na di - dissertation

7/30/2019 Na Di - Dissertation

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Photoelastic and Electro-Optic Effects: Study of

PMN-29%PT Single Crystals

by

Na Di

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor David J . Quesnel

Department of Mechanical Engineering

Arts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of Rochester

Rochester, New York

2009


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Curriculum Vitae

The author was born in Shenyang, Liaoning province, China on November 28,

1977. She attended Liaoning Key High School and graduated in 1996. Sheenrolled at Fudan University in 1996 and finished her B.S. degree program in

Theory and Applied Mechanics in 2000. Thereafter she continued her graduate

study at Fudan University and graduated with a Masters degree in Engineering

Mechanics in 2003.

In fall 2003, she was accepted into the doctoral program at the University of

Rochester under the supervision of Professor David J . Quesnel. She received

her second Masters degree in Mechanical Engineering from the University of

Rochester in 2005.

In May of 2005 she attended the U.S. Navy Workshop on Acoustic

Transduction Materials and Devices where she became familiar with the issues

constraining the behavior of next generation piezoelectric single crystals. Shortly

thereafter, she conceived of the idea of using photoelastic methods to

characterize the stress distributions in these materials from which this thesis

developed. While pursing her thesis research, she regularly participated in the

U.S. Navy Workshop on Acoustic Transduction Materials and Devices by making

the presentations that are listed below.

z Photoelastic study of PMN-29%PT single crystals, U.S. Navy Workshop on

Acoustic Transduction Materials and Devices, May 2006.

z Photoelastic study of PMN-29%PT single crystals, U.S. Navy Workshop on

Acoustic Transduction Materials and Devices, May 2007.

z Photoelastic study of PMN-PT single crystals under electric fields, U.S. Navy

Workshop on Acoustic Transduction Materials and Devices, May 2008.


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Acknowledgements

I would like to thank my advisor, Professor David J . Quesnel first. I am

thankful for his diligent guidance and constant encouragement. I have learned a

lot from him, from academic knowledge to language and life. Without his

financial and academic support throughout my graduate studies, I would never

have been able to finish my thesis.

Next, I would like to thank Mr. J ohn C. J ace Harker and Mr. Stephen R.

Robinson. J ace is a fantastic lab mate, who always has a lot of brilliant ideas,

and will share them with me without reservation. We have held many meaningful

discussions over my research problems, and he helped a lot with my writing.

The strong technical skills of Stephen, who prepared samples and took the

photographs, are very much appreciated. He also helped me to improve my

English writing. Without J ace and Stephens help, I also would never have been

able to write out my thesis.

Many thanks to Professor Sheryl M. Gracewski, Professor Paul D.

Funkenbusch, Professor James C. M. Li, Professor J ohn C. Lambropoulos,

Professor Stephen J . Burns, Professor Renato Perucchio and Professor Ahmet T.

Becene for their guidance and the knowledge I have learned from their classes.

I would like to thank Chris Pratt for helping me in conducting X-Ray

experiments. Thank you also to J ill Morris and Carla Gottschalk for all their help

along the way.

Final words of thanks go to my parents for their love and support.

Portions of this thesis are derived from publications that appear in the archival

literature. In particular, Chapter 2 draws from: Na Di and David J . Quesnel,

Photoelastic effects in Pb(Mg1/3Nb2/3)O3-29%PbTiO3 single crystals investigated

by three-point bending technique, J. Appl. Phys. 101, 043522 (2007); and

Chapter 3 is derived from: Na Di, J ohn C. Harker, and David J . Quesnel,


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Photoelastic effects in Pb(Mg1/3Nb2/3)O3-29%PbTiO3 single crystals investigated

by Hertzian contact experiments, J. Appl. Phys. 103, 053518 (2008); In this

work, J ohn Harkers contribution was through editing of the initial draft to a form

suitable for publication, with the technical discussion necessary to get the

meanings as intended.

Chapter 4 and Chapter 5 will be submitted for consideration as journal

articles. This is reflected in the format selected for these chapters. They will be

coauthored with my advisor, David J . Quesnel.


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Abstract

Relaxor ferroelectrics PMN-PT single crystals exhibit extra-high dielectric and

piezoelectric properties compared with conventional piezoelectric ceramics.

They are becoming widely used in high performance electromechanical devices.

However, PMN-PT single crystals are elastically softer than PMN-PT

polycrystalline ceramics. Mechanical loads and electric fields interact to produce

fractures at relatively low stresses, and cracks grow under both AC and DC

electric fields. To prevent the failure of the electromechanical devices, we need

to have a better understanding of the mechanisms of fracture in this material

when it is subjected to mechanical and electrical loadings.

Photoelasticity is an efficient and effective method to measure the internal

stress distributions of materials that result from both internal residual stress and

external loading. I report the exploration of the use of this classic technique to

study internal stresses inside PMN-PT single crystals through bending and

Hertzian contact experiments. Effects under electric field loading were also

investigated using birefringence techniques.


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Table of Contents

Chapter 1 Introduction - piezoelectric single crystals & photoelasticity

1.1 PMN-29%PT single crystals....1

1.2 Photoelasticity... ...7

1.2.1 Discovery of the phenomenon of Photoelasticity7

1.2.2 Mathematical formulation of Photoelasticity..................9

1.2.3 Plane polariscope and circular polariscope. 10

1.3 Preliminary three-point bending experiments... .13

1.3.1 Experimental setup.... ..13

1.3.2 Fringe pattern..........................15

1.3.3 Deflection versus fringe order........17

1.3.4 Summary ...18

1.4 References... ....................20

Chapter 2 Photoelastic study using three-point bending technique

2.1 Introduction.... .26

2.2 Experimental procedure...... .29

2.3 Results and discussion .32

2.3.1 Fringe pattern....32

2.3.2 Loading force versus deflection................34

2.3.3 Stress-optical coefficient.....36

2.3.4 Youngs modulus...38

2.4 Summary................ ...39

2.5 References.............40


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Chapter 3 Photoelastic study using Hertzian contact experiments

3.1 Introduction........ 42

3.2 Experimental procedure..... .....44

3.3 FEM modeling methods... 46

3.4 Results and discussion. 50

3.5 Conclusions..........................53

3.6 References..54

Chapter 4 Photoelastic study using four-point bending technique

4.1 Introduction................57

4.2 Experimental procedure........60

4.3 Results and discussion.............63

4.3.1 Fringe pattern63

4.3.2 Fiber stress versus fringe order............... 64

4.3.3 Fringe-stress coefficient................ 66

4.3.4 Mechanical poling effect. 69

4.4 Conclusions. ....................... 69

4.5 References. .. 70

Chapter 5 Electrical field induced optical effects in PMN-29%PT single crystal

5.1 Introduction ...............73

5.2 Experimental procedure..................77

5.3 Results and discussion...................79

5.3.1 Hertzian Contact electric field loading effects...79

5.3.2 Electric poling effects................. 85

5.3.3 Mechanical poling versus electrical poling...89

5.4 Conclusions...................91


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5.5 References. .... ...92

Chapter 6 Summary

6.1 Summary....................95

6.2 References......99

Appendices

Appendix Basic theory of optical properties of crystals101

Appendix Basics ofphotoelasticity.....104Appendix Calibration ofin-situ loading frame...108

References for Appendices.... . ..111


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List of Tables

Table 3.1 Elastic stiffness constants of PMN-30%PT single crystals (10Dij

c10

N/m2)...47

Table 3.2 Input parameters used in ANSYS. The elastic stiffness constants:

(10ijc

10 N/m2). Young's modulus of glass:E(1010 N/m2). Poisson's ratio

of glass: .48


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List of Figures

Figure 1.1 (a) ABO3 Perovskite Structure. If one shifts the A-site ions to the

center of the cell, the structure will appear differently with 12 oxygen

atoms at the center of each cell edge, an arrangement often shown ingeology texts, (b) Spontaneous polarization for the R phase in unpoled

PMN-29%PT; Illustration redrawn from a similar figure in reference

[5]...................................................................................4

