Calculation of the Magnitude of Relative Shear in Metal Cutting

G. L. Kufarev

(Presented by Professor, Doctor A. M. Rozenberg)

The process of cutting metals is a process of plastic deformation. The question of the type of deformed state during cutting remains unresolved up to the present time. Some authors liken the cutting process to the process of compression, others—to simple shear.

It can be shown experimentally that the type of stress–strain state in the cutting process is simple shear adjacent to compression. According to V. M. Rozenberg [1], the type of deformed state is characterized by the angle \(\beta\):

\[ \frac{e_1+e_2-2e_3}{e_1-e_2} = \frac{\operatorname{tg}(30^\circ-\beta)}{\operatorname{tg}30^\circ}. \tag{1} \]

Here \(e_1, e_2, e_3\) are the principal strains; \(e_1 > e_3 > e_2\);
\(\beta\) is the angle characterizing the type of stress–strain state.

According to the conclusions of V. M. Rozenberg, under tension \(\beta = 0\), under simple shear \(\beta = 30^\circ\), and under simple compression \(\beta = 60^\circ\). In Figs. 1 and 2 are shown deformation patterns of a grid of circles, obtained during free cutting of a copper specimen by tools having rake angles \(\gamma = 27^\circ\) and \(37^\circ\). The crosses in these figures indicate the centers of coni-

Fig. 1

cal ...

… impressions of diameter 0.03 mm, previously applied in a definite system to the carefully polished surface of the specimen.

Fig. 2

Measurement of the semiaxes of the ellipses into which the circles were transformed after deformation makes it possible to find the magnitudes of the principal deformations from the formulas:

\[ \left. \begin{aligned} e_1 &= \ln \frac{a}{r};\\ e_2 &= \ln \frac{b}{r};\\ e_3 &= -e_1 - e_2, \end{aligned} \right\} \tag{2} \]

where \(a\) and \(b\) are the semiaxes of the ellipse,

\(r\) is the radius of the circle before deformation.

The obtained magnitudes of the principal deformations are given in Table 1.

Table 1

Calculation of the angle \(\beta\) by formula (1) for cutting cases with rake angles \(\gamma = 17^\circ\), \(27^\circ\), and \(37^\circ\) gives, respectively, the values \(34^\circ30'\), \(31^\circ03'\), and \(36^\circ22'\), which also indicates the presence in the cutting process of deformation of simple shear combined with compression. At the same time, it is evident that the share of compression in this type of deformed state is insignificant, since the deviation of the angle \(\beta\) from the value \(\beta = 30^\circ\) is small.

The fact that compression deformation also occurs in the cutting process—deformation in the direction of the third principal axis—is objectively confirmed by the experimentally observed widening of the chip. At the same time, the widening of the chip is usually small (5–10%). If there were no widening of the chip in the process of its removal, such a process would be a process of simple shear only.

As shown by the works of A. M. Rozenberg [2] and V. V. Kuzyushin [3], the degree of deformation in the cutting of plastic metals must be characterized by the magnitude of the relative shear.

Recently, two relations have been proposed for determining the relative shear in cutting. The first of them is based on the assumption that all deformation in the cutting process occurs by shear in the direction of plane AB (Fig. 3), which makes a certain angle \(\beta_1\) with the direction of motion of the cutter; the magnitude of this angle may be calculated from the chip shrinkage:

Fig. 3

\[ \tg \beta_1 = \frac{\cos \gamma}{\zeta - \sin \gamma}; \]

where \(\gamma\) is the rake angle of the cutter, \(\zeta\) is the chip shrinkage, determined by the ratio of the chip thickness to the thickness of the layer being removed.

Under this assumption, the following formula is obtained for determining the relative shear:

\[ \varepsilon = \ctg \beta_1 + \tg(\beta_1 - \gamma). \tag{3} \]

This relation was first obtained by Merchant [4], and later by a number of Soviet researchers [5, 2, 3].

