Stress State During Shear Chip Formation

M. G. Goldshmidt, G. D. Del, G. L. Kufarev

(Presented by the scientific seminar of the Department of Machine Tools and Metal Cutting)

One of the principal tasks in studying the process of chip formation is the investigation of the stress state in the cutting zone. Owing to the very small dimensions of the plastic region and the considerable gradient of stress variation, none of the experimental methods has yielded a solution to this problem. In the Metal-Cutting Laboratory of TPI, the possibility was tested of applying the method for investigating plastic deformation by measuring hardness [1, 2] to the study of the cutting process, and a method was developed for determining the stress state during the formation of shear chips.

The results presented below concern an investigation of stresses in the process of shear-chip formation and of the influence of the thickness of the cut on the stress state. The investigation was carried out under free cutting at microspeeds \((V = 23.5\ \text{mm/min})\). The cutting process was performed with transverse feed of the table of a 6N13P milling machine, on whose bed a dynamometer of VNII design was mounted. The magnitude of the vertical feed was monitored by an indicator. The components of the cutting force were recorded by means of a four-channel RFT oscillograph; the chip roots were fixed by stopping the table, the moment of stopping being marked on the oscillogram by disconnecting one of the two channels connected to the time marker. In view of the specific nature of the process under study and of the experimental method, the following requirements are imposed on the properties of the metals investigated: smooth variation of hardness, absence of microcracks, and periodicity of the cutting process. Periodicity was checked by comparing oscillograms of the components of the cutting force corresponding to the formation of a series of successive elements.

Microhardness was measured on a PMT-3 instrument under a load of 100 g. The surface on which the measurements were made was carefully polished and etched. Removal of the work-hardened layer was checked by comparing the results of Vickers hardness and microhardness measurements. Hardness measurements were made in sections located in the middle of the specimen width, where plane deformation occurs. In the case of plane deformation, the hardness-measurement method makes it possible to determine the stress state.

Calibration curves relating the hardness \(H_v\) to the shear-stress intensity \(k\) were obtained by testing cylindrical specimens of the materials under investigation in axial compression with lubrication. The shear-stress intensity was determined from the formula

\[ k=\frac{4P}{\sqrt{3}\pi D^2}, \]

where \(P\) is the deforming force and \(D\) is the diameter of the compressed specimen. The microhardness was measured in the meridional section of the compressed specimens. From the hardness distribution in the chip root, the distribution of \(k\) was determined with the aid of the calibration curve. The interpretation was performed according to the procedure set forth in paper [2]. The boundary conditions adopted in this case were as follows: boundary \(AB\) (Fig. 1) is free of load.

Fig. 1. Network of slip lines in the chip root.

Friction at point \(B\) was neglected. In the region \(ABC\), the Cauchy problem was solved; in the region \(CBD\), the initial characteristic problem. Owing to the insufficiency of the boundary conditions, the stresses in the zone \(OBD\) remained unknown. The integrals \(A_\alpha\) and \(A_\beta\) were determined at the nodes of the network of slip lines from the formulas:

\[ A_{\alpha 16'}=A_{\alpha 5}-\frac{k_a-k_b}{2k_{16'}\,l_{a-b}}\,l_{5-16'}, \]

\[ A_{\beta 16'}=A_{\beta 4}-\frac{k_c-k_d}{2k_{16'}\,l_{c-d}}\,l_{4-16'}. \]

From these integrals, the hydrostatic pressure \(\sigma\) and the angle \(\Theta\) of inclination of the tangent to the slip line \(\alpha\) were determined:

\[ \sigma=-k(A_\alpha+A_\beta), \qquad \Theta=\frac{1}{2}(A_\alpha-A_\beta). \]

In individual cases the positions of points were refined by the mean angles \(\frac{1}{2}(\Theta_5+\Theta_{16'})\) and \(\frac{1}{2}(\Theta_4+\Theta_{16'})\). Knowing \(\Theta\) at the points of the second row, one can

obtain a third row of points, etc. Thus, slip-line networks were constructed for each of the cases studied. The stresses along the lines of the future fracture of the element were determined from the following formulas:

\[ \sigma_x=\sigma-k\sin 2\Theta';\qquad \sigma_y=\sigma+k\sin 2\Theta';\qquad \tau_{xy}=k\cos 2\Theta', \]

where \(\Theta'\) is the angle between the tangent to the slip line \(\alpha\) and the fracture line,

\[ \Theta'=\Theta+\beta_1-180^\circ . \]

The study of the stress state by stages of element formation was carried out on Kh18N9T steel with hardness \(H_v=190\text{--}200\). Cutting was performed with a feed \(a=0.15\) mm using a tool with rake angle \(\gamma=+8^\circ\). The specimen width was 3 mm. Owing to the high work-hardening capacity of Kh18N9T steel and the considerable depth of the plastic zone, reaching 3–4 times the chip thickness, careful preparation of the machined surface proved necessary. This preparation was carried out by removing, with carbon tetrachloride as lubricant, thin chips using a tool with \(\gamma=+30^\circ\). Under the adopted cutting conditions, an element forms in 12–13 sec. The fixed chip roots correspond to 0.5, 1.9, 2.8, 4.1, and 6.2 sec of element formation. The fracture of a chip element is preceded by the initiation and development of a defining crack; therefore it was not possible to study the stress state immediately before separation of the element, since the presence of a substantial crack probably excludes a significant near-tool layer of the chip from the region in a state of plastic flow.

