photoelastic and electro-optic effects: study of pmn-29%pt
TRANSCRIPT
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. She
enrolled 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 Master’s 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 Master’s 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.
“Photoelastic study of PMN-29%PT single crystals”, U.S. Navy Workshop on
Acoustic Transduction Materials and Devices, May 2006.
“Photoelastic study of PMN-29%PT single crystals”, U.S. Navy Workshop on
Acoustic Transduction Materials and Devices, May 2007.
“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. John C. “Jace” Harker and Mr. Stephen R.
Robinson. Jace 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 Jace and Stephen’s 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 John 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 Jill 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, John 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, John Harker’s 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 Photoelasticity…………7
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 Young’s 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 pattern………………………………………………63
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 Ⅰ Basic theory of optical properties of crystals…………………101
Appendix Ⅱ Basics of photoelasticity……………...………………..………104
Appendix Ⅲ Calibration of in-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 (10Dijc 10
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: ν ………………………………………………………………….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 in geology 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-up………………………………14 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 retardation…………………………………………………………………31
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…………………………………………………………33
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 is
calculated 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 illumination………………46
Figure 3.3 The 3 differently oriented samples relative to {001}-oriented pseudo-cubic axes. Arrows a and c represent compression along <100> direction; arrows b and d represent compression along <110> 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 <100> direction on sample 1. (b) Stress
intensity contour from ANSYS®…………………………………………..50 Figure 3.6 (a) Hertzian indentation along <110> direction on sample 2. (b) Stress
intensity contour from ANSYS®……………………………………………51
Figure 3.7 Hertzian indentation along <100> direction on sample 3 is shown in (a); Hertzian indentation along <110> 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, cσ and tσ 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 stress………………68 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/cm and 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 electric field 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 - piezoelectric single crystals and
photoelasticity
1.1 PMN-29%PT single crystals
In 1880, the famous brothers Pierre and Jacques Curie first 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 1950’s,
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
2
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(B1/3B’2/3)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
<111> and that of the T phase is <001>. 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 (μm scale), there are micro-domains (0.1 μm 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 μm 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]:
321321ricitypiezoelectconverseelasticity
kkijklijklij EesC −=σ (1-1)
321321ricitypiezoelectconverseelasticity
kkijklijklkl EdSs += σ (1-2)
321321typermittiviricitypiezoelectdirect
kikklikli EseD ε+= (1-3)
5
where ijσ , , and are stress, strain, electric field and electric
displacement tensors, respectively. , ,
kls kE iD
ijklC ijklS ikε , and are the elastic
constant tensor, elastic compliance tensor, the dielectric constants, the
piezoelectric 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
6
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
7
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
8
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.
9
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 σσλ
−= Ctn ) (1-4)
Here 1σ and 2σ are the maximum and minimum principal stresses, n is
fringe order, t is the sample’s thickness, and λ is the wavelength of the incident
light. C is the stress-optic coefficient which is a constant. From Eq. (1-4), we
have:
Ctn
2221 λσσ
=− (1-5)
Hence
Ctn
2maxλτ = (1-6)
The stress-optic coefficient C is useful to quantitatively analyze the internal
stresses, such as maxτ as a function of position for any fringe pattern if we know
the fringe order n.
Eq. (1-5) can also be rearranged to:
tnf
Ctn
==−λσσ 21 (1-7)
Here, we introduce another concept: the fringe-stress optical coefficient , where f
Cf /λ= 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 allowing f
10
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 <100> 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 processing’s 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,
11
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; Illustrations
redrawn 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 ω= (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 δ is generated. Let be the slow axis and be the
fast axis, we have:
2E 3E
)2
cos(cos
)2
cos(sin
3
2
δωθ
δωθ
+=
−=
tkE
tkE (1-9)
After the light passes out through the analyzer, only the y axis component of the
light is visible, so:
12
)sin(2
sin)2sin(
)2
cos()2
cos(cossin
sincos 324
tk
ttk
EEE
ωδθ
δωδωθθ
θθ
=
⎥⎦⎤
⎢⎣⎡ +−−=
−=
(1-10)
The intensity of the light we see thus dependent on the orientationθ , the time
and the phase difference t δ for each point in the image. Regions where
)2sin( θ or 2
sin δ or )sin( tω 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 δθap 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
δ 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 πθ = .
Use of a circular polariscope eliminates isoclinics. Two quarter-wave plates
are added to the plane polariscope with their axes at 450 and 1350 to one of the
polarizers 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 retardationδ , thus only isochromatics will be seen. Circular
13
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.
14
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
15
were polished down to 0.05 μm grit size. The dimension of the bar is 11X1.8X1
mm as 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.
16
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 of
three glass rods, which show a two-lobed fringe pattern. This will be further
discussed in Chapter 3.
17
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
Def
lect
ion
( μm
)
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 reference
to 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
18
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 the
first 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
19
first estimated. The apparent Young’s modulus along <100> 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.
20
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26
2 Photoelastic study using three-point bending technique
Abstract: 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<001> ~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
27
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.
28
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 σσλ
−= Ctn . (2-1)
Here 1σ and 2σ are the maximum and minimum principal stresses, is the
fringe order, and is the sample's thickness along the optical path.
n
t λ 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
n22
21 λσσ=
− , (2-2)
where both sides are divided by 2 to produce the form of the maximum shear
stress,
Ct
n2max
λτ = (2-3)
Once the stress-optical coefficient C is known for a given material, it can be
used to quantitatively evaluate maxτ 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 order
fringes 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
29
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)
30
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
31
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 μm 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
32
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
33
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 xσ on the outer surface at point A is the maximum tension stress
maxσ , known as the fiber stress, and yσ equals zero as a result of the free
surface boundary condition. Thus, the fringe order is directly proportional to fiber
stress xσ 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.
34
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.
35
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], leading
to 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,
36
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:
2maxmax
max6LthPab
IyM
=−=σ (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
maxσ calculated from Eq. (2-4)
represents the principal stress 1σ at point A since 2σ is zero there.
37
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
38
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=λtC (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
λ 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 Young’s modulus
From beam bending theory, the load P and the displacement Aδ of the point
A are related by
)(6
222 baLabLEIP A
−−=
δ (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 <001> is reported to be 2X1010 N/m2, much
lower than the ~15X1010 N/m2 value reported for the <111> direction or the
~7.5X1010 N/m2 value reported for polycrystalline material of the same chemistry.
39
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 1010 N/m2. Since
annealing removes all the residual fringes, the inhomogeneously distributed
domain switching responsible for the residual fringes must be reversible.
40
2.5 References 1. S.-E. Park and T. R. Shrout, “Ultrahigh strain and piezoelectric behavior in
relaxor based ferroelectric single crystals”, J. Appl. Phys. 82, No. 4, 1804, 1997.
2. Y. Yamashita, “Large electromechanical coupling factors in perovskite
binary material system”, Jpn. J. Appl. Phys. 33, 5328, 1994. 3. Z.-G. Ye, B. Noheda, M. Dong, D. Cox and G. Shirane, “Monoclinic phase
in the relaxor-Based piezoelectric/ferroelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 system”, Phys. Rev. B, 64, 184114, 2001.
4. C. S. Tu, F.-T. Wang, R. R. Chien, B. Hugo Schmidt and G. F. Tuthill,
“Electric-field effects of dielectric and optical properties in Pb(Mg1/3Nb2/3)0.65Ti0.35O3 crystal”, J. Appl. Phys. 97, No. 6, 064112, 2005.
5. X. Zhao, B. Fang, H. Cao, Y. Guo, and H Luo, “Dielectric and piezoelectric
performance of PMN-PT single crystals with compositions around the MPB: influence of composition, poling field and crystal orientation”, Materials Science and Engineering: B, 96, 254-262, 2002.
6. X. M. Wan, J. Wang, H. L. W. Chan, C. L. Choy, H. S. Luo and Z. W. Yin,
“Growth and optical properties of 0.62 Pb(Mg1/3Nb2/3)O3-0.38PbTiO3 single crystals by a modified Bridgman technique”, Journal of Crystal Growth, 263, 251, 2004.
7. X. J. Zheng, J. Y. Li and Y. C. Zhou, “X-ray diffraction measurement of
residual stress in PZT thin films prepared by pulsed laser deposition”, Acta Materialia, 52, 3313–3322, 2004.
8. Z. L. Yan, X. Yao and L. Y. Zhang, “Analysis of internal-stress-induced
phase transition by thermal treatment”, Journal of Ceramics International, 30, 1423, 2004.
9. H. C. Liang, Y. X. Pan, S. Zhao, G. M. Qin and K. K. Chin,
“Two-dimensional state of stress in a silicon wafer”, J. Appl. Phys. 71, No. 6, 2863-2870, 1992.
41
10. M. Lebeau, G. Majni, N. Paone and D. Rinaldi, “Photoelasticity for the investigation of internal stress in BGO scintillating crystals”, Nuclear Instruments & Methods in Physics Research A, 397, 317-322, 1997.
11. K. Higashida, M. Tanaka, E. Matsunaga and H. Hayashi, “Crack tip stress
fields revealed by infrared photoelasticity in silicon crystals”, Materials Science & Engineering A, 387-389, 377-380, 2004.
12. M. M. Frocht, Photoelasticity, VI, New York, John Wiley & Sons, Inc.
1941.
13. K. Ramesh, Digital Photoelasticity Advanced Techniques and Applications, Springer, 2000.
14. A. Sehirlioglu, D. A. Payne, and P. Han, “Thermal expansion of phase
transformations in (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3: evidence for preferred domain alignment in one of the (001) directions for melt-grown crystals”, Phys. Rev. B 72, 214110, 2005.
15. D. Viehland and J. F. Li, “Stress-induced phase transformations in
<001>-oriented Pb(Mg1/3Nb2/3)O3–PbTiO3 Crystals: bilinear coupling of ferroelastic strain and ferroelectric polarization”, Philosophical Magazine, 83, 53-59, 2003.
16. M. Abplanalp, D. Barosva, P. Bridenbaugh, J. Erhart, J. Fousek, P. Guter,
J. Nosek, and M. Sulc, “Ferroelectric domain structures in PZN-8/%PT single crystals studied by scanning force microscopy”, Solid State Commun. 119, 7, 2001.
17. G. D. Shyu, A. I. Isayev, and C. T. Li, “Photoviscoelastic behavior of
amorphous polymers during transition from the glassy to rubbery state”, J. Polym. Sci., Part B: Polym. Phys. 39, 2252, 2001.
42
3 Photoelastic study using Hertzian contact experiments
Abstract: Photoelastic effects in small single-crystal PMN-29%PT samples
were investigated by Hertzian contact experiments. The experiments were
performed on samples having various crystallographic orientations. The
resulting photoelastic fringe patterns were observed to be strongly dependent on
the orientation of the samples, showing that pseudo-cubic unpoled PMN-29%PT
single crystals have highly anisotropic elastic properties. Annealing above the
Curie temperature was found to completely remove the fringe patterns created by
the Hertzian indentation experiments. In addition, three-dimensional simulations
of the Hertzian contact experiments were performed using ANSYS®. The
simulations used cubic-form elastic constants calculated from data on poled
PMN-30%PT single crystals. The ANSYS® modeling results were comparable to
the experimentally observed fringe patterns, suggesting that the elastic properties
of pseudo-cubic unpoled PMN-PT single crystals may resemble those of
pseudo-tetragonal poled PMN-PT single crystals. This resemblance is
considered significant because of the uniqueness of the growth direction during
seeded crystal growth. ANSYS® provided a reliable method for qualitative
simulation of photoelastic effects in unpoled PMN-PT single crystals under
mechanical loading.
3.1 Introduction
After being poled in the pseudo-cubic [001] direction, domain-engineered
PMN-PT single crystals exhibit much greater electromechanical properties than
conventional piezoelectric ceramics. Because of this, bulk single crystals of
relaxor ferroelectric PMN-PT show great promise as a replacement for ceramics
in many applications, such as sensors, actuators, and motors, and they are the
43
focus of intense research activities [1-5]. However, PMN-PT single crystals are
mechanically softer than PMN-PT polycrystalline ceramics [6]. Therefore
PMN-PT single crystals more easily develop residual internal stresses induced by
preparation processes. In particular, when unpoled PMN-PT single crystals are
initially machined through cutting and polishing, the resulting damage can lead to
significant cracking problems later, reducing system performance and reliability.
