FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 18, No 4, 2020, pp. 595 - 610 https://doi.org/10.22190/FUME200416041B © 2020 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND Original scientific paper MATHEMATICAL MODELING OF THE INFLUENCE PARAMETERS DURING FORMATION AND PROPAGATION OF THE LÜDERS BANDS Tin Brlić1, Stoja Rešković1, Zoran Jurković2, Gordan Janeš3 1University of Zagreb, Faculty of Metallurgy, Croatia 2University of Rijeka, Faculty of Engineering, Croatia 3University of Rijeka, Center for Advanced Computing and Modeling, Croatia Abstract. In this study, an analysis of the influence parameters measured by the static tensile test, thermography and digital image correlation was performed during formation and propagation of the Lüders bands. A new approach to the prediction of stresses, maximum temperature changes and strains during the Lüders band formation and propagation is proposed in this paper. Application of the obtained mathematical models of influence parameters gives a clear insight into the behavior of niobium microalloyed steel at the beginning of the plastic flow, which can improve product quality and reduce costs during the forming of microalloyed steels with the appearance of the Lüders bands. The obtained models of influential parameters during formation and propagation of the Lüders bands have been developed by the regression analysis method. The proposed mathematical models showed low deviations of calculated results ranging from 1.34% to 12.37%. The local stress amounts, important in the forming of microalloyed steels since indicating surface roughness and plastic flow possibilities during the Lüders band propagation, are obtained by the mathematical model. It was found that stress amounts increase during the Lüders band propagation in the area behind the Lüders band front. The difference in stress amount between the start of the Lüders band propagation and advanced Lüders band propagation is 25.53 MPa. Key Words: Lüders Bands, Regression Analysis, Microalloyed Steel, Thermography, Digital Image Correlation Received April 16, 2020 / Accepted September 02, 2020 Corresponding author: Tin Brlić University of Zagreb Faculty of Metallurgy, Aleja narodnih heroja 3, 44 000 Sisak, Croatia E-mail: tbrlic@simet.unizg.hr 596 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ 1. INTRODUCTION Mathematical modeling [1] and optimization of process parameters [2] are often used for improving the quality of products as well as reducing costs of the production process for various metallic materials. The appearance of the Lüders bands presents a problem for the product quality during the forming process [3]. Consequently, mathematical modeling was used in order to predict behavior of the Lüders bands during cold deformation of metallic materials [4]. It is well known that niobium microalloyed steels show inhomogeneous deformations, i.e. Lüders bands, at the start of plastic flow during cold deformation [5], which can be a significant problem during the deformation process of microalloyed steels. It is possible to improve product quality and reduce production process costs by developing mathematical models of influential parameters. Given that, it is not necessary to carry out additional experimental tests that require additional costs since the values of significant parameters obtained by the mathematical models can be predicted. The formation and propagation of the Lüders bands in metallic materials showed a difference in the values of influential parameters of stress, temperature changes [6] and strains behind the Lüders band front [7]. Since changes of influence parameters can be more or less pronounced, it is important to analyze them in detail and describe with mathematical models in order to give a clear insight of value changes of the parameters during the formation and propagation of the Lüders bands. Static tensile test, thermography and digital image correlation are confirmed as reliable methods for determining the stress, temperature changes and strain amounts [8] during the formation and propagation of the Lüders bands. It is well known that stress values can be measured from the force-elongation diagram by static tensile test. Values of temperature changes and their distribution in Lüders bands can be determined with high accuracy using thermography. Strain values and their distribution can be determined by digital image correlation during static tensile testing. Therefore, it can be