Work Hardening During Hole Rolling

V. D. Varlakov

According to data from a number of investigators, it has been established that, as a result of work hardening, the fatigue strength of parts subjected to alternating dynamic loads increases. This phenomenon can be explained on the basis that, during work hardening, surface compressive stresses are created, whereas fatigue failure of parts occurs only as a result of tensile stresses. The compressive stresses created by work hardening reduce the resultant tensile stress (from the load and the work hardening) and thereby significantly increase fatigue strength. In addition, work hardening imparted to the surface of a part by means of special tools increases the wear resistance and the finish of the surface being processed. Thus, it may be stated with confidence that surface work hardening can, in a number of cases, be used as a positive factor.

The task of the present article is to clarify certain questions connected with the work hardening of the surface obtained when holes are rolled with the aid of a ball mandrel¹.

Change in Surface Hardness

The specimens (rings) obtained after rolling treatment were cut along a chord (Fig. 1), and the hardness of the treated surface was measured by the Rockwell method on scale “B.” The hardness readings determined in this way were taken as the values of the surface hardness. The Rockwell method was chosen because its hardness readings are readily comparable with Brinell hardness readings—the standard method of determining hardness. The direct use of the Brinell method for determining the hardness of the surface layer is excluded, since this method employs excessively large loads capable of pressing through the work-hardened layer.

The question of the change in surface hardness under work hardening accompanying the treatment of external surfaces has been considered by a number of authors (Thum A., Herger, Wittmann, Romanov, Rosenberg A. M.). In all these authors the investigations of the work-hardened layer were of secondary importance. For internal surfaces, however, questions of work hardening still remain insufficiently elucidated.

Investigations of the change in surface hardness as a result of work hardening on the internal surface were carried out by us on a series of carbon steels and with variation of the conditions of its formation. In our investigations, work was carried out to determine surface hardness as a function of: 1) the difference between the diameters of the “rolling” and the hole, with a constant hole diameter (fluctuations in hole dimensions amounted to 10–12 microns); 2) rolling speed, feed, and number of passes over the layer, with a constant difference in diameters (the diameters being treated varied from 38 to 85 mm);

3) the initial hardness of the steel; 4) the forces arising on the balls of the mandrel.

The first series of experiments consisted in establishing the dependence of the surface hardness on the setting diameter of the mandrel balls \((d_y)\). The experiments were carried out on steel 45, having a Brinell hardness of 197. The hole diameter remained practically constant. By changing \(d_y\), we thereby changed the thickness of the deformed layer and, consequently, also the magnitude of the

Fig. 1. Specimen for determining surface hardness and the depth of penetration of work hardening

Fig. 2. Influence of the mandrel setting diameter on the surface hardness after rolling

forces arising on the balls. It was established that the hardness of the surface layer does not remain constant with a change in the thickness of the deformed layer. The results of these investigations are given in Table 1 and Fig. 2.

Table 1

Influence of the mandrel setting diameter on the surface hardness after rolling
\((n\) and \(s\)—constant)

From the table presented it is seen that the magnitude of the surface hardness increases with an increase in the setting diameter and in the deformation force (an increase in the setting diameter is analogous to an increase in pressure on the surface being treated). It is also seen here that this increase in the hardness of the surface layer is observed only up to a certain limit. After this

there is no increase in the limit of surface hardness; instead, there is observed (as we shall see below) an increase in the depth of penetration of deformation. A considerable increase in the degree of deformation is accompanied by destruction of the surface layer—“peeling.”

Similar phenomena of attaining a limiting hardness by a metal undergoing work hardening were also observed by G. O. Neiman and Moser. In our investigations it has been established that the limit of increase in hardness of the surface layer is a setting diameter of 37.88 mm. We adopted this setting diameter as optimal, since deformation at a larger value did not give an increase in surface hardness. This setting diameter was adopted by us as optimal also because smaller values

Fig. 3. Influence of speed and feed during rolling on the magnitude of surface hardness after rolling

leave traces of the preceding machining, which have no practical significance. A setting diameter of 37.88 mm, or a difference between the rolling diameter and the hole diameter before rolling of 0.16 mm, was taken by us in investigations of the influence of speed, feed, and initial hardness of the material. At the same time, after carrying out the main experiments with a diameter of 37.88 mm, check experiments were performed on other diameters, varying up to 85 mm while maintaining a constant difference of diameters (0.16 mm), which fully confirmed the correctness of the conclusions drawn.

