Investigation of Structural Cast Iron Hardness for Castings of Automobile Industry on the Basis of Construction and Analysis of Regression Equation in the Factor Space «carbon (C) – Carbon Equivalent (C eq)»

The object of research is structural iron with lamellar graphite, in which the carbon equivalent (C<sub>eq</sub>) is in the range (4.2–4.4) %, and the carbon content (C) in the range (3.42–3.57) %. The aim of research is description of the distribution of the hardness value of structural cast iron of serial meltings in the C–C<sub>eq</sub> factor space at fixed values of the Cr–Ni–Cu–Ti alloy content in narrow intervals. It is shown that a polynomial regression equation of the form HB=HB(C, C<sub>eq</sub>) can be used to obtain a workable analytical description. It is shown that such structure of the equation and those obtained by the method of least squares corresponding coefficient estimates provide 92 % accuracy of the forecast even with a small sample of data.<br><br>On the basis of the canonical transformation of the response surface and its ridge analysis, it is established that it is possible in principle to satisfy different requirements for hardness. So, if the range of the planning area C=(3.42–3.57) % and C<sub>eq</sub>=(4.2–4.4) % is chosen as the imposed constraint, then several sub-optimal solutions are possible. This is the case if the task of minimizing hardness is not set and the range HB=180–250 satisfies the quality requirements specified by the production conditions. If the priority is hardness minimization, then the sub-optimal solution is one and it is like the point of intersection of the constraint line (r=1.414) and the lower ridge line y=y(r). On the basis of this, it is concluded that the sub-optimal solutions are multi-variant, depending on the requirements of production. A nomogram has been constructed, which makes it possible to select in a rational way the technological regimes of out-of-furnace treatment in the part concerning the adjustment of the chemical composition of the alloy.


Introduction
In the technologies of foundry production, priority is given to the quality management of obtained castings. Ta king into account the fact that the quality of castings is formed by two components of technological processes -the metallurgical component and the technology of the moldthere are basically two main approaches to research. The first of them is based on a comprehensive study of the design and manufacture of casting technology, the second on the study of smelting and outoffurnace processing [1][2][3]. Among the considered quality criteria, the surface cleanli ness, the presence of surface defects, the correspondence of dimensional and geometric accuracy specified in the design documentation, mechanical and special properties can be distinguished. And if the first are formed by the techno logical components of the process -the mold technology, the latter is formed by the metallurgical component, which depends on the melting and outoffurnace processing. It is the mechanical properties -the ultimate tensile strength and hardness -that are controlled by government stan dards. At the same time, it should be noted that when comparing samples of constructional cast iron with general machinebuilding design, preference should be given to those with a maximum tensile strength and a minimum hardness. With regard to hardness, it is necessary to take into ac count that the lower level of the hardness should provide the specified performance properties in case the surface is contacting. The upper level of hardness should ensure the possibility of highquality machining and do not cause a reduction in the reliability of the metalworking tool. It is also important to take into account that sudden increase in hardness indicates changes in the microstructure and the formation of carbides. In particular, we can talk about the most dangerous for structural iron carbon carbide -Fe 3 C. Thus, hardness as a regulated quality indicator of structural iron is important from the technological point of view, and with information -as an indirect indicator, indicating undesirable changes in the microstructure. There fore, the research areas devoted to the study of the effect of physical, chemical, technological and structural factors on the hardness of structural iron are relevant.

The object of research and its technological audit
The object of research is structural cast iron for commer cial castings for automotive castings, in which the carbon ISSN 2226-3780 equivalent (C eq ) is in the range (4.2-4.4) %, and the car bon content (C) in the range (3.42-3.57) %. Previous studies on the effect of carbon and carbon equivalent on the ultimate tensile strength [4] required additional hardness studies. This production need is caused by the importance of evaluating the effect of C and C eq on hard ness, in order to study the possibilities of its reduction for free surfaces of castings without adversely affecting the strength parameters. Such decrease in the hardness of structural iron within acceptable limits can also contrib ute to improvement of the machining processes for the surfaces. Finally, the availability of sound technological solutions, based on an adequate analytical description of the effect of C and C eq on hardness, can help reduce costs in the smelting of structural cast iron. This is ensured by minimizing them on the basis of the choice of the optimum composition of the charge for the minimum cost.
From the point of view of theoretical results, it is of interest to investigate the joint effect of carbon and carbon equivalent on hardness as an indirect indicator for the subsequent study of the mechanisms of the formation of cast iron microstructure.
Technological audit was carried out in the foundry of JSC «Kremenchuk plant of road machines» (Kremenchuk, Ukraine). The technological audit was aimed at reveal ing real quantitative characteristics of hardness (HB) in the specified range of С-С eq at realization of the basic technological process of induction melting of synthetic cast iron СЧ20 GOST141285. The melting was carried out in an induction crucible furnace with an acid lining ИСТ1/0.8М5. As a modifier, ferrosilicon FeSi75 was used. Detailed technological process regimes are described in [4].
In accordance with the technological instructions in force at the enterprise, samples were taken for chemical analysis, and wedge samples were cast to determine the hardness of cast iron in accordance with GOST 141285.

