%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Literature Review} \label{chap:review}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This chapter is divided into two major parts. The first part provides an overview of metal
cutting physics, a description of the main zones of deformation, and a review of the major
cutting models. The second part is dedicated to the acoustic emission (AE) signal and, in
particular, to its generation during metal cutting.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Metal Cutting Physics} \label{sec:cut_physics}

There are three zones of interest in the cutting process
(Figure~\ref{fig:deformation_zones}). The first area, called the primary zone, is the
boundary between the deformed and undeformed material. Once the material has been
deformed through the primary zone to form the chip, a secondary zone of deformation
is observed at the interface between the tool rake face and the chip as it flows over
the tool. The tertiary zone corresponds to the contact between the tool clearance
face and the newly machined surface of the workpiece.
\begin{figure}[h]
   \center{\includegraphics{deformation_zones.eps}}
   \caption{Zones of interest in cutting.}
   \label{fig:deformation_zones}
\end{figure}

In Figure~\ref{fig:deformation_zones}, the chip is represented as a continuous band of
uniform thickness~$t_2$ and width~$w$. However, this is a simplification, for there are
three major types of chip formation \cite{asm54,shaw84}:
\begin{description}

   \item [Continuous] Chips are smooth continuous ribbons with reasonably constant
   thickness produced during machining at high speeds of ductile materials
   (Figure~\ref{fig:chip_types}a).

   \item [Continuous with built-up edge] At speeds where the temperature at the tool-chip
   interface is relatively low, a stagnant mass of chip material may adhere to the tool
   rake face. This detached material acts as the cutting edge, increasing the effective
   rake angle, and is called built-up edge (BUE) (Figure~\ref{fig:chip_types}b).

   \item [Discontinuous] This type of chip is formed during the machining of relatively
   brittle materials at low cutting speeds where individual segments are formed by
   alternate deformation and fracture of the metal (Figure~\ref{fig:chip_types}c).

   \item [Segmented] As shown in Figure Figure~\ref{fig:chip_types}d, this type of chip is
   only partially fractured, and the back of the chip is very wavy. Two formation
   mechanisms have been proposed for the generation of segmented chips~\cite{shaw84}:
   adiabatic shear, and fracture and
   rewelding.

\end{description}
\begin{figure}[h]
   \center{\includegraphics{chip_types.eps}}
   \caption[Types of chip formation.]
   {Types of chip formation.
   (a)~Continuous. (b)~Continuous with BUE\@. (c)~Discontinuous. (d)~Segmented.}
   \label{fig:chip_types}
\end{figure}

For the purpose of studying chip formation, it is useful to start with the simplest
possible conditions, and, hence, some restrictions have to be made. Throughout the next
sections, the following conditions apply unless specifically stated:
\begin{itemize}

   \item Cutting is orthogonal and can, therefore, be represented in two dimensions.

   \item The chip does not spread to either side (plane strain). This condition is
   practically encountered if the width~$w$ is large compared with the uncut chip
   thickness $t_1$.\footnote{For example, Shaw~\cite{shaw84} states that $w/t_1 \geq
   5$, whereas Oxley~\cite{oxley89} states that $w/t_1 \geq 10$, so that plane strain
   conditions can be assumed.}

   \item The tool is perfectly sharp. With this condition, the tertiary zone is
   neglected, since there is no contact along the clearance face.

   \item The generation of chips is assumed continuous and without BUE\@.

   \item Relative vibration between the tool and workpiece, referred as chatter, are
   disregarded.

   \item There is no externally applied coolant or lubricant.

\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Zones of Deformation}

\subsubsection{Primary Zone}

Experimental observation of photomicrographs\footnote{Photomicrographs are photographs
that are taken through a microscope of pre-prepared surface sections of partially
formed chips from a cutting process that has been abruptly stopped.} of partially
formed chip sections have shown that chip formation occurs in a narrow zone of intense
shear that extends from the tool edge to the work surface
\cite{asm54,kececioglu58,shaw84}. This zone can be approximated to a plane
\cite{ernst41}, called the shear plane and represented by plane AB in
Figure~\ref{fig:primary_zone}a, across which the work velocity $V$ is instantaneously
changed to the chip velocity $V_\mr{C}$. This requires a discontinuity in the
tangential component of velocity across AB equal to $V_\mr{S}$ (shear velocity), as
shown by the velocity diagram in Figure~\ref{fig:primary_zone}b. This velocity
discontinuity only applies to ideal rigid-plastic materials of constant flow stress.
\begin{figure}[h]
   \center{\includegraphics{primary_zone.eps}}
   \caption[Primary zone of deformation.]
   {Primary zone of deformation.
   (a)~Shear plane and parallel-sided shear zone models. (b)~Velocity diagram.}
   \label{fig:primary_zone}
\end{figure}

