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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the objective of the present thesis is to study the generation of acoustic emission
(AE) during metal cutting, in this chapter, the experimental results presented in
Chapter~\ref{chap:results} (Tests~I--IV) are analysed, and the mechanics of the metal
cutting process is carefully studied. To start with, the experimental data is compared
with published results from other authors, followed by the developing of a rather simple
empirical model that relates AE power with the cutting parameters cutting speed and feed
rate, so that the overall relationship between AE power and cutting speed and feed rate
can be easily visualized; then published models that relate AE power to the cutting power
are evaluated against Test~I experimental results, and it is concluded that these models
are somewhat inadequate.

Moreover, in order to study the suitability of the theory formulated by Oxley and
co-workers~\cite{oxley89} to model metal cutting, predicted forces and temperatures are
compared with the experimental ones from Test~I and~IV, and since the experimental
temperature results were measured some distance bellow the cutting zone, the
finite-difference technique is used to determine the relationship between measured and
cutting temperatures from the tool-chip interface. It is demonstrated that Oxley's theory
is adequate to model metal cutting of carbon steel work materials.

Since it is demonstrated that plastic deformation is the main process by which
continuous-type AE is generated, Oxley's model is used to predict the basic parameters
governing plastic deformation in metal cutting, i.e\ strain, strain rate, flow stress, and
temperature. In order to retain only continuous-type AE, a technique called AE
mode~\cite{blum90}, which avoids the influence of bust-type AE, is used to calculate AE
power. Furthermore, it is concluded that AE power increases with primary strain and
secondary strain rate, but it decreases with cutting temperature. In addition, a measure
of the frequency content of AE, the mean frequency~\cite{rangwala91b}, is utilized, and it
is concluded that the frequency content of AE increases with strain rate, and it decreases
with temperature.

Finally, since dislocation motion is the basic process governing plastic deformation in
metals~\cite{hull84}, a qualitative model that predicts that AE is generated when
dislocations surmount obstacles at the typical high strain rates and temperatures
encountered in metal cutting is proposed. Good agreement is obtained between predictions
of the proposed model and experimental results from Test~I and~IV.


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

\subsection{Orthogonality}

The analysis of the experimental data from Tests~I--IV (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
(Chapter~\ref{chap:introd}). 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 of
tubular type, as required to ensure orthogonality in turning
(Chapter~\ref{chap:introd}). 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~I--IV ($0.4$~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 both AE level 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~I--IV 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 main cutting and feed forces obtained from the
orthogonal model (Figure~\ref{fig:forces}). Nevertheless, since $\cos{C_\mr{S}} \approx 1$
($\cos{(-5^\circ)} = 0.996$), and  $\sin{C_\mr{S}} \approx 0$ ($\cos{(-5^\circ)} =
-0.087$), the following simplifications were made: $t'_1 = t_1$, $w' = w$, $F'_\mr{Y} =
F_\mr{Y}$, and $F'_\mr{Z} = 0$; or, in other words, the side cutting angle of the tool was
assumed to be $0^\circ$.

\subsection{Comparison of Test I Results with Published Data}
\label{sec:compar_test_i}

As shown in Section~\ref{sec:procedure_test_i}, the importance of Test~I
is that it provides a general overview of the variation of AE, cutting
forces, vibrations, 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~I 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 $V$ bellow 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 type 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~I.

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 $T_1$ measure 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_i}, only one thermocouple was used during the course of
Test~I.} (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 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 $T_1$ values are
computed by using the finite-difference method, so that these values can compared with the
experimental $T_1$ values from Test~I.

\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~I 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 be attributed to the phenomenon of built-up edge
(BUE). Theses results are supported by published material that 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 for the lower
$w$ values, 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~I data with $V$,
$t_1$, and $w$. This parameter, which gives an indication of the AE 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~I 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 bellow the minimum $V$ value utilized in Test~I (50~m$/$min), and, therefore, an
agreement between Rangwala and Dornfeld and Test~I 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~I (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~I 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{Test II and Test III Results}

As stated in Sections~\ref{sec:procedure_test_i} and~\ref{sec:procedure_test_ii}, the main
objective of Tests~II and~III was to obtain AE data for continuous variation of $t_1$ and
$w$, respectively. Another objective was to investigate the generation of AE as both $t_1$
and $w$ take up very low values. However, the AE data obtained was noticed, although more
or less in agreement with Test~I data results, to be quite affected by random noise
signals. As discussed in Section~\ref{sec:procedure}, a different method from Test~I was
used to acquire the AE data from Tests~II and~III, and, in this section, the relative
failure of the results of AE from Test~II and~III is examined.

