Numerical Simulation as a Tool for the Study, Development, and Optimization of Rolling Processes: A Review

Abstract

Rolling is one of the most important processes in the metallurgical industry due to its versatility. Despite its inherent advantages, design and manufacturing by rolling still rely on trial-and-error-based optimizations, which reduces its efficiency. To minimize the cost and time spent on the development of new rolling schedules, various analytical and numerical methods have been used in recent years. Among other alternatives, simulations based on the finite element method (FEM) are the most widely used. This allows for the analysis of the feasibility of new rolling schedules considering metal alloys with different characteristics, process conditions, or the creation of new operations, as well as the optimization of existing ones. This paper presents a literature review including the latest developments in the field of numerical simulation of rolling processes, which have been classified according to the type of rolling into the following categories: flat rolling, shape rolling, ring rolling, cross-wedge rolling, skew rolling, and tube piercing.

1. Introduction

Rolling is a metal-forming process in which products are obtained by subjecting the metal material to high compressive forces as it passes through a series of rolls [1]. In this process, the material undergoes severe plastic deformation [2], either symmetrically or asymmetrically [3], at different temperatures [4]. Consequently, parts subjected to rolling experience a change in cross-section, affecting their shape and dimensions and modifying their mechanical properties and microstructure [5]. Examples of changes observed in the rolling process involve hardening due to dislocations, recovery, static and/or dynamic recrystallization, and phase transformation [6].

The term rolling encompasses a wide number of processes that can be classified according to different criteria. Depending on the working temperature, rolling processes can be classified into hot and cold rolling [6]. Hot rolling allows processing large pieces and increasing the ductility and toughness of different alloys. On the other hand, cold rolling enables greater dimensional control, smoother surfaces, and higher strength. The choice between hot or cold rolling, or a combination of both, depends on the specific application [7]. Other lesser-used processes are warm [8] and cryogenic rolling [9].

Rolling processes can also be classified according to the shape of the final products. Using this criterion, flat, section, and hollow rolling processes can be found. These processes allow us to obtain plates, sheets, strips, foils, rails, beams, channels, angles, bars, rods, wires, and seamless pipe geometries, among others [10].

Other criteria are the rolling direction, which differentiates between lengthwise, transverse, and skew rolling; and the mode of rolling mill operation, either continuous or reverse rolling [11].

The versatility of rolling and the characteristics of its products make this process one of the most important in the metallurgical industry. It is estimated that at least 95% of both ferrous and non-ferrous alloys have been processed at some stage of their production life by rolling [10,11]. Despite the inherent advantages and efficiency of the rolling process, the design and manufacture of new products through rolling still depend on trial-and-error-based optimizations, which reduces its efficiency. This is particularly pronounced in hot rolling due to the complexity of the involved thermomechanical metallurgical phenomena. Additionally, during rolling, the formed parts are susceptible to unforeseen deformations that complicate optimization, such as springback, warpage, buckling, and cracking induced by residual stresses [6,12].

To reduce costs and time spent on the development of new rolling schedules, various analytical and numerical methods have been used [13,14]. They allow the analysis of modifications of the conditions of a rolling schedule and the behavior of the material, as well as the creation of new operations and optimization of the existing ones. They also contribute to increasing the speed of existing production processes and the quality of products [15]. Some of them include the slab method, the slip-line field method, the upper bound method, the boundary element method, and the finite element method (FEM) [16]. Compared to other methods, FEM simulations are the most widely used due to their accuracy and capacity to deal with the increasing complexity of the process and the diversity of shapes [17]. For this type of calculation, there are different commercial programs available, for example, ABAQUS [18], ANSYS [19], DEFORM [20], and MSC Marc [21].

The interest in FEM-based simulation has grown significantly in recent years. For example, a Scopus search conducted on 5 June 2024 using the keywords “simulation rolling process” and limiting the search to articles published in the last thirty years for the fields of engineering and materials science returns a total of 3666 results, Figure 1. In this figure, it can be observed that there is a continuous increase in the number of publications over the years. From 2012 onwards, there is a significant increase. The highest peak in publications occurred in 2023, with a total of 345 articles published based on the specified search criteria. At the time of the search, a total of 192 articles have been published so far in 2024.

Table 1 includes a compilation of the ten most relevant papers from each of the last three years obtained from a bibliographic search following the criteria indicated above. In this table, the articles have been ordered, firstly, by year of publication and, secondly, by decreasing number of citations, the first being the most cited at the date of consultation. For each of these papers, classification has been made according to the rolling temperature, the rolling process, the material employed, and the numerical method applied.

Table 1. Classification of selected papers based on year of publication, number of citations, rolling temperature, rolling process, material studied, and simulation method utilized.

Refs.	Year	Citations	Temperature	Rolling Process	Material	Numerical Method
[22]	2022	15	Hot rolling	Cross-wedge rolling	AISI 1045	FEM
[23]	2022	14	Hot rolling	Ring rolling	AA2219	FEM
[24]	2022	13	Hot rolling	Ring rolling	TC4 Ti alloy	FEM
[25]	2022	9	Hot rolling	Ring rolling	IN718	FEM
[26]	2022	9	Hot rolling	Ring rolling	AA1050	FEM
[27]	2022	9	Hot rolling	Skew rolling	LZ50 steel	FEM
[28]	2022	8	Hot rolling	Flat rolling	AZ80 Mg alloy	FEM
[29]	2022	7	Hot rolling	Combined rolling-extrusion	Al-Zr-Hf alloy	FEM
[30]	2022	7	Hot rolling	Flat rolling	AISI 1045	FEM
[31]	2022	7	Hot rolling	Flat rolling	AA2XXX; AA7XXX	FEM
[32]	2023	10	Cold rolling	Corrugated roll bonding	T2 Cu/AA1060 clad plate	FEM
[33]	2023	9	Hot rolling	Flat rolling	AA7A04	FEM
[34]	2023	5	Hot rolling	Ring rolling	Ti-6Al-4V alloy	FEM
[35]	2023	5	Hot rolling	Flat rolling	Commercially pure Ti/ Q345 steel clad plate	FEM
[36]	2023	4	Hot rolling	Shape rolling	SAE 52100	FEM
[37]	2023	3	Cold rolling	Equal-channel angular rolling	AA6061	FEM
[38]	2023	3	Cold rolling	Equal-channel angular rolling	Commercially pure Cu	FEM
[39]	2023	3	Hot rolling	Flat rolling	AZ31 Mg alloy	FEM
[40]	2023	3	Hot rolling	Flat rolling	Commercially pure Mo	FEM
[41]	2023	3	Cold rolling	Flat rolling	Q235 steel	FEM
[42]	2024	6	Hot rolling	Ring rolling	GH738 Ni-based superalloy	FEM
[43]	2024	4	Warm and hot rolling	Asymmetrical rolling	Ti-6Al-4V alloy	FEM
[44]	2024	1	Hot rolling	Ring rolling	42CrMo4 steel	FEM
[45]	2024	1	Hot rolling	Flat rolling	Commercially pure Al/ commercially pure Mg clad plate	FEM
[46]	2024	1	Cold rolling	Hollow embossing rolling	AISI 316L	FEM
[47]	2024	1	Hot rolling	Skew rolling	100Cr6 steel	FEM
[48]	2024	1	Hot rolling	Flat rolling	Q235 steel/SS 1Cr13 clad plate	FEM
[49]	2024	1	Hot rolling	Shear rolling	Al-3Ca-2La-1Mn Al alloy	FEM
[50]	2024	0	Hot rolling	Ring rolling	AA2219	FEM
[51]	2024	0	Hot rolling	Flat rolling	701, 705 and 706 Al alloys	FEM

Table 1. Classification of selected papers based on year of publication, number of citations, rolling temperature, rolling process, material studied, and simulation method utilized.

