INVESTIGATION OF THE STRESS STATE IN THE CHIP

M. F. POLETIKA

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

Figure 1 shows a diagram of the interaction of the cutter, the chip, and the workpiece being machined. The cutter, moving with velocity \(v\) relative to the workpiece, cuts off a layer of thickness “\(a\)” and turns it into a chip. The chip continuously flows along the cutter, touching it over the segment \(DB\), whose length we shall denote by \(c\), calling it the contact length.

The diagram in Fig. 1 corresponds to the particular, but most widespread in modern production conditions, case of the formation of a continuous chip without noticeable built-up edge formation. At a constant cutting speed \(v\) and with the volumetric properties of the machined material unchanged, the process of continuous-chip formation is characterized by extremely high stability in time, and the pattern of force interaction between the cutter, the chip, and the workpiece is thereby constantly preserved. The level of cutting speeds used in practice is such that the mass forces caused by the acceleration acquired by the chip relative to the workpiece during deformation are negligibly small. All this gives grounds for considering the process of continuous-chip formation as a static problem.

Experimental investigation of the deformation zone during the formation of a continuous chip shows that the right-hand boundary of this zone (\(AD\) in Fig. 1) may, with a sufficiently high degree of approximation, be regarded as a plane and, consequently, the line \(AD\) as a straight line. In the absence of built-up edge, the volume of the zone of secondary plastic deformation in the chip is small. The principal volume of the chip is occupied by material that has already undergone plastic deformation and is in an elastically stressed state.

This means that the chip, together with the workpiece being machined, may be regarded as an elastically stressed body. On this body, from the right, another elastic body—the cutter—presses; and since at some point \(B\) the chip necessarily leaves the surface of the cutter, the area of their contact is limited. All this makes it possible to reduce the problem of the interaction of the cutter with the chip to a contact problem of the theory of elasticity.

Under conditions of orthogonal free cutting, the third principal stress in the cutting zone, acting perpendicular to the plane of the drawing in Fig. 1, is relatively small, which is not difficult to show experimentally. This makes it possible to adopt, as a first approximation,

that the stressed state in the cutting zone is plane. Consequently, the problem of the interaction of the cutter with the chip is a plane problem and can be solved by the methods of the plane theory of elasticity.

Since the modulus of elasticity of the tool material is usually several times greater than the modulus of elasticity of the material being machined, the error will be small if the cutter is regarded as rigid (absolutely solid). The material in the cutting zone will be regarded as elastic and isotropic. Strictly speaking, this is incorrect, since the chip has a clearly expressed texture, but as a first approximation it is quite admissible.

With this formulation we obtain the problem of contact of a rigid profile with an elastic half-plane. The most effective solutions

Fig. 1.

of these problems are provided by methods of functions of a complex variable, which we shall use below.

The shape of the cutting zone as an elastic region is complex (Fig. 1), and the solution of the posed problem in general form encounters mathematical difficulties. Therefore, below a special case is considered, when the rake angle is equal to zero. In Fig. 2, in the plane of the complex variable \(z=x+iy\), the corresponding scheme of contact of a rigid profile—the cutter—with an elastic body—the cutting zone—is shown.

The force pressing the cutter against the elastic region is applied not at the middle of the length \(C\) of its contact with the chip, but closer to the cutter tip. This gives grounds for considering the cutter as an asymmetrically loaded punch, one of whose edges (point \(B\)) is so conjugated with the elastic region that the contact pressure at this point is equal to zero.

As is known, the basic equations of the plane theory of elasticity characterizing the stressed state of a body reduce to the following:

\[ \sigma_y-i\tau_{xy}=\Phi_0(z)+\overline{\Phi_0(z)}+z\Phi_0'(z)+\overline{\Psi_0(z)}, \tag{1} \]

\[ 2\mu (u_0 - i v_0) = \varkappa \varphi_0(z) - z\overline{\Phi_0(z)} - \overline{\psi_0(z)} . \tag{2} \]

where \(\sigma_y, \tau_{xy}\) are the stress components in the elastic region under study, that is, in the region \(TOCLAD\);

\(u_0, v_0\) are the displacement components in this region \((u_0\) in the direction of the \(x\)-axis, \(v_0\) in the direction of the \(y\)-axis);

\(\varphi_0(z), \psi_0(z), \Phi_0(z), \Psi_0(z)\) are functions of the complex variable \(z\), holomorphic in the indicated region, with

\[ \Phi_0(z) = \varphi_0'(z) \quad \text{and} \quad \Psi_0(z) = \psi_0'(z), \]

and, finally,

\(G, \varkappa\) are elastic constants, determined by the formulas, where \(E\) is the modulus of elasticity, \(\nu\) is Poisson’s ratio.