During Test~I data acquisition, AE data was acquired in sets of 8388608 samples; each of
these sets was posteriorly divided into eight subsets of 1048576 data points. During the
acquisition of Test~II and~III data, sets of 65536 AE data points were taken. In times
units, since the sampling rate of Test~I was 10~MS$/$s, and of Tests~II and~III was
5~MS$/$s, Test~I data sets corresponded to approximately 105~ms, whereas Test~II and~III
data sets correspond to approximately 13~ms. Therefore, the initial AE raw data sets from
Test~I contained 16 times more samples than Test~II and~III data sets. Moreover, as stated
in Sections~\ref{sec:results_ae}, \ref{sec:results_test_ii},
and~\ref{sec:results_test_iii}, the main method to process the AE raw data was by
computing the $I_\mr{mode}$. If the data sets are two small, the peak of the probability
density function of the previously full-wave rectified and low-pass filtered data will not
represent the continuous AE data level if the influence of a random 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, the $I_\mr{mode}$ will represent the continuous-type AE data
which is normally associated with the plastic deformation of the cutting process. This is
the reason why the AE results from Test~II and~III, presented in
Figures~\ref{fig:AE_feed_var} and~\ref{fig:AE_profile}, respectively, 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 about an incident with Test~III data results is 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 Figure~\ref{fig:AE_profile}d shows that, within a range of
width of cut values, the value of $I_\mr{mode}$ abruptly rises 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 $w$.]
   {Example of AE generation during chatter for varying $w$;
    080M40, $V = 200$~m$/$min, $t_1 = 0.1$~mm$/$rev.}
   \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 vibrations are 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~I experimental data
(Section~\ref{sec:results_test_i}, 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 AE mode amplitude ($I_\mr{mode}$) for different cutting conditions and work
materials. In this section, a more meaningful physical representation of the AE level was
utilized, the AE power amplitude ($I_\mr{power}$), 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 AE power and cutting speed in logarithmic scale.]
   {Relationship between AE 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 $I_\mr{power}$ with cutting speed ($V$) for feed rates ($t_1$)
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 as a
straight line of positive slope for each material, 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 AE power and feed rate in logarithmic scale.]
   {Relationship between AE 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 speed 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 a straight line of negative slope
for each material, suggesting a power function relationship between $I_\mr{power}$ and
$V$. 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 ($r^2$)\footnote{$r^2 =
{\sum_{i = 1}^{n} (\hat{y}_i - \overline{y})^2}/{\sum_{i = 1}^{n} (y_i -
\overline{y})^2}$, 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.} 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 $t_1$.
\begin{figure}[h!]
   \center{\includegraphics{3D_plots.eps}}
   \caption[Three-dimensional representation the AE power as a
   function of cutting speed and feed rate.]
   {Three-dimensional representation the AE 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}
Figure~\ref{fig:3d_ae_power} shows plots of the relationship stated by
Equation~\ref{eq:empirical_ae_power} for the four work materials.


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

\begin{figure}[h!]
   \center{\includegraphics{[AE_power][speed].eps}}
   \caption[Relationship between AE power and cutting power for
   varying cutting speed.]
   {Relationship between AE 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 AE power and cutting power for
   varying feed rate.]
   {Relationship between AE 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 AE power and cutting power for varying width of cut.]
   {Relationship between AE 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} states that the total cutting power is
proportional to the main cutting force and cutting speed, i.e.\ $\dot U = F_\mr{X}V$, and
that $\dot U$ is essentially the summation to 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 verify this relationship, the
experimental data presented in Section~\ref{sec:results_test_i} (Test~I 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
(\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, the 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 data 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
Equation~\ref{eq:ae_cut_power} relationship. 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. 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 test 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 the 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 $n
> 1$, and no clear relationship was observed between
$I_\mr{power}$ and $t_1$. 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} depended on $\alpha$. 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. 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 that the
range of the current Test~I experimental values (\mbox{0.01--0.2 mm$/$rev}), and,
therefore, as it can be seen in Figures~\ref{fig:[mode][feed]}, the level of AE tends
normally do increase with decreasing $t_1$ when $t_1$ is low, whereas the experimental
data provided by Rangwala and Dornfeld exhibited more or less constant AE level 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 bellow 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 technic 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; and, 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 guessed based on published values~\cite{holman02}; hence,
\mbox{$\kappa = 50$ W$/$m-$^\circ$C} and \mbox{$h = 100$ W$/$m$^2$-$^\circ$C}. 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 technic 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 the geometry, cutting forces, stresses,
temperatures, etc.\ of metal cutting \cite{oxley89}. The model requires as inputs the
cutting conditions $V$, $t_1$, $w$, $\alpha$, and $T_\mr{W}$ (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~I
(Section~\ref{sec:results_test_i}). However, only results for the carbon steel work
materials (080A15 and 080M40) were obtained since no flow stress data is given 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 calculated by
extrapolation. Besides, experimental results from Test~IV
(Section~\ref{sec:results_test_iv}) for variable workpiece surface temperature
($T_\mr{WS}$) are compared with prediction from Oxley's model, where, in this case,
$T_\mr{W}$ was assumed to be equal to values of $T_\mr{WS}$.

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

\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}
\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}

Figure~\ref{fig:[Oxley_R][speed]} shows the variation of the resultant cutting force ($R$)
with $V$; Figure~\ref{fig:[Oxley_R][feed]} show the variation of $R$ with $t_1$. The
predictions of $R$ are plotted along with the experimental results from Test~I. 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, it was noticed the occurrence of BUE, 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 criterium 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 criterium 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 criterium 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 bellow predictions occur mainly in the range of $V$ and $t_1$ that lead to
the prediction of BUE according to the above criterium.

\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:[temp][R_theor_exper]} shows the variation of $R$ with $T_\mr{WS}$.
Experimental data from Test~IV is plotted along with 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; 080A15.]
   {Variation of $T_\mr{int}$ and $T_1$ with width of cut; 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; 080M40.]
   {Variation of $T_\mr{int}$ and $T_1$ with width of cut; 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 bellow 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 thought 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 increase $T_1$. 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.
