%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 mostly for
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}, 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 manly affected by the secondary zone of deformation. Since all current
experimental results of AE (Tests I--IV) presented in Chapter~\ref{chap:results} were
performed with an AE sensor 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 expect that the values
of $k_\mr{int}$, $\dot{\gamma}_\mr{int}$, and $v_\mr{int}$ 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~I data (Figures~\ref{fig:[mode][speed]}
and~\ref{fig:[mode][feed]}), 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 before (Section~\ref{sec:oxley_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 is discussed in Section~\ref{sec:width_AE}. By
observing Figure~\ref{fig:SR_k_v_AE}, 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 do 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~I 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 $V$,
$\dot\gamma_\mr{AB}$ 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~I 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
apparent 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~I 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 total 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_AR_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_AR_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}), but this will not have any effect on any
   conclusion 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 and secondary
   ($k_\mr{int}$) deformation zones play an important role in the macroscopic
   physics of cutting (i.e.\ geometry, stresses, cutting power, etc.), it seems
   not to have a direct effect impact on the generation of AE. However, the two
   materials utilized are two carbon steels and, therefore, similar in
   structure. Therefore, it is possible 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}

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_i}, 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 $f_\mr{mean}$.

\begin{figure}[h!]
   \center{\includegraphics{Meanfreq.eps}}
   \caption[Variation of AE mean frequency with $\dot\gamma_\mr{int}$ and $T_\mr{int}$.]
   {Variation of AE mean frequency 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}
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 $V$ (bellow 60~m$/$min), where the cutting temperatures are low, the effect
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 cutting parameters analysed previously for AE power ($k_\mr{int}$,
$v_\mr{int}$, and $\gamma_\mr{AB}$), where also analysed, but no clear trends could be
distinguished: $f_\mr{mean}$ remains more or less unchangeable with $k_\mr{int}$,
$v_\mr{int}$, and $\gamma_\mr{AB}$, but 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~I (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 bellow 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 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 and AE sensor), and when AE waves cross theses 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{Dislocation Motion as the Origin of Acoustic Emission}
\label{sec:disloc_AE}

In Section~\ref{sec:ae_theory}, it is stated 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~I cutting
experiments, though continuous chips were always formed, BUE was observed for low values
of $V$ and $t_1$; 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~I AE signal, which is 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 dislocations 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~I AE data 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~I cutting condition (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~I cutting conditions and work materials 080A15 and 080M40.]
   {Minimum and maximum values of $\dot\gamma_\mr{int}$ and $T_\mr{int}$ for Test~I
   cutting conditions (Table~\ref{tab:speed_feed}) and 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, 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~I 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 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 the latter is a
linear function of strain rate (Equation~\ref{eq:disloc_stress_2})~\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~I conditions involve secondary zone deformation that is
governed by dislocations moving in the viscous damping regime.

\subsection{Acoustic Emission due to Dislocation Motion}
\label{sec:ae_disloc}

As briefly described in Section~\ref{sec:disloc_flow_stress} and shown in
Figure~\ref{fig:disloc_dynamics}, the resisting force $F_\mr{B}$ of an obstacle against a
dislocation trying to surpass it can be modelled as a force that increases from an initial
position $x_1$, reaching a maximum $F_\mr{B}^\mr{max}$ at an intermediate point
$x_\mr{max}$ between $x_1$ and $x_2$, and then decreasing until position $x_2$ is reached.
According to Equation~\ref{eq:disloc_obstacle}, a high enough shear stress must be applied
to the slip plane of the dislocation if the dislocation is going to travel between $x_1$
and $x_2$. Furthermore, between $x_1$ and $x_\mr{max}$, as the dislocation forces its way
against the obstacle, strain energy is stored, and as the dislocation position exceeds
$x_\mr{max}$, this strain energy is rapidly released between $x_\mr{max}$ and $x_2$. This
will be considered here as the basis for the generation mechanism of an AE event due to
the passage of a dislocation over an obstacle.
\begin{figure}[h]
   \center{\includegraphics{disloc_dynamics.eps}}
   \caption{Resisting force of obstacle on a mobile dislocation.}
   \label{fig:disloc_dynamics}
\end{figure}