Let us note that the stresses vanish at infinity and determine the behavior of the functions \(\Phi_0(z), \Psi_0(z)\) at an infinitely distant point by the following relations:

Fig. 2.

\[ \left. \begin{aligned} \Phi_0(z) &= \frac{C_1}{z} + O\!\left(\frac{1}{z}\right),\\[4pt] \Psi_0(z) &= \frac{C_2}{z} + O\!\left(\frac{1}{z}\right). \end{aligned} \right\} \tag{4} \]

The base of the cutter-die, according to Fig. 2, is a straight line, and its equation has the form

\[ x=-\rho y+\text{const}, \]

where \(\rho\) is the angular displacement of the die.

Thus the boundary condition for horizontal displacements will be determined as follows:

\[ u_0=-\rho y+\text{const}. \tag{5} \]

If the contact pressure is denoted by \(p(x)\), then the stresses on the segment \(AB\) under the cutter will be expressed as:

\[ \left. \begin{aligned} \sigma_x&=-p(x),\\ \tau_{xy}&=\mu p(x) \end{aligned} \right\}, \tag{6} \]

where

\[ \mu=\frac{F}{N} \tag{7} \]

the coefficient of friction of the chip on the cutter is assumed to be the same at all points along the contact length.

On the remaining portions of the boundary of the elastic region under consideration, we shall have

\[ \text{on } \quad OG \text{ and } BL:\quad \sigma_x=0;\ \tau_{xy}=0; \]

\[ \text{on } \quad OT \text{ and } AD:\quad \sigma_y=0;\ \tau_{xy}=0. \]

As the method for solving the problem posed, we use a conformal mapping of the upper half-plane \(s=\xi+i\eta\) onto the region under study. The rectangular coordinates \(\xi\) and \(\eta\) of the half-plane are thereby transformed into curvilinear orthogonal coordinates inside the region under study, and, expressing through them the functions of the complex variable that appear on the right-hand sides of equations (1) and (2), we shall have

\[ \left. \begin{aligned} \varphi_0(z)&=\varphi_0[\omega(s)]=\varphi(s),\\ \psi_0(z)&=\psi_0[\omega(s)]=\psi(s),\\ \Phi_0(z)&=\Phi_0[\omega(s)]=\Phi(s),\\ \Psi_0(z)&=\Psi_0[\omega(s)]=\Psi(s), \end{aligned} \right\} \tag{8} \]

where \(z=\omega(s)\) is the function by means of which the mapping is carried out. In addition, it also follows from (8) that

\[ \left. \begin{aligned} \varphi'(s)&=\omega'(s)\Phi(s),\\ \psi'(s)&=\omega'(s)\Psi(s). \end{aligned} \right\} \tag{9} \]

The equations (1) and (2) themselves, after transition to the curvilinear coordinates \(\xi\) and \(\eta\), take the form

\[ \sigma_\eta-i\tau_{\xi\eta} = \Phi(s)-\overline{\Phi'(s)} + \frac{\omega(s)}{\omega'(s)}\,\overline{\Phi'(s)} + \frac{\overline{\omega'(s)}}{\omega'(s)}\,\psi(s); \tag{10} \]

\[ 2\mu(u+iv) = \frac{\overline{\omega'(s)}}{|\omega'(s)|} \left[ \varkappa \varphi(s) - \omega(s)\overline{\Phi(s)} + \overline{\psi(s)} \right]. \tag{11} \]

The elastic region \(TOGLAD\) under study is a quadrilateral, two vertices of which lie at an infinitely distant point—

i.e. A conformal mapping of the upper half-plane onto a domain of this kind is carried out by the function

\[ z=\omega(s)=\frac{a_1}{\pi i}\int_{1}^{s}\sqrt{(s-1)(s-d)}\,\frac{ds}{s^2}, \tag{12} \]

\[ \omega'(s)=\frac{a_1}{\pi i}\cdot \frac{\sqrt{(s-1)(s-d)}}{s^2}. \tag{13} \]

According to Fig. 2, the tool tip in the transformed domain will correspond to the point \(A'(\xi=1)\), and the point \(B\) at which the chip separates from the tool—to the point \(B'(\xi=m)\). The orientation of the loaded segment \(A'B'\) with respect to the new coordinate axes will already be different, and, consequently, instead of (6) we obtain

\[ \begin{aligned} \sigma_\eta &= p(s),\\ \tau_{\xi\eta} &= \mu p(s), \end{aligned} \tag{14} \]

where \(p(s)\) is the contact pressure expressed in terms of the new variable \(s\).

