%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Influence of Width of Cut on Acoustic Emission}
\label{sec:width_AE}

In orthogonal and semi-orthogonal cutting, the dimension of $w$ is totally independent
of the simplified two-dimensional geometry of the cutting process presented in
Figure~\ref{fig:cutting}, and, therefore, a cutting process with width of cut $Kw$
should act as if $K$ independent and equivalent cutting processes with width of cut $w$
were occurring alongside. Consequently, variables that are not directly dependent on
$w$, such as velocities, stresses, strains, strain rates, and temperatures, should not
be affect by $w$; on the other hand, parameters that are directly dependent of $w$,
such as forces and cutting power, should be a direct proportion of $w$. Following the
same reasoning, if an energetic level equivalent to $I_\mr{power}$ were detected by a
remote AE sensor for a process of width of cut $w$, and if the width of cut were
increased to $Kw$, intuitively, an energetic level $KI_\mr{power}$ should be detected
by the sensor. However, as discussed in Section~\ref{sec:preliminary_discussion}, this
is clearly not in agreement with Test~A results
(Figures~\ref{fig:[mode][080A15][width]}--\ref{fig:[mode][6082-T6][width]}) and also
and with published results~\cite{lan85,messaritis86,inasaki81,lan86,blum90}, since,
generally, although it is true that, for low values of $w$, the value of $I_\mr{power}$
increases with increasing $w$, its rate of increase decreases as $w$ increases, and for
high values of $w$, the value of $I_\mr{power}$ becomes more or less constant with
further increase in $w$. Moreover, Lan and Naerheim~\cite{lan85} proposed a pulsing
test which employed one transmitting and one receiving AE transducer at each end of a
tool holder, and they concluded that due to the phase interference between AE events at
high rates, the amplitude of continuous-type AE may increase (constructive
interference) or decrease (destructive interference), depending on the duration of the
AE event, event rate, frequency, and characteristic of the wave propagation path.
Messaritis and Borthwick~\cite{messaritis86} also suggested that an increase in volume
of deforming material, due to an increase in $w$, should result in a proportional
increase in the number of AE sources, and, consequently, in a proportional increase of
AE activity; however, Messaritis and Borthwick reasoned that a remotely mounted
transducer would not receive a signal proportional to the volume increase of AE
activity due to the effects of attenuation, scattering, and mode
conversions\footnote{Mode conversion occurs when a wave encounters an interface between
materials of different acoustic impedances and the incident angle is not normal to the
interface, as described in Section~\ref{sec:ae_theory}.} and combinations.
Consequently, it seems reasonable to conjecture that an increase in $w$ will lead to an
equivalent increase of AE activity at the generation source, but, at a remote location,
this increase in AE activity may not be sensed proportionally. It is not the aim of the
model that is going to be developed next to provide a truthful representation of the
process of AE generation and transmission that occurs in metal cutting, but only to
show how it may be possible that an increase in $w$ may not correspond to a
proportional increase of AE at a remote location in the cutting tool.

\begin{figure}[h]
   \center{\includegraphics{AE_simul_diagram.eps}}
   \caption{Diagram of simulation of AE generation and propagation.}
   \label{fig:AE_simul_diagram}
\end{figure}
The two-dimensional model presented in Figure~\ref{fig:AE_simul_diagram} assumes that
the generation source of AE is a line with length $w$ (corresponding to the width of
cut), and a transducer is at a point S, located perpendicular to one end of the
generation line at distance $l$. Since this model is just an oversimplification of
the real situation encountered in metal cutting, the following assumptions were made:
\begin{itemize}

   \item All AE events occur in phase along the generation line. This is agreement
   with Gillis~\cite{gillis72} (Section~\ref{sec:ae_theory}), who conjectured that if
   dislocation motion is the main generation mechanism of AE, each dislocation tends
   to move according to the stress field imposed by its moving neighbours, and,
   consequently, all dislocation lines should move cooperatively, so that all
   vibration waves produced by all moving dislocations are in phase.

   \item It is considered that the transmission medium (cutting tool) is infinite in
   size. As discussed in Section~\ref{sec:ae_theory}, when the AE waves meet the
   boundary of the cutting tool, the waves are reflected back into the tool,
   according to the law defined by Equation~\ref{eq:reflec_transm}. Therefore, since
   the AE waves are reflected every time they reach a boundary, not only the direct
   waves represented in Figure~\ref{fig:AE_simul_diagram} will arrive at point S, but
   also all reflected waves that get to point S\@. As a result, any attempt to
   realistically model AE propagation becomes extremely complex.

   \item Geometrical losses and attenuation are neglected. Since, as assumed
   previously, only direct waves are considered, propagation distances becomes less
   relevant, and, hence, geometrical losses and attenuation can be ignored because
   they are functions of propagation distance (Equations~\ref{eq:geometrical_lossses}
   and~\ref{eq:attenuation}).

\end{itemize}

Moreover, at every point of the generation zone, an AE wave (per unit length) is
generated according to the following equation:
\begin{eqnarray}
   I(t) = I_\mr{max}\sin{(2 \pi f t)}
   \label{eq:simul_AE_1}
\end{eqnarray}
where $I_\mr{max}$ is the maximum amplitude of the wave (per unit length), $t$ is the
time, and $f$ is the frequency of the wave. Every wave generated at a location $y$ in
the generation zone will travel a distance $d$ before reaching point S, where the
integrated effects of $I(t)$ between 0 and $w$ can be computed as follows:
\begin{eqnarray}
   I_\mr{S}(t) = I_\mr{max}\int_{y = 0}^{y = w}
   \sin \left[ 2 \pi f \left(t  - \frac{d}{c_\mr{L}} \right) \right]dy
   \label{eq:simul_AE_2}
\end{eqnarray}
where
\begin{eqnarray}
   d = \sqrt{l^2 + y^2}
   \label{eq:simul_AE_3}
\end{eqnarray}
and $c_\mr{L}$ is the speed of longitudinal waves travelling in the medium. A computer
routine produced in \mbox{MATLAB} (software from \mbox{The MathWorks}) was used to
solve the integral defined by Equation~\ref{eq:simul_AE_2} numerically. Furthermore,
numerical values had to be assigned to the constants defined in
Equations~\ref{eq:simul_AE_2} and~\ref{eq:simul_AE_3}, and this was done with
quantities in the same order of magnitude of those encountered in the real situation:
$c_\mr{L} = 6000$~m$/$s, which is the speed of longitudinal waves travelling in
steel~\cite{blitz71} (material of cutting tool); $l = 10$~mm, which is of the same
order of magnitude of the distance between the cutting zone and the two AE sensors
utilized in Tests~A and~B (Section~\ref{sec:instrument}).

\begin{figure}[h]
   \center{\includegraphics{AE_simul.eps}}
   \caption[Simulation of detected AE at a point with varying $w$.]
   {Simulation of detected AE at a point with varying $w$.
   (a)~$f = 100$~kHz. (b)~$f = 500$~kHz.~(c)~$f = 1000$~kHz.}
   \label{fig:AE_simul}
\end{figure}
Figure~\ref{fig:AE_simul} shows the root mean square value of the resultant wave at
point $S$ ($I_\mr{RMS}$) for different values of $w$ and three different frequencies
(100, 500, and 1000~kHz). It can be observed that when $f = 500$~kHz
(Figure~\ref{fig:AE_simul}b), although the value of $I_\mr{RMS}$ increases with $w$,
its rate of increase decreases with $w$, in a very similar fashion to the
experimental AE trends due to varying $w$. Basically, as the wave generation location
$y$ increases from 0 to $w$, the resultant wave arriving at point $S$ will be delayed
by $d/c_\mr{L}$ (Equation~\ref{eq:simul_AE_2}), since distance $d$ increases with $y$
(Equation~\ref{eq:simul_AE_3}). At a high enough $y$ value, the term $d/c_\mr{L}$
becomes significant, for, at point S, the phase difference of waves originating next
to $y = 0$ and next to $y = w$ starts to be high enough, so that their interaction
will not result in $I_\mr{RMS}$ values that are proportional to any further increase
in $w$, resulting, ultimately, in a decrease of $I_\mr{RMS}$ with increasing $w$, as
shown for $f = 1000$~kHz in Figure~\ref{fig:AE_simul}c.

