Effect of Strain Rate and Temperature on Stress During Cutting of a Brittle Metal
Abstract Generated abstract
This paper examines whether strain rate and temperature affect shear stress during cutting of a brittle metal, using gray cast iron as the main example. The authors argue from deformation theory that, because brittle metals undergo little plastic strain hardening and therefore little recovery, strain rate should have no direct effect on stress, while temperature may have only a secondary influence through changes in material properties. They calculate specific shear work and shear-zone temperature from measured cutting forces, friction conditions, rake angle, and shear-plane angle, and compare cutting tests over 2 to 550 m/min with compression fracture tests at much lower speed. The results indicate that shear stress is governed mainly by cast-iron hardness and is nearly independent of cutting speed, rake angle, and strain rate within the studied range.
Full Text
Effect of Strain Rate and Temperature on Stress During the Cutting of a Brittle Metal
A. M. Rozenberg and Yu. A. Rozenberg
As shown in the work of A. M. Rozenberg and A. N. Eremin, the effect of strain rate on stress during the cutting of ductile metals is exactly the same as in other types of deformation \[1\]; therefore it is possible to predict what this effect will be in the cutting of brittle metals.
The influence of rate on stress in the deformation process depends on the phenomena of strain hardening and recovery. Strain hardening increases the stress; recovery, by removing strain hardening to one degree or another, thereby lowers the stress.
Since recovery proceeds over time and with different intensities at different temperatures, the strain rate also affects the course of recovery and, through recovery, the stress, in different ways in different temperature ranges. The whole variety of effects of strain rate on stress depends only on the effect of rate on the completeness of the recovery process.
In the deformation of brittle metals there is no need to speak of strain hardening, since strain hardening is the result and consequence of plastic change of shape, which in brittle metals is not observed to any appreciable degree. Since brittle deformation is not accompanied by hardening and by an increase in stress that is the result of hardening, the possibility of a reduction in stress as a result of recovery is thereby also excluded.
Since strain rate affects stress only insofar as this rate affects the completeness of the recovery process and the removal of strain hardening, it is natural to conclude that, for brittle metals, there should be no appreciable effect of strain rate on stress. Thus, it should be expected that, in the cutting of brittle metals, the cutting speed should not in itself affect the stress in the chip shear zone, and that this stress should remain unchanged under all the various cutting conditions, provided that it is not influenced by any additional circumstances. Such an additional circumstance may be the heating temperature of the deformation zone, which changes with a change in cutting speed. If, during cutting, this temperature changes the initial mechanical characteristics of the metal being machined, then this may change the stress in the chip shear zone. However, this will already be a consequence not of the strain rate but of the temperature.
When cutting a ductile metal, the effect of strain rate on stress could be revealed by experiments at very low cutting speeds (microspeeds), at which a wide range of changes in cutting speed (for example, from 0.001 to 1 m/min) does not change the temperature in the deformation zone to any noticeable extent. In solving this question as applied to brittle metals, it does not appear possible to make use of microspeeds, since under these conditions the chip from a brittle metal separates in individual elements, and the cutting force is subject to strong fluctuations.
tions, which does not make it possible to obtain reliable values of it, necessary for calculating stresses.
Therefore, in the present case, an experimental solution of the question of the influence of strain rate on stress is possible only in the range of such speeds at which the process proceeds more uniformly, the chip comes off the tool in the form of separate elements, and the cutting force can be reliably recorded. At these speeds, the temperature in the chip shear zone rises appreciably with an increase in cutting speed and may affect the initial properties of the metal.
Temperature in the Chip Shear Zone
The temperature in the chip shear zone is a function of the amount of heat released in this zone as a result of the shear work performed, and also a function of the intensity of heat removal.
In some cases it may be assumed that heat flows into the shear zone from the rake face of the tool, where the temperatures in most cases are considerably higher than in the shear zone. Such a heat inflow from the rake face can occur only through the chip. The probability of heat inflow from the rake face through the chip is the smaller, the higher the cutting speed, since with an increase in cutting speed the velocity of chip motion along the rake face of the tool increases and, beginning with some value, becomes greater than the velocity of heat propagation. From this moment the possibility of heating the shear zone by heat from the rake face of the tool ceases, since the directions of chip motion along the rake face and of the heat flow from the rake face to the shear zone are opposite.
As applied to the case of cutting a brittle metal (gray cast iron), which does not produce a continuous flow chip, the possibility of heating the shear zone by heat from the rake face is generally unlikely.
