THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE

DEPARTMENT OF ENGINEERING SCIENCE AND MECHANICS

Nonlinear Ultrasonic Measurements for the Characterization of Fracture Toughness in Steel Alloys

COLIN WILLIAMS SPRING 2021

A thesis

submitted in partial fulfillment of the requirements

for a baccalaureate degree in Engineering Science

with honors in Engineering Science

Reviewed and approved* by the following:

Dr. Parisa Shokouhi Associate Professor of Engineering Science and Mechanics

Thesis Supervisor

Dr. Andrea P. Arguelles Assistant Professor of Engineering Science and Mechanics

Honors Adviser

Judith A. Todd Department Head

P.B. Breneman Chair and Professor of Engineering Science and Mechanics

* Electronic approvals are on file.

i

ABSTRACT - Technical

The knowledge of “fracture toughness,” a mechanical strength parameter, is essential to ensure the

operational safety of fracture-critical systems and components such as pressure vessels and pipelines.  Loss

of fracture toughness can be an early indicator of catastrophic failure by rapid brittle fracture. However, it

is not possible to quantify fracture toughness in service for structural health monitoring (SHM). Traditional

fracture toughness testing is destructive and cannot be completed in-situ. We seek to investigate the utility

of nonlinear ultrasonic testing as a nondestructive alternative to traditional fracture toughness testing

procedures. This research is motivated by the high sensitivity of nonlinear ultrasonic parameters to a

material’s microstructure when compared to conventional (linear) ultrasonic tests. Therefore, because of

their mutual dependence on microstructure, we hypothesize a correlation between the measurable nonlinear

ultrasonic parameters and fracture toughness characteristics of a material. In this thesis, we investigate the

existence of such a correlation in 4130 steel samples with different heat treatments and hardness

values. Using the technique of Second Harmonic Generation (SHG), both surface and bulk waves were

used to estimate the nonlinearity parameters of eight different tempered steel samples. These same samples

were also tested destructively for their fracture toughness characteristics using Charpy V-Notch testing. We

report two sets of results pertaining to bulk and surface waves. Results of nonlinear bulk

wave testing indicate a correlation between the nonlinearity parameter and plate hardness values. Results

for surface waves show trends between nonlinearity parameters and plate hardness values as well as wave

velocity and plate hardness values. The continuation of this project will utilize numerical simulations

to explore how various microstructural features influence both fracture toughness values and nonlinearity

parameters with the ultimate goal of quantitative in-situ fracture toughness inspection.

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ABSTRACT – Non-Technical

Fracture toughness is a critical parameter for evaluating the quality and strength of systems and components

such as pipelines, pressure vessels, and more. Fracture toughness represents the resistance to a crack or flaw

growing within a sample. Because fracture toughness evaluation requires destructive laboratory testing -

meaning the sample becomes destroyed during the examination - it cannot be completed on systems that

are in real service situations. For example, a pipeline that is actively carrying natural gas cannot be

destructively tested. Fracture toughness is important because monitoring its value can be an early indicator

of dangerous failure, such as a crack which becomes too large and causes destruction of the sample.

Nonlinear ultrasonic testing suggests a novel solution to nondestructively characterize fracture toughness.

Using ultrasonic techniques, two types of ultrasonic waves were used to estimate the “nonlinearity

parameters” of eight different steel samples. The nonlinearity (which represents the non-uniformity and

microscopic flaws within the sample) measured using ultrasound allows the understanding of how the

sample behaves at the micro-scale, which has relations to fracture toughness behaviors. These same samples

were additionally tested destructively for their fracture toughness characteristics. This study presents

experimental relationships between destructive (fracture toughness) and nondestructive (ultrasound) test

parameters. The ultimate goal is to create a framework which can use ultrasound to measure fracture

toughness “in the field” (such as an active pipeline) without destroying the sample, with the intent to use

the knowledge to mitigate the dangerous failure of these systems.

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TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................... v

ACKNOWLEDGEMENTS ........................................................................................ vii

Chapter 1: Introduction ................................................................................................ 1

Chapter 2: Literature Review ...................................................................................... 4

2.1: Fracture Toughness ................................................................................................... 4 2.2: Fundamentals of Linear Ultrasonic Testing ............................................................. 7 2.3: Nonlinear Ultrasonic Testing .................................................................................... 8 2.4: Existing Research ..................................................................................................... 13 2.5: Conclusion ................................................................................................................ 16

Chapter 3: Materials and Methods .............................................................................. 17

3.1: Steel Alloy Samples .................................................................................................. 17 3.2: Experimental Set-Up ................................................................................................ 17

3.2.1 Bulk Wave Measurements .............................................................................. 17 3.2.2 Surface Wave Measurements .......................................................................... 18 3.2.3 Destructive Mechanical Testing ..................................................................... 19

3.3: Bulk Wave Transducer Holder Design ..................................................................... 20

Chapter 4: Ultrasonic Signal Analysis Techniques ..................................................... 21

4.1: Ultrasonic Signal Processing .................................................................................... 22

Chapter 5: Design of Test Protocol ............................................................................. 27

5.1: Choice of Ultrasonic Couplant ................................................................................. 27 5.1.1: Vacuum Grease .............................................................................................. 28 5.1.2: Molasses ........................................................................................................ 29 5.1.3: Ultrasonic Gel ................................................................................................ 30

5.2: Time-Dependency of Measurements ........................................................................ 31 5.3: Surface Preparation ................................................................................................... 31 5.4: Repeatability ............................................................................................................. 32

Chapter 6: Results and Discussion .............................................................................. 34

6.1: Bulk Wave Measurements ........................................................................................ 34

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6.2: Surface Wave Measurements ................................................................................... 37 6.3: Discussion ................................................................................................................. 39

Chapter 7: Conclusions and Future Outlook ............................................................... 41

Appendix A ................................................................................................................. 43

MATLAB routines for analysis and figure generation .................................................... 43

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LIST OF FIGURES

Figure 1: Configurations for three common fracture toughness tests: (a) compact tension test, (b) single edge-notched bend test, and (c) middle-cracked tension test [3]. ......................... 5

Figure 2: Computer-generated layout for predicting fracture toughness in ductile materials with microstructure [8]. ........................................................................................................... 6

Figure 3: An example SHG experimental set-up for measuring higher harmonic generation. The schematic is from [15]. .................................................................................................... 10

Figure 4: Correlation between nonlinearity parameter and fracture toughness in CrMoV rotor steel [15]. ......................................................................................................................... 14

Figure 5: Schematic of SHG experimental set-up. .................................................................. 18

Figure 6: The assembly utilized by GuidedWave for nonlinear surface wave measurements. 19

Figure 7: Example Charpy V-Notch specimen and test set-up [23]. ....................................... 20

Figure 8: Custom 3D printed holder in use for bulk wave testing. ......................................... 21

Figure 9: An ultrasonic signal before and after applying the Hanning Window. .................... 22

Figure 10: An example ultrasonic signal in the frequency domain obtained using Fast Fourier Transform. The data shown here corresponds to a test conducted on plate HC40. ......... 23

Figure 11: An example plot of A2vs. A12. The slope of this plot represents the relative

nonlinearity parameter. .................................................................................................... 24

Figure 12: A log-log plot for full amplitude range. This data corresponds to plate HC40. .... 25

Figure 13: A log-log plot example for the first 10 input amplitudes. ...................................... 26

Figure 14: Two-Hour test results for vacuum grease couplant in terms of A2/A12 over time. 28

Figure 15: Two-Hour test results for molasses couplant in terms of A2/A12over time. ........... 29

Figure 16: Two-Hour test results for ultrasonic gel couplant in terms of A2/A12over time. ... 30

Figure 17: A picture of the plate HC36 surface (a) before and (b) after preparation. ............. 32

Figure 18: These 7 runs are plotted together to show evidence of repeatability. .................... 32

Figure 19: Estimated relative bulk wave nonlinearity parameter for each plate. .................... 34

Figure 20: Bulk wave arrival time delay for different plates. ................................................. 35

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Figure 21: Low amplitude arrivals for each plate signal. ........................................................ 36

Figure 22: Relative nonlinearity parameter vs. absorbed energy in bulk wave samples. ........ 36

Figure 23: Estimated relative surface wave nonlinearity parameter for each plate hardness. . 37

Figure 24: Surface wave arrival time delay for different plates. ............................................. 38

Figure 25: Correlations between absorbed energy and relative nonlinearity of bulk waves. .. 38

Figure 26: Correlations between absorbed energy and relative nonlinearity of surface waves.39

vii

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Parisa Shokouhi, for her guidance and support throughout

the duration of this project. I would also like to thank Dr. Jacques Rivière, the Shokouhi/Rivière lab

group, and all the members of the PennSUL Lab who have assisted me during this project, with special

thanks to Pedro Lama and Jared Gillespie. Additional thanks goes out to GuidedWave for their generosity

in allowing me to assist this project. Finally, I would like to thank those involved with the Erickson

Discovery Grant program for funding my work during the Summer of 2020.

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Chapter 1: Introduction

The goal of this research is to find methods to evaluate the critical strength properties of metals

non-invasively without compromising their integrity. The objective is to investigate relationships between

the strength parameter “fracture toughness” and measurable nonlinear parameters. Finding this

relationship will enable the evaluation of fracture toughness nondestructively using nonlinear ultrasonic

testing procedures.

Fracture toughness is a material property representing a material’s resistance to rapid, brittle

fracture in the presence of a pre-existing flaw. Fracture toughness is critical to the safety of components

and structures because it controls crack stability and propagation. The capability to nondestructively

characterize fracture toughness in fracture critical components has the potential to transform the field of

nondestructive evaluation (NDE) and increase the effectiveness and reliability of SHM.

Conventionally, fracture toughness is determined through the use of destructive testing, such as

the Single Edge Bend or Charpy Impact tests. Destructive tests are expensive, time-consuming,

geometrically limited, and ultimately can result in the loss of the sample examined, severely limiting their

application to in-situ evaluation. If a solution could be developed for evaluating fracture toughness in-situ

and nondestructively, it would be a significant advancement in the field of NDE.

The ultimate goal and impact of this research is to improve the safety of fracture-critical

structures and components through NDE. For example, fracture toughness is used to predict the brittle

failure of pressure vessels in the oil and gas industry [1]. Evaluating fracture toughness with destructive

testing is impossible for in-situ applications. In addition, because of its sensitivity to microscopic

heterogeneities including voids and grain boundaries, fracture toughness is already a difficult parameter to

characterize and can vary significantly within a sample. Consequently, fracture toughness measured in a

laboratory setting may be non-negligibly different from the same material’s fracture toughness when

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measured in-situ due to changes in microstructure from service, aging, and environmental conditions. In

high-pressure vessels, pipelines, reactors, and other fracture-critical structures, such an extreme variation

from design values can lead to unforeseen catastrophic failures. A nondestructive method to evaluate

fracture toughness will enable testing of fracture-critical structures and components under realistic

loading scenarios. This will help identify compromised structures and components for repair, resulting in

improved safety and mitigation of potential failures.

In this study, we propose the use of ultrasonic evaluation. Investigating the feasibility of using

nonlinear ultrasound techniques to characterize fracture toughness is a novel concept in the field of

NDE. Earlier efforts used the more traditional linear ultrasonic methods, but have not succeeded due to

the low sensitivity of these methods to microstructural variations. One example is the study by Sinclair

and Eng [2], who were unable to find a reliable correlation between fracture toughness and linear

ultrasonic parameters including attenuation and wave velocity. The researchers detailed issues in picking

appropriate frequency ranges and signal processing difficulty when measuring these values.

