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Introduction

At the present time in the technical literature, there are extensive experimental data on the flow stress of the metal from the G strain e, strain rate U and temperature T, presented in the form of hardening curves. In several cases, including the development of computer programs is necessary to develop empirical formulas, which are necessary for the calculation of the flow stress of metals G.

1. Relevance of the topic

An urgent task is to obtain empirical formulas for calculating the flow stress of metal for construction, tool and stainless steels based on the available experimental plastometric information.

2. The purpose and objectives of the study

The aim is to develop a method for determining the constants of empirical formulas for calculating the flow stress of metal for construction, tool and stainless steels based on the available experimental data plastometric.

After selecting the type of empirical formula is required to determine the constants occurring in it, based on available experimental information on the work-hardening curves. In this case there are two urgent tasks:

  1. To ensure high accuracy of definition of G, depending on e, U, T on the basis of hardening curves;
  2. Implement evidence-based selection of the most rational points in the range of the factors e, U, T to determine the appropriate value G.

3. Development of a method of determining the constants of empirical formulas for calculating the flow stress of the metal

3.1 Ensuring high accuracy of definition of G, depending on e, U, T on the basis of work-hardening curves

To solve the first problem it is advisable to develop a computer program for determining the values of G by spline - interpolation of experimental data [2], [3].

Since the experimental information can be presented in different forms, has developed several windows presentation of experimental data (Fig. 1, 3).

The program window for a set of graphs at different strain rates and the fixed temperature

Figure 1 - The program window for a set of graphs at different strain rates and the fixed temperature.

The program window for a set of graphs with ke, ku, kT

Figure 2 - The program window for a set of graphs with ke, ku, kT.

The program window for a set of graphs at different temperatures and a fixed value of strain rate.

Figure 3 - The program window for a set of graphs at different temperatures and a fixed value of strain rate.

Determination of the quantities G, depending on the arbitrary values of e, U, T invited to perform as follows. In the first stage of the window a computer program recorded the scanned curves of hardening. The original data set (Fig. 1, 3).

Once given all the necessary background information necessary to determine the coordinates of nodal points on the axes in units of raster images.

In the program window (Figure 4) for all nodal points of the coordinate axes are mapped to values of G and e in units as specified on the coordinate axes, as well as in units of raster images, which are defined by software. Running a graphical visualization of the constructed lines, which is necessary to ensure the most accurate matching network built in a different color from the original grid. In the program window (Figure 4) for all nodal points of the coordinate axes are mapped to values of G and e in units as specified on the coordinate axes, as well as in units of raster images, which are defined by software. Running a graphical visualization of the constructed lines, which is necessary to provide the most accurate matching network built in a different color from the original grid.

The window construction of the grid

Figure 4 - The window construction of the grid

Based on this information for any point on the graph, we can determine the abscissa and ordinate in raster units, and then calculate them in the units specified on the coordinate axes. The program calculates the values of the flow stress of the metal G (e, U, T) and writes them to the table.

Performed stepwise change in the value of factors e, U, T, values obtained flow stress of the metal stored in a table. You must completely fill in the table of experimental values.

Handled all the curves of hardening in the whole range of factors, e, U, T.

Next, spline interpolation is performed to the information received and the construction of spline - the curves in the window (Figure 5). If the initial course of the hardening curve is rather complicated, for example, there are excesses and spline - curve is not accurately placed on the original curve, it is possible to increase the number of points and achieve a complete coincidence of the interpolation curve and the original one.

Window removal of experimental information and control the construction of spline - curves.

Figure 5 - Window removal of experimental information and control the construction of spline - curves.

In Table. 1 shows an experimental digital data from the curves of hardening in the whole range of factors, e, U, T.