A very important angle is the one formed between the shear plane and the direction of $V$,
called the shear angle~($\phi$). Although the shear angle can be measured directly from
photomicographs, $\phi$ is normally determined with the following relationship:
\begin{eqnarray}
   \tan{\phi} = \frac{r\cos{\alpha}}{1-r\sin{\alpha}}
   \label{eq:shear_angle}
\end{eqnarray}
where $r$ is the ratio of uncut to cut chip thickness $t_1/t_2$. Although the chip
thickness~$t_2$ can be obtained by direct measurement, this is not very precise due to
the roughness on the back surface of the chip. In practical tests, the mean chip
thickness can be obtained by taking the weight ($W_\mr{C}$) of a piece of chip of
length $l_\mr{C}$. The mean chip thickness is then
\begin{eqnarray}
   t_2 = \frac{W_\mr{C}}{\rho w l_\mr{C}}
   \label{eq:chip_thickness}
\end{eqnarray}
where $\rho$ is the density of the work material. Moreover, the shear strain
occurring in crossing the tangential velocity discontinuity~$V_\mr{S}$ is given by
\begin{eqnarray}
   \gamma_\mr{AB} = \frac{V_\mr{S}}{V_\mr{N}}
                  = \frac{\cos{\alpha}}{\sin{\phi}\cos{(\phi - \alpha})}
   \label{eq:shear_strain_AB}
\end{eqnarray}
where $V_\mr{N}$ is the velocity normal to AB\@.

The instantaneous change in velocity across AB, assumed in the shear plane model, leads
to an infinite value of the shear strain rate~($\dot{\gamma}_\mr{AB}$). Consequently, a
parallel-sided shear zone model has emerged \cite{kececioglu58,oxley63,wright77}, which
is still in accordance with the observation that chips are formed in a narrow zone. The
parallel-sided shear zone is represented by zone CDEF in
Figure~\ref{fig:primary_zone}a. The change in speed from~$V$ to~$V_\mr{C}$ occurs now
along smooth stream lines from CD to EF\@. Stevenson and Oxley~\cite{stevenson70}
concluded that the tangential velocity discontinuity~$V_\mr{S}$ was still a good
representation of the overall change in velocity in the primary zone. The mean value of
the shear strain rate can then be obtained from
\begin{eqnarray}
   \dot{\gamma}_\mr{AB} = \frac{V_\mr{S}}{\Delta s_1}
   \label{eq:shear_strain_rate_AB1}
\end{eqnarray}
where $\Delta s_1$ is the thickness of the parallel-sided shear zone. By analysing
experimental results of the shear zone size from Kececiouglu~\cite{kececioglu58},
Stevenson and Oxley~\cite{stevenson70} realized that $\Delta s_1$ should be dependent
only on scale and not on velocity. Moreover, using printed grids to measure the
plastic flow in chip formation, Stevenson and Oxley showed that the strain rate
distribution was approximately symmetrical about AB with the maximum value occurring
at AB\@. Consequently, the following empirical equation was derived:
\begin{eqnarray}
   \dot{\gamma}_\mr{AB} = C\frac{V_\mr{S}}{t_1}
   \label{eq:shear_strain_rate_AB2}
\end{eqnarray}
The material based constant $C$ was found to be $2.59$ for a $0.13$\% carbon steel.
However, as pointed out by Oxley~\cite{oxley89},
Equation~\ref{eq:shear_strain_rate_AB2} should be changed to a more accurate
equation:
\begin{eqnarray}
   \dot{\gamma}_\mr{AB} = C\frac{V_\mr{S}}{l_\mr{AB}}
   \label{eq:shear_strain_rate_AB3}
\end{eqnarray}
where $l_\mr{AB}$ is the length of AB\@. Hastings~\etal~\cite{hastings74} found $C$ to be
approximately $5.9$ for a $0.16$\% carbon steel. Furthermore, the total shear strain
occurring in the shear zone (along EF) is given by
\begin{eqnarray}
   {\gamma}_\mr{EF} = \frac{\cos{\alpha}}{\sin{\phi}\cos{(\phi - \alpha})}
   \label{eq:shear_strain_EF}
\end{eqnarray}

\subsubsection{Secondary Zone}

The importance of the secondary zone of deformation must be realized by the fact that the
primary chip formation process depends upon the secondary process as much as the secondary
deformation process depends upon the primary process \cite{zorev63}. Due to the existence
of this correlation, the primary and secondary processes can not be examined isolated from
each other.

The earliest cutting process analysis treated the tool-chip interface as a classical
sliding friction situation~\cite{ernst41}. In this analysis, the force required to
slide the chip along the tool rake face ($F_\mr{int}$) is proportional to the force
normal to the interface ($N_\mr{int}$):
\begin{eqnarray}
   F_\mr{int} = \mu N_\mr{int}
   \label{eq:friction_coefficient}
\end{eqnarray}
where $\mu$ is the coefficient of friction, which is only dependent on forces
$F_\mr{int}$ and $N_\mr{int}$ and is independent on the size of the apparent
tool-chip contact area.

However, experimental observations of chip photomicrographs and worn tools have led
to the conclusion that, over most of the area of the tool-chip interface, sliding is
impossible under most cutting conditions \cite{trent84}. Zorev~\cite{zorev63} also
showed that friction on the tool face is so great that, at least over a part of the
contact area closest to the cutting edge, the contact layer of the chip is retarded
or seized by the tool, and the process of friction between the two surfaces is
replaced by plastic shear of the chip. Therefore, the tool-chip interface was divided
into two distinct regions (Figure~\ref{fig:secondary_zone}). The first region
(nearest to the cutting edge), called the seizure or sticking zone, mainly involves
shearing of the chip; the second region (nearest to the point at which the chip
leaves the tool), called the friction zone, is governed by the classical laws of
sliding friction. Shaw~\cite{shaw84} pointed out that there should be a zone of
transition between the sticking and the friction zones. Stevenson and
Oxley~\cite{stevenson70} assumed that the tool-chip interface can be represented by a
rectangular plastic zone with no sliding. Assuming that the velocity at the tool chip
increases from zero at the tool face to the rigid chip velocity~$V_\mr{C}$ in a
distance~$\Delta s_2$, the shear strain rate at the interface can be calculated as
follows:
\begin{eqnarray}
   \dot{\gamma}_\mr{int} = \frac{V_\mr{C}}{\Delta s_2}
   \label{eq:strain_rate_int}
\end{eqnarray}
Stevenson and Oxley~\cite{stevenson70} found $\Delta s_2$ to be
approximately $\frac{1}{8}$ of $t_2$.
\begin{figure}[h]
   \center{\includegraphics{secondary_zone.eps}}
   \caption{Secondary zone of deformation.}
   \label{fig:secondary_zone}
\end{figure}

