INFLUENCE OF THE CHEMICAL COMPOSITION OF CHROMIUM–MANGANESE STEELS ON THEIR MACHINABILITY

G. L. KUFAREV, V. I. LIVSHITS

(Presented by the Joint Scientific Seminar of the Departments of Mechanical-Engineering Technology, Machine Tools, and Metal Cutting)

The question of the optimal chemical composition of chromium–manganese steels is central when replacing nickel with manganese in complex-alloy steels. Usually the service properties of the steel remain unchanged under considerable variation in manganese content, and the main criterion of optimality becomes machinability by cutting.

At the Sibtyazhmash plant (Krasnoyarsk), in order to replace heat-resistant chromium–nickel steels in cement-kiln structures, steel Kh13G19TL [1] was developed. However, the question of its machinability by cutting remained unclear. The fact is that the melting technology for this steel gives a considerable fluctuation in manganese content from heat to heat, while the other elements remain relatively stable. Since the heat resistance of the steel changes hardly at all with considerable fluctuation in manganese content, the technical specifications permit manganese variations within the limits of 17–21%, while in fact castings entering the machine shop contain manganese within the limits of 16–25%. Naturally, it is impossible to answer the question of the machinability of the steel under such fluctuation of its chemical composition without setting up special investigations. A further complicating circumstance is that the mechanical properties and, evidently, the machinability of high-manganese and chromium–manganese steels depend to a considerable degree on the melting conditions, the composition and granulation of the charge, etc. [2]. At the Sibtyazhmash plant the opinion had become established that increasing the percentage of manganese improves the machinability of steel Kh13G19TL; therefore, foundry workers often tried to issue manganese at the upper limit of its content.

It is known from the theory of metal cutting that the machinability of various materials by cutting changes under the influence of a whole series of factors. In other words, variations (fluctuations) in machinability are determined by the action of a number of causes, each of which can be represented by a special argument. Between the distributions of two quantities \(x_i\) and \(y_i\) there may exist [3] two types of dependence (relationship): 1) a functional, or complete, relationship; 2) a correlational, or incomplete, relationship.

In an experiment set up to answer a question concerning some production process, it is most often found that each definite value of \(x\) corresponds not to one, but to several different values of \(y\), and the distribution of the values of the characteristic \(y\) changes together with the change in \(x\). In mathematical statistics

Such an incomplete relationship, in contrast to a functional one, is called a correlational relationship.

In production processes, correlational dependencies operate everywhere, since the result of a process is influenced by numerous technological factors, and some of them dominate under the given conditions. By applying the methods of mathematical statistics, it is possible to determine the degree of influence of one or another factor on the result of a process. In particular, methods for studying relationships between random variables constitute the subject of correlation analysis. In random processes related to machine-building production, rectilinear correlation [4] most often occurs; it presupposes a joint normal distribution of the quantities \(x_i\) and \(y_i\) [i.e., the sample under investigation is taken from such a common (general) population in which both quantities are each distributed according to the normal law]. Further, random processes characterizing machine-building production are, for the most part [5], stationary (here the probability distribution functions do not depend on the initial time reference point). For stationary random processes the ergodic hypothesis is valid. According to this hypothesis, a large number of observations of one system at arbitrarily chosen instants of time has the same statistical properties as the same number of observations of a set of similar and arbitrarily chosen systems at one and the same instant of time. Thus, according to the ergodic hypothesis, any one of a set of systems can be used to analyze the behavior of the entire set of systems.

Taking into account all that has been stated above, for investigations whose purpose was to determine the influence of the percentage manganese content on machinability, two samples were selected. The first sample, of 24 blanks, was selected from different heats of one order (one production batch). The second sample consisted of blanks cast for different orders; it included the entire first sample—24 blanks of one order—and, in addition, 10 blanks from different heats of another order.

Chemical analysis of all blanks selected for the experiment showed that the contents of carbon, silicon, and chromium in them varied within very narrow limits, which, according to the data of A. D. Vershinskaya [6], do not exert any appreciable influence on machinability. Fluctuations in manganese content were within the limits of 16.3–24.3%. To eliminate the influence of instability in the properties of carbide cutting inserts, all experiments were carried out with a single VK8 carbide insert. Machining conditions were: \(V = 60\) m/min, \(s = 0.31\) mm/rev, \(t = 3\) mm. Tool wear occurred only on the flank face. The blunting criterion adopted was \(h_{з} = 0.8\) mm.

Experiments on the blanks of the first sample showed that, for one and the same manganese content (in all, the sample contained 8 different values of manganese content), there is observed not a constant value of tool life, but a variation (several unequal values) of tool life. Therefore, in order to answer the question posed, it was necessary to turn to correlation analysis of the experimental results [3]. For the further discussion, let us denote the percentage manganese content by the characteristic \(x\), and tool life by the characteristic \(y\).