To better understand and control the cracking problem, photoelastic
techniques have been applied to study the internal stresses of unpoled
PMN-29%PT single crystals. In a previous article (Chapter 2), we have reported
the photoelastic study of [001]-oriented unpoled PMN-29%PT single crystals
using three-point bending techniques [7]. The fringe pattern was observed to be
comparable to that of typical isotropic materials, so it was hypothesized that
unpoled PMN-PT single crystals might possess isotropic mechanical properties.
To verify this, in the current work photoelastic techniques were applied to
study the internal stress field of unpoled PMN-29%PT single crystals during
Hertzian contact experiments. We used several samples with different
crystallographic orientations to explore whether the elastic properties were
orientation dependent. The differently oriented samples showed very different
fringe patterns, especially when loaded from different directions. For example,
when loaded in the <100> direction, the samples displayed a two-lobed fringe
pattern, but when loaded in the <110> direction, the samples displayed a
one-lobed fringe pattern. This implies that the unpoled crystals possess highly
anisotropic elastic properties.
There are two possible explanations for the anisotropy of elastic properties.
First, Sehirlioglu, et al. found that, in unpoled multidomain PMN-PT single crystals,
the thermal expansion coefficient in one of the <100> directions is slightly larger
than in the other two <100> directions [8]. They attributed this effect to the
44
seeded crystal growth method, in which the growth direction exhibits unique
properties. The uniqueness of the growth direction may also explain the highly
anisotropic elastic properties of unpoled PMN-PT single crystals observed here.
Second, Zhang, et al. reported that the domain wall motion may contribute to the
effective elastic constants, especially affecting the response of PMN-PT in <110>
directions [9]. This may explain why unpoled PMN-PT single crystals show
different fringe patterns when loaded from <100> and <110> directions.
In parallel with the experiments, ANSYS® simulations were created to allow a
qualitative interpretation of the experimental fringe patterns. Using cubic-form
elastic properties calculated from pseudo-tetragonal poled PMN-30%PT single
crystals, the ANSYS® modeling results were comparable to the fringe patterns of
unpoled PMN-29%PT single crystals. This implies that ANSYS® can provide a
reliable method for qualitative simulation of unpoled PMN-PT single crystals
under mechanical loading, and that the elastic properties of unpoled PMN-PT
single crystals are close to those of poled PMN-PT crystals.
3.2 Experimental procedure
To perform the photoelastic experiments, a mini-loading frame using a
pneumatic cylinder was designed to apply the load. The mini-loading frame with
the sample inside was placed in a ZeissTM optical microscope for in situ
observation of fringes during loading. The frame was thin enough to fit under the
10X objective lens with the sample in focus. The mini-loading system was
calibrated so that the applied force could be obtained directly from the reading of
a pressure gauge, with accuracy of ±2.5%. A slice from a glass rod with 1 mm
radius was used to exert the Hertzian cylinder load. Figure 3.1 shows the top
view of the loading frame with the Hertzian contact experiment set-up. The
maximum force applied in each experiment was approximately 10 N, while
45
illustrations and simulations show typical loads around 4 N.
To explore the orientation effect, three samples were prepared as shown in
Figure 3.2: One with all six faces having {100} orientation; another one with all
four side faces having {110} orientation; the third one with two side faces {100}
and the other two side faces {110}. Figure 3.3 displays models for the three
samples in a cubic coordinate system, with the two loading directions <100> and
<110> labeled a, b, c and d corresponding to each experiment performed. The
same orientations were also used for ANSYS® modeling. The samples were
rectangular plates approximately 3 mm X 4 mm and 1 mm thick. The samples
were annealed at 400 oC for one hour before doing the experiments, in order to
remove any residual stresses and to reduce the initial birefringence.
Figure 3.1 (a) Top view of the loading frame. (b) Hertzian contact experimental
set-up as indicated by the arrow.
The optical microscope was configured as a circular polariscope, so only
isochromatic fringes were recorded. Fringe patterns observed in this way are
directly representative of the principal stress difference, according to
photoelasticity theory [10]. This will be discussed in detail below. Samples were
illuminated with green light of wavelength ~535 nm, obtained using a band-pass
filter.
46
Figure 3.2 Initial birefringence patterns of three samples in three different
orientations under circularly polarized illumination.
Figure 3.3 The 3 differently oriented samples relative to {001}-oriented pseudo-cubic axes. Arrows a and c represent compression along <100> direction; arrows b and
d represent compression along <110> direction.
3.3 FEM modeling methods
The ANSYS® simulation was simplified using several assumptions, based on
the experimental parameters. Phase transformations at the contact point stress
singularity were neglected. The low loading force (around 4 N) and the resulting
47
low stresses made phase transformations unlikely to occur in other regions [11].
The converse piezoelectric effect was also neglected, with the assumption that
the randomly oriented micro- and nanodomains in the unpoled sample would
produce zero net electric field on the macro-scale.
Therefore, with phase transformations and electric charge generation
neglected, the unpoled PMN-PT single crystals were treated as simple crystals.
Solid 5 (coupled-field eight-node brick) elements were used to model the PMN-PT
crystal, with the piezoelectric coefficient tensor set to zero. This was for later
convenience in developing the same finite element method (FEM) simulations for
poled PMN-PT single crystals. Solid 45 (eight-node brick) elements were used
to model the glass rod. Because of the above simplifying assumptions, only
elastic constants were required as inputs for the ANSYS® simulation. Although
the elastic, piezoelectric, dielectric, and even electro-optic properties of poled
PMN-PT single crystals have been studied and reported by several researchers
[12-15]; the elastic properties of unpoled PMN-PT single crystals are seldom, if
ever, reported. One possible reason for this is that samples must be poled for
the elastic properties to be measured using resonance methods. Consequently,
material properties reported by Cao et al. [16] for <001>-poled PMN-30%PT
single crystals (see Table 3.1) will be adapted in the present work to represent the
elastic properties of unpoled material.
Table 3.1: Elastic stiffness constants of PMN-30%PT single crystals
(10
Dijc
10 N/m2). Cao et al. [16]
Dc11 Dc12 Dc13 Dc33 Dc44 Dc66
11.8 10.4 9.5 17.4 7.8 6.6
48
At room temperature, the PMN-29%PT multidomain single crystals are in the
rhombohedral phase. In each lightly skewed cubic unit cell, the dipole is along
one of eight <111> directions, thus there are eight possible domains with different
spontaneous polarization within the multidomain system. After being poled
along the <100> direction, four degenerate polarization orientations remain.
Therefore, from a macroscopic view, poled PMN-PT single crystals may be
treated as pseudo-tetragonal crystals with 4mm symmetry [12-14]. Cao et al.
treated <001>-poled PMN-30%PT single crystals as pseudo-tetragonal [16] and
their elastic constants (Table 3.1) take a tetragonal form. Similarly, in this
chapter we assume that unpoled PMN-29%PT single crystals can be treated as
having a pseudo-cubic structure. This pseudo-cubic assumption has been used
previously in the literature for measurement of electro-optical properties [15].
For cubic symmetry, there are three distinct elastic constants (Table 3.2),
while for tetragonal symmetry; there are six (Table 3.1) [17]. For this
ANSYS
cubijc
tetijc
® simulation, approximate cubic-form elastic constants were created by
averaging as follows: , , and
. The resulting elastic stiffness constants, and the elastic
data (
),,( 33111111tettettetcub cccavgc = ),,( 13131212
tettettetcub cccavgc =
),,( 66444444tettettetcub cccavgc =
E ,ν ) used for soda-lime-silica glass rods [18], are listed in Table 3.2.
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: ν .
11c 12c 44c E ν
13.7 9.8 7.4 7.4 0.21
In ANSYS®, three-dimensional (3D) models were built with the following
49
symmetry-based constraints: Nodes on the bottom surface (X-Z plane) were
constrained in Y, nodes on the central surface (Y-Z plane) were constrained in X,
and nodes on the middle surface (parallel to X-Y plane) were constrained in Z.
The X and Y constraints are shown in Figure 3.4. The constraints on the middle
surface were omitted for clarity. To reduce the number of elements and
calculating time, the circular glass rod was modeled as a half-circular glass rod,
with the top surface nodes coupled to behave as one node. The dimensions of
the block sample were 3x4x1, and the half glass rod had radius of 1. These
proportions corresponded 1 to 1 with the experimental samples.
The model for compression along the [001] direction on a {001}-oriented
model is shown in Figure 3.4. The other three orientations shown in Figure 3.3
were built as ANSYS® models by rotating the first model's local coordinate system
relative to the global coordinate system. With this technique, the material
property input parameters did not need to be changed.
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 piezocrystal. Triangle
symbols represent displacement constraints. Top arrow indicates the force applied to all of the coupled nodes (nodes on top surface).
The ANSYS® simulation output was plotted as a stress intensity contour plot,
where the “stress intensity” is defined by ANSYS® as the largest difference
50
between the three principal stresses. We chose the stress intensity contour on
the front/back surface which is free of surface traction and therefore
representative of a two-dimensional (2D) stress state. For a 2D stress state, the
stress-optic law is expressed as [10]:
)( 21 σσλ
−= ChN (3-1)
where is the fringe order, h is the thickness of the material, and N λ is the
wavelength of the incident light. C is known as the relative stress-optic
coefficient. 1σ and 2σ are the maximum and minimum principal stresses.
Therefore, the stress intensity contour on the top surface should directly
correspond to the experimental fringe pattern.
3.4 Results and discussion
The value of the relative stress-optic coefficient is unknown for unpoled
PMN-PT single crystals with different orientations, and the exact value of the
elastic stiffness constants for unpoled PMN-29%PT are (as mentioned above)
also unknown. Therefore, we must perform a qualitative analysis of the
simulated and experimental results, as data required for a more precise
quantitative analysis is unavailable.
Figure 3.5 (a) Hertzian indentation along <100> direction on sample 1, and (b)
Stress intensity contour from ANSYS®.
51
Figure 3.5, Figure 3.6 and Figure 3.7 display experimental fringe patterns with
the comparable ANSYS® results for the three samples, respectively. Each fringe
picture was taken under a different loading force, but simulation results all used
the same simulated loading force 4 N.
Figure 3.6 (a) Hertzian indentation along <110> direction on sample 2, and (b)
Stress intensity contour from ANSYS®.
Figure 3.7 Hertzian indentation along <100> direction on sample 3 is shown in (a);
Hertzian indentation along <110> 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.
52
The pictured fringe patterns are not like the typical water-drop-shaped fringes
of isotropic materials [19]. The fringes are all different for the three samples with
two loading directions: On sample 1 (loaded in the <100> direction), the fringes
have a two-lobe shape; on sample 2 (loaded in the <110> direction), the fringes
have a narrow single-lobe shape. On sample 3, when loaded in the <100>
direction, the fringes have a wider two-lobe shape compared to sample 1; when
loaded in the <110> direction, the fringes have a wider single-lobe shape
compared to sample 2. The different fringe patterns imply that unpoled
PMN-29%PT single crystals are highly anisotropic materials.
The anisotropy ratio can be calculated as [20]:
1211
442cc
cA−
= (3-2)
An anisotropy ratio of 1 indicates a perfectly isotropic material. Using the
estimated elastic constants from Table 3.2, the anisotropy ratio is 3.79, which
indicates fairly high anisotropy.
The ANSYS® simulation results were scaled to have dimensions comparable
with the fringe pattern pictures. It is easy to see that the contours from ANSYS®
simulation strongly resemble the experimental fringe patterns. This suggests
that the estimated cubic-form elastic constants are reasonable approximations to
the true values. As the constants were calculated from poled PMN-PT single
crystal properties, one possible conclusion could be that unpoled PMN-29%PT
single crystals have tetragonal-like elastic properties, which may be explained by
the seeded crystal growth method [8].
Finally, residual fringes remained in the crystal after the loading force was
released, as shown in Figure 3.8(a). It seems obvious that plastic deformation
occurred in these samples. The deformation mechanism can be explained as
mechanical-loading-induced domain switching. After one hour annealing at
53
400 oC, the residual fringes were totally removed, as shown in Figure 3.8(b).
This means that this mechanically induced domain switching can be removed by
annealing.
Figure 3.8 Residual butterfly fringes are fully annealed out at 400 oC for one hour.