On the other hand, due to their high sensitivity to microstructural imperfections within a

material, nonlinear ultrasonic techniques have a strong potential to evaluate fracture toughness. While

conventional linear ultrasonic techniques are capable of detecting features on the same order of magnitude

as the wavelength used during evaluation, nonlinear methods are capable of detecting microstructural

features which are orders of magnitude smaller than the wavelength [4]. When conducting nonlinear

ultrasonic testing, a material’s response can be quantified by so-called “nonlinearity parameters.” These

parameters respond to microscopic structural variations and show sensitivity to microstructural flaws and

features. Comparatively, fracture toughness is also notoriously sensitive to minute microstructural

variations. Because of this mutual dependence of the nonlinearity parameter and fracture toughness on

microstructure, we hypothesize a correlation between the two.

This thesis presents an experimental investigation of the hypothesized correlation between fracture

toughness and nonlinear ultrasonic parameters. In the laboratory, a series of 4130 steel alloy plates that

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have been subjected to various heat treatments will be tested. Through the use of prior destructive testing,

the impact resistance of these plates - a proxy for fracture toughness - is known and documented. In this

study, the nonlinear ultrasonic technique of SHG is used to determine the nonlinearity parameters for each

of the plates, with the goal of relating nonlinearity parameters and fracture toughness in the context of in-

situ NDE of fracture-critical structures.

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Chapter 2: Literature Review

2.1: Fracture Toughness

In its most general definition, fracture toughness is a parameter used to quantify a material’s

resistance to the propagation of a crack. The ability to evaluate fracture toughness is key to the study of

fracture mechanics. The knowledge of fracture toughness of components and systems is essential to

damage tolerance design, and residual strength analysis.

According to a review of fracture toughness by Zhu and Joyce [3], the behavior of a metal during

fracture toughness testing can be described by three factors: the fracture charateristics of the material, the

strength and deformation behavior, and the effects of geometric constraints. The fracture behavior can be

broken down into the micro-behavior of the fracture – typically described as either a “ductile” or “brittle”

fracture. While ductile fracture is characterized by a slow, uniform crack growth process, brittle fracture

is rapid, unstable, and often results in unexpected failure. Brittle fracture toughness can be studied from

the crack initiation site, which is often a clearly defined location on a fracture surface. Ductile fracture

toughness requires the use of a fracture curve to quantify fracture toughness due to its steady, measurable

progression. The strength and deformation behavior of a material can most often be classified as either

linear-elastic, nonlinear-elastic, or elastic-plastic [3].

In order for fracture toughness to be effectively used in the engineering design process, it would

ideally be characterized as a material property, which can be easily transferred from controlled laboratory

environments to in-situ applications. In reality, the fracture toughness value of a given material can vary

drastically between these two environments. The challenge of transferring fracture toughness has resulted

in a strict set of testing standards created by the American Society for Testing and Materials (ASTM).

These standards require specific geometrical boundaries on laboratory specimens with the objective of

generating conservative fracture toughness results to apply to real environments. Tests following these

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standards generate conservative results because in-situ components and structures have no geometric

constraints to govern their fracture behavior, and therefore additional factors of safety are necessary.

Additional environmental factors such loading, temperature, and corrosion also have adverse effects on

the evolution of fracture toughness within a given material in-situ.

The current ASTM standard for fatigue and fracture testing is ASTM E1823-20a [4]. Within this

manual are definitions and standards of fatigue and fracture testing, as well as references to other ASTM

standards for specific tests including Linear Elastic Plain Stain Fracture Toughness Testing, Crack-Tip

Opening Displacement Testing, Chevron-Notch Testing, and other testing procedures. Examples of

common fracture toughness laboratory test configurations are shown in Figure 1: (a) The compact tension

test, (b) single edge-notched bend test, and (c) middle-cracked tension test [3].

Figure 1: Configurations for three common fracture toughness tests: (a) compact tension test, (b) single edge-notched bend test, and (c) middle-cracked tension test [3].

The mutual connection between fracture toughness and nonlinearity parameters on microstructure

is critical to the hypothesis driving this research. In brittle materials such as ceramics, many experiments

and simulations ([5], [6]) have correlated crack propagation and behaviors quantitively with

microstructural features. One main objective of this area of research is improving the performance of

engineered materials such as ceramics and composites. For example, one such study explores the potential

to predict the behavior of engineered materials through grain boundaries and inclusions [7]. The paper

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includes a discussion of influencing factors including orientation of grain boundaries and inclusion of

voids to influence brittle fracture toughness.

In contrast, there is not an extensive amount of research that quantify the effects of

microstructural grains and grain boundaries on the fracture measures of metals. However, some simplified

computational approaches have been recently explored in metals, such as the work of Roy & Zhou on

Molybdenum (Mo) [8]. Figure 2 shows the two-dimensional configuration that is simulated, representing

a realistic laboratory-scale fracture toughness test following ASTM guidelines.

Figure 2: Computer-generated layout for predicting fracture toughness in ductile materials with microstructure [8].

This configuration adheres to the guidelines of ASTM compact tension testing for plain strain

fracture toughness. The polycrystalline microstructure shown in red is intended to simulate single phase

pure Mo, assuming a homogenous and isotropic in-grain model with randomly oriented grain structure

generated by the Voronoi tessellation function. This simulation was created using ABAQUS. Using this

model and these microstructural simulations, fracture toughness estimates were generated that maintain

consistency with experimental values for Mo. For this reason, conclusions are drawn which allow fracture

toughness to be represented as functions of microstructural changes such as grain size and grain boundary

strength. This model shows promise to be extended to three-dimensions, as well as to other materials.

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2.2: Fundamentals of Linear Ultrasonic Testing

Before the concepts of nonlinear ultrasonic testing pertinent to this study can be discussed, it is

important to explore the fundamentals of linear ultrasonic testing. One primary assumption that governs

the theory of linear ultrasonic testing is that the shape of the emitted wave does not distort as it propagates

through a material [9]. When considering an ultrasonic wave propagating through a one-dimensional,

homogenous, linear-elastic medium, its behavior is governed by the wave equation shown in Equation 1:

"!#"$! =

1'!"!#"(! (1)

where # represents particle displacement, $is the material coordinate, ( is time, and ' is the wave

velocity of the material – a function of material properties. In the event that the emitted waveform

becomes distorted during propagation due to heterogeneities within the sample, the linear wave equation

is no longer suitable and nonlinear theory must be utilized.

Conventional ultrasonic testing typically involves the use of a piezoelectric transducer pair. Due

to the piezoelectric effect, these transducers are able to convert mechanical changes in pressure or force

into an electrical signal. In order to obtain a useable signal, a coupling agent is applied between the

transducer face and sample. Coupling removes the air gaps which can cause unwanted energy reflection

from around the contact area, ensuring the expected transmission of energy between transducer and

sample.

Ultrasonic testing is typically conducted in two configurations: through-transmission and pulse-

echo testing. Through-transmission testing, such as the bulk-wave ultrasonic testing conducted in this

study, requires access to both sides of the sample with the receiving and emitting transducers. In pulse-

echo testing, access to only one side of the test sample is required as a single transducer sends and

receives the ultrasonic signal after its reflection on the sample’s back wall. Pulse echo testing can be

conducted in geometrically limited situations, and can provide information about the thickness of a

sample that through transmission testing does not provide.

8

Materials characterization using ultrasonic evaluation is based on the measurement of several

common linear parameters, including wave velocity and attenuation. Because ultrasound is a form of

volumetric inspection, it can be useful for correlating the bulk properties of a material. From measuring

the shear wave and longitudinal wave velocities of a material, important material properties such as

elastic modulus, shear modulus, and Poisson’s Ratio can be determined. It is also important to note that

temperature and environmental conditions may have an effect on these velocity responses [10].

Another important parameter measured using ultrasonic techniques us attenuation, or the loss of

energy a wave experiences during its propagation. Attenuation results from both absorption and scattering

during propagation, and measuring wave attenuation can provide information about a material’s grain

size, inclusions, porosity, or distribution of discontinuities [11].

There are several practical setbacks to the use of linear ultrasonic techniques. One major issue

with pulse velocity measurements is their relative lack of sensitivity. These methods are practical for

sensing large defects only, not subtle micro-cracks or voids within a material [9]. In the context of

fracture toughness and its strong dependency on microstructure, this lack of sensitivity is a substantial

issue. The linear bulk wave ultrasonic method has other limitations including the necessity for

information regarding the thickness of a sample to determine pulse velocity, which may not be available

depending on the sample’s location and geometry. This limits the utility of linear ultrasonic evaluation

techniques for the nondestructive evaluation of fracture toughness in-situ.

2.3: Nonlinear Ultrasonic Testing

The fundamental theory of nonlinear ultrasound hails from the concept of a material’s elastic

nonlinearity. Elastic nonlinearity refers to a material’s inherently nonlinear constitutive (stress-strain)

relationship, resulting from microscopic heterogeneities such as dislocations, grain boundaries, and

micro-cracks. When impinged by an ultrasonic wave, interactions with these heterogeneities result in

9

higher harmonic generation [12]. The result is a waveform that becomes cumulatively more distorted as it

propagates, accompanied by the generation of higher harmonic components visible in the frequency

spectrum of the signal. Incident waveforms become distorted by the nonlinear elastic response, generating

harmonic waves of frequencies at double, triple, etc. the fundamental frequency.

Nonlinearity can be measured by introducing finite-amplitude stress waves into a sample,

however it exists inherently within a material no matter what amplitude of wave is emitted. At small

amplitudes, the emitted and received waveforms have similar frequency content because the stress-strain

relationship behaves linearly. However, as the amplitude of the incident wave increases, the material’s

nonlinearity is activated and the received wave is distorted cumulatively along its path. Consequently,

higher harmonic components appear more clearly in the frequency spectrum as the amplitude of the

emitted wave increases. Typically, when studying nonlinearity, only the second-order higher harmonic

wave component is analyzed. Orders beyond the second are present, however, their nonlinearity effects

are considerably smaller and more difficult to measure, and therefore not as frequently studied.

The nonlinear ultrasonic technique utilized in this research is SHG. The SHG technique is a

suitable choice for these experiments due to its proven capability to detect microstructural changes within

metals [13]. In SHG, as an ultrasonic wave travels through a material and interacts with its

microstructural features in such a way that higher harmonic generation occurs. From measuring the

amplitude of these higher harmonics, the acoustic nonlinearity parameter (+) can be measured

quantitatively. SHG has been studied since the 1960s, with early research focusing on materials such as

aluminum and copper. Today, SHG is utilized as a method to investigate the behavior of a wide range of

materials under realistic loading conditions reliably and efficiently.

Unlike the more basic ultrasonic testing which relies on the simplified linear wave equation

(Equation 1), SHG is governed by nonlinear wave equation. Considering a longitudinal wave propagating

through a medium in one-dimension, the nonlinear wave equation can be expressed as [14]:

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(1 − +"#"$)

"!#"$! =

1'!"!#"(! (2)

where + represents the (classical) nonlinearity parameter, a function of the second and third order elastic

constants of the material. Note that Equation 2 does not consider higher order terms. Under the

assumptions of quadratic nonlinearity and plane wave propagation, Equation (2) can be solved with the

method of perturbation [14]:

# = ." sin(2$ − 3() ++."!2!$

8 cos(22$ − 23()(3)

where ." represents the amplitude of the first harmonic wave, and 2 = #$ is the wavenumber, where 3 is

the frequency, and ' is the speed of light. The value preceding the cosine term, %&!"'"(

) , is commonly

denoted as .!, representing the amplitude of the second harmonic wave in the absence of attenuation and

damping. .!can be rearranged to solve for a relative measurement of +, denoted as +′:

+′ =8.!."!2!$

(4)

Using Equation 4, we now have a method to quantify the “relative” nonlinearity parameter (+′)

through the relation between the first and second harmonic amplitudes, which can be measured using

SHG. An experimental SHG testing set-up is shown in Figure 3 [15].