In conclusion, the oversimplified model developed previously shows that when the
sources of different AE events are located some distance away from each other, as
when the generation volume is increased by increasing $w$ in metal cutting, since the
distance between the generation source and transducer may be different for different
AE events, the resultant waves may arrive at the sensor out of phase, and, hence, the
integration of all AE waves arriving at the transducer may be lower than what
intuitively expected. The values of $I_\mr{RMS}$ shown in Figure~\ref{fig:AE_simul}
were produced for simulated waves with a single frequency, whereas the experimental
results of $I_\mr{RMS}$ were computed from real waves with multiple frequencies
within a specific frequency range; however, as it can be seen from
Figure~\ref{fig:AE_simul}, the higher the frequency, the less linear becomes the
relationship between $I_\mr{RMS}$ and $w$, and consequently, if a multiple-frequency
wave were simulated, similar conclusions would be arrived at. Moreover, if the
boundaries of the cutting tool were considered, the reflected waves reaching the
sensor would also travel different distances if they were generated at different
locations, contributing, in the same manner, to a difference in phase of the arriving
waves. This model would be very complex, and geometrical losses and attenuation would
have to be taken into account.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Influence of Basic Cutting Parameters on Acoustic Emission}
\label{sec:parameters_AE}

In Section~\ref{sec:cut_ae_power}, it was shown that, in metal cutting, the energy
models that relate $I_\mr{power}$ proportional to the work rate of deformation, though
indicating some of the factors that may affect the generation of AE, generally fail to
give an universal relationship that corroborates well with experimental data.
Essentially, all the work of deformation ends up as thermal energy, but if considering
plastic deformation as a source of AE, a minor part of this work must be released in
the form of AE\@. Therefore, there must be some sort of relationship between AE and
$\dot{U}$. By examining Equation~\ref{eq:plastic_work}, one may conclude that the basic
parameters that influence $\dot{U}$ are: applied stress ($\sigma$), strain rate
($\dot{\epsilon}$), and volume of material participating into deformation ($v$). In
this section, a relationship between AE and these basic parameters is investigated, and
the parameters that contribute the most to the generation of AE are evaluated. First,
in Section~\ref{sec:energy_analysis}, AE is analysed in terms of its energetic
representation ($I_\mr{power}$), and followed, in Section~\ref{sec:frequency_analysis},
by an analysis of the frequency content of AE ($f_\mr{mean}$).

In Section~\ref{sec:ae_metal_cut}, it was concluded that if cutting is performed with
continuous chip formation, without BUE, and with a perfectly sharp tool, AE is assumed
to be mainly generated due to plastic deformation from the primary and secondary zones
of deformation (Figure~\ref{fig:ae_zones}a). However, Uehara and Kanda~\cite{uehara84}
measured AE signals simultaneously from both tool and workpiece sides and concluded
that AE measured at the tool side is mainly affected by the secondary zone of
deformation. Since all current experimental results of AE (Tests~A and~B) presented in
Chapter~\ref{chap:results} were performed with AE sensors located at the tool side
(Figure~\ref{fig:ae_sensors_location}), one can expect that the major contribution for
the current measured AE signals is also due to the effect of secondary zone
deformation. Besides, according to Oxley's theory~\cite{oxley89}, the tool-chip
interface is represented by a rectangular plastic zone with uniform shear stress
distribution, as represented in Figure~\ref{fig:secondary_oxley_zone}
\begin{figure}[h]
   \center{\includegraphics{secondary_oxley_zone.eps}}
   \caption[Simplified secondary shear zone geometry]
   {Simplified secondary shear zone geometry~\cite{oxley89}.}
   \label{fig:secondary_oxley_zone}
\end{figure}
for a tool with a 0$^\circ$ rake angle. Therefore, in such a simplified case, the
basic cutting parameters can be reduced to three constants: shear flow stress
($k_\mr{int}$), shear strain rate ($\dot{\gamma}_\mr{int}$), and deformation volume
($v_\mr{int} = l_\mr{int} \delta t_2 w$, where $\delta t_2$ is the thickness of the
secondary deformation zone).\footnote{In this case, the secondary work rate produced
by plastic deformation at the tool-chip interface can be derived from
Equation~\ref{eq:plastic_work}: $\dot{U}_\mr{int} = k_\mr{int} \dot{\gamma}_\mr{int}
v_\mr{int}$.} Moreover, as discussed in Section~\ref{sec:oxley_model}, Oxley's model
seems to give predictions of cutting forces and temperatures that are reasonably
accurate, i.e.\ similar trends are obtained between predictive and experimental
results, and the predictive values lie within the same order of magnitude of the
experimental ones. As a result, one can reasonably expect that the values of
$k_\mr{int}$, $\dot{\gamma}_\mr{int}$, and $v_\mr{int}$ (and the other cutting
parameters as well) can also be predicted by means of Oxley's model.

\subsection{Energy Analysis} \label{sec:energy_analysis}

Figure~\ref{fig:SR_k_v_AE} shows the variation of $I_\mr{power}$ with $k_\mr{int}$,
$\dot{\gamma}_\mr{int}$, and $v_\mr{int}$ for Test~A data, but only for work materials
080A15 (Figures~\ref{fig:SR_k_v_AE}a, \ref{fig:SR_k_v_AE}c, and \ref{fig:SR_k_v_AE}e)
and 080M40 (Figures~\ref{fig:SR_k_v_AE}b, \ref{fig:SR_k_v_AE}d, and
\ref{fig:SR_k_v_AE}f), since, as discussed in Section~\ref{sec:oxley_model}, Oxley's
model predictions are solely possible for carbon steels. No data for variable width of
cut was employed ($w = 1.2$~mm), and account for the influence of $w$ on the level of
AE was discussed in Section~\ref{sec:width_AE}. By observing Figure~\ref{fig:SR_k_v_AE}
and as discussed next, it can be concluded that none of the individual parameters can
be used alone to model $I_\mr{power}$:
\begin{figure}[h!]
   \center{\includegraphics{SR_k_v_AE.eps}}
   \caption[Variation of $I_\mr{power}$ with basic cutting parameters $k_\mr{int}$,
   $\dot{\gamma}_\mr{int}$, and $v_\mr{int}$.]
   {Variation of $I_\mr{power}$ with basic cutting parameters $k_\mr{int}$,
   $\dot{\gamma}_\mr{int}$, and $v_\mr{int}$.
   (a)~$I_\mr{power}$ against $k_\mr{int}$; 080A15.
   (b)~$I_\mr{power}$ against $k_\mr{int}$; 080M40.
   (c)~$I_\mr{power}$ against $\dot{\gamma}_\mr{int}$; 080A15.
   (d)~$I_\mr{power}$ against $\dot{\gamma}_\mr{int}$; 080M40.
   (e)~$I_\mr{power}$ against $v_\mr{int}$; 080A15.
   (f)~$I_\mr{power}$ against $v_\mr{int}$; 080M40.}
   \label{fig:SR_k_v_AE}
\end{figure}
\begin{description}

   \item [AE and strain rate (Figures~\ref{fig:SR_k_v_AE}a and~\ref{fig:SR_k_v_AE}b)]
   The level of AE increases with increasing $\dot{\gamma}_\mr{int}$, however, the data is
   very scattered, and it seems that there is another factor affecting the generation of
   AE\@.

   \item [AE and flow stress (Figures~\ref{fig:SR_k_v_AE}c and~\ref{fig:SR_k_v_AE}d)]
   The data is very scattered and no clear trend is observed between $I_\mr{power}$ and
   $k_\mr{int}$.