Table 1 - Experimental digital information from the curves of hardening

e 5 10 15 20 25 30 35 40 45 50
T=900, U=0,5 7.933 8.828 9.586 10.207 10.793 11.207 11.517 11.655 11.724 11.690
T=900,U=5 10.400 11.511 12.489 13.333 14.087 14.783 15.391 15.826 16.261 16.609
T=900,U=50 13.333 15.373 17.098 18.588 19.843 21.067 21.981 22.743 23.352 23.810
T=1000,U=0,5 5.546 6.319 6.891 7.395 7.798 8.103 8.345 8.483 8.586 8.552
T=1000,U=5 7.565 8.609 9.435 10.178 10.800 11.378 11.867 12.267 12.489 12.667
T=1000,U=50 10.231 12.000 13.490 14.824 16.000 17.020 17.882 18.431 18.980 19.216
T=1100,U=0,5 3.322 3.966 4.471 4.908 5.210 5.445 5.613 5.748 5.782 5.782
T=1100,U=5 5.455 6.304 7.043 7.609 8.130 8.565 8.913 9.217 9.391 9.478
T=1100,U=50 8.154 9.615 10.769 11.769 12.549 13.255 13.725 14.039 14.275 14.353
T=1200,U=0,5 2.034 2.542 2.983 3.322 3.593 3.831 3.966 4.034 4.000 3.932
T=1200,U=5 4.045 4.591 5.136 5.591 5.955 6.217 6.391 6.522 6.565 6.478
T=1200,U=50 6.000 7.154 8.077 8.846 9.308 9.692 10.000 10.154 10.154 9.923

Created window (Figure 6) allows to determine the values of the flow stress of the metal at fixed values of e, U, T. These values are calculated as follows.

In the first phase, the spline interpolation of the initial information on the basis of third-degree polynomials.

In the next stage when e = e * is calculated first array G for given values of the information in the original factors of U and T. The calculation results are shown in the table at the top of the window.

After performing a spline interpolation of the data given by the value of an additional U = U * is calculated, and the second array G as defined in the original data values of the factor T. The calculation results are displayed in another table, below.

In the final step is the interpolation of the data and calculates the required value of G at t = T *.

Figure - 6 program window spline interpolation curves of hardening

Figure - 6 program window spline interpolation curves of hardening

3.2 Scientific and informed choice of the most rational points in the range of the factors e, U, T to determine the appropriate values of G

To solve the second problem, it is proposed to apply the method of calculation of the planned experiment [5]. Created window (see Figure 7) where a table located in the upper part, passed the limits of variation factors e, U, T. In the same window formed a table of code values and natural factors. In accordance with the theory of planned experiment, the plan is the matrix for the 3 factors e, U, T always contains 15 lines to determine the values of G. The planned experiment covers the entire range of the factors e, U, T, and determines the most rational point for determining the values of G on the basis of experimental data. And it is a science-based theory of planned experiment a minimum of experiments.

To become 15СХНД shows the values of the flow stress of the metal seksp obtained by spline interpolation curves of hardening. Under the proposed method are found in the constant in the formula of Professor. V.I. Zyuzin [1], and on this basis calculated the values of Gp. The values of the constants are presented in the right pane, calculated by the method of least squares. Also found an average relative deviation of calculated values of gp, from the corresponding experimental values Geksp equal to 2.5%.

Windows program for calculation of the constants in the formula V.I. Zyuzin

Figure 7 - Windows program for calculation of the constants in the formula V.I. Zyuzin

4. The results of the Conclusions

By using the proposed method of determining the constants of empirical formulas for calculating the flow stress of the metal and the developed computer program was carried out calculations of the constants in the formula V.I. Zyuzin [1] for 36 grades of steel. The average relative error of approximation of experimental data for all grades according to the formula of Professor. Zyuzin [1] was 4.7%. The constants are presented in Table 2.