Figure 3: An example SHG experimental set-up for measuring higher harmonic generation. The schematic is from [15].

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In order to measure an “absolute” +, the absolute physical displacements of the first and second

harmonic waves must be measured. One such example in literature is that of Yost and Cantrell [16], who

measured the absolute nonlinearity parameter of aluminum 2024 under artificial aging. Because the

change in both relative and absolute nonlinearity measurements are proportional, the microscopic

heterogeneities they seek to explain can be quantified with either technique. One common way to quantify

+′ is by plotting values of .! versus ."! through a range of amplitudes. If the nonlinearity of the material

follows the quadratic assumption, the slope of this linear plot represents +′. Using the simplified +′ allows

a quicker comparison of results relative to a particular experimental design and initial state, assuming all

other experimental parameters are held constant [13].

To understand the parameter +, it is important to recognize the possible microstructural features

that can contribute to its variations. Among the most notable of these features are dislocations,

precipitates, and micro-cracks [13]. To model the dislocation motion contributing to acoustic nonlinearity,

a model developed by Suzuki et al. [17] considers a dislocation as a line pinned between two points,

behaving as if it were a fixed-fixed vibrating string. The “pinning” points defining this model could be

microstructural features such as a point defect. An incident ultrasonic wave would set this “string” into

vibrational motion, creating a non-negligible strain on the material which can be combined with the lattice

strain to create a total material strain. This total strain can then be combined with other geometrical and

material properties to create an equation for the change in nonlinearity parameter due to dislocation

pinning, as detailed in Suzuki’s paper.

The direct effect of a precipitate on the magnitude of + is negligible. However, precipitates

interacting with dislocations in a microstructure can have an effect on the changes in +.A precipitate

within a microstructure creates an additional local stress on pinned dislocations, which alters the total

strain of a material and therefore + following the model presented by Suzuki [13], [17].

Nonlinear ultrasonic testing methods and + have also been tied to detecting early signs of

microcracking in steel [18]. Using two transducers in contact with a specimen at a fundamental frequency

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of 30 kHz (="), the amplitudes of the second (=!,60 kHz) and third (=*, 90 kHz) higher harmonics were

used to obtain two nonlinearity parameters, denoted as + = =!/="! and ++ = =*/="*. To assess the

nonlinearity parameters ability to indicate the formation of microcracking, nonlinear ultrasonic tests were

repeated on samples undergoing forced corrosion. To that end, ultrasonic pulses were emitted every ten

minutes over the span of three weeks. Once the variation of either nonlinearity parameter rose above a

given threshold, the time was recorded as the time necessary to create microcracks through forced

corrosion. In the frequency spectrum, the variation of higher harmonics over time is concluded to be a

sign of onset microcracking in steel rebar. Under the appearance of micro-cracking, + measurements rose

to as much as ten times their initial values in these experiments. Alongside more traditional methods of

visual or microscope inspection, the use of nonlinear ultrasonic testing has the ability to detect onset

cracking of steel-reinforced concrete reliably with the described methods [18]. Due to the preliminary

nature of these results, a mathematical relationship between + and microcrack features is not developed

here. It is also worth noting the automation used in this testing procedure can further reduce the amount of

experimental effort required to determine cracking when compared to traditional methods.

Along with the high sensitivity to microstructural changes nonlinear ultrasonic testing provides, it

also possesses a set of limitations which must be accounted for during experiments and discussion of

results. When designing an SHG experiment, it is important to prepare a clean sample surface, consider

the best choice of gel coupling, and ensure consistent transducer orientation and contact against the

sample. Currently, the measurement of absolute nonlinearity parameters is only possible in a laboratory

environment, a barrier which must be overcome before the transition to in-situ applications. Because the

relative nonlinearity parameter is only capable of measuring changes in nonlinearity, a baseline absolute

value or “calibration system” [13] inevitably will need to be established for any sort of predictive

capability in-situ.

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2.4: Existing Research

The feasibility of correlating ultrasonic parameters and material strength properties has been

hypothesized for several decades. An early research publication authored by Alex Vary in 1979 proposed

a theoretical mathematical relationship between ultrasonic and fracture toughness factors within metal

materials. Specifically, Vary [19] proposes parameters and notation which appear to mirror the current

standards used in ultrasonic testing and research such as ? (Young’s Modulus) and @, (longitudinal wave

velocity). He also establishes a parameter denoted as +. At a glance, this may appear to be an early

representation of the nonlinearity parameter as it is known in our research. However, it can be seen that

Vary defined this + instead as the “attenuation slope” or the change in attenuation of the ultrasonic signal

versus change in frequency.

It is an interesting learning opportunity to go “back in time” to understand the early evolution of

ultrasonic NDE from our modern perspective. Through derivations and mathematical models, Vary

concludes that both fracture toughness and the yield strength of a material are functions of ultrasonic

wave factors in metal materials. Also, he states that ultrasonic NDE measurements can potentially be used

to correlate with fracture properties of metal polycrystalline materials. Finally, a hypothesis for future

“ranking” of fracture toughness is stated, as well as use of “purely ultrasonic techniques” to determine

fracture toughness once correlation curves have been established for a given material. These correlation

curves are what future research including this study seeks to understand and create.

In 2003, an original contribution by Jeong et al. [15] presents for the first time a nonlinear

ultrasonic method for estimation of fracture toughnessfor CrMoV rotor steel. Several CrMoV samples

were heat-treated to various degrees, similar to the plates that are tested in this project. Unlike our

proposed experiment, the fracture toughnessvalues of the CrMoV steel samples were estimated in a two-

step process. First, the fracture appearance transition temperature (FATT) was determined as a function of

aging time using Charpy V-Notch Impact testing. FATT is the temperature at which a sample will display

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a half-brittle and half-ductile fracture surface. Because ductile and brittle characteristics are correlated to

the fracture toughness of a material (ductile materials are tougher than brittle), the FATT is a suitable

parameter for correlating strength characteristics. FATT was then correlated to the nonlinearity parameter

calculated from bulk wave SHG analysis of the second-order higher harmonics. The next step used

existing literature correlating FATT and excess temperature to estimate fracture toughness.

The results of this source determined a relationship between +and FATT as a function of aging

time. The conclusion indicates that a correlation exists between +and fracture toughness, created with the

link of FATT. This result can be shown in Figure 4. These results are encouraging and provide support

for the conjecture of correlating +and fracture toughnessin heat-treated steel samples. In the results of

this study (Section 6), Figure 22 shows a comparative figure to Figure 4 using data collected from the

bulk wave tests of our heat treated steel samples. The trends in this data appears to be in opposition to

Figure 4.

Figure 4: Correlation between nonlinearity parameter and fracture toughness in CrMoV rotor steel [15].

Up to this point, when discussing the correlation between nonlinearity parameters and fracture

toughness, their shared relationship to microstructure has been stated, but not explored in further detail. In

2007, a research paper published by Zheng [20] provided more detail on the relationship between

microstructure and fracture toughness so crucial to this hypothesis. In this report, titanium-aluminum

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alloys subjected to various microstructure-altering heat treatments were explored with both fracture

toughness experiments and scanning electron microscope (SEM) imaging. The ultimate goal in this work

is to investigate several relationships that characterize a microstructure – notably grain size and lamellar

spacing – and fracture toughness. Through careful heat treatment processes, two series of alloys were

designed. In one, grain size was kept constant while varying lamellar spacing, and the other kept the

lamellar spacing constant varying the grain size. To estimate fracture toughness, ASTM standard plane-

strain Chevron-Notch specimens were used. They reported an increase in fracture toughness with

increasing grain size up to approximately 700 microns, followed by a decrease in fracture toughness as

the grain size continued increasing. For the second set of samples, fracture toughness increased as

lamellar spacing decreases, consistent with the Hall-Petch relation. It is worth noting that only a small

number of data points were used to establish these relationships (only eight were used for the fracture

toughness/grain size model). Additionally, the titanium alloy is not the same as the steel alloys tested

here. With these points in mind, Zheng outlined a reasonable methodology to examine the relationships

between fracture toughness and microstructure systematically which could be applied to other materials.

One of the broader impacts of this research concerns pipeline fracture safety. Azeddine et. al [21]

studied degradation in API 5L X52 pipeline steel to calculate fracture toughness under dynamic loading

conditions. The relationship between microstructure and fracture toughness is once again in focus. To

better understand the nature of pipeline failures due to aging, this research used a more complex fracture

toughness model known as “notch fracture toughness.” This model uses more realistic geometric

assumptions than a common linear fracture toughness testing model. They found that the fracture

initiation energy in pipeline steel is the same across both welded and unwelded areas, and the required

energy for crack propagation is “conditioned” [21] by the microstructure of the material. Although less

empirical than the conclusions of Zheng, conclusions from Azeddine et al. [21] are more pertinent to the

material and potential industrial applications of this research. These studies provide additional evidence

16

to support our research hypothesis. Pipeline steel is only one example of the many in-situ applications of

nondestructive fracture toughness testing.

Because the nonlinearity exhibited in materials can be attributed to a wide list of physical features

on the macro, meso, and micro scales, understanding the main sources contributing to higher harmonic

generation is valuable to NDE research. One such example of this work is that of Kamali [22] on

plastically deformed Aluminum 1100, where changes in nonlinearity parameters were studied under

different strain levels both experimentally and with numerical simulations.

A study of nonlinearity with varying microstructural features will provide insight into

quantitatively relating nonlinear parameters to damage in order to prevent potential fracture in deformed

structures. This would require numerical modeling of microstructure, simulation of nonlinear ultrasonic

testing and higher harmonic generation. This is the direction for continuation of this research.

2.5: Conclusion

Engineers rely on NDE techniques such as SHG and linear ultrasound to ensure safe and reliable

operation of components and structures found in everyday life. The increasing capability to characterize

materials and assess incipient damage before any macro-scale failure occurs is an evolving study in the

field of NDE. Fracture toughness is a critical parameter to these SHM efforts, and researchers across the

world are working to bolster its detection in-situ to mitigate the chances of catastrophic failure.

Nonlinear ultrasonic techniques such as those discussed here are cost-effective, portable,

repeatable, and offer solutions for a range of NDE applications. There are many sources that suggest the

nonlinearity parameter +measured through SHG can be used to monitor microstructural changes in a

material which may be indicative of changes in fracture toughness values, and ultimately the chances of

failure through brittle crack propagation. Understanding these relationships is the primary objective of

this research.

17

Chapter 3: Materials and Methods

3.1: Steel Alloy Samples

The selected material for this experiment is 4130 steel alloy, chosen for its range of achievable

hardness through heat treatment and tempering, as well as its availability and common use for in-situ

applications. Each plate was heat treated and tempered to Rockwell C Hardness values ranging from 22

through 54. Specifically, the plates tested have approximate Rockwell C Hardness values of 22, 24, 28,

32, 36, 40, 52, and 54. The dimensions of the plates for ultrasonic testing are 6” x 10” x 0.394.”