Figure 1.2 Phase diagram of PMN-PT single crystals; Illustration redrawn from

a similar figure in reference [33].......5

Figure 1.3 PMN-29%PT single crystal as received.8

Figure 1.4 (a) Dark field plane polariscope set-up. (b) Light vector

representation; Illustrations redrawn from a similar figure in reference

[38].11

Figure 1.5 ZeissTM Microscope set-up... .14

Figure 1.6 Preliminary three-point bending set-up14

Figure 1.7 Three-point bending image at 450 to both the polarizer and the

analyzer... ..15

Figure 1.8 Principal Stress Vectors from ANSYS simulation of three-point

bending. Only left half of sample is shown.16

Figure 1.9 Three-point bending image at zero degree to both the polarizer and

the analyzer....16

Figure 1.10 (a) Three-point bending of unpoled PMN-29%PT; (b) Three-point

bending image of isotropic materials [39].17

Figure 1.11 Deflection versus fringe order....17

Figure 1.12 3D CAD model of loading frame. BimbaTM cylinder is mounted

through a hole in the aluminum frame....19

Figure 2.1 (a) in situ loading frame working under microscope and (b) in situ


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loading frame with three-point bending set-up as indicated by the

arrow. ...30

Figure 2.2 3-point bending schematic. P is the loading force; c and t are the

compression and tension fiber stress. and are the reaction

loads....30

1 2R R

Figure 2.3 Unpoled PMN-29%PT single crystal beam viewed in crossed polars:

(a) (100) face, as-received; (b) (100) face, after annealing; and (c) (010)

face, after annealing. Polarizer and analyzer are horizontal and vertical,

respectively. Strong colors in (a) indicate regions of net birefringent

retardation31

Figure 2.4 (a) Initial fringe pattern showing 0.5 fringe at point A on the free

surface opposite the loading point. (b) Second-order fringes at A, (c)

Sixth-order fringes at A, and (d) first-order fringe remaining at A after the

load is released33

Figure 2.5 Force versus deflection during increasing load for three experimental

runs...................................34

Figure 2.6 Force versus deflection with polynomial fit curve.35

Figure 2.7 Fringe order versus fiber stress. The stress-optical coefficient iscalculated from the slope of the proportional region.37

Figure 3.1 (a) Top view of the loading frame. (b) Hertzian contact experimental

set-up as indicated by the arrow..45

Figure 3.2 Initial birefringence patterns of three samples in three different

orientations under circularly polarized illumination46

Figure 3.3 The 3 differently oriented samples relative to {001}-oriented

pseudo-cubic axes. Arrows a and c represent compression along direction; arrows b and d represent compression along

direction...46

Figure 3.4 ANSYS model for use in computation of fringe pattern images.

Boundary conditions are shown. Contact elements are used at the

interface between the Hertzian cylinder indenter and the rectangular


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piezocrystal (light gray). Cyan symbols represent displacement

constraints. Red arrow indicates the force applied to all of the coupled

nodes (green)49

Figure 3.5 (a) Hertzian indentation along direction on sample 1. (b) Stress

intensity contour from ANSYS..50

Figure 3.6 (a) Hertzian indentation along direction on sample 2. (b) Stress

intensity contour from ANSYS51

Figure 3.7 Hertzian indentation along direction on sample 3 is shown in

(a); Hertzian indentation along direction is shown in (c). The

initial birefringence is responsible for the asymmetric fringe in (a) and

the layers along the surface in (c). Stress intensity contour from ANSYS

are shown in (b) and (d) correspondingly.....51

Figure 3.8 Residual butterfly fringes are fully annealed out at 400 oC for one

hour..53

Figure 4.1 (a) Overview of the in situ loading frame and (b) Four-point bending

set-up; A represents the tilting bar, and B represents the beam

sample.........61

Figure 4.2 Beam 1 (a) and beam 2 (b) after one hour annealing at 400 oC.61

Figure 4.3 Four-point bending layout. P is the loading force,c

andt

are

compression and tension stresses respectively. The diagram under the

sample shows the absolute value of the bending moment..62

Figure 4.4 (a) 3.5 order of fringes and (b) 2 order of fringes left after the load is

released...64

Figure 4.5 Maximum fiber stress versus fringe order..65

Figure 4.6 Maximum fiber stress versus fringe order with polynomial fit curve...65

Figure 4.7 From the light intensity plot, displacement between fringes can be

measured. Each valley of the intensity curves represents a fringe

(darkest field), and each peak of the intensity curves represents the half

order of fringe (brightest field)..66


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Figure 4.8 Stress versus fringe order for different load level. The number label

represents the maximum fringe order obtained for each load level. The

slope of each data line represents the fringe-stress coefficient..67

Figure 4.9 Fringe stress coefficient versus maximum fiber stress68

Figure 4.10 Fringe patterns of pure bending region at different load levels. (a)

Totally 11 order of fringes; (b) totally 16 order of fringes..69

Figure 5.1 Initial birefringence patterns of four differently oriented samples under

circularly polarized illumination after one hour annealing at

400 oC..77

Figure 5.2 (a) Overview of the in situ electrical loading frame and (b) Top view of

Hertzian contact electrical loading set-up....78

Figure 5.3 Hertzian contact electrical loading (electrical point load) experiments

on {100}-oriented beam 1: (a) -2.3 KV/cm DC were applied from the top

rod electrode for 2 minutes; (b) 2.3 KV/cm for 2 minutes; (c) 2.3 KV/cm

applied to resulting fringes of (a) for another 2 minutes; (d) 2.3 KV/cm for

additional 2 minutes after (c). The arrows in the pictures represent the

electric field direction 81

Figure 5.4 Fringe pattern comparison: (a), (c) and (e) are fringes induced by DC

electric field loading for beam 1 with 2.3KV/cm, sample 3 with 1.8KV/cmand sample 4 with 1.8KV/cm. (b), (d) and (f) are fringes under Hertzian

mechanical loading for comparably oriented samples, as shown in

Chapter 3.....82

Figure 5.5 Hertzian contact electrical loading experiments on differently

oriented samples using square waveform voltage with 0.5 Hz and 500

Hz, respectively: (a), (b), and (c) (top row) resulted from square

waveform voltage of 0.5 Hz. (d), (e), and (f) (bottom row) were from

square waveform voltage of 500 Hz. (a) and (d) are from beam 1; (b)

and (e) are sample 3; (c) and (f) are sample 4. The magnitude of electricfield is 2.3 KV/cm for beam 1 and 1.8KV/cm for both sample 3 and 4...84

Figure 5.6 From top to bottom, beam 2 is electrical poled with increment DC

voltage. The experiment set-up is with two block electrodes; both the top

and bottom surfaces are plated with gold..86


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Figure 5.7 (a) 2.8 KV/cm square waveform cyclic electric field with 0.5 Hz was

applied to beam 2. (b) Birefringence of beam 2 after annealing. The

arrow points at crack generated during the experiment..88

Figure 5.8 (a) 2.5 KV/cm DC electric field was applied to sample 4 using

electrical Hertzian contact experimental set-up. (b) Hertzian

mechanical loading on poled region....89

Figure 5.9 Mechanical poling and electrical poling representation...90

Figure A1 Representation of optical index ellipsoid; Illustration redrawn from a

similar figure in reference [1].........101

Figure A2 Circular polariscope set-up, reproduced from a similar figure in

reference [3]..104

Figure A3 Top view of calibration stage. 108

Figure A4 Side view of calibration stage.108

Figure A5 Overview of loading frame..109

Figure A6 Loading force versus pressure......110


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1 Introduction - piezoelectr ic single crystals and

photoelasticity

1.1 PMN-29%PT single crystals

In 1880, the famous brothers Pierre and Jacques Curiefirst discovered direct

piezoelectric effects in quartz crystals [1, 2]. They found that when a weight is

placed on the surface of a quartz plate, electric charges are generated on both

surfaces of the quartz plate. The charge was measured to be linearly

proportional to the weight placed. Following the discovery of the direct

piezoelectric effect, Lippmann in 1881 theoretically predicted the converse

piezoelectric effect, which says a voltage applied to a piezoelectric crystal

produces elastic strains in the crystal [2, 3]. Later, general theory of

piezoelectricity was thoroughly accounted by Voigt [2, 4]. For the next 60 years,

extensive characterization was performed on BaTiO3 ceramics. In the 1950s,

Pb(Zn1/3Nb2/3)O3 (PZT) ceramics were found to exhibit an exceptionally strong

piezoelectric response. Since then, modified PZT ceramics and PZT-based solid

solution systems have become the dominant piezoelectric ceramics for various

applications [5].

This defining characteristic of the piezoelectric materials is due to the fact that

the centers of positive and negative charges do not coincide. Namely the crystal

structure does not have a center of symmetry. Such materials possess a

spontaneous polarization. When the spontaneous polarization can be reversed

by an applied electric field, the material is called a ferroelectric. Thus

ferroelectrics are a subset of piezoelectric materials.