Refs.	Year	Citations	Temperature	Rolling Process	Material	Numerical Method
[22]	2022	15	Hot rolling	Cross-wedge rolling	AISI 1045	FEM
[23]	2022	14	Hot rolling	Ring rolling	AA2219	FEM
[24]	2022	13	Hot rolling	Ring rolling	TC4 Ti alloy	FEM
[25]	2022	9	Hot rolling	Ring rolling	IN718	FEM
[26]	2022	9	Hot rolling	Ring rolling	AA1050	FEM
[27]	2022	9	Hot rolling	Skew rolling	LZ50 steel	FEM
[28]	2022	8	Hot rolling	Flat rolling	AZ80 Mg alloy	FEM
[29]	2022	7	Hot rolling	Combined rolling-extrusion	Al-Zr-Hf alloy	FEM
[30]	2022	7	Hot rolling	Flat rolling	AISI 1045	FEM
[31]	2022	7	Hot rolling	Flat rolling	AA2XXX; AA7XXX	FEM
[32]	2023	10	Cold rolling	Corrugated roll bonding	T2 Cu/AA1060 clad plate	FEM
[33]	2023	9	Hot rolling	Flat rolling	AA7A04	FEM
[34]	2023	5	Hot rolling	Ring rolling	Ti-6Al-4V alloy	FEM
[35]	2023	5	Hot rolling	Flat rolling	Commercially pure Ti/ Q345 steel clad plate	FEM
[36]	2023	4	Hot rolling	Shape rolling	SAE 52100	FEM
[37]	2023	3	Cold rolling	Equal-channel angular rolling	AA6061	FEM
[38]	2023	3	Cold rolling	Equal-channel angular rolling	Commercially pure Cu	FEM
[39]	2023	3	Hot rolling	Flat rolling	AZ31 Mg alloy	FEM
[40]	2023	3	Hot rolling	Flat rolling	Commercially pure Mo	FEM
[41]	2023	3	Cold rolling	Flat rolling	Q235 steel	FEM
[42]	2024	6	Hot rolling	Ring rolling	GH738 Ni-based superalloy	FEM
[43]	2024	4	Warm and hot rolling	Asymmetrical rolling	Ti-6Al-4V alloy	FEM
[44]	2024	1	Hot rolling	Ring rolling	42CrMo4 steel	FEM
[45]	2024	1	Hot rolling	Flat rolling	Commercially pure Al/ commercially pure Mg clad plate	FEM
[46]	2024	1	Cold rolling	Hollow embossing rolling	AISI 316L	FEM
[47]	2024	1	Hot rolling	Skew rolling	100Cr6 steel	FEM
[48]	2024	1	Hot rolling	Flat rolling	Q235 steel/SS 1Cr13 clad plate	FEM
[49]	2024	1	Hot rolling	Shear rolling	Al-3Ca-2La-1Mn Al alloy	FEM
[50]	2024	0	Hot rolling	Ring rolling	AA2219	FEM
[51]	2024	0	Hot rolling	Flat rolling	701, 705 and 706 Al alloys	FEM

Table 1 shows that in twenty-five of the thirty publications selected, different hot rolling processes have been studied, which represents more than 80% of the articles included in the table. Hot rolling is a metal manufacturing process extensively used to obtain finished and semi-finished products. Use of this type of rolling often involves thermomechanical complexities [6] due to phenomena such as viscoplastic deformation, recrystallization, and recovery [52]. Given that these are complex processes widely used in the metallurgical industry, there is a great interest in their study using numerical simulation.

Cold rolling has been investigated in five of the articles included in Table 1. Although thermomechanical complexities do not occur in cold rolling, other challenges, such as the lack of flatness of the rolled products [41] or springback [46], suffered by the materials may occur. The prediction of these phenomena can be a challenge in order to optimize the manufacturing process, which justifies the presence of papers focused on cold rolling.

As regards warm rolling, it has only been dealt with in one article in conjunction with hot rolling.

In terms of the type of manufacturing process, ten different types of rolling can be identified in Table 1. The most studied process was flat rolling, found in eleven of the thirty articles, which represents 36% of the selected papers. This is consistent with the fact that according to [53], about 40–60% of rolled products are manufactured by flat rolling. Among the eleven articles on flat rolling, ten of them have been performed for hot rolling conditions, whereas only one has focused on cold rolling.

According to Table 1, the second most studied rolling process is ring rolling, as eight of the thirty articles focus on the study of this process, which corresponds to 27%. The interest in the study of this process is based on its complexity when performing the rolling in a stable way, mainly in the manufacture of large rings [23], the variety of geometries that can be adopted by ring sections, and the intrinsic complexity of hot rolling processes, which is the method used in all the ring rolling processes reported.

The remaining eleven articles included in Table 1 are focused on the study of other rolling processes. Thus, two articles are concerned with the study of skew rolling [27,47], one about shape rolling [36], and one focused on the study of cross-wedge rolling [22]. Other papers are related to the study of other processes such as hot shear rolling [49], cold corrugated rolling bonding [32], cold hollow embossing rolling [46], asymmetrical rolling [43], as well as hybrid processes such as combined hot rolling-extrusion [29] and cold equal-channel angular rolling [37,38].

In terms of the materials studied, aluminum alloys and steels are the most tested materials, being present in eleven and ten of the articles included in Table 1, respectively. In other studies, simulations of rolling processes have been carried out for titanium [24,34,35,43], copper [32,38], magnesium [28,39,45], and molybdenum [40] alloys and nickel-based superalloys [25,42]. In four papers, clad plates consisting of two different materials [32,35,45,48] have been studied.

Finally, with regard to the simulation methods used, Table 1 shows that in recent years the use of finite elements has become the simulation method of choice, being the method used in all the papers included in Table 1.

In this article, a review of the literature found using the search criteria mentioned above is presented. Selected articles are analyzed in Section 2 and classified according to the type of rolling, as follows: flat rolling, shape rolling, ring rolling, cross-wedge rolling, skew rolling, and tube piercing.

In addition to the articles published in the last three years included in Table 1, others published previously have been included because of their relevance or for containing data that contribute to an overview of the state of the art in the field.

2. Classification Based on Type of Rolling Process

In this section, using the type of rolling as criteria, the selected papers have been classified into flat rolling, shape rolling, ring rolling, cross-wedge rolling, skew rolling, and tube piercing. Papers have been briefly discussed, and the main rolling parameters, as well as the most relevant details of the numerical method used, have been provided. The review was aimed at highlighting the benefits obtained from the use of numerical simulation as a tool for the study and improvement of rolling processes.

2.1. Flat Rolling

Flat rolling is a metal-forming process used in the manufacturing of flat products with a rectangular cross-section. In this process, the material is fed by two cylindrical rolls rotating in opposite directions, which are known as working rolls [54]. Depending on the type of rolling mill used, backup rolls may be involved in the process. Figure 2 shows a schematic representation of flat rolling.

Flat rolling is one of the most important types of rolling processes. It is estimated that about 40–60% of rolled products are manufactured with this process. Due to its relevance, there are many publications that try to identify how rolling parameters affect product quality and optimize the process [16].

Phaniraj et al. [55,56] simulated the hot rolling process of flat carbon steel products using DEFORM, a FEM-based software. Eighteen sheets with thicknesses ranging from 2.0 to 4.0 mm of low-carbon steels were employed. The recrystallization and grain growth phenomena were modeled using semi-empirical equations taken from the literature. By comparing the predictions of their model with the literature data, Phaniraj et al. concluded that during the hot rolling process of this type of steel, the austenite-ferrite transformation occurs. On the other hand, the simulations performed allowed to predict the rolling loads with an error of ±15% and the material temperatures with an error of ±15 °C.

Sun et al. [57] developed a FEM model to study the thermal and mechanical behavior of 304 stainless steel during the hot rolling of flat products. The effect of the rolling parameters on the appearance of the edge seam defect was studied, as well as the conditions that reduce or eliminate this type of defect. The behavior of the material in a non-stationary state using a rigid-viscoplastic model was analyzed. Experimental tests were conducted using a pilot plant, and the results were compared with the predictions obtained by simulation. From this comparison, the authors demonstrated and validated the ability of the model to simulate the studied process. Subsequently, four hot rolling cases of four passes each were analyzed using preforms of 1260 mm initial length and thicknesses of 200–201 mm. For the first case, a furnace discharge temperature of 1257 °C was used, while for the remaining three cases this temperature was 1262 °C. The geometrical deformation suffered by the sheet, the edge seam defects, the temperature distribution in the thickness direction, and the temperature of the sheet at the head, center, and tail throughout the process were analyzed. It was demonstrated that the developed numerical model was effective for analyzing the appearance of the defect known as an edge seam defect. The reduction of this type of defect by reducing the rolling speed of the upper roll and the temperature difference between the upper and lower surfaces of the sheet was also proved.

Yu et al. [58] investigated the behavior of spherical Al2O3 inclusions in 304 stainless steel strips during multi-pass cold rolling using 3D FEM simulations in LS-DYNA. The study was focused on analyzing the deformation of inclusions under different sizes and positions of inclusions. In addition, the relationship between the deformation of inclusions and crack generation was analyzed. The results are shown in Figure 3. According to the obtained results, the increase in size of the inclusion and its proximity to the surface led to an increase in the deformation of the inclusion (Figure 3a,b). It was also observed that the deformation in the front of the inclusion was greater than in the rear of the inclusion, suggesting the possibility of fatigue cracks.

Robert-Núñez et al. [59] simulated in ABAQUS the cold rolling of aluminum alloy 6063 plates of 100 mm length, 9 mm thickness, and widths of 10 and 30 mm to acquire a better understanding of the influence of the process variables on the stress and strain distributions. From the results obtained, authors observed a hardening on the surface of the plates and a heterogeneous deformation of the material.

In [60], Sherstnev et al. investigated the kinematics of precipitate formation in flat 5083 aluminum alloy products obtained by hot rolling. Their research aimed to integrate precipitation kinetics with microstructure models. To achieve this, authors simulated the rolling process of a 100 mm × 100 mm × 17 mm preform using FORGE 2008 and calculated the kinetics of the precipitate formation process using the thermodynamic simulation program MatCalc. The developed model enabled the prediction of dislocation density, crystalline structure during hot rolling, and the volume fraction of recrystallized material after rolling. Figure 4 represents the results obtained in the roll gap. In the laminated part, the dislocation density, Figure 4a, decreases from the surface towards the center, while the sub-grain size, Figure 4b, increases from the surface towards the center in response to the temperature and strain rate distribution. To validate the simulation, a series of tests were conducted using a laboratory-scale rolling mill. The results indicated that the developed model accurately described the evolution of the microstructure of the material during hot rolling.

