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\chapter{Oxley's Theory}
\label{chap:oxley}


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This appendix is a complement to Section~\ref{sec:var_mat_prop}, where the
calculation of parameters, such as the geometry, cutting forces, stresses, and
temperatures of orthogonal cutting is described according to a predictive machining
model formulated by Oxley and his co-workers \cite{oxley89}. The inputs for the model
are the material properties, cutting speed ($V$), uncut chip thickness ($t_1$), width
of cut ($w$), tool rake angle ($\alpha$), and initial work material temperature
($T_\mr{W}$). The following section describes the necessary material properties model
inputs, and, in the subsequent section, the calculation required to attain the
machining prediction.


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\section{Material Properties} \label{sec:mat_prop}

An important assumption is that the plastic stress-strain properties of the work
material are represented by the empirical equation
\begin{eqnarray}
   \sigma = \sigma_1\epsilon^n
   \label{eq:stress_strain}
\end{eqnarray}
where $\sigma$ and $\epsilon$ are the uniaxial stress and strain respectively, and
$\sigma_1$ and $n$ are material constants that depend on strain rate and temperature.
Uniaxial flow stress, strain and strain rate ($\sigma$, $\epsilon$ and $\dot{\epsilon}$
respectively) are related to plane stress, strain and strain rate ($k$, $\gamma$, and
$\dot{\gamma}$, respectively) by the following relationships:
\begin{eqnarray}
   \sigma & = & \sqrt{3}k
   \label{eq:uniaxial_stress} \\
   \epsilon & = & \frac{\gamma}{\sqrt{3}}
   \label{eq:uniaxial_strain} \\
   \dot{\epsilon} & = & \frac{\dot{\gamma}}{\sqrt{3}}
   \label{eq:uniaxial_strain_rate}
\end{eqnarray}
The values of $\sigma_1$ and $n$ were obtained from stress-strain data of high speed
compressions tests for a range of carbon steels\cite{oyane67}. Since $\sigma_1$ and
$n$ depend on strain rate and temperature, a set of functions was developed to relate
$\sigma_1$ and $n$ to a single variable that combines the effects of strain rate and
temperature, called velocity-modified temperature ($T_\mr{mod}$)~\cite{macgregor46}.
These functions are stated in reference~\cite{oxley89}, where $\sigma_1$ and $n$ are
related to $T_\mr{mod}$ and carbon content. Figure~\ref{fig:flow_stress} shows a plot
of these functions for both 080A15 and 080M40 carbon steels.
\begin{figure}[h]
   \center{\includegraphics{flow_stress.eps}}
   \caption[Flow stress properties for work materials 080A15 and 080M40.]
   {Flow stress properties for work materials 080A15 and 080M40.
   (a)~$\sigma_1$ (b)~$n$.}
   \label{fig:flow_stress}
\end{figure}
Moreover, $T_\mr{mod}$ can be found from the following
relationship:
\begin{eqnarray}
   T_\mr{mod} =
   T \left( 1 - \nu\log_{10}\frac{\dot\epsilon}{\dot\epsilon_0} \right)
   \label{eq:vel_mod_temp}
\end{eqnarray}
where $T$ is the temperature in K, and $\nu$ and $\dot\epsilon_0$ are constants
($\dot\epsilon_0 = 1/$s and $\nu = 0.09$).

