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# Submerged Sonic Sledhammers Sharply Shape Synthesized Stellite Specimens, Showcasing Superior Strength

* Preamble :ignore_heading:
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** Packages :ignore_heading:


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** Info :ignore_heading:

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** Packages 2 :ignore_heading:

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** Abstract :ignore_heading:

# remove this line if you don't want pagination on preliminary pages
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\noindent
In accordance with the Academic Regulations the thesis must contain an abstract preferably not exceeding 200 words, bound in to precede the thesis. The abstract should appear on its own, on a single page.  The format should be the same as that of the main text. The abstract should provide a synopsis of the thesis and shall state clearly the nature and scope of the research undertaken and of the contribution made to the knowledge of the subject treated. There should be a brief statement of the method of investigation where appropriate, an outline of the major divisions or principal arguments of the work and a summary of any conclusions reached. The abstract must follow the Title Page.
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** Dedication & Acknowledgements :ignore_heading:

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If a dedication is included then it should be immediately after the Abstract page.\par
I don't what it is actually.

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\LARGE\textbf {Acknowledgements}
\end{center}
\vspace{5pt}

\noindent I wanna thanks all coffee and tea manufacturers and sellers that made the completion of this work possible.
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** COMMENT Declaration :ignore_heading:
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** TOC, Tables, Figures, Glossary :ignore_heading:
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*** Glossary :ignore_heading:

# #+LaTeX_HEADER: \newacronym{gcd}{GCD}{Greatest Common Divisor}
# #+LaTeX_HEADER: \newacronym{lcm}{LCM}{Least Common Multiple}

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#+LaTeX_HEADER: \newacronym{hv}{HV}{Hardness Vickers Scale}
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#+LaTeX_HEADER: \newacronym{fcc}{FCC}{Face Centred Cubic}
#+LaTeX_HEADER: \newacronym{hcp}{HCP}{Hexagonal Close Packed}
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*** COMMENT Our own publications
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** End of Preliminaries :ignore_heading:

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* Chapters

** Introduction

*** Cavitation

# Cavitation erosion (CE) is a common mechanism of material deterioration in hydrodynamic environments, and many components are subjected to serious CE, such as propellers, impellers, turbines, centrifugal-chambers and valves [1], [2], [3]. \cite{houCavitationErosionMechanisms2020}
# Cavitation is generally induced by rapid pressure variations or high-frequency vibrations in liquid environments, and is associated with the formation, growth and collapse of bubbles [4], [5], [6], [7]. \cite{houCavitationErosionMechanisms2020}
# When such bubbles implode in the proximity of a solid surface, powerful micro-jets and/or shock waves with high speeds and impact pressures are produced that can cause the fatigue, fracture and material depletion [8], [9], [10]. \cite{houCavitationErosionMechanisms2020}
# As a result, pits can form on component surfaces that are exposed to repeated loading and then can coalesce and form deep cavities, thereby leading to CE and the eventual failure of components [11], [12], [13]. \cite{houCavitationErosionMechanisms2020}


*** TODO Paragraph: Cavitation :ignore_heading:
*** Paragraph: Introduction to Stellite Alloys for Hostile Environments :ignore_heading:

#+BEGIN_COMMENT
- [X] What they are and where they came from.
  + [X] Identification as Cobalt-base superalloys
  + [X] Core beneficial properties
        high strength, corrosion resistance, high-temperature hardness
  + [X] Pioneering figure (Elwood Haynes) and seminal alloy
        Stellite 6 with nominal composition
#+END_COMMENT

Stellites are a family of cobalt-base superalloys used in aggresive service environments due to retention of strength, wear resistance, and oxidation resistance at high temperature \cite{ahmedStructurePropertyRelationships2014, shinEffectMolybdenumMicrostructure2003}. Starting with Elwood Haynes's development of alloys like Stellite 6 in the early 1900s \cite{hasanBasicsStellitesMachining2016}, stellites became critical to components used in medical implants & tools, machine tools, and nuclear components, and new variations on the original CoCrWC and CoCrMoC alloys see expanding use in sectors like oil & gas and chemical processing \cite{malayogluComparingPerformanceHIPed2003, ahmedStructurePropertyRelationships2014, raghuRecentDevelopmentsWear1997}.

*** Paragraph: Impact of Composition, Microstructure, and Processing on Corrosion and Cavitation Performance :ignore_heading:

#+BEGIN_COMMENT
- [X] What they are and where they came from.
  + [ ] Main alloying elements and ref to tab:stellite_composition
  + [ ] Describe the microstructure briefly
  + [ ] Carbon grades
#+END_COMMENT

The main alloying elements in Stellite alloys are cobalt (Co), chromium (25-33 wt% Cr), tungsten (0-18 wt% W), molybdenum (0-18 wt% Mo), carbon (0.1-3.3 wt% C), and trace elements iron (Fe), nickel (Ni), silicon (Si), phosphorus (P), sulphur (S), boron (B), lanthanum (La), & manganese (Mn); \tref{tab:stellite_composition} summarizes the nominal and measured composition of commonly used Stellite alloys \cite{ahmedMappingMechanicalProperties2023, alimardaniEffectLocalizedDynamic2010, ashworthMicrostructurePropertyRelationships1999, bunchCorrosionGallingResistant1989, davis2000nickel, desaiEffectCarbideSize1984, ferozhkhanMetallurgicalStudyStellite2017, pacquentinTemperatureInfluenceRepair2025, ratiaComparisonSlidingWear2019, zhangFrictionWearCharacterization2002}. Stellite alloys possess a composite-like microstructure, combining a cobalt-rich matrix strengthened by solid solutions of chromium, tungsten, & molybdenum, with embedded hard carbide phases with carbide formers Cr (of carbide type $\textrm{M}_{7}\textrm{C}_{3}$ & $\textrm{M}_{23}\textrm{C}_{6}$) and W/Mo (of carbide type $\textrm{MC}$ & $\textrm{M}_{6}\textrm{C}$), that impede wear and crack propagation \cite{ahmedSlidingWearBlended2021a, crookCobaltbaseAlloysResist1994}.


#+begin_src latex :tangle noweb :noweb yes :exports results
\afterpage{%
\begin{landscape}
\begin{ThreePartTable}

%\renewcommand\TPTminimum{\pageheight}
%% Arrange for "longtable" to take up full width of text block
%\setlength\LTleft{0pt}
%\setlength\LTright{0pt}
%\setlength\tabcolsep{0pt}

\begin{TableNotes}
    \small
    \item[a] Hot Isostatic Pressing
    \item[b] Inductively coupled plasma atomic emission spectroscopy
    \item[c] Shielded metal arc welding
    \item[d] Gas tungsten Arc Welding
    \item[e] Plasma transfered Arc Welding
\end{TableNotes}

\caption{Stellite Compositions}
\label{tab:stellite_composition}

\begin{longtable}{l|ll|ll|l|llllllll|lll}

\toprule
\textbf{Alloy} &
\textbf{Co} & \textbf{Cr} & \textbf{W} & \textbf{Mo} & \textbf{C} & \textbf{Fe} &
\textbf{Ni} & \textbf{Si} & \textbf{P} & \textbf{S}  & \textbf{B} & \textbf{La} &
\textbf{Mn} & \textbf{Ref} & \textbf{Process Type} & \textbf{Observation} \\

<<stellite1>>
<<stellite3>>
<<stellite4>>
<<stellite6>>
<<stellite7>>
<<stellite12>>
<<stellite19>>
<<stellite20>>
<<stellite21>>
<<stellite31>>
<<stellite190>>


\end{longtable}
\end{ThreePartTable}
\end{landscape}
}
#+end_src

**** Table: Show the table of stellite compositions :ignore:noexport:

***** Stellite 1 :ignore:noexport:
#+NAME: stellite1
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{4}{*}{Stellite 1}
& 47.7 & 30 & 13 & 0.5 & 2.5 & 3 & 1.5 & 1.3 & & & & & 0.5 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 48.6 & 33 & 12.5 & 0 & 2.5 & 1 & 1 & 1.3 & & & & & 0.1 & \cite{alimardaniEffectLocalizedDynamic2010} & & \\
& 46.84 & 31.7 & 12.7 & 0.29 & 2.47 & 2.3 & 2.38 & 1.06 & & & & & 0.26 & \cite{ahmedMappingMechanicalProperties2023} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
#+end_src

***** Stellite 3 :ignore:noexport:

#+NAME: stellite3
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 3}
& 50.5 & 33 & 14 & & 2.5 & & & & & & & & & \cite{bunchCorrosionGallingResistant1989} & & \\
& 49.24 & 29.57 & 12.07 & 0.67 & 2.52 & 2.32 & 1.07 & 1.79 & & & & & 0.75 & \cite{ratiaComparisonSlidingWear2019} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
#+end_src

***** Stellite 4 :ignore:noexport:

#+NAME: stellite4
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{5}{*}{Stellite 4}
& 45.43 & 30 & 14 & 1    & 0.57 & 3    & 3 & 2 & & & & & 1 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 51.5  & 30 & 14 &      & 1    & 1    & 2 & 0.5 & & & & & & \cite{zhangFrictionWearCharacterization2002} & & \\
& 51.9  & 33 & 14 &      & 1.1  &      &   &     & & & & & & \cite{bunchCorrosionGallingResistant1989} & & \\
& 49.41 & 31 & 14 & 0.12 & 0.67 & 2.16 & 1.82 & 1.04 & & & & & 0.26 & \cite{ahmedMappingMechanicalProperties2023} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
& 50.2  & 29.8 & 14.4 & 0 & 0.7 & 1.9 & 1.9 & 0.8 & & & & & 0.3 & \cite{ashworthMicrostructurePropertyRelationships1999} & HIPed\tnote{a} & \\
#+end_src

