2. Research on Iterative Algorithm of Thermal-Mechanical Coupling in Orthogonal Cutting
Ji Xia [
8] proposed an iterative algorithm of thermal-mechanical coupling in orthogonal cutting. The cutting force predicted by the Oxley’s cutting force model is used as the input of the cutting temperature prediction model to obtain the cutting temperature, and then put it into the Oxley’s cutting force model to obtain a new cutting force. After several iterations, when the output cutting force converges to a certain value, at this time, it is considered that the cutting force and the cutting temperature have reached a balance, which is the process of thermal-mechanical coupling.
The Oxley’s cutting force model considers that the strain rate on the shear zone is almost constant. This view was confirmed by Stevenson and Oxley [
9]. Furthermore, it can be considered that the flow velocity V
S on the shear zone is constant. In addition, it is assumed that the flow stress on the shear zone is uniformly distributed at a certain instant. So the cutting force can be calculated based on the uniformly distributed stress on the shear zone, and the cutting mechanism is analyzed based on the geometric relationship. The assumptions of the model are as follows:
- (1)
There is no tool wear and cutting vibration, and no built-up edge is generated.
- (2)
Plastic deformation in the cutting layer only occurs in a plane perpendicular to the cutting edge.
- (3)
The primary and secondary deformation zones are assumed to be narrow area.
- (4)
The temperature and strain on the shear plane are uniformly distributed.
The geometric relationship of force and material flow speed in three different directions (along the cutting direction, along the shear plane AB and along the tool–chip interface) is shown in
Figure 1. The resultant cutting force
Ptotal can be expressed by the flow stress
σAB on the shear plane AB by the geometric relationship as:
where
hc is the cutting thickness,
w is the width of the cutting edge participating in cutting,
Φ is the shear angle of the primary shear zone, and
θ is the angle between the force
Ptotal and the shear plane AB. The friction
F and normal force
N at the rake face of the tool, the force
Fc in the cutting direction and the force
Ft in the feed direction can be expressed as:
where
α is the rake angle of the tool and
λ is the friction angle of the rake face.
Shear flow stress
σAB can be obtained from various constitutive models, such as the Johnson–Cook constitutive model, Zerilli–Armstrong constitutive model, Bammann–Chiesa–Johnson constitutive model, etc. Hyperbolic tangent (TANH) constitutive model is applied to predict the shear flow stress at shear zone. The constitutive model is proposed by Calamaz [
10]. It considers the strain softening effect of Ti6Al4V on the basis of J-C constitutive model. According to the study of Ducobu [
11], the constitutive parameters are shown in the
Table 1.
Since many parameters in the Oxley’s cutting force model need to be determined by experiments, Ji Xia [
8] proposed an iterative algorithm to capture the parameters that need to be experimentally defined in the Oxley′s cutting force model. This algorithm makes the input parameters of the cutting force model include only material parameters, process parameters and tool geometry parameters, which improves the predictive efficiency.
The model developed by Huang and Liang [
12] is used in the present study to predict the cutting temperature. This model considers the combined effect of the heat source in primary and secondary deformation zones, the non-uniform heat distribution ratio along the chip contact surface to obtain the temperature along the tool–chip interface. In the present study, the temperature generated by the friction in the tertiary deformation zone is considered. The heat source in the tertiary deformation zone is calculated based on the model proposed by Su [
1]. It is assumed that the heat distribution ratio in the tertiary deformation zone is constant. The temperature of chips and tools is calculated by the following equation.
where
Ri,
R′I,
R″i are the distances between any point on the workpiece and a certain differential unit on the heat source.
B(x) is the percentage of heat generated at the tool–chip interface entering the chip. 1 −
γ is the percentage of heat generated by the heat source
qrub entering the tool.
Assuming that the temperature at the side of the tool and the temperature at the side of the chip are consistent on the adiabatic boundary, the following relationship exists at the tool–chip interface.
B(x) is calculated by Equation (7) and the temperatures in the primary shear zone and the temperature at the tool–chip interface are obtained by Equations (3) and (4), respectively.