Table 2 - Constants in the formula of Professor. V.I. Zyuzin

Steel A , МПа n1 n2 n3 Error, %
У8,[6],стр.156, рис.107 1821 0,233 0,196 0,00294 2,2
У12А,[6],стр.159, рис.111 1447,9 0,24025 0,15444 0,0024765 8,3
У12А,[7],стр.83, рис.33 5951,3 0,18979 -0,15356 0,003304 10
X17H2,[6],стр.200, рис.164 6453,8 0,25152 0,06584 0,003656 3,2
Х12,[6],стр.185, рис.139 2882,3 0,22104 0,0765 0,0025331 3,1
ХВГ,[6],стр.137, рис.79 3472,5 0,25561 0,13761 0,0029445 4,2
ХВГ,[7],стр.85, рис.35 4279,3 0,28837 0,13308 0,0030085 5,1
Р18,[6],стр.168, рис128 4834,5 0,1629 0,0675 0,0030983 5,1
Р18,[6],стр.169, рис.130 3118,4 0,20879 0,12924 0,0028369 2,1
Cт3,[6],стр.101, рис.22 1846,1 0,23057 0,1521 0,0028402 2,3
Сталь 45,[6],стр.105, рис.28 1935,6 0,27336 0,17505 0,0028004 16,4
Сталь 45,[6],стр.105, рис.29 1733,1 0,23969 0,14375 0,0027614 3,1
Сталь 55,[6],стр.108, рис.37 2250,6 0,23481 0,15406 0,0029966 2
12ХН3А,[6],стр.146, рис.97 1955,2 0,24089 0,13244 0,0027751 2,6
14ГН, [6],стр.119, рис.49 2055,7 0,24508 0,15734 0,0028744 2,5
15СХНД,[6],стр.133, рис.71 1871,5 0,25049 0,16055 0,002806 2,5
18ХНВА,[6],стр.137, рис.80 3126,2 0,29523 0,10937 0,0027974 3,9
18ХНВА,[7],стр.87, рис.37 12113,6 0,25072 -0,11248 0,003671 11,5
40X,[6],стр.122, рис.52 2183,9 0,24376 0,14499 0,0029576 3,5
60C2,[6],стр.161, рис.114 2174,9 0,20983 0,15854 0,0028432 2,7
60С2,[6],стр.161, рис.113 3546,3 0,21555 0,08984 0,0032892 3,8
60С2[7],стр.84, рис.34 4148,2 0,247 0,07593 0,0032819 3,5
ШХ15,[6],стр.163, рис.118 1855,2 0,21926 0,15687 0,0028206 2,7
2Х18Н9,[7],стр.89, рис.39 2365,2 0,2643 0,11194 0,0022885 4,1
4Х13,[7],стр.86, рис. 36 2146,6 0,25424 0,07646 0,001976 4,3
10Х17Н13М2Т,[6],стр.219,рис.192 7018,7 0,27233 0,03964 0,0030591 6
10Х17Н13М2Т,[6],стр.221,рис.195 2685 0,23885 0,14783 0,0027323 1,9
12X13,[6],стр.186, рис.141 11889,9 0,29699 0,08867 0,0041241 6,6
12X13,[6],стр.187, рис.142 3491,1 0,25718 0,16121 0,0031423 4,6
12Х18Н9Т,[6],стр.207, рис.177 2394,2 0,25237 0,07633 0,0025765 2,8
12Х18Н9Т,[6],стр.211, рис.181 4234.0 0,25968 0,07041 0,0026974 4,7
20Х23H18,[6],стр.223, рис.199 9230,2 0,26303 0,09778 0,0036406 8
40Х13,[6],стр.190, рис.149 5602,1 0,24724 0,06111 0,0035183 3,2
40Х13,[6],стр.191, рис.150 3394,2 0,20741 0,10326 0,0028074 4,5
Х18Н9Т,[7],стр.88, рис.38 4017,1 0,21782 0,1013 0,0029308 5,2
Х18Н25С2,[6],стр.225, рис.202 6969,8 0,17122 0,05129 0,0032477 6,2

The calculation of the constants included in the second degree polynomial [4]. The average relative error of approximation of experimental data for all grades of 5.2% formula. The constants are presented in Table 3.

