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THEBIS

20': 0

 

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Unlversity

 

 

 

This is to certify that the

dissertation entitled

Analysis of Composite Laminates

By Moire Interferometry
presented by

Elias Ruda Shakour

has been accepted towards fulﬁllment
of the requirements for

Ph.D. Engineering Mechanics

 

 

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11/00 mm.“

ANALYSIS OF COMPOSITE LAMINATES BY MOIRE INTERFEROMETRY

BY

Elias Ruda Shakour

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Materials Science and Mechanics

2000

ABSTRACT

ANALYSIS OF COMPOSITE LAMINATES BY MOIRE INTERFEROMETRY

By

Elias Ruda Shakour

Composite laminates are organized by laminae with various ﬁber
orientations. The sudden change of material properties across laminate
interfaces is of an inherent characteristic of composite laminates. As a result of
material property mismatch through the laminate thickness, high stresses and
strains could take place on laminate interfaces. They are especially signiﬁcant
around free edges of composite laminates. In order to examine the stresses and
strains with high gradients in composite laminates, an accurate measuring
method is required. Due to its high sensitivity, moire interferometry has been
useful for in-plane displacement measurements. Since moire interferometry is a
surface method, its measurements are purely of surface conditions. The
feasibility of the surface method for general structural analysis has become a
serious question. This study was focused on validation and applications of moire
interferometry. It was divided into three sections. In the ﬁrst section, the existing
two-layer grating technique was simpliﬁed to one-layer technique aiming at
reducing the reinforcement effect of moire grating on testing specimens. A further
study on no-layer grating was also explored for a possible combination of moire

interferometry and photoelasticity. It was believed that the combined technique

could provide two independent measurements for mutual veriﬁcation. The one-
layer grating technique was then used in the second section of free-edge
investigation. Composite beams with various widths were subjected to four-point
bending. Experimental results revealed that the normal strains were independent
of the beam width. However, the shear strains were signiﬁcantly increased as the
beam width decreased, signaling the free-edge effect. Comparisons based on
composite beams made of different materials and Iaminations veriﬁed that free-
edge effect was more signiﬁcant in composite laminates with higher material
property mismatch among laminae. In fact, the depth of boundary layer was
proportional to the level of material properties mismatch. Based on the
conclusion that the normal strains were insensitive to the free-edge effect, many
assembled composite beams were investigated by moire interferometry in the
third section. Various assembling techniques such as mechanical riveting,
adhesive bonding, stitching joining and their combinations were used for joining
two composite beams together. Experimental results combined with analytical
modeling investigated, i.e. the interface model and the interphase model,
presented an effective way to express the joining force of each assembled
composite beam. It was concluded that mechanical riveting was an efﬁcient

assembling technique, at least for the composite laminates used in the study.

Copyright by
Elias Ruda Shakour
2000

To my wife Sana and my daughter Yasmeen, whose affection has been a
continuous energy during of these courses of studies. To my parents Ruda and
Miriam who taught me responsibility and courage.

ACKNOWLEDGMENTS

The author whishes to express his sincerest appreciation and gratitude to
his academic and major advisor Professor Dahsin Liu for his valuable guidance,
encouragement and advise in personal as well as professional matters during the
course of this research. Professor Liu has unselﬁshly shared with the author his
greatest attribute, creativity, and an invaluable tool that few possess. The author
also extends his appreciation to the guidance committee members Dr. Tim

Hogan, Dr. Patrick Kown, and Dr. Stanley Flegler.

Special appreciation and love is extended to the author‘s wife, Sana, for
consistent encouragement, understanding, patience and love throughout his
graduate career; the author's daughter, Yasmeen for spiritually strengthening the
author; his parents for inspiration and support; his brothers and sisters Hania,

Gabriel, Nasser, Amira, and Laila for the patience, love, and support.

vi

TABLE OF CONTENTS

ACKNOWLEDGMENTS ...................................................................................... vi
TABLE OF CONTENTS ..................................................................................... vii
LIST OF TABLES ................................................................................................. x
LIST OF FIGURES .............................................................................................. xi
Chapter 1 ............................................................................................................. 1
MOIRE INTERFEROMETRY ................................................................................ 1
Introduction ........................................................................................................ 1
Background ....................................................................................................... 2
Geometric moire ............................................................................................ 2
Moire interferometry ....................................................................................... 4
Literature review ................................................................................................ 8
Diffraction grating ........................................................................................... 8
Applications of moire interferometry ............................................................. 10
Objectives ........................................................................................................ 12
Organization .................................................................................................... 13
References ...................................................................................................... 14
Chapter 2 ........................................................................................................... 17
COMBINING MOIRE INTERFEROMETRY AND PHOTOELASTICITY ............. 17
Abstract ........................................................................................................... 1 7
Introduction ...................................................................................................... 18
Grating techniques .......................................................................................... 20
Two-layer grating ......................................................................................... 20
Conventional technique ............................................................................ 20
Modiﬁed technique ................................................................................... 21
One-layer grating ......................................................................................... 21
Zero-layer grating ......................................................................................... 24
Combining moire and photoelasticity ............................................................... 26
Transmission photoelasticity ........................................................................ 26
Reﬂection photoelasticity ............................................................................. 30
Single coating for both moire and photoelasticity ......................................... 30
Case studies .................................................................................................... 33
Effects of coating thickness .......................................................................... 33
Stress concentration .................................................................................... 36
Summary ......................................................................................................... 41

vii

Acknowledgements ......................................................................................... 41

References ...................................................................................................... 43
Chapter 3 ........................................................................................................... 46
MISMATCH OF MATERIAL PROPERTIES AND FREE-EDGE EFFECT .......... 46

Abstract ........................................................................................................... 46

Free-edge effect .............................................................................................. 47

Moire interferometry ........................................................................................ 49

Experimental investigations ............................................................................. 52

Moire on composite laminates ..................................................................... 52
Materials and testing .................................................................................... 53
Effects of specimen width ............................................................................ 54
Mismatch of material properties ................................................................... 55
Results and discussions .................................................................................. 56
Strain Calculations ....................................................................................... 56
7,, based on strain-displacement relations ................................................... 58
M Based on vector analysis ......................................................................... 60
Depth of Boundary Layer ............................................................................. 63
Effect of material type .................................................................................. 67
Effect lamination type ................................................................................... 72

Conclusions ..................................................................................................... 73

Acknowledgements ......................................................................................... 74

References ...................................................................................................... 75
Chapter 4 ........................................................................................................... 77
JOINING FORCES IN ASSEMBLED COMPOSITE BEAMS ............................. 77

Abstract ........................................................................................................... 77

Introduction ...................................................................................................... 78

Analytical modeling .......................................................................................... 79

The interface model ..................................................................................... 79
The interphase model .................................................................................. 84
Experimental investigations ............................................................................. 87
Composite laminates ................................................................................... 87
Joining techniques ....................................................................................... 88
Moire interferometry ..................................................................................... 92
Results and conclusions .................................................................................. 96
Moire fringe patterns .................................................................................... 96
Maximum joining force ............................................................................... 102

Singular points ............................................................................................... 109

Joining forces ................................................................................................ 1 12

Conclusions ................................................................................................... 1 14

Acknowledgments ......................................................................................... 1 15

References .................................................................................................... 117
Chapter 5 ......................................................................................................... 118

viii

CONCLUSIONS AND RECOMENDATIONS .................................................... 118

Major conclusions .......................................................................................... 119
Chapter two ................................................................................................ 1 19
Chapter three ............................................................................................. 120
Chapter four ............................................................................................... 121

Recommendations for future work ................................................................. 121

APPENDIX A .................................................................................................... 124
Procedures for aluminum evaporation ......................................................... 124
APPENDIX B .................................................................................................... 126
Moire fringe patterns used for subtractions ................................................. 126

Combining moire interferometry and photoelasticity ...................................... 126

Joining forces in assembled composites ....................................................... 127

Mismatch of materials and the free edge effect ............................................. 130

LIST OF TABLES

Chapter 4

Table 4. 1- Comparisons of joining forces ................................................................... 113

LIST OF FIGURES

Chapter 1
Figure 1.1- Moire fringe patterns from various shapes of patterns overlapping one
another ......................................................................................................... 3
Figure 1.2- Schematic diagram for geometrical moire formation ...................................... 5
Figure 1.3 - Schematic diagram of moire interferometry .................................................. 7
Chapter 2

Figure 2.1- Schematic diagram of optical setup for moire interferometry (horizontal two
beams only). ............................................................................................... 19

Figure 2.2- Schematic diagrams of specimen with (a) two-layer grating, (b) one-layer
grating, and (c) zero-layer grating ............................................................... 22

Figure 2. 3 - U-displacement moire fringe patterns from four-point bend tests for (a)
polished aluminum specimen and (b) transparent PSM-5A specimen ......... 25

Figure 2. 4 - Schematic diagram of processing setup for casting a specimen with zero-
layer grating. ............................................................................................... 27

Figure 2.5 - Schematic diagram of optical setup for combining moire interferometry and
transmission photoelasticity. ....................................................................... 28

Figure 2.6 - Fringe patterns of PSM-5A beam under four-point bending: (a) transmission
isochromatic fringes, (b) U-displacement fringes, and (c) simultaneous
recording of photoelasticity and moire fringes ............................................. 29

Figure 2.7 - Schematic diagram of optical setup for combining moire interferometry and
reﬂection photoelasticity .............................................................................. 31

Figure 2.8 - Fringe patterns of a PSM-5A beam under four point bending: (a) reﬂection
isochoromatic fringes, (b) U-displacement fringes, and (c) simultaneous
recording of photoelasticity and moire fringes. ............................................ 32

Figure 2.9 - Fringe patterns of a glass/epoxy beam with PL8 moire photoelasticity
coating (a) reﬂection isochromatic fringes, (b) U-displacement fringes, and
(c) simultaneous recording of photoelasticity and moire fringes. ................. 34

Figure 2.10 - Moire fringe patterns of glass/epoxy beams with PL8 grating thickness of
(a) 0.55 mm, (b) 1.36 mm, and (c) 3.17 mm ................................................ 35

xi

Figure 2.11 - Photoelasticity fringe patterns of glass/epoxy beams with PL8 grating
thickness of (a) 0.55 mm (b) 1.36 mm, and (c) 3.17 mm ............................. 37

Figure 2.12 - Fringe patterns of a notched glass/epoxy beam with PL8 moire-
photoelasticity coating (a) reﬂection isochromatic fringes, (b) U -
displacement fringes, and (0) simultaneous recording of photoelasticity and

moire fringes ............................................................................................... 38
Figure 2.13 - Schematic diagram of ﬁnite element model for the beam with central hole;
it has 1482 elements, the element type is 8 node element of CPSBR, ........ 39
Figure 2.14 - Comparisons of results from moire interferometry, photoelasticity, and ﬁnite
element analysis. ........................................................................................ 40
Chapter 3
Figure 3.1 - Schematic diagram of optical setup for moire interferometry (horizontal two
beams only). ............................................................................................... 51
Figure 3.2 - Moire fringe patterns for graphite/epoxy laminates with three different widtg;
Figure 3.3 - Calculation of shear strain on the laminate interface based on least square
method ........................................................................................................ 59
Figure 3.4 - Shear strain calculations based on vector analysis ..................................... 61
Figure 3.5 - Distribution of ex, through laminate thickness ............................................. 64
Figure 3.6 - Distribution of 8,, through thickness ............................................................ 65
Figure 3.7 - The distribution of yxy through the laminate ................................................. 66
Figure 3.8 - Moire fringes of glass/epoxy laminates ....................................................... 68
Figure 3.9 - Distribution of exx through laminate thickness ............................................. 69
Figure 3.10 -Distribution of 8w through thickness ........................................................... 70
Figure 3.11 - Distribution of yxy through thickness .......................................................... 71
Chapter 4
Figure 4.1 - Free body diagrams for: (a) the interface model, and (b) the interphase
model .......................................................................................................... 81
Figure 4.2 - Schematic diagram of various joining techniques ....................................... 90

xii

Figure 4.3 - Schematic diagram of optical setup for moire interferometry (horizontal two
beams only) ................................................................................................ 94

Figure 4.4 - Moire fringe pattern for: (a) ONE (one laminate beam), (b) 2B (2 for two-
laminate and B for bonding), (c) TWO (for two laminates), and (d) 2T (2 for
two-laminate and T for taping) .................................................................... 97

Figure 4.5 - Moire fringe pattern for: (a) 2R2 (the ﬁrst 2 for two-laminate, R for riveting
and the second 2 for two rivets per 50 mm), (b) 2R3 (R for riveting and 3 for
three rivets per 50 mm), (c) 2R4 (R for riveting and 4 for four rivets per 50
mm), and (d) 2BR3 (B for bonding, R for riveting and 3 for three rivets per 50
mm) ............................................................................................................ 99

Figure 4.6.- Moire fringe patterns of various joinig subjected to four point bending: (a)
283 (90N) (S for stitching and 3 for three stitches per 50 mm), (b) 283 (200
N) , (c) 285 (S for stitching and 5 for 5-by-2 stitching pattern), and (d) 2B83
(B for bonding, S for stitching and 3 for three stitches per 50 mm) ............ 101

Figure 4.7 - Normal strain calculations, (a) moire fringe pattern for ONE case in ﬁve
designated locations, (b) relation between u-fringes and N)( in ﬁve locations,
and (c) strains in the ﬁve locations along y-axis ........................................ 105

Figure 4.8 - Comparison of normal strain for perfect bonding ONE and imperfect bonding
2B ............................................................................................................. 107

Figure 4.9 - Schematic diagram for: (a) prefect bonding, (b) imperfect bonding, and (c)
no bonding ................................................................................................ 108

Figure 4.10 - Comparison between analytical predictions and experimental results 1 11
APPENDIX A

Figure A.1- Fringe patterns of a notched glass/epoxy beam with PL8 moire-
photoelasticity coating (344.72 N) (a) reﬂection iscochromatic fringes, (b) U-
displacement fringes, and (c) simultaneous recording of photoelasticity and
moire fringes. ............................................................................................ 126

Figure A.2 - Moire Fringe pattern for 67.5 N, (a) 2B, (b) TWO, (c) 2T, and (d) 2R2 ..... 127
Figure A.3 - Moire fringe patterns (67.5 N) (a) 2R3, (b) 2R4, (c) 2BR3, and (d) 283....128

Figure A. 4 - Moire fringe patterns (67.5 N), (a) 285, and (b) 2383 ............................. 129
Figure A.5 - Moire fringe patterns for graphite/epoxy laminates with three different widths

................................................................................................................. 130
Figure A.6 - Moire fringes of glass/epoxy laminates at a zero loaf ............................... 131

xiii

Chapter 1

MOIRE INTERFEROMETRY

Introduction

When one pattern of repeating lines overlaps with another similar pattern,
a third pattern called moire fringe could be created. This phenomenon can be
seen in many aspects of daily life. For example, when one looks through a fence
towards another fence, a pattern of light and dark lines is composed from the two
fences. The meaning of the word moire derives from the French word of silk in
which the threads form patterns when overlapping one another [1 .1]. Figure 1.1

demonstrates two different moire patterns.

When two sets of grating made from straight lines, their superposition will
create moire fringe patterns that are straight lines as well. This phenomenon is
called geometric moire. In geometric moire, the moire fringes are governed by
mathematical equations. The mathematical equations represent displacements
that occur due to overlapping of two sets of grating. The direct relationship
between the two sets of grating and the moire fringes enables geometrical moire
to be used for deformation measurements. However, if the deformation is too
small, the geometric moire, due to its low grating frequency, will be unable to

measure the deformed displacements.