3.5 Conclusions
Photoelastic effects in small PMN-29%PT block samples were investigated by
Hertzian contact experiments. It was found that unpoled PMN-29%PT single
crystals display extremely different fringe patterns under Hertzian compression for
differently oriented samples, which implies that this crystal has extremely
anisotropic elastic properties. The anisotropy ratio may be as high as 3.90,
using cubic-form elastic constants calculated from data on poled PMN-30%PT
pseudo-tetragonal single crystals. Annealing above the Curie temperature can
completely remove the residual fringe patterns caused by
mechanical-loading-induced domain switching. Also, 3D simulations of the
Hertzian contact experiments were performed using ANSYS®. Stress intensity
contours from the ANSYS® models were strikingly similar to the experimentally
54
observed fringe patterns of corresponding experimental samples, showing that
the elastic properties of unpoled PMN-PT single crystals are close to the
properties of tetragonal poled PMN-PT single crystals. This similarity may be
caused by the seeded crystal growth method and the possible tetragonal
symmetry a special direction would imply. Also, ANSYS® is a reliable method for
qualitative simulation of the photoelastic response of unpoled PMN-PT single
crystals under mechanical loading.
3.6 References 1. S.-E. Park and T. R. Shrout, “Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals”, J. Appl. Phys. 82, No. 4, 1804, 1997. 2. Y. Yamashita, “Large electromechanical coupling factors in perovskite binary material system”, Jpn. J. Appl. Phys. 33, 5328, 1994.
3. S.-E. Park, P. Lopath, K. Shung, and T. R. Shrout, “Ultrasonic transducers using piezoelectric single crystal perovskites”, Ferroelectrics 2, 543, 1996.
4. S.-E. Park and T. R. Shrout, “Characteristics or relaxor-based piezoelectric single crystals for ultrasonic transducers”, IEEE Trans. UFFC 44, 1140-1147, 1997.
5. T. R. Shrout, Z. P. Chang, N. Kim, and S. Markgraf, “Dielectric behavior of single-crystals near the (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 morphotropic phase-boundary”, Ferroelectr. Lett., 12, 63, 1990.
6. D. Viehland and J. F. Li, “Young’s modulus and hysteretic losses of 0.7 Pb(Mg1/3Nb2/3)O3-0.3PbTi O3”, J. Appl. Phys. 94, 7719, 2003.
7. N. Di, and D. J. Quesnel, “Photoelastic effects in Pb(Mg1/3Nb2/3)O-29%PbTiO3 single crystals investigated by three-point bending technique”, J. Appl. Phys. 101, 043522, 2007.
55
8. A. Sehirlioglu, D. A. Payne, and P. Han, “Thermal expansion of phase transformations in (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3: evidence for preferred domain alignment in one of the (001) directions for melt-grown crystals”, Phys. Rev. B 72, 214110, 2005.
9. R. Zhang, W. Jiang, B. Jiang, and W. Cao, “Elastic, dielectric and piezoelectric coefficients of domain engineered 0.7Pb(Mg1/3Nb2/3)O-0.3PbTiO3 single crystal”, Fundamental Physics of Ferroelectrics 2002, 188-197, 2002.
10. K. Ramesh, Digital Photoelasticity Advanced Techniques and Applications, Springer, 2000.
11. D. Viehland and J. F. Li, “Stress-induced phase transformations in <001>-oriented Pb(Mg1/3Nb2/3)O3–PbTiO3 Crystals: bilinear coupling of ferroelastic strain and ferroelectric polarization”, Philosophical Magazine, 83, 53-59, 2003.
12. R. Zhang, B. Jiang and W. W. Cao, “Elastic, piezoelectric, and dielectric properties of multidomain 0.67Pb(Mg1/3Nb2/3)1-0.33TixO3 single crystals”, J. Appl. Phys. 90, 3471-3475, 2001.
13. R. Zhang, B. Jiang, and W. Cao, “Orientation dependence of piezoelectric properties of single domain 0.67Pb(Mg1/3Nb2/3)O3-0.3PbTiO3 crystals”, Appl. Phys. Lett. 82, 3737, 2003.
14. W. Jiang, R. Zhang, B. Jiang, and W. Cao, “Characterization of piezoelectric materials with large piezoelectric and electromechanical coupling coefficients”, Ultrasonics 41, 55-63, 2003.
15. X. Wan, H. Xu, T. He, D. Lin, and H. Luo, “Optical properties of tetragonal Pb(Mg1/3Nb2/3)O30.62-PbTiO30.38 single crystal”, J. Appl. Phys. 93, 4766, 2003.
16. H. Cao, V. H. Schmidt, R. Zhang, W. Cao, and H. Luo, “Elastic, piezoelectric, and dielectric properties of 0.58Pb(Mg1/3Nb2/3)O3 -0.42PbTiO3 single crystal”, J. Appl. Phys. 96, 549, 2004.
17. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1985
56
18. D. G. Holloway, The Physical Properties of Crystals, Oxford University Press, New York, 1985.
19. K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985.
20. J. P. Hirth and J. Lothe, Theory of Dislocations, McGraw Hill, 1968.
57
4 Photoelastic study using four-point bending technique
Abstract: Photoelastic effects in unpoled PMN-29%PT single crystals have
been investigated using four-point bending experiments. Photoelastic fringes in
the constant moment region were observed to be uniformly parallel to the edge of
the beam, corresponding to a state of pure bending. The fiber stress versus
fringe order plot is consistent with results reported earlier using three-point
bending experiments and in addition, the fringe-stress coefficient has been
evaluated for several load levels. Fringe-stress coefficients varied from 3.5X103
N/m to 5.5X103 N/m. From an initial maximum of 5.5X103 N/m, the fringe-stress
coefficient decreases monotonically and nearly linearly with increasing stress until
~45 MPa. At this point, the fringe-stress coefficient begins to rise. The initial
decrease of the fringe-stress coefficient may be explained by mechanical poling,
i.e. plastic yielding via domain switching and/or phase transformation. The
subsequent increase of the fringe-stress coefficient at higher loading levels can
be explained as the saturation of these mechanical poling effects. The increased
fringe-stress coefficient also correlates with an increase in the observed optical
transmittance of the crystal. This chapter will discuss implications of these
observed mechanical poling effects.
4.1 Introduction
Domain-engineered (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT) single crystals
possess extra-high piezoelectric coefficients, electromechanical coupling factor,
and field induced strain response compared with conventional piezoelectric
ceramics [1-6]. Thus relaxor ferroelectric PMN-PT single crystals are promising
to replace ceramics in high performance applications, such as higher sensitivity
ultrasonic transducers and large strain actuators. However, PMN-PT single
58
crystals are very fragile compared with ceramic materials of the same chemistry,
and more easily develop cracks. This will reduce the performance and reliability
of devices. Because cracks are initiated more easily in samples containing
residual stresses that are induced during crystal growth [7-8] and preparation
processes, it is necessary to monitor and control the residual stresses throughout
the entire preparation processes. Careful attention to cutting and polishing
processes can limit residual stresses and subsurface damage, leading to
substantial improvements in the mechanical robustness of piezoelectric single
crystals.
As is well known, photoelasticity is an effective and easily implemented
technique to measure the internal stresses in transparent materials. It offers
both quantitative determination and qualitative observation of the stress
distribution in a sample [9-11]. However, implementation of quantitative
photoelastic stress analysis requires that the fringe-stress coefficient f be
determined by direct experiment. Once this key parameter is known for a given
material, it directly links the observed fringe patterns to numerical values of the
internal stress. The fringe-stress coefficient is contained within the stress-optic
law which, for a two-dimensional stress state, can be written as:
)( 21 σσλ
−= Ctn . (4-1)
Here 1σ and 2σ are the maximum and minimum principal stresses at the
point of observation. is the fringe order, is the sample thickness through
which the light travels, and
n t
λ is the wavelength of the incident light. C is the
stress-optical coefficient. Eq. (4-1) can be rearranged to
,21 ntfn
tC==−
λσσ (4-2)
where the fringe-stress coefficient Cf /λ= represents the principal stress
difference necessary to produce a one fringe order change in a crystal of unit
59
thickness. The fringe-stress coefficient depends on the stress-optical coefficient
C of the material and the wavelength of the incident light. To allow easy
comparison and stress calculation via the fringe-stress coefficient, a single
standard wavelength should be used. While any color monochromatic light is
acceptable, ~535 nm green light was selected for current and previously reported
experiments. The engineering units of the fringe-stress coefficient are N/m.
In a previous paper (Chapter 2), we have reported the stress-optical
coefficient of [001]-oriented unpoled PMN-29%PT single crystals evaluated using
three-point bending technique [12]. However, in three-point bending
experiments, only the stress corresponding to the fringe at the bottom edge of the
beam can be calculated correctly using beam bending theory. Information from
other fringes is not useful because the stress state is a mixture of bending and
shear. In one experimental run with increasing load, for each load level we can
obtain only one datum. The stress-optical coefficient must then be estimated
from the linear region of the fringe order vs. fiber stress plot. As a result, from
one experimental run only one stress-optical coefficient can be calculated, and to
reduce the error, the same run needs to be repeated many times. Furthermore,
in three-point bending experiments, it is difficult to accurately measure how and
whether the stress-optical coefficient changes with increasing stress.
All of these problems can be solved in four-point bending experiments.
Four-point bending produces a pure bending region without any shear stress,
resulting in multiple parallel fringes that are aligned with the sample edges. For
each fringe, the corresponding tension (or compression) stress can be calculated
using beam bending theory. Since the sample contains multiple usable (pure
bending) fringes at each load, it is possible to plot multiple data points and
calculate the stress-optical coefficient once for each load level. Variations of the
stress-optical coefficient with increasing load can then be observed.
60
The purpose of this chapter is to measure and report the stress optical
coefficient and the fringe stress coefficient of PMN-29%PT single crystals. The
four point bending method allows us to further determine and report the stress
dependence of these coefficients. A secondary purpose of this chapter is to
introduce the hypothesis of mechanical poling to characterize and explain the
complex sequence of crystallographic changes induced by mechanical loads
which comprises two states in sequence: a first state where domains freely
switch towards preferred directions to form larger domains, and a second state
where mechanical loads induce a rhombohedral to monoclinic phase
transformation. Mechanical poling can induce permanent, yet reversible,
changes in the domain structure as evidenced by residual photoelastic fringes
which can be recovered by annealing. The two states and their transitions
correspond to observed changes of the values of the fringe stress coefficient.
4.2 Experimental procedure
An in situ loading frame using a pneumatic cylinder as shown in Figure 4.1
was built to allow us to perform photoelastic measurements on small beam
samples under a ZeissTM microscope. The frame was calibrated so that the
applied force could be known directly from the reading of an air pressure gauge to
within 2.5%. The loading frame was designed with a low profile to allow a 10X
objective lens to be used to observe the experiments. Four glass rods with 1 mm
radius were used to exert the loading force. A tilting bar (labeled “A” in Figure
4.1 (b)) was used to provide symmetrical loading. A removable alignment fixture
was designed to position the glass rods before each experiment so that the upper
and lower glass rod spans were reliably set at 5 mm and 15 mm respectively.
±
Two (001)-oriented unpoled PMN-29%PT single crystal beam samples were
prepared as shown in Figure 4.2. Beam 1 with dimensions 1mm x 1mm x 17mm
61
was previously subjected to three-point bending experiments, and was obtained
from H.C. Materials [12]; beam 2 with dimensions 1mm x 2.2mm x 18mm was
obtained from TRS Ceramics. During the experiments the samples were always
oriented so that light was transmitted through the 1 mm thick dimension. The
surfaces for light transmission were polished to optical quality using a graded
series of polishing papers down to 1um grit, then finished with
chemical-mechanical polishing using 0.05um colloidal silica. Before each
experiment, the beam samples were annealed at 400 oC for 1 hour to reduce any
residual fringes and minimize the initial fringe order. Figure 4.2 shows typical
fringe patterns after annealing; the fringe orders were close to 0th-order (black)
and typically less than 1st-order (rose) over the majority of the sample area.
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.
Figure 4.2 Beam 1 (a) and beam 2 (b) after 1 hour annealing at 400 oC.
62
Figure 4.3 Four-point bending layout. P is the loading force, cσ and tσ are compression
and tension stresses respectively. The diagram under the sample shows the absolute value of the bending moment.