In this case the temperature in the shear zone will depend only on the heat released in this zone and on the intensity of its removal, mainly into the workpiece.
Heat removal will decrease with an increase in cutting speed, which will cause a relative rise in the temperature in the shear zone as the cutting speed increases, all other conditions being constant.
If, in calculating the temperature in the shear zone, heat removal from this zone is neglected, then only an error in the direction of overestimating the temperatures relative to the actual ones is possible.
Without taking into account the possibility of heat removal, it is not difficult to calculate the temperature in the shear zone, since it must depend only on the specific work performed during the cutting process in this zone. The specific work determines the amount of heat released per unit volume of metal, and this latter, together with the heat capacity of the metal being machined, gives the temperature. Thus, to determine the temperature in the shear zone it is necessary to determine the specific shear work.
The total specific work performed by the rake face of the tool is determined by the expression
\[ A_w = \frac{P_1 \cdot v}{abv} = \frac{P_1}{ab}\ \text{kg mm}/\text{mm}^3, \tag{1} \]
where \(A_w\) is the total specific work performed by the rake face of the tool,
\(a, b\) are the dimensions of the cross section of the cut—the thickness and width of the cut,
\(v\) is the cutting speed,
\(P_1\) is the force applied on the rake face in the direction of the cutting speed.
The total specific work performed by the rake face is the sum of the specific work of friction on the rake face \(A_{w_F}\) and the specific work in the chip shear zone \(A_{w_t}\).
\[ A_w = A_{w_F} + A_{w_t}. \]
Hence the specific work in the shear zone is
\[ A_{w_t} = A_w - A_{w_F}. \tag{2} \]
The specific work of friction on the rake face is
\[ A_{w_F} = \frac{F v_F}{a b v}, \]
where \(F\) is the friction force on the rake face,
\(v_F\) is the velocity of chip displacement along the rake face.
Fig. 1
In turn,
\[ F = P_1 \frac{\sin \eta}{\cos \omega}, \]
\[ v_F = v \frac{\sin \beta_1}{\cos(\beta_1 - \gamma)}, \]
as follows from Figs. 1 and 2.
Fig. 2
Here \(\eta\) is the angle of friction between the chip and the rake face of the cutter,
\(\beta_1\) is the angle defining the direction of the plane of maximum shear stresses,
\(\omega\) is the angle between the forces \(P_1\) and \(R\), with
\[ \tg \omega = \frac{P_2}{P_1}. \]
Hence, for the specific work of friction we obtain
\[ A_{wF}=\frac{P_1}{ab}\, \frac{\sin\eta\cdot\sin\beta_1}{\cos\omega\cdot\cos(\beta_1-\gamma)} . \tag{3} \]
Substituting (1) and (3) into (2), we obtain an expression for the specific work of shear:
\[ A_{wt}=\frac{P_1}{ab} \left\{ 1-\frac{\sin\eta\cdot\sin\beta_1}{\cos\omega\cdot\cos(\beta_1-\gamma)} \right\}. \tag{4} \]
It follows from equation (4) that, in order to determine the specific work in the shear zone of the chip, it is necessary to know the quantities \(P_1, P_2, \eta, \beta_1, \gamma, a, b\). All these quantities are determined experimentally. The forces \(P_1\) and \(P_2\), as forces acting on the rake face of the cutter, are determined by subtracting from the forces measured by the dynamometer the forces applied to the flank face of the cutter. In free cutting, the force \(P_2\) is located perpendicular to the cutting surface; in non-free cutting it is the resultant of forces lying in the plane perpendicular to the cutting surface, and is obtained as the difference between the forces measured by the dynamometer and the forces located on the flank face in this plane.
Fig. 3
With known \(P_1\) and \(P_2\), the angle \(\omega\) is determined from the expression
\[ \tg \omega = \frac{P_2}{P_1}. \]
The friction angle \(\eta\) can be determined, for a known rake angle \(\gamma\), from the expression
\[ \eta=\omega+\gamma. \]
The angle \(\beta_1\), which determines the position of the shear plane when cutting a brittle metal, is found by direct measurement during cutting through a microscope with crosshairs and a graduated scale in the field of view.
Figure 3 gives the results of measurements of the angle \(\beta_1\) when cutting cast irons of different hardness with cutters having different rake angles \(\gamma\).
