Accurate prediction algorithm of rolling force in slab gradient temperature rolling process

The temperature gradient rolling (GTR) method effectively addresses the issue of uneven deformation of surface and core in the rolling of extra-thick plate. Different from the traditional near uniform temperature rolling (UTR) process, GTR involves a significant temperature gradient along the thickness direction of the rolled piece. As a result, the rolling force prediction model used in uniform temperature rolling cannot be directly applied in GTR. Rolling force prediction model is crucial for process control models of plate mill and serves as the foundation for plate thickness control. However, there has been limited research on rolling force prediction methods specifically for GTR. In order to predict the change of rolling force in GTR process and enhance the control of the rolling process, the current paper proposed a simplified method based on the deformation resistance model, temperature distribution fitting, differential method, and loop iteration. The precision of the method was examined by using a finite element model and plane strain experiment, demonstrating a mean error between the method and FEM was less than 5%.


Introduction
The heavy plate is widely used in oceanography engineering, architecture, and shipbuilding as the critical structure which requires high strength and good ductility.Additionally, the ultra-heavy plate is also required to exhibit excellent toughness or corrosion resistance based on the applied environment.Currently, the most significant challenge in rolling ultra-heavy plate is the serious deformation inhomogeneity between its surface and center.The deformation inhomogeneity also induces the difference in microstructure and mechanical properties between the surface and the center.
To improve that, Yu Wei et al. proposed a novel process called gradient temperature rolling (GTR) which can homogenize the strain distribution [1,2].By increasing the strain in the plate center, the central deformation is improved, and refines the central grain size after γ-α transformation [3].However, there are still two problems to be solved.Firstly, the particles precipitated before rolling may influence the stability of mechanical properties, particularly toughness.Secondly, the manufacturer requires a new method to forecast the roll force due to the changes of strain, stress, and temperature induced by the gradient temperature.
At present, two main methods have been developed as the core of the neural network for rolling force calculation since the Karman's differential equation, the regression method based on big data, and the theoretical calculation based on plastic theory [4].Li  firstly applied cosine velocity field in analytical approach to predict rolling torque and rolling force [7].The finite element method is gradually used to estimate the roll force with the cyber development.Guo et al. proposed a computational model combining a finite element method (FEM) with an artificial neural network (ANN) to predict the rolling force in the hot rolling of Mg alloy plates [8].Kim et al. developed an online model using finite element to forecast the rolling force in a round-oval-round pass rolling [9].Lin corrected the error of elasticity-plasticity model by using a neural network [10].However, all of the studies focused on improving the precision of the existing methods for uniform temperature rolling by introducing new assumption, updated algorithm, or considering detailed condition.Regarding the rolling force during GTR, Wang et al. proposed a method based on the rolling theory, which emphasizes deformation and involves a complex process, and its error increases as the reduction increases [11].Therefore, further research is needed to develop methods for calculating the rolling force during GTR or in processes with heterogeneous temperature distributions.
The present study aims to determine the temperature distribution in the thickness direction of the billet according to the finite element software, assuming a known cooling process prior to rolling.On this basis, a simplified method for accurately predicting GTR rolling force is proposed.The method was verified through the result of a finite element model created with MSC Marc software and a plane strain experiment.The temperature distribution across the thickness during GTR was calculated using MSC Marc software and verified by measuring with K-type thermocouple.

The FEM simulation of GTR
Plate rolling is a complicated process involving large displacements.In this study, a tridimensional (3-D) model for a quarter of a rolled piece was constructed using MSC Marc software, assuming the validity of the law of symmetry.The roller was defined as a rigid body and with a diameter of 400 mm and a speed of 0.5 m/s.The rolled piece which has a size of 80 mm (thickness) × 160 mm (width) × 500 mm (length) was meshed with plane strain solid 11 elements, with the element size set to 2.5 mm × 15 mm × 5 mm, for temperature and rolling force calculations.Various thermo-mechanical coupling boundary and contact conditions were considered, including thermal radiation, heat conduction, and frictional heat between roller and piece, and were applied to the model.The parameters used in these conditions are presented in Table 1.The rolling schedule is provided in Table 2. Schindler et al. proposed a mathematical model to calculate the deformation resistance of a C-Mn-Nb-V-based high strength low alloy [12], Siciliano et al. investigated the influence of microalloying elements on mean flow stresses during hot rolling with 0.09~0.16%C-Mn-Si-Ni-V-Ti steel and indicated that the effect of single microalloying element was small [13], and Dobatkin et al. (2009) studied plastic deformation of 0.09% C-Mn-Si-Nb-V-Ti steel [14].Based on these studies, the deformation resistance was formulated as [15]: where σ is the stress, ε is the strain, ε is the strain rate, and T is the temperature.
The rolling geometry model established by Marc is shown in Fig. 1a.The initial temperature distribution for the gradient temperature rolling (GTR) is illustrated in Fig. 1b.The temperature distribution is achieved through the cooling method depicted in Fig. 1c, which involves starting at 1200 °C and then subjecting the surface temperature to water cooling at an average rate of approximately 10 °C/s for 60 s, followed by air cooling within 10 s. Figure 1c illustrates the temperature curves at various positions within the slab during the cooling process, while Fig. 1d displays the temperature distribution along the thickness of the slab prior to each pass.Throughout the GTR process, there is a temperature gradient of approximately 300 °C between the center and the surface.

