2.3. Numerical Modelling

The programs FORGE 2011^® (Transvalor, Sophia-Antipolis, France) and QTSteel^® (ITA-Technology and Software, MSL-Metaltech Services Ltd., Ostrava, Czech Republic) were used for the numerical modelling of the rolling process of the wire rod. The deformation parameters calculated in the FORGE 2011^® program were used to perform numerical modelling of microstructure development using the QTSteel^® program and to perform physical modelling using the GLEEBLE 3800 simulator (Dynamic Systems Inc. Poestenkill, NY, USA).

In the FORGE 2011^® program, a mathematical model is used for the numerical modelling of the three-dimensional plastic flow of metal during rolling in grooves, in which the mechanical state of the deformed material is described using the Norton–Hoff law [44,45]:

where S_ij—strain tensor deviator, ε˙i—strain rate intensity, ε˙ij—strain rate tensor, εi—strain intensity, T—temperature, K—consistency depending on yield stress σ_p, m_m—coefficient characterizing hot metal deformation (0 < m_m < 1).

The friction conditions prevailing on the contact surface of the material with the tools are described using the Coulomb and Treska friction models, in which the appropriate coefficient values are assumed [46]:

where τ_j—unit friction force vector, σ_p0—base yield stress, σ_n—normal stress, μ—coefficient of friction, m—friction factor.

The temperature fields are calculated on the basis of differential equations describing temperature changes with transient heat flow [46]:

where k_x, k_y, k_z—distribution functions of anisotropic thermal conductivity coefficients in x, y, z directions, T_s—function describing the temperature in the zone in question, Q—rate distribution function for generating strain heat, c_p—specific heat distribution function of the deformed material, ρ—density distribution function.

The boundary conditions were adopted as the combined boundary conditions of the second and third types, in the form [46]:

where l_x, l_y, l_z—directional cosines normal to the surface of the deformed strip, q—heat flow rate on the surface of the cooled zone, α_k—convection losses.

Equations (4) and (5) clearly determine the heat exchange during modelling of the rolling process.

The input data for numerical modelling of the analysed rolling process are given in Table 3. These data were adopted on the basis of the technical literature and previous experience.

Boundary conditions for numerical modelling of rolling process of 5.5 mm wire rod ¹.

¹ Table based on data published in work [3].

The initial and boundary conditions necessary for numerical modelling of the analysed rolling process of the 5.5 mm wire rod are given in Table 3. These data were determined on the basis of the technical literature [43,47,48,49,50,51]. Other data necessary to carry out the numerical modelling of the rolling process (initial temperature, relative rolling reduction, strain times and break times, roller rotational rates and linear rate of the strip) were adopted on the basis of the industrial data. In the analysed process, the average temperature on the cross-section of the charge before the first stand of the rolling mill was 1130 °C and the average temperature of the side surface was 1075 °C. The thermophysical properties of 20MnB4 steel (Table 4) were taken from material database of FORGE 2011^® software.

Thermophysical properties of 20MnB4 steel ¹.

¹ Data based on material base of FORGE 2011^® program.

The Hensel–Spittel equation was used to describe the rheological properties of the studied steel [46]:

where σ_p—yield stress, MPa, T—temperature, °C, ε—true strain, ε˙—strain rate, s⁻¹, A, m₁–m₉—coefficients.

During the numerical modelling of the rolling process of 5.5 mm wire rod in the NTM and RSM blocks of the rolling mill, the rheological properties of the studied steel were defined using Equation (6) and the coefficients given in Table 5.

Equation coefficients (6) used during numerical modelling of 5.5 mm wire rod rolling in NTM and RSM blocks of rolling mill [3].

When determining the rheological properties of the investigated steel for the rolling conditions in the wire rod rolling mill (NTM and RSM blocks), the results of published studies, among others, in works [31,32] were taken into account, in which the values of yield stress for steels with a similar chemical composition to 20MnB4 steel were determined, in terms of the strain rate and temperature occurring during the rolling of 5.5 mm diameter wire rods. By extrapolating the values of yield stress of the examined steel for the strain rate occurring during the rolling of the 5.5 mm diameter wire rod, values consistent with those published in papers [31,32] were obtained.

In the QTSteel^® program, when forecasting the microstructure and mechanical properties of heat treated or thermoplastically processed steel, data from the cooling curves on the TTT chart are used. Calculating the percentage content of the microstructure components is performed step by step for the relevant sections of the cooling curve. To describe the kinetics of the transformation of individual components of the microstructure, the program uses the Avrami equation (7) [52,53]:

where: Xi(T,t)—volume fraction of individual components of the microstructure: ferrite, perlite, bainite, k(T) and n(T)—parameters depending on the transformation mechanism and places of privileged nucleation and on the cooling rate, calculated on the basis of TTT charts for a given temperature, T—temperature, t—time, Xγ—volume fraction of residual austenite.

The volume fraction of martensite during martensitic transformation is calculated on the basis of the Koistinen–Marburger equation [53]:

where: Xm—volume fraction of martensite, b, n—constant, Tms—martensitic transformation start temperature, T—temperature, Xγ—volume fraction of residual austenite.

Vickers HV hardness is determined by means of a regression equation [52,53]:

where: HV—Vickers hardness, Xf, Xp, Xb, Xm—volume fractions: ferrite, perlite, bainite, martensite, C₀, D_i, E_i, F_i, G_i—constant, c_i—percentage of alloying additions.

The tensile strength was determined based on Equation (10) [52]:

where: UTS—Ultimate tensile strength, HV—Vickers hardness, and a, b—constant.

Yield strength YS is determined by Equation (11) [52,53]:

where: D_α—ferrite grain size, C_r—cooling rate, Xf, Xp, Xb, Xm—volume fractions: ferrite, perlite, bainite, martensite.

Detailed results of research carried out using the DIL 805 A/D dilatometer [54], the aim of which was to determine the phase transition temperatures, develop TTT and DTTT (Deformation Time Temperature Transition) charts and to determine the most favourable cooling conditions for 20MnB4 steel, were published, among others, in [55]. Taking into account the obtained results, the DTTT graph (Figure 1) was used to determine the impact of the cooling conditions on the forming of the wire rod microstructure immediately after the deformation process. The characteristic temperatures of phase transitions and hardness of 20MnB4 steel are presented in Table 6.

Real DTTT diagram for 20MnB4 steel [55]. Reproduced with permission from Laber, K., Koczurkiewicz, B., Determination of optimum conditions for the process of controlled cooling of rolled products with diameter 16.5 mm made of 20MnB4 steel, Proceedings of the 24th International Conference on Metallurgy and Materials—METAL 2015; published by Tanger Ltd., 2015.

Characteristic temperatures of phase transitions of 20MnB4 steel ¹.

¹ Table based on data published in work [55]. Reproduced with permission from Laber, K., Koczurkiewicz, B., Determination of optimum conditions for the process of controlled cooling of rolled products with diameter 16.5 mm made of 20MnB4 steel, Proceedings of the 24th International Conference on Metallurgy and Materials—METAL 2015; published by Tanger Ltd., 2015.

It was found that in order to obtain a ferritic–pearlitic microstructure in the finished product, the cooling rate should not exceed 15 °C/s. Increasing the cooling rate above 15 °C/s causes the formation of bainite, bainitic-martensitic and martensitic structures in the material, which results in deterioration of the ability of the investigated steel for further cold working, or in extreme cases prevents it.