Phase- -Stepped Digital Speckle Pattern Interferometry (DSPI)
and Its Application to Mechanical Joining of Composite

                   

 

 

 

 

 

 

 

 

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PHASE-STEPPED DIGITAL SPECKLE PATTERN INTERFEROMETRY (DSPI)
AND ITS APPLICATION TO MECHANICAL JOINING OF COMPOSITE
MATERIALS

By

Francesco Lanza di Scalea

A THESIS

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

MASTER OF SCIENCE

Department of Materials Science and Mechanics

1996

ABSTRACT

PHASE-STEPPED DIGITAL SPECKLE PATTERN INTERFEROMETRY (DSPI)
AND ITS APPLICATION TO MECHANICAL JOINING OF COMPOSITE
MATERIALS

By

Francesco Lanza di Scalea

In this work an in-plane sensitive setup of a Digital Speckle Pattern Interferometer
(DSPI) together with a commercial four-bucket phase stepping algorithm is used for high-
precision measurements. The displacement maps are fitted with least-square polynomials
which are eventually differentiated along different directions to obtain strains. A
comparison with results from electrical resistance strain gages in a cantilever beam
experiment is used to verify the accuracy of DSPI for strain analysis.

The technique is then applied to the study of mechanically fastened composites. The
strain/stress field around a single pin in a [0°/9O°]S glass fiber-reinforced epoxy laminate
loaded in tension through the pin is determined. The effect of an interference fit between
the pin and the hole on the stress distribution is investigated for a static load. Results for a
case of perfect fit and two cases of interference fit are compared. A substantial decrease,

up to 50%, in stress concentration factor is observed when an interference fit is used.

ACKNOWLEDGMENTS

The author wishes to offer his sincerest appreciation and gratitude to his advisor and
major professor Dr. Gary Lee Cloud for his continuous and valuable guidance during the
course of this research.

He also wishes to thank Dr. Robert Soutas-Little for his encouragement and support
throughout the author’s course of study at Michigan State University.

The author extends his gratitude to Henry Wede and Xiaolu Chen for their helpful

assistance and discussions.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vi
LIST OF FIGURES ............................................................................... vii
CHAPTER I INTRODUCTION ............................................................ 1
1.1 Problem outline and choice of experimental technique .................................. 1
1.2 Objective and scope ........................................................................... 3
1.3 Relevant literature ............................................................................. 4
1.3.1. Application of Speckle Interferometry to metrology ......................... 4
1.3.2. Study of pin-loaded holes in mechanical structures .......................... 8
1.3.3. Study of the effect of an interference fit ....................................... 11

CHAPTER II DIGITAL SPECKLE PATTERN INT ERFEROMETRY (DSPI)..15

2.1 Introduction ................................................................................... 15
2.2 The speckle effect ............................................................................. 15
2.3 Speckle size .................................................................................... 17
2.4 Speckle Correlation Interferometry ......................................................... 18
2.4.1 Out-of-plane sensitive setup ...................................................... 21

2.4.2 In-plane sensitive setup ........................................................... 23

2.4.3 Recording speckle in speckle correlation interferometry .................... 27

2.4.4 Correlation fringe visibility ...................................................... 28

2.5 Digital Speckle Pattern Interferometry (DSPI) ........................................... 28
2.5.1 Introduction ........................................................................ 28

2. 5.2 Optical setup for in-plane sensitive DSPI ....................................... 29

2. 5.2. 1 The image acquisition system in DSPI ................................... 31

2.5.3 Fringe formation in in-plane sensitive DSPI ................................... 32

2.6 Automatic quantitative fringe analysis ..................................................... 36
2.6.1 Introduction ........................................................................ 36

2.6.2 The phase-shitting technique .................................................... 38

2.6.3 Some phase extraction algorithms .............................................. 42

2.6.4 Comparison of algorithms ........................................................ 45

2.6.5 Phase-shitting devices ............................................................ 45

2.6.6 Phase unwrapping ................................................................. 46

iv

2.6.7 Sources of error in phase measurement and remedies ........................ 48

2.6.8 Issues on spatial resolution in DSPI ............................................. 51

2.6.9 The displacement calculation ..................................................... 52

2.7 Advantages and disadvantages of DSPI .................................................... 53

CHAPTER III EXPERIMENTAL PROCEDURE: PRELIMINARIES ........... 55

3.1 Introduction .................................................................................... 55

3.2 Calibration of the technique .................................................................. 56

3.3 Experimental apparatus and system “setup” ............................................... 61

3.4 Strain calculation from the displacement maps ............................................ 64
3.4.1 Verification of the accuracy of the method: the cantilever beam

experiment ........................................................................... 66

CHAPTER IV STUDY OF A SINGLE PIN-LOADED HOLE IN FIBER-

REINFORCED PLASTIC ................................................. 72
4.1 Introduction .................................................................................... 72
4.2 The specimen: geometry and mechanical properties ..................................... 72
4.3 Experimental procedure ...................................................................... 77
4.4 Experimental results and discussion ........................................................ 80

CHAPTER V STRESS CONCENTRATION RELIEF BY THE USE OF

INTERFERENCE-FIT PINS ............................................... 95
5.1 Introduction .................................................................................... 95
5.2 Experimental results and discussion ....................................................... 100
CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS .................. 105

APPENDIX A Displacement sensitivity in a dual-beam illumination DSPI setup

with cylindrical illumination wavefronts ................................ 108
APPENDIX B Displacement field in rigid in-plane rotation of a plate ................ 113
LIST OF REFERENCES ..................................................................... 115

LIST OF TABLES

Table 1 - Comparison of SCF ’s determined in the present work, by Matthews
at al. (I 982) and by De Jong (I 977). F GRP=fiber glass
reinforced plastic; CF RP=carbon fiber reinforced plastic ........................... 93

Table 2 - Maximum stress concentration factors for ligament, bearing and
shear-out regions for (1 ) perfect fit, (2) low interference
fit and (3) high interference fit ........................................................... 104

vi

LIST OF FIGURES

Figure 1 - A laser speckle pattern (fiom Cloud 1995) ........................................ 16
Figure 2 - The formation of a laser speckle pattern .......................................... 16

Figure 3 - Combination of a speckle field with a reference field in speckle
correlation interferometry ........................................................... 19

Figure 4 - Speckle interferometer sensitive to out-of-plane component of
displacement ........................................................................... 21

Figure 5 - Speckle interferometer sensitive to in-plane component ofdisplacement. . 24

Figure 6 - Schematic of a typical in—plane sensitive DSPI system with phase shifiing. .. 30
Figure 7 - Phase fiinges for a rigid in-plane rotation. Angle of rotation 6:1 x10'4rad.. 35

Figure 8 - Phase fringes for a rigid in-plane rotation. Angle of rotation 0=2x10'4rad.. 35

Figure 9 - Brightness and phase distribution on a line of a fiinge pattern ................. 40
Figure 10 - (a) phase-stepping technique; (b) phase-integration technique ............... 40
Figure 11 - Determination of the phase “modulo 27:” ....................................... 47
Figure 12 - Schematic of phase unwrapping ................................................... 48
Figure 13 - Sampling of a speckle pattern with finite-sized detector elements.

(a) good sampling; (b) bad sampling .............................................. 49
Figure 14 - In-plane rotating plate used to induce known values of dx ..................... 60
Figure 15 - Formation of correlation fringes for rigid in-plane rotation of a plate ...... 60
Figure 16 - Setup for the cantilever beam experiment ........................................ 66

Figure 17 - Strain and stress distributions along a vertical section for a beam

vii

in pure bending with two diflcrent Young ’s moduli
in traction and compression ........................................................ 68

Figure 18 - Phase fiinges obtained in the cantilever beam experiment .................... 69

Figure 19 - Plots of strain an obtained with I) DSPI and 2) strain gages,

along section S-S of the cantilever beam ........................................ 70
Figure 20 - Specimen used for the test: dimensions and loading condition ............... 74
Figure 21 - Schematic of specimen and loading apparatus .................................. 77
Figure 22 - Location of main failure modes in mechanically fastened joints .............. 80

Figure 23 - DSPI phase fi'inges for displacement dy parallel to the direction of load
- Isotropic specimen ................................................................ 81

Figure 24 - DSPI phase fringes for displacement dy parallel to the direction of load
- Composite specimen ............................................................... 81

Figure 25 - DSPI phase fiinges for displacement dx perpendicular to the direction of
the load - Composite specimen .................................................... 83

Figure 26 - Strains 8y parallel to the direction of the load along vertical diameter
in the bearing region ................................................................ 85

Figure 27 - Ejfect of fiiction on bearing strain ................................................. 85

Figure 28 - Strain 8y along lines parallel to the direction of the load located
in the ligament area ................................................................ 86

Figure 29 - Strains ex along lines perpendicular to the direction of the load
located above the hole ............................................................. 88

Figure 30 - Strains ax along lines perpendicular to the direction of the load

located below the hole (bearing zone) ............................................ 90
Figure 31 - Shear strain yxy along location of shear-out failure ............................ 91
Figure 32 - Location of maximum stresses near the hole ..................................... 92

Figure 33 - Eflect of the prestress due to an interference fit on the stress
distribution near the hole for a case of isotropic strip subjected
to axial pulsing load (from Regalbuto and Wheeler 1970) .................... 95

viii

Figure 34 - Schematic of variation of maximum shear stress near the hole
with applied static load, for perfect fit and two cases of
interference fit (fiom Jesspo et al. 1 95 6) ......................................... 97

Figure 35 - Normal stress concentration factor cry/0N along the ligament area for
(1) perfect fit, (2) low interference fit, (3) high interference fit .............. 101

Figure 36 - Bearing stress concentration factor 03/01, along the bearing region for
(1) perfect fit, (2) low interference fit, (3) high interference fit .............. 102

Figure 37 - Shear stress concentration factor 239/01, along the shear-out region for
(1) perfect fit, (2) low interference fit, (3) high interference fit .............. 103

Figure A1 - Schematic of dual-beam illumination DSPI with cylindrical
illumination wavefronts ........................................................... 108

Figure A2 - Computer-simulated measurement result for the dual-beam
illumination DSPI setup using cylindrical illumination wavefronts

with ur=30 ° for a displacement field dx=10,wn, dy=dz.=0 ................... 1 1 1

Figure A3 - Computer-simulated measurement result for the dual-beam
illumination DSPI setup using cylindrical illumination wavefionts

with w=30 ° for a displacement field dz=10pm, dx=dy=0 .................... 1 12
Figure B] - Displacement field in an in-plane rotation of a plate, for a small

angle of rotation 6 .................................................................. 1 13
Figure BZ - dx component for a 10x10 mm2 plate rotated by an angle 0:5 ° ............ 1 14

Figure B3 - dy component for a 10x10 mm‘2 plate rotated by an angle 955 ° ............ 1 14

CHAPTER I

INTRODUCTION

1.1 Problem outline and choice of experimental technique

Due to their high strength-to-weight ratio, composite materials have been used
extensively in aircraft, space vehicles and ground transportation vehicles, as well as in
sporting goods, medical equipment, prostheses and others. Like many other structural
designs, members made of composite must also be joined together. Because of the
anisotropic and heterogeneous nature, the joint problem in composites is more difficult to
analyze than is the case with isotropic materials. Two methods frequently used are
bonded joints and mechanically fastened joints. The former is preferred for the joining of
major structural members. On the other hand, the latter approach is less expensive but
less widely used. One of the possible reasons is the lack of a comprehensive
understanding of the detailed stress distribution in such mechanically fastened joints. For
anisotropic materials the stress distribution is strongly dependent upon several parameters
such as material properties, laminate lay-up, ply orientation, friction, clearance and
interference between mechanical fastener and hole, etc. A further understanding of the
stress distribution around pin-loaded holes in composites would be very useful for
estimating the strength of such a joint.

Interference fit pins are used to lower the stress concentration factors and to reduce

the magnitude of the local oscillatory stress component in the case of oscillatory loads

2
and thereby increase the fatigue life of the component. Not many studies have been done

on the effects of interference on composite materials and, as mentioned earlier, several
parameters contribute to complicate the stress field around the hole in this case.

Most of the studies done on this application in the past have been numerical and
theoretical. More experimental results are needed in order to gain a better understanding
of this complex problem. A full-field analysis of the deformation field on the surface of
the object seems to be the most suitable approach.

Transmission photoelasticity has been used successfully to study stress—
concentrations around holes in isotropic material plates (Jessop, Snell and Holister 1956
and 1958, Lambert and Brailey 1962, Frocht and Hill 1940), but it cannot be used for
opaque composite materials.

Strain gages have been used to study the same problem (F rocht and Hill 1940,
Regalbuto and Wheeler 1970), but they have the obvious disadvantage of being a
pointwise technique and not being able to represent accurately high strain spatial
gradients such as those around a loaded hole.

Speckle photography has been used for interference fit fastener investigations (Ford et
al. 1975), but the main disadvantages are that this technique requires photographic films
and gives rise to a loss of definition resulting from speckle effects in the fringe
photograph. Furthermore this technique has sensitivity limits determined by film
resolution and camera aperture (Cloud 1995).

Moire interferometry has been used successfully to study stress concentrations around
loaded and non-loaded holes in composite tensile members (Herrera Franco 1985, Cloud

et al. 1987, Czarnek, Post and Guo 1987), but this method requires the replication of a

3
diffraction grating on the specimen and the use of photographic plates.

In the present work, phase-stepped Digital Speckle Pattern Interferometry (DSPI) has
been used. Since, in the case of a pin-loaded hole in a tension plate, the displacements are
predominantly parallel to the surface of the object, the in-plane sensitive arrangement of
DSPI has been employed. DSPI is a variation of Electronic Speckle Pattern
Interferometry (ESPI) which processes speckle data inside a computer rather than in an
analog device (Creath 1984). It is an accurate, non-invasive, quick and simple technique
for full-field stress analysis. No photographic processing is involved, and the method
doesn’t require particularly stable environments. This makes DSPI a potential substitute
for Moire interferometry and strain gages.

Phase-stepped DSPI has been used in the recent years for measurement of both the
out-of-plane and the in-plane components of the displacement of the surface of an object
for static loading (mechanical or thermal) and dynamic loading. This method has also
been used for contouring of complicated structures. Only a few studies with this

technique have been conducted for strain/stress analysis.

1.2 Objective and scope

The main objectives in this research are:

a) To verify that phase-stepped Digital Speckle Pattern Interferometry is a quick and
accurate method for full-field strain/stress analysis.

b) To study the strain/stress field around a single pin in fiber-reinforced plastic laminate

loaded in tension through the pin.

4

c) To study the reduction in stress concentration factors resulting from the introduction of

an interference fit between pin and hole in composite materials.

1.3 Relevant literature
1.3.1 Application of Speckle Interferometry to metrology

Digital Speckle Pattern Interferometry (DSPI) is a well-established tool in the
mapping of vibration, displacement or deformation fields. A review paper covers some of
the latest developments in DSPI (Joenathan 1991).

In dynamic analysis, time-averaged, pulsed laser, phase modulation and stroboscopic
methods have been employed. In-plane vibration of objects has been studied by Mendoza
Santoyo, Shellabear and Tyrer (1991) using DSPI. In the in-plane sensitive setup of
DSPI, they used a pulsed laser, which illuminated the object at different points in the
vibration cycle, and the single phase step technique, which only requires two fringe
patterns at 0° and 90° phase difference between the beams, to obtain quantitative
information about the amplitude, direction and phase of vibration over the field.

Joenathan and Khorana (1992) investigated the effect of speckle averaging in a study
of the out-of-plane vibration modes of a plate. They used a fiber-optic out-of-plane-
sensitive DSPI setup with 1: and «I2 phase-stepping methods. They found an optimum
number of acquired frames to be superimposed for a certain state of the object, which
gives maximum contrast of the resultant correlation fringes.

DSPI has also been used for generating contours of diffuse objects. This technique

overcomes some of the drawbacks suffered by the already existing optical methods, such

5
as holographic methods. The advantage of DSPI is that intermediate photographic

processing is eliminated. Joenathan, Pfister and Tiziani (1990) studied contouring of an
object by employing a dual beam illumination DSPI technique. They analyzed the
sensitivity and the orientation of the contour planes. Two exposures were made with the
object tilted between the exposures, and the subtracted speckle pattern consisted of
fringes representative of the shape depth of the test surface. To obtain quantitative data
about the shape of the surface of the object, they used a three-step phase-shifting
technique.

Instead of tilting the test object, a new method for DSPI contouring has recently been
reported. It involves shifting the illumination beams in the case of either one illumination
beam with one reference beam (Winther and Slettemoen 1984) or dual-beam illumination
(Zou et al. 1992). The two images before and after shifting are subtracted from one
another to obtain interference contour fringes across the object, similar to the projected
fringes of projection Moire.

