%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Influence of Basic Cutting Parameters on AE}
\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 $I_\mr{power}$ 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, afterwards, the relative contribution of
each of these parameters to the generation of AE is evaluated.

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 the 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; 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.

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:w_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 $AE_\mr{power}$:
\begin{figure}[p]
   \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 (Figure~\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 either 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, as defined in Section~\ref{sec:var_mat_prop} and Appendix~\ref{chap:oxley}, and it
was found that both $\dot\gamma_\mr{AB}$ and $\dot\gamma_\mr{int}$ increase with $V$,
which is is in agreement with Messaritis and Borthwick; but $\dot\gamma_\mr{AB}$ decreases
with $t_1$, which is in agreement with Messaritis and Borthwick, while
$\dot\gamma_\mr{int}$ increases with $t_1$. As a result, it was concluded that the values
of $\dot\gamma$ presented by Messaritis and Borthwick must represent values from the
primary zone of deformation. However, as stated before, AE is thought to be mainly due to
deformation from the secondary zone, and, therefore, a relationship between $I_\mr{power}$
and $\dot\gamma_\mr{int}$ (and not with $\dot\gamma_\mr{AB}$) is of interest, but, as
noted before, Figures~\ref{fig:SR_k_v_AE}a and \ref{fig:SR_k_v_AE}b show that no clear
function can be established between $I_\mr{power}$ and $\dot\gamma_\mr{int}$.

\begin{figure}[h]
   \center{\includegraphics{SR_AB_int_delta.eps}}
   \caption{Relationship between $\dot\gamma_\mr{AB}$ and
   $\dot\gamma_\mr{int}\delta$}
   \label{fig:SR_AB_int_delta}
\end{figure}
An investigation of the relationship between the basic cutting parameters from the primary
and secondary zones of deformation indicated, as shown in
Figure~\ref{fig:SR_AB_int_delta}, that the primary parameter $\dot\gamma_\mr{AB}$ is more
or less proportion to the secondary parameters' product $\dot\gamma_\mr{int}\delta$. As a
result, it was decided to look for a possible relationship between $I_\mr{power}$ and the
product $\dot\gamma_\mr{int}\delta$. In Section~\ref{sec:ae_empirical_model}, it was
observed that the relationship between $I_\mr{power}$ and both $V$ and $t_1$ can be
approximated to a straight line when plotted on a logarithmic scale; therefore, the
relationship between $I_\mr{power}$ and $\dot\gamma_\mr{int}$ was set between logarithmic
values as well, i.e $\log_{10}{I_\mr{power}}$ and
$\log_{10}{(\dot\gamma_\mr{int}\delta)}$, as shown in Figure~\ref{fig:Oxley_AE1}.
\begin{figure}[h]
   \center{\includegraphics{Oxley_AE1.eps}}
   \caption[Attempt to model $I_\mr{power}$ with $\dot{\gamma}_\mr{int}\delta$.]
   {Attempt to model $I_\mr{power}$ with $\dot{\gamma}_\mr{int}\delta$.
   (a)~Separate work materials 080A15 and 080M40.
   (b)~Joined work materials 080A15 and 080M40.}
   \label{fig:Oxley_AE1}
\end{figure}
Figure~\ref{fig:Oxley_AE1}a shows that when the data for both work materials 080A15 and
080M40 is plotted separately, the relationship between $\log_{10}{I_\mr{power}}$ and
$\log_{10}{(\dot\gamma_\mr{int}\delta)}$ can be approximated to two straight lines.
However, the two straight lines are quite unexceptional fits due to their corresponding
relatively low coefficients of determination ($r^2 = 0.873$ for 080A15, and $r^2 = 0.834$
for 080M40). When a single straight line is fitted between the experimental data of both
work materials, an even lower coefficient of determination is obtained ($r^2 = 0.773$),
indicating that an independent relationship for each work material ought exist:
\begin{eqnarray}
   I_\mr{power} = K(\dot\gamma_\mr{int}\delta)^n
   \label{eq:Oxley_AE1}
\end{eqnarray}
where $K$ and $n$ are dependent on work material and sensor setup configuration. The
values of $K$ and $n$, found by linear regression, are presented in
Table~\ref{tab:Oxley_AE1}.
\begin{table}[h]
   \caption{Equation~\ref{eq:Oxley_AE1} constants found
   by linear regression.}
   \label{tab:Oxley_AE1}
   \center{\begin{tabular}{cccc}
      \hline
      material & $K$  & $n$   & $r^2$ \\
      \hline
      080A15   & 46.5 & -6.42 & 0.873 \\
      080M40   & 42.7 & -6.76 & 0.834 \\
      both     & 24.8 & -5.77 & 0.773 \\
      \hline
   \end{tabular}}
\end{table}