Additionally, for the purposes of destructive testing, five Charpy V-Notch specimens were available for

each hardness value.

3.2: Experimental Set-Up

The test methods used in this study include bulk-wave ultrasound, surface-wave ultrasound, and

Charpy V-Notch testing. The following sections provide detail regarding each test method.

3.2.1 Bulk Wave Measurements

The test was conducted in a through-transmission mode with the two transducers held on either

side of each plate. A transducer with a center frequency of 5 MHz (NdtXducer DPR300-900V) was used

as the transmitter, and a 10 MHz (Olympus V544-SM) transducer was used to receive and record the

ultrasonic signal with sampling frequency of 250 MHz. The pulse repetition frequency was set to 100 Hz.

A ten-cycle tone burst pulse with a center frequency of 5 MHz and amplitude ranging from 25 to 500 V is

sent to the transmitter in increasing 25 V increments. The experimental setup also includes a Ritec high-

powered amplifier (RAM-10000, Ritec), 50-Ohm load (Ritec RT-50), Olympus Pulser/Receiver (5072PR,

18

Olympus), and National Instruments 5-Slot Chassis (PXIe-1073, National Instruments) with the National

Instruments PXIe-5170R acquisition card. A schematic is shown in Figure 5. To collect data, the chassis

is connected to a Dell laptop equipped with a custom-designed Lab-View program for signal acquisition

and real-time processing/visualization.

Figure 5: Schematic of SHG experimental set-up.

3.2.2 Surface Wave Measurements

To complete the surface wave testing on these steel samples, GuidedWave (Bellfonte, PA), an

ultrasonic innovation company, utilized their nonlinear surface wave testing equipment to collect

nonlinear data. The surface wave fixture and custom wedge design shown in Figure 6 ensure consistent

and repeatable coupling for surface wave measurements. For each sample, a nonlinear surface wave test

was conducted, similarly to the bulk wave procedure. The choice of coupling was pure mineral (baby) oil.

To ensure consistent coupling and repeatability, 10 pounds of pressure was applied to the fixture. These

measurements utilized a sampling frequency of 50 MHz, pulse repetition frequency of 200 Hz, and

voltage range increasing from 40 to 300 V in 10 V increments. The emitting transducer was a narrowband

19

2.5 MHz probe, while the receiver was a broadband 5 MHz. The incident points on the transducers were

separated by 1 inch.

Figure 6: The assembly utilized by GuidedWave for nonlinear surface wave measurements.

3.2.3 Destructive Mechanical Testing

The destructive mechanical testing procedure chosen for testing the steel samples was the Charpy

V-Notch test. From this testing procedure, the absorbed impact energy (J) for each plate was estimated.

This test involves striking a notched specimen with a predetermined amount of force from a pendulum at

a chosen height. Measuring the impact sustained by the sample during this test helps estimate the energy

absorbed by the sample during its fracture. These tests were conducted according to ASTM E23-16b. An

example testing schematic is shown in Figure 7.

20

Figure 7: Example Charpy V-Notch specimen and test set-up [23].

At each hardness, the five notched specimens were tested using this destructive method, and their

results were averaged for each hardness value. While traditional fracture toughness testing results are

reported in SI units of ABC√E , the results of the Charpy V-Notch destructive test report the absorbed

energy in the SI unit of Joules. Charpy V-Notch testing is simpler to perform than fracture toughness

testing, and significant correlations between the two exist, such as those studied by Yu [24] and Barbosa

[25]. In this study, absorbed energy is reported as an equivalent to fracture toughness values.

3.3: Bulk Wave Transducer Holder Design

One of the practical challenges in designing an ultrasonic through-transmission bulk wave testing

procedure is the design and construction of a custom fixture to hold the emitting and receiving transducers

during the data acquisition process. During the early stages of this project, several design criteria were

established to govern the design and functionality of this holder. Among these criteria included the ability

for the holder to be quicky adjustable for testing a plate at multiple locations, to be simple to assemble

and set up, and to consistently hold the transducers directly in line with each other for accurate

21

transmission and reception of the ultrasonic signal. Figure 8 shows this holder being used in the

laboratory.

Figure 8: Custom 3D printed holder in use for bulk wave testing.

Through the use SolidWorks, this holder was designed and iterated to fulfill the stated design

requirements. The holder was then 3D printed. This was an iterative design process, but resulted in a

better fitting and more robust holder.

Chapter 4: Ultrasonic Signal Analysis Techniques

The recorded SHG signals are analyzed using signal processing techniques to extract test

parameters and interpret results. Using MATLAB, a data analysis script was developed specifically for

this project which streamlines the transition between raw data and interpretable results for both nonlinear

bulk wave and surface wave testing.

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4.1: Ultrasonic Signal Processing

The first processing step was “windowing” the recorded response using a Hanning Window. In

the time domain, the windowed signal transitions more smoothly to zero on both ends. This translates into

better-defined peaks and less side lobes in the frequency domain. An example of a raw output signal and

the corresponding windowed version is shown in Figure 9.

Figure 9: An ultrasonic signal before and after applying the Hanning Window.

After windowing, it is necessary to transform the time domain signal into the frequency domain

using the Fast Fourier Transform (FFT). The result of this procedure, shown in Figure 10, is the

frequency spectrum of the signal. From this spectrum, the amplitudes of the fundamental (A1) and 2nd

harmonic wave components (A2) were measured to estimate the relative nonlinearity (++ = =!/="!). As

expected, the fundamental amplitude was measured at 5 MHz, and the second harmonic at 10 MHz. To

more accurately measure the peaks of these harmonics, a polynomial was fitted to the top 5 points of each

peak, and the maximum value of this polynomial was chosen as the value for A1 and A2 respectively. For

bulk waves, this process was repeated 20 times at a range of input voltage amplitudes from 25V to 500V.

For surface waves, the process was repeated 27 times for signals corresponding to input voltages varying

23

between 40V and 300V. Different transducers, wave propagation modes, and testing procedures account

for the differences in these testing parameters.

Figure 10: An example ultrasonic signal in the frequency domain obtained using Fast Fourier Transform. The data shown here corresponds to a test conducted on plate HC40.

As the input voltage increases, the nonlinearity of the sample is activated and measured. After

extracting the amplitude of fundamental and second harmonic peaks at each voltage, the relative

nonlinearity parameter can be estimated from the slope of =!/="!. Figure 11 shows an example of =!

plotted vs. ="! for plate HC40.

According to the theory (Section 2.3), we expect a quadratic relationship between the frequency

of the fundamental and second harmonic wave components. In practice, this relationship may deviate

from the expectations. Other forms of nonlinearity were introduced during the data collection process,

such as nonlinearity in the data acquisition system and nonlinearity introduced during experimental

procedures, can alter this relationship. During data analysis, we utilized novel techniques to minimize the

influence of these unwanted forms of nonlinearity and ensure that the estimated relative nonlinearity

parameter reflects the materials nonlinearity.

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Figure 11: An example plot of A2vs. A12. The slope of this plot represents the relative nonlinearity parameter.

The relationship between =! and =" is ideally quadratic:

=! = ="!(5)

If a perfect quadratic relationship no longer holds, the relationship will be defined through an arbitrary

exponent $.

=! = ="((6)

Moving from exponential to logarithmic form, it follows that:

log"- =! = $ log"- =" (7)

Therefore, to find the true scaling exponent, one can plot the logarithm of =! vs. the logarithm of

=" and calculate the slope. Ideally, the slope of this line should be 2 (Equation 5). If the slope differs from

2 or varies, further data analysis was done to select a range of input voltages, where the log-log slope is

nearly 2. Using data in this range assured a nearly quadratic relation between =! and =" , which allowed

the estimation of the relative nonlinearity parameter.

For example, considering plate HC40, the log-log plot shown in Figure 12 for the full amplitude

range yields an exponent of about 2.39.

25

Figure 12: A log-log plot for full amplitude range. This data corresponds to plate HC40.

From visual inspection of the right half of the figure, it appears that higher amplitude input

signals result in a steeper slope. This higher slope could be related to nonlinearity or ‘saturation’ in the

data acquisition system affecting the SHG measurements. Considering only the first half of this data (the

first 10 data points) as shown in Figure 13, the slope changes from 2.39 to approximately 1.99, which is

much closer to 2. In other words, using this range of data would ensure an almost quadratic relationship

between =" and =!, and therefore was used when estimating the relative nonlinearity parameter. The

above procedure was completed for every nonlinearity parameter estimation presented in this research

providing confidence that test results represent the nonlinearity of the material, and not other sources.

26

Figure 13: A log-log plot example for the first 10 input amplitudes.

To quantify the measurement variability, three bulk wave tests were conducted at various

locations across each plate. At each of these locations, seven independent SHG tests were conducted and

analyzed using the process previously described. In between each test, the transducers were uncoupled

from the sample, and all surfaces were cleaned according to the test protocol (Section 5). New ultrasonic

gel was reapplied prior to every trial. Finally, to calculate the estimated relative nonlinearity parameter for

a given location, the average of the five most repeatable trials was recorded at each location. This was to

minimize the adverse effect of differences in couplant application or transducer alignments between

different trials. Finally, the results at these three location averages were once again averaged to arrive at a

final estimate for each of the eight plates.

For surface wave data provided by GuidedWave, the experimental procedure was fairly similar to

those described for bulk wave testing and a very similar data analysis approach was taken. However, for

surface waves, only one trial was completed at each of the three test locations on a given plate. The final

estimated relative nonlinearity parameter for surface waves was calculated as the average of data acquired

at these three locations.

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Chapter 5: Design of Test Protocol

Due to the sensitivity of nonlinear ultrasonic measurements to experimental parameters and

procedures, extensive preliminary experiments were conducted to establish effective and repeatable test

procedures. The investigated experimental parameters included the type of ultrasonic couplant used

during testing, surface preparation of the plates, and timing of each experiment.

5.1: Choice of Ultrasonic Couplant

Due to the sensitivity of SHG measurements, the choice of coupling agent (or couplant) between

the transducer face and the test material is an important consideration. The purpose of the couplant is to

ensure the effective transmission of ultrasonic energy from the transducer to the sample [26]. However,

the introduction of a couplant can add variability into ultrasonic measurements, and such effects are

amplified in a nonlinear experimental set-up. Application thickness, couplant viscosity and trapped air

bubbles are among the parameters which may lead to measurement variations.

In this project, three typical couplants – vacuum grease, molasses, and ultrasonic gel – were

considered for bulk wave testing. The goal of investigating the influence of different couplants was to

find the most suitable choice for repeatable data. Several parameters were considered when evaluating

couplant choices including, the ease of application, received signal quality, and changes in the frequency

content of the signal over time. In all three cases, the couplants “set” after some passage of time. This

setting process typically involved the couplants drying and further attaching the transducers to the

surface, thus influencing the nonlinear responses. The most desirable couplant for nonlinear ultrasonic

measurements is the one that provides a consistent relationship between the first and second harmonic

peaks (=" and =!) over time. Using repeated 375 V input signals over a period of two hours, the evolution

of this relationship over time was observed and used to determine the most suitable coupling.

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5.1.1: Vacuum Grease

The first couplant tried in the preliminary testing was high vacuum grease (Dow Corning).

Compared to the other two couplants tested, vacuum grease was the thickest and appears most viscous.