\subsubsection{Tertiary Zone}

It was stated previously that the effects of the tertiary zone are normally neglected
if a sharp tool is used. However, in practice, and especially when the clearance face
shows some degradation, there is always some contact of the work material with this
face \cite{trent84}. This contact can be reduced by increasing the clearance angle, but
this results in weakening of the tool edge. Evidence from photomicrographs
\cite{trent84} and experiments with a controlled clearance face contact
area~\cite{kobayashi60} demonstrate that seizure also occurs on the clearance surface,
particularly in the region closest to the cutting edge.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mechanics of Cutting} \label{sec:mech_cut}

\subsubsection{Forces}

Knowledge of the forces acting on the tool is an important aspect for the design and
optimization of machines and cutting tools, and also for the scientific investigation of
the cutting process.

The classic way of analysing the cutting forces in orthogonal cutting \cite{ernst41}
is by isolating the chip as a free body and placing it in static equilibrium. This is
achieved if the resultant force $R_1$ on the chip along the shear plane is equal,
opposite, and co-linear to the resultant force $R_2$ on the chip along the tool-chip
interface (Figure~\ref{fig:forces}).
\begin{figure}[h]
   \center{\includegraphics{forces.eps}}
   \caption{Cutting forces.}
   \label{fig:forces}
\end{figure}
The forces $R_1$ and $R_2$ can be resolved into three sets of components:
\begin{itemize}

   \item Along the cutting speed and feed rate directions, tangential force ($F_\mr{X}$)
    and feed force ($F_\mr{Y}$), respectively.

   \item Along and normal to the shear plane AB, $F_\mr{AB}$ and $N_\mr{AB}$,
   respectively.

   \item Along and normal to the tool-chip interface, $F_\mr{int}$ and $N_\mr{int}$,
   respectively.

\end{itemize}
Normally, it is important to obtain the shear ($F_\mr{AB}$ and $N_\mr{AB}$) and frictional
($F_\mr{int}$ and $N_\mr{int}$) components of the forces in terms of $F_\mr{X}$ and
$F_\mr{Y}$. The following relationships apply:
\begin{eqnarray}
   F_\mr{AB}  & = & F_\mr{X}\cos\phi - F_\mr{Y}\sin\phi
   \label{eq:force1} \\
   N_\mr{AB}  & = & F_\mr{X}\sin\phi + F_\mr{Y}\cos\phi
   \label{eq:force2} \\
   F_\mr{int} & = & F_\mr{X}\sin\alpha + F_\mr{Y}\cos\alpha
   \label{eq:force3} \\
   N_\mr{int} & = & F_\mr{X}\cos\alpha - F_\mr{Y}\sin\alpha
   \label{eq:force4}
\end{eqnarray}
The mean angle of friction ($\lambda$) is an important angle that describes the frictional
conditions at the tool-chip interface, which can be determined by the following ratio:
\begin{eqnarray}
   \tan\lambda = \frac{F_\mr{int}}{N_\mr{int}}
   \label{eq:friction_angle}
\end{eqnarray}

For a semi-orthogonal turning operation, the resultant cutting force acting on the tool
can be measured in three directions (Figure~\ref{fig:turning_forces}):
\begin{figure}[h]
   \center{\includegraphics{turning_forces.eps}}
   \caption{Cutting force components in semi-orthogonal turning.}
   \label{fig:turning_forces}
\end{figure}
the first component, the tangential force ($F_\mr{X}$), acting along the vertical
direction of the cutting speed, is normally the largest of the three force components;
the second component, the force component acting on the horizontal direction of the
feed rate, is referred to as the feed force ($F_\mr{Y}$); the third component, the
radial force ($F_\mr{Z}$), acting in a direction normal to $F_\mr{X}$ and $F_\mr{Y}$
simultaneously, tends to push the tool away from the workpiece; this is the smallest of
the three force components in semi-orthogonal cutting, being usually ignored, thus,
reducing the force analysis to two dimensions, as presented in Figure~\ref{fig:forces}.

Measurement of the cutting forces can be achieved with dynamometers. Some deformation is
associated with the operation of every dynamometer, and the deflection produced is then
measured and converted into force units, e.g.\ piezoelectric transducers and strain
gauges. Shaw~\cite{shaw84} gives a general overview of metal cutting dynamometry.

\subsubsection{Stresses}

High shear and normal stresses occur both on the primary zone of deformation and on the
rake face of the cutting tool. Moreover, the stresses acting on the chip from both primary
and secondary deformation zones are related to each other by the conditions of static
equilibrium of the chip.