Figure 2 presents the distribution of \(k\) at various stages of element formation and the corresponding stress diagrams. It is evident from the figure that, as the dimensions of the forming element increase, the region of plastic deformation continually increases. The position of the zones of increased work hardening also does not remain constant. Special experiments showed that after the appearance of a crack, both the increase in the plastically deformed region and the growth of work hardening in the main volume of the element cease. The phenomena accompanying crack development and fracture are accompanied by a substantial increase in hardness in the zones adjoining the fracture surface. Analysis of micrographs of chip roots and separated elements showed that fracture occurs at an angle \(\beta_1=8^\circ\) along a line close, in the section under consideration, to a straight line.

The distribution of stresses along lines parallel to the fracture line is characterized by considerable nonuniformity and, despite the change in the boundary of the element, has a qualitatively identical character for all stages of its formation. The greatest nonuniformity is characteristic of the stresses \(\sigma_y\), which in all the cases studied change sign. The distribution of the stresses \(\tau_{xy}\) and \(\sigma_x\) is more uniform.

The contour of the region in which the hardness is equal to the initial hardness is the elastic-plastic boundary. It is clear that, as this contour is approached, the isostatics, in their outline, increasingly approach the boundary of the plastic region. Comparison of the pattern of isostatics with the slip-line network showed that, as the boundary is approached, the slip lines of the \(\beta\) family practically merge with the isostatics. This shows that the boundary of the plastic region during formation of a shear chip is a slip line.

Of great interest is the study of the stress state when the chip thickness is varied, since it is known that the latter largely determines the type of chip formed. This part of the work

Fig. 2. Stress state during the formation of a chip element when cutting steel Kh18N9T
\((v = 23.5\ \text{mm/min},\ a = 0.15\ \text{mm})\).

was carried out when cutting LS59-1 brass with variable chip thickness \((a = 0.4;\ 0.6;\ 0.8\ \text{mm})\) using a tool with rake angle \(\gamma = +29^\circ\). At a chip thickness \(a \leq 0.2\ \text{mm}\) under these conditions a continuous chip is formed. An increase

Fig. 3. Stress diagrams in the shear plane as a function of chip thickness (cutting of LS59-1 brass, \(v=23.5\ \text{mm/min}\), \(b=2.8\ \text{mm}\)).

in chip thickness leads to the formation of a shear chip. The chip roots were fixed at the moment preceding crack formation. When cutting LS59-1 brass, the chip thickness does not affect the shear angle of the element; the shear surface is close to a plane, and deformation over the entire width of the specimen may be regarded as plane [3].

Figures 3 and 4 show the stress diagrams acting in the shear plane of the element and the distribution of hydrostatic pressure (the relative position of the tool and the shear plane is preserved). The shear stresses here are distributed relatively uniformly. This made it possible to extrapolate the value of the average shear stress to the cutter tip and to compare it with the average shear stress calculated from dynamometer measurements of the cutting force.

Fig. 4. Hydrostatic pressure in the shear plane of the chip element.

In this case, the forces on the flank face were separated by A. M. Rosenberg’s method; the area over which the force \(P_\tau\) acts was measured directly at the chip root. The results of the verification are given in the table.

Table 1

The comparison showed that in all three cases the difference did not exceed 5%, which indicates the high accuracy of the experimental determination of the stress state from the hardness distribution.

Conclusions

1. The method of studying plastic deformation by measuring hardness is suitable for determining the stress state in the cutting zone.

2. The distribution of normal stresses along the shear plane of the element is characterized by substantial nonuniformity. The shear stresses are distributed more uniformly. The degree of nonuniformity depends on the properties of the material being machined. During element formation, the stress diagrams are qualitatively identical.

3. A change in the depth of cut has no significant effect on the stress state in the cutting zone.

4. The boundary of the plastic region during the formation of a chip element of shearing is a slip line.

References

1. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, 1962.

2. G. D. Del’. Study of plastic deformation by measuring hardness. Proceedings of TPI, vol. 138, 1965.

3. Metal Cutting and Tools. Edited by A. M. Rozenberg, Mashinostroenie, 1964.