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Photoelastic and Electro-Optic Effects: Study of
PMN-29%PT Single Crystals
by
Na Di
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor David J . Quesnel
Department of Mechanical Engineering
Arts, Sciences and Engineering
Edmund A. Hajim School of Engineering and Applied Sciences
University of Rochester
Rochester, New York
2009
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Curriculum Vitae
The author was born in Shenyang, Liaoning province, China on November 28,
1977. She attended Liaoning Key High School and graduated in 1996. Sheenrolled at Fudan University in 1996 and finished her B.S. degree program in
Theory and Applied Mechanics in 2000. Thereafter she continued her graduate
study at Fudan University and graduated with a Masters degree in Engineering
Mechanics in 2003.
In fall 2003, she was accepted into the doctoral program at the University of
Rochester under the supervision of Professor David J . Quesnel. She received
her second Masters degree in Mechanical Engineering from the University of
Rochester in 2005.
In May of 2005 she attended the U.S. Navy Workshop on Acoustic
Transduction Materials and Devices where she became familiar with the issues
constraining the behavior of next generation piezoelectric single crystals. Shortly
thereafter, she conceived of the idea of using photoelastic methods to
characterize the stress distributions in these materials from which this thesis
developed. While pursing her thesis research, she regularly participated in the
U.S. Navy Workshop on Acoustic Transduction Materials and Devices by making
the presentations that are listed below.
z Photoelastic study of PMN-29%PT single crystals, U.S. Navy Workshop on
Acoustic Transduction Materials and Devices, May 2006.
z Photoelastic study of PMN-29%PT single crystals, U.S. Navy Workshop on
Acoustic Transduction Materials and Devices, May 2007.
z Photoelastic study of PMN-PT single crystals under electric fields, U.S. Navy
Workshop on Acoustic Transduction Materials and Devices, May 2008.
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Acknowledgements
I would like to thank my advisor, Professor David J . Quesnel first. I am
thankful for his diligent guidance and constant encouragement. I have learned a
lot from him, from academic knowledge to language and life. Without his
financial and academic support throughout my graduate studies, I would never
have been able to finish my thesis.
Next, I would like to thank Mr. J ohn C. J ace Harker and Mr. Stephen R.
Robinson. J ace is a fantastic lab mate, who always has a lot of brilliant ideas,
and will share them with me without reservation. We have held many meaningful
discussions over my research problems, and he helped a lot with my writing.
The strong technical skills of Stephen, who prepared samples and took the
photographs, are very much appreciated. He also helped me to improve my
English writing. Without J ace and Stephens help, I also would never have been
able to write out my thesis.
Many thanks to Professor Sheryl M. Gracewski, Professor Paul D.
Funkenbusch, Professor James C. M. Li, Professor J ohn C. Lambropoulos,
Professor Stephen J . Burns, Professor Renato Perucchio and Professor Ahmet T.
Becene for their guidance and the knowledge I have learned from their classes.
I would like to thank Chris Pratt for helping me in conducting X-Ray
experiments. Thank you also to J ill Morris and Carla Gottschalk for all their help
along the way.
Final words of thanks go to my parents for their love and support.
Portions of this thesis are derived from publications that appear in the archival
literature. In particular, Chapter 2 draws from: Na Di and David J . Quesnel,
Photoelastic effects in Pb(Mg1/3Nb2/3)O3-29%PbTiO3 single crystals investigated
by three-point bending technique, J. Appl. Phys. 101, 043522 (2007); and
Chapter 3 is derived from: Na Di, J ohn C. Harker, and David J . Quesnel,
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Photoelastic effects in Pb(Mg1/3Nb2/3)O3-29%PbTiO3 single crystals investigated
by Hertzian contact experiments, J. Appl. Phys. 103, 053518 (2008); In this
work, J ohn Harkers contribution was through editing of the initial draft to a form
suitable for publication, with the technical discussion necessary to get the
meanings as intended.
Chapter 4 and Chapter 5 will be submitted for consideration as journal
articles. This is reflected in the format selected for these chapters. They will be
coauthored with my advisor, David J . Quesnel.
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Abstract
Relaxor ferroelectrics PMN-PT single crystals exhibit extra-high dielectric and
piezoelectric properties compared with conventional piezoelectric ceramics.
They are becoming widely used in high performance electromechanical devices.
However, PMN-PT single crystals are elastically softer than PMN-PT
polycrystalline ceramics. Mechanical loads and electric fields interact to produce
fractures at relatively low stresses, and cracks grow under both AC and DC
electric fields. To prevent the failure of the electromechanical devices, we need
to have a better understanding of the mechanisms of fracture in this material
when it is subjected to mechanical and electrical loadings.
Photoelasticity is an efficient and effective method to measure the internal
stress distributions of materials that result from both internal residual stress and
external loading. I report the exploration of the use of this classic technique to
study internal stresses inside PMN-PT single crystals through bending and
Hertzian contact experiments. Effects under electric field loading were also
investigated using birefringence techniques.
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Table of Contents
Chapter 1 Introduction - piezoelectric single crystals & photoelasticity
1.1 PMN-29%PT single crystals....1
1.2 Photoelasticity... ...7
1.2.1 Discovery of the phenomenon of Photoelasticity7
1.2.2 Mathematical formulation of Photoelasticity..................9
1.2.3 Plane polariscope and circular polariscope. 10
1.3 Preliminary three-point bending experiments... .13
1.3.1 Experimental setup.... ..13
1.3.2 Fringe pattern..........................15
1.3.3 Deflection versus fringe order........17
1.3.4 Summary ...18
1.4 References... ....................20
Chapter 2 Photoelastic study using three-point bending technique
2.1 Introduction.... .26
2.2 Experimental procedure...... .29
2.3 Results and discussion .32
2.3.1 Fringe pattern....32
2.3.2 Loading force versus deflection................34
2.3.3 Stress-optical coefficient.....36
2.3.4 Youngs modulus...38
2.4 Summary................ ...39
2.5 References.............40
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Chapter 3 Photoelastic study using Hertzian contact experiments
3.1 Introduction........ 42
3.2 Experimental procedure..... .....44
3.3 FEM modeling methods... 46
3.4 Results and discussion. 50
3.5 Conclusions..........................53
3.6 References..54
Chapter 4 Photoelastic study using four-point bending technique
4.1 Introduction................57
4.2 Experimental procedure........60
4.3 Results and discussion.............63
4.3.1 Fringe pattern63
4.3.2 Fiber stress versus fringe order............... 64
4.3.3 Fringe-stress coefficient................ 66
4.3.4 Mechanical poling effect. 69
4.4 Conclusions. ....................... 69
4.5 References. .. 70
Chapter 5 Electrical field induced optical effects in PMN-29%PT single crystal
5.1 Introduction ...............73
5.2 Experimental procedure..................77
5.3 Results and discussion...................79
5.3.1 Hertzian Contact electric field loading effects...79
5.3.2 Electric poling effects................. 85
5.3.3 Mechanical poling versus electrical poling...89
5.4 Conclusions...................91