Let us consider the stages of correlation analysis using the machining of the first sample as an example. Table 1 presents the observations \(y\) corresponding to a definite value of \(x\), i.e., grouped according to \(x\).

In addition, the values of \(x\) are arranged in a ranked series, i.e., in increasing order. In each row, \(\Sigma y\) is calculated—the sum of all values of \(y\)—and \(n_i\) (\(m\)) is the number of observations of \(y\) for the given value of \(x\),

Next, the mean values of the observations \(y\) in each row are determined by the formula

\[ \bar y=\frac{\Sigma y}{m}. \]

Table 1

Using the first and last columns of Table 1, we construct a graph (Fig. 1). The broken line on the graph is called the empirical regression line of \(y\) on \(x\). The numbers near the points on the graph correspond to the number of observations of \(y\) for the given value of \(x\). The next stage is a qualitative analysis of the empirical regression line. Examination of the graph in Fig. 1 makes it possible to note a tendency for \(y\) to decrease as \(x\) increases. This tendency (a decrease in tool life with increasing manganese content) appears quite plausible if one compares data on the machinability of steels with low manganese content and high-manganese steels.

Fig. 1

We shall determine the form of the relationship between the variables \(x\) and \(y\) for two cases at once: a relationship in the form of a straight line and a relationship in the form of a hyperbola.

Equation of the straight line:

\[ \bar y_x = a_0 + a_1 x. \]

Equation of the hyperbola:

\[ \bar y_x = a_0 + \frac{a_1}{x}. \]

Systems of normal equations for finding the parameters \(a_0\) and \(a_1\):

for the straight line

\[ a_0\Sigma m + a_1\Sigma xm = \Sigma ym, \]

\[ a_0\Sigma xm + a_1\Sigma x^2m = \Sigma xym; \]

for the hyperbola

\[ a_0\Sigma m + a_1\Sigma \frac{m}{x} = \Sigma ym, \]

\[ a_0\Sigma \frac{m}{x} + a_1\Sigma \frac{m}{x^2} = \Sigma \frac{ym}{x}. \]

To solve the systems of normal equations, on the basis of the data in Table 1 we compile Table 2.

Table 2

Summing the columns of Table 2 vertically, we obtain the data for substitution into the systems of normal equations:

Solving the systems of normal equations gives the following results:

Hence the equations of the correlation relationship are obtained in the following form:

The next stage is the compilation of Table 3 of the aligned values $\bar{y}_x$, which are obtained by substituting the actual values of the variable $x$ from Table 1 into the equation of the correlation relationship.

Plotting the aligned values $\bar{y}_x$ on the graph (Fig. 1), we obtain the theoretical regression lines of $y$ on $x$.

To assess the closeness of the relationship between the variables $x$ and $y$, the empirical correlation coefficient $r$ is calculated. For its calculation it is necessary to know the variance of the aligned values $\sigma_{\mathrm{в}}^2$, calculated by the formula

Table 3

\[ \sigma_{\mathrm{v}}^{2}=\frac{\sum(\bar{y}_{x}-Y)^{2}m}{\sum m};\qquad Y=\frac{\sum ym}{\sum m}. \]

From Table 2: \(\sum ym=90.1;\ \sum m=24;\ Y=\frac{90.1}{24}=3.75.\)

The data for calculating \(\sigma_{\mathrm{v}}^{2}\) are given in Table 4.

Variance of the fitted values:

Table 4

| \multicolumn{4}{c}{Equation of a straight line} | \multicolumn{4}{c}{Equation of a hyperbola} |
|---:|---:|---:|---:|---:|---:|---:|---:|
| \(\bar{y}_{x}\) | \(\bar{y}_{x}-Y\) | \((\bar{y}_{x}-Y)^{2}\) | \((\bar{y}_{x}-Y)^{2}m\) | \(\bar{y}_{x}\) | \(\bar{y}_{x}-Y\) | \((\bar{y}_{x}-Y)^{2}\) | \((\bar{y}_{x}-Y)^{2}m\) |
| 7.0 | 3.25 | 10.6 | 10.6 | 6.29 | 2.54 | 6.45 | 6.45 |
| 6.5 | 2.75 | 7.6 | 15.2 | 5.84 | 2.09 | 4.57 | 8.74 |
| 5.9 | 2.15 | 4.6 | 4.6 | 5.24 | 1.49 | 2.23 | 2.23 |
| 4.7 | 0.95 | 0.9 | 0.9 | 4.21 | 0.46 | 0.21 | 0.21 |
| 3.6 | −0.15 | 0.0 | 0.0 | 3.54 | −0.21 | 0.04 | 0.52 |
| 2.8 | −0.95 | 0.9 | 0.9 | 3.06 | −0.69 | 0.47 | 0.47 |
| 2.1 | −1.65 | 2.7 | 5.4 | 2.69 | −1.06 | 1.12 | 2.24 |
| 1.9 | −1.85 | 3.4 | 10.2 | 2.56 | −1.19 | 1.42 | 4.26 |
| | | \(\sum\) | 47.8 | | | \(\sum\) | 25.12 |

In addition, to calculate the correlation coefficient it is necessary to know the variance of the actual values

\[ \sigma_{\mathrm{f}}^{2}=\frac{\sum(y_{\mathrm{f}}-Y)^{2}}{\sum m}. \]

The data for calculating \(\sigma_{\mathrm{f}}^{2}\) are given in Table 5.