The aim and objectives of research
The aim of research is description of the hardness dis tribution of structural cast iron for series castings for au tomotive castings in the C-C eq factor space in the ranges C = (3.42-3.57) % and C eq = (4.2-4.4) %, for fixed in nar row intervals, the values of the Cr-Ni-Cu-Ti content of the alloying complex. This would give an opportunity to choose the optimal values for C and C eq for the selected criteria for the subsequent evaluation of the possibility of minimizing costs in the smelting of cast iron.
To achieve this aim, it is necessary to solve the fol lowing tasks.
1. Construction of a workable analytical description of the influence of the selected input variables on cast iron hardness.
2. Investigation of the response surface for the pre sence of optimal or suboptimal values of input variables.

Research of existing solutions of the problem
Hardness, as one of the quality indicators of structural iron, depends on a variety of technological factors, but on the other hand, its magnitude is related to macro and microstructure. Theoretically, any impact in the techno logical process can affect the change in hardness, precisely as one of the quality indicators. Therefore, researchers, as a rule, approach the solution of problems in a complex manner. Thus, a method for estimating microstructural inhomogeneities: graphite, foundry defects, and the struc ture of a metal matrix were proposed in [5]. The latter, as is known, is the determining factor for the formation of hardness of structural cast iron. The proposed method allows, in the opinion of the authors, to predict the es timation of the output characteristic on the basis of in formation on microstructural inhomogeneities and loading conditions.
The studies described in [6] allowed to establish the qualitative influence of the new modifier containing SiC, in combination with ferrosilicon FeSi75, on the morphol ogy of graphite, the matrix structure and the mechanical properties of cast iron. It has been established that it is possible to form a large number of microzones with a high content of carbon and a silicon concentration that promote a favorable course of the graphitization process. However, the assessment was carried out only at a qualitative level. The influence of the modifier composition in the mold on the microstructure and fatigue strength of castings made from cast iron EN GJS 7002 is described in [7]. In this paper, the authors talk about the possibility of managing properties through effective modification. In this case, ef ficiency is considered in the sense of a specified targeted effect on the microstructure of the alloy. However, the problem is investigated primarily on the basis of a material science, rather than a technological approach.
In a number of works, for example, [8,9], it was noted that regression analysis or modified Griffiths and Hall Petch equations can be used to study the formation of a metal matrix. Here it should be noted once again that if the strength of cast iron depends mainly on the amount, shape, size and distribution of graphite, then the hardness is determined mainly by the metal matrix.
The influence of the alloying parameters, together with the regulation of the C/Si ratio in cast iron, on the micro structure and the mechanical properties of cast iron, is described in [10]. In this work, it is established that the output characteristics of the cast iron modifier in combi nation with antimony (Sb) influence the data. However, the results described in this paper refer to highstrength cast iron and the possibility of spreading the findings in it to cast iron with plate graphite requires additional stu dies. The problem of globular graphite formation during the modification of cast iron by magnesium is given at tention in [11], and the choice of the modifier type and the development of modifying technology as technological factors for controlling mechanical properties are discussed in [12][13][14]. Among the most highly developed modifiers, for example, the Superseed ® Extra Inoculant [12], Re seed ® Inoculant [13] and SMZ ® Inoculant [14] modifiers can be noted. The Superseed ® Extra Inoculant modifier minimizes bleaching in cast iron castings, promotes the formation of evenly distributed graphite, neutralizes the harmful effects of nitrogen and promotes the formation of small graphite inclusions, reducing the graphite chipping during machining. This effect of the modifier is explained by the authors in the presence of zirconium and stron tium in its composition, which improve the nucleation with a minimum degree of supercooling and reduce the risk of formation of supercooled graphite and ferrite. The Reseed® Inoculant modifier is designed for highstrength TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 3/1(41), 2018 ISSN 2226-3780 and gray cast iron with low sulfur content and contri butes, in particular, to the formation of globular graphite with a good degree of globularity in the thick sections of castings from highstrength cast iron. Also, this modifier helps to prevent the formation of microshrinkage porosity in the castings. This effect is ensured by the presence in the modifier of a balanced number of active elementscalcium and cerium. Obviously, due to these effects, one should expect an increase in the cast iron hardness. The SMZ ® Inoculant modifier can be used for graphitizing modification of gray cast iron and vermicular graphite cast iron and is suitable for late modification in a metal stream (MSI process). This ensures stabilization of devia tions in chemical composition and regulation of nitrogen content in cast iron. Such effect, as the authors of [12] note, can be explained by a carefully balanced amount of calcium and aluminum, which ensures maximum control over bleaching. Despite the qualitative assessment of the expected effects of the modification, the lack of quantita tive estimates, which can only be made on the basis of an analytical description, does not allow making sound technological decisions. In particular, we can talk about the selection of the chemical composition that provides the specified properties, and allows the possibility of optimi zing the charge by the criterion of minimum costs. Certain exceptions in considering the problem from this point of view have the works [15,16], in which the accent is made precisely on the methods of quantitative evaluation. The authors of these works have investigated the use of methods for constructing «compositionproperties» models under conditions of uncertainty, with hardness chosen as the output variable.
The described work allows to conclude that there are no ready solutions for the reasonable choice of the com position of cast iron from the point of view of ensuring a given hardness. As for the questions of the effect of the chemical composition of cast iron in a specific range of variation of the input C-C EQ on the hardness of struc tural iron, the corresponding work has not been found. Therefore, to solve the emerging practical issues on the choice of the chemical composition of cast iron, which provides the given values of its hardness and which al lows further minimizing the cost of its production, special studies are necessary.