The schematic four-point bending loading system is represented in Figure
4.3. The symmetrical loading scheme stresses the beam with a constant
bending moment 6PLM = , producing a pure bending stress state in the loading
region. The resulted axial stress ( )yσ has a uniform stress gradient through
the beam height:
( ) 3
2thPLy
IMyy ==σ , (4-3)
where is the moment of inertia of the beam, and and are the
thickness and the height of the beam, respectively.
12/3thI = t h
L is 15 mm in all of the
experiments performed. is the perpendicular distance to the neutral axis. In
the pure bending region,
y
( )yσ is the principal stress 1σ at , and the principal
stress
y
02 =σ at , allowing us to simplify Eq. (4-2) to: y
( ) ntfy =σ . (4-4)
Therefore, fringe-stress coefficient can be obtained from the slope of
stress versus fringe order plot.
f
63
To use Eq. (4-3) to calculate the stress for an observed fringe, the distance
between the fringe and the 0
y
th order fringe (neutral axis) needs to be measured.
During the experiments, pictures of fringe patterns were taken with the 10X
objective lens at several different load levels. The displacement of each
fringe could then be directly measured based on variations in light intensity, using
a method that is described in more detail below.
y
The experiments were performed using a dark field circular polariscope setup
to eliminate the isoclinic fringes and show only the isochromatic fringes, which
depend only on the internal principal stress difference. A narrow band-pass
green filter with central wavelength ~535 nm was used because this wavelength
produced a suitable number of fringes throughout the applied load range. Three
four-point bending experiments were performed on beam 1. Between the
experiments, beam 1 was heated to remove all of the residual fringes produced
by bending, allowing it to be reused; this “annealing” effect is discussed in more
detail below. Unfortunately, beam 2 was used only once, and was broken during
that experiment.
4.3 Results and discussion
4.3.1 Fringe Patterns
In the pure bending region, perfectly parallel fringes were observed,
comparable to those typical of isotropic materials [14]. As the applied load was
increased, fringes were generated from the top and bottom edges of the beam
and migrated towards the stress-free neutral axis where the 0th order fringe
always lies. Fringes formed in pairs (top and bottom), producing symmetric
patterns about the neutral axis. The orders of the fringes were thus typically
counted from the center as L，2,1 ±±=n . The fringes for each load level were
64
found to be distributed uniformly along the height of the beam, corresponding to
the linear variation of the principal stress shown in Figure 4.3.
Figure 4.4 (a) and (b) show photoelastic images of beam 1 obtained while a
load was applied and after the load was released, respectively. After the load
was released, there were usually many residual fringes left. These residual
fringes could be fully removed by 1 hour of heating at 400 oC. The beam
sample could then be reused. This implies that any optical or mechanical effects
caused by the bending experiments were reversible.
Figure 4.4 (a) 3.5 order of fringes and (b) 2 order of fringes left after the load is released.
4.3.2 Fiber stress versus fringe order
Figure 4.5 shows a plot of maximum fiber stress versus maximum fringe order
for a range of different loads. Maximum fiber stress (in tension side) was
calculated using Eq. (4-3) with 2/ty = , the same method was used in our
three-point bending work [12]. Included in this figure are data from three runs for
beam 1 and one run for beam 2. In addition, one run of data for beam 1 under
three-point bending is included for comparison. The results are highly consistent
and independent of any changes in bending scheme or beam dimension. The
slope in Figure 4.5 was obtained in terms of least-square fit based on data below
the optical yield stress of 20 MPa obtained from Figure 4.6, that is comparable to
the value obtained from Figure 2.6 in Chapter 2. The correlation coefficient is
0.95.
Below 20-30 MPa, a linear relationship can be seen between the fiber stress
65
and the fringe order. Above that limit some nonlinear behavior occurred. This
corresponds to results from our three-point bending work which demonstrated
mechanical yielding at higher load levels [12]. Using Eq. (4-4), from the slope of
the linear region of Figure 4.5 we calculated the fringe-stress coefficient to be
4.8X103 N/m, which equals to a stress-optical coefficient of 111.5X10-12 Pa-1.
Figure 4.5 Maximum fiber stress versus fringe order.
Figure 4.6 Maximum fiber stress versus fringe order with polynomial fit curve.
66
4.3.3 Fringe-stress coefficient
The calculated fringe-stress coefficient of 4.8X103 N/m is only an approximate
result for low level stresses or loads. As discussed previously, four-point
bending provides a method to obtain the fringe-stress coefficient more accurately
at different levels of applied load. This method is illustrated in Figure 4.7. For
each load level, a microscope image is captured, and then a vertical section is
taken through the image. Using ImageProTM software, the green pixel intensity
along the section can be plotted as a function of distance. The vertical
displacement between each fringe and the 0
y
th order fringe can then be measured
in ImageProTM with an error of +/- 3 μm.
Given the displacement of each fringe from the center line, the corresponding
stress for each fringe order can then be calculated using Eq. (4-3). The result for
each microscope image is a line of stress vs. fringe order data points representing
the behavior of the sample at that load level.
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).
Four of these data lines are plotted in Figure 4.8. The four load levels shown
had maximum fringe orders of 4, 7, 13, and 16, respectively. For each load the
data are essentially linear, and the slope of each line is the fringe-stress
coefficient at that load. It can be seen that with increasing load the fringe-stress
67
coefficient (slope) decreases first, and then increases. This trend can be seen
more clearly in Figure 4.9, which shows the fitted slopes (fringe-stress coefficient)
for many load levels and includes both beam 1 and beam 2 data sets. The
fringe-stress coefficient decreases as the applied load increases, and reaches a
minimum at a max fiber stress of 45-50 MPa (with fringe order 14). Beyond this
point the fringe-stress coefficient begins to increase. The fringe-stress
coefficient varied in the range of (3.5~5.5) X 103 N/m.
In our previous three-point bending experiments [12] we observed mechanical
yielding as the beam was bent. This was evidenced by nonlinearity in the
force-displacement data. We observed similar nonlinearity in the stress vs.
fringe order data. Further, the max fringe order was linearly proportional to beam
displacement over the entire range of loads during the three-point bending
experiments as mentioned in the preliminary three-point bending experiment.
Therefore, we hypothesize that changes in the fringe-stress coefficient must
correlate with changes in the Young's modulus of the material.
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.
68
Figure 4.9 Fringe stress coefficient versus maximum fiber stress.
With this interpretation, the optical four-point bending data presented in Figure
4.8 clearly shows initial yielding of the bar, followed by stiffening at stresses above
~50 MPa. This behavior may be the result of internal stress-induced domain
switching/phase transformation. In unpoled PMN-29%PT single crystals with
rhombohedral structure, there are eight possible domain orientations along one of
eight <111> directions. Under a static stress state, domains are free to reorient.
Domain rotation may involve transformations from rhombohedral to monoclinic B
or tetragonal phase occurring locally according to the local stress level. This
domain switching partially absorbs the mechanical work of the externally applied
load, reducing the influence of the load and causing the material to behave as if
the elastic constant has decreased (softening). As the applied load increases,
eventually most domains have reoriented and the effect is saturated, causing the
material to resist further deformation and effectively stiffen. Similar effects
resulting from uniform compression stress have been reported by Viehland, et al.
[15] and Viehland and Li [16].
69
4.3.4 Mechanical poling effect
During the experiments, when the applied load increased to the point where
the maximum fiber stress was ~45 MPa, the compression region began to
brighten. As shown in Figure 4.10(a), the brightness was relatively uniform
throughout the whole sample while the maximum fringe order was under 14.
However, once the fringe order went higher than 14 (Figure 4.10(b)), the
compression region became much brighter than other regions. This implies that
the transmission of that region was highly enhanced. As the applied load was
increased further, this brightening effect began to occur in the tension region as
well, and moved towards the neutral axis along with the fringes. Wan, et al. [7,
13] suggests that enhanced transmission is correlated with poling, so this effect
may indicate that the bright regions have become poled in some way. This
contributes to the hypothesis that PMN-PT undergoes mechanical poling - an
effect induced by mechanical loading.
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.
4.4 Conclusions
Photoelastic effects in unpoled PMN-29%PT single crystals were investigated
using four-point bending experiments. Perfectly parallel fringes were distributed
uniformly through the height of the beam, implying a pure bending stress state
provided by four-point bending experiments. The fiber stress vs. fringe order plot
70
is highly consistent with the result from previous three-point bending experiments.
Two methods were used to calculate the fringe-stress coefficient: first from the
slope of the linear region in Figure 4.5, was found to be 4.8X10f 3 N/m; Second,
the fringe-stress coefficient was measured from each birefringence image for
multiple loading levels, and varied in the range (3.5~5.5)X103 N/m depending on
the load. As shown in Figure 4.9, with increasing load, the fringe-stress
coefficient decreases to a minimum and then increases. This effect is
interpreted as evidence of internal domain switching and phase transformation, i.e.
a mechanical poling process. Increased optical transmission in high-stress
regions was also observed, a behavior which supports the hypothesis that a
poling process occurs.
Finally, we observed that after the samples were unloaded, many fringes and
the higher optical transmission effects remained. We found that heating above
the Curie temperature restored the samples to their initial optical state. This
implies that all of the mechanical poling effects that we observed are reversible.
4.5 References 1. S.-E. Park and T. R. Shrout, “Ultrahigh strain and piezoelectric behavior in
relaxor based ferroelectric single crystals”, J. Appl. Phys. 82, No. 4, 1804, 1997.
2. Y. Yamashita, “Large electromechanical coupling factors in perovskite
binary material system”, Jpn. J. Appl. Phys. 33, 5328, 1994.
3. Z G Ye, B Noheda, M Dong, D Cox and G Shirane, “Monoclinic phase in the relaxor-based piezo-/ferroelectric Pb(Mg1/3Nb2/3)O3-29%PbTiO3 system”, Physics Review B, 64, 184114, 2001.
4. X. Zhao, B. Fang, H. Cao, Y. Guo and H. Luo, “Dielectric and piezoelectric
performance of PMN-PT single crystals with compositions around the MPB: influence of composition, poling field and crystal orientation”, Materials Science and Engineering: B, 96, 254, 2002.
71
5. R. Zhang, B. Jiang and W. Cao, “Elastic, piezoelectric, and dielectric properties of multidomain .67Pb(Mg1/3Nb2/3)O3 -0.33PbTiO3 single crystals”, J. Appl. Phys. 90, 3471, 2001.
6. M. Dong and Z. G. Ye, “High-temperature solution growth and
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and optical properties of 0.62 Pb(Mg1/3Nb2/3)O3-0.32 PbTiO3 single crystals by a modified Bridgman technique”, Journal of Crystal Growth, 263, 251, 2004.
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5 Electric field induced optical effects in PMN-29%PT single
crystals
Abstract: Stress fields induced by electric fields in unpoled PMN-29%PT are
compared with those induced by mechanical loading using optical birefringence
visualization. Photoelastic experiments explore the differing effects of square
waveform cyclic electric field and DC field, as well as cyclic electric field frequency
effects. Experiments are performed on differently oriented samples to examine
the effects caused when electric fields are applied in different crystallographic
directions for comparison. The causes of crack initiation are examined.
Results show that it is easy for crystals to be locally poled around their edges by
cyclic electric field, and these edge regions are highly susceptible to crack
formation under a continuously applied cyclic electric field. Finally, the
mechanisms of mechanical poling and electrical poling are discussed.
5.1 Introduction
Piezoelectric single crystals possess stronger dielectric and piezoelectric
properties than those of conventional piezoelectric ceramics. They have
promising potential to replace ceramics and can be used in wide-ranging
applications, like transformers, actuators or sensors [1-6]. The working
condition of those devices is either under high magnitude cyclic or direct electric
field, under mechanical loading, or sometimes a mix of electrical loading and
mechanical loading. Unfortunately, piezoelectric single crystals are very
susceptible to fracture. This is why single crystals have not yet seen wide
implementations. They are elastically softer than ceramics and easily develop
cracks under even low levels of stress concentration. As is well known for
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piezoelectric materials, stress can be induced by both mechanical loading and
electrical loading, and there are numerous of processes that can cause stresses
in these single crystals. For example, during growth, an inhomogeneous
temperature field exists which may induce residual stress inside the crystals [7-8].
Then, during the machining process, mechanical loading causes increased
residual stresses inside the crystals, especially the surfaces, edges and corners.