***** Stellite 6 :ignore:noexport:
#+NAME: stellite6
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{10}{*}{Stellite 6}
& 51.5 & 28.5 & 4.5 & 1.5 & 1 & 5 & 3 & 2 & & & 1 & & 2 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 63.81 & 27.08 & 5.01 & & 0.96 & 0.73 & 0.87 & 1.47 & & & & & 0.07 & \cite{ratiaComparisonSlidingWear2019} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
& 60.3 & 29 & 4.5 & & 1.2 & 2 & 2 & 1 & & & & & & \cite{zhangFrictionWearCharacterization2002} & & \\
& 61.7 & 27.5 & 4.5 & 0.5 & 1.15 & 1.5 & 1.5 & 1.15 & & & & & 0.5 & \cite{bunchCorrosionGallingResistant1989} & & \\
& 58.46 & 29.5 & 4.6 & 0.22 & 1.09 & 2.09 & 2.45 & 1.32 & & & & & 0.27 & \cite{ahmedMappingMechanicalProperties2023} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
& 58.04 & 30.59 & 4.72 & & 1.24 & 2.03 & 1.87 & 0.80 & 0.01 & 0.01 & & & & \cite{ferozhkhanMetallurgicalStudyStellite2017} & PTAW\tnote{e} & OES \\
& 55.95 & 27.85 & 3.29 & & 0.87 & 6.24 & 3.63 & 1.23 & 0.01 & 0.01 & & & 0.45 & \cite{ferozhkhanMetallurgicalStudyStellite2017} & GTAW\tnote{d} & OES \\
& 52.40 & 30.37 & 3.57 & & 0.96 & 6.46 & 3.93 & 1.70 & 0.01 & 0.01 & & & 0.3 & \cite{ferozhkhanMetallurgicalStudyStellite2017} & SMAW\tnote{c} & OES \\
& 60.3  & & 31.10 & 4.70 & 0.30 & 1.10 & 1.70 & 1.50 & 1.30 & & 0.00 & & 0.3 & \cite{pacquentinTemperatureInfluenceRepair2025} & LP-DED & ICP-AES \& GDMS \\
& 60.6 & 27.7 & 5 & 0 & 1.2 & 1.9 & 2 & 1.3 & & & & & 0.3 & \cite{ashworthMicrostructurePropertyRelationships1999} & HIPed\tnote{a} & \\
#+end_src

***** Stellite 7 :ignore:noexport:
#+NAME: stellite7
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 7
% & 64 & 25.9 & 4.9 & 0 & 0.5 & 1.5 & 1.1 & 1.1 & & & & & 1 & \cite{ashworthMicrostructurePropertyRelationships1999} & HIPed\tnote{a} & \\
#+end_src

***** Stellite 12 :ignore:noexport:

#+NAME: stellite12
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 12}
& 53.6 & 30 & 8.3 & & 1.4 & 3 & 1.5 & 0.7 & & & & & 1.5 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 55.22 & 29.65 & 8.15 & 0.2 & 1.49 & 2.07 & 2.04 & 0.91 & & & & & 0.27 & \cite{ahmedMappingMechanicalProperties2023} & HIPed\tnote{a} & ICP-OES\tnote{b} \\

#+end_src

***** Stellite 19 :ignore:noexport:

#+NAME: stellite19
#+begin_src latex :tangle noweb :exports none
\midrule
Stellite 19
  & 50.94 & 31.42 & 10.08 & 0.79 & 2.36 & 1.82 & 2 & 0.4 & & & 0.09 & & 0.1 & \cite{desaiEffectCarbideSize1984} & & \\

#+end_src

***** Stellite 20 :ignore:noexport:

#+NAME: stellite20
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 20}
& 41.05 & 33 & 17.5 & & 2.45 & 2.5 & 2.5 & & & & & & 1 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 43.19 & 31.85 & 16.3 & 0.27 & 2.35 & 2.5 & 2.28 & 1 & & & & & 0.26 & \cite{ahmedMappingMechanicalProperties2023} & HIPed\tnote{a} & ICP-OES\tnote{b} \\
#+end_src

***** Stellite 21 :ignore:noexport:
#+NAME: stellite21
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 21}
& 59.493 & 27 & & 5.5 & 0.25 & 3 & 2.75 & 1 & & & 0.007 & & 1 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
& 60.6 & 26.9 & 0 & 5.7 & 0.2 & 1.3 & 2.7 & 1.9 & & & & & 0.7 & \cite{ashworthMicrostructurePropertyRelationships1999} & HIPed\tnote{a} & \\
#+end_src

***** Stellite 22 :ignore:noexport:
#+NAME: stellite22
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 22
% & 54 & 27 & & 11 & 0.25 & 3 & 2.75 & 1 & & & & & 1 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\

#+end_src

***** Stellite 23 :ignore:noexport:
#+NAME: stellite23
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 23
% & 65.5 & 24 & 5 & & 0.4 & 1 & 2 & 0.6 & & & & & 0.3 & \cite{davis2000nickel} & \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 25 :ignore:noexport:
#+NAME: stellite25
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 25
% & 49.4 & 20 & 15 & & 0.1 & 3 & 10 & 1 & & & & & 1.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 27 :ignore:noexpoer:
#+NAME: stellite27
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 27
% & 35 & 25 & & 5.5 & 0.4 & 1 & 32 & 0.6 & & & & & 0.3 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 30 :ignore:noexport:
#+NAME: stellite30
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 30
% & 50.5 & 26 & & 6 & 0.45 & 1 & 15 & 0.6 & & & & & 0.6 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 31 :ignore:noexport:
#+NAME: stellite31
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 31}
& 57.5 & 22 & 7.5 & & 0.5 & 1.5 & 10 & 0.5 & & & & & 0.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
& 52.9 & 25.3 & 7.8 & 0 & 0.5 & 1.1 & 11.4 & 0.6 & & & & & 0.4 & \cite{ashworthMicrostructurePropertyRelationships1999} & HIPed\tnote{a} & \\
#+end_src

***** Stellite 80 :ignore:noexport:
#+NAME: stellite80
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 80
% & 44.6 & 33.5 & 19 & & 1.9 & & & & & & 1 & & & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 188 :ignore:noexport:

#+NAME: stellite188
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 188
% & 37.27 & 22 & 14 & & 0.1 & 3 & 22 & 0.35 & & & & 0.03 & 1.25 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 190 :ignore:noexport:

#+NAME: stellite190
#+begin_src latex :tangle noweb :exports none
\midrule
\multirow{2}{*}{Stellite 190}
& 46.7 & 27 & 14 & 1 & 3.3 & 3 & 3 & 1 & & & & & 1 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
& 48.72 & 27.25 & 14.4 & 0.2 & 3.21 & 2.1 & 2.81 & 1 & & & & & 0.31 & \cite{ahmedMappingMechanicalProperties2023}
& HIPed\tnote{a} & ICP-OES\tnote{b} \\
#+end_src

***** Stellite 300 :ignore:noexport:
#+NAME: stellite300
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 300
% & 44.5 & 22 & 32 & & 1.5 & & & & & & & & & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 694 :ignore:noexport:
#+NAME: stellite694
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 694
% & 45 & 28 & 19 & & 1 & 5 & & 1 & & & & & 1 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 703 :ignore:noexport:
#+NAME: stellite703
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 703
% & 44.6 & 32 & & 12 & 2.4 & 3 & 3 & 1.5 & & & & & 1.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 706 :ignore:noexport:
#+NAME: stellite706
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 706
% & 55.8 & 29 & & 5 & 1.2 & 3 & 3 & 1.5 & & & & & 1.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 712 :ignore:noexport:
#+NAME: stellite712
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 712
% & 51.5 & 29 & & 8.5 & 2 & 3 & 3 & 1.5 & & & & & 1.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src

***** Stellite 720 :ignore:noexport:
#+NAME: stellite720
#+begin_src latex :tangle noweb :exports none
% \midrule
% Stellite 720
% & 37.2 & 33 & & 18 & 2.5 & 3 & 3 & 1.5 & & & 0.3 & & 1.5 & \cite{davis2000nickel} &  \multicolumn{2}{c}{Nominal composition} \\
#+end_src


*** Paragraph: Co phases :ignore_heading:


Understanding the cobalt phase is crucial for studying structural changes in Co-based alloys widely used in industry. The fcc cobalt phase, especially its delayed transition to hcp at ambient and moderate temperatures \cite{DUBOS2020128812}, is of particular interest due to its impact on material properties in Co-based alloys \cite{Rajan19821161}. As the cobalt phase in stellite alloys is observed to consist of the fcc phase \cite{Rajan19821161}, the potential for strain-induced fcc to hcp transformation is of interest under the mechanical loading of cavitation erosion.


Cobalt exhibits a hexagonal close-packed (hcp) structure above 700 K \footnote{the theoretical transition temperature was determined to be 825 K by Lizarraga et al \cite{Lizarraga2017}} and shifts to a face-centered cubic (fcc) structure above this temperature.