The thermal properties of the work material are also required for the model
computation: specific heat ($c$) and thermal conductivity ($\kappa$). Empirical
temperature dependent equations for the computation of $c$ and $\kappa$ are presented
in reference~\cite{oxley89}. For carbon steels, $c$ is given as independent from
chemical composition:
\begin{eqnarray}
   c = 420 + 0.504T
   \label{eq:specific_heat}
\end{eqnarray}
where $c$ is in \mbox{J$/$kg-K}, and the temperature $T$ is in $^\circ$C\@. However,
chemical composition dependent functions are given for $\kappa$. For example, for the
080A15 material composition,
\begin{eqnarray}
   \kappa = 52.78 - 0.0283T
   \label{eq:specific_heat1}
\end{eqnarray}
and for the 080M40,
\begin{eqnarray}
   \kappa = 49.35 - 0.0245T
   \label{eq:specific_heat2}
\end{eqnarray}
where $\kappa$ is in \mbox{W$/$m-K}, and the temperature $T$ is in $^\circ$C\@.
Moreover, since the effect of temperature on work material density ($\rho$) is
negligibly small, $\rho$ was assumed constant and equal to 7862~kg$/$m$^3$.


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\section{Computation Procedure}

As it can be observed in Figure~\ref{fig:oxley}, the main computation procedure of
Oxley's model is structured by three nested loops: in the outer loop, $\delta$ (ratio
of secondary plastic zone thickness to chip thickness $t_2$) is varied until the
tangential force ($F_\mr{X}$) reaches a minimum value; in the middle loop, for each
$\delta$, $C$ (empirical primary strain rate constant) is varied until $\sigma_\mr{N}
\approx \sigma'_\mr{N}$, where both $\sigma_\mr{N}$ and $\sigma'_\mr{N}$ represent the
normal stress at the cutting edge (B in Figure~\ref{fig:primary_zone}a) found by two
different methods; and in the inner loop, for each $C$, $\phi$ (shear angle) is varied
until $\tau_\mr{int} \approx k_\mr{int}$, where $\tau_\mr{int}$ and $k_\mr{int}$
represent the shear flow stress of the chip at the tool-chip interface found by
analysis of the stress distributions along the primary shear plane (AB in
Figure~\ref{fig:primary_zone}a) and secondary tool-chip interface, respectively.
Therefore, in order to find $\phi$, the calculations are divided into two stages:
analysis of primary zone and secondary zone stresses.