Nalawade et al. [61] used the commercial software FORGE to conduct three-dimensional FEM simulations to understand the impact of rolling parameters on the deformation behavior of a 38MnVS6 micro-alloyed steel bloom. The simulations predicted various aspects, including rolling load, torque, temperature distribution, material flow, microstructural phase constitution, and grain size distribution. Figure 5 shows the simulated and experimental results for torque, rolling load, and surface temperature. As can be seen in Figure 5a,b, comparisons between predicted and experimental values for torque and load revealed good agreement. Similarly, the predicted phase constitution matched the experimentally determined microstructure, indicating that the developed deformation model effectively predicted the behavior of the material during the hot rolling process. However, some variations in surface temperature values between the predicted and experimental results were observed (Figure 5c). Nalawade et al. attributed this discrepancy to variations in emissivity caused by scale formation.

Tamimi et al. [62] performed numerical simulations with the FEM software ABAQUS/Explicit of asymmetric cold rolling (ASR) of aluminum alloy 5182. Two-pass simulations were performed at room temperature to investigate the impact of processing parameters on the initiation and growth of shear deformation in the sheet thickness. Simulations allowed to define the optimum conditions of the ASR process. Finally, experimental tests were carried out, and it was found that the mechanical behavior and texture evolution predicted by the numerical models agreed with the experimental results. Thus, it was demonstrated that the shear deformation extended through the entire thickness of the sheet during ASR and was the cause of the shear texture.

Yu et al. [63] analyzed a cold flat rolling process using a 3D fast multipole boundary element method (FM-BEM). Numerical simulations were performed using a 2030 four-high mill for a fictitious elastoplastic strip material with a width-thickness ratio of 1850. The results showed that FM-BEM required fewer artificial assumptions and provided more accurate results in a shorter time than the traditional boundary element method (BEM), FEM, and finite difference method (FDM).

Pourabdollah and Serajzadeh [64] employed an upper-bound solution coupled with thermal FEM analysis to predict the thermomechanical behavior of an AISI 304L stainless steel strip under hot and warm rolling. A two-dimensional FEM model was used to forecast the temperature field inside the rolls. Additionally, an artificial neural network (ANN) analysis was applied to enhance result accuracy. Simulations were conducted under identical rolling conditions and temperatures of 1000 °C for hot rolling and 600 °C for warm rolling. The analysis of results on the work roll surface revealed an increase in the temperature of the deformation zone due to strip contact. For warm rolling, the maximum temperature of the work roll reached approximately 480 °C, whereas for hot rolling, it was about 640 °C. Water-spray cooling caused a significant temperature drop on the surface of the roll, although temperature variations beneath the surface exhibited a smoother profile compared to the surface region. Finally, comparison between measured and predicted values confirmed the accuracy of the simulations, showing good agreement with experimental results.

In [65], Nomoto et al. proposed a microstructure-based multiscale simulation framework to analyze the hot rolling of duplex stainless steels, employing various commercial simulation software. This paper established a method to link different simulation software for different length scales, ranging from nanometric to macroscopic. Initially, simulations of microstructure evolution were conducted using the Multi-Phase Field (MPF) method by MICRESS coupled with the Calculation of Phase Diagrams (CALPHAD) database by Thermo-Calc. The temperature distribution within the slab was calculated using FEM. Following this, macroscopic elastoplastic mechanical properties were determined via a virtual material test using ABAQUS/Explicit 6.14 and HOMAT software. Subsequently, the hot rolling of the slab was simulated by ABAQUS, using as input data the values of the mechanical properties obtained. Finally, the MPF method was applied via MICRESS to simulate static recrystallization within the slab. The MPF method was used to simulate microstructure evolution and estimate resulting mechanical properties. It was observed that there was still some room for improvement in the quantitative results.

Soulami et al. [66] generated a FEM model with LS-DYNA software to refine the hot rolling of uranium alloyed with 10 wt.% molybdenum (U-10Mo) foils encased in a metallic roll pack to prevent rolling-associated defects. Three main types of defects were considered: thickness non-uniformity, “dog-boning”, and waviness of the rolled sheet-pack. Validation of the model was achieved by comparing the separation force values after each pass using 1018 low-carbon steel, with an error not higher than 14%. Afterward, the occurrence of defects in the simulated models was analyzed. The following four cases were considered: rolling of a U-10Mo coupon inside a 1018 steel can, rolling of a U-10Mo coupon inside a 304 stainless steel can, rolling of a U-10Mo coupon inside a Zircaloy-2 can, and bare rolling of a U-10Mo coupon. The results are shown in Figure 6. When analyzing the “dog-boning” and waviness results, Figure 6a,b, the largest defects occurred when using 1018 steel can, whereas a significant reduction was observed when using Zircaloy-2 can. Additionally, bare-rolling simulations exhibited a defect-free rolled coupon. From these results, the authors concluded that the “dog-boning” defect arises due to the difference in yield strength between the can and the U-10Mo fuel coupon, while the waviness defect is a consequence of a sudden change in material resistance. Hence, this research inferred that reducing the strength mismatch between the coupon and materials can enhance the quality of the rolled sheet.

These authors supplemented the study [67] by analyzing initial temperatures ranging from 600 to 1000 °C with intervals of 100 °C. Simulated values of the temperature were compared with the experimental results, while the prediction of the effective strain and effective strain rate were analyzed. When analyzing the effective strain, it was observed that it was not homogeneously distributed in the thickness direction due to the geometry of the deformed zone and the effect of friction. However, the strain distribution was more uniform when using an initial temperature of 700 °C. According to the authors, this could be due to a lower temperature gradient along the thickness direction during rolling, resulting in less inhomogeneity in the strain field.

Faini et al. [68] investigated the impact of primary hot rolling parameters on the elimination of cavity defects in slabs of 316L stainless steel produced by continuous casting. FEM simulations of experimental tests were conducted to analyze the effect of the integral of stress triaxiality ratio, Q, and equivalent strain, ε_eq, on void closure. The numerical model was validated through experimental tests conducted in an industrial plant, utilizing samples measuring 280 mm × 340 mm × 1000 mm. These samples were rolled on both the short and long sides in a reversible duo rolling mill, with thickness reduction percentages of 14, 21, and 28% and an initial temperature of 1250 °C. Simulations were executed by replicating the conditions of each experimental test using DEFORM-3D v11 software. A correlation was established between the Q and ε_eq indices and the equivalent void diameter, suggesting that they were related to void crushing. Additionally, FEM simulations allowed to establish the relationship between process parameters and the values of Q and ε_eq. Thus, low Q values and high ε_eq values were observed for high reduction percentages and long cooling times. Because of this, the probability of the internal defect closure was increased.

Rout et al. [69] used DEFORM-3D to compute the variations in temperature, strain, and strain rate, as well as differences in microstructure, in small hot-rolled samples of austenitic 304LN stainless steel measuring 78 mm × 10 mm × 10 mm. Samples were rolled in one pass to a thickness of 5 mm at temperatures of 900, 1000, and 1100 °C. The results revealed that higher strain rate distributions and lower temperatures led to a partially recrystallized microstructure in the center of the rolled samples.

In [70], Mancini et al. analyzed the origins of edge defects occurring during the hot rolling of 1.4512 ferritic stainless steel flat bars. The objective of the study was to enhance the quality of finished products by reducing jagged border defects. For this purpose, thermomechanical and metallurgical models were integrated into the proprietary FEM software MSC Marc. These models were employed to examine defects in the final products at both macroscopic and microscopic scales. The results indicated that the defect stemmed from process conditions resulting in abnormal heating, leading to uncontrolled grain growth at the edges. These grains, which were work-hardened and elongated, did not undergo recrystallization during hot deformation. Consequently, they tended to displace the surrounding matrix toward the edges of the bar, resulting in fractures.

Kumar et al. [20] investigated the deformation in the roll bite during a plate rolling process of Nb-bearing micro-alloyed steel using DEFORM-3D. The Norton-Hoff constitutive equation was employed, with coefficients obtained through multivariable optimization techniques using experimental data from a dynamic thermomechanical simulator, Gleeble-3500. By inputting these data into the simulation software along with other process variables, the results of strain, stress, roll force, and temperature were obtained. The simulated roll force results were compared with values obtained experimentally using a load cell, and a good agreement was achieved. Finally, authors discussed the effect of temperature and friction coefficient on the stress distribution in the roll bite. Figure 7 illustrates the predicted results of the stress distribution for temperatures of 1150 °C, Figure 7a, and 1250 °C, Figure 7b. It was observed that the peak stress decreased with increasing temperature at a rate of 0.163 MPa·°C⁻¹. Kumar et al. attributed this decrease to the reduction in flow stress with rising temperature. Additionally, the stress distribution appeared similar for both studied temperatures. The effect of the friction coefficient on the stress distribution was compared for values of 0.45 and 0.80. Although no significant differences in peak values were observed, variations in the stress distribution were detected, presumably due to the contribution of the friction component.