In order to measure small deformations accurately, geometrical moire has
been modiﬁed by a technique called moire interferometry. Moire interferometry is
an optical method that is based on a laser light. In performing moire
interferometry, a grating made of furrows with equal spacing must be attached on
the surface of a specimen. When the grating is illuminated by two laser beams
with speciﬁc incident angles, the two beams will be reﬂectively diffracted from the
grating. If the specimen is undeformed, the two beams will be normal to the
specimen surface and parallel to each other. If the specimen is deformed, the
grating will be altered, and so will be the direction of the diffraction beams.
Consequently, the two diffracted beams will interfere with each other and form a
set of fringes, i.e. the moire fringes. A good summary of moire interferometry can

be found in Post et al. [1.2] and Post [1.3]

Background

Geometric moire

Geometric moire, as shown in Figure 1.2, is based on two sets of grating
with a relatively low frequency, eg. 50 lines/mm [1 .4-1 .6]. One set of grating is
strictly attached to the surface of a ﬂat specimen. Another set of grating identical

grating is used as reference and is required to create a moire fringe pattern. [1.6]

A moire fringe pattern consists of contour lines of in-plane displacements

in the directions normal to the reference grating lines. A recorded fringe pattern

i“‘

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another

can be are related to in mathematical equations to determine the deformation of
a tested specimen. An equation combining displacement and moire fringe is

given below:

U=iN, (in

f
where U is the displacement in the direction perpendicular to the reference
grating lines, f is the frequency of the reference grating, and N is an integer value
assigned for the contour lines and is known as fringe order. Strains of the tested
specimen can then be obtained by differentiating Equation (1 .1) In the case that
the grating lines are normal to the x-axis, the strains in the x-directions should

become

. 1mm. «2)
fax

 

Apparently, the mathematical equation is also dependent on the frequency
of the reference grating [1.5]. Geometric moire has been used in, many
applications, however, its sensitivity is too low for measurements involving small

deformation.

Moire interferometry

In an effort to increase the sensitivity of displacement measurements, the
geometric moire technique was extended to an interferometric technique. This

new technique was called moire interferometry. Moire interferometry uses a light

     
  

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lllllllw

l
l

l .
K=-5-4-3'2-1 O 1 2 34 56

Figure 1.2- Schematic diagram for geometrical moire formation

source capable of high sensitivity. Moire interferometry is based on the diffraction
of two coherent beams from specimen and the interference of the diffracted
beams as well. One of the primary elements of moire interferometry is the
diffraction specimen grating. When a load is applied to a specimen, the specimen
will deform along with 'the grating that is attached on the specimen surface. This
deformation changes the frequency and orientation of the grating. Subsequently,
when two coherent light beams illuminate on the deformed specimen in equal but
opposite angles, the wave-front of the diffraction becomes warped The two
warped wave-fronts will interfere with one another to create a contour map

known as the moire fringe patterns as shown in Figure 1.3.

Moire interferometry can be explained in a casual way that the specimen
gratings interferes with a virtual reference grating created by interference of two
coherent laser beams [1.2]. The specimen grating is usually of a cross-line
grating, meaning that it has two sets of mutually perpendicular grating lines and
allowing for measurements in two directions normal to the grating lines. Most
often, the cross-line grating is positioned in such a way to measure
displacements in the x-direction and the y-direction. Hence, displacements are
related to two images (x-displacement fringe pattern and y-displacement fringe

pattern) by an expression similar to that of Equation (1 .1), i.e.

1

U =—Nx,and V =—1—N (1.3)
f f

y

Diffraction of beam1

 

 

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Figure 1.3 - Schematic diagram of moire interferometry

where U, and V are displacements in x and y- axis respectively, and Nx and Ny

are integer values representing the fringe orders for the U and V, respectively.

The complete two-dimensional strains can then be expressed by

combining Equation (1.3) with strain-displacement relations,

 

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Literature review

Diffraction grating

The fundamental and essential tool needed for moire interferometry to
work is the high frequency diffraction grating. There are two methods of making
grating [1.7-1.8]: the direct exposure method onto the specimen and the
replication method. In the direct method, a photoresist material is spread on the
specimen surface. The specimen is then exposed to an interference patterns
formed by two coherent beams for direct grating. In the replication method, a

submaster grating is used to transfer the grid onto the specimen surface.

The direct exposure method is applied in a high temperature environment.

It is to prepare a high temperature grating [1 .9-1 .13]. lfju et al [1.13] coated a

specimen with a ﬁlm of reﬂective metal and then applied a photoresist layer. The
specimen was then exposed to an interference pattern. lon etching was then
applied to remove the photoresist layer while retaining the grating of the reﬂective
metal. This technique results in a zero thickness grating that can withstand high
temperature environments. Cloud et al [1.10] and Kang et al [1.11] created high
temperature grating by exposing a photoresist layer to radiation from a mercury
light through a submaster grating. Kang et al [1.11] used the same procedures as
Cloud et al [1.10], to generate grating except substituting the reﬂective metal by a

special photoresist painting.

The replication method utilizes the following procedures. A layer of
photoresist material is applied to a glass substrate and is then exposed to an
interference pattern grating to generate a holographic grating. The gratings is
then produced via replication, which is well described by Basehore et al. [1.14],
Post at al. [1 .2], Walker et al. [1.16], and Rowlands et al. [1.17]. The most usable
method of replication is presented by lﬁu [1.18]. His procedures use the following
steps (a) A master grating made of silicone is replicated onto glass substrate
using an epoxy resin, the two layers are then separated. (b) Two successive
layers of reﬂective metals separated by Photo-ﬂo (from Eastman Kodak
Company) are evaporated on the epoxy grating. The upper layer of the metal is
separated and transferred to the specimen by a thin ﬁlm of epoxy. Guo et al. [1.7]

also gave a brief descriptions of replication method.

Boeman [1.20] presented a new technique to replicate grating on semi-

cylindrical surfaces and other curved surfaces. By using a ﬂexible substrate he

replicated the deformed grating from the circumference of a hole. The ﬂexible
substrate was then be ﬂattened and placed on the interferometer for interference
with the virtual reference grating. Mollenhauer [1.20] used a replication technique
similar to Boeman's technique, however, instead of ﬂattening the deformed
grating from the cylindrical shape, he kept the deformed grating on the circular
sector. He then recorded the moire fringe pattern in discrete sections with an

increment of ﬁve degrees

Applications of moire interferometry

Moire interferometry has become a widely adopted technique for
measuring displacements and hence strains due to its whole-ﬁeld and high-
sensitivity capability. It has many applications in areas ranging from conventional

metals to composite materials.

Barbat et al. [1.21] utilized moire interferometry to measure the shear
strains along the contact zone between a die and the metallic material. They
veriﬁed the frictional force and identiﬁed the frictional condition. Klein et al. [1.22]
measured strains for polymer- based composites subjected to compressive
loading. The strains were used to determine the viscoelastic behavior of the
composites under compressive loading. Bowles et al. [1.23] investigated moire
interferometry in a thermal environment. In their study, Moire was used in high
temperature for measuring coefﬁcients of thermal expansion of graphite/epoxy

laminates.

1O

Moire and microscopic moire were studied extensively by Han [1.24-1.26]
to determine thermal deformation for microelectronic devices. Han introduced a
frequency with a higher grating of 4800 lines/mm by passing the laser light

through a medium with a higher index of refractions.

McKelvie [1.27] used the technique as a tool to satisfy the boundary
condition of a crack due to delamination in multi-layered composites. He used
moire interferometry as a tool to calculate the J-lntegral in a multi—layered
multidirectional composite structure around the tip of crack on a free-edge
surface. He then substituted the experimental deformation into a ﬁnite element

model to ﬁnd the G. and G...

An important investigation on the micromechanical behavior of thick-
laminated composite was conducted by Post [1.4] and Guo et al [1.29]. Their
studies showed that there was a strong free-edge effect when graphite/epoxy
was under compression. Molenhauer [1.20] used a grating technique modiﬁed
from Bemoan [1.19] to measure the strain concentration of around a drilled hole

in multi-layered graphite/epow composites.

Moire interferometry had been used for measuring surface displacements
in structures with complex geometry. As technology advanced, there was a need

to integrate results from more measurements with a computational models.

11

Objectives

The objectives of this research are

1. To modify the moire grating technique, which will extend the moire
interferometry to more versatile measurements and to obtain more
information in each measurement. Effort is centered in reducing the coating
layers from the conventional two-layer grating to a modiﬁed one-layer grating.

2. To combine the moire interferometry and the photoelasticity technique, which
will be capable of presenting both moire and photoelasticity fringe patterns
simultaneously. Effort is centered on modifying the moire interferometer to
incorporate the polariscopes in two different systems: Transparent
photoelasticity, and reﬂection photoelasticity.

3. To investigate the free-edge effect on distributions of surface strains. This
method of high interlaminar shear stresses due to free-edge effect is an
important concern in designs of composites laminates. This investigation will
also evaluate the feasibility of experimental methods based on surface
measurements such as moire interferometry.

4. To explore the joining force of assembled composite beams. Effort is
centered on quantifying the joining conditions, both analytical and
experimental for different joining techniques. The joining techniques will
include various combinations such as adhesive bonding, mechanical riveting,
stitching joining and combinations of these techniques. The joining techniques

will be considered as alternatives to thick-laminated composite plates.

12

Organization

This work is composed of ﬁve chapters: Chapter one is an introduction,
which describes brieﬂy the historical background, and the objectives of the
research. Chapter two deals ﬁrst with modifying the moire grating, and second
with combining moire interferometry and photoelasticity. Chapter three
investigates the mismatch of materials properties and the free edge effect on
distributions of surface strains. Chapter four focuses on the joining forces in
assembled composite beams. Chapter ﬁve draws the conclusions from the

studies presented.

13

References

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10

1.11

1.12

Walker, C. "A Historical Review of Moire Interferometry" Experimental
Mechanics, 281-299, (December 1994).

Post, D., Han, B., and Ifju, P., High Sensitivity Moire, Springer-Vedag,
1993

Post, D. "Chapter 7, Moire Interferometry" Handbook on Experimental
Mechanics, Second Edition, Edited by Kobayashi, A, SEM press, 297-
329.1993.

Cloud, 6.,” Optical methods of engineering analysis,” Cambridge
University Press 1995.

Parks, V, "Chapter 6, Geometric Moire" Handbook on Experimental
Mechanics, Second Edition, Edited by Kobayashi, A, SEM press, 267-
296,1993.

Sciammarella, C., "The Moire Method - A Review" Experimental
Mechanics, 418-433, November 1982

Guo, Y., Ifju, P., Boeman, R., and Dai, F., ”Formation of Specimen Grating
for Moire Interferometry Applications,” Experimental Techniques, 28-32,
(September/October 1999.)

Guo, Y., Ifju, P., Boeman, R., and Dai, F., ”Formation of Specimen Grating
for Moire Interferometry Applications,” Abstracts Proceeding of SEM
Spring Conference, 291-292, Iellevue, WA, (June 1997.)

Huimin, P., Dietz, P., Fulong D. and Schmid, A.,” Fabrication of High
Temperatures Grating for High Temperature, “Experimental Techniques,
28-29, (September/October 1995.)

Cloud, 6., Radake, R. and Peiffer, J., ”Moire Grating for High
Temperatures and Long Times,” Experimental Mechanics, 19 (10), 1-3,
(October 1979.)

Kang, B., Zhang, G., Jenkins, M., Ferber, M., Ifju, P., “Development of
Moire Interferometry for In Situ Material Surface Deformation
Measurement At High Temperature,” proceedings of the ATEM 93
Conference on Advanced Conference on Advanced Technology in
Experimental Mechanics, Kanazawa, Japan, 964-976, (August, 1993.)

Guo, Y., Chen, W., and Lim, K., ”Experimental Determinations of Thermal

14

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.24

1.25

Strains in Semiconductor Packaging Using Moire interferometry,”
Advances in Electronic Packaging, ASME, 779-784, (1992.)

Ifju, P., Post, 0.,” Zero-Thickness Specimen Grating For Moire
Interferometry,” Experimental Techniques, 45-47, (March/April 1991 .)

Basehore, M. and Post, 0., “High-frequency, High-reﬂectance
Transferable Moire Gratings,” Experimental Techniques, 29-31 (May
1984.)

Rowalds, R.,Vallem, J., ”On Replication for Moire-Fringe Multiplication,”
Experimental Mechanics, 167-169, (May 1980.)

Walker, 0., and Mcklevie, J.,"A Practical Multiplied-Moire System,”
Experimental Mechanics, 316-320, (August 1978.)

Rowalds, R.,Vallem, J., ”On Replication for Moire-Fringe Multiplication,”
Experimental Mechanics, 167-169, (May 1980.)

Ifju, P, ”Replication of Master Grating," Special report.

Boeman, R. "lnterlaminar Strains at The Free-Edge of a Hole for Thick
Composites in Compression," Proceeding of SEM Spring Conference
Milwaukee, Wisconsin, 203-315, June 1991.

Mollenhauer, D. " Interlaminar Deformation at Hole in Laminated
Composites: A detailed Experimental Investigation Using Moire
Interferometry," Ph.D. Dissertation, Virginia Polytechnic Institute and State
University, Blacksburg, VA, August 1997.

Barbat, S., "Metal-Forming studies Iy Moire Interferometry," Experimental
Techniques, 25-28, (June 1989.)

Klien, R.,Tuttle, M., ”Strain Measurements ln Composites Subjected To
Compressive Loading Using Moire Interferometry,” Proceeding of SEM
Spring Conference Huston, Texas, 649-656, June 1987.

Bowels, D. and Post, D., and Herakovich, C. T., and Tenney, D., R.” Moire
Interferometry for Thermal Expansion of Composites," Experimental
Mechanics, 441-447, (December 1981 .)

Han, B, ”Higher Sensitivity Moire Interferometry for Micromechanics
Studies," Optical Engineering, Vol. 31, N07, 1517-1526, (1992.)

Han, B. "Extension of Moire Interferometry Into The Ultra-High Sensitivity
Domain", Ph.D. Dissertation, Virginia Polytechnic Institute and State

15

 

1.26

1.27

1.28

University, Blacksburg, VA, March 1991.

Han, 8., "Recent Advancements of Moire and Microscopic Moire
Interferometry for Thermal Deformation Analysis of Microelectronic
Devices,” Experimental Mechanics, 278-288, (December 1998.)

Mcklevie, J., and Perry, K." Moire Interferometry as a Detailed Validator
for Computational modeling of Composites," Composite Structures, 299-
305, (1998).

Guo, Y., Post, 0., and Han, B. ”Thick Composite in Compression:

Micromechanical Behavior and Smeared Engineering Properties,” Journal
of Composite Materials, Vol. 26, No. 13, 1930-1944, (1992)

16

Chapter 2

COMBINING MOIRE INTERFEROMETRY AND PHOTOELASTICITY

Abstract

Both moire interferometry and photoelasticity have been used for stress
analysis of composite structures. In this study, polariscopes were incorporated
into a moire interferometer to form a system capable of presenting both moire
and photoelasticity fringe patterns. However, obtaining both types of fringe
patterns simultaneously requires a transparent moire grating and a reﬂection
photoelasticity coating (also transparent) on the surface of a testing specimen. In
this study, an epoxy material named PL8 commonly used for photoelasticity
coating was used for both purposes. In other words, both the moire grating and
reﬂection photoelasticity coating were made of the same epoxy material and
molded simultaneously. In order to develop such a one-layer grating technique
for two different types of measurement, a systematic study was conducted on
reduction of the layer number of coating from conventional two-layer grating to a
modiﬁed one-layer. To verify the feasibility of the new grating technique, a
glass/epoxy specimen was coated with the single-layer moire-photoelasticiy
grating. A hole was prepared at the center of the specimen, which was then
subjected to four-point bending. Strain distributions obtained from moire
interferometry and reﬂection photoelasticity were found to have a fair agreement

with those from ﬁnite element analysis.