Zorev [6] derives an equation for the relative shear on the basis of different ideas about the cutting process. He assumes that the deformation of the metal during cutting in the zone ABC (Fig. 4) proceeds by means of successive shears along planes fanning out from the cutting edge. Plane AC, whose position is determined by the angle \(\varphi_1\), is the boundary at which the deformation of the metal ends. Zorev determines the magnitude of the final relative shear on plane AC (and therefore also in the chip) by the formula

\[ \varepsilon_{\varphi_1} = \ctg \varphi_1 + \tg(\varphi_1 - \gamma). \tag{4} \]

Fig. 4

It should be noted that this relation contradicts the author’s basic premises. This would be precisely the relative shear in the case where all deformation proceeded in the plane determined by the angle \(\varphi_1\). But this means that formula (4) makes it possible to determine the magnitude of the relative shear occurring only in plane AC, but does not take into account the deformation that has occurred in zone ABC.

Thus, to evaluate the degree of deformation of the metal in the chip by the relative shear calculated from formula (4) is, in princi-

...fundamentally incorrect. The same idea can be explained differently. If, when removing a layer of thickness $t$, all deformation occurs in the plane $AC$ (Fig. 5), then the chip thickness obtained as a result of such deformation will be equal to $t_1'$, which is less than the actually occurring chip thickness $t_1$. It follows from what has been said that formula (4) must give underestimated values of the relative shear.

Formula (3) can be accepted unconditionally only when the actual picture of deformation is close to the scheme shown in Fig. 3, that is, in the case when the zone of preliminary deformation $ABC$ (Fig. 4) is small and can be reduced to a plane. However, often (especially in cutting at speeds less than $1\ \text{m/min}$) the zone of preliminary deformation proves to be substantial. In these cases the validity of formula (3) may be called into question.

Fig. 5

Similarly, any other expression for relative shear derived on the basis of one or another approximate scheme of deformation in cutting will have all the shortcomings and limitations inherent in that approximate scheme. The actual picture of the deformed state in the process of cutting metals, because of its complexity, is at present not amenable to exact mathematical analysis.

It therefore seems advisable to obtain such a dependence for the relative shear as would make it possible to determine it independently of the scheme of deformation in cutting.

Such a dependence can be derived on the basis of the following experimental facts.

1. The type of deformed state in cutting is extremely close to simple shear. A circle drawn on the side surface of the specimen, after deformation, becomes an ellipse. The area of the ellipse differs only insignificantly from the area of the original circle.

2. Lines which before deformation were parallel to the motion of the cutter, after deformation prove to be parallel to the front face of the tool. This proposition is valid both in the absence and in the presence of a zone of preliminary deformation (Fig. 3)\[^1\].

3. The direction of the chip texture is the direction of the greatest tensile deformation, that is, it coincides with the major semiaxis of the ellipse [11].

Let, when a layer of thickness $t$ is removed (Fig. 6), a chip be formed whose thickness is $t_1$. Thus the chip shrinkage is equal to

\[ \zeta = \frac{t_1}{t}. \]

In this case a circle with center at point $O$ was transformed into an ellipse equal to it in area, whose major axis made with the direction—

\[^1\]: See Fig. 7 in the article “The Rate of Deformation in the Process of Cutting Ductile Metal,” included in this collection.

…with the front face of the cutter the angle \(\psi_{\mathrm{э}}\). Points \(M\) and \(N\) of the circle prove to lie on straight lines parallel to the front face, separated from one another by a distance \(CD\), equal to:

\[ CD = MN \cdot \zeta . \]

Fig. 6

If the radius of the circle is taken as unity, then

\[ O'D = r \cdot \zeta = \zeta . \]

Let us choose a coordinate system with its origin at the point \(O'\), the axes \(x\) and \(y\) of which coincide with the principal axes of the ellipse. The straight line \(AB\) is tangent to the ellipse at the point \(M'\), and therefore in the adopted coordinate system is expressed by the equation

\[ \frac{x \cdot x_{M'}}{a^2} + \frac{y \cdot y_{M'}}{b^2} = 1, \tag{5} \]

where \(x\) and \(y\) are the current coordinates of the tangent,

\(x_{M'}\) and \(y_{M'}\) are the coordinates of the point of tangency,

\(a\) and \(b\) are the major and minor semiaxes of the ellipse.