   \item [AE and volume (Figures~\ref{fig:SR_k_v_AE}e and~\ref{fig:SR_k_v_AE}f)]
   Surprisingly, a quite defined relationship seems to exist between $I_\mr{power}$ and
   $v_\mr{int}$: $I_\mr{power}$ decreases with increasing $v_\mr{int}$, like the
   relationship $I_\mr{power} \propto v_\mr{int}^n$, where $n < -1$, or, in other words,
   $I_\mr{power}$ increases very rapidly as $v_\mr{int}$ tends to zero. This is a very
   disconcerting relationship, since if the only parameter affecting AE generation was
   $v_\mr{int}$, when $v = 0$, no material was involved in cutting, and, therefore, no AE
   should be generated. For example,
   Figures~\mbox{\ref{fig:[mode][080A15][width]}--\ref{fig:[mode][6082-T6][width]}}, show
   that when $w$ decreases ($\dot{\gamma}_\mr{int}$ and $k_\mr{int}$ are kept constant, and
   only $v_\mr{int}$ is affected), $v_\mr{int}$ decreases, and $I_\mr{power}$ is also
   observed to decrease, especially when $w$ tends to zero, opposing the trend observed in
   Figures~\ref{fig:SR_k_v_AE}e and~\ref{fig:SR_k_v_AE}f. However, since $v_\mr{int}$ is
   also dependent on $l_\mr{int}$, $t_2$ (chip thickness), and $\delta$ (ratio between
   thickness of secondary deformation zone and $t_2$), each of these three variables must
   be investigated in relation to its contribution to AE generation.
\end{description}

A strong relationship between AE energy and strain rate has been evidenced by many
publications \cite{dornfeld80,messaritis86,blum90,rangwala91a}, i.e.\ $I_\mr{power}$
increases with increasing strain rate. This relationship can also be acknowledged by
the fact that $I_\mr{power}$ increases with increasing speed, which has been reported
by numerous authors
\cite{dornfeld80,kannatey-asibu81,messaritis86,blum90,rangwala91a,saini96} and also
confirmed by Test~A data results (Figure~\ref{fig:[mode][speed]}), since strain rate is
strongly dependent on speed. Messaritis and Borthwick~\cite{messaritis86} utilized
Oxley's model to predict shear strain rate ($\dot\gamma$) and concluded that since both
$\dot\gamma$ and $I_\mr{power}$ present similar trends for varying values of $V$,
$t_1$, and $w$, $I_\mr{power}$ might be mainly dependent on $\dot\gamma$, and the
relationship held by Equation~\ref{eq:messaritis_ae_model} was derived. However,
Messaritis and Borthwick did not specify whether the value of $\dot\gamma$ represents
the shear strain rate from the primary ($\dot\gamma_\mr{AB}$) or from the secondary
($\dot\gamma_\mr{int}$) zone of deformation. Therefore, it was decided to run Oxley's
model, and it was found that both $\dot\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$
increase with $V$, and $\dot\gamma_\mr{AB}$ decreases with $t_1$, which is in agreement
with Messaritis and Borthwick, whereas $\dot\gamma_\mr{int}$ increases with $t_1$,
which is in disagreement with Messaritis and Borthwick. As a result, it was concluded
that the value of $\dot\gamma$ presented by Messaritis and Borthwick must represent
primary deformation strain rate $\dot\gamma_\mr{AB}$.
\begin{figure}[h!]
   \center{\includegraphics{SR_AB_AE.eps}}
   \caption[Relationship between $\dot\gamma_\mr{AB}$ and $I_\mr{power}$.]
   {Relationship between $\dot\gamma_\mr{AB}$ and $I_\mr{power}$.
   (a)~080A15. (b)~080M40.}
   \label{fig:SR_AB_AE}
\end{figure}
Consequently, it was decided to plot $I_\mr{power}$ results from Test~A against
$\dot\gamma_\mr{AB}$, and, as shown in Figure~\ref{fig:SR_AB_AE}, though $I_\mr{power}$
increases with $\dot\gamma_\mr{AB}$, the results are very scattered. The reason for the
apparently well-behaved relationship between $I_\mr{power}$ and $\dot\gamma_\mr{AB}$
presented by Messaritis and Borthwick is due to the fact that their experiments were
performed for a more reduced range of cutting conditions ($V$ varied for an unique
$t_1$ value, and $t_1$ varied for an unique $V$ value) than the wider range of cutting
conditions utilized in Test~A experiments (Table~\ref{tab:speed_feed}).

In any case, as indicated before, the detected AE signals are thought to be mainly due to
secondary zone deformation, and, therefore, a relationship between $I_\mr{power}$ and
$\dot\gamma_\mr{int}$ (and also other secondary parameters) is of interest. Consequently,
any apparent relationship between AE and primary parameters, as proposed by Messaritis and
Borthwick~\cite{messaritis86}, is observed because there must be some sort of relationship
between primary and secondary parameters, which is evidenced by the two relationships
shown in Figure~\ref{fig:primary_secondary}.
\begin{figure}[h!]
   \center{\includegraphics{primary_secondary.eps}}
   \caption[Relationship between primary and secondary parameters.]
   {Relationship between primary and secondary parameters.
   (a)~$\dot\gamma_\mr{int}\delta$ against $\dot\gamma_\mr{AB}$.
   (b)~$\delta$ against~$\gamma_\mr{AB}$.}
   \label{fig:primary_secondary}
\end{figure}
Consequently, as an approximately linear and material independent relationship seems
to exist between the primary parameter $\dot\gamma_\mr{AB}$ and the product of the
secondary parameters $\dot\gamma_\mr{int}\delta$
(Figure~\ref{fig:primary_secondary}a), and as $\delta$ appears to form a material
dependent relationship with the strain occurring at AB ($\gamma_\mr{AB}$) ($\delta$
and its rate of increase increase with $\gamma_\mr{AB}$, as shown in
Figure~\ref{fig:primary_secondary}b), it was decide to investigate an empirical
relationship of the type
\begin{eqnarray}
   I_\mr{power} = K\gamma_\mr{AB}^{n_1}\dot\gamma_\mr{int}^{n_2}
   \label{eq:AE_S_AB_SR_int}
\end{eqnarray}
The values of constants $K$, $n_1$, and $n_2$ were calculated by non-linear regression
with the aid of computer software DataFit from Oakdale Engineering and are presented in
Table~\ref{tab:AE_S_AB_SR_int}.
\begin{table}[h]
   \caption{Equation~\ref{eq:AE_S_AB_SR_int} constants found
   by non-linear regression analysis.}
   \label{tab:AE_S_AB_SR_int}
   \center{\begin{tabular}{cccc}
      \hline
      $K$                     & $n_1$ & $n_2$ & $r^2$ \\
      \hline
      $4.421 \times 10^{-14}$ & 4.511 & 2.385 & 0.994 \\
      \hline
   \end{tabular}}
\end{table}

\begin{figure}[h!]
   \center{\includegraphics{AE_exper_predic.eps}}
   \caption{Relationship between experimental AE values and predictive empirical AE
   values.}
   \label{fig:AE_exper_predic}
\end{figure}
The resultant coefficient of multiple determination $r^2$ is very close to one,
meaning that Equation~\ref{eq:AE_S_AB_SR_int} is a good empirical model, especially
because this relationship is material independent and, therefore, valid for both
carbon steel work materials, as shown in Figure~\ref{fig:AE_exper_predic}. From the
preceding relationship, the following conclusions can be attained:
\begin{itemize}

   \item As discussed before and presented by several authors, AE is strongly dependent
   on strain rate; $I_\mr{power}$ and its rate of increase increase with
   $\dot\gamma_\mr{int}$.