concluded that using thermography and digital image correlation can determine temperature changes and strain values which can be compared with the stress values obtained by the static tensile test. Previous research has shown that mathematical modeling is often used to describe processes during deformation under various metal forming technologies [9]. Bhirud and Gawande [10] used the regression analysis method to determine the mathematical model for temperature increase of work piece during final milling. Results of the obtained mathematical model are compared with the measured results of temperature increase of the work piece and a confirmation test is conducted for the validation of the predicted results of temperature increase. The regression analysis method proved to be suitable for a better insight into the tensile behavior of AISI 316 stainless steel during which temperature change and strain rate have been associated by corresponding empirical equations [11]. Nasri et al. [12] have shown usability of the multiple regression method during sheet metal forming formability for comparison between experimental results and model predictions. Jurković et al. [13] conducted the tests of rolling steel sheets where the output parameter, forming force-roll load, are mathematically modeled by the regression analysis of various input parameters such as tensile strength, sheet thickness and sheet width where regression analysis proved to be a reliable and accurate method. The research [14] confirms the regression analysis method as a very good method for obtaining mathematical models in determining the influence of cutting speed, feed rate and depth of cut on cutting tool vibration in different directions on carbon steel. The obtained regression mathematical models for vibration of cutting tool with machining parameters have shown a very good match between the results predicted from the developed models and the experimental ones. Other studies [15] Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 597 confirmed regression analysis as a reliable method for obtaining appropriate mathematical models of selected output parameters. The significance of the process parameter influence of spindle rotation speed, feed speed and rolling-beating mode during cold rolling-beating forming process on ASTM 1045 steel for obtaining mathematical models of forming forces and forming qualities objectives by regression analysis is determined in [16]. Mathematical multiple linear regression analysis of mechanical properties of potential weld joints, such as yield strength, ultimate tensile strength and percentage elongation, on aluminum alloy AA 6061 during manufacturing conditions is conducted in [17]. Models have proven to be very accurate in prediction of mechanical properties which can greatly assist engineers in proper planning and simultaneously improve efficiency of aluminum alloy welding operations. The phenomenon of the Lüders bands tends to be mathematically described for different steels [18]. Studies [4, 19] used developed models by regression analysis for different influencing parameters in low carbon steel with Lüders bands. Geometrical properties such as length, width and depth of the Lüders bands were examined which appeared on the tinplates during the cold stamping process. The predicted model results of influencing parameters showed good agreement with the measured results obtained by experimental tests. The tendency of Sadowski et al. [20] to model the phenomenon of Lüders bands by regression analysis is manifested where they sought to model different yield plateau gradients for different structural carbon steels. Rešković et al. [21] used factorial experiment with repeated measurements to determine influence of different factors at the proportionality limit after which Lüders bands appear in niobium microalloyed steel during cold deformation. They concluded that the deformation degree has the greatest influence on the proportionality limit during cold drawing of niobium microalloyed steel tubes. In the local area of maximum temperature changes and maximum strains behind the Lüders front detailed influence of individual influencing parameters is not mathematically described. Therefore, the aim of this paper is to develop mathematical models for influential parameters ΔTmax (maximum temperature change), σ (stress) and ɛmax (maximum strain) to predict the behavior of niobium microalloyed steel in the area behind the Lüders front which shows the greatest temperature and strain changes during cold deformation. 