In contrast to conventional piezoelectric ceramics, single crystal relaxor

ferroelectrics Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT) and


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Pb(Zn1/3Nb2/3)O3-xPbTiO3 (PZN-xPT) exhibit extra-high dielectric and

piezoelectric properties and have become a new generation of piezoelectric

materials, attracting constant attention in recent years [6-10]. Both of them are

widely used in high performance applications such as medical imaging, active

noise suppression, and acoustic signature analysis.

Because PMN-PT has relatively high field-induced strain response and a

small hysteresis loop compared to PZN-PT, PMN-PT is more attractive than

PZN-PT [10]. Furthermore, relaxor-based ferroelectric single crystals PMN-PT,

with compositions near the morphotropic phase boundary (MPB) between the

ferroelectric rhombohedral and tetragonal phases, have ultimate

electromechanical coupling factors (k33 >90%), high piezoelectric coefficients

(d33>2000 pC/N) and high strain levels up to 1.7% [11-12]. They also have

potential to be used in electro-optical technology for their high electro-optical

coefficients [13-14]. Thus my study focuses on PMN-29%PT (close to MPB)

single crystals. The origins of PMN-PT single crystals excellent performance

have been attributed to the polarization rotation induced by the external electric

field [15].

However, these materials face crack problems which will reduce the

performance of the devices. Some researchers have studied the fracture

problems of piezoelectric materials, both theoretically [16-21] and experimentally

[22-26], however, most of them focused on using an AC electric field to drive the

growth of existing cracks. Because internal stress plays a significant role in

causing cracks and also in the propagation of cracks, further study of these

internal stresses induced either by mechanical loading or by electrical loading is

an important research topic. This will enable us to better understand and control

the internal stresses in relaxor ferroelectrics devices.

What are the possible sources of internal or residual stress in PMN-29%PT?


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Residual stresses are induced by inhomogeneous strain. Inhomogeneous strain

may be produced by thermal gradients during crystal growth [27-28], by phase

transitions during cooling [27, 29], and by mechanical operations, such as cutting,

grinding and polishing during each step of the machining processes [29]. Each

step, therefore, has the potential to produce more residual stresses in the single

crystals.

The topic is significant, but also very difficult due to the complicated internal

domain structures and the intrinsic coupling effects between mechanical and

electric fields. PMN-xPT single crystals have a simple perovskite ABO3 structure

above Curie temperature (for PMN-29%PT, the Curie temperature is about 135

C), pictured in Figure 1.1(a), and it may readily have complex perovskite

structure A(B1/3B2/3)O3, as well. X-ray diffraction (XRD) shows unpoled

PMN-xPT single crystals have a tetragonal-rhombohedral MPB (morphotropic

phase boundary). When x is under 30%, PMN-xPT is in rhombohedral (R) R3m

phase at room temperature; when x is above 33%, it begins to transform to

tetragonal (T) P4mm phase through monoclinic (M) or orthorhombic (O)

symmetries [30-33]. The spontaneous polarization direction of the R phase is

and that of the T phase is . The piezoelectric effect is observed to

peak at the morphotropic phase boundary. The enhancement in the

piezoelectric effect at the morphotropic phase boundary has been attributed to the

coexistence of the different phases, whose polarization vectors become more

readily aligned by an applied electric field when mixed in this manner than may

occur in either of the single phase regions.

PMN-29%PT is in the R phase at room temperature and there are eight

possible directions for the spontaneous polarization as shown in Figure 1.1(b).

After an electric field poling, PMN-29%PT will transform from R to M phase first

[32]. With increasing poling field, M to T phase transition may occur. Phase


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diagram is shown in Figure 1.2 [33]. The coercive electric field is about 5 KV/cm.

With the fluctuation of chemical composition, macro-domains formed with different

polarization directions. Some researchers even found that within

macro-domains (m scale), there are micro-domains (0.1m scale), and within

micro-domains, there are nano-domains (nm scale). This is called the domain

hierarchy [34]. Accordingly, internal stress study has different levels. In this

thesis, I examine internal stress at the m level through the use of optical

methods.

Figure 1.1 (a) ABO3 Perovskite Structure. If one shifts the A-site ions to the center of the cell,

the structure will appear differently with 12 oxygen atoms at the center of each cell edge, an

arrangement often shown in geology texts, (b) Spontaneous polarization for the R phase in

unpoled PMN-29%PT; Illustration redrawn from a similar figure in reference [5].

The constitutive equations of the piezoelectric materials are [35-36]:

321321

ricitypiezoelectconverseelasticity

kkijklijklij EesC = (1-1)

321321

ricitypiezoelectconverseelasticity

kkijklijklkl EdSs += (1-2)

321321

typermittiviricitypiezoelectdirect

kikklikli EseD += (1-3)


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where ij , , and are stress, strain, electric field and electric

displacement tensors, respectively. , ,

kls kE iD

ijklC ijklS ik , and are the elastic

constant tensor, elastic compliance tensor, the dielectric constants, thepiezoelectric stress coefficients and the piezoelectric strain coefficients,

respectively; these tensors are material specific. So, both the external

mechanical loading and electrical loading will induce internal stress/strain and

polarization, accompanied by domain switching. If the loading is large enough, it

can even induce phase transitions. This coupling between electrical and

mechanical field variables in the constitutive equations will bring serious

mathematical difficulty to the internal stress analysis problem.

kije kijd

Figure 1.2 Phase diagram of PMN-PT single crystals; Illustration redrawn from a similar figure

in reference [33].

Traditionally in materials research any of several types of strain gages can be

employed to help measure the internal strain and, further, to analyze the internal

stress. Since the available samples are too small to use strain gages, optical

methods were adopted, i.e. photoelasticity techniques. Compared with other

stress measurement techniques, photoelasticity can offer efficient quantitative

determination as well as qualitative observation of the stress distributions


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resulting from both internal stress and external loading [37-46]. Initial

explorations show that PMN-29%PT single crystals can be polished to be optically

transparent and the application of external loads produces an extremely large

number of well-defined fringes when observed with a polarizing microscope.

This work is novel because it explores the usage of optical techniques in

measuring stresses in next generation piezoelectric materials, which will be an

important quality assurance tool to produce robust and reliable devices in the

years ahead.

Optical methods can help to analyze electrical loading effect in piezoelectric

materials as well. An electric field applied to the piezoelectric single crystals will

cause at least three effects. First, the refractive index changes in proportion to

the electric field. This is known as linear electro-optic (EO) effect. Second, the

electric field induces internal stress/strain; this is known as the converse

piezoelectric effect. These internal stresses will induce photoelastic effects.

When the electric field is large enough, it can also pole the sample (align domains)

to induce phase transformations. Third, the refractive index changes in

proportion to the square of the electric field. This is known as the quadratic

electro-optic or Kerr effect. All of these three effects contribute to the observed

birefringence. Recently the optical properties of piezoelectric materials such as

the refractive indices have been reported [47-50]. Unpoled PMN-29%PT singe

crystals have many domains with different orientation, retaining an optically

isotropic pseudocubic state. Under this assumption, the refractive index of

unpoled PMN-30%PT single crystals is reported to be 2.501 [47]. Poled

tetragonal PMN-38%PT single crystals have an effective EO coefficient of

42.8 pm/V as reported in reference [48]. However, because applied field will

cause domain rotation and phase transformation in unpoled PMN-PT single

crystals, precise determination of the pure EO coefficients is neither possible, nor


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useful. Thus EO coefficient determination is not included in this thesis.

Besides the electric-optic effect, several researchers studied the cyclic electric

field induced effects in ferroelectric ceramics or piezoelectric single crystals

[51-56], which including domain switching, phase transformation, micro-cracking,

and fracture. The observation and study are normally carried out through TEM,

dielectric measurements, and optical microscopy, etc. Crack growth is directly

observed under the optical microscope and micro-crack growth under TEM.

Phase transformation is studied by measuring change of the dielectric properties.

These studies help reveal what is going on when the piezoelectric materials are

under electrical loading. However, if we can get to know the internal stress state

of the materials during the electrical loading, we can better understand the crack

initiation condition and crack growth. Fortunately, unpoled PMN-PT single

crystals can be polished to be transparent and show colorful birefringence. I

focused on AC/DC field-induced birefringence of samples originally without a

crack. Crack initiation caused by electric fields and phase transformations were

examined. Optical observation of phase transformations is a field where not

much research has been done and further study is necessary.

