%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Analysis and Discussion} \label{chap:discuss}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this chapter, the experimental results presented in Chapter~\ref{chap:results} are
analysed, and from the theory expounded in Chapter~\ref{chap:review} (and
Appendices~\ref{chap:oxley}--\ref{chap:finite_diff}), the generation mechanism of
acoustic emission (AE) detected during the course of metal cutting are discussed. To
begin with, a preliminary discussion regarding the experimental results, including a
comparison with published material, is undertaken; a basic empirical AE model is
derived, and some published AE models are discussed against the experimental data.
Additionally, a theory needed for the analysis of the metal cutting process (a
semi-empirical theory for metal cutting modelling), as well as the finite-difference
theory (for the analysis of the measured temperatures) are discussed. Afterwards, the
relationship between AE and the basic cutting parameters governing plastic
deformation during metal cutting are evaluated. Finally, a model that relates AE to
the motion of dislocations at the typically high strain rates and temperatures
involved in metal cutting is proposed; the experimental results are revealed to be in
fairly good agreement with model predictions, indicating the appropriateness of the
model.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminary Discussion of Results}
\label{sec:preliminary_discussion}

\subsection{Orthogonality}

The analysis of the experimental data from Tests~A and~B (Section~\ref{sec:procedure}),
which is presented throughout this chapter, was done by considering that cutting was
orthogonal and, therefore, could be reduced into a two-dimensional system
(Section~\ref{sec:cut_basics}). However, in reality, cutting was not fully orthogonal,
but, as it is demonstrated in this section, the effects of this departure from
orthogonality can be neglected for the present tooling system.

Cutting was semi-orthogonal since the workpieces were solid cylinders and not tubular,
as required to ensure orthogonality in turning (Section~\ref{sec:cut_basics}). One
condition to avoid great discrepancies from orthogonality is that the tool nose is
small in relation to the width of cut. The radius of the nose of the cutting tool used
in Tests~A and~B ($0.4$~mm) was relatively large (33\%) in relation to the width of cut
mostly used during the tests ($1.2$~mm). However, Messaritis and
Borthwick~\cite{messaritis86} presented, for similar conditions (tool nose radius of
$0.4$~mm and widths of cut between 0.1 and 3~mm), results of AE level and cutting
forces for both orthogonal and semi-orthogonal conditions, and they concluded that
although the level of AE and cutting forces were slightly higher for semi-orthogonal
cutting, semi-orthogonal cutting could still be used in place of orthogonal cutting
with minor differences of results.

Another important departure from orthogonality is that the side cutting angle
($C_\mr{S}$) of the cutting tool used in Tests~A and~B was $-5^\circ$, which, in order
to guarantee orthogonality, should be $0^\circ$. As stated by Lin~\etal~\cite{lin82},
an orthogonal model can be used if the feed rate ($t_1$) and width of cut ($w$) are
changed into the new values $t'_1$ and $w'$, respectively, according to the following
equations:
\begin{eqnarray}
   t'_1 & = & t_1 \cos{C_\mr{S}}
   \label{eq:new_feed} \\
   w'   & = & \frac{w}{\cos{C_\mr{S}}}
   \label{eq:new_width}
\end{eqnarray}
Moreover, new values of the cutting forces in the cutting speed, feed
rate, and radial directions ($F'_\mr{X}$, $F'_\mr{Y}$, and $F'_\mr{Z}$,
respectively) must be defined:
\begin{eqnarray}
   F'_\mr{X} & = & F_\mr{X}
   \label{eq:new_F_X} \\
   F'_\mr{Y} & = & F_\mr{Y}\cos{C_\mr{S}}
   \label{eq:new_F_Y} \\
   F'_\mr{Z} & = & -F_\mr{Y}\sin{C_\mr{S}}
   \label{eq:new_F_Z}
\end{eqnarray}
where $F_\mr{X}$ and $F_\mr{Y}$ are the tangential and feed forces obtained from the
orthogonal model (Figure~\ref{fig:forces}). Nevertheless, since \mbox{$\cos{C_\mr{S}}
\approx 1$} (\mbox{$\cos{(-5^\circ)} = 0.996$}), and  \mbox{$\sin{C_\mr{S}} \approx 0$}
(\mbox{$\cos{(-5^\circ)} = -0.087$}), the following simplifications were made:
\mbox{$t'_1 = t_1$}, \mbox{$w' = w$}, \mbox{$F'_\mr{Y} = F_\mr{Y}$}, and
\mbox{$F'_\mr{Z} = 0$}; or, in other words, the side cutting angle of the tool was
assumed to be $0^\circ$.

\subsection{Comparison of Test A Results with Published Data}
\label{sec:compar_test_A}

As shown in Section~\ref{sec:procedure_test_A}, the importance of Test~A is that it
provides a general overview of the variation of AE, cutting forces, vibration, and
temperature signals generated during the turning process of four different work
materials and for a wide range of cutting conditions. The cutting force results are
shown in Section~\ref{sec:results_forces}, temperature results in
Section~\ref{sec:results_temp}, and AE results in Section~\ref{sec:results_ae}. The
vibration signals were not presented because they are not relevant for the present
study. In this section, the results from Test~A are compared with data from published
material.

\subsubsection{Cutting Forces}

Figure~\ref{fig:[forces][speed]} shows that, as a general trend, $F_\mr{X}$ and
$F_\mr{Y}$ decrease with increasing cutting speed ($V$). Trent~\cite{trent84}
confirms these results by stating that it is common experience that the cutting
forces decrease as $V$ is raised when cutting most metal and alloys. Wright and
Robinson~\cite{wright77} showed similar trends for cutting force results of a copper
workpiece. Stevenson and Oxley~\cite{stevenson70} presented the variation of
$F_\mr{X}$ and $F_\mr{Y}$ with $V$ for a 0.13\% carbon steel, and it was shown that
both $F_\mr{X}$ and $F_\mr{Y}$ decreased slightly with increasing $V$ for high values
of $V$, but $F_\mr{X}$ and $F_\mr{Y}$ decreased abruptly with decreasing $V$ below a
certain value of $V$. These results are also observed in the present results for both
carbon steel materials
(Figures~\ref{fig:[forces][speed]}a--\ref{fig:[forces][speed]}d). For carbon steels,
these types of trends are also confirmed by other
publications~\cite{hastings74,hastings80}.