At ambient conditions, the metastable FCC retained phase can be transformed into HCP phase by mechanical loading, although any HCP phase is completely transformed into a FCC phase between 673 K and 743 K \cite{DUBOS2020128812}.

The main allotropes of the Co solid solution in stellite are the hexagonal close-packed (hcp) $\epsilon−Co$ (ICDD# 01-071-4239) and the face-centered cubic (fcc) $\gamma−Co$ (ICDD 00-015-0806) \cite{wuMicrostructureEvolutionCrack2019} with the $\epsilon−Co$ phase being more thermodynamically stable below 700K \cite{lizarragaFirstPrinciplesTheory2017}. However the $\gamma$\textrightarrow$\epsilon$ transformation is difficult to achieve under normal condition, leading to cobalt alloys often retaining the metastable $\gamma$ phase \cite{davis2000nickel, frenkMicrostructuralEffectsSliding1994}. At ambient conditions, the metastable fcc retained phase in stellites can undergo a strain-induced martensitic-type transition involving partial dislocation movement \cite{HUANG2023106170}, such as during cavitation erosion as observed experimentally by Woodford \cite{woodfordCavitationerosionlnducedPhaseTransformations1972}.

The fcc to hcp transition is related to the very low stacking fault energy of the fcc structure (7 mJ/m2) of cobalt \cite{Tawancy1986337}
# The
# with a thermally induced

# The $\gamma$\textrightarrow$\epsilon$ transformation can be induced at temperatures between 673 K and 743 K


# cobalt alloys often possess a majority $\gamma−Co$ phase because the $\gamma$\textrightarrow$\epsilon$ transformation rarely occurs under normal cooling conditions,

This promotes the FCC→HCP martensitic transformation and improves the mechanical properties of the alloy. Phase composition seriously affects the properties of alloys. The FCC phase is soft and plastic, while the HCP phase is hard and not plastic. Therefore, when an alloy undergoes strain-induced FCC→HCP martensitic transformation, the martensite HCP phase plays a “secondary hardening” effect, which greatly improves the overall hardness of the alloy.




This fcc to hcp transition is linked to the very low stacking fault energy of the fcc structure (approximately 7 mJ/m²). The addition of other elements influences these characteristics: solid-solution strengthening increases the fcc cobalt matrix strength by distorting the atomic lattice and can decrease the low stacking fault energy by adjusting the electronic structure. Solute atoms like molybdenum (Mo) and tungsten (W), due to their large atomic sizes, discourage dislocation motion in stellites. Since dislocation cross-slip is a primary deformation mode in imperfect crystals at elevated temperatures and dislocation slip is a diffusion-enhanced process at high temperatures, this contributes to high-temperature stability. Furthermore, elements like nickel (Ni), iron (Fe), and carbon (C) stabilize the fcc structure of cobalt (a = 0.35 nm), while chromium (Cr) and tungsten (W) tend to stabilize the hcp structure (a = 0.25 nm and c = 0.41 nm).



The $\epsilon-\textrm{Co}$ (ICDD# 01-071-4239)and $\gamma-\textrm{Co}$ (ICDD 00-015-0806) phases are the main phases of the Co solid solution in stellite
Although the $\epsilon-\textrm{Co}$ phase is more stable at room temperature according to the phase diagram, cobalt alloys often posses a majority $\gamma-\textrm{Co}$ phase; the $\gamma$\textrightarrow$\epsilon$ transformation rarely occurs under normal cooling condiyions and the metastable $\gamma$ phase is generally retained



# Understanding the matrix phase
# Understanding the cobalt phase is crucial for studying structural changes in Co-based alloys widely used in industry.
Cobalt and Co-Cr alloys undergo thermally induced phase transformation from the high temperature face-centered cubic (fcc) $\gamma$ phase to low temperature hexagonal close-packed (hcp) $\epsilon$ phase at 700 K and strain induced fcc-hcp transition through maretensitic-type mechanism (partial movement of dislocations) \cite{HUANG2023106170}. At ambient conditions, the metastable FCC retained phase in stellites can be transformed into HCP phase by mechanical loading, although any HCP phase is completely transformed into a FCC phase between 673 K and 743 K \cite{DUBOS2020128812}; the metastable fcc cobalt phase in stellite alloys \cite{Rajan19821161} absorbs a large part of imparted energy under the mechanical loading of cavitation erosion. The fcc to hcp transition is related to the very low stacking fault energy of the fcc structure (7 mJ/m2) \cite{Tawancy1986337}.

# Let's talk about the addition of other elements
Solid-solution strengthening leads to increase of the fcc cobalt matrix strength (due to distortion of the atomic lattice with the addition of elements of different atomic radii), and decrease of low stacking fault energy \cite{Tawancy1986337} due to the adjusted electronic structure of the metallic lattice. Dislocation motion in stellites is discouraged by solute atoms of Mo and W, due to the large atomic sizes. Given that dislocation cross slip is the main deformation mode in imperfect crystals at elevated temperature, as dislocation slip is a diffusion process that is enhanced at high temperature, this leads to high temperature stability \cite{LIU2022294}. In addition, nickel (Ni), iron (Fe), and carbon (C) stabilize the fcc structure of cobalt (a = 0.35 nm), while chromium (Cr) and tungsten (W), stabilize the hcp structure (a = 0.25 nm and c = 0.41 nm) \cite{Vacchieri20171100, Tawancy1986337}.


# The Co solid solution in Stellites is a metastable fcc crystal with a very low stacking fault energy.


# One of the main applications of Stellite alloys can be wear and erosion resistance, which results from various carbides and Co solid solution [1], [7]. Cobalt imparts to its alloys an unstable fcc crystal structure with a very low stacking fault energy.

# The cubic FCC phase of cobalt (ICDD 00-015-0806)
# The hexagonal HCP phase of cobalt (ICDD 01-071-4239)

# The wear resistance of Stellite alloys benefits from the strain-induced face-centered cubic (FCC) to hexagonal closepacked (HCP) transformation of Co \cite{collierTribologicalPerformanceMolybdenum2020, ahmedFrictionWearCobaltBase2017}


# I *love* this paragraph
# https://doi.org/10.1016/j.surfcoat.2018.11.011
# The ε-Co and γ-Co phase are the main phases in Co alloys [39]. According to the phase diagram, the ε-Co phase is more stable at room temperature [40]. However, the γ-Co to ε-Co transformation rarely occurs under normal cooling conditions, and as a result, the γ-Co phase is generally retained at room temperature [39], as we found in the as-deposited Stellite 6 coatings. Nevertheless, this transformation can be triggered athermally (e.g. quenching from temperature of γ-Co) [25,41], isothermally (e.g. aging at temperatures between 650 °C and 950 °C) [35,42,43], or by strain [30,31]. In this study, the thermal fatigue process was achieved by quenching, with heating temperatures of up to 650 °C, and the crack tip introduced severe plastic deformation, all conditions that promote the γ-Co to ε-Co transformation.



\cite{woodfordCavitationerosionlnducedPhaseTransformations1972}

confirmed that $\gamma$\textrightarrow$\epsilon$ transformation occur on the surface of cobalt-base alloys during cavitation erosion, a relationship a direct relationship between . While the extent of this deformation-induced transformation was observed to increase with the severity of erosion and reach a steady state corresponding with the weight loss rate, subsequent experiments on Stellite 6B with varying initial hcp phase content and different aging treatments failed to establish a direct correlation between the transformation characteristics and erosion resistance.



Solid-solution strengthening is provided by elements not tied in secondary phases, leading to increase of the fcc cobalt matrix strength.

With the addition of elements with different atomic radiuses, the atomic lattice of the fcc cobalt matrix is distorted leading to increased strength. The already low stacking fault energy of the fcc cobalt structure (7 mJ/m2) \cite{Tawancy1986337} is further decreased, inhibiting dislocation cross slip.
Given that dislocation cross slip is the main deformation mode in imperfect crystals at elevated temperature, as dislocation slip is a diffusion process that is enhanced at high temperature, this leads to high temperature stability \cite{LIU2022294}.

The addition of nickel (Ni), iron (Fe), and carbon (C) stabilize the fcc structure of cobalt, while chromium (Cr) and tungsten (W), stabilize the hcp structure. Cr guarantees hot corrosion resistance and forms M23C6 carbides, while form MC carbides \cite{Vacchieri20171100}. The fcc cobalt phase has lattice constant a = 0.35 nm while the hcp cobalt phase has lattice constant a = 0.25 nm and c = 0.41 nm \cite{Tawancy1986337}.






*** Paragraph: Carbides and the grades of stellite alloy :ignore_heading:

The proportion and type of carbides depend on carbon content and the relative amounts of carbon with carbide formers (Cr, W, Mo), which greatly influence alloy performance and intended applications as carbides provide higher strength and but may reduce corrosion resistance due to localized corrosion at carbide boundaries. High carbon alloy (>1.2 wt%) have higher hardness due to greater carbide formation and are primarily used for wear resistance, low carbon alloys (<0.5 wt%) are used for enhanced corrosion resistance, and medium carbon alloys (0.5 wt% - 1.2 wt%) are used in applications requiring a combination of wear and corrosion resistance \cite{davis2000nickel}.

The carbide and grain size can be controlled by the rate of freezing, with larger carbide sizes indicating slower freezing rates \cite{yuInfluenceManufacturingProcess2008}.