Commercial FEM software ANSYS was applied to model the experiments

and offer theoretical/computational results, helping to interpret the experiment

results that I obtained.

1.2 Photoelasticity

1.2.1 Discovery of the phenomenon of photoelasticity

Photoelasticity is a well-known efficient method to measure the internal

stresses in a variety of transparent materials. This phenomenon was first

discovered by Sir David Brewster in the year 1815 [40]. He presented a paper

before the Royal Society of London where he reported the effect. In his


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experiments, he placed a piece of glass in between two crossed polarizers. He

found that when the glass was stretched transversely to the direction of

propagation of light, the field of view grew brighter, therefore showing that an

artificial birefringence is induced in the glass by the mechanical stress.

Furthermore, he found in the case of solids which are initially birefringent, the

initial birefringence is altered by the stress. Thereafter, photoelastic techniques

were developed to study crystals and other transparent solids. It has become an

important experimental method for the measurement of internal stress.

Figure1.3 PMN-29%PT single crystal as received.

Employing this method is very simple. Namely using a crossed polarizer

set-up, we can see stresses. Bright colors such as magenta and green, as well

as closely spaced fringes imply high level stresses. With this simple rule, we can

already tell that the samples as received have large birefringence. Figure 1.3

shows an example of fringe pattern for the as received sample. This fringe

pattern results from the net birefringence associated with combining the initial

domain distribution and residual stresses. Broad faces are polished transparent

and the four edges are in as received condition. The big lobes on four corners

with many little bumps along the edges are most likely due to residual stresses

from machining operations. Usually machining stresses are compressive near

the surface and tensile inside the sample.


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1.2.2 Mathematical formulation of Photoelasticity

The stress-optic law of photoelasticity states that if there is a difference in

principal stresses along two perpendicular directions in an otherwise optically

isotropic material, the refractive index in these directions is different and the

induced birefringence is proportional to the difference. For a two-dimensional

stress state, the law simplifies to [38-39]:

( 21

= Ct

n ) (1-4)

Here 1 and 2 are the maximum and minimum principal stresses, n is

fringe order, t is the samples thickness, and is the wavelength of the incident

light. C is the stress-optic coefficient which is a constant. From Eq. (1-4), we

have:

Ct

n

22

21 =

(1-5)

Hence

Ct

n

2max

= (1-6)

The stress-optic coefficient C is useful to quantitatively analyze the internal

stresses, such as max as a function of position for any fringe pattern if we know

the fringe ordern.

Eq. (1-5) can also be rearranged to:

t

nf

Ct

n==

21 (1-7)

Here, we introduce another concept: the fringe-stress optical coefficient , wheref

Cf /= represents the principal stress difference necessary to produce a one

fringe order change in a crystal of unit thickness. The fringe-stress coefficient

depends on the stress-optical coefficient C of the material and the wavelength of

the incident light. Therefore, C is more general and convenient than allowingf


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the use of incident light with different wavelength. However, to allow easy stress

calculation and also easy comparison of optical properties with mechanical

properties, need to be evaluated. When is used for stress calculation, a

single standard wavelength should be used. While any color monochromatic light

is acceptable, ~535 nm green light was selected for the experiments reported in

this thesis. The engineering units of the fringe-stress coefficient are N/m.

f f

For anisotropic crystals, the mathematical formulation of photoelasticity is

more complex. The relevant equations are in Appendix I. As shown in Figure

1.1(b), unpoled PMN-29%PT single crystals have eight possible polarization

directions. When the number of domains is large enough, the global structure of

the unpoled sample can be treated as pseudocubic for unpoled PMN-29%PT

single crystals [51] and pseudotetragonal for poled crystals [58]. These

assumptions enable photoelastic methods to be applicable to these materials in

theory, though the experimental results may turn out differently due to the

complex domain hierarchy structures that can develop as a result of

thermal/mechanical processings history of the samples.

1.2.3 Plane polariscope and circular polariscope

The general arrangement of light fields to perform photoelastic experiments

consists of two typical arrangements: The plane polariscope and the circular

polariscope. In photoelasticity, stress fields are displayed through the use of

light. The basic arrangement of a plane polarized microscope includes a

polarizer and an analyzer, mounted with a 900 rotation between them to minimize

the transmission of light through the pair. If an isotropic material is placed

between the plates, it will not affect the intensity of the transmitted light regardless

of its angle to the polarizers. This setup, with polarizers crossed is called a dark

field plane polariscope, as shown in Figure 1.4(a). The other arrangement,


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called a bright field plane polariscope, features the polarizer and analyzer parallel

to one another and was not used in this analysis.

(a) (b)

Figure 1.4 (a) Dark field plane polariscope set-up. (b) Light vector representation; Illustrationsredrawn from a similar figure in reference [38].

In Figure 1.4(b), consider polarized light coming out the polarizer aligned with

E1 parallel to the x axis:

)cos(1

tkE = (1-8)

When entering the sample, the light vector splits to two vectors along the principal

stress axes. As the two components of light propagate through the sample, a

phase difference of is generated. Let be the slow axis and be the

fast axis, we have:

2E

3E

)2

cos(cos

)2

cos(sin

3

2

+=

=

tkE

tkE

(1-9)

After the light passes out through the analyzer, only the y axis component of the

light is visible, so:


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)sin(2

sin)2sin(

)2

cos()2

cos(cossin

sincos 324

tk

ttk

EEE

=

+=

=

(1-10)

The intensity of the light we see thus dependent on the orientation , the time

and the phase differencet for each point in the image. Regions where

)2sin( or2

sin

or )sin( t are zero are dark. The overall appearance is

similar to a contour map. These black bands in the stress patterns are known as

fringes. Namely when intensity of the light is zero, there is a fringe. Intensity is

proportional to the square of the amplitude and the time dependent term is usually

not considered:

)2

(sin)2(sin 22

ap II = (1-11)

Here represents the amplitude of the incident light and other factors

affecting the transmission light intensity. From Eq. (1-11), we can see, there are

two set of fringes superimposed over each other, isochromatics and isoclinics.

Isochromatics are caused by the incident light phase difference

aI

of 2m (here

m is an integer), or as is often said, a retardation caused by the principal stress

difference at the point. Isoclinic fringes are contours of constant inclination,

when the polarizer axis coincides with one of the principal stress directions at the

point of interest, 2/,0 = .

Use of a circular polariscope eliminates isoclinics. Two quarter-wave plates

are added to the plane polariscope with their axes at 45

0

and 135

0

to one of thepolarizers to achieve circular polarized microscopy. The details of circular

polariscope are described in Appendix II with the basic set up illustrated in Figure

A2. The result is that the intensity of light transmitted for circular dark-field only

depends on the retardation, thus only isochromatics will be seen. Circular


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polarized microscopy is used in most experiments conducted here. Plane

polarized microscopy can help to define the phase of PMN-PT single crystals by

measuring the extinction angle of the light through the crystal relative to the

direction of the polarizer.

1.3 Preliminary three-point bending experiments

1.3.1 Experimental setup

A ZeissTM Axioskop2MAT microscope was used for all the photoelasticity

experiments conducted, as shown in Figure 1.5. This microscope was modified

by addition of a rotation stage normally found on polarizing microscopes to

facilitate rotation of the sample. Light goes straight up from the bottom. A 2

megapixel camera is used to transfer the images to the connected computer, so

we can observe the images on a large monitor. When quarter wavelength

retardation plates are applied, the microscope is configured as a circular

polariscope. In preliminary three-point bending experiment, the microscope was

configured as a plane polariscope without quarter wavelength plates.

For preliminary three-point bending experiments, a parallel clamp with jaws

only 1mm tall was designed to apply the force to a sample while observing with a

polarizing microscope. This device is shown in Figure 1.6. Screws A and B are

adjusted individually to keep the loading faces parallel to each other as they are

brought together. Three semi-cylinder shaped glass rods were cut and polished,

each with 1mm height and 1 mm radius to exert loading and supporting forces.


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Figure 1.5 ZeissTM

Microscope set-up.

Figure 1.6 Preliminary three-point bending set-up.

An unpoled [001]-oriented PMN-29%PT single crystal bar was used in these

bending experiments. This crystal was obtained from H.C. Materials Corporation,

Bolingbrook, Illinois. Unless otherwise noted, crystals used for this research

were grown by H. C. Materials Corporation. The surfaces for light transmission


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were polished down to 0.05 m grit size. The dimension of the bar is 11X1.8X1

mmas measured directly from micrographs taken under a 2.5X objective lens.