Moreover, for the very high strain rates encountered during Test~I, viscous damping
becomes the dominant mechanism governing dislocation motion. In this regime, another
force, called damping force, will oppose the displacement of a dislocation; this force is
proportional to the dislocation velocity ($V_\mr{D}$)~\cite{kumar68,kumar69}:
\begin{eqnarray}
   F_\mr{D} = BV_\mr{D}
   \label{eq:F_D}
\end{eqnarray}
where $B$ is the dislocation drag coefficient, which is related to coefficient $m$ in
Equation~\ref{eq:disloc_stress_2} according to the following relationship:
\begin{eqnarray}
   B = b^2 \rho_\mr{MD} m
   \label{eq:B}
\end{eqnarray}
where $b$ is the Burgers vector. Therefore, these conditions are somewhat different from
the ones described in Section~\ref{sec:disloc_obstacles} (where thermal and damping
effects are neglected), since, in this case, the total force acting on the dislocation
will be given by
\begin{eqnarray}
   F = F_\mr{B} + F_\mr{D}
   \label{eq:F}
\end{eqnarray}
and now the shear stress acting on the slip ($\tau$) plane must be sufficient to overcome
the overall effect of $F$, and not just $F_\mr{B}$. Moreover, as the damping mechanism are
prevailing, $F_\mr{D}$ must be considerably higher than $F_\mr{B}$, so that the effect of
$F_\mr{B}$ due to the presence of an obstacle should not affect $V_\mr{D}$ to a large
extent, and, therefore, $V_\mr{D}$ must remain more or less invariant during the flight of
the dislocation. Consequently, as a result of Equation~\ref{eq:F_D}, as long as $V_\mr{D}$
remains constant, $F_\mr{D}$ must also remain constant and not depend on the position of
the dislocation. Clearly, the external load applied to the crystal must be sufficiently
high to allow the dislocations to overtake these two opposing forces that act on every
mobile dislocation, and, consequently, an expression for the flow stress ($k$) will also
be influenced by these two forces (see Equation~\ref{eq:disloc_stress_2}):
\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.
Furthermore, the power required to produce plastic deformation is given by:
\begin{eqnarray}
   \dot{U} = (\tau_\mr{B} + \tau_\mr{D})\dot{\gamma}v
   \label{eq:power}
\end{eqnarray}
where $\dot{\gamma}$ is the shear strain rate, and $v$ is the material participating into
deformation.\footnote{Simplified conditions were considered: plastic deformation occurs in
a rectangular volume of constant shear stress and constant strain rate, such as the
conditions presented in Section~\ref{sec:parameters_AE}
(Figure~\ref{fig:secondary_oxley_zone}).}

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 a 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. It is proposed that this phenomenon also takes place when a
dislocation surmounts an obstacle, since, as discussed before, a force $F_\mr{B}$ (rising
from $x_1$, reaching a maximum $F_\mr{B}^\mr{max}$ at $x^\mr{max}$, and decreasing until
$x_2$, as shown in Figure~\ref{fig:disloc_dynamics}) will be imposed by the obstacle due
to to the passage of the dislocation, and since, as verified previously, $V_\mr{D}$
remains constant, a reaction to $F_\mr{B}$ (having the same strength but opposite
direction) will be impinged by the dislocation on the obstacle; as a consequence,
potential strain energy will be accumulated by the obstacle, but, as $x^\mr{max}$ is
reached, this energy gets released (because $F_\mr{B}$ is removed between $x^\mr{max}$ and
$x_2$), 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 $F_\mr{B}$ between
$x_1$ and $x_2$. Intuitively, the fastest $F_\mr{B}$ decreases between $x^\mr{max}$ and
$x_2$, the higher the frequency content of the emitted AE wave. For this reason, it is
thought that the damping force $F_\mr{D}$ will not contribute to the generation of AE,
since, as AE is a high frequency wave (Section~\ref{sec:ae_theory}) and $V_\mr{D}$ does
not change, $F_\mr{D}$ remains constant during the displacement of the dislocation, and,
as shown before, the quickest the change, the highest the frequency of the emission.
Furthermore, the energy of the emitted AE pulse will be dependent on the area under the
curve defined by relationship between $F_\mr{B}$ and $x$, between $x_1$ and $x_2$
(Figure~\ref{fig:disloc_dynamics}), which is the energy involved to surmount the obstacle:
\begin{eqnarray}
   U_\mr{B} = \int_{x_1}^{x_2}F_\mr{B}dx
   \label{eq:power}
\end{eqnarray}

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, in metal cutting, the
detected AE will be a function of the integrated contribution of all AE pulses generated
due the release of localized strain energy when dislocations surpass obstacles. However,
as shown next, the detected AE will be a function of many factors:
\begin{description}