17

Introduction

Due to its high accuracy and high sensitivity, moire interferometry has
been commonly used in both macroscopic and microscopic strain
measurements. Many applications can be found in the literature [2.2-2.18].
Details of the technique for moire interferometry has been well summarized in a
book authored by Post, Han, and Ifju [2.19]. Figure 2.1 shows one of the
common setups of moire interferometry. It is also the fundamental setup used in

the present study.

As shown in Figure 2.1, the point laser beam is diverged through a spatial
ﬁlter and then redirected and collimated by a parabolic mirror. The enlarged
collimated light is then illuminated onto four mirrors (M) located in the moire
interferometer. For reasons of clarity, only the two horizontal ones (the right and
the left) are shown in the ﬁgure. The light beams reﬂected from these four mirrors
are directed onto another four mirrors for further beam angle adjustment. (Again,
only the two horizontal mirrors are shown in the diagram.) The moire
interferometry is based on optical diffraction and interference. As the two beams
(eg. the right and the left) are emitted from the second set of mirrors, the beams
are reﬂectively diffracted by the moire grating on the testing specimen. Under a
no-loading condition, the two diffracted beams should point in the direction
normal to the specimen surface and cause no interference between them.
However, if the specimen is deformed due to external loading, the pitch of

moire grating will be altered. And the diffracted beams will no longer be

18

 

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Laser
M01re
Interferometer ,
Spatlal
Filter
I I M
/ -
0: ﬂ Double-Sided Mirror
\ ,
\L.‘ .
Specimen $ Parabohc
................ M _ Mirror
Computer I anqgal
& Image ‘
Processmg

 

 

Figure 2.1- Schematic diagram of optical setup for moire interferometry
(horizontal two beams only).

parallel to each other and normal to the specimen surface. Interference between
the beams will then take place and form a moire fringe pattern representing the
contours of constant displacement. The two horizontal beams (the right and the
left) can give horizontal displacement (U-displacement) contours while the two
vertical beams (not shown in the diagram) gives vertical displacement (V-
displacement) contours. The deformation of the specimen can be calculated from

the contours (moire fringe patterns).

The objectives of the present study is to extend moire interferometry to
more versatile measurements and to obtain more information in each
measurement, possible modiﬁcations in both grating technique and the optical

setup need to be explored.

Grating techniques
Two-Ia yer grating
Conventional technique

In addition to the moire interferometer and optic setup in Figure 2.1,
another important element of moire interferometry is moire grating. As a
conventional practice [2.6], the preparation of moire grating for a testing
specimen can be achieved by the following procedures. First of all, a mother
grating made of epoxy is required. A layer of aluminum is then coated on the
surface of the mother grating. The aluminum layer is then transferred onto a

testing specimen via a sticky coating layer of epoxy on the specimen surface.

20

Once the epoxy layer is cured, there exist two layers, epoxy and aluminum, on

the specimen. The technique is then called two-layer grating technique.

Modiﬁed technique

Many studies aiming at improving the moire grating can be found in the
literature [2.14-2.19]. The following procedures for grating preparation were
developed in the present study. Instead of the epoxy mother grating used in the
conventional technique, the present technique started from a mother grating
made of silicon rubber. An epoxy was then spin-coated on the surface of the
silicon rubber grating. In addition to duplicity the moire grating, the epoxy layer
was also used as a bonding medium to transfer the grating onto a testing
specimen. Once the epoxy is cured, it can be coated with aluminum to enhance

the capability optical reﬂection.

Figure 2.2(a) shows the schematic diagram of a specimen with two-layer
grating for both the conventional and simpliﬁed techniques. In an effort to further
simplify the grating technique but retain good quality of moire fringe pattern for

strain analysis, a few studies listed below were explored.

One-Ia yer grating

In an effort to reduce the number of layers, the aluminum coating layer

was omitted from the two-layer grating technique, resulting in one-layer grating,

i.e. epoxy layer only as shown in Figure 2. 2(b) schematically. It was expected

21

aluminum W

epoxy l
specimen

 

 

(a)

 

IEpoxy

 

Spechnen

(b)

spechnen
(C)

Figure 2.2- Schematic diagrams of specimen with (a) two-layer grating, (b) one-
layer grating, and (c) zero-layer grating

22

that the optical reﬂection would be hindered up to some extent. In order to
investigate the effects of aluminum coating on the image enhancement, the
following studies representing specimens with different degrees of reﬂection were

performed:

A. Black Specimen — In this study, an aluminum specimen with dimensions of
100 mm x 12.5 mm x 3 mm was coated with a layer of black paint. An
epoxy layer was then spin-coated on the surface of the black paint layer

before being printed with moire grating.

B. White Specimen - Instead of the black paint mentioned in A, a layer of
white paint was coated on the surface of the aluminum specimen before

the epoxy-grating layer was introduced to the specimen.

C. Bright Specimen - The aluminum specimen was polished carefully to have
a very smooth and shining surface before being coated with the epoxy

grating.

D. Transparent Specimen - Instead of aluminum specimen, a photoelasticity
sensitive material PSM-5A, was coated with epoxy grating, which is also

transparent.

These four specimens represented four different degrees of reﬂection. The
bright specimen gave the best reﬂection followed by the white specimen, which
had better reﬂection than the black specimen. Being transparent, the specimen

had the lowest capability of reﬂection.

23

All four specimens were loaded by four-point bending. Experimental
results of the two extreme cases of reﬂection, C and D, are shown in Figures
2.3(a) and 2.3(b), respectively. (The image from specimen C was specially
prepared by using a ﬁlter to reduce the strength of the shining background.)
Apparently, both specimens show details of moire fringe patterns. And, in fact, by
adjusting the light intensity and exposure time, it was possible to obtain images
with similar quality. In other words, It was possible to obtain good quality of fringe

pattern with less efforts without using the aluminum coating layer.

Zero-Ia yer grating

To further reduce the number of grating layers, a technique called zero-
layer grating was developed. In fact, ﬁrst of all, a silicon rubber mold was built for
casting the epoxy specimen. The mold was essentially a container for
accommodating liquid polymer. In order to cast an epoxy specimen with moire
grating on its surface in one single process, an aluminum specimen having the
same in-plane dimensions as the specimen-to-be was coated with a silicone
rubber moire grating on the surface and the specimen was ﬁxed at the bottom of
the silicon rubber mold. The thickness of the aluminum specimen was chosen so
that the remaining height of the silicon rubber mold was equal to the thickness of

the specimen-to-be. A liquid epoxy was then poured into the mold. And a dead

24

. . . . _~ ,
. .:‘ -.'
‘ .' ,. t. "
- .l .I
- ..«I‘ > ‘ .
, “
... ,_,.

.1..._\. ,

mm. ‘

 

Figure 2. 3 - U-displacement moire fringe patterns from four-point bend tests for
(a) polished aluminum specimen and (b) transparent PSM-5A specimen

25

weight was placed on top of the liquid polymer. Figure 2.4 shows the schematic
diagram of the processing setup. Once the epoxy specimen was cured, it could
be popped out easily due to the use of silicon rubber as the mold. Since both the
specimen and the coating were made of the same material and formed
simultaneously, i.e. there was no independent layer for grating, this type of
specimen was called no-layer grating. A schematic diagram for no-layer grating

is shown in Figure 2.2(c).

Combining moire and photoelasticity

Transmission photoelasticity

Applying four-point bending to the specimen of no-layer grating resulted in
high quality moire fringe patterns. In addition, since the specimen was made of
high-sensitivity photoelasticity material PSM-5A, clear transmission
photoelasticity fringes could be obtained by adding a polarizer and an analyzer to
the existing moire interferometry system. Figure 2.5 shows the schematic
diagram of the experimental setup. The dotted lines are used in the diagram to
enhance the optical path for photoelasticity investigation whereas the solid lines
for moire interferometry. Figures 2.6(a), 2.6(b) and 2.6(c) show the results of
photoelasticity fringe pattern (isochromatic), moire fringe pattern (U-

displacement) and the simultaneous recording of the two, respectively.

26

 

4—— dead weight

 

 

<—-

—-— silicon rubber mold

 

A

liquid polymer

 

wwwvwmmwxcf

 

‘i

silicon rubber grating
aluminum mold

 

 

 

 

 

 

Figure 2. 4 - Schematic diagram of processing setup for casting a specimen with
zero-layer grating.

27

   
 
   

Mirror

Light Parabolic

A 1 .
na yzer error

Figure 2.5 - Schematic diagram of optical setup for combining moire
interferometry and transmission photoelasticity.

28

 

(C)

Figure 2.6 - Fringe patterns of PSM-5A beam under four-point bending: (a)
transmission isochromatic fringes, (b) U-displacement fringes, and (c)
simultaneous recording of photoelasticity and moire fringes

29

Reﬂection photoelasticity

Transmission photoelasticity can present high-quality isochromatic fringe
patterns and has been commonly used in engineering applications.
Unfortunately, most engineering materials are opaque or have low sensitivity of
photoelasticity phenomenon. Hence, it is necessary to use the technique of
reﬂection photoelasticity for those materials {2.20-2.22}. Reﬂection photoelasticity
fringe pattern could be obtained by coating the PSM-5A specimen of one-layer
grating with a reﬂective material, such as gold, on the surface opposite to that
having moire grating. A few setups were explored. Figure 2.7 shows the position
of the polarizer and analyzer of the experimental setup found to be the most
convenient and gave the highest quality of image among the techniques
explored. The dotted lines again show the optical path of the photoelasticity
investigation. Experimental results of photoelasticity fringe pattern (isochromatic),
moire fringe pattern (U-displacement), and the combination of the two fringe

patterns are given in Figures 2.8(a), 2.8(b), and 2.8(c), respectively.

Single coating for both moire and photoelasticity

The technique combining moire interferometry and reﬂection
photoelasticity can be applied to both transparent and opaque specimens. In
order to simplify the technique requiring two sets of coating, a single coating
technique was developed for both moire interferometry and reﬂection
photoelasticity. A procedure similar to casting zero-layer grating was followed, in

casting the single moire-photoelasticity coating. Firstly, an aluminum specimen

3O

 

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Laser |
Spatial Filter P olarizer
M [E Monochromatic
.......... light
\I L4 .
M ' H'M
Double-Sided Mirror
\P1 b 1'
Specimen ara 0 1c
. Analyzer Mirror
CCD
Camera
Computer
& Image
processmg

 

 

 

Figure 2.7 - Schematic diagram of optical setup for combining moire
interferometry and reﬂection photoelasticity.

31

 

(C)

Figure 2.8 - Fringe patterns of a PSM-5A beam under four point bending: (a)
reﬂection isochoromatic fringes, (b) U-displacement fringes, and (c) simultaneous
recording of photoelasticity and moire fringes.

32

having the same in-plane dimensions as the mold and coated with silicon rubber
moire grating was ﬁxed at the bottom of the silicon rubber mold as shown in
Figure 2.4. Liquid epoxy PL8 was then poured into the mold until the height was
equal to the desired thickness. The testing specimen, followed by dead weight,
was then placed on the surface of the liquid epoxy. The entire setup was left for

curing at room temperature for about 24 hours.

A glass/epow specimen coated with the PL8 moire-photoelasticity layer
was prepared and subjected to four-point bending. Results of the photoelasticity
fringe pattern, moire fringe pattern, and the simultaneous recording of the two

patterns are given in Figure 2.9(a), 2.9 (b), and 2.9(c), respectively.

Case studies

Effects of coating thickness

Glass/epoxy specimens with thickness of 6.4 mm were used in the study.
In order to investigate the thickness effects of the moire-photoelasticity coating
on the measurements for both moire and photoelasticity fringes, three thickness
of coating were prepared in the study. They were 0.55 mm, 1.36 mm and 3.17
mm. In order to prepare the 1.36 mm and 3.17 mm coatings, two silicon molds of
different thickness were fabricated; whereas the 0.55 mm coating was prepared
via the technique of one-layer grating. Figures 2.10(a), 2.10(b) and 2.10(c) show

the moire fringe patterns of the three specimens subjected to four point bending.

33

 

Figure 2.9 - Fringe patterns of a glass/epoxy beam with PL8 moire photoelasticity
coating (a) reﬂection isochromatic fringes, (b) U-displacement fringes, and (c)
simultaneous recording of photoelasticity and moire fringes.

34

 

Figure 2.10 - Moire fringe patterns of glass/epoxy beams with PL8 grating
thickness of (a) 0.55 mm, (b) 1.36 mm, and (c) 3.17 mm

35

Apparently, the differences among the three moire fringe patterns are not
signiﬁcant, indicating the good bonding between the coatings and the specimens.
The moire fringe patterns also imply the effect of coating reinforcement on
specimen deformation. However, due to the fact that the sensitivity of a
photoelasticity coating is proportional to its thickness, the 3.17 mm coating gave
a denser and sharper fringe pattern than the 1.36 mm one under the same
loading condition. Figures 2.11(a), 2.11(b) and 2.11(c) show the testing results.
Apparently, the 3.17 mm case has the highest photoelasticity fringe density and

is suitable for further stress analysis.

Stress concentration

In order to evaluate the feasibility of using the combined technique to
common stress analysis, an aluminum beam with dimensions of 100 mm x 12.5
mm x 6.4 mm was coated with a moire-photoelasticity layer of 3.17 mm. A hole of
3mm in diameter was introduced to the center of the specimen by a drilling
process of 300 rpm and using water as coolant. The specimen was then loaded
in a four point bending ﬁxture. Figures 2.12(a) and 2.12(b), and 2.12(c) show
photoelasticity, moire and combined fringe patterns, respectively. In addition, a
ﬁnite element analysis Figure 2.13 based on ABAQUS was performed to
compare the experimental results. Figure 2.14 shows the experimental results
and numerical simulations of the bending strain distributions along the line across
the hole center (shown by the dotted line). Note that because of the low

photoelasticity sensitivity of the coating, the results from the reﬂection

36

 

(C)

Figure 2.11 - Photoelasticity fringe patterns of glass/epoxy beams with PL8
grating thickness of (a) 0.55 mm (b) 1.36 mm, and (c) 3.17 mm

37

-.

is...

.- .....~.-. ——..~..—..»...... ... . . .. .... .. .. .... -
mu ~o.mv.n 1am—5- .. . -
t

”I W4, 144%.:er .
\ .v “if

“in-Inu— a .7..ﬂ>-u‘w~ v-cyg-v. w... “u”...

 

Figure 2.12 - Fringe patterns of a notched glass/epoxy beam with PL8 moire-
photoelasticity coating (a) reﬂection isochromatic fringes, (b) U -displacement
fringes, and (0) simultaneous recording of photoelasticity and moire fringes

38

Load

\11’

   
  

f

     

Boundary
conditions

Boundary
conditions

4.

   

Figure 2.13 - Schematic diagram of ﬁnite element model for the beam with
central hole; it has 1482 elements, the element type is 8 node element of
CPSBR,

39

P/2
:25.4mm_l_ 25.4mm

P/2

 

 

 

 

 

 

 

125.4 mm
1 Y I
i
O 12.7 mm
P/2 P/2
y(mm)
. D 13 "
QB 12 ' O Moire
D A 11 '
El 10 . EIFEA
U
[[315 9 - A Photoelasticity
u 8 r
O 7 .
6 u
5 C
4 - ”A: D
3 .
2 - E9130
1 . D q—O-t

 

-400.0 -300.0 -200.0 -100.0 0.0

Saw)
100.0 200.0 300.0 400.0

Figure 2.14 - Comparisons of results from moire interferometry, photoelasticity,
and ﬁnite element analysis.