The experiments are performed between crossed polarizers under plane

polarized dark field. A green light filter with 535nm wavelength was applied to

show well-defined fringes.

1.3.2 Fringe pattern

Figure 1.7 is taken at an angle of 450 to both polarizer and analyzer. This

image clearly shows isoclinic fringes on the neutral axis, caused because

principal stresses on the neutral axis are perpendicular to each other and at an

angle of 450 to the parallel and perpendicular edge directions of the three-point

bending specimen. To verify this, the principal stress field in a beam under

three-point bending was calculated using ANSYS software. Figure 1.8 shows

the resulting principal stress field displayed as vectors modeled in an isotropic

material. It is evident that the principal stress direction is at 450 to both the

polarizer and analyzer inside the solid line circles, which should be a bright region,

and is 00 to both the polarizer and analyzer inside the dash line circle, which

should be all dark under the microscope according to the photoelastic theory.

This matches the fringe pattern shown in Figure 1.7, implying that the basic

photoelastic technique works on PMN-29%PT single crystals. This supporting

result is also verified in subsequent four-point bending experiments.

Figure 1.7 Three-point bending image at 450

to both the polarizer and the analyzer.


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Figure 1.8 Principal stress vectors from ANSYS

simulation of three-point bending. Only left

half of sample is shown.

Fringe picture taken at zero degree to both polarizer and analyzer is shown in

Figure 1.9, and it shows isoclinic fringes as well. This may be easily verified by

rotating the analyzer and polarizer coordinate 450 clockwise in Figure 1.8.

Figure 1.9 Three-point bending image at zero degree to both the polarizer and the analyzer.

When the applied loading force is increased, fringes were observed

generated at the central portions of the top and bottom edges and move towards

the neutral axis. This process continues until we cause the bar to snap, usually

with as many as 25 or more fringes. The fringe pattern seen in PMN-PT single

crystals is similar to that seen in typical isotropic materials, as illustrated in Figure

1.10. The exception is the fringes resulting from Hertzian contact loading ofthree glass rods, which show a two-lobed fringe pattern. This will be further

discussed in Chapter 3.


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Figure 1.10 (a) Three-point bending of unpoled PMN-29%PT; (b) Three-point bending image

of isotropic materials [39].

1.3.3 Deflection versus fringe order

Stress vs. Fringe Order

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

Fringe Order

Deflection(m)

Figure 1.11 Deflection versus fringe order

In preliminary three-point bending experiments, we purposely put a thin glass

plate (a cover slip) on the glass rods, to obtain an edge to be used as a referenceto measure the deflection during the bending process, as shown in Figure 1.7.

Deflection versus fringe order was plotted in Figure 1.11. It is obvious that

deflection is linearly proportional to the fringe order. This implies for unpoled

PMN-29%PT single crystals, the mechanical properties characterized by the


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deflection are linearly proportional to the optical properties which control fringe

order. If the samples were isotropic and homogeneous in their optical properties

and mechanical properties, then the birefringence would be proportional to strain.

1.3.4 Summary

Through preliminary three-point bending experiments, isoclinic fringes were

observed; fringe patterns were also comparable to those of typical isotropic

materials. This means the photoelastic technique is useful and can be further

used to study internal stresses of unpoled PMN-29%PT single crystals.

However, loading force was unknown so that quantitative calculations could not

be performed; fringe patterns were observed only qualitatively while the loading

was increased. Because the elastic constants of unpoled PMN-29%PT single

crystals are also unknown, there was no way to analyze the internal stress. The

only quantitative result directly obtained from preliminary three-point bending is

the linear relationship between deflection and the fringe order. It was necessary

to design a new device which provides the same function while also allowing a

known force to be applied. To solve this problem, a BimbaTM 5/16 bore air

cylinder is used to design an in situ loading frame as shown in Figure 1.12.

Details of calibration of the loading system are provided in Appendix III. The

calibration result is that the loading force obtained from reading of the pressure

gauge is within 2.5% of the applied value.

In the following three chapters, birefringence response and internal stresses of

unpoled PMN-29%PT single crystals under mechanical loading are studied for thefirst time using photoelastic techniques in a series of sequential experiments

comprising: three-point bending experiments, four-point bending experiments,

and Hertzian contact loading experiments. In the three-point bending

experiments, the numerical value of the stress-optical coefficient of PMN-PT was


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first estimated. The apparent Youngs modulus along direction of unpoled

PMN-PT single crystals was calculated. In Hertzian contact loading experiments,

orientation dependences of fringe patterns were observed, showing the

anisotropic properties of unpoled PMN-PT single crystals. ANSYS simulations

of piezoelectric single crystals were performed, verifying that the anisotropic

elastic properties indeed cause the orientation dependence of fringe patterns that

were observed. The results were published in two papers, references [58] and

[59] respectively. To further examine the variations of stress-optical coefficients

with incremental mechanical stresses, four-point bending experiments were

performed. A paper has recently been submitted to report the results. Finally,

electric field loading experiments were performed; the results of which are

reported in Chapter 5.

Figure 1.12 3D CAD model of loading frame. BimbaTM

cylinder is mounted through a

hole in the aluminum frame.


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1.4 References

1. J. Curie, and P. Curie, Development by pressure of polar electricity in

hemihedral crystals with inclined faces, Bull. Soc. Min. de France, 3,

90-93, 1880.

2. S. Katzir, Measuring constants of nature: confirmation and determination

in piezoelectricity, Studies in History and philosophy of modern physics,

34, 579-606, 2003.

3. G. Lippmann, Principle de la conservation de l'lectricit, Ann. Chim.

Phys. Ser. 5, 24, 145-178, 1881.

4. W. Voigt, Lehrbuch der Kristallphysik, B. G. Teubner, Leipzig and Berlin,

1928.

5. K. Uchino, and J. R. Giniewicz, Micromechatronics , Marcel Dekker, Inc.,

2003.

6. J. Kuwata, K. Uchino, S. Nomura, Phase transition in the

Pb(Zn1/3Nb2/3)O3-PbTiO3 system, Ferroelectrics, 37, 579-582, 1981.

7. 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.

8. J. Zhao, Q. M. Zhang, N. Kim and T. Shrout, Electromechanical

properties of relaxor ferroelectric lead magnesium niobate-lead titanate

ceramics, Jpn. J. Appl. Phys, 34, 5658, 1995.

9. 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.

10. 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.


35/125

21

11. D. Viehland, and J. Powers, "Electromechanical Coupling Coefficient of

-Oriented Pb(Mg1/3Nb2/3)O3-PbTiO3 Crystals: Stress and

Temperature Independence", Appl. Phys. Lett. 78, 3112, 2001.

12. S.-E. Park and T. R. Shrout, "Characteristics of relaxor-based

piezoelectric single crystals for ultrasonic transducers", IEEE Trans.

Ultrason. Ferroelectr. Freq. Control. 44, 1140, 1997.

13. X. Wan, H. Xu, T. He, D. Lin, and H. Luo, "Optical properties of tetragonal

Pb(Mg1/3Nb2/3) 0.62Ti0.38O3 single crystal, J. Appl. Phys. 93, 4766, 2003.

14. Y. Barad, Y. Lu, Z.-Y. Cheng, S.-E. Park, and Q. M. Zhang, Composition,

temperature, and crystal orientation dependence of the linear

electro-optic properties of Pb(Zn1/3Nb2/3)O3-PbTiO3 single crystals, Appl.

Phys. Lett. 77, 1247, 2000.

15. H. Fu, and R. Cohen, Polarization rotation mechanism for ultrahigh

electromechanical response in single-crystal piezoelectrics, Nature 403,

281, 2000.

16. H. Y. Wang and R. N. Singh, Crack propagation in piezoelectric ceramics:

effects of applied electric fields, J. Appl. Phys. 81, No. 11, 7471, 1997.

17. S. Kumar and R. N. Singh, Crack propagation in piezoelectric materials

under combined mechanical and electrical loadings, Acta Mater., 44, No.1, 173, 1996.

18. Z. Suo, C. M. Kuo, D. M. Barnett, and J. R. Willis, Fracture mechanics for

piezoelectric ceramics, J. Mech. Phys Solid, 40, No. 4, 739, 1992.

19. T. Y. Zhang and J. E. Hack, Mode-III cracks in piezoelectric materials, J.

Appl. Phys. 71, No. 12, 5865, 1992.