Chen et al. [28] studied the deformation behavior during hot rolling of AZ80 magnesium alloy plates subjected to ultrasonic processing during casting. Numerical 3D FEM simulations were carried out using the DEFORM-3D software. Preforms with a length of 200 mm, width of 120 mm, and thickness of 13 mm were used, which were rolled at 300 and 400 °C. The simulations were repeated for preforms obtained with and without ultrasonic processing. Simulation results revealed that AZ80 samples obtained by ultrasonic processing required less effective stress and had less damage during rolling.

Gravier et al. [31] studied the effect of rolling parameters on the evolution of pore volume on aluminum samples processed by hot rolling, since the pore volume affects the properties of the final product. For this purpose, the authors proposed to use in situ mechanical tests, characterized by X-ray microtomography, to obtain experimental data on the actual pore volume evolution under a representative hot rolling deformation. To ensure that the load states accessible in the uniaxial tests are representative of the load states during rolling, they performed finite element simulations of both industrial rolling at meter scale and uniaxial tests at millimeter scale. The simulations of the rolling process were carried out using LAM3 software, which considers the thermomechanical phenomena that take place during rolling and uses a stationary Eulerian formulation. On the other hand, simulations of the uniaxial tests were carried out using LS-DYNA R9.1.0 software. The results obtained indicate that although the proposed method allows studying the evolution of pore closure, the conclusions obtained are limited to its evolution during hot rolling. Nevertheless, the method could be used to study the kinetics of pore closure or opening or to study the evolution of the morphology during hot rolling.

Zhou et al. [71] investigated the temperature and equivalent strain distribution of SUS436L stainless steel slabs using a two-dimensional FEM model. The study considered different surface temperatures for finish rolling, with slabs initially 90 mm thick and surface temperatures of 800, 850, and 900 °C. After three rolling passes, a thickness of 5 mm was achieved. To validate the simulations, the maximum rolling force determined on each simulated pass was compared with experimental values. Despite measured values being slightly higher than predicted ones, no significant differences were observed, thus confirming the validity of the model. The analysis of the results revealed that the reduction of the surface temperature improved deformation permeability and uniformity due to the increased temperature difference between the surface and the center of the strip.

Sun et al. [33] used DEFORM-3D software to study the influence of deformation parameters on the uniformity of equivalent strain distribution in as-cast 7A04 aluminum alloy after being processed by hot rolling. The microstructure of the deformed samples was analyzed by optical microscopy, scanning electron microscopy with energy dispersive spectroscopy, X-ray diffraction, and microhardness. The authors performed FEM simulations at temperatures ranging from 330 to 480 °C, using 20 × 30 × 10 mm samples. The results obtained by FEM simulations indicated the existence of a certain influence of the deformation temperature on the equivalent strain distribution. However, this influence disappears when working at temperatures above 380 °C.

Wang et al. [39] developed a multiscale coupled dislocation density model to predict the microstructure of rolled sheets of AZ31 magnesium alloy. The dislocation density model was inserted into a subroutine of the ABAQUS program, and the effect of temperature and rolling speed on the dislocation density and volume fraction of dynamic recrystallization was investigated by FEM. Simulation results revealed that the dislocation density increased rapidly in the first rolling pass but decreased significantly with time. According to the authors, the increase in dislocation density was influenced by work hardening, dynamic recovery, and dynamic recrystallization. As for dynamic recrystallization, the highest values belonged to the surface of the plate, while the lowest values corresponded to the center of the plate. This was due to the difference in temperature distribution, strain, and strain rate. Furthermore, it was determined that the rolling strength decreased with temperature and increased with rolling rate. Results of the rolling force were verified with experimental data, and a good agreement was achieved with relative errors between 10 and 15%.

Han et al. [40] simulated unidirectional and hot cross rolling operations of commercially pure molybdenum plates in order to predict and analyze the temperature, stress, and strain distributions. Using MSC Marc software, an elastoplastic FEM model was established with the updated Lagrange method. Rolling of plates of 100 mm length, 50 mm width, and 13.2 mm thickness were simulated. When analyzing the results, it was observed that the distribution of the temperature was non-uniform due to the joint action of surface cooling caused by the contact with the rollers, the generation of heat inside the material by plastic deformation, and the surface reheating after rolling. In addition, it was concluded that the non-uniformity of the stress and strain fields was due to the joint influence of rolling stress, contact friction, and external resistance. By comparing the results of the simulations with the experimental data, the authors found that the numerical model was well aligned with the actual process.

Wang et al. [41] investigated the causes of an atypical type of defect called inclined wave defects, which appear during cold rolling of strips. For this purpose, numerical simulations were carried out using a 3D elastoplastic FEM model. The simulations were performed using ABAQUS/Standard, and the distribution of deformations and stresses in the three-stand, two-roll cold rolling process was analyzed. The results showed that the obtained load distributions were consistent with the conditions for the generation of inclined wave defects and were used for the suppression of inclined wave defects in strips. According to the authors, the proposed methodology provided theoretical support for the establishment of inclined wave control strategies.

On the other hand, in the bibliography it is possible to find works that study the process called flat cross rolling, which consists of a variant of the conventional flat rolling process in which samples are rotated 90° in the rolling plane after each pass, interchanging width and length. Thus, in [17], the behavior of AISI 304 stainless steel plates was studied, while in [72], an attempt was made to predict the edge profile of 304 stainless steel plates.

In addition, simulation has also been used to study the behavior of clad plates under rolling. Thus, in [73] the phenomenon of plastic instability in the cold rolling of clad plates of different flow stresses was studied, in [74] the effect of hot rolling on the microstructure and properties of 2205/Q235 clad plates was studied, in [75] analyzed the microstructure evolution and mechanical behavior of Mg/Al sheets manufactured by a new corrugated rolling process, in [76] the vacuum hot rolling of 2205/NI/EH40 clad plates was simulated, in [77] the deformation mechanism and microstructure evolution of 316L/Q235B/316L clad plates manufactured by hot corrugated rolling were studied, in [32] analyzed the deformation behavior and bonding properties of Cu/Al coated plates fabricated by cold corrugated rolling, in [45] the upper bound method was applied to the modeling of the asymmetric rolling of double layered Al/Mg clad plate, in [48] the layer thickness and strain of Q235/1Cr13 clad plates manufactured by rolling with different roll diameters were modeled, while in [51] the hot rolling of 7000 series aluminum alloy clad sheets was simulated.

In addition to the rolling operations, in the literature it is possible to find other papers in which both pre- and post-rolling operations are simulated. Thus, in [78] the cold charge rolling and hot charge rolling processes are simulated; in [79] the cooling stage of slabs in run-out table; in [80] investigated the phase transformation behavior of a steel during the coil cooling process after hot rolling.; in [21] the effect of the descaling stage; in [81] the behavior of the scale.

Furthermore, research focused on the study of rolling mill rolls by means of numerical simulations has been carried out. For example, in [82,83,84] the temperature in the rolls of the rolls was studied; in [85] the abrasive wear of the work rolls; in [86] the mechanical behavior of the backup rolls; while in [87] it was investigated the flatness control ability of a 6-high continuous variable crown control rolling mill.

As shown in Table 2, steel is the main alloy simulated, being studied in eleven of the twenty-four papers considered, while aluminum is present in five of them. Uranium alloy [66], magnesium alloy [28,39], and commercially pure molybdenum [40] have been considered in the other articles. Hot flat rolling is the process to which most attention is paid, as it has been simulated in nineteen of the twenty-five articles considered, varying the working temperature between 300 and 1460 °C, depending on the material studied. In second place are the cold rolling processes, studied in five articles, and, finally, there are the warm rolling processes, which are studied only in [64], working with stainless steel at a temperature of 600 °C. As far as the number of simulated passes is concerned, multi-pass simulations of between 2 and 15 passes have been studied in half of the articles discussed. Regarding the workpiece size, the lengths ranged from 30 to 4000 mm, the width from 10 to 1850 mm, and the thickness from 1.2 to 500 mm. The reductions applied to the workpieces range from 5 to 80% depending on the desired product characteristics. The dimensions of the work rolls vary, ranging from 65 to 1095.1 mm; rolling speeds vary between 50.5 and 9860 mm·s⁻¹; and, finally, friction coefficients range from 0.06 to 1.

Table 2. Main rolling parameters used in the numerical simulations of the flat rolling process.

