Developing a Non-Destructive EvaluationTechnique Using Resonance UltrasoundSpectroscopy for Fission Based Target

Thesis presented to the Faculty of the Graduate School at theUniversity of Missouri-Columbia

In partial Fulfillment of the Requirements of the Degree Master ofScience

byAbu Rafi Mohammad Iasir

Dr. Gary Solbrekken, Thesis SupervisorDecember 2015

The Undersigned, appointed by the Dean of the Graduate School,have examined the thesis entitled

Developing a Non-Destructive EvaluationTechnique Using Resonance UltrasoundSpectroscopy for Fission Based Target

presented by Abu Rafi Mohammad Iasir,

a candidate for the degree for Master of Science,

and hereby certify that, in their opinion, it is worthy of acceptance.

Professor Gary Solbrekken

Professor John Gahl

Professor Patrick Pinhero

Acknowledgements

I would like to thank my supervisor Dr. Gary Solbrekken for his assistance and ad-

vice throughout this project. Dr. Solbrekken challenged me to learn new technique

and also taught me how to cope with new challenges of research. As an advisor

he has allowed all the freedom I needed, and providing necessary guidance.

I am extremely appreciative of my Friends and family who have supported me

and suffered through my detailed monologues on the new era of Nuclear Engineer-

ing. My thanks goes to my colleagues Dr. Shrisharan Govindarajan, Annemarie

Hoyer and Casey Jesse who provided much needed camaraderie and assisted me

to retain my sanity in our windowless laboratory in the basement through all the

rain and sun we never saw. I would like to thank John Kennedy for his help

throughout this project regarding Abaqus simulation.

Research is more than designing experiment or simulation, it is also discovering

your ability to push the boundary. Thanks to all my friends for making my time

outside classroom and lab so interesting.

ii

Contents

Acknowledgements ii

List of Figures v

List of Tables vii

Nomenclature viii

Abstract xi

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Non-Destructive testing of Irradiated Materials: Literature Re-view 52.1 Radiation Effects in Materials . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Radiation Effects on Elastic Constants . . . . . . . . . . . . 8

3 RUS: Mathematical Development 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Choice of Basis Function . . . . . . . . . . . . . . . . . . . . 173.2.2 Lagrangian in Matrix Form . . . . . . . . . . . . . . . . . . 183.2.3 Generalized Eigenvalue Equation . . . . . . . . . . . . . . . 203.2.4 Example:Calculation of Resonance frequencies of Rectangu-

lar parallelepiped . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Basics of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Elastic Isotropy Conditions . . . . . . . . . . . . . . . . . . 263.3.2 Engineering Constants . . . . . . . . . . . . . . . . . . . . . 28

3.4 Hollow Tube Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Classical Solution of Vibrating rings and tubes . . . . . . . . . . . . 32

4 RUS simulation and experiment 354.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.3 Why use a Lock-in amplifier? . . . . . . . . . . . . . . . . . 38

4.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

iii

5 Results and Analysis 425.1 Identification of Natural Frequencies . . . . . . . . . . . . . . . . . 425.2 Calculation of the Elastic Constant . . . . . . . . . . . . . . . . . . 455.3 Mode Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Contact Pressure and RUS 536.1 Thermal Contact Resistance . . . . . . . . . . . . . . . . . . . . . . 536.2 Contact Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3.1 Target Assembly . . . . . . . . . . . . . . . . . . . . . . . . 566.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Conclusions, Recommendations and Future Work 637.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A XYZ Algorithm 70

iv

List of Figures

2.1 Effect of fast-neutron irradiation on the tensile properties of reactorsteels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Fractional change in Young’s modulus of 99.99% copper as a func-tion of exposure to 1-Mev electrons [9] . . . . . . . . . . . . . . . . 8

2.3 Variation of elastic constants with neutron dose for isotropic graphite[15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Variation of poisson’s with neutron dose for isotropic graphite [15] . 102.5 Dimensional change of neutron irradiated graphite [35] . . . . . . . 112.6 Young’s modulus change of neutron irradiated graphite versus neu-

tron fluence [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Change in velocities of shear and longitudinal waves in steel(A533B-

1) before and after irradiation [21] . . . . . . . . . . . . . . . . . . . 13

3.1 Flow chart of the mathematical development of a typical RUS problem 213.2 Dimension associated with rectangular parallelepiped(RP) . . . . . 223.3 Stress component on a particular point . . . . . . . . . . . . . . . . 243.4 Symmetry logic proves that ε61 = −ε61 = 0 in an isotropic medium . 263.5 Isotropic object subjects to pressure P . . . . . . . . . . . . . . . . 29

4.1 RUS setup to detect defects in ball bearing of several centimetersin diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 RUS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Schematic Diagram of the RUS experiment for the data acquisition 384.4 RUS measurements taken for a wide range of frequency from 1 KHz

to 3 MHz for hollow cylindrical shape . . . . . . . . . . . . . . . . 384.5 Block diagram of a lock-in amplifier . . . . . . . . . . . . . . . . . . 394.6 Mesh of the cylindrical shell . . . . . . . . . . . . . . . . . . . . . . 41

5.1 An example of measuring Q factor using Lorentzian distribution . . 435.2 Resonance spectrum of a perfect bearing element(a), one with small

flaw (b), and one with large splitting (c) . . . . . . . . . . . . . . . 445.3 Algorithm used to invert the resonance frequencies to elastic constants 465.4 Behavior of the modes with the change of the Poisson’s ratio . . . . 495.5 Mode shapes and their harmonics for the lowest modes for L/D=1.56(ν =

0.33). The shading intensity is proportional to the element displace-ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.6 Mode shapes and their harmonics for the lowest modes for L/D =1.65(ν = 0.33). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.7 Mode shapes for the lowest mode for L/D=6.4(ν = 0.33). . . . . . . 51

v

5.8 Behavior of mode depend on ν(Poisson’s ratio) normalized to thefirst torsional mode(LB1, first longitudinal bending mode; SA 1,Symmetric axial mode; F1, flexural mode ). . . . . . . . . . . . . . 52

6.1 Temperature drop due to thermal contact resistance . . . . . . . . . 536.2 Magnified profile of two surfaces in contact with changing contact

pressure(A)less contact pressure (B)higher contact pressure . . . . . 556.3 Draw plug Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4 a)Inner and Outer tubes together b)Two concentric cylinders before

deformation c)After assembling the target . . . . . . . . . . . . . . 576.5 Dimensions(inches) of a draw plug(48) . . . . . . . . . . . . . . . . 586.6 Plot of pressure during swagging using two different plug sizes . . . 586.7 RUS spectrum of the outer tube . . . . . . . . . . . . . . . . . . . . 596.8 RUS spectrum of the inner tube . . . . . . . . . . . . . . . . . . . . 606.9 RUS spectrum of the targets . . . . . . . . . . . . . . . . . . . . . . 616.10 RUS spectra of outer tube, inner tube and one of the targets . . . . 62

vi

List of Tables

5.1 Experimentally determined frequencies and quality factor . . . . . 425.2 Measured and Computed natural frequencies in MHz for the alu-

minum cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1 Weights of the three different targets . . . . . . . . . . . . . . . . . 576.2 Highest amplitudes in the interested frequency zone . . . . . . . . 61

vii

Nomenclature

α attenuation coefficient of ultrasonic wave

δii′ Kronecker delta

εij strain tensor

η unit vector normal to surface

Γ stiffness matrix

Λ inner to outer diameter ratio

λ basis set, Lame constant

Z+ positive integer

L Lagrangian

S free surface

Sijkl complience constants

µ Lame constant

ν Poisson’s ratio

ω angular frequency

ωr reference frequency

ωs signal frequency

φ radiation fluence

Φλ shape function

< real number

ρ density

σ RMS surface roughness

σij stress component

σY S yield stress

viii

A Area

A regression coefficient

a center line radius of the ring

aiλ generalized constant

B barreling

B regression coefficient

C velocity of ultrasonic wave through a metal

c radius of the cross-section

c2 Vickers correlation coefficient

Cl longitudinal sound wave velocity

Ct transverse sound wave velocity

Cijkl stiffness tensor

D diameter

d1, d2, d3 sides of a rectangular parallelepiped

E elastic modulus

Ek kinetic energy

Ep potential energy

F flexural

f0 first torsional mode

fext extensional vibration frequency

fflex flexural vibration frequency

ftorsion torsional frequency

FB flexura bending

h tube wall thickness

h1, h2 ultrasonic pulse amplitude from wall 1 and 2

Hc contact microhardness, MPa

hc contact conductance, W/m2K

k harmonic mean conductivities

ksh shape factor

ix

L length

LB longitudinal bending

n power factor

n wave length on the circumference

P applied pressure

p pressure

Q quality factor

q′′

heat transfer

R thermal contact resistance

S longitudinal shearing

SA symmetric axial

T thickness, temperature

T torsional

t time

t1, t2 propagation time of the wall 1,2

tanθ mean absolute slope of the surface profile

V volume

Vr reference voltage

Vs signal voltage

Vpsd voltage output from phase sensitive detector

s surface vector

u displacement vector

x

Abstract

The purpose of this thesis is to study Resonance Ultrasound Spectroscopy(RUS)

and it’s potential to evaluate the change in interfacial thermal resistance due to ir-

radiation. Resonant Ultrasound Spectroscopy is conventionally used to determine

the material properties of elastic bodies. It is a nondestructive technique that is

very capable of extracting the elastic constants for a complete anisotropic mate-

rial. Finite Element Method(FEM) is used to determine the natural frequency of

a hollow cylinder. FEM was used due to the shape of the object. An experimental

system was developed to capture the resonant frequencies of a hollow cylinder

which is similar to Molybdenum-99 target. After successfully determining the res-

onance frequencies from the spectra, the frequencies were inverted to the elastic

constants using the finite element model. Radiation effects on elastic constants

was also studied. An investigation was made to assess the usefulness of RUS in in

evaluating radiation damage of materials. An experimental study was also com-

pleted to analyze the differences in RUS spectra in a contact pressure analysis

between two cylinders of a Molybdenum-99 target.

xi

Chapter 1

Introduction

1.1 Introduction

Resonant Ultrasound Spectroscopy (RUS) is a measurement technique for mate-

rial properties of an elastic object that uses resonant frequencies of the material.