Zou et al. (1992) proposed an arrangement for contouring by DSPI with four
illumination beams, thereby making it unnecessary to move anything during
measurements.

Since DSPI is non-contacting, fast and sensitive, it has also been used for non-
destructive inspections. Cloud and Nokes (1993) conducted investigations to detect
internal defects in graphite epoxy materials and examined the relative sensitivity of DSPI
and another interferometric technique, Electronic Shereography (ES). Both thermal and
mechanical loading were used. Damages caused anomalies in the correlation fringe

patterns and therefore could be detected.

6
Ratnam, Evans and Tyrer (1992) used DSPI to measure the radial expansion of a

heated diesel engine piston. A simple “mirror” concept enabled simultaneous
measurement of in-plane and out-of-plane components of displacements. A three-stage
phase-stepping technique was used to measure the displacements. Before being
processed, the images were low-pass filtered to remove the high-frequency noise.

Gulker et al. (1990) used DSPI for in situ deformation monitoring on buildings. They
determined the out-of-plane component of the displacement by evaluating the correlation
fringes generated in a transportable speckle interferometer. To determine the direction of
the displacements in the object, they used continuous fringe observation.

Preater (1984) used pulsed laser DSPI to measure in-plane displacement on rotating
components. He was able to record interference subtraction fringes for component
tangential velocities up to 15ms".

DSPI coupled with the phase-shifting technique can give quantitative detailed
information on the displacement fields in an object surface. When the strain/stress fields
are to be determined, a further step must be taken: the displacement maps must be
differentiated along different directions. Vrooman and Maas (1991) used phase-stepped
Speckle Interferometry to determine surface strains in a T-shaped aluminum test object.
They first determined the three components of the displacement vector at each object
point by performing three measurements with different sensitivity vectors and using a
four-step phase shifting technique. They then calculated the three components of the
surface strain (the two normal strains and the shear strain) by differentiating the

displacement maps along the two in-plane axes. The upper limit of the measuring range

7
of the strain in their setup was ~200ustrain for a 100x100 mm2 object region. The

application of a smoothing filter to reduce phase errors caused by speckle decorrelation,
gave a repeatability of the measured strains at each pixel of ~0.3 ustrain rms.

Jia and Shah (1994) used in-plane DSPI to observe the displacement and strain fields
in the fracture of carbon-fiber-reinforced concrete specimens subjected to tension. Their
setup was a two—dimensional in-plane displacement-sensitive interferometer consisting of
two single DSPI systems, each to measure independently one of the the two in-plane
components of the displacement. Qualitative information about fracture processes, crack
initiation and crack propagation were extracted by monitoring the correlation fringes
continuously and in real time. The quantitative analysis was carried out through an
analysis of the sensitivity vector for two-dbensional DSPI. After determining the
displacements, the strains were calculated using a least-square surface-fitting method.
The discrete two components of displacement were fitted with two plane surface
functions which were eventually differentiated along two in-plane directions to obtain
normal and shear strains.

DSPI has been used by Vikhagen and Malmo (1991) to measure deformations in
polymer laminates and concrete. They used a max-min scanning phase-shifting procedure
to measure the deformation. Before calculating the spatial gradients of the deformation,
the phase images were smoothed to reduce the noise level. They were able to detect
cracks with length less than 1mm in a concrete specimen in a pressure test, using the fact
that cracks spatially shift the deformation gradient (strain) displayed on the video monitor

and therefore are easily detected. The crack propagation could be also monitored.

8
1.3.2 Study of pin-loaded holes in mechanical structures

Several studies on mechanical joints and specifically pin-loaded holes have been
conducted in the past, and some of them are mentioned here.

A review paper on the strength of mechanically fastened joints in fiber-reinforced
plastics was written by Godwin and Matthews (1980). Effects of material parameters,
fastener parameters and design parameters are summarized and discussed.

Chang, Scott and Springer (1984) presented methods for sizing composite laminates
containing one or several pin loaded holes in order to obtain maximum failure load .

The stress field around pin-loaded holes in composite plates has been studied
numerically in the past. Pradhan and Ray (1984) did a finite element investigation of the
stress distribution around pin-loaded holes and gave results for isotropic as well as fiber
reinforced plastic composite materials. They considered the case of full contact (contact
angle=180°) and that of partial contact (contact angle<180°) between pin and hole. They
found that the maximum circumferential stress at the edge of the hole depends strongly
on the material properties as well as the ratio of hole diameter to width of the plate. This
maximum stress was found to be high in the case of graphite/epoxy and boron/epoxy
unidirectional laminae, with respect to glass/epoxy. Furthermore, this maximum
circumferential stress was shown to depend strongly upon the contact angle between pin
and hole. They also concluded that the rate of decay of stresses along the axis through the
pin, perpendicular to the loading force, is higher in orthotropic plates then in isotropic
plates, and the reverse is true for the stresses along the axis parallel to the loading force.

Matthews, Wong and Chryssafitis (1982) applied a three-dimensional finite element

9
analysis and showed that the stress distribution around a loaded hole in fiber-reinforced

laminates depends on whether the load is applied via a pin or a bolt. Values of stress
concentration factors (SCF) for different geometries of the specimen are given and they
are compared to results from other references.

Eriksson (1986) used finite element analysis to calculate contact stresses and stresses
in the vicinity of the hole boundary, taking account of the contact problem. He studied
effects of laminate elastic properties, clearance, friction and load magnitude and
concluded that these parameters affect the stress distribution significantly.

Wong and Matthews (1981) conducted a two-dimensional finite element analysis of
bolted joints in fiber reinforced plastic. They also compared the strain pattern in a two-
hole joint with that in a single-hole joint.

Cohen et al. (1995) used finite element analysis to predict the strength of
multifastener, thick composite joints. The stress fields around the fastener-loaded hole in
both single and multifastener joints were determined. They then used the maximum strain
criterion to locate possible sites of failure initiation and utilized this information in
conjunction with the average stress criterion to predict the strength of the joint.

Some studies have also been done using analytical methods. Theocaris (1956) gave
an analytical exact solution for the stress distribution resulting from loading a hole with a
pin in isotropic materials. He examined different values of diameter of the hole.

Zhang and Ueng (1984) found an analytical solution of stresses around a pin-loaded
hole in orthotropic plates using the complex stress functions which satisfy the
displacement boundary conditions along the hole. They observed that the stress field is

strongly affected by the elastic properties of the material and the presence of friction,

10
which increases steadily the shear stress along the edge of the hole and concluded that

decreasing the friction between pin and hole and keeping the pin load along the principal
axis of higher Young’s modulus will increase the ability to support the pin-load.

De Jong (1977) studied analytically the stress field near the hole in orthotropic and
isotropic plates using Lekhnitskii’s method of complex functions. He considered the hole
loaded without friction on only a part of its edge. He showed that the stresses depend on
the material properties.

The effect of pin elasticity, clearance and friction on the stresses in a pin-loaded
orthotropic plate has been studied by Hyer, Klang and Cooper (1987) with the complex
variable approach. They found that friction changes the sign of the circumferential stress
in the bearing region and reduces the bearing stress. It also increases the peak
circumferential stress. They also found that the contact angles change as the load level
increases.

Collings (1977 and 1982) studied the strength of composite laminate joints for the
different failure modes. He suggested that the inclusion of i45° plies strengthens the
laminate.

There have also been experimental studies of the pin-loaded hole problem. Cloud et
al. (1987) studied experimentally the pin-loaded hole problem in orthotropic composites
using high-sensitivity interferometric Moire and numerically with boundary element
techniques. They examined multi-fastener arrays, load-spreading washers, bolt preload
effects and field measurements.

Herrera Franco (1985) determined strain fields around a pin-loaded hole in isotropic

11
and orthotropic specimens using high-sensitivity interferometric Moire. He observed a

substantial reduction in stress concentration factors when an isotropic material insert is
used between the hole and the pin.

Frocht and Hill (1940) used strain gages and photoelasticity to determine the stress
concentration factors in isotropic plates loaded through pin in the hole. They studied also
the effect of geometrical parameters of the joint.

Nisida and Saito (1966) used an interferometric method to study the stress

distribution around a pin-loaded hole in isotropic materials.

1.3.3 Study of the effect of an interference fit

The effect of an interference-fitted pin on the stress distribution has been studied for
both isotropic and orthotropic materials. Nevertheless, a complete understanding of the
behaviour of such joints, especially in the case of anisotropic materials, is yet to be
attained. It is hoped that this paper is a further step in the right direction.

Jessop et al. studied photoelastically the stress field near the hole for the cases of (1)
load applied to the plate only (1956) , load applied (2) to pin only and (3) to pin and plate
simultaneously (1958). Their studies were on isotropic materials and showed a reduction
in stress concentration factors (SCF’s) resulting from the introduction of an interference-
fit pin in the hole. This effect was shown to be qualitatively the same for the case of load
applied to the plate only and load applied to the pin. They also found that the decrease of
SCF’s depends on the ratio of interference stress to applied tension. When varying loads
are applied to the pin or the plate, the effect of interference is to produce a rise in the

mean stress level at critical points on the hole boundary with a marked fall in the

12
oscillatory stress. This phenomenon leads to an increase of the fatigue life of the joint.

There appears to be an optimum value of interference for given applied load at which the
endurance of the joint will have its highest value.

Lambert and Brailey (1962) studied photoelastically the influence of the coefficient of
friction on the stresses in a pin-joined connection with interference for isotropic
materials. They found non-linearities in load transfer mechanism from the pin to the plate
in the case of an interference fit (non-linear relation between applied load and maximum
stresses in the plate), except at very high initial interference. They showed further that
this behavior is due to relative slip between the two surfaces and therefore depends upon
both the initial interference and the coefficient of friction between pin and hole. They
concluded that, in order to fully obtain the benefits of the use of high interference fits in
reducing the increment in stresses and enhancing the fatigue life of a joint under pulsating
load, a high coefficient of friction between pin and hole coupled with a high initial
interference fit is needed.

F rocht and Hill (1940) used strain gages and photoelasticity on isotropic material to
obtain stress concentrations around the pin-loaded hole. They investigated the effect of
geometry and clearance between pin and hole. In the case of clearance, they found
variations in SCF’s with the load.

Stress distributions from interference fits and uniaxial tension in isotropic plates were
determined theoretically and experimentally using strain gages by Regalbuto and Wheeler
(1970). They observed that interference prestress and load-induced stress cannot be
simply added to calculate the resultant stress field, but they should be nonlinearly

superimposed. The result is that application of a tensile load to the prestressed assembly

13
decreases the change in stress level due to the applied load with respect to the case of no

prestress: this smaller excursion contributes to increase the fatigue life of the member in
the case of a varying pulsating load.

Numerical studies on interference fit pins in composite plates subjected to both pull
and push type of loads have been performed by Ramarnurthy (1989). He showed that
interference increases fatigue life. Using finite element analysis, he found that the
inteference fit pin exhibits partial contact/separation under the loads and the contact
region is a non-linear function of the load magnitude. The same author, in another paper
(1990), studied interference fit pin joints subjected to bearing bypass loads, finding
similar non-linearities.

Sendeckyj and Richardson (1974) studied the fatigue behaviour of graphite/epoxy
joints loaded through an interference fit pin and showed that increasing the level of
interference increases the fatigue life.

Ford et al. (1975) gave a prescription for fatigue-life analysis and prediction for
isotropic joints assembled with interference fit fasteners, using analytical and
experimental techniques (speckle photography, strain gages and dislocation etching).
They were able to determine stress fields due to interference along different directions in
the specimen and observed the decrease in cyclic strain amplitude in fatigue tests when
the degree of interference was increased, for three values of interference. They used loads
that were large enough to induce plastic deformations around the hole.

In a survey paper on fasteners for composite structures, Cole, Batch and Potter (1982)
noted that interference fit pins are desirable not only to increase fatigue life but also to

increase load-sharing in joints with multiple rows of fasteners and to retard fuel leakage

14
in tank areas. They also pointed out that it is the low interlaminar strength of composites

that restricts the use of interference fit for these materials.

CHAPTER H

DIGITAL SPECKLE PATTERN INTERFEROMETRY (DSPI)

2.1 Introduction
A comprehensive presentation of the main issues concerning Speckle Interferometry
techniques has been given by Cloud (1995). His book has been the main source of

information for this chapter.

2.2 The speckle effect

When an object with rough surface (on the scale of an optical wavelength, ~0.6 pm)
is illuminated with a coherent light such as laser light, its visual or photographic image
appears covered with a grainy structure as shown in Figure 1: this is what is known as
“the speckle effect”.

(The origin of this effect is explained in Figure 2. When laser light is reflected or
scattered from the “rough” surface of the object, the optical wave at any receiving point
results from a combination of different waves coming from different points of the
illuminated surface. The optical path lengths of these waves, from source to object point
to receiving point, differ in general depending on surface roughness and the geometry of
the system. These coherent waves interfere at the receiving point, giving rise at an

irradiance that can be anything from dark to fully bright.

15

 

Figure 1. A laser speckle pattern (from Cloud 1995).

laser source

destructive

interference

O

constructive

interference

 

     

detector plane

Figure 2. The formation of a laser speckle pattern.

17
In Figure 2 the two extreme cases of irradiance equal to zero (“destructive”

interference —> dark speckle) and maximum irradiance (“constructive” interference -—)
bright speckle) are shown. This type is called “objective speckle”, as opposed to
“subjective speckle” which involves the presence of a lens to create an image of the
object, in which case the diffraction limit of the lens must be considered. As a result of
this, the waves creating an image do not travel from object point to image point. Rather
they go from object “cell” to image “cell”. The results are substantially the same as the
case of “objective” speckle . As an example, any speckle observed by the eye involves the
imaging optics of the eye and therefore is “subjective” speckle.

The speckle, regardless of whether is subjective or objective, has a known brightness
probability density function that shows that the most probable brightness for a speckle is

zero, i.e. there are more dark speckles in the field than bright speckles.

2.3 Speckle size

For subjective speckle the size S of the individual speckles is related to the aperture
ratio F =focal length/diameter of aperture=f/a of the imaging lens (the f/No.), the
magnification M of the lens and the wave length of the light A.
The speckle size in the image is:
S.-':-122(1+M),1F (1)

From simple lens theory, the speckle size on the object is given by

F
Sobj 5 1.22(1 + MM 1!— (2)

This is defined as the resolution element on the object. The speckle size usually ranges

18
between 5 and 50 um.

Sometimes two speckle patterns are superimposed, such as with the in-plane sensitive
setup. In this case the spatial frequency distribution of the combination of the two beams
will be similar to that of either one of the two beams.

The minimum speckle size in this case is given by:
Smin s 122(1+ MMF (3)
The resolution limit of the detector array must be higher than Smin in order to resolve the

speckle pattern.

2.4 Speckle Correlation Interferometry

Speckle Correlation Interferometry is a technique that uses the phase information
within a speckle and the coherent combination of speckle fields as the basis of
measurement. The phase data of the waves coming from different object points and
interfering coherently to create a speckle are already converted to amplitude information.
In this case, only the individual speckle irradiance can be recorded. In order to somehow
record the phase data, specifically the average phase within one speckle cell, the speckle
field must be combined with a reference beam (or just another speckle beam). In this way
the phase changes between two images of the object corresponding to two different states
of the object (undeforrned and deformed state or two different deformed states) can be
seen as correlation fringes which are representative of object displacement components.
Figure 3 is useful to understand what happens when a speckle field is combined with a

reference field.

l9

illuminating beam

  
 

image lens detector plane

 

 

 

ODjCCt ".v'.‘ F,-

reference beam

Figure 3. Combination of a speckle field with a reference field in speckle correlation
interferometry.

The effect of the reference beam is to make the speckles very dependent on the phase
difference between reference and object beams. The short speckle-producing waves give
the idea that the average optical phase within a speckle becomes sensitive to z-axis
displacements of the object when a reference beam with a nearly zero incidence angle is
used. Sensitivity to out-of-plane motions is equal to that of holographic interferometry.
The three modes of fringe observation of real-time, time-average and double-exposure are
conceptually the same for the two techniques.

Speckle metrology involves recording and/or combining two speckle patterns for two
states of the specimen. In order for Speckle Correlation Interferometry to function, the
two speckle patterns have to be “correlated” with one another. “Correlated” means that

these two fields have to be superimposed. There are two causes of loss of correlation in

20
speckle metrology: phase decorrelation and memory loss.

Phase decorrelation: a speckle can be seen as a signature of a microscopic object
surface element. If the displacements of the object are relatively large, an alteration of
relative phase relationships within each speckle during speckle movement occurs, altering
the fundamental speckle signature. It seems that a phase change of 21: from edge to edge
across a given speckle results in total phase decorrelation (Cloud 1995).