Differential method of rolling force for GTR
The model used in manufactured field can be described as Formula (2).
where K f = 1.15•σ,F is the rolling force/N, W is the width of deformation area/mm, R ' is the roller radius with elasticity/ mm, Q p is the influence function of deformation area, K f is the resistance stress of material/MPa, and σ is the yield stress of material/MPa.In order to accommodate the changed strain, stress, temperature distributions, and the field situation, a series of assumptions and simplified conditions were established as follows.
For problem 6, the finite element components are divided into solid and rigid body.The solid will deform under the action of elastic force, which has a weak effect on the roll diameter, while the rigid body is not affected by deformation.Therefore, in order to reduce the reference of additional parameters, the default roll is rigid body material.The effect of elasticity on the roller radius was ignored in this study since the FEM model involved elasticity, which may induce a large error during rolling force calculation.Additionally, the elasticity of roller can be reflected by iteration algorithm in the practical application.Misaka Kasuke's model for Q p was applied in this paper as follows: (3) where l d is the length of deformation zone, R is the radius of roll, h is the thickness at the entrance of rolled piece, h' is the thickness at the exit of rolled piece, and h is the average thickness.A cyclic differential method was employed to calculate the rolling force, taking into account gradient temperature distribution.As a result, the total rolling force is therefore equal to the sum of all differential elements, expressed as: where n is the element number.
The strain rate at different positions of slab thickness was assumed to be accorded with A. H. Cxopxoдoв (Formula (5)).
Figure 2 provides an illustration of the differential method.T s and T c represent the temperatures of the slab surface and its center, respectively; the cyclic parameter i is the cyclic parameter.When i=1, the slab thickness (h) is divided into two differential parts by a proportionality coefficient (a 11 ).The proportionality coefficient makes the area sum of the two parts (shadow area) equal to the integral area of the temperature distribution, which can guarantee that the total thermal energy of the slab is in a consistent status before and after differentiation.Similarly, the thickness after rolling (h ' ) is divided in the same proportion.These processes can be described by Formulas ( 6), (7), and (8).
Here, According to Formula (6).After the i-th difference, the total thickness h is divided into h i,1 to h i,j by the proportional parameter a.After the next difference, h i,1 is decomposed by a i,1 into h (i + 1),1 and h (i + 1),2 , and so on and so forth, h i,j is decomposed by a i,j into h (i + 1), (2j-1) and h (i + 1),2j .Therefore, Formula (10) is obtained.At the same time, Formula (6) can be reformulated as Formula (10), and Formula (8) can be reformulated as Formula (11).Each slab thickness is divided into two new thickness parameters by proportionality coefficient.The formula for temperature is expressed as Formula (12).
During the difference process, the proportion of each difference element to the total rolling force is determined by Formula (13).Because the numerator ranges in value from , so Formula (13) can be reduced to Formula ( 14).
(12) Figure 3 illustrates the complete process of loop iteration, which was implemented using Visual C++ 2013 and Gaussian elimination method.The loop was terminated when the standard deviation reached 4%.The temperature distributions presented in Fig. 1b and d were transformed into continuous curves through five-order polynomial fittings.
The large specimen plane strain experimenter [16] was used to verify the result of the prediction method.The equipment is controlled by a computer system and has pressure recorder with pressure spring.Figure 4a-c depict the experimental setup, the illustration of specimen (80 mm 3 ), and the specimen after deformation, respectively.The specimen was equipped with three thermocouples (surface, 1/4 thickness, and 1/2 thickness) and was heated to 1200 °C in a medium-frequency induction furnace and subjected to the cooling process as shown in Fig. 1c.The temperatures tested by thermocouples and their comparison with Fig. 1b are shown in Fig. 5.

Results
The rolling force of FEM is shown in Fig. 6 whose horizontal section during stable rolling was taken as the average to compare with the calculated results.Table 3    average value of the simulated rolling force and the calculated results after the process as shown in Fig. 3, and the standard deviation between them.It can be observed that the average error between FEM and the method is less than 5% and the maximum error is 6%.The precision reaches the general error standard of rolling force model.Figure 7 gives the results of plane strain experiment.The profile of pressure curves is quite different with the simulated results because of the difference between compressing and rolling.To represent the pressure when the compression travel reaches the set value, the data from 10 ms (millisecond) before unloading per pass were averaged.The comparison between experimental and calculated pressures which can be obtained by dividing the data of Table 1 by deformation zone area of rolling ( W • √ RΔh ) can be seen in Fig. 8.The errors between experimental and calculated results are small (<5%) in the first three passes but exceed 8% in fourth and fifth passes.This discrepancy is attributed to the smaller volume of the experimental specimens compared to the simulated piece.The central temperature had such an abrupt decrease staged that the resistance force of the specimen was increased.The central temperature curves of simulated piece and experimental specimen are presented in Fig. 9.

presents the
When the initial temperature field is known prior to deformation, the rolling force model provided in this paper can significantly reduce the calculation time in the finite difference iterations.Table 4 presents the time and standard deviation used in calculating the first roll pass.However, only 3 differences are needed to reach the standard deviation of 4%.When the mass scaling factor is 1000, the CPU time required to calculate the FEM model is 434 s.