20. S. B. Park and C. T. Sun, Effect of electric field on fracture of

piezoelectric ceramics, International Journal of Fracture, 70, 203, 1995.

21. Y. E. Pak, Crack extension force in a piezoelectric material, J. Appl.

Mech. 57, 647, 1990.


36/125

22

22. F. X. Li, S. Li and D. N. Fang, Domain switching in ferroelectric single

crystal/ceramics under electromechanical loading, Materials Science &

Engineering B, 120, 119-124, 2005.

23. Z. K. Xu, "In situ tem study of electric field-induced microcracking in

piezoelectric single crystals", Materials Science & Engineering B, 99,

106-111, 2003.

24. C. S. Lynch, W. Yang, L. Collier, Z. Suo and R. M. McMeeking, Electrical

field induced cracking in ferroelectric ceramics, Ferroelectrics, 166, 11-30,

1995.

25. H. Makino and N. Kamiya, Effects of DC electric field on mechanical

properties of piezoelectric ceramics, Jpn. J. Appl. Phys, 33, 5323, 1994.

26. H. C. Cao and A. G. Evans, Electric-field-induced fatigue crack growth in

piezoelectrics, J. Am. Ceram. Soc, 77, No. 7, 1783, 1994.

27. 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.

28. 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.

29. 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.

30. S. W. Choi, T. R. Shrout, S. J. Jang and A. S. Bhalla, Morphotropic phase

boundary in Pb (Mg1/3Nb2/3) O3-PbTiO3 system, Mater. Lett., 8, 253-255,

1989.

31. X. M. Wan, X. Y. Zhao, H. L. W. Chan, C. L. Choy and H. S. Luo, Crystal

orientation dependence of the optical bandgap of (1-x) Pb(Mg1/3Nb2/3)

O3-x PbTiO3 single crystals, Materials chemistry and physics, 92,

123-127, 2005.


37/125

23

32. C. S. Tu, R. R. Chien, F. T. Wang, V. H. Schmidt and P. Han, Phase

stability after an electric-field poling in Pb(Mg1/3Nb2/3)1-xTixO3 crystals,

Physical review B, 70, 220103(R), 2004.

33. Y. Guo, H. Luo, D. Ling, H. Xu, T. He, and Z. Yin, The phase transition

sequence and the location of the morphotropic phase boundary region in

(1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 single Crystal, J. Phys.: Condensed

Matter. 15, 77-82, 2003.

34. F. Bai, J. Li, and D. Viehland, "Domain hierarchy in annealed

(001)-oriented Pb(Mg1/3Nb2/3)O3-x%PbTiO3 single crystals", Appl. Phys.

Letters, 85, No. 12, 2313-2315, 2004.

35. C. M. Landis, Non-linear constitutive modeling of ferroelectrics, Current

opinion in Solid State and Materials Science, 8, 59-69, 2004.

36. H. Berger, U. Gabbert, H. Koppe, R. R.-Ramos, J. B.-Castillero, R.

G.-Diaz, J. A. Otero and G. A. Maugin, Finite element and asymptotic

homogenization methods applied to smart composite materials,

Computational Mechanics, 33, 61-67, 2003.

37. J. W. Dally and W. F. Riley, Experimental Stress Analysis, McGraw-Hill,

Inc., 1991.

38. K. Ramesh, Digital Photoelasticity Advanced Techniques andApplications, Springer, 2000.

39. M. M. Frocht, Photoelasticity, VI, New York, John Wiley & Sons, Inc.

1941.

40. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of

Crystals, Plenum Press, 1981.

41. J. Beynon, Introductory University Optics, Prentice Hall, 1996.

42. F. L. Pedrotti, S.J., L. S. Pedrotti, Introduction to Optics, Prentice Hall,

1993.

43. 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, 1991.


38/125

24

44. 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.

45. 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, 377-380, 2004.

46. Y. Lu, Z. Y. Cheng, Y. Barad and Q. M. Zhang, Photoelastic effects in

tetragonal Pb(Mg1/3Nb2/3)O3-x%PbTiO3 single crystals near the

morphotropic phase boundary, J. Appl. Phys. 89, No. 9, 5075-5078,

2001.

47. X. M. Wan, H. L. W. Chan and C. L. Choy, Optical properties of

(1-x)Pb(Mg1/3

Nb2/3

)O3-xPbTiO

3single crystals studied by spectroscopic

ellipsometry, J. Appl. Phys. 96, No. 3, 1387-1391, 2004.

48. X. M. Wan, H. Q. Xu, T. H. He, D. Lin and H. S. Luo, Optical properties of

tetragonal Pb(Mg1/3Nb2/3)0.62Ti0.38O3 single crystal, J. Appl. Phys. 93, No.

8, 4766-4768, 2003.

49. Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park and Q. M. Zhang, Composition,

temperature, and crystal orientation dependence of the linear

electro-optic properties of Pb(Zn1/3Nb2/3)O3-PbTiO3 single crystals,

Applied Physics Letters, 77, No. 9, 1247-1249, 2000.

50. X. M. Wan, D.Y. Wang, X. Y. Zhao, H. S. Luo, H.L.W. Chan and C.L. Choy,

Electro-optic characterization of tetragonal

(1x)Pb(Mg1/3Nb2/3)O3xPbTiO3 single crystals by a modified snarmont

setup, Solid State Communications,134, No. 8 , 547-551, 2005.

51. 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.

52. 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.
http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=5545&_auth=y&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5f86fc6f24dbd3a66b604fd6939d07e6http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235545%232005%23998659991%23593370%23FLA%23&_auth=y&view=c&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5afdd3541d8d9cd752cbde6ccd139186http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235545%232005%23998659991%23593370%23FLA%23&_auth=y&view=c&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5afdd3541d8d9cd752cbde6ccd139186http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235545%232005%23998659991%23593370%23FLA%23&_auth=y&view=c&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5afdd3541d8d9cd752cbde6ccd139186http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235545%232005%23998659991%23593370%23FLA%23&_auth=y&view=c&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5afdd3541d8d9cd752cbde6ccd139186http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=5545&_auth=y&_acct=C000022660&_version=1&_urlVersion=0&_userid=483663&md5=5f86fc6f24dbd3a66b604fd6939d07e6


39/125

25

53. Z. R. 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)-0.32Ti0.38O3 single crystal under electric field, J.

Appl. Phys. 104, 024112, 2008.

54. Z. K. Xu, In situ TEM study of electric field-induced microcracking in

piezoelectric single crystals, Materials Science and Engineering B, 99,

106-101, 2003.

55. E. T. Keve and K. L. Bye, Phase identification and domain structure in

PLZT ceramics, J. Appl. Phys. 46, 87, 1975.

56. F. X. Li, Shang Li, and D. N. Fang, Domain switching in ferroelectric

single crystal/ceramics under electromechanical loading, Materials

Science and Engineering B, 120, 119-124, 2005.

57. 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.

58. 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.

59. N. Di, J. C. Harker and D. J. Quesnel, Photoelastic effects inPb(Mg1/3Nb2/3)O-29%PbTiO3 single crystals investigated by Hertzian

contact experiments, J. Appl. Phys. 103, 053518, 2008.


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2 Photoelastic study using three-point bending technique

Abst ract: Photoelastic effects in an unpoled PMN-29%PT single crystal

beam have been investigated using three-point bending experiments. A linear

relationship between the applied load and the measured displacement was

observed up to a proportional limit of ~30 MPa. Beyond this proportional limit,

yielding was observed. Samples were loaded as high as 77 MPa without fracture.

Young's modulus Y ~1.9X1010 N/m2 was determined directly from the initially

linear region using beam theory. The photoelastic fringe order versus fiber

stress plot also displays an initially linear region up to a proportional limit of ~20

MPa, suggesting that optical measurements are a more sensitive measure of the

onset of microplasticity than mechanical measurements. Residual photoelastic

fringes associated with yielding were completely removable by annealing above

the Curie temperature, implying that plastic deformation occurs by reversible

processes such as domain switching and phase transformation. The

stress-optical coefficient for unpoled PMN-29%PT determined from the initially

linear region of the fringe order versus fiber stress curve is 104X10-12

Pa-1

. This

value is large and comparable with the stress-optical coefficient of polycarbonate,

making unpoled PMN-29%PT single crystal a good candidate for optical stress

sensors and acousto-optic modulators.

2.1 Introduction

Relaxor ferroelectric single crystals exhibit ultrahigh dielectric and

piezoelectric properties compared with conventional piezoelectric ceramics.