The coordinates of point \(A\):

\[ x_a = 0; \]

\[ y_a = O'A = \frac{O'D}{\cos \psi_{\mathrm{э}}} = \frac{\zeta}{\cos \psi_{\mathrm{э}}}. \]

The coordinates of point \(B\):

\[ x_b = O'B = \frac{O'D}{\sin \psi_{\mathrm{э}}} = \frac{\zeta}{\sin \psi_{\mathrm{э}}}; \]

\[ y_b = 0. \]

Substituting successively the coordinates of points \(A\) and \(B\) into the equation of the tangent (5), to which they belong, we find the coordinates of the point of tangency \(M'\):

\[ x_{M'} = \frac{a^2 \cdot \sin \psi_{\mathrm{э}}}{\zeta}; \]

\[ y_{M'} = \frac{b^2 \cdot \cos \psi_{\mathrm{э}}}{\zeta}. \]

But the point \(M'\) belongs to the ellipse; hence

\[ \frac{x_{M'}^2}{a^2} + \frac{y_{M'}^2}{b^2} = 1. \tag{6} \]

Substituting into this equation the values found for the coordinates of the point \(M'\) and noting that, when the areas of the circle and the ellipse are equal, \(a \cdot b = r^2 = 1\), that is, \(b = \dfrac{1}{a}\), we obtain

\[ a^4 - \frac{\zeta^2}{\sin^2 \psi_{\mathrm{э}}}\, a^2 + \operatorname{ctg}^2 \psi_{\mathrm{э}} = 0 . \]

Solving this equation makes it possible to determine the value of \(a\)

\[ a = \sqrt{ \frac{\zeta^2}{2 \cdot \sin^2 \psi_{\mathrm{э}}} + \sqrt{ \frac{\zeta^4}{4 \cdot \sin^4 \psi_{\mathrm{э}}} - \operatorname{ctg}^2 \psi_{\mathrm{э}} } }. \tag{7} \]

Knowing the magnitude of the major semiaxis of the ellipse, one can calculate the magnitude of the principal deformation in the direction of this axis

\[ e_1 = \ln \frac{a}{r} = \ln a = \ln \sqrt{ \frac{\zeta^2}{2 \cdot \sin^2 \psi_{\mathrm{э}}} + \sqrt{ \frac{\zeta^4}{4 \cdot \sin^4 \psi_{\mathrm{э}}} - \operatorname{ctg}^2 \psi_{\mathrm{э}} } }. \tag{8} \]

Taking into account the indications available in the literature [7] on the calculation of the maximum relative shear in simple-shear deformation, we obtain:

\[ \gamma_{\mathrm{k}}' = 2 \cdot \operatorname{sh} e_1 = 2 \cdot \operatorname{sh} \left[ \ln \sqrt{ \frac{\zeta^2}{2 \cdot \sin^2 \psi_{\mathrm{э}}} + \sqrt{ \frac{\zeta^4}{4 \cdot \sin^4 \psi_{\mathrm{э}}} - \operatorname{ctg}^2 \psi_{\mathrm{э}} } } \right]. \tag{9} \]

Here \(\gamma_{\mathrm{k}}'\) is the maximum relative shear in cutting,
\(\zeta\) is the chip shrinkage,
\(\psi_{\mathrm{э}}\) is the experimentally determined angle between the direction of the texture of the chip material and the rake face of the cutter.

Thus, this method makes it possible to determine the magnitude of the relative shear in metal cutting from two experimentally found quantities—\(\zeta\) and \(\psi_{\mathrm{э}}\); the derivation of the relative-shear equation is based only on entirely reliable experimental facts and is free from the shortcomings of any of the approximate deformation schemes in cutting.