Discussion
The error in the prediction method primarily stems from several factors.
(1) The error is attributed to Q p .The rolling force is the force acting on the cambered surface of the roller.Q p makes a vertical component of the rolling force equivalent to the rolling force acting on the roller.At present, most models for Q p are based on statistical regression using large datasets, focusing on various temperature ranges, deformation levels, and slab thickness.However, because of the complexity of the theoretical arithmetic, although Misaka Kasuke's model for Q p is well conformable to the situation of a small reduction, it cannot correctly reflect all conditions during rolling.(2) The approximate processes of strain and strain rate in the calculated method lead the unavoidable system error.The current method preferentially guaranteed the veracity of temperature data by using the process shown in Fig. 2 because the stress described in Formula 1 is highly sensitive to temperature variations.To avoid the complicated mathematical derivation and promote to the manufacturers, the strain and strain rate are simply treated to the same value in GTR, despite the fact that high temperatures in the center lead to significant deformation while the surface with lower temperatures undergoes less deformation.(3) The experimental force values of the fourth and fifth pass are higher than the measured values, which can be attributed to the contact conditions between the workpiece and the hammerhead.During deformation, the deformed workpiece and the wedge-shaped side of the hammer generate sidewall normal pressure force F t and friction force F f .Additionally, pressure components will be generated in the pressing direction ΔF c , while the pressure on the contact surface with the die and workpiece F c constitutes the total pressure.The magnitude of the additional pressure component ΔF c increases with greater deformation and lower deformation temperature, as illustrated in Fig. 10.The molecule of Formula ( 14) used in the current method resulted in calculated results that were lower than the experimental result.Formula ( 14) is a convenient approach for predicting rolling force, but it lacks accuracy due to the experimental hammerhead design that prioritizes easy demolding.Additionally, the direct relationship between rolling force and exp(1/T) introduces calculation errors.
In fact, the existing rolling models used in practical processes are constructed based on a large amount of production data and further refined through self-learning system and artificial neural network system.These systems help to reduce errors arising from factors such as the elasticity of the roller, Q p , and approximation processes.The algorithm kernel, which governs the convergence conditions, calculation points, and loop iteration methods, is the most crucial component of a rolling force model.The current method utilized the differential element method and focused on the rolling force distribution generated by the temperature distribution along the thickness direction of slab.Therefore, if the slab temperature distribution can be obtained by fast and accurate algorithm, the rolling force could be predicted by the present method, and furthermore, the accuracy could be enhanced by the self-learning algorithm.
The new method presented in this study successfully solves the challenging issue of the traditional rolling force model in the application of GTR, making it highly valuable for steel plate production.Additionally, in comparison to the FEM model, the new model significantly reduces the calculation time while still maintaining accuracy.This improvement in efficiency is of great significance in practical applications.

Conclusion
A simplified model is proposed for accurate prediction of rolling force based on gradient temperature rolling (GTR) method.The GTR rolling force model is developed through iterative calculations and approximations of strain and strain rate, taking into account the temperature distribution along the thickness of the slab and the material's deformation resistance.
Moreover, in practical industrial conditions, the mass of rolled products is generally larger than that of test samples, resulting in potential discrepancies between the results obtained from computer simulations, laboratory validations, and actual production.Therefore, in addition to accounting for the influence of additional friction and normal pressure between the mold sidewall and the workpiece, the GTR rolling force prediction model can be further optimized and improved through the implementation of self-learning algorithms.
et al. increased the prediction accuracy of hot strip rolling force model through incorporating a new friction model into the traditional Sims model [5].Gao et al. proposed a method based on bisector yield criterion and Pavlov principle to calculate the rolling force during vertical rolling [6].Zhang et al.

Fig. 1
Fig. 1 FEM model by MSC Marc and temperature conditions: a FEM model; b temperature distribution of fist pass; c temperature curves during cooling; d temperature distributions of passes 2~4

Fig. 7 Fig. 8
Fig. 7 Pressure of deformation zone during plane strain compress

Fig. 9
Fig. 9 Comparison of central temperature between FEM and experiments

Table 2
Rolling schedule of FEM modelh is the thickness before rolling, h' is the thickness after rolling, Δh is the redaction, r is the reduction rate

Table 3
Comparison of simulation and calculation

Table 4
Time and standard deviation as the number of different increases