In order to transform the boundary condition (5) for the displacements, we note that the \(\xi\) axis, as is easily seen from Fig. 2, under the mapping is rotated counterclockwise through an angle \(\varepsilon=\dfrac{\pi}{2}\), which is equivalent to multiplication by \((-i)\):

\[ u+iv=-i(u_0+iv_0)=v_0-iu_0, \]

whence it follows that

\[ u=v_0,\qquad v=-u_0, \]

and, consequently, condition (5) takes the form

\[ v=\rho \xi+\text{const.} \tag{15} \]

For the subsequent presentation we shall use equation (10), giving it the following form:

\[ (\sigma_\eta-i\tau_{\xi\eta})\,\omega'(s) = \omega'(s)\Phi(s)+\overline{\omega'(s)\Phi(s)} + \overline{\omega(s)}\Phi'(s)+\omega'(s)\Psi(s). \tag{16} \]

We set ourselves the task of expressing, through the boundary conditions (14), (15), the function \(\omega'(s)\Phi(s)\). According to (13), this function is defined in the upper half-plane and is holomorphic in it, with the exception of the point \(O'(s=0)\), where it has a pole. The indicated point is the image of the infinitely distant point of the plane \(z=x+iy\). Therefore, in order to estimate the behavior of the function \(\omega'(s)\Phi(s)\) near the point \(O'\), that is, as \(s\to 0\), we must use formula (4), which leads to the following condition:

\[ [\omega'(s)\Phi(s)]_{s\to 0}=-\frac{\Gamma}{s^2}+\frac{C}{s}+O(1), \tag{17} \]

where \(O(1)\) is a function holomorphic as \(s\to 0\).

Analytically continuing the function \(\omega'(s)\Phi(s)\) into the lower half-plane through the unloaded portions of the boundary, one can express through it the function \(\omega'(s)\Psi(s)\) in the following manner:

\[ \overline{\omega'(s)\Psi(s)} = -\omega'(s)\Phi(s)-\overline{\omega'(s)\Phi(s)}-\omega(s)\overline{\Phi'(s)}. \tag{18} \]

Substitution of expression (18) into equality (16) gives

\[ (\sigma_{\eta_1}-i\tau_{\xi\eta_1})\,\omega'(s) = \omega'(s)\Phi(s)-\omega'(\bar{s})\Phi(\bar{s})+ \]
\[ +[\omega'(s)-\omega'(\bar{s})]\overline{\Phi(s)} +[\omega(s)-\omega(\bar{s})]\overline{\Phi'(s)}. \tag{19} \]

Since

\[ \left[\frac{\omega'(s)}{|\omega'(s)|}\right]'=0, \]

equation (11), after differentiation and analogous transformations, can be put in the form:

\[ 2G(u'+iv')= \frac{\overline{\omega'(s)}}{|\omega'(s)|} \left\{\chi\omega'(s)\Phi(s)+\omega'(\bar{s})\Phi(\bar{s})+\right. \]
\[ \left. +[\omega'(\bar{s})-\omega'(s)]\overline{\Phi(s)} +[\omega(\bar{s})-\omega(s)]\overline{\Phi'(s)} \right\}. \tag{20} \]

In order now to use the boundary conditions, equations (19) and (20) must be applied to points lying on the contour, that is, on the real axis \((\eta=0,\ s=\xi)\). Doing this, we obtain:

\[ (\sigma_{\eta_1}-i\tau_{\xi\eta_1})_{\eta_1=0} = \Phi(s)-\Phi(\bar{s}), \tag{21} \]

\[ [2G(u'+iv')]_{\eta_1=0} = i[\chi\omega'(s)\Phi(s)+\omega'(\bar{s})\Phi(\bar{s})]. \tag{22} \]

The first of these equalities, after passing on its right-hand side to the limiting values of the functions and applying condition (14) on its left-hand side, takes the form

\[ (1-i\mu)p(\xi)=\Phi^+(\xi)-\Phi^-(\xi), \tag{23} \]

which can be transformed as follows:

\[ \Phi^-(\xi)=-\frac{1+i\mu}{1-i\mu}\,\Phi(s). \tag{24} \]