Applying the vacuum grease uniformly to the transducer surfaces was difficult. Rather than spreading out

evenly across the transducer, the vacuum grease often remained stuck to the applicator and layered

unevenly. Once applied, a secondary issue of generating repeatable signal with vacuum grease emerged.

Due to the thickness of the grease and its inability to be applied evenly, mounting the emitter and receiver

parallel within the holder was a challenge. Ultimately, once an acceptable signal was generated, a two-

hour test pulsing every two minutes was conducted. Given the importance of the relationship between =!

and =" when calculating the relative nonlinearity parameter β’, we evaluate each couplant by studying the

evolution of =!/="! over time. When analyzing each couplant, a consistent value of =!/="! is desired as

time progresses. Figure 14 shows the two-hour test results for vacuum grease. This figure shows that the

=!/="!value rises continuously throughout the duration of the test, and never reaches a desired plateau.

The application difficulty and the two-hour test results eliminated vacuum grease from the experimental

protocol.

Figure 14: Two-Hour test results for vacuum grease couplant in terms of A2/A12 over time.

29

5.1.2: Molasses

The next coupling considered for the SHG testing was molasses (Grandma’s Original). Compared

to vacuum grease, molasses was a thinner substance. Consequently, generating a quality signal using the

molasses couplant was a quicker process than that for vacuum grease. However, as the two-hour test

progressed, some issues became apparent, which were later reflected in the results shown in Figure 15.

Due to the experimental setup and vertical orientation of the transducers, there was an effect of gravity on

the couplant-transducer system. While the sensors were held in place firmly by the holder, the thinner

molasses was not. As time passed, molasses was observed to slowly run down the side of the plate. By the

end of the two-hour test period, molasses was pooling on the surface underneath the holder.

Figure 15: Two-Hour test results for molasses couplant in terms of A2/A12over time.

Figure 15 shows the =!/="!relationship captured during the preliminary test using molasses as

the couplant. The observed trend is different than what was seen with vacuum grease, but still do not

satisfy the desired consistency over time. The nonlinearity parameter appears to settle after roughly 60

minutes, which may have been due to the excess molasses running out of the contact surface below the

30

transducer. However, waiting 60 minutes to do a test is not practical. Due to this lack of consistency,

molasses was not chosen for future SHG testing.

5.1.3: Ultrasonic Gel

The third ultrasonic couplant considered for SHG testing was an ultrasonic gel (Magnaflux

Soundsafe 20-012). Application and signal processing with ultrasonic gel was a streamlined process

compared to molasses and vacuum grease. Also, the effects of gravity on this couplant were negligible

over time. Figure 16 shows the results of time-dependency test for plate HC52 using the ultrasonic gel

couplant.

Figure 16: Two-Hour test results for ultrasonic gel couplant in terms of A2/A12over time.

The range of variability in the ultrasonic gel coupling is notably smaller than those seen in both

vacuum grease (Figure 14) and molasses (Figure 15). Moving forward with testing, this ultrasonic gel

couplant was used.

31

5.2: Time-Dependency of Measurements

To better understand the “stability” of SHG measurements over time, the nonlinear response was

measured over three time intervals. In the first set of experiments, 20 signals of the same 375 V amplitude

were emitted every 30 seconds over a ten-minute period. In the second set, 20 signals were sent every 60

seconds over a 20-minute period. Finally, in the third, 60 amplitudes were sent every two minutes during

a 120-minute period. For all of these tests, the same input voltage of 0.75 V was used. Besides the total

duration of each experiment and frequency of pulsing, all other parameters were held constant for these

three experiments.

To analyze time dependency, the time evolution if the first and second harmonic amplitude

(="and =!), as well as =!/="!were studied. From the results of these tests at a constant voltage (not

shown here), it is determined that the long-term response when using ‘ultrasonic gel’ as the couplant is

consistent. Therefore, the remaining experiments were conducted confidently using shorter-duration

measurements.

5.3: Surface Preparation

A notable advantage of ultrasonic NDE is that it can be done using little or no surface

preparation. Samples can typically be prepared by hand using simple low-cost methods such as polishing

with emery papers or a wire brush [27]. Surface preparation prior to NDE testing helps to eliminate

roughness, remove rust, and create a better surface for uniform coupling between the sample and the

transducer. All these factors remove measurement variability.

To prepare the plates, hand-sanding with 300 grit paper was first used to remove rust and other

stains. Afterwards, rubbing alcohol was also used to clean the plates and remove debris. Figure 17 shows

an example of one sample before and after this surface preparation process. We used the same sample

preparation protocol before testing all the plates.

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Figure 17: A picture of the plate HC36 surface (a) before and (b) after preparation.

5.4: Repeatability

It is important to ensure SHG tests taken on each plate produce repeatable results. When

conducting bulk measurements, three locations were chosen on each plate. At each of these three

locations, seven trial runs were conducted (Run 1-7). Figure 18 shows an example of the repeatability

measured for plate HC40. Due to the repetition of each ++measurement, we are able to incorporate error

bars in later figures to show the variability during testing. Although seven trials were recorded each time,

the most repeatable five runs were used for the subsequent analysis. This is to exclude the effect of

inconsistent sensor placement and couplant application from the results.

Figure 18: These 7 runs are plotted together to show evidence of repeatability.

a b

33

To summarize, the final testing procedure determined from these preliminary tests included the

use of ultrasonic gel couplant, short-duration tests, and removal of surface roughness and debris using

300 grit emery paper. This combination of test protocol allows for repeatable measurements moving

forward into the collection of nonlinear data.

34

Chapter 6: Results and Discussion

6.1: Bulk Wave Measurements

Before considering the destructive testing results, correlations between relative nonlinearity

parameter (+′) and plate hardness values were investigated. Figure 19 shows a summary of this data.

Figure 19: Estimated relative bulk wave nonlinearity parameter for each plate.

As previously described, each “Location Average” represents the mean of five repeated runs. The

final “Average Value” is the mean of the three Location Averages. From these results, a trend is observed

between the relative nonlinearity parameter and increasing plate hardness. As the plate harness increases,

the estimated nonlinearity values decrease. This trend is more pronounced for the higher hardness plates

HC52 and HC54.

When conducting nonlinear ultrasonic testing, we also collected linear parameters such as wave

velocity. Using the arrival time for the lowest hardness plate (HC22) as a reference value, cross

correlation of recorded waveforms was conducted to calculate the changes in bulk wave velocity with

35

plate hardness. When calculating this delay, the second-lowest incident amplitude signal was selected.

This choice is to eliminate the possibility of higher amplitude nonlinear factors interfering with linear

wave velocity measurements. Figure 20 shows the results of these velocity measurements at three plate

locations.

Figure 20: Bulk wave arrival time delay for different plates.

The bulk wave velocity measurements do not show a discernable trend. Also, it appears that the

HC24 wave velocity is an outlier. Further plotting of the signals help to visualize this delay. Figure 21

shows the first arriving packet of each plate response at a low amplitude. It can be seen that every arrival

time except for HC24 was before the 3 KL marker.

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Figure 21: Low amplitude arrivals for each plate signal.

Referring back to the existing research presented in this paper (Section 2.5), Figure 22 shows the

trends between absorbed energy and ultrasonic testing. These axis are the same as those presented in

Figure 4, and a contrasting trend can be seen in these two figures.

Figure 22: Relative nonlinearity parameter vs. absorbed energy in bulk wave samples.

37

6.2: Surface Wave Measurements

When considering surface wave results, it should be noted that in realistic NDE applications,

pulse-echo and surface wave measurements are more common than through-transmission bulk wave

testing due to their practicality and the geometric limitations of through-transmission testing. For

example, when testing a section of pipe, it is often difficult or impossible to access both the outer and

inner walls with transducers to generate a through-transmission bulk wave measurement. Companies such

as GuidedWave seek surface wave (and guided wave) solutions when considering in-situ applications.

Figure 23 shows the results of surface wave SHG testing. The estimated relative nonlinearity

parameter at each plate hardness values shows a different relationship than that observed in bulk waves.

Figure 23: Estimated relative surface wave nonlinearity parameter for each plate hardness.

A non-monotonic trend is observed in these surface wave results. Although the trend is not the

same as that observed in Figure 19, the higher hardness plates HC52 and HC54 show nonlinearities that

are clearly different from the rest of the plates.

Similar to bulk wave testing, velocity measurements were also conducted on surface wave data

using cross correlation. Again, a low amplitude signal recorded on HC22 was chosen as reference and the

time delay was calculated at each of the three surface wave testing locations using cross-correlation of the

38

signals. Figure 24 shows the results. The two plates with the largest hardness clearly show slower surface

wave velocities than the rest.

Figure 24: Surface wave arrival time delay for different plates.

Finally, to investigate the relationship between fracture toughness values and nonlinearity

parameters, the results of the Charpy V-Notch and SHG testing were compared. Figure 25 shows this

bulk wave result.

Figure 25: Correlations between absorbed energy and relative nonlinearity of bulk waves.

39

Similarly, Figure 26 displays this relationship for the surface wave data.

Figure 26: Correlations between absorbed energy and relative nonlinearity of surface waves.

6.3: Discussion

The bulk wave velocities shown in Figure 20 do not show any trend with plate hardness. It

follows that traditional linear bulk wave ultrasonic testing is not feasible for detecting differences in plate

hardness and most likely fracture toughness. This conclusion agrees with the results of past research

efforts [2].

In contrast to the bulk wave results, a trend can be observed in the surface wave velocity plots. At

each location, the wave velocity remains relatively constant for the lower hardness plates. As hardness

increases into the ranges beyond HC40, a clear delay in wave arrival is observed. The observed trend in

this linear ultrasonic parameter suggests that a correlation may be present between linear surface wave

measurements and hardness for a certain range of hardness values. Consequently, this result suggests

there may be a trend present between linear surface wave measurements and fracture toughness values.

40

When comparing surface and bulk wave results, one notable difference is the order of magnitude

of the relative nonlinearity parameters. Surface wave nonlinearities measure two orders of magnitude

larger than bulk wave results. Being relative values for two different wave modes, a direct comparison of

the values is not possible, although still a point to consider for further analysis. The two measurements

also show different trends with plate hardness. One hypothesis for these differences is the potentially

nonuniform distribution of thermal damage across the thickness of the heat-treated plates, leaving the

near-surface portion of the plate to be more heterogenous. Because surface waves only propagate a small

depth into the plate, cumulative nonlinear effects may be higher than bulk waves which travel through the

less damaged core.

Also, surface and bulk waves are not the same physically, and have different modes of

propagation. For this reason, their responses through the plate may vary. Future microstructural analysis,

such as use of a SEM or X-ray diffraction, will be necessary to understand the nature of this difference in

nonlinearity.

Despite the results for surface and bulk wave testing differing, they both offer evidence of a

correlation with fracture toughness. When considering the results shown in Figures 25 and 26, is can be

seen that relationships not only exist between plate hardness and nonlinearity, but also between

nonlinearity an absorbed energy. For in-situ applications involving complex geometries and limited

access, surface wave testing methods are often utilized. When considering the error associated with these

results, surface wave data exhibits a reduced amount of variability in comparison to bulk wave

measurements. Potential sources of error include the differences in experimental procedures between the

bulk wave and surface wave tests conducted in different laboratories, as well as the nature of the different

wave propagation modes and plate responses. The type of transducers chosen, testing parameters (such as

sampling frequency), and transducer fixtures utilized in each experiment are potential sources of error.

