Fast-Moving Heat Sources and Average Contact Temperature in Metal Cutting
Abstract Generated abstract
This paper develops an approximate heat conduction treatment for fast-moving heat sources relevant to metal cutting. It reduces the quasi-stationary temperature field of a half-plane to a semi-infinite rod model, derives a criterion under which lateral heat redistribution may be neglected, and defines an average contact temperature by equating heat input to that for a constant boundary temperature. The method is then applied to frictional contact, the chip-formation shear zone, and the tool rake face, yielding formulas for average contact and cutting temperatures in terms of heat flux, velocity, contact dimensions, and thermal properties.
Full Text
FAST-MOVING HEAT SOURCES AND AVERAGE CONTACT TEMPERATURE IN METAL CUTTING
V. S. KUSHNER
(Presented by the scientific seminar of the Department of Higher Mathematics)
Formulation of the problem and method of investigation
Let a linear heat source of uniform intensity move along the boundary of a half-plane with constant velocity \(v\). In the rectangular coordinate system \(x, y\), fixed with respect to the source \(q\), Fig. 1, let us isolate a rectangular element of width \(\Delta y\)
Fig. 1.
and regard it as a semi-infinite rod whose lateral sides are parallel to the \(x\)-axis. The position of the rod is determined by the formula
\[ y = vt, \tag{1} \]
where \(t\) is the time of heating of the rod. Let the temperature field of the half-plane be quasi-stationary [1] in the coordinate system \(x, y\). Then the temperature field \(U(x,y)\), independent of the time \(t_0\), for the moving rod may be written in the form \(U(x,t)\). The time \(t\) in no way depends on \(t_0\) and is a parameter of the spatial coordinate \(y\); \(t\) and \(y\) cannot simultaneously be independent variables. Thus, the problem of determining the field \(U(x,y)\) is reduced to determining the temperature of the rod \(U(x,t)\).
Let the lateral sides of the rod under consideration be thermally insulated. Then \(\operatorname{grad} U\) of the rod coincides with the derivative in the direction \(x\), and the gradient of the temperature field reproduced by such a rod will make an angle \(\varphi_y\) with the \(x\)-axis. The real rod cannot be regarded as thermally insulated, since there is always a redistribution of heat in the \(y\) direction, tending to level the temperature in this direction. In this case the gradient of the temperature field \(U(x,y)_{\phi}\) will make an angle \(\varphi_{\phi}\) with the \(x\)-axis, with \(\varphi_{\phi}<\varphi_y\).
If the angle \(\varphi_y\) is small, the rod under consideration may be regarded as thermally insulated at the sides.
For a semi-infinite rod thermally insulated at the sides, to the end of which a constant heat flux \(q\) is supplied, the temperature is known [2]:
\[ U(x,t)=\frac{2q}{\lambda}\sqrt{\bar a t}\,\operatorname{ierfc}(z), \tag{2} \]
where
\[ z=\frac{x\sqrt{v}}{2\sqrt{\bar a y}}, \]
\(\bar a\) and \(\lambda\) are the coefficients of thermal diffusivity and thermal conductivity.
Using (1) and (2), we find
\[ \cos\varphi_y= \frac{1}{ \sqrt{ 1+\frac{\bar a}{vy} \left[ \frac{\operatorname{ierfc}(z)}{\operatorname{erfc}(z)}-z \right]^2 } }. \tag{3} \]
From analysis of formula (3) it follows that the angle \(\varphi_y\) is small if the inequality
\[ \frac{\bar a}{vy}<0.5 \tag{4} \]
is satisfied.
In what follows we shall assume that condition (4) is always satisfied, and the rod under consideration will be regarded as thermally insulated at the sides.