   \item The amount of deformation work previously done to the material plays an
   important role on the generation of AE\@. Equation~\ref{eq:AE_S_AB_SR_int} shows
   that $I_\mr{power}$ and its rate of increase increase with the amount of strain
   that has been done to the material due to primary deformation (represented by
   $\gamma_\mr{AB}$).\footnote{Actually, according to the parallel shear zone
   theory~\cite{oxley63}, only half of the total primary shear strain occurs at AB
   (Figure~\ref{fig:primary_zone}), so that the total shear strain occurring in the
   primary shear zone of deformation equals $2\gamma_\mr{AB}$, but this does not
   alter the conclusions derived from Equation~\ref{eq:AE_S_AB_SR_int}.}

   \item Equation~\ref{eq:AE_S_AB_SR_int} indicates that the amount of AE generated is
   not dependent on work material properties. Therefore, although the flow stress
   properties in the primary ($k_\mr{AB}$) and secondary ($k_\mr{int}$) deformation
   zones play an important role in the macroscopic physics of cutting (such as geometry,
   stresses, and cutting power), it seems not to have a direct impact on the
   generation of AE\@. However, the two materials utilized are two carbon steels and,
   therefore, similar in structure. Therefore, it is very likely that
   Equation~\ref{eq:AE_S_AB_SR_int} will not be valid when AE data from a wider range
   of dissimilar materials is utilized.

\end{itemize}

\subsection{Frequency Analysis} \label{sec:frequency_analysis}

\begin{figure}[h!]
   \center{\includegraphics{Meanfreq.eps}}
   \caption[Variation of $f_\mr{mean}$ with $\dot\gamma_\mr{int}$ and $T_\mr{int}$.]
   {Variation of $f_\mr{mean}$ with $\dot\gamma_\mr{int}$ and $T_\mr{int}$.
   (a)~$f_\mr{mean}$ against $\dot\gamma_\mr{int}$.
   (b)~$T_\mr{int}$ against $\dot\gamma_\mr{int}$.
   (c)~$f_\mr{mean}$ against $T_\mr{int}$}.
   \label{fig:MeanFreq}
\end{figure}
The plots shown is Figures~\ref{fig:[freq][speed]}--\ref{fig:[freq][6082-T6][width]}
show that although $f_\mr{mean}$ remains more or less unchanged with $t_1$ and $w$, it
decreases with $V$. At a first glance, this decrease of $f_\mr{mean}$ with $V$ was
quite surprising, since, as $V$ increases the speed by which deformation takes place
(strain rate), it was expected to move the spectra of the AE signals towards higher
frequency levels. However, as pointed out by Rangwala and Dornfeld~\cite{rangwala91b}
and briefly mentioned in Section~\ref{sec:compar_test_A}, increasing $V$ results in
more heat generation and, thus, in a rise of tool-chip interface temperature; as this
rise in temperature tends to decrease $f_\mr{mean}$, at a high enough value of $V$, the
effect of the temperature rise offsets the effect of strain rate. As a result, it was
decided to investigate the effect of both $\dot\gamma_\mr{int}$ and $T_\mr{int}$
(values obtained by means of Oxley's theory~\cite{oxley89}) on the value of
$f_\mr{mean}$.

Figure~\ref{fig:MeanFreq}a shows the variation of $f_\mr{mean}$ with
$\dot\gamma_\mr{int}$ for work materials 080A15 and 080M40. The plot shows that,
though the data is very scattered, generally, there is a slight decrease of
$f_\mr{mean}$ with $\dot\gamma_\mr{int}$. According to the previous reasoning, the
increase of $\dot\gamma_\mr{int}$ must also be accompanied by an increase of
$T_\mr{int}$, whose trend is confirmed by the plot shown in
Figure~\ref{fig:MeanFreq}b. Therefore, as confirmed by the relationship shown in
Figure~\ref{fig:MeanFreq}c, an increase of $T_\mr{int}$ will lead to a decrease of
$f_\mr{mean}$, independently on the variation of $\dot\gamma_\mr{int}$. However,
according to Rangwala and Dornfeld~\cite{rangwala91b}, for low values of $V$ (below
60~m$/$min), where the cutting temperatures are low, the effects of strain rate
dominate, and thus, at this range of $V$ values, $f_\mr{mean}$ is expected to
increase with $\dot\gamma_\mr{int}$. Moreover, the relationship between $f_\mr{mean}$
and the other basic secondary parameters previously analysed for AE power
($k_\mr{int}$ and $v_\mr{int}$) were also analysed, but no clear trends could be
distinguished: $f_\mr{mean}$ remains more or less unchangeable with $k_\mr{int}$ and
$v_\mr{int}$, and the data is very scattered.

\begin{figure}[h!]
   \center{\includegraphics{Meanfreq_S_AB.eps}}
   \caption{Variation of $f_\mr{mean}$ with $\gamma_\mr{AB}$.}
   \label{fig:MeanFreq_S_AB}
\end{figure}
Figure~\ref{fig:MeanFreq_S_AB} shows that $f_\mr{mean}$ is almost unaffected by
different values of primary strain $\gamma_\mr{AB}$; however, the data is very
scattered. In fact, any relationship between $f_\mr{mean}$ and any cutting cutting
parameter leads always to very high scatter. This is due to the fact that as AE sensors
work in their range of resonance, the frequency content of the detected AE signals are
dominated by this resonance frequency. The sensor used during the course of Test~A
(MICRO~80S) works with a peak frequency frequency of 324~kHz (from calibration
certificate). The examples shown in Figures~\ref{fig:[080M40][spectrum]}
and~\ref{fig:[304S15][spectrum]}a show that there is a small peak at about this
frequency. However, most AE energy is detected on the low frequency range of the
spectrum (approximately below 200~kHz), and one likely explanation for this occurrence
may be to the fact that between the generation source of AE (tool-chip interface) and
the sensor, there are three interfaces of dissimilar materials (interface between
cutting insert and shim, between cutting insert and tool holder, and between tool
holder and AE sensor), and when AE waves cross these interfaces, the higher frequencies
are the most attenuated (according to Equation~\ref{eq:attenuation_coeff}, the
attenuation coefficient $a$ increases sharply with increasing frequency).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Temperature Effect on  Acoustic Emission} \label{sec:ae_temp}

Figure~\ref{fig:T_int_AE} shows that when $I_\mr{power}$ from Test~A experiments is
plotted against predictions of $T_\mr{int}$ from Oxley's theory~\cite{oxley89}, no
clear trend can be distinguished, and the results are very scattered. However,
Figures~\ref{fig:[temp_var][080A15][mode]} and~\ref{fig:[temp_var][080M40][mode]} in
Section~\ref{sec:results_test_B} show that, by using Test~B experimental data from both
material 080A15 and 080M40, the level of AE is observed to decrease with workpiece
surface temperature ($T_\mr{WS}$), and since as shown in Figure~\ref{fig:T_W_T_int},
\begin{figure}[h!]
   \center{\includegraphics{T_int_AE.eps}}
   \caption{Variation of $I_\mr{power}$ with $T_\mr{int}$.}
   \label{fig:T_int_AE}
\end{figure}
a linear relationship is obtained between $T_\mr{WS}$ and predictions of $T_\mr{int}$
from Oxley's model (as discussed in Section~\ref{sec:oxley_model}, $T_\mr{W} =
T_\mr{WS}$), $I_\mr{power}$ in Figure~\ref{fig:T_int_AE} should also decrease with
$T_\mr{int}$. Consequently, in this section, it was decided to study the influence of
cutting temperature on AE using both Test~A and Test~B results and also to find out the
apparent disagreement between Test~A and Test~B results.
\begin{figure}[h!]
   \center{\includegraphics{T_W_T_int.eps}}
   \caption[Variation of $T_\mr{int}$ with $T_\mr{WS}$.]
   {Variation of $T_\mr{int}$ with $T_\mr{WS}$.
   (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:T_W_T_int}
\end{figure}