Next, adding equation (22) to another one obtained from it by passing to the conjugate values of the functions, we shall have

\[ \chi\omega'(\bar{s})\Phi(\bar{s}) +\omega'(\bar{s})\Phi(\bar{s}) +\overline{\omega'(\bar{s})\Phi(\bar{s})} +\omega'(s)\overline{\Phi}(s). \tag{25} \]

Finally, passing in equalities (24) and (25) to the limiting values of the functions and solving them jointly, after some transformations and application of boundary condition (15), we obtain

\[ \left(\chi+\frac{1+i\mu}{1-i\mu}\right)\omega'(\xi)\Phi^+(\xi) + \left(1+\chi\frac{1+i\mu}{1-i\mu}\right)\omega'(\xi)\Phi^-(\xi) = -4G\rho\omega'(\xi), \]

or

\[ [\omega'(\xi)\Phi(\xi)]^+ - g[\omega'(\xi)\Phi(\xi)]^- = f(\xi), \tag{26} \]

where

\[ g = - \frac{ 1+\chi\frac{1+i\mu}{1-i\mu} }{ \chi+\frac{1+i\mu}{1-i\mu} } = -\frac{1+i\tan\pi\alpha}{1-i\tan\pi\alpha} = -e^{2\pi i\alpha}, \tag{27} \]

\[ f(\xi) = -\frac{4G\rho\omega'(\xi)} {\chi+\frac{1+i\mu}{1-i\mu}} = -\frac{4G\rho(1-i\mu)\omega'(\xi)\cos\pi\alpha\,e^{\pi i\alpha}} {\chi+1}. \tag{28} \]

\[ \alpha = \frac{1}{\pi}\operatorname{arctg}\mu\,\frac{z+1}{z-1}. \tag{29} \]

Equation (26) is a formulation of the so-called problem of linear conjugation. The solution of this problem is given by the formula

\[ \omega'(s)\Phi(s)=\frac{X(s)}{2\pi i}\int_m^{d_0} \frac{f(\xi)\,d\xi}{X^+(\xi)(\xi-s)}+R(s)X(s), \tag{30} \]

where \(X(s)\) is the solution of the homogeneous problem of linear conjugation

\[ X^+(\xi)-gX^-(\xi)=0, \tag{31} \]

and \(R(s)\) is a rational function.

We take the form of the solution of the homogeneous problem of linear conjugation as

\[ X(s)=(s-m)^{-\gamma}(s-d_0)^{\gamma-1}, \]

where

\[ \gamma=\frac{\ln g}{2\pi i} =\frac{\ln(-e^{2\pi i\alpha})}{2\pi i} =\frac{1}{2}+\alpha \tag{32} \]

and, consequently,

\[ X(s)=(s-m)^{-\frac12-\alpha}(s-d_0)^{-\frac12+\alpha} = \frac{1}{(s-m)^{\frac12+\alpha}(s-d_0)^{\frac12-\alpha}} \tag{33} \]

and

\[ X^+(\xi)=X(\xi)= \frac{1}{(\xi-m)^{\frac12+\alpha}(\xi-d_0)^{\frac12-\alpha}}, \tag{34} \]

We define the rational function \(R(s)\), in accordance with the conditions (17) imposed on the function \(\omega'(s)\Phi(s)\), as follows:

\[ R(s)=\frac{D_1}{s^2}+\frac{D_2}{s}. \tag{35} \]

If now formula (28) is represented as

\[ f(\xi)=A\omega'(\xi) \tag{36} \]

and the corresponding substitution is made in equation (30), then this latter, taking (35) into account, is written as

\[ \omega'(s)\Phi(s)=\frac{AX(s)}{2\pi i} \int_m^{d_0}\frac{\omega'(\xi)\,d\xi}{X^+(\xi)(\xi-s)} +\left(\frac{D_1}{s^2}+\frac{D_2}{s}\right)X(s). \tag{37} \]

In accordance with expressions (13) and (34), the integral entering formula (37) has the form

\[ I=\frac{a_1}{\pi i}\int_m^{d_0} \frac{ \sqrt{(\xi-1)(\xi-d_0)}(\xi-m)^{\frac12+\alpha} (\xi-d_0)^{\frac12-\alpha}\,d\xi }{ \xi^2(\xi-s) }. \tag{38} \]

Integral (38) cannot be evaluated in elementary functions. Therefore, leaving the expression for the solution of the problem in terms of this integral

subject of further investigations, below we consider only the special case when $\mu=0$.