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5.5 References. .... ...92
Chapter 6 Summary
6.1 Summary....................95
6.2 References......99
Appendices
Appendix Basic theory of optical properties of crystals101
Appendix Basics ofphotoelasticity.....104Appendix Calibration ofin-situ loading frame...108
References for Appendices.... . ..111
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List of Tables
Table 3.1 Elastic stiffness constants of PMN-30%PT single crystals (10Dij
c10
N/m2)...47
Table 3.2 Input parameters used in ANSYS. The elastic stiffness constants:
(10ijc
10 N/m2). Young's modulus of glass:E(1010 N/m2). Poisson's ratio
of glass: .48
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List of Figures
Figure 1.1 (a) ABO3 Perovskite Structure. If one shifts the A-site ions to the
center of the cell, the structure will appear differently with 12 oxygen
atoms at the center of each cell edge, an arrangement often shown ingeology texts, (b) Spontaneous polarization for the R phase in unpoled
PMN-29%PT; Illustration redrawn from a similar figure in reference
[5]...................................................................................4
Figure 1.2 Phase diagram of PMN-PT single crystals; Illustration redrawn from
a similar figure in reference [33].......5
Figure 1.3 PMN-29%PT single crystal as received.8
Figure 1.4 (a) Dark field plane polariscope set-up. (b) Light vector
representation; Illustrations redrawn from a similar figure in reference
[38].11
Figure 1.5 ZeissTM Microscope set-up... .14
Figure 1.6 Preliminary three-point bending set-up14
Figure 1.7 Three-point bending image at 450 to both the polarizer and the
analyzer... ..15
Figure 1.8 Principal Stress Vectors from ANSYS simulation of three-point
bending. Only left half of sample is shown.16
Figure 1.9 Three-point bending image at zero degree to both the polarizer and
the analyzer....16
Figure 1.10 (a) Three-point bending of unpoled PMN-29%PT; (b) Three-point
bending image of isotropic materials [39].17
Figure 1.11 Deflection versus fringe order....17
Figure 1.12 3D CAD model of loading frame. BimbaTM cylinder is mounted
through a hole in the aluminum frame....19
Figure 2.1 (a) in situ loading frame working under microscope and (b) in situ
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loading frame with three-point bending set-up as indicated by the
arrow. ...30
Figure 2.2 3-point bending schematic. P is the loading force; c and t are the
compression and tension fiber stress. and are the reaction
loads....30
1 2R R
Figure 2.3 Unpoled PMN-29%PT single crystal beam viewed in crossed polars:
(a) (100) face, as-received; (b) (100) face, after annealing; and (c) (010)
face, after annealing. Polarizer and analyzer are horizontal and vertical,
respectively. Strong colors in (a) indicate regions of net birefringent
retardation31
Figure 2.4 (a) Initial fringe pattern showing 0.5 fringe at point A on the free
surface opposite the loading point. (b) Second-order fringes at A, (c)
Sixth-order fringes at A, and (d) first-order fringe remaining at A after the
load is released33
Figure 2.5 Force versus deflection during increasing load for three experimental
runs...................................34
Figure 2.6 Force versus deflection with polynomial fit curve.35
Figure 2.7 Fringe order versus fiber stress. The stress-optical coefficient iscalculated from the slope of the proportional region.37
Figure 3.1 (a) Top view of the loading frame. (b) Hertzian contact experimental
set-up as indicated by the arrow..45
Figure 3.2 Initial birefringence patterns of three samples in three different
orientations under circularly polarized illumination46
Figure 3.3 The 3 differently oriented samples relative to {001}-oriented
pseudo-cubic axes. Arrows a and c represent compression along direction; arrows b and d represent compression along
direction...46
Figure 3.4 ANSYS model for use in computation of fringe pattern images.
Boundary conditions are shown. Contact elements are used at the
interface between the Hertzian cylinder indenter and the rectangular
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piezocrystal (light gray). Cyan symbols represent displacement
constraints. Red arrow indicates the force applied to all of the coupled
nodes (green)49
Figure 3.5 (a) Hertzian indentation along direction on sample 1. (b) Stress
intensity contour from ANSYS..50
Figure 3.6 (a) Hertzian indentation along direction on sample 2. (b) Stress
intensity contour from ANSYS51
Figure 3.7 Hertzian indentation along direction on sample 3 is shown in
(a); Hertzian indentation along direction is shown in (c). The
initial birefringence is responsible for the asymmetric fringe in (a) and
the layers along the surface in (c). Stress intensity contour from ANSYS
are shown in (b) and (d) correspondingly.....51
Figure 3.8 Residual butterfly fringes are fully annealed out at 400 oC for one
hour..53
Figure 4.1 (a) Overview of the in situ loading frame and (b) Four-point bending
set-up; A represents the tilting bar, and B represents the beam
sample.........61
Figure 4.2 Beam 1 (a) and beam 2 (b) after one hour annealing at 400 oC.61
Figure 4.3 Four-point bending layout. P is the loading force,c
andt
are
compression and tension stresses respectively. The diagram under the
sample shows the absolute value of the bending moment..62
Figure 4.4 (a) 3.5 order of fringes and (b) 2 order of fringes left after the load is
released...64
Figure 4.5 Maximum fiber stress versus fringe order..65
Figure 4.6 Maximum fiber stress versus fringe order with polynomial fit curve...65
Figure 4.7 From the light intensity plot, displacement between fringes can be
measured. Each valley of the intensity curves represents a fringe
(darkest field), and each peak of the intensity curves represents the half
order of fringe (brightest field)..66
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Figure 4.8 Stress versus fringe order for different load level. The number label
represents the maximum fringe order obtained for each load level. The
slope of each data line represents the fringe-stress coefficient..67
Figure 4.9 Fringe stress coefficient versus maximum fiber stress68
Figure 4.10 Fringe patterns of pure bending region at different load levels. (a)
Totally 11 order of fringes; (b) totally 16 order of fringes..69
Figure 5.1 Initial birefringence patterns of four differently oriented samples under
circularly polarized illumination after one hour annealing at
400 oC..77
Figure 5.2 (a) Overview of the in situ electrical loading frame and (b) Top view of
Hertzian contact electrical loading set-up....78
Figure 5.3 Hertzian contact electrical loading (electrical point load) experiments
on {100}-oriented beam 1: (a) -2.3 KV/cm DC were applied from the top
rod electrode for 2 minutes; (b) 2.3 KV/cm for 2 minutes; (c) 2.3 KV/cm
applied to resulting fringes of (a) for another 2 minutes; (d) 2.3 KV/cm for
additional 2 minutes after (c). The arrows in the pictures represent the
electric field direction 81
Figure 5.4 Fringe pattern comparison: (a), (c) and (e) are fringes induced by DC
electric field loading for beam 1 with 2.3KV/cm, sample 3 with 1.8KV/cmand sample 4 with 1.8KV/cm. (b), (d) and (f) are fringes under Hertzian
mechanical loading for comparably oriented samples, as shown in
Chapter 3.....82
Figure 5.5 Hertzian contact electrical loading experiments on differently
oriented samples using square waveform voltage with 0.5 Hz and 500
Hz, respectively: (a), (b), and (c) (top row) resulted from square
waveform voltage of 0.5 Hz. (d), (e), and (f) (bottom row) were from
square waveform voltage of 500 Hz. (a) and (d) are from beam 1; (b)
and (e) are sample 3; (c) and (f) are sample 4. The magnitude of electricfield is 2.3 KV/cm for beam 1 and 1.8KV/cm for both sample 3 and 4...84