By virtue of this, the given method for determining the magnitude of the relative shear in the cutting process is the most objective of all existing methods.

In the case when the angle \(\psi_{\mathrm{э}}\) proves to be greater than \(75^\circ\), equality (7) may, with a sufficiently high degree of accuracy, be reduced to the form:

\[ a = \frac{\zeta}{\sin \psi_{\mathrm{э}}}. \]

And then

\[ \gamma_{\mathrm{k}}' = 2 \operatorname{sh} \left[ \ln \frac{\zeta}{\sin \psi_{\mathrm{э}}} \right]. \]

The angle \(\psi_{\mathrm{э}}\) may be determined from a microsection of the chip.

The method for determining chip shrinkage in cutting at high speeds has at present been developed sufficiently fully and needs no explanation.

As for determining shrinkage in cutting at microspeeds (up to 1 m/min), here it is necessary to point out one circumstance that until now has not been taken into account by anyone.

As follows from Figs. 1 and 2, the pattern of deformation in cutting at low speeds (these roots were obtained at \(V = 19\) mm/min) is such that not only the material located above the theoretical cutting line passes into the chip, but also part of the material located below this line.

The nature of the distortion of the square grid in the studies of other authors [8, 9] testifies to the same thing.

Consequently, shrinkage determined from the ratio

\[ \zeta = \frac{t_1}{t}, \]

in the given case has no physical meaning. The correct value of chip shrinkage is determined as follows:

\[ \zeta = \frac{t_1}{t'}, \]

where \(t'\) is the thickness of the layer of metal actually going into the chip; \(t' > t\) (Figs. 1 and 2). It is also permissible here to determine shrinkage from the relation:

\[ \zeta = \frac{l}{l_1} \cdot \frac{b}{b_1}, \]

where \(l_1\) and \(b_1\) are the length and width of the chip,
\(l\) is the path of the cutter over which a chip of length \(l_1\) was removed,
\(b\) is the width of cut.

In this case the value \(l\) can be determined only by direct measurement. Determination of the cutter path by the method of weighing the chip is inadmissible. In this method the cutter path is determined by the formula:

\[ l = \frac{Q}{t \cdot b \cdot d}, \]

where \(Q\) is the weight of the chip removed over the path \(l\),
\(t\) and \(b\) are the thickness and width of the layer being removed,
\(d\) is the density of the metal.

Since in this relation the value of \(t\) will prove to be smaller than the actual one, \(l\), and then \(\zeta\), will be obtained as definitely overestimated.

Thus, as applied to the chips shown in Figs. 1 and 2, the magnitude of the maximum relative shear can be calculated by three methods:

\[ 1.\quad \varepsilon = \operatorname{ctg}\beta_1 + \operatorname{tg}(\beta_1 - \gamma), \]

where

\[ \operatorname{tg}\beta_1 = \frac{\cos\gamma}{\zeta - \sin\gamma}. \]

\[ 2.\quad \varepsilon_{\varphi_1} = \operatorname{ctg}\varphi_1 + \operatorname{tg}(\varphi_1 - \gamma), \]

where \(\varphi_1\) is the angle between the upper boundary of the plastic-deformation zone and the direction of motion of the cutter (determined directly from a photograph of the chip root).

3.

\[ \gamma'_k = 2\,sh\left[\ln \sqrt{\frac{\zeta^2}{2\cdot \sin^2 \psi_\mathrm{e}}} + \sqrt{\frac{\zeta^4}{4\cdot \sin^4 \psi_\mathrm{e}}-\operatorname{ctg}^2 \psi_\mathrm{e}}\right] \]

or

\[ \gamma'_k \simeq 2\,sh\left[\ln \frac{\zeta}{\sin \psi_\mathrm{e}}\right], \]

where \(\psi_\mathrm{e}\) is the angle between the direction of the texture of the chip material and the plane of the front face of the cutter (determined experimentally from a macrosection of the chip or of its root).