# In addition to the solid solution toughness and carbide hardness, the stress-induced FCC \textrightarrow{} HCP phase transformation of the Co-based solid solution further increases wear resistance through work hardening.


# Cr has a dominant role in the formation of carbide type $\textrm{M}_23\textrm{C}_6$



*** Role of HIPping vs as Cast

# The HIPed alloy had a much finer microstructure Fig. 2 d with Cr-rich carbides dark phase uniformly distributed in the matrix. The typical carbide size was 1 – 3 m, which was much finer than the cast counterpart. There was no bright W-rich phase observed in the HIPed microstructure, which could be attributed to the fast solidification in the powder manufacturing process, restricting the segregation of W-rich zones. Subsequently during HIPing of the powder, tungsten remained evenly distributed throughout the alloy because its large atomic radius hinders diffusion. This evolution of the HIPed microstructure was therefore fundamentally different from the dendritic microstructure of the cast alloy, which was caused by the rejection of elemental species in the melt during the crystal growth of Co-rich dendrites. Hence above the liquidus line of this complex Co alloy, elemental species were free to arrange themselves depending on the thermal kinetics of the mold without any dependency on diffusion, and hence a truly three-dimensional network of carbides was formed.

# Contrary to this, in the case of HIPed microstructure, the primary dendrites formed on the alloy powder Fig. 2 a , and the carbides in the powder particles Fig. 3 a promoted carbide growth due to the diffusion of carbon and other elemental species within and across the individual powder particle boundaries. As this diffusion process is time, temperature, and pressure dependent during HIPing, and the HIPing temperature 1200° C in this investigation was lower than the melting point of the powder, carbide growth was sluggish when compared with casting. Hence the size of individual carbide particles was much smaller than the cast counterpart. Although not reported in Sec. 3, authors also found that re-HIPing the HIPed alloy under similar conditions as were reported earlier in Sec. 2.1 did not substantially increase the average carbide size, indicating that carbide growth was more dependent on temperature than time during the HIPing process.


*** Paragraph: Stellite 1 :ignore_heading:

# Stellite 1 is a high-carbon and high-tungsten CoCrWMoCFeNiSiMn alloy, making it suitable for tribological applications such as valve seating, wear pads in gas turbines, bearing sleeves, slurry pumps, ball bearings and expeller screws. Stellite 1 alloy is labeled as CoCrW alloy in this paper. Stellite 21 is a low-carbon, high-molybdenum alloy used in applications such as forging and hot-stamping dies and valve trims in the chemical industry (Ref 1, 2).

*** Paragraph: Tungsten and Molybdenum carbides

# Tungsten is primarily  utilized in Stellite alloys to provide solid solution strength and carbide formation but can be  replaced by Mo which also partitions to the carbides.

# Tungsten (W) and molybdenum (Mo) have a similar function in providing additional strength to Co solid solution matrix of Stellite alloys due to large atomic size, that is, they impede dislocation flow when present as solute atoms


#  Thermodynamic Modeling of the C-Co-Mo and C-Mo-Ni Ternary Systems
# zhangThermodynamicModelingCCoMo2016

# Studies [18,19,20] show that the solid solubility of W in the Co phase can be increased through an appropriate heat treatment process which leads to the strengthening of the Co phase. The dissolution of W and C in the Co phase increases the stacking fault energy and martensitic transformation temperature, inhibits the martensitic transformation, and stabilizes the cobalt phase with a face-centered cubic crystal structure (fcc Co phase) which increases the toughness of the cemented carbide [14,18].

Tungsten (W) and molybdenum (Mo) are refractory elements that provide solid solution strengthening to the matrix, by virtue of their large atomic size that impedes dislocation flow when present as solute atoms \cite{boeckRelationshipsProcessingMicrostructure1985}, and also form $\textrm{M}_6\textrm{C}$ and $\textrm{M}_12\textrm{C}$ carbides along with $\textrm{MC}$ carbides and $\textrm{Co}_3\textrm{M}$ & $\textrm{Co}_7\textrm{M}_6$ intermetallics during solidification.

In carbon-rich regions, the $\textrm{MC}$ phase (of type $\textrm{WC}$ and $\textrm{MoC}$) is observed \cite{zhangThermodynamicInvestigationPhase2019}, which ca


In carbon-poor regions, ternary $\textrm{M}_6\textrm{C}$ and $\textrm{M}_{12}\textrm{C}$ carbides have been identified, where the $\textrm{M}_6\textrm{C}$ carbide (of type $\textrm{Co}_3\textrm{Mo}_3\textrm{C}$) is stable in the temperature ranges of 900C to 1300C and can vary in composition from Mo_40_Co_46C_14 to Mo56Co30C14, while the $Mo_12C$ carbide of type ($Co6Mo6C$ carbide decomposes into $Mo_6C$ and $\mu-Mo$ phases above 1100C \cite{zhangThermodynamicModelingCCoMo2016}.

When present in large quantities, W and Mo also participate in formation of W-rich or Mo-rich carbides during alloy solidification \cite{davis2000nickel, raghuRecentDevelopmentsWear1997},

leading to generation of Topologically Close-Packed (TCP) phases, such as the \mu phase (of type Co_7W6 and Co7Mo6) and \sigma phase (pf type Co3W and Co3Mo) \cite{zhangThermodynamicInvestigationPhase2019}, which are intermetallic brittle phases that add strength to the material \cite{yuTriboMechanicalEvaluationsCobaltBased2007, ishidaIntermetallicCompoundsCobase2008} while also promoting crack initiation and propagation \cite{zhaoFirstprinciplesStudyPreferential2023}. Previous work on the Stellite 1 sample by Ahmed et al \cite{ahmedMappingMechanicalProperties2023} indicate that Co6W6C is identified as the main W-rich carbide in Stellite 1, although Co3W3C was also identified inaddition to Co3W and Co7W intermetallics.




# The carbide formed by W and Mo tends to be the M6C carbide
# \cite{klarstromMetallographyMicrostructuresCobalt2004, antonyWearResistantCobaltBaseAlloys1983}

# #+CAPTION: Experimental and calculated phase diagram for the Co-Mo system. Reprinted with permission \cite{davydovThermodynamicAssessmentCoMo1999}
# [[file:Figures/Co_Mo_phasediagram_davydovThermodynamicAssessmentCoMo1999.png]]

# #+CAPTION: Experimental and calculated phase diagram for the Co-Mo-C system. Reprinted with permission \cite{zhangThermodynamicModelingCCoMo2016}
# [[file:Figures/Co_Mo_C_phasediagram_zhangThermodynamicModelingCCoMo2016.png]]



There are two main phases in the tungsten-carbon system: the hexagonal monocarbide $\textrm{WC}$ (ICDD Card# 03-065-4539, COD:2102265), denoted as $\delta-\textrm{WC}$, and multiple variations of hexagonal-close-packed subcarbide $\textrm{W}_2\textrm{C}$ (ICDD:00-002-1134, COD:1539792) \cite{kurlovPhaseEquilibriaWC2006, tulhoffCarbides2000}

WC carbides precipitate as discrete particles distributed heterogeneously throughout the alloy intragranularly
# There are two hexagonal carbides in the tungsten – carbon system (Fig. 2): the monocarbide, WC, and the subcarbide [12070-13-2], W 2 C. The hexagonal WC, also called a-WC, decomposes at its incongruent melting point of 2776 C. Its range of homogeneity is extremely narrow: from 49.5 to 50.5 mol % C

# Several stable tungsten carbide phases in the WC phase diagram. They consist of the S phase with simple hexagonal structure (prototype: WC), γ phase with face-centred-cubic structure (prototype: NaCl), and β phase with hexagonal-close-packed structure (prototype: PbO2). Both β and β phases are nonstoichiometric with a solubility of C wt.% in approximate ranges of 2.2∼3.0 and 3.6∼3.9, respectively. The S phase is a stoichiometric compound. For the sake of convenience, we represent the β, γ and δ phases by their corresponding compounds W2C, WC1-x and WC, respectively. \cite{gubischTribologicalCharacteristicsWC1x2005}

The precipitation of the tungsten-rich phase $M_6C$ is closely related to the decomposition of the MC carbide, and the $M_6C$ only occurs in the vicinity of the MC \cite{jiangSecondaryM6CPrecipitation1999}, as $M_6C$ carbides form only when the tungsten and.or molybdenum content exceeds 4-6 a/o.

*** Paragraph: Chromium carbide :ignore_heading:


# The primary M7C3 carbide, found mostly in hypereutectic alloys, has a high melting point among the carbides and therefore, precipitates first during solidification if the carbon content is high enough to favor its formation.
# \cite{klarstromMetallographyMicrostructuresCobalt2004, antonyWearResistantCobaltBaseAlloys1983}


Chromium carbides have high hardness and wear resistance, as well as excellent resistance to chemical corrosion, making them often used in surface coatings \cite{liElectronicMechanicalProperties2011}

In the Cr-C binary phase diagram, there are three phases : cubic Cr23C6 (space group , melting point 1848 K), orthorhombic Cr3C2 (space group Pnma, melting point 2083 K) and Cr7C3 (space group Pnma, melting point 2038 K) \cite{medvedevaStabilityBinaryTernary2015} \cite{liElectronicMechanicalProperties2011}

The M_23C6 carbides are formed during heat treatment of carbides with a lower M/C ratio or from solid solution close to boundaries \cite{medvedevaStabilityBinaryTernary2015}. Fine M23C6 carbides act as obstacles to gliding of mobile dislocations, which result in long-term creep strength \cite{godecCoarseningBehaviourM23C62016}.

