531a
——-< 2 >

d sinaO

Two servo mirrors located at two stationary observation points
reflect the interference pattern onto slotted plates covering photomul-
tiplier tubes (PMT's). The mirror position is controlled by a mini-
computer which divides each sweep of the fringes into 256 consecutively
numbered positions and stores the accompanying PMT voltage for each
position. These data are converted to the analog strain (voltage) by
using equation (12). A more detailed discussion of the I.S.G. may be
found in Reference (14).

(f) Application of the I.S.G. Technique for an Epoxy Composite:

The interference pattern is formed by reflecting the laser beam
from a very smooth surface. Unlike metallic materials, the polymeric
composite materials are non-reflecting. Thus it was decided to metallize
the surface by vapor deposition. A1, Au and Cu vapors were tried as
the coating materials. Of these, Cu produces the best reflecting
coating. However, after making indentations to obtain the interference
patterns, there were several problems:

(1) The interference patterns were very abnormal.
(2) The reflecting angle was too small to make the fringe
patterns appear on the servo mirrors.

(3) Sometimes the indentations failed to produce fringe patterns.

21

After much trial and error and after microscopic inspection, it was
concluded that problem (3) is due to the.special property of glass—
fibers. Because the glass-fiber deforms almost elastically as the
indentations are applied to the sample, they bounce back after the inden-
tation load is removed (Figure 12). The net effect causes the surface
of indentation to be irregular which in turn destroys a regular and
periodic fringe pattern.

In order to avoid this problem, we tried to make indentations in
the regions that contain only epoxy» These results show that although.
it is possible to get fringe patterns,prohlems (1) and (2) still exist.
From.Figure 13, it is obvious that, because of the viscoelastic re—
covery of epoxy, the surface of the indentation bounces back with time.
This would decrease the angle a to a’ (where a’ is a function of time).
This is the reason why the angle of the indented surface is too small
to produce a fringe pattern on the servo mirror. Also at various por—
tions of an indentation there are different degrees of relaxation.

Thus a slightly curved and rough.vapor deposited surfaces are produced.
An attempt to make a stable indentation shape by allowing creep flow-
(i.e. to leave the major indentation load longer) also failed to
solve this problem.

Knowing these reasons, we believed that the only way to solve
these problems was to avoid the indentation into the composite ma-
terial. The next step was, then, to increase the thickness of the

deposition layer. From Figure 14, the thickness of the layer should

22

 

Figure 12 : The Indentation in the Region

That Contains Both Fibers and

Epoxy

23

 

 

Figure 13 : Indentations in the Region That

Contains Epoxy Only

24

Laser Beam

./1

/
22cr/rr/

1

l."

 

 

 

38 cm

 

Figure 14 : Geometry of the Indentation

25

be over 8 microns. Unfortunately as the thickness of the deposition
increases, the surface of the deposition becomes rough and less re-
flecting. We thus have to polish the surface after vapor deposition.
But the bonding between the deposited layer and the sample is not very
strong and hence the layer tends to peel off during metallographic
polishing.

Several attempts were made to circumvent these problems. The final
solution was to bond an aluminum foil on the sample. Using "M-Bond",
an aluminum foil can be bonded to the Sample. Since the surface of
the aluminum foil is not very smooth, it was necessary to polish the
sample surface by metallographic techniques after bonding the aluminum
foil. Microhardness indentations were then successfully and easily
produced on the metallized surface (Figure 9). Interferencr fringe
patterns from such surfaces are well formed and quite reproduceable. A
clip-on extensometer was used to check the validity of the I.S.G.
technique for this composite material.

(g) Cyclic Stress-Strain Test:

A computer controlled closed—loop MTS testing machine was used to
perfom all of the cyclic tests. Woods metal grip was used for the sample
grip so that sample could be mounted in a stress free condition.

Before testing, the aluminum foil was bonded on the sample. After
polishing the foil surface indentations were applied. One set of
indentations were across the width of the sample in order to determine

the distribution of ox across the top layer. For the calibration

26

test, strain was measured with.the I.S.G. as well as by-using a MTS

extensometer (model #632, 133 20).