41

Chapter 7: Conclusions and Future Outlook

In this study, the elastic nonlinearity present within steel materials was investigated as a

measurable parameter for an indirect assessment of fracture toughness noninvasively. Understanding the

correlation between ultrasonic nonlinearity parameters and fracture toughness values provides a

foundation which could enable in-situ evaluation of fracture toughness on fracture-critical components

without compromising their integrity or functionality. Here, the feasibility of both through transmission

bulk wave and surface wave tests was investigated.

A nonlinear ultrasonic technique known as SHG was utilized for both bulk and surface wave

measurements. Eight 4130 steel plates, each heat treated and tempered to different Rockwell C Hardness

values, were tested for their mechanical properties and ultrasonic responses. From these results, we

observed trends between bulk wave nonlinearity and plate hardness, as well as surface wave nonlinearity

and plate hardness. However, these trends were not the same. In addition, consideration was given to the

magnitudes of the relative nonlinearity parameters for the two wave types are on different orders of

magnitude, although their direct comparison is not possible.

In addition to nonlinear ultrasonic measurements, the wave velocities of both bulk waves and

surface waves were measured. From these velocity measurements, a decreasing trend in wave speed with

increasing plate hardness was observed in the surface wave data. For bulk wave velocities, no discernable

trend was observed across the full range of available hardness values. Considering the observed trends for

both velocities and nonlinearity parameters, the plates with higher hardness exhibited more pronounced

changes than those with lower hardness.

The differences between nonlinearity parameter trends in surface and bulk wave measurements

are a key direction for the continuation of this research. It is hypothesized that the thermal “damage” due

to heat treatment in each steel plate is not uniform across its depth. Due to the different particle motions

of surface and bulk waves during propagation, this nonuniformity could be accentuated. Bulk wave

42

particles travel in the same direction as the wave front, while particle motion in a surface wave follows an

elliptical path during wave propagation. Future microstructural analysis is necessary to examine the

validity of this hypothesis. Potential parameters of interest include microhardness measured at various

depths through the plate, an SEM analysis of the cross sectional area of the plates, and measurement of

any residual stresses using X-ray diffraction present along the plate surface that can contribute to the

nonlinearity. Additionally, numerical modeling and simulations will be conducted to model the nonlinear

behaviors observed in these tests. Through modeling the micro-scale heterogeneities that contribute to the

macro-scale nonlinearity parameters for each plate, we can investigate which microstructural feature

contributes to material nonlinearity, and to what extent. These simulations are planned to be conducted

using a commercial finite element package ABAQUS.

The outcome of this research can lead to novel methods for the SHM of bridges, pipelines,

pressure vessels, and other safety critical systems and components. This will result in improved public

safety and will prevent potential economic loss and environmental damage associated with the failure of

such systems.

43

Appendix A

MATLAB routines for analysis and figure generation

%% Script to Process Bulk Wave Ultrasonic Data Binary Files % Colin Williams, Pedro Lama % This script takes a given set of ultrasonic data and info of .dat format. % and analyzes to produce useful information. % Inputs include the data and info binary files from ultrasonic testing, % and several relevant variables to characterize the signal % Outputs include plots of data, spectra, and nonlinearity parameter, as % well as automatically saving the figures to a folder. If the desired folder % doesn't exist, program will create it named "Spectrum_Figures" % Figure Save currently OFF % Initialization steps. clc; % Clear the command window. close all; % Close all figures clear; % Erase all existing variables. set(0,'defaulttextinterpreter','latex') % PLATE NUMBER plate = 1; % LOCATION NUMBER location = 1; A1 = []; j = 1; polyno = 4; % For the polynomial fitting of peak maxes % Variables to Adjust (User Input) center_freq_1 = 5e6; center_freq_2 = 2*center_freq_1; % For experiment time_beg = 2.2; time_end = 5.4; % Read data file. [file,path] = uigetfile('*.dat'); if isequal(file,0) disp('User selected Cancel');

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else disp(['User selected ', fullfile(path,file)]); end fid = fopen(file,'r'); signal = fread(fid,inf,'double'); % Full Data Binary File fclose('all'); %Read Info File. [file2,path2] = uigetfile('*.dat'); if isequal(file2,0) disp('User selected Cancel'); else disp(['User selected ', fullfile(path,file2)]); end INFO = importdata(file2); % Info on data % Testing Parameters f_s = INFO(1,2); % Sampling Frequency recordtime = INFO(1,5); recordtime_us = recordtime*10^6; t_s = 1/f_s; % Time Step N = round(recordtime/t_s); % Number of Points time = (0:N-1)*t_s; % N should be for time of one acquisition time_us = time*10^6; % Reshape time vector to micro seconds signalmatrix = reshape(signal,N,[]) ; % N x 100 matrix, % Command Learned from Prof. Riviere to shape the 100 pulses into a coherent signal % Plot Data vs Time figure plot(time_us,signalmatrix) title("Reshaped Signal vs. Time") xlabel('Time') ylabel('Amplitude') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; signal_mean = mean(signalmatrix,2); % Average %signal_Norm = signal_mean/max(signal_mean); %% Normalizing the signal signal_mean = signal_mean - mean(signal_mean); % DC offset hanning_w=[zeros(round(time_beg*N/recordtime_us),1); ... % Create zeros hanning(round((time_end-time_beg)*N/recordtime_us)); ... % Signal Range zeros((round((recordtime_us-time_end)*N/recordtime_us)),1)]; % More zeros signalhanning=signal_mean.*hanning_w; % Create Hanning Signal

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figure plot(time_us,signal_mean,time_us,signalhanning) title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' Signal vs. Time Run ',sprintf('%d',j)]) xlabel('Time (microseconds)'); ylabel('amplitude (V)') legend('Signal','Han. Window Signal','location','best') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; N=5000; % Resolution df = 1/(t_s*(N-1)); % max time scale f_nyq = f_s/2; % Nyquist Frequency % FFT Spectrum of Data With Window S = fft(signalhanning,N); % choose number of points to fft w (add more zeros) abs_S = abs(S); f_vector = 0:df:f_nyq; % FFT - Spectrum Plot (log y axis) half_S = abs_S(1:N/2); % This is the signal with no window Q = fft(signal_mean,N); abs_Q = abs(Q); half_Q = abs_Q(1:N/2); % Peakfinder [pks_1,locs_1] = findpeaks(half_S(1:750)); max_peak_1 = max(pks_1); max_loc_1 = find(half_S == max_peak_1); peak_find_f_vec_1 = f_vector(max_loc_1); % Begin Polyfit for most precise max peak fit index_1 = [max_loc_1-5 max_loc_1+5]; [spect_value_1,index_max_1]=max(half_S(index_1(1):index_1(2))); % Find the maximum value of the function half_S over the range index(1) to index(2). % Half_S is the spectrum. index_max_1=index_max_1+index_1(1)-1; % Converts the found index to the real position of the index in strainfilt poly_1=polyfit(transpose(f_vector(index_max_1-polyno:index_max_1+polyno)),half_S(index_max_1-polyno:index_max_1+polyno),2); % Fit a second order polynomial to points within (polyno) number of points % of the previously found maximum. % In this case, x (f_vector) is a frequency vector, and y is the spectrum max_pos_1 = (-poly_1(2)/(2*poly_1(1))); % equivalent to -b/2a

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better_amp_1 = polyval(poly_1,max_pos_1); A1(j) = better_amp_1; y1 = polyval(poly_1, f_vector); % Peak Finder 2 [pks_2,locs_2] = findpeaks(half_S(190:225)); max_peak_2 = max(pks_2); max_loc_2 = find(half_S == max_peak_2); peak_find_f_vec_2 = f_vector(max_loc_2); index_2 = [max_loc_2-5 max_loc_2+5]; [spect_value_2,index_max_2]=max(half_S(index_2(1):index_2(2))); index_max_2=index_max_2+index_2(1)-1; poly_2=polyfit(transpose(f_vector(index_max_2-polyno:index_max_2+polyno)),half_S(index_max_2-polyno:index_max_2+polyno),2); max_pos_2 = (-poly_2(2)/(2*poly_2(1))); better_amp_2 = polyval(poly_2,max_pos_2); A2_poly(j) = better_amp_2; %% note we dont use the poly fit for run #1 due to error in final fitting y2 = polyval(poly_2, f_vector); % Peak Finder 3 [pks_3,locs_3] = findpeaks(half_S(250:330)); max_peak_3 = max(pks_3); max_loc_3 = find(half_S == max_peak_3); peak_find_f_vec_3 = f_vector(max_loc_3); index_3 = [max_loc_3-5 max_loc_3+5]; [spect_value_3,index_max_3]=max(half_S(index_3(1):index_3(2))); index_max_3=index_max_3+index_3(1)-1; poly_3=polyfit(transpose(f_vector(index_max_3-polyno:index_max_3+polyno)),half_S(index_max_3-polyno:index_max_3+polyno),2); max_pos_3 = (-poly_3(2)/(2*poly_3(1))); better_amp_3 = polyval(poly_3,max_pos_3); A3_poly(j) = better_amp_3; y3 = polyval(poly_3, f_vector); % Thing to add some markers to the peaks figure semilogy(f_vector/1e6,half_S,'bo') % Only half of data points necessary for FFT grid on

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title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' FFT Signal Spectrum Run ',sprintf('%d',j)]) xlabel("Frequency (MHz)") xlim([0 25]) % need to window ylabel("Amplitude (a.u)") hold on plot(max_pos_1/1e6,better_amp_1, 'k*') plot(max_pos_2/1e6,better_amp_2, 'm*') plot(max_pos_3/1e6,better_amp_3, 'r*') hold on plot(f_vector(97:104)/1e6,y1(97:104)) plot(f_vector(197:204)/1e6,y2(197:204)) plot(f_vector(297:304)/1e6,y3(297:304)) legend('Signal', 'Peak 1', 'Peak 2','Peak 3','Fit Line') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; hold off % % LINES 154 - 164 ARE TO SAVE IMAGES % % Check to see if the directory exists, if not, make the directory % if ~ exist('Sprectrum_Figures','dir') % mkdir('Spectrum_Figures') % end % % Procedure to save spectrum figure to directory % q = char(j); % Converts j from double to character for sprintf file naming % x = [pwd '/Spectrum_Figures/Spectrum']; % y = sprintf('%d.png',q); % z = [x y]; % Combine non-changing directory location with incrementing file number each loop % saveas(gcf, z) % save current figure to specified location % Prompt to enter number of tests to run the rest on auto prompt = {'Enter the total number of tests, program will run remaining data automatically:'}; dlgtitle = 'Input'; dims = [1 35]; definput = {'0'}; % Default input answer = inputdlg(prompt,dlgtitle,dims,definput); totalruns = str2double(answer); % The next loop auto runs the rest of tests, taking amplitudes, saves figures. % Does not ask for new range of frequencies, because they remain the same % peaks for j = 2:totalruns % For experimental data filenamedata = sprintf('LANL_Sample21_Test3_run%d_data.dat',j); filenameinfo = sprintf('LANL_Sample21_Test3_run%d_info.dat',j); INFO = importdata(filenameinfo); fid = fopen(filenamedata, 'r');