In the primary zone, one can model the shear and normal stresses acting on the shear plane
AB as mean quantities. Therefore, uniform shear shear stress ($\tau_\mr{AB}$) and uniform
normal stress ($\sigma_\mr{AB}$) distributions are assumed to act over the entire area of
primary zone ($A_\mr{AB}$):
\begin{eqnarray}
      \tau_\mr{AB}   & = & \frac{F_\mr{AB}}{A_\mr{AB}}
      \label{eq:primary_stress1} \\
      \sigma_\mr{AB} & = & \frac{N_\mr{AB}}{A_\mr{AB}}
      \label{eq:primary_stress2}
\end{eqnarray}
where
\begin{eqnarray}
   A_\mr{AB} = \frac{t_1 w}{\sin{\phi}}
   \label{eq:primary_area}
\end{eqnarray}

Fenton and Oxley~\cite{fenton68} stated that the shear stresses along AB could be assumed
as constant, but that the normal stresses along AB could only be assumed constant for a
material of uniform flow stress. For non-uniform flow stress conditions, Fenton and Oxley
modelled the normal stress distribution to change linearly along AB, so that $N_\mr{AB} =
(\sigma_\mr{A} + \sigma_\mr{B})/2$, where $\sigma_\mr{A}$ and $\sigma_\mr{B}$ are the
normal stresses at points A and B\@.

The nature of the tool-chip interface and the distribution of the
shear~($\tau_\mr{int}$) and normal~($\sigma_\mr{int}$) stresses are critical in
understanding the cutting process. Very high stresses act on the secondary zone,
combined with high temperatures and large strains in the chip adjacent to the tool
face. The classical analysis of stresses on the tool face assumes that sliding
friction is present and that the stresses are uniformly distributed. Hence, the mean
values of $\tau_\mr{int}$ and $\sigma_\mr{int}$ can be estimated as follows:
\begin{eqnarray}
      \tau_\mr{int}   & = & \frac{F_\mr{int}}{A_\mr{int}}
      \label{eq:secondary_stress1} \\
      \sigma_\mr{int} & = & \frac{N_\mr{int}}{A_\mr{int}}
      \label{eq:secondary_stress2}
\end{eqnarray}
where $A_\mr{int}$, the tool-chip interface area, is given by
\begin{eqnarray}
   A_\mr{int} = wl_\mr{int}
   \label{eq:secondary_area}
\end{eqnarray}
where $l_\mr{int}$ is the length of the tool-chip contact zone.

However, experimental results suggest that the stresses are not uniformly distributed
along the rake face~\cite{zorev63,trent84,xiaoping97,buryta98}.
\begin{figure}[h]
   \center{\includegraphics{secondary_stresses.eps}}
   \caption[Distribution of stresses on the tool rake face.]
   {Distribution of stresses on the tool rake face~\cite{zorev63}.}
   \label{fig:secondary_stresses}
\end{figure}
Zorev~\cite{zorev63} presented a stress distribution model in which the normal stress
is assumed to take a maximum value~($\sigma_\mr {max}$) at the tip of the tool and
then to decrease, as a power function, down to the point at which the chip leaves the
tool, where $\sigma_\mr{int}$ is zero (Figure~\ref{fig:secondary_stresses}). The
normal stress is, thus, defined by
\begin{eqnarray}
   \sigma_\mr{int} = \sigma_\mr{max} \left( \frac{x}{l_\mr{int}} \right)^n
   \label{eq_secondary_normal_stress}
\end{eqnarray}
where $x$ is the distance to the left from which the chip leaves the tool, and $n$ is the
exponent. In the sliding zone (length $l_2$), the shear and normal stresses are related:
\begin{eqnarray}
   \tau_\mr{int} = \mu\sigma_\mr{int} =
   \mu\sigma_\mr{max} \left( \frac{x}{l_\mr{int}} \right)^n
   \label{eq_secondary_shear_stress}
\end{eqnarray}
where $\mu$ is the coefficient of friction, as defined in
Equation~\ref{eq:friction_coefficient}. However, as $\tau_\mr{int}$ reaches the value
of the shear flow stress ($k_\mr{int}$)\footnote{The shear flow stress ($k$) is the
state of pure shear that will induce yielding or the onset of plastic deformation
\cite{hosford83}.} of the interface chip material, instead of sliding against the
rake surface, it takes less energy for the material adjacent to the interface to
shear internally. This can be described as sticking friction. Therefore, in the
sticking zone (length $l_1$), $\tau_\mr{int}$ assumes a constant value $k_\mr{int}$,
as it can be observed in Figure~\ref{fig:secondary_stresses}. The frictional force
components $N_\mr{int}$ and $F_\mr{int}$ can be obtained by integrating the stresses
of the area of contact, resulting in
\begin{eqnarray}
   N_\mr{int} & = & \frac{\sigma_\mr{max}wl_\mr{int}}{1 + n}
   \label{eq:secondary_normal_force} \\
   F_\mr{int} & = & k_\mr{int}w \left( l_1 + \frac{l_2}{1 + n} \right)
   \label{eq:secondary_friction_force}
\end{eqnarray}

Other stress distributions have been proposed. For example, Xiaoping~\cite{xiaoping97},
and Buryta and Sowerby~\cite{buryta98} presented stress distributions where
$\sigma_\mr{int}$ increases with increasing $x$, but at locations near the cutting
edge, $\sigma_\mr{int}$ stops rising and assumes a constant value. Fenton and
Oxley~\cite{fenton68} assumed that the normal stress distribution is triangular with
the maximum value at the cutting edge. Moreover, Fenton and Oxley neglected the sliding
zone at the tool-chip interface and assumed that the material at the interface is in a
state of plastic shear with an uniform shear stress distribution~$k_\mr{int}$.