From Figs. 3 and 4 it is evident that the influence of speed, feed, and number of passes over the layer on surface hardness is absent.

Fig. 4. Influence of the number of passes over the layer on the magnitude of surface hardness after rolling

Figures 5 and 6 show the change in surface hardness as a function of the initial hardness of the steel. Fig. 5 gives the absolute values of this hardness, and Fig. 6 shows its relative increase as the initial hardness of the material varies. From the latter graph (Fig. 6) it is evident that, as the hardness of the steel increases, the relative increase in surface hardness during rolling decreases. On average, for carbon

Fig. 5. Change in surface hardness after rolling with variation in the initial hardness of the steel

structural steels it lies within the range 28–48%. This circumstance is extremely important, since it makes it possible, by using surface rolling, to obtain in low-carbon steel a wear resistance equal to that of high-carbon steel; wear resistance depends strongly on the hardness of the material.

Fig. 5. Relative change in surface hardness after rolling

Depth of Work Hardening and Rolling Conditions

Our investigations, as well as investigations by other authors, have established that the work hardening obtained on the surface of a part by the action

under certain forces applied to it, spreads from the periphery into the depth of the metal. It has been established that the hardness of the work-hardened layer changes from its maximum value at the surface to the normal hardness of the undeformed metal. Consequently, in the work-hardened layer there is a gradual change in hardness, and hence also in the degree of deformation. We set ourselves the task of determining the depth of penetration of deformation as a function of a number of factors.

Determining the regularity of the change in the depth of penetration of work hardening is of considerable scientific and practical interest, since this question has not been sufficiently studied.

We believed that rolling can and should become one of the methods of finishing holes, and for this it is necessary to know which factors influence the depth of penetration of work hardening. In a number of cases, owing to the design of a part, strengthening of the surface by work hardening cannot be carried out as the final operation in machining the given surface, and then it will be necessary, after rolling, to subject this surface to additional cutting. In the latter case it is especially important to create work hardening of a predetermined magnitude in advance, so that in the course of subsequent machining (chip removal) the entire work-hardened layer is not removed. Such machining may most often be required in the repair of parts after some wear. From what has been said, the reasons that compelled us to undertake a detailed study of the question of the depth of penetration of work hardening during hole rolling become clear. The need to study this question is further aggravated by the fact that all investigations available in this field concern only external surfaces.

To solve the problem posed, it was necessary to adopt a method for determining hardness at a certain depth of the specimen that would give us sufficiently accurate results. The difficulty of solving the question consisted in the fact that we were dealing with internal surfaces, the filing or etching away of which does not eliminate possible errors owing to the difficulty of controlling the magnitude of the removed layer on a sectioned specimen. For this reason we chose the “oblique cut” method. The advantages of this method over existing methods are that here we deal with somewhat increased distances, owing to which the relative accuracy of our method is undoubtedly higher. A second advantage of the method we chose should be considered the simplicity of preparing the specimens and of measuring hardness on them, as well as the distances at which hardness readings were taken.

The essence of the adopted method is that, for the purpose of determining the depth of penetration of work hardening, the specimens were cut along a certain chord in such a way that a narrow strip of the work-hardened internal surface remained. Parallel to this cut, the base of the specimen was cut off, thereby creating a sufficiently reliable supporting surface. On the specimen thus obtained (Fig. 1), at some arbitrary distance from the remaining strip of the internal rolled surface, hardness measurements were made with a Rockwell apparatus. As a result of these measurements, ball impressions remained on the surface. Using a Brinell microscope, we measured the distances from the center of the ball impression to the edge of the remaining strip and the strip itself. The readings obtained in this way were then converted into distances along the radius.