Figure 5.6 From top to bottom, beam 2 is electrical poled with increment DC
voltage. The experiment set-up is with two block electrodes; both the top
and bottom surfaces are plated with gold..86
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Figure 5.7 (a) 2.8 KV/cm square waveform cyclic electric field with 0.5 Hz was
applied to beam 2. (b) Birefringence of beam 2 after annealing. The
arrow points at crack generated during the experiment..88
Figure 5.8 (a) 2.5 KV/cm DC electric field was applied to sample 4 using
electrical Hertzian contact experimental set-up. (b) Hertzian
mechanical loading on poled region....89
Figure 5.9 Mechanical poling and electrical poling representation...90
Figure A1 Representation of optical index ellipsoid; Illustration redrawn from a
similar figure in reference [1].........101
Figure A2 Circular polariscope set-up, reproduced from a similar figure in
reference [3]..104
Figure A3 Top view of calibration stage. 108
Figure A4 Side view of calibration stage.108
Figure A5 Overview of loading frame..109
Figure A6 Loading force versus pressure......110
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1 Introduction - piezoelectr ic single crystals and
photoelasticity
1.1 PMN-29%PT single crystals
In 1880, the famous brothers Pierre and Jacques Curiefirst discovered direct
piezoelectric effects in quartz crystals [1, 2]. They found that when a weight is
placed on the surface of a quartz plate, electric charges are generated on both
surfaces of the quartz plate. The charge was measured to be linearly
proportional to the weight placed. Following the discovery of the direct
piezoelectric effect, Lippmann in 1881 theoretically predicted the converse
piezoelectric effect, which says a voltage applied to a piezoelectric crystal
produces elastic strains in the crystal [2, 3]. Later, general theory of
piezoelectricity was thoroughly accounted by Voigt [2, 4]. For the next 60 years,
extensive characterization was performed on BaTiO3 ceramics. In the 1950s,
Pb(Zn1/3Nb2/3)O3 (PZT) ceramics were found to exhibit an exceptionally strong
piezoelectric response. Since then, modified PZT ceramics and PZT-based solid
solution systems have become the dominant piezoelectric ceramics for various
applications [5].
This defining characteristic of the piezoelectric materials is due to the fact that
the centers of positive and negative charges do not coincide. Namely the crystal
structure does not have a center of symmetry. Such materials possess a
spontaneous polarization. When the spontaneous polarization can be reversed
by an applied electric field, the material is called a ferroelectric. Thus
ferroelectrics are a subset of piezoelectric materials.
In contrast to conventional piezoelectric ceramics, single crystal relaxor
ferroelectrics Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT) and
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Pb(Zn1/3Nb2/3)O3-xPbTiO3 (PZN-xPT) exhibit extra-high dielectric and
piezoelectric properties and have become a new generation of piezoelectric
materials, attracting constant attention in recent years [6-10]. Both of them are
widely used in high performance applications such as medical imaging, active
noise suppression, and acoustic signature analysis.
Because PMN-PT has relatively high field-induced strain response and a
small hysteresis loop compared to PZN-PT, PMN-PT is more attractive than
PZN-PT [10]. Furthermore, relaxor-based ferroelectric single crystals PMN-PT,
with compositions near the morphotropic phase boundary (MPB) between the
ferroelectric rhombohedral and tetragonal phases, have ultimate
electromechanical coupling factors (k33 >90%), high piezoelectric coefficients
(d33>2000 pC/N) and high strain levels up to 1.7% [11-12]. They also have
potential to be used in electro-optical technology for their high electro-optical
coefficients [13-14]. Thus my study focuses on PMN-29%PT (close to MPB)
single crystals. The origins of PMN-PT single crystals excellent performance
have been attributed to the polarization rotation induced by the external electric
field [15].
However, these materials face crack problems which will reduce the
performance of the devices. Some researchers have studied the fracture
problems of piezoelectric materials, both theoretically [16-21] and experimentally
[22-26], however, most of them focused on using an AC electric field to drive the
growth of existing cracks. Because internal stress plays a significant role in
causing cracks and also in the propagation of cracks, further study of these
internal stresses induced either by mechanical loading or by electrical loading is
an important research topic. This will enable us to better understand and control
the internal stresses in relaxor ferroelectrics devices.
What are the possible sources of internal or residual stress in PMN-29%PT?
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Residual stresses are induced by inhomogeneous strain. Inhomogeneous strain
may be produced by thermal gradients during crystal growth [27-28], by phase
transitions during cooling [27, 29], and by mechanical operations, such as cutting,
grinding and polishing during each step of the machining processes [29]. Each
step, therefore, has the potential to produce more residual stresses in the single
crystals.
The topic is significant, but also very difficult due to the complicated internal
domain structures and the intrinsic coupling effects between mechanical and
electric fields. PMN-xPT single crystals have a simple perovskite ABO3 structure
above Curie temperature (for PMN-29%PT, the Curie temperature is about 135
C), pictured in Figure 1.1(a), and it may readily have complex perovskite
structure A(B1/3B2/3)O3, as well. X-ray diffraction (XRD) shows unpoled
PMN-xPT single crystals have a tetragonal-rhombohedral MPB (morphotropic
phase boundary). When x is under 30%, PMN-xPT is in rhombohedral (R) R3m
phase at room temperature; when x is above 33%, it begins to transform to
tetragonal (T) P4mm phase through monoclinic (M) or orthorhombic (O)
symmetries [30-33]. The spontaneous polarization direction of the R phase is
and that of the T phase is . The piezoelectric effect is observed to
peak at the morphotropic phase boundary. The enhancement in the
piezoelectric effect at the morphotropic phase boundary has been attributed to the
coexistence of the different phases, whose polarization vectors become more
readily aligned by an applied electric field when mixed in this manner than may
occur in either of the single phase regions.
PMN-29%PT is in the R phase at room temperature and there are eight
possible directions for the spontaneous polarization as shown in Figure 1.1(b).
After an electric field poling, PMN-29%PT will transform from R to M phase first
[32]. With increasing poling field, M to T phase transition may occur. Phase
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diagram is shown in Figure 1.2 [33]. The coercive electric field is about 5 KV/cm.
With the fluctuation of chemical composition, macro-domains formed with different
polarization directions. Some researchers even found that within
macro-domains (m scale), there are micro-domains (0.1m scale), and within
micro-domains, there are nano-domains (nm scale). This is called the domain
hierarchy [34]. Accordingly, internal stress study has different levels. In this
thesis, I examine internal stress at the m level through the use of optical
methods.
Figure 1.1 (a) ABO3 Perovskite Structure. If one shifts the A-site ions to the center of the cell,
the structure will appear differently with 12 oxygen atoms at the center of each cell edge, an
arrangement often shown in geology texts, (b) Spontaneous polarization for the R phase in
unpoled PMN-29%PT; Illustration redrawn from a similar figure in reference [5].