Memory loss: when the object deforms, the resolution element also moves. If the
movement is large enough so that each displaced speckle does not appreciably overlap
the corresponding original speckle position, the displaced speckle cannot be compared
with its first version. The maximum allowable motion of each speckle in speckle
correlation interferometry is some fraction of the speckle diameter. If electronic detectors
(photographic devices, sensing element in a CCD video device, etc.) are used, the
movement of each speckle cannot be so large that the speckle is shifted across the
detector element, in which case the detector would “lose memory” of the speckle which
contains the displacement information. An implication of this is that speckle size should
be large for speckle correlation interferometry.

Speckle decorrelation puts a limit on the range of measurements with speckle
correlation interferometry.

Is should be mentioned that there is another technique based on the speckle effect
called “Speckle Photography”, where the speckles before and after the deformation must
form a pair that is separated by a small distance in order to form a sort of grating. This

method is complementary to Speckle Correlation Interferometry in terms of range and

21

sensitivity of measurements.
Two arrangements of Speckle Correlation Interferomery have been developed in the

past to measure in-plane and out-of-plane components of displacement.

2.4.1 Out-of-plane sensitive setup
A setup sensitive to the out-of-plane component of displacement is shown in Figure 4

(Jones and Wykes 1983). This setup has been used for several applications in speckle

 

 

 

 

 

 

interferometry.
x
A Illuminating beam 1 reference beam
‘ imaging system
\ _ z
l l fl
i beam splitter
obdject detectrng plane

Figure 4. Speckle interferometer sensitive to out-of-plane component of displacement.

The z—axis is chosen to be the optical axis. The object is illuminated at an incidence
angle 0i (angle between illumination direction and normal to the object surface). A
reference wave, which has usually either spherical or planar wave front and nominally

normal incidence, is added to the image using a beam splitter. Applying the theory of the

22
sensitivity vector (Vest 1979) with a viewing angle 0,=0, the expression of the change Air

in the phase of the object beam relative to the reference beam, point by point on the

detecting plane is :
2n
A¢=(7)(l+cosei)dz (4)

where 0i is the incidence angle of the illuminating beam, 3. the wave length of the light
used to illuminate the object and dithe out-of-plane component of the displacement.

It should be noticed that for ordinary applications the incidence angle of the
illuminating beam is considered constant across the illuminated object even if the beam
has a spherical wave front (spherical wave front is easier to achieve because it doesn’t
require a collirnating lens and also it’s not easy and economical to obtain a large
collirnated beam to cover a large area). In the same fashion, a spherical wave front
reference beam can be considered incident at a constant angle across the detecting plane.
This approximation is more acceptable the further the object is from the expanding lens
of the illuminating beam and the further the detecting plane is from the expanding lens of
the reference beam. A quantitative analysis of the measurement errors induced by a
variable sensitivity vector across the field has been done by Chen (1995). The case of
cylindrical wave fronts in a dual-beam illumination DSPI setup is examined in Appendix
A.

Assume that the system is operating in real-time mode. The brightness of the image,
which is the negative of the original speckle pattern, varies as the specimen moves in the
z-direction. A dark fringe in the image will occur in those points where Ad) =2n1r, n being

an integer, i.e. wherever

23

n).
= __ 5
dz (1 +cosai) ( )

n is the fringe order.

This means that the sensitivity of this method is

dz ,1 (6)

7 (1+ cos 9i)
As a practical example, for A=O.633 pm (He-Ne laser light) and 0i=10° , it is

iii = 0.32m (7)
n

This means that there is a fringe every 0.32 um of displacement, which is a very high
sensitivity. The maximum measurable displacement is limited by decorrelation and/or

memory loss as explained earlier.

2.4.2 In-plane sensitive setup

In order to measure the in-plane component of the displacement, a dual-beam
illumination setup is needed (Leendertz 1970). In this arrangement two object speckle
patterns are combined coherently.

Referring to Figure 5, the object lies in the xy-plane, and the z-axis, normal to the
object surface, is the viewing axis. The object is illuminated by two plane wave fronts
inclined at equal and opposite angles 0 with respect to the surface normal. Let I.(x,y) and
I,(x,y) be respectively the light irradiances of the first and of the second object beam at
the imaging plane. At any image point P, the irradiance resulting from the combination of

the two beams is:

24

X

imaging system

 

 

 

 

 

 

detect-ing plane

Figure 5. Speckle interferometer sensitive to in-plane component of displacement.

I=Il+12+2‘/711—2cos¢ (8)
where d) is the phase difference between the two waves contributing to point P in the
image at its initial state (1 ).
This is the typical interference equation, stating that the intensity in each point of an
interferograrn is a cosine firnction of the phase difference between the two light
contributions arriving at the considered position.
If the object is displaced by a displacement vector d having generally three

components d," d, and d,, the irradiance at P becomes:

(1) The irradiance I of a beam is the time average of the square of the wave amplitude A
of the beam. It represents the time average of the power density of the light.

25

1=Ir+12+241112008(¢-4¢) (9)
where Ab is the added phase difference induced by the displacement.

Using again the sensitivity vector theory, the total phase change in beam 1 is:

51=(k3-k1)'d
_27r k . 0 . 9 k . . k
—7[ -(—sm -1—cos - )]-(dx-l+dy-j+dz- )

= 27”[dxsina+dz(1+cos9)] (10)
The total phase change in beam 2 is:

62=(k3-k2)-d
= gf-[k—(sinfl-i—cosB-k)]~(dx-i+dy-j+dz-k)

=-27”[_dxsin0+dz(l+cost9)] (11)

In equations (10) and (11): k1 and R2 are the unit vectors along the two illurrrinating
directions, k3 is the unit vector along the viewing direction; i, j and k are the unit vectors
along the (x,y,z) reference system.

The Ab in equation (9) is the difference of 8, and 52, i.e.
A¢=(47”) dxsinB (12)

where 9 is the illumination angle (angle between the direction of the illuminating beam
and the normal to the object surface) and it is the wave length of the light used.

A dark fiinge in the image will occur in those points where Ab =2n1t, n being an integer,

26
i.e. wherever

 

 

n2
= (13)
dx 2 sing
The parameter n in Equation (13) represents the fringe order.
The sensitivity of this setup is:
Q= .1 (14)
n 2 sm 6
As a practical example, for 350.633 um (He-Ne laser light) and 0=30° , it is
d x
— = 0.63 3pm (15)
n

This means that there is a fringe every 0.633 pm of displacement. The maximum
measurable displacement is again limited by decorrelation and/or memory loss.

For a complete analysis of the deformation of an object, the two in-plane components
of the displacement need to be determined, in addition to the out-of-plane component. To
measure the second in-plane component, either the optical setup or else the specimen
must be rotated by 90° about the z-axis. Another solution is the use of a two-dimensional
interferometer (J ia and Shah 1994) where the two components of the in-plane
displacement are measured basically at the same time, avoiding errors due to imperfect
repositioning of the elements after rotation by 90°.

The recent introduction of optical fiber technology in speckle interferometers (e. g.
Joenathan and Khorana 1992) gives the system more flexibility and makes it easier to

rotate the optical paths to measure both components of the displacement.

27
2.4.3 Recording speckle in speckle correlation interferometry

As an image recording device, a CCD (charge-coupled device) camera is usually used
in speckle correlation interferometry. The camera is used in combination with an imaging
lens.

The aperture of the imaging lens should be such that it matches the resolution of the
camera, i.e. gives speckles which can be resolved by the camera detector array, which is
typically a 512x512 pixel array. Increasing the aperture decreases the speckle size and
therefore the correlation fringes are less grainy and smoother. Moreover, the required
laser power can be reduced. Also, since more light illuminates the object, the speckle
contrast and the speckle modulation increase. On the other hand, when the light levels are
good, closing the aperture has the advantage of increasing the speckle size and thus
increasing both contrast and modulation on the detector array (see Section 2.6.7). A
larger speckle size means also an increase in range of measurements in correlation
speckle interferometry. In adjusting the aperture there is therefore always a trade off . As
a general rule of the thumb, the aperture of the viewing system should be set to the
smallest diameter possible, consistent with a reasonable bright object beam.

The techniques to record the specklegrarn are the same as those used in holographic
interferometry, i.e. real-time, time-average or double-time modes.

The system is sensitive to vibrations, when vibration amplitudes are larger than
speckle size. Vibration isolated tables are usually used.

Specklegrams for correlation interferometry can be recorded in high-contrast

photoplates. The resolution needed in the emulsion is not as high as the one required in

28
holographic interferometry, especially if the system is designed to have large speckles.

The possibility of recording specklegrams with electronic means avoids the use of

photographic materials. This subject is discussed in Section 2.5.

2.4.4 Correlation fringe visibility

Fringe contrast in speckle correlation interferometry is low as compared to other types
of interferometry. The highest contrast is achieved in real-time fringes, and the theoretical
maximum visibility is on the order of 14%. Double exposure and time-average processes
give very low contrast fiinges, and the reason is that two speckle patterns have been
incoherently added before fihn development. Areas where the two speckle patterns are
correlated will have high-contrast speckles, whereas areas where the speckle patterns are
uncorrelated will exhibit poorly contrasted speckles. Optical spatial filtering can improve
fringe contrast.

With the advent of electronic image acquisition and computer fringe processing,
contrast enhancement and digital filtering are used to substantially improve the visibility
of correlation fringes. These new techniques are known as Digital Speckle Pattern

Interferometry.

2.5 Digital Speckle Pattern Interferometry (DSPI)
2.5.1 Introduction
The basic idea of DSPI was developed in England by Butters and Leendertz (1971)

and in the United States by Macovsky, Ramsey and Schaefer (1971). The use of

29

television image acquisition and computer image processing has revolutionized optical
methods of metrology. DSPI is a variation of Electronic Speckle Pattern Interferometry
(ESPI), which processes speckle data inside a computer rather than in an analog device
(Creath 1984). It measures components of displacement of an object surface. DSPI is also
known as video holography or TV holography.

DSPI is an electronic version of Holographic Interferometry which uses a CCD
camera instead of a photographic fihn to record the images. The resolution of the
recording medium need not be high compared to that required for holography. This is due
to the fact that only the speckle pattern must be resolved, the speckles being now the
carriers of the phase information and not the very fine fringes formed by interference of
object and reference beam in holography. The speckle size can be varied by adjusting the
magnification M and/or the aperture ratio F=focal length/aperture=f/a of the viewing lens,
as explained in Section 2.3 and is typically in the range of 5 to 50 um. Usually a standard
television camera can resolve the speckle patterns. Video processing is then used to

obtain correlation fringes and perform quantitative analysis to calculate displacements.

2.5.2 Optical setup for in-plane sensitive DSPI

Figure 6 shows the schematic of a typical in-plane sensitive DSPI system which uses
a phase-shifting technique for quantitative analysis.

For most work, a He-Ne laser or a diode laser is used. In the diagram, a beam splitter
is used to obtain the two illuminating beams which have to be coherent with each other in

order to interfere. The prism is used to compensate the optical path of one beam so that

30

 

 

 

    
 
 
  

 

 

beam splitter .
7 mrrror
I He-Ne Laser R 5 \
,, . mirror
" <14) . 7 PZT
prism .
mrrror
+——-9 :

......................... . object

 

mirror \

collirnating lens

 

microscope objective+spatial filter

 

I Ple controller I

 

 

high resolution display

 

 

 

 

 

 

 

 

....................... —| . ........... . :

 

Figure 6. Schematic of a typical in-plane sensitive DSPI system with phase shifting.

31
the difference between the optical path lengths of the two beams can be made less than

the coherence length of the laser light and therefore satisfy the coherence condition. The
two spatial filters are used to remove extraneous optical noise. Two microscope
objectives expand the beams in a spherical wavefiont. The plano-convex collirnating
lenses then give collirnated beams. This is necessary in order for the sensitivity vector to
be the same over the field, as noted in Section 2.4.1. A piezo-electric transducer (PZT),
controlled by a computer, is used to shift the phase of one of the two illuminating beams.
This procedure is known as “phase-shifting” or “phase-stepping” technique and will be
explained later in this work. It provides quantitative information on the displacement
field. A CCD camera with a typical resolution of 512x512 pixels is used to record the
speckle patterns, which are eventually stored in the RAM memory of a computer and
processed. A high resolution monitor displays the correlation fringes.

In the out-of-plane sensitive setup of DSPI a reference beam is directed into the
detector plane of the camera and only one object beam is used. In this work the attention
is focused on the in-plane sensitive DSPI and the discussion is valid for the out-of-plane
setup provided that one of the two illuminating (“object”) beams is substituted with a
reference beam with nominally 0=0° angle of incidence on the camera detector plane (see

Section 2.4.1).

2.5.2.1 The image acquisition system in DSPI
The CCD camera is used because it has wide range of linearity, low noise at low light

intensity and high signal-to-noise ratio. When light falls on the CCD camera array, an

32
output is formed in terms of a voltage signal which ideally is proportional to the image

irradiance. The signal is then transferred to the computer monitor, whose brightness
should vary ideally linearly with the irradiance of the original image. The word “ideally”
is used to emphasize that in reality the linear range is limited. Attention should be given

to camera and monitor linearities for precise measurements.

2.5.3 Fringe formation in in-plane sensitive DSPI

In DSPI at least two images of the object are taken in order to create fringes. They can
be undeforrned and deformed states or two different deformed states. A fiinge pattern is
obtained using video processing by storing the initial reference image of the object in a
frame grabber and then subtracting it from the one being acquired, displaying the
modulus of the result on a TV monitor. Fringes, similar to those obtained in holographic
interferometry, then appear when a phase change is introduced between the reference
image and the subsequent ones. The detector plate of the camera is located in the image
plane of the speckle interferometer.

At the first step, the coherent combination of the two illuminating beams at a point or

pixel in the CCD sensor plane results in the following intensity of light:

lbefore= Ai+A3+2A1A2008¢ (16)
where A,(x,y) and A,(x,y) are the amplitudes of the first and second object beams and
b(x,y) is the phase difference between the two beams at each point of the detector plane.

Equation (16) is obtained from Equation (8) with the simple substitution I=A2 (this

substitution is usually done wherever the amplitude of the electric field of the light A has

33
to be used instead of the brightness I). This is what is called the reference image. Usually

the undeforrned state of the object is taken as reference. This image is digitized by a
frame grabber in a computer and it is then stored in a frame buffer.

The second step consists of deforming the object by applying some kind of load and
taking an image of the deformed object. The intensity of light on the CCD camera

becomes:

[after = A? + A% +2A1Azcos(¢-A¢) (17)
where Ab is the additional phase change due to the deformation of the object. It is
important to keep in mind the assumption that the deformation of the object does not
displace the speckles: only a phase change within the speckles occurs; the speckle fields
have to be correlated, as noted earlier in this work.

In the real-time mode, the subsequent frames are subtracted continuously from the
stored reference frame. If the output camera signals Vbefore and V,,,,, are proportional to the

input image intensities, the subtracted signal is given by:

Vs = (Vbefore - Vafler) °C (1 before - I after)
=2A1A2[cos¢—cos(¢—A¢)] (18)
This signal has positive and negative values. However, the television monitor will
display negative values as areas of blackness, and, to avoid this loss of signal, V, needs to
be rectified before being displayed on the monitor. To do this V, is either squared or the
modulus is taken. The brightness on the monitor is then proportional to the absolute value

of V, and, point-by-point in the monitor, is given by:

34

B = KA1A2 cosb- cos(b-Ab)| (19)

 

where K is a constant (2).

The resultant brightness on the monitor B varies between maximum and minimum values

B,,,,, and Bmi, given by:
BM =KA1A2 where Ab=(2n+1)1t with 11 an integer—-) bright point
B“, =0 where Ab=2n1r with n an integer—) dark point.

For any other Ab¢(2n+1)1r or 2m, the subtraction gives a nonzero value, and the
monitor shows a spot whose gray level varies accordingly to the phase change. These are
called “phase fringes”. Thus the monitor shows a set of hinges immersed in speckle noise
and following the zones of constant phase change. The signal should be high-pass filtered
to improve fringe visibility by removing low-fiequency noise.

Phase fringe patterns in a plate rotated rigidly, originated by in-plane sensitive DSPI
using video signal subtraction, are shown in Figures 7 and 8 for two different angles 0 of

rotation of the plate.

(2) Creath (1985) observed that by subtracting the two intensities, taking the modulus
squared and taking an ensemble average over many realizations of this fiinge fimction
the resultant brightness at the detector plane becomes :

(I1 I2) = 8(A112XAg)sin2 (% 41¢) (20)

This fiinge function is similar to fi'inges in a classical interferometer where the fiinges
are sinusoidally dependent on a relative phase difierence. However Equation (20)
depends on the phase diflerence between exposures rather than the phase difference
between object and reference beams.

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Figure 8.Phase fiinges for a rigid in-plane rotation. Angle of rotation 0=2x104 rad.

