The geometrical parameters associated to oblique machining are illustrated in Fig.1.
The inclination angle 2, is measured between the cutting edge, modelled as a straight line, and the normal to the cutting velocity direction. The normal rake angle, the undeformed chip thickness and the width of cut are given respectively by a11 , t1 and w . When the inclination angle 2, is different from zero, the chip flow direction is not normal to the cutting edge. This direction is characterised by the angle 'le which has to be determined.
Stationary oblique cutting is analysed by using a thermomechanical modelling of the material flow within the primary shear zone combined with a temperature dependent
MODELLING OF CUTTING AND EXPERIMENTAL VALIDATION 53
friction law at the tool-chip interface, see Moufki et al. (2000). This friction law was introduced to consider the extreme conditions of pressure, velocity and temperature during machining, at the tool-chip interface, Moufki et al. ( 1998). The proposed oblique cutting model assumes that the primary shear zone is a thin band of constant thickness h. Its inclination with respect to the workpiece free surface is given, in a plane normal to the cutting edge, by the angle ¢" called normal shear angle. The plastic deformation of the workpiece is supposed to be limited to this band. The secondary shear zone, due to the contact and friction at the tool-chip interface, and the complex material flow near the tool edge are not considered. Therefore, all the variables introduced to describe the flow through the band depend only on the coordinate z along the normal to the band (one- dimensional formulation), see Fig. 2. In addition, the shearing in the band is supposed to be adiabatic. This assumption is reasonable when the cutting speeds are large enough.
Therefore, this approach is especially appropriate for high speed machining.
It is well known that strain rates larger than 103 sã1 can be attained in the primary shear zone, even for conventional cutting velocities. In this zone, the deformations are large and the temperature may be of the order of 300 - 500 °C. Then, to model the cutting process, the thermomechanical behaviour of the workpiece material at these conditions has to be identified. In the proposed model, the workpiece material is considered to be isotropic, rigid viscoplastic and its behaviour is given by a Johnson-Cook law:
(1)
The variables y , y , and () represent respectively the shear strain, the shear strain rate and the absolute temperature. Characteristics of the material behaviour are defined by the strain hardening exponent n, the strain rate sensitivity m , the thermal softening coefficient v and by A, B, Yo, and the temperatures T, (reference temperature) and
T1 (melting temperature).
In the primary shear zone, the plastic deformation is supposed to follow the 12 -flow theory. Then, it can be demonstrated that the shearing direction in the band, characterised by the unit vector x, (Fig. 2.), is independent of the coordinate z and is given by:
V -V ( )
x, = jjv: _Vii= - cos(17,) x + sin(17,) y (2)
where (x, y, z) is the orthonormal basis associated to the primary shear band. Ve and V are respectively the chip velocity and the cutting speed vectors, see Fig. 1. The chip velocity magnitude Ve is calculated by considering the incompressibility condition:
54 A. MOUFKI ET AL.
V = V cosA., sin¢,,
c cos17ccos(¢11 -a11) (3)
where Vis the cutting speed intensity.
(a)
Shear zone
Workpiece
Xn Shearing direction in the band
Yn
/Views
(b)
Culling edge
Figure. 2. The primary shear zone.
(a) View in a plane normal to the cutting edge, the orthonormal basis (x. y, z) is associated to the band.
(b) View in a plane parallel to the band, x, is the shearing direction independent of the coordinate z.
The angle 1], , characterising the direction of shearing is then deduced from the following equation:
_1 ( tan('lc) sin(¢,,) - tan( A.,) cos(¢,, - a,,))
17, =tan
cos( a,,)
where the chip flow angle on the tool rake face 'le will be determined later.
(4)
The conservation equations of momentum, of energy (assuming adiabatic conditions), and the constitutive law (I) can be respectively written as, Dudzinski and Molinari (1997):
r = p (v cos( A.,) sin(¢,, ))2 y + <0 (5)
MODELLING OF CUTTING AND EXPERIMENTAL VALIDATION 55
B = B,v + Pcl _pj p (v cos(A.s) sin(t,1)11))2 i_ + r0 rJ
2
(6)
(7)
where p , c and fJ represent respectively the material density, the heat capacity and the fraction of the plastic work converted into heat (Taylor-Quinney coefficient). ()w is the absolute temperature of the workpiece. In addition, the compatibility condition gives:
dy f(r,ro)
dz V cos(A.,) sin(t,1)11) (8)
The plastic deformation is supposed to be limited to the primary shear zone, then we have two boundary conditions:
y=O at the entry of the shear band
y - __ I _ ( cos( a") ) at the outflow of the shear band
1 - COS('f/5) sin(¢11) cos(¢11 -a11)
(9)
Integration ofEq. (8) with use of the conditions (9) leads to:
JV cos(A.s) sin(t,1)11 ) dy _ h = 0
0 i(r,ro) (10)
Thus, Eq. ( 10) is a non linear equation which can be solved to calculate the unknown shear stress r 0 at the entry of the band.