By using the approximated relationship stated by
Equation~\ref{eq:specific_power3}, representing the phenomenon called the
size effect~\cite{shaw84}, the following relationship can be deduced:
\begin{eqnarray}
   F_\mr{X} \propto t_1^{0.8}
   \label{eq:F_X_t_1}
\end{eqnarray}
Although the exponent $0.8$ is a rough estimation, one can conclude that $F_\mr{X}$
increases with $t_1$, but with a rate of increase that decreases as $t_1$ is raised.
This trend is visible in all experimental results shown in
Figure~\ref{fig:[forces][feed]}, where both $F_\mr{X}$ and $F_\mr{Y}$ are observed to
increase with $t_1$, but presenting a less significant rate of increase as $t_1$
increases. Experimental results showing the relationship between cutting forces and
$t_1$ are not normally explicitly presented due to its well-known and regular
relationship. In Section~\ref{sec:oxley_forces}, results  from Oxley's
model~\cite{oxley89} for varying $t_1$ are compared with the experimental cutting force
results from Test~A\@.

Experimental results showing the relationship between cutting forces and $w$ are also
not normally explicitly presented, since this relationship is also well-known and
regular. It is normally assumed that changing $w$ does only affect the scale of the
cutting process~\cite{shaw84}, meaning, therefore, that both $F_\mr{X}$ and
$F_\mr{Y}$ are directly proportional to $w$. This trend is also confirmed by the
computation method used by Oxley's model~\cite{oxley89} to predict cutting forces
(Section~\ref{sec:oxley_forces}). The experimental results shown in
Figures~\ref{fig:[forces][080A15][width]}--\ref{fig:[forces][6082-T6][width]} are
consistent with this relationship, where $F_\mr{X}$ and $F_\mr{Y}$ are observed to
increase linearly with increasing $w$.

\subsubsection{Cutting Temperatures}

Figures~\mbox{\ref{fig:[temp][speed]}--\ref{fig:[temp][6082-T6][width]}} shows the
variation of temperature $T_1$ measured by thermocouple~1\footnote{Although there is a
reference to two thermocouples in Section~\ref{sec:instrument}, as stated in
Section~\ref{sec:procedure_test_A}, only one thermocouple was used during the course of
Test~A\@.} (Section~\ref{sec:instrument}), placed 5~mm bellow the tool-chip interface.
Although $T_1$ is the result of the heat conducted between the higher temperature at
the tool-chip interface ($T_\mr{int}$) and the location of thermocouple~1, there are
some difficulties when one needs to calculate $T_\mr{int}$ based on $T_1$ measurements.
First of all, as evidenced by published material~\cite{chow88,silva99}, once
$T_\mr{int}$ is established at the tool-chip interface, there will be a delay until
$T_1$ settles to is final value. However, as stated in Section~\ref{sec:results_temp},
this problem was partially overcome, since, for each test, the maximum value of $T_1$
was chosen, and by the end of each test, the value of $T_1$ was expected to be nearly
settled down to its maximum final value. Another problem is that though the value of
$T_1$ is dependent on $T_\mr{int}$, the relationship between the two is dependent on
many variables. Techniques to predict $T_\mr{int}$ based on temperature measurements
from a remote location have been addressed by some authors~\cite{chow88,silva99}.
Moreover, in Section~\ref{sec:temp_predict}, it is shown how the finite-difference
technique can be used to compute $T_1$ from known $T_\mr{int}$ values; in
Section~\ref{sec:oxley_temp}, Oxley's model~\cite{oxley89} is used to predict
$T_\mr{int}$, and then these values are used to compute $T_1$ by using the
finite-difference method, so that predictive $T_1$ values can be compared with the
experimental $T_1$ values from Test~A\@.

\subsubsection{Acoustic Emission}

Figure~\ref{fig:[mode][speed]} shows the relationship between the level of AE
($I_\mr{mode}$, as defined in Section~\ref{sec:ae_theory}) and $V$. Both
$I_\mr{mode}$ and its rate of increase are observed to increase with increasing $V$.
This relationship is repeatedly observed with small divergences in many published
papers~\cite{dornfeld80,kannatey-asibu81,inasaki81,messaritis86,blum90,rangwala91a,saini96}.
However, though the AE level is normally presented as the root mean square of AE
($I_\mr{RMS}$), instead of $I_\mr{mode}$, the two hold equivalent energetic units
(Section~\ref{sec:ae}). Some authors~\cite{kannatey-asibu81,inasaki81,saini96}
sugested that the level of AE is directly proportional to $V$; however, when the data
is analysed closely, the rate of increase of the level of AE tends to increase with
increasing $V$. Moreover, Blum and Inasaki~\cite{blum90} explicitly stated that
$I_\mr{mode}$ and its rate of increase both increase with increasing $V$, and,
although not explicitly stated, the data presented by Messaritis and
Borthwick~\cite{messaritis86} exhibit the same relationship.

Generally, the level of AE from Test~A experiments is observed to decrease with
increasing feed rate, especially when \mbox{$V = 200$ m$/$min}
(Figures~\ref{fig:[mode][feed]}b, \ref{fig:[mode][feed]}d, \ref{fig:[mode][feed]}f, and
\ref{fig:[mode][feed]}h). When \mbox{$V = 75$ m$/$min}, the results are quite scattered
and no clear relationship can be determined (Figures~\ref{fig:[mode][feed]}a,
\ref{fig:[mode][feed]}c, \ref{fig:[mode][feed]}e, and \ref{fig:[mode][feed]}g), and, as
shown in Section~\ref{sec:oxley_forces}, this can perhaps be attributed to the
phenomenon of built-up edge (BUE). These results are supported by published material,
which shows that for low cutting speeds (below around 100~m$/$min), the level of AE is
more or less insensitive to $t_1$~\cite{dornfeld80,inasaki81,lan85, lan86}, and, for
higher cutting speeds, the AE level decreases with $t_1$, tending to a constant value
as $t_1$ increases~\cite{messaritis86,blum90}.