2. EXPERIMENTAL WORK Measured experimental results of influencing parameters during the formation and propagation of the Lüders bands were obtained on microalloyed steel with 0.048% Nb. The chemical composition of niobium microalloyed steel is shown in Table 1. Influential parameter of the stress was obtained by static tensile testing at the stretching rates of 5 mm/min, 20 mm/min and 50 mm/min corresponding to strain rates of 0.0018 s-1, 0.007 s-1 and 0.0185 s-1. Static tensile tests were carried out on the tensile machine WPM EU 40 mod with a nominal force value 400kN. Determination of the local maximum temperature changes and maximum strain values was carried out by infrared camera JENOPTIK VarioCAM®M82910 and digital camera Blackfly S Color (FLIR). Values of influential parameters were measured using the aforementioned research methods during formation and propagation of the Lüders bands. The parameters of maximum localized temperature change (ΔTmax), maximum strains (ɛmax) and stress values (σ) were measured using thermography, digital image correlation and the static tensile test. 598 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ In addition to the influential parameters, different strain rates during static tensile testing were also considered. The measurements of parameters ΔTmax and ɛmax were performed in the area behind the Lüders band front during formation and propagation of the Lüders bands since the largest changes of the analyzed parameters were measured in this area. Other parameters during the experimental tests that may have influenced the accuracy of the mathematical model were kept constant during the static tensile test. The levels of the minimum and maximum values of the input parameters were determined during formation and propagation of the Lüders bands when determining σ as output parameter, Table 2, and ΔTmax, ɛmax as output parameters in Tables 3 and 4. 2.1 Regression analysis method This paper presents modeling of output parameters σ, ɛmax and ΔTmax using a mathematical statistical method of regression analysis. It is possible to define the relationship between the output effects of process (y) and input variables (xi) by regression analysis. Mathematical modeling of influential parameters during the formation and propagation of the Lüders bands by the regression analysis is performed by the second order polynomial model using Eq. (1): y=b0x0+b1x1+b2x2+b3x3+b11x1 2+b22x2 2+b33x3 2+ b12x1x2+b13x1x3+b23x2x3+b123x1x2x3 (1) where xi are the variable parameters of process i= 0, 1, 2,..., k and bi are the model coefficients. Table 1 Chemical composition of niobium microalloyed steel (wt%) Element C Mn Si P S Al Nb N Nb steel 0.12 0.78 0.18 0.011 0.018 0.020 0.048 0.0080 Table 2 Physical values of input parameters for determining output parameter σ during the formation and propagation of the Lüders bands at v = 0.007 s-1 Parameters Minimum value Mean value Maximum value Formation of the Lüders band ΔTmax (°C) -0.47 1.26 2.05 ɛmax (mm/mm) 0.0072 0.0204 0.0336 Propagation of the Lüders band ΔTmax (°C) 1.37 2.64 3.91 ɛmax (mm/mm) 0.0300 0.0396 0.0492 Table 3 Physical values of input parameters for determining output parameter ɛmax during the formation and propagation of the Lüders bands Parameters Minimum value Mean value Maximum value Formation of the Lüders band ΔTmax (°C) -0.59 0.98 2.55 σ (MPa) 484.70 524.60 564.50 v (s-1) 0.00180 0.01015 0.01850 Propagation of the Lüders band ΔTmax (°C) 0.350 2.285 4.220 σ (MPa) 498.30 529.95 561.60 v (s-1) 0.00180 0.01015 0.01850 Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 599 Table 4 Physical values of input parameters for determining output parameter ΔTmax during the Lüders bands propagation Parameters Minimum value Mean value Maximum value ɛmax (mm/mm) 0.02790 0.03855 0.04920 σ (MPa) 498.30 529.95 561.60 v (s-1) 0.00180 0.01015 0.01850 Multiregression coefficient, R, determined by Eq. (2) is very important for the quality of the mathematical models obtained by the regression analysis method: R=√1- ∑ (Yj E -Yj R ) 2 N j=1 ∑ (Yj E -Y̅j E ) 2 N j=1 (2) where 𝑌𝑗 𝐸 are the experimental values, 𝑌𝑗 𝑅 the calculated values and �̅�𝐸 = ∑ 𝑌𝑗 𝐸𝑁 𝑗=1 𝑁 the arithmetic mean of all experimental results. Determination coefficient, R2, is determined by calculated multiregression coefficient R. The general form of a second order polynomial model with interacting factors will take the following