The constitutive equations of the piezoelectric materials are [35-36]:
321321
ricitypiezoelectconverseelasticity
kkijklijklij EesC = (1-1)
321321
ricitypiezoelectconverseelasticity
kkijklijklkl EdSs += (1-2)
321321
typermittiviricitypiezoelectdirect
kikklikli EseD += (1-3)
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where ij , , and are stress, strain, electric field and electric
displacement tensors, respectively. , ,
kls kE iD
ijklC ijklS ik , and are the elastic
constant tensor, elastic compliance tensor, the dielectric constants, thepiezoelectric stress coefficients and the piezoelectric strain coefficients,
respectively; these tensors are material specific. So, both the external
mechanical loading and electrical loading will induce internal stress/strain and
polarization, accompanied by domain switching. If the loading is large enough, it
can even induce phase transitions. This coupling between electrical and
mechanical field variables in the constitutive equations will bring serious
mathematical difficulty to the internal stress analysis problem.
kije kijd
Figure 1.2 Phase diagram of PMN-PT single crystals; Illustration redrawn from a similar figure
in reference [33].
Traditionally in materials research any of several types of strain gages can be
employed to help measure the internal strain and, further, to analyze the internal
stress. Since the available samples are too small to use strain gages, optical
methods were adopted, i.e. photoelasticity techniques. Compared with other
stress measurement techniques, photoelasticity can offer efficient quantitative
determination as well as qualitative observation of the stress distributions
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resulting from both internal stress and external loading [37-46]. Initial
explorations show that PMN-29%PT single crystals can be polished to be optically
transparent and the application of external loads produces an extremely large
number of well-defined fringes when observed with a polarizing microscope.
This work is novel because it explores the usage of optical techniques in
measuring stresses in next generation piezoelectric materials, which will be an
important quality assurance tool to produce robust and reliable devices in the
years ahead.
Optical methods can help to analyze electrical loading effect in piezoelectric
materials as well. An electric field applied to the piezoelectric single crystals will
cause at least three effects. First, the refractive index changes in proportion to
the electric field. This is known as linear electro-optic (EO) effect. Second, the
electric field induces internal stress/strain; this is known as the converse
piezoelectric effect. These internal stresses will induce photoelastic effects.
When the electric field is large enough, it can also pole the sample (align domains)
to induce phase transformations. Third, the refractive index changes in
proportion to the square of the electric field. This is known as the quadratic
electro-optic or Kerr effect. All of these three effects contribute to the observed
birefringence. Recently the optical properties of piezoelectric materials such as
the refractive indices have been reported [47-50]. Unpoled PMN-29%PT singe
crystals have many domains with different orientation, retaining an optically
isotropic pseudocubic state. Under this assumption, the refractive index of
unpoled PMN-30%PT single crystals is reported to be 2.501 [47]. Poled
tetragonal PMN-38%PT single crystals have an effective EO coefficient of
42.8 pm/V as reported in reference [48]. However, because applied field will
cause domain rotation and phase transformation in unpoled PMN-PT single
crystals, precise determination of the pure EO coefficients is neither possible, nor
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useful. Thus EO coefficient determination is not included in this thesis.
Besides the electric-optic effect, several researchers studied the cyclic electric
field induced effects in ferroelectric ceramics or piezoelectric single crystals
[51-56], which including domain switching, phase transformation, micro-cracking,
and fracture. The observation and study are normally carried out through TEM,
dielectric measurements, and optical microscopy, etc. Crack growth is directly
observed under the optical microscope and micro-crack growth under TEM.
Phase transformation is studied by measuring change of the dielectric properties.
These studies help reveal what is going on when the piezoelectric materials are
under electrical loading. However, if we can get to know the internal stress state
of the materials during the electrical loading, we can better understand the crack
initiation condition and crack growth. Fortunately, unpoled PMN-PT single
crystals can be polished to be transparent and show colorful birefringence. I
focused on AC/DC field-induced birefringence of samples originally without a
crack. Crack initiation caused by electric fields and phase transformations were
examined. Optical observation of phase transformations is a field where not
much research has been done and further study is necessary.
Commercial FEM software ANSYS was applied to model the experiments
and offer theoretical/computational results, helping to interpret the experiment
results that I obtained.
1.2 Photoelasticity
1.2.1 Discovery of the phenomenon of photoelasticity
Photoelasticity is a well-known efficient method to measure the internal
stresses in a variety of transparent materials. This phenomenon was first
discovered by Sir David Brewster in the year 1815 [40]. He presented a paper
before the Royal Society of London where he reported the effect. In his
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experiments, he placed a piece of glass in between two crossed polarizers. He
found that when the glass was stretched transversely to the direction of
propagation of light, the field of view grew brighter, therefore showing that an
artificial birefringence is induced in the glass by the mechanical stress.
Furthermore, he found in the case of solids which are initially birefringent, the
initial birefringence is altered by the stress. Thereafter, photoelastic techniques
were developed to study crystals and other transparent solids. It has become an
important experimental method for the measurement of internal stress.
Figure1.3 PMN-29%PT single crystal as received.
Employing this method is very simple. Namely using a crossed polarizer
set-up, we can see stresses. Bright colors such as magenta and green, as well
as closely spaced fringes imply high level stresses. With this simple rule, we can
already tell that the samples as received have large birefringence. Figure 1.3
shows an example of fringe pattern for the as received sample. This fringe
pattern results from the net birefringence associated with combining the initial
domain distribution and residual stresses. Broad faces are polished transparent
and the four edges are in as received condition. The big lobes on four corners
with many little bumps along the edges are most likely due to residual stresses
from machining operations. Usually machining stresses are compressive near
the surface and tensile inside the sample.
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1.2.2 Mathematical formulation of Photoelasticity
The stress-optic law of photoelasticity states that if there is a difference in
principal stresses along two perpendicular directions in an otherwise optically
isotropic material, the refractive index in these directions is different and the
induced birefringence is proportional to the difference. For a two-dimensional
stress state, the law simplifies to [38-39]:
( 21
= Ct
n ) (1-4)
Here 1 and 2 are the maximum and minimum principal stresses, n is
fringe order, t is the samples thickness, and is the wavelength of the incident
light. C is the stress-optic coefficient which is a constant. From Eq. (1-4), we
have:
Ct
n
22
21 =
(1-5)
Hence
Ct
n
2max
= (1-6)
The stress-optic coefficient C is useful to quantitatively analyze the internal
stresses, such as max as a function of position for any fringe pattern if we know
the fringe ordern.
Eq. (1-5) can also be rearranged to:
t
nf
Ct
n==
21 (1-7)
Here, we introduce another concept: the fringe-stress optical coefficient , wheref
Cf /= represents the principal stress difference necessary to produce a one
fringe order change in a crystal of unit thickness. The fringe-stress coefficient
depends on the stress-optical coefficient C of the material and the wavelength of
the incident light. Therefore, C is more general and convenient than allowingf
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the use of incident light with different wavelength. However, to allow easy stress
calculation and also easy comparison of optical properties with mechanical
properties, need to be evaluated. When is used for stress calculation, a
single standard wavelength should be used. While any color monochromatic light
is acceptable, ~535 nm green light was selected for the experiments reported in
this thesis. The engineering units of the fringe-stress coefficient are N/m.
f f
For anisotropic crystals, the mathematical formulation of photoelasticity is
more complex. The relevant equations are in Appendix I. As shown in Figure
1.1(b), unpoled PMN-29%PT single crystals have eight possible polarization
directions. When the number of domains is large enough, the global structure of
the unpoled sample can be treated as pseudocubic for unpoled PMN-29%PT
single crystals [51] and pseudotetragonal for poled crystals [58]. These
assumptions enable photoelastic methods to be applicable to these materials in
theory, though the experimental results may turn out differently due to the
complex domain hierarchy structures that can develop as a result of
thermal/mechanical processings history of the samples.