As shown in
Figures~\ref{fig:[mode][080A15][width]}--\ref{fig:[mode][6082-T6][width]}, as a
general trend, the level of AE increases with $w$ for low values of $w$, but its rate
of increase decreases as $w$ increases, tending to a constant value for high $w$
values. Many publications show that the level of AE remains constant as $w$ is
varied~\cite{inasaki81,lan86,blum90}; however it can be observed that this trend is
observed for quite high values of $w$. When results are present for lower $w$
values~\cite{lan85,messaritis86}, the level of AE is observed to decrease with
decreasing $w$, in the same fashion as the experimental data shown in
Figures~\ref{fig:[mode][080A15][width]}--\ref{fig:[mode][6082-T6][width]}.

Figures~\ref{fig:[freq][speed]}--\ref{fig:[freq][6082-T6][width]} show the variation of
the mean frequency ($f_\mr{mean}$) of the AE power spectrum of Test~A data with $V$,
$t_1$, and $w$. This parameter, which gives an indication of the frequency content of
the AE signals, was introduced by Rangwala and Dornfeld~\cite{rangwala91b} and is
defined in Section~\ref{sec:ae_theory}. According to Rangwala and Dornfeld's results,
$f_\mr{mean}$ increases with $V$ up to a certain value of $V$ (60~m$/$min), from which
$f_\mr{mean}$ starts to decrease with further increase in $V$. Test~A results of
$f_\mr{mean}$ are observed to decrease with $V$ for the whole range of $V$ values
(Figure~\ref{fig:[freq][speed]}). However, it must be noted that the range of $V$
values utilized by Rangwala and Dornfeld that exhibit an increase of $f_\mr{mean}$ with
$V$ are mostly below the minimum $V$ value utilized in Test~A (50~m$/$min), and,
therefore, an agreement between Rangwala and Dornfeld and Test~A results must be
concluded when considering the variation of $f_\mr{mean}$ with $V$. Moreover, an
agreement is also obtained between Rangwala and Dornfeld and Test~A
(Figure~\ref{fig:[freq][feed]}) results, for the variation of $f_\mr{mean}$ with $t_1$,
where $f_\mr{mean}$ seems to be more or less insensitive with  $t_1$. Finally, Rangwala
and Dornfeld supply no results for the variation of $f_\mr{mean}$ with $w$, but
$f_\mr{mean}$ is presumed to be insensitive to the changes of $w$, since $w$ is
expected to affect solely the volume of material participating into deformation, and
this should have no effect on the frequency content of the AE signals; this trend is
also observed for Test~A results, where $f_\mr{mean}$ remains more or less insensitive
with $w$ (Figures~\ref{fig:[freq][080A15][width]}--\ref{fig:[freq][6082-T6][width]}).

\subsection{Feed Rate Variation and Width of Cut Variation Tests}
\label{sec:feed_width_variation_tests}

As mentioned in Section~\ref{sec:procedure_test_B}, two other types of tests, namely
feed rate variation and width of cut variation tests, were carried out, whose AE
results were, unfortunately, highly affected by random noise signals. The same method
as of Test~B, and thus a different method from Test~A, was used to acquire data during
the feed rate and width of cut variation tests, and, in this section, the relative
failure of the AE results from these tests is examined.

As described in Section~\ref{sec:results_ae}, during Test~A acquisition, although sets
of 8388608 samples were acquired, each set was subsequently divided in smaller sets of
1048576 data points (105~ms, since the sampling rate was 10~MS$/$s). During the feed
rate and width of cut variation tests, sets of only 65536 samples (13~ms, since the
sampling rate was 5~MS$/$s) were acquired. Therefore, the effective length of Test~A
data sets was 16 times larger than that of the data sets from the feed rate and width
of cut variation tests. Moreover, as stated in Sections~\ref{sec:results_ae}
and~\ref{sec:results_test_B}, the main method to process the AE raw data was by
computing $I_\mr{mode}$, and if the data sets are too small, the peak of the
probability density function of the previously full-wave rectified and low-pass
filtered data will not represent the level of continuous AE if the influence of a
random bust-type signal is colouring the data set. On the other hand, if the data set
is large enough, a single bust-type signal will not be sufficient to move the peak of
the probability density function, and, therefore, $I_\mr{mode}$ will represent the
continuous-type AE data, which is normally associated with the plastic deformation of
the cutting process. Consequently, one may conclude that the small size of the the data
sets from the feed rate and width of cut variations tests is the main reason why the AE
results from these tests did not provide much additional knowledge on the generation of
AE from metal cutting. Therefore, it can be concluded that large samples of AE data
(more than 100~ms of duration) must be acquired in order to obtain meaningful results.

One interesting observation with the width of cut variation results was the
generation of AE when high chatter vibrations\footnote{Chatter is the relative motion
between the cutting tool and workpiece that arrives when not enough energy is
dissipated by either the damping of the structure or the friction of the cutting
process. As a result, relative motion between the tool and the workpiece, at one of
the natural frequencies of the system, grows beyond acceptable limits. This leads to
poor surface finish, reduced dimensional accuracy, increased tool wear, tool
fracture, and even damage to the machine tool itself.} are present. As some results
showed that, within a range of width of cut values, the value of $I_\mr{mode}$
abruptly rose to very high values, it was decided to investigate if this was due to
the phenomenon of chatter, and, therefore, as shown in
Figure~\ref{fig:profile_chatter},
\begin{figure}[h!]
   \center{\includegraphics{profile_chatter.eps}}
   \caption{AE generation during chatter for varying width of cut.}
   \label{fig:profile_chatter}
\end{figure}
both vibrational and AE curves were plotted together. Approximately between widths of
cut of 1 and 1.7~mm, the amplitude of the vibration signals is very high, indicating
the onset of chattering, which is also accompanied by an abrupt rise of $I_\mr{mode}$.
The maximum $I_\mr{mode}$ value without chatter is 19~mV, whereas the maximum
$I_\mr{mode}$ value when chatter is present is 382~mV, resulting in an increase of
20~times of $I_\mr{mode}$ (which is actually higher than the increase of the vibration
signals). Therefore, it has to be concluded that chatter has an important impact on the
generation of AE\@.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Empirical Modelling of Acoustic Emission Energy}
\label{sec:ae_empirical_model}