forms with respect to the parameters analyzed in this paper. Model for output parameter σ: ▪ for the formation and propagation of the Lüders band at v = 0.007 s-1: y (σ)=b0+b1∆Tmax+b2εmax+b11∆Tmax 2 +b22ɛmax 2 +b12∆Tmaxεmax (3) The general form of a mathematical model for output parameters ΔTmax and ɛmax denoted by z: ▪ for the formation and propagation of the Lüders band: y(z)=b0+b1z+b2σ+b3v+b11zmax 2 +b22σ 2+b33v 2+ b12zσ+b13zv+b23σv+b123zσv (4) 3. RESULTS AND DISCUSSION Mathematical modeling of influence parameters (ɛmax, ΔTmax, σ) using regression analysis was performed during the formation and propagation of the Lüders bands at the start of the plastic flow in niobium microalloyed steel during cold deformation. The experimental results of the maximum local temperature changes and strains in the area behind the Lüders band front during formation and propagation of the Lüders bands were performed in the area according to Fig. 1. Experimental results obtained by using static tensile testing, thermography and digital image correlation are shown in Table 5 for output parameter σ, and in Tables 6-8 for output parameters (ɛmax, ΔTmax). 600 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ a) b) Fig. 1 The area of measurements of: a) maximum temperature change and b) maximum strain in the area behind the Lüders front Table 5 Experimental results of input parameters (ΔTmax, ɛmax) and output parameter σ obtained by measured values during formation and propagation of the Lüders bands at v = 0.007 s-1 N u m b e r o f m e a su re m e n t Formation of the Lüders band Propagation of the Lüders band Input parameters Output parameter Input parameters Output parameter ΔTmax (°C) ɛmax (mm/mm) σ (MPa) ΔTmax (°C) ɛmax (mm/mm) σ (MPa) 1 -0.47 0.0076 543.3 1.37 0.0336 498.3 2 -0.35 0.0072 530.6 1.72 0.0374 498.3 3 0.17 0.0137 497.5 1.97 0.0378 500.6 4 0.17 0.0094 535.8 2.01 0.0316 531.1 5 0.42 0.0113 552.0 2.05 0.0300 522.6 6 0.67 0.0211 506.9 2.16 0.0412 512.3 7 1.07 0.0267 499.0 2.19 0.0388 510.0 8 1.23 0.0205 523.2 2.22 0.0398 510.8 9 1.37 0.0336 498.3 2.46 0.0386 540.7 10 1.67 0.0234 536.0 2.71 0.0373 521.3 11 2.01 0.0316 531.1 2.73 0.0416 545.9 12 2.05 0.0300 522.6 2.76 0.0434 549.3 13 2.80 0.0441 554.4 14 3.04 0.0427 528.5 15 3.30 0.0479 558.7 16 3.42 0.0439 524.5 17 3.65 0.0481 526.7 18 3.91 0.0492 521.2 Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 601 Table 6 Experimental results of input parameters (ΔTmax, σ, v) and output parameter ɛmax obtained by measured values during formation of the Lüders bands N u m b e r o f m e a su re m e n t Input parameters Output parameter N u m b e r o f m e a su re m e n t Input parameters Output parameter ΔTmax (°C) σ (MPa) v (s-1) ɛmax (mm/mm) ΔTmax (°C) σ (MPa) v (s-1) ɛmax (mm/mm) 1 -0.59 489.4 0.0018 0.0057 19 0.67 506.9 0.0070 0.0211 2 -0.55 484.7 0.0018 0.0073 20 0.74 555.9 0.0185 0.0199 3 -0.47 493.3 0.0018 0.0060 21 0.80 512.6 0.0018 0.0308 4 -0.47 543.3 0.0070 0.0076 22 0.81 539.8 0.0018 0.0350 5 -0.44 508.3 0.0018 0.0069 23 0.99 545.3 0.0185 0.0227 6 -0.35 530.6 0.0070 0.0072 24 1.07 499.0 0.0070 0.0267 7 -0.31 545.6 0.0185 0.0059 25 1.23 523.2 0.0070 0.0205 8 -0.29 515.9 0.0018 0.0127 26 1.37 498.3 0.0070 0.0336 9 -0.28 516.9 0.0185 0.0080 27 1.51 539.5 0.0185 0.0325 10 -0.24 509.4 0.0185 0.0086 28 1.66 549.1 0.0185 0.0324 11 0.02 510.9 0.0018 0.0172 29 1.67 536.0 0.0070 0.0234 12 0.17 497.5 0.0070 0.0137 30 1.69 564.5 0.0185 0.0324 13 0.17 535.8 0.0070 0.0094 31 2.01 531.1 0.0070 0.0316 14 0.34 536.7 0.0018 0.0198 32 2.05 522.6 0.0070 0.0300 15 0.39 509.1 0.0018 0.0279 33 2.06 535.0 0.0185 0.0325 16 0.42 552.0 0.0070 0.0113 34 2.16 536.4 0.0185 0.0380 17 0.5 512.1 0.0018 0.0215 35 2.53 528.1 0.0185 0.0400 18 0.62 536.1 0.0018 0.0280 36 2.55 540.3 0.0185 0.0412 Mathematical models for the influence parameters were obtained by using measured experimental values of ΔTmax, ɛmax and σ during formation and propagation of the Lüders bands. The experimental measured values were used to develop the equations of the second order polynomial model. The results of the data processing by regression analysis performed in Microsoft® Excel are shown in Tables 9-12. Regression analysis of influence parameters (ɛmax, ΔTmax, σ) based on measured values during the formation and propagation of the Lüders bands revealed high values of multiregression coefficient (R) and determination coefficient (R2) which confirms that the obtained mathematical models can be considered reliable. The results of the