1.2.3 Plane polariscope and circular polariscope
The general arrangement of light fields to perform photoelastic experiments
consists of two typical arrangements: The plane polariscope and the circular
polariscope. In photoelasticity, stress fields are displayed through the use of
light. The basic arrangement of a plane polarized microscope includes a
polarizer and an analyzer, mounted with a 900 rotation between them to minimize
the transmission of light through the pair. If an isotropic material is placed
between the plates, it will not affect the intensity of the transmitted light regardless
of its angle to the polarizers. This setup, with polarizers crossed is called a dark
field plane polariscope, as shown in Figure 1.4(a). The other arrangement,
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called a bright field plane polariscope, features the polarizer and analyzer parallel
to one another and was not used in this analysis.
(a) (b)
Figure 1.4 (a) Dark field plane polariscope set-up. (b) Light vector representation; Illustrationsredrawn from a similar figure in reference [38].
In Figure 1.4(b), consider polarized light coming out the polarizer aligned with
E1 parallel to the x axis:
)cos(1
tkE = (1-8)
When entering the sample, the light vector splits to two vectors along the principal
stress axes. As the two components of light propagate through the sample, a
phase difference of is generated. Let be the slow axis and be the
fast axis, we have:
2E
3E
)2
cos(cos
)2
cos(sin
3
2
+=
=
tkE
tkE
(1-9)
After the light passes out through the analyzer, only the y axis component of the
light is visible, so:
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)sin(2
sin)2sin(
)2
cos()2
cos(cossin
sincos 324
tk
ttk
EEE
=
+=
=
(1-10)
The intensity of the light we see thus dependent on the orientation , the time
and the phase differencet for each point in the image. Regions where
)2sin( or2
sin
or )sin( t are zero are dark. The overall appearance is
similar to a contour map. These black bands in the stress patterns are known as
fringes. Namely when intensity of the light is zero, there is a fringe. Intensity is
proportional to the square of the amplitude and the time dependent term is usually
not considered:
)2
(sin)2(sin 22
ap II = (1-11)
Here represents the amplitude of the incident light and other factors
affecting the transmission light intensity. From Eq. (1-11), we can see, there are
two set of fringes superimposed over each other, isochromatics and isoclinics.
Isochromatics are caused by the incident light phase difference
aI
of 2m (here
m is an integer), or as is often said, a retardation caused by the principal stress
difference at the point. Isoclinic fringes are contours of constant inclination,
when the polarizer axis coincides with one of the principal stress directions at the
point of interest, 2/,0 = .
Use of a circular polariscope eliminates isoclinics. Two quarter-wave plates
are added to the plane polariscope with their axes at 45
0
and 135
0
to one of thepolarizers to achieve circular polarized microscopy. The details of circular
polariscope are described in Appendix II with the basic set up illustrated in Figure
A2. The result is that the intensity of light transmitted for circular dark-field only
depends on the retardation, thus only isochromatics will be seen. Circular
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polarized microscopy is used in most experiments conducted here. Plane
polarized microscopy can help to define the phase of PMN-PT single crystals by
measuring the extinction angle of the light through the crystal relative to the
direction of the polarizer.
1.3 Preliminary three-point bending experiments
1.3.1 Experimental setup
A ZeissTM Axioskop2MAT microscope was used for all the photoelasticity
experiments conducted, as shown in Figure 1.5. This microscope was modified
by addition of a rotation stage normally found on polarizing microscopes to
facilitate rotation of the sample. Light goes straight up from the bottom. A 2
megapixel camera is used to transfer the images to the connected computer, so
we can observe the images on a large monitor. When quarter wavelength
retardation plates are applied, the microscope is configured as a circular
polariscope. In preliminary three-point bending experiment, the microscope was
configured as a plane polariscope without quarter wavelength plates.
For preliminary three-point bending experiments, a parallel clamp with jaws
only 1mm tall was designed to apply the force to a sample while observing with a
polarizing microscope. This device is shown in Figure 1.6. Screws A and B are
adjusted individually to keep the loading faces parallel to each other as they are
brought together. Three semi-cylinder shaped glass rods were cut and polished,
each with 1mm height and 1 mm radius to exert loading and supporting forces.
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Figure 1.5 ZeissTM
Microscope set-up.
Figure 1.6 Preliminary three-point bending set-up.
An unpoled [001]-oriented PMN-29%PT single crystal bar was used in these
bending experiments. This crystal was obtained from H.C. Materials Corporation,
Bolingbrook, Illinois. Unless otherwise noted, crystals used for this research
were grown by H. C. Materials Corporation. The surfaces for light transmission
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were polished down to 0.05 m grit size. The dimension of the bar is 11X1.8X1
mmas measured directly from micrographs taken under a 2.5X objective lens.
The experiments are performed between crossed polarizers under plane
polarized dark field. A green light filter with 535nm wavelength was applied to
show well-defined fringes.
1.3.2 Fringe pattern
Figure 1.7 is taken at an angle of 450 to both polarizer and analyzer. This
image clearly shows isoclinic fringes on the neutral axis, caused because
principal stresses on the neutral axis are perpendicular to each other and at an
angle of 450 to the parallel and perpendicular edge directions of the three-point
bending specimen. To verify this, the principal stress field in a beam under
three-point bending was calculated using ANSYS software. Figure 1.8 shows
the resulting principal stress field displayed as vectors modeled in an isotropic
material. It is evident that the principal stress direction is at 450 to both the
polarizer and analyzer inside the solid line circles, which should be a bright region,
and is 00 to both the polarizer and analyzer inside the dash line circle, which
should be all dark under the microscope according to the photoelastic theory.
This matches the fringe pattern shown in Figure 1.7, implying that the basic
photoelastic technique works on PMN-29%PT single crystals. This supporting
result is also verified in subsequent four-point bending experiments.
Figure 1.7 Three-point bending image at 450
to both the polarizer and the analyzer.
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Figure 1.8 Principal stress vectors from ANSYS
simulation of three-point bending. Only left
half of sample is shown.
Fringe picture taken at zero degree to both polarizer and analyzer is shown in
Figure 1.9, and it shows isoclinic fringes as well. This may be easily verified by
rotating the analyzer and polarizer coordinate 450 clockwise in Figure 1.8.
Figure 1.9 Three-point bending image at zero degree to both the polarizer and the analyzer.
When the applied loading force is increased, fringes were observed
generated at the central portions of the top and bottom edges and move towards
the neutral axis. This process continues until we cause the bar to snap, usually
with as many as 25 or more fringes. The fringe pattern seen in PMN-PT single
crystals is similar to that seen in typical isotropic materials, as illustrated in Figure
1.10. The exception is the fringes resulting from Hertzian contact loading ofthree glass rods, which show a two-lobed fringe pattern. This will be further
discussed in Chapter 3.