Table 5

The empirical correlation coefficient is calculated by the formula

\[ r=\sqrt{\frac{\sigma_{\mathrm{v}}^{2}}{\sigma_{\phi}^{2}}}: \]

by the equation of a straight line

\[ r=\sqrt{\frac{1.99}{6.8}}=0.54, \]

by the equation of a hyperbola

\[ r=\sqrt{\frac{1.05}{6.8}}=0.39, \]

The value of the correlation coefficient \(r\) by itself is still insufficient to judge the existence or absence of correlation between the variables \(x\) and \(y\). It is necessary to assess the significance (reliability) of the obtained correlation coefficient. For this purpose its root-mean-square error is calculated:

\[ \sigma_r=\frac{1-r^2}{\sqrt{n}}, \]

where \(n\) is the total number of observations.

By the equation of a straight line

\[ \sigma_r=\frac{1-0.54^2}{\sqrt{24}}=0.145, \]

by the equation of a hyperbola

\[ \sigma_r=\frac{1-0.39^2}{\sqrt{24}}=0.173. \]

Next the ratio \(\frac{|r|}{\sigma_r}\) is calculated. If it is greater than 3, then the obtained correlation coefficient indicates the presence of a significant relationship between \(x\) and \(y\); if the ratio is less than 3, the relationship is not real (is absent) [4].

For the case under consideration:

by the equation of a straight line

\[ \frac{r}{\sigma_r}=\frac{0.54}{0.145}=3.75>3, \]

the relationship exists

by the equation of a hyperbola

\[ \frac{r}{\sigma_r}=\frac{0.39}{0.173}=2.25<3, \]

the relationship is absent.

Consequently, it can be stated with confidence that the machinability of blanks made of steel Kh13G19TL depends linearly on the manganese content in the castings (within the heats of one order), and, with an increase in manganese content, machinability worsens (tool life decreases). A hyperbolic relationship between these variables is absent. It should be noted that the conclusion obtained does not coincide with the opinion existing at the plant that machinability improves with an increase in manganese content. This discrepancy is evidently explained by the result obtained below.

The value of the correlation coefficient, with probability 0.997, will lie within the limits

\[ r-3\sigma_r<r<r+3\sigma_r. \]

\[ 0.105<r<0.975. \]

In the opinion of A. M. Dlin [7], if the lower bound of \(r\) is small, then there are no grounds for considering that a correlation relationship exists between the quantities \(x\) and \(y\) in the general population from which the sample under consideration was taken. To check this circumstance, a second sample was considered, including blanks cast under different orders. The basic data for the second sample are presented in Table 6. The results of the search for rectilinear and curvilinear correlation relationships are given

in Table 7. The empirical and theoretical regression lines are presented in Fig. 2.

Fig. 2.

Studies of the second sample confirmed the opinion that there is no correlation between the variables \(x\) and \(y\)

Table 6

in the case of a broader population, which includes blanks smelted under different orders. It may be assumed that the existence of a relationship in the case of a small sample and the absence of a relationship in the case of a large sample are connected with those differences that are caused by different orders (launch batches). In the case of the large sample, the intrusion of a more substantial factor—the change in the properties of castings in heats of another order—disrupted the variation in tool life and made the correlation relationship unrealistic. The validity of such an assumption is also confirmed by literature data [2] on a strong depen-

…dependence of the properties of chromium–manganese steels on the melting and casting technology, the charge composition, the condition of the furnace, and even the skill of the steelmakers.

Table 7

REFERENCES

1. “Quality, Reliability, Durability,” collection of reports, TsBTI, Krasnoyarsk, 1964.
2. “Technology of Mechanical Engineering,” scientific and technical collection, Moscow, TsINTIMASH, 1960.
3. B. Kh. Belov et al. A Practical Methodology of Correlation Analysis. Collection of works of VNII, issue 4, Moscow, 1964.
4. A. K. Kutai, Kh. B. Kordonskii. Accuracy Analysis and Quality Control in Mechanical Engineering, Moscow–Leningrad, Mashgiz, 1958.
5. I. M. Malikov et al. Fundamentals of the Theory and Calculation of Reliability. Sudpromgiz, Leningrad, 1960.
6. “Improvement of the Technology for the Production of Turbine Blades,” collection of articles. Moscow, Mashgiz, 1956.
7. A. M. Dlin. Mathematical Statistics in Engineering. “Sovetskoe Radio,” 1949.