Resonant frequencies are usually obtained through a very simple experiment us-

ing two or more ultrasonic transducers. For anisotropic materials, it is capable

of extracting all 21 elastic constants. Exciting the object and recording the re-

sponses are the two major parts of the experiment. The main advantage of RUS

is the simplicity of the experiment. Theoretically, the vibration spectrum of an

elastic object contains much information about the object, both microscopic and

macroscopic. Thermodynamic properties e.g. entropy and Helmholz free energy

can also be measured using RUS [39] [1].

Obtaining the elastic constants from the spectrum is usually referred as the

”inverse problem” in RUS. The ”forward problem” is the solution of the math-

ematical model which depends on the shape of the object. This means that a

numerical approximation is used to predict the resonant spectrum of the object,

and this approximation is compared to the measured spectrum. If the measured

and calculated spectrum do not agree within the acceptable limits, the parameters

used in the calculation are adjusted and the computation is redone until the error

reaches to a tolerable limit. This iterative process is the basis of the RUS method.

Initially the RUS method was limited to particular types of shapes, with high

symmetry such as cylinders, spheres, and rectangular parallelepipeds (RP) [28].

1

But more recently attempts have been made to study arbitrarily shaped objects

by Maynard [29]. Study of carbon nanotubes was the first attempt to study the

hollow geometry [27].

1.1.1 History

It was known before the industrial revolution that sound carries information of

the vibrating object. Before any scientific development, bell makers around the

world knew that different alloys produce different sound, and the shape of the

object is important in the quality of the sound. This is an early example how

shape and material influence the sound quality. Presence of any kind of crack or

imperfection changes the sound quality. Knowledge of the relationship between

shape, material, and resonant frequency was used in a variety of fields. Before

World War I, British railroad engineers had a similar type of technique. They

would tap the wheels of the train and use the sound to detect cracks. This may

be the first engineering use of sound to test material performance [38]. Later

attempts were made to understand the resonance of solids mathematically based

on the shape and composition. Nobel laureate Lord Rayleigh’s contribution in

the late eighteenth century was significant. In World War II German engineers

utilized a violinist to rub his bow over different turbine blade designs to verify that

the resonant frequencies of the blades would be very different from the angular

speed of the engine itself to avoid the catastrophic failure that could result from

the resonant frequency of the blade [16]. The knowledge of natural resonance is

also important in structure design.

Interest in elastic properties goes back to the 17th century when Galileo and

other philosophers studied the static equilibrium of bending beams. In 1660

Robert Hooke first developed the law of elasticity. With the help of mathemati-

cians such as Leonhard Euler, Joseph Lagrange, George Green, Simeon-Denis

Poisson and others, the mathematical understanding of the elasticity became very

strong. Augustus Love summarized all of these theories in his quintessential book,

2

”Treatise on the Mathematical Theory of Elasticity”, in 1927. The theory of elas-

ticity suggests that the elastic constants of a material could be determined through

measurement of the sound velocities in the material. This gave rise to the idea of

conventional time-of-flight measurement with ultrasonic pulses.

Natural frequency measurements were used as early as 1935, but these early

methods could only find approximate solutions [8]. Gabriel Lame and Horace

Lamb found analytic solutions to the forward problem for a spherical shape with

isotropic and noncrystalline materials [30]. These investigations were focused

mainly on the Earth after a large earthquake. The geophysics community con-

tributed in this particular problem with a view to determining the Earth’s interior

structure, and to measure accurately the elastic moduli of materials believed to

be earth’s constituents [36]. These investigators did not solve the inverse problem

for sphere.

In 1964 Fraiser and LeCrew [11] used the solution of the sphere and inverted

graphically, which may be the first RUS measurement. Geophysicists Orson An-

derson, Naohiro Soga and Edward Schreiber who worked to improve Fraiser and

LeCrew’s method, introduced the term Resonant Ultrasound Technique (RST).

Anderson and Schreiber used RST to measure spherical lunar samples in 1970.

In their paper, an observation about the composition of the moon was made and

some ancient claims about moon being made of cheese were studied. In Erasmus’s

collection of Latin proverbs, Adagia’s(1452) phrase, ”his friends believed the moon

to be made of Green cheese” was quoted [2]. Sound speed of different cheeses and

lunar rocks were also studied.

Harod Demarest, who was a graduate student of Anderson at Columbia Uni-

versity, first used RST for a cubic shape. He is the first one to approach the

forward problem numerically rather than analytically, based on variational princi-

ples. The Rayleigh-Ritz technique and integral equations were used by Demarest

to define the free vibrations. He referred this technique as cube resonance. Mi-

neo Kumazawa, a post-doctoral researcher at Columbia university was pursuing

3

ultrasonic measurements of minerals during the lunar rocks experiments. He stud-

ied Demarest’s method and when he returned to Nagoya University, established

a laboratory on resonant ultrasound. His graduate student, Ichiro Ohno worked

with him to develop the technique. In 1976 Ohno published a paper with some

significant development of Demarest’s work [42]. In 1988 while working with small

piezoelectric film transducers, Migliori and Maynard faced problem with the res-

onance frequencies and Migliori tracked down the geophysics literature for RUS.

Migliori extended the technique and introduced the term Resonant Ultrasound

Spectroscopy. Even with the limitation of geometry choice, this technique soon

proved its usefulness in material science. In the early years the biggest problem

was the limitation of computational resources. With the advancement of comput-

ing power, RUS is becoming more useful in the field of material research.

4

Chapter 2

Non-Destructive testing ofIrradiated Materials: LiteratureReview

2.1 Radiation Effects in Materials

To understand the non-destructive evaluation of irradiated materials, one has

to understand the radiation effects on materials. How radiation creates damage

plays an important role in designing the non-destructive technique. In reactor

environments, varying types of intense nuclear radiation are responsible for the

damage. Radiation, mainly alpha and beta particles (from radioactive decay),

gamma rays, neutrons and fission products usually cause the damage. As far

as nonfuel materials, which are mainly metals, neutrons are the most active in

producing radiation effects. High energy fission products and alpha particles are

restricted to the fuel because of their short range. Beta and gamma radiation

can cause ionization and excitation of electrons, but they are mainly limited to

nonmetallic substances. The effects of neutrons on materials mainly result from

the transfer of kinetic energy to the atomic nuclei. This transfer of energy may

cause a reaction e.g. (n, γ) or other neutron reactions or elastic scattering. If the

energy of the recoil nucleus is sufficient enough to displace it from its equilibrium

position in the lattice of a solid, then a physical change will be observed which

is irreversible. The effect of nuclear radiation in disrupting the crystal lattice by

the displacement of the atoms is commonly known as ”radiation damage”. Four

5

major categories of mechanical behavior are impacted by damage.

1. Radiation Hardening

2. Embrittlement and fracture

3. Swelling

4. Irradiation Creep

Radiation hardening usually increases yield stress and the ultimate tensile stress

as a function of fast-neutron fluence and temperature. Extensive research has been

done to establish the fact that exposure of all metals to fast-neutron irradiation

results in an increase in the yield strength. For both the austenitic and ferritic

steels, irradiation increases the yield strength much more than it does the ultimate

tensile strength. Irradiation causes an increase in the 0.2% offset yield strength of

austenitic steels [54]. Typical engineering stress-strain curves for the two types of

steels are shown in Figure 2.1.

(a) Face-centered cubic structure (316-type stainless steel)

(b) Body-centered cubic structure(A533B RPV steel)

Figure 2.1: Effect of fast-neutron irradiation on the tensile properties of reactorsteels

In addition to the increasing yield strength with irradiation, the ductility is

reduced. Radiation hardening in both fcc and bcc metals is attributed to the

production of various defects by radiation within the grains. Defects produced by

neutron irradiation of metals include [43]:

1. Point defects (vacancies and interstitial)

6

2. Impurity atoms (atomically dispersed transmutation products)

3. Small vacancy clusters (depleted zones)

4. Dislocation loops (faulted or unfaulted, vacancy or interstitial type)

5. Dislocation lines (unfaulted loops that have joined the dislocation network

of the original microstructure)

6. Cavities (voids and helium bubbles)

7. Precipitates ( in the case of stainless steel, M23C6 carbides or inter metallic

phases)

Radiation strengthens a metal in two different ways: (1) It can increase the

stress required to start a dislocation moving on its glide plane. Resistance to dis-

location start up is called source hardening. The applied stress required to release

a dislocation into its slip plane is called the unpinning or unlocking stress. (2)

Once moving , dislocation can be slowed down by natural or irradiation-produced

obstacles close to or lying in the slip plane. This is called friction hardening [43].

There are numerous hypotheses regarding the mechanisms of irradiation hard-

ening, the dose-dependence of the increase in the yield stress, ∆σY S, has been

studied frequenty [43] [49]. There are a lot of models that correlate irradiation

hardening with the dose [51]. Among the models for irradiation hardening, the

simplest is in the form of the power-law expression:

∆σY S = hφn (2.1)

Where φ is the radiation fluence, which can be replaced by a factor of defect

or by dpa (displacement per atom) [6]. At low doses, the exponent n has been

measured to be about 1/2 for many metals and 1/4-1/3 for copper and gold

[6] [41]. After a certain higher dose the hardening shows a saturation behavior

7

mostly because of cascade overlap [43]. In the saturation region Makin and Minter

suggested a two-parameter equation [34].