3. Milling Force Prediction Based on Analytical Method
When the milling force model is established, the cutting-edges of the tool are discretized along the axis of the tool, and the cutting process of each discrete cutting edge is regarded as an oblique cutting to calculate separately, as shown in
Figure 2a,b. Milling force is mainly divided into two parts: chip forming force and ploughing force. The chip forming force mainly stems from the normal pressure
N and the friction force
F exerted by the chip on the tool. The ploughing force mainly stems from the force
Pcut in the cutting direction and the force
Pthrust perpendicular to the cutting direction caused by the relative movement between the tool tip and the workpiece, and the detailed calculations refer to the study of Su [
1]. The cutting thickness and direction of the cutting force are different at different positions on the cutting-edge during milling. Considering the influence of chip side flow on force and temperature is challenging, so it is considered that the chip side flow angle is zero during milling which makes it easier to introduce the iterative algorithm of thermal-mechanical coupling in orthogonal cutting.
Figure 2b illustrates the geometric relationship of the oblique cutting process without considering the phenomenon of chips side-flow. Obviously, the angle between the frictional force
F and the radial force
Fr is the rake angle
α. Make a section that is perpendicular to the cutting edge, as shown in
Figure 2c, the rotation matrix
M1 can be obtained as:
Then the radial force
Fr and the circumferential force
Ft′ perpendicular to the cutting edge are expressed as:
As shown in
Figure 2b, the rotation matrix
M2 considering the inclination angle
β can be obtained as:
Then the radial force
Fr, the circumferential force
Ft, and the axial force
Fa are expressed as:
Figure 3 shows the trajectory of a point on the cutting edge with an axial height
z during milling. The rotation matrix
M3 considering the rotation angle of the tool can be obtained as:
where
is the tool speed,
t is the cutting time,
z is the axial height of the tool,
is the tool helix angle,
R is the tool radius,
can be expressed as:
Then the cutting force
dFX along the feed direction, the cutting force
dFY in the direction perpendicular to the workpiece surface, and the tool axial cutting force
dFZ on the micro cutting edge are expressed as:
Assuming that the milling process is down milling, the total cutting time
Tc is divided into three stages, as shown in
Figure 3. The first stage is the cut-in stage. A part of the cutting edge is in contact with the workpiece, which occurs during the point on the end of the tool moves from A to B. The second stage is the stable cutting stage. The cutting edge is completely in contact with the workpiece, which occurs during the point on the end of the tool moves from B to C. The third stage is the cut-out stage. A part of the cutting edge is in contact with the workpiece, which occurs during the point on the end of the tool moves from C to D.
The total milling force is calculated based on three cutting stages, and its expression is:
where
n is the number of micro cutting edges involved in the cutting process.
The cutting thickness
hc during milling can be considered as a function of the independent variable
t.
Figure 4 is a geometric schematic of the cutting thickness, where points O
1 and O
2 are the positions of the tool axis at time
t1 and
t2 respectively, and point O is the position of the tool axis when the cutting edge cuts into the workpiece. A reference coordinate system XOY is established with the point O as the origin. The cutting thickness
hc is defined as the length of the workpiece passing between the point on the cutting edge and the axis of the tool, and its expression is:
where
is the coordinate of the cutting edge,
is the coordinate of a point on the workpiece surface.
As shown in
Figure 4, when the coordinate of a point on the cutting edge is
, the cutting thickness
hc is the distance between the cutting edge and the surface which is to be machined, and when the coordinate of a point on the cutting edge is
, the cutting thickness
hc is the distance between the cutting edge and the machining surface. Coordinate point
is used as the judgment standard, which can be expressed by the cutting amount and the geometric angle of the tool as:
where
is the feed per tooth,
is the cut-in angle, which is obtained from the geometric relationship and its expression is:
.
Assuming that the cutting process is ideal without vibration and tool wear, it can be considered that the distance in the horizontal direction between the cutting path of the previous tooth and the cutting path of the current tooth is
. In the reference coordinate system XOY, the current path can be expressed as:
As shown in
Figure 4, A and B are on the same straight line. If the slope of the straight line is
k, the equation can be obtained:
where
t1 is the time for the cutting edge to move to
on the cutting path of the previous cutter tooth.
Calculate
t1 by Equation (19) and substitute it into Equation (20) to calculate
.
Finally,
and
are substituted into Equation (16) to calculate the cutting thickness
hc. When the cutting thickness
hc is the distance between the cutting edge and the surface which is to be machined, since
is known,
can be obtained by Equation (21).
The calculation process of the milling force is shown in
Figure 5.