These machining processes include cutting, grinding, and polishing. Finally,
crystals are normally used under either high frequency cyclic electric field or
under pulsed DC electric field loading. These working conditions can also
increase the internal stresses in the single crystals. Given the relatively high
residual stresses induced previously from crystal growth and machining process,
it is very easy for the crystals to fracture and finally fail to function. To strengthen
the crystals, we can develop new polishing techniques, and also apply annealing
to crystals to reduce the residual stresses. It is not clear if this will be sufficient
to allow crystals with small residual stresses to survive harsh working conditions
and obtain a long useful service life. Thus it is indispensable to investigate the
electric field effects. In this chapter, we will focus on studying electric field
induced internal stress in unpoled PMN-29%PT single crystals using optical
birefringence techniques.
Several researchers have already studied the cyclic electric field effects in
ferroelectric ceramics or piezoelectric single crystals such as Lynch et al. [9-10]
and other researchers [11-14]. Topics include domain switching, phase
transformation, microcracking, and fracture. The observation and study are
normally conducted through means such as TEM, dielectric measurements, optic
microscopy, and other similar techniques. Crack growth is directly observed
under an optical microscope, and microcrack growth is observed under TEM.
Phase transformation is studied by measuring changes in dielectric properties.
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These studies have helped gain increased understanding of what is going on
when the piezoelectric materials are under electrical loading. However, if we
can evaluate the internal stress state of the materials during the electrical loading,
we can better understand crack initiation and growth conditions. Fortunately,
unpoled PMN-PT single crystals can be polished to be transparent and show
colorful birefringence. Previously, using well-known photoelastic technique, we
already studied the internal stress fields of PMN-PT single crystals under
mechanical loading such as bending and Hertzian indentation [15-16]. Similarly,
we can use birefringence techniques to study the internal strain of PMN-PT single
crystals induced by electrical loading.
When electric field is applied to PMN-29%PT, several electro-optic effects are
induced. First, all transparent solids become birefringent when subjected to an
electric field, this phenomenon being known as the Kerr effect [17]. The
mechanism behind this phenomenon is the same as that of photoelasticity: the
electric field changes the index of refraction of the materials in the solid state. The
difference between the photoelastic phenomenon and the Kerr effect is that the
photoelastic birefringence is linearly proportional to the mechanical stress, while
the Kerr electro-optic birefringence is proportional to the square of the electric
field. That is why the Kerr effect is also called the quadratic electro-optic effect.
Second, there is a phenomenon known as the Pockels effect, which applies only
to crystals that are noncentrosymmetric such as piezoelectric crystals [17]. The
external electric field alters the dipole moment of the single crystals, causing the
Pockels effect. Similar to the photoelastic effect, birefringence of the Pockels
effect is linearly proportional to the electric field so that the Pockels effect is also
known as the linear electro-optic effect. Third, coming with the Pockels effect
and owing to the piezoelectric nature of PMN-29%PT single crystals, the external
electric field will induce strain and in turn cause a photoelastic effect. Thus, in
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general, the observed birefringence is induced by the combination of three effects,
one quadratic electro-optic effect and two linear electro-optic effects. Compared
to the linear electro-optic effects, the nonlinear Kerr effect is much weaker than
the Pockels effect in piezoelectric crystals, though it responds quickly to changes
in electric field, even for frequencies as high as 10 GHz. In our experiments, the
applied field is relatively small (under 3 KV/cm), so the response is primarily
induced by the linear electro-optic effects, which directly affects internal domain
switching and internal strain. This strain is constrained by the surrounding
material, resulting in a stress field. Therefore, the electric field induced
birefringence represents the internal stress field. This chapter is aimed at
gaining a fundamental understanding of the internal stress fields induced by
electric fields. Birefringence is used as a means to visualize the internal stress
state.
Unpoled PMN-PT single crystals were used in our experiments for two
reasons: first, unpoled PMN-PT single crystals are transparent while poled
PMN-PT single crystals are nearly opaque under microscopy; second, the
electrical poling process may induce residual stress, which is an important topic
studied here. As we know, even unpoled PMN-PT single crystals possess
anisotropic mechanical and optical properties [16]. To explore the orientation
dependence of electro-optic effects, several samples of different orientations
were used in this study. “Hertzian contact” electrical loading (the electrical
equivalent of mechanical point loading) and electrical poling experiments were
performed. Fringe patterns induced from “Hertzian contact” electrical loading
are highly orientation dependent, and are comparable with those induced by
Hertzian contact mechanical loading. In electrical poling experiments, cyclic
electric fields were found to induce cracks much more easily than DC electric
fields. The possible reason is that cyclic electric field can easily pole the crystals
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locally, thus degrading the crystal properties by introducing boundaries between
poled and unpoled regions where large strain gradients can cause cracking .
5.2 Experimental procedure
Figure 5.1 Initial birefringence patterns of four differently oriented samples under
circularly polarized illumination after 1 hour annealing at 400 oC. Single-crystal samples of unpoled PMN-29%PT in four different orientations
were prepared as shown in Figure 5.1. Samples were obtained from H.C.
Materials. Beam 1 with dimensions of 9mm x 1.7mm x1mm was “Hertzian
contact” electrically loaded. Beam 2 with dimensions of 8mm x 1.8mm X1mm
was used in electrical poling experiments. Both of these beams are
{100}-oriented. Figure 5.1(c) and (d) show sample 3 and sample 4 respectively,
which are used in “Hertzian contact” electrical loading experiments. Sample 3
has two side faces {100} and the other two side faces {110}, while sample 4 has
all four side faces having {110} orientation. When observed under the microscope,
light was always transmitted through the 1 mm thickness of all of the samples.
The surfaces for light transmission were polished to be optically transparent using
a sequence of graded abrasives with average grit size down to 0.05 μm. Before
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experiments, all samples were annealed at 400 oC for one hour to reduce residual
stresses.
As mentioned above, two different experiments have been performed to
explore the electric field loading effects. The first one is the “Hertzian contact”
electrical loading experiment. It was designed to imitate the mechanical
Hertzian contact experiments to explore whether an equivalent fringe pattern will
emerge to that observed in mechanical loading of unpoled PMN-29%PT single
crystals. An equivalent fringe pattern would allow a comparison of electrical and
mechanical loading in these crystals. The other experiment is an electrical
poling experiment.
Figure 5.2 (a) Overview of the in situ electrical loading frame and (b) Top view of
“Hertzian contact” electrical loading set-up. The “Hertzian contact” electrical loading experiment arrangement is shown in
Figure 5.2(b). The electrical poling experiments were performed with the same
experimental setup except the rod electrode was replaced by a block electrode.
The white loading frame was made using high density polyethylene. Different
size metal blocks and a 1mm radius metal rod were used as electrodes. In
“Hertzian contact” electrical loading experiments, the rod electrode was used to
apply drive voltage and block electrodes to provide ground. Furthermore, the
sample surface contacting the block electrode was coated with gold. In the
electrical poling experiment, both electrically contacted surfaces were coated with
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gold to provide a relatively uniform electric field through the samples. All
experiments are conducted at room temperature. The polyethylene load frame
allows a 10X lens to be used to observe small cracks as they initiate in the
crystals. A circular polariscope employing a green filter was used to eliminate
the isoclinic fringes and thus obtain well-defined isochromatic fringes in order to
characterize the electric field induced strains.
In the experiments, we applied DC and square waveform cyclic (AC) electric
field to the crystals with varying magnitude of the electric field and varying
frequency of the square waveform AC. Unless otherwise noted, voltage was
applied for a fixed 2 minute period for each experiment to allow simple
comparison of the resulting fringes. This is because the fringes will grow bigger
with time that the sample is exposed to the test voltage. We found the fringe
pattern is different for DC vs. square waveform AC electric field, and also, the
fringe pattern is dependent on the orientation of the sample. For a given test
time, the size of the fringe pattern is proportional to the magnitude of the electric
field; it is also related to the frequency of the AC field. Between the experiments,
samples are annealed to remove all the residual fringes so that they can be used
again.
5.3 Results and discussions
5.3.1 “Hertzian Contact” electric field loading effects
Hertzian contact loading experiments were conducted to explore
birefringence of unpoled PMN-29%PT single crystals under an electrical point
loading conditions. The experimental set-up, as shown in Figure 5.2(b), is
almost the same as the mechanical Hertzian contact loading set-up, except the
driving force: one is mechanical force, the other is electrical loading. Since
piezoelectric materials respond to both mechanical and electrical loading, to do
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the same type experiments, we can compare the different driving force effects on
strain generation by observing the birefringence with optical methods. These
types of experiments are also of practical importance because they characterize
single point electrical loading effects which may happen in several in-service
cases. During electrical poling, if the electrical-plating materials are non-uniform
or the surfaces are rough, the crystals will have an applied electrical Hertzian
contact loading. Also, during service, when cracks are generated there will be
electrical Hertzian contact loading near the cracks. This is the electrical
equivalent at a stress concentration representing an electric field concentration.
All pictures shown for electrical loading represent the residual fringes present
after the electric field is removed. Because there was almost no motion or
relaxation of the fringe pattern, these photos characterized the electrically loaded
state as well. In short, all the fringes stayed where they had grown to. This is a
substantial difference with the residual fringes observed under mechanical
loading, which relaxed significantly after the load was released.
In “Hertzian contact” electrical loading experiments, separate experiments
were performed where the DC electric field was applied in opposite directions to
examine if the internal domains have a preference for the DC field direction.
Figure 5.3 show the birefringence obtained from beam 1. There are two
experiments shown here: Figure 5.3(a), (c) and (d) are from one run of the
experiment, while Figure 5.3(b) is from another run of the experiment. In the first
experiment, first a -2.3 KV/cm electric field was applied for 2 minutes, then the
electric field was reversed to +2.3 KV/cm and applied for another 2 minutes
resulting in fringes as shown in Figure 5.3(c). Figure 5.3(d) shows fringe pattern
after additional 2 minutes of applying +2.3 KV/cm. For comparison, Figure 5.3(b)
was taken in another experiment after +2.3 KV/cm had been applied for 2
minutes.
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It is obvious that DC field induces two-lobed fringe patterns of very similar
shape regardless of the electric field direction. However, there are more fringes
from +2.3 KV/cm than those from -2.3 KV/cm field. This difference seems likely
to be induced by the initial internal domains switching preference.
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.
Figure 5.3(c) shows that after the electric field was reversed, the fringe pattern
immediately develop two bumps around the region contacting the rod electrode
making the two-lobed shape more like butterfly wings. When voltage is reversed
(compare Figure 5.3(c) and 5.3(a)), parts of the fringes seem to pull back toward
the origin, which upper portions of the fringe moved outwards, resulting in the
butterfly wing shape. Figure 5.3(d) shows that 2 more minutes of reversed DC
voltage grew the fringes larger and maintained the butterfly wing shape. It was
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observed that the spot under the rod electrode became extremely bright. This
enhanced transmission implies the region has been electric poled with a very
large domain size [18-19]. It is thought that larger domains with same
polarization of rhombohedral phase or monoclinic phase are formed around the
region so that the light scattering losses caused by discontinuous refractive index
at the boundaries of different domains is highly reduced [20].
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/cm and 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. Figure 5.4 shows the comparison of electric field induced fringe pattern with
those induced by Hertzian contact mechanical loading. For a sample of the
same orientation, it is evident the fringe patterns are similar although they are not
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exactly the same. The {100}-oriented samples generate “two-lobed” fringe
pattern under either mechanical loading or electrical loading when loaded from
[100] direction. When loaded from [110] direction, “single-lobed” fringe patterns
are formed as shown in Figure 5.4(c)-(f), independent of which direction we view
the pattern from, [110] or [001].
In Di et al. [16], the orientation and loading direction dependence of fringe
patterns are explained as results from the inherent elastic property of the material
itself. Fringe patterns resulting from mechanical loading were the same as those
fringe patterns which were obtained using ANSYS simulations; this verified that
the elastic properties of unpoled PMN-PT single crystals are highly anisotropic.
To conduct ANSYS simulations of these electrical loading experiments,
piezoelectric strain coefficients need to be considered along with the elastic
properties. However, the samples are unpoled PMN-PT crystals; any
piezoelectric properties would be very weak and not constant for different
samples. Furthermore, measuring the properties of samples uses the
piezoelectric response to induce the strains so that the conventional
measurement methods are not available for unpoled piezoelectric material [21].
