Determination of the Maximum Shear Stress During Cutting Based on the Characteristics of the Compression Process

G. L. Kufarev, V. A. Govorukhin

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

In [1] it is shown that, for the cutting process (plane strain state) and the compression process (linear stress state), there is a single flow curve plotted in the coordinates stress intensity \((\sigma_i)\)—strain intensity \((e_i)\). Coincidence of these flow curves is observed in all cases of cutting at microspeeds, when the temperature–speed regime of the deformation process during cutting may be considered coincident with that during compression, and also in those cases of high-speed cutting when the relation \(\sigma_i = f(e_i)\) in the final stage of deformation satisfies the equation of the flow line under compression. Usually such coincidence occurs at values of \(e_i\) close to 1.5.

The existence of proof of the coincidence of the flow curves in compression and cutting makes it possible, for these cases, to give a new interpretation of the problem of determining the maximum shear stress during cutting on the basis of the specific work of deformation.

For compression, the specific work is determined by the area ABG bounded by the flow curve in the coordinates compressive stress \((\sigma_{1c})\), greatest principal strain \(\left(e_{1c} = \ln \frac{h_0}{h}\right.\), where \(h_0\) and \(h\) are the initial and current height of the specimen). Figure 1a gives the graph of such a dependence for brass L62. For the simple-shear scheme, which describes the deformation process during cutting, the specific work is determined by the area ANK in Fig. 1b. Here, \(\tau_p\) is the maximum shear stress, and \(\gamma_p\) is the maximum relative shear. As E. G. Osipov [2] showed, in the process of simple shear the maximum relative shear is equal to the true one: \(\gamma_p = g_p = 2e_{1p}\), where \(e_{1p}\) is the magnitude of the greatest principal strain.

Since in compression \(\sigma_i = \sigma_{1c}\) and \(e_i = e_{1c}\), the graph in Fig. 1a also represents the flow curve \(\sigma_i = f(e_i)\), which is common to compression and cutting. The unity of the flow curve \(\sigma_i = f(e_i)\), and of the relations between the stresses \(\sigma_i\) and \(\tau\), as well as between the various

![Figure 1]

Figure 1. Flow curves under plastic deformation of L62 brass: (a) in compression; (b) in cutting.

Visible labels in the figure:

• Vertical axis: stresses, kg/mm²
• Left scale:

\[ \sigma_i,\qquad \tau_p=\frac{\sigma_i}{\sqrt{3}} \]

• Horizontal axis: deformations
• Bottom scales: \(0,\ 0.1,\ 0.2,\ 0.3,\ 0.4,\ 0.5,\ 0.6\) and \(0,\ 0.1,\ 0.2,\ldots,\ 1.1\)
• Right-side relations:

\[ q_p=\sqrt{3}\,\varepsilon_i, \qquad \varepsilon_i=\varepsilon_n\frac{L_c}{h} \]

• Curve and construction labels: \(A\), \(a\), \(b\), \(\delta\), \(H\), \(h\), \(\sigma_i\), \(\tau_p\), \(d\varepsilon_i\), \(dq_p\).

the deformation characteristics make it possible to write the following expressions:

\[ \sigma_i = 2\tau_c = \sqrt{3}\tau_p;\qquad \tau_p=\frac{\sigma_i}{\sqrt{3}} \tag{1} \]

\(\tau_c, \tau_p\) — the maximum shear stresses in compression and cutting;

\[ e_i=e_{1c}=\ln\frac{h_0}{h}=\frac{2}{\sqrt{3}}\,e_{1p};\qquad e_{1p}=\frac{\sqrt{3}}{2}\ln\frac{h_0}{h} \]

\(e_{1c}, e_{1p}\) — the greatest principal true strains in compression and cutting;

\[ g_c=1.5e_{1c}=1.5\ln\frac{h_0}{h};\qquad g_p=2e_{1p}=\sqrt{3}\ln\frac{h_0}{h} \tag{2} \]

\(g_c, g_p\) — true shears in compression and cutting. Prof. A. M. Rosenberg [3] proposed the following approximation of the flow curve

\[ \sigma_{1c}=B\cdot g_c^\kappa \tag{3} \]

\(B\) is the value of \(\sigma_{1c}\) at \(g_c=1\), and he took the value of the true shear as the relative shear.