The dependence of the distance readings along the chord and the radius is shown in Fig. 7 and can be expressed by the following mathematical relation:

\[ b = \frac{1}{2}\sqrt{4(r_1 - h_1)^2 + c_2^2} - r_1 . \tag{1} \]

Using equation (1), we determined the depth at which the hardness measurement was made, measured along the radius. On the basis of the calculations obtained, curves were plotted for the change in hardness as a function of depth (Fig. 8). From the graphs obtained in this way, the depth of penetration of work hardening was determined.

The depth of deformation or work hardening was taken to be the depth corresponding to the inflection points of the curves plotted for each specimen, from the constant hardness of the base material to the rising branch.

Fig. 7

Fig. 8. Determination of the depth of penetration of work hardening

As a result of the investigations carried out, a quite definite regularity was obtained: a gradual decrease in hardness from the surface layer, where it is constant, to the base material, which also has a constant hardness. Considering the regularities obtained for the change in hardness through the thickness of the rolled layer, it is interesting to note that it decreases more smoothly with an increase in the degree of deformation, as is seen from the graphs in Fig. 8, where the upper curve corresponds to the maximum degree of deformation, and the lower to the minimum.

A gradual decrease in hardness was also noted in the works of Horger and V. A. Keldyushev [2], who determined the depth of penetration of work hardening by the gradual mechanical removal of surface layers.

The phenomenon of a gradual decrease in the hardness of the work-hardened layer from the maximum to the hardness of the base material is a valuable quality of the proposed treatment method, since, along with the increased hardness of the surface layer, there is no sharp boundary in hardness. This eliminates the possibility of overstressing at the transition point from the work-hardened layer to the base metal and the possible spalling of the work-hardened layer during service of the part, which sometimes occurs with surface hardening.

Influence of the rolling conditions on the depth of penetration of work hardening

In the investigations of A. M. Rozenberg [3], in experiments conducted on 1 GPZ [4], and in the X-ray structural investigations of Thomassen and Mac-Ketchen [4], it is noted that the depth of work hardening changes with a change in the rate of deformation. An increase in the rate of deformation is accompanied by a decrease in the depth of work hardening.

Data fully confirming the correctness of the conclusions of the above-mentioned researchers were also obtained in our experiments.

The results of investigations of the influence of deformation rate on the depth of work-hardening penetration are given in Table 2 and in Fig. 9.

Table 2

Influence of deformation rate on the depth of work-hardening penetration

It is evident from the table and the graph that, when the feed is varied from \(0.575\) mm/rev to \(0.1\) mm/rev and the number of revolutions from \(500\) rev/min to \(320\) rev/min, the depth of work-hardening penetration changes from \(1.0\) mm at \(n = 500\) rev/min and \(s = 0.575\) mm/rev—the maximum deformation rate in this series of experiments—to \(2.25\) mm at \(n = 320\) rev/min and \(s = 0.145\) mm/rev.

As can be seen from the graph in Fig. 9, the dependence of the depth of work-hardening penetration “\(b\)” on the speed and feed is expressed by parallel lines. Consequently, it can be expressed by the equation

\[ b = f(s) \tag{2} \]

or

\[ b = \frac{C}{s^x}, \tag{3} \]

where \(C\) is a constant depending on the rolling speed, the material, and the experimental conditions,

\(x\) is an exponent characterizing the material being processed.

Fig. 9. Influence of deformation rate on the depth of work-hardening penetration

Specific values of the coefficient “\(C\)” and the exponent “\(x\)” can be determined from the graph in Fig. 9.

Determination of their values:

\[ \tg \alpha = x = \frac{2A}{B}. \tag{4} \]

The ordinate along the “\(y\)” axis is taken with a coefficient of 2 because in the drawing the scale of the values of the depth of penetration of work hardening is half the scale of the feed,

\[ \tg \alpha = \frac{2.7.0}{60} = 0.233. \]

The coefficient “\(C\),” expressed through the depth of penetration of work hardening and the feed, can be determined from the following relation

\[ C = bS^x, \tag{5} \]

for \(v_1 = 59.8\ \text{m/min}\),

\[ C_1 = b_1 S^x = 1.4 \cdot 0.1^{0.233} = 0.836, \]

\(v_2 = 38.3\ \text{m/min}\),

\[ C_1 = b_2 S^x = 2.1 \cdot 0.1^{0.233} = 1.26. \]

From the obtained values of the coefficient “\(C\)” we see that it depends on the rolling speed.