Therefore, there is no piezoelectric strain coefficient data reported for unpoled
PMN-PT single crystals. Thus the comparable fringe patterns observed between
electrical and mechanical loading must indicate that the elastic properties affect
the anisotropic response primarily through one of the linear electric-optical effects:
the electric field induced strain conversely causes photoelastic effects.
On the other hand, the difference between the electrical and mechanical
fringe patterns, such as the two lobes growing along different angles, must be
caused by the Pockels effect directly inducing a switching of dipole moments by
the electric field. Furthermore, the DC field induced much larger fringe size
without generating cracks than mechanical loading did, and residual fringes
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induced by the DC field didn’t relax much, while those induced by mechanical
loading did. This is also consistent with the Pockels effect: once a dipole
moment is switched, it is hard for them to switch back.
Next the cyclic electric field frequency effect will be discussed. During the
experiments, we observed that as the electric field is cycled at low frequency; the
fringes grow and shrink, giving a ratchet-like incremental growth. When the
frequency is raised to over 120 Hz, the fringes appear to grow at a constant rate.
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 electric field is 2.3 KV/cm for beam 1 and 1.8 KV/cm for both sample 3 and 4.
As shown in Figure 5.5, the top three pictures show fringes induced by a
square waveform voltage with a frequency of 0.5 Hz; the bottom three pictures
show fringes from squarewave voltage with a frequency of 500 Hz. No matter
what orientation the sample is, the low frequency squarewave voltage induced
large fringes while the high frequency voltage only induced small fringes.
Furthermore, if compared with Figure 5.4 (a), (c) and (e), under the same
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magnitude of electric field, obviously the DC field was found to have deeper
penetration than the cyclic electric field. There is one possible explanation for
this phenomenon: as observed in reversed DC experiments, the reversed voltage
didn’t grow the fringe pattern immediately; instead it worked on reversing the
polarization directions which originally aligned to an opposite direction.
Therefore the fringe shape was changed to develop “bumps” near the electrode
side, however the size of the fringe didn’t increase much as shown in Figure 5.3
(a) and (c). In other words, a reversal voltage needs time to grow fringes.
When the frequency of cycled electric field is higher than 0.5 Hz, which means
there is less than one second for positive or negative fields to act on the domains.
Apparently, there is not enough time for fringes to grow large. The small fringes
and bright regions under the electrode imply that electrical energy is mainly
stored close to the surface, and poles (orients) the domains locally, as shown in
Figure 5.5 (d), (e) and (f).
Furthermore, through the linear Pockels effect, a cyclic electric field tends to
switch the internal domain’s polarization as fast as its own frequency. This is
actually cyclic straining process similar to a fatigue process. It is reasonable to
assure that the elastic properties of the “bright” regions are degraded so that
continually applying a cyclic electric field to these regions may easily initiate
cracks. The hypothesis of the degradation of properties in the “bright” region will
be experimentally examined in the following section.
5.3.2 Electrical poling effects
Beam 2 and sample 4 were used in electrical poling experiments. In these
experiments, both top and bottom surfaces were coated with gold. The
experiments were performed with incremental DC voltage starting with ~+1 KV/cm
and increased to ~+6 KV/cm. For comparison, square wave form voltage was
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applied to the samples after incremental DC field was applied.
As shown in Figure 5.6, beam 2 was electrically poled by applying incremental
DC voltage at room temperature for one minute intervals. Two sequential stages
were observed in the poling process: first electric field driven fringes growing
inwards from the top and bottom surfaces. This phenomenon only occurred at
very low electric field at a range of (1.1-1.3) KV/cm. Second, when the electric
field was above 1.3 KV/cm, most of the internal bulk region was slightly poled,
appearing dark and cloudy, except the two edge regions near the coated surfaces
which are fully poled and begin to become very bright. The higher the electric
field was, the stronger the cloudiness was and the brighter the edges became.
Figure 5.6 From top to bottom, beam 2 is electrical poled with incremental DC voltage.
The experiment set-up is with two block electrodes; both the top and bottom surfaces are plated with gold.
These poling effects are due to the internal domain switching and phase
transformation. We know that in unpoled {100}-oriented PMN-PT single crystals,
there are eight possible polarization directions along <111> directions. When a
low level electric field is applied, four of the polarization directions that are
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contrary to the electric field direction will be removed by switching states so that
only the other four dipole directions remain. The four-dipole domain structure is
called an engineered domain structure. As the magnitude of electric field is
increased, those four polarizations will first switch from <111> direction to <110>
direction, then to <100> direction which is parallel to the electric field direction.
Correspondingly, the phase transforms from rhombohedral to monoclinic to
tetragonal (or orthorhombic) phase. This phase transition path of R →M → T is
reported by Bai and Li using XRD and dielectric properties methods respectively
[22, 23]. Furthermore, in reference [22], it is found that at room temperature, it is
easy for the phase transition from R → M, however, it is difficult for the M → T to
occur because there is a high energy-barrier between the M and T phases. The
dark cloud internal region may be thought of as a mixture of both R and M phases.
Internal boundaries contribute to the cloudiness. Small domains are aligning
themselves to form larger domains. This will induce lots of strain in between
those domains. Furthermore, refractive index discontinuities arise due to the
variation of the orientation of successive domains. This results in considerably
higher light scattering than in the thermally depoled optically-uniform state,
causing the whole area to be less transparent. These are similar phenomena to
those explained with the light scattering theory of piezoelectric ceramics by [13,
24]. For the fully-poled regions which have formed a large domain along the
edge, the transmission of light will be highly promoted, so they become extremely
bright [18, 19]. The cloudiness is only getting darker and darker under
incremental DC field without becoming bright, because the M → T phase
transition is difficult.
In another experiment, beam 2 was first poled with incremental DC field up to
3.4 KV/cm, and then with a low frequency squarewave cyclic electric field of 2.8
KV/cm. The resulting fringes are shown in Figure 5.7(a). It appeared that the low
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frequency cyclic electric field immediately poled beam 2: generating dark strips
parallel with the electric field in the middle region and extremely bright areas along
the edge. The bright region is much larger than that formed under DC field. In
addition, cracks were initiated parallel to the electric field from under the block
electrode where displacements are constrained, and crack growth along the
edges was observed. The edge region is extremely bright because domain
switching saturation (large number of small domains which have consolidated into
fewer larger domains) and phase transformation which occurred there.
Consequently, the bright region will lose much of its elastic flexibility and the
elastic constant is increased. Elastic constant of [001]-oriented PMN-32PT
single crystal is seen to increase with increasing electric field reported by Viehland
[25]. Furthermore, a continually applied cyclic electric field has a strain switching
fatigue effect, which will generate cracks much more easily than a DC field, as we
describe below.
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.
To verify that bright region has lost much of its elastic flexibility, Hertzian
mechanical loading experiments using the parallel clamp fixture were performed
on sample 4 which was first poled with DC field. The experiment results are
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shown in Figure 5.8. In Figure 5.8(a), with a DC electric field of 2.5 KV/cm
applied, semi-circular fringes developed in the bright region near both the upper
and lower electrodes, which are very different from those shown in Figure 5.4(e).
In Figure 5.8(b), a glass rod was used to exert mechanical loading directly on the
bright region. Cracks developed along the loading direction without generating
any fringes, implying that the mechanical properties of the bright region have
changed and can no longer produce the strain responsible for the mechanical
fringes. It is evident that the bright region has different optical and elastic/plastic
properties than the original material.
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.
5.3.3 Mechanical poling versus electrical poling
As discussed above, cyclic electric field was shown to pole PMN-PT single
crystals more easily than DC field. The elastic properties of bright regions poled
by electric field are degraded leading to a loss of resistance to cracking which was
verified experimentally. These bright regions are very fragile to cracks if
subjected to electric field or mechanical loading. There is yet another poling
effect induced by external mechanical loading called mechanical poling.
Compared to electrical poling, mechanical poling is less likely to produce bright
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regions because of the different poling mechanism. However, large strain testing
in bending has demonstrated a bright region developing as a result of purely
mechanical loading. Let us examine this in detail.
Figure 5.9 Mechanical poling and electrical poling representation.
For <001>-oriented PMN-PT single crystals, mechanical poling is a
two-dimensional effect compared to electrical poling, which is a one-dimensional
effect, as shown in Figure 5.9. The original eight possible polarization directions
will switch to find themselves in a single plane under mechanical loading. Under
compression, the plane is perpendicular to the stress direction, while under
tension, the plane is parallel to the stress direction. It is obvious that electrical
poling can align domain polarization in one direction parallel to the electric field.
Therefore, there are at least four polarization directions left after mechanical
poling, and only one polarization direction left after thorough electrical poling.
We already know that the mechanical property and the optical property are highly
dependent on the internal domain structure. After poling, large numbers of small
domains are aligned and form fewer large domains. During this process, the
elastic modulus is stiffened. The fewer domains of different orientations which
are left, the more stiffened the elastic modulus becomes. This means that after
electrical poling, the elastic stiffness is higher than that after mechanical poling,
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and it is higher because the density of the domains still available to switch to
accommodate strain gradients is reduced. Thus electric field loading which
gives rise to a more completely poled structure may initiate cracks more easily
than mechanical loading. Essentially, electrical poling eliminates the domain
switching capability of the material that is responsible for its apparent plastic
deformation capability so that poled crystals are more prone to crack.
5.4 Conclusions
Birefringence induced by electric fields on unpoled PMN-29%PT single
crystals was investigated using optical techniques. There are a total of three
electro-optic effects: the Kerr quadratic effect, the Pockels linear effect and the
piezoelectric strain-induced photoelastic effect. For piezoelectric materials, the
observed birefringence is primarily induced by the later two linear electric-optical
effects. “Hertzian contact” electrical loading experiments and electrical poling
experiments were performed. In Hertzian electrical loading experiments, the DC
field induced fringe pattern is highly orientation-dependent, which is comparable
to the Hertzian mechanical loading fringe pattern. Cyclic electric fields with high
frequency induced fringes which are much smaller than those induced by a DC
field or a low frequency cyclic electric field. However, a cyclic electric field,
regardless of frequency, may easily pole PMN-PT single crystals, especially the
local region close to the conducting coated surfaces, thus making these areas
likely to generate cracks. These phenomena are observed in electrical poling
experiments, as well. The explanation is that the time for each sign of field to act
to orient domains becomes extremely short with increment of the frequency of
cyclic electric field, so there is not enough time for fringe to grow big. Also
domain switching follows the frequency of the electric field, which reduces the
ability of the material to use domain switching to accommodate strain gradients.
92
This, in effect, embrittles the material. To verify this, “Hertzian contact”
mechanical loading experiments were performed on a {110}-{110} oriented
sample that was first electrically poled. Cracks were observed to form
immediately along the loading direction under the loading cylinder: This implies
that the bright region poled by the electric field has less elastic flexibility because
the internal domains are fully-poled and, in effect, form a larger single domain with
one polarization direction. Reduction of the number of multiple domains
degrades the ability of the PMN-PT single crystals to accommodate strain and
strain gradients, reducing the toughness of the material. Finally, the
mechanisms of mechanical poling and electrical poling were discussed.
5.5 References 1. S.-E. Park and T. R. Shrout, “Ultrahigh strain and piezoelectric behavior in
relaxor based ferroelectric single crystals”, J. Appl. Phys. 82, No. 4, 1804, 1997.
2. Y. Yamashita, “Large electromechanical coupling factors in perovskite
binary material system”, Jpn. J. Appl. Phys. 33, 5328, 1994.
3. Z.-G. Ye, B. Noheda, M. Dong, D. Cox and G. Shirane, “Monoclinic phase in the relaxor-Based piezoelectric/ferroelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 system”, Phys. Rev. B, 64, 184114, 2001.
4. X. Zhao, B. Fang, H. Cao, Y. Guo, and H Luo, “Dielectric and piezoelectric
performance of PMN-PT single crystals with compositions around the MPB: influence of composition, poling field and crystal orientation”, Materials Science and Engineering: B, 96, 254-262, 2002.
5. R. Zhang, B. Jiang and W. W. Cao, “Elastic, piezoelectric, and dielectric
properties of multidomain 0.67Pb(Mg1/3Nb2/3)1-0.33TixO3 single crystals”, J. Appl. Phys. 90, 3471-3475, 2001.
6. M. Dong, and Z. G. Ye, “High-temperature solution growth and
characterization of the piezo-/ferroelectric (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 single crystals”, Journal of Crystal Growth, 209, 81-90, 2000.