Refs.	Material	Passes	Condition	Workpiece Temp. [°C]	Workpiece Size [mm]	Reduction [%]	ø Roll [mm]	Speed [mm·s−1]	Friction Coef.
[55]	0.34%C steel	6	HR	N/S	N/S × N/S × 28	N/S	627.5	1300–9860	0.25–0.5
[56]	0.34%C steel	6	HR	N/S	N/S × N/S × 28	N/S	627.5	1300–9860	0.25–0.5
[57]	AISI 304	4	HR	1257–1262	N/S × N/S × 201	8.2–10.9	100; 1100	2430–2600	N/S
[58]	AISI 304	3	CR	RT	N/S × 30 × 3	N/S	400	N/S	0.15
[59]	AA6063	1	CR	RT	100 × 10; 30 × 9	80	65	63	N/S
[60]	AA5083	1	HR	550	100 × 100 × 17	17.6	250	65.4	0.35
[61]	38MnVS6 steel	8	HR	1235	3480 × 400 × 320	5–13	925	2905.8	0.5
[62]	AA5182	3	CR	RT	60 × N/S × 1.2	30; 50	180	18.9–75.8	0.1; 0.4
[63]	N/S	N/S	CR	RT	N/S × 1850 × 1.25	20	600	N/S	0.1
[64]	AISI 304L	1	HR; WR	600; 1000	140 × 40 × 4	25; 40	150	394.7	0.3; 0.8
[65]	AISI 304	1	HR	1460	50 × 40 × 20	40	160	300	1.0
[66]	U-10Mo	15	HR	591–650	48.5 × 37.7 × 9.4	5–10	254	133	0.35
[68]	AISI 316L	1	HR	1250	1000 × 340 × 280	14–28	980; 985	4951.6	0.7
[69]	AISI 304	1	HR	900–1100	78 × 10 × 10	50	320	50.5	0.7
[70]	Steel 1.451	5	HR	N/S	N/S	N/S	N/S	N/S	N/S
[20]	AISI 1015	14	HR	859–1250	3196 × N/S × 220	N/S	1095.1	2819.8	0.45–0.8
[28]	AZ80 Mg alloy	1	HR	300; 400	200 × 120 × 13	40	320	N/S	0.39; 0.53
			HR
[31]	AA2XXX; AA7XXX	1	HR	N/S	4000 × 1500 × 500	N/S	N/S	N/S	N/S
[71]	AISI 436L	7	HR	800–1200	140 × N/S × 90	24.2–40.0	450	3000	0.35
[33]	AA7A04	3	HR	330–480	30 × 20 × 10	20–60	N/S	N/S	0.5
[39]	AZ31 Mg alloy	1	HR	300–500	150 × 40 × 5.6	35	200	174–367	0.3
[40]	CP Mo	2	HR	1260–1350	100 × 50 × 13.2	5–30	400	520	0.3
[41]	Q235	3	CR	RT	2000 × 1200 × 3	30–36.7	440	420–700	0.06–0.08

N/S: not specified; CP: commercially pure; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Table 2. Main rolling parameters used in the numerical simulations of the flat rolling process.

Refs.	Material	Passes	Condition	Workpiece Temp. [°C]	Workpiece Size [mm]	Reduction [%]	ø Roll [mm]	Speed [mm·s−1]	Friction Coef.
[55]	0.34%C steel	6	HR	N/S	N/S × N/S × 28	N/S	627.5	1300–9860	0.25–0.5
[56]	0.34%C steel	6	HR	N/S	N/S × N/S × 28	N/S	627.5	1300–9860	0.25–0.5
[57]	AISI 304	4	HR	1257–1262	N/S × N/S × 201	8.2–10.9	100; 1100	2430–2600	N/S
[58]	AISI 304	3	CR	RT	N/S × 30 × 3	N/S	400	N/S	0.15
[59]	AA6063	1	CR	RT	100 × 10; 30 × 9	80	65	63	N/S
[60]	AA5083	1	HR	550	100 × 100 × 17	17.6	250	65.4	0.35
[61]	38MnVS6 steel	8	HR	1235	3480 × 400 × 320	5–13	925	2905.8	0.5
[62]	AA5182	3	CR	RT	60 × N/S × 1.2	30; 50	180	18.9–75.8	0.1; 0.4
[63]	N/S	N/S	CR	RT	N/S × 1850 × 1.25	20	600	N/S	0.1
[64]	AISI 304L	1	HR; WR	600; 1000	140 × 40 × 4	25; 40	150	394.7	0.3; 0.8
[65]	AISI 304	1	HR	1460	50 × 40 × 20	40	160	300	1.0
[66]	U-10Mo	15	HR	591–650	48.5 × 37.7 × 9.4	5–10	254	133	0.35
[68]	AISI 316L	1	HR	1250	1000 × 340 × 280	14–28	980; 985	4951.6	0.7
[69]	AISI 304	1	HR	900–1100	78 × 10 × 10	50	320	50.5	0.7
[70]	Steel 1.451	5	HR	N/S	N/S	N/S	N/S	N/S	N/S
[20]	AISI 1015	14	HR	859–1250	3196 × N/S × 220	N/S	1095.1	2819.8	0.45–0.8
[28]	AZ80 Mg alloy	1	HR	300; 400	200 × 120 × 13	40	320	N/S	0.39; 0.53
			HR
[31]	AA2XXX; AA7XXX	1	HR	N/S	4000 × 1500 × 500	N/S	N/S	N/S	N/S
[71]	AISI 436L	7	HR	800–1200	140 × N/S × 90	24.2–40.0	450	3000	0.35
[33]	AA7A04	3	HR	330–480	30 × 20 × 10	20–60	N/S	N/S	0.5
[39]	AZ31 Mg alloy	1	HR	300–500	150 × 40 × 5.6	35	200	174–367	0.3
[40]	CP Mo	2	HR	1260–1350	100 × 50 × 13.2	5–30	400	520	0.3
[41]	Q235	3	CR	RT	2000 × 1200 × 3	30–36.7	440	420–700	0.06–0.08

N/S: not specified; CP: commercially pure; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Concerning the main characteristics of the numerical model included in Table 3, in fifteen articles a 3D FEM model has been used, whereas in four articles a 2D discretization has been used. Only in [63,64] a fast multipole boundary element method and an upper-bound finite element solution have been used, respectively. For the resolution of the model, an explicit method was used in [62,65,66], whereas in the rest of the studies the method used was not indicated. Regarding the definition of the model, in [31] a Eulerian definition was used, while in the rest of the articles consulted a Lagrangian definition was used. In most of the studies included in Table 3, a transient regime simulation was carried out, whereas in [31,55,56,64] a steady-state simulation was chosen. As far as the type of analysis is concerned, thermomechanical analysis is predominant, with mechanical analysis having been performed in [41,58,59,63]. The programs used are varied, finding studies in which ABAQUS, DEFORM, FORGE, MSC Marc, LS-DYNA, and LAM3 have been utilized. In terms of mesh, the use of quadrangular elements predominates in the two-dimensional simulations, as well as hexahedral elements in the three-dimensional ones. Only one study has been reported in which tetrahedral elements were applied [68]. With respect to mesh size and finite element size, not much information has been provided in the studies included in Table 3, with meshes ranging from 4000 to 100,000 elements and 0.5 mm elements in [66].

Table 3. Main characteristics of the numerical models applied in the numerical simulations of the flat rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[55]	FEM	N/S	La	2D	St	Th-Me	DEFORM	Quad	N/S	N/S
[56]	FEM	N/S	La	2D	St	Th-Me	DEFORM	Quad	N/S	N/S
[57]	FEM	N/S	La	N/S	Tr	Th-Me	N/S	N/S	N/S	N/S
[58]	FEM	N/S	La	3D	Tr	Me	LS-DYNA	Hexa	43,520	N/S
[59]	FEM	N/S	La	2D	Tr	Me	ABAQUS	Quad	4000	N/S
[60]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	N/S	N/S
[61]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	83,973	N/S
[62]	FEM	Ex	La	N/S	N/S	N/S	ABAQUS	N/S	N/S	N/S
[63]	FM-BEM	N/S	La	3D	Tr	Me	N/S	N/S	N/S	N/S
[64]	UBFES	N/S	N/S	2D	St	Th-Me	N/S	Quad	N/S	N/S
[65]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[66]	FEM	Ex	La	3D	Tr	Th-Me	LS-DYNA	Hexa	69,649	0.5
[68]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	50,000	N/S
[69]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Hexa	19,000	N/S
[70]	FEM	N/S	La	3D	Tr	Th-Me	MSC Marc	N/S	N/S	N/S
[20]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	32,000	N/S
[28]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	1,000,000	N/S
[31]	FEM	N/S	Eu	3D	St	Th-Me	LAM3	Hexa	N/S	N/S
[71]	FEM	N/S	La	2D	Tr	Th-Me	N/S	Quad	N/S	N/S
[33]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	N/S	N/S
[39]	FEM	N/S	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[40]	FEM	N/S	La	3D	Tr	Th-Me	MSC Marc	Hexa	19,200	N/S
[41]	FEM	N/S	La	3D	Tr	Me	ABAQUS	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; FM-BEM: fast multipole boundary element method; UBFES: upper-bound finite element solution; Ex: explicit; Im: implicit; La: Lagrangian; Eu: Eulerian; Tr: transient; St: steady-state; Me: mechanical; Th-Me: thermomechanical.