40

photoelasticity are not as detailed as the moire counterpart. And the results

based on all three techniques seem to agree with one another fairly.

Summary

Based on the experimental results and stress analysis, the following

conclusions can be drawn:

A simpliﬁed two-layer grating technique was developed. Two-layer grating
could be further simpliﬁed to one-layer grating by simply omitting the enhancing
layer, i.e. aluminum layer. The one-layer grating technique was found to be

feasible for moire measurements.

It was worthwhile to develop the technique of no-layer grating since the
“layer" did not have thickness; neither was there mismatch of material properties

between the specimen and the “layer.”

The technique combining moire and photoelasticity together was capable
of presenting two independent sets of experiment and result. It provides a self-

veriﬁcation capability or double—check system.

Acknowledgements

The author wishes to express his sincere thanks to Professor Daniel Post

of Virginia Tech for designing the moire interferometer, Professor Peter Ifju of

41

University of Florida for providing the master moire grating, and Professor Andre

Lee of Michigan State University for valuable discussions on spin coating.

42

References

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

Liu, 0., Shakour, E.” Combining Moire Interferometry and Photoelasticity"
Submitted to Experimental Mechanics.

Basehore, M. and Post, D., “High-frequency, High-reﬂectance
Transferable Moire Gratings,” Experimental Techniques, 29-31 (May
1984).

Ifju, P.,Kilday, B.,Niu, X., “Residual Stress Measurements in AS4/350-6
Laminates,” Abstract Proceedings of SEM Spring Conference, 337-338,
Bellevue, WA, (June 1997).

Bowels, D. and Post, D., and Herakovich, C. T., and Tenney, D., R.” Moire
Interferometry for Thermal Expansion of Composites," Experimental
Mechanics, 441-447, (December 1981).

Ifju, P.," The Shear Gage: For Reliable Shear Modulus Measurements of
Composite Materials," Experimental Mechanics, 369-378, (December
1994)

Czarnek, R., Post, D., "Strain Concentration Factors in Composite Tensile
Members with Central Holes," Proceedings of the 1987 SEM Spring
Conference on Experimental Mechanics, Houston, TX, 657-663,(June
1987)

Czarnek, R., and Guo, Y. “ Surface Contouring Using Virtual Grating,”
Experimental Techniques, 35-37, (May/June 1991).

Bradley, D., Adams, 0., Gascoigine" Interlaminar Strains and
Compressive Strength Reductions Due to Nested Layer Waviness in
Composite Laminates," Journal of Reinforced Plastics and Composites, 17
(11), 989-1011, 1998.

Shaw, M., "The Effects of Strength Probabilistics on Fracture Mode of
Ceramic/metal Multilayers", Engineering of Fracture Mechanics, 61, 49-74,
1998.

Huimin, P., Dietz, P., Fulong D. and Schmid, A.," Fabrication of High
Temperatures Grating for High Temperature, “Experimental Techniques,
28-29, (September/October 1995).

Cloud, 6., Radake, R. and Peiffer, J., ”Moire Grating for High

Temperatures and Long Times,” Experimental Mechanics, 19 (10), 1-3,
(October 1979).

43

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2.20

2.21

2.22

2.23

2.24

Kang, B., Zhang, G., Jenkins, M., Ferber, M., Ifju, P., “Development of
Moire Interferometry for In Situ Material Surface Deformation
Measurement At High Temperature,” proceedings of the ATEM 93
Conference on Advanced Conference on Advanced Technology in
Experimental Mechanics, Kanazawa, Japan, 964-976, (August, 1993).

Guo, Y., Chen, W., and Lim, K., ”Experimental Determinations of Thermal
Strains in Semiconductor Packaging Using Moire interferometry,”
Advances in Electronic Packaging, ASME, 779-784, (1992).

Guo, Y., Ifju, P., Boeman, R., and Dai, F., ”Formation of Specimen Grating
for Moire Interferometry Applications,” Abstracts Proceeding of SEM
Spring Conference, 291-292, Bellevue, WA, (June 1997).

Walker, 0., and Mcklevie, J.,”A Practical Multiplied-Moire System,”
Experimental Mechanics, 316-320, (August 1978).

Ifju, P., Post, D.,” Zero-Thickness Specimen Grating For Moire
Interferometry,” Experimental Techniques, 45-47, (March/April 1991).

Rowalds, R.,Vallem, J., ”On Replication for Moire-Fringe Multiplication,”
Experimental Mechanics, 167-169, (May 1980).

Post, D., Han, B., and Ifju, P., High Sensitivity Moire, Springer-Verlag,
1993.

Cloud, 6.,” Optical methods of engineering analysis,” Cambridge
University Press,1995.

Hung, Y., and Pottinger, M., “An Improved Oblique-incidence technique for
Principal-strain Separation in Photoelastic Coating,” Experimental
Mechanics, 55-63, (February 1962).

Zandman, F., Render, 8., and Riegner, E., “Reinforcing Effect of
Birefringent Coating,” Experimental Mechanics, 55-63, (February 1962).

Theocaris, P., and Daferrnos, K. “A Critical Review on the Thickens Effect
of Birefringent Coating,” Experimental Mechanics, 271-276, (September
1964)

Zandman, F., and Post, 0., “ Photoelastic-coating Analysis in thermal
Fields,” Experimental Mechanics, 215-221, (September 1963).

Daniel, M., Rowards, R., and Post, D. “Strain Analysis of Composite by
Moire Methods,” Experimental Mechanics, 246-252, (June 1973).

2.25

2.26

2.27

Pipes, B. ”lnterlaminar Stresses in Composite Laminates Under Uniform
Axial Extension," Journal of Composite Materials, Vol. 4, 538-459,
(October 1970).

Han, B., and Post, 0., ”Immersion Interferometer for Microscopic Moire
Interferometry," Experimental Mechanics, 38-41, (March 1992).

Han, B., "Recent Advances of Moire and Microscopic Moire Inerferometry

for Thermal Deformation Analyses of Microelectronic Devices,"
Experimental Mechanics, 278-288, (December 1998).

45

Chapter 3

MISMATCH OF MATERIAL PROPERTIES AND FREE-EDGE EFFECT

Abstract

The free-edge effect is a unique and important phenomenon in composite
laminates because it could cause high interlaminar stresses on laminate
interfaces along free edges and eventually would result in delamination in
composite laminates. This study investigated the free-edge effect in
graphite/epoxy and glass/epoxy laminates. Moire interferometry was used in the
investigations. Experimental results revealed that the shear strains on free edges
increased as the width of a composite laminate decreased. However, the normal
strains on free edges were not affected by the width change. In addition, the
depth of boundary layer was found to be greater than the laminate thickness,
especially in the graphite/epoxy laminate, as usually assumed. The free-edge
effect, in fact, was associated with mismatch of material properties. Since
graphite/epoxy had higher anisotropy than glass/epoxy, it experienced a stronger
free-edge effect than the glass/epoxy counterpart. Similarly, a composite
laminate consisting of thicker laminae had a higher inﬂuence from free edges

than that consisting of thinner laminae.

46

Free-edge effect

Fiber-reinforced polymer-matrix composites are excellent materials for
high-performance structures. They are usually used in thin-laminate form
because their properties in the plane of laminates can be excellent. However,
their properties in the thickness direction of laminates are not outstanding
because composite laminates are weakly bonded by matrix through the
thickness. In addition to the poor bonding strength, sudden changes of material
properties across laminate interfaces also pose a serious concern on laminate
strength. The sudden changes of material properties could be due to different

ﬁber orientations or different material types across the laminate interfaces.

When a composite laminate is subjected to loading, some stress
components on the laminate interfaces, so-called interlaminar stresses, could
become tremendously large due to sudden changes of material properties across
the laminate interfaces. As a result of the high interlaminar stresses and the low
interlaminar strengths mentioned earlier, the composite laminate might be

damaged along the laminate interfaces, so-called delamination.

The onset of delamination of a composite laminate subjected to uniaxial
tension always starts from free edges of the composite laminate. This is because
the highest interlaminar stresses are located on laminate interfaces along the
free edges. The phenomenon of high interlaminar stresses associated with free
edges is a unique phenomenon in composite laminates and is called free-edge

effect.

47

Free-edge effect has been well investigated analytically [3.1-3.10]. There
are essentially two major modes of free-edge effect; one is concerning the

interlaminar shear stress r,“ (x and y are in-plane coordinates while 2 is

thickness coordinate) in angle-ply laminates and the other is the interlaminar
normal stress on in cross-ply laminates. When an angle-ply laminate, such as
[+ 6/— 613, is subjected to uniaxial tension, the interface between the +6-Iamina
and the -6?-lamina will have high interlaminar shear stress In because of the
opposite tension-shearing effects between the two laminae. Similarly, when a
cross-ply laminate, such as [0/90]s, is subjected to uniaxial tension, the interface
between the 0-lamina and the 90-lamina will have high interlaminar normal

stress on because of the different Poisson's ratios between the two laminae.

The free-edge effect can cause high interlaminar stresses (both shear and
normal), which are important concerns in composite laminate designs. It is also
critically important to the evaluations of experimental methods based on surface
measurements, e.g. moire, holography, speckle interferometry, etc. The

objectives of this study are

1. to investigate the free-edge effect on distributions of surface strains and

2. to evaluate the feasibility of using moire interferometry for two-

dimensional analysis of composite laminates.

48

Moire interferometry

Owing to its high sensitivity, moire interferometry has been commonly
used in investigations of composite laminates. Like many other optical methods,
moire interferometry is a surface technique. A moire grating (lines with periodic
spacing) should be coated on the interested surface of a testing specimen for
moire investigations. When the grating is illuminated by two laser beams with
special incident angles, the two beams will be reﬂectively diffracted from the
grating, normal to the specimen surface and parallel to each other. As a result,
there is no interference between them and an image of no fringe, so-called null
ﬁeld, should be obtained. However, when the specimen is subjected to loading,
the spacing among the grating lines will be altered , so will be the directions of the
diffracted beams. Consequently, the two beams will interfere each other and form

a set of fringes, so-called moire fringes.

The moire fringe pattern resulting from the interference of the two
diffracted beams is actually a pattern of contours representing constant levels of
displacement in the direction normal to the grating lines. In order to obtain
constant levels of displacement in another direction, another two laser beams
should be used along with a moire grating with lines normal to that direction.
Hence, combining the two sets of displacement contours of moire fringes, i.e. u-
fringes and v-fringes, and the linear strain-displacement relations (for small

deformation), i.e.

49

0v an 6v
— ’ = — + — 301
the strain ﬁeld of the loaded specimen can be found. In the above equations, u

and v are displacement components in the x- and y-direction, respectively, while

6'

x, 5, and 7,0, are the strain components of the surface under moire
investigation.

The four laser beams required for the moire interferometry are generated
from a moire interferometry system. The top view of the setup is shown in Figure
3.1. As can be seen, the point laser beam is diverged through a spatial ﬁlter and
then redirected and collimated by a parabolic mirror. The enlarged collimated
light is then illuminated onto four mirrors (designated as M in the diagram)
located in the moire interferometer (outlined by a rectangle in the diagram
because all components involved are attached to a ﬁxture). The four mirrors, two
horizontally positioned and two vertically positioned, form a cross pattern. (For
clarity, only the two horizontal mirrors and beams are shown in the diagram.) The
laser beams reﬂected from the four mirrors are redirected onto another four
mirrors, which also form a cross pattern, for further beam angle adjustment.
(Again, only the two horizontal mirrors and beams are shown in the diagram.)

Details of the optical technique for moire interferometry can be found in a book

authored by Post, Han and lﬁu [3.12].

50

 

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Laser
M01re
Interferometer .
Spat1al
Filter
/
\ ‘
Specimen $ Parabolic
-.-..- M; _ Mirror
Computer Cgrgga
& Image ‘
Processmg

 

 

Figure 3.1 - Schematic diagram of optical setup for moire interferometry
(horizontal two beams only).

51

Experimental investigations

Moire on composite laminates

Most studies on free-edge effect are based on analytical approaches. Very
few studies are of experimental investigations. Pipes and Daniel [3.11] are the
ﬁrst to use a geometrical moire method to study the free-edge effect. The depth
from free edges of composite laminates and inﬂuenced by the free edges is
usually called the depth of boundary layer and has long been assumed to be
approximately equal to the thickness of composite laminates. Mollenhaur [3.13]
have identiﬁed the interlaminar shear strains around an open hole of a composite
laminate subjected to uniaxial tension based on moire interferometry. The

experimental results agree very well with analytical predictions.

Free edges of composite laminates have high interlaminar strains and
high strain gradients due to the free-edge effect. Hence, they are critical locations
in composite laminates. Since moire interferometry is an excellent method for
surface measurements because of high sensitivity, it seems to be an ideal
technique for studies of free-edge effect. However, moire interferometry has also
been used for two-dimensional measurements based on plane strain or plane
stress assumption. The validity of using strains obtained from moire
interferometry for domains beyond free edges is an important issue for moire

interferometry.

52

Materials and testing

In order to investigate the free-edge effect, both graphite/epoxy and
glass/epoxy were used to prepare cross-ply laminates. The graphite/epoxy
laminates were of 7-Iamina plates and had a stacking sequence of
[0/90/0/90/0/90/0]. Their thickness was 6mm. The glass/epoxy laminates,
however, had three different stacking sequences, i.e. [07/907/07],
[03/903/03/903/03/903/03] and [0/90/0l. . .121, These stacking sequences were
designated as 3-lamina, 7-lamina and 21-lamina. The three types of glass/epoxy

laminates were fabricated together and had identical thickness of 4.29mm.

All testing specimens were machined to have a length of 100mm but
different widths. They were then subjected to four-point bending. The outer two
loading points of the four-point bending ﬁxture were located 75mm apart while
the inner two loading points were 25mm apart. Both the outer loading points and
the inner loading points were located symmetric with respect to the midspan of
the composite laminates. Hence, there was an overhang of 12.5mm on each side

of the specimens.

For moire interferometry, each specimen was coated with a moire grating
along one edge. The moire grating included two sets of lines with constant
spacing and parallel to the x- and y-axis, for measurements of displacement in y-
and x-direction, respectively. Both sets of line had a frequency of 1200 lines/mm.
The edge was then investigated by the moire interferometry system shown in

Figure 3.1 when the specimen was under four-point bending.

53

Effects of specimen width

The free-edge effect, by name, was due to the existence of free edges in
composite laminates. It actually covered the free edges of composite laminates
and their vicinity. Hence, it was of a local phenomenon. The cause of free-edge
effect was associated with mismatch of material properties due to sudden
changes of ﬁber orientation or material through the laminate thickness. Since the
depth of boundary layer was an important concern in laminate designs and a
critical parameter for surface measuring techniques, composite laminates with

various widths were investigated.

In this study, the aforementioned graphite/epoxy laminates were used for
investigations of depth of boundary layer. Three widths, 25.4mm, 12.7mm and
6.35mm, were chosen. A 100mm-long laminate was ﬁrst machined to have a
width of 25.4mm. Once investigated by moire interferometry, the width was cut
into half, i.e. 12.7mm, for a second moire investigation. It was again out into half
to have a width of 6.35 mm for a third moire investigation. During the second and
the third cut on the specimen, the moire grating was protected from

contamination by a thin-layer coating of silicon rubber.

The width of 25.4mm was about four times the laminate thickness.
Because the boundary layer of a composite laminate was commonly assumed to
be approximately equal to the thickness of the composite laminate [1], a
composite laminate should have a minimum width twice the thickness to have a

zone free from free-edge effect. Hence, 25.4mm was considered wide enough for

the graphite/epoxy laminate, which had a thickness of 6mm, to avoid interference
of free-edge effects from both free edges. The 12.7mm specimen, however, had
about the minimum width to avoid interference between the two free edges since
it was about twice the laminate thickness. The 6.35mm specimen was expected
to experience interference between the two free edges because it was only one-

half of the minimum required width.