93
7. X. Wan, J. Wang, H. L. W. Chan, C. L. Choy, H. Luo and Z. Yin, “Growth and optical properties of 0.62 Pb(Mg1/3Nb2/3)O3-0.32 PbTiO3 single crystals by a modified Bridgman technique”, Journal of Crystal Growth, 263, 251, 2004.
8. Z. Yan, X. Yao, and L. Zhang, “Analysis of internal-stress-induced phase
transition by thermal treatment”, Ceramics International, 30, 1423, 2004.
9. C. S. Lynch, W. Yang, L. Collier, Z. Suo, and R. M. McMeeking, “Electric field induced cracking in ferroelectric ceramics”, Ferroelectrics, 166, 11 – 30, 1995.
10. C. S. Lynch, L. Chen, Z. Suo, R. M. McMeeking, and W. Yang,
“Crack-growth in ferroelectric ceramics driven by cyclic polarization switching”, Journal of intelligent material systems and structures, 6, 191-198, 1995.
11. Z. Li, Z. Xu, X. Yao, and Z.-Y. Cheng, “Phase transition and phase stability
in [110]-, [001]-, and [111]- oriented 0.68 Pb(Mg1/3Nb2/3)O3-0.32 PbTiO3 single crystal under electric field”, J. Appl. Phys. 104, 024112, 2008.
12. Z. Xu, “In situ TEM study of electric field-induced microcracking in
piezoelectric single crystals”, Materials Science and Engineering B, 99, 106-101, 2003.
13. E. T. Keve and K. L. Bye, “Phase identification and domain structure in
PLZT ceramics”, J. Appl. Phys. 46, 87, 1975.
14. F.-X. Li, S. Li, D.-N. Fang, “Domain switching in ferroelectric single crystal/ceramics under electromechanical loading”, Materials Science and Engineering B, 120, 119–124, 2005.
15. N. Di, and D. J. Quesnel, “Photoelastic effects in
Pb(Mg1/3Nb2/3)O-29%PbTiO3 single crystals investigated by three-point bending technique”, J. Appl. Phys. 101, 043522, 2007.
16. N. Di, J. C. Harker and D. J. Quesnel, “Photoelastic effects in
Pb(Mg1/3Nb2/3)O-29%PbTiO3 single crystals investigated by Hertzian contact experiments”, J. Appl. Phys. 103, 053518, 2008.
94
17. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals, Plenum Press, 1981.
18. X. Wan, J. Wang, H. L. W. Chan, C. L. Choy, H. Luo, and Z. Yin, “Growth
and optical properties of 0.68 Pb(Mg1/3Nb2/3)O3-0.32 PbTiO3 single crystals by a modified Bridgman technique”, Journal of Crystal Growth, 263, 251–255, 2004.
19. X. Wan, H. Luo, J. Wang, H. L. W. Chan, and C.L. Choy, “Investigation on
optical transmission spectra of (1-x) Pb(Mg1/3Nb2/3)O-xPbTiO3 single crystals”, Solid State Communications, 129, 401-405, 2004.
20. Z. Feng, X. Zhao, and H. Luo, “Effect of poling field and temperature on
dielectric and piezoelectric property of <001>-oriented 0.7Pb(Mg1/3Nb2/3)O3-0.3PbTiO3 crystals”, Materials Research Bulletin, 41, 1133–1137, 2006.
21. W. Jiang, R. Zhang, B. Jiang, and W. Cao, “Characterization of
piezoelectric materials with large piezoelectric and electromechanical coupling coefficients”, Ultrasonics, 41, 55-63, 2003.
22. F. M. Bai, N. G. Wang, J. F. Li, D. Viehland, G. Xu, and G. Shirane, J. Appl.
Phys. 96, 1620, 2004.
23. Z. Li, Z. Xu, X. Yao and Z.-Y. Cheng, “Phase transition and phase stability in [110]-, [001]-, and [111]-oriented 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3 single crystal under electric field”, J. Appl. Phys. 104, 024112, 2008.
24. E. T. Keve and A. D. Annis, “Studies of phases, phase transitions and
properties of some PLZT ceramics”, Ferroelectrics, 5, 77-89, 1973.
25. D. Viehland, J. Powers, L. Ewart, and J. F. Li, “Ferroelastic switching and elastic nonlinearity in <001>-oriented Pb(Mg1/3Nb2/3)O3-PbTiO3 and Pb(Zn1/3Nb2/3)O3-PbTiO3 crystals”, J. Appl. Phys. 88, 4907-4909, 2000.
95
6 Summary
6.1 Summary
Relaxor-based ferroelectric single crystals of composition PMN-29%PT have
ultimate electromechanical coupling factors (k33 >90%), high piezoelectric
coefficients (d33>2000 pC/N) and high strain levels up to 1.7%, and therefore are
promising as replacements for conventional ceramics in wide range of
applications. However, because of the nature of single crystals, PMN-PT single
crystals are mechanically softer than PMN-PT polycrystalline ceramics [1].
Therefore PMN-PT single crystals more easily develop residual internal stresses
as a result of preparation processes, poling processes, and working loads, both
electrical and mechanical. The probability of crack initiation is strongly related to
the residual internal stresses. For this reason, it is important to investigate the
internal stress field during mechanical/electrical loading to better understand and
control the cracking problem. This thesis provides fundamental study of
birefringence induced by mechanical or electric field loading of PMN-29%PT
single crystals using optical methods. According to classical photoelasticity theory,
the birefringence is directly related to the internal stress field though its
proportionality to strain, therefore providing a visualization method to observe the
internal strain field. Strains induced from both electrical and mechanical loading
cause birefringent effects.
In Chapters 2, 3 and 4, classical photoelastic experiments including bending
and Hertzian contact loading experiments were conducted, and the results were
reported. In bending experiments, not only was the stress-optical coefficient C
estimated from three-point bending experiments, but also the load dependent
variation of the fringe-stress coefficient Cf /λ= was obtained through
four-point bending experiments. In all experiments, incident light wavelength λ
96
is 535 . C has a value of 104X10nm -12 Pa-1, equivalent to an of value 5.14
X10
f
3 N/m, while varied in a range of 3.5X10f 3 N/m to 5.5X103 N/m. Notice
that the change of isn't monotonic: as load is increased, first decreases,
then at around the 14
f f
th fringe obtained (~45 MPa fiber stress), reached its
minimum value and began to increase with increasing load, as shown in Figure
4.8.
f
As reported in preliminary three-point bending experiments, the optical
property (characterized by fringe order) is linearly proportional to the mechanical
property (characterized by deflection) of unpoled PMN-29%PT single crystals. In
three-point bending experiments, Young’s modulus along the <001> direction was
calculated as 1.9X1010 N/m2, which is comparable to that obtained by Viehland
and Li [2]. However, this is only an average value. From observing the slope of
Figure 2.5, force versus deflection, the elastic property represented by the
nominal Young’s modulus is found first to have softened, then stiffened as load is
increased, which is consistent with the change of . Similar results were
reported by Viehland, et al. [3]. The explanations for these macroscopic
phenomena are microscopic strain-driven deformation processes, including
domain switching and phase transformation. This process consists of two
stages: first many small domains will freely switch in response to loadings to form
fewer large domains, and phase transformation from Rhombohedral to Monoclinic
phase may occur locally where internal strain is sufficiently high. During this
stage, the elastic modulus becomes softened. In the second stage, domain
switching and phase transformation saturate, thus the elastic modulus becomes
stiffened.
f
Because internal domain switching and phase transformation is induced by
external mechanical loading, the phenomenon is called mechanical poling.
97
Regions being mechanically poled were observed to become very bright because
the transmission is greatly enhanced [4-5]. Notice that mechanical poling is a
2-dimensional effect compared to electrical poling, which is a 1- dimensional
effect, as shown in Figure 5.9. This means that after electrical poling, the
elastic stiffness is higher than that after mechanical poling. Thus electric field
loading may initiate or trigger cracks more easily than mechanical loading. This
is verified in Chapter 5.
In Chapter 5, electric field effects were investigated using birefringence
techniques. “Hertzian contact” electric field loading and electrical poling
experiments were performed with a DC field and a cyclic electric field, respectively.
Also for the cyclic electric field, frequency effects were explored.
From “Hertzian contact” experiments, the DC field was found to have a
deeper penetration than the cyclic electric field. In addition, the higher the
frequency of the cyclic field, the less penetration was observed in the fringe
pattern. This is due to extremely short time that each sign of the field is applied
to the material. Most of the electrical induced changes were close to the surface
and poled the local region to be very “bright”. The “bright” region has lost most of
its elastic flexibility and become susceptible to cracks. Thus cyclic field were
found more easily to initiate cracks in PMN-PT single crystals than DC field when
these samples were later subjected to mechanical loads.
From the electrical poling experiments, the DC field poling was observed to
first drive fringes to grow inside the sample at very low field magnitude of (1.1-1.3)
KV/cm, then, turning the entire sample turned dark and cloudy except regions
close to the surface as the field approached 5.4 KV/cm. This means that after
poling, a good practice would be to trim off a thin layer from the surface which has
been poled. When applying cyclic electric field to a sample previously poled with
DC field, the cyclic field immediately turned the sample to stripe patterns parallel
98
to the electric field. These stripes are aligned domains. Furthermore cracks
were observed to develop from the edge towards the inside of the sample,
especially at highly strained regions, such as under the electrode, as shown in
Figure 5.7. From these results, we can draw the conclusion that a cyclic field
has a fatigue effect on PMN-PT single crystals, which may over-pole and crack
the material easily compared to DC field; thus a cyclic field shouldn’t be used for
electrical poling purposes.
Finally, orientation dependent optical and mechanical properties were
studied through Hertzian contact mechanical loading experiments and “Hertzian
contact” electrical loading experiments. Samples with three different orientations
were selected to perform the experiments: One is {100}-oriented on all six faces,
one with two side faces {100} and the other two side faces {110}, the last one with
all four side faces having {110} orientation. In Chapter 3, results of Hertzian
contact mechanical loading experiments were reported. Fringe pattern for all
three samples were totally different, and the loading direction was found to be an
important factor controlling the fringe pattern as well. Generally speaking, when
loaded from a <100> direction, the resulting fringe pattern is a two-lobed pattern;
when loaded from <110> direction, the resulting fringe pattern is a narrow
one-lobed pattern. Similar results were observed from “Hertzian contact”
electrical loading experiments. This shows that pseudo-cubic unpoled
PMN-29%PT single crystals have highly anisotropic elastic properties and that the
strain gradients giving rise to birefringence from mechanical and electrical field
loadings are similar.
Using cubic-form elastic constants calculated from data on poled
PMN-30%PT single crystals, ANSYS® simulation results were comparable to the
experimentally observed fringe patterns. This suggests that the elastic
properties of pseudo-cubic unpoled PMN-PT single crystals may resemble those
99
of pseudo-tetragonal poled PMN-PT single crystals. There are two possible
explanations for the anisotropy of elastic properties. First, the seeded crystal
growth method with a unique growth direction may induce the highly anisotropic
elastic properties of unpoled PMN-PT single crystals, as reported by Sehirlioglu,
et al. [6]. Second, Zhang, et al. reported that domain wall motion may
contribute to the effective elastic constants, especially affecting the response of
PMN-PT in <110> directions [7]. This may explain why unpoled PMN-PT single
crystals show different fringe patterns when loaded from <100> and <110>
directions. That is, the fringe patterns depend not only on the elastic strain but
also on strains caused by domain switching.
As for the similar fringe pattern observed in the “Hertzian contact” electrical
loading experiments, because there is no piezoelectric strain coefficient reported
for unpoled PMN-PT single crystals, it is impossible to conduct ANSYS simulation.
Based on the similar fringe pattern, it is reasonable to conclude that the elastic
properties affect the anisotropic response through one of the linear electric-optical
effects: the electric field induced strain causes photoelastic effects which reflect
the anisotropy of the elastic properties that were explored by mechanical loading.
6.2 References 1. D. Viehland and J. F. Li, “Young’s modulus and hysteretic losses of 0.7
Pb(Mg1/3Nb2/3)O3-0.3PbTi O3”, J. Appl. Phys. 94, 7719, 2003. 2. D. Viehland and J. F. Li, “Stress-induced phase transformations in
<001>-oriented Pb(Mg1/3Nb2/3)O3–PbTiO3 Crystals: bilinear coupling of ferroelastic strain and ferroelectric polarization”, Philosophical Magazine, 83, 53-59, 2003.