In this section, an empirical model of AE energy, based on Test~A experimental data
(Section~\ref{sec:results_test_A}, Figures~\ref{fig:[mode][speed]}
and~\ref{fig:[mode][feed]}) is developed. Figures~\ref{fig:[mode][speed]}
and~\ref{fig:[mode][feed]} show, for constant width of cut (\mbox{$w = 1.2$ mm}), the
variation of $I_\mr{mode}$ for different cutting conditions and work materials. In this
section, a more meaningful physical representation of the level of AE was utilized, the
AE power ($I_\mr{power}$), which is equivalent to the energy rate of the generated AE
signals: $I_\mr{power} = I_\mr{mode}^2$.

\begin{figure}[h!]
   \center{\includegraphics{[log][speed].eps}}
   \caption[Relationship between $I_\mr{power}$ and cutting speed in logarithmic scale.]
   {Relationship between $I_\mr{power}$ and cutting speed in
   logarithmic scale. (a)~\mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~\mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[log][speed]}
\end{figure}
Figures~\ref{fig:[log][speed]}a and~\ref{fig:[log][speed]}b show logarithmic scale
plots representing the variation of $I_\mr{power}$ with cutting speed for feed rates of
0.02 and 0.1 mm$/$rev, respectively. It can be observed in both plots that, in the
logarithmic scale, the relationship between $I_\mr{power}$ and $V$ can be represented,
for each material, as a straight line of positive slope, suggesting the existence of a
power function relationship between $I_\mr{power}$ and $V$.

\begin{figure}[h!]
   \center{\includegraphics{[log][feed].eps}}
   \caption[Relationship between $I_\mr{power}$ and feed rate in logarithmic scale.]
   {Relationship between $I_\mr{power}$ and feed rate in
   logarithmic scale. (a)~\mbox{$V = 75$ m$/$min}.
   (b)~\mbox{$V = 200$ m$/$min}.}
   \label{fig:[log][feed]}
\end{figure}
Figures~\ref{fig:[log][feed]}a and~\ref{fig:[log][feed]}b show logarithmic scale plots
of $I_\mr{power}$ against $t_1$ for cutting speeds of 75 and 200 m$/$min, respectively.
When $V = 200$ m$/$min, it can be observed that, in the logarithmic scale, the
relationship between $I_\mr{power}$ and $t_1$ can be approximated to straight lines of
negative slope for the different materials, suggesting a power function relationship
between $I_\mr{power}$ and $t_1$. However, when $V = 75$ m$/$min, $I_\mr{power}$ seems
to be more or less insensitive to $t_1$, and a clear relationship is not exhibited.

Due to the power function nature of the relationship between $I_\mr{power}$, $V$, and
$t_1$, the following empirical relationship was investigated:
\begin{eqnarray}
   I_\mr{power} = KV^{n_1}t_1^{n_2}
   \label{eq:empirical_ae_power}
\end{eqnarray}
where $K$, $n_1$, and $n_2$ are material dependent constants. Furthermore, the data shown
in Figures~\ref{fig:[mode][speed]} and \ref{fig:[mode][feed]} was utilized to compute
constants $K$, $n_1$, and $n_2$ by non-linear regression analysis with the aid of the
computer software DataFit from Oakdate Engineering. The resultant constants are displayed
in Table~\ref{tab:empirical_constants}.
\begin{table}[h!]
   \caption{Equation~\ref{eq:empirical_ae_power} constants found
   by non-linear regression analysis.}
   \label{tab:empirical_constants}
   \center{\begin{tabular}{ccccc}
      \hline
      material & $K$                    & $n_1$ & $n_2$   & $r^2$ \\
      \hline
      080A15   & $9.489\times 10^{-11}$ & 3.676 & -0.5502 & 0.937 \\
      080M40   & $3.076\times 10^{-11}$ & 3.942 & -0.6311 & 0.964 \\
      304S15   & $8.201\times 10^{-12}$ & 4.643 & -0.2555 & 0.965 \\
      6082-T6  & $7.641\times 10^{-11}$ & 3.773 & -0.0103 & 0.946 \\
      \hline
   \end{tabular}}
\end{table}
For all materials, the coefficient of multiple determination\footnote{The coefficient
of multiple determination $r^2 = {\sum_{i = 1}^{n} (\hat{y}_i -
\overline{y})^2}/{\sum_{i = 1}^{n} (y_i - \overline{y})^2}$, where $\hat{y}_i$ and
$y_i$ are the predicted and experimental response values for the $i^\mr{th}$ data
point; and $\overline{y}$ is the mean of the experimental responses for all $n$ data
points. A good fit will result in $r^2$ values close to one, whereas a poor fit will
result in $r^2$ values close to zero.} ($r^2$) presents values fairly close to one,
meaning that Equation~\ref{eq:empirical_ae_power} is a good empirical model to
represent the relationship between $I_\mr{power}$, $V$, and $t_1$.
Figure~\ref{fig:3d_ae_power} shows plots of the relationship stated by
Equation~\ref{eq:empirical_ae_power} for the four work materials.
\begin{figure}[h!]
   \center{\includegraphics{3D_plots.eps}}
   \caption[Three-dimensional representation of $I_\mr{power}$ as a
   function of cutting speed and feed rate.]
   {Three-dimensional representation of $I_\mr{power}$ as a
   function of cutting speed and feed rate.
   (a)~080A15. (b)~080M40. (c)~304S15. (d)~6082-T6.}
   \label{fig:3d_ae_power}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cutting Power and Acoustic Emission Power}
\label{sec:cut_ae_power}

\begin{figure}[h!]
   \center{\includegraphics{[AE_power][speed].eps}}
   \caption[Relationship between $I_\mr{power}$ and cutting power for
   varying cutting speed.]
   {Relationship between $I_\mr{power}$ and cutting power for varying
   cutting speed; \mbox{$w = 1.2$ mm}.
   (a)~\mbox{$t_1 = 0.02$ mm$/$rev}. (b)~\mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[AE_power][speed]}
\end{figure}

\begin{figure}[h!]
   \center{\includegraphics{[AE_power][feed].eps}}
   \caption[Relationship between $I_\mr{power}$ and cutting power for
   varying feed rate.]
   {Relationship between $I_\mr{power}$ and cutting power for
   varying feed rate; \mbox{$w = 1.2$ mm}.
   (a)~\mbox{$V = 75$ m$/$min}. (b)~\mbox{$V = 200$ m$/$min}.}
   \label{fig:[AE_power][feed]}
\end{figure}

\begin{figure}[h!]
   \center{\includegraphics{[AE_power][width].eps}}
   \caption[Relationship between $I_\mr{power}$ and cutting power for varying width of cut.]
   {Relationship between $I_\mr{power}$ and cutting power for
   varying width of cut. \mbox{$V = 75$ m$/$min},
   \mbox{$t_1 = 0.02$ mm$/$rev}.
   (a)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (d)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[AE_power][width]}
\end{figure}