The influence of the rolling speed on the depth of penetration of work hardening at constant feed varies according to a straight-line law, for which the equation may be written:

\[ b = A - mv, \tag{6} \]

where \(A\) is a constant depending on the experimental conditions and the material being machined,

\(m\) is the coefficient at the rolling speed, corresponding to the tangent of the angle of inclination of the straight line.

From the experimental data the values of the coefficients \(m\) and \(A\) will be

\[ m = -\frac{1.4 - 2.1}{59.8 - 38.3} = 0.0325, \]

\[ A = b + mv = 2.1 + 0.0325 \cdot 38.3 = 3.345, \]

then

\[ b = 3.345 - 0.0325\,v. \]

It was indicated above that the lines corresponding to the depth of penetration of work hardening at different speeds are parallel in logarithmic coordinates and that the values of “\(C\)” are a function of the rolling speed.

The values of the coefficient “\(C\)” at different speeds can be determined from equation (5), which we do for \(S = 0.1\ \text{mm/rev}\) and different values of the speed. The calculation results are collected in Table 3.

Table 3

Values of the coefficient “\(C\)” at different rolling speeds

The data of the table are plotted on the graph in Fig. 10. From the graph it is seen that the values of the coefficient “\(C\),” just like the depths of penetration of work hardening, are expressed by a rectilinear dependence, for which an equation of the following form may be written

\[ C = A_1 - m_1 v. \]

The corresponding values of \(A_1\) and \(m_1\) will be

\[ m_1=\frac{1.26-0.836}{59.8-38.3}=0.197, \]

\[ A_1=C+m_1v=1.26+0.0197\cdot 38.3=2.015. \]

Fig. 10. Effect of rolling speed on the depth of penetration of work hardening

Substituting the values of “\(C\)” into formula (3), we obtain the expression for the depth of penetration of work hardening as a function of speed and feed in the following form

\[ b=\frac{A_1-m_1v}{s^x}. \tag{8} \]

For our material, equation (8) may be written as:

\[ b=\frac{2.015-0.0197\,v}{s^{0.233}}. \tag{9} \]

Let us check the correctness of the derived dependence

\[ b=\frac{2.015-0.0197\cdot 59.8}{0.3^{0.233}}=1.16, \]

which corresponds exactly to the graph in Fig. 9.

The dependence derived here for the depth of penetration of work hardening on speed and feed is valid for steel 45.

In the works of Herger and Foppel [5], the relationship between deformation pressure and penetration depth is given. Both authors believe that, with an increase in the deformation force, the depth of penetration of work hardening increases.

In the present investigations, complete agreement was obtained between the test results and the dependence expressed by the cited authors. We have established that, with an increase in the difference between the diameters of the mandrel and the hole before machining, which is equivalent to an increase in the degree of deformation, the depth of penetration of work hardening increases significantly. The results of the investigations are given in Table 4 and represented graphically in Fig. 11.

From the graph and the table it is evident that the depth of penetration of work hardening can vary considerably depending on the thickness of the layer deformed during rolling. Thus, in our experiments, when the difference between the diameters of the mandrel and the machined hole was changed from \(0.12\) mm to \(0.38\) mm, that is, by a factor of 3.15, the depth of work hardening changed from \(0.9\) mm to \(4.5\) mm, that is, by a factor of 5.

Schmalz [6] arrives at analogous conclusions, stating: “The depth of the deformed zone increases strictly in proportion to the increase in cutting pressure.”