multiregression coefficients and determination coefficients obtained by regression analysis of the influencing parameters are shown in Table 9. High values of the multiregression coefficients and determination coefficients from the regression analysis were obtained. Mathematical models for the influential parameters during formation and propagation of the Lüders bands are shown by the following equations: ▪ mathematical model of the influential parameter σ during formation of the Lüders bands at v = 0.007 s-1: σ=622.9334+75.945∆Tmax-12078.3εmax+26.281∆Tmax 2 +281771ɛmax 2 -4063.16∆Tmaxεmax (5) ▪ mathematical model of influential parameter σ during propagation of the Lüders bands at v = 0.007 s-1: σ=640.894-90.392∆Tmax-1049.07εmax-86.8013∆Tmax 2 -399310ɛmax 2 +13425.7∆Tmaxεmax (6) 602 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ ▪ mathematical model of influential parameter ɛmax during formation of the Lüders bands: ɛmax=1.0398+0.178ΔTmax-0.004σ+4.866v-0.00078ΔTmax 2 +0.00000401σ2+144.558v2- 0.00031ΔTmaxσ-8.807ΔTmaxv-0.01577σv+0.0169ΔTmaxσv (7) Table 7 Experimental results of input parameters (ΔTmax, σ, v) and output parameter ɛmax obtained by measured values during propagation of the Lüders bands N u m b e r o f m e a su re m e n t Input parameters Output parameter N u m b e r o f m e a su re m e n t Input parameters Output parameter ΔTmax (°C) σ (MPa) v (s-1) ɛmax (mm/mm) ΔTmax(° C) σ (MPa) v (s-1) ɛmax (mm/mm) 1 0.35 508.6 0.0018 0.0309 28 2.27 510.8 0.0018 0.0405 2 0.39 509.1 0.0018 0.0279 29 2.46 540.7 0.0070 0.0386 3 0.61 509.8 0.0018 0.0325 30 2.55 535.6 0.0185 0.0392 4 0.75 511.4 0.0018 0.0331 31 2.71 521.3 0.0070 0.0373 5 0.80 512.6 0.0018 0.0308 32 2.73 545.9 0.0070 0.0416 6 0.81 539.8 0.0018 0.0350 33 2.76 549.3 0.0070 0.0434 7 0.98 543.3 0.0018 0.0379 34 2.80 554.4 0.0070 0.0441 8 1.00 509.7 0.0018 0.0336 35 2.86 529.0 0.0185 0.0378 9 1.11 510.1 0.0018 0.0344 36 3.03 526.7 0.0185 0.0414 10 1.18 549.0 0.0018 0.0432 37 3.04 528.5 0.0070 0.0427 11 1.19 547.3 0.0018 0.0409 38 3.08 539.2 0.0185 0.0408 12 1.37 498.3 0.0070 0.0336 39 3.30 558.7 0.0070 0.0479 13 1.46 545.3 0.0018 0.0448 40 3.35 540.5 0.0185 0.0429 14 1.50 510.6 0.0018 0.0363 41 3.39 533.4 0.0185 0.0424 15 1.55 510.5 0.0018 0.0368 42 3.42 524.5 0.0070 0.0439 16 1.72 498.3 0.0070 0.0374 43 3.60 527.9 0.0185 0.0432 17 1.82 509.7 0.0018 0.0377 44 3.62 545.5 0.0185 0.0449 18 1.91 510.8 0.0018 0.0386 45 3.65 526.7 0.0070 0.0481 19 1.97 500.6 0.0070 0.0378 46 3.74 531.3 0.0185 0.0442 20 2.01 531.1 0.0070 0.0316 47 3.84 535.7 0.0185 0.0429 21 2.05 522.6 0.0070 0.0300 48 3.86 557.2 0.0185 0.0476 22 2.06 534.6 0.0185 0.0402 49 3.91 521.2 0.0070 0.0492 23 2.07 549.0 0.0018 0.0455 50 4.05 546.0 0.0185 0.0463 24 2.16 512.3 0.0070 0.0412 51 4.08 561.6 0.0185 0.0480 25 2.16 528.1 0.0185 0.0324 52 4.12 537.4 0.0185 0.0436 26 2.19 510.0 0.0070 0.0388 53 4.18 530.9 0.0185 0.0473 27 2.22 510.8 0.0070 0.0398 54 4.22 544.7 0.0185 0.0447 ▪ mathematical model of influential parameter ɛmax during propagation of the Lüders bands: ɛmax=3.036+0.0732ΔTmax-0.01168σ-16.120v+0.00107ΔTmax 2 +0.000011σ 2 + 50.721v 2 -0.00014ΔTmaxσ+4.2917ΔTmaxv+0.0287σv-0.00836ΔTmaxσv (8) ▪ mathematical model of influential parameter ΔTmax during propagation of the Lüders bands: Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 603 ΔTmax=-309.167+48.762ɛmax+1.1811σ+2310.75v+1210.47ɛmax 2 -0.00113σ2- 10906.6v2-0.03997ɛmaxσ-77240.3ɛmaxv-3.848σv+146.69ɛmaxσv (9) Table 8 Experimental results of input parameters (ɛmax, σ, v) and output parameter ΔTmax obtained by measured values during propagation of the Lüders bands N u m b e r o f m e a su re m e n t Input parameters Output parameter N u m b e r o f m e a su re m e n t Input parameters Output parameter ɛmax (mm/mm) σ (MPa) v (s-1) ΔTmax (°C) ɛmax (mm/mm) σ (MPa) v (s-1) ΔTmax (°C) 1 0.0309 508.6 0.0018 0.35 28 0.0405 510.8 0.0018 2.27 2 0.0279 509.1 0.0018 0.39 29 0.0386 540.7 0.0070 2.46 3 0.0325 509.8 0.0018 0.61 30 0.0392 535.6 0.0185 2.55 4 0.0331 511.4 0.0018 0.75 31 0.0373 521.3 0.0070 2.71 5 0.0308 512.6 0.0018 0.80 32 0.0416 545.9 0.0070 2.73 6 0.0350 539.8 0.0018 0.81 33 0.0434 549.3 0.0070 2.76 7 0.0379 543.3 0.0018 0.98 34 0.0441 554.4 0.0070 2.80 8 0.0336 509.7 0.0018 1.00 35 0.0378 529.0 0.0185 2.86 9 0.0344 510.1 0.0018 1.11 36 0.0414 526.7 0.0185 3.03 10 0.0432 549.0 0.0018 1.18 37 0.0427 528.5 0.0070 3.04 11 0.0409 547.3 0.0018 1.19 38 0.0408 539.2 0.0185 3.08 12 0.0336 498.3 0.0070 1.37 39 0.0479 558.7 0.0070 3.30 13 0.0448 545.3 