In Brown and Armarego (1964), it is shown that the normal shear angle ¢11 is independent of the inclination angle A5 • In this work, we adopt this assumption and we suppose that ¢11 is given by the empirical Zvorykin law ( 1893):
tPn =A,+ Ai(a" -A.) with A. = tan -i Cfi)
( 11)
56 A. MOUFKI ET AL.
where a,,, à and A. designate respectively the normal rake angle, the mean friction coefficient and the mean angle of friction at the tool-chip interface. The constants A1 and
A2 , depending on the workpiece material, are determined from experimental data.
At the tool-chip interface, an important heating can be induced by the large values of pressure and sliding velocity. The friction condition at the tool rake face is surely affected by this heating. In Moufki et al. ( 1998), a Coulomb friction law, with a mean friction coefficient à dependent on the mean temperature at the interface 7;"' , was introduced:
(12)
the coefficients /]0 and q are identified from experimental data; T1 is the melting temperature of the workpiece material.
The shear force exerted at the outflow of the primary shear band is collinear to the material velocity jump between the entry and the exit of the band. In addition, on the tool rake face, the frictional force and the chip flow directions are supposed to be collinear.
Therefore, the following equation is obtained which determines implicitly the chip flow angle 1Jc :
cos(¢. - a.) sin(¢.) sin(17J - tan( A.,) cos\¢11 - a,,) cos(17J + (cos( a.) - sin(¢. - a,,) sin(¢.) )tan( A.) sin(17J cos(17J + tan( A.,) tan( A.,) sin(¢. - a11) cos(¢,, - a11) cos2(17J= 0
(13)
To calculate the temperature distribution within the chip beyond the primary shear zone, the approach of Moufki et al. (1998) is used. The chip heating is due to the viscoplastic deformation in the primary shear band and to the contact and friction at the tool-chip interface. Experimental data show that the pressure distribution, exerted by the chip on the tool rake face, is not uniform and is a decreasing function of the position from the cutting edge, Usui and Takeyama (1960), Zorev (1963), Kato et al. (1972), Buryta et al,. (1994). To consider this fact, we choose the following pressure distribution p(z f):
(14)
where le is the tool-chip contact length and p0 represents the pressure on the tool tip which is determined by considering the equilibrium of the forces applied to the chip. To
MODELLING OF CUTTING AND EXPERIMENTAL VALIDATION 57
calculate le, the normal stress distribution in the primary shear zone is assumed to be uniform. The coordinate z 1 represents the distance from the tool cutting edge, measured along the chip flow direction, see Fig. 1. The mean temperature T;"' at the tool-chip interface is finally obtained as, Moufki et al. ( 1998):
- t • (f ã - ; )
f - à Po V l - 2- C' -1 j Cj 2 + ()
int - ~ FJ: 2i l ~ ( ) ~ -i 2(. ') 3 I
7f p C i=O + j=O l + ) +
. i q!
with C. > = ( ;:: - . ) I ã1 ':> l . l.
(15)
It has been shown that q = 2 is an acceptable approximation of q . ()1 is the absolute temperature at the outflow of the primary shear band. Due to the fact that à is depending on T;n,, the relationship (15) is an implicit equation in terms of f;n,.
To use the present model, the thermomechanical response of the workpiece material has to be identified through the constitutive law ( 1 ). Ir. the same way, the parameters of the friction law and of the Zvorykin faw have to be determined. The known cutting conditions are: the cutting speed V, the inclination angle A., , the normal rake angle a,, and the undeformed chip thickness t1 • The variables: Ji, ¢,,, T/e, T/s, Ve, y1 , r0 ,
and T;n, are respectively given by the Eqs. (12), (11), (13), (4), (3), (9), (10) and (15).
The shear r1 stress aud the temperature 01 at the outflow of the primary shear band are obtained from Eqs. (5) and (6):
' = 1 p ( v cos A S sin A )S ' r I + T O (16)
(17)
The equilibrium of the forces applied on the chip and of their moments about the cutting edge, gives the tool-chip contact length le and, using the relationship (14), the pressure po:
1 =ti q+2 (sin(¢,,-a,,)+tan(A.)cos(¢,,-an)cos(77J)
e 2 sin(</Jn) COS(T/e) (18)
4 cos(77,) T1 q + 1 (l
9) Po {1-(tan(A.) cos(77J )2) sin(2(</Jn -an))+ 2 tan( A.) cos(77J cos(2(</Jn -an)) q + 2
58 A. MOUFKI ET AL.
To solve the preceding set of equations we proceed as follows. Let us consider that at the j'h iteration f;:/> has been estimated. From this estimation we can calculate àu>, Â,'.1> ,
l]UJ lJu> ru> ,uJ ,uJ eu> VU> I UJ and p u> Note that lJuJ and ,uJ are
c ' s ' I ' 0 ' I ' I ' c ' c 0 ã c 0
respectively the solutions of implicit Eqs. (13) and (10). A new estimate of f;:t'> is obtained from the implicit Eq. ( 15). Once the accuracy on the estimate of 1';0, is good enough, the calculations are stopped.