Under this assumption, formulas (29), (32), (33), and (34) are simplified and take the form

\[ \alpha=0;\qquad r=\frac{1}{2},\qquad g=-1; \tag{39} \]

\[ \left. \begin{aligned} X(s)&=\frac{1}{\sqrt{(s-d_0)(s-1)}},\\[4pt] X^{+}(\xi)&=X(\xi)=\frac{1}{\sqrt{(\xi-d_0)(\xi-1)}} , \end{aligned} \right\} \tag{40} \]

and integral (38) is transformed as follows:

\[ I=\frac{a_1}{\pi i}\int_m^{d_0} \frac{\sqrt{(\xi-1)(\xi-m)(\xi-d_0)}}{\xi^2(\xi-s)}\,d\xi . \tag{41} \]

The computation of integral (41) is carried out by elementary methods with the aid of the substitution

\[ q=\sqrt{\frac{\xi-1}{\xi-m}}, \]

which, after cumbersome transformations and calculations, leads to the following final expression:

\[ \begin{aligned} I={}&A_1+\frac{A_2}{s}+\frac{A_3(d_0-d_0s-s)}{s^2}+\\ &+\frac{(s-d_0)\sqrt{(m-s)(s-1)}}{s^2} \left[ 2\operatorname{arctg}\sqrt{\frac{(1-d_0)(s-m)}{d_0-m}}-\pi \right], \end{aligned} \tag{42} \]

in which the values of the constant coefficients $A_{1,2,3}$ are given by the formulas:

\[ \left. \begin{aligned} A_1&=\frac{a_1}{\pi} \left( 2\operatorname{arctg}\sqrt{\frac{1-d_0}{d_0-m}}-\pi \right); \\[6pt] A_2&=-\frac{a_1}{\pi} \left[ \sqrt{(1-d_0)(d_0-m)} +\frac{d_0(m-1)}{2\sqrt{-m}} \ln \frac{\sqrt{m(d_0-1)}+\sqrt{d_0-m}} {\sqrt{m(d_0-1)}-\sqrt{d_0-m}} \right]; \\[6pt] A_3&=\frac{a_1\sqrt{-m}}{\pi} \ln \frac{\sqrt{m(d_0-1)}+\sqrt{d_0-m}} {\sqrt{m(d_0-1)}-\sqrt{d_0-m}} . \end{aligned} \right\} \tag{43} \]

Substituting the obtained expression (42) into equation (37), we shall have

\[ \begin{aligned} \omega'(s)\Phi(s)={}& \frac{AA_1X(s)}{2\pi i} +\frac{AA_2X(s)}{2\pi i s} +\frac{AA_3(d_0-d_0s-s)X(s)}{2\pi i s^2} +\\ &+\frac{A(s-d_0)\sqrt{(m-s)(s-1)}}{2\pi i s^2} \left[ 2\operatorname{arctg}\sqrt{\frac{(1-d_0)(s-m)}{d_0-m}}-\pi \right]\times \end{aligned} \]

\[ \times X(s)+\left(\frac{D_1}{s^2}+\frac{D_2}{s}\right)X(s). \tag{44} \]

To determine the constants \(D_{1,2}\), we use relation (17), which estimates the behavior of the function \(\omega'(s)\Phi(s)\) in the neighborhood of the point \(s=O\). It is not difficult to show that the function \(X(s)\) near this point is represented by the series

\[ \left. X(s)\right|_{s\to 0} =\left[(s-d_0)(s-m)\right]^{-\frac12}_{s\to 0} =a_0+a_1s+a_2s^2+\ldots = \tag{45} \]

\[ = -\frac{1}{\sqrt{md_0}} -\frac{(m+d_0)s}{2md_0\sqrt{md_0}} -\frac{(3d_0^2+2md_0+3m^2)s^2}{8m^2d_0^2\sqrt{md_0}} +\ldots \]