The phase difference Ab is related to the displacement component by the sensitivity

vector, which depends on the optical geometry of the system. Recalling the relation

36
between phase difference and displacement component for an in-plane sensitive setup

(Equation (13)), a dark fringe will appear wherever:

d _ n).
x 23in6

(21)
Analogous relations are present for an out-of-plane sensitive setup.

Fringes in DSPI can also be created with video signal addition, where the light fields
corresponding to the two states of the object are added at the image plane of the camera
instead of being subtracted. The addition process has the disadvantage that is very
sensitive to noise produced by dust particles for example, which tend to collect on the
beam splitter and act as light scattering centers. This degrades the quality of the fiinges.
In the subtraction mode, because the noise is the same for both images, its effects cancel

out in the resulting fringe pattern, and, therefore, the subtraction fringes are qualitatively

better.

2.6 Automatic quantitative fringe analysis
2.6.1 Introduction

Interferometric fringes are a map of a warped wavefront which is in turn indicative of
something useful, such as components of displacement or the shape of a surface. To
interpret these fringes, one can count the fringes and perform other calculations using
some parameters depending on the particular technique used. The results are then
interpolated between the fiinges to give the variable of interest anywhere in the field.

This process presents some difficulties. It is slow and tedious. Sometimes there are

not enough fringes to provide accurate results from interpolation, especially when the

37
spatial gradients of the variable of interest are high. Although techniques of fiinge

multiplication have been developed, this is not the best way to obtain an accurate map
over the field of the variable of interest. Moreover, in the case of DSPI, the correlation
fiinges are not clearly defined because they are composed of variations in speckle
contrast (see Figures 7 and 8). They are good for qualitative measurements but have not
produced good quantitative results, since it is hard to determine the fringe centers and the
fringe spatial resolution is generally pretty low.

A technique of fiinge multiplication in DSPI has been developed by Floureux (1993).
He subdivided the total deformation into small increments, recorded the phase fringes for
each increment and digitally added them together to obtain fringes representing the total
deformation. But the point is that multiple fringes don’t actually represent a higher
amount of information than regular fringes, since theoretically even a low-frequency
phase-fringe pattern contains detailed local displacement information in the form of phase
values: this is the property that is used in phase-shifting techniques.

Quantitative data can be obtained using phase-shifting interferometry (PSI). The
whole point is purposely shifting the phase of one beam in the interferometer with respect
to the other. The phase of the test wavefront relative to the reference wavefront can be
simply calculated form the interferogram intensities measured for multiple phase shifts.
This phase measurement technique has been known for almost thirty years (Carre’ 1966,
Creath 1985).

In speckle interferometry the phase maps are calculated from sets of phase-shifted
intensity data taken before and after the deformation and then combined to produce the

phase of the difference between exposures. Each set of phase-shifted intensity data is

38
obtained in the same way as PSI. To find the phase of the object deformation, the phases

of the speckle patterns before and after the deformation are simply subtracted. The
deformation of the object is then calculated from the phase data using the sensitivity

vector theory.

2.6.2 The phase-shifting technique
Recalling Equation (8), which expresses the irradiance resulting from interference of
two speckle patterns in an in-plane sensitive setup at any image point on the detector

plane of the interferometer, it can be written:

 

1(x.y) = 110w) + 12(x,y)+ 2J11(x,y)12(x,y) COS¢ (22)
where I,(x,y) and I,(x,y) are the light intensities of, respectively, beam 1 and beam 2, and
b is the phase difference between the two waves contributing to point P in the image.

For convenience, equation (22) can be rewritten in the following form:
1(x.y)= 10(x.y>[1+rocos¢(x,y)] (23)

where the intensity 10(x,y) is the sum of the intensities of the two illuminating beams

 

2,/
(IO=I,+12) or the dc intensity, 70(x, y) = I 11:2 is the modulation depth of the intensity
1 2

in the interference fringe pattern, and b(x,y) is the phase difference between the two
illuminating beams.

In equation (23) there are three unknowns, namely 10,7, and b . Therefore a minimum
of three intensity measurements (called buckets) at known values of phase shifts is

needed to calculate the phase b(x,y).

39
If a known phase shift ori is introduced in one of the beams in the inteferometer, equation

(23) becomes:

m = 10(x.y)[1+7ocos[¢<x.y>+a.-]] (24)

where a, is the average value of the relative phase shifi for the ith exposure.

The present discussion is made for the in-plane setup i.e. double illumination. For
the out-of-plane sensitive setup, i.e. single illumination and reference beam, the concepts
are the same and the reference beam is the one usually shifted by known amounts.

Equation (24) is sometimes written in the literature (e.g. Morimoto and F ujisawa

1993 and 1994, Perry and McKelvie 1993) as:

f (x,y) = a(x,y)005[¢(x,y) + a] + b(x,y) (25)
where f(x,y) is the brightness distribution, a(x,y) is the amplitude of brightness or
modulation strength, b(x,y) is the average brightness or background intensity variation,
b(x,y) is the phase that one has to analyze at any point (x,y) on the fiinge pattern and or is
the amount of the phase shift. A schematic of the resulting brightness distribution on a
line of a fringe pattern is shown in Figure 9.

There are two ways of introducing phase changes, as shown in Figure 10 :
1) Discrete phase steps are used so that the phase is constant during the integration time
of each TV-frame (remember that detectors always measure time-averaged irradiance).
This is called phase-stepping.
2) The phase is linearly and continuously shifted during the integration time of each TV-

frame. This is called phase-integration or phase-shifting.

40

Fringe center

/1\

 
    

Brightness

 

 

 

 

0 21: 411: 611 Phase

 

 

Position

Figure 9. Brightness and phase distribution on a line of a fringe pattern.

ai cri

‘ 1 l

=' l ‘l'V-fram' tun' -
I E 5 mg e : E 1 'l'V-framing time

 

 

 

 

 

L
7

time time

(b)

V

(a)

Figure 10. (a) phase-stepping technique; (b) phase-integration technique.

In phase-stepping, or, in equation (24) is a constant for each ith bucket. In phase-

integration instead, ori is the average phase shift during the ith TV frame integration time.

41
In phase-integration, the average recorded intensity integrated during one TV frame is

given by (Greivenkamp 1984):

a(x,y) = gilagjjglzowil + yocos(¢(x.y) + a(t))]da(t)

a

 

sinA / 2
= Io(x.y){1+yo( A ,2 )cos(¢(x.y)+a,-)} (26)
where A is the amount of change in relative phase of the beams within each exposure.
The integration over A makes this expression applicable for any phase-shifting technique.

For the case of phase stepping it is A=0.

The term

 

sinA/2)

21/2 (27)

7(x.y) = ro(x,y)(

is the recorded intensity modulation or fiinge visibility.

The only difference between phase stepping and phase integration is that the latter
yields a smaller modulation in the measured interference pattern per bucket than the
former (Van Wingerden, Frankena and Smorenburg 1991). In other words, the fringe
visibility in phase integration is lower than in phase stepping. This is clearly seen by
noting that y(x,y) is always smaller than yo(x,y) in equation (27). The two terms coincide
only for A=0.

For A=0 equation (26) becomes equation (24) and the term y(x,y) coincides with yo(x,y).
Phase stepping is therefore a simplification of the integration method. For A=21t there
would be no modulation intensity. In order to have modulation of the intensities as the

phase is shifted, the phase shift per exposure needs to be between 0 and 1t.

42
2.6.3 Some phase extraction algorithms

As stated earlier, a minimum of three buckets has to be recorded in order to calculate
the phase distribution for a given interferometric fringe pattern. Many phase calculation
algorithms have been developed. They generally use different numbers N of buckets and
different amounts of phase shifts a,. The same formulas for calculation of phase from the
buckets apply in the phase stepping and the phase integration techniques. Some of the
common algorithms used are the following:

(1) Three buckets (90): N=3, A=1t/2 , shifis a: 1r/4, 31t/4, and Sit/4.

The three intensity measurements are expressed as (Creath 1988):

11(x.y) = Io(x,y)(l +7COS[¢(x.y) +lflD (28)
12 (LY) = 10(an’)[1 + 7003[¢(X,.V) + §7ID (29)
[3(X,y)= [a(x,y)(l+7COS[¢(x,y)+§7rD (30)

The phase at each point is given by:

= 13(x’y)-12(x9y)]
My) m4li(x,y)-12(x.y) (3 1)

 

The intensity modulation is:

 

y) = J[n(x. y) —12(x. y)]2 +[12(x. y) —13(x. y>]2
. 210

 

70¢ (32)

From equation (27), for A=1t/2 it is: y=0.9yo. This means that with a phase-integration

method the fringe visibility reduction with respect to the phase-stepped method in this

43

case is 10%.

(2) Three buckets (120): N=3, A=21t/3 , shifts at: -21t/3, 0, and +2‘l't/3.

In this case the phase is given by:

 

¢(x,y) = arc J3(13(x,y)_12(x,y)) ]
211(an’)-12(xa}’)‘13(xJ)
The intensity modulation is:

 

(l3[13(x,y)—12(x,y)]2 +[211(x,y) — 12(x,y)—13(x,y)]2

 

r(x,y) =
210

For this case, y=0.83y,.

(3) Four buckets (90): N=4, A=n/2 , shifts a=0, n/2, 1t and +31t/2.

(33)

(34)

This is the most common algorithm. The intensities for the four exposures are given by:

11(x.y) = 10(x.y)(1+ rcos[¢<x.y>])
12 (x,y) = Io(x.y)(1+ 7 c05[¢(x.y) + 3D
b(x,y) = Io(x,y)(1+ r COS[¢(x,y) + 27])

r
14(x.y) = 10(JC.y)(1+ Nova,” + 32751]

The phase is given by:

= I 4000—12000)
my) mailman-Isa.»

 

The modulation is then expressed as:

(35)

(36)

(37)

(38)

(39)

44

J[14(x,y)—12(x,y)]2 + [11(x,y)—13(x,y)]2
210

 

(40)

 

7(x,y) =

Here, y=0.9y0.

It is useful to notice that the reduction in fringe visibility in the case of phase-
integration is pretty small, and this leads one to think that this method is generally better
than the phase-stepped technique, which is slower because a time delay is required
between the steps to damp out any oscillation in the PZT and settle down the shifted
beam.

(4) Carre’ technique .

In all the previous algorithms the amounts of phase shift or has to be known. The
knowledge of this parameter requires a precise calibration of the phase shifter, which is
not an easy task to perform. A technique which is independent from the magnitude of the
phase changes was developed by Carre’ (1966). He proposed that the phase is shifted by a
constant value or between consecutive intensity measurements, assuming the phase shift
linear with time.

The expression of the phase is:

 

([3(12—13)—(11-I4)][(12-13)+(11‘14)]

¢(x,y) = arctan (12 + I3)-(11+ I4)

 

(41)

An advantage of the Carre’ technique, other than the obvious that the phase shifter
doesn’t need to be calibrated, is that it can work with converging or diverging beams
where the amount of phase shift varies across the beam. The disadvantage is that the

phase calculation is more complicated.

45

2.6.4 Comparison of algorithms

Van Wingerden et al. (1991) give a comprehensive survey on the possible errors
occuring in the different phase measurement techniques. As sources of system errors,
they consider light source instability, imperfect beam phase shifting, mechanical
vibrations, nonlinearity of the detector and quantization of the detector signal. They
conclude that if the system is not well calibrated and 4 buckets are measured, the Carre’
formula gives the best error reduction. If the system is properly calibrated, the most
accurate formula is that for 4 buckets. For phase-stepping, however, the measurement
error still decreases (to a nonvanishing asymptotical value) for an increasing number N of
buckets. Therefore, techniques which use more than 4 buckets (e. g. 5 buckets) are
preferable. The disadvantage is, of course, an increase in computational analysis. A large
number of buckets reduces also the effect of the random phase-shift errors.

Cloud (1995) points out that the 4 bucket technique is the best for eliminating effects
caused by second- and third-order detection nonlinearities. He also notes that there is
little difference in results between phase-stepping and phase-integration. Only in the case

of non-linear phase-shift errors is the integration method superior.

2.6.5 Phase-shifting devices

The most common way of perfonning the phase shift in a beam is the piezoelectric
transducer (PZT) mirror. Other ways have been used, such as tilting glass plate (Creath
1988) or polarization phase-shift device (Jin and Tang 1992). In optical fiber speckle
interferometry, the phase shifting is usually performed by stretching the optical fiber

wrapped around a PZT (e. g. Joenathan and Khorana 1992).

46
2.6.6 Phase unwrapping

The formulas given in Section 2.6.3 for the calculation of the phase b(x,y) involve an
arctan function. Therefore, they yield results “modulo 1t”. This means that the calculated
phase has an uncertainty of lift, where n is an integer.

The first step is to obtain a phase “modulo 21:”. This is done considering the sign of
the numerator and the denominator of the tangent function, i.e. the sine and cosine
functions respectively .

_ sin¢ _ Num

tan¢
cos¢ Den

 

(42)

Doing this, one can exactly locate the phase b in one of the four quadrants, as shown
in Figure 11. This method can be applied to all the algorithms treated earlier, except for
the Carre’, which involves a more complicated process (Creath 1985).

A similar procedure is commonly used for data reduction in strain gage rosette
analysis. In this case an arctan function is involved for the calculation of the orientation
of the principal strains with respect to the rosette axes.

After evaluating the phase modulo 21:, the phase ambiguities are then removed by
comparing the values between two adjacent detector elements (phase unwrapping). If the
absolute difference between two adjacent detector elements exceeds a certain threshold
(which is usually 1:, for reliable removal of discontinuities) a fiinge edge is detected and
an offset equal to a multiple of 21: is added or subtracted until this difference is less than
1:. A fringe order number N is then recorded such that the phase at that point is given by
“old phase+2N1t”. The deformation phases are then usually displayed as different gray

levels (normally, gray level 255 is equal to one fringe period). This calculation restricts

47
the total number of resolvable fringes, in other words the sensitivity of the system, since

the phase difference between adjacent points should not vary by more than 1: (Chang et

al. 1985).

 

 

 

 

 

 

a
‘coa

Figure 11. Determination of the phase “modulo 2a” .

Figure 12 shows a schematic of a phase unwrapping.

48
(5 ¢ A ¢ '
unwrapped phase distribution

6a-

41:—

wrapped phase distribution

 

 

 

pixels pixels

Figure 12. Schematic of phase unwrapping.

This calculation needs a point in the detector array to start the unwrapping. The final
phase map is therefore relative to this reference location. The only case where it yields
absolute phase values is when the chosen starting reference point is a fixed point in the
object. Sometimes it is not possible to locate a fixed point in a structure and the result is
only relative to a point. Usually the starting point is chosen to be in one of the four
comers or at the center of the image (e.g. Vrooman and Maas 1991); the detector array is
then scanned point by point until the whole image is unwrapped. Region by region

unwrapping is also possible.

2.6.7 Sources of error in phase measurement and remedies
Noise, low modulation, physical holes and speckle decorrelation can induce errors in
the phase unwrapping. Good regions, e.g. regions with small phase gradients, low noise

and correlated speckles, should be unwrapped before other regions. Data-conditioning is

49

necessary before performing the unwrapping. Operations like detection of low-modulated
pixels, median filtering and smoothing filtering are very useful for this purpose.

Noise can arise from particles in the air and in the optical components, electronic
noise and round-off-errors in the frame grabber system. The total noise term has usually a
continuous spectrum with a predominance of low frequency component (V ikhagen 1991).

Low modulation is a fundamental problem in phase shifting technique (Creath, Cheng

and Wyant 1985). This problem is schematized in Figure 13.

(a) 0))
good sampling bad sampling

high modulation low modulation

 

 

 

 

 

= detector element . = speckle

Figure 13. Sampling of a speckle pattern with finite-sized detector elements. (a) good
sampling; (b) bad sampling.
Because speckle is sampled with a finite-sized detector element, the measured
intensity is an average over the detector area. To resolve the speckles in the image plane,
the f/No. of the viewing system is set to adjust the speckle size according to the pixel

size. Yet, for only one speckle to influence each detector and therefore for having perfect

50
sampling of the speckle pattern, the detector size would have to be about a tenth of the

speckle size (Creath 1985). If the aperture were made this small, very little light would
get through. With a detector element of the same size as the speckle, the intensity will not
modulate during phase shifting as it would for point sampling. This is due to averaging
over a pixel with more than one speckle influencing it. This problem could be corrected
using point detector elements.

When the modulation y(x,y) at a pixel as the phase is shifted is less than some 7,,” the
pixel is considered “bad” and is usually removed from the data sets before performing
any phase calculation. ymin is.called modulation threshold and is set by the user.

Median filtering is an operation which takes all the data points contained within a
window, sorts them until the median is fourid and then replaces the values of the center of
the window with the calculated median. The window is then moved by one pixel and the
procedure is repeated until the whole array has been processed. The size of the window
depends on the context. This technique is good in filling up bad data points but does not
change the pixel values with phase errors.