Although M23C6 can precipitate as primary carbide during solidification, it is most commonly found in secondary carbides along grain boundaries.

M7C3 is a metastable pseudo-eutectic carbide that typically forms at lower carbon-chromium ratios and effectively transforms into secondary M23C6 upon heat treatment.

In addition to being a carbide former, chromium provides solid solution strengthing and corrosion/oxidation resistance to the cobalt-based matrix.


The Cr7C3 carbide is unstable at high temperatures and transforms to M_{23}C_{6} upon heat treatment. Under further temperature and time, Cr_{23}C_{6} partially transforms to Cr_6C \cite{mohammadnezhadInsightMicrostructureCharacterization}.

$$ 2Cr_{7}C_{3} + 9Cr \rightarrow Cr_{23}C_{6} $$
$$ Cr_{23}C_{6} + 13Cr \rightarrow 6Cr_{6}C $$

$$ 23Cr_{7}C_{3} \rightarrow 7Cr_{23}C_6 + 27C $$
$$ 6C + 23Cr \rightarrow Cr23C6 $$


# \ce{M_3C -> M_2C -> M_{23}C_6}
# \ce{M_3C -> M_7C_6 -> M_{23}C_6}
# \ce{M_3C -> M_6C}
# \ce{M_3C -> M_23C_6}


*** Paragraph 2: Fundamental Mechanisms of Corrosion and Cavitation Resistance :ignore_heading:

#+BEGIN_COMMENT
- [ ] Fundamental Strengthening Mechanisms
  + [ ] Primary mechanism: Hard carbide precipitation (e.g., M$_{7}$C$_{3}$, M$_{23}$C$_{6}$), with dependence on carbon content and processing.
  + [ ] Secondary mechanism: Solid solution strengthening by specific elements (W, Mo, Cr), also linked to carbon content.
  + [ ] Additional mechanism: Stress-induced phase transformation (fcc to hcp) contributing to wear resistance via work hardening.
- [ ] How they are made and modified.
- [ ] Current research directions and future outlook.
#+END_COMMENT

The remarkable ability of Stellite alloys to withstand these specific challenges stems from key metallurgical features. Their corrosion resistance is primarily attributed to a high chromium content, typically 20-30 wt.%, which promotes the formation of a highly stable, tenacious, and self-healing chromium-rich passive oxide film on the material's surface; this film acts as a barrier isolating the underlying alloy from the corrosive environment. Alloying elements such as molybdenum and tungsten can further enhance this passivity, particularly improving resistance to localized corrosion phenomena like pitting and crevice corrosion in aggressive media. Concurrently, their outstanding cavitation resistance is largely derived from the unique behavior of the cobalt-rich matrix, which can undergo a stress-induced crystallographic transformation from a face-centered cubic (fcc) to a hexagonal close-packed (hcp) structure. This transformation, often facilitated by mechanical twinning, effectively absorbs the intense, localized impact energy from collapsing cavitation bubbles and leads to significant work hardening, thereby impeding material detachment and erosion.

Antony suggests that the cavitation-erosion resistance of Stellites derives from the matrix phase and is enhanced by the strain-induced fcc \textrightarrow hcp allotropic transformation \cite{antonyWearResistantCobaltBaseAlloys1983}.

*** Corrosion resistance of Stellites



The cavitation erosion of stellites has been investigated in experimental studies \cite{Wang2023, Szala2022741, Mitelea2022967, Liu2022, Sun2021, Szala2021, Zhang2021, Mutascu2019776, Kovalenko2019175, E201890, Ciubotariu2016154, Singh201487, Hattor2014257, Depczynski20131045, Singh2012498, Romo201216, Hattori20091954, Ding201797, Guo2016123, Ciubotariu201698}, along with investigations into cobalt-based alloys \cite{Lavigne2022, Hou2020, Liu2019, Zhang20191060, E2019246, Romero2019581, Romero2019518, Lei20119, Qin2011209, Ding200866, Feng2006558}.



Stellites achieve oxidation resistance through the formation of a passivating external Cr2O3 scale, due to the high proportion of Cr in their chemical composition \cite{pettitOxidationHotCorrosion1984}.

as seen by Zhang et al in Green Death solution \cite{zhangPittingCorrosionCharacterization2014}.

However, Cr-based carbides may be preferentialy oxidized below the external Cr2O3 scale, particularly at the boundary of carbides which are depleted of Cr \cite{zhangPittingCorrosionCharacterization2014}, where preferential attack of carbides proceed until they have been consumed \cite{pettitOxidationHotCorrosion1984}.




Lemaire et al \cite{lemaireEvidenceTribocorrosionWear2001} investigated the behavior of Stellite 6 in pressurized high temperature water and proposed an oxidative wear mechanism where wear proceeds by repeated detachment of the surface oxide spontaneously forming on the stellite surface.

Di Martino et al \cite{dimartinoCorrosionMetalsAlloys2004} also found that the protective chromium-rich film are abraded easily, leading to further corrosion.


 In such lower-temperature regimes, the passive films formed are typically very thin (in the nanometer range, rather than the micrometer scale observed at high temperatures)


Molybdenum and tungsten have favorable effects on the selective oxidation of chromium until chromium has been depleted, at which point molybdenum and tungsten result in increased oxidation due to development of less protective phases \cite{pettitOxidationHotCorrosion1984}.


** Analytical Investigations
# \chapter{Analytical Investigations}
# \section{Introduction}
# \section{Finite Element Model (FEM)}
# \section{Model description}
# \section{Model Validation}
# \section{Result Analysis of Typical Load Case}

*** Strain hardening



# The present paper is a contribution to this subject. It presents a model of prediction of the erosion damage applicable to ductile materials only. Other limitations of the model will be pointed out along the presentation. The originality of this work lies in the fact that the proposed model is fully predictive and involves no parameters to be adjusted on the basis of experimental data. It is based upon the original work of Karimi and Leo @5#.

# Interesting tidbit? Maybe not actually something to include
# comparable to that obtained in conventional peening \cite{swietlickiEffectsShotPeening2022},



Cavitation bubble collapse induce a work hardening of the ductile material's surface, characterized by the thickness of the hardened layers and the shape of the strain profile below the surface.
# \cite{berchicheCavitationErosionModel2002}

This analysis was first proposed by Karimi & Leo in 1987 \cite{karimiPhenomenologicalModelCavitation1987} and adapted by Berniche et al in 2002 \cite{berchicheCavitationErosionModela}, and Franc \cite{francIncubationTimeCavitation2009} in 2009.

The strain profile within the material can usually be modeled by the following power law:

\begin{equation}
\epsilon\left(x\right) = \epsilon_s {\left(  1 - \frac{x}{L}  \right)}^{\theta}
\end{equation}

where $\epsilon\left(x\right)$ is the strain at depth $x$ from the eroded surface, $\epsilon_s$ is the failure rupture strain on the eroded surface, $L$ is the thickness of the hardened layer, $\theta$ is the shape factor of the power law. The parameters $L$ and $\theta$ are determined from the microhardness measurements on cross sections of the cavitation affected region.


# Tensile tests done by Rehan, in order to give us K and n
# \cite{ahmedMappingMechanicalProperties2023}
# The tensile tests were carried out on an Instron tensometer following the BS EN 10,002 standard [29]. The dumbbell-shaped specimens, with 25 mm gauge length and 4 mm diameter, were used in this investigation. The tests were conducted at 0.05 mm/min, equalling a strain rate of 0.000033 s−1. Three tests were conducted on each alloy. The fracture sections were examined via SEM.



The strain hardening effect after erosion tests was calculated from the following formula:

\begin{equation}
\Delta{}HV = \dfrac{{HV}_{x} - {HV}_{0}}{{HV}_{0}} \cdot 100 \%
\end{equation}

where ${HV}_{x}$ is the hardness at a distance below the cavitation crater, while HV_0 is the initial hardness.

# After each cycle, the thickness of the hardened layer $L$ and the surface strain $\epsilon_s$ will increase continuously until damage is initiated at the surface ($\epsilon_s$ reaches the failure rupture strain $\epsilon_R$), at which point the strain profile is in steady-state.

# \begin{equation}
# \epsilon_R = \epsilon_{mean} {\left(  1 - \frac{\Delta L }{L+ \Delta L}  \right)}^{\theta}
# \end{equation}







# Let's talk more about Woodford
In Woodford's investigations on the $\gamma$\textrrightarrow$\epsilon$ transformation on the surface of cobalt-base alloys during cavitation erosion, the transformed layer was found to extend to a depth of 25 to 50 \um \footnote{Converted from the original 1e-3 to 2e-3 \cite{woodfordCavitationerosionlnducedPhaseTransformations1972}}.