The chip shrinkage, appearing in the first and third methods, is determined by the relation:

\[ \zeta=\frac{t_1}{t'}. \]

For the three chips considered, the shrinkage can also be calculated from the ratio of the size of the ellipse after deformation, measured in the direction perpendicular to the front face of the cutter, to the diameter of the circle before deformation. Both of these shrinkage calculations led to completely identical results.

The validity of each method for calculating the relative shear was assessed by comparing their results with the actual maximum relative shear, determined experimentally from the distortion of a circle during deformation. Here the principal deformations were determined from formulas (2). As for the calculation of relative shear, there are two points of view. Many authors believe that the maximum relative shear is the difference between the two principal deformations determined from the distortion of a circle, that is:

\[ g' = e_1 - e_2 . \tag{10} \]

At the same time, V. G. Osipov [7] and A. Nadai [10] believe that, when applied to the process of simple shear, the shear determined by formula (10) is pseudo-maximum, and that the corresponding maximum relative shear should be determined from the relation

\[ \gamma_k = 2\,sh\,\frac{g'}{2}. \tag{11} \]

In our calculations we used expression (11). The results of calculating the relative shear by the different methods are summarized in Table 2.

Table 2

Analysis of the data in the table permits the following conclusions to be drawn.

1. The first method gives values of the relative shear that are extremely close to reality, but systematically overestimated.

2. The results of calculation by the second method give substantially underestimated values of the relative shear, and therefore this method should be considered unacceptable.

3. The third method gives exact agreement with the experimentally determined values of the relative shear, and may be considered the most correct of the three methods examined.

When working at high cutting speeds, the zone of preliminary deformation, as experience shows, is quite insignificant. In this case, calculations by the first and third methods lead to identical results. Since the third method can be classified as experimental, this agreement of results must be regarded as proof of the applicability of the first method for calculating the relative shear in the cutting process at high speeds. As for the second method, its application to work at high speeds proves impossible. In this case the upper boundary of the deformation zone¹ is situated quite differently than is assumed by the scheme of the second method (Fig. 4).

REFERENCES

1. Smirnov-Alyaev G. A. Resistance of Materials to Plastic Deformations. Mashgiz, 1949.

2. Rozenberg A. M. and Eremin A. N. Plastic deformation of the chip in metal cutting. Izvestiya TPI, vol. 75, 1954.

3. Kuzyushchin V. V. Plastic deformation in cutting. Stanki i instrument, No. 4, 1951.

4. E. Merchant — Mechanics of the metal cutting process. Journal of Applied Physics, vol. 16, No. 0, 5, 6, 1945.

5. Nekrasov S. S. On the maximum value of the specific cutting pressure. Podshipnik, Nos. 2 and 4, 1953.

6. Zorev N. N. Problems in the Mechanics of the Metal-Cutting Process. Mashgiz, 1956.

7. Osipov V. G. On the characteristics of finite deformations. In: Problems of Metallurgy, Academy of Sciences of the USSR, 1953.

8. W. Okoshi and S. Fukui — Researches on the cutting action of planing tool, by microkinematographic, photoelectric and piezoelectric methods. Scientific Papers of the Institute of Physical and Chemical Research, No. 455—456, October 25, 1933, v. 22, pp. 97—166.

9. D. A. Oliver, T. S. Lister, M. D. Kinman, D. Fitzgeorge — Machining research. Aircraft Production, vol. 18, No. 3, March 1956, pp. 118—124.

10. Nadai A. Plasticity and Fracture of Solids. IL Publishing House, 1954.

11. Rozenberg A. M. and Eremin A. N. Structure of the chip in cutting plastic metal. Izvestiya TPI, vol. 75, 1954.

¹ See Fig. 7 in the article “Rate of Deformation in the Process of Cutting Plastic Metal,” included in this collection.