Smoothing filtering is a routine used to smooth data points. It compares each data
point to usually eight nearby pixels in the form of a plus sign. If most of the nearby points
have phase values different form the center pixel, its value is replaced with the average of
the good nearby pixels, otherwise it is left alone. If the dimension of the array smoothing
window is nxn, the random noise is reduced by a factOr of 1/sqrt(n).

Detector nonlinearity is another potential source of error in phase measurement. For

CCD (charge-coupled-device) detectors the nonlinearity “is negligible (7 z 1), but for

51
vidicon tubes the nonlinearity can be considerable (e.g. y = 0.7) . The measurement errors

caused by detector nonlinearity can be approximated linearly as a function of the intensity
deviation (difference between measured intensity and actual light intensity at the
detector) (Van Wingerden et al. 1991).

Environment disturbances such as air turbulence and floor vibration can also cause
measurement errors. A simple check with live correlation fringes can give one a feeling
of the influence of these factors for a particular experiment. When possible, tests are

carried out on vibration-isolated tables.

2.6.8 Issues on spatial resolution in DSPI

Phase-shifting techniques increase sensitivity and spatial resolution of the
measurements in DSPI. The spatial resolution limit, i.e. the resolution of small sized
details in the deformation pattern and the sensitivity limit (or signal resolution), i.e. the
resolution of the magnitude of deformations/displacements of the object surface, are
connected to one another. As an example, smoothing of the images results in an increase
in sensitivity but a decrease in spatial resolution.

To investigate the spatial resolution of a DSPI system, one can test an object which
gives a periodic deformation pattern (V ikhagen 1991): the resolution limit is then defined
by the minimum resolved spatial period measured in units of the pixel distance. The
theoretical limit for spatial resolution is, however, given by the pixel size in the flame
grabber system. The frame grabber boards normally used in PC’s for DSPI have a spatial

resolution of 512x512 pixels. This theoretical limit represents a deformation pattern with

52
a spatial period of two pixel distances and can be achieved only if the spatial resolution of

the camera is equal to or better than the spatial resolution of the frame grabber board in
the computer.

Noise has a big influence on resolution limits of the DSPI system. Because of the
noise factor, it is unrealistic to operate with a spatial resolution limit of one pixel size for
a deformation experiment: the high spatial frequency components of the noise may then
give large errors in the phase unwrapping. In order to reduce the effect of noise and to
increase the signal-to-noise ratio (SNR), the image is smoothed before proceeding for the
phase calculation as explained in Section 2.6.7. The smoothing inevitably reduces the
spatial resolution of the measurements, so the point is to choose the amount of smoothing

so that one gets a satisfactory signal-to-noise ratio without losing too much resolution.

2.6.9 The displacement calculation
To find the deformation of an object, the phases calculated with the phase-shifting
technique before and after the displacement are subtracted from one another. They are

unwrapped after performing the subtraction.

A¢(x9y) = ¢bef0re — ¢afler = ¢(x9y) - [¢(x9y) _ A¢(x,y)] (43)

The phase variation Ab(x,y) is then related to the displacement component by
Equations (4) and (12) for an out-of-plane and an in-plane sensitive setup respectively.

In the in-plane arrangement, for example, the displacement is calculated as:

53

dx(x,y) = 47rsin6

where the symbols are explained in Section 2.4.2.

2.7 Advantages and disadvantages of DSPI

Advantages of DSPI are:
1) Computer controlled fast data acquisition (1/30 sec. per frame), signal processing and

graphical result presentation.

2) Precise quantitative analysis with the use of optical phase measurement techniques
such as phase-shifting, which give spatial resolutions on the order of a few pixels and

high sensitivity, even in cases where only few fringes are observed.

3) Since typically only 1/30 see. is needed to record a frame of specklegrarn, the vibration
isolation is relaxed a bit. Furthermore, ideally an acceptable vibration should have
amplitude smaller than the speckle size and this is always a smaller restriction
compared to the requirement for vibration isolation in holographic interferometry. This
means that the system does not require the highly stable enviromnent necessary for

conventional holographic interferometry.

4) The system can be used in brightly lighted conditions. No dark room is needed.

54
5) No photographic processing, optical spatial filtering or plate relocation are needed;

material cost is then very low.

6) The technique is non-invasive.

Disadvantages of DSPI are:

1) Limited maximum range of measurements (~50 um) without special measures.

2) Low spatial resolution of correlation fiinges. Necessity of phase measurement

techniques when quantitative analyses are needed.

3) Sensitive to rigid-body motions.

4) Usually, necessity of plane illuminating wave fronts which implies difficulty in

studying large areas.

CHAPTER HI

EXPERIMENTAL PROCEDURE: PRELIMINARIES

3.1 Introduction

In this study Digital Speckle Pattern Interferometry (DSPI) was used to study
mechanically fastened composites. High precision strain measurements were requested.
In order to accomplish this, a phase-stepping algorithm had to be used. The setup
employed throughout this work was the in-plane sensitive one, and it is schematized in
Figure 6, page 29. From this point on we will always refer to this setup.

At the first stage, three goals had to be accomplished:
1) A commercial four-bucket phase-stepping algorithm by Ealing Electro-Optics was
available. This algorithm was designed by the company to be used in an out-of-plane
sensitive DSPI setup. In other words, the software displayed values of out-of-plane
components of displacement. This software wasn’t editable, being already compiled. The
first goal was therefore understanding how such an algorithm could be used in an in-
plane sensitive DSPI setup. This basically turned into a “calibration” problem.
2) The system had to be “set-up”, in terms of aperture ratio of the viewing system, image
filtering, etc. Also sensitivity and maximum range of the measurements had to be
determined.
3) The measured displacements had to be converted to strain values. The procedures to

convert displacements into strains are always a challenging part in this kind of study,

55

56

because it is easy to introduce errors. The third goal was therefore to make sure that the
procedure used to get strains from displacements was accurate enough for the purposes of

the present research.

3.2 Calibration of the technique

The commercial phase-stepping algorithm involved was a four-bucket phase-stepping
(Section 2.6.3). It was designed to be used along with a commercial unit for out-of-plane
DSPI by Ealing Electro-Optics. Four images of the undefonned object and four images of
the deformed object were taken at different phase-shifts. The program showed then the
out-of-plane component of the displacement of the object as 2D and 3D plots on the
computer monitor. I .

The idea that this result could also be used in an in-plane setup after some
modification was the following. In any phase-stepping or phase-shifting algorithm, the
calculation of the phase does not depend on which kind of setup (in-plane sensitive or
out-of-plane sensitive) is used. The phases “before” and “after” the deformation are
simply subtracted to obtain the phase Ab(x,y) due to the deformation (see Section 2.6.9).

The phase Ab(x,y) is then related to the displacement component by equations which

depend on the optical setup. Namely, for an out-of-plane sensitive setup it is:

A¢(X.y)4
27r(cos 0,- + cos 9v)

 

dz<x9y) = (45)

where it is the wavelength of illumination and 0i and 0,, the angles of illumination and

viewing with respect to the surface normal.

57

For an in-plane sensitive setup it is:

A¢(x.y)l (46)
472' sint9

d x (x, y) =
where 3. is the wavelength of illumination and 0 the illumination angle with respect to the
surface normal.

Because of the phase-unwrapping procedure, the results obtained by a phase-

calculation algorithm are not absolute values, but relative to the point were the phase

unwrapping itself starts, as explained in Section 2.6.6. It can be written:
Aiécalc (x’y) : A¢abs (x’y) _ A¢ref (x0 ’y0) (47)

where: Abw,(x,y) is the calculated phase due to the deformation, point by point in the
specimen;
Ab,b,(x,y) is the absolute phase due to the deformation, point by point in the
specimen;
Abn,(x0,yo) is the phase due to the deformation at the reference point (x0,y0) in the
specimen where the unwrapping procedure starts.

This is generally true unless the starting point is a fixed point in the object. In our pin-
loaded composite experiment any point in the pin was a fixed point; the pin being
“infinitely” rigid, it remained static during deformation. But starting the unwrapping from
these points would have caused large errors, because the pin was a low modulated region
of the image. Therefore, in the experiments performed in the present work, the
displacements obtained were always relative to a point.

From equations (45) and (47), the calculated out-of-phase component of the displacement

can be expressed as:

58

d (x )- A¢ca,(x,y)it -
zcal ,y 2n(cost9,-+cost9v)

 

— A¢abs(x,y)l _ A¢ref(x°9y°)l
27r(cos 0,- + cos 6v) 2zr(cos 6,- + cos 9v)

 

(48)

From equation (46), the absolute in-plane component of the displacement can be

expressed as:

 

 

 

 

A ¢abs (x,y)l
dxabs (x,y) - 47rsint9 (49)
Combining equations (48) and (49) we get:

. A ¢ (X0, yo)/Ir

c080 + c059 ref
dxabs(xay) = 2'sin0 vdzcalc(xay)+ 47rsin0 =Adzcalc(xsy)+B (50)
where: A = 0086i + c056,, (51)

2 sm 0
A ¢ (X0, yo)/1
B = ref (52)

 

47: sin 9
Equation (50) relates the out-of-plane component of the displacement calculated by the
commercial phase-stepping algorithm to the absolute in-plane component of the
displacement.

A and B are constants over the specimen. Their expressions are given in equations
(51) and (52). The multiplying constant A is some kind of “calibration” constant; it only
depends on the optical setup used, namely on illumination and viewing angles. The
additive constant B depends also on the reference point in the specimen where the phase

unwrapping starts.

59
Nevertheless, in the subsequent calculation of strains, the displacement maps are

differentiated along certain directions on the specimen and therefore the additive constant
B in equation (50) does not influence the strain values, which are always representing
“absolute” quantities (not relative to the value at a reference location). Since the results of
this work are in terms of strains/stresses, we can simply ignore the constant B in this
discussion and think always in terms of absolute quantities.

Equation (50) can be therefore simply written as:

dx (x,y) = Adz(x.y) (53)

The knowledge of the “calibration” constant A made it possible to convert the results
given by the commercial software, namely d,(x,y), into the needed quantities, namely
d,(x,y), for the in-plane sensitive setup.

The determination of A from equation (51) would have involved the measurement of
the angles 0,, 0,, and 0, which is not an easy task to perform with a high degree of
accuracy. Furthermore, measurement errors would be amplified in the subsequent
calculation of strains. For these reasons, instead of performing a “theoretical” calibration
according to equation (51), an “overall” calibration was performed to calculate the value
of the constant A which had to be used in the setup. For this purpose, known values of in-
plane displacements were induced using a rigidly-rotating plate equipped with a
micrometer accurate to within 0.0001”, as shown in Figure 14. This device was
developed by Cloud (1975) for use in quantitative speckle photography.

This experiment gave horizontal correlation fringes (loci of constant component d, of

displacement), as expected for small rotations of the plate (Appendix B). Figure 15

60
illustrates the formation of the hinges.

rotating plate
y L I
x
micrometer accurate
to within 0.0001”

 

II/V'". S .'."\I,

viewing system

 

computer I

 

Figure 14. ln-plane rotating plate used to induce known values of dx

 

I
I
I
I
l
I
I
I
\ I
I
I
I
I
l
I

Center of Rotation

I

W1}.

 

 

Figure 15. Formation of fiinges for rigid in-plane rotation of a plate.

61
Figures 7 and 8 in Section 2.5.3 show the correlation fi'inges obtained in this

experiment for two different angles of rotation 0=1x10" rad and 0=2x10" rad.

Values of the displacement obtained by the commercial algorithm were then
compared to the known values of induced displacements. As expected, they differed by a
multiplication constant and an additive constant. The value of the multiplication constant
A was easily determined by this comparison, providing a satisfactory calibration of the
system.

Its worth noting that in this “overall” calibration no measurements of optical angles

were involved.

3.3 Experimental apparatus and system “setup”

The optical system used was presented in Figure 6 (Section 2.5.2). The light source
was a Helium-Neon laser by Siemens with a maximum output of 15 mW. Its wave length
was 0.6328um.

The imaging system (see Figure 5, Section 2.4.2) consisted of a CCD 512x512 pixel
resolution black and white video camera and a viewing lens. The camera was connected
to a 12 volt DC power supply. After different tests, the optimum aperture ratio of the
viewing system was determined to be F=f/a=4. This gave maximum speckle contrast and
was appropriate for the light source power used. This aperture ratio was used throughout
the work. The focal length of the viewing lens was f=50mm and the magnification
M=1:1.4. The camera detector array was connected to a Data Translation DT2851 frame-

grabber card installed in a 286 Hewlett Packard Personal Computer, where the image

62
processing was performed. A high resolution monitor was used for result display.

The expanding lenses used for the two illuminating beams were two microscope
objectives with magnification M=40X and numerical aperture 0.65. To remove the
optical noise, two spatial filters with pin hole diameter equal to 10 um were also
employed. After being expanded, the two illuminating beams were collirnated by two
plano-convex lenses with focal length f=305 mm and diameter D=89 mm. The incidence
angle of the illuminating beams (angle 0 in Figure 5) was 0=30°. This angle determined
the sensitivity of the correlation fringes.

With this system we were able to investigate a circular area on the specimen of
approximately 80 mm of diameter. All the specimens used had to be coated with a matte
white paint in order to have a reasonably diffusive surface. All the experiments were
performed on an optical table with pneumatic isolator legs by Technical Manufacturing
Corporation (TMC).

The rotating plate experiment discussed in Section 3.2 was also used to determine
sensitivity, maximum range of the measurements and spatial resolution of the correlation
fiinges.

The experimental sensitivity of the correlation fiinges was found to be
z0.675um/fiinge. The theoretical sensitivity was calculated to be 0.6328um/fringe
through the use of equation (14). The small difference between theoretical and
experimental sensitivity is due to approximations in the value of the angle of illumination
0 used in equation (14). Nearly zero sensitivity was observed for the other two

components of the displacement, namely d, and d,.

63
The maximum range of the measurements was determined by inducing large rotations

in the plate until speckle decorrelation occurred, which determined the disappearance of
the correlation fiinges. This maximum measurable displacement was determined to be
d,m,,z17um. The theoretical upper limit of the measurement range is determined by the
pixel size in the detector plane and the speckle size on the object, which is in turn
dependent on aperture ratio and magnification of the viewing system (see Section 2.3).
For our case, the theoretical maximum range of the measurements calculated fi'om
equation (2) was dmu=7.4um. It is not clear why the difference with the experimental
value is so large. Surely the speckle phenomenon is a random one in itself, being
originated by random light scattering from an object surface. Therefore it is believed that
the theoretical speckle diameter given in equation (2) does not truly represent the
dimension of each speckle in a specklegram. Also, imprecision in determination of the
values of viewing system aperture ratio and magnification used in equation (2)
contributes to this mismatch. The experimental result was considered more reliable than
the theoretical one.
In the following experiments we never approached speckle decorrelation caused by too
large displacements; the deformation of the test specimens was always within the
measurable range.

The maximum spatial resolution of the correlation fringes was ~1 fiinge per
millimeter of the object surface and we were able to observe up to 25 fringes for a
~19x19 mm2 area of the object.

Before performing the phase calculation, the images were filtered with a 5x5 pixel

64
array smoothing window at least three times. Each filtering took about 22 seconds. This

filtering provided a good signal-to-noise ratio without losing too much signal spatial
resolution. A pixel modulation threshold (see Section 2.6.7) for the minimum modulation
acceptable for a pixel was set, so that pixels with low modulation were ignored during
phase calculation. It is better that no value of phase be calculated than that a random one
be obtained. After the phase calculation was performed, the resolution of small sized
details in the deformation pattern (signal spatial resolution) was of the order of a few

pixels.

3.4 Strain calculation from the displacement maps
Strain components are related to displacement components of a deformation by the

following relations :

 

“if (54)
dd

gy=—oyl (55)
-5dx ddy

rxy- o} + a (56)

where : d x and d y are the displacement components along the two in-plane directions x
and y; s, and e y are the longitudinal strains along the same directions and y ,y the shear
strain.

In order to obtain both in-plane components d,, cly of the displacement, the specimen

was rotated by 90° without moving the optical elements.

65
The displacement maps were imported in a commercial spreadsheet as arrays of

numbers. This was accomplished through the use of a digitizing tablet.

The procedure for calculating spatial gradients of a deformation is always a delicate
one, because in the differentiation process possible errors are amplified, as opposed to an
integration process. Two different ways of obtaining strains were examined:

1) The finite-difference method (e.g. Hombeck 1975), which evaluates point by point the
derivative of the displacement plot using finite intervals of width h. These derivatives are
accurate to within an error of the order of h. More terms in the Taylor series expansion
can be taken in order to reduce this error (e. g. Herrera Franco 1985 ).