From formula (2) let us determine the temperature of the end of the rod on the boundary of the half-plane:
\[ U(\tau)=\frac{1.13q}{\lambda}\sqrt{\bar a\,\tau}. \tag{5} \]
For a semi-infinite rod thermally insulated at the sides, whose end is maintained at the constant temperature \(U_k\), the temperature \(U(x,t)\) is equal to [2]
\[ U(x,t)=U_k\operatorname{erfc}(z), \tag{6} \]
whence
\[ \operatorname{grad} U(0,t)=-\frac{U_k}{\sqrt{\pi \bar a\,t}}. \tag{7} \]
The quantity of heat entering the rod under the given conditions during time \(\tau\) is equal to
\[ Q(\tau)=\int_0^\tau \lambda\,\frac{U(\tau)}{\sqrt{\pi \bar a\,t}}\,dt =2\lambda\,\frac{U(\tau)}{\sqrt{\pi \bar a}}\cdot\sqrt{\tau} =1.27\,q\tau. \tag{8} \]
In the real process the temperature of the end of the rod during time \(\tau\) changes from \(0\) to \(U(\tau)\). In this case, in the interval \((0,\tau)\) there will be found a point \(\xi\) \((0<\xi<\tau)\) and the corresponding temperature, constant over the time \(\tau\),
temperature \(U_k\) \([0 < U_k < U(\tau)]\), such that the following equality will hold:
\[ Q_\phi = Q^*(\xi)=2\lambda \frac{U_k}{\sqrt{\pi a}}\sqrt{\tau}=q\tau . \tag{9} \]
The temperature \(U_k\) is that conventional calculated temperature, constant over the time \(\tau\), for which equality (9) is satisfied. In what follows we shall call the temperature \(U_k\) the average contact temperature. From (8) and (9) it follows that \(U(\tau)=1.27\,U_k\).
Determination of the average contact temperature in friction
Let body 1 move over the surface of body 2 with constant velocity \(v\), Fig. 2, \(a\), and let it be possible to calculate the heat flux \(Q\), equivalent to the friction power,
\[ Q=Fv=\sigma b\delta v, \tag{10} \]
Fig. 2.
where \(F\) is the friction force, \(\sigma\) is the specific tangential force, \(b\) is the contact width, and \(\delta\) is the contact length.
\[ Q=Q_1+Q_2, \tag{11} \]
where \(Q_1\) and \(Q_2\) are the heat fluxes into bodies 1 and 2. To determine \(Q_1\) and \(Q_2\), according to the method described above, at the instant \(t_0>\dfrac{\delta}{v}\) we isolate an elementary volume enclosed between two planes perpendicular to the \(y\)-axis, and shall regard this volume as two semi-infinite rods \(A\) and \(B\), thermally insulated on the sides, Fig. 2. The heat flux through the area \(dF=b\,dy=bv\,dt\), according to Fourier’s law, is equal to
\[ Q=\int_{t_1}^{t_2} -\lambda\,\operatorname{grad} U(0,t)\,bv\,dt = \int_{t_1}^{t_2} \frac{U_k \varepsilon bv}{\sqrt{\pi t}}\,dt = \tag{12} \]
\[ =2\frac{U_k\varepsilon bv}{\sqrt{\pi}}\left(\sqrt{t_1}-\sqrt{t_2}\right), \]
where \(\varepsilon=\sqrt{\lambda C_v}\) is the coefficient of heat accumulation.
For body 1,
\[ t_1=t_0-\frac{\delta}{v};\qquad t_2=t_0;\qquad \varepsilon=\varepsilon_1. \tag{13} \]
For body 2,
\[ t_1=0;\qquad t_2=\frac{\delta}{v};\qquad \varepsilon=\varepsilon_2. \tag{14} \]
Writing (11) with allowance for (10), (12), (13), (14) and expressing the temperature \(U_k\), we obtain
\[ U_k=\frac{\sigma\delta\sqrt{\pi}} {2\left[\varepsilon_1\left(\sqrt{t_0}-\sqrt{t_0-\frac{\delta}{v}}\right)+\varepsilon_2\sqrt{\frac{\delta}{v}}\right]}. \tag{15} \]
It is not difficult to verify that
\[ \lim_{\frac{vt_0}{\delta}\to\infty}\frac{Q_1}{Q_2}=0. \]
Consequently, for sufficiently large \(v\), the heat flow into body 1 may be neglected even at small values of the time \(t_0\). Then formula (15) takes the form
\[ U'_k=\frac{\sigma}{2\varepsilon}\sqrt{\pi\delta v}. \tag{16} \]
If bodies 1 and 2 are regarded as semi-infinite in the \(y\) direction, Fig. 2, b, the contact length \(\delta^*\) will depend on the time \(t_0\), and
\[ t_0=\frac{\delta^*}{v}. \tag{17} \]
Substituting (17) into (15), we obtain the formula
\[ U_k=\frac{\sigma}{2(\varepsilon_1+\varepsilon_2)}\sqrt{\pi\delta^* v}. \tag{18} \]
Determination of the Average Temperature in the Chip-Formation Zone
Let the entire primary deformation of the chip be concentrated in a narrow rectilinear zone and be uniform over the thickness and width of the chip. Such a deformation zone may be idealized as a shear plane. In the shear plane as a whole, per unit time, an amount of heat \(Q\) is liberated, equivalent to the chip-formation power,
\[ Q=A_{\mathrm{sh}}abv_0, \tag{19} \]
where \(A_{\mathrm{sh}}\) is the specific work of chip formation, \(a\) is the depth of cut, \(b\) is the width of cut, and \(v_0\) is the cutting speed.