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Figure 1.10 (a) Three-point bending of unpoled PMN-29%PT; (b) Three-point bending image
of isotropic materials [39].
1.3.3 Deflection versus fringe order
Stress vs. Fringe Order
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30
Fringe Order
Deflection(m)
Figure 1.11 Deflection versus fringe order
In preliminary three-point bending experiments, we purposely put a thin glass
plate (a cover slip) on the glass rods, to obtain an edge to be used as a referenceto measure the deflection during the bending process, as shown in Figure 1.7.
Deflection versus fringe order was plotted in Figure 1.11. It is obvious that
deflection is linearly proportional to the fringe order. This implies for unpoled
PMN-29%PT single crystals, the mechanical properties characterized by the
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deflection are linearly proportional to the optical properties which control fringe
order. If the samples were isotropic and homogeneous in their optical properties
and mechanical properties, then the birefringence would be proportional to strain.
1.3.4 Summary
Through preliminary three-point bending experiments, isoclinic fringes were
observed; fringe patterns were also comparable to those of typical isotropic
materials. This means the photoelastic technique is useful and can be further
used to study internal stresses of unpoled PMN-29%PT single crystals.
However, loading force was unknown so that quantitative calculations could not
be performed; fringe patterns were observed only qualitatively while the loading
was increased. Because the elastic constants of unpoled PMN-29%PT single
crystals are also unknown, there was no way to analyze the internal stress. The
only quantitative result directly obtained from preliminary three-point bending is
the linear relationship between deflection and the fringe order. It was necessary
to design a new device which provides the same function while also allowing a
known force to be applied. To solve this problem, a BimbaTM 5/16 bore air
cylinder is used to design an in situ loading frame as shown in Figure 1.12.
Details of calibration of the loading system are provided in Appendix III. The
calibration result is that the loading force obtained from reading of the pressure
gauge is within 2.5% of the applied value.
In the following three chapters, birefringence response and internal stresses of
unpoled PMN-29%PT single crystals under mechanical loading are studied for thefirst time using photoelastic techniques in a series of sequential experiments
comprising: three-point bending experiments, four-point bending experiments,
and Hertzian contact loading experiments. In the three-point bending
experiments, the numerical value of the stress-optical coefficient of PMN-PT was
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first estimated. The apparent Youngs modulus along direction of unpoled
PMN-PT single crystals was calculated. In Hertzian contact loading experiments,
orientation dependences of fringe patterns were observed, showing the
anisotropic properties of unpoled PMN-PT single crystals. ANSYS simulations
of piezoelectric single crystals were performed, verifying that the anisotropic
elastic properties indeed cause the orientation dependence of fringe patterns that
were observed. The results were published in two papers, references [58] and
[59] respectively. To further examine the variations of stress-optical coefficients
with incremental mechanical stresses, four-point bending experiments were
performed. A paper has recently been submitted to report the results. Finally,
electric field loading experiments were performed; the results of which are
reported in Chapter 5.
Figure 1.12 3D CAD model of loading frame. BimbaTM
cylinder is mounted through a
hole in the aluminum frame.
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2 Photoelastic study using three-point bending technique
Abst ract: Photoelastic effects in an unpoled PMN-29%PT single crystal
beam have been investigated using three-point bending experiments. A linear
relationship between the applied load and the measured displacement was
observed up to a proportional limit of ~30 MPa. Beyond this proportional limit,
yielding was observed. Samples were loaded as high as 77 MPa without fracture.
Young's modulus Y ~1.9X1010 N/m2 was determined directly from the initially
linear region using beam theory. The photoelastic fringe order versus fiber
stress plot also displays an initially linear region up to a proportional limit of ~20
MPa, suggesting that optical measurements are a more sensitive measure of the
onset of microplasticity than mechanical measurements. Residual photoelastic
fringes associated with yielding were completely removable by annealing above
the Curie temperature, implying that plastic deformation occurs by reversible
processes such as domain switching and phase transformation. The
stress-optical coefficient for unpoled PMN-29%PT determined from the initially
linear region of the fringe order versus fiber stress curve is 104X10-12
Pa-1
. This
value is large and comparable with the stress-optical coefficient of polycarbonate,
making unpoled PMN-29%PT single crystal a good candidate for optical stress
sensors and acousto-optic modulators.
2.1 Introduction
Relaxor ferroelectric single crystals exhibit ultrahigh dielectric and
piezoelectric properties compared with conventional piezoelectric ceramics.
Materials such as PMN-PT single crystals have become a next generation of
piezoelectrics that have attracted constant attention in recent years [1-5]. These
materials are finding wide-ranging applications in medical imaging, active noise
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suppression, and acoustic signature analysis. Residual stresses and internal
stresses in PMN-PT single crystals, however, can reduce the performance of
devices and lead to the initiation of cracks.
Residual stresses are induced by inhomogeneous strain. Inhomogeneous
strain may be produced by thermal gradients during crystal growth [6-7], by phase
transitions during cooling [6-8], and by mechanical cutting and finishing
operations during device fabrication [9]. When the size scale of the residual
stress distribution approaches the size scale of the microstructure, residual
stresses are often referred to as internal stresses or microstresses. Clearly, the
presence of a stress distribution within a component will influence its response to
applied loadings. To better understand and control stresses in relaxor
ferroelectrics devices, it is necessary to monitor internal stresses and residual
stresses.
Photoelasticity is an efficient and effective method to measure the residual
stresses and applied stresses in many transparent materials. It offers both
quantitative determination and qualitative observation of the stress distribution in
a sample [9-12]. Simply by examining a sample between crossed polarizers in
either the loaded or unloaded state, we can observe the influence of stress as a
result changes in optical birefringence. High-order pastel colors from the
Michel-Levy interference color chart, such as magenta and green, in combination
with closely spaced fringes, imply high stress levels and high stress gradients.
Quantitative evaluation of stress level requires that we measure the retardation
caused by stress and relate this to the stress-optical properties of the material
and the length of the optical path.
The stress-optic law of photoelasticity states that if there is a difference in
principal stresses along two perpendicular directions in an otherwise optically
isotropic material, the refractive index in these directions is different.
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Fundamentally, this effect arises from a change in the spacing between the atoms
due to strains induced by the principal stresses. The difference in refractive
index is an induced birefringence which is proportional to the difference in
principal stresses. The maximum and minimum refractive index directions are
aligned with the principal stresses.
For a two-dimensional plane stress state, the stress-optic law for an isotropic
material can be expressed as [12, 13]:
)( 21
= Ct
n . (2-1)
Here 1 and 2 are the maximum and minimum principal stresses, is the
fringe order, and is the sample's thickness along the optical path.
n
t is the
wavelength of the incident light and is a constant known as the stress-optical
coefficient. From Eq. (2-1), we have
C
Ct
n
22
21 =
, (2-2)
where both sides are divided by 2 to produce the form of the maximum shear
stress,
Ct
n
2max
= (2-3)
Once the stress-optical coefficient C is known for a given material, it can be
used to quantitatively evaluate max for a fringe pattern, provided we know the
fringe order. Photoelastic fringe patterns suitable for stress analysis are easily
recorded with the use of a circular polariscope and a monochromatic filter. The
fringe order may be found by locating a zero-order fringe and counting. Zero orderfringes in bending samples occur along the neutral axis. More complete details
of photoelastic methods for isotropic materials can be found in references [12,
13].