**** Data for strain hardening :ignore:

#+CAPTION: Microhardness HV_0.01 of Al0.1CoCrFeNi HEA \cite{nairExceptionallyHighCavitation2018a}
|                  x |                  y |
| 15.025380710659899 |  339.3103448275862 |
| 29.949238578680202 |  277.2413793103448 |
|  44.87309644670051 |  212.0689655172414 |
| 59.974619289340104 | 230.34482758620692 |
|  74.89847715736042 | 203.10344827586206 |
|  89.82233502538071 | 203.44827586206895 |
| 104.74619289340102 | 195.86206896551724 |
|  120.0253807106599 | 195.86206896551724 |
| 135.12690355329948 | 187.58620689655174 |
|  150.2284263959391 | 155.86206896551724 |
|  164.9746192893401 | 153.10344827586206 |

#+CAPTION: Microhardness HV_0.01 of 316LSS \cite{nairExceptionallyHighCavitation2018a}
|                  x |                  y |
| 14.847715736040609 | 288.62068965517244 |
|  29.77157360406091 | 251.72413793103448 |
|   45.0507614213198 |                240 |
| 59.974619289340104 | 219.31034482758622 |
|  74.89847715736042 | 227.24137931034483 |
|  89.82233502538071 |  228.9655172413793 |
|  104.9238578680203 | 221.72413793103448 |
|  120.0253807106599 | 218.27586206896552 |
| 135.48223350253807 | 224.82758620689657 |
| 149.87309644670052 | 225.86206896551727 |
|  165.1522842639594 | 224.48275862068965 |

*** Correlative empirical methods

Empirical methods are common for addressing complex cavitation erosion, involving lab tests to correlate cavitation erosion resistance with mechanical properties.

**** Karimi and Leo

The Karimi and Leo phenomenological model describes cavitation erosion rate as a function of

Karimi and Leo


**** Noskievic

Noskievic formulated a mathematical relaxation model for the dynamics of the cavitation erosion using a differential equation applied to forced oscillations with damping:

\begin{equation}
\frac{\mathrm{d}^2 v }{\mathrm{d}t^2} + 2 \alpha \frac{\mathrm{d} v }{\mathrm{d}t} + \beta^2 v = I
\end{equation}

where $I$ is erosion intensity, which can vary linearly with time, $v = \frac{\mathrm{d} v }{\mathrm{d}t}$ is erosion rate, $\alpha$ is strain hardening or internal friction of material during plastic deformation, and $\beta$ is coefficient inversely proportional to material strength. The general solution of equation can be written as:

#+BEGIN_EXPORT latex
\begin{equation}
v = a f_0 \left( \delta, \tau \right) + b f_1 \left( \delta, \tau \right)
\end{equation}

\begin{equation}
f_0\left(\ \delta,\tau \right) = \left\{ \begin{array}{@{}lr@{}}
1 - \mathrm{exp}{ \left(  - \delta \tau \right) } \left[ \dfrac{\delta}{\omega} \mathrm{sin} \left(\omega \tau\right) + \mathrm{cos}{\left( \omega \tau \right)} \right]
& \text{if } -1 < \delta < 1; \delta \neq 0 \\
1 - \dfrac{1}{ {{\delta_0}^2} - 1} \left[ \delta_0^2 \mathrm{exp}{\left( -\dfrac{\tau}{\delta_0}}\right) - \mathrm{exp}{\left(- \delta_0 \tau\right)} \right]
&  \text{if } \delta > 1 \\
1 - \mathrm{cos}{\left( \tau \right)}
& \text{if } \delta = 0 \\
1 - \left(1 + \tau \right) \mathrm{exp} \left( - \tau \right)
& \text{if } \delta = 1
\end{array} \right\} \\
\end{equation}
\begin{equation}
f_1\left(\ \delta,\tau \right) = \left\{ \begin{array}{@{}lr@{}}
1 - \dfrac{2\delta}{\tau} \left[ 1 - {\mathrm{exp}\left(-\delta \tau\right)} {\left[ {\mathrm{cos} \omega \tau} + {\epsilon \mathrm{sin} \omega \tau} \right]} \right]
& \text{if } -1 < \delta < 1; \delta \neq 0 \\
1 - \dfrac{1}{\tau} \left( 2 \delta - \dfrac{1}{\delta_0 \left( \delta_0^2 - 1 \right)} \left[\mathrm{exp}{\left( -\delta_0 \tau \right) - \delta^4 \mathrm{exp}{\left( \dfrac{-\tau}{\delta_0} \right)} \right] \right)
&  \text{if }  \delta > 1 \\
1 - \dfrac{ \mathrm{sin}{\left( \tau \right)} }{\tau}
& \text{if }  \delta = 0 \\
1 - \dfrac{2 \left[ 1 - \mathrm{exp}\left(-\tau\right) \right]}{\tau} + \mathrm{exp} \left( - \tau \right)
& \text{if }  \delta = 1
\end{array} \right\}
\end{equation}
\begin{equation}
\delta = \dfrac{\alpha}{\beta},\quad
\tau = \beta t,\quad
\epsilon = \dfrac{\delta^2 - 0.5}{\delta \sqrt{ 1 - \delta^2  } },\quad
\omega = \sqrt{1 - \delta^2},\quad
\delta_0 = \delta + \sqrt{ \delta^2 - 1 }
\end{equation}
#+END_EXPORT

***** Noskievic python function :noexport:ignore:

#+NAME: noskievic
#+BEGIN_SRC python :exec no :exports none
def noskievic(t, alpha, beta, a, b):
    """
    Calculates the erosion rate (v) based on Noskievic's relaxation model.

    This model describes the dynamics of cavitation erosion using a differential
    equation applied to forced oscillations with damping. The general solution
    for the erosion rate (v) is given by:
    v = a * f0(delta, tau) + b * f1(delta, tau)

    where delta = alpha / beta and tau = beta * t. The functions f0 and f1
    have different forms depending on the value of delta.

    Args:
        t (float): Time.
        alpha (float): Material parameter representing strain hardening or
                       internal friction of the material during plastic
                       deformation (α).
        beta (float): Material parameter, a coefficient inversely proportional
                      to material strength (β).
        a (float): Coefficient for the f0 component in the general solution.
        b (float): Coefficient for the f1 component in the general solution.

    Returns:
        float: The calculated erosion rate (v) at the given time t.

    Notes:
        The intermediate parameters used in the model are:
        - delta (δ) = alpha / beta
        - tau (τ) = beta * t
        - omega (ω) = sqrt(1 - delta^2)  (for -1 < delta < 1, delta != 0)
        - delta_0 (δ₀) = delta + sqrt(delta^2 - 1) (for delta > 1)
        - epsilon (ε) = (delta^2 - 0.5) / (delta * sqrt(1 - delta^2)) (for -1 < delta < 1, delta != 0)

        The functions f0(delta, tau) and f1(delta, tau) are piecewise
        functions dependent on the value of delta:
        - Case 1: -1 < delta < 1 and delta != 0
        - Case 2: delta > 1
        - Case 3: delta = 0
        - Case 4: delta = 1
        This implementation should handle these cases internally to compute
        f0 and f1 correctly.
    """
    import numpy as np
    d = alpha/beta
    tau = beta*t
    if np.isclose(d,0.0):
        print("d=0")
        f0 = 1 - np.cos(tau)
        f1 = 1 - ((np.sin(tau))/(tau))
    elif np.isclose(d,1.0):
        print("d=1")
        f0 = 1 - (1 + tau)*np.exp(-tau)
        f1 = 1 - 2*((1 - np.exp(-tau))/(tau)) + np.exp(-tau)
    elif -1 < d and d < 1:
        e = (d**2 - 0.5)/(d*np.sqrt(1 - d**2))
        w = np.sqrt(1 - d**2)
        f0 = 1 - np.exp(-d*tau)*((d/w)*np.sin(w*tau) + np.cos(w*tau))
        f1 = 1 - ((2*d)/(tau))*(1 - np.exp(-d*tau)*( np.cos(w*tau) + e*np.sin(w*tau) ))
    elif d > 1:
        d_0 = d + np.sqrt(d**2 - 1)
        f0 = 1 - (1/(d_0**2 - 1))*(d_0**2 * np.exp(-tau/d_0) - np.exp(-d_0 * tau))
        f1 = 1 - (1/tau)*(2*d - (1/(d_0*(d_0**2-1)))*( np.exp(-d_0*tau) - d**4*np.exp(-tau/d_0) )  )
    else:
        raise ValueError("d is not within bounds")

    return a*f0 + b*f1
#+END_SRC


**** Hoff and Langbein equation

Hoff and Langbein proposed a simple exponential function for the rate of erosion, representing the normalized erosion rate requiring only the
A simple exponential function for the rate of erosion was proposed by Hoff and Langbein,

$$ \frac{ \dot{e} }{ \dot{e_{max}} } = 1 - e^{\frac{-t_i}{t}} $$

$\dot{e}$ - erosion rate at any time t
$\dot{e_{max}}$ - Maximum of peak erosion rate
$t_i$ - incubation period (intercept on time axis extended from linear potion of erosion-time curve)
$t$ - exposure time

**** L Sitnik model

$$ V = V_o {\left[ ln\left( \frac{t}{t_o} + 1  \right) \right]} ^ {\beta} $$


$$ \dot{V} = \frac{\beta V_o}{t + t_o} {\left[ ln \left( \frac{t}{t_o} + 1 \right) \right]}^{\beta - 1}$$

V_o > 0
t_o > 0
\beta >= 1




** Experimental Investigations


*** Materials and Microstructure

# Alternatively say industrially optimized parameters rather than saying the temp amd pressure
The HIPed Stellite 1 alloys were manufactured by canning the gas-atomized powders, manufactured by Deloro Stellite (UK), at a temperature and pressure of 1200C and 100 MPa, for 4 hours in a HIPing vessel where the chemical composition and sieve analysis have been reported in previous work by Ahmed et al \cite{ahmedInfluenceAlloyComposition2025}.