Table 4

Dependence of the depth of penetration of work hardening on the difference between the diameters of the mandrel and the hole before rolling

From the graphs expressing the dependence of hardness on the depth of penetration of work hardening, we see that, besides the change in the depth of penetration of work hardening, the character of the hardness distribution within the work-hardened layer also changes. The hardness obtained with a greater difference in diameters \((\delta d)\) is higher than the hardness obtained at the same depth with a smaller difference in diameters. This gives us grounds to assert that parts work-hardened with greater deformation will have higher mechanical strength not only because of the increase in the thickness of the work-hardened layer, but also because of the increase in hardness at a smaller depth. Thus, for example, if we take the hardness

Fig. 11. Dependence of the depth of penetration of work hardening on the difference between the diameters of the mandrel and the hole before rolling

at a depth of 0.5 mm for \(\delta d\) equal to 0.12, 0.23, and 0.38, then it is respectively equal to 91.5, 97.5, and 99.5 units on the Rockwell “B” scale. Consequently, with a greater deformation force, the volume of stronger metal increases and the strength of the entire product rises appreciably.

The dependence presented in Fig. 11 can be expressed by the equation

\[ b = C_2 \delta d^y . \tag{10} \]

From the graph, \(y=\operatorname{tg}\alpha=1.41\); then the coefficient \(C_2\) is determined by taking the specific values of “\(b\)” and “\(\delta d\)” at \(b=2\) mm and \(\delta d=0.21\) mm.

\[ C_2=\frac{b}{\delta d^y}=\frac{2}{0.21^{1.41}}=18.3. \]

If the depth of penetration of work hardening at \(\delta d=0.16\) is taken as unity, then for other differences in diameters it can be expressed by the coefficient \(K_1\). The values of the coefficient \(K_1\) are given in Table 5 and in Fig. 11 (lower line).

Table 5

Values of the coefficient \(k_1\).

From Fig. 11 we see that the dependence of \(K_1\) is analogous to the dependence of “\(b\)” on the difference in diameters, that is,

\[ K_1=C_2^1\delta d^y. \tag{11} \]

The lines expressing \(b=f(\delta d)\) and \(K_1=f(\delta d)\) are parallel; consequently, the exponent at \(\delta d\) is the same in both cases:

\[ C_2^1=\frac{K}{\delta d^y}=\frac{1.51}{0.21^{1.41}}=14.6. \]

Next, the author determines the dependence of the depth of penetration of work hardening on the initial hardness of the steel. The experiments showed that the depth of penetration of work hardening depends on the hardness of the steel. The graph in Fig. 12 clearly reflects this dependence.

Fig. 12. Dependence of the depth of penetration of work hardening on the initial hardness of the steel.

In the graph: vertical axis \(b\), mm; horizontal axis \(H_b\); \(n=320\) rpm; \(S=0.275\) mm/rev.

Taking the depth of penetration of work hardening at \(H_b=197\) as unity, the following values were obtained for the coefficient \((K_2)\), which takes into account the influence of \(H_b\) on the depth of penetration of work hardening (Table 6).

Table 6

Values of the coefficient \(k_2\).

The influence of the number of passes (\(i\)) along the layer is presented in Fig. 13. From the graph in Fig. 13 it is evident that, with an increase in the number of passes along the layer, the relative increase in the depth of penetration of work hardening decreases.

Fig. 13. Influence of the number of passes along the layer on the depth of penetration of work hardening

The values of the coefficient \((K_3)\), which accounts for the influence of the number of passes along the layer on the depth of penetration of work hardening, are presented in Table 7.

Table 7

Values of the coefficient \(k_3\)

To obtain a complete picture of the dependence of the depth of penetration of work hardening on the rolling conditions, experiments were carried out to establish the influence of the number of rolling balls. The experiments were conducted with 3, 4, 6, and 8 balls. The experimental results, presented in Fig. 14, show that with an increase in the number of balls in the mandrel, the depth of penetration of work hardening decreases. A certain fluctuation in the feed per ball is considered insignificant, since it does not exceed 0.006 mm and is caused by the impossibility of stepless adjustment of the feed on the machine.

In logarithmic coordinates, the dependence of the penetration depth on the number of balls is expressed by a straight line and may be represented by the equation

\[ b = C_4 n_{\sh}^{-z}, \tag{12} \]

where \(n_{\sh}\) is the number of balls,
\(C_4\) is a coefficient depending on the experimental conditions and on the material being processed,
\(z\) is the exponent for \(n_{\sh}\).