\subsubsection{Energy Considerations}

In machining operations, the total energy per unit time (power) can be calculated as
follows:
\begin{eqnarray}
   \dot U = F_\mr{X}V
   \label{eq:power}
\end{eqnarray}
However, this energy can be normalized by dividing it by the volume of material being
removed ($v$). This is called specific power ($u$), and since $v = t_1wV$, $u$ can be
calculated as follows:
\begin{eqnarray}
   u = \frac{F_\mr{X}}{t_1w}
   \label{eq:specific_power1}
\end{eqnarray}
The specific power can be partitioned into four components \cite{shaw84}:
\begin{itemize}

  \item Primary shear specific power ($u_\mr{S}$).

  \item Secondary frictional specific power ($u_\mr{C}$) on the tool-chip interface.

  \item Surface specific power ($u_\mr{A}$) due to the formation of new surface areas in
  cutting.

  \item Momentum or kinetic specific power ($u_\mr{M}$) due to the change in momentum of
  the chip.

\end{itemize}
Normally, $u_\mr{A}$ and $u_\mr{M}$ are negligible relative to the other two components so
that
\begin{eqnarray}
   u = u_\mr{S} + u_\mr{C}
   \label{eq:specific_power2}
\end{eqnarray}
However, $u_\mr{M}$ takes on increasing importance with very high speed machining.
Furthermore, the primary and secondary components of the total specific power can be
calculated as follows:
\begin{eqnarray}
   u_\mr{S} & = & \frac{F_\mr{AB}V_\mr{S}}{t_1wV}
   \label{eq:specific_primary_power} \\
   u_\mr{C} & = & \frac{F_\mr{int}V_\mr{C}}{t_1wV}
   \label{eq:specific_secondary_power}
\end{eqnarray}
The specific power tends to remain approximately constant for a given work material
operating under different cutting conditions. However, $u$ has been observed to
decrease with $t_1$ (size effect), according to the following approximate
relationship~\cite{shaw84}:
\begin{eqnarray}
   u \propto \frac{1}{t_1^{0.2}}
   \label{eq:specific_power3}
\end{eqnarray}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Heat Generation and Cutting Temperatures}

Practically all the work performed in machining is converted into
heat~\cite{trent84}, resulting in a temperature rise of the chip, tool, and work
material. Three zones of heat generation can be distinguished: the primary
deformation zone, the secondary tool-chip interface, and the tertiary zone; the
latter source is normally neglected if a sharp tool is used. The importance of
knowing the temperature distribution in metal cutting is very important, since
temperature affects the mechanical properties of the work material, chip, and,
therefore, the mechanics of cutting. Moreover, the rate of tool wear is also affected
by the temperature at the tool.

There are different methods for the measurement of cutting temperatures
\cite{bickel63,trent84,silva99}; the main methods are:

\begin{description}

   \item [Tool-work thermocouple method] This method employs the thermocouple\footnote{A
   thermocouple is formed when two dissimilar metals touch each other. When the
   temperature of this junction is different to the temperature of other parts of the
   metals, a voltage that is dependent on this temperature difference is generated.} effect
   formed by the contact of the two dissimilar metals at the tool-chip interface. Although
   this method is easy to implement, there is no certainty as to which interfacial
   temperature is actually being measured. However, it is normally assumed that the method
   measures the average temperature at the tool-chip interface.
   Stephenson~\cite{stephenson93} presented the theory and some implementation issues for
   the tool-work thermocouple method.

   \item [Inserted thermocouples] Small diameter thermocouples are inserted in different
   places in the interior of the tool, so that the temperature field within the tool can be
   obtained and the temperatures at the surfaces can be found by extrapolation. This
   technique is, however, tiresome because different tools are normally prepared for each
   individual thermocouple location. Chow and Wright~\cite{chow88} developed a technique
   in which the interfacial maximum, minimum, and tool edge temperatures are predicted
   from the signal of a thermocouple located at the bottom of a turning tool insert.

   \item [Radiation methods] These methods are based on the measurement of the infrared
   radiation of the temperature from accessible surfaces of the tool, work, and chip.
   Chao~\etal~\cite{chao61} used a special technique to measure the temperature
   distribution at the tool clearance face. Jaspers~\etal~\cite{jaspers98} developed a
   method to compute the tool-chip interface temperature from the temperature distribution
   at the top free side of the chip, obtained with an infra-red camera. Radiation methods
   are complex and are best suited for laboratory studies.

   \item [Tool hardness] The temperature distribution in the tool may be obtained by
   observing the changes in hardness of steel tools after machining. This method may
   provide more information about the temperatures near the cutting edge, but it is
   arduous, difficult, and only applies to steel tools.