The purpose of this chapter is to explore photoelastic techniques for the
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investigation of stress distributions in unpoled PMN-29%PT single crystals. We
present the results of investigations performed using three-point bending
experiments. While the optical properties of unpoled PMN-29%PT are not
isotropic, fringe patterns are comparable with those typical of isotropic materials.
Experimental results show that there exists a linear relationship between loading
force and displacement and between fringe order and fiber stress within a
proportional limit. Beyond the proportional limit, yielding takes place. Yielding is
interpreted as stress-induced domain switching. Residual stress remaining after
unloading can be removed by annealing above the Curie temperature suggesting
that these switches are reversible. The linear relationships observed suggests
that photoelastic methods can be used more generally for these materials. The
use of optical techniques to measure stresses in next generation piezoelectrics
will be an important quality assurance tool to produce robust and reliable devices.
2.2 Experimental procedure
An in situ loading frame built to perform photoelastic measurements on
small-size beams is shown in Figure 2.1. Figure 2.1(a) illustrates the loading
frame below the objective of a ZeissTM microscope configured as a circular
polariscope while Figure 2.1(b) provides a top view of the three-point bending
set-up. Mechanical loading is applied using a BimbaTM pneumatic cylinder
shown in Figure 2.1(a), and three 1mm radius glass rods illustrated in Figure
2.1(b). The in situ load frame was calibrated so that the applied force could be
obtained directly from the reading of a pressure gauge within 2.5% accuracy.The loading frame allows a 10X objective lens to be used to make deflection
measurements of the beam during bending.
A schematic of the three-point bending loading system is presented in Figure
2.2. Principal faces of the beam are parallel to the (100), (010), and (001)
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planes, respectively. The thickness and height are approximately equal at
t=1.06 mm and h=1.07 mm, enabling the beam to be bent in either direction by
changing the direction of the applied load P. An experiment in which P is aligned
with [100] is called [100] bending as shown in Figure 2.2. The small size of the
experimental set-up means that the exact placement of the loading rods will vary
from one run to the next. Specific values of the overall span length and the
numerical values of a and b were measured directly from micrographs taken
using a 5X objective lens.
Figure 2.1 (a) in situ loading frame working under microscope and (b) in situ loading
frame with three-point bending set-up as indicated by the arrow.
Figure 2.2 three-point bending schematic. P is the loading force, c and t are the
compression and tension fiber stress. and are the reaction loads.1R 2R
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The unpoled PMN-29%PT single crystal beam (H.C. Materials Corporation)
was polished using an Allied High Tech Multi-PrepTM polishing system following
the rule of threes. Fixed abrasive diamond films in progressively finer sizes,
each removing a thickness of three times the diameter of the previous abrasive,
were followed by a final polish using 0.05 m colloidal silica on each of the four
major faces. Sufficient material was removed between each abrasive step so
that no damaged material from the previous abrasive remained after each
polishing step. The final surfaces were of optical quality.
Figure 2.3 Unpoled PMN-29%PT single crystal beam viewed in crossed polars: (a) (100)
face, as-received; (b) (100) face, after annealing; and (c) (010) face, after annealing.
Polarizer and analyzer are horizontal and vertical, respectively. Strong colors in (a) indicate
regions of net birefringent retardation.
Residual stresses, net birefringence from initial domain distributions, or a
combination of these are apparent in the as-received sample as indicated by the
bright, low order color fringe patterns shown in Figure 2.3(a). For proper
photoelastic measurements, an initially stress-free sample with no net
birefringence is desired. It was found that annealing at 400 oC for 1 hour
substantially reduced the residual stresses and the apparent initial birefringence.
The observed fringe order looking through the (100) face was reduced to ~0.45
for the white regions [13] and 0 for black regions, as shown in Figure 2.3(b). The
fringe order of the (010) face shown in Figure 2.3(c) was reduced to ~0.28, the
fringe order associated with gray color as indicated in reference [13]. It is
important to note that the entire sample does not show uniform extinction as
would be the case for an isotropic material. Different faces show different
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extinction levels at nearly equal thicknesses implying the crystal is optically
anisotropic, perhaps a result of crystal growth [14].
Photoelastic experiments were performed using a ZeissTM optical microscope
configured as a dark field circular polariscope. This configuration eliminates
isoclinic fringes and thus produces only isochromatic fringes. Isochromatic
fringes depend only on the magnitude of the principal stress differences at each
point, greatly simplifying the analysis. A monochromatic green filter with
wavelength ~535 nm was used to record photographs for the evaluation of the
fringe order. Fringe order was counted from 5X objective lens images at the
point of maximum tensile stress on the free surface of the sample (point A) as
shown in Figure 2.2. Fractional fringes were estimated to the nearest 0.3 using
graphical intensity information from image analysis software. The displacement
of the sample relative to a fixed reference was measured directly from 10X
objective lens images at the same location A. Bending was performed on both
(100) and (010) faces for comparison, even though crystallographic symmetry
consideration suggests the results should be identical. Between each
experiment, the beam was annealed to remove all residual fringes, allowing the
same beam to be used again and again.
2.3 Results and discussion
2.3.1 Fringe patterns
Figure 2.4 shows photoelastic images obtained at different load levels for [100]
bending. Figure 2.4(a) shows the unloaded state, Figure 2.4(b) and Figure 2.4(c)
show examples of well-defined fringes obtained under increasing load, and Figure
2.4(d) shows residual fringes when the load is removed. These fringe patterns,
obtained using monochromatic green illumination under circular polariscope, are
comparable to those typical of isotropic materials [12]. During the experiments, it
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could be seen that fringes originated at the central portions of the top and bottom
edges and moved inward toward the neutral axis with increasing load. The
fringes formed in pairs, producing the relatively symmetric patterns shown in
these figures. Fringes of increasing fringe order distribute uniformly along the
height of the beam, corresponding to a linear variation of the principal stress along
the thickness, as shown in Figure 2.2. The upper half of the beam is in
compression, while the lower half is in tension. The zero-order fringe always lies
along the neutral axis which is stress free according to elementary beam theory.
We can see clearly from Figure 2.4(b) that the observed fringe pattern is
asymmetric at low loads: the zero- order fringe exists only on the right portion of
the neutral axis, corresponding to the region showing exactly zero fringe order in
the unloaded state. Thus the fringe patterns we observe from applied loading
are qualitatively consistent with fringe patterns that would have been obtained
from an isotropic sample. Note that in three-point bending experiments, the
principal stress x on the outer surface at point A is the maximum tension stress
max , known as the fiber stress, and y equals zero as a result of the free
surface boundary condition. Thus, the fringe order is directly proportional to fiber
stress x according to Eq. (2-1).