Along the abscissa axis here is plotted the temperature on the front face of the tool, measured with a natural thermocouple. It would have been possible to plot the angle \(\beta_1\) not as a function of the temperature on the front face, but as a function of the cutting speed; however, in that case, for each hardness of cast iron separate curves would have been obtained, and the graph would have been less illustrative. Fig. 3 shows that the angle \(\beta_1\), changing with changes in the temperature on the front face, and consequently also with the cutting speed, according to curves of a complex character, at the same time has the same value for different cast irons at identical temperatures on the front face of the tool. At identical temperatures it also does not depend on the cross section of cut.
Fig. 4
In Fig. 4 is presented a graph of the variation of the specific shear work for different rake angles when cutting hard cast iron \(H_B = 207\text{–}217\). Here, too, the temperatures on the front face are plotted along the abscissa axis. In order to give an idea of the influence of the cutting speed on the specific shear work, Fig. 5 gives the same results in the coordinates speed—specific work.
Fig. 5
Comparison of Figs. 3 and 4 shows that the specific work is related to the angle \(\beta_1\), which is quite natural, since the latter determines the magnitude of the relative shear.
Fig. 4 gives the values of the specific work in the shear zone for cast iron with hardness \(H_B = 207\text{–}217\). For cast irons of other hardnesses the specific works will vary in proportion to the hardness, since the investigations show—
show that, under conditions of constant temperature on the rake face of the tool, the forces applied on the rake face are directly proportional to the hardness of the cast iron, while the friction angles \(\eta\) and the shear angles \(\beta_1\) do not depend on the hardness of the cast iron.
Having the value of the specific work \(A_{wt}\) and the character of its change with changes in the parameters of the cutting process, it is not difficult to determine the temperature in the shear zone from the expression
\[ t^\circ_{\mathrm{sh}}=\frac{1000\cdot A_{wt}}{427\cdot \rho \cdot c}+t^\circ_0 . \tag{5} \]
Here \(427\) is the mechanical equivalent of heat,
\(\rho\) is the specific weight of the material being machined,
\(c\) is the mean specific heat capacity of the material being machined,
\(t^\circ_0\) is the temperature of the blank.
For calculation, the following numerical values may be adopted for gray cast iron:
\[ \begin{aligned} \rho &= 7.0\ \text{g/cm}^3,\\ c &= 0.13\ \text{cal}/\text{g}^\circ\text{C},\\ t^\circ_0 &= 20^\circ . \end{aligned} \]
Using these values, from equation (5) we calculated the temperatures in the shear zone for the case of cutting cast iron of hardness \(H_B=207\text{—}217\) with tools having different rake angles. The results are presented in Fig. 6 as a function of the temperature on the rake face, and in Fig. 7 as a function of the cutting speed. Figs. 6 and 7 show that
Fig. 6
Fig. 7
the temperature in the shear zone, even when machining hard cast iron and when using a tool with angle \(\gamma=-10^\circ\), reaches only \(400^\circ\). At larger rake angles it is considerably lower. For cast irons of lower hardness it will also be lower, decreasing almost directly in proportion to the hardness of the cast iron, since the specific work in the shear zone \(A_{wt}\), at constant temperature on the rake face, is directly proportional to the hardness of the cast iron.
Figs. 6 and 7 were constructed without taking into account heat removal from the shear zone. This led to certain errors. Thus, Fig. 6 shows that at temperatures on the rake face below \(300^\circ\), the temperature in the shear zone rises, and in its values exceeds the temperature on the rake face of the tool. The temperature on the rake face of the tool was obtained by measurement using the natural thermocouple method, whereas the temperature in the shear zone was calculated from the specific work, without taking into account the possibility of heat removal. It is difficult to imagine such a situation in which the temperature in the shear zone would be higher than on the
of the front face of the cutter, since on the front face the chip is additionally heated by the heat released as a result of friction of the chip against the front face.
When the cutting speed is reduced to zero, the temperatures both in the shear zone and on the front face should decrease to room temperature. Therefore, in the left-hand part of Figs. 6 and 7 we drew tentative broken lines for the decrease in temperature in the shear zone.
Thus, the temperature in the shear zone, as the cutting speed is increased, rises rapidly, reaches its highest value at those cutting speeds at which the temperature on the front face is equal to \(500^\circ\), and then, with a further increase in cutting speed, decreases noticeably.