Table 3. Main characteristics of the numerical models applied in the numerical simulations of the flat rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[55]	FEM	N/S	La	2D	St	Th-Me	DEFORM	Quad	N/S	N/S
[56]	FEM	N/S	La	2D	St	Th-Me	DEFORM	Quad	N/S	N/S
[57]	FEM	N/S	La	N/S	Tr	Th-Me	N/S	N/S	N/S	N/S
[58]	FEM	N/S	La	3D	Tr	Me	LS-DYNA	Hexa	43,520	N/S
[59]	FEM	N/S	La	2D	Tr	Me	ABAQUS	Quad	4000	N/S
[60]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	N/S	N/S
[61]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	83,973	N/S
[62]	FEM	Ex	La	N/S	N/S	N/S	ABAQUS	N/S	N/S	N/S
[63]	FM-BEM	N/S	La	3D	Tr	Me	N/S	N/S	N/S	N/S
[64]	UBFES	N/S	N/S	2D	St	Th-Me	N/S	Quad	N/S	N/S
[65]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[66]	FEM	Ex	La	3D	Tr	Th-Me	LS-DYNA	Hexa	69,649	0.5
[68]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	50,000	N/S
[69]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Hexa	19,000	N/S
[70]	FEM	N/S	La	3D	Tr	Th-Me	MSC Marc	N/S	N/S	N/S
[20]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	32,000	N/S
[28]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	1,000,000	N/S
[31]	FEM	N/S	Eu	3D	St	Th-Me	LAM3	Hexa	N/S	N/S
[71]	FEM	N/S	La	2D	Tr	Th-Me	N/S	Quad	N/S	N/S
[33]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	N/S	N/S
[39]	FEM	N/S	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[40]	FEM	N/S	La	3D	Tr	Th-Me	MSC Marc	Hexa	19,200	N/S
[41]	FEM	N/S	La	3D	Tr	Me	ABAQUS	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; FM-BEM: fast multipole boundary element method; UBFES: upper-bound finite element solution; Ex: explicit; Im: implicit; La: Lagrangian; Eu: Eulerian; Tr: transient; St: steady-state; Me: mechanical; Th-Me: thermomechanical.

2.2. Shape Rolling

Shape rolling is a process used to transform blooms and billets into various products with different cross sections through multiple passes [88]. This process can produce simple sections like rounds, squares, and rectangles, as well as more complex sections such as U-, L-, I-, T-, H-shaped, or other irregular structural shapes. The rolling process involves defining intermediate shapes or passes to achieve the desired final geometries. There is no singular sequence for this process, and numerous combinations can be used to obtain the desired final shape. Therefore, it is important to have a proper pass design to obtain high-quality sections with maximum productivity and minimum production costs.

Numerous researchers have conducted experimental studies on the flow of metal in shape rolling. Due to the involvement of multiple process variables and the complex nature of material flow in shape rolling, the use of numerical techniques as an engineering tool becomes highly attractive for analysis [89].

Studies on the simulation of the shape rolling process can be divided into two categories, i.e., hot rolling and cold rolling. Hot shape rolling is mainly concerned with the production of simple geometric sections or structural shapes. Figure 8 shows a schematic representation of a hot shape rolling process for the creation of structural products.

Table 4 and Table 5 summarized the main rolling parameters and characteristics of the numerical simulations performed in the papers analyzed in this section.

Yuan et al. [90] used MSC Marc software to develop a 3D FEM model for studying the thermal behavior of rods and wires during continuous thirty-pass hot rolling of both 304 stainless steel and GCr15 steel. The authors achieved a good agreement between the predicted temperature field results and the experimentally obtained values, indicating the effectiveness and efficiency of the developed model.

The study of Yuan et al. [15] was extended in order to investigate the hot continuous rolling process using both static and dynamic procedures. The simulated temperature field results showed good agreement with the experimental values. The study revealed that the static procedure was more accurate and suitable for simulating the rolling process at lower speeds, such as the roughing mill. In contrast, the dynamic procedure was more efficient and better suited to simulate higher-speed rolling processes, for example, the finishing mill.

Li et al. [18] presented a study about the hot shape rolling of large H-beam Q235 steel. They developed a FEM model and used kernel scripts and custom applications of the ABAQUS GUI Toolkit to simulate the process. After analysis, it was determined that the developed model was able to accurately predict the stress and temperature fields, as well as the material flow during rolling. An error lower than 6% was obtained when comparing simulated temperature fields to the experimentally measured ones. These findings indicate the potential of the model to provide essential information to optimize the rolling process and develop new types of H-beams.

In the study conducted by Wang et al. [52], three plasticity models (Johnson-Cook, Zerilli-Armstrong, and a combined one) were analyzed through numerical simulation. The models were created using data from experimental compression tests carried out on micro-alloyed medium carbon steel at various temperatures and strain rates. The combined model showed better agreement with the experimental data than the other models. The authors implemented the combined model in ABAQUS 6.12 to simulate the hot rolling of steel bars and analyzed the effect of temperature and strain rate on stress and torque. The results indicated that temperature had a significant impact on the stress distribution, while the effect of strain rate was limited. The torque increased with decreasing temperature and increasing strain rate.

Kurt and Yasar [1] compared the results of hot shape rolling of S275JR steel sections using experimental, analytical, and simulated approaches. They used a 3D FEM model generated in Simufact Forming to simulate the production of HEA 240 profiles and compared the geometric dimensions obtained in the first three passes with experimental results. A high agreement was found, with a similarity ratio of 95–99.1%. After that, the analytically calculated dimensions of a calibrated IPE 140 profile were compared to the validated numerical model. A comparison of the dimensions of the first five passes showed a close relationship in the results, with a deviation lower than 5%. According to the authors, numerical simulations significantly reduced losses in the production process and helped achieve production targets.

Pérez-Alvarado et al. [13] simulated using Simufact Forming the complete rolling schedule of I-shaped rectangular skate beams of AISI E52100 steel to predict the final length, cross-section geometry, stress, plastic deformation, and rolling power. The simulations were carried out for a constant temperature of 1200 °C and the actual temperature of the beam in each pass. By comparing both simulations, it was possible to determine the critical passes with the highest power requirement, which corresponded to those passes with the highest cross-section reduction and lowest material temperature. A new rolling schedule has been proposed with a reduced number of passes. This new schedule omits passes with lower power requirements that do not significantly impact the geometry of the beam. The results obtained have demonstrated that this new methodology allows better control of the process.

Most recently, Singh and Singh [36] studied the effect of rolling process parameters on the response parameters during hot rolling of SAE 52100 steel bars. Specifically, the process parameters studied were rolling speed, billet temperature, reduction ratio, billet size, and roll diameter. The response parameters studied were roll separation force, driving torque, and end crop length. The study was conducted by numerical simulations using the FEM-based program FORGE NxT 1.1. The numerical model was validated by comparing it with experimental data. It was observed that there were no significant differences between the simulated and experimental values of the response parameters, with a coincidence level of 95%. Subsequently, a convergence study of the simulation results with respect to the level of discretization of the finite element mesh was carried out and the optimum element size was determined. By comparing the results of the simulations, Singh and Singh observed that the five process parameters studied significantly affected the roll separating force and driving torque, while the end crop length was only significantly influenced by roll diameter, billet cross section, and reduction.

Cold shape rolling is primarily used to produce long, thin-walled metal products with a constant cross-section and tight tolerances. This is achieved by progressively bending and folding long strips of metal through a series of roll stations (Figure 9). During this process, the thickness of the material is not significantly altered, except in the localized bend areas. As a result, only its geometry is affected.

The geometry of the final cross-section can vary from a simple open-channel shape to a closed tube section or a complex profile with multiple bends. To achieve these shapes, the strip must pass through a variable number of roller stations, depending on the complexity of the section and the design of the rolling schedule. Even the roll-forming of simple open-channel sections requires meticulous design and control to ensure a high-quality product with the necessary geometrical accuracy [91].

Bui and Ponthot [92] used Metafor proprietary software to simulate the cold rolling process of a U-channel to measure the development of strain and to identify potential forming problems. The results were compared with experimental data from the literature. A parametric study was conducted, and it was noted that the yield limit and work-hardening exponent had a significant impact on product quality. However, the forming speed and friction did not appear to have a significant effect on the outcomes.

Chen [93] used DEFORM-3D to investigate the plastic deformation behavior of internal cavity defects during cold shape rolling of V-sectioned 6062 aluminum alloy sheets. The study aimed to simulate the closure of the internal voids around the roll gap. Numerical results indicated that void closure increased with decreasing thickness. As a result of this research, it was demonstrated the capability of DEFORM-3D to model the shape rolling of sheets that contain internal voids.

Hanoglu and Sarler designed in [94] non-symmetric products that would be manufactured through cold rolling. The authors used a 2D simulation system developed previously in [95] to investigate the shape rolling of two complex non-symmetric groove types. The simulation process used the meshless local radial basis function collocation method. Due to the complexity of the process, the solution system was carried out through multiple slices aligned perpendicularly to the rolling direction. The results were analyzed for temperature, displacement, strain, and stress fields, as well as rolling force and torques. Finally, the authors created a computer application for industrial use based on C# and .NET.

Table 4. Main rolling parameters used in the numerical simulations of the shape rolling process.