Most AE energy models presented in Section~\ref{sec:ae_cont_energy} regard
$I_\mr{power}$ to be proportional to the plastic work rate of
deformation~\cite{dornfeld80,kannatey-asibu81,schmenk84,blum90,rangwala91a,saini96}. In
Section~\ref{sec:mech_cut}, Equation~\ref{eq:power} suggests that the total cutting
power is proportional to the tangential force and cutting speed, i.e.\ $\dot U =
F_\mr{X}V$, and that $\dot U$ is essentially the summation of the work rate produced in
the primary and secondary deformation zones. Consequently, the following equation can
be derived:
\begin{eqnarray}
   I_\mr{power} = K\dot{U} = KF_\mr{X}V
   \label{eq:ae_cut_power}
\end{eqnarray}
where $K$ is a constant of proportionality. In order to investigate this relationship,
the experimental data presented in Section~\ref{sec:results_test_A} (Test~A results)
were used to calculate values of $I_\mr{power}$ and $\dot{U}$. Due to the power
function nature of the AE data, as described in Section~\ref{sec:ae_empirical_model},
it was decided to plot the relationship between $I_\mr{power}$ and $\dot{U}$ on a
logarithmic scale as well
(\mbox{Figures~\ref{fig:[AE_power][speed]}--\ref{fig:[AE_power][width]}}), in order to
improve its visualization. On the logarithmic scale, the relationship between
$I_\mr{power}$ and $\dot{U}$, as defined in Equation~\ref{eq:ae_cut_power}, is
represented by a straight line with $45^\circ$ inclination (\mbox{slope $= 1$}) and an
intercept that depends on constant $K$. This straight line is plotted in
\mbox{Figures~\ref{fig:[AE_power][speed]}--\ref{fig:[AE_power][width]}}, but since $K$
is unknown, a convenient vertical location of the line was chosen, so that it could be
visualized together with the experimental data for comparison.

Figure~\ref{fig:[AE_power][speed]} presents the relationship between $I_\mr{power}$
and $\dot{U}$ for varying $V$ (\mbox{$t_1 = 0.02$} and 0.1~mm$/$rev, and \mbox{$w =
1.2$ mm}). If the experimental data was in accordance with
Equation~\ref{eq:ae_cut_power}, there would be a value for $K$ that made all the
experimental data points fall on the $45^\circ$ inclination straight line. Although
the experimental data seem to fall on a straight line, it shows a slope larger than
$45^\circ$. However, it seems that the experimental values form a relationship
between $I_\mr{power}$ and $\dot{U}$ that is, within a certain margin of error,
independent of the work material.

Figure~\ref{fig:[AE_power][feed]} presents the relationship between $I_\mr{power}$ and
$\dot{U}$ for varying $t_1$ (\mbox{$V = 75$ m$/$min} and 200~m$/$min, and \mbox{$w =
1.2$ mm}). For \mbox{$V = 75$ m$/$min} (Figure~\ref{fig:[AE_power][feed]}a),
$I_\mr{power}$ seems to be more or less insensitive to $\dot{U}$ for varying $t_1$,
contradicting the relationship defined by Equation~\ref{eq:ae_cut_power}. Moreover, the
relationship held by Equation~\ref{eq:ae_cut_power} is also not verified when \mbox{$V
= 200$ m$/$min}, since $I_\mr{power}$ experimental values approximate to straight lines
of negative slope and different $K$ values for different work materials. As a result,
Equation~\ref{eq:ae_cut_power} is not confirmed by the experimental data when $t_1$ is
varied.

Figure~\ref{fig:[AE_power][width]} show the relationship between $I_\mr{power}$ and
$\dot{U}$ for varying $w$. It can be observed that, in certain cases, the relationship
approximates to a straight line with $45^\circ$ inclination, e.g.\ when \mbox{$V = 75$
m$/$min} and \mbox{$t_1 = 0.02$} for work materials 080A15, 080M40, and 304S15
(Figure~\ref{fig:[AE_power][width]}a). However, in most cases, the relationship between
$I_\mr{power}$ and $\dot{U}$ fall on a line with an inclination less that $45^\circ$.
Moreover, it can also be observed that the experimental data of $I_\mr{power}$ against
$\dot{U}$, when approximated to straight lines, present different values of $K$ for
different work materials, opposing the relationship held by
Equation~\ref{eq:ae_cut_power}.