0.0018 1.46 40 0.0429 540.5 0.0185 3.35 14 0.0363 510.6 0.0018 1.50 41 0.0424 533.4 0.0185 3.39 15 0.0368 510.5 0.0018 1.55 42 0.0439 524.5 0.0070 3.42 16 0.0374 498.3 0.0070 1.72 43 0.0432 527.9 0.0185 3.60 17 0.0377 509.7 0.0018 1.82 44 0.0449 545.5 0.0185 3.62 18 0.0386 510.8 0.0018 1.91 45 0.0481 526.7 0.0070 3.65 19 0.0378 500.6 0.0070 1.97 46 0.0442 531.3 0.0185 3.74 20 0.0316 531.1 0.0070 2.01 47 0.0429 535.7 0.0185 3.84 21 0.0300 522.6 0.0070 2.05 48 0.0476 557.2 0.0185 3.86 22 0.0402 534.6 0.0185 2.06 49 0.0492 521.2 0.0070 3.91 23 0.0455 549.0 0.0018 2.07 50 0.0463 546.0 0.0185 4.05 24 0.0412 512.3 0.0070 2.16 51 0.0480 561.6 0.0185 4.08 25 0.0324 528.1 0.0185 2.16 52 0.0436 537.4 0.0185 4.12 26 0.0388 510.0 0.0070 2.19 53 0.0473 530.9 0.0185 4.18 27 0.0398 510.8 0.0070 2.22 54 0.0447 544.7 0.0185 4.22 Table 9 Calculated values of multiregression coefficient (R) and coefficient of determination (R2) of mathematical models of influential parameters during Lüders bands formation and propagation Influential parameters σ (MPa) ɛmax (mm/mm) ΔTmax (°C) Formation of the Lüders band R 0.881 0.977 R2 0.776 0.954 Propagation of the Lüders band R 0.871 0.948 0.968 R2 0.759 0.899 0.938 604 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ Table 10 Calculated results of influence parameter σ during formation and propagation of the Lüders bands at v = 0.007 s-1 Number of measurement Formation of the Lüders band Propagation of the Lüders band Output parameter Output parameter σ (MPa) σ (MPa) 1 532.0 525.6 2 539.7 520.9 3 535.9 533.8 4 530.4 516.6 5 514.6 537.3 6 498.8 525.0 7 496.6 486.1 8 501.5 494.5 9 537.5 515.5 10 541.5 525.6 11 524.5 524.4 12 523.4 514.4 13 529.4 14 532.7 15 537.3 16 540.7 17 542.2 18 553.1 Deviation 1.34% 1.65% Experimental values of the influence parameters during formation and propagation of the Lüders bands are added into the obtained mathematical models. The following calculated values of the influence parameters from the predicted models are shown in Tables 10-12. Mathematical models of the influence parameters show low deviations of calculated values compared to the experimental results. This confirms the applicability of the obtained mathematical models in monitoring the observed influence parameters during the formation and propagation of the Lüders bands during cold deformation in niobium microalloyed steel. The functionality and reliability of mathematical models of influential parameters during formation and propagation of the Lüders bands were verified by selecting random values of the influencing parameters in the area of formation and propagation of the Lüders bands. Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 605 Table 11 Calculated results of influence parameter ɛmax during formation and propagation of the Lüders bands Formation of the Lüders band Propagation of the Lüders band N u m b e r o f m e a su re m e n t Output parameter N u m b e r o f m e a su re m e n t Output parameter N u m b e r o f m e a su re m e n t Output parameter N u m b e r o f m e a su re m e n t Output parameter ɛmax (mm/mm) ɛmax (mm/mm) ɛmax (mm/mm) ɛmax (mm/mm) 1 0.0060 19 0.0276 1 0.0369 28 0.0373 2 0.0133 20 0.0328 2 0.0390 29 0.0373 3 0.0242 21 0.0037 3 0.0416 30 0.0363 4 0.0266 22 0.0106 4 0.0426 31 0.0345 5 0.0283 23 0.0231 5 0.0408 32 0.0388 6 0.0050 24 0.0292 6 0.0443 33 0.0415 7 0.0096 25 0.0055 7 0.0298 34 0.0430 8 0.0179 26 0.0207 8 0.0297 35 0.0457 9 0.0242 27 0.0334 9 0.0309 36 0.0490 10 0.0079 28 0.0364 10 0.0315 37 0.0357 11 0.0257 29 0.0408 11 0.0333 38 0.0391 12 0.0303 30 0.0103 12 0.0369 39 0.0401 13 0.0053 31 0.0291 13 0.0316 40 0.0438 14 0.0153 32 0.0354 14 0.0339 41 0.0441 15 0.0251 33 0.0080 15 0.0365 42 0.0473 16 0.0324 34 0.0230 16 0.0392 43 0.0374 17 0.0122 35 0.0311 17 0.0396 44 0.0418 18 0.0188 36 0.0403 18 0.0427 45 0.0441 19 0.0340 46 0.0454 20 0.0388 47 0.0453 21 0.0405 48 0.0447 22 0.0441 49 0.0388 23 0.0457 50 0.0410 24 0.0500 51 0.0421 25 0.0341 52 0.0436 26 0.0376 53 0.0471 27 0.0390 54 0.0485 Deviation 12.37% Deviation 3.37% Therefore, a confirmation test was performed to compare randomly selected experimental values of the influencing parameters with the calculated results obtained by predicted mathematical models, Tables 13-15. Conducted confirmation tests have established high accuracy of the mathematical models in obtaining calculated values of the influence parameters from randomly selected experimental values during formation and propagation of the Lüders bands. 