Therefore, the principal part of the pole of the function \(\omega'(s)\Phi(s)\), according to (44) and (45), will be determined by the expression

\[ \frac{D_1a_0}{s^2}+\frac{D_1a_1+D_2a_0}{s} = -\frac{D_1}{\sqrt{md_0}\,s^2} -\frac{(m+d_0)D_1+2md_0D_2}{2md_0\sqrt{md_0}} \tag{46} \]

and from comparison of (46) and (17) it follows that

\[ \left. \begin{aligned} D_1&=\sqrt{md_0}\,\Gamma,\\ (m+d_0)D_1+2md_0D_2&=-2Cmd_0\sqrt{md_0}. \end{aligned} \right\} \tag{47} \]

The constant \(\Gamma\) is proportional to the value of the stress at an infinitely distant point. We have assumed that the stresses vanish at infinity. Consequently, it must be that \(\Gamma=0\), and then equations (47) give:

\[ D_1=0, \]

\[ D_2=-C\sqrt{md_0}, \]

and substitution of the computed values of the constants into equation (44) transforms it to the form:

\[ \omega'(s)\Phi(s) = \frac{AA_1X(s)}{2\pi i} + \frac{AA_2X(s)}{2\pi i s} + \]

\[ +\frac{AA_3(d_0-d_0s-s)X(s)}{2\pi i s^2} + \frac{A(s-d_0)\sqrt{(m-s)(s-1)}\,f_1(s)X(s)}{2\pi i s^2} - \]

\[ -\frac{C\sqrt{md_0}\,X(s)}{s}. \tag{48} \]

The substitution

\[ 2\operatorname{arctg}\sqrt{\frac{(1-d_0)(s-m)}{d_0-m}}-\pi=f_1(s) \tag{49} \]

has been introduced to shorten the notation.

The function \(\Phi(s)\), on the basis of (13) and (48), is expressed as

\[ \Phi(s)= -\frac{AA_1X(s)s^2}{2a_1\sqrt{(s-d_0)(s-1)}} + \frac{AA_2X(s)s}{2a_1\sqrt{(s-d_0)(s-1)}} + \]

\[ +\frac{AA_3(d_0-d_0s-s)X(s)}{2a_1\sqrt{(s-d_0)(s-1)}} + \frac{\pi i A_1\sqrt{(s-d_0)(m-s)}\,f_1(s)X(s)}{2a_1s^2} - \]

\[ -\frac{\pi i C\sqrt{md_0}\,sX(s)}{a_1\sqrt{(s-d_0)(s-1)}}. \tag{50} \]

Recall that, according to (28) and (36), the constant \(A\) contains the unknown value \(\rho\) of the elastic rotation of the cutter die. In order to eliminate \(A\) from equation (50), we require that the function \(\Phi(s)\) be bounded at the point \(s=m\). For this purpose we set equal to zero the coefficient of

\[ (s-m)^{-\frac{1}{2}} \]

on the right-hand side of equality (50), first putting \(s=m\).

It is not difficult to verify that, upon carrying out this operation, we arrive at the expression

\[ \frac{AA_1m^2}{2}+\frac{AA_2m}{2}+\frac{AA_3(d_0-md_0-m)}{2} -\pi iCm\sqrt{md_0}=0, \]

from which we find the constant \(A\):

\[ A=\frac{2\pi iCm\sqrt{md_0}}{A_1m^2+A_2m+A_3(d_0-md_0-m)} . \tag{51} \]

To determine the contact pressure \(p(\xi)\), in equation (50) we pass to the limiting values of the function \(\Phi(s)\) as \(s\) tends to \(\xi\), first from the upper and then from the lower half-plane. Taking (39) into account, the difference of the limiting values will be equal to:

\[ p(\xi)=\Phi^+(\xi)-\Phi^-(\xi) =\frac{AA_1\xi^2\chi(\xi)}{a_1\sqrt{(\xi-d_0)(\xi-1)}}+ \]

\[ +\frac{AA_1\xi\chi(\xi)}{a_1\sqrt{(\xi-d_0)(\xi-1)}} +\frac{AA_3(d_0-d_0\xi-\xi)\chi(\xi)} {a_1\sqrt{(\xi-d_0)(\xi-1)}}- \]

\[ -\frac{2\pi iC\sqrt{md_0}\,\xi\chi(\xi)} {a_1\sqrt{(\xi-d_0)(\xi-1)}} . \tag{52} \]

If now in the equality obtained we replace \(A\) according to (51), then, after elementary transformations, we arrive at the following formula:

\[ p|\xi|= \frac{2\pi C\sqrt{md_0}\,(A_1m\xi-A_3d_0)\sqrt{\xi-m}} {a_1D(d_0-\xi)\sqrt{1-\xi}}, \tag{53} \]

where

\[ D=A_1m^2+A_2m+A_3(d_0-md_0-m). \tag{54} \]