The flow curves in other coordinates are well approximated by analogous relationships.

\[ \sigma_i=D\cdot e_i^\kappa \tag{4} \]

\(D\) is the value of \(\sigma_i\) at \(e_i=1\)

\[ \tau_p=E\cdot g_p^\kappa \tag{5} \]

\(E\) is the value of \(\tau_p\) at \(g_p=1\)

Confirmation of this is provided by the experimental relationships shown in Fig. 2, obtained in the compression of L62 brass. In constructing the dependence \(\tau_p=f(g_p)\), these quantities were determined from equations (1) and (2).

A characteristic point is that the exponent in the deformation characteristic in each of equations (3), (4), and (5) proved to be unchanged.

Having these equations, one can also calculate the specific work. In compression (Fig. 1a):

\[ \begin{aligned} A_{w_c} &=\int_{e_{i0}}^{e_i}\sigma_i\cdot de_i =\int_{e_i}^{e_i}D\cdot e_i^\kappa\cdot de_i =D\cdot\int_{h_0}^{h}\left(\ln\frac{h_0}{h}\right)^\kappa\cdot d\left(\ln\frac{h_0}{h}\right) = \\[4pt] &=\frac{D}{\kappa+1}\left(\ln\frac{h_0}{h}\right)^{\kappa+1} =\frac{D\cdot e_i^{\kappa+1}}{\kappa+1} =\frac{\sigma_i\cdot e_i}{\kappa+1} \end{aligned} \tag{6} \]

In cutting (Fig. 1b):

\[ \begin{aligned} A_{w_p} &=\int_{g_{p0}}^{g_p}\tau_p\cdot dg_p =\int_{g_{p0}}^{g_p}E\cdot g_p^\kappa\cdot dg_p = \\[4pt] &=E\cdot\int_{h_0}^{h}\left(\sqrt{3}\ln\frac{h_0}{h}\right)^\kappa \cdot d\left(\sqrt{3}\ln\frac{h_0}{h}\right) = \\[4pt] &=\frac{E\cdot\left(\sqrt{3}\cdot\ln\frac{h_0}{h}\right)^{\kappa+1}}{\kappa+1} =\frac{E\cdot g_p^{\kappa+1}}{\kappa+1} =\frac{\tau_p\cdot g_p}{\kappa+1} \end{aligned} \tag{7} \]

Fig. 2. Approximation of flow curves in compression and cutting of L62 brass

Vertical axis: $\sigma_i$, $\tau_p$, $\eta$; stresses, kg/mm².

Horizontal axis: deformations, $q_c$, $g_p$, $e_i$.

Curves:

1. $\sigma_i = D e_i^{k}$, $D = 76$, $k = 0.415$
2. $\sigma_i = B q_c^{k}$, $B = 64$, $k = 0.445$
3. $\tau_p = E g_p^{k}$, $E = 34.5$, $k = 0.445$

It is not difficult to verify that, in the case where the flow curves in compression and cutting coincide, when \(\tau_p\) and \(g_p\) are determined by equations (1) and (2),

\[ A_{w_p}=A_{w_c};\quad \frac{\tau_p\cdot g_p}{\kappa+1}= \frac{\sigma_i\cdot e_i}{\kappa+1}. \tag{8} \]

A. M. Rosenberg determined the specific work in cutting by the product \(\tau_p\cdot g_p\), although from equation (7) it follows that this will be valid only for \(\kappa=0\), i.e., in machining an ideally plastic metal.

Nevertheless, he gives experimental confirmation of his calculations by superposing the experimental points obtained in cutting on the curve \(A_w=f(g_c)\), constructed from compression tests. In this, the true shear in cutting \(g_p\) was taken to be equal to the relative shear and was calculated by the formula:

\[ g_p=\frac{\xi^2-2\xi\cdot \sin\gamma+1}{\xi\cdot \cos\gamma}; \]

where \(\xi\) is the chip compression ratio by its thickness,
\(\gamma\) is the rake angle of the cutter.