2) The least-square fitting method, which consists in fitting the displacement map with a
4th or 5th order least-square polynomial and thendifferentiating the polynomial along the
in-plane directions. The displacement maps obtained with a phase-calculation algorithm
are often affected by noise. Therefore a smoothing procedure such as the use of a least-
square polynomial is often used to reduce noise levels (V rooman and Maas 1991, Jia and
Shah 1994). Moreover, this method is quick and gives a continuous plot of the strain
values.

The latter method was used in this work. Since high spatial gradients of strains were
expected near the hole, the displacement maps were fitted “piecewise”. Different
polynomials were used where the displacement maps presented marked variations in
slope. Also, preliminary tests showed that for our kind of strain spatial gradients, the

difference in resulting strain values between the two mentioned methods was within 5%.

66
3.4.1 Verification of the accuracy of the method: the cantilever beam experiment

In order to verify the accuracy of the described technique for strain analysis, a simple
cantilever beam experiment was performed and results from electrical resistance strain
gages were compared to the ones obtained with DSPI. A rectangular cross-section
cantilever beam of a photoelastic material (length=6.5”, width=0.75”, thickness=0.5”)
was subjected to a load Pz0.5 lb. The specimen setup is shown in Figure 16. The Young’s
Modulus of the material was unknown; knowledge of it was unnecessary for the purposes
of this experiment. Two strain gages of type EA-13-240LZ-120 by Measurements Group
were installed in the upper and lower side of the beam. The two gages were taken from
the same lot. Their properties were: R=120.0i0.3%, Gage Factor at 24°C=2.110i0.5%,

K, at 24°C=(0.2i0.2)%, Lot number R-A48AF154.

 

 

 

s
e P

/ E ER#1
' ...... c
E x K

/ 3 ----- r [I gH=o.75
' . ...... x2.

Li; 2 s s

/- 31311142 5 5 .

325” S 8:059

<. ................................................................... §
6.5’ 3'.

Figure 16. Setup for the cantilever beam experiment.

Before perfomring the test, the specimen was manually cycled in order to eliminate

67
possible residual stresses and hysteresis.

The strain gages were connected to a P-3500 strain indicator, in a 1/4 bridge
configuration. The two readings from the gages were taken for eight loading cycles. Zero
readings were observed when load was removed, except for the first loading cycle which
was therefore considered a “training” one. The gages were self-compensated (STC) for
2024-T4 Aluminum. Effects of thermal apparent strains were neglected because the tests
were quick and no appreciable variation of temperature was observed during the
measurements. The results showed an average value of 48urn/m and a standard deviation
of 2um/m for the upper gage (ER#1) and an average value of -38p.m/m and a standard
deviation of 211an for the lower gage (ER#2).

The two different values of the indicated strains for upper and lower gages were at
first unexpected. For a homogeneous beam, the two readings should have been equal and
opposite. To be sure that this was not a systematic error due to the gages, some
measurements were taken with the beam rotated by 180° about its longitudinal axis so
that upper and lower gages resulted interchanged. Also in this configuration the results
for the gage in tension and the one in compression were different, the latter showing
always the lower absolute value of strain. It was concluded that the material tested had
two different Young’s moduli in compression and in tension. This is not unusual in
plastic materials. In a case of a beam subjected to “pure bending”, when the bending
moment is constant in each section of the beam and no shear forces are present, the usual
assumption that the sections remain “flat” after deformation implies that the strains are

linear along a vertical direction (e.g. Sharnes 1964). When different moduli for traction

68
and compression are present, in order to maintain equilibrium the stresses along a vertical

direction have to be linear but with two different slopes for the part of the beam in tension
and the part in compression. What happens in this case is therefore a “shift” of the neutral

axis, as shown in Figure 17.

 

 

 

Figure 17. Strain and stress distributions along a vertical section for a beam in pure

bending with two dififerent Young ’s moduli in tension and compression.

It should be noticed that our configuration was not a case of pure bending (see Figure
16). In our case the bending moment was not constant along the length of the beam. With
reasonable approximation, the condition of linear strain distribution can be assumed in
cases like this where the applied loads are small.

An area of 0.75x0.75 in2 of the beam was then analyzed with the in-plane phase-
stepped DSPI technique. The same load as in the strain gage test was applied. The phase

fringes obtained are shown in Figure 18.

69

 

 

 

 

 

Figure 18. Phase fiinges obtained in the cantilever beam experiment.

Note that the fiinges in Figure 18 are not horizontal; rather they tend to converge
towards the beam mid-plane as expected for a cantilever bending load condition. The
slope of the fringes is opposite in the upper and lower part of the beam, showing that the
upper part is in tension (positive 8,“) whereas the lower part is in compression (negative
8,0,). Also, it can be noted that the slope of the fringes in the upper part is larger (in
absolute value) than the slope of the fiinges in the lower part. This means that the strains
8,, in the part in tension are larger than those in the part in compression, at the same

distance form the beam mid-plane. This observation confirmed the strain gage readings.

70
The strains were then calculated along the vertical section S-S (see Figure 16) as

explained in Section 3.4, and the results are presented in Figure 19.

 

 

 

S
/l Eer—i—J l
/| ; I
/l 1:,”ng
S
6::

uexx
1’ I
g 1

 

 

 

    

section S-S

 

 

 

Figure 19. Plots of strain Exx obtained with I) DSPI and 2) strain gages, along section
S-S of the cantilever beam.

The distribution of strains obtained with DSPI along this vertical section was very
close to linear, as expected. In Figure 19 the experimental points from DSPI are fitted
with a least-square straight line. The outputs from the two strain gages are also connected
with a straight line. The agreement between DSPI and strain gage results is shown to be

to within 7%. These results established confidence about the accuracy of this method for

71
strain measurements by phase-stepped DSPI.

CHAPTERIV

STUDY OF A SINGLE PIN-LOADED HOLE IN
FIBER-REINFORCED PLASTIC

4.1 Introduction

In most composite mechanical structures, joints are the weakest elements. Among the
mechanical fasteners, pins are widely used for simplicity of installation and reliability.
The mechanical behavior of pin-loaded joints in composite materials is complicated by a
number of factors such as fiber orientation, lay-up, friction, etc. More study is needed in
this area to have a full understanding of this Subject. DSPI is a quick and accurate
technique for experimental mechanics and has been used for this purpose in the present
work.

Strain and stress distributions are determined in critical locations near the hole in a
pin-loaded joint of a fiber-glass reinforced epoxy laminate for a case of static tensile load.
In this chapter a case of perfect fit (“push-fit”) between pin and hole is examined. The
pin has nominally the same diameter as the hole.

Stress concentration factors are determined and compared with results from similar

studies present in literature.

4.2 The specimen: geometry and mechanical properties
The material studied in this research was a glass-reinforced-epoxy [O°/90°]S laminate

(R1500/ 1581, 13 plies, 0.14” thick) supplied by CIBA-GEIGY, Composite Material
72

73
Department, 10910 Talbert Avenue, Fountain Valley, California 92708.

The dimensions of the test coupon had to be chosen very carefully, because many
studies have proved that geometric factors influence failure modes and elastic behavior of
the specimen (see, e.g. review paper by Godwin and Matthews 1980). Two geometrical
factors were determinant and they are shown in Figure 20:

I- 1) The ratio e/d between the distance “e” between hole center and free end perpendicular
to the loading axis and the diameter “d” of the hole. The effect of end distance was
investigated by Collings (1977) and it was determined that a minimum ratio e/d=3 was
necessary to develop full bearing strength and to avoid end effects.

2) The ratio w/d between width of the specimen and diameter of the hole. Collings (1977)
also determined that a minimum w/d=8 was required to have full bearing strength.

Our specimen had e/d=4 and w/d=8. It was cut in a previous research work (Herrera
Franco 1985). The drilling of the through-the-thickness hole was carried out using two
steel templates to avoid delamination. The pins used to load the specimen were stainless
steel cylindrical pins 2” long. They were considered “perfectly rigid” throughout the
work.

The laminate was constituted of 13 unidirectional plies. Each ply (lamina) could be
considered orthotropic with two in-plane directions of material symmetry. Also the whole
laminate could be considered “macroscopically” orthotropic: in fact, being a “cross-ply”
laminate (0° and 90° fiber orientations), it had three mutually orthogonal planes of
material symmetry. Therefore two principal in-plane material axes were present in the

specimen and they are represented by axes “1” and “2” in Figure 20. These axes were

74

also aligned with the natural axes of the specimen “x” and “y”. For this reason, from now

on it will be referred indifferently to reference system (1,2) or (x,y).

 

 

 

 

 

 

 

“l i 11“”
.1 ,
t=0.l4”
||| '
L X y
L=8.0 drill x
thru
d=0.25”
P |e=l.0”
V
1' |
w=2.0”

Figure 20. Specimen used for the test: dimensions and loading condition.

Because all the laminae were positioned symmetrically with respect to the laminate
mid-plane, the specimen was a “symmetrical” laminate. The laminate was also
“balanced”, meaning that it contained an equal number of laminae at +0 degrees and -0
degrees fiber orientation with respect to its longitudinal axis. In this special case of a
“symmetrical” and “balanced” laminate, there is no coupling between extensional and

shear behavior and between extensional and curvature behavior; that is, extension forces

75

will produce only extensions and bending moments will produce only curvatures.

The mechanical behavior of this specimen can be described by four independent
“engineering” (also called “macroscopic” or “effective”) constants (c. g. Whitney, Daniel
and Pipes 1982), namely:

0

E1, Young’s modulus in direction 1; E]: o,°/ a, when oy°,r,y°=0

0

E2, Young’s modulus in direction 2; E2= oy°/ a,

when o,°,r,y°=0
G12, shear modulus in plane 1-2; G”: t,y°/ y,y° when o,°, oy°=0
V12, Poisson’s ratio for transverse strain in direction 2 and stress applied in direction 1,
v12= - ay° / s,° when oy°,r,y°=0.
where o,°, of , and txy° are the far-field applied stresses and e,°, ey° , and yxy° are the
longitudinal and shear mid-plane strains of the laminate along the (x,y) directions.
Therefore, the whole laminate could be macroscopically considered as an
homogeneous and orthotropic lamina characterized by the “effective” mechanical
properties mentioned above.
These properties were determined experimentally in the previous work by Herrera Franco
(1985) and were found to be:
E1= 3.188x10" psi (21.9804 GPa)
F1: 3.0824x106 psi (21.2523 Gpa) (57)
v”: 0.11
Looking at the two Young’s moduli, it appears that the degree of anisotropy of the

specimen is very low. Ideally, in a “cross-ply” laminate with the same number of plies at

0° and 90° orientation, it is E1=E2. Because our specimen was constituted by an odd

76
number of plies (13), it is believed that the ply corresponding to the laminate mid-plane

was at 0° orientation, giving the specimen a slightly larger stiffness in that direction. The
laminate under investigation could be reasonably considered “quasi-isotropic”.
The shear modulus was unknown. An “engineering” shear modulus can be

determined using the known relation for isotropic materials G=E/2(1+v) with an average

of El and E2:
0., = “V—’(E"—EZ)= 1.41x106 psi (9.72 GPa) (58)
2(1 + V12)

Whitney et al. (1982) proposed an expression for G12 analogous to equation (58) for a
[i45°]S laminate.
The second in-plane Poisson’s ratio is determined by the known relation for orthotropic
materials v21=v,2(E2/E,), giving in our case vzlzv,2=0.l 1.

For a generalized plane stress condition, the stress-strain relations (generalized
Hooke’s law) for an homogeneous and orthotropic laminate along the 1-2 directions can

be expressed as (Lekhnitskii 1968):

E1 V12E
O'1=—_£1+———2-£2
1-V12 V21 1-V12 V21
V12E2 E2
0' =-——e +———e 59
2 l-V12V21 1 1-V12V21 2 ( )
712=G12712

 

These equations relate the average values of stress and strain components with respect
to the thickness of the laminate, using the “effective” elastic constants for the laminate as

a whole.

77
4.3 Experimental procedure

The specimen was loaded in tension using a stainless steel loading fi'ame equipped
with a hydraulic pump by Scott Engineering Sciences Corporation, as illustrated in Figure
21.

The loading frame was especially built to be very stiff. This was necessary in order to
avoid any rigid body rotation of the specimen during the tests. Out-of-plane rotations
didn’t affect the measurements because the method was not sensitive to out-of-plane
displacements. In-plane rigid body rotations, on the other hand, had a large influence on
the calculation of the shear strain (which involves the crossed derivatives of the
displacements) and therefore had to be minimized. The specimen was mounted in the
loading frame using grips especially designed to allow rotations, thus avoiding any end

effects.

double-acting piston

 

 

 

 

 

J hydraulic pump

 

l1

\ loading cell
\—— grip

\

N l .
specrmen

 

 

 

 

 

 

 

Figure 21 . Schematic of specimen and loading apparatus.

78
Before taking the measurements the coupon was preloaded with about 10 lb so that it

would self-align along the loading direction: this precaution reduced the chance of in-

plane rigid body rotations during measurements. The applied load was measured using a

loading cell by Photoelastic Division, Measurements Group, which was connected to a P-

3500 strain indicator also by Measurements Group. The load values directly in pounds

appeared in the strain indicator display. The net applied load was P=27 lb. This load

provided a nominal far-field tensile stress o,z96.4 psi. This small load was applied in
order to avoid any plastic deformation in the specimen. The absence of plastic
deformation was confirmed by the disappearance of the real-time correlation fiinges
when load was removed.

The procedure followed to determine the elastic behavior of the laminate near the pin-
loaded hole was:

a) Measurement of displacements d, and dy using DSPI. The entire load fi'ame with the
specimen was rotated by 90° to measure both in-plane components.

b) Calculation of in-plane strain components 8,, e, and 7,, from equations (54)-(56).

0) Calculation of in-plane stress components 0,, cry and 1,, from equation (59), using the
elastic properties of the laminate given in (57) and (5 8): determination of stress
distribution near the hole and stress concentration factors (SCF’s).

The final step of the analysis was the determination of the stress concentration factors

(SCF’s). Different ways of expressing the SCF’s for mechanically fastened joints have

appeared in the literature. In this work the maximum stresses at the critical locations of

the specimen were related to the following “nominal “or “reference” stresses:

 

 

P
Far-field tensile stress ow = (60)
w x t
Net tensile stress through the hole - ——£—— (61)
O'net (w - d) x t
. P
Average bearing stress a), = d t (62)
X

where w, t, d and P are shown in Figure 20.

The DSPI technique measures displacements on the outside surface of the specimen.
Therefore the strains calculated in step b) refer to the outside ply of the laminate. With a
reasonable degree of approximation these strains can be referred to as average values for
the laminate as a whole, and the stresses and SCF’s calculated in step c) can also be
considered average or “engineering” values referring to the whole specimen. This is
considered reasonable also because this laminate is “symmetrical” with respect to its mid-
plane; therefore, no curvature effects are present under tensile applied load.

For both isotropic and composite materials, there are three main failure modes in
mechanically fastened joints (Matthews 1987): ligament tension, bearing and shear-out.
These modes of failure are caused respectively by tensile normal stresses, compressive
normal stresses and shear stresses. Figure 22 shows the location of these three types of
failure.

The failure mechanisms of composites depend upon many factors such as fiber type,
orientation, matrix, etc. In this work strains and stresses are determined along the critical
locations shown in Figure 22. Some other locations near the loaded hole are also

considered. All the analyses are carried out in the elastic range of the material.

80
Before testing the composite specimen, a preliminary experiment was performed with

an isotropic specimen to check visibility and appearance of the DSPI correlation fringes.

In this test no quantitative analyses were performed.

[—0— Q,

‘1'

 

 

 

 

 

 

 

 

 

1garnent tension bearing shear-out

Figure 22. Location of main failure modes in mechanically fastened joints.

4.4 Experimental results and discussion

Figures 23 and 24 show the phase fringes obtained with DSPI for the component (I, of
the displacement parallel to the direction of the load for the isotropic and the composite
specimen respectively. Each fringe represents a displacement d,z0.67um. By following
the evolution of the fringes in real time while increasing the load, it was determined that
the fringe order increased going from the center of the hole towards the lower portion of
the specimen and from the center of the hole towards the upper portion of the specimen.
This was an indication of positive normal strains 8, in the direction of the load in the
upper part of the specimen and negative normal strains 8, in the direction of the load in
the lower part (bearing region). A high number of fringes per unit of length in the
bearing/shear-out region indicates (1) large negative normal strains 8, and (2) large shear
strains y,,. Fringes almost perpendicular to the direction of the load in the upper region of

the specimen indicate shear strains 7,, nearly equal to zero.

 

 

 

 

 

Figure 23. DSPI phase fi'inges for displacement d, parallel to the direction of load -
Isotropic specimen.