606 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ Table 12 Calculated results of influence parameter ΔTmax during propagation of the Lüders bands Propagation of the Lüders band N u m b e r o f m e a su re m e n t Output parameter N u m b e r o f m e a su re m e n t Output parameter ΔTmax (°C) ΔTmax (°C) 1 0.90 28 2.41 2 1.09 29 2.54 3 1.25 30 2.75 4 1.46 31 1.81 5 1.89 32 2.41 6 1.78 33 2.63 7 0.42 34 2.75 8 0.69 35 2.58 9 0.89 36 2.96 10 0.99 37 2.03 11 1.00 38 2.68 12 1.37 39 3.09 13 0.78 40 3.36 14 1.09 41 3.59 15 1.31 42 4.05 16 1.50 43 3.05 17 1.58 44 3.36 18 1.81 45 3.46 19 1.74 46 3.59 20 2.49 47 3.79 21 3.18 48 4.08 22 3.33 49 2.90 23 3.93 50 3.14 24 4.01 51 3.48 25 1.38 52 3.82 26 1.71 53 4.18 27 1.88 54 4.13 Deviation 11.90% Based on the previously obtained results and validation of the mathematical models for the influential parameters, it can be concluded that the mathematical models showed high reliability since high coefficients of multiregression and determination coefficients and low deviations of calculated results were obtained. Mathematical models are evidence that output parameter stress (σ) can be described by maximum temperature change (ΔTmax) and maximum strain (ɛmax) considering good agreement between the experimental results and the calculated ones from polynomial models during the formation and propagation of the Lüders bands. Therefore, maximum temperature change (ΔTmax) can be considered as a stress change that occurs during the formation and propagation of the Lüders bands. Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 607 Table 13 Validation of mathematical models by the confirmation test for σ during formation and propagation of the Lüders bands at v = 0.007 s-1 N u m b e r o f m e a su re m e n t Formation of the Lüders band Propagation of the Lüders band E x p e ri m e n ta l re su lt s C a lc u la te d re su lt s o f th e m o d e l E x p e ri m e n ta l re su lt s C a lc u la te d re su lt s o f th e m o d e l σ (MPa) σ (MPa) σ (MPa) σ (MPa) 1 557 558.7 525.2 523.8 2 533.4 515.5 514.2 522.4 3 534.6 560.4 557.3 547.0 4 535.6 524.5 Deviation 2.64% 1.24% Table 14 Validation of mathematical models by the confirmation test for ɛmax during formation and propagation of the Lüders bands N u m b e r o f m e a su re m e n t Formation of the Lüders band Propagation of the Lüders band E x p e ri m e n ta l re su lt s C a lc u la te d re su lt s o f th e m o d e l E x p e ri m e n ta l re su lt s C a lc u la te d re su lt s o f th e m o d e l ɛmax mm/mm ɛmax mm/mm ɛmax mm/mm ɛmax mm/mm 1 0.0166 0.0215 0.0455 0.0417 2 0.0199 0.0232 0.0355 0.0341 3 0.0415 0.0441 0.0393 0.0403 4 0.0392 0.0362 0.0491 0.0485 5 0.0404 0.0363 6 0.0457 0.0478 7 0.0467 0.0466 8 0.0450 0.0450 9 0.0480 0.0485 Deviation 15.04% 3.59% Mathematical models have shown that the values of other influencing output parameters (ɛmax, ΔTmax) during the formation and propagation of the Lüders bands can be described with high accuracy. It means that the behavior of niobium microalloyed steel can be predicted with models of influencing parameters in the region behind the Lüders front where maximum local temperature changes, i.e. stress changes, and strain changes are present during the cold deformation. 608 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ Table 15 Validation of mathematical models by the confirmation test for ΔTmax during propagation of the Lüders bands N u m b e r o f m e a su re m e n t Propagation of the Lüders band E x p e ri m e n ta l re su lt s C a lc u la te d re su lt s o f th e p o ly n o m ia l m o d e l ΔTmax (°C) ΔTmax (°C) 1 2.02 2.07 2 1.15 1.22 3 2.07 1.71 4 3.91 4.06 5 2.22 2.72 6 3.07 2.66 7 4.05 3.92 8 4.12 3.83 9 4.01 4.15 Deviation 8.82% Lüders bands, i.e. surface roughness, can often occur due to the effort of continuously improving of the mechanical properties (increasing yield strength and ductility) of microalloyed steels. Therefore, in order to modify and improve the metal forming process, it is important to determine the stress amounts during the Lüders band propagation. Stress changes at different deformation degrees are visible during static tensile test in the area behind the Lüders band front by applying a mathematical model according to Eq. (6) during the propagation of the Lüders band. From the obtained results of the mathematical model, it can be concluded that the stress amount in the area behind the Lüders band front increases with increasing of deformation degree from point 