The tangential stress was determined from the cutting forces recorded by the dynamometer:

\[ \tau_p= \frac{P_z\cdot \cos(\Phi+\omega)\cdot \sin\Phi} {a\cdot b\cdot \cos\omega} \]

where: \(P_z\) is the main component of the cutting force;
\(\Phi\) is the angle determining the direction of shear in cutting relative to the direction of cutting velocity;
\(\omega\) is the angle determining the direction of the resultant force relative to the direction of cutting velocity;
\(a,b\) are the thickness and width of the layer being cut.

Fig. 3. Comparison of the specific work in cutting and compression of L62 brass.

Thus \(\tau_p\) and \(g_p\) were determined correctly; however, the specific work determined by the product \(\tau_p\cdot g_p\) exceeded the actual value, determined by equation (7), by \((\kappa+1)\) times. This is well illustrated by curves 1 and 2 in Fig. 3, constructed

in the coordinates \(A_w = f(e_i)\), of which the first corresponds to compression tests (equation 6), and the second is drawn through the experimental points obtained in cutting.

A. M. Rozenberg compared the specific works in cutting and compression as follows:

\[ \tau_p \cdot g_p = \frac{B \cdot g_c^{\kappa+1}}{1.5(\kappa+1)} = \frac{B \cdot g_c^\kappa \cdot g_c}{1.5(\kappa+1)} = \frac{\sigma_i \cdot g_c}{1.5(\kappa+1)} \tag{9} \]

It was assumed here that this equality is satisfied under the condition \(g_p = g_c\). The latter condition, as follows from equation (2), will be satisfied if the actual value of the strain in the left-hand side of equation (9) is reduced by a factor of \(\dfrac{1.5}{\sqrt{3}}\). Consequently, A. M. Rozenberg in fact checked not equation (9), but the equality

\[ \frac{1.5}{\sqrt{3}} \, \tau_p \cdot g_p = \frac{\sigma_i \cdot g_c}{1.5(\kappa+1)} \tag{10} \]

Substituting \(\tau_p = \dfrac{\sigma_i}{\sqrt{3}}\) and putting \(g_p = g_c\), we obtain:

\[ \frac{1.5}{3} = \frac{1}{1.5(\kappa+1)} \quad \text{or} \quad 1 = \frac{1.33}{\kappa+1}. \]

Thus, in the adopted coordinates \(A_w = f(g_c)\), with \(g_p = g_c\), we have

\[ A_{w p} = \tau_p \cdot g_p = A_{w c} \cdot \frac{1.33}{\kappa+1}. \tag{11} \]

It is obvious that equality (11) is fulfilled only in the case when \(\kappa = 0.33\). If \(\kappa > 0.33\), the curve \(\tau_p \cdot g_p = f(g_p)\) will be located below the compression curve (see curves 3 and 4 in Fig. 3). For \(\kappa < 0.33\) the picture is reversed. For the six steels for which the value of \(\kappa\) ranged from 0.09 to 0.385 [3], A. M. Rozenberg would have had to obtain errors in determining the specific work in cutting within the limits from \(+12\) to \(-4\%\).

In accordance with equation (9), and equating \(g_p = g_c\), A. M. Rozenberg proposes determining the shear stress in cutting by the formula:

\[ \tau_p = \frac{B \cdot g_p^\kappa}{1.5(\kappa+1)} \tag{12} \]

This equation, although obtained from incorrect premises, gives values close to the true ones. We shall show this. The transformation of the valid equality

\[ \frac{\tau_p \cdot g_p}{(\kappa+1)} = \frac{B \cdot g_c^{\kappa+1}}{1.5(\kappa+1)} \]

or, otherwise,

\[ \tau_p = \frac{B \cdot g_c^{\kappa+1}}{1.5(\kappa+1)} \cdot \frac{(\kappa+1)}{g_p} = \frac{B \cdot g_c^\kappa \cdot g_c \cdot (\kappa+1)} {1.5(\kappa+1) \cdot g_p} \tag{13} \]

into equation (12) requires the following operations:

1. It is necessary to divide equation (13) by \(\kappa+1\). This was done by the author in deriving the equation for the specific work in cutting.