Table 3 - The constants included in the second degree polynomial

Stell a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 Error, %
У8,[6],стр.156,рис.107 971,79 -117,73 -0,019373 0,000681 380,1 3,523 -1,6063 6,732 -0,23269 -0,0016206 -0,0047786 4,7
У12А,[6],стр.159,рис.111 1013,38 -179,65 -0,002575 0,000684 299,47 1,204 -1,6464 5,664 -0,10009 -0,000287 -0,0047221 6,4
У12А,[7],стр.83,рис.33 361,11 -1934,6 0,005728 -0,000002 867,6 -1,794 -0,281 -0,152 -0,03788 0,0005897 -0,0006526 15,2
X17H2,[6],стр.200,рис.164 911,22 -530,42 -0,012062 0,000517 781,26 2,203 -1,3648 3,733 -0,4002 -0,0011977 -0,0025991 3,2
Х12,[6],стр.185,рис.139 2361,99 -68,81 -0,004431 0,001669 667,65 0,363 -3,9161 1,992 -0,41832 0,0005091 -0,0014959 5,4
ХВГ,[6],стр.137,рис.79 2957,95 -200,75 -0,006177 0,002298 1053,76 1,68 -5,1827 0,049 -0,73653 -0,0003474 0,0005761 9,8
ХВГ,[7],стр.85,рис.35 3227,27 -241,81 -0,005607 0,002511 1264,07 1,39 -5,6576 1,562 -0,90406 -0,0002568 -0,0001453 8,8
Р18,[6],стр.168,рис128 1536,33 -390,54 -0,062706 0,000919 431,37 13,648 -2,3422 18,893 -0,1389 -0,008926 -0,0095226 0,8
Р18,[6],стр.169,рис.130 1206,42 -195,12 -0,029542 0,000717 695,37 3,677 -1,8295 3,776 -0,42954 -0,0010825 -0,0025322 4,4
Cт3,[6],стр.101,рис.22 806,16 -135,6 -0,016365 0,000508 397,36 2,382 -1,2601 4,098 -0,23401 -0,0008692 -0,0026443 4,8
Сталь 45,[6],стр.105,рис.28 783,58 -437,71 -0,001826 0,000479 757,46 1,718 -1,2327 -0,43 -0,39957 -0,0007308 0,0010493 8,8
Сталь 45,[6],стр.105,рис.29 758,1 -246,46 -0,012169 0,000474 482,88 1,559 -1,1833 4,413 -0,26031 -0,0002575 -0,003208 4,4
Сталь 55,[6],стр.108,рис.37 920,66 -132,88 -0,017421 0,000594 396,35 2,677 -1,4614 5,093 -0,22086 -0,0009506 -0,0039443 4,5
12ХН3А,[6],стр.146,рис.97 623,28 -368,64 -0,014707 0,000359 549,81 3,176 -0,9344 3,492 -0,2631 -0,001618 -0,0024437 4,3
14ГН, [6],стр.119,рис.49 844,26 -117 -0,016021 0,000527 415,08 2,279 -1,3167 5,675 -0,24186 -0,0006316 -0,0043089 4,4
15СХНД,[6],стр.133,рис.71 773,95 -129,56 -0,015768 0,000473 403,76 1,869 -1,1936 6,883 -0,22978 -0,0003161 -0,0052415 4,9
18ХНВА,[6],стр.137,рис.80 2026,1 -427,65 -0,004213 0,001479 615,59 1,419 -3,4401 2,681 -0,23134 -0,0004947 -0,0017936 8,7
18ХНВА,[7],стр.87,рис.37 1063,21 -1728,48 0,003461 0,000568 1230,83 -1,526 -1,5319 3,135 -0,39259 0,0005711 -0,003358 8,9
40X,[6],стр.122,рис.52 647,31 -130,87 -0,018208 0,00036 316,75 3,606 -0,951 4,513 -0,15923 -0,0020224 -0,002841 5,1
60C2,[6],стр.161,рис.114 869,3 -125,3 -0,018206 0,000502 395,81 2,751 -1,301 6,733 -0,22807 -0,0008681 -0,004979 4,1
60С2,[6],стр.161,рис.113 1766,6 -153,81 -0,002393 0,001276 371,41 0,979 -2,977 1,123 -0,19894 -0,0003961 -0,0006674 4,8
60С2[7],стр.84,рис.34 1572,59 -298,19 -0,001933 0,001123 575,59 1,033 -2,6333 1,637 -0,30368 -0,0005012 -0,0012188 3,7
ШХ15,[6],стр.163,рис.118 792,46 -162,11 -0,012802 0,000487 316,46 2,319 -1,2236 9,614 -0,14116 -0,0008038 -0,007971 4,2
2Х18Н9,[7],стр.89,рис.39 2521,91 -369,18 -0,005893 0,001792 591,16 1,294 -4,2186 0,179 -0,15333 0,0000523 0,0001038 8,7
4Х13,[7],стр.86, рис. 36 2755,96 -749,54 -0,003478 0,002081 1061,77 0,094 -4,7245 6,823 -0,48056 0,0006962 -0,0054642 4,1
10Х17Н13М2Т,[6],стр.219,рис.192 735,59 -506,13 -0,002576 0,00015 1236,58 1,735 -0,7566 -1,965 -0,69106 -0,0010672 0,0022645 3,3
10Х17Н13М2Т,[6],стр.221,рис.195 1257,16 -184,28 -0,02875 0,000791 605,52 3,023 -1,9635 8,573 -0,35592 -0,0005825 -0,0061543 4,4
12X13,[6],стр.186,рис.141 1727,97 -515,86 -0,066632 0,001244 627,32 24,993 -2,9165 21,584 -0,27793 -0,020722 -0,0080364 2
12X13,[6],стр.187,рис.142 1138,39 -163,12 -0,025339 0,00067 483,53 1,799 -1,7426 9,773 -0,2348 0,0006653 -0,0086492 4,3
12Х18Н9Т,[6],стр.207,рис.177 1003,69 -615,22 -0,012176 0,000658 884,41 3,44 -1,6001 -3,65 -0,42473 -0,0020676 0,0036643 3,3
12Х18Н9Т,[6],стр.211,рис.181 1889,19 -532,68 -0,003032 0,00121 921,48 1,179 -2,9902 1,919 -0,40154 -0,0003034 -0,0013529 4,6
20Х23H18,[6],стр.223,рис.199 2986,63 -426,26 -0,156398 0,002202 725,38 -7,246 -5,1001 60,347 -0,38106 0,0102688 -0,0389154 2,3
40Х13,[6],стр.190,рис.149 857,42 -544,18 -0,009708 0,000482 812,35 2,432 -1,2781 0,584 -0,41586 -0,0014521 -0,0001192 3,7
40Х13,[6],стр.191,рис.150 2052,57 -393,28 -0,003381 0,001418 575,52 2,26 -3,3689 0,74 -0,25155 -0,0012538 0,0000705 5,7
Х18Н9Т,[7],стр.88,рис.38 1919,82 -408,98 -0,003145 0,001285 616,15 2,574 -3,1008 -0,57 -0,28577 -0,0015789 0,0014448 5,2
Х18Н25С2,[6],стр.225,рис.202 1810,36 -639,99 0,100446 0,001078 1257,14 2,269 -2,7517 -70,439 -0,80171 0,0000092 0,0761921 0,9