The sieve analysis of these powders indicate that the majority of powder particles were in the size range of 45 to 180 um.

The cast alloy samples were produced via sand casting process.

|                  | Co   |    Cr |     W |   Mo |    C |   Fe |   Ni |   Si |   Mn |
| HIPed Stellite 1 | Bal. | 31.70 | 12.70 | 0.29 | 2.47 | 2.30 | 2.38 | 1.06 | 0.26 |



|                  | +250 | +180 |  +125 |   +45 |  -45 |
| HIPed Stellite 1 | 0.10 | 2.40 | 47.90 | 49.50 | 0.10 |


*** Materials and Microstructure

The HIPed alloy was produced via canning the gas-atomized powders at 1200C and 100 MPa pressure for 4h, while the cast alloys were produced via sand casting.
% Sieve analysis and description of powders

% Refer to Table of chemical compositions of both cast and HIPed alloys.

The microstructure of the alloys were observed via SEM in BSE mode, and the chemical compositions of the identified phases developed in the alloys were determined via EDS as well as with XRD under Cu $K_{\alpha}$ radiation.

Image analysis was also conducted to ascertain the volume fractions of individual phases.

# \chapter{Experimental Investigations}

# \section{Introduction}


# \section{X-ray diffraction technique of residual stress measurement}

# \section{Surface Roughness Measurements}
# \section{Microhardness measurements}


The Vickers microhardness was measured using a Wilson hardness tester under loads of BLAH. Thirty measurements under each load were conducted on each sample.

*** Paragraph XRD: Experimental determination of $\epsilon-Co$ :ignore_heading:



The microstructure phase identification was investigated out using X-ray diffraction technique with Cu-K$\alpha$ radiation ($\lambda = 1.5406 \AA{}$).
The volume fraction of \epsilon-Co can be determined using the intensity of the $(200)_{\gamma}$ and $(10\bar{1}1)_{hcp}$ peaks, using the following equation proposed by Sage and Guillaud:

\begin{equation}
\textrm{hcp} (\textrm{vol}\%) = \frac{I(10\bar{1}1)_{\epsilon}}{I(10\bar{1}1)_{\epsilon} + 1.5I(200)_{\gamma}}
\end{equation}


# HCP (vol\%)  the volume fraction of the martensitic $\epsilon-Co$ phase
# I(10\bar{1}1)_{\epsilon} is the integrated intensity of the martensite peak
# I(200)_{\gamma} is the integrated intensity of the austenite peak


*** Experimental determination of SFE

To experimentally determine the SFE, the XRD method proposed by Reed and Schramm was employed \cite{reedRelationshipStackingfaultEnergy1974}:

\begin{equation}
SFE = \frac{K_{111} \omega_0 G_{111} a_0}{\pi \sqrt{3}} \frac{{<\epsilon_{50\AA}^2>}_{111}}{\alpha} A^{-0.37}
\end{equation}

where:


\begin{equation}
K_{111}\omega_0 &= 6.6 \\
G_{111} = \frac{1}{3}\frac{1}{C_{44} + C_{11} - C_{12}}
A &= \frac{1 C_{44}}{C_{11}-C_{12}}
\end{equation}


ployed \cite{reedRelationshipStackingfaultEnergy1974}:

$SFE$ = stacking fault energy $\frac{mJ}{m^2}$
$K_{111}\omega_0$ = 6.6, as obtained by
$A$ is the Zener elastic anisotropy
$C_{ij}$ are elastic stiffness coefficients
$G_{111}$ is the shear modulus of the (111)-plane, in which stacking faults are formed
$a_0$ is the lattic constant of the fcc-metal matrix
${<\epsilon^2_{111}>}_{50\AA}$ is thr root mean square microstrain in the <111> direction averaged over the distance of $50 \AA$
$\alpha$ is stacking fault probability

**** Elastic constant




*** Microhardness :ignore_heading:

# Yeah, I'm heavily assuming that the University of Sharjah lab is still open, ooof.
Microhardness measurements were taken on the surfaces of the as-cast and HIPed samples. The Wilson Tukon 1102 hardness tester was used for Vickers microhardness testing with a load of 300 grams (HV_{0.3}) for 10s, and averaged by using ten individual indentations. The specimen surface was prepared in the same fashion as for microstructural analysis.

Previous work

*** Indentation fracture toughness :ignore_heading:

The indentation fracture toughness was made with hardness equipment (AVK-A, AKASHI) at a load of 49 N for 10 s, and the value was obtained from five measurements on the cross section. The fracture toughness was evaluated based to the Evans-Wilshaw equation [21, 22].

\begin{equation}
K_{IC} = 0.079 {\left( \frac{P}{a^{\frac{3}{2}}} \right)}log{\left(\frac{4.5 a }{c}\right)}
\end{equation}

where $P$ is indenter load $[\textrm{mN}]$, $2c$ is the crack length $\left[\mu\textrm{m}\right]$, and $2a$ is the length of indentation diagonal $\left[\mu\textrm{m}\right]$


*** Experimental electrochemical - polarize electrode to -1.5 to remove oxides

# Prior to each measurement the electrodes were polarized at −1.50 V vs. SCE to remove oxides that were air-formed at the surface.
# \cite{metikos-hukovicPassivationCorrosionBehaviours2007}

Passivation and corrosion behaviours of cobalt and cobalt–chromium–molybdenum alloy
Author links open overlay panelM. Metikoš-Huković
, R. Babić
https://www.sciencedirect.com/science/article/pii/S0010938X07000819

*** Description of constant phase element

# \cite{metikos-hukovicPassivationCorrosionBehaviours2007}
# Because the measured capacitive response is not generally ideal, due to a certain heterogeneity of the electrode surface [34], [35], a constant phase element (CPE) is introduced for fitting the spectra, instead of an ideal capacitance element. Its impedance can be defined by Z(CPE) = [Q(jω)n]−1, where Q is a constant, ω is the angular frequency, and n is the CPE power. The factor n is an adjustable parameter which has values between −1 and 1; a value of −1 is characteristic for an inductance, a value of 1 corresponds to a capacitor, a value of 0 corresponds to a resistor, and a value of 0.5 can be assigned to diffusion phenomena [34].

*** Open circuit potential measurement


# In general, I love the format of the following paper
# https://link.springer.com/article/10.1007/s11665-025-10951-x
# ogunlakinMicrostructuralElectrochemicalCorrosion2025

# Great phrasing https://link.springer.com/article/10.1007/s11665-025-10951-x
# ogunlakinMicrostructuralElectrochemicalCorrosion2025
# The open circuit potential (OCP) measurement observed the specimen’s unaltered corroding potential in the absence of any applied external current. The system is anticipated to reach a stable state when the potential is independent of the immersion duration. Consequently, the OCP indicates the electrochemical environment’s stability before system polarization, offering insights into the samples’ thermodynamic inclination towards corrosion.

# Reference to Figure, along with whether all specimens achieve equilibrium
# https://link.springer.com/article/10.1007/s11665-025-10951-x
# ogunlakinMicrostructuralElectrochemicalCorrosion2025
# Figure 6 illustrates the OCP values of all samples after immersion in 3.5wt% NaCl for 1 h, revealing that the specimens achieved equilibrium after the 1 h monitoring period.

# Comparison of alloys, with specifc numbers
# https://link.springer.com/article/10.1007/s11665-025-10951-x
# ogunlakinMicrostructuralElectrochemicalCorrosion2025
# Both the IN718 and the innovative 50 IN718-50 Co212 alloys reached a stable state, displaying a nobler potential in contrast to the Co212 alloy. Specifically, the Co212 alloy exhibited a potential of approximately − 243.7 mV, whereas the IN718 and 50 IN718-50 Co212 alloys maintained stable potentials of − 139.3 mV and − 146.1 mV, respectively, after the 1 h monitoring period.

# Final conclusion stuff
# https://link.springer.com/article/10.1007/s11665-025-10951-x
# ogunlakinMicrostructuralElectrochemicalCorrosion2025
# Notably, the 50 IN718-50 Co212 alloy, despite indicating an early formation of a protective film, demonstrated a continual increase in OCP over time, with the potential to surpass that of the IN718 alloy. In general, the degree of nobility of the OCP after reaching equilibrium reflects the inertness of the sample, indicating a lower propensity for corrosion in the test environment.


*** Electrochemical instrument and experiments

A Corrtest potentiostat was used for electrochemical experiments in a conventional three electrode cell, with the sample as working electrode with exposed area 2cm2, a saturated calomel electrode (SCE) as reference electrode, and a Pt plate as counterelectrode.

All electrochemical experiments were performed at room temperature.

# Need to convince Dr Rehan to use cyclic voltametry
The open circuit potential was continuously recorded for 1 h, before the electrical impedence spectroscopy (EIS), LPR, and cyclic voltametry experiments.