To compute the constant \(C\), we shall use the obvious circumstance that the integral sum of the contact pressures over the length of contact must be equal to the normal force \(N\) acting on the cutter. In the original region \((z=x+iy)\), with the notation of Fig. 2, this condition is expressed as

\[ \int_{a-c}^{a} p(y)\,dy=N . \tag{55} \]

Since \(p(y)=p(\xi)\), and from the same Fig. 2 it follows that

\[ a_1+iy=\omega(\xi), \]

whence

\[ i\,dy=\omega'(\xi)\,d\xi, \]

then, making the corresponding substitutions in expression (55) and changing

properly the limits of integration, we obtain in the transformed domain

\[ \int_m^{d_0} p(\xi)\,\omega'(\xi)\,d\xi=-iN \tag{56} \]

or, taking (53) and (54) into account,

\[ \frac{2C\sqrt{m d_0}}{D}\int_m^{d_0} \frac{(A_1m\xi-A_3d_0)\sqrt{\xi-m}}{\xi^2\sqrt{d_0-\xi}}\,d\xi=-N. \tag{57} \]

The integral in the last equation is evaluated by elementary methods and leads to the expression

\[ I_1=\int_m^{d_0} \frac{(A_1m\xi-A_3d_0)\sqrt{\xi-m}}{\xi^2\sqrt{d_0-\xi}}\,d\xi = \]

\[ = \frac{\pi}{2\sqrt{m d_0}} \left[ 2mA_1(\sqrt{m d_0}-m)+A_3(m-d_0) \right]. \]

Substituting this expression into equality (57), we find the value of the constant \(C\)

\[ C=\frac{DN}{\pi\left[2mA_1(\sqrt{m d_0}-m)+A_3(m-d_0)\right]}, \tag{58} \]

with allowance for which formula (53) takes the form:

\[ p(y)=p(\xi)=- \frac{2N\sqrt{m d_0}\,(A_1m\xi-A_3d_0)\sqrt{\xi-m}} {a_1\left[2mA_1(\sqrt{m d_0}-m)+A_3(m-d_0)\right]\sqrt{1-\xi}}. \tag{59} \]

Substituting (40), (51), (54), and (58) into equation (50), we obtain the final expression for the stress function:

\[ \Phi(s)= \frac{N\sqrt{m d_0}\left[A_1ms^2+A_2ms+A_3m(d_0-ds-s)-D\right]} {\pi\left[2mA_1(\sqrt{m d_0}-m)+A_3(m-d_0)\right](s-d_0)\sqrt{(s-1)(s-m)}} + \]

\[ +\frac{Nm\sqrt{m d_0}(m-s)f_1(s)} {\pi\left[2mA_1(\sqrt{m d_0}-m)+A_3(m-d_0)\right]s^2\sqrt{s-1}}. \tag{60} \]

Let us recall that the constants \(A_{1,2,3}\) entering equalities (59) and (60) are computed from formulas (43), while the function \(f_1(s)\) is determined by expression (49).

On the basis of formula (60) and equation (18), an expression may be obtained for the function \(\Psi(s)\), which makes it possible, by means of relation (10) and analogous relations, and also by means of the representation (12) of the mapping function, to compute the stress components at all points of the elastic region under consideration. The corresponding formulas are not given here because of their cumbersomeness.

To compute the contact pressures it is sufficient to have formula (59) and expression (12), which, for points lying on the \(\xi\)-axis, has the form:

\[ y=a+\frac{a_5}{\pi} \left[ \frac{\sqrt{(1-\xi)(d_0-\xi)}}{\xi} +\frac{2}{\xi}\operatorname{arctg} \sqrt{\frac{d_0-\xi}{d_0(\xi-1)}}- \right. \]

\[ -\ln \frac{\sqrt{1-\xi}+\sqrt{d_0-\xi}}{\sqrt{1-\xi}-\sqrt{d_0-\xi}}\biggr], \tag{61} \]

where \(a\) is the thickness of the cut (Fig. 2),
\(\xi\) is the chip shrinkage;
\(d_0\) is determined by the formula

\[ d_0=-\left(\frac{1-\sqrt{1+\xi^2}}{\xi}\right)^2 . \]