The analysis of the accuracy of "self-contained" methods of prof. Nikolayev [8] and the method of prof. Andreyuk [9].

The planned experiment involving 15 settlements quantities G, covers the entire range of the factors e, U, T, and determines the most rational point for the comparison of experimental and calculated values of G [5].

Calculations carried out for 27 grades of steel. Found that the average relative error of the method of Nikolaev VA. [8] was 14.5% (the maximum relative error (for steel R18, see Chart 4) is equal to 32.3%). The average relative error of the method Andreyuk L. et al [9] was 21.2% (the maximum relative error (for steel R18, see Chart 4) is equal to 67%). In carrying out the calculations for the considered steels was determined by a number of constants appearing in the formulas of methods, which are presented in the table 4.

Table 4 - Constants included in the method of Nikolaev V.A. and the method of L.V. Andreyuk

Steel [6] The limits of change factors Method of Nikolaev V.A Method of L.V. Andreyuk
e U, с-1 G0, МПа Error, % N A B C Error, %
Cт3, стр.101, рис.22 0,05-0,5 0,5-50 88,353 6,56 74,777 0,134 0,186 -2,957 4,97
Сталь 45, стр.105, рис.28 0,05-0,5 0,05-150 91,313 18,2 75,195 0,148 0,186 -3,369 18,39
Сталь 45, стр.105, рис.29 0,05-0,4 0,5-50 88,353 5,35 74,691 0,144 0,193 -3,003 7,83
Сталь 55, стр.108, рис.37 0,05-0,5 0,5-50 90,46 7,56 75,783 0,143 0,199 -2,977 4,34
12ХН3А, стр.146, рис.97 0,05-0,4 0,5-50 104,924 12,19 100,273 0,116 0,185 -2,806 24,95
14ГН, стр.119, рис.49 0,05-0,5 0,5-50 98,928 6,23 90,933 0,124 0,19 -3,065 13,19
15СХНД, стр.133, рис.71 0,05-0,5 0,5-50 98,274 7,54 86,713 0,117 0,185 -2,943 10,04
18ХНВА, стр.137, рис.80 0,05-0,45 0,05-150 111,419 14,52 100,72 0,119 0,206 -2,954 8,9
40X, стр.122, рис.52 0,05-0,5 0,5-50 97,992 9,86 88,577 0,136 0,208 -3,125 21,05
60C2, стр.161, рис.114 0,05-0,5 0,5-50 101,825 12 76,032 0,149 0,207 -3,166 12,54
60С2, стр.161, рис.113 0,05-0,5 0,05-150 100,711 9,57 72,959 0,154 0,203 -3,211 12,45
ШХ15, стр.163, рис.118 0,05-0,5 0,5-50 100,05 7,78 94,082 0,152 0,202 -3,173 26,87
У8, стр.156, рис.107 0,05-0,5 0,5-50 91,769 10,05 77,8 0,15 0,198 -2,992 12,47
У12А, стр.159, рис.111 0,05-0,4 0,05-150 91,542 11,49 80,509 0,158 0,173 -2,987 10,94
X17H2, стр.200, рис.164 0,05-0,4 0,5-50 112,357 12,52 123,742 0,116 0,118 -3,597 38,14
Х12, стр.185, рис.139 0,05-0,4 0,05-150 111,227 30,03 140,38 0,148 0,144 -3,711 21,88
ХВГ, стр.137, рис.79 0,05-0,5 0,05-150 104,619 28,97 82,604 0,157 0,222 -3,432 22,17
Р18, стр.168, рис128 0,05-0,5 0,05-7,5 114,492 32,32 195,135 0,151 0,117 -3,985 29,89
Р18, стр.169, рис.130 0,05-0,5 0,5-50 115,059 23,7 210,405 0,122 0,076 -2,409 66,96
10Х17Н13М2Т, стр.219,рис.192 0,05-0,5 0,05-150 158,669 13,91 179,823 0,103 0,107 -3,14 17,7
10Х17Н13М2Т, стр.221,рис.195 0,05-0,5 0,5-50 139,018 7,96 168,776 0,097 0,09 -2,716 28,5
12X13, стр.186, рис.141 0,05-0,4 0,05-7,5 111,212 17,79 126,52 0,116 0,161 -3,681 22,32
12X13, стр.187, рис.142 0,05-0,5 0,5-50 111,389 10,54 126,11 0,11 0,162 -3,657 15,74
12Х18Н9Т, стр.207, рис.177 0,05-0,4 0,5-50 123,604 3,69 179,336 0,078 0,142 -3,226 55,16
12Х18Н9Т, стр.211, рис.181 0,05-0,5 0,05-150 127,422 30,01 185,08 0,066 0,121 -3,344 11,56
40Х13, стр.190, рис.149 0,05-0,4 0,5-50 111,214 11,54 124,682 0,127 0,178 -3,713 44,75
40Х13, стр.191, рис.150 0,05-0,4 0,05-150 111,253 29,53 126,592 0,127 0,18 -3,72 9,39