III. EXPERIMENTAL RESULTS AND DISCUSSION

(a) Static Tensile Properties:
The fibers are the principal reinforcing or load-carrying agents.
They are typically strong and stiff and generally exhibit a linear
elastic behavior (Figure 15). The function of the epoxy matrix is to
support and protect the fibers, and to provide a means of distributing
the load among the fibers. The epoxy itself, generally exibits a
viscoelastic behavior (Figure 16). Because the fibers provide the
majority of the strength and stiffness, the fiber—reinforced composite
is treated as a linear elastic material. The results of the stress-
strain curves (Figure 17) elucidate this phenomenon.
The experimental stress-strain curves of 0° directional sample can
be easily separated into three regions (Figure 18).
Region I: (the linear portion at lower stresses)
In this region, both the glass-fiber and the epoxy
are within the elastic limits.
Region II: (the curved region)
In this region, the stress has already reached the
yield strength of the epoxy, i.e. the fiber conti-
nues to deform elastically, but the epoxy deforms
plastically.
Region III: (the linear portion at higher stresses)

In this region, because of the high stress, the

27

28

Q

 

_
fl

C

Figure 15: Schematic Stress-Strain Behavior of
a Linear Elastic Material

O1,

62

 

 

C

Figure 16: Schematic Stress- Strain Behavior
of a Viscoelastic Material (61> 62

>63)

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31

epoxy is no more able to carry the load and the stress
is carried by the glass—fibers only. This region is
again linear and reflects the linear elastic behavior
of glass-fibers.

(b) Failure Criterion:

Figure 19 has shown that there are essentially two different
failure modes in these samples. For smaller angles of e (approxi-
mately 0 0- 10° ), the fracture surface is "brush—like" and a severe
delamination occurs which extends inside the grip section of the sam-
ple. For larger angles, 6 greater than 100 , the fracture surface is
clean and very little delamination occurs.

These two modes can be explained by the scanning electron micro-
scopic pictures of the fracture surface. Under the tension test, the
strain is the same for the glass-fiber and the epoxy. However, the
glass-fiber has a higher elastic modulus from that of the epoxy, there-
fore, the stress of the fiber would be larger than that of epoxy.

Thus a matrix shear stress would result at the interface. For the 00
samples, all the load is carried by the fibers. The matrix shear
stress could not exceed the strength of the fiber, but could exceed

the allowable matrix shear stress. Therefore, delamination would occur
before the fibers are broken (Figure 20). After the stress reaches

the fracture strength some of the fibers would be broken, followed by
cumulative fiber failure across the sample, and hence cause the overall

fracture of the composite. Therefore, the fracture surface would

32

 

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34

appear brush—like and delaminated surface. This failure mode is called
fiber—failure mode. For the 900 samples, the load is not all carried
by the unidirectional fibers, the only fibers which can carry the load
are those in the random layers and lie parallel to the loading direc—
tion. Since those fibers are relatively few in number, the sample
will be broken by means of debonding between the fiber and the epoxy
matrix followed by the propagation of this crack perpendicular to the
tensile axis (Figure 21, 22). Therefore, the fracture surface would
be very clean. This failure mode is called matrix-failure mode.
Consider the lamina as shown in Figure 23. There, the coordinates
of the applied stress axes are X and Y; whereas the principal coordi—
nates (with respect to the unidirectional fiber orientation) are l and
2. The transformation equations for expressing stresses in an 1—2
coordinate system in terms of stresses in an X—Y coordinate system

(See appendix A) are

01 c0526 sinze 25inecose ox
02 = sinze cosze —23inecose 0y (13)
012 —sin6cose sinecose cosze—sinze 0

X7
where 6 is the angle from the X—axis to the l—axis.
Since a uniform uniaxial stress 0 was applied in the direction of

X-axis, the state of stress is then

(14)

 

35

 

Figure 21 : SEM Micrograph of a 900 Sample

36

 

Figure 22: SEM Micrograph of a 90° sample
(750 x )

37

 

 

 

Z:
_ //////

 

 

/\

Figure 23: Positive Rotation of Principal
Material Axes from Arbitrary
X-Y Axes

 

38

Hence equation (13) can be simplified as

._. 2
ol Ocos 9
02 = Usinze (15)
112 = cainecose

Since these two failure modes have different failure mechanisms, it
may be assumed that they are independent. For the fiber-failure mode,
we shall assume that the fracture strength of equates to oocosze,
where 00 is the fracture strength of 0 degree orientation because for
small angles, the factor of c0520 will not change very much. For the
matrix—failure mode, we shall assume that it is a function of 02 and

r

12. Thus, for fiber failure mode and matrix failure modes respective-

ly, we have:

of = oocosze (l6.a)

F(02° r12) = l (16.b)