The first stage of the calculations begins by calculating the shear strain
($\gamma_\mr{AB}$) and the shear strain rate ($\dot{\gamma}_\mr{AB}$) at AB:
\begin{eqnarray}
   \gamma_\mr{AB} & = & \frac{\cos{\alpha}}{2\sin{\phi}\cos{(\phi - \alpha})}
   \label{eq:SS_AB} \\
   \dot{\gamma}_\mr{AB} & = & C\frac{V_\mr{S}}{l_\mr{AB}}
   \label{eq:SSR_AB}
\end{eqnarray}
where the length of AB
\begin{eqnarray}
   l_\mr{AB} = \frac{t_1}{\sin{\phi}}
   \label{eq:l_AB}
\end{eqnarray}
and the shear velocity
\begin{eqnarray}
   V_\mr{S} = \frac{V \cos{\alpha}}{\cos{(\phi - \alpha)}}
   \label{eq:V_S}
\end{eqnarray}
Then, the uniaxial strain ($\epsilon_\mr{AB}$) and strain rate ($\dot\epsilon_\mr{AB}$)
are calculated from Equations~\ref{eq:uniaxial_strain}
and~\ref{eq:uniaxial_strain_rate}. Afterwards, since the temperature at AB
($T_\mr{AB}$) is unknown, $T_\mr{AB}$ is initially equated to $T_\mr{W}$, so that
$T_\mr{mod}$ at AB can be calculated (Equation~\ref{eq:vel_mod_temp}) in order to
obtain $\sigma_1$ and $n$, as described in Section~\ref{sec:mat_prop}. The shear force
along AB is then calculated:
\begin{eqnarray}
   F_\mr{AB} = \frac{k_\mr{AB}t_1w}{\sin{\phi}\cos{\phi}}
   \label{eq:F_AB}
\end{eqnarray}
where $k_\mr{AB}$ can be found from Equation~\ref{eq:uniaxial_stress}. In order to
calculate the rise in temperature at AB ($\Delta T_\mr{AB}$), the proportion of heat
conducted into the work must be determined first:
\begin{eqnarray}
   \beta = \left\{
   \begin{array}{lll}
   0.5 - 0.35\log_{10}{(R_\mr{T}\tan{\phi)}}
   & \mbox{for} & 0.04 \leq R_\mr{T}\tan{\phi} \leq 10.0 \\
   0.3 - 0.15\log_{10}{(R_\mr{T}\tan{\phi})}
   & \mbox{for} & R_\mr{T}\tan{\phi} \geq 10.0
   \end{array}
   \right.
   \label{eq:heat_cond_work}
\end{eqnarray}
where $R_\mr{T}$ is a non-dimension thermal number given by
\begin{eqnarray}
   R_\mr{T} = \frac{\rho cVt_1}{\kappa}
   \label{eq:R_T}
\end{eqnarray}
and $c$, $\kappa$ and $\rho$ are obtained with the equations stated in
Section~\ref{sec:mat_prop}. At this point, it is possible to calculate $\Delta
T_\mr{AB}$:
\begin{eqnarray}
   \Delta T_\mr{AB} = \eta \Delta T_\mr{SZ}
   \label{eq:rise_T_AB}
\end{eqnarray}
where the rise in temperature in the primary shear zone
\begin{eqnarray}
   \Delta T_\mr{SZ} = \frac{1 - \beta}{\rho c t_1 w} \times
                          \frac{F_\mr{AB}\cos{\alpha}}{\cos{(\phi - \alpha)}}
   \label{eq:rise_T_SZ}
\end{eqnarray}
and $\eta$ is a correction factor because not all plastic work of chip formation occurs
at AB ($\eta = 0.7$). The temperature $T_\mr{AB}$ can then be recalculated:
\begin{eqnarray}
    T_\mr{AB} = T_\mr{W} + \Delta T_\mr{AB}
    \label{eq:T_AB}
\end{eqnarray}
The previous temperature dependent calculations (Equations~\ref{eq:vel_mod_temp}
and~\ref{eq:F_AB}--\ref{eq:rise_T_AB}) are then repeated until the temperature rise
$\Delta T_\mr{AB}$ converges to a constant value. Afterwards, the resulting cutting
force is calculated:
\begin{eqnarray}
    R = \frac{F_\mr{S}}{\cos{\theta}}
    \label{eq:R}
\end{eqnarray}
where $\theta$ is the angle between $R$ and AB:
\begin{eqnarray}
    \tan{\theta} = 1 + 2 \left(\frac{\pi}{4} - \phi \right) - Cn
    \label{eq:theta}
\end{eqnarray}
Before calculating $k_\mr{int}$, the friction force at the tool-chip interface must be
evaluated:
\begin{eqnarray}
    F_\mr{int} = R\sin{\lambda}
    \label{eq:F_int}
\end{eqnarray}
where the friction angle
\begin{eqnarray}
    \lambda = \theta - \phi + \alpha
    \label{eq:lambda}
\end{eqnarray}
Finally,
\begin{eqnarray}
    \tau_\mr{int}= \frac{F_\mr{int}}{lw}
    \label{eq:k_int}
\end{eqnarray}
where the tool-chip contact length
\begin{eqnarray}
    l_\mr{int} = \frac{t_1 \sin{\theta}}{\cos{\lambda\sin{\phi}}}
    \left[ 1 + \frac{Cn}{3(1 + \frac{\pi}{2} - 2\phi - Cn)} \right]
    \label{eq:l}
\end{eqnarray}