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signal = fread(fid,'double'); status = fclose(fid); % Testing Parameters f_s = INFO(1,2); % Sampling Frequency recordtime = INFO(1,5); t_s = 1/f_s; % Time Step N = round(recordtime/t_s); % Number of Points time = (0:N-1)*t_s; % N should be for time of one aquisition time_us = time*10^6; % Reshape time vector to micro seconds signalmatrix = reshape(signal,N,[]) ; % N x 100 matrix, % Command Learned from Prof. Riviere to shape the 100 pulses into a coherent signal signal_mean = mean(signalmatrix,2); % Average of many pulsed signals signal_mean = signal_mean - mean(signal_mean); % DC offset hanning_w=[zeros(round(time_beg*N/recordtime_us),1); ... % Windowing hanning(round((time_end-time_beg)*N/recordtime_us)); ... zeros((round((recordtime_us-time_end)*N/recordtime_us)),1)]; signalhanning=signal_mean.*hanning_w; figure plot(time_us,signal_mean,time_us,signalhanning) title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' Signal vs. Time Run ',sprintf('%d',j)]) xlabel('Time (microseconds)'); ylabel('amplitude (V)') legend('Signal','Han. Window Signal','location','best') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; % figure plot(time_us,signal_mean) title('Signal Mean vs. Time') xlabel('time (\mus)'); ylabel('amplitude (log scale) (V)') set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; N=5000; % Resolution df = 1/(t_s*(N-1)); % max time scale f_nyq = f_s/2; % Nyquist Frequency % FFT Spectrum of Data S = fft(signalhanning,N); abs_S = abs(S); f_vector = 0:df:f_nyq;

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% FFT - Spectrum Plot (log y axis) half_S = abs_S(1:N/2); % % Procedure to save spectrum figure to directory % q = char(j); % Converts j from double to character for sprintf file naming % x = [pwd '/Spectrum_Figures/Spectrum']; % y = sprintf('%d.png',q); % z = [x y]; % Combine non-changing directory location with incrementing file number each loop % saveas(gcf, z) % save current figure to specified location % Find the amplitude of each harmonic [xA1] = [(center_freq_1 - 0.2*center_freq_1),(center_freq_1 + 0.2*center_freq_1)]; % Input variable method % Peakfinder [pks_1,locs_1] = findpeaks(half_S(1:750)); max_peak_1 = max(pks_1); max_loc_1 = find(half_S == max_peak_1); peak_find_f_vec_1 = f_vector(max_loc_1); index_1 = [max_loc_1-5 max_loc_1+5]; [spect_value_1,index_max_1]=max(half_S(index_1(1):index_1(2))); % find the maximum value of the function half_S over the range % index(1) to index(2) % half_S is the spectrum index_max_1=index_max_1+index_1(1)-1; % converts the found index to the real position of the index in strainfilt poly_1=polyfit(transpose(f_vector(index_max_1-polyno:index_max_1+polyno)),half_S(index_max_1-polyno:index_max_1+polyno),2); % fit a second order polynomial to points within (polyno) number of % points of the previously found maximum % in this case, x (f_vector) is a frequency vector, and y is the spectrum max_pos_1 = (-poly_1(2)/(2*poly_1(1))); % equivalent to -b/2a % you may end up needing to round this depending on what you want to use % it for better_amp_1 = polyval(poly_1,max_pos_1); A1(j) = better_amp_1; y1 = polyval(poly_1, f_vector); % Amplitude 2: Picking second order frequency range % Peak Finder 2 [pks_2,locs_2] = findpeaks(half_S(185:235)); % Come back here later (index problem) max_peak_2 = max(pks_2);

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max_loc_2 = find(half_S == max_peak_2); peak_find_f_vec_2 = f_vector(max_loc_2); index_2 = [max_loc_2-5 max_loc_2+5]; [spect_value_2,index_max_2]=max(half_S(index_2(1):index_2(2))); index_max_2=index_max_2+index_2(1)-1; poly_2=polyfit(transpose(f_vector(index_max_2-polyno:index_max_2+polyno)),half_S(index_max_2-polyno:index_max_2+polyno),2); max_pos_2 = (-poly_2(2)/(2*poly_2(1))); better_amp_2 = polyval(poly_2,max_pos_2); A2_poly(j) = better_amp_2; y2 = polyval(poly_2, f_vector); % Peak Finder 3 [pks_3,locs_3] = findpeaks(half_S(290:330)); %come back here later (index problem) max_peak_3 = max(pks_3); max_loc_3 = find(half_S == max_peak_3); peak_find_f_vec_3 = f_vector(max_loc_3); index_3 = [max_loc_3-5 max_loc_3+5]; [spect_value_3,index_max_3]=max(half_S(index_3(1):index_3(2))); index_max_3=index_max_3+index_3(1)-1; poly_3=polyfit(transpose(f_vector(index_max_3-polyno:index_max_3+polyno)),half_S(index_max_3-polyno:index_max_3+polyno),2); max_pos_3 = (-poly_3(2)/(2*poly_3(1))); better_amp_3 = polyval(poly_3,max_pos_3); A3_poly(j) = better_amp_3; %% note we dont use the poly fit for run #1 due to error in final fitting y3 = polyval(poly_3, f_vector); % Thing to add some markers to the peaks figure semilogy(f_vector/1e6,half_S,'ko') % Only half of data points necessary for FFT title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' FFT Signal Spectrum Run ',sprintf('%d',j)]) xlabel("Frequency (MHz)") xlim([0 25]) % need to window ylabel("Amplitude (a.u)") hold on plot(max_pos_1/1e6,better_amp_1, 'm*') plot(max_pos_2/1e6,better_amp_2, 'm*') plot(max_pos_3/1e6,better_amp_3, 'm*')

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hold on legend('Signal') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; % This does the same thing for every remaining run end %% The Plotting Section % With the amplitudes of the runs recorded, graph A1^2 vs A2 to calculate nonlinearity % parameter from the slope A1_squared = A1.^2; % Plot A1^2 vs A2 from each of the 20 runs using Polyfit A2 figure plot(A1_squared,A2_poly,'o') title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A1 Squared vs. A2']) xlabel('A1 Squared') ylabel('A2') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; grid on % Plot log10(A1) vs log10(A2), the slope of which should be ~2 to represent % quadtratic relationship. The y-intercept is Beta. figure plot(log10(A1),log10(A2_poly),'o') title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' logA1 vs logA2']) xlabel('log(A1)') ylabel('log(A2)') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; %% Look at A2/A1^2 and A3/A1^2 vs various time intervals % time_interval = linspace(0,9,20); % 10 min (1 min intervals)

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% time_interval = linspace(1,20,20); % 20 min (1 min intervals) % time_interval = linspace(1,120,60); % 2 Hr (2 min intervals) qq = A2_poly./A1_squared; % figure % plot(time_interval,qq,'bo','MarkerSize',15) % % title(['(HC',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A2/A1^2 at 0.75V repeated over time']) % ylabel('A2/A1^2','FontSize', 20) % ylim([2.5e-4 4.5e-4]) % xlim([0 125]) % xlabel('Time (Min)','FontSize', 20) % ax = gca; % ax.YRuler.Exponent = 0; % ax.XAxis.FontSize = 18; % ax.YAxis.FontSize = 18; % A1_cubed = A1.^3; % qq3 = A3_poly./A1_cubed; % figure % plot(time_interval,qq3,'bo') % title(['(HC',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A3/A1^3 at 0.75V repeated over time']) % ylabel('A3/A1^3') % % ylim([5.4e-7 1.4e-4]) % xlabel('time (min)') % set(gca,'FontName','cmr12') % set(gca,'Fontsize',16); % ax = gca; % ax.YRuler.Exponent = 0; % ax.XRuler.Exponent = 0; % figure % plot(time_interval,A1,'bo') % title(['(HC',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A1 at 0.75V repeated over time']) % ylabel('A1') % xlabel('time (min)') % set(gca,'FontName','cmr12') % set(gca,'Fontsize',16); % ax = gca; % ax.YRuler.Exponent = 0; % ax.XRuler.Exponent = 0; % figure % plot(time_interval,A2_poly,'bo') % title(['(HC',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A2 at 0.75V repeated over time']) % ylabel('A2')

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% xlabel('time (min)') % set(gca,'FontName','cmr12') % set(gca,'Fontsize',16); % ax = gca; % ax.YRuler.Exponent = 0; % ax.XRuler.Exponent = 0; % figure % plot(time_interval,A3_poly,'bo') % title(['(HC',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A3 at 0.75V repeated over time']) % ylabel('A3') % xlabel('time (min)') % set(gca,'FontName','cmr12') % set(gca,'Fontsize',20); % ax = gca; % ax.YRuler.Exponent = 0; % ax.XRuler.Exponent = 0; %% Looking at different sections of log-log or A2 vs A1^2 a = 2; % Lower bound b = 7; % Upper bound % Next few lines find the linear slope of the plots on a specified range logA1 = log10(A1); logA2 = log10(A2_poly); log_coef = polyfit(logA1(a:b),logA2(a:b),1) ; log_slope = log_coef(1) ; A1A2_coef = polyfit(A1_squared(a:b),A2_poly(a:b),1) ; A1A2_slope = A1A2_coef(1) ; %% Section to plot A2/A1^B, where B is the slope of the log-log graph and ~= 2 many times. % In this case, I am using the slope from the interval a:b to make the % exponent. A1_ExpB = A1.^(log_slope); % Plot A1^B and A2 figure plot(A1_ExpB,A2_poly,'o') title(['(LANL ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' A1 B vs. A2']) xlabel('A1 B') % Figure out how to type ^ in latex ylabel('A2') set(gca,'FontName','cmr12') set(gca,'Fontsize',20);

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ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; %% Save Vectors to File in This Folder % save('Loc3_Trial7_Test1_A2','A2_poly') % save('Loc3_Trial7_Test1_A1^2','A1_squared') % save('Loc3_Trial7_Test1_A3', 'A3_poly') % save('Loc3_Trial7_A1^B', 'A1_ExpB') Bulk Wave Analysis and Plotting Different Ranges of Amplitude Data:

% Putting A2 vs A1^2 or anything else on the same plots % Initialization steps. clc; % Clear the command window. close all; % Close all figures clear; % Erase all existing variables. set(0,'defaulttextinterpreter','latex') % Plate Info plate = 40; loc = 2; a = 1; b = 7; % Load A1^2 load('Loc2_Trial1_A1^2'); A1_SQ_1 = A1_squared; load('Loc2_Trial2_A1^2'); A1_SQ_2 = A1_squared; load('Loc2_Trial3_A1^2'); A1_SQ_3 = A1_squared; load('Loc2_Trial4_A1^2'); A1_SQ_4 = A1_squared; load('Loc2_Trial5_A1^2'); A1_SQ_5 = A1_squared; load('Loc2_Trial6_A1^2'); A1_SQ_6 = A1_squared; load('Loc2_Trial7_A1^2'); A1_SQ_7 = A1_squared; % Load A2 load('Loc2_Trial1_A2'); A2_1 = A2_poly;

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load('Loc2_Trial2_A2'); A2_2 = A2_poly; load('Loc2_Trial3_A2'); A2_3 = A2_poly; load('Loc2_Trial4_A2'); A2_4 = A2_poly; load('Loc2_Trial5_A2'); A2_5 = A2_poly; load('Loc2_Trial6_A2'); A2_6 = A2_poly; load('Loc2_Trial7_A2'); A2_7 = A2_poly; % Plotting Section figure(1) plot(A1_SQ_7,A2_7,'o') title(['(HC',sprintf('%d ',plate) 'Loc',sprintf('%d)',loc) 'Unaltered A2 Vs. A1 Squared Data']) xlabel('A1^2') ylabel('A2') set(gca,'Fontsize',16); grid on grid minor hold on legend('Run 1','Run 2','Run 3','Run 4','Run 5','Run 6','Run 7','location','best') set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; %% figure(2) plot(A1_SQ_7(a:b),A2_7(a:b),'o') title(['(HC',sprintf('%d ',plate) 'Loc',sprintf('%d)',loc) ' A2 Vs. A1 Squared ', '(Points ',sprintf('%d',a) '-',sprintf('%d)',b)]) xlabel('A1 Squared','interpreter','latex') ylabel('A2','interpreter','latex') set(gca,'Fontsize',16); grid on grid minor hold on plot(A1_SQ_2(a:b),A2_2(a:b),'o') hold on plot(A1_SQ_3(a:b),A2_3(a:b),'kx') hold on plot(A1_SQ_4(a:b),A2_4(a:b),'o') hold on