 

 

 

 

 

Figure 24. DSPI phase fringes for displacement d, parallel to the direction of load -
Composite specimen.

82

Figure 25 shows the phase fringes for the component (1, of the displacement
perpendicular to the direction of the applied load in the composite specimen. The fringes
in Figure 25 were obtained by applying a large load to the specimen. They do not refer to
the 27 lb load experiment. The large load was applied to obtain an appreciable number of
fringes for the (1, component.

For the strain/stress calculation, a load equal to 27 lb was applied, as explained
earlier. Even if this load produced only one or two correlation fringes in the specimen, the
phase-stepping technique was able to give results with a resolution of a few pixels: the
biggest advantage of the use of phase measurement techniques such as phase-stepping is
that there is no need for the use of fringe multiplication techniques even in cases where
only few fringes are observed.

Figure 25 can be used for qualitative analysis. Fringes almost perpendicular to the
direction of the applied load indicate high shear strains 7,, in the shear-out region. On the
other hand, fiinges almost parallel to the direction of the applied load indicate shear
strains 7,, nearly equal to zero in the region above the hole. The non-symmetrical
distribution of fiinges with respect to the longitudinal axis of symmetry “y” of the
specimen is probably due to variations of mechanical properties over the specimen itself.
A possible cause of this is a variation in fiber volume fraction in the plies and/or a
variation in fiber orientation.

Plots of strains were obtained for three regions around the loaded hole: (1) the region
below the hole (bearing region); (2) the region above the hole; (3) the net-section region

(ligament region).

 

 

 

 

 

r11~,
Llnv

 

Figure 25. DSPI phase fiinges for displacement d, perpendicular to the direction of
the load- Composite specimen.

Figure 26 shows strains 8, parallel to the direction of the load along the vertical
diameter of the hole in the bearing region.

The high value of the bearing strain near the hole vanishes at a distance from the edge
of the hole approximately equal to the radius r of the hole. Since a monotonic decrease in
bearing strain is expected from the hole to the free edge of the specimen, the slight
increase in compressive strain observed at a distance from the edge of approximately 1.5
times the radius r of the hole is believed to be caused by nonhomogeneous distribution of
mechanical properties of the specimen.

The maximum compressive strain is shown to be not at the edge of the hole, but at a

certain distance from it. This effect was not reported by some authors (e.g. Pradhan and

84
Ray 1984 and De Jong 1977) who always found maximum bearing stress at the edge of

the hole. These researchers used finite element analysis without considering friction
effects. The behavior observed in the present work was, however, detected by Herrera
Franco (1985) and it is probably due to the effect of fiiction between hole and pin.
Eriksson (1986) and Hyer et al. (1987) proved independently that friction decreases the
radial bearing stress at the edge of the hole. This is physically correct because frictional
forces will transfer axial load through tangential stresses and, owing to equilibrium, the
radial stress must decrease. They also proved that this effect is more marked on the
vertical diameter of the hole than in any other diametrical locations around the edge of
the hole. Moreover, the frictional forces at the edge of the hole on the vertical diameter
give a positive bearing strain a, due to the Poisson effect (8mm in Figure 27). The
superimposition of the negative bearing strain due to the axial load (8mm) in Figure 27)
and the positive strain due to fiiction produces a lower resultant negative bearing strain
on the vertical diameter at the edge of the hole. This effect is illustrated in Figure 27.
Figure 28 shows strains 8, along lines parallel to the axis of the load for the ligament
area. The highest strain values are obtained along the line tangent to the hole. A decrease
in strain by a factor of about 2 is observed at a distance from the hole equal to 0.5*r along
the “x” axis, r being the radius of the hole. This is representative of the high spatial
stress/strain gradient near the hole. Pradhan and Ray (1984) determined that the rate of
decay of stresses along the axis through the pin perpendicular to the loading force is

higher in orthotropic plates as compared to isotropic plates.

85

distance from center of the hole/radius r of the hole ,_

, 3 2 1
1 o l 1 1

 

 

  
 
 

   

-o.ooom ..
-o.ooooz ..
-o.ooooe ._
-o.oooo-1..
, -o.00005 ..
, -0.00006 .. [m/ml
. -o.oooor ..

 

l -o.oooos ..
‘ -o.oooos ..
l

 

 

 

o0.0001

 

Figure 26. Strains 6); parallel to the direction of the load along vertical diameter in the
bearing region.

 

pin loaded hole

   
 

axial
1 load

.1.

> - 0.....- +

shearinml)>0

      

 

 

 

Figure 27. Eflect of fiiction on bearing strain.

86

 

 

 

 

 

distance from center of the hole/radius r of the hole

 

 

 

acme I4 I-3 1-2 I-1 l0 I1 12 L3 I4
-—+ )l
—> a I
H I

 

 

 

 

 

 

 

 

 

e, [m/m]
0.0115 ..
norm --
0m ,
0 I am, . _
| ”l. i . I
l ‘7 hole center

 

 

i came

Figure 28. Strain ey along lines parallel to the direction of the load located in the
ligament area.

87
Also in Figure 28, the high strain near the hole decreases to an approximately

constant average value far from the hole. Above the hole the normal strain is positive
(tensile area), and below the hole it is negative (compression area) owing to the bearing
effect.

It is observed that the maximum value of the strain does not occur at the end of the
horizontal diameter of the hole, i.e. in the minimum net area of the plate, but slightly
below the hole. The same effect was observed by Herrera Franco (1985). Also Nisida and
Saito (1966) showed that the maximum value of circumferential stress at the edge of the
hole does not occur in the minimum net section but at an angle (p of about 85° from the
vertical diameter below the hole. Frocht and Hill (1940) observed that the maximum
stresses do not occur at the ends of the horiZontal diameter when there is clearance
between pin and hole. These two studies were on isotropic plates. Eriksson (1986) found
that the maximum circumferential stress occurs at an angle (p of about 80° in a composite
plate. Hyer et al. (1987) found this angle to range fi'om 70° and 88° for different laminate
lay-ups and attributed this to effects of friction, clearance and contact angle between pin
and hole. De Jong (1977) found that the maximum normal stress in the direction of the
applied load occurs at an angle (p of about 80°. Zhang and Ueng (1984) found the
maximum circumferential stress angle to be about 75° for a [0°/45°] laminate and about
45° for a [90°/45°] laminate.

Figure 29 shows strains a, perpendicular to the direction of the load along parallel
lines located above the hole. The highest transverse strain is shown to be along the

vertical diameter of the hole. The strain becomes small away from the hole. The negative

 

 

 

 

distance from center of the hole/radius r of the hole

1'5 (-4 I-3 7-2

H 0 ll '2 I3 I4 '5

 

 

 

 

 

 

 

 

 

 

 

- 0.00003 11
i a, [m/m] I hole center
1 0.000020 0
l 0.000011. I
. \ I
l O 4 r- .- —- - -—-~ I w.
l a \ \ I 1"“ 9"—
) 000001) , ‘ I 4 / d
‘ "‘-h t
l v.
1 ....... 1 1 r 1
0,0000% 3 ‘y
c f
| 000004.. 2 3
000005.. .
: a
1 000006--
0.00007 I

 

 

 

Figure 29. Strain ex along lines perpendicular to the direction of the load located above

the hole.

 

 

 

 

89
sign of the strain on the vertical diameter of the hole under tensile applied load is caused

by the Poisson effect. A compressive (negative) circumferential stress is present on the
upper edge of the hole along its vertical diameter. This finding is consistent with results
from several authors (e.g. De Jong 1977). Away from the hole, where the stress
concentration is small, the strain 8, assumes a constant average value. A slight asymmetry
with respect to the specimen vertical axis is noticed in the plots of Figure 29: this can be
attributed to variation in mechanical properties across the specimen.

Figure 30 shows strains a, perpendicular to the direction of the load along parallel
lines located below the hole (bearing zone). The highest value of strain is observed along
the vertical diameter of the hole. Its positive sign is due to the Poisson effect and the
bearing load induced by the pin. The circumferential stress at the lower edge of the hole
along its vertical diameter has a positive sign (tensile). This is confirmed by results
obtained by other authors (e. g. De Jong 1977). The strain value diminishes away from the
hole. The general behavior of these plots is similar to that found by Herrera Franco
(1995) for the same material. The asymmetry with respect to the vertical axis of the
specimen in the plots of Figure 30 can be again attributed to variation in mechanical
properties of the specimen.

Figure 31 shows the shear strain 7,, along a vertical tangent to the hole in the lower
part of the specimen. This is the usual location of the shear-out failure mode. The value of
7,, on the horizontal diameter of the hole (“x” axis) is shown to be different from zero.
When no fiiction between pin and hole is present, the shear stress at the edge of the hole

along its horizontal and vertical diameters is shown to be nearly zero by several

90

 

l distance from center of the hole/radius r of the hole

 

fi

l 0mm 1—5 1-4 03 1-2 1-1 10 11 12 1,214 ,5

 

 

 

0%-- A T I

 

001m .. ‘Y 1

, 0011115- ’ .
3 d o
. 01mm- ; b '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘7, a"
I e in: «w . Ir 1
“m " 'u at!
G R I ‘7' ,
411nm- * v 1 ’
-0 *8‘ [m/m] hole center
00112.

 

 

Figure 30. Strain ax along lines perpendicular to the direction of the load located below
the hole (bearing zone).

91
investigators (e.g. Matthews et al. 1982, Pradhan and Ray 1984). The value different

from zero of shear strain in the horizontal diameter on the edge of the hole in Figure 31 is
due to friction, as showed by Zhang and Ueng (1984).

7,, is shown to be maximum at a distance equal to one radius r from the edge of the
hole along the shear-out-direction. At points located far from the hole, the shear strain
becomes smaller. This is qualitatively consistent with results from other authors; for a
case of carbon fiber reinforced plastic (CFRP) unidirectional composite laminate,
Pradhan and Ray (1984) showed that the maximum shear stress occurs in the shear-out
area at an angle of about 60° from the horizontal diameter (“x” axis) of the hole. De Jong

(1977) found this angle to be about 70° for the same laminate.

 

0.00025

 

 

0.0002 .

 

0.00015 3

0.0001

0.00005 3

 

 

 

 

 

 

 

[ distance from center of the hole/radius r of the hole i

Figure 31. Shear strain yxy along location of shear-out failure.

92
Figure 32 shows qualitatively the locations of the maximum stresses found in the

specimen, for the three possible failure modes: normal tensile stress cm, in the ligament

in the bearing region and shear stress r,,,,,, in the

region, normal compressive stress om,

shear-out region.

 

Oymax

 

 

 

ymin xymax

 

 

 

 

Figure 32. Location of maximum stresses near the hole.

Table 1 shows the values of stress concentration factors (SCF’s) calculated for this
case of perfectly fitted pin in the present research. Results fiom two other studies by De
Jong (1977) and Matthews et al. (1982) are also presented for comparison. The reference
stresses used to express the SCF’s have been presented in Section 4.3 in equations (61)

and (62).

93

Table 1. Comparison of SCF ’s determined in the present work, by Matthews et al. (1982)
and by De Jong (1977). F GRP =fiber glass reinforced plastic; CF RP=carbon

 

 

 

 

 

 

 

 

 

fiber reinforced plastic.
Present Matthews et De Jong (1977)
analysis al. (1982)
Material tested [0°/90°ls [0°/i45°/0°]s [904°li45°]s I [0°/i60°]s
F GRP CFRP CFRP elastically isotropic CF RP
Phase-
Method used Stepped 3-D FEM Complex Functions
DSPI
0' y max
K =
l 0b 0.63 1.79 1.2 0.8
0' y max
K liet = 4.38 5.37 4.95 4.89
0' net
Txy max
K? = 0b 0.37 0.56 . 0.5 0.6

 

 

 

 

 

 

 

Table 1 shows that the SCF’s found in the present analysis are all lower than the ones
obtained by the other authors. This mismatch can be attributed to differences in
mechanical properties of the materials tested, which highly affect the stress concentration
factors. In the other two analyses mentioned, carbon-fiber reinforced plastic was tested
and this material is generally stiffer than fiber-glass reinforced plastic which was used in
this work (to give an idea, for a unidirectional lamina, tensile longitudinal modulus is 40
GPa for a 60% volume fraction E Glass/Epoxy, as opposed to 210 GPa for a 62% volume
fraction High Modulus Carbon/Epoxy, according to Chawla (1987)). Rowland et al.
(1982) pointed out that softer materials always exhibit lower stress concentration factors

than stiffer ones, because they are more able to wrap around the pin and therefore to

94
distribute the pressure more evenly over the edge of the hole. Pradhan and Ray (1984),

using FEM, found that the maximum stress around the hole was high in the case of
graphite/epoxy and boron/epoxy unidirectional laminae compared to the case of
glass/epoxy.

As observed in Section 4.2, the laminate tested in this research can be considered
quasi-isotropic, because the ratio E,/E2 is very close to 1. Indeed, our results are closer to
those presented in the last column of Table 1, which refers to an elastically isotropic

material, the larger mismatch being in the shear stress concentration factor.

CHAPTER V

STRESS CONCENTRATION RELIEF BY THE USE OF
INTERFERENCE-FIT PINS

5.1 Introduction

Machine assemblies are often prestressed to eliminate stress gradients and to increase
fatigue life.

Let us suppose that a strip with a boltedconnection is subject to an axial cyclic

pulsing load as shown in Figure 33.

applied load+prestress cyclic change

111114 ””5““ \iiiiiv

    

 

 

 

 

 

 

 

 

 

3p
. ..... 9.....
' — - I ...:.°.: ::: a:
U 2 P
I———l\—~_-'_. W
l l l l l l l l l l
(a) applied tensile load only (b) interference fit only (c) applied load + interference

Figure 33. Eflect of the prestress due to an interference fit on the stress distribution near
the hole for a case of isotropic strip subjected to axial pulsing load (fi'om
Regalbuto and Wheeler 1 970).

95

96

If no prestress due to interference between pin and hole is present, the axial stress at
the edge of the hole will rise from zero to roughly 3 times the far field stress applied to
the ends of the plate. When an interference fit exists between pin and hole, a state of
initial biaxial stress is created near the hole. When the tensile load is applied, the resultant
stress is not given by the simple addition of the two stress distributions caused by the
interference and the applied load. What has been observed (Love 1963) is a profile
obtained by adding the average tensile stress to the initial prestress distribution. Even if
the average stress is increased, the change in stress level is now only p, as opposed to 3p
for the case of no prestress. This decrease in change of stress causes an increase in
fatigue life of the component, the fatigue life being influenced more by the alternating
component of the stress than by its mean value. 1

In a case of a static load the effect of the prestress induced by an interference fit is to
decrease the change in peak tensile or shear stress produced by the applied tension. This
effect is illustrated in Figure 34, which refers to the study by Jessop et al. (1956) of an
isotropic plate with hole subjected to static tensile load at the ends. Figure 34 shows the
variation of maximum shear stress at the edge of the hole with the applied load, for the
case of perfect fit and two cases of interference; the effect of interference is to decrease
the slope of these lines.

For a plate loaded in tension through the pin, the beneficial effect of an interference
fit between pin and hole is qualitatively the same as the case of a plate loaded at the ends
(Jessop et al. 1956 and 195 8). The effect of interference has been studied by some authors

(see Section 1.3.3) for both isotropic and composite materials. Especially for the latter, a

97
complete understanding of the problem is yet to be completed.

maximum shear stress at the edge of the hole

‘1
// high interference

1111

 

 

 

low interference

 

1111

 

 

 

(0,0) : applied load

Figure 34. Schematic of variation of maximum shear stress near the hole with applied
static load, for perfect fit and two cases of interference fit (fiom Jessop et al.
1 956).

In this work two degrees of interference fit between pin and hole were investigated.
“Low” interference : I = (d’-d)/d = 0.0002
“High” interference : I = (d’-d)/d = 0.0004
where d’ and d are the diameters of pin and hole respectively.

In order to obtain these values of interference, pins measured to an accuracy of

0.0001” were used. Before installation, the larger pin was cooled down with liquid

98

nitrogen to cause a thermal contraction. The smaller pin was simply pushed into the hole
manually. Particular care was used in this process in order to avoid delamination of the
specimen.

The specimen tested was the same as the one used for the perfect-fit case (Capter IV).
The axial loads used in these tests were P=27lb for the low interference test (far field
stress o,=96.43psi) and P=34lb for the high interference test (far field stress
o,=121 .43psi).