∆σY S = A[1− exp(−Bφ)]1/2 (2.2)

Where A and B are regression coefficients. Equation 2.2 shows a continuous

transition from low-dose regime, with φ1/2 dependence, to the saturation regime

with an asymptotic value A at very high doses.

2.1.1 Radiation Effects on Elastic Constants

Neutron bombardment has a profound effect on the Young’s modulus. Theoretical

studies were done by Dienes to predict the change the Young’s modulus in early

sixties [10]. Dieckamp and Sosin studied the effect of electron irradiation on the

Young’s Modulus of copper at 250 C showing the the modulus increases very

sharply and saturates early [9]. Figure 2.2 shows three different line for three

different specimens based on the heat treatment: A was slightly cold worked, B

was well annealed and C was heavily cold-worked.

Figure 2.2: Fractional change in Young’s modulus of 99.99% copper as a functionof exposure to 1-Mev electrons [9]

8

Radiation effects on zirconium’s Young’s modulus were studied by Ozkan [44].

The change of elastic constants of reactor graphite’s was studied by Goggin and

Reynolds [15]. For their experiment isotropic graphite (grade A) was used. All

sets of elastic constant were determined before and after irradiation. Poisson’s

ratio was also determined and compared with the unirradiated graphite.

Figure 2.3: Variation of elastic constants with neutron dose for isotropic graphite[15]

For the isotropic graphite in Figure 2.3, initially the Young’s modulus is ap-

preciably higher than the static modulus because of the lower stress level at which

the the dynamic measurement is made. After a certain irradiation the dynamic

modulus increases more rapidly than the static and eventually reaches a satura-

tion value. Poisson’s ratio measurements (Figure 2.4) show a downward trend

from a small hump in the same dose region where the static modulus exceeds the

dynamic.

9

Figure 2.4: Variation of poisson’s with neutron dose for isotropic graphite [15]

Maruyama et al. [35] used thirteen different isotropic graphites with differ-

ent density and maximum grain size and those were irradiated in different flu-

ences from 2.11 to 2.86 × 1026 ncm−2 at different temperatures. Dimensional

changes, elastic modulus and thermal conductivity were measured after irradi-

ation. Young’s modulus and Poisson’s ratios were calculated from the velocity

of ultrasonic waves travelling longitudinally and transversely to the direction of

cylindrical axis. For isotropic material the relationship between the sound waves

and the elastic modulus is [24].

E = 2ρ(1 + ν)C2t (2.3)

ν =2− (Cl/Ct)

2{1− (Cl/Ct)}(2.4)

Here Cl is the longitudinal sound wave velocity, and Ct is the transverse sound

wave velocity.

10

Figure 2.5: Dimensional change of neutron irradiated graphite [35]

The dimensional change in Figure 2.5 shows shrinkage to expansion with in-

creasing neutron fluences regardless of differences in material parameters. This di-

mensional instability impacted the reactor design in early reactors where graphite

was used as a moderator [14].

Figure 2.6: Young’s modulus change of neutron irradiated graphite versus neutronfluence [35]

11

Figure 2.6 shows the change of Young’s modulus in the neutron irradiated

graphites. It shows an increase of the initial Young’s modulus which satisfies the

theoretical prediction [23].

Non-destructive testing for the evaluation of irradiation embrittlement in nu-

clear materials was studied by Ishii et al. [21]. Even though they used traditional

sound velocity measurement to characterize the irradiated material, it shows that

ultrasonic technique can be used to qualitatively measure the embrittlement in

the nuclear materials. The changes in velocity and attenuation coefficient of the

ultrasonic wave propagating in the materials were observed after irradiation. The

velocity and attenuation coefficient of both shear and longitudinal waves propa-

gating in the material were calculated by the following equation [25].

C =2T

t2 − t1(2.5)

α =20log10(h1/h2)

2T(2.6)

Where C and α are the velocity and the attenuation co-efficient of the ultrasonic

wave, respectively. T is the thickness, t is the propagation time of the back wall

echo, h is the pulse amplitude of the back wall. The subscript 1 and 2 represent

the first and second back wall echoes.

12

Figure 2.7: Change in velocities of shear and longitudinal waves in steel(A533B-1)before and after irradiation [21]

Velocities of both shear and longitudinal waves in the irradiated specimen

were lower than those in the unirradiated one. Both waves velocities increase

after annealing at 673 k. Since the longitudinal and the shear waves depend on

the Young’s, shear and bulk modulus, the change of the modulus can be calculated

from the change of the velocities using Equation 2.3 and 2.4.

13

Chapter 3

RUS: Mathematical Development

3.1 Introduction

Many methods are available to measure the elastic constants of solids. The reso-

nance method was limited to specific shape and crystallographic symmetry. Using

the calculus of variations the integral equations for free vibrating elastic bodies

are developed. Lets assume an arbitrarily shaped object with elastic tensor Cijkl

and density ρ with a free surface S surrounding a volume V . Both quantities(ρ

and Cijkl) may vary with the position in V . Let ω be a non-negative real number,

u(r) be a real-valued function of position r in V . Then the combination of {ω,u}

is a free oscillation or resonance if the real-valued displacement field satisfies

s(r, t) = <(u(r)eiωt) (3.1)

the elastic equations of motion in volume V . The analytical solution of this

mechanical behavior is next to impossible [56]. In fact for a given volume, for all

values of ω there will be no displacement (u) value that will satisfy the boundary

condition. To solve the problem a numerical approach is used.

14

3.2 Formulation of the Problem

We consider conservative systems in which the kinetic and potential energy is

function of position only. And the kinetic and potential energy is a function of

resonance. Using classical mechanics the Lagrangian of the system is

L =

∫V

(Ek − Ep)dV (3.2)

Here Ek and Ep are the kinetic and potential energy respectively. The potential

energy associated with the displacement u is given by the strain energy

Ep =1

2

∫V

Cijkl∂ui∂xj

∂uk∂xl

(3.3)

Where ui is the ith (i = 1, 2, 3 corresponding to the x, y or z directions in coor-

dinate space) component of the displacement vector u and we have assumed that

the displacements’ time dependence is eiwt. The corresponding kinetic energy will

be

Ek = ω2K (3.4)

where

K =1

2

∫V

ρuiuidV (3.5)

Now the Lagrangian becomes (using Eq. 3.2)

L = ω2K − Ep (3.6)

The kinetic and potential energy in the above equation can be calculated for any ω

and u. Now it is necessary to find a condition where we can use the above equation

for resonance only. In this situation classical mechanics helps by assuming that

the quantity L is stationary if and only if ω and u are a resonance of V. The term

stationary implies that L does not change within the volume V and surface S if

15

the displacement u is replaced by u + ∂u.

L → L+ ∂L (3.7)

L+ ∂L =1

2

∫V

[ρω2(ui + ∂ui)

2 − Cijkl∂(ui + ∂ui)

∂xj

∂uk∂xl

]dV (3.8)

Keeping terms to first order in ∂ui, the change in lagrangian is

∂L =

∫V

[ρω2ui∂ui − Cijkl

∂(∂ui)

∂xj

∂uk∂xl

]dV (3.9)

After integration by parts it yields,

∂L =

∫V

(ρω2ui + Cijkl

∂2uk∂xj∂xl

)i

∂uidV +

∫S

(~ηjCijkl

∂uk∂xl

)i

∂uidS (3.10)

Here, ~ηj is the normal vector to the surface. The above equation (3.10) can be

written in the following form:

∂L =

∫V

(elastic wave equation)i∂uidV +

∫S

(surface traction)i∂uidS (3.11)

Due to the stationary condition the change of Lagrangian has to be zero(i.e, ∂L =

0). To satisfy the condition each of the two terms inside the Equation 3.10 have

to be independently zero. Setting the first term of the equation to zero yields the

wave equation:

ρω2ui +∑j,k,l

Cijkl∂2uk∂xj∂xl

= 0 (3.12)

The second bracketed term is the expression of the free-surface boundary condi-

tion. ∑j,k,l

ηjCijkl∂uk∂xl

=∑j

~ησij = 0 (3.13)

The above equation represents the ith component of the surface traction vector.

Where ηi is the unit vector normal to surface and σij is the ij component of

16

the stress tensor. So the displacement vector ui which satisfies these conditions

(Equation 3.12, 3.13) will represent a singular value of ω; which is one of the

normal mode frequencies of free vibration of the system.

3.2.1 Choice of Basis Function

The next step is to turn the displacement vector u into some mathematical func-

tion. One way is to use a simple expansion that can satisfy a range of geometry.

Using Rayleigh-Ritz method [47], the displacement vector can be expanded in the

following way:

ui =∑λ

aiλΦλ (3.14)

Where aiλ are the generalized constants and Φλ are a set of shape functions.

Both of these are linearly independent and satisfying the global kinematic bound-

ary condition. The choice of Φλ is arbitrary. Historically, Demarest and Ohno

used normalized Legendre polynomials [8] [42]. This had the advantage of dealing

with simpler matrices, but it also limits the choice of object shape. Visscher used

the following method to expand the shape function Φλ [52].