Methods of research
According to the results of industrial tests, described in detail in [4], a sample of input and output variables is formed. This sample includes the results of 200 serial meltings, of which the data are selected that fell within the range C = (3.42-3.57) % and C eq = (4.2-4.4) %. The data are preliminarily filtered and normalized by a standard procedure [4], the input variable «carbon content (C)» is denoted by x 1 , the input variable «carbon equivalent (C eq )» is denoted by x 2 . Fig. 1 shows the hardness values of 12 samples selected for further study.
where а i -the estimated coefficients. , the least squares (OLS) method is used: minimizing the leastsquares functional of the form: where F -the matrix of the experimental design, which has the form: .
. -matrix of experimental hardness values.

Research results
Using the OLS, the values of the coefficients of the regression equations of the form (1) are calculated from (2): Electronic copy available at: https://ssrn.com/abstract=3691024 ISSN 2226-3780 Considering the fact that the coefficients of the regression equations are estimated on the basis of a passive experiment that does not allow parallel measurements of the output vari able at each point of the plan, the possibility of testing the homogeneity of the experimental plan is not available. The adequacy of the model, based on Fisher's Ftest, or testing the hypothesis that the variance of experimental errors is equal, and the model's inadequacy, is not therefore possible. Therefore, the potential performance of the model us evalu ated on the basis of checking the number of experimental points that fell within a given confidence interval (Fig. 2).
From Fig. 2 it follows that 11 test points (92 %) fell into the confidence interval. Therefore, there is reason to believe that the regression equation of the form (1) is operable for further analysis.
Since the most interesting is the identification of sta tionary points and the description of the response surface in their vicinity, the canonical transformation of the re sponse surface is performed, similarly to the procedure described in [4] As a result of the transfer and rotation of the axes and the transition from the coordinate system (x 1 ; x 2 ) to the coordinate system (ξ 1 ; ξ 2 ), the initial equation of the response surface is transformed to: where B -the rotation matrix, ′ = B B I, and the difference between the values of the output variable at an arbitrary and stationary point is described by the equation: The following values are obtained by the realization of the procedure 1-4: λ 1 = -15.1685, λ 2 = 21.89937. This means that the equation describing the response surface in the canoni cal form has the form: Since the ratio of the eigenvalues in mag nitude and sign determines the form of the response surface, and: the response surface, just as in the case of a tensile strength test [4], is a hyperbolic parabaloid. However, the position of the saddle point in it is not so pronounced (Fig. 3).  Fig. 3. The response surface, which describes the distribution of hardness values of structural iron in the factor space C-C eq (input variables are given in the normalized form) Fig. 4 is a top view of the response surface, from which it can be seen that with the increase in the carbon equiva lent, the hardness of the cast iron is reduced. The effect of the carbon content on hardness is more complex -it is described by a parabolic dependence.  Electronic copy available at: https://ssrn.com/abstract=3691024