The first concern was to be sure that these values of interference and the applied load
would not cause plastic deformation in the specimen. The theoretical solution for pins
with interference fit in a circular hole in an infinite isotropic plate is given by AC.
Stevenson. For a rigid pin of diameter (d+e) inSerted in a hole of diameter d in an infinite
plate, the stresses at the hole boundary are given by

099,, = -o,,,,,=2Ge/d, (63)

 

where G=shear modulus of the plate=
2(1+ v)

To a first approximation, as explained in Section 4.2, the laminate tested in this research
can be considered quasi-isotropic with a shear modulus G=1.41x106 psi. The values of
the stresses at the edge of the hole due to interference are given by equation (63) which in
our case yields:

“Low” interference: 090,5 «rm-ma: 560 psi

“High” interference: 00011.1: «rm-me 1,120 psi

The highest nominal tensile stress in the net section due to the applied load was

o.,,d=140psi (case of high interference: applied load P=34 lb). Multiplying this value by

99

the stress concentration factor K°M=4.38 from Table 1 in Section 4.4, gives the maximum
tangential stress at the edge of the hole caused by only the applied load 099m=481psi It
has been mentioned (see Figure 33) that the resultant stress at the edge of the hole due to
interference and applied tensile load is always less than the simple addition of the two
stresses 099,, and 099.0“. The value o,, = 099,, + 006.0,, = 1,120 + 481 = 1,601psi constitutes
therefore an excessive estimate of the actual resultant maximum stress in the plate.

The minimum tensile strength in a fiber glass reinforced epoxy unidirectional lamina
is the strength in the direction perpendicular to the fibers. Chawla (1987) reports for this
parameter a value of (5,,Id as 4,000 psi for an E-glass reinforced epoxy. Since this value is
much larger than the calculated o,,,=1,601psi, we had a large margin of confidence that
the material would never undergo plastic deformations during the tests. This was
confirmed experimentally by the disappearance of the real-time correlation fringes upon
removal of the load.

It is important to realize that in the present research the specimen already prestressed
with an interference fit was tested with DSPI. The initial (“undefonned”) state was
therefore the prestressed state and the final (“deformed”) state was the resultant state of
prestress+applied-load-induced stress. The results obtained are therefore representative of
the change in stress distribution due to the applied load. They do not represent the
resultant, actual distribution of strain/stress in the specimen. It is the diminution of this
change due to the applied load that is observed when an interference fit is used. Of

course, in the case of perfect fit, the measured stress/strain distributions are the actual

distributions in the specimen.

100

5.2 Experimental results and discussion

Plots of stress distributions in critical locations near the hole are obtained following
the procedure explained in Section 4.3. The two cases of interference fit are compared to
the case of perfect fit between the pin and the hole. For comparison, the stress values are
normalized with the reference stresses illustrated in equations (60)-(62) in Section 4.3.

Figure 35 shows the distribution of the normal stress 0, parallel to the direction of the
load along the ligament area for the case of perfectly fitted pin and the two cases of
interference fit. The values of the stress 0, are normalized with the far field nominal
stress oN=P/(th). In all three cases the stress assumes an almost constant value at a
distance of about 1.5xr from the edge of the hole, r being the radius of the hole. A marked
decrease in stress concentration factors is observed for the cases of interference fit. This
decrease is bigger for the case of high interference.

Figure 36 shows the distribution of the normal compressive stress a, parallel to the
direction of the load in the bearing region for the case of perfectly fitted pin and the two
cases of interference fit. The stresses are normalized with the bearing reference stress
ob=P/(dxt). The bearing effect almost vanishes at a distance from the edge of the hole
equal to lxr, r being the radius of the hole, for the perfect fit and the low interference
cases. For the case of high interference this distance is found to be about 2xr. The
maximum stress concentration factor decreases when the level of interference is

increased.

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 6

I . perfect-fit b b b b ‘

5 .. '
1 1 low interference I
1 4 1 A y .
, o ’10" high interference x 1
- ------ .. z 1
2 4’ \\\ i l I
1 ‘” hole ‘.‘ n— =5 :: 1
n 1
7 o . . . f 3, i
l 0 0.5 1 1:5 2 2.5 73 3:5 41 1

1 1 distance from center of the hole/radius r of the hole ,
L 1

Figure 35. Normal stress concentration factor 03/037 along the ligament area for (1)
perfect fit, (2) low interference fit, (3) high interfierence fit.

Figure 36 shows that the maximum stress never occurs at the edge of the hole, but
rather occurs at a certain distance from it for all the three cases examined, the largest
distance being for the high interference case (lxr). As discussed in Section 4.4 (Figures
26 and 27), this can be attributed to friction effects between the pin and the hole, which
decrease the radial bearing stress at the edge of the hole (Hyer et al. 1987). This
hypothesis is confirmed by the plots in Figure 36, where this behavior is more marked for
the case of high interference fit, where larger friction forces than the other cases are
present.

Also, it can be noted in Figure 36 that the stress value at the edge of the hole is

smaller in the low interference case than it is in the high interference case. A similar

102
behavior was observed by Jessop et al. (1958), in their studies on pin-loaded joints for

isotropic materials. They concluded that there is an optimum value of interference, for a
given applied load, at which the reduction of the stress in certain locations at the edge of

the hole is maximum.

 

 

 

0.45

0.44 = ,.
y .

0.35 i“ I

0,10,, 1 ‘r'
0.3 .. i <-

 

H
1

 

 

 

 

low interference

 

 

0.251:

 

 

 

0.21

 

high interference

 

 

0.15” ~s

 

0.0511 \i 1

 

I ‘ .
1 0.1“ . :
I
1

 

O 0.5 1 1.5 2 2.5 3 3.5 4 1
, distance from center of the hole/radius r of the hole

 

Figure 36. Bearing stress concentration factor 03/02, along the bearing region for (1)
perfect fit, (2) low interference fit, (3) high interference fit.

Figure 37 shows the distribution of the shear stress t,, along the shear-out region for
the case of perfectly fitted pin and the two cases of interference fit. The stresses are

normalized with the bearing reference stress 0,,=P/(dxt). A decrease in maximum shear

103

 

 

 

Tar/Oh
0.4

 

 

 

 

 

0.35 .. low mterference perfect fit

0.3 .,

 

 

 

 

ALIA
I

 

 

 

0.25 ..

0.2 ..

 

l
0.15 ..

 

0.1 ..

0.05 . ’ hole ‘

1 0 1 1. — + e '
0.5 1.5 2 2.5 :1 7
-0.05 1
distance from center of the hole/radius r of the hole ,

1 -04

Figure 37. Shear stress concentration factor rx/az, along the shear-out region for (1 )
perfect fit, (2) low interference fit, (3) high interference fit.

 

high interference

 

 

 

 

 

 

 

 

 

 

stress concentration factor is observed for the interference fits, this effect being larger for
the high interference case. The maximum stress along the shear-out failure zone occurs at
a distance from the horizontal diameter of the hole (“x” axis) equal to 2xr for the perfect
fit case, 0.5xr for the low interference and lxr for the high interference case.

As mentioned in Section 4.4 (Figure 31), the value of shear stress different from zero
in the horizontal diameter on the edge of the hole (“x” axis) is due to friction effects
(Zhang and Ueng 1984). In the case of perfect fit, relative slip between pin and hole can
occur and this causes nonlinearities in load transfer mechanism from the pin to the plate,
in the sense that doubling the load more than doubles the stress values near the hole

(Lambert and Brailey 1962). This effect could contribute to the high value of shear stress

104

at the horizontal diameter of the hole (“x” axis) in the case of perfect fit, as compared to
the cases of interference fit, as shown in Figure 37.

Table 2 presents a smnmary of stress concentration factors (SCF’s) obtained for the
critical locations of ligament, bearing and shear-out regions, for the three cases of fit
between pin and hole examined. It is worth noting that the absolute maximum stresses in
the plots of Figures 35-37 are considered for the calculation of the SCF’s in Table 2;
these maximum values refer to different locations around the hole, as shown in Figures
35-37.

A decrease in all SCF’s is observed going from perfect fit to low interference and to

high interference fit.

Table 2. Maximum stress concentration factors for ligament, bearing and shear-out
regions for (1) perfect fit, (2) low interference fit and (3) high interference fit.

 

STRESS CONCENTRATION FACTORS

 

 

 

 

 

perfect fit “low” interference “high” interference
ligament
region 0,,,,,/0N 5.00 4.25 2.70
bearing
region 0,,,,,,,/0b 0.4 0.25 0.18
Sheff“ 1.‘ “/6 0 37 0 29 0 24
region "Ym b ' ' '

 

 

 

 

 

 

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

An in-plane sensitive phase-stepped Digital Speckle Pattern Interferometer (DSPI)
has been developed for high-precision strain/stress analysis. Parameters such as viewing
camera aperture and lens magnification are adjusted in order to obtain maximum speckle
contrast. The setup is calibrated using a rigidly-rotating plate. The DSPI correlation
fringes obtained have sensitivity z0.67um/fringe and spatial resolution limit zlfringe/mm
of the object surface. The use of the phase-stepping technique and image filtering routines
makes it possible to obtain a spatial resolution limit of the order of a few pixels and a
very high sensitivity limit , which is virtually only subject to the limitation of the
sampling of the images by the detector array of the viewing system. The maximum range
of the measurable displacement, before speckle decorrelation occurred, is determined to
be zl 7pm.

The displacement maps are fitted with least-square polynomials, which are eventually
differentiated along in-plane directions to obtain strains. The accuracy of this method for
strain calculation is checked by comparing the results from DSPI with outputs from
electrical resistance strain gages for the simple case of a cantilever beam, showing
agreement to within 7%.

The technique has been applied to the study of single-pin joint in a glass-reinforced-

epoxy laminate loaded through the pin. Strain and stress distributions near the hole are

105

106

determined and stress concentration factors are calculated. A case of perfect (“push”) fit

and two cases of interference fit between pin and hole are investigated to study the effect

of an interference fit on the stress distributions.

1)

2)

3)

4)

The major conclusions of this work are:

Digital Speckle Pattern Interferometry (DSPI) is a quick, simple and accurate method
for experimental stress analysis. The use of a phase-stepping technique gives a spatial
resolution limit of the order of a few pixels and a sensitivity limit only subject to the
limitation of the sampling of the images by the imaging detector array.

A phase-stepping algorithm designed to be used in an out-of-plane sensitive setup can
be also employed in an in-plane sensitive setup, if the system is properly calibrated.
The stress concentration near a pin loaded hole in a composite joint can be accurately
determined through the use of phase-stepped DSPI.

Reduction in stress concentration factors up to a factor of 50% is observed when an
interference fitted pin is used.

Although this study involves static loads, the results suggest that, in the case of

varying pulsating load, the effect of interference is to decrease the oscillatory component

of the stress at critical points near the hole and therefore increase the fatigue life of the
1

component. The results found are limited to the case of elastic deformation of the

specimen. Care must be used in choosing the right value of interference, because the

combination of prestress and applied-load-induced stress can result in plastic yielding of

the specimen. When this occurs, it is expected that fatigue strength is reduced (Jessop et

al. 1956).

107
The measurement of the diameter of the hole is complicated by irregularities of the

hole surface. The determination of the exact degree of interference between pin and hole
used is probably affected by these errors.

The influence of an interference fit on the stress distribution of mechanically fastened
joints needs more study, especially for the case of composite materials where several
parameters such as fiber orientation, lay-up and mechanical properties contribute to
complicate the stress field near the hole.

Possible future research areas are:

1) Investigation of reduction of stress concentration factors due to interference for
different composite materials, lay-ups and/or fiber orientations.

2) Analytical and/or numerical studies to acquire a better understanding of the effect of
interference on stress distribution near loaded holes.

3) Systematic investigation of different values of interference to find an optimum value
which gives maximum stress concentration relief, for given material and geometry.

4) Studies on prediction of fatigue endurance of pin-loaded components under given
loading conditions for different values of interference.

5) Development of a quick method for determination with high accuracy of the exact

value of interference between pin and hole.

APPENDIX A

108
APPENDIX A

Displacement sensitivity in a dual-beam illumination DSPI setup with cylindrical

illumination wavefronts

Let us suppose that a dual-beam illumination DSPI setup employs cylindrical

illumination wavefronts, as shown in Figure A1.

km beam #1

0-111/2

0+w/2

beam #2

 

 

 

Figure Al. Schematic of dual-beam illumination DSPI with cylindrical illumination
wavefi'onts.

109

The object lies in the (x,z) plane. The “2” direction is the viewing direction. The
incidence angle of the illuminating beams changes across the x direction but is constant
along the y direction coming out of the page (“cylindrical” wavefront). Suppose that this
angle is 0 at the center of the illruninated area (point “0” in Figure 38). Suppose also that
the angle of illumination of each of the two beams (angle between unit vectors k1A and
k,B and between k2,, and km in Figure A1) is 11;.

Let us focus the attention on point A on the object and follow the same procedure
used in Section 2.4.2 to calculate the sensitivity of this setup to the displacement
components.

The total phase change in beam #1 at point A is:

c51=(k3-k1A)°¢i

2 . . . .
=7fl[k—(—srn(0—%)-1-cos(0-%)-k)]~(dx-1+dy-j+dz-k)

2n .
=7[dxsrn(B-%)+dz(1+005(9‘12/')] (A1)

The total phase change in beam #2 at point A is:

62=(k3-k2A)°d
=-2:,’-’-[k-(sin(e+%)-i-cos(e+%).k)]-(d,-i+d,-j+dz-k)

2 .
=—:—[—d,sm(e+ %) + d,(1+ cos(6+ %)1 (A2)

where: i, j and k are the unit vectors along the reference system (x,y,z);
k3, k1,, k2,, 0, 01/2 are shown in Figure A1;

(I is the displacement vector of point A on the object.

110
The phase change at point A due to the displacement d is the difference between 8, and

82, i.e.:
A¢ A = 211M742 sin 9cos—2V1) + dz(2 sin 03in 131)] (A3)

Equation (A3) shows that the phase change Ab at point A is a function of both
components of displacement d, and d,. The system is therefore sensitive not only to the
in-plane component d, but also to the out-of-plane component d,.

For pure in-plane displacement (d,=0), the sensitivity of the system is:

 

913 = ’1 (A4)
2 sin 6 cos K
2
where n is the fiinge order.
For pure out-of-plane displacement (d,=0), the sensitivity of the system is:
Q = ____’1_ (A5)
" 2 sinesinig-

By following the same procedure, the phase change at point B in Figure A] results:
A¢ B = 2,51de sinBcosg) — dz(2 sinflsin 131)] (A6)

In a generic point P on the object different from A and B, the expression of the phase
change Ab is given by:

- equation (A3) for P laying between A and O in Figure A1;

- equation (A6) for P laying between 0 and B in Figure A1;

where the angle 111/2 must be substituted with the actual illumination angle for point P

(referring to Figure A1, angle between line H2-O and line H2-P or angle between line

Ill
Hl-O and Hl-P, being these two angles equal).

The sensitivity of the system to the displacement is therefore changing across the
object surface. Figure A2 shows a computer-simulated measurement result for the dual-
bearn illumination DSPI setup using cylindrical illumination wavefronts with w=30° for a

displacement field d,=10um, d,=d,=0.

 

Figure A2. Computer-simulated measurement result for the dual-beam illumination DSPI
setup using cylindrical illumination wavefronts with I/I=30 ° for a
displacement field dx=10prn, dy=dz=0.

A sensitivity compensation algorithm could be used to correct the measurement errors

shown in Figure A2.

Figure A3 shows a computer-simulated measurement result for the dual-beam

112
illumination DSPI setup using cylindrical illumination wavefronts with w=30° for a

displacement field d,=10um, d,=d,=0.

 

Figure A3. Computer-simulated measurement result for the dual-beam illumination DSPI
setup using cylindrical illumination wavefronts with I//=30 ° for a
displacement field dz=10,um, dx=dy=0.

If the illuminating beams have plane wave fronts, w=0, and equations (A3) and (A6)
coincide with equation (12) in Section 2.4.2, which is the case of the setup sensitive only

to the in-plane component d, of displacement.

APPENDIX B

l 13
APPENDIX B

Displacement field in rigid in-plane rotation of a plate

Let us suppose that a rectangular plate is rotated rigidly by an angle 0 in the plane
(x,y), as shown in Figure B1. Let us also assume that 0 is small, so that the displacement

vector at any point on the plate can be considered tangent to the trajectory of the point.

 

‘7
X

 

 

 

 

 

 

Figure B 1. Displacement field in an in-plane rotation of a plate, for a small angle of
rotation 0.

For a generic point P distant r from the center of rotation O and whose position vector

forms an angle or with the “y” axis, the two in-plane components of the displacement are:

d,=d"'cosor=r0cosor (Bl)

d,=d"'sin01=r0sinor (B2)

114
Figures B2 and B3 show respectively the d, and the d, component of the displacement for

a 10x10 mm2 plate rotated by an angle 0=5°. Note that the d, component is linear along
the “y” direction and constant along the “x” direction. The opposite is true for the d,

component.

 

Figure B2.dx component for a 10x10 mm2 plate rotated by an angle 0=5 ".

 

Figure B3. dy component for a 10x10 mm2 plate rotated by an angle 0=5 ‘3

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