Mismatch of material properties

Since the free-edge effect was caused by mismatch of material properties
through the laminate thickness, the signiﬁcance of material type in the free-edge
effect became an important issue. Hence, a glass/epoxy laminate also of 7-
lamina stacking sequence, i.e., [03/903/03/903/03/903/03], was investigated to
compare with the graphite/epoxy counterpart. In addition, the signiﬁcance of
different stacking sequences in the free-edge effect was also of interest. In the
study, glass/epoxy laminates with stacking sequences of [07/907/07] and
[0/90/0/...]21, namely 3-lamina and 21-Iamina, were also investigated to compare
with the 7-Iamina counterpart, i.e. [03/903103/903/03/903/03]. In order to compare
the three types of glass/epoxy laminate accurately, the composite specimens
were fabricated together from a curing bag to warrant that an identical curing
cycle was imposed and all laminates had similar thickness and ﬁber volume

fraction.

55

 

 

 

Results and discussions

Strain Calculations

The photographs of moire fringe patterns for displacement components in
x-direction, i.e. u-fringes, and y-direction, i.e. v-fringes, for graphite/epoxy
laminates with all three widths were shown in Figure 3.2. They were the images
between the two inner loading points, i.e. within the constant moment zones. The
magniﬁcation of the photographs could be obtained from comparing the height of
the photographs with the thickness of graphite/epoxy laminates, i.e. 6mm. During
the moire investigations, the photographs were taken by a CCD camera and

recorded in a computer

For comparison, the force exerted on each loading point was set to be
proportional to the specimen width. A force of 1112N was used for the 25.4mm
specimen, 556N was used for the 12.7mm specimen and 278M for the 6.35mm
specimen. By examining the photographs in Figure 3.2, the difference among the
three cases was noticeable. However, it was also noted that the fringe patterns of
all three cases were not of null ﬁeld when there was no loading. Hence, a
quantitative comparison of strain distributions was preferred over a qualitative

description based on moire fringe patterns.

In order to recover the real strains, the initial fringes (at no loading) must
be subtracted from the loaded counterparts. The procedures for ﬁnding the strain

components ax, lay and 7” could be based on the strain-displacement relations.

56

 

7 ' nv-fringes

ﬁlm” .
1...“...2llllelllj

ll
v-fringes

 
  

 

 

u

 

 

.-».-‘.'.'—Z_‘.-.-. .- - .5;

”"93ﬁﬁﬁéé‘
(c) 6.35 mm, 278 N

Figure 3.2 - Moire fringe patterns for graphite/epoxy laminates with three different
widths

57

A technique named vector analysis was also introduced brieﬂy for ﬁnding shear
strains. For demonstration purpose, only calculations of shear strains based on

these two techniques were given below.

7,, based on strain-displacement relations

The shear strain 7,, could be obtained from combining the displacement-

fringe relations, i.e.

 

x N)’ (3 2)
u: , v=—— .
f

and the linear strain-displacement relations given in Equation (3.1). Hence, it

could be expressed by the following strain-fringe equation

 

 

7., = ﬂag: + 6;] (3.3)

In the study, the frequency f of the moire interferometry system was 2400
lines/mm. It was twice the frequency of moire grating. Due to the nature four-
point bending, the ﬁrst component of Equation (3.3) was negligible. Accordingly,

Equation (3.3) could be simpliﬁed to be

1 6N),
7 f ax

 

yxy (3.4)

The distribution of shear strain at a cross-section 6mm right to the midspan, i.e.

x = 6mm, of the 25.4mm-wide graphite/epoxy composite laminate was of

58

 

 

 

 

 

 

L7

 

 

0 2L44 .l-ie 2% 1b 1'2““)

h=6 mm
6hl7 0
Sb” -
4h” - L5.
3hl7 - .L4
2hl7 -
1hl7 - O

 

 

. . . wot)
0 1 2 3 4
(c)

Figure 3.3 - Calculation of shear strain on the laminate interface based on least
square method

59

interest. To begin with, the moire fringe orders, i.e. Ny, were assigned to the v-
fringes as shown in Figure 3.3(a). Figure 3.3(a) was, in fact, exactly the same as
Figure 3.2(a). The moire fringes were attributed to a force of 1112N exerted at
each loading point. Since the graphite/epoxy laminate had seven laminae, the
non-zero shear strains at the six laminate interfaces were of interest. These
interfaces were designated as L2 to L7 in the diagram. Relations between the
fringe order Ny and the x-coordinate for the six interfaces were then established
and given in Figure 3.3(b) along with their least-squares curves. The shear strain
at each interface was, in fact, the slope of each Ny-x curve at x = 6mm. The six
shear strains on the six laminate interfaces were shown in Figure 3.3(c) by solid
circles. Apparently, the shear strain reached maximum around the midplane of

the composite laminate.

The same procedures were used for laminates with widths of 12.7mm and
6.35mm. And similar procedures based on strain-displacement relations were

also used for distributions of normal strains 6‘. and 5y.

3,, Based on vector analysis

Another technique to ﬁnd 7... was called vector analysis. Shown in Figure

3.4(a) were the v-fringes identical to Figure 3.3(a). As a demonstration, the
procedures to identify the shear strains at the intersecting point between the third
laminate interface from the bottom of the composite laminate, i.e. the L4 line, and

a vertical line located 6mm right to the midspan were performed.

60

y (mm)

L8
L7
L6
L5
L4
L3

L2
L1

      
 

x (mm)

tangential

Ny=1

tangential

h=6 mm -
6hl7 -
5hl7 '
4hl7-
3hl7'l

2hl7. L4
L2
1hl7. L3

L7

L6

 

 

. . . . . . 7.,(104)
0 1 2 3 4 5 6

(C)

Figure 3.4 - Shear strain calculations based on vector analysis

61

To begin with, fringe orders were assigned to the fringes around the point
of interest as shown in Figure 3.4(a). The point of interest and the moire fringes
in the neighborhood were isolated from the remaining of Figure 3.4(a) and shown
in Figure 3.4(b). Apparently, the moire fringe of order 1 passed right through the
point of interest. Due to the kinky nature of the fringe Ny = 1, two lines tangent to
the fringe were drawn from the point of interest. Another two lines normal to the
two tangential lines were also drawn from the point of interest to the center of the

neighboring fringe Ny = 2. The lengths of the two normal lines were designated

as F; and Fy" as shown in Figure 3.4(b). The superscripts a and b represented

for "above" and "below" the laminae interface, i.e. the L4 line, respectively, while

the subscript y indicated that v-fringes were involved. The horizontal components

of the two normal lines were named Fy‘; and Fy’; while the vertical components
were named F; and F} For clarity, only Fy‘; and F}; were shown in the
diagram. The shear strain 7” for the lamina above the third interface and below

the interface could be found as follows

 

 

F: F’;
7k}. rfy=f’- (3.5)

If a point of interest did not coincide with a moire fringe, interpolation
procedures should be imposed.
The shear strains on the laminate interfaces for the cross-section 6mm right
to the midspan were shown in Figure 3.4(c). There were two shear strains on

each laminate interface because shear strains were not continuous across the

62

laminate interface (though shear stresses were). It was noted that only one shear
strain could be found for each laminate interface, if the technique based on the

strain-displacement relations was used.

If a composite laminate had non-zero strains at no loading, the subtraction
of initial strains from the loaded counterpart should be performed. Otherwise, the
strains due to lower loading should be subtracted from those due to higher

loading to recover the true strains.

Depth of Boundary Layer

Based on the techniques for strain calculations given above, the

distributions of ex, 5}, and fry for the three widths of graphite/epoxy beams were

found and shown in Figures 3.5, 3.6 and 3.7, respectively. Because initial fringes
were not of null ﬁeld in the graphite/epoxy laminates, subtractions of strains due
to lower loading from higher loading were imposed. In the study, the fringes due
to 890N, 445N and 222.5N for 25.4mm, 12.7mm and 6.35mm, respectively, were
subtracted from those due to 1112N, 556N and 278N, respectively, resulting in
net forces of 222N, 111N and 55.5N at each loading point, respectively. The

distribution of a, through the laminate thickness was given in Figure 3.5.

Apparently, there was hardly any difference among the three cases. A similar

result was also found for the distribution of 63 through the laminate thickness, as

shown in Figure 3.6.

63

 

 

h=6 mm .
m 6h/7-
m 5h/7.
23:) - 4h/7 Cl 6.35 mm
3h/7-m A 12.7 mm
o 25.4 mm
2h/7- ADJ
h/7- ND
. . . If}. .
-2 -1 O 1 2
ex, (10“)

Figure 3.5 - Distribution of exx through laminate thickness

64

 

 

-1 -0f5 0 0.5

Figure 3.6 - Distribution of eyy through thickness

65

ector
-A— 12.7 mm

-9- 25.4 mm

—B— 6.35 mm
}analysis

 

I”.

30'635m:}ﬂmm-50

 

 

 

A 12 7 mm dIsplacement
O 25 4 mm relations

 

 

Figure 3.7 - The distribution of M through the laminate

66

The distribution of 73y, however, had a signiﬁcant increase as the laminate

width decreased as shown in Figure 3.7 This result was consistent with the fact

that my was dependent on the free-edge effect. The maximum shear strain in the

12.7mm specimen showed a signiﬁcant increase (actually more than doubled)
from that in the 25.4mm one, implying that the boundary layer was greater than
the laminate thickness as usually assumed. The maximum shear strain of the
6.35mm specimen was even higher than that of 12.7mm, indicating a much
stronger effect from the free edges. The results from the strain-displacement

relations agreed very well with those from the vector analysis.

Effect of material type

The moire fringes of glass/epoxy laminates with dimensions of 100mm by
25.4mm and thickness of 4.29mm were given in Figure 3.8. There were three
types of stacking sequence, namely 3-lamina, 7-lamina and 21-lamina. All the u-
fringes and v-fringes were due to a force of 111N at each loading point. However,
the initial fringes at no loading were not of null ﬁeld and must be subtracted from

the loaded counterparts.

The distributions of a, and g were about the same in all three cases as

shown in Figures 3.9 and 3.10. These results were similar to those of
graphite/epoxy case and again indicated that the normal strains were not

subjected to the free-edge effect. However, the distributions of 7.9, in the three

67

l

' v-fringes

v-fringes

 
  

   
    

  

i '1'.
Wig
s3 is;

u-fringes - i I A I - v-fringes
(c) 21-lamina, 111N

Figure 3.8 - Moire fringes of glass/epoxy laminates

68

y (mm)

 

 

cm 4.29. D 3-lamina
4.0.
00 A7-lamina
A0 3.5.
O o 021-Iamina
430 3.0-
O I
8 02.5
2.0‘ 0 LP
- o
1.0 040
0.5 O
I i I T I I I I I I I I H
-7-6-5-4-3-2-100123456
8.410”)

Figure 3.9 - Distribution of exx through laminate thickness

69

 

 

4.29
4.0::| on m
3.5. 0 up
30 O O
2.51 O 0 DA .
El 3-lamna
O C) "' 2 O
O 1_5- A 7-lanina
,5) G]
o o 0 10' 021-larnina
o 05‘
l—B I I I —0 I I r
-3 -2 -1 0 1 2 3
3w (104)

Figure 3.10 -Distribution of eyy through thickness

70

D3-Iamina
A7-Iamina } 25-4 mm
0 21 -Iamina

El 3-Iamina

A 7-Iamina }12-7 mm
y (mm) 0 21 -Iamlna

I 3-Iamina

A 7-lamina } 6-35 mm
.21-Iamina

 

 

 

 

' 10‘
97)!“ )

Figure 3.11 - Distribution of M through thickness

71

cases as shown in Figure 3.11 were different. Apparently, the higher the number
of laminae, the smoother the distribution. However, the maximum shear strains of

the three cases were very similar.

The difference of shear strains could also be found from comparing the 7-
lamina glass/epoxy and the 7-lamina graphite/epoxy laminates. After considering
the difference of thickness, the shear strain in the graphite/epoxy laminate was
much higher than that in the glass/epoxy laminate. This result was believed to be
attributed to the higher mismatch of material properties in graphite/epoxy than in
glass/epoxy. In other words, graphite/epoxy had higher anisotropy than
glass/epoxy; and the higher interlaminar strains due to the higher mismatch of
material properties (between O-Iamina and 90—Iamina) resulting from the higher

anisotropy were responsible for the stronger free-edge effect.

Effect lamination type

According to Figure 3.11, the distribution of 7,0, in the 21-lamina specimen

was much smoother than that in the 7-lamina one since the former had a much
higher number of laminae through the thickness than the latter. In other words,
the difference of shear strains across the laminate interfaces of the 21-lamina
specimen was less than that in the 7-lamina one because the material mismatch
between O-Iamina and 90-lamina was smaller than that between 03-lamina and
903-lamina. The 3-lamina specimen had the highest difference of shear strains
among the three cases due to the highest material mismatch between 07-lamina

and 907-lamina.

72

The distributions of yxy for the glass/epoxy beams with width of 12.7mm

and 6.35mm were also given in Figure 3.11. They were obtained from loading the
specimens with a force of 55.5N and 27.75N at each loading point, respectively.
Surely, the initial strains at no loading should be subtracted from the loaded
counterparts to reveal the true strains. Again, the changes of maximum shear
strains due to the changes of width (from 25.4mm to 12.7mm and then to
6.35mm) were signiﬁcant. However, the change in the 21-lamina was not as

large as that in the 7-lamina, which was not as large as that in the 3-lamina.

Conclusions

The moire interferometry is a surface technique for displacement
measurements. Experimental results based on moire interferometry have
revealed that the shear strains on free edges increase signiﬁcantly as the width
of a composite laminate decreases. However, the normal strains on free edges
are not inﬂuenced by the free-edge effect. Accordingly, the normal strains based
on moire interferometry could be used to represent the normal strains beyond the
surface for plane-strain and plane stress problems. However, the shear strains
based on moire interferometry should not be used beyond the surface at all
because of the free-edge effect. Based on the results that shear strains change
signiﬁcantly with width changes, it has been found that the depth of boundary

layer is greater than the thickness of composite laminates as usually assumed.

The free-edge effect is associated with mismatch of material properties.

Since graphite/epoxy has higher anisotropy than glass/epoxy, a composite

73

laminate made of graphite/epoxy experiences stronger free-edge effect than the
glass/epoxy counterpart. Similarly, a composite laminate consisting of thicker
laminae has higher mismatch of material properties among laminae and hence is

more strongly inﬂuenced by free edges than that consisting of thinner laminae.

Acknowledgements

The author wishes to express his sincere thanks to Professor Daniel Post
of Virginia Tech for providing the blue prints of building the moire interferometry
system used in the study and many valuable discussions on moire
interferometry. Thanks also extended to Professor Peter lfju of University of
Florida for Providing the original moire grating and many valuable discussions on
grating preparations. The ﬁnancial support from Michigan Research Fund is also

greatly appreciated.

74

References

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

Pipes, R, Pagano, N. "Interlaminar Stresses in Composite Laminates
Under Uniform Axial Extension," Journal of Composite Materials, Vol. 4,
538-548, (October 1970).

Becker, W.,” Closed-form Solution for the free-edge effect in cross-ply
laminates," Composite Stmctures, Vol. 26, 39-45, (1993).