Refs.	Material	Condition	Passes	Workpiece Temp. [°C]	Workpiece Size [mm]	ø Roll [mm]	Rev. [rpm]	Friction Coef.
[90]	AISI 304; GCrl5 steel	HR	30	N/S	400 × 150 × 150	N/S	N/S	N/S
[15]	AISI 304	HR	30	N/S	1300 × 150 × 150	N/S	N/S	N/S
[18]	Q235 steel	HR	N/S	N/S	N/S	N/S	N/S	N/S
[52]	Medium carbon steel	HR	1	1000–1100	4000 × ø 235	606	5.75	0.5; 0.6
[1]	S275JR steel	HR	3; 5	1200	N/S × 360 × 280; N/S × 150 × 150	N/S	55–74	0.36–0.72
[13]	AISI E52100	HR	25	869–1200	3911 × 812 × 203	1104.9	60–65	0.3–0.4
[36]	SAE 52100	HR	1	1170–1260	N/S × 100 × 100–N/S × 200 × 200	100–1000	20–65	0.3
[92]	N/S	CR	3	RT	1200 × 236 × 4	N/S	N/S	0; 0.2
[93]	AA6062	CR	1	RT	N/S × 40 × 10	200	N/S	0.6
[94]	N/S	CR	1	RT	N/S	N/S	N/S	N/S

N/S: not specified; HR: hot rolling; CR: cold rolling; RT: room temperature.

Table 4. Main rolling parameters used in the numerical simulations of the shape rolling process.

Refs.	Material	Condition	Passes	Workpiece Temp. [°C]	Workpiece Size [mm]	ø Roll [mm]	Rev. [rpm]	Friction Coef.
[90]	AISI 304; GCrl5 steel	HR	30	N/S	400 × 150 × 150	N/S	N/S	N/S
[15]	AISI 304	HR	30	N/S	1300 × 150 × 150	N/S	N/S	N/S
[18]	Q235 steel	HR	N/S	N/S	N/S	N/S	N/S	N/S
[52]	Medium carbon steel	HR	1	1000–1100	4000 × ø 235	606	5.75	0.5; 0.6
[1]	S275JR steel	HR	3; 5	1200	N/S × 360 × 280; N/S × 150 × 150	N/S	55–74	0.36–0.72
[13]	AISI E52100	HR	25	869–1200	3911 × 812 × 203	1104.9	60–65	0.3–0.4
[36]	SAE 52100	HR	1	1170–1260	N/S × 100 × 100–N/S × 200 × 200	100–1000	20–65	0.3
[92]	N/S	CR	3	RT	1200 × 236 × 4	N/S	N/S	0; 0.2
[93]	AA6062	CR	1	RT	N/S × 40 × 10	200	N/S	0.6
[94]	N/S	CR	1	RT	N/S	N/S	N/S	N/S

N/S: not specified; HR: hot rolling; CR: cold rolling; RT: room temperature.

Table 5. Main characteristics of the numerical models applied in the numerical simulations of the shape rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[90]	FEM	Im	La	3D	Tr	Th-Me	MSC Marc	Hexa	N/S	N/S
[15]	FEM	Im	La	3D	Tr	Th-Me	MSC Marc	Hexa	5850	N/S
[18]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[52]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[1]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	3.8–8
[13]	FEM	N/S	N/S	3D	Tr	Th-Me	Simufact Forming	Hexa	270,673	38
[36]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	20
[92]	FEM	N/S	La	3D	Tr	Me	Metafor	Hexa	4560	5
[93]	FEM	N/S	La	3D	St	Me	DEFORM	N/S	114,000	N/S
[94]	MLRBFCM	N/S	N/S	2D	Tr	Th-Me	N/S	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; MLRBFCM: meshless local radial basis function collocation method; Ex: explicit; Im: implicit; La: Lagrangian; Tr: transient; St: steady-state; Me: mechanical; Th-Me: thermomechanical.

Table 5. Main characteristics of the numerical models applied in the numerical simulations of the shape rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[90]	FEM	Im	La	3D	Tr	Th-Me	MSC Marc	Hexa	N/S	N/S
[15]	FEM	Im	La	3D	Tr	Th-Me	MSC Marc	Hexa	5850	N/S
[18]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[52]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[1]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	3.8–8
[13]	FEM	N/S	N/S	3D	Tr	Th-Me	Simufact Forming	Hexa	270,673	38
[36]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	20
[92]	FEM	N/S	La	3D	Tr	Me	Metafor	Hexa	4560	5
[93]	FEM	N/S	La	3D	St	Me	DEFORM	N/S	114,000	N/S
[94]	MLRBFCM	N/S	N/S	2D	Tr	Th-Me	N/S	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; MLRBFCM: meshless local radial basis function collocation method; Ex: explicit; Im: implicit; La: Lagrangian; Tr: transient; St: steady-state; Me: mechanical; Th-Me: thermomechanical.

As shown in Table 4, steel is the main alloy simulated in the considered papers, with aluminum alloy being used only in [93]. In seven of the ten articles included in Table 4, hot rolling of shapes for temperatures between 869 and 1260 °C has been simulated, while cold rolling has been treated in three of them. This may indicate that there is a greater interest in the study of hot rolling of long products. Similar to what was observed in Section 2.1, in half of the articles discussed, multi-pass processes have been studied, which for the cases analyzed in this section range from 3 to 30 passes. As regards the initial dimensions of the workpieces, in four of the articles square section billets with a side between 100 and 200 mm have been used, while in [52] a circular section billet with a diameter of 235 mm has been considered. The initial length of the workpieces in those cases where indicated ranged from 400 to 4000 mm. The data on the work rolls are diverse, with roll diameters ranging from 100 to 1104.9 mm and roll speeds from 5.75 to 74 rpm. As far as friction coefficients are concerned, a variety of values between 0 and 0.72 have been found.

Concerning the main characteristics of the numerical model included in Table 5, it can be seen that in most of the articles a 3D FEM model has been used. Only in [94] a 2D meshless local radial basis function collocation method has been used. For the resolution of the model, an implicit method was used in [15,90], while in [18,52], an explicit method was employed. Regarding the definition of the model, in all cases a Lagrangian definition was used. In most of the studies included in Table 5, a transient regime simulation was carried out, whereas in [93] a steady-state simulation was chosen. As far as the type of analysis is concerned, thermomechanical analysis is predominant, with mechanical analysis having been performed in [92,93]. The programs used are varied, finding studies in which MSC Marc, ABAQUS, Simufact Forming, FORGE NxT, Metafor, and DEFORM have been used. In terms of mesh, the use of hexahedral elements predominates, and the use of tetrahedrons has been recorded only in [36]. The meshes used have between 4560 and 270,673 elements, and their size ranges between 3.8 and 8 mm. The choice of these parameters depends mainly on the dimension of the model and the accuracy required.

2.3. Ring Rolling

Ring rolling is a metal-forming process for producing seamless ring-shaped parts. Three sets of rolls are used in the process. The first consists of a main drive roll and a mandrel. A donut-shaped preform is placed on the mandrel, and the gap between the mandrel and the main drive roll is slowly reduced, causing the radial cross-section of the ring to decrease. As the ring rotates, it undergoes circumferential extrusion, resulting in an increase in its diameter. The second group of rings is made up of axial rolls that limit the expansion of the ring in the axial direction and control its height. Finally, a group of guide rolls keeps the circular shape intact and offers support during the rolling process [96]. A schematic representation of the parts involved in ring rolling is shown in Figure 10.

This process produces parts with high dimensional accuracy, close tolerances, smooth surfaces, uniform quality, and favorable grain orientation. It also saves material and energy while reducing production times [97]. Ring rolling is used in the manufacture of a wide range of products, such as train tires, gear rims, slewing rings, bevel ring gears, sheaves, valve bodies, food processing dies, chain master links, and rotating and nonrotating rings for jet engines and other aerospace applications [98].

Despite its advantages, the ring rolling process, especially hot ring rolling, is characterized by a complex coupled thermomechanical deformation behavior, which affects the quality of the final product [99]. During the process, there are various sources of heat, such as plastic work, friction, and contact between the workpiece and the rolls. On the other hand, many physical and mechanical properties of materials depend on temperature. Therefore, when creating FEM models to simulate the process, all these factors must be considered. These models not only provide insight into the mechanics of ring rolling and defect formation but also offer a quick and affordable way to optimize various process parameters without the need for experimental testing [100]. Table 6 and Table 7 summarized the main rolling parameters and characteristics of the numerical simulations performed in the papers analyzed in this section.

Li et al. proposed in [23] a 3D FEM thermomechanical numerical model to describe the actions of the rolls on the ring during the hot radial-axial rolling process of 2219 aluminum alloy ultra-large rings with four guide rolls. The objective of this paper was to provide a basis for determining the guide force and guide roll position to realize the stable rolling of ultra-large rings. ABAQUS/Explicit 6.4 was used in the development of the simulations. Based on the results obtained in this paper, a plastic instability criterion was developed for the hot radial-axial rolling process of ultra-large rings with four guide rolls. According to the authors, based on this criterion, the guide force and layout of the guide rolls could be optimally determined.