It has been proposed (2) that F( 02 , 012) = 1 can be written in the

form:

n T n
{—39 + (7&3) = 1 (17)
2 12

where S and s are material constants.
02 012

39

Substituting equation (15) into equation (17), it follows that

2
of = 00c03*9

(a

where of is the fracture strength in the fiber failure region and cm

n n
o . n n =
) sinzne + ( 2 > Sln 9 cos 9 l (18)

T12

30

NU)

is the fracture strength in the matrix failure region. By the Newton
interpolation polynomial method and equation (17), it is found that n
is equal to 1.16 (Figures 24, 25). At the critical angle ec, which is

the transition point, both failure criteria should hold at the same

 

time; namely, of = om = cc. Then,
a = oocosze
c c
o n 2n 0c n n n (19)
c sin e + <' ) sin 6 cos 9 = l
-—- c s c c
03 T12

Eliminating oc from equation (19),

 

0o n 2n 2n 00 n n 3n
(—-—> sin 6 cos 6 + C > sin 6 cos 6 = l (20)
s c c s c c

T12

 

40

 

O' 539. : fl
(MPOLME \‘1 j I ‘

\\ ___ n =1.2o
Curve Fitting
\ n ExperimentalDah

mm] ‘
JVU
2'23 \
_ \
\
_ r
139 Q
-

I l l l l
(I 20 4G 60 80 10 0

 

41m -
wall

 

 

 

 

 

\g.._________g

 

 

 

 

 

 

 

Figure 24: n = 1.20

(MPa)

 

Figure 25: n = 1.16

 

41

Since
n = 1.16
00 = 537 Mpa
5—
02 — 71 Mpa
5—
112— 119.6 Mpa

6 can be calculated by using a computer. The value of ecdetermined

this way is:

This angle shows a very good agreement with the experimentally observed
value (Figure 19). Both curves of different failure criteria are plot-
ted in Figure 26.and a good correspondence between the experimental
points and the calculated curves are seen.

(c) Validity of I.S.G. Technique for Composites:

From the extensometer, the Young's Modulus of the sample is de—
termined to be 37040 Mpa (Figure 27) and the average value of the mo-
dulus obtained by using the I.S.G. is 37550 Mpa. Since the composite
materials are a heterogeneous and elastically nonlinear material, the
Young's Modulus will have a different value as the location of the
micro—gauge section changes. Hence the difference of 1.76% between
the average value of modulus from the I.S.G. and that from the clip—

on gauge is within the resonable limit. Thus the strain in aluminum

42

 

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Figure 27: Extensometer Strain vs. Stress

44

foil is the same as that in the composite materials (no aluminum foil
was used while the clip—on gauge was employed). The thickness of the
aluminum foil used in these experiments is very small ( =10 microns)
and thus the reinforcement effect can be neglected. Since the aluminum
foil does not extend to the grip sections, the load is not directly
transmited via the aluminum foil. Hence the response of the aluminum
foil by the grip will also be neglected.

(d) Stress Distribution Across the Top Surface:

Theoretical predictions of Pagano (8), Wang and Crossman (9) state
that surface stress along the tensile axis decreases as the parameter
Y/b increases, where b is the half width of the sample, Y is the dis—
tance along the Y axis from the central vertical plane of the sample.
Figure 28 shows the nature of this decrease as predicted by Pagano (8),
Wang and Crossman (9). Although we do not have sufficient number of
experimental points, it is interesting to note that the ox value near
the edge of the sample is lower than that near the central stress axis.
Two experimental data points (full circles) are shown in Figure 28.
Considering the experimental uncertainties and the limited data, the
agreement between the theoretical curve of Wang and Crossman, and the
two experimental points seems to be interesting. Detailed and sys—

tematic experiments, however, are needed to confirm this observation.

45

 

 

  

 

 

 

 

2 5.5
T7: 3.0; xQ
q \
x —
a") '""P°9°"° m 5.0 0c»
2 . m
\ 25 _ Wang and Crossman ( | ) f
x 'u
b 2.
0 Experimental Data (II) _ 4.5 E
2.0 I L I I 1 I. I I 1 "’
O . .2 .4 .6 .8 LO
Y/b

Figure 28 2 Distribution of O'xalong Surface

of the Sample for a Constant
Strain 10-6

IV. CONCLUSIONS

Based on the information obtained, the following conclusions can

be reached:

1. For this special material, the failure criterion can be
expressed by two simple equations. For fiber failure mode

and matrix failure mode respectively, we have:

a = o c0526
f o

n n
o 2 T
< > sin n9 +( m) sinnecosne = 1
Is
12

where n is 1.16 for this material.