Lessard, L., Schmidt, A, and Shokrieh, M,"Three-Dimensional Stress
Analysis of Free-Edge Effects in Simple Composite Cross-Ply Laminate,"
lntemational Journal of Solid Structures, Vol. 33, No. 15, 2243-2259,
(1996).

Becker, W., Jin, P., Neuser, P.," Interlaminar Stresses at the Free Comers
of a Laminate," Compostie Structures, Vol. 45, 155-162, (1999).

Altus, E, Rotem, A, Shmueli, M,"Free Edge Effect in Angle Ply Laminates-
A new Three Dimensional Finite Diffrenece Solution," Journal of
Composite Materials, Vol. 14, 21-30, (Januarry, 1980).

Wang, A, Crossman, F .," Some New Results of Edge-Effect in Symmetric
Composite Laminates," Journal of Composite Materials, Vol. 11, 92-106,
(January, 1977).

Chaudhuri, R, Xie, M, ”Free-Edge Stress Singularity in a Bimaterial
Laminate," Composite Structures, Vol. 2, 129-136, (1998).

Becker, W.,” Closed-form solution for the Free-Edge Effect in Cross-Ply
Laminates,” Composite Structures, Vol 26, 39-45, (1993).

Zhang, C., and Yeh, H.,"A Variation Approach for Straight Edge Stresses
in Laminated Composites," Composite structures, Vol. 43, 243-254,
(1998)

Oplinger, W., Parker B., and Chiang, F., ”Edge-Effect Studies in Fiber-
reinforced Laminates," Experimental Mechanics, 347-354, (September
1974)

Pipes, B., and Daniel, I,” Moire Analysis of the interlaminar Shear Edge-
Effect in Laminated Composites," Journal of Composite Materials, Vol. 5,
255-259, (April 1971).

Post, D., Han, B., and lﬁu, P., High Sensitivity Moire, Springer-Verlag,

75

3.13

3.14

3.15

1993.

Mollenhauer, D. " Interlaminar Deformation at Hole in Laminated
Composites: A detailed Experimental Investigation Using Moire
Interferometry," Ph.D. Dissertation, Virginia Polytechnic Institute and State
University, Blacksburg, VA, August 1997.

Herakovich, C., Post, D., Buczek, M., and Czarnek, R., "Free-Edge Strain
Concentrations in Real Composite Laminates: Experimental-Theoretical
Correlation,” Journal of Applied Mechanics, Vol. 52, 787-793, (December
1985)

Post, D., "Moire Interferometry; Advances and Applications," Experimental
Mechanics, 276-280, (September 1991)

76

Chapter 4

JOINING FORCES IN ASSEMBLED COMPOSITE BEAMS

Abstract

This study investigated the joining force of assembled composite beams.
Two identical composite laminates made of cross-ply glass/epoxy were joined via
various joining techniques such as adhesive bonding, mechanical riveting,
stitching joining and combinations of these techniques. The assembled two-
laminate composite beams were loaded by four-point bending. The
corresponding normal strains were identiﬁed by moire interferometry. Depending
on the joining condition between the laminates, the moire patterns of u-fringes
and v-fringes may be continuous or discontinuous across the joining interface. To
quantify the joining condition, two analytical models based on the Classical Beam
Theory, the interface model and the interphase model, were presented. The
joining forces of the assembled composite beams were identiﬁed. Mechanical
riveting was found to be an efﬁcient joining technique for assembling the thin

composite laminates.

77

Introduction

Due to their high stiffness-to-weight and high strength-to-weight ratios,
ﬁber-reinforced polymer matrix composite materials have found many
applications in high-performance structures. These composite materials are
usually used in thin-laminate form. As the technologies of composite materials
advance, more and more thick-section composite laminates are manufactured
[4.1] and used in various industries. However, the manufacturing cost for thick
composite laminates could become unaffordable. This is especially true when

uniform properties through the laminate thickness are of primary concern.

In an effort to reduce the manufacturing cost and to avoid non-uniforrn
properties through the laminate thickness, assembled composite plates have

been considered as alternatives to thick laminated composite plates [4.2].

Assembled composite plates are constructed by assembling multiple thin
composite laminates together via various joining techniques. In a previous study
[4.2], techniques including mechanical riveting, adhesive bonding, stitching
joining [4.3.4.4], and their combinations have been used in assembling thin
composite laminates. Those experimental results revealed that joining stiffness
plays an important role in impact resistance. The results suggest that a better

joint gives higher joining stiffness and subsequently higher perforation threshold.

78

As an effort to investigate the joining conditions of assembled composite
laminates based on various joining techniques and to quantify the joining
conditions, both analytical modeling and experimental investigation are

presented in this study.

Analytical modeling

In order to quantify the joining conditions of assembled composite beams
based on various joining techniques, two analytical models were presented. They
were the interface model and the interphase model. Although these models could
be applied to assembled composite beams consisting of multiple laminates, only
those beams made up of two laminates were of interest in the study. The
interface model assumed that the joining interface between the two composite
laminates had no thickness. However, the interphase model considered that
there was a ﬁnite thickness, i.e. an interphase layer, between the laminates. All
the assembled composite beams investigated in the study had an effective length
of 75 mm (between the two outer rollers of the four-point bend ﬁxture) and a
nominal thickness about 6.4 mm, resulting in an aspect ratio of length-to-
thickness around 11.7. Accordingly, both the interface and interphase models

were developed based on the Classical Beam Theory.

The interface model

This model considers that assembled composite beams consist of two

layers and a joining interface. Each layer could be made up of a composite

79

laminate and the joining interface between the layers is assumed to have no
thickness. Accordingly, the ﬁnite dimensions of the bonding interfaces due to
adhesive and double-coated tape in real joining practice are neglected. As shown
in Figure 4.1(a). the free-body diagram of an interface model is subjected to
transverse loading because the transverse loading can help to characterize the
joining condition of the assembled composite beam. In addition, two resultant
forces F1 and F2 and two resultant bending moments m1 and m2 are also added
to Figure 4.1(a). It should be noted that these resultant forces and moments are
of internal force components and, for convenience, they are shown on the
generic cross-sections along the longitudinal axes (x1 and x2) of the composite
layers. The internal force components should not be mixed up with the external
forces, such as the transverse loading, usually depicted on the surfaces of a free-

body diagram.

When the two composite layers are not assembled by any joining
technique, each layer is expected to behave independently if there is no friction
between them. The resultant forces F1 and F2 should vanish because there is no
force transferring between the two layers. The primary stress, i.e. the normal
stress along the x-direction, in each layer can then be calculated directly from the
ﬂexure formula. However, if the two layers are assembled together by any joining
technique, there will be load transferring between them, and F1 and F2 will no
longer be zero. The resultant bending moments m1 and m2, however, are not
directly tied to the joining condition. They are more related to the external force,

i.e. the transverse loading.

80

 

 

Y1
h, layer (1) 0L x, ' 2 Fl
1
y
h, layer (2) 03—1 x2 -:§->F2
m

 

 

 

 

 

(a)

y I
11. layer (1) oI—m

h3 h2 layer (3) 033 X3

 

(b)

Figure 4.1 - Free body diagrams for: (a) the interface model, and (b) the
interphase model

81

The geometry and the coordinate systems of the interface model are
shown in Figure 4.1(a). According to the diagram, x,o,-y,- is the local coordinate
system of layer (i) that has a thickness of h,-. If layer ( 1) and layer (2) are made of
the same material and have identical dimensions, the following equations should

hold:

’71: ’72, A1 = A2, E1 = 52, [1:12, (4.1)

where A,- is the cross-sectional area, E,- is the Young’s modulus and I,- is the
second moment of area of layer (i). Based on strength-of-materials analysis, the
normal stresses along the longitudinal axes (x1 and x2) of the composite layers

can be expressed by the following equations:

a“) = _5 _ﬂ
x A I l
1 ' (4.2)
0"” = 5; m2

According to equilibrium of forces, the two resultant forces have opposite sense

but identical magnitude, i.e.

F, = F2. (4.3)

When the two layers are joined together by any means, both the individual

layers and the assembled beam have an identical bending curvature x, i.e.

82

1": ml 2 m2 : ml+m2 (4.4)
15,1, 15,12 15,1,+15,12

Since E,I, = E212 (from Equation (4.1)). it yields
m, = m2. (4.5)

From equilibrium of moments, the sum of the two resultant bending
moments and the moment generated by the two resultant forces should be equal
to the bending moment due to the transverse loading. By deﬁning

+Fl(hl+h2)

2 (4.6)

M=m,+m2

as the total bending moment due to the transverse loading for the generic cross-
section of the composite beam, the normal stresses given in Equation (4.2) can

be rewritten as

bit—Fl”! , 1y
" A, 21, ‘ 21, ‘

(4.7)

F Fh M
<2>=_I_+_H_ __
x Al 21l y2 21] y2

When there is no joining between the two layers, the resultant force F1
should vanish because the two layers are not related to each other and there is

no load transferring between them. Equation (4.7) can then be simpliﬁed to be
0'“) = a”) = ——A1- y, . These results imply that the normal stresses in the two

1: x
I

composite layers are equal to each other and each is attributed to 50% of the

83

total bending moment M (i.e. m1 = m2 = M/2) due to the transverse loading.
However, as the joining condition improves, the load transferring between the
layers also increases, and a ﬁnite resultant force will exist. Hence, the resultant
force is actually the joining force between the two layers and can be used to

quantify the joining condition of assembled composite beams.

The interphase model

This model considers that assembled composite beams consist of three
layers as shown in Figure 4.1 (b). The two outer layers are of composite
laminates; whereas the interphase layer is a joining layer, which could be either
ﬁctitious or real. The assumption of an interphase layer seems to be real for
composite beams assembled by adhesive or double-coated tape, which have a
uniform thickness between the composite layers. However, the assumption of an
interphase layer is only ﬁctitious for joints that are based on mechanical riveting

and stitching.

The geometry and coordinate systems for the interphase model subjected
to transverse loading are shown in Figure 4.1(b) along with the resultant force F,-
and the bending moment m,- at the generic cross-sections along the longitudinal
axes of the layers. In the cases where layer ( 1) and layer (3) are identical in both

material and dimensions, the following equations hold:

’71 = ’73, A1 '-' A3, E1: E3, ,1 = [3. (4.8)

84

And the normal stresses in the three layers can be expressed by the

following equations:

(,0): 5-1%. I
AI II

of.” bﬂy. (4.9)
12

Based on equilibrium of forces, the two resultant forces have opposite sense but

the same magnitude, i.e.

F, = F3. (4.10)
Because all three layers are assembled together, they have an identical bending

curvature K, i.e.

m, __ m2 _ m3 __ m,+m2-l-m3

 

 

 

x = — — - (4.11)
19,1, 5,1, E313 E,I, + 15,12 + 13,13
Since E,I, = E313 (based on Equation (8)). it yields
m1 = m3 . (4.12)
By deﬁning
M=m,+m2+m3+F,(h,+h2) (4.13)

85

as the total bending moment due to the transverse loading for the generic cross
sections shown in Figure 4.1 (b), the normal stresses given in Equation (4.9) can
be rewritten as

a(,,=__F_,+F,(h,+h2)E, _ ME,
’ A, 215,1,4-15212 ‘ 2E,I,+E212

 

 

yr

_ 4.14
" 25,1,4-15212 2 213,1,+1~:,1,y2 ( )

 

 

,,,=f,_+F,(h,+h,)E, _ ME,
" A, 21E,1,+15,12 3 21E,1,+15,12

 

 

Y3

According to Equation (4.14), the resultant force F,- and the bending rigidity
of the interphase layer E2I2 (or even the Young’s modulus of the interphase layer
E2) can be identiﬁed, if the following are known: the normal stresses, the
dimensions and the Young's modulus of the composite layers. As mentioned
earlier, the resultant force is related to the load transferring between the
composite layers. Therefore, the resultant force is really the joining forces of the
assembled composite beam. Similarly, since the bending rigidity is associated
with the property of the joining layer, that bending rigidity can represent the
joining condition of the assembled composite beam. However, if the bending
rigidity of the interphase layer E212 can be neglected (either because the
interphase layer is too thin or the modulus is too low), the ﬁrst and the third

equations of Equations (4.14) can be simpliﬁed as

86

a“) =_f,_+F,(h,+h2) AL

" A, 21, 1’21,);1

(4.15)
0"” =£+F,(h, +h2)y —£—y
" A, 21, 3 21, 3

These two equations are identical to those presented in the interfacial

model, i.e. Equation (4.7). The second equation of Equation (4.14), however,

becomes trivial.

Experimental investigations

Composite laminates

In experimental investigations, assembled composite beams were made
up of glass/epoxy laminates which had a stacking sequence of [0/90/0l...]13 and
a nominal thickness of 3.2 mm. The dimensions of the composite laminates used
as building blocks for the assembled composite beams were 100 mm x 25 mm.
Every two composite laminates were joined to form a two-laminate beam. The

assembled composite beams were then subjected to four-point bending.

As shown in Figure 4.2(a), the distance between the inner rollers of the
bending ﬁxture was 25 mm while that between the outer rollers was 75 mm.
Accordingly, the assembled composite beams had an effective length of 75 mm
(the distance between the outer rollers) and a total thickness around 6.4 mm
(twice the laminate thickness although the true thickness of an assembled

composite beam was also dependent on the thickness of the joining layer). In

87

addition, it should be noted that the bending moment M between the inner rollers

was constant and equal to 25P Nmm (where P was the force exerted on the

composite beam by each roller) and the shearing force within the interval

vanished.

Joining techniques

Many joining techniques such as mechanical riveting, adhesive bonding,

stitching joining, and their combinations were performed in the study. Figures

4.2(a)-4.2(f) showed schematic diagrams (either side view or top view) of some

assembled composite beams used in the study. Details of the designations for all

the assembled beams investigated in the study were given below.

A.

ONE (for one-laminate beam) - Shown in Figure 4.2(a) was the side view of a
single composite laminate used as the building block for assembled
composite beams. The single laminate had a nominal thickness of 3.2 mm.
Details of the four-point bending conﬁguration was also shown in the diagram.
ZB (2 for two-laminate beam and B for bonding) — As shown in Figure 4.2(b),
two composite laminates were bonded together by a room-temperature curing
epoxy. The thickness of the bonding layer was around 0.2 mm. Thus, the total

thickness of the assembled composite beam was around 6.6 mm.

. TWO (for two laminates) - Two composite laminates were simply piled up

together without any form of joining between them. However, a small amount
of epoxy used as moire grating applied to one edge of the assembly was
found to inﬁltrate the interface between the laminates and might effectively

join the laminates up to some extent.

88

D. 2T (2 for two-laminate and T for taping) — Two composite laminates were
joined together by a double-coated tape. The tape was about 0.1 mm thick.
E. 2R2 (the ﬁrst 2 for two-laminate, R for riveting and the second 2 for two rivets

per 50 mm) — Two composite laminates were joined together through the

89

 

 

 

(a)0NE (b)ZB
25m - I' m
—I -l
(c)2R2 (d)2R3

 

Figure 4.2 - Schematic diagram of various joining techniques

90

laminate thickness by two rivets. Figure 4.2(c) showed the top view of the
assembly. During riveting, two holes were prepared by a #30 drilling bit before
4x580 steel rivets (1/8” in diameter and 1/4” in grip) were push in by a riveting
hand gun. The two rivets were located along the centerline of the beam and
the distance between them was 50 mm. They were also symmetric with
respect to the mid-span of the assembled beam.

. 2R3 (R for riveting and 3 for three rivets per 50 mm) - Instead of two rivets
used in 2R2, three rivets evenly separated within the same 50 mm interval
were employed. Figure 4.2(d) showed the top view of the joining
conﬁguration.