Since the empirical relationship for $I_\mr{power}$ derived in
Section~\ref{sec:energy_analysis} (Equation~\ref{eq:AE_S_AB_SR_int}) is only dependent
on $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$, and not directly dependent on cutting
temperature $T_\mr{int}$, it was decided to find out how Test~A values of
$\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ vary with $T_\mr{int}$; this is shown in
Figure~\ref{fig:T_int_S_AB_SR_int} for both carbon steel work materials. Although the
data is very scattered, it can be observed that $\gamma_\mr{AB}$ and its rate of
decrease reduce with increasing $T_\mr{int}$, and $\dot\gamma_\mr{int}$ and its rate of
increase rise with increasing $T_\mr{int}$. Therefore, according to
Equation~\ref{eq:AE_S_AB_SR_int}, although increasing $T_\mr{int}$ should result in a
decrease of $I_\mr{power}$ due to the reduction of $\gamma_\mr{AB}$, in reality, the
effect of increasing $\dot\gamma_\mr{int}$ with $T_\mr{int}$ should dominate, and
$I_\mr{power}$ should, therefore, increase, since the rate by which
$\dot\gamma_\mr{int}$ is increased rises with $T_\mr{int}$, whereas the rate by which
$\gamma_\mr{AB}$ is decreased falls with $T_\mr{int}$.
\begin{figure}[h!]
   \center{\includegraphics{T_int_S_AB_SR_int.eps}}
   \caption[Variation of $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ with $T_\mr{int}$.]
   {Variation of $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ with $T_\mr{int}$.
   (a)~080A15. (b)~080M40.}
   \label{fig:T_int_S_AB_SR_int}
\end{figure}

It must be noted that the value of $T_\mr{int}$ is a result of the heat produced due to
plastic deformation during metal cutting, and, therefore, $T_\mr{int}$ is dependent on
the parameters which control plastic deformation, which clearly include
$\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$; however, $T_\mr{int}$ itself is also an
important parameter governing the physics of plastic deformation. This is a consequence
of the fact that in metal cutting, all variables are interrelated. Hence, if $T_\mr{W}$
is kept constant, it is believed that although increasing $T_\mr{int}$ contributes to a
lowering of $I_\mr{power}$, this effect is masked by the other two variables that are
also observed to vary with increasing $T_\mr{int}$, i.e.\ $\gamma_\mr{AB}$ and
$\dot\gamma_\mr{int}$. However, with Test~B results, when the value of $T_\mr{W}$ is
varied, an extra amount of heat is added to the cutting process, and $T_\mr{int}$ is
not exclusively dependent on plastic deformation, and, therefore, as it can be seen in
Figures~\ref{fig:[temp_var][080A15][mode]} and~\ref{fig:[temp_var][080M40][mode]}, the
level of AE is observed to decrease with increasing $T_\mr{W}$, and with $T_\mr{int}$
as well, since $T_\mr{int}$ is proportional to $T_\mr{W}$. Moreover, as shown in
Figure~\ref{fig:T_W_S_AB_S_int}, it should be noted that when $T_\mr{W}$ increases,
$\gamma_\mr{AB}$ decreases and $\dot\gamma_\mr{int}$ remains constant
(Figure~\ref{fig:T_W_S_AB_S_int}a), $\dot\gamma_\mr{int}$ increases and
$\gamma_\mr{AB}$ remains constant (Figure~\ref{fig:T_W_S_AB_S_int}c), or both
$\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ decrease (Figure~\ref{fig:T_W_S_AB_S_int}b).
In all these three cases, the variations of $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$
also contribute to the lowering of $I_\mr{power}$ together with that due to the
increase of $T_\mr{int}$.
\begin{figure}[h!]
   \center{\includegraphics{T_W_S_AB_S_int.eps}}
   \caption[Variation of $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ with $T_\mr{WS}$.]
   {Variation of $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ with $T_\mr{WS}$.
   (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:T_W_S_AB_S_int}
\end{figure}

In conclusion, three variables seem to be the most significant factors affecting the
generation of AE: $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$, which are expected to
increase the level of AE, resulting in Equation~\ref{eq:AE_S_AB_SR_int} derived in
Section~\ref{sec:energy_analysis}; and cutting temperature $T_\mr{int}$, which is
expected to lower the level of AE\@. However, since $T_\mr{int}$ is due to heat
generated during the process of plastic deformation, $T_\mr{int}$ is related to the
other process parameters like $\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$, and,
therefore, its effect on the generation of AE is masked by the influence of
$\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$, resulting in the indistinguishable
relationship shown in Figure~\ref{fig:T_int_AE}. However, if $T_\mr{int}$ is
externally modified by cutting with different values of $T_\mr{W}$, since
$\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ are not the only parameters affecting the
value of $T_\mr{int}$, the level of AE is very visibly seen to decrease with
$T_\mr{int}$, as, in this case, when $T_\mr{W}$ is varied,
Equation~\ref{eq:AE_S_AB_SR_int} should not be valid any longer, since the effects of
$T_\mr{int}$  due to varying $T_\mr{W}$ must also be taken into account.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dislocation Motion as the Origin of Acoustic Emission}
\label{sec:disloc_AE}

In Section~\ref{sec:ae_theory}, it is postulated that the two major sources of AE in
metals are the motion of dislocations and also the development of cracks;
continuous-type AE is associated with dislocation motion, whereas higher amplitude
burst-type AE with cracking processes. Moreover, as discussed in
Section~\ref{sec:ae_metal_cut}, since metal cutting does not only involve plastic
deformation, but also cracking due to chip and cutting tool fracture, as well as other
burst-type AE emission processes, such as collision and entangling of chips, the
resulting AE signal should be a combination of continuous-type emissions with randomly
appearing high amplitude bursts. In principle, continuous AE should be generated if the
cutting process produces continuous chips without BUE, and if the cutting tool is
perfectly sharp. However, during the course of Test~A cutting experiments, though
continuous chips were always formed, BUE was observed for low values of $V$ and $t_1$
(as illustrated in Section~\ref{sec:oxley_forces}, BUE was also predicted by means of
Oxley's model); besides, other burst-type generating processes, like accidental
collision, entangling, and fracture of chips, as well as minor tool fracture, could not
be fully prevented from happening. The example presented in
Figure~\ref{fig:[080M40][time]} shows a typical Test~A AE signal that was formed by a
continuous-type wave superimposed by two large bursts. Consequently, since the
technique described in Section~\ref{sec:ae_theory} to calculate the mode of the
full-wave rectified and low-pass filtered AE signal is insensitive to outlying values,
like randomly appearing bursts, the outcome ($I_\mr{mode}$) should only reflect the
energy due to the mechanisms that generate continuous emissions. Therefore, if
dislocation motion is the mechanism responsible for plastic deformation in metals
(Appendix~\ref{chap:disloc}), and if continuous-type AE is due to plastic deformation,
there must be a relationship between $I_\mr{mode}$ and dislocation motion. In this
section, the process by which dislocation movement occurs during plastic deformation in
metal cutting are firstly investigated; afterwards, the process by which dislocation
movement may cause AE is analysed; finally, experimental AE data from Test~A is used to
evaluate and discuss the relationship between AE and dislocation motion during metal
cutting.