Φλ = xl(λ)ym(λ)zn(λ) (3.15)

Where (l,m, n) are discrete element of basis set λ and l,m, n ∈ Z+. Equation

(3.14) can be written in the following form:

ui =∞∑

(l,m,n)

ai(l,m,n)Φ(l,m,n) (3.16)

The above equation cannot be summed to infinity, thus a truncation condition

will be imposed. The truncation condition is:

l +m+ n ≤ N (3.17)

17

The truncation condition will introduce truncation error since Equation 3.16 is

not being summed to infinity. In practice N has a number that will compromise

between computational time and accuracy. It is a combination problem to find

out how many polynomials will appear in the expansion of the displacement. The

correct number of combinations is

(N + 3

3

)=

(N + 3)!

3!(N + 3)!=

(N + 1)(N + 2)(N + 3)

6(3.18)

3.2.2 Lagrangian in Matrix Form

For a given number of N there are basis elements counting all three components of

u. Now replacing the displacement vector (Equation 3.14) with the basis function

in Equation 3.2 yields the following form.

L =1

2

∫V

[ρω2aiλai′λ′ΦλΦλ′δii′ −

1

2Cijklaiλai′λ′

∂Φλ

∂xj

∂Φ′λ∂xl

]dV (3.19)

Here δii′ is a Kronecker delta, which provides following condition

δii′ =

1 if i=i’

0 if i 6= i’

(3.20)

λ and λ′ represent two different basis sets. To keep track of all the subscripts

in Equation 3.19 the following conditions are required.

i the Cartesian component of ul(λ) the component of x in Φλ

m(λ) the component of y in Φλ

n(λ) the component of z in Φλ

Equation 3.19 can be written in the following matrix equation.

L =1

2ω2aTEa− 1

2aTΓa (3.21)

18

Where a is a vector with elements ai whose transpose is aTand where E and Γ

are matrices whose order R (number of columns and rows) is determined by the

truncation condition in Equation 3.17. The matrix E has elements

Eλiλ′i′ = δii′

∫V

ΦλρΦλ′dV (3.22)

And the matrix Γ has elements

Γλiλ′i′ = Cijkl

∫V

∂Φλ

∂xj

∂Φλ′

∂xj′dV (3.23)

If Φλ was chosen to be an orthonormal set with respect to the density ρ (Demarest

and Ohno did using Legendre polynomials), E would have been a unit matrix and

our calculation would be simple. Since our choice of Φλ provides a wide range of

shapes, it has computational complexity. To simplify the matrices, the following

calculation can be followed. Using the K matrix from Equation 3.5:

K = 12

∫VρuiuidV

Kλλ′ = 12

∫Vραλx

l(λ)ym(λ)z(λ)δi(λ)i′(λ) × αλ′xl(λ′)ym(λ′)zn(λ

′)dV

= 12

∫Vρxl(λ+λ

′)ym(λ+λ′)zn(λ+λ′)dV

(3.24)

There is a similar but complicated expansion of matrix E. In both cases the result

has similar integral form as follows.

∫V

f(x, y, z)xlymzndV (3.25)

Where f(x, y, z) is either density for matrix E or one of the Cartesian components

of the elastic stiffness tensor. In any region (constant S and volume V) in which

f(x, y, z) can be written as a monomial basis function, a condition that includes

the nearly universal case of being constant in S, all of the integrals are in the

19

following form:

I(l,m, n) =

∫S

xlymzndV (3.26)

All of these problems reduce to the evaluation of the integral from the above

equation, which is sometimes referred as geometric integral. Matrix E and Γ can

be re-written in the following way.

Eλiλ′i′ =

∫V

ρxlymzndV = ρI (3.27)

Γλiλ′i′ =

∫V

Cijklf(x, y, z)xlymzndV = Cijklf(x, y, z)I (3.28)

3.2.3 Generalized Eigenvalue Equation

Equation 3.2 for Lagrangian is stationary if the displacements ui are solutions

of the free-vibration problem. These solutions may be obtained by setting the

derivatives of Equation 3.21 to zero with respect to each of the R amplitude aiλ.

Since δL = 0, ∂L/∂aiλ = 0, this yields the following equation:

ω2Ea = Γa (3.29)

Equation 3.29 is the generalized symmetric eigenvalue problem and is readily solved

with modern software. This is the ultimate equation that is solved to get the

resonance frequencies. Here eigenvalues are ω2 and the eigenvectors are a. Thus,

calculating the resonant frequencies for a 3D elastic body is essentially one of

calculating the matrices Γ and E. The special vectors a are the mode shapes of

the system. These are the special initial displacements that will cause the mass

to vibrate harmonically.

Now the problem is reduced to correctly determining the matrices E and Γ.

To explain how to generate these matrices a simple geometry with high symmetry

will be used. The most frequently used sample geometry for RUS is the rectangu-

lar parallelepiped (RP). Here we are assuming that the crystallographic axes align

20

with the sample faces and with orthorhombic or higher crystal symmetry. The

mathematical development is summarized in the following flow chart Figure 3.1.

Figure 3.1: Flow chart of the mathematical development of a typical RUS problem

21

3.2.4 Example:Calculation of Resonance frequencies of Rect-

angular parallelepiped

Figure 3.2: Dimension associated with rectangular parallelepiped(RP)

For a RP of sides 2d1, 2d2, 2d3 (Figure 3.2), the integral in Equation 3.26 can be

evaluated explicitly to give

I(l,m, n) =8dl+1

1 dm+12 dn+1

3

(l + 1)(m+ 1)(n+ 1)(3.30)

To get the above integral form some assumptions were made. In order to sim-

plify matrix Γ, we have to understand the elastic constant Cijkl and how it can be

minimized using isotropic assumptions. To generate matrix E and Γ for rectangu-

lar parallelepiped, a Matlab code is given in appendix A. This code is written based

on Visscher’s XYZ algorithm [52]. And the generalized eigen function equation

solved using Matlab’s eig function.

3.3 Basics of Elasticity

When a force, either internal or external, is applied to a continuum every point of

this continuum is influenced by this force. Internal force is referred as body force

22

and external force as contact force. Gravity is an example of body force. Body

forces are proportional to the volume of the medium and to its density. Contact

forces depend on the surface they are acting on.

If an external force acts on a continuum, it will lead to deformation (changes

of size and shape) of the medium. Internal forces acting within the medium will

try to resist this deformation. As a consequence the medium will return to its

original shape and volume when the external forces are removed. If this recovery

of this original shape is perfect, the medium is called is elastic. The law relating

the applied force to the deformation is called Hooke’s Law. It is defined by stress

and strain. Hooke’s law states that the strain is linearly proportional to stress or

conversely, that the stress is linearly proportional to stress [3]. The second form

is stated by writing each component of stress (elastic restoring force) as a general

linear function of all the strain components. The generalized form will be

σxx = Cxxxxεxx + Cxxxyεxy + Cxxxzεxz+

Cxxyxεyx + Cxxyyεyy + Cxxyzεyz+

Cxxzxεzx + Cxxzyεzy + Cxxzzεzz (3.31)

The above equation can be written in the following form

σij = Cijklεkl with i, j, k, l = x, y, z (3.32)

Here Cijkl are the components of the fourth-order stiffness tensor of material

properties or Elastic moduli. The fourth-order stiffness tensor has 81 constants and

16 components for three-dimensional problem. The stress tensor σij completely

describes the state of stress at any point p of the continuum.

23

Figure 3.3: Stress component on a particular point

σij =

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

with i, j = 1, 2, 3 (3.33)

Equation 3.32 contains nine equations (corresponding to all possible combinations

of the subscripts ij) and each equation contains nine strain variables. Due to

the symmetry of strain and stress tensors the stiffness tensor reduces to a smaller

number. Using the symmetry of stress tensor:

σij = σji =⇒ Cijkl = Cjikl (3.34)

This reduces the number of stiffness constants from 81 = 3 × 3 × 3 × 3 → 54 =

6× 3× 3. Similarly using the symmetry of strain tensor:

εij = εji =⇒ Cijkl = Cjikl (3.35)

This reduces the material constants to 36 = 6 × 6. To reduce the size of the

material constants we can use the symmetry property of the tensor itself. Which

24

is

Cijkl = Cklij (3.36)

This further reduces the number of material constants to 21. The most general

anisotropic (triclinic crystal structure) linear elastic material therefore has 21 elas-

tic constants. Now the stress-strain relationship can be written as follows

σ11

σ22

σ33

σ23

σ13

σ12

=

C1111 C1122 C1133 C1123 C1113 C1112

C2222 C2233 C2223 C2213 C2212

C3333 C3323 C3313 C3312

symm C2323 C2313 C2312

C1313 C1312

C1212

ε11

ε22

ε33

2ε23

2ε13

2ε12

(3.37)

Since the 3×3×3×3 tensors become 6×6, we can use general matrix notation to

explain the stress-strain relation. With these constraints on the stiffness constants

the four subscript on the tensor may be reduced to two by using abbreviated

subscript notation, where

I ij1 xx2 yy3 zz4 yz, zy5 xz, zx6 xy, yx

[C] =

C11 C12 C13 C14 C15 C16

C22 C23 C24 C25 C26

C33 C34 C35 C36

symm C44 C45 C46

C55 C56

C66

(3.38)

25

The above equation shows the elastic constants for a complete anisotropic mate-

rial, which has 21 elastic constants. When the material has symmetry within its

structure the number of elastic constants reduces even further.