Materials such as PMN-PT single crystals have become a next generation of

piezoelectrics that have attracted constant attention in recent years [1-5]. These

materials are finding wide-ranging applications in medical imaging, active noise


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suppression, and acoustic signature analysis. Residual stresses and internal

stresses in PMN-PT single crystals, however, can reduce the performance of

devices and lead to the initiation of cracks.

Residual stresses are induced by inhomogeneous strain. Inhomogeneous

strain may be produced by thermal gradients during crystal growth [6-7], by phase

transitions during cooling [6-8], and by mechanical cutting and finishing

operations during device fabrication [9]. When the size scale of the residual

stress distribution approaches the size scale of the microstructure, residual

stresses are often referred to as internal stresses or microstresses. Clearly, the

presence of a stress distribution within a component will influence its response to

applied loadings. To better understand and control stresses in relaxor

ferroelectrics devices, it is necessary to monitor internal stresses and residual

stresses.

Photoelasticity is an efficient and effective method to measure the residual

stresses and applied stresses in many transparent materials. It offers both

quantitative determination and qualitative observation of the stress distribution in

a sample [9-12]. Simply by examining a sample between crossed polarizers in

either the loaded or unloaded state, we can observe the influence of stress as a

result changes in optical birefringence. High-order pastel colors from the

Michel-Levy interference color chart, such as magenta and green, in combination

with closely spaced fringes, imply high stress levels and high stress gradients.

Quantitative evaluation of stress level requires that we measure the retardation

caused by stress and relate this to the stress-optical properties of the material

and the length of the optical path.

The stress-optic law of photoelasticity states that if there is a difference in

principal stresses along two perpendicular directions in an otherwise optically

isotropic material, the refractive index in these directions is different.


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Fundamentally, this effect arises from a change in the spacing between the atoms

due to strains induced by the principal stresses. The difference in refractive

index is an induced birefringence which is proportional to the difference in

principal stresses. The maximum and minimum refractive index directions are

aligned with the principal stresses.

For a two-dimensional plane stress state, the stress-optic law for an isotropic

material can be expressed as [12, 13]:

)( 21

= Ct

n . (2-1)

Here 1 and 2 are the maximum and minimum principal stresses, is the

fringe order, and is the sample's thickness along the optical path.

n

t is the

wavelength of the incident light and is a constant known as the stress-optical

coefficient. From Eq. (2-1), we have

C

Ct

n

22

21 =

, (2-2)

where both sides are divided by 2 to produce the form of the maximum shear

stress,

Ct

n

2max

= (2-3)

Once the stress-optical coefficient C is known for a given material, it can be

used to quantitatively evaluate max for a fringe pattern, provided we know the

fringe order. Photoelastic fringe patterns suitable for stress analysis are easily

recorded with the use of a circular polariscope and a monochromatic filter. The

fringe order may be found by locating a zero-order fringe and counting. Zero orderfringes in bending samples occur along the neutral axis. More complete details

of photoelastic methods for isotropic materials can be found in references [12,

13].

The purpose of this chapter is to explore photoelastic techniques for the


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investigation of stress distributions in unpoled PMN-29%PT single crystals. We

present the results of investigations performed using three-point bending

experiments. While the optical properties of unpoled PMN-29%PT are not

isotropic, fringe patterns are comparable with those typical of isotropic materials.

Experimental results show that there exists a linear relationship between loading

force and displacement and between fringe order and fiber stress within a

proportional limit. Beyond the proportional limit, yielding takes place. Yielding is

interpreted as stress-induced domain switching. Residual stress remaining after

unloading can be removed by annealing above the Curie temperature suggesting

that these switches are reversible. The linear relationships observed suggests

that photoelastic methods can be used more generally for these materials. The

use of optical techniques to measure stresses in next generation piezoelectrics

will be an important quality assurance tool to produce robust and reliable devices.

2.2 Experimental procedure

An in situ loading frame built to perform photoelastic measurements on

small-size beams is shown in Figure 2.1. Figure 2.1(a) illustrates the loading

frame below the objective of a ZeissTM microscope configured as a circular

polariscope while Figure 2.1(b) provides a top view of the three-point bending

set-up. Mechanical loading is applied using a BimbaTM pneumatic cylinder

shown in Figure 2.1(a), and three 1mm radius glass rods illustrated in Figure

2.1(b). The in situ load frame was calibrated so that the applied force could be

obtained directly from the reading of a pressure gauge within 2.5% accuracy.The loading frame allows a 10X objective lens to be used to make deflection

measurements of the beam during bending.

A schematic of the three-point bending loading system is presented in Figure

2.2. Principal faces of the beam are parallel to the (100), (010), and (001)


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planes, respectively. The thickness and height are approximately equal at

t=1.06 mm and h=1.07 mm, enabling the beam to be bent in either direction by

changing the direction of the applied load P. An experiment in which P is aligned

with [100] is called [100] bending as shown in Figure 2.2. The small size of the

experimental set-up means that the exact placement of the loading rods will vary

from one run to the next. Specific values of the overall span length and the

numerical values of a and b were measured directly from micrographs taken

using a 5X objective lens.

Figure 2.1 (a) in situ loading frame working under microscope and (b) in situ loading

frame with three-point bending set-up as indicated by the arrow.

Figure 2.2 three-point bending schematic. P is the loading force, c and t are the

compression and tension fiber stress. and are the reaction loads.1R 2R


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The unpoled PMN-29%PT single crystal beam (H.C. Materials Corporation)

was polished using an Allied High Tech Multi-PrepTM polishing system following

the rule of threes. Fixed abrasive diamond films in progressively finer sizes,

each removing a thickness of three times the diameter of the previous abrasive,

were followed by a final polish using 0.05 m colloidal silica on each of the four

major faces. Sufficient material was removed between each abrasive step so

that no damaged material from the previous abrasive remained after each

polishing step. The final surfaces were of optical quality.

Figure 2.3 Unpoled PMN-29%PT single crystal beam viewed in crossed polars: (a) (100)

face, as-received; (b) (100) face, after annealing; and (c) (010) face, after annealing.

Polarizer and analyzer are horizontal and vertical, respectively. Strong colors in (a) indicate

regions of net birefringent retardation.

Residual stresses, net birefringence from initial domain distributions, or a

combination of these are apparent in the as-received sample as indicated by the

bright, low order color fringe patterns shown in Figure 2.3(a). For proper

photoelastic measurements, an initially stress-free sample with no net

birefringence is desired. It was found that annealing at 400 oC for 1 hour

substantially reduced the residual stresses and the apparent initial birefringence.

The observed fringe order looking through the (100) face was reduced to ~0.45

for the white regions [13] and 0 for black regions, as shown in Figure 2.3(b). The

fringe order of the (010) face shown in Figure 2.3(c) was reduced to ~0.28, the

fringe order associated with gray color as indicated in reference [13]. It is

important to note that the entire sample does not show uniform extinction as

would be the case for an isotropic material. Different faces show different


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extinction levels at nearly equal thicknesses implying the crystal is optically

anisotropic, perhaps a result of crystal growth [14].

Photoelastic experiments were performed using a ZeissTM optical microscope

configured as a dark field circular polariscope. This configuration eliminates

isoclinic fringes and thus produces only isochromatic fringes. Isochromatic

fringes depend only on the magnitude of the principal stress differences at each

point, greatly simplifying the analysis. A monochromatic green filter with

wavelength ~535 nm was used to record photographs for the evaluation of the

fringe order. Fringe order was counted from 5X objective lens images at the

point of maximum tensile stress on the free surface of the sample (point A) as

shown in Figure 2.2. Fractional fringes were estimated to the nearest 0.3 using

graphical intensity information from image analysis software. The displacement

of the sample relative to a fixed reference was measured directly from 10X

objective lens images at the same location A. Bending was performed on both

(100) and (010) faces for comparison, even though crystallographic symmetry

consideration suggests the results should be identical. Between each

experiment, the beam was annealed to remove all residual fringes, allowing the

same beam to be used again and again.

2.3 Results and discussion

2.3.1 Fringe patterns

Figure 2.4 shows photoelastic images obtained at different load levels for [100]

bending. Figure 2.4(a) shows the unloaded state, Figure 2.4(b) and Figure 2.4(c)

show examples of well-defined fringes obtained under increasing load, and Figure

2.4(d) shows residual fringes when the load is removed. These fringe patterns,

obtained using monochromatic green illumination under circular polariscope, are

comparable to those typical of isotropic materials [12]. During the experiments, it


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could be seen that fringes originated at the central portions of the top and bottom

edges and moved inward toward the neutral axis with increasing load. The

fringes formed in pairs, producing the relatively symmetric patterns shown in

these figures. Fringes of increasing fringe order distribute uniformly along the

height of the beam, corresponding to a linear variation of the principal stress along

the thickness, as shown in Figure 2.2. The upper half of the beam is in

compression, while the lower half is in tension. The zero-order fringe always lies

along the neutral axis which is stress free according to elementary beam theory.