\subsection{Mechanics of Dislocation Motion in Metal Cutting}
\label{sec:mech_disloc_cut}

During metal cutting, plastic deformation occurs at very high strain rates and
temperatures. By applying Test~A cutting conditions (Table~\ref{tab:speed_feed}) to
Oxley's model~\cite{oxley89}, predictions of strain rates and temperatures were
observed to be in the ranges presented in Table~\ref{tab:max_min_SR_T}.
\begin{table}[h!]
   \caption[Minimum and maximum values of $\dot\gamma_\mr{int}$ and $T_\mr{int}$ for
   Test~A cutting conditions; work materials 080A15 and 080M40.]
   {Minimum and maximum values of $\dot\gamma_\mr{int}$ and $T_\mr{int}$ for Test~A
   cutting conditions (Table~\ref{tab:speed_feed}); work materials 080A15 and 080M40.}
   \label{tab:max_min_SR_T}
   \center{\begin{tabular}{ccc}
      \hline
      material & $\dot\gamma_\mr{int}$, s$^{-1}$ & $T_\mr{int}$, K \\
      \hline
      080A15   &                      700--62400 &       780--1170 \\
      080M40   &                    2900--146200 &       810--1260 \\
      \hline
   \end{tabular}}
\end{table}

According to Section~\ref{sec:disloc_flow_stress} in Appendix~\ref{chap:disloc}, at
temperatures approximately above $0.5T_\mr{M}$, where $T_\mr{M}$ is the melting
temperature in degrees~K, diffusion controlled mechanisms become significant, movement
of dislocations by climb becomes possible, and certain dislocations previously hindered
at obstacles start to move, resulting in a higher density of mobile dislocations
($\rho_\mr{MD}$). In this regime, named diffusion controlled, flow stress decreases
with increasing temperature and decreasing strain rate~\cite{kumar68,kumar69}. Since
for carbon steels \mbox{$T_\mr{M} \approx 1730$ K}, and by observing that only for the
lowest values of $V$ and $t_1$ ($T_\mr{int}$ values in
Figures~\ref{fig:[Oxley_T][speed]} and~\ref{fig:[Oxley_T][feed]} must be converted into
degrees~K), $T_\mr{int}$ does not reach $0.5T_\mr{M}$, it can be concluded that, for
practically all Test~A cutting conditions with work materials 080A15 and 080M40,
secondary zone plastic deformation results from dislocation movement where diffusion
mechanisms are present.

Moreover, in Section~\ref{sec:disloc_flow_stress}, it is also stated that at high
strain rates (above 1000~s$^{-1}$~\cite{kumar69}), opposing viscous damping forces
become the dominant mechanism governing dislocation movement. In this regime, named
viscous damping, the flow stress is a combination of the stress needed to overcome
dislocation obstacles like forest dislocations plus the stress needed to overcome the
damping forces, where, according to Equation~\ref{eq:disloc_stress_2}, the latter is a
linear function of strain rate~\cite{kumar68,kumar69}. By analysing the very high
$\dot\gamma_\mr{int}$ values presented in Table~\ref{tab:max_min_SR_T}, one can
conclude that cutting operations with the 080A15 and 080M40 work materials and Test~A
conditions involve secondary zone deformation that is governed by dislocations moving
in the viscous damping regime.

\subsection{Model of Acoustic Emission due to Dislocation Motion}
\label{sec:ae_disloc_model}

According to Section~\ref{sec:disloc_flow_stress}, in the viscous damping regime, the flow
stress can be expressed as:
\begin{eqnarray}
   k = \tau_\mr{B} + \tau_\mr{D}
   \label{eq:k}
\end{eqnarray}
where $\tau_\mr{B}$ is the stress required to overcome the forces posed by
dislocation obstacles, and $\tau_\mr{D}$ is the stress required to overcome the
damping forces. Therefore, on a microscopic level, two forces oppose the movement of
a dislocation: the damping force $F_\mr{D}$ and the force due to dislocation
obstacles like forest dislocations $F_\mr{B}$. Force $F_\mr{D}$ is proportional to
dislocation speed ($V_\mr{D}$) and, per unit length of dislocation line, is given by
\begin{eqnarray}
   F_\mr{D} = BV_\mr{D}
   \label{eq:F_D}
\end{eqnarray}
where $B$ is the dislocation damping coefficient. As shown in
Figure~\ref{fig:disloc_dynamics},
\begin{figure}[h]
   \center{\includegraphics{disloc_dynamics.eps}}
   \caption[Forces opposing dislocation movement.]
   {Forces opposing dislocation movement~\cite{hull84}.}
   \label{fig:disloc_dynamics}
\end{figure}
$F_\mr{B}$ increases when the dislocation position ($x$) gets near an obstacle:
$F_\mr{B}$ starts to rise at some position $x^1$ in the vicinity of the obstacle,
reaching a maximum $F_\mr{B}^\mr{max}$ at an intermediate point $x^\mr{max}$ between
$x^1$ and $x^2$; after $x^\mr{max}$ has been reached, $F_\mr{B}$ decreases until
position $x^2$ is arrived at. As a result, the total force acting on the dislocation is
given by
\begin{eqnarray}
   F = F_\mr{B} + F_\mr{D}
   \label{eq:F}
\end{eqnarray}
If there were no barriers to dislocation motion, the only force opposing the movement
of a dislocation would be $F_\mr{D}$, which is proportional to $V_\mr{D}$. Since,
according to Kumar~\cite{kumar69}, at very high strain rates, $V_\mr{D}$ is governed
by constant $B$ in Equation~\ref{eq:F_D}, and  if the conditions of deformation
(e.g.\ applied stress and temperature) are kept constant, $V_\mr{D}$ will remain
steady, and so will $F_\mr{D}$. However, in real crystals, dislocation obstacles are
always present (Section~\ref{sec:disloc_obstacles}) and do oppose dislocations as
they interact with them. However, according to Kumar, in the viscous damping region,
the time a dislocation is hindered by an obstacle is negligible when compared to the
time spent to travel between obstacles (the applied stress is high enough to overcome
the obstacles instantaneously), and, therefore, $V_\mr{D}$ can still be assumed as
constant. Moreover, since an obstacle imposes a force $F_\mr{B}$ against a moving
dislocation, a reaction force, with the same amplitude but opposite direction, will
be imposed by the dislocation on the obstacle, and, consequently, at every instant,
the energy transmitted by the dislocation to the obstacle per unit length of
dislocation line is given by:
\begin{eqnarray}
   \dot{U}_\mr{B} = F_\mr{B}V_\mr{D}
   \label{eq:U_B}
\end{eqnarray}
As $V_\mr{D}$ remains constant, $\dot{U}_\mr{B}$ will be proportional to $F_\mr{B}$.
Therefore, every time a dislocation surmounts an obstacle, potential strain energy
$\dot{U}_\mr{B}$ will be stored until position $x^\mr{max}$ is reached, and, between
$x^\mr{max}$ and $x^2$, $\dot{U}_\mr{B}$ will be released; this process is very
rapid, since $V_\mr{D}$ is high and dislocation obstacles are narrow~\cite{hull84}.

As stated in Section~\ref{sec:ae_theory}, the generation of an AE event is attributed
to the rapid release of strain energy when some part of the material is stressed up to
its maximum strength, so that the resulting AE wave will be formed due to the
combination of a series of such events. The theory proposed by Gillis~\cite{gillis72},
presented in Section~\ref{sec:ae_theory}, states that an AE wave is generated when a
dislocation that is moving between two minimum energy position in the lattice reaches
its middle position of maximum energy, since strain energy is then rapidly released;
however, this can only be valid for perfect crystals, without dislocation obstacles,
and when dislocations do not intercept other dislocations. For real crystals, like the
materials utilized in Test~A, other mechanism has to be suggested. Consequently, it is
proposed here that the release of strain energy when a dislocation surmounts an
obstacle is the main mechanism by which the detected AE is generated during plastic
deformation of metal cutting operations, since as a dislocation surpasses the obstacle,
as $\dot{U}_\mr{B}$ decreases, strain energy is released, the lattice vibrates, and an
AE pulse is emitted. The amplitude and frequency content of the emitted pulse will be
dependent on the force profile between $x^1$ and $x^2$ shown in
Figure~\ref{fig:disloc_dynamics}. Intuitively, the faster $F_\mr{B}$ changes, the
higher the frequency content of the emitted AE wave, and the higher
$F_\mr{B}^\mr{max}$, the higher the amplitude of AE\@. Moreover, since, as proposed by
Ivanov~\cite{ivanov84} (Section~\ref{sec:ae_theory}), AE is formed due to a series of
discrete pulses, the characteristic and shape of which is dependent on the physical
mechanisms of each individual event, the detected AE in metal cutting will,
consequently, be a function of the average effect of a large number of AE pulses
generated due to the release of localized strain energy as dislocations surpass
obstacles. However, the AE signal detected at the sensor will be a function of several
factors, such as the density of obstacles ($\rho_\mr{B}$), relationship of $F_\mr{B}$
against $x$ between $x^1$ and $x^2$, velocity $V_\mr{D}$, transmission wave path
between AE source and sensor, and instrumentation. A qualitative description of the
most relevant factors that affect the resulting acquired AE is carried out next:

\begin{description}

   \item [Relationship between dislocation and obstacle] As stated before, the shape of
   the $F_\mr{B}$-curve versus dislocation position $x$ will dictate the frequency content
   and energy for each AE event, i.e.\ the steeper the curve, the higher the frequencies,
   and the larger $F_\mr{B}^\mr{max}$, the larger the energy content of the emitted AE
   pulse. It is out of the scope of the current work to investigate the exact relationship
   between $F_\mr{B}$ and $x$; however, as discussed in Section~\ref{sec:mech_disloc_cut},
   since the temperatures involved in metal cutting are high, diffusion mechanisms are
   present, and, according to Appendix~\ref{chap:disloc}, in this regime, increasing
   temperature will facilitate dislocations to surmount obstacles, resulting in a lowering
   of $F_\mr{B}^\mr{max}$. A direct consequence will be the reduction of the energy level
   of the AE pulses, and, since the $F_\mr{B}$-curve becomes less steep, the frequency
   content of the AE events will be reduced as well.

   \item [Density of obstacles] Since all AE pulses contribute to the total detected AE
   wave, if more obstacles are present (higher $\rho_\mr{B}$), more AE pulses are
   generated, and, hence, more AE is generated. Moreover, according to
   Appendix~\ref{chap:disloc}, as plastic work contributes to the generation of new
   dislocations, the density of dislocations ($\rho_\mr{D}$) is increased and thus the
   number of forest dislocations. As forest dislocations are the main source of
   dislocation obstacles, increasing plastic strain ($\gamma$) will lead to an increase in
   the energy level of AE\@. However, as the shape of the $F_\mr{B}$-curve is not changed,
   $\gamma$ is not expected to directly alter the frequency content of AE\@.

   \item [Dislocation speed] As $V_\mr{D}$ increases, according to Equation~\ref{eq:U_B},
   $\dot{U}_\mr{B}$ also increases, resulting in a direct rise in the energy content of
   the generated AE pulses. As strain rate ($\dot{\gamma}$) is a linear function of
   $V_\mr{D}$ (Equation~\ref{eq:disloc_strain_rate}), increasing $\dot{\gamma}$ is
   expected to increase the energy of AE\@. Moreover, as increasing $V_\mr{D}$ will result
   in a faster release of strain energy, the frequency content of the generated AE pulses
   is also expected to increase.

   \item [Transmission path, sensor, and instrumentation] An important factor to take into
   consideration is the distortion caused by the transmission medium, sensor coupling and
   sensitivity, and instrumentation
   to the propagating AE waves. As discussed in Section~\ref{sec:ae_theory}, amplitude
   attenuation is a function of frequency, and since a high frequency wave will be more
   attenuated than a lower one, if, at the source, the AE pulses shift towards higher
   frequencies, a lower AE level may be detected at the sensor. Sensor sensitivity is
   another factor that may be taken into account, since amplitudes of AE waves of
   different frequencies arriving at the sensor may be detected in different ways (waves
   whose frequency content is closer to the resonance of the sensor will be more
   amplified). Moreover, the instrumentation is also of major importance, since, for
   instance, if the energy level of the AE pulses within a certain frequency range increases,
   and if there is an electronic filter excluding signals within that frequency range, the frequency
   shift of the AE pulses will not be detected.

\end{description}

In summary, dislocation movement during a typical metal cutting processes is controlled by
the viscous damping regime, and by assuming that the energy that is released when a
dislocation surmounts an obstacle is the basic mechanism for AE generation, the energy
level of the detected AE is expected to increase with $\gamma$ and $\dot{\gamma}$, but it
is expected to decrease with temperature. Moreover, the frequency content of AE is
predicted to increase with $\dot{\gamma}$, to decrease with temperature, and to remain
unchanged with $\gamma$. However, due to the complex effects of the transmission path,
sensor, and instrumentation to the AE signal, at this stage, only a qualitative analysis
is accomplished.

\subsection{Evaluation of Dislocation Model}
\label{sec:eval_disloc_model}

In this section, the conclusions achieved from the qualitative model proposed in
Section~\ref{sec:ae_disloc_model} are going to be evaluated against the energy and
frequency analysis carried out with the experimental data from Test~A
(Section~\ref{sec:parameters_AE}). Afterwards, the influence of temperature on the
generation of AE is evaluated with the aid of Test~B results
(Section~\ref{sec:ae_temp}). Finally, published material is evaluated and compared with
the model derived in Section~\ref{sec:ae_disloc_model}.

\subsubsection{Acoustic Emission Energy}

In Section~\ref{sec:energy_analysis}, according to Equation~\ref{eq:AE_S_AB_SR_int},
and also as shown in Figure~\ref{fig:AE_S_AB_SR_int}, the energetic component of Test~A
AE data, $I_\mr{power}$, and its rate of increase rise with both primary strain
$\gamma_\mr{AB}$ and secondary strain rate $\dot{\gamma}_\mr{int}$.
\begin{figure}[h]
   \center{\includegraphics{AE_S_AB_AR_int.eps}}
   \caption[Variation of $I_\mr{power}$ with $\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$.]
   {Variation of $I_\mr{power}$ with $\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$
   (Equation~\ref{eq:AE_S_AB_SR_int}).
   (a)~$I_\mr{power}$ against $\gamma_\mr{AB}$
   for $\dot{\gamma}_\mr{int} = 5$, 10, 50, and~$100 \times 10^3$~s$^{-1}$.
   (b)~$I_\mr{power}$ against $\dot{\gamma}_\mr{int}$
   for $\gamma_\mr{AB} = 1.5$, 2, 4, and~7.}
   \label{fig:AE_S_AB_SR_int}
\end{figure}
Moreover, for the range of cutting conditions and work materials utilized in Test~A,
Equation~\ref{eq:AE_S_AB_SR_int} is sufficient to empirically model $I_\mr{power}$, and
$I_\mr{power}$ is not, therefore, dependent on other basic parameters. This strong
dependence of $I_\mr{power}$ on $\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$ can be
explained in terms of the AE generation mechanisms proposed in
Section~\ref{sec:ae_disloc_model}:
\begin{description}

   \item [Primary strain (Figure~\ref{fig:AE_S_AB_SR_int}a)] Since half of the total
   plastic strain occurs at the shear plane AB~\cite{oxley63}
   (Figure~\ref{fig:primary_zone}), $\gamma_\mr{AB}$ can be used as a measure of the total
   plastic work done to the material before it enters the secondary zone of deformation.
   Furthermore, in Section~\ref{sec:ae_disloc_model}, it was discussed that as plastic
   work increases the density of dislocation obstacles $\rho_\mr{B}$, the resulting value
   of $I_\mr{power}$ will also increase; this conclusion is in agreement with
   Equation~\ref{eq:AE_S_AB_SR_int}, since $I_\mr{power} \propto \gamma_\mr{AB}^{n_1}$,
   where $n_1 = 4.511$. Index $n_1$ is dependent on the combination of two factors:
   firstly, it is dependent on the relationship between $\gamma_\mr{AB}$ and $\rho_\mr{B}$
   (the generation of dislocation obstacles due to plastic strain), and, secondary,
   it is also dependent on the combined effect of the transmission path, sensor, and
   instrumentation on the resulting acquired AE data.

   \item [Secondary strain rate (Figure~\ref{fig:AE_S_AB_SR_int}b)] In
   Section~\ref{sec:ae_disloc_model}, it was also concluded that strain rate was directly
   related to increasing $I_\mr{power}$. This is also in agreement with
   Equation~\ref{eq:AE_S_AB_SR_int}, since $I_\mr{power} \propto
   \dot{\gamma}_\mr{int}^{n_2}$, where $n_2 = 2.385$. The value of $n_2$ is a function of
   the combined characteristics of the transmission path, sensor, and instrumentation.