1 to point 2 according to: σpoint 1, Lüders band propagation =640.894-90.392∙0.76-1049.07∙0.0234-86.8013∙0.76 2 - 399310∙0.0234 2 +13425.7∙0.76∙0.0234=517.63 MPa σpoint 2, Lüdersbandpropagation=640.894-90.392∙3.42-1049.07∙0.047-86.8013∙3.42 2 - 399310∙0.047 2 +13425.7∙3.42∙0.047=543.16 MPa Comparing the stress amounts at the start of the Lüders band propagation (point 1) and during the Lüders band propagation (point 2), a higher stress amount of 543.16 MPa was determined at point 2 at a higher deformation degree. A higher stress amount indicates a more pronounced surface roughness in this area during Lüders band propagation which is a significant problem during forming of microalloyed steels. The possibility of the stress amounts prediction with the obtained mathematical models is proven. For example, during cold drawing and bending of microalloyed pipes with inhomogeneous deformations, the significance of the surface roughness can be predicted and so can the possibilities of plastic flow of niobium microalloyed steel from obtained Mathematical Modeling of the Influence Parameters During Formation and Propagation of the Lüders... 609 stress amounts. Predicting of the stress amounts in steels, with the Lüders band appearance, can also be significant during sheet forming, especially sheet metal drawing, and deep drawing in automobile production. It has been proved that the obtained mathematical models can improve the product quality and reduce the costs of the forming process of microalloyed steels since the calculation of the values of the influential parameters by using predicted models do not necessarily require additional experimental tests requiring additional costs. 4. CONCLUSION Mathematical models of the influence parameters during the formation and propagation of the Lüders bands in the area behind the Lüders band front in niobium microalloyed steel are developed in this paper. The proposed mathematical models give a new approach to the prediction of stresses, maximum temperature changes and strains during the Lüders band formation and propagation in niobium microalloyed steel. High reliability of the obtained mathematical models are determined with high coefficients of multiregression within the range from 0.871 to 0.977, low deviations of calculated results from 1.34% to 12.37% and low deviations of confirmation results from 1.24% to 15.04%. It has been proved that output parameter stress (σ) can be described by maximum temperature change (ΔTmax) and maximum strain (ɛmax) where maximum temperature change (ΔTmax) can be considered as a stress change during the formation and propagation of Lüders bands. Stress amounts at different deformation degrees, in the deformation zone during Lüders band propagation, were determined for modification and improvement of steel forming processes. Using the obtained mathematical model, it was observed that the stress changes at different deformation degrees during static tensile tests in the area behind the Lüders band front. At the start of the Lüders band propagation, a lower stress amount of 517.63 MPa, is determined with respect to the advanced Lüders band propagation stress amount of 543.16 MPa. A higher stress amount, obtained by the proposed mathematical model, indicates a more pronounced surface roughness in the area of the Lüders band propagation. The proposed mathematical models can find their application in the industry for improving the forming process by prediction of the surface roughness and plastic flow of the tested steel using the obtained amounts of influence parameters, especially stress amounts. The significance of the surface roughness as well as the possibilities of plastic flow of niobium microalloyed steel can be predicted by the proposed mathematical models during sheet forming and deep drawing in automobile production as well as in cold drawing and bending of microalloyed pipes for pipelines with inhomogeneous deformations. Application of the obtained mathematical models of influence parameters can reduce costs in the forming process while no additional experimental tests are required. Acknowledgements: This work is financially supported by the Croatian Science Foundation under project number IP-2016-06-1270 (Principal Investigator: Prof.dr.sc. Stoja Rešković) and by the University of Rijeka, Croatia, contract number uniri-tehnic-18-100-1235. 610 T. BRLIĆ, S. REŠKOVIĆ, Z. JURKOVIĆ, G. JANEŠ REFERENCES 1. 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