. 2R4 (R for riveting and 4 for four rivets per 50 mm) — Similar to the 2R2 and
2R3 cases, two composite laminates were joined together. However, four
rivets were employed in the 50-mm interval. As a result, the inner rollers of
the bending ﬁxture were located right on top of two inner rivets.

. 2BR3 (B for bonding, R for riveting and 3 for three rivets per 50 mm) - Two
composite laminates were ﬁrst bonded by the epoxy used in the 28 case.
They were then further joined together by three rivets with a pattern identical
to the 2R3 case.

283 (S for stitching and 3 for three stitches per 50 mm) — Two composite
laminates were joined together by hand stitches through the laminate
thickness as shown in Figure 4.2(e). Before performing the stitches, three

holes with locations identical to those in the 2R3 case were prepared. The

91

holes were formed by a drilling bit of 1/16” in diameter. A steel wire of 28
gauge was used as the stitching line.

J. 285 — (S for stitching and 5 for 5-by-2 stitching pattern) Two composite
laminates were joined together with a 5-by-2 stitching pattern (totally ten
stitching holes) within a 50mm-by—12.5mm area as depicted in Figure 4.2(f).
For each four nodes forming a square unit, there were totally six stitching
lines, i.e. two horizontal, two vertical and two diagonal lines, on each side of
the assembly.

K. 2383 (B for bonding, S for stitching and 3 for three stitches per 50 mm) — Two
composite laminates were ﬁrst bonded by the epoxy used in the ZB case and

then stitched together by the stitching pattern used for 283.

Moire interferometry

The interface and the interphase models presented earlier were based on
two-dimensional analysis. Eexamining Equation (4.7) of the interface model and
Equation (4.14) of the interphase model revealed that the information regarding
the normal stress a,‘ through the laminate thickness y,- was required for
identifying the resultant force F,- (for both models) and the bending rigidity E2l2 of
the interphase layer (for the interphase model only). The resultant force and the
bending rigidity, in fact, represented the joining force and the joining condition,

respectively, of the assembled composite beams.

Accordingly, an experimental technique that could provide the needed

stress information through the laminate thickness would be useful in ﬁnding the

92

joining force and the joining condition. Due to its high sensitivity and high

accuracy, moire interferometry was chosen for the stress analysis.

Moire interferometry is a displacement-based technique. To perform moire
interferometry, assembled composite beams must be coated with a layer of
moire grating on one edge. Details of the one-layer grating technique used in the
study are found in Reference [4.5]. The grating-coated assembled beams were
then loaded by four-point bending. The deformation of each beam could then be

investigated by a moire interferometer.

The top view of the moire interferometry setup used in the study was
shown in Figure 4.3. However, details of the optical technique are found in a
book authored by Post, Han and ltju [4.6]. As shown in Figure 4.3, the point laser
beam was diverged through a spatial ﬁlter and then redirected and collimated by
a parabolic mirror. The enlarged collimated light was then illuminated onto four
mirrors (designated as M in the diagram) located in the moire interferometer. The
four mirrors, two horizontally positioned and two vertically positioned, formed a
cross pattern. For clarity, only the two horizontal mirrors and beams were shown
in the diagram. The laser beams reﬂected from the four mirrors were redirected
onto another four mirrors (also formed a cross pattern) for further beam angle
adjustment. Again, only the two horizontal mirrors and beams were shown in the

diagram.

Moire interferometry is based on optical diffraction and interference. As

the two horizontal beams reﬂected from the second set of mirrors, the beams

93

 

M Laser

 

Moire

Interferometer .
Spat1al

Filter

 

 

 

 

 
   
 

_ Double-Sided Mirror

\
Specimen ﬁ Parabolic
M

 

 

 

 

 

 

 

 

 

 

 

 

_ Mirror
Computer | Cgrggal

& Image '
Processmg

 

 

 

Figure 4.3 - Schematic diagram of optical setup for moire interferometry
(horizontal two beams only)

94

were illuminated onto the specimen surface (actually edge) that was coated with
a layer of moire grating. Due to the high frequency (1200 lines/mm) of the moire
grating used in the study and its mirror-like surface, the two horizontal beams
were then diffracted reﬂectively by the grating. Under no-loading condition, two
(one from each horizontal beam) diffracted beams of the ﬁrst-order should emit
from the specimen edge. If the angles of the second set of mirrors were adjusted
property according to the grating frequency, the two diffracted beams should be
both normal to the specimen surface and cause no interference between them.
However, if the specimen is deformed due to transverse loading, the frequency
and orientation of the moire grating lines on the specimen edge would be altered.
In this state the diffracted beams would be neither parallel to each other nor
normal to the specimen edge any more. As a result, interference between them
would take place and a moire fringe pattern representing the contours of constant

displacement would be formed right on the specimen edge.

The two horizontal beams could provide contour lines for displacement
along the longitudinal direction (x-direction), namely u-fringes, whereas the two
vertical beams (not shown in the diagram) could give contour lines for
displacement through the thickness direction (y-direction), namely v-fringes. The
strain ﬁeld of the deformed composite beam could then be calculated from the
displacement contour lines, i.e. moire fringe patterns, with the use of linear strain-
displacement relations. Because the assembled composite beams had an aspect

ratio of 11.7 and they were subjected to small transverse loading in the study, the

95

normal stress a" was the primary component in the stress analysis. Accordingly,

only u-fringes were required for stress analysis.

Results and conclusions

Moire fringe patterns

The assembled composite beams were loaded by four-point bending.
Moire patterns of u-fringes and v-fringes at several loading levels were recorded
by a CCD camera and stored in a computer for later analysis. The u-fringes
played a primary role on joining analysis for the purpose of calculation. Figures
4.4(b) — 4.4(l) displayed both u-fringes and v-fringes for the two-laminate beams
due to a force of 45 N at each loading point (roller), i.e. P= 45 N. This loading
level resulted in a constant bending moment of 1125 Nmm between the two inner
rollers. Hence, only the moire patterns within the pure bending zone were
presented. For comparison, the moire fringes of the ONE case based on P = 22.5
N were also given in Figure 4.4(a). Note also that the fringe patterns at no-
loading condition were not of null ﬁeld due possibly to misalignment of the optical

system and shrinkage of moire grating during replication process.

A. ONE — The moire patterns of u-fringes and v-fringes were shown in Figure
4.4(a). Both were symmetric with respect to the longitudinal and thickness
axes. It was believed that an assembled two-laminate composite beam
with perfect joining should have fringe patterns identical to the patterns in

Figure 4.4(a).

96

I ',"'."",w ,\"‘.‘ ,1 ‘_
. . q ),.’ .
v 53.11: MA!

v-fringes

112111.181)- "-.-..éssilllll

t

\4 lllll‘ttzza‘fswill

v-fringes
“t

Liiltijilllllllul I}; :1 ll lllfl“: . .

. l qrr'ﬂr " f ‘ main.“

v-fringes

 

  
    

 
      
  

 
 

   

Figure 4.4 - Moire fringe pattern for: (a) ONE (one laminate beam), (b) 28 (2 for
two-laminate and B for bonding), (c) TWO (for two laminates). and (d) 2T (2 for
two-laminate and T for taping)

97

. 28 — As shown in Figure 4.4(b), both u-fringes and v-fringes revealed
good bonding in the 23 case since the discontinuity of u-fringes across the
bonding interface was insigniﬁcant. The difference of v-fringes between
the 2B case and the ONE case was believed to result from the difference
of initial fringes at no-loading condition.

. TWO — The u-fringes and v-fringes were shown in Figure 4.4(c). Both
fringes in each laminate of the beam seemed to be very independent.
Each pattern seemed to resemble the corresponding pattern of the ONE
case up to some extent.

. 2T — As shown in Figure 4.4(d), the fringe patterns seemed to be similar to
those of the TWO case. It implied that the bonding due to the double-
coated tapes used in the study was very poor.

. 2R2 —Although the v-fringes were kinky across the joining interface,
implying imperfection of joining, the u-fringes (shown in Figure 4.5(a))
were like those of the ZB case. The results seemed to indicate that two
rivets per 50 mm were sufﬁcient for joining the composite laminates, at
least for the 45 N used in the investigation.

. 2R3 — Shown in Figure 4.5(b), the u-fringes were like those of the 2R2
case except some isolated irregularities. Since the v-fringes of the 2R3
case were less kinky across the joining interface than those of the 2R2

case, the former seemed to be a better joint than the latter.

98

‘   7 “ j...:i((((

u-fringes ‘ l ‘ .... -fringes

 

. 0..an -
illuunllllllllfull‘ll

‘1‘ mm" ’
' «Illllllullhl

Inn “11111.. "willllnumnnqmmh ....mm

 

 

 

 

 

 

 

 

 

 

u-fringes L l _. I, v-fringes

(d)

Figure 4.5 - Moire fringe pattern for: (a) 2R2 (the ﬁrst 2 for two-laminate, R for
riveting and the second 2 for two rivets per 50 mm), (b) 2R3 (R for riveting and 3
for three rivets per 50 mm), (c) 2R4 (R for riveting and 4 for four rivets per 50
mm), and (d) ZBR3 (B for bonding, R for riveting and 3 for three rivets per 50
mm)

99

G. 2R4 - Shown in Figure 4.5(c), the moire fringe patterns of the 2R4 case
were similar to those of the 2R3 case, implying a similarjoining condition
in the two cases.

H. 28R3 - Based on Figure 4.5(d), 28R3 apparently was a very good joint.

I. 283 - Similar to other cases, 283 was loaded by four-point bending.
Before the breakage of the stitching lines, e.g. at 45 N, the u-fringes,
shown in Figure 4.6(a), resembled those of the ONE case. The v-fringes,
however, were of two separated patterns though not as independent as
those of the TWO case. As compared with 2R2, the joining condition of
283 seemed to be relatively poor. Once the stitching lines broke, e.g. at
200 N, the u-fringes as shown in Figure 4.6(b) clearly became two
independent sets. However, the v-fringes became more dependent on
each other possibly due to the high contact-friction between the laminates
at the high loading level.

J. 285 - The u-fringes shown in Figure 4.6(c) revealed that 285 was a good
joint. The v-fringes, however, showed a great deal of discontinuity across
the joining interface. According to the fringe patterns, the two laminates in
the 285 case seemed to have a closer interaction than those in the 283
case, indicating that 285 was a better joint than 283.

K. 2883 —As shown in Figure 4.6(d), the fringe patterns of 2883 were very
similar to those of 28. Apparently, epoxy bonding outperformed the

stitching joining and played the dominant role in the 2883 case.

100

p

'.
.-
.. -

.‘1'.

\

u-fringes

 

 

 
  

 

 

 

 

 

u-fringes

   

L- .
u-fringes v-fringes

((0
Figure 4.6.- Moire fringe patterns of various joinig subjected to four point
bending: (a) 283 (90N) (S for stitching and 3 for three stitches per 50 mm), (b)

283 (200 N) , (c) 285 (S for stitching and 5 for 5-by-2 stitching pattern), and (d)
2883 (B for bonding, S for stitching and 3 for three stitches per 50 mm)

101

Maximum joining force

Before the joining forces in various assembled composite beams are
calculated, the ONE case should be carefully examined because it could be used
as basis for comparison of the joining forces. To begin with, moire fringe orders
were assigned to the u-fringes, i.e N2, as shown in Figure 4.7(a). The fringes in
Figure 4.7(a) were identical to those in Figure 4.7(a). Both were obtained from
P = 22.5 N. Five locations through the laminate thickness were then selected and
horizontal lines at the ﬁve locations were added to the Figure 4.7(a). Relations
between the u-fringe order N, and the x-coordinate for the ﬁve horizontal lines

were established and given in Figure 4.7(b). The normal strain, however, could

 

 

be obtained from combining the displacement-fringe relation (u = N’ ) and the
. . . . au .
linear strain-displacement relation (1:,r = 5; ), Le.
.9, =16“: (4.16)
f 6x

In the study, the system frequency f was 2400 lines/mm. It was twice the

frequency of moire grating.

The normal strain given in Equation (4.16) was, in fact, the slope of the N,-
x relation. From Figure 4.7(b). the strains at the ﬁve locations along the y-axis
could be identiﬁed. They were shown in Figure 4.7(c) with solid circles. A solid
line based on least-squares analysis was also presented. Apparently, the solid

circles matched very well with the least-squares line, indicating that the strain

102

distribution was linear through the laminate thickness. This result was consistent
with the assumptions of the Classical Beam Theory imposed in the analytical

modeling and the small loading utilized in the experimental investigations.

As mentioned earlier, the moire fringes were not of null ﬁeld at no-loading
condition. By subtracting the residual strains at P = 0 from those at P = 22.5 N
(Figure 4.7(c)), the real strains due to external loading could be obtained. Based
on the real strains and P = 22.5 N, the Young's modulus of the composite
laminate was found to be 22.27 GPa. This value was used in subsequent

calculations.

In addition to the perfect-joining case ONE, the experimental result of an
imperfect-joining case 28 was also obtained. For comparison, they were both
depicted on Figure 4.6. Since the thickness of the ONE case was 3.2 mm and
that of 28 was 6.6 mm, a magniﬁcation factor of 4 according to the ﬂexure

formula was imposed on the ONE case.

As shown in Figure 4.8, the normal strains in the perfect-joining case
could be represented by a single straight line through the laminate thickness
whereas those of the imperfect-joining case two straight lines, one on each layer.
Apparently, the strain distribution of an assembled two-layer composite beam
was strongly affected by the joining condition. Figures 4.9(a), 4.9(b), and 4.9(c)
showed schematic diagrams of the normal strains in three assembled beams

with perfect joining, imperfect joining and no joining, respectively.

103

The normal strains represented by a solid line in Figure 4.7(a) were
continuous across the joining interface in the perfect-joining case. They could be
obtained from the ﬂexure formula by assuming that the perfectly joined beam

was a single beam with double thickness.

The normal strains in the imperfect-joining case then should be between
the two extreme cases, as concluded from the 28 case shown in Figure 4.8. The

schematic diagram for the imperfect-joining case was shown in Figure 4.9(b).

The normal strains in the no-joining case as shown in Figure 4.9(c),
however, were of two identical but independent dashed lines, indicating that each
layer behaved independently. Accordingly, the joining force F,- presented in the
analytical modeling was created for joining the composite layers together. The
higher the joining force, the less the mismatch of the normal strains, and the

normal strains, across the joining interface.

In addition to identifying the Young's modulus of the composite laminate,
the u-fringes from ONE case could also be used to identify the theoretical value
of maximum joining force in perfectly joined composite beams. The ONE case,
on one hand, can be considered as being made up of two composite layers
(each has a thickness equal to one half of the total thickness of the ONE beam)

with perfect joining. Recalling Equation (4.7), the normal stress in layer (1) is

F Fh M
0.11) =__I+__L __ 4.7
x A, 21,” 21,”1 ( )

104

 

 

 

Figure 4.7 - Normal strain calculations, (a) moire fringe pattern for ONE case in
ﬁve designated locations, (b) relation between u-fringes and Nx in ﬁve locations,
and (c) strains in the ﬁve locations along y-axis

105

3

where 1, = 1’1”; and h, is one half of the total thickness h of the ONE beam. oh

 

the other hand, the ONE case is actually a single laminated composite beam.
And the normal stress in the beam can be calculated from the ﬂexure formula,

i.e.

__ 4.17
. 1 ( )
bh3
where Y=y1+ h/4 and l= 3.
By equating Equations (4.7) and (4.17), it yields
F = 311
211 (4.18)

Apparently, the joining force in the two composite layers perfectly joined
together is dependent on the total bending moment M due to the transverse
loading and the total thickness of the assembly h. In other words, there is a
theoretical joining force for perfectly joined composite beams. For example, by
substituting M = 562.5 Nmm (based on P = 22.5 N and 25 mm between the outer
roller and the inner roller as shown in Figure 4.2(a)) and h = 6.4 mm (for a two-

laminate beam) into Equation (4.18), a joining force of 139 N is obtained.