3.3.1 Elastic Isotropy Conditions

It is necessary to determine the restrictions imposed by elastic isotropy on stiffness

matrices. In an isotropic medium the three coordinates axes, x,y,z and the three

coordinate planes yz, xz, xy are equivalent. This means the elastic constants are

independent of the direction of the axes. Consequently, the response of the medium

must be the same for a compressive stress applied along any axis. Therefore:

ε11 = ε22 = ε33ε12 = ε13 = ε23

Also, the shear strain produced in a coordinate plane by a shear stress applied

in that plane must be the same for all coordinate planes,

ε44 = ε55 = ε66

Figure 3.4: Symmetry logic proves that ε61 = −ε61 = 0 in an isotropic medium

Now, dealing with other shear strains, suppose that a compressive stress T1

26

applied along the x axis produces a shear strain ε6 in the xy plane. Due to the

isotropic property if the plane is rotated 1800 about the x axis, an equal valid

solution to the applied stress T1 will exist. This reverses the sign of the ε6, which

can therefore only be zero. Hence

ε16 = 0

Similarly

ε14 = ε15 = ε24 = ε25 = ε26 = ε34 = ε35 = ε36 = 0

Similarly using the same argument in a different coordinate plane it can be shown

that

ε45 = ε46 = ε56 = 0

The above conditions can be applied to the elastic stiffness matrix (equation 3.38).

When these conditions are used the matrix becomes

[C] =

C11 C12 C12 0 0 0

C11 C12 0 0 0

C11 0 0 0

symm C44 0 0

C44 0

C44

(3.39)

The above expression for the elastic constants may be sufficient for the isotropic

material. And it is also similar to the stiffness matrix of the cubic material.

But using the rotation over the z axis it can be shown that[3] the condition for

invariance of matrix [C] for an isotropic material is

C12 = C11 − 2C44 (3.40)

This same condition can be obtained using rotation about the y axis or x axis.

27

From equation 3.44 and 3.40 it is clear that for an isotropic medium there are only

two independent elastic constants. These are often called Lame constants λ and

µ, defined by

λ = C12

µ = C44

Using the Lame constants the stiffness matrix can be represented in the fol-

lowing way:

[C] =

λ+ 2µ λ λ 0 0 0

λ+ 2µ µ 0 0 0

λ+ 2µ 0 0 0

symm µ 0 0

µ 0

µ

(3.41)

Lame constant µ is referred to as shear modulus, which is the measure of the

shear stress. The second Lame constant λ is important as a combination with

other elastic constants, e.g., the Young’s modulus E, the bulk modulus K, and

Poisson’s ratio ν.

3.3.2 Engineering Constants

Equation 3.32 provides the linear relationship between stress (σij) and strain (εkl).

Alternatively, the strains may be expressed as a general linear function of the

stresses,

εij = Sijklσkl (3.42)

Here Sijkl is called compliance constants, which is the inverse of the stiffness

matrix (Equation 3.38). That is:

[S] = [C]−1 (3.43)

28

The compliance constants for an isotropic medium can be obtained by taking

inverse of the stiffness matrix of the isotropic medium (3.44). Matrix C has di-

mensions of 6× 6, so the inverted matrix will also have the same dimensions. So

the compliance matrix will be

[S] =

S11 S12 S12 0 0 0

S11 S12 0 0 0

S11 0 0 0

symm S44 0 0

S44 0

S44

(3.44)

with

S11 =C11 + C12

(C11 − C12)(C11 + 2C12)

S12 =−C12

(C11 − C12)(C11 + 2C12)

S44 =1

C44

Figure 3.5: Isotropic object subjects to pressure P

Now, an isotropic parallelepiped (Figure 3.5) is subject to static pressure P on

29

one end and is unconstrained on its sides. Then the strain field will be:

ε11 = S12p

A

ε22 = S12p

A

ε33 = S11p

A

Poisson’s ratio, ν, provides information about the ratio between lateral and lon-

gitudinal strain in uniaxial tensile stress. Using the condition in Figure 3.5, ν can

be written in the following form using strains:

ν = −ε11ε33

= −ε22ε33

=S12S11

=C12

C11 + C12

=λ

2(λ+ µ)(3.45)

And Young’s modulus is defined by

Y =σ33ε33

=1

S11=

(C11 − C12)(C11 + 2C12)

C11 + C12

= 2µ(1 + ν) (3.46)

3.4 Hollow Tube Geometry

The shape of the object for RUS experiment has an important impact on the for-

ward calculation, as it influences the mode selection. Mode shape identification

and frequency analysis for different compact geometries have been done for cylin-

ders [17], spheres [26], cubes [8] and rectangular parallelepiped [52]. It is also done

for arbitrary shapes which is called ’potato’ [52]. The regular shapes were stud-

ied as these are the obvious choice for fundamental mechanical property studies.

However, hollow cross sections were also studied in the context of carbon nano

tubes, both analytically [32] and numerically [27]. Jaglinski and Wang [22] did a

study to calculate the resonance frequencies for the hollow geometry using RUS.

But they kept their experiment limited to very small object. Even though the

Rayleigh-Ritz method creates mathematical complexity for the arbitrarily shaped

geometry, finite element method (FEM) provides a very good solution. The fun-

30

damental assumptions of RUS is the stress free surface of the object. But a hollow

geometry creates another surface inside the object. This condition makes the in-

tegration part difficult to calculate with proper boundary condition. On the other

hand defining a geometry with FEM can be less arduous. Since FEM converts the

system of partial differential equations to linear algebraic system, proper boundary

conditions can be provided in the interested region. So for this thesis FEM was

used for the calculation of the resonance frequencies. Commercial finite element

package, Abaqus, is used for this purpose.

31

3.5 Classical Solution of Vibrating rings and tubes

The equations of motion for an isotropic elastic medium are,

(λ+ µ)∂2uj∂xj∂xi

+ µ∂2ui∂xj∂xj

= ρ∂2ui∂t2

(3.47)

Here λ and µ are Lame constants, subscripts i and j are labels for spatial di-

mensions, ranging from 1 to 3, t denotes time, and ρ is mass density. To derive

the resonant frequencies for the free vibration problem, the required boundary

conditions are σzz = σzr = σzθ on the z = 0 and z = L surfaces, σrr = σrθ = 0

on the r = a and r = b cylindrical surfaces. There is no analytical solution for

the general problem. Analytical solutions to some of the mechanical systems are

presented in the following discussions.

For thin, narrow, annular rings the resonant frequencies were obtained by

Hoppe in 1872 [18]. It was done for flexural vibrations, in which flexural vibrations

travel around the ring. Later, in 1889, Rayleigh [48] found the solution of the finite

frequency spectrum regarding extensional (rod) waves. For a thin disk the in-plane

vibrations are radial and torsional waves, and each of the problems can be solved

separately. When these vibrations occur together the simultaneous solution of

both wave equations is necessary. For short isotropic cylinders with length equal

to diameter the elastic constant can be found in the following way. As torsion

being the fundamental resonant mode [53].

ftorsion = ksh√µ/ρ (3.48)

Where frequency in hertz (f = ω/2π), ftorsion is the resonant frequency, ksh is

the shape factor with units of 1/m, µ is the isotropic shear modulus in pa, and ρ

is the material density in kg/m3. For cubes [8] ksh =√

2/(πL) and for cylinders

of length equal to diameter [17], ksh = 1/(2L). In both cases, L is the sample

32

length. Flexural vibrations include both displacement at right angles to the plane

and twist. Flexural vibrations can be determined using following equation [30].

f 2flex =

Ec4

16πma4n2(n2 − 1)2

(n2 + 1)(3.49)

Where the radius of the cross-section is c and the central-line radius of the ring is a.

Torsional vibrations, similar to torsional vibrations of a straight rod, correspond

to twist of the cross-section about its centerline. For a complete circular ring

there are vibrations of this type with n wave-lengths to the circumference, and the

frequency ftor is given by the equation [30].

f 2tor =

µc2

16πma2(1 + ν + n2) (3.50)

Extensional vibrations are again similar to the extension of a straight rod and are

given by the following equation.

f 2ext =

Ec2

4πma2(1 + n2) (3.51)

When n=0, the tube vibrates radially so that the central-line forms a circle of

periodically variable radius appearing as ring breathing, and the cross sections

move without rotation. The propagation of the free harmonic waves in an infinitely

long cylindrical rod has been discussed by Pochhammer [46] on the basis of the

linear theory of elasticity. Gazis determined the frequency equations for several

modes for long thick walled tubes under plane strain assumptions which is similar

to ring bending [12]. The analytical solutions are only presented for the shell

approximation, or h/a� 1 where h is the tube wall thickness and a is the center-

line radius of the ring. Extensional mode (different from previous extensional

vibration) frequencies can be approximated by the following equation.

fext′ =n

2h

√λ+ 2µ

ρ

[1 +

7− 15ν

8(1− ν)(nπ)2

(h

a− h/2

)2]

(3.52)

33

The resonant frequency for ring extensional vibration is given by the following

equation.

fring−ext =1

π(2a− h)

õ

ρ

√n2 + 1

2(1− ν)(3.53)

34

Chapter 4

RUS simulation and experiment

4.1 Experiment

4.1.1 Equipment

A general RUS experiment consists of multiple piezoelectric transducers, one to

excite and others to detect the responses [40] [37]. No supporting clamp is used

since it reduces the surface area and creates obstacles to the free vibration. No

bonding agent between the transducers and the sample was used. In a typical RUS

experiment the object is held between two transducers and a network analyzer is

used to excite one of them. Using two transducers is usually good for shapes

like a cube, rectangular parrallelepiped etc. If the object is large and round a

sophisticated technique has to be used to detect all the frequencies. The process

is more complicated if the sample has higher mass and length. The technique

to excite and detect a sample gets more complicated depending on the choice of

the material’s shape and environment. Figure 4.1a shows an example using three

transducers for a sphere of several centimeters in diameter [36].

35

(a) ball bearing of sev-eral centimeters on threetransducers

(b) Schematic of the three transducers setupfor a ball bearing

Figure 4.1: RUS setup to detect defects in ball bearing of several centimeters indiameters

Since one of the objectives of this thesis is to measure the elastic constants of

a hollow cylinder with RUS, a study was done to make an optimum design for the

hollow cylinder. The cylindrical object has an inner diameter of 0.025 m , outer

diameter of 0.029 m and length of 0.161 m. For the purpose of simplicity it is

assumed that the inner and outer diameter are constant over the whole length.