 

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2. The I.S.G. can be successfully applied to the composite materials.
This is a very powerful technique which can determine the stress
distribution by measuring the local stress-strain behaviors.

3. The free edge effect has been shown by experimental results.

4. The stress distribution that was measured by the I.S.G. has
been compared well with the theoretical prediction of Wang and

Crossman (9), and Pagano (8).

46

APPENDIX

The transformation of stresses state from X,Y and Z coordinates to

1,2 and 3 coordinates can be expressed as following:

T
“11 c’12 °13 “xx °xy c'xz
021 022 023 = A ny Oyy oyz A
031 032 033 Ozx ozy 022

From Figure 23, it is an in-plane stresses state and rotated an

angle about Z—axis. Therefore, the relationship (a) can be simplified

as:
all 012 0 c056 sine O oxx oxy O cose —Sin6 0
021 022 0 = -Sine cose 0 oyx Uyy 0 sine cose O
0 O 0 0 0 1 O 0 0 0 0 1
2 . 2
o o 0 0 cos 9 + 20 Sinecose + a sin e
11 12 xx xy yy
2 . 2 .
021 022 0 —oxxsinecose + oxy(cos 6—31n e) + oyysinecose
0 O 0 O

—o cosesine + o (cosze-sinze) + o sinecose
xx xy YY

o sinze—Z sinecose + o c0326
xx xy yy

0

47

0

0

0

48

Hence,

0 c0329 sinze 23inecose o

11 xx]\
0 = sinze cosze -23in6cose o

22 ny’
o -sinecose sinecose cosze-sinze o

12 xy

10.

ll.

12.

REFERENCES

Tsai, S. W. and wu, E. M., "A General Theory of Strength.for
Anisotropic Materials." Journal of Composite Material, Vol. 5
(1971), p. 58.

Hashin, Z. and Rotem, A., "A Fatigue Failure Criterion for Fiber
Reinforced Materials," Journal of Composite Materials, Vol. 7
(1973), p. 448.

Rotem, A. and Hashin, 2;, " Fatiguewof Angle Ply Laminates,"
AIAAJournal, Vol. 14 (1970), p.271.

Rotem, A., "Fatigue Failure of Multidirectional Laminate," AIAA
Journal, Vol. 17 (1979), p. 271.

Puppo, A. H. and Evenson, H. A., " Interlaminar Shear in Laminated
Composites Under Generalized Plane Stress," Journal of Composite
Materials, Vol. 4 (1970), p. 204.

Pipes, R. B. and Pagano, N. J., "The Influence of Stacking Sequence
on Laminate Strength," Journal of Composite Materials, Vol. 5 (1971)
, p. 50.

Pipes, R. B., "Moire Analysis of the Interlaminar Shear Edge Effect
in Laminated Composites," Journal of Composite Materials, Vol. 5
(1971), p. 255

Pagano, N. J., "Stress Fields in Composite Laminates," International
Journal of Solid Structure, Vol. 14 (1978), p. 385.

Wang, A. S. D. and Crossman, F.W., "Some New results on Edge Effect
in Symmetric Composite Laminates," Journal of Composite Materials,
V01. 11 (1979), p. 92

Pipes, R. B., and Pagano, N. J., "Interlaminar Stress in Composite
Laminates Under Axial Extension," Journal of Composite Material,
Vol. 4 (1970), p. 538.

wang, S. S. and wang, H. T., "Interlaminar Crack Growth in Fiber
Reinforced Composites During Fatigue," Journal of Engineering Materials
and Technology, Vol. 101 (1979), p. 34.

Sharpe, W. N. Jr., "The Interferometric Strain Gage," Experimental
Mechanics, Vol. 8, No. 4 (1968), p. 164.

49

l3.

14.

15.

50

Sharpe, W. N. Jr., "A Short Gage Length.0ptical Gage for Small
Strain," Experimental Mechanics, Vol. 14, No. 9 (1974), p. 373.

Bofferding, C. H. "A Study of Cycle Stress and Strain Concentration
Factor at Notch Roots Throughout Factors Life," Master Thesis,
Michigan State University, 1980.

Lucas, L. J., "Experimental Verification of the Neuber Relation
at Room and Elevated Temperature," Master Thesis, Michigan State
University, 1982.

 

 

 

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