\end{description}

There have been several attempts to predict cutting temperatures analytically. Silva and
Wallbank~\cite{silva99} presented a review of the analytical methods used to estimate
cutting temperatures. Stephenson~\cite{stephenson91} compared calculations from four
cutting temperature models with experimental data from infrared and tool-work thermocouple
measurements. Shear plane temperatures were obtained from the infrared data and tool-chip
interface temperatures from the tool-work thermocouple data. It was concluded that
although most cutting temperature models are qualitatively accurate, they tend to
overestimate cutting temperatures. Moreover, no model was accurate when predicting
tool-chip temperatures during a discontinuous chip formation process.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Metal Cutting Modelling}

Most metal cutting models deal with the prediction of the shear angle $\phi$ because,
for a given rake angle $\alpha$, undeformed chip thickness $t_1$, and width of cut $w$,
the geometry of the cutting process (Figure~\ref{fig:primary_zone}a) is not completely
defined unless $\phi$ is known. Therefore, no estimation of cutting forces, stresses,
and temperatures can be made without first evaluating $\phi$. Two types of models are
described next: the first type considers that deformation is performed at constant flow
stress, i.e.\ the effects of strain, strain rate, and temperature on material
properties are neglected; in contrast, the second type of model described makes use of
experimental flow stress data to develop an approximate machining theory in which
account is taken of the effect of strain, strain rate, and temperature.

\subsubsection{Constant Material Properties}
\label{sec:const_mat_prop}

One of the first attempts to quantitatively model metal cutting was achieved by Ernst
and Merchant~\cite{ernst41} who considered primary deformation to take place on a
single plane (AB in Figure~\ref{fig:primary_zone}a). Furthermore, the chip was assumed
to slide over the rake face, following the classical laws of friction, being the
coefficient of friction ($\mu = \tan{\lambda}$) expressed as in
Equation~\ref{eq:friction_coefficient}. The shear stress on plane AB ($\tau_\mr{AB}$)
was assumed to be uniform, as defined in Equation~\ref{eq:primary_stress1}, which can
also be expressed as follows:
\begin{eqnarray}
   \tau_\mr{AB} = \frac{R\cos{(\phi - \lambda - \alpha)}\sin{\alpha}}{t_1 w}
   \label{eq:primary_shear_stress}
\end{eqnarray}
where $R$ is the resultant cutting force ($R_1$ or $R_2$ in Figure~\ref{fig:forces}).
Ernst and Merchant reasoned that $\phi$ should be such that $\tau_\mr{AB}$ would be a
maximum, and a relationship for $\phi$ was obtained by differentiating
Equation~\ref{eq:primary_shear_stress} with respect to $\phi$ and equating the resulting
expression to zero. Differentiation was performed by considering both $R$ and $\lambda$
independent of $\phi$. As a result, the following relation for $\phi$ was derived:
\begin{eqnarray}
   \phi = \frac{\pi}{4} - \frac{1}{2}(\lambda - \alpha)
   \label{eq:shear_angle_ernst_merchant}
\end{eqnarray}
where $\phi$ is expressed in radians.

\begin{figure}[h]
   \center{\includegraphics{slip_line.eps}}
   \caption[Slip line field solution for machining.]
   {Slip line field solution for machining \cite{lee51}.
   (a)~Without BUE\@. (b)~With BUE\@.}
   \label{fig:slip_line}
\end{figure}
Lee and Shaffer~\cite{lee51} proposed a constant stress slip line~\footnote{In plane
strain deformation, slip lines consist of two families of orthogonal curves (I and
II), whose directions coincide with the direction of maximum shear stress ($k$)
acting on the plane of flow. Therefore, the two principal stresses ($\sigma_1$ and
$\sigma_3$) acting on this plane make $45^\circ$ with both slip lines. The normal to
the plane of flow is a principal direction with principal stress $\sigma_2 =
\frac{1}{2}(\sigma_1 + \sigma_3)$. Furthermore, $k = \frac{1}{2}|\sigma_1 -
\sigma_3|$ and $p = -\sigma_2$, where $p$ is the compressive stress that acts normal
to the slip line. In classical slip line field theory, materials have constant shear
flow stress $k$, and the following stress equilibrium equations apply: $p + 2k\psi =
\mbox{constant along a I line}$, and $p - 2k\psi = \mbox{constant along a II line}$,
where $\psi$ is the angular rotation of the I lines from a fixed reference axis. Slip
lines meet free surfaces and frictionless interfaces at $45^\circ$. In sticking
friction interfaces, one type of slip line meets the interface tangentially and the
other type at $90^\circ$. Reference~\cite{hosford83} may be consulted for further
information.} field, represented as triangle ABC in Figure~\ref{fig:slip_line}a, for
continuous chip formation processes without BUE\@. As in the previous
model~\cite{ernst41}, the shear plane AB was assumed to be a direction of maximum
shear stress. It should also be noted that in the previous model, AB also represents
a slip line, since it is a direction of maximum shear stress. It was further assumed
that BC represents the full tool-chip contact length, so that AC is a free surface
(angle between AB and CA is $45^\circ$). From analysis of the Mohr's circle diagram
of the field ABC represented in Figure~\ref{fig:slip_line}a, the following shear
angle expression was derived:
\begin{eqnarray}
   \phi = \frac{\pi}{4} - (\lambda - \alpha)
   \label{eq:shear_angle_lee_shaffer1}
\end{eqnarray}
where $\phi$ is expressed in radians.