Effect of temperature on hardness
In order to check whether a decrease in the mechanical properties of cast iron is possible under the influence of the temperatures occurring in the shear zone, we tested the hardness of cast iron at various temperatures. The test was carried out by pressing in a cone made of the hard alloy VK-8.
The cone had an apex angle of \(136^\circ\). The indentation was carried out under a Brinell press at a load of \(187.5\ \mathrm{kg}\).
The impressions obtained were measured under a toolmaker’s microscope. To obtain Brinell hardness numbers, the cone was first calibrated by making its impressions on metals of various known hardnesses. Fig. 8 presents the influence of temperature on the hardness of cast iron.
Fig. 8
It follows from this figure that, with increasing temperature, the hardness of cast iron decreases; up to a temperature of \(450^\circ\) the decrease in hardness is insignificant, while with a further increase in temperature a rapid decrease in hardness takes place.
We are interested in the decrease in hardness within the temperature range up to \(400^\circ\), since, as we have shown, the temperature in the shear zone when cutting cast iron does not exceed this value. Fig. 8 shows that at a temperature of \(400^\circ\) the hardness of cast iron, in comparison with the initial value, falls by 15%. Within these limits an influence of temperature on the stress in the chip shear zone is also possible. At the same time it should be pointed out that the heat and the temperature in the shear zone are the result of the work performed in this zone, and since this work has already been performed, the heat released can no longer affect the shear stress, because the shear has already taken place. The shear stress can be reduced by heating the metal lying ahead of
the shear zone, i.e., that volume of metal which enters the shear zone. Heating of this volume of metal may occur by the transfer of heat from the shear zone by thermal conduction. The temperature in this volume of metal will, of course, be lower than in the shear zone, especially in cutting a brittle metal, when the heated chip, in brittle shear, is completely separated from the machined piece of metal, which makes heat transfer into the volume of metal located ahead of the shear zone difficult. For all these reasons, one should expect an entirely insignificant effect of temperature on the stress in the cutting process, which apparently may be neglected.
Effect of the Strain Rate on the Stress in Cutting Cast Iron
Having determined, in the cutting process, the cutting forces applied on the front face of the tool, and having measured the angle $\beta_1$ which establishes the position of the plane of shear of the chip, we were able to calculate the shear stress $\tau$ in the shear plane. This shear stress was obtained for cast irons of different hardness, different rake angles of the tool, over a wide range of cutting speeds from 2 to 550 m/min, and with a wide variation of temperatures on the front face and in the shear zone.
The values of the shear stresses as a function of cutting speed, tool rake angle, and hardness of the cast iron are shown in Figs. 9 and 10. These figures show that the stresses depend on the hardness of the cast iron and almost not at all on the cutting speed or on the rake angle of the tool. The scatter of points within approximately 10%, observed in Figs. 9 and 10, may be the result of experimental errors both in determining the cutting forces and the angles $\beta_1$. Figs. 9 and 10 show that, over a wide range of variation in cutting speed, and consequently also in strain rate, the shear stress in the shear plane when cutting cast iron is a constant quantity and does not depend on the strain rate.
Fig. 9
This same proposition may be confirmed by comparing the shear stresses in cutting and in compression at the moment of failure of compressed specimens. Specimens in the form of cylinders with a height-to-diameter ratio
\[ \frac{h_0}{d_0}=2, \]
were subjected to compression under a press, with the load recorded at the moment of their fracture. The shear stresses in compression were determined from the relation
\[ \tau=\frac{\sigma}{2}, \]
\[ \sigma = \frac{P_{\text{comp}}}{f}, \]
where \(P_{\text{comp}}\) is the compressive force,
\(f\) is the cross-sectional area of the specimen at the moment of fracture.
Figure 11 presents a comparison of the shear stresses in cutting and in compression of cast irons of different hardness. It may be stated that the shear stresses in both cases are situated fairly well along one common straight line, characterizing the influence of cast-iron hardness on the shear stress.
Fig. 10
Thus, one may speak of equality of the shear stresses in cutting and in compression. Cutting, as we have shown, was performed over a wide range of speeds from 2 to 550 m/min; compression was carried out at a speed of the order of 0.0001 m/min. The ratio of the speeds in cutting and in compression ranged from 20,000 to 5,500,000, and in this case equality of the shear stresses was obtained, which also indicates the absence of an effect of strain rate on the stress.
Fig. 11
REFERENCES
- Rosenberg A. M. and Eremin A. N. Effect of strain rate on stress in the cutting of plastic metals. The article is being published in the present collection.