Lv et al. [24] investigated the rolling of Ti-6Al-4V titanium alloy profiled rings in order to achieve multi-objective optimization of the main process parameters. For this purpose, a 3D FEM-based thermomechanical model was established in Simufact Forming. Simulation of the rolling and cooling processes was performed, and the variation of the residual stress with the rolling parameters was analyzed. It was determined that the optimum values of the main rolling process parameters were an initial temperature of 967 °C, a mandrel feed speed of 0.65 mm·s⁻¹, and a main roll speed of 20.7 rpm. Using these parameters, the residual stress on the inner face was reduced by 25%. Good agreement was observed between simulated and experimentally measured values.

Liang et al. [25] simulated the formation mechanism and control method of multiple geometrical defects in the rolling process of Inconel 718 profiled conical section rings. A 3D thermomechanical model was developed in the ABAQUS/Explicit package and using an arbitrary Lagrangian–Eulerian definition. Based on the simulation results, methods to avoid defect formation were proposed, which included improved target ring design, mandrel feed rate, and ring blank. Finally, the defect control methods were applied to the manufacture of rings in industrial experiments, and the results were verified with the simulation results, obtaining relative errors of 5.2% in the geometries.

Tian et al. [26] proposed an innovative hot constrained ring rolling process for the manufacture of conical rings with thin sterna and high ribs with application for the aerospace industry. In order to evaluate the proposed process and achieve the manufacture of near-net-shape rings, 3D thermomechanical FEM simulations were performed in DEFORM-3D software. Simulations were verified and compared with experimental results, and a good match was obtained, with an error of 5.8% in the height of the ribs produced. Simulations revealed that the friction factor between the ring and the tooling influenced the rolling of constrained rings, as well as the diameter and feed rate of the idler roll. Obtained results showed that as these parameters increased, the height difference of two ribs gradually increased, and so did the degree of inhomogeneous deformation and the maximum rolling force.

Nayak et al. [34] studied the effect of feed rate on the development of heterogeneities during hot rolling of Ti-6Al-4V alloy rings using FEM simulations. The research was carried out for rings with initial and target diameters of 150 and 170 mm rolled at 880 °C. Feed rates of 1 and 2.5 mm·s⁻¹ were considered. Moreover, 3D FEM-based simulations were performed using ABAQUS/Explicit, and the temperature and strain distribution in the cross section of the rings were compared for both values of feed rate (Figure 11). By analyzing the results shown in this figure, it was observed that the strain and temperature distributions were more homogeneous when laminating with a higher feed rate, while for the lower feed rate the strain was mainly concentrated on the outer surface of the ring and did not penetrate into its core.

Most recently, Deng et al. [42] studied the deformation behavior and filling characteristics of rings with an outer groove during hot rolling using FEM-based numerical simulation. The objective of this paper was to provide guidelines for the design and optimization of outer grooved rings for industrial production. Simulations were carried out using ABAQUS software for a GH738 stainless steel. To validate the model, the manufacturing of a ring was performed experimentally, and, by comparing the diameter and height with the predicted values, a good agreement was achieved. The proposed process was simulated using three ring-shaped preforms with different dimensions, and it was determined that there were three deformation behaviors.

Gröper et al. [44] describe a measurement procedure for determining the ring position on the basis of circumferential measurements with Hall sensors for the manufacture of eccentric rings by hot forming, radial-axial rings. This method was tested by means of FEM simulations using the commercial software ABAQUS/Explicit, which allowed us to drastically reduce the number of experimental tests. For this purpose, a virtual sensor was included in the FEM model to simulate the measurement system implemented in the real model. In addition, the robustness of the numerical model was tested by introducing artificially generated errors. As a result of the numerical simulations, measurements of position and wall thickness of the rings were obtained with a difference of 3.9% with respect to the experimental measurements, which demonstrated that the application of the proposed measurement method was possible.

Ge et al. [50] proposed a profiled ring rolling method based on position/force feedback aimed at minimizing the instability and forming quality problems present in the manufacture of large rings by hot rolling. A FEM model was established using ABAQUS to analyze this method and optimize the process control parameters. From the results predicted by simulation, it was determined that under the rolling stability condition, the ring cross section and outer radius met the design requirements, and the central displacement and roundness errors of the ring were reduced by 75%.

Table 6. Main rolling parameters used in the numerical simulations of the ring rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	ø Main Roll [mm]	Feed Rate [mm·s−1]	Friction Coef.	ø0 Workpiece [mm]	øf Workpiece [mm]
[23]	AA2219	HR	420	25.7	900	N/S	0.3	2683	5040
[24]	Ti-6Al-4V	HR	N/S	14.3	N/S	0.8	0.8	555	976
[25]	IN718	HR	1000	15.2	1816	0.25–1.0	0.3	1141.8	N/S
[26]	AA1050	HR	450	208	N/S	0.5	0.3	120	N/S
[34]	Ti-6Al-4V	HR	880	26	N/S	1; 2.5	N/S	150	170
[42]	SS GH738	HR	1100	57	400	1	0.3	N/S	856–866
[44]	42CrMo4 steel	HR	1200	N/S	N/S	N/S	0–0.5	300	367
[50]	AA2219	HR	450	6.65	950	N/S	0.3	1592	3300

N/S: not specified; HR: hot rolling.

Table 6. Main rolling parameters used in the numerical simulations of the ring rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	ø Main Roll [mm]	Feed Rate [mm·s−1]	Friction Coef.	ø0 Workpiece [mm]	øf Workpiece [mm]
[23]	AA2219	HR	420	25.7	900	N/S	0.3	2683	5040
[24]	Ti-6Al-4V	HR	N/S	14.3	N/S	0.8	0.8	555	976
[25]	IN718	HR	1000	15.2	1816	0.25–1.0	0.3	1141.8	N/S
[26]	AA1050	HR	450	208	N/S	0.5	0.3	120	N/S
[34]	Ti-6Al-4V	HR	880	26	N/S	1; 2.5	N/S	150	170
[42]	SS GH738	HR	1100	57	400	1	0.3	N/S	856–866
[44]	42CrMo4 steel	HR	1200	N/S	N/S	N/S	0–0.5	300	367
[50]	AA2219	HR	450	6.65	950	N/S	0.3	1592	3300

N/S: not specified; HR: hot rolling.

Table 7. Main characteristics of the numerical models applied in the numerical simulations of the ring rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[23]	FEM	Explicit	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[24]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	26,500	5
[25]	FEM	Ex	ALE	3D	Tr	Th-Me	ABAQUS	Hexa	50,000	N/S
[26]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	230,000	0.3–0.6
[34]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	1.85
[42]	FEM	N/S	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[44]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[50]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	5460	N/S

N/S: not specified; FEM: finite element method; Ex: explicit; La: Lagrangian: ALE: arbitrary Lagrangian–Eulerian: Tr: transient; Th-Me: thermomechanical.

Table 7. Main characteristics of the numerical models applied in the numerical simulations of the ring rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[23]	FEM	Explicit	La	3D	Tr	Th-Me	ABAQUS	N/S	N/S	N/S
[24]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	26,500	5
[25]	FEM	Ex	ALE	3D	Tr	Th-Me	ABAQUS	Hexa	50,000	N/S
[26]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	230,000	0.3–0.6
[34]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	1.85
[42]	FEM	N/S	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[44]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	N/S	N/S
[50]	FEM	Ex	La	3D	Tr	Th-Me	ABAQUS	Hexa	5460	N/S

N/S: not specified; FEM: finite element method; Ex: explicit; La: Lagrangian: ALE: arbitrary Lagrangian–Eulerian: Tr: transient; Th-Me: thermomechanical.

As can be seen in Table 6, the papers discussed in the present section address the numerical simulation of different aluminum alloys [23,26,50], titanium [24,34], steels [42,44], and nickel-based superalloys [25]. Hot rolling conditions have been used in all the cases, with temperatures ranging from 420 to 1200 °C depending on the material used. Diameters and speeds of the main rolls vary, with speeds ranging from 6.65 to 208 rpm and diameters from 400 to 1816 mm. Mandrel feed rate values are more homogeneous and range from 0.25 to 2.5 mm·s⁻¹. As regards the friction coefficient, its values range from 0 to 0.8, depending on the used model. Concerning the diameter of the rings, preforms with diameters between 120 and 2683 mm and target diameters between 170 and 5040 mm have been used.

As can be seen in Table 7, the simulations have been performed using 3D FEM thermomechanical models. Where indicated, an explicit resolution method suitable for the simulation of dynamic events has been used. Lagrangian definition has been used in most of the cases studied, with the exception of [25], where an arbitrary Lagrangian–Eulerian approximation has been used. Regarding the regime, all the simulations have been developed in a transient regime due to the fact that it is an incremental manufacturing process in which a steady state is not reached. ABAQUS has been the most used simulation program, while Simufact Forming and DEFORM have been used in [24,26], respectively. By using these programs, variable-sized meshes have been created. Depending on the scale of the model under study and the desired accuracy, between 5460 and 230,000 finite elements have been employed. The sizes of these elements vary between 0.3 and 5 mm.

2.4. Cross-Wedge Rolling

Cross-wedge rolling, or transverse rolling, is a manufacturing process that involves the use of wedge-shaped rolls that rotate in the same direction to reshape a cylindrical billet into a different axisymmetric