We can see clearly from Figure 2.4(b) that the observed fringe pattern is

asymmetric at low loads: the zero- order fringe exists only on the right portion of

the neutral axis, corresponding to the region showing exactly zero fringe order in

the unloaded state. Thus the fringe patterns we observe from applied loading

are qualitatively consistent with fringe patterns that would have been obtained

from an isotropic sample. Note that in three-point bending experiments, the

principal stress x on the outer surface at point A is the maximum tension stress

max , known as the fiber stress, and y equals zero as a result of the free

surface boundary condition. Thus, the fringe order is directly proportional to fiber

stress x according to Eq. (2-1).

Figure 2.4 (a) Initial fringe pattern showing 0.5 fringe at point A on the free surface opposite

the loading point. (b) Second-order fringes at A, (c) Sixth-order fringes at A, and (d) first-order

fringe remaining at A after the load is released.


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2.3.2 Loading force versus deflection

Figure 2.5 illustrates the applied compressive load as a function of the

displacement measured at the center of the beam. Two data sets for [100]

bending and one data set for [010] bending are shown. The results are highly

repeatable, independent of the orientation of the bending, indicating that the

mechanical properties are the same for both orientations as we would expect

given the nearly identical dimensions of the samples. The force depends linearly

on the displacement over the initial portion up to a proportional limit which implies

that the loading induces elastic deformation in this regime. The proportional limit

is approximately 2 N, which corresponds to a fiber stress of 25 - 30 MPa, for the

sample geometries and span lengths used in these tests. Beyond the

proportional limit there is yielding after which the data appears to continue upward

with a reduced slope.

Figure 2.5 Force versus deflection during increasing load for three experimental runs.


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Figure 2.6 Force versus deflection with polynomial fit curve.

The slope in Figure 2.5 was obtained in terms of least-square fit based on

data below the proportional limit. The correlation coefficient is 0.988. As shown

in Figure 2.6, with a fourth-order polynomial fit for all the data, the resulting curve

provides a simple and reliable way to determine the proportional limit rather than a

simple visual inspection. The yield stress of 25 - 30 MPa is comparable to 20

MPa reported by Viehland for PMN-32%PT single crystals [15].

We interpret this yielding effect as the result of stress induced domain

switching that occurs throughout the sample, spreading from the high stress

surfaces as a result of the stress gradients. Plastic deformation represents a

reorientation of the polarization of the nanodomains distributed throughout this

otherwise unpoled sample. Essentially, the stress is changing the population of

the dipoles of the eight possible orientations of the unpoled sample [16], leadingto strain. At these modest stresses, the sample does not undergo large scale

mechanical poling or stress induced phase transformations that are possible in

this system. We can say this because the yielding was not accompanied by the

massive changes in optical properties expected from phase transitions. Rather,


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the development of the photoelastic fringe patterns proceeded smoothly through

the yielding region as described below. Thus, it seems appropriate to assign the

yielding phenomena shown in Figure 2.5 to distributed domain reorientation (i.e.

domain switching). By this process, the yielded portions of the sample have

been mechanically poled.

It is also possible that the apparent yielding we see is, in part, the result of

concentrated strains that occur at the loading points as a result of the Hertzian

contact stresses. These large contact stresses could be sufficient to trigger

stress induced phase transformations in the neighborhood at the loading points.

More work is needed to assess the relative importance of this contribution.

2.3.3 Stress-optical coefficient

From elementary beam theory, the fiber stress during three point bending is

expressed as:

2

maxmaxmax

6

Lth

Pab

I

yM== (2-4)

Here is maximum bending moment at the location of point A and

is area moment of inertia of the beam cross section. P is the

loading force, while a , , , and represent dimensions as shown in Figure 2.2.

maxM

12/3thI =

b h t

L is the span length, namely ( +b ). is negative with a numerical value

equal to half the height h at location A. The stress

a maxy

max calculated from Eq. (2-4)

represents the principal stress 1 at point A since 2 is zero there.


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Figure 2.7 Fringe order versus fiber stress. The stress-optical coefficient is calculated

from the slope of the proportional region.

Figure 2.7 displays the fringe order versus the fiber stress. The slope was

obtained in terms of a least-square fit based on data below the yield stress

obtained from Figure 2.6. The correlation coefficient of the linear fit is 0.9774.

Figure 2.7 shows the same trends between the fringe order and the fiber stress as

that between the loading force and the deflection, only the proportional limit

occurs earlier in the data set. The fringe order, characterizing the optical

properties, is a more sensitive indicator of the deviation from linearity than the

displacement. The proportional limit in fringe order versus stress is 20 MPa

compared to 25-30 MPa discussed earlier for load versus displacement.

Here again, the data is highly repeatable, particularly the data from the same

type of bending. Difference between the two orientations of the bending may be

attributed to the initially different birefringence at zero applied load. Namely, the

optical properties are different for [100] and [010] experiments because the initial

domain distributions are different for these two cases. At stresses below the


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proportional limit, fringe order is linear with the fiber stress showing a photoelastic

effect. The explanation for the optical yielding should be the same as for the

mechanical yielding. Namely, it should be the result of distributed domain

switching and possibly locally phase transformation at the loading points.

From the slope of the linear region in Figure 2.7, 0.2077 MPa-1, we calculate

the stress-optical coefficient using Eq. (2-1):

MPa

12077.0=

tC(2-5)

Here is the thickness of the beam through which the light passes, 1.06 mm

to 1.07 mm, depending on the orientation.

t

is the wavelength of the green

filter, ~535 nm. The stress-optical coefficient C is calculated as 104X10-12

Pa-1

.

Stated another way, approximately 2.4 MPa of shear stress (4.8 MPa of principal

stress difference) will induce one order of fringe in the nominally 1 mm thick

samples reported here. Fringes represent regions of constant shear stress for

each fringe order whose values can be determined using Eq. (2-3).

2.3.4 Youngs modulus

From beam bending theory, the load P and the displacement A of the point

A are related by

)(

6222 baLab

LEIP A

=

(2-6)

where E is Young's modulus along the [001] direction. From the slope of

Figure 2.5 below the proportional limit, the Young's modulus is 1.8-1.9X1010 N/m2.

Similar results were obtained by Viehland and Li [15], where the Young's modulus

for PMN-30%PT single crystal along is reported to be 2X1010 N/m2, much

lower than the ~15X1010 N/m2 value reported for the direction or the

~7.5X1010 N/m2 value reported for polycrystalline material of the same chemistry.


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After bending, it was usual to observe residual fringes remaining in the beam

after release of the loading force, as shown in Figure 2.4(d). The higher the

loading force applied, the higher the fringe order remaining after unloading. It is

hypothesized that domain switching and perhaps local phase transformations lock

the stresses inside the beam by producing inhomogeneous strains which are

larger in those regions further from the neutral axis.

Experimentally, it was found that annealing can remove the residual fringes.

Annealing at 400 oC for one hour was sufficient to remove all residual fringes and

restore the initial fringe pattern. This means that any stress induced domain

switching or possible phase transformations caused by the bending experiments

are reversible.

2.4 Summary

Three-point bending experiments were performed on an unpoled

PMN-29%PT single crystal. The crystal was restored to its initial condition

between bending experiments by annealing for one hour at 400 oC. The

relationship between the load and the displacement and between the fringe order

and the fiber stress is linear below a proportional limit. Beyond that proportional

limit, stress induced domain switching (mechanical poling) can explain the

apparent yielding. The stress-optical coefficient of the unpoled PMN-29%PT is

approximately 104X 10-12

Pa-1

, higher than the values for materials used in

photoelastic stress analysis such as polycarbonate, 82X 10-12 Pa-1 [17]. Young's

modulus determined from the present experiment is 1.8 - 1.9X 10

10

N/m

2

. Sinceannealing removes all the residual fringes, the inhomogeneously distributed

domain switching responsible for the residual fringes must be reversible.


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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)O3-0.38PbTiO3single 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, 33133322, 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.


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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, 37

na di - dissertation

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