Figure 2.4 (a) Initial fringe pattern showing 0.5 fringe at point A on the free surface opposite
the loading point. (b) Second-order fringes at A, (c) Sixth-order fringes at A, and (d) first-order
fringe remaining at A after the load is released.
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2.3.2 Loading force versus deflection
Figure 2.5 illustrates the applied compressive load as a function of the
displacement measured at the center of the beam. Two data sets for [100]
bending and one data set for [010] bending are shown. The results are highly
repeatable, independent of the orientation of the bending, indicating that the
mechanical properties are the same for both orientations as we would expect
given the nearly identical dimensions of the samples. The force depends linearly
on the displacement over the initial portion up to a proportional limit which implies
that the loading induces elastic deformation in this regime. The proportional limit
is approximately 2 N, which corresponds to a fiber stress of 25 - 30 MPa, for the
sample geometries and span lengths used in these tests. Beyond the
proportional limit there is yielding after which the data appears to continue upward
with a reduced slope.
Figure 2.5 Force versus deflection during increasing load for three experimental runs.
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Figure 2.6 Force versus deflection with polynomial fit curve.
The slope in Figure 2.5 was obtained in terms of least-square fit based on
data below the proportional limit. The correlation coefficient is 0.988. As shown
in Figure 2.6, with a fourth-order polynomial fit for all the data, the resulting curve
provides a simple and reliable way to determine the proportional limit rather than a
simple visual inspection. The yield stress of 25 - 30 MPa is comparable to 20
MPa reported by Viehland for PMN-32%PT single crystals [15].
We interpret this yielding effect as the result of stress induced domain
switching that occurs throughout the sample, spreading from the high stress
surfaces as a result of the stress gradients. Plastic deformation represents a
reorientation of the polarization of the nanodomains distributed throughout this
otherwise unpoled sample. Essentially, the stress is changing the population of
the dipoles of the eight possible orientations of the unpoled sample [16], leadingto strain. At these modest stresses, the sample does not undergo large scale
mechanical poling or stress induced phase transformations that are possible in
this system. We can say this because the yielding was not accompanied by the
massive changes in optical properties expected from phase transitions. Rather,
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the development of the photoelastic fringe patterns proceeded smoothly through
the yielding region as described below. Thus, it seems appropriate to assign the
yielding phenomena shown in Figure 2.5 to distributed domain reorientation (i.e.
domain switching). By this process, the yielded portions of the sample have
been mechanically poled.
It is also possible that the apparent yielding we see is, in part, the result of
concentrated strains that occur at the loading points as a result of the Hertzian
contact stresses. These large contact stresses could be sufficient to trigger
stress induced phase transformations in the neighborhood at the loading points.
More work is needed to assess the relative importance of this contribution.
2.3.3 Stress-optical coefficient
From elementary beam theory, the fiber stress during three point bending is
expressed as:
2
maxmaxmax
6
Lth
Pab
I
yM== (2-4)
Here is maximum bending moment at the location of point A and
is area moment of inertia of the beam cross section. P is the
loading force, while a , , , and represent dimensions as shown in Figure 2.2.
maxM
12/3thI =
b h t
L is the span length, namely ( +b ). is negative with a numerical value
equal to half the height h at location A. The stress
a maxy
max calculated from Eq. (2-4)
represents the principal stress 1 at point A since 2 is zero there.
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Figure 2.7 Fringe order versus fiber stress. The stress-optical coefficient is calculated
from the slope of the proportional region.
Figure 2.7 displays the fringe order versus the fiber stress. The slope was
obtained in terms of a least-square fit based on data below the yield stress
obtained from Figure 2.6. The correlation coefficient of the linear fit is 0.9774.
Figure 2.7 shows the same trends between the fringe order and the fiber stress as
that between the loading force and the deflection, only the proportional limit
occurs earlier in the data set. The fringe order, characterizing the optical
properties, is a more sensitive indicator of the deviation from linearity than the
displacement. The proportional limit in fringe order versus stress is 20 MPa
compared to 25-30 MPa discussed earlier for load versus displacement.
Here again, the data is highly repeatable, particularly the data from the same
type of bending. Difference between the two orientations of the bending may be
attributed to the initially different birefringence at zero applied load. Namely, the
optical properties are different for [100] and [010] experiments because the initial
domain distributions are different for these two cases. At stresses below the
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proportional limit, fringe order is linear with the fiber stress showing a photoelastic
effect. The explanation for the optical yielding should be the same as for the
mechanical yielding. Namely, it should be the result of distributed domain
switching and possibly locally phase transformation at the loading points.
From the slope of the linear region in Figure 2.7, 0.2077 MPa-1, we calculate
the stress-optical coefficient using Eq. (2-1):
MPa
12077.0=
tC(2-5)
Here is the thickness of the beam through which the light passes, 1.06 mm
to 1.07 mm, depending on the orientation.
t
is the wavelength of the green
filter, ~535 nm. The stress-optical coefficient C is calculated as 104X10-12
Pa-1
.
Stated another way, approximately 2.4 MPa of shear stress (4.8 MPa of principal
stress difference) will induce one order of fringe in the nominally 1 mm thick
samples reported here. Fringes represent regions of constant shear stress for
each fringe order whose values can be determined using Eq. (2-3).
2.3.4 Youngs modulus
From beam bending theory, the load P and the displacement A of the point
A are related by
)(
6222 baLab
LEIP A
=
(2-6)
where E is Young's modulus along the [001] direction. From the slope of
Figure 2.5 below the proportional limit, the Young's modulus is 1.8-1.9X1010 N/m2.
Similar results were obtained by Viehland and Li [15], where the Young's modulus
for PMN-30%PT single crystal along is reported to be 2X1010 N/m2, much
lower than the ~15X1010 N/m2 value reported for the direction or the
~7.5X1010 N/m2 value reported for polycrystalline material of the same chemistry.
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After bending, it was usual to observe residual fringes remaining in the beam
after release of the loading force, as shown in Figure 2.4(d). The higher the
loading force applied, the higher the fringe order remaining after unloading. It is
hypothesized that domain switching and perhaps local phase transformations lock
the stresses inside the beam by producing inhomogeneous strains which are
larger in those regions further from the neutral axis.
Experimentally, it was found that annealing can remove the residual fringes.
Annealing at 400 oC for one hour was sufficient to remove all residual fringes and
restore the initial fringe pattern. This means that any stress induced domain
switching or possible phase transformations caused by the bending experiments
are reversible.
2.4 Summary
Three-point bending experiments were performed on an unpoled
PMN-29%PT single crystal. The crystal was restored to its initial condition
between bending experiments by annealing for one hour at 400 oC. The
relationship between the load and the displacement and between the fringe order
and the fiber stress is linear below a proportional limit. Beyond that proportional
limit, stress induced domain switching (mechanical poling) can explain the
apparent yielding. The stress-optical coefficient of the unpoled PMN-29%PT is
approximately 104X 10-12
Pa-1
, higher than the values for materials used in
photoelastic stress analysis such as polycarbonate, 82X 10-12 Pa-1 [17]. Young's
modulus determined from the present experiment is 1.8 - 1.9X 10
10
N/m
2
. Sinceannealing removes all the residual fringes, the inhomogeneously distributed
domain switching responsible for the residual fringes must be reversible.
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