Conclusions

The solution of the problems indicated above provides a method for determining the constants of empirical formulas for calculating the flow stress of metals G.

Developing a new method based on the planned experiment and computer program allowed by the experimental information on plastometric hardening curves to determine the constants appearing in the formula for calculating the flow stress of the metal, depending on e, U, T. Received more than 120 new formulas for calculating the flow stress of metal construction, tool and stainless steels.

List of sources

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  2. Яковченко А.В. Определение напряжения течения металла с учетом истории процесса нагружения на основе уравнения А.Надаи/ А.В.Яковчеко, Н.И.Ивлева, А.А.Пугач// Наукові праці ДонНТУ. Металургія, 2011.-Вип.12(177). - С.181 - 193.
  3. Яковченко А.В. Анализ точности известных методов расчета напряжения течения металла в зависимости от химического состава стали / А.В. Яковченко, А.А. Пугач, Н.И. Ивлева // Вісник Приазовського державного технічного університету. Сер.: Технічні науки: Зб. наук. праць. – Маріуполь: ДВНЗ «Приазов. держ. техн. ун-т», 2011. - Вип.2(23). - С. 69 - 80.
  4. Данилов А.В. Анализ и усовершенствование методов расчета напряжения течения металла в процессах горячей пластической деформации. Металлургия и обработка металлов (выпуск 12) / Материалы научно-исследовательских работ студентов и молодых ученых физико-металлургического факультета ДонНТУ. – Донецк: ДонНТУ, 2009. – С. 42,43.
  5. Винарский, М.С. Планирование эксперимента в технологических исследованиях : учеб. пособие / М.С. Винарский, М.В Лурье. – К.: Техника, 1975. – 168 с.
  6. Полухин П.И. Сопротивление пластической деформации металлов и сплавов: Справочник / П.И. Полухин, Г.Я. Гун, А.М. Галкин. – М.: Металлургия, 1983. - 352с.
  7. Примение теории ползучести при обработке металлов давлением. Поздеев А.А., Тарновский В.И., Еремеев В.И., Баакашвили В.С. Изд-во «Металлургия», 1973, 192с.
  8. Николаев В.А. Теория прокатки: Монография. - Запорожье: Издательство Запорожской государственной инженерной академии, 2007. - 228с.
  9. Андреюк Л.В. Аналитическая зависи¬мость сопротивления деформации сталей и сплавов от их химического состава / Л.В. Андреюк, Г.Г. Тюленев, Б.С. Прицкер // Сталь. – 1972. – № 6. – C. 522, 523.