**** Cavitation Erosion Test Apparatus :ignore_heading:

A magnetostrictive vibratory apparatus, operating in general accordance with ASTM Standard G32, was utilized. The system functioned at an ultrasonic frequency of 20 kHz, with a peak-to-peak displacement amplitude of the horn tip maintained at 96 µm. The horn, fabricated from a cavitation-resistant titanium alloy, featured a flat tip of 16 mm diameter. Experiments were conducted using a stationary specimen configuration, with the specimen positioned 0.5 mm below the vibrating horn tip, this distance being precisely set using a dial gauge.


**** Paragraph: Lit review of corrosion :ignore_heading:

The aqueous oxidation of Stellite 6 alloy was investigated in a 1979 study using X-ray Photoelectron Spectroscopy (XPS) \cite{mcintyreXRayPhotoelectronSpectroscopic1979}. Specimens were exposed to pH 10 water at 285°C. To understand the oxidation behavior, the study measured dissolved oxygen concentration against exposure duration.


The high-temperature corrosion resistance of stellite coatings is attributable to the formation of cobalt & chromium surface \cite{cesanekDeteriorationLocalMechanical2015}.


Heathcock et al found that carbides are selectively eroded, with the carbide-matrix interface acting as initiating erosion site \cite{heathcockCavitationErosionCobaltbased1981}.



**** Paragraph 4: Synergistic Challenges in Applications Prone to Corrosion and Cavitation :ignore:
**** Paragraph 5: Research and Development for Enhanced Corrosion and Cavitation Performance :ignore:
**** Paragraph 6: Influence of HIPing :ignore:

Compared with the case alloys, the HIPed alloys had relatively finer, rounded, and distributed carbides.


**** Paragraph: Cavitation Erosion Resistance


The primary result of an erosion test is the cumulative mass loss versus time, which is then converted to volumetric loss and mean depth of erosion (MDE) versus time for the purposes of comparison between materials of different densities. The calculation of the mean depth of erosion for this test method should be performed in conformity with ASTM G-32.


**** General Background
# \section{General Background}

# Analysis: The paragraph effectively introduces the challenge of cavitation erosion in fluid-handling systems and discusses the need for materials with improved resistance or proposes potential mitigation strategies.

# Problem Statement: It starts by clearly stating the prevalence and detrimental effects of cavitation (e.g., material erosion, reduced efficiency, noise, vibration, component failure) in key applications (e.g., pumps, propellers, turbines, valves), establishing the necessity for effective solutions.

# Focus Area / Proposed Solution: It highlights the goal of developing or utilizing materials/coatings/treatments with enhanced cavitation resistance, or introduces specific approaches intended to combat the damage.

# Mechanism: It explains the fundamental mechanism of cavitation damage (e.g., vapor bubble formation and violent collapse, generation of shockwaves and micro-jets) leading to material erosion, and potentially discusses how a proposed solution resists this mechanism.

# Context/Validation: It grounds the issue by referencing specific industries or critical components where cavitation erosion is a significant operational problem (e.g., marine propulsion, hydropower generation, hydraulic machinery, chemical processing) and underscores the importance of resistant materials in these contexts.

# Relevant Properties: It lists specific material characteristics or properties deemed crucial for resisting cavitation erosion (e.g., toughness, hardness, fatigue strength, work-hardening capacity, corrosion resistance, grain structure, phase stability).

# Knowledge Gap: Critically, it may point out limitations in current materials, testing standards, predictive models, or fundamental understanding, such as predicting erosion rates accurately, performance under combined erosion-corrosion conditions, or the behavior of novel materials.

# Call for Research/Development: Consequently, it emphasizes the need for further research, development of new materials/coatings, improved testing protocols, or advanced modeling techniques to better predict and mitigate cavitation erosion.

# Potential Applications: It suggests specific components (e.g., impellers, propellers, valve seats, cylinder liners) or systems that would directly benefit from advancements in cavitation-resistant materials, improving reliability and performance across various sectors.


# \chaptermark{Cavitation Erosion} % optional for veryy long chapter, you can rename what appear in the header

# %% have a mini table of content at the start of the chapter
# {
# \hypersetup{linkcolor=black}
# \minitoc
# }

%cite:@Franc2004265, @Romo201216, @Kumar2024, @Kim200685, @Gao2024, @20221xix, @Usta2023, @Cheng2023, @Zheng2022

Cavitation erosion presents a significant challenge in materials degradation in various industrial sectors, including hydroelectric power, marine propulsion, and nuclear systems, stemming from a complex interaction between fluid dynamics and material response \cite{francCavitationErosion2005, romoCavitationHighvelocitySlurry2012}. Hydrodynamically, the phenomenon initiates with the formation and subsequent violent collapse of vapor bubbles within a liquid, triggered by local pressures dropping to the saturated vapor pressure. These implosions generate intense, localized shockwaves and high-speed microjets that repeatedly impact adjacent solid surfaces \cite{gevariDirectIndirectThermal2020}. From a materials perspective, these impacts induce high stresses (100-1000 MPa) and high strain rates, surpassing material thresholds and leading to damage accumulation via plastic deformation, work hardening, fatigue crack initiation and propagation, and eventual material detachment. Mitigating this requires materials capable of effectively absorbing or resisting this dynamic loading, often under demanding conditions that may also include corrosion.


% Martensitic transformation
Crucially, the cobalt matrix often possesses a low stacking fault energy, facilitating a strain-induced martensitic transformation from a metastable face-centered cubic $\gamma$ phase to a hexagonal close-packed $\epsilon$ phase under the intense loading of cavitation. This transformation is a primary mechanism for dissipating impact energy and enhancing work hardening, contributing significantly to Stellite's characteristic cavitation resistance \cite{huangMicrostructureEvolutionMartensite2023, tawancyFccHcpTransformation1986}.

HIPing is a thermo-mechanical material processing technique which involves the simultaneous application of pressure (up to 200 MPa) and temperature (2000 C), which results in casting densification, porosity closure, and metallurgical bonding. \cite{yuComparisonTriboMechanicalProperties2007}

While commonly applied via casting or weld overlays, processing routes like Hot Isostatic Pressing (HIP) offer potential advantages such as microstructure refinement \cite{stoicaInfluenceHeattreatmentSliding2005} finer microstructures and enhanced fatigue resistance \cite{ahmedInfluenceReHIPingStructure2013, yuComparisonTriboMechanicalProperties2007}.

HIPing of surface coatings results in microstructure refinement, which can yield improved fatigue and fracture resistance.

HIPing leads to carbide refinement, which can yield improved impact toughness \cite{yuInfluenceManufacturingProcess2008}, and reduce carbide brittleness \cite{yuComparisonTriboMechanicalProperties2007}.

Furthermore, HIP facilitates the consolidation of novel 'blended' alloys created from mixed elemental or pre-alloyed powders, providing a pathway to potentially tailor compositions or microstructures for optimized performance. However, despite the prevalence of Stellite alloys and the known influence of processing on microstructure and properties, the specific cavitation erosion behavior of HIP-consolidated Stellites, particularly these blended formulations, remains underexplored in academic literature. Given that erosion mechanisms in Stellites often involve interactions at the carbide-matrix interface \cite{szalaEffectNitrogenIon2021}, understanding how HIP processing and compositional blending affect these interfaces and the matrix's transformative capacity under cavitation, especially when potentially coupled with corrosion, constitutes a critical knowledge gap addressed by this research.


% Need to describe Stellite 1
\section{Stellite 1}

Stellite 1 is a high-carbon and high-tungsten alloy, making it suitable for demanding applications that require hardness & toughness to combat sliding & abrasive wear \cite{crookCobaltbaseAlloysResist1994}




\section{Stellites}
\section{Objectives and Scope of the Research Work}
\section{Thesis Outline}
\section{Literature Survey}
\section{Cavitation Tests}


*** COMMENT Cavitation erosion mechanisms based on erosion particles





** Discussion
# \chapter{Discussion}



# \chapter{Cavitation Erosion}

\section{Experimental Test Procedure}
\subsection{Hardness Tests}
\subsection{Cavitation}

\section{Relationships between cavitation erosion resistance and mechanical properties}
\section{Influence of vibratory amplitude}

% Insert the whole spiel by that French dude about displacement and pressure (and then ruin it)
The pressure of the solution depends on the amplitude of the vibratory tip attached to the ultrasonic device. Under simple assumptions, kinetic energy of cavitation is proportional to the square of the amplitude and maximum hammer pressure is proportional to A.

\begin{align}
x        &= A sin(2 \pi f t) \\
v        &= \frac{dx}{dt} = 2 \pi f A sin(2 \pi f t) \\
v_{max}  &= 2 \pi f A \\
v_{mean} &= \frac{1}{\pi} \int^\pi_0  A sin(2 \pi f t) = 4 f A \\
\end{align}

However, several researchers have found that erosion rates are not proportional to the second power of amplitude, but instead a smaller number.
Thiruvengadum \cite{thiruvengadamTheoryErosion1967} and Hobbs find that erosion rates are proportional to the 1.8 and 1.5 power of peak-to-peak amplitude.
Tomlinson et al find that erosion rate is linearly proportional to peak-to-peak amplitude in copper [3].
Maximum erosion rate is approximately proportional to the 1.5 power of p-p amplitude [4].
The propagation of ultrasonic waves may result in thermal energy absorption or into chemical energy, resulting in reduced power. For the purposes of converting data from studies that do not use an amplitude of 50um, a exponent factor of 1.5 has been applied.



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* COMMENT Publications :ignore_heading:

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* Bibliography :ignore_heading:


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