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plot(A1_SQ_5(a:b),A2_5(a:b),'x') hold on plot(A1_SQ_6(a:b),A2_6(a:b),'+') hold on plot(A1_SQ_7(a:b),A2_7(a:b),'*') hold on plot(A1_SQ_8(a:b),A2_8(a:b),'*') hold on plot(A1_SQ_9(a:b),A2_9(a:b),'*') hold on plot(A1_SQ_10(a:b),A2_10(a:b),'*') hold on plot(A1_SQ_11(a:b),A2_11(a:b),'*') hold on plot(A1_SQ_12(a:b),A2_12(a:b),'*') hold on legend('Run 1','Run 2','Run 3','Run 4','Run 5','Run 6','Run 7','Run 8','Run 9','Run 10','Run 11', 'Run 12', 'location','best') set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; %% Finding Slope of Each Run for Beta and 'B' % Adjust to include the 5 best runs A1_All = [sqrt(A1_SQ_2);sqrt(A1_SQ_3);sqrt(A1_SQ_4);sqrt(A1_SQ_5);sqrt(A1_SQ_6)]; A1_SQ_All = [ A1_SQ_2; A1_SQ_3; A1_SQ_4; A1_SQ_5; A1_SQ_6]; A2_All = [ A2_2; A2_3; A2_4; A2_5; A2_6]; logA1 = log10(A1_All); logA2 = log10(A2_All); for n = 1:5 log_coef = polyfit(logA1(n,a:b),logA2(n,a:b),1) ; log_slope(n) = log_coef(1); A1A2_coef = polyfit(A1_SQ_All(n,a:b),A2_All(n,a:b),1) ; A1A2_slope(n) = A1A2_coef(1) ; end figure plot(logA1(5,a:b),logA2(5,a:b),'o') % title(['(HC ',sprintf('%d ',plate) 'Loc ',sprintf('%d)',location) ' logA1 vs logA2']) xlabel('log(A1)') ylabel('log(A2)') set(gca,'FontName','cmr12') set(gca,'Fontsize',20); ax = gca;

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ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0; % Mean of slopes shows average of the parameters log_slope_mean = mean(log_slope); A1A2_slope_mean = mean(A1A2_slope); % Std of slopes shows deviaton of parameters stand_dev_log = std(log_slope); stand_dev_A1A2 = std(A1A2_slope); %% Finding Slope of A2 vs A1^B where B~=2 (Using same ranges and kept runs as normal slope calculation above for j=1:5 A1_B_All(j,:) = A1_All(j,:).^log_slope(j); % Populate A1^B vector for each of the 5 kept runs on the given interval end n = 0; for n = 1:5 A1A2_B_coef = polyfit(A1_B_All(n,a:b),A2_All(n,a:b),1) ; A1A2_B_slope(n) = A1A2_B_coef(1) ; end % Mean of B slope A1A2_B_slope_mean = mean(A1A2_B_slope); % Standard Deviation of B Slope stand_dev_B = std(A1A2_B_slope); % Plotting A2 vs A1^B figure plot(A1_B_All(1,:),A2_All(1,:),'o') title(['(HC',sprintf('%d ',plate) 'Loc',sprintf('%d)',loc) 'Unaltered A2 Vs. A1 B Data']) xlabel('A1 Squared') ylabel('A2') set(gca,'Fontsize',16); grid on grid minor hold on plot(A1_B_All(2,:),A2_All(2,:),'o') hold on plot(A1_B_All(3,:),A2_All(3,:),'o') hold on plot(A1_B_All(4,:),A2_All(4,:),'o') hold on plot(A1_B_All(5,:),A2_All(5,:),'o') legend('Run 1','Run 2','Run 3','Run 4','Run 5', 'location','best') set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; ax.YRuler.Exponent = 0; ax.XRuler.Exponent = 0;

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Bulk Wave Relative Nonlinearity Parameter and Velocity and Plate Hardness

%% FOR ORIGINAL RANGE CHOICE HC = [ 22 24 28 32 36 40 52 54]; plate22 = [6.2983E-5 6.9493E-5 6.3279E-5]; mean22= mean(plate22); std22= std(plate22); plate24 = [6.3812E-5 6.6216E-5 6.1937E-5]; mean24 = mean(plate24); std24 = std(plate24); plate28 = [5.3198E-5 6.8827E-5 7.0739E-5]; mean28 = mean(plate28); std28 = std(plate28); plate32 = [6.0215E-5 6.4888E-5]; mean32 = mean(plate32); std32 = std(plate32); plate36 = [5.7050E-5 6.0315E-5 6.5358E-5]; mean36 = mean(plate36); std36 = std(plate36); plate40 = [5.3010E-5 6.1813E-5 6.2476E-5]; mean40 = mean(plate40); std40 = std(plate40); plate52 = [5.2931E-5 5.2584E-5 5.6145E-5]; mean52 = mean(plate52); std52 = std(plate52); plate54 = [5.2126E-5 4.1666E-5 4.8166E-5]; mean54 = mean(plate54); std54 = std(plate54); Beta = [mean22 mean24 mean28 mean32 mean36 mean40 mean52 mean54]; err = [std22 std24 std28 std32 std36 std40 std52 std54]; figure (1) scatter(HC(1),mean22,'ro','filled') hold on scatter(HC(1),plate22(1),'ko') hold on scatter(HC(1),plate22(2),'ko') hold on scatter(HC(1),plate22(3),'ko') hold on scatter(HC(2),mean24,'ro','filled')

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hold on scatter(HC(2),plate24(1),'ko') hold on scatter(HC(2),plate24(2),'ko') hold on scatter(HC(2),plate24(3),'ko') hold on scatter(HC(3),mean28,'ro','filled') hold on scatter(HC(3),plate28(1),'ko') hold on scatter(HC(3),plate28(2),'ko') hold on scatter(HC(3),plate28(3),'ko') hold on scatter(HC(4),mean32,'ro','filled') hold on scatter(HC(4),plate32(1),'ko') hold on scatter(HC(4),plate32(2),'ko') hold on scatter(HC(5),mean36,'ro','filled') hold on scatter(HC(5),plate36(1),'ko') hold on scatter(HC(5),plate36(2),'ko') hold on scatter(HC(5),plate36(3),'ko') hold on scatter(HC(6),mean40,'ro','filled') hold on scatter(HC(6),plate40(1),'ko') hold on scatter(HC(6),plate40(2),'ko') hold on scatter(HC(6),plate40(3),'ko') hold on scatter(HC(7),mean52,'ro','filled') hold on scatter(HC(7),plate52(1),'ko') hold on scatter(HC(7),plate52(2),'ko') hold on scatter(HC(7),plate52(3),'ko') hold on scatter(HC(8),mean54,'ro','filled') hold on scatter(HC(8),plate54(1),'ko') hold on scatter(HC(8),plate54(2),'ko') hold on scatter(HC(8),plate54(3),'ko') hold on title('Bulk Wave Plates and Nonlinearity Values') xlabel('Plate Hardness Number') set(gca,'XTick',0:4:56); ylabel('A2/A1^2')

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ylim([3E-5 7.1E-5]) legend('Average Value','Location Averages'); grid on grid minor set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; legend('Average Value','Location Averages'); grid minor set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca; %% Arrival Time Bulk Plots (amplitude 2) HC = [ 22 24 28 32 36 40 52 54]; Location1 = [0 -0.028 -0.032 0.016 -0.032 -0.148 -0.004 -0.088]; Location2 = [0 -0.180 0.004 -0.012 -0.028 -0.024 -0.048 -0.104]; Location3 = [0 -0.180 0.012 0.004 -0.080 -0.016 -0.028 -0.072]; figure plot(HC,Location1,'ro--') hold on plot(HC,Location2,'bo:') hold on plot(HC,Location3,'k*-') title('Bulk Wave Arrival Time Delay from HC22') xlabel('Plate Hardness Number') set(gca,'XTick',0:4:56); ylabel('Arrival Time Difference (Microseconds)') ylim([-0.25 0.1]) legend('Location 1','Location 2','Location 3','location','best'); grid minor set(gca,'FontName','cmr12') set(gca,'Fontsize',16); ax = gca;

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Col in Wi l l i ams Honors engineering leader in Penn State’s Integrated Undergraduate-Graduate Program and Schreyer Honors

College seeking R&D opportunities in the field of manufacturing and nondestructive evaluation (NDE).

E D U C A T I O N The Pennsylvania State University

• M.S. Engineering Science and Mechanics (Spring 2022) • B.S. with Honors, Engineering Science, Schreyer Honors College (Spring 2021)

o Minors: Engineering Leadership Development & Engineering Mechanics E X P E R I E N C E Systems Engineering Intern: Lockheed Martin RMS (Beginning May 2021)

• Incoming position as an intern in electronic systems and avionics in Owego, NY Research Assistant: Los Alamos National Laboratory (February 2021 – Present)

• Utilizing nonlinear ultrasonic techniques for the project titled “Ultrasonic Measurements of Additively Manufactured Materials”

Undergraduate Researcher: Penn State Ultrasonic Lab (Aug 2019 – Present)

• Researching characterization of metal alloys using nonlinear ultrasonic NDE • Creating signal analysis programs in MATLAB and designing custom 3D

printed components for experiments in SolidWorks • Utilize laboratory equipment with Lab-View and simulations in ABAQUS

Engineering Coach: Engineering Leadership Development (Jan 2020 – Aug 2020)

• Manager of four engineering teams during semester-long design challenges Engineering R&D Intern: Solar Innovations, Inc. (May 2019 – Aug 2019)

• Programmed with Visual Studio to automate AutoCAD shop drawings • Modeled and analyzed structures using civil engineering software (STAAD) • Worked in a test lab to pass products with national testing certifications

Part-Time Lending Desk: Penn State Libraries (Jan 2017 – Present)

S K I L L S S U M M A R Y

MATLAB – SolidWorks – Microsoft Office Suite – Visual Studio – STAAD – InDesign Virtual Bench – AutoCAD – Lab-view – ABAQUS – COMSOL – C++

A C T I V I T I E S

The Society of Engineering Science – Social Chair (Aug 2017 – Present)

Engineering Orientation Network – Head Mentor (Aug 2017 – Present) Phi Sigma Pi National Honor Fraternity – Alumni Chair (Nov 2018 – Present)

FOTO THON Organization (August 2018 – Present) A W A R D S A N D H O N O R S

NSF Graduate Research Fellowship Program – Honorable Mention (March 2021) Engineering Science & Mechanics Today – 2nd Prize Oral Presentation (March 2021)

Erickson Discovery Grant (May – Aug 2020) O’Halla Honors Scholarship in Engineering (August 2020)

Theodore Holden Thomas, Jr. Memorial Scholarship (August 2019, 2020) Vernon H. Neubert Dynamics Award in Engineering Science (May 2019)