   \item [Relationship between dislocation and obstacle] As stated before, the shape of
   the curve $F_\mr{B}$ versus dislocation position $x$ will dictate the frequency content
   and energy of each AE event: the steeper the curve, the higher the frequencies, and,
   according to Equation~\ref{eq:power}, the larger the area under the curve, the larger
   the energy. 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
   Section~\ref{sec:disloc_obstacles} and~\ref{sec:disloc_flow_stress}, in this regime,
   increasing temperature will help dislocations to surmount obstacles, and, therefore,
   $F_\mr{B}^\mr{max}$ is believed to decrease, resulting in a decrease in the frequency
   content of the AE events (since the $F_\mr{B}$ curve becomes less steeper), and a also
   in a energetic decrease of the AE events (since the area under the $F_\mr{B}$ curve
   is reduced).

   \item [Density of obstacles] Since all AE pulses contribute to the total detected AE
   wave, if more obstacles are present, more AE pulses are generated, and, therefore, more
   AE is generated. Moreover, according to Section~\ref{sec:multipl_disloc}
   and~\ref{sec:disloc_obstacles}, 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 affect
   the frequency content of AE.

   \item [Dislocation speed] As $V_\mr{D}$ increases, more obstacles are surmounted by a
   dislocation per unit time, and, therefore, more AE events are generated. Actually, the
   number of AE events per unit time is proportional to $V_\mr{D}$. As $\dot{\gamma}$ 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 by the obstacles, the
   frequency content of the resulting AE event is expected to increase, and thus that of
   the detected AE wave.

   \item [Transmission path and instrumentation] The amount of AE detected by the sensor,
   will be related to the contribution of energy $U_\mr{B}$ from all AE events. However,
   the original AE signal will be distorted from its source to the acquired signal due to
   the influence of the transmission path, sensor sensitivity, and signal condition system
   (Section~\ref{sec:ae_theory}), i.e.\ some frequencies will be more attenuated than
   others. An example of an extreme case is that if a band-pass filter with a frequency
   band outside that of the original AE pulses is employed, no AE will be detected, though
   AE is being generated. As a result, no simple relationship can be derived between the
   sum of the energy $U_\mr{B}$ from all AE events and the detected AE signal.

\end{description}
In summary, during a typical metal cutting operation, the energy level of the detected AE
is expected to increase with $\gamma$ and $\dot{\gamma}$, but 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 sensitivity, and
instrumentation on the AE signal, at this stage, only a qualitative analysis is possible.
In the following section, the relationship between dislocation motion and Test~I AE data
is evaluated according to the analysis performed in Section~\ref{sec:parameters_AE}.

\subsection{Analysis with Experimental Data}

The empirical equation derived in Section~\ref{sec:energy_analysis},
(Equation~\ref{eq:AE_S_AB_SR_int}) equates the AE energetic component $I_\mr{power}$ to
the primary strain $\gamma_\mr{AB}$ and secondary strain rate $\dot{\gamma_\mr{int}}$,
according to the plots presented in Figure~\ref{fig:AE_S_AB_AR_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_AR_int}
\end{figure}
It can be seen that $I_\mr{power}$ and its rate of increase increase with both
$\gamma_\mr{AB}$ and $\dot{\gamma}_\mr{int}$, as  shown in
Figures~\ref{fig:AE_S_AB_AR_int}a and~\ref{fig:AE_S_AB_AR_int}b, respectively. As it will
be concluded from the following discussion, the relationship between $I_\mr{power}$ and
$\gamma_\mr{AB}$ and $\dot{\gamma_\mr{int}}$ is in conformity with the analysis presented
in Section~\ref{sec:mech_disloc_cut}, where it is concluded that AE detected during
plastic deformation of metal cutting is generated due to dislocation movement:
\begin{description}

   \item [Primary strain (Figure~\ref{fig:AE_S_AB_AR_int}a)] Since half of the total
   plastic strain occurs at the shear plane AB
   (Figure~\ref{fig:primary_zone})~\cite{oxley63}, $\gamma_\mr{AB}$ can be used as a
   measure of the total plastic work done to the material before
   it enters the secondary zone of deformation. Furthermore, as plastic work is
   responsible for the

   \item [Secondary strain rate (Figure~\ref{fig:AE_S_AB_AR_int}b)]

\end{description}






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

reasons: see \cite{messaritis86}, \cite{gillis72}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Influence of Temperature on AE}