The relative agreement found in the published energy models
\cite{dornfeld80,kannatey-asibu81,schmenk84,blum90,rangwala91a,saini96} is mainly due
to the fact that the AE data was given from cutting tests where normally only $V$ was
changed, i.e.\ $t_1$ and $w$ were kept constant. Figure~\ref{fig:[AE_power][speed]}
also shows that when only $V$ is varied, the relationship between $I_\mr{power}$ and
$\dot{U}$, when plotted on a logarithmic scale, seems to fall on a single straight line
for different work materials. However, this line exhibits a slope higher than one, so
that in order to fit the experimental data, Equation~\ref{eq:ae_cut_power} may be
changed as follows: $I_\mr{power} = K\dot{U}^n$, where $n$ is the slope of the line in
the logarithmic scale ($n > 1$), and $K$ and $n$ depend on $t_1$ and $w$ but are work
material independent. Dornfeld and Kannatey-Asibu~\cite{dornfeld80} presented AE data
for varying $V$ and $t_1$, but although their energy model equated $I_\mr{power}$
proportional to $V$ and $t_1$, $I_\mr{power}$ seemed to be proportional to $V^n$, where
\mbox{$n > 1$}, and no clear relationship was observed between $I_\mr{power}$ and
$t_1$. Lan and Dornfeld~\cite{lan86} pointed out that although the relationship
presented by Dornfeld and Kannatey-Asibu~\cite{dornfeld80} provided some indication of
the cutting parameters upon which AE depends, the effects of $t_1$ and $w$ have not
clearly been evaluated experimentally. Kannatey-Asibu and
Dornfeld~\cite{kannatey-asibu81} presented experimental AE data for varying $V$ and
tool rake angle ($\alpha$), and although, in their model, $I_\mr{power}$ was
proportional to $V$, straight lines were plotted for the relationship between
$\sqrt{I_\mr{power}}$ and $V$, suggesting that $I_\mr{power}$ was proportional to
$V^2$. Moreover, different relationships were observed between $\sqrt{\dot{U}}$
(theoretical $\sqrt{I_\mr{power}}$) and $I_\mr{RMS}$ (experimental
$\sqrt{I_\mr{power}}$) for different $\alpha$ values, suggesting that constant $K$ of
Equation~\ref{eq:ae_cut_power} was dependent on $\alpha$. Saini and Park~\cite{saini96}
presented a more complex model, and although it showed good agreement with experimental
data, $t_1$ and $w$ were, again, kept constant. The results presented by Blum and
Inasaki~\cite{blum90} exhibited some agreement between theoretical and experimental
$I_\mr{power}$ when $V$ was varied, but opposite trends were observed between
experimental and theoretical $I_\mr{power}$ when $t_1$, $\alpha$, and $w$ were varied.
Nevertheless, the model presented by Rangwala and Dornfeld~\cite{rangwala91a} presented
an excellent agreement between $I_\mr{power}$ and the power of dislocation damping
($\dot{U}_\mr{D}$) for experimental data where both $V$ and $t_1$ were varied. However,
the range of $t_1$ values (\mbox{0.1--0.2 mm$/$rev}) was much narrower than the range
of Test~A experimental values (\mbox{0.01--0.2 mm$/$rev}), and, therefore, as can be
seen in Figures~\ref{fig:[mode][feed]}, the level of AE tends normally to increase with
decreasing $t_1$ when $t_1$ is low, whereas the experimental data provided by Rangwala
and Dornfeld exhibited more or less a constant level of AE for varying $t_1$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Temperature Measurement Analysis}
\label{sec:temp_predict}

\begin{figure}[h!]
   \center{\includegraphics{temp_model.eps}}
   \caption{Geometry of finite-difference temperature model.}
   \label{fig:temp_model}
\end{figure}

\begin{figure}[h!]
   \center{\includegraphics{temp_simul.eps}}
   \caption[Prediction of sensor temperatures $T_1$ and $T_2$ for known
   $T_\mr{int}$ and tool-chip interface dimensions.]
   {Prediction of sensor temperatures $T_1$ and $T_2$ for known
   $T_\mr{int}$ and tool-chip interface dimensions.
   (a)~Variation of $T_1$ with $T_\mr{int}$;
   \mbox{$w = 1$ mm} and \mbox{$l_\mr{int} = 0.1$ mm}.
   (b)~Variation of $T_1$ with $w$;
   \mbox{$T_\mr{int} = 1000^\circ$C} and \mbox{$l_\mr{int} = 0.1$ mm}.
   (c)~Variation of $T_1$ with $l_\mr{int}$;
   \mbox{$T_\mr{int} = 1000^\circ$C} and \mbox{$w = 1$ mm}.}
   \label{fig:temp_simul}
\end{figure}

As described in Section~\ref{sec:instrument} (Figure~\ref{fig:thermocouples}), two
thermocouples were installed some distance below the top of the cutting tool (5 and
7~mm, respectively), so that the temperatures at points 1 and 2 ($T_1$ and $T_2$)
were a result of the heat transfer caused by the high temperature at the tool-chip
interface. Hence, the finite-difference numerical technique was employed, so that the
relationship between the actual mean temperature at the tool-chip interface
$T_\mr{int}$ and temperatures $T_1$ and $T_2$ could be determined. The
finite-difference model was defined as follows (Figure~\ref{fig:temp_model}): the
tool was simplified to a rectangular block (20~mm in all three dimensions), where the
tool-chip contact area is a rectangular zone with dimensions $w$ (width of cut) and
$l_\mr{int}$ (tool-chip interface length), placed at a corner of the block;
temperature $T_\mr{int}$ was assumed constant over the whole tool-chip contact area;
heat was assumed to flow from the tool-chip area to the rest of the tool due to
conduction; finally, convective heat was assumed to flow between the tool and the
outside surrounding air (temperature $T_\infty$). Since the thermal conductivity of
the tool material ($\kappa$) and the coefficient of heat convection ($h$) were
unknown, these values were based on published values (\mbox{$\kappa = 50$
W$/$m-$^\circ$C} and \mbox{$h = 100$ W$/$m$^2$-$^\circ$C})~\cite{holman02}. It must
be added that very accurate values of $\kappa$ and $h$ are not so important, since
the objective of this model is an evaluation of the relative effect of $T_\mr{int}$,
$w$, and $l_\mr{int}$ on the measurement of $T_1$ and $T_2$.
Appendix~\ref{chap:finite_diff} shows the finite-difference technique in more detail.