The second stage of the calculations starts by assuming that the initially unknown mean
chip temperature ($T_\mr{C}$) is equal to $T_\mr{W} + \Delta T_\mr{SZ}$, so that $c$
can be determined (Section~\ref{sec:mat_prop}), and thus the mean temperature rise in
the chip:
\begin{eqnarray}
    \Delta T_\mr{C} =
    \frac{F_\mr{int}\sin{\phi}}{\rho c t_1 w \cos{(\phi - \alpha)}}
    \label{eq:rise_T_C}
\end{eqnarray}
The temperature $T_\mr{C}$ is then recalculated:
\begin{eqnarray}
    T_\mr{C} = T_\mr{W} + \Delta T_\mr{SZ} + \Delta T_\mr{C}
    \label{eq:T_C}
\end{eqnarray}
and the calculations of $c$ and $\Delta T_\mr{C}$ are then repeated until $\Delta
T_\mr{C}$ converges to a constant value. Thereafter, $\kappa$ must be calculated
(Section~\ref{sec:mat_prop}) in order to determine $R_\mr{T}$ (Equation~\ref{eq:R_T}),
so that the maximum temperature rise in the chip ($\Delta T_\mr{max}$) can be computed:
\begin{eqnarray}
    \log_{10}{\frac{\Delta T_\mr{max}}{\Delta T_\mr{C}}} =
    0.06 - 0.195\delta \sqrt{\frac{R_\mr{T}t_2}{l}} +
    0.5\log_{10}{\frac{R_\mr{T}t_2}{l}}
    \label{eq:rise_T_max}
\end{eqnarray}
The chip thickness
\begin{eqnarray}
    t_2 = \frac{\cos{(\phi - \alpha)}}{\sin{\phi}}t_1
    \label{eq:t_2}
\end{eqnarray}
The mean tool-chip interface temperature can then be determined:
\begin{eqnarray}
    T_\mr{int} = T_\mr{W} + \Delta T_\mr{SZ} + \psi \Delta T_\mr{max}
    \label{eq:T_int}
\end{eqnarray}
where $\psi$ is a correction factor since $T_\mr{int}$ is an average value ($\psi =
0.7$). The mean shear strain rate at the interface must also be calculated:
\begin{eqnarray}
    \dot{\gamma}_\mr{int} = \frac{V_\mr{C}}{\delta t_2}
    \label{eq:SSR_int}
\end{eqnarray}
where the chip velocity
\begin{eqnarray}
    V_\mr{C} = \frac{V \sin{\phi}}{\cos{\phi - \alpha}}
    \label{eq:V_C}
\end{eqnarray}
The uniaxial strain rate at the interface ($\dot\epsilon_\mr{int}$) is calculated from
Equation~\ref{eq:uniaxial_strain_rate}. Hence, having found $T_\mr{int}$ and
$\dot\epsilon_\mr{int}$, $T_\mr{mod}$ can then be calculated from
Equation~\ref{eq:vel_mod_temp}, and hence $\sigma_1$, as described in
Section~\ref{sec:mat_prop}. Finally, $k_\mr{int}$ can be calculated as follows:
\begin{eqnarray}
    k_\mr{int} = \frac{\sigma_1}{\sqrt{3}}
    \label{eq:k'_int}
\end{eqnarray}
This equation neglects the influence of strain on flow stress.

In conclusion, a unique value for $\phi$ is found, for each $C$ value, when
$\tau_\mr{int} \approx k_\mr{int}$. Next, an unique value for $C$ and $\phi$ is found,
for each $\delta$ value, when $\sigma_\mr{N} \approx \sigma'_\mr{N}$, where
\begin{eqnarray}
    \sigma_\mr{N}  & = & \frac{R \cos{\lambda}}{l w}
    \label{eq:sigma_N} \\
    \sigma'_\mr{N} & = & \left( 1 +  \frac{\pi}{2} - 2\alpha - 2Cn \right) k_\mr{AB}
    \label{eq:sigma'_N}
\end{eqnarray}
And, finally, an unique value for $\delta$, $C$, and $\phi$ is found, when $F_\mr{X}$
is minimum, so that an unique solution for the cutting forces, stresses, and
temperatures is determined.

\drawingscale{0.35mm} \placedrawing{oxley.tex}{Oxley's model computation
diagram.}{fig:oxley}