\[ Q=Q_1+Q_2, \tag{20} \]
where \(Q_1\) and \(Q_2\) are the heat flows into the chip and into the workpiece.
\[ Q_1=C_vabv_0 V_g, \tag{21} \]
where \(C_v\) is the volumetric heat capacity, and \(V_g\) is the average temperature in the shear plane. In the rectangular coordinate system \(x,y\), shown in Fig. 3 and rigidly connected with the tool, let us single out a volume of metal enclosed between two planes perpendicular to the shear plane, and let us regard this volume as a semi-infinite rod thermally insulated at the sides. The heat flow \(Q_2\)—
is the amount of heat passing per unit time through the cutting line \(x = y \tg \beta_1\), where \(\beta_1\) is the angle of inclination of the shear plane. Taking \(y = v_2 t\), we determine the heat flux through the cutting plane.
Fig. 3. 1 — tool, 2 — workpiece, 3 — chip.
We determine \(\operatorname{grad} U(x,t)\) by differentiating expression (6) with respect to \(x\).
\[ Q_2=\int_{t_1}^{t_2}\lambda\,\frac{U_g}{\sqrt{\pi \bar a t}}\, \exp\left(-\frac{v_2^2 \tg^2\beta_1}{4\bar a}\,t\right)\,dt. \]
Changing the variable \(t\),
\[ t=\frac{4\bar a}{v_2^2\tg^2\beta_1}\,w^2,\qquad dt=\frac{4\sqrt{\bar a t}}{v_2\tg\beta_1}\,dw, \]
we obtain
\[ Q_2=2\lambda b U'_g\,\ctg\beta_1\,\operatorname{erf}(z_1), \tag{22} \]
where
\[ z_1=\tg\beta_1\sqrt{\frac{a v_2}{4\bar a\sin\beta_1}}. \]
Substituting (21) and (22) into (20) and expressing \(U_g\), we obtain
\[ U_g=\frac{A_{\varkappa}\tau}{C_r+\dfrac{2\lambda\,\operatorname{erf}(z)}{a v_0\tg\beta_1}}. \tag{23} \]
Determination of the average temperature on the rake face of the tool
It is known that for small \(z\) the propagation of heat in a plate occurs in the same way as in a semi-infinite body [2]. Therefore, in calculating the average temperature on the rake face of the tool, we shall regard the chip thickness as unbounded. Let the secondary-deformation zone have thickness \(m\), Fig. 3. We shall assume the temperature in the secondary-deformation zone to be the same throughout the whole zone and equal to \(U(l)\). The tem-
the heat flux \(Q\), equivalent to the friction power on the front face of the tool, is divided into the heat flux \(Q_1 = mbv_1 C_v U(l)\), carried away with the so-called “flowing” layer of thickness \(m\), and the heat flux \(Q_2\), carried away along the contact length \(c\) from the flowing layer into the chip. The heat flux \(Q_2\) is determined from (12) and (14), and in the new notation will have the form
\[ Q_2 = 2b\varepsilon_2 \sqrt{\frac{v_1 c}{\pi}} . \]
Substituting the expressions obtained into the equation \(Q = Q_1 + Q_2\) and expressing \(U(l)\), we obtain
\[ U(l) = \frac{F}{ b\left( 2\varepsilon_2 \sqrt{\frac{c}{\pi v_1}} + mC_v \right) }. \]
The cutting temperature, as the average over the length of contact of the chip with the tool and the machined surface, is equal to
\[ U_p = U_0 + \frac{[U_g + U(l)] \cdot c + U_k \cdot \delta}{c + \delta}. \]
REFERENCES
- G. Parkus. Unsteady Temperature Stresses. GIFML, Moscow, 1963.
- A. V. Lykov. Theory of Heat Conduction. GITTL, Moscow, 1960.