A lock-in amplifier is used to generate and read the frequencies to and from the

transducers. The lock-in amplifier used for the experiment is Zurich Instrument’s

HF2LI. A Labview program is used to interface the ultrasonic transducers with the

lock-in amplifier. The ultrasonic transducers used in the experiment are VP-1093,

also known as a pinducer by the manufacturer, Velpy Corporation.

In order to get the resonant frequencies with sufficient accuracy, one must min-

imize the loading of the sample by the transducers. The weight of the sample can

be highest load on the transducers [36]. The error that comes from the frequency

measurement influences the calculation of the elastic constant. The contact pres-

sure between the transducers and the sample have the ability to shift the natural

frequency up to more than 1KHz [55]. The experimental setup is pictured below.

36

(a) Three pinducers setup(without the object) for RUS

(b) Setup with hollow cylinder

Figure 4.2: RUS setup

4.1.2 Data Collection

Using a lock in amplifier provides a great deal of flexibility in data handling.

Modern lock-in amplifiers have both a resonator (generate frequency) and sweep

function (generate a range of frequencies) that are helpful for RUS. The schematic

of the data collection is shown in Figure 4.3. The sweep function was used to

generate a range of frequencies of interest. Usually a wide range of frequencies is

generated to excite the object. To find the frequencies of interest a wide range

37

of frequencies were used, starting at 1 Hz through 3 MHz. Figure 4.4 shows the

response of the excitation.

Figure 4.3: Schematic Diagram of the RUS experiment for the data acquisition

It can be seen from the Figure 4.4 that after 1.8 MHz, the response is relatively

small. The response amplitude is in DC voltage.

Figure 4.4: RUS measurements taken for a wide range of frequency from 1 KHzto 3 MHz for hollow cylindrical shape

The quality factor Q, is calculated over the interested frequency range. The

positions of the peaks occur at multiples of the natural frequency and the quality

factor for each resonance provides information about the dissipation of elastic

energy [36]. To get the resonance frequency and the quality factor at a particular

location the Breit-Wigner [5] distribution is used.

4.1.3 Why use a Lock-in amplifier?

Lock-in amplifiers are used to measure and detect very small AC signals all the

way down to a few nanovolts. Lock-in amplifiers use a technique known as phase

38

sensitive detection to single out the component of the signal at a specific refer-

ence frequency and phase. Noise signals at frequencies other than the reference

frequency are rejected and do not affect the measurement. The block diagram of a

lock-in amplifier is shown in Figure 4.5. The lock-in consists of five stages: (1) an

ac amplifier, called the signal amplifier; (2) a voltage controlled oscillator (VCO);

(3) a multiplier, called phase sensitive detector (PSD); (4) a low-pass filter; and

(5) a dc amplifier. The signal to be measured is fed into the signal amplifier.

Figure 4.5: Block diagram of a lock-in amplifier

The ac amplifier is a voltage amplifier sometimes combined with variable fil-

ters. The voltage controlled oscillator is an oscillator, except it can synchronize

with an external reference signal both in frequency and phase. The phase sensitive

detector is a circuit which takes two voltages as inputs and produces an output

which is the product of the two input voltages. The low pass filter is an RC filter

whose time constant may be varied by the user. Some of the amplifiers have more

than one RC filter. The dc amplifier is a low frequency amplifier. The output of

the PSD is simply the product of two sine waves.

Vpsd = VsVrsin(ωst+ θs)sin(ωrt+ θr) (4.1)

Here Vs is the signal voltage, ωs is the signal frequency , Vr is the reference

voltage and ωr is the reference frequency. And θ provides the phase informa-

tion. Using the trigonometrical identity the above equation can be treated in the

39

following way.

Vpsd =1

2VsVrcos((ωr − ωs)t+ (θs − θr))−

1

2VsVrcos((ωr + ωs)t+ (θs + θr))

The two AC signals produced from PSD have two frequencies, one is the difference

of the frequencies (ωr−ωs) and other is the summation of the frequencies (ωr+ωs).

If the output of PSD is passed through a low pass filter, the AC signal will be

removed, in ideal situation, no signal will be availabe after low pass filter. However,

if ωr = ωs, the difference frequency component will be a DC signal. In this case,

the filtered PSD output will be

Vpsd =1

2VsVrcos(θs − θr) (4.2)

The above dc signal (Equation 4.2) is proportional to the signal amplitude that is

being fed in the amplifier.

4.2 Numerical Analysis

A hollow cylindrical numerical model was created using the commercial finite

element code ABAQUS on a PC. Three dimensional deformable solid models were

created using length of 0.161 m, inner diameter of 0.025 m and outer diameter

of 0.029 m. No symmetries were assumed in the model to avoid exclusion of

asymmetric deformation modes. Swept meshes of reduced integration, linear 3D

stress hex element meshes were applied from global seeds (with curvature control)

of a minimum of 0.001. All meshes contained around 980,000 elements. For the

isotropic case, a Young’s modulus of 70 GPa, Poisson’s ratio of 0.3 and density

of 2700 kg/m3 were initially assumed, then the Young’s modulus was varied.

Additionally an anisotropic model with 5% anisotropy was studied. Frequencies

and mode shapes were determined using the linear perturbation and frequency

analysis module. Eigenvalues were computed using the Lanczos solver [20], with

40

acoustic-structural coupling. Those results will be shown in future chapters. A

complete mesh of the tube is shown in Figure 4.6 which has total of 958,763

elements.

(a) Figure of mesh that was used for cylindrical shell

(b) Mesh structure at the edge of the cylinder

Figure 4.6: Mesh of the cylindrical shell

41

Chapter 5

Results and Analysis

5.1 Identification of Natural Frequencies

The experimental findings of the resonance frequencies and the quality factor (Q)

Table 5.1: Experimentally determined frequencies and quality factor

frequency (MHz) quality factor (Q)0.249761819 877312.770.2526230818 3167.5921280.2857712602 1901080.3019532575 55356.902350.31732144539 1498.891480.3460458839 2933310.665954311 2685.0390.7051525159 3425.080.7051581863 3713.7624770.7051629068 4064.337310.7051843006 5951.7924460.7304325851 327404.790.7305864031 10031233

0.74005 241.8464060.7677167502 4341.860.7677167546 4341.30960.7731016531 4776.8106470.8186399228 2726.077070.8186410207 3299.9742050.8200538496 4022.981970.8569976883 3169.8338791.002378015 22994476.4621.102426855 377.34081.201872122 4495.6231.255168378 905.498

42

for a hollow cylinder (length = 0.161 m, inner diameter = 0.025 m, outer diame-

ter=0.029 m) are in Table 5.1.

The quality or Q factor associated with a frequency is defined as

Q =f

∆f(5.1)

Figure 5.1: An example of measuring Q factor using Lorentzian distribution

Where ∆f is the full width at half maximum (FWHM), or the frequency width

at the 3 dB points on the power spectrum around frequency. Q is inversely related

to attenuation, so low Q materials have high losses [39]. Finding the resonance

frequency based on the Q value is very arduous for a wide range of frequency.

The curve fitting was done using a matlab code from Zadler and Le Rousseau

[57] which is availabe at reference [7]. After getting all the Q values, sorting the

frequencies based on the Q values is also critical. This Q value is sometime called

intrinsic Q, since the object is loosely coupled with apparatus.

Splitting of the resonance frequencies is shown in Figure 5.2, which shows spec-

tra for a good peak and two defective peaks. When the peaks overlap strongly,

it may be impossible to detect splittings without the phase information. To solve

that problem selective excitation is used in the particular frequency ranges. For-

43

tunately some of the modern machines (Lock-in amplifier) can also detect phase

information along side the amplitude response. Phase information can help to

separate closely attached peaks.

Figure 5.2: Resonance spectrum of a perfect bearing element(a), one with smallflaw (b), and one with large splitting (c)

There are many reasons for splitting of peaks. It is usually happens for the non-

symmetry or anisotropy in the sample. Phase information gives some flexibility

in identifying the overlapping peaks. The sensitivity of the phase information

provides good information when the Q value is low or an additional mode is

present [39].

44

5.2 Calculation of the Elastic Constant

Calculation of elastic constants from resonant frequencies is usually called the

inverse process. This is an essential part of RUS. For isotropic crystal structure

the relation between the Young’s modulus (E) and shear modulus (µ) can be

found in Equation 3.45 and 3.46. The values for C11 and C44 can then be obtained

through the relation

C11 =E(1− ν)

(1 + ν)(1− 2ν)(5.2)

and

C44 = µ (5.3)

For the iteration process the value of C11 was varied until the smallest error was

found. For this process the Poisson’s ratio is kept constant at 0.33 and the other

values are changed (Figure 5.3). Young’s and shear modulus is related by the

following equation

E = 2µ(1 + ν) (5.4)

The two elastic moduli C11 and C44 are related by

C44 =C11(1− 2ν)

2(1− ν)(5.5)

Since C44 is dependent on C11, only C11 was varied from a guessed value.

For the Rayleigh-ritz method, the inverse problem (resonance frequency→ elastic

moduli) uses Levenberg-Marquardt algorithm [39], which is known as the damped

least-squares method for non-linear system. But for FEM the inverse problem

becomes complicated. A good solution has been developed by Plesek et al [45] for

FEM. Another modified application of FEM of the inverse problem would be to

use initial guesses for the elastic constants and FEM to calculate the resonance

frequencies. This process is highly time consuming. The result depends on a

good initial guess of the elastic constants. The process of inverting the resonance

45

frequencies to elastic constants is shown in Figure 5.3.