Lee and Shaffer~\cite{lee51} presented a second slip line field solution for machining
operations with continuous chip formation and BUE\@. Figure~\ref{fig:slip_line}b shows
that the slip line field consists of a uniform stress region ABC and a region of varying
stress ABD representing the BUE\@. The size of the BUE is defined by the angle $\omega$.
Analysis of the Mohr's circle diagrams of both fields ABC and ABD resulted in the
following equation:
\begin{eqnarray}
   \phi = \frac{\pi}{4} + \omega - (\lambda - \alpha)
   \label{eq:shear_angle_lee_shaffer2}
\end{eqnarray}
Angle $\phi$ is expressed in radians. Equation~\ref{eq:shear_angle_lee_shaffer2}
differs from Equation~\ref{eq:shear_angle_lee_shaffer1} only by $\omega$ (size of
BUE). When there is no BUE, $\omega$ is zero, and, consequently,
Equation~\ref{eq:shear_angle_lee_shaffer2} results in
Equation~\ref{eq:shear_angle_lee_shaffer1}.

Kobayashi and Thomsen~\cite{kobayashi59} presented experimental results for a range of
work materials by plotting the shear angle~$\phi$ against $\lambda - \alpha$. It was
observed that, generally, for a given material, an increase in $\lambda - \alpha$ results
in a decrease of $\phi$. However, for a given value of $\lambda - \alpha$, different
values of $\phi$ were observed for different materials. Moreover,
Equations~\ref{eq:shear_angle_ernst_merchant},
\ref{eq:shear_angle_lee_shaffer1}~and~\ref{eq:shear_angle_lee_shaffer2} represent straight
lines of type
\begin{eqnarray}
   \phi = B - m(\lambda - \alpha)
   \label{eq:shear_angle_general_solution}
\end{eqnarray}
Therefore, since  $m$ and $B$ are known constants in
Equations~\ref{eq:shear_angle_ernst_merchant}, \ref{eq:shear_angle_lee_shaffer1},
and~\ref{eq:shear_angle_lee_shaffer2}, an unique solution for $\phi$ is obtained for a
given value of $\lambda - \alpha$ (Equation~\ref{eq:shear_angle_lee_shaffer2} may
actually give different $\phi$ values if $\omega$ is allowed to vary), which is clearly
unsuitable to represent the actual variation of $\phi$ with $\lambda - \alpha$. Better
agreement with experimental results is achieved if
Equation~\ref{eq:shear_angle_general_solution} is utilized, but the values of $B$ and
$m$ have to be provided. The problem lies in the simplifications assumed for each
analytical model, which may be a considerable cause of errors, such as: in the first
model~\cite{ernst41}, tool-chip sliding friction is assumed, and differentiation is
performed independently of $R$ and $\lambda$; and the work material is assumed
perfectly plastic in the slip line field models~\cite{lee51}. Shaw~\cite{shaw84}
reviewed the better known shear angle analysis and concluded that it is unlikely that a
relatively simple model will possibly be established. Oxley~\cite{oxley98} also stated
that, at this stage, a purely analytical solution to the machining problem seems
unachievable. Nevertheless, although oversimplified, analytical models have undoubtedly
added to the understanding of the cutting process.

\subsubsection{Variable Material Properties}
\label{sec:var_mat_prop}

The machining theory briefly presented next was derived by Oxley and his co-workers;
it is described in detail in~\cite{oxley89} and is briefly reviewed in
Appendix~\ref{chap:oxley}. Chip formation is assumed to be formed in the narrow
plastic zone represented by the parallel sided shear zone CDEF in
Figure~\ref{fig:primary_zone}a. In the centre of this zone, the shear plane AB is
assumed to be a direction of maximum shear stress and maximum shear strain rate. The
tool-chip interface is represented by a rectangular plastic zone with uniform shear
stress distribution and triangular normal stress distribution \cite{fenton68}. In
order to predict the shear angle~$\phi$, cutting forces, and temperatures for a given
set of cutting conditions ($V$, $t_1$, and $w$) and tool rake angle $\alpha$, an
iterative process was developed. Basically, this process starts by selecting a range
of $\phi$ values upon which, for each value of $\phi$, the shear stress at the
tool-chip interface ($\tau_\mr{int}$) is calculated from the stresses at the shear
plane AB, and for the same range of $\phi$ values, the interfacial temperatures and
strain rates are calculated and, subsequently, the shear flow stress of the chip at
the interface ($k_\mr{int}$). The solution is taken for the value of~$\phi$ at which
$\tau_\mr{int} = k_\mr{int}$.

The work material flow stress properties were obtained from high speed compression
tests for a range of plain carbon steels and are represented by the two material
constants $\sigma_1$ and $n$, which define the empirical stress-strain equation $\sigma
= \sigma_1\epsilon^n$ for a given temperature and strain rate, where $\sigma$ and
$\epsilon$ are the uniaxial\footnote{The uniaxial stress ($\sigma$) is the stress
necessary to initiate yielding in uniaxial tension or compression tests and is related
to the shear flow stress $k$ by the equation $\sigma = \sqrt{3}k$. Furthermore,
$\epsilon = \gamma/\sqrt{3}$ and $\dot{\epsilon} = \dot{\gamma}/\sqrt{3}$, where
$\epsilon$ and $\dot{\epsilon}$ are the uniaxial strain and strain rates, and $\gamma$
and $\dot{\gamma}$ are the maximum shear strain and shear strain rate in plane strain.}
flow stress and strain, respectively. Therefore, this model has only been applied to
plain carbon steels, and in order to obtain $\sigma_1$ and $n$, the material
temperature ($T$), the uniaxial strain rate ($\dot{\epsilon}$), and the percentage of
carbon in the steel are required quantities.