Due to the somewhat unsatisfactory relationship just derived
(Equation~\ref{eq:Oxley_AE1}), since constants $K$ and $n$ depende on work material, and
since no account on flow stress data was taken, it was decided to include the parameter
$k_\mr{int}$ in the relationship derived above (Equation~\ref{eq:Oxley_AE1}). For a given
value of $\dot\gamma_\mr{int}$, work material must be the parameter most influential on
$k_\mr{int}$ (though $k_\mr{int}$ is also influenced by $T_\mr{int}$, this will be
neglected here). Therefore, if all the influence of work material can be represented by
$k_\mr{int}$, by including it in Equation~\ref{eq:Oxley_AE1}, so that $I_\mr{power}$ is a
function of $k_\mr{int}$ and $\dot\gamma_\mr{int}\delta$, one can expect that the new
relationship is independent of work material and, therefore, more universal. As it can be
seen in Figure~\ref{fig:Oxley_AE2},
\begin{figure}[h]
   \center{\includegraphics{Oxley_AE2.eps}}
   \caption[Attempt to model $I_\mr{power}$ with
   $\dot{\gamma}_\mr{int}\delta$ and $k_\mr{int}$.]
   {Attempt to model $I_\mr{power}$ with
   $\dot{\gamma}_\mr{int}\delta$ and $k_\mr{int}$.
   (a)~Separate work materials 080A15 and 080M40.
   (b)~Joined work materials 080A15 and 080M40.}
   \label{fig:Oxley_AE2}
\end{figure}
the method to achieve the new relationship was to plot the term $F\log_{10}{k_\mr{int}} +
\log_{10}{I_\mr{power}}$ against $\log_{10}{\dot\gamma_\mr{int}\delta}$ and then, with the
method of linear regression, to find the best fit straight line. In this process, the data
for both work materials was used together since a relationship independent of work
material was sought. The relative contribution of $k_\mr{int}$ is given by constant $F$,
and since it was unknown, its value was obtained by looking for the value of $F$ that
gives the maximum value of $r^2$ for the best fit straight line. As shown in
Figure~\ref{fig:BestF},
\begin{figure}[h]
   \center{\includegraphics{BestF.eps}}
   \caption{Value of $F$ that gives a maximum coefficient of determination.}
   \label{fig:BestF}
\end{figure}
$F$ was found to be 6.04 for a maximum value of $r^2 = 0.944$. This value of $r^2$ is
relatively high, which provides a good indication of the suitability of the proposed
relationship. This relationship can be stated as follows:
\begin{eqnarray}
   I_\mr{power} = K\frac{(\dot\gamma_\mr{int}\delta)^n}{k_\mr{int}^F}
   \label{eq:Oxley_AE2}
\end{eqnarray}
Constants $K$, $n$, and $F$ are work material independent and rely upon sensor setup
configuration only; for $F = 6.04$, the values of $K$ and $n$ were found to be 70.4 and
8.96, respectively.

It is interesting to note that when Equation~\ref{eq:Oxley_AE1} is used to model AE data,
a much lower $r^2$ value is obtained when the data of the two work material is used
together (Figure~\ref{fig:Oxley_AE1}b), instead of finding a model each work material
(Figure~\ref{fig:Oxley_AE1}b). Whereas, when Equation~\ref{eq:Oxley_AE2} is used
separately (Figure~\ref{fig:Oxley_AE2}a), or the data for both work materials is used
together (Figure~\ref{fig:Oxley_AE2}b). This indicates, adding the fact that the value of
of $r^2$ is much closer to one when using Equation~\ref{eq:Oxley_AE2}, that there is an
obvious improvement when moving from Equation~\ref{eq:Oxley_AE1} to \ref{eq:Oxley_AE2}.
Moreover, although the model defined by Equation~\ref{eq:Oxley_AE2} is purely empirical,
it provides quite valuable information about the generation of AE in metal cutting: the
main parameters that affect the generation of AE ($\dot\gamma_\mr{int}$, $\delta$, and
$k_\mr{int}$) have been identified, and their relative contribution to the generation of
AE can be computed.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Role of Dislocation Motion on AE}
\label{sec:disloc_AE}


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

reasons: see \cite{messaritis86}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Frequency Content of AE}