Figure 5.3: Algorithm used to invert the resonance frequencies to elastic constants

Since C11 and C44 are the two elastic moduli being calculated and are related

by equation 5.4 only one of the elastic constants was guessed by guessing the

Young’s modulus. After several guesses of the Young’s modulus the satisfactory

values of the error and the calculated frequencies were obtained and are shown in

the following table. In Table 5.2 the calculation is shown for C11 = 102.826× 109

Pa and C44 = 26.1× 109 Pa which corresponds to Young’s modulus of 69.39 GPa

and Poisson’s ratio of 0.33.

46

Table 5.2: Measured and Computed natural frequencies in MHz for the aluminumcylinder

fexpt fcalc |%err| = |fexpt−fcalcfexpt

|0.249761819 0.249137414 0.250.2526230818 0.251966262 0.260.2857712602 0.283227842 0.890.3019532575 0.298148646 1.260.31732144539 0.327634392 3.250.3460458839 0.349125692 0.890.665954311 0.661625608 0.650.7051525159 0.699088204 0.860.7051581863 0.706004376 0.120.7051629068 0.702906385 0.320.7051843006 0.701869934 0.470.7304325851 0.725684773 0.650.7305864031 0.728248527 0.320.7400500000 0.74108607 0.140.7677167502 0.761037614 0.870.7677167546 0.769712821 0.260.7731016531 0.763824433 1.20.8186399228 0.817739419 0.110.8186410207 0.805542764 1.60.8200538496 0.823334065 0.4

47

5.3 Mode Identification

The mode identification process was carried out for a complete FEM model of a

hollow cylinder. The first model was a small cylinder with L/D (outer diameter)

ratio of 1.56 with the same inner and outer diameter of the cylinder that we are

interested. The resulting modes and frequencies are compared with Jaglinski and

Wang [22] and Li et. al.[27]. The identified modes are described in the following

paragraph. After identiying the fundamental modes, the length of the cylinder

was increased (L/D = 6.4).

The deformed shape obtained from the FEM analysis can be classified into the

following categories. S modes are longitudinal shearing, LB modes are longitudinal

bending, B modes are barreling or breathing modes. F modes are circumferential,

sometimes called flexural, which can be viewed as sinusoidal curve around the

surface [27]. Symmetric and antisymmetric F modes exist, in symmetry cases the

sinusoids at the end of the tube are in phase and for anti-symmetry they are out

of phase. Flexural Bending (FB) modes are both symmetric and antisymmetric

ring bending, combined with a barreling deformation. Some of these modes are

highly dependent upon the length of the tube. To understand the length effect on

the mode identification, four different simulations were done base on four different

L/D ratios. The experiment was done for L/D ratio of almost 6.4, which is a

fairly long length (0.161 m). Some of the modes are hard to identify for a long

length. For a small L/D ratio it is hard to identify any mode below torsional. If

the L/D ratio increases, after a certain limit some of the modes create a ”wave

on string” effect.

48

Figure 5.4: Behavior of the modes with the change of the Poisson’s ratio

Figure 5.4 shows the change of the location of the modes with the change of the

Poission’s ratio for small ring (L/D = 1.56). Fundamental torsional mode (T1) and

shear mode (s1) do not change with the change of the Poison’s ratio. Barreling

mode or breathing mode shows a decreasing trend. First and second flexural

bending mode (FB-1 and FB-2) do not change with the change of Poisson’s ratio

too.

49

(a) F1-1 (b) F1-2

(c) F2-1 (d) F2-2

(e) FB-1 (f) FB-2

(g) FB-3 (h) T1

Figure 5.5: Mode shapes and their harmonics for the lowest modes forL/D=1.56(ν = 0.33). The shading intensity is proportional to the element dis-placements.

50

(a) B1 (b) F2

(c) LB1 (d) S1

Figure 5.6: Mode shapes and their harmonics for the lowest modes for L/D =1.65(ν = 0.33).

It is known that lowest vibrational modes of finite length and high aspect ratios

are dominated by bending and ring bending, or, ”wave-on-a-string, [27]” modes,

similar to infinitely long tubes [13][12]. Some of the lowest modes are hard to

distinguish for long tube. Flexural behaviour of the infinitly long tube influences

the lowest modes. The ratio of the inner to outer diameter (Λ = ID/OD) also

influences the mode shapes and the frequency location.

(a) LB1 (b) SA-1 (c) T1

Figure 5.7: Mode shapes for the lowest mode for L/D=6.4(ν = 0.33).

51

Jaglinski and Wang [22] studied the effect of Λ = ID/OD in the mode shape

and frequency. For the present study the value of Λ is 0.881. According to Jaglinski

and Wang when 0 < Λ < 0.3 the fundamental mode is T1. For Λ > 0.3 the

fundamental mode becomes the antisymmetric ring bending mode F1 [22]. The

first torsional mode is independent of Λ. For higher values of Λ some of the modes

overlap and the harmonics of F and FB modes shift to above 250 kHz. The change

of modes with respect to Poisson’s ratio is shown in Figure 5.8. The three modes

for the long tube(target size, Λ = 0.881) were studied in this case. To understand

a wide range change of mode change, Figure 5.4 provides a very good example.

Figure 5.8: Behavior of mode depend on ν(Poisson’s ratio) normalized to thefirst torsional mode(LB1, first longitudinal bending mode; SA 1, Symmetric axialmode; F1, flexural mode ).

Change of Young’s modulus has a profound impact on the mode frequency. To

see how the mode changes with respect to Young’s modulus, five different simula-

tions were completed with different Young’s moduli. The result of the simulation

is shown in Figure ??. For these simulations, the actual target size cylinder was

used. For the initial three modes the change shows a similar pattern. To under-

stand the impact of changing only the Young’s modulus, the shape and Poisson’s

ratio were unchanged.

52

Chapter 6

Contact Pressure and RUS

6.1 Thermal Contact Resistance

When two bodies come in contact, heat flows from the hotter to the colder body.

A temperature difference is observed at the interface between the two surfaces in

contact. The magnitude of the temperature drop is related to the thermal contact

resistance between the contacting surfaces. Thermal contact resistance is an effect

due to surface roughness causing a temperature drop across the interface. These

surface irregularities creates intermittent points of contact and air gap in the

interface. The heat transfer is actually governed by both the conduction through

the contact spots and conduction or convection/radiation across the gaps.

Figure 6.1: Temperature drop due to thermal contact resistance

53

q′′

x = q′′

contact + q′′

gap (6.1)

Where q′′x is total heat transfer across the contacting surfaces. Figure 6.1 shows

temperature drop across the contact surface[4]. Thermal contact resistance is a

function of the temperature difference due to surface imperfection and the heat

transfer rate. This relationship can be shown in the following way.

R =∆T

q′′x(6.2)

Where R is the thermal contact resistance and ∆T is the temperature difference

across the interface of the two surfaces.

6.2 Contact Pressure

As discussed in the previous section heat transfer through the interfaces formed

by the mechanical contact of two solids occurs in three forms: conduction through

contacting spots, conduction through the gas-filled voids and radiation. In nor-

mal situations, radiation effects are small compared to the other two parameters.

Another geometric parameter that controls the heat transfer through the con-

tacting spots is the ratio of actual to apparent areas of contact. This ratio is

called contact pressure which is determined by relative contact pressure. The rel-

ative contact pressure is defined as the ratio of applied pressure to the contact

microhardness(P/Hc). This relative contact pressure also plays an important role

in the thickness of the air gap between the contacting interfaces. The ratio P/Hc

controls three geometric factors that control the heat transfer: contact spot den-

sity, mean contact spot radius, and separation distance of the mean planes of

the two contacting surfaces [50]. Contact microhardenss, Hc, depends on several

parameters: mean surface roughness, method of surface preparation and applied

54

pressure. An explicit relation was found in reference [50] for the relative contact

pressure

P

Hc

=

[P

(1.62× 106σ/m)c2

] 11+.071c2

(6.3)

Here σ is the RMS surface roughness and c2 is the Vickers correlation coefficients.

Figure 6.2: Magnified profile of two surfaces in contact with changing contactpressure(A)less contact pressure (B)higher contact pressure

Figure 6.2 shows the change of contacting surface based on the contact pressure.

Consequently it will also change the thermal contact resistance. In the context of

Molybdenum-99 targets contact pressure plays an important role in design [33],

and choosing appropriate forming pressure. Reducing the contact resistance is one

of the major challenges of designing the target. Low contact resistance will help to

remove heat throughout the irradiation process which is necessary for the safety

of the target. It is not possible to experimentally determine the actual contact

pressure between the surfaces. For plastically deformed surfaces Yovanovich [31]

proposed the following correlation for thermal contact conductance :

hcσ

ktanθ= 1.25

(P

Hc

)0.95

(6.4)

Here hc is solid spot conductance, tanθ is the mean absolute slope for the surface

profile and k is the harmonic mean conductivities of the two contacting surfaces.

55

6.3 Experiment

6.3.1 Target Assembly

This particular experiment was designed to investigate the effect of different con-

tact pressures effect on the RUS spectrum. To characterize the contact pressure

between two concentric cylinders, three different targets were made using three

different draw plug sizes without any foil. Since three different plug sizes were

used to make three different targets, the contact pressures are different in each

target. The draw plug assembly procedure was used to make the targets (Figure

6.3). The draw plug target manufacturing method or drawing process uses a die

and plug to mechanically expand the inner tube while holding the outer tube in

steady. The two aluminum tubes cut to a specified length are pre-manufactured

(with proper tolerance) so that inner tube can slide freely into the outer tube.