The method to determine $\tau_\mr{int}$ for a given value $\phi$ starts by
calculating the shear strain ($\gamma_\mr{AB}$) and shear strain rate
($\dot{\gamma}_\mr{AB}$) at AB (Figure~\ref{fig:primary_zone}a).
Equation~\ref{eq:shear_strain_rate_AB3} is used to obtain $\dot{\gamma}_\mr{AB}$.
Since the temperature at AB ($T_\mr{AB}$) is still unknown, at a first approach,
$T_\mr{AB}$ is assumed to be equal to the initial workpiece temperature ($T_\mr{W}$),
and this value is used to determine the specific heat and thermal conductivity of the
material. With the values of $T_\mr{AB}$ and $\dot{\gamma}_\mr{AB}$, the work
material flow stress properties $\sigma_1$ and $n$ can be determined, and, thus, the
shear flow stress at AB ($k_\mr{AB}$). Subsequently, the shear force $F_\mr{AB}$ is
computed by substituting $\tau_\mr{AB}$ by $k_\mr{AB}$ in
Equation~\ref{eq:primary_stress1}. At this point, it is possible to recalculate
$T_\mr{AB}$ by considering that all the work produced in the primary zone is
converted into heat and that a known proportion of this heat is conducted into the
work material. The calculations are then repeated with the new value of $T_\mr{AB}$,
and this procedure is repeated until $T_\mr{AB}$ converges to a specific value.
Finally, the cutting forces at this converged temperature value are taken as the
appropriate values for the assumed $\phi$, and the shear stress $\tau_\mr{int}$ is
calculated from Equation~\ref{eq:secondary_stress1}.

In order to compute $k_\mr{int}$ for a given angle $\phi$, an iterative procedure is
again necessary to calculate the mean chip temperature ($T_\mr{C}$). Initially,
$T_\mr{C}$ is taken as the temperature in the primary shear zone, and this is then used
to calculate the specific heat of the material. The temperature $T_\mr{C}$ is then
recalculated assuming that all the work done by the chip at the interface is converted
into heat. This procedure is repeated until $T_\mr{C}$ converges to a specific value.
Afterwards, an empirical relation is used to find the average temperature at the
interface ($T_\mr{int}$). The average shear strain rate ($\dot{\gamma}_\mr{int}$) in
the plastic zone adjacent to the interface is determined from an empirical equation.
Finally, in order to calculate $k_\mr{int}$, a stress-strain relation that neglects the
influence of strain on flow stress is used, and, therefore, only the flow stress
property $\sigma_1$ needs to be determined.

If $\tau_\mr{int}$ and $k_\mr{int}$ are plotted against $\phi$, the solution for
$\phi$ is taken at the intercept of $\tau_\mr{int}$ with $k_\mr{int}$. However, when
there is more than one intercept, Hastings~\etal~\cite{hastings74} reasoned that the
solution should be taken for the intercept corresponding to the highest value of
$\phi$.

Hastings~\etal~\cite{hastings80} applied this theory for two plain carbon steels and
for a range of cutting conditions. Excellent agreement was obtained between predicted
and experimental cutting forces for machining without BUE ($F_\mr{X}$ and $F_\mr{Y}$
decreasing with increasing $V$ and $\alpha$). The specific power $u$ was observed to
decrease with $t_1$ (size effect). Furthermore, $\phi$ was predicted and experimentally
found to increase with $V$. Hastings~\etal\ proposed that the decrease in the cutting
forces and the increase in $\phi$ with $V$ are due to the decrease in $k_\mr{int}$
caused by the increase of the interfacial temperatures. The cutting forces were hardly
affected by the difference in carbon content of the two work materials; however, the
carbon content was observed to increase $\phi$. The secondary shear zone thickness
$\Delta s_2$ was predicted to decrease with increasing $V$, $\alpha$, and carbon
content, and with decreasing $t_1$; these trends were confirmed by photomicrographs of
chip sections.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tool Wear}

Although the present work does not consider the wear of the cutting tool, a brief
overview of the types of tool wear encountered is given here.

\begin{figure}[h]
   \center{\includegraphics{tool_wear.eps}}
   \caption{Flank and crater tool wear.}
   \label{fig:tool_wear}
\end{figure}

During metal cutting, the tool is subjected to high stresses and temperatures, the action
of cutting gradually changes the shape of the tool edge, resulting in ineffective cutting
or total tool failure. The main causes of tool deterioration are described next
\cite{trent59,trent84,shaw84}:
\begin{description}

   \item [Flank or clearance face wear] As shown in Figure~\ref{fig:tool_wear}, this type
   of wear results in a loss of relief angle on the clearance face and is formed due to
   the frictional contact between the clearance face and the newly machined surface of the
   workpiece.

   \item [Crater wear] At high cutting speeds, due to the high temperatures and stresses
   present at the rake face, a characteristic crater develops at a short distance from the
   cutting edge (Figure~\ref{fig:tool_wear}), caused by the sticking contact of the chip
   with the rake face.

   \item [Plastic deformation] When the cutting edge reaches a sufficiently high
   temperature, plastic deformation starts due to the weakening of the mechanical strength
   of the tool  material.

   \item [Fracture and chipping] Tool fracture and chipping result due to excessive
   loading or thermal-mechanical shock at the cutting edge. Chipping causes a small damage
   of the tool edge, whereas fracture results in a significant loss of the cutting zone.
   Tool failure due to fracture must be avoidable due to its unpredictable nature.

\end{description}