1. To reduce the quantities \(g_p\) and \(g_c\), which are not equal to each other. In doing so, the coefficient \(\dfrac{\sqrt{3}}{1.5}\) is introduced in hidden form, since the reduction is possible only under the condition

\[ g_c \cdot \sqrt{3}=g_p \cdot 1.5 \]

1. To replace the quantity \(g_c^k\) by the quantity \(g_p^k\). This, again, is possible only by introducing the coefficient

\[ \left(\frac{\sqrt{3}}{1.5}\right)^k . \]

Thus, in obtaining equation (12), A. M. Rozenberg introduces in hidden form the coefficient

\[ \left[\frac{\sqrt{3}}{1.5(k+1)}\right]^{k-1}. \]

This coefficient is not equal to unity, but is close to it.

For

\[ \begin{array}{rcl} \kappa=0.1 & \text{it is equal to} & 1.06,\\ \kappa=0.2 & & 0.985,\\ \kappa=0.4 & & 0.875, \end{array} \]

and consequently the shear stress determined from equation (12), for different values, will fluctuate about its actual value, proving equal to it only for one particular value of \(\kappa\). This can also be shown in another way. Equation (12) is nothing other than an approximation of the flow curve in the coordinates \(\tau_p=f(g_p)\)—(5). Consequently, the coefficients standing in these equations before \(g_p^k\) must be equal:

\[ E=\frac{B}{1.5(k+1)} \]

Let us check this equation for \(\sigma_{i_p}=\sigma_{i_c}\) and \(e_{i_p}=e_{i_c}\)

\[ \frac{\sigma_i}{1.5(k+1)g_c^k} = \frac{\tau_p}{g_p^k} = \frac{\sigma_i}{\sqrt{3}\cdot g_p^k} ;\qquad \frac{1}{1.5(k+1)\cdot(1.5\cdot e_i)^k} = \frac{1}{\sqrt{3}\cdot \sqrt{3}^{\,k}\cdot e_i^k} \]

\[ 1=\frac{1}{k+1}\left(\frac{\sqrt{3}}{1.5}\right)^{k+1} \]

Thus, the left- and right-hand sides of this equation are not equal to each other, and their ratio is determined by the same coefficient as was obtained earlier. Since equation (8) does not contain the shortcomings inherent in expression (11), and is ideally satisfied when the flow curve \(\sigma_i=f(e_i)\) is the same in cutting and compression, it is precisely this equation that should be used in determining the maximum stress in cutting from the characteristics of the compression process. From equation (8) it follows that:

\[ \tau_p=\frac{\sigma_i\cdot e_i}{g_p} = \frac{D\cdot e_i^k\cdot g_o}{\sqrt{3}\cdot g_p} = \frac{D\cdot g_p^k}{\sqrt{3}^{\,k+1}} \tag{14} \]

Here,

\[ g_p=\operatorname{ctg}\Phi+\operatorname{tg}(\Phi-\gamma) = \frac{\xi^2-2\xi\cdot\sin\gamma+1}{\xi\cdot\cos\gamma} \]

Equation (14) can be obtained without resorting to the equality of specific works (8). It is well known [4] that the specific work of plastic deformation is proportional to the intensity of the stress state.

Consequently, the equality \(\sigma_{i_p}=\sigma_{i_c}\) already reflects the condition of equality of the specific works in cutting and compression.

\[ \tau_p=\frac{\sigma_{i_p}}{\sqrt{3}}=\frac{\sigma_{i_c}}{\sqrt{3}}=\frac{D\cdot e_i^{\kappa}}{\sqrt{3}}=\frac{D\cdot g_p^{\kappa}}{\sqrt{3^{\kappa+1}}} \]

In conclusion, it should once again be emphasized that determining \(\tau_p\) from the characteristics of the compression process is possible only in the case of a single flow curve for these processes.

REFERENCES

1. G. L. Kufarev, V. A. Govorukhin. “On the coincidence of flow curves in cutting and compression processes,” Izvestiya TPI, in press.

2. V. G. Oskolov. “On the characteristics of finite deformations,” in the collection Problems of Metallurgy, Moscow, 1953.

3. Edited by A. M. Rozenberg. Cutting of Metals and Tools, Mashinostroenie, Moscow, 1964.

4. N. M. Belyaev. Strength of Materials, OGIZ, Moscow–Leningrad, 1945.