\end{description}

\subsubsection{Acoustic Emission Frequency}

In Section~\ref{sec:ae_disloc_model}, it was concluded that if the main mechanism of AE
generation is the release of strain energy due to the opposition of obstacles to the
displacement of dislocations, then the frequency content of AE should increase with
strain rate but decrease with temperature. This is in agreement with the frequency
analysis carried out in Section~\ref{sec:frequency_analysis} for Test~A AE data, where
it was established that although $f_\mr{mean}$ increases with $\dot{\gamma}_\mr{int}$,
this is not observed because $\dot{\gamma}_\mr{int}$ results in heat generation and
thus higher $T_\mr{int}$ values, and, for the levels of strain rate encountered during
Test~A metal cutting, the effects of $T_\mr{int}$ are dominant, and, therefore,
$f_\mr{mean}$ is seen to decrease with $T_\mr{int}$ even when $\dot{\gamma}_\mr{int}$
increases. In Section~\ref{sec:ae_disloc_model}, it was also concluded that the
frequency content of AE should be independent of strain, whose predictions are clearly
supported by the Test~A AE result shown Figure~\ref{fig:MeanFreq_S_AB}.

\subsubsection{Temperature and Acoustic Emission}

In Section~\ref{sec:ae_disloc_model}, it was concluded that another parameter that
should have an impact on the level of AE was temperature. However, by plotting Test~A
$I_\mr{power}$ against $T_\mr{int}$ (Figure~\ref{fig:T_int_AE}), high scatter was
observed and no clear relationship could be established between $I_\mr{power}$ and
$T_\mr{int}$. Moreover, as concluded in Section~\ref{sec:ae_temp}, the apparent
insensitivity of $I_\mr{power}$ to varying $T_\mr{int}$ is due to the disguising effect
of the other two parameters $\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$. By analysing
Test~B AE results, where the workpiece temperature was changed, it was concluded that
the level of $I_\mr{power}$ is reduced with increasing $T_\mr{int}$ if both
$\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$ are kept constant, which is in agreement
with the AE generation mechanism proposed in Section~\ref{sec:ae_disloc_model}.

\subsubsection{Published Dislocation Models}

As stated in Section~\ref{sec:ae_cont_energy}, Rangwala and Dornfeld~\cite{rangwala91a}
also suggested that, at the high strain rates typically encountered in metal cutting,
viscous damping governs the motion of dislocations and, consequently, proposed that
dislocation damping is responsible for AE generation in metal cutting. As a result,
according to the relationship held by Equation~\ref{eq:damping_power}, they stated that
$I_\mr{power}$ should be proportional to the damping power ($\dot{U}_\mr{D}$), which
resulted in the following relationship:
\begin{eqnarray}
   I_\mr{power} \propto \dot{\gamma}^2
   \label{eq:rangwala_ae_power}
\end{eqnarray}
Rangwala and Dornfeld followed by noting that there was a linear relationship between
$V$ and experimental values of $I_\mr{RMS}$, and since $\dot{\gamma}$ is basically
proportional to $V$, Equation~\ref{eq:rangwala_ae_power} was assumed to be
demonstrated. However, as previously discussed in
Section~\ref{sec:preliminary_discussion}, by analysing the results from Test~A shown in
Figure~\ref{fig:[mode][speed]} and other experimental results from published material,
a relationship of the type $I_\mr{mode} \propto \dot{\gamma}^n$ (or $I_\mr{RMS} \propto
\dot{\gamma}^n$), where $n > 1$ is always obtained, and since $I_\mr{power} =
I_\mr{mode}^2$, a more general relationship of the type $I_\mr{power} \propto
\dot{\gamma}^m$, where $m > 2$, instead of Equation~\ref{eq:rangwala_ae_power}, should
be obtained. Moreover, at every instant, per unit length of dislocation line,
$\dot{U}_\mr{D}$ is given by
\begin{eqnarray}
   \dot{U}_\mr{D} = F_\mr{D}V_\mr{D} = BV_\mr{D}^2
   \label{eq:U_D}
\end{eqnarray}
Therefore, since $B$ and $V_\mr{D}$ remain more or less constant for steady plastic
deformation, $\dot{U}_\mr{D}$ will also remain constant during the flight of
dislocations. As the high frequency AE signals are generated due to the rapid release
of of strain energy, a rapid energy varying process, like the one described for
$\dot{U}_\mr{B}$ in Section~\ref{sec:ae_disloc_model} (Equation~\ref{eq:U_B}), can
justify  the generation of AE\@. Besides, the model proposed by Rangwala and Dornfeld,
does not account for the increase level of AE due to the effect of plastic strain which
is, undoubtedly, a critical parameter contributing to the generation of AE, as
demonstrated by the empirical model derived from Test~A AE data
(Equation~\ref{eq:AE_S_AB_SR_int}, Figure~\ref{fig:AE_S_AB_SR_int}). Consequently,
although the dynamics of dislocation motion in the viscous damping regime is dependent
on the two energy components $\dot{U}_\mr{D}$ and $\dot{U}_\mr{B}$, only the generation
mechanisms responsible for $\dot{U}_\mr{B}$ (strain energy released due to dislocations
transposing obstacles) can justify the variation of AE power with strain and strain
rate.

In a subsequent paper, Rangwala and Dornfeld~\cite{rangwala91b} explained the frequency
content of AE according to the theory proposed by Rouby~\etal~\cite{rouby77}, described
in Section~\ref{sec:ae_theory} (Equation~\ref{eq:rouby_ae_model}), in which the
frequency content of the AE waves was proportional to $V_\mr{D}$ but also proportional
to the inverse of the distance between obstacles ($d_\mr{B}$). In
Section~\ref{sec:ae_disloc_model} it was also concluded that $f_\mr{mean}$ increases
with $V_\mr{D}$; however, $f_\mr{mean}$ was predicted to be unaffected by variations of
$\rho_\mr{B}$ and, consequently, by variations of $d_\mr{B}$, which is in agreement
with Test~A results, since, as shown in Figure~\ref{fig:MeanFreq_S_AB}, $f_\mr{mean}$
was observed to be more or less unaltered by $\gamma_\mr{AB}$. Moreover, the frequency
model proposed by Rangwala and Dornfeld does not explain the strong dependence of
$f_\mr{mean}$ on temperature, shown by their own experimental data and also by Test~A
data (Figure~\ref{fig:MeanFreq}c). On the other hand, the dislocation model presented
in Section~\ref{sec:ae_disloc_model} takes into account the shift of $f_\mr{mean}$
towards lower values when the temperature is decreased.

In conclusion, during metal cutting, plastic deformation is governed by the motion of
dislocations at high strain rates and temperatures. In this regime, named as viscous
damping, two types of forces oppose the motion of dislocations: damping forces, which
are proportional to strain rate, and forces due to dislocation obstacles, which are
dependent on the density of the obstacles and on the strength of the obstacles against
the passage of dislocations. However, although these two forces are responsible for the
low frequency parameters, such as forces, stresses, and temperatures, in
Section~\ref{sec:ae_disloc_model} it is suggested that only the forces due to the
opposition of obstacles to the passage of dislocations may be responsible for the
detected high frequency AE\@. This is contrary to the model proposed by Rangwala and
Dornfeld~\cite{rangwala91a,rangwala91b}, who proposed that the energy due to damping
was responsible for the detected AE energy. Furthermore, according to the model
proposed in Section~\ref{sec:ae_disloc_model}, the following relationships were
derived: the energy content of AE should increase with increasing strain and strain
rate, and decrease with increasing temperature; the frequency content of AE should
increase with strain rate, decrease with temperature, and remain unchanged with strain.
These relationships were confirmed by the experimental data provided by Tests~A and~B,
indicating that, during metal cutting, although plastic deformation is mainly due to
dislocation damping, AE should be generated mainly due to the opposition of obstacles
to the movement of dislocations.