106

y (mm)

. ONE

 

 

 

 

 

Figure 4.8 - Comparison of normal strain for perfect bonding ONE and imperfect
bonding 28

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e X
(a)
751251
{(252,
(b)
yl
A
‘x
\ _X,
\
v \
\ Y2
\1‘ .X2
‘ ,
x
\

 

 

 

(C)

Figure 4.9 - Schematic diagram for: (a) prefect bonding, (b) imperfect bonding,
and (o) no bonding

108

Singular points

Equation (4.7) is valid for all joining conditions, good joining as well as
poor joining. Both equations are organized by two parts: one associated with the
joining force F,- and the other associated with the total bending moment M. Since
only the joining force is dependent on the joining condition, only the part
associated with the joining force is affected by the joining condition. M is actually

calculated from the ﬂexure formula.

By setting the part associated with the joining force equal to zero, i.e.

F, F,h,
__+____ = 0, 4.19
A, 21, y, ( )

it can be found that the normal stress is only dependent on the total

bending moment M when

 

,, = 5, (4.20)
6
Substituting Equation (4.20) back into Equation (4.7), yields
0,0) = _ M
x 2
bit, (21)

Equation (4.21) gives an invariant normal stress at y1 = h1/6. Similarly, it

at y2 = -h2/6. These two locations, i.e. y1= h1/6

 

can also be found that of,” = A42

2

109

and y2 = -h2/6, are called singular points since the normal stresses of assembled

two-laminate beams at these two points are independent of joining condition.

As a resultant component, the joining force F,- should be constant in each
joining case. However, by closely examining Equation (4.7), it was found that F,-
ﬂuctuated around the singular points due to numerical sensitivity. Figure 4.10
showed the comparison between the analytical predictions based on Equation
(4.7) and the experimental results for the ONE case. The joining force became
singular at the singular points. It then was suggested that the experimental
measurements should avoid the neighborhood of the singular points. Since the
density of moire fringes was usually quite low close to the joining interface, e.g.
Figure 4.4(b) for the 28 case, the beam surfaces were thus concluded as better
locations for joining analysis. In addition, it should be noted that good
agreements between the analytical predictions and the experimental results, as
shown in Figure 4.8, for many locations through the beam thickness were

obtained.

The analytical modeling was based on two-dimensional approach and the
experimental investigation was actually of three-dimensional analysis. As a
result, the good agreements between them suggested that the free-edge effect,
which was usually considered as an important phenomenon in laminated

composite materials [4.7], should not be of a major concern in the study.

110

Y (mm)

 

 

4
3 . .‘f Omodeling
O 2 . ‘ Aexpenment
1 .
joining
' ' ' ' ' force (N)
-400 -300 -200 -100 _1. 200
O
O -2 - . 5 ‘
1 b.

Figure 4.10 - Comparison between analytical predictions and experimental
results

111

Joining forces

The top and bottom surfaces were better locations than the mid-plane for
joining analysis because of their high density of moire fringes. They were also
better than points close to the singular points because of stable stress
distribution. To identify the joining forces of various assembled beams, the
normal strains at the surfaces were obtained by subtracting the strains due to P =
45 N from those due to P = 67.5 N. This procedure was based on the fact that
the moire fringe patterns in the assembled beams were not of null ﬁeld at no-

loading condition.

For the ONE case, the joining force based on Equation (4.18) was 132 N
while those obtained from experiments based on both surfaces were 139 N. They
agreed quite well. By using experimental results and Equation (4.7), the joining
forces based on both surfaces for various types of joining were also obtained and
listed in Table (4.1). Results from the top surfaces seemed to agree with those
from the bottom surfaces. All the joining forces were also normalized by that of
the perfect joining case, i.e. ONE, for comparison. A similar procedure was also
performed for the interphase model based on Equation (4.14). The results from
the interphase model were, in fact, the averages of the corresponding values

obtained from the top and bottom surfaces based on the interface model.

112

 

Interface Model

Interphase Model

 

(based on top and

 

 

 

 

 

 

 

 

 

 

 

 

 

top surface bottom surface bottom surfaces)
jomtng jounlng

‘11:: 21:: Warm
ONE 139 100 139 100 139 100
28 123 88 125 89 124 89
TWO -139 -100 -137 -98 -138 -99
2T -31 -22 -7 -5 -1 9 -1 3
2R2 124 89 128 92 126 90
2R3 1 38 99 139 100 1 39 1 00
2R4 142 102 144 103 143 102
28R3 132 95 141 101 137 98
283 97 70 77 55 87 62
285 111 80 110 79 111 79
2883 124 89 122 87 123 88

 

 

 

 

 

 

 

Table 4. 1- Comparisons of joining forces

113

 

According to Table (4.1), the joining force of the 28 case was not as high
as the perfect bonding case, ONE, indicating that the epoxy used in the study
gave an imperfect-joining result. The joining forces of 2R2 and 2R3 were very
close to that of ONE while the joining force of 2R4 was actually higher than that
of ONE. It was believed that the high joining forces were attributed to the
excellent joining conditions based on riveting and the rigidity of the rivets
themselves. Comparing the 2R2, 2R3 and 2R4 cases reveals that the joining
force increased as the density of rivets increased. A similar result could be found

from comparing the 283 and 285 cases.

The comparison of the joining forces of 2BR3, 2R3 and 28 revealed that
the mechanical riveting was more efﬁcient than the epoxy bonding. The
mechanical riveting alone was sufﬁcient for joining the composite beams. The
epoxy bonding, in fact, played a redundant role in the 28R3 case. The
comparison of the 2883, 283 and 28 cases indicated that epoxy bonding was
more efﬁcient than stitching joining. And gain, the less efﬁcient technique, i.e.
stitching joining, played a redundant role in the case based on combined
techniques, i.e. 2883. The negative joining forces of the TWO and 2T cases

were not clear although they were both of poor joining cases.

Conclusions

1. Although it is a surface technique, moire interferometry is able to give a

whole-ﬁeld analysis, e.g. through-the-thickness measurement. Due to

114

transverse loading, the free-edge effect can be neglected in the investigation
of normal strains. In addition, the surface points are found to be better than
other locations through the thickness of a composite beam for normal strain
analysis because the density of the moire fringes are higher and the stresses
are relatively uniform close to the surfaces.

2. Though based on the Classical Beam Theory, both the interface model and
the interphase model are capable of presenting joining forces of assembled
composite beams with the use of moire interferometry. Good agreements
between analytical predictions and experimental investigations seem to
indicate that it is feasible to use the technique combining analytical models
and moire interferometry for identifying joining forces.

3. Among the joining techniques investigated in the study, mechanical riveting
seems to be an efﬁcient joining technique because it is easy to employ and

can give high joining force.

Acknowledgments

The author wishes to express his sincere thanks to Professor Dan Post of
Virginia Polytechnic Institute and State University for help in constructing the
moire interferometer used in this study and many valuable discussions on the
fundamentals of moire interferometry. Thanks are also extended to Professor

Peter ltju of University of Florida for providing the original moire gratings from

115

which many gratings used in this study were derived. The ﬁnancial support from

the Michigan Research Excellence Fund is also greatly acknowledged.

116

References

4.1 Y. D. S. Rajapakse, ed., Mechanics of Thick Composites, AMD-Vol. 162,
The American Society of Mechanical Engineers, 1993.

4.2 Liu, D. and 8.8. Raju, “Effects of Joining Techniques on Impact
Perforation Resistance of Assembled Composite Plates,” to appear in
Experimental Mechanics.

4.3 Liu, Dahsin, “Delamination Resistance in Stitched and Unstitched
Composite Plates Subjected to Impact Loading,” J. Reinforced Plastics
and Composites, 9 (1 ), 59-69, 1990.

4.4 Lee, Chienhom and Liu, Dahsin, ”Tensile Strength of Stitching Joint in
Woven Glass Fabrics,” J. Engineering Materials and Technology, 112 (2),
125-130, 1990.

4.5 Liu, Dahsin and Shakour, Elias, “Combining Photoelasticity and Moire
Interferometry,” submitted to Experimental Mechanics.

4.6 Post, 0,. Han, B. and Ifju, P., High Sensitivity Moire, Springer-Verlag,
1991.

4.7 Pipes, RB. and Pagano, N.J., ”lnterlaminar Stresses in Composite

Laminates under Uniform Axial Extension," J. Composite Materials, 4,
538-548.

117

.11

Chapter 5

CONCLUSIONS AND RECOMENDATIONS

This thesis work was composed of four chapters. Chapter 1 introduced the
moire technique and gave a literature review on the moire grating and its
advancement towards the sensitivity enhancement of the measurement
technique. Then, a various applications were introduced in experimental

mechanics that utilized the moire interferometry technique.

Chapter 2 described the modiﬁcation of the grating layers from a
conventional two-layer grating, to a one-layer grating, and ﬁnally a zero-layer
grating (self grating). The optical moire interferometer setup was also modiﬁed to
include either transparent photoelasticity setup or reﬂective photoelasticity setup,
namely combining moire interferometry with photoelasticity. A photoelastic
sensitive epoxy named PL8 was molded as a zero-layered grating and served as
a coating layer to measure the stress concentrations of a drilled hole in a

glass/epoxy specimen.

Chapter 3 presented a comprehensive study on the relationship between
the mismatches of material properties and the free edge effect. In the
investigations, the one-layer grating was bonded on the surface of a
graphite/epoxy, a composite laminates with a stacking sequence of

[0/90/0/90/0/90/0] and on the surface of glass/epoxy composite laminates with

118

 

stacking sequences of [07/907/07], [03/903/03/903/03/903/03], and [0/90/0,...]21. The
investigations identiﬁed: the depth of the boundary layer; it also veriﬁed the effect
of material type, between graphite/epoxy and glass/epoxy, and the effect of

lamination type, among the 3-lamina, 7-lamina, and 21-lamina laminates.

Chapter 4 described using the one-layer grating to conduct a study on the
joining forces of assembled thin laminated composites. The one-layer grating
was bonded onto the edge of the laminated beams. The beams were assembled
by using various joining techniques, such as mechanical riveting, adhesive
bonding, stitching joining and their combination. Based on the Classical Beam
Theory, the interface model and the interphase model were compared with those

of experimental results.

Major conclusions

Chapter two

1. A simpliﬁed two-layer grating technique was developed. Two-layer grating
could be further simpliﬁed to one-layer grating by simply omitting the
enhancing layer, i.e. aluminum layer. The one-layer grating technique was

found to be feasible for moire measurements.

2. A zero-layer grating technique was developed. The advantage of the one-
layer technique is that it does not have thickness; neither is there a
mismatch of material properties between the specimen and the zero-

layen

119

3. A technique combining moire and photoelasticity was capable of
presenting two independent sets of experiment and result. This technique

provides a self-veriﬁcation capability by comparing both sets of results.

Chapter three

1. Experimental results based on moire interferometry have revealed that
the shear strain on the free-edge increase signiﬁcantly as the width of a

composite laminate decreases.

2. Because of the free edge-effect, shear strain based on moire

interferometry should not be used beyond the surface at all.

3. The depth of the boundary layer was found based on moire interferometry

is greater than the thickness of composite laminates.

4. The free edge-effect is associated with mismatch of material properties.
Since graphite/epoxy has higher anisotropy than glass/epoxy, a
composite laminate made of graphite/epoxy experiences stronger free-

edge effect than the glass/epoxy counterpart.

5. Normal strains on free edges are not inﬂuenced by the free edge effect.
Accordingly, the normal strains based on moire interferometry could be
used to represent the normal strains beyond the surface for plane-strain

and plane stress problems.

120

Chapter four

1.

Moire interferometry is able to give a whole-ﬁeld analysis, e.g. the edges
through the laminate thickness, although it is a surface measurement
technique. The free-edge effect can be neglected in the investigation of

the normal loading.

Both the interface model and the interphase model present joining forces
of assembled composite beams based on the Classical Beam Theory with
the use of moire interferometry. It is feasible to use the technique of
combining analytical models and moire interferometry for identifying

joining forces.

Mechanical riveting seems to be an efﬁcient joining technique because it

is easy to employ and can give a high joining force.

Recommendations for future work

1.

Investigators should explore modifying the diffraction grating. A technique
must be able to attach the diffraction grating onto the surface of the
specimen in a technique similar to the strain-gage installment. Using this
modiﬁcation will allow the moire nterferometry to be less complicated and

more friendly especially when conducting testing in a ﬁeld position.

121

2. Investigators should explore building a micro-scale moire interferometer
that can be positioned in front of an optical microscope to measure within

the micro mechanics ﬁeld, such as the interface interphase between

ﬁbers and matrices.

122

APPENDICES

123

APPENDIX A

PROCEDURES FOR ALUMINUM EVAPORATION

The machine that was used for aluminum evaporation is Bendex- Balzer

Vacuum Inc. (Rochester, NY). The operation was as follows:
1. Position the Tungsten wire in a 17 inches height above the grating.
2. Lay down the master grating.

3. Close the bell jar.
4. Pump down the diffusion pump.
a. gauge to V2.
b. lever to 1.
c. pump to 0.05 torr.
5. Pump down the bell jar.
a. Lever to 3.
b. gauge to H..
c. pump to 5x107 torr.

6. Turn the Quartz Crystal Thickness monitor on.

124

 

7. Turn the current switch on.

8. Monitor the Quartz Crystal till it reaches the desire thickness (for

aluminum; 1.3 hz-)1 A on the quartz Crystal monitor.)

125

APPENDIX B

MOIRE FRINGE PATTERNS USED FOR SUBTRACTIONS

Combining moire interferometry and photoelasticity

1414“ Hum ,»- . .., 1.. .

- we. .... .(thnlﬁwm‘m~

‘ 1 w-‘v'vmtwu-w:““ J
\ ll. ﬂ "

- ,.1114!Salute-«~1rw 12¢th . ‘v - ': 7 n

1- '- «1 , Mn).- (”It , ,‘ (”I
$110581» - . »

Ive-Nu... tor-11.

 

 

(C)

Figure A.1- Fringe patterns of a notched glass/epoxy beam with PL8 moire-
photoelasticity coating (344.72 N) (a) reﬂection iscochromatic fringes, (b) U-
displacement fringes, and (c) simultaneous recording of photoelasticity and moire
fringes.

126

 

Joining forces in assembled composites

 

Figure A.2 - Moire Fringe pattern for 67.5 N, (a) 28, (b) TWO, (c) 2T, and (d) 2R2

127

 

(d)

a) 2R3, (b) 2R4, (c) 28R3, and (d)

(

283
128

Figure A.3 - Moire fringe patterns (67.5 N)

 

Figure A. 4 - Moire fringe patterns (67.5 N), (a) 285, and (b) 2883

129

Mismatch of materials and the free edge effect

 

 

v- fringes

 

  
     
   

     

‘~- h) “ ( ll
. 11.11.?) 111111

u-frlnges H l ’ v- fringes

    

\____A

 

 

(c) 6.35 mm, 222.4 N

Figure A.5 - Moire fringe patterns for graphite/epoxy laminates with three
different widths

130

r,‘ V 1.11“” .. .. 1"]. :1 7| V. ' ...,“
$917136 1"?“‘4 lff‘..\.':<‘-)"rf*'l . 1,1) t. . .th .

   
    

       

 

 

W \‘m N; s NW”
v-fringes

(c) 21-lamina, zero load

Figure A.6 - Moire fringes of glass/epoxy laminates at a zero loaf

131

U

llIlllllllllllllllllll