The values of the coefficient \(K_4\), which takes into account the influence of the number of mandrel balls on the depth of penetration of work hardening, are presented in Table 8 and in Fig. 14.

Table 8

Values of the coefficient \(K_4\).

From this same graph the coefficient \(K_4\) can be expressed by the equation

\[ K_4=\frac{C'_4}{n_{\mathrm{sh}}^{z}}, \tag{13} \]

whence \(C'_4=1.26\cdot 5^{0.534}=3.02\).

Fig. 14. Influence of the number of balls in the mandrel on the depth of penetration of work hardening.

The diameter of the balls has no influence on the depth of penetration of work hardening. Taking into account all the conditions of rolling, equation (8) may be written as follows:

\[ b=K_1\cdot K_2\cdot K_3\cdot K_4 \frac{(A_1-m_1v)}{S^x}. \tag{14} \]

Substituting the values of the coefficients \(K_1\) and \(K_4\), we obtain

\[ b=K_2\cdot K_3\cdot C'_2\cdot C'_4 \frac{(A_1-m_1v)\,\delta\, d^y}{S^x n_{\mathrm{sh}}^{z}}. \tag{15} \]

Since \(C'_2\) and \(C'_4\) are constant quantities, we may replace them by a single coefficient \(C\)

\[ C=C'_2\cdot C'_4=14.6\cdot 3.02=44.092. \]

In final form, equation (15) may be written

\[ b=C\cdot K_2\cdot K_3 \frac{(A_1-m_1v)\,\delta\, d^y}{S^x n_{\mathrm{sh}}^{z}}. \tag{16} \]

Figure 15 gives a check of the correctness of the derived formula. Here the lines correspond to the theoretical calculation, while the points located around them were obtained experimentally. As can be seen from the graph, complete agreement was obtained.

Fig. 15. Control experiments to reveal the influence of the rolling speed on the depth of penetration

Conclusions

1. The influence of feed, rolling speed, and the number of passes over a layer on the value of the surface hardness is absent in those cases when the amount of deformation has reached a certain limit.

2. The value of the surface hardness depends on the initial hardness of the material. The relative increase in hardness decreases with increasing initial hardness.

3. The depth of penetration of deformation is a function of speed, feed, number of passes over a layer, the difference between the diameters of the tool and the hole, the initial hardness of the steel being worked, and the number of balls in the mandrel.

4. With a decrease in the degree of deformation, whether due to a change in the number of passes or in the difference between the diameters of the mandrel and the hole, the depth of penetration of work hardening decreases.

5. The influence of the rolling speed and feed is opposite to the influence of the degree of deformation; namely, an increase in the deformation speed is accompanied by a decrease in the magnitude of its penetration into the depth of the metal.

6. The initial hardness of the material and the number of balls affect the depth of work hardening similarly to the speed factor; that is, with an increase in hardness and in the number of balls, it decreases.

7. The value of the depth of penetration of work hardening, as a function of a number of factors, can be expressed by a mathematical dependence of the following form:

\[ b = C \cdot K_2 \cdot K_3 \frac{(A_1 - m_1 v)\delta\, dy}{S^x n_{\mathrm{sh}}^z}. \]

References

1. Witman F. F. The influence of surface deformation on the impact cold brittleness of steel. ZhTF, vol. VII, no. 9, 1937.

2. O’Neill G. Hardness of Metals and Its Measurement. Metallurgizdat, 1940.

3. Rosenberg A. M. Experimental study of the chip-formation process. Izvestiya STI, 1929.

4. Collection of Abstracts of Papers of the Conference on Metal Cutting. Publishing House of the Academy of Sciences of the USSR, 1937.

5. Foppl. Long-Term Testing of Metals, 1929.

6. Shmalts G. Surface Quality. Mashgiz, 1941.

1. See the author’s article “Rolling Holes with a Ball Mandrel as a Method of Finishing,” printed in this volume. ↩