3. D. Viehland, J. Powers, L. Ewart, and J. F. Li, “Ferroelastic switching and
elastic nonlinearity in <001>-oriented Pb(Mg1/3Nb2/3)O3-PbTiO3 and Pb(Zn1/3Nb2/3)O3-PbTiO3 crystals”, J. Appl. Phys. 88, No. 8, 4907-4909, 2000.
100
4. X. Wan, J. Wang, H. L. W. Chan, C. L. Choy, H. Luo and Z. Yin, “Growth and optical properties of 0.62 Pb(Mg1/3Nb2/3)O3-0.32 PbTiO3 single crystals by a modified Bridgman technique”, Journal of Crystal Growth, 263, 251, 2004.
5. X. Wan, H. Luo, J. Wang, H. Chan, and C. Choy, “Investigation on optical
transmission spectra of (1-x) Pb(Mg1/3Nb2/3)O3-x PbTiO3 single crystals”, Solid State Communications, 129, 401, 2004.
6. A. Sehirlioglu, D. A. Payne, and P. Han, “Thermal expansion of phase
transformations in (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3: evidence for preferred domain alignment in one of the (001) directions for melt-grown crystals”, Phys. Rev. B 72, 214110, 2005.
7. R. Zhang, W. Jiang, B. Jiang, and W. Cao, “Elastic, dielectric and
piezoelectric coefficients of domain engineered 0.7Pb(Mg1/3Nb2/3)O-0.3PbTiO3 single crystal”, Fundamental Physics of Ferroelectrics 2002, 188-197, 2002.
101
Appendix Ι Basic theory of optical properties of crystals
This Appendix is to provide the background of basic theory of optical
properties of crystals, which is well documented in references [1] and [2].
Consider plane-polarized light passing through a crystal. There are in general
two waves, with different velocity, propagating through the crystal. The value of
for each wave is called the refractive index for that wave. Most
generally, if there are three principal axes
vc / n
zyx ,, of the dielectric constant tensor,
we can describe the refractive index by the refractive index ellipsoid, as shown
in Figure A1(a).
n
Figure A1: Representation of optical index ellipsoid; Illustration redrawn from a similar figure
in reference [1].
Now, for plane waves which propagate in the direction of any radius vector,
like the Z’ arrow in Figure A1(b), the two refractive indices are given by the
principal axes of the ellipse in which the ellipsoid is intersected by a plane
perpendicular to this radius vector, as the X’ and Y’ arrows shown in Figure A1(b).
The directions of these principal axes are the directions of the corresponding
electric vectors. The refractive index ellipsoid is defined by the equation:
102
1＝2
2
2
2
2
2
zyx nz
ny
nx
++ (A1-1)
xn , , are called the principal refractive indices, and yn zn ),,( zyxiKn ii == .
is the principal dielectric constant. Note the refractive index isn’t a tensor,
but the reciprocals of the square of can be treated as a tensor. If
iK n
n 2
1
ijn is
denoted by , the optical parameters, we have: ijB
(A1-2) 1＝2332
222
11 zByBxB ++
which is called the optical index ellipsoid. In its most general form, it is given by:
(A1-3) 1＝222 1231232
332
222
11 xyBzxByzBzByBxB +++++
Under an applied stress klσ , the Eq. (A1-3) changed to:
(A1-4) 1＝222 1231232
332
222
11 xyBzxByzBzByBxB ′+′+′+′+′+′
So the changes are given by: ijBΔ
1,2,3),,,( ==−′=Δ lk jiBBB klijklijijij σπ (A1-5)
Here ijklπ is called the stress optical constant, which is a forth rank tensor with
units of m2/N. Since and ijB klσ are both symmetric tensors of second rank,
the ijklπ are not fully independent. ijlkijkljiklijkl ππππ == ; , therefore the number of
independent coefficients ijklπ is reduced from 81 to 36. Correspondingly,
because the strain is linear with the stress, photoelastic effects may also be
expressed in terms of the strains:
)3,2,1,,,( ==−′=Δ srjipBBB rsijrsijijij ε (A1-6)
Here is called the elasto-optical constant, and is dimensionless. and ijrsp ijrsp
103
ijklπ are interrelated through and , the elastic constants and the
compliance constants. Using only one suffix let 11=1, 22=2, 33=3, 23=4, 31=5
and 12=6, Eq. (A1-5) and (A1-6) can be written as:
ijklC ijkls
)6...,,2,1,( ==−′=Δ nmBBB nmniim σπ (A1-7)
)6...,,2,1,( ==−′=Δ nmpBBB nmniim ε (A1-8)
For uniaxial (tetragonal, hexagonal and trigonal) crystals, the number of
refractive indices is two, which are those along the x and y axis; the refractive
indices are both the , while for the z axis, the refractive index is . and
are called ordinary and extraordinary refractive indices. Thus, the Eq. (A1-2)
can be written as:
on en on
en
12e
2
20
2
20
2
=++nz
ny
nx (A1-9)
If we let light travel along the z axis, i.e. the optical axis, the uniaxial crystals may
behave like an isotropic material, because the section of the ellipsoid
perpendicular to the light path is a circle.
104
Appendix ΙΙ Basics of photoelasticity
This appendix is to provide the background of basic theory of photoelasticity,
which is also documented in references [3-4]. I will also summarize why a
circular polariscope can eliminate the isoclinic fringes. In photoelasticity, stress
fields are displayed through the use of light. Typically, plane-polarized light or
circular-polarized light are incident on the sample. The basic arrangement of
polarized microscope has already been discussed in Chapter 1. Here the
circular polariscope will be described, as shown in Figure A2(a). Two
quarter-wave plates, one with its axis at 450 and the other at 1350 were introduced
in the plane polarized microscopy configuration to achieve circular polarized
microscopy.
(a) (b)
Figure A2 Circular polariscope set-up, reproduced from a similar figure in reference [3].
To explain how a circular polariscope eliminates the isoclinic fringes, Jones
calculus needs to be introduced [3]. Generally speaking, an optical component
in a polariscope introduced both a rotation and retardation. In Jones calculus,
these can be represented as matrices.
First for a rotation, imagine an incident light ( , ), after passing through an
optical component, has been rotated by an angle
u v
θ . Then:
105
θθ sincos vuu +=′ (A2-1)
θθ cossin vuv +−=′ (A2-2)
Namely, the light after rotation:
⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡−=
⎭⎬⎫
⎩⎨⎧
′′
vu
vu
θθθθ
cossinsincos (A2-3)
The following matrix is referred to as the rotation matrix:
(A2-4) ⎥⎦⎤
⎢⎣⎡− θθ
θθcossinsincos
Second, for representing retardation, let’s still assume an incident light (u , ).
To be general, assume:
v
)cos()cos(
22
11
αωαω
+=+=
tavtau
(A2-5)
If is the slow axis and is the fast axis, then the light coming out of the
medium with a retardation of
u v
δ can be represented as:
)
2cos(
)2
cos(
22
11
δαω
δαω
++=′
−+=′
tav
tau (A2-6)
Using complex number notations, the emerging light can be obtained as:
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎭⎬⎫
⎩⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
′′ −
tii
i
i
i
eeaea
eeRv
u ωα
α
δ
δ
2
1
2
12
2
00
(A2-7)
Here R represents the real part, which we will deal with. Now, for a retarder,
with both rotation and retardation, we can represent it as follows:
⎥⎦⎤
⎢⎣⎡−⎥⎦
⎤⎢⎣
⎡ −
θθθθ
δ
δ
cossinsincos
00
2
2
i
i
ee (A2-8)
If we want to represent the light vector with respect to the original reference axes,
then:
106
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−
−−=
⎥⎦⎤
⎢⎣⎡−⎥⎦
⎤⎢⎣
⎡⎥⎦⎤
⎢⎣⎡ − −
θδδθδ
θδθδδ
θθθθ
θθθθ
δ
δ
2cos2
sin2
cos2sin2
sin
2sin2
sin2cos2
sin2
cos
cossinsincos
00
cossinsincos
2
2
ii
ii
ee
i
i
(A2-9)
Namely any retarder can be represented by the matrix:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−
−−
θδδθδ
θδθδδ
2cos2
sin2
cos2sin2
sin
2sin2
sin2cos2
sin2
cos
ii
ii (A2-10)
The retarder matrix can be used to represent a quarter-wave plate by substituting
and 0135=θ2πδ = , since quarter-wave plate provides a retardation of 2
π :
⎥⎦⎤
⎢⎣⎡
11
21
ii (A2-11)
In Figure A2(a), consider polarized light coming out the polarizer aligned parallel
to the x axis:
ti
y
x keEE ω
⎭⎬⎫
⎩⎨⎧=
⎭⎬⎫
⎩⎨⎧
01 (A2-12)
After entering the quarter-wave plate, we have:
ti
y
x keii
EE ω
⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡=
⎭⎬⎫
⎩⎨⎧
10
11
21 (A2-13)
After entering the sample, a phase difference of δ is generated for the light.
ti
y
x keii
ii
ii
EE ω
θδδθδ
θδθδδ
⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−
−−=
⎭⎬⎫
⎩⎨⎧
01
11
2cos2
sin2
cos2sin2
sin
2sin2
sin2cos2
sin2
cos
21
(A2-14)
Just multiply the individual matrices of the various optical elements from the left
side of the original matrix in the order they are placed in the polariscope, we can
obtain the final output light. After another quarter-wave plate we have:
107
ti
y
x keii
ii
ii
ii
EE ω
θδδθδ
θδθδδ
⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−
−−
⎥⎦⎤
⎢⎣⎡−
−=⎭⎬⎫
⎩⎨⎧
01
11
2cos2
sin2
cos2sin2
sin
2sin2
sin2cos2
sin2
cos
11
21
(A2-15)
Upon simplification one gets:
ti
iy
x kee
EE ω
θδ
δ
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎭⎬⎫
⎩⎨⎧
− 2
2sin
2cos
(A2-16)
In the dark field, when the analyzer is crossed with polarizer, the intensity of light
transmitted is obtained as the product of , which is simplified as: yy EE
2
sin 2 δad II = (A2-17)
Here represents the amplitude of the incident light. While in a bright field
with the analyzer parallel to the polarizer, we have the intensity of light , one
gets
aI
bI
2cos2 δ
ab II = (A2-18)
It is to be noted that no matter for dark field or bright field, the intensity equations
are independent of θ and hence the extinction condition is only a function of δ
and thus only isochromatics will be seen. The separating of isoclinics and
isochromatics is a significant achievement and greatly simplifies the analysis of
photoelastic fringe patterns.
108
Appendix Ⅲ Calibration of in-situ loading frame
In this Appendix, I report the calibration of the cylinder loading system. To
calibrate the cylinder loading force from reading the pressure gauge, a calibration
system was designed and fabricated with all components as shown in Fig A3-5.
Figure A3 Top view of calibration stage
Figure A4 Side view of calibration stage
109
Figure A5 Overview of loading frame
The loading system with the cylinder is first clamped on the stage; the loading
bar with a steel ball pinned on the head is put inside the front “hole” of the loading
system. As the pressure is increased, the cylinder pushes the brass to touch
the steel ball, thus the force is transferred to the load cell. Data for loading force
versus incremental pressure are obtained and plotted in Figure A5. Data clearly
shows a “straight line” as the pressure increases. All the data with pressure
increasing are displayed by black dots each with ± 2.5% error line. The fit line
falls within the error line, implying the force calculated
from function is within an error range of
5874.23254.0 −= xy
± 2.5%.
110
Figure A6 Loading force versus pressure
Notice that the loading system can exert force over a range of 0-25 Newton.
In all of our experiments, the force exerted normally fell within a range of 0-15
Newton indicating the pressure we applied is within a range of 0-50 Psi. In
addition, to eliminate any hysteresis effects, we performed experiments with only
increasing pressure. Pressure was never decreased until the experiment was
completed.
111
References for Appendices
1. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals, Plenum Press, 1981.
2. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford,
1985.
3. K. Ramesh, Digital Photoelasticity Advanced Techniques and Applications, Springer, 2000.
4. M. M. Frocht, Photoelasticity, VI, New York, John Wiley & Sons, Inc.
1941.