Once the tubes have been slid together, they are placed in a die that confines

the outer tube in the drawing process. A hydraulically driven rod with a plug

attached to the end is then forced through the inner tube. The size of the plug is

important because it deforms the inner tube plastically, and thus reduces the gape

between the two cylinders. Increasing the plug size will increase the deformation

that occurs in the inner tube (aluminum) leading to an increase in contact pressure.

Figure 6.3 shows the draw pug set up for target manufacturing.

Figure 6.3: Draw plug Rig

56

As mentioned earlier, the inner tube deforms plastically as the plug is drawn

though. The outer tube deforms elastically. During the manual process of drawing

the plug through the inner tube, the formation (drawing) pressure was recorded.

This is the required pressure to drive the plug through the target which is not an

actual forming pressure, but it provides a qualitative idea about the deformation.

Higher forming pressure means higher deformation.

Table 6.1: Weights of the three different targets

Draw plug ID Plug Outer Diameter (m) Tube Length (m) Target Mass (g)48 0.02662 0.161 61.47049 0.02664 0.161 60.94250 0.02667 0.161 61.213

Figure 6.4: a)Inner and Outer tubes together b)Two concentric cylinders beforedeformation c)After assembling the target

Forming pressure for the two plugs (48 and 50 ) is shown in Figure 6.6. As can

be seen from the plot that forming pressures are also different for different plugs

along the distance of the target, but they have a similar trend. At the beginning

of the process the required pressure is higher.

57

Figure 6.5: Dimensions(inches) of a draw plug(48)

(a) plug 48 (b) plug 50

Figure 6.6: Plot of pressure during swagging using two different plug sizes

58

6.3.2 Results

The purpose of this experiment was to detect variations in RUS spectrum in

different contact pressures which is related to thermal contact resistance. Since

three different draw plugs were used (plug 48, 49 and 50), it is assumed that the

targets have three different contact pressures. The outer diameter of the plugs are

shown in Table 6.1.

To evaluate the RUS responses of the targets, the spectrum of the individual

tube (inner and outer) is studied. The spectrum of the outer tube is shown

in Figure 6.7. The inner tube also shows the same type of response (Figure

6.8). All three tubes show a similar pattern. At around 8.3 × 105 Hz there is a

response around 0.9×10−4 volts. The amplitude of the peaks is considered for this

experiment. Since the geometry is a hollow cylindrical and the mass is constant

before and after making the target, the only variable will be the amplitude.

Figure 6.7: RUS spectrum of the outer tube

59

The diameter of the plug varies from 0.02662 m to 0.02667 m . These different

plug sizes will exert different pressures in the inner tubes. As the plug size in-

creases, it will cause more plastic deformation on the inner tube. Given the sizes

of the plugs it is assumed that only the inner tube is plastically deformed [19].

Outer tube’s deformation is elastic. Mass of the object plays an important role in

response of the of the amplitude. The masses of the three different targets are in

Table 6.1 .

Figure 6.8: RUS spectrum of the inner tube

The experimental findings of the RUS of the three different assembled targets

are shown in Figure 6.9.

60

Figure 6.9: RUS spectrum of the targets

Figure 6.9 shows the RUS response for three different targets. The frequency

range that we are interested in is 6.5 to 10 × Hz. In this frequency range the

amplitudes vary significantly. The result can be summarized in Table 6.2.

Table 6.2: Highest amplitudes in the interested frequency zone

Draw plug size frequency range (×105)Hz Amplitude (Vrms × 10−5)volts48 6.5-9 149 6-10 3.550 6-10 5

The differences in the amplitudes of spectrum can be explained using the con-

tact pressure between the two cylinders. Since the inner tube plastically deformed

with three different radii, contact pressures are different in these targets. The

RUS setup used to excite and measure the frequency spectrum is same as Figure

61

4.2. As the draw plug size increases the response amplitudes increases. From

Figure 4.2, only the outer tube is excited and the receiving transducers are receiv-

ing signals from the outer tube. Since the masses of these targets are almost the

same (Table 6.1), the amplitude depends on the modes that are being produced.

Since aluminum is used for both inner and outer tube, surface roughness does not

change for the deformed targets. The only variable that can influence the response

in RUS has to be contact pressure (Equation 6.4).

Figure 6.10: RUS spectra of outer tube, inner tube and one of the targets

Figure 6.10 shows that outer tube has a response around 1 × 10−4V and the

inner is around 0.8× 10−4V . When they are pressed together with draw plug 49,

the response reduced to around 0.4× 10−4V . All the individual tubes’ (inner and

outer) response were examined before they were deformed to a target.

62

Chapter 7

Conclusions, Recommendationsand Future Work

The goals of this thesis were to explore the potential of RUS methods for long

cylindrical tubes, study the contact pressure between the two cylinders in target,

and the usefulness of this technique in assessing target damage due to radiation.

The results obtained from RUS are highly dependent on the choice of the geome-

try. A finite element model was made to predict the frequency, based on geometry,

density and elasticity matrix. After predicting the appropriate elastic constants,

the RUS spectrum amplitude on different targets with different draw plug was

studied. Experimental results from the contact pressure analysis was discussed.

Theoretical study was completed to see the effect of radiation on mechanical prop-

erties of the material. After carefully studying the radiation effect on material,

some recommendations can be made to use this technique properly inside a hot

cell.

7.1 Conclusion

Experimentally detected frequency based on the quality factor is shown in Figure

5.1. Minimum frequency was detected around 250 kHz. Theoretically, below 250

kHz some modes existed, but due to the length and weight of the sample all

these mode frequencies had undetectable responses. In RUS, a Demarest plot is

usually used to measure the experimental Poisson’s ratio. However, to generate a

63

complete Demarest plot, it is necessary to experimentally detect first 20 modes. A

time-of-flight method can be handy in this situation to measure Poisson’s ratio for

an isotropic material. Even though the finite element method used in this study

was highly time consuming, a simple mesh study was done to see the effect of the

meshing on the eigen frequencies. A two pinducer system with a different holding

mechanism was used to detect the frequencies below 250 kHz. But because of

the size and shape of the sample the experiment was done in ”surface to surface”

manner. To extract all the frequencies below 300 kHz a study should be done

to make a sophisticated detection system. To see the effect on the spectrum

three different targets were made with three different plug sizes, which produced

distinguishable responses.

7.2 Recommendations

Based on the result discussed above, a couple of recommendations can be made.

The importance of sophisticated measurement system has already been estab-

lished. Machining of the sample is very important, any non-uniformity will pro-

duce discrepancies in the frequency response. Before using RUS to measure ra-

diation effects, some parameters should be considered. Dimensional change can

happen after irradiation (Figure 2.5). Before making any RUS measurement it is

necessary to measure the dimension properly. Temperature has profound effect on

the elastic constants. So cooling down of the irradiated material, and proper heat

treatment might be necessary for the sample. An experiment can be designed to

measure the contact resistance of the assembled targets. A numerical model of the

assembled targets might help to understand the behavior of the RUS responses.

7.3 Future Work

Future work in extending RUS method should involve looking at the development

of a quantitative measurement system for contact pressure between two cylinders.

64

Creating a correlation between the thermal contact resistance and amplitude of the

target would help to mathematically explain the phenomena. This idea is based

on the experimental results from target. To use RUS in radiation environment,

inserting the whole setup in the hot cell is an important step. The transducer

response in the radition field should be studied. Study of non-uniformity and

anisotropy in the field of RUS may help in studying radiation damage with RUS.

65

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Appendix A

XYZ Algorithm

c l cc l e a r a l lNN=input ( ’ Input the value o f N’ ) ;%−−−−−next 13 l i n e s a s s i g n an index IG to each b a s i s f unc t i onC=ze ro s ( 3 , 3 , 3 , 3 ) ;IG=0;f o r I =1:3

f o r L=1:NN+1f o r M=1:NN+1

f o r N=1:NN+1i f (L+M+N > NN+3) , break , endIG=IG+1;IC (IG)=1;LB(IG)=1;MB(IG)=1;NB(IG)=1;

endend

end

endrank =0.5∗(NN+1)∗(NN+2)∗(NN+3);NR=IG ;Gamma=ze ro s ( rank , rank ) ;f o r IG=1:NR

f o r JG=IG :NRI=IC (IG ) ;J=IC (JG ) ;LS=LB(IG)+LB(JG ) ;MS=MB(IG)+MB(JG ) ;NS=NB(IG)+NB(JG ) ;

Gamma(IG ,JG)=C( I , 1 , J , 1 )∗LB(IG)∗LB(JG)∗ func (LS−2,MS,NS) + . . .

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C( I , 2 , J , 2 )∗MB(IG)∗MB(JG)∗ func (LS ,MS−2,NS) + . . .C( I , 3 , J , 3 )∗NB(IG)∗NB(JG)∗ func (LS ,MS,NS−2)+. . .C( I , 1 , J , 2 )∗LB(IG)∗MB(JG) + . . .C( I , 2 , J , 1 )∗MB(IG)∗LB(JG)∗ func (LS−1,MS−1,NS) + . . .C( I , 1 , J , 3 )∗LB(IG)∗NB(JG) + . . .C( I , 3 , J , 1 )∗NB(IG)∗LB(IG)∗ func (LS−1,MS,NS−1)+. . .C( I , 2 , J , 3 )∗MB(JG)∗NB(IG ) + . . .C( I , 3 , J , 2 )∗NB(IG)∗MB(JG)∗ func (LS ,MS−1,NS−1);

Gamma(JG, IG)=Gamma(IG ,JG ) ;i f ( I==J ) E(IG , IG)=func (LS ,MS,NS) ;

endendend

[ vec t s v a l s ]= e i g (E\Gamma) ;

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