ISSN 2226-3780
This means that the maximum hardness is reached at about the average level of carbon content (3.495 %). Before this value, the hardness increases with increasing carbon content, and then decreases.
In other words, from the point of view of using car bon as a factor in reducing the hardness of cast iron, its allowable range is limited to an interval (3.42-3.495) %. A more significant factor from this point of view is the carbon equivalent, which needs to be increased. This means that in order to reduce the hardness value it is necessary to increase the width of the C-C eq interval in the Fe-C state diagram. To find suboptimal points, it is advisable to use the ridge analysis of the received response surface [17]. To do this, it is necessary to obtain a parametric descrip tion of the type: . . There may be several suboptimal solutions. They are defined as the inter section points of the ridges and the constraints r = 1.414. In this case, the tasks of minimizing hardness are not set and the range HB = 180-250 satisfies the quality requirements specified by the production conditions. If the priority is to minimize hardness, then the suboptimal solution is found as the point of intersection of the restriction r = 1.414 and the lower branch of the ridge line II-III. Consequently, the resulting results in the form of Fig. 7 allow to draw a number of important conclusions from the practical point of view concerning the satisfaction of different requirements for hardness. This indicates the multivariance of the obtained solutions, the choice of the most preferable of which is de termined by the requirements of the production conditions.  Obviously, there are many sub optimal solutions given by the first equation of system (8). For the case when the requirements HB = 180-250 are sufficient, such solutions are shown in Fig. 8.
From a practical point of view, the transformation of the solutions obtained in a normalized form to a natural form is of special interest -obtained descrip tion is a nomogram. The nomogram, as is known, is a convenient tool in the hands of a technologist [18][19][20], and allows choosing rational modes of the technological process. Fig. 9 shows such nomogram for the investigated range of values of input variables.
As follows from the above descrip tion, in order to select the necessary correction value for the carbon or car bon equivalent, it is possible to estimate the distance between the point of the factor space corresponding to the actual values of C and C eq and the nearest of the two curves. Obviously, the best choice will be the one providing the minimum consumption of corrective additives, that is, one for which the distance from the current point (cor responding to the actual values of C and C eq ) to the cor responding curve will be minimal.

SWOT analysis of research results
Strengths. Among the strengths of this research, it is necessary to note the possibility of using the resulting regression equation to solve two key problems: -predicting the hardness by the actual chemical com position, obtained during the melting process; -selection of the composition providing a given level of hardness. In the first case, it becomes possible to reduce the number of laboratory hardness tests by re ducing the corresponding costs. In the second case, the prospect of minimizing the cost of the burden opens, that is, reducing the cost of 1 ton of good casting. It should also be noted that the possibility of an indirect evaluation of the appearance of undesirable carbides in the microstructure, in particular cementite, is possible. This can contribute to the selection of more ratio nal solutions with regard to the technological regimes of secondary treatment. Finally, a targeted choice of chemical composition, which provides minimum, but acceptable from the standpoint of strength, hardness of structural iron should contribute to improving the reliability of the cutting tool.
Weaknesses. The weaknesses of this research are re lated to the fact that the regression equation is built on the basis of an arbitrary area of experiment planning. This means that the obtained estimates of the coeffi cients are far from optimal and there is a principal pos sibility of increasing the accuracy. This is possible, for example, by optimizing the experimental design. However, this desire to improve quality will require the need for additional fusions, which is associated with significant additional costs.
Opportunities. Additional opportunities for using the above results in industrial conditions are related to the optimization of the chemical composition of cast iron or the optimization of the charge composition. In the latter case, the initial data can be obtained suboptimal solutions. Ad ditional opportunities are also opened during the offfurnace treatment -thanks to the use of obtained nomograms, the rational selection of corrective additives is simplified. In this case, there is a principal possibility to minimize costs precisely at the expense of the most acceptable option.
Threats. The obvious risks when using the results are due to the fact that consumers prefer to purchase cas tings from highstrength cast iron with nodular graphite or highquality gray cast iron with vermicular graphite. This is completely justified, since the mechanical or spe cial properties of such cast irons are much higher. From the point of view of the manufacturer of cast iron for castings, if the operating conditions of cast iron parts are nonrigid, typical, there is no need to spend extra money in pursuit of increasing mechanical properties. And if the manufacturer's costs are one of the criteria for mini mization, then they are not interested in the consumer. From the point of view of using the obtained solutions in production, there is a management risk -changing the composition of the charge requires a revision of the consumption rates, and possibly suppliers of charge ma terials. This, in turn, requires the presence of progressive management and especially the corresponding level of top managers of production.

Conclusions
1. It is shown that a polynomial regression equation can be used to obtain a workable analytical description of the effect of carbon (C) and the carbon equivalent (C eq ) Even with a small sample of data, this accuracy is 92 %. 2. On the basis of the canonical transformation of the received response surface, the presence of a saddle point is revealed, which, however, is not as pronounced as for a response surface that describes the magnitude of the tensile strength in the same range of input variables. The ridge analysis of the described response surface shows that there is a principal possibility of satisfying different requirements for hardness. So, if the range of the planning area C = (3.42-3.57) % and C eq = (4.2-4.4) % is chosen as the imposed constraint, then several suboptimal solutions are possible. This is the case if the task of minimizing hardness is not set and the range HB = 180-250 satisfies the quality requirements specified by the production con ditions. If the priority is hardness minimization, then the suboptimal solution is one. Thus, there are many suboptimal decisions, depending on the requirements of production. It is shown that such solutions, in fact, are a nomogram that allows to select in a rational way the technological regimes of outoffurnace processing in the part concerning the correction of the chemical composition of the alloy.