On Briks's Scheme of the Plane Chip Formation Process
Abstract Generated abstract
This paper examines Briks’s proposed scheme for plane chip formation, in which the cut metal layer is assumed to deform through successive straight slip lines converging at the tool tip. Using Lévy’s stress equations, the Mises yield condition, and boundary conditions on the free outer boundary of the plastic region, the author derives the required geometry of that boundary and shows that the scheme implies a convex boundary relative to the tool tip. This result conflicts with experimental observations, where the boundary is concave, and the associated velocity field is also shown not to satisfy the slip-line velocity equations and to imply a positive volume change. The paper concludes that Briks’s scheme is unsuitable for analyzing the stress and deformation state in chip formation.
Full Text
ON BRICKS’ SCHEME OF THE PROCESS OF PLANE CHIP FORMATION
V. I. SADCHIKOV
(Presented by the scientific seminar of the Department “Machine Tools and Metal Cutting”)
Figure 1 shows Bricks’ scheme of the process of plane chip formation. Here it is assumed [2] that the layer of metal being cut is deformed into a chip by successive shears along the slip lines \(OM\), converging at the tool tip \(O\) and forming an angle \(\beta\) with the direction of the cutting speed. The equation of the slip lines of this family in the rectangular coordinate system \(x, y\), associated with the tool tip, is written as
\[ y = x \operatorname{tg}\beta . \tag{1} \]
The state of stress on the external boundary \(AB\) of the plastic region \(OAB\) will be described in the rectangular coordinate system \(t, n\), whose axes coincide with the tangent and the normal at each point of the boundary \(AB\), using Lévy’s equations [1]
Fig. 1.
\[ \left. \begin{array}{l} \sigma_t\\ \sigma_n \end{array} \right\} = 2k\omega + k\cos 2(\varphi+\alpha) \]
\[ \tau_{tn}=k\sin 2(\varphi+\alpha). \tag{2} \]
Here \(\varphi\) is the angle formed by the first principal direction of the stresses with the \(x\)-axis, \(\alpha\) is the angle of rotation of the coordinate axes \(tn\) relative to the axes \(xy\), \(k\) is the yield limit, and \(\omega\) is the dimensionless quantity equal to
\[ \frac{\sigma_t-\sigma_n}{4k}. \]
There are no external loads on the boundary \(AB\); therefore along \(AB\) the tangential stresses \(\tau_{tn}\) and the normal stresses \(\sigma_n\) are equal to zero. But the boundary \(AB\) belongs to the plastic region; therefore on it
the yield condition must be satisfied, for example, in the form given by Mises,
\[ (\sigma_t-\sigma_n)^2+4\tau_{tn}^{\,2}=4k^2, \]
from which it follows that \(\sigma_t=\pm 2k\). Along \(AB\) there must be compressive stresses and \(\sigma_t<0\), therefore
\[ \sigma_t=-2k. \]
Under these conditions equations (2) have the solution
\[ \varphi+\alpha=\frac{\pi}{2};\qquad \omega=-\frac{1}{2}, \]
which determines the stressed state on the boundary \(AB\). It follows from this solution that at every point of the boundary \(AB\) the normal coincides with the first principal direction of the stresses, which forms an angle \(\varphi\) with the \(x\)-axis, while the tangent coincides with the second principal direction, which forms an angle \(\alpha\) with the \(x\)-axis.
The slip lines form angles of \(45^\circ\) with the principal directions; therefore the slip lines of the Briks scheme at the boundary \(AB\) must form angles of \(45^\circ\) or \(135^\circ\) with the second principal direction, which coincides with the tangent to the boundary \(AB\). It follows from this that the tangents to the outer boundary \(AB\) of the Briks scheme form, with the \(x\)-axis, in the first case the angle \(45^\circ+\beta\), and in the second case the angle \(135^\circ+\beta\). This condition determines the differential equation of the boundary \(AB\): in the first case
\[ \frac{dy}{dx}=\operatorname{tg}(45^\circ+\beta)=\frac{x+y}{x-y}, \]
in the second case
\[ \frac{dy}{dx}=\frac{y-x}{x+y}, \]
the solution of which gives the equation of the boundary \(AB\) in the first case
\[ x^2+y^2=Ce^{2\operatorname{arctg}\frac{y}{x}}, \]
in the second case
\[ x^2+y^2=Ce^{-2\operatorname{arctg}\frac{y}{x}}. \]
The lines described by the obtained equations are convex with respect to the origin of coordinates, since in the first case
\[ \frac{d^2y}{dx^2}=2\frac{x^2+y^2}{(x-y)^3}>0, \]
\[ \frac{dy}{dx}=\frac{x+y}{x-y}>0 \qquad \text{for } x>y, \]
in the second case
\[ \frac{d^2y}{dx^2}=-2\frac{x^2+y^2}{(x+y)^3}<0. \]
\[ \frac{dy}{dx}=\frac{y-x}{x+y}<0 \quad \text{for} \quad x>y. \]
Consequently, the assumption in Briks’s scheme that the slip lines are straight lines converging at the tool tip leads to a convex outer boundary \(AB\) with respect to the tool tip, which contradicts experiment: the boundary \(AB\) is always concave.
It is easy to verify that the components of the velocity field (polar coordinate system \((r,\beta)\) with the pole at the tool tip)
\[ v_r=f(\beta), \qquad v_\beta=0 \]
of this scheme do not satisfy the equations for the velocities
\[ dv_r-v_\beta\,d\varphi=0 \quad \text{along a radial slip line,} \]
\[ dv_\beta+v_r\,d\varphi=0 \quad \text{along a circular slip line,} \]
where \(\varphi\) is the angle formed by the slip line with the polar axis, and the deformation of the metal in the chip-formation zone is supposedly accompanied by an increase in its volume, since the divergence of the velocity vector is greater than zero
\[ \operatorname{div}\,\vec v=\frac{f(\beta)}{r}>0. \]
It follows from this that Briks’s scheme is unsuitable for the analysis of the stressed and deformed state of chip formation.
LITERATURE
- Sokolovskii V. V. Theory of Plasticity. Publishing House of the Academy of Sciences of the USSR. Moscow, 1946.
- Zorev N. N. Questions of the Mechanics of the Process of Cutting Metals, Mashgiz, 1956.