Figure~\ref{fig:temp_simul} exhibits the variation of $T_1$ and $T_2$ with
$T_\mr{int}$, $w$, and $l_\mr{int}$ (Figures~\ref{fig:temp_simul}a,
\ref{fig:temp_simul}b, and \ref{fig:temp_simul}c, respectively). It can be observed
that $T_1$ is predicted to be higher than $T_2$, as expected from practical experience,
since thermocouple 1 is further than thermocouple 2 from the tool-chip interface.
Moreover, $T_1$ and $T_2$ increase linearly with increasing $T_\mr{int}$, which is a
sound relationship, since in order to calculate $T_\mr{int}$, one simply needs to
multiply the measured values of $T_1$ or $T_2$ by some constant value. However, it is
also observed that $T_\mr{int}$ also varies with both $w$ and $l_\mr{int}$; this makes
the prediction of $T_\mr{int}$ from measured values difficult because, although $w$ is
known, $l_\mr{int}$ is dependent on the cutting process and, therefore, unknown in
advance.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Oxley's Model Results and Analysis}
\label{sec:oxley_model}

Section~\ref{sec:var_mat_prop} and Appendix~\ref{chap:oxley} describe a machining
theory, formulated by Oxley and co-workers, that predicts, for example, the geometry,
cutting forces, stresses, and temperatures of metal cutting \cite{oxley89}. The model
requires as inputs the cutting conditions $V$, $t_1$, $w$, $\alpha$, $T_\mr{W}$
(initial workpiece temperature), the thermal material properties $c$ (specific heat)
and $\kappa$ (thermal conductivity), the density of the material ($\rho$), and,
finally, the flow stress properties of the work material. In this section, the
predictive results from the model are analysed and compared with the experimental
results from Test~A (Section~\ref{sec:results_test_A}). However, only results for the
carbon steel work materials (080A15 and 080M40) were obtained, since no reliable flow
stress data could be found for the other two work materials. It must be added that
since the model did not converge for some of the lowest values of $V$ and $t_1$, the
model was run for the higher values of $V$ and $t_1$, and then the results for the
lower values of $V$ and $t_1$ were estimated by extrapolation. Besides, experimental
results from Test~B (Section~\ref{sec:results_test_B}) for the variable workpiece
surface temperature ($T_\mr{WS}$) are compared with predictions from Oxley's model,
where, in this case, $T_\mr{W}$ was assumed to be equal to $T_\mr{WS}$.

\begin{figure}[h!]
   \center{\includegraphics{[Oxley_R][speed].eps}}
   \caption[Variation of $R$ with cutting speed.]
   {Variation of $R$ with cutting speed; \mbox{$w = 1.2$ mm}.
   (a)~080A15; \mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~080A15; \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~080M40; \mbox{$t_1 = 0.02$ mm$/$rev}.
   (d)~080M40; \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[Oxley_R][speed]}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{[Oxley_R][feed].eps}}
   \caption[Variation of $R$ with feed rate.]
   {Variation of $R$ with feed rate; \mbox{$w = 1.2$ mm}.
   (a)~080A15; \mbox{$V = 75$ m$/$min}.
   (b)~080A15; \mbox{$V = 200$ m$/$min}.
   (a)~080M40; \mbox{$V = 75$ m$/$min}.
   (b)~080M40; \mbox{$V = 200$ m$/$min}.}
   \label{fig:[Oxley_R][feed]}
\end{figure}

\subsection{Cutting Forces}
\label{sec:oxley_forces}

\begin{figure}[h!]
   \center{\includegraphics{BUE_Boundary.eps}}
   \caption[Prediction of BUE for varying cutting speed and feed rate.]
   {Prediction of BUE for varying cutting speed and feed rate. (a)~080A15. (b)~080M40.}
   \label{fig:bue_boundary}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{[temp][R_theor_exper].eps}}
   \caption[Variation of $R$ with work surface temperature.]
   {Variation of $R$ with work surface temperature.
   (a)~080A15; \mbox{$V = 75$ m$/$min}; \mbox{$t_1 = 0.1$ mm$/$rev}.
   (b)~080A15; \mbox{$V = 200$ m$/$min}; \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~080M40; \mbox{$V = 200$ m$/$min}; \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[temp][R_theor_exper]}
\end{figure}
Figure~\ref{fig:[Oxley_R][speed]} shows the variation of the resultant cutting force
($R$) with $V$; and Figure~\ref{fig:[Oxley_R][feed]} shows the variation of $R$ with
$t_1$. The predictions of $R$ are plotted along with the experimental results from
Test~A\@. It can be observed that there is a fairly good agreement between predictive
and experimental values, especially when $V$ and $t_1$ are high. However, for low
values of $V$, especially when $t_1 = 0.02$ (Figures~\ref{fig:[Oxley_R][speed]}a
and~\ref{fig:[Oxley_R][speed]}c), predictions of $R$ seem to be substantially lower
than the experimental values of $R$. Moreover, during the experimental procedure, the
occurrence of BUE was noticed, since traces of material attached to the tool were
observed after cutting for the lowest values of $V$ and $t_1$. As discussed previously
in Section~\ref{sec:cut_physics}, BUE tends to increase the effective rake angle,
resulting in a reduction of the cutting forces~\cite{trent84}. In addition,
Hastings~\etal~\cite{hastings80,oxley89} introduced the following criterion to predict
BUE: BUE will be formed if \mbox{$T_\mr{int} < 1000$ K} and \mbox{$T_\mr{mod} < 700$
K}, where $T_\mr{int}$ and $T_\mr{mod}$ are the mean and velocity-modified temperatures
at the tool-chip interface, respectively. With values of $T_\mr{int}$ and $T_\mr{mod}$
obtained from Oxley's model, the criterion of BUE predictions were calculated for
different values of $V$ and $t_1$; these results are shown in
Figure~\ref{fig:bue_boundary}, where the line that represents the onset of BUE is
plotted against $V$ and $T_1$. This criterion was also used to mark (as a vertical
dashed line) the onset of BUE in the plots shown in Figures~\ref{fig:[Oxley_R][speed]}
and \ref{fig:[Oxley_R][feed]}. As a result, it can be observed that the experimental
values of $R$ that fall below predictions occur mainly in the range of $V$ and $t_1$
that lead to the prediction of BUE according to the above criterion.

Figure~\ref{fig:[temp][R_theor_exper]} shows the variation of $R$ with $T_\mr{WS}$.
Experimental data from Test~B are plotted along with the prediction from Oxley's model.
It can be observed that experimental and predictive values present similar trends and
are in the same order of magnitude; however, experimental result from the 080A15 work
material (Figures~\ref{fig:[temp][R_theor_exper]}a
and~\ref{fig:[temp][R_theor_exper]}b) fall slightly bellow predictions, whereas
experimental result for the 080M40 work material fall slightly above predictions.

\subsection{Cutting Temperatures}
\label{sec:oxley_temp}

\begin{figure}[h!]
   \center{\includegraphics{[Oxley_T][speed].eps}}
   \caption[Variation of $T_\mr{int}$ and $T_1$ with cutting speed.]
   {Variation of $T_\mr{int}$ and $T_1$ with cutting speed;
   \mbox{$w = 1.2$ mm}.
   (a)~080A15; \mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~080A15; \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~080M40; \mbox{$t_1 = 0.02$ mm$/$rev}.
   (d)~080M40; \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[Oxley_T][speed]}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{[Oxley_T][feed].eps}}
   \caption[Variation of $T_\mr{int}$ and $T_1$ with feed rate.]
   {Variation of $T_\mr{int}$ and $T_1$ with feed rate;
   \mbox{$w = 1.2$ mm}.
   (a)~080A15; \mbox{$V = 75$ m$/$min}.
   (b)~080A15; \mbox{$V = 200$ m$/$min}.
   (a)~080M40; \mbox{$V = 75$ m$/$min}.
   (b)~080M40; \mbox{$V = 200$ m$/$min}.}
   \label{fig:[Oxley_T][feed]}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{[Oxley_T][080A15][width].eps}}
   \caption[Variation of $T_\mr{int}$ and $T_1$ with width of cut; work material 080A15.]
   {Variation of $T_\mr{int}$ and $T_1$ with width of cut; work material 080A15.
   (a)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (d)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[Oxley_T][080A15][width]}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{[Oxley_T][080M40][width].eps}}
   \caption[Variation of $T_\mr{int}$ and $T_1$ with width of cut; work material 080M40.]
   {Variation of $T_\mr{int}$ and $T_1$ with width of cut; work material 080M40.
   (a)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (b)~\mbox{$V = 75$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.
   (c)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.02$ mm$/$rev}.
   (d)~\mbox{$V = 200$ m$/$min}, \mbox{$t_1 = 0.1$ mm$/$rev}.}
   \label{fig:[Oxley_T][080M40][width]}
\end{figure}
\begin{figure}[h!]
   \center{\includegraphics{temp_exper_predic.eps}}
   \caption{Relationship between experimental $T_1$ values and
   predictive $T_1$ values from finite-difference computations.}
   \label{fig:exper_predic_T1}
\end{figure}

Another quantity of interest is the interface temperature $T_\mr{int}$. However, as
indicated in Sections~\ref{sec:instrument} and~\ref{sec:temp_predict}, although this
temperature was measured during the experiments, it was measured indirectly by means of
a thermocouple installed some distance below the interface (5~mm). The experimental
results from thermocouple 1 (temperature $T_1$) are shown in
\mbox{Figures~\ref{fig:[Oxley_T][speed]}--\ref{fig:[Oxley_T][080M40][width]}} along
with Oxley's model predictions of $T_\mr{int}$. Predictions of $T_\mr{int}$ are much
higher than experimental values of $T_1$, and they also seem to exhibit fairly
different trends. This difference in trends is more obvious in
Figures~\ref{fig:[Oxley_T][080A15][width]} and~\ref{fig:[Oxley_T][080M40][width]},
where $T_\mr{int}$ remains constant with $w$, but $T_1$ experimental increases with
increasing $w$. In Section~\ref{sec:temp_predict}, predictions of $T_1$ using the
finite-difference method showed that $T_1$ depends not only on $T_\mr{int}$, but also
on $w$ and $l_\mr{int}$. Subsequently, the same method was used here to determine $T_1$
as a function of $T_\mr{int}$, $l_\mr{int}$, and $w$ ($l_\mr{int}$ was predicted with
Oxley's model). Figure~\ref{fig:exper_predic_T1} shows the experimental values of $T_1$
plotted against the finite-difference predictions of $T_1$. Although with some scatter,
a linear relationship was observed between predicted and experimental values of $T_1$,
and, hence, a best-fitting straight line was plotted through the data points, giving
the following relationship:
\begin{eqnarray}
   \mbox{experimental }T_1 =
   0.586\times \mbox{predicted }T_1 - 31.4^\circ\mbox{C}
   \label{eq:T1_exper_predic}
\end{eqnarray}
The slope of the resulting line is not equal to one, meaning that the predictive
$T_1$ values are different but proportional to the experimental $T_1$ values. This is
due to the inaccuracy of the thermal constants $\kappa$ and $h$ (50~W$/$m-$^\circ$C
and 100~W$/$m$^2$-$^\circ$C, respectively) used during the finite-difference
computations. Subsequently, it was decided to correct $T_1$ predictions by applying
Equation~\ref{eq:T1_exper_predic}. These corrected $T_1$ predictions are plotted in
\mbox{Figures \ref{fig:[Oxley_T][speed]}--\ref{fig:[Oxley_T][080M40][width]}} along
with $T_1$ experimental values and $T_\mr{int}$ predictions. A fairly good agreement
is observed between predictive and experimental values of $T_1$.
Figure~\ref{fig:[Oxley_T][speed]} show that although $T_\mr{int}$ increases with $V$,
$T_1$ increases much slower with $V$. This is due to the fact that $l_\mr{int}$
decreases with increasing $V$, and, as shown in Section~\ref{sec:temp_predict}, this
tends to decrease $T_1$. Moreover, Figure~\ref{fig:[Oxley_T][feed]} show that $T_1$
seems to increases more rapidly with $t_1$ than $T_\mr{int}$ does. This is due to the
fact that $l_\mr{int}$ increases with $t_1$, and as shown in
Section~\ref{sec:temp_predict}, this contributes to the increase in $T_1$. Finally,
Figures~\ref{fig:[Oxley_T][080A15][width]} and~\ref{fig:[Oxley_T][080M40][width]}
show that although $T_\mr{int}$ remains constant with $w$, $T_1$ increases
considerably with $w$, since, as shown in Section~\ref{sec:temp_predict}, increasing
$w$ tends to increase $T_1$.

In conclusion, the previous analysis shows that the predictions of $R$ and $T_1$ by
means of Oxley's model (and finite-difference in the case of $T_1$) are in fairly
good agreement with their respective experimental counterparts. This gives a good
indication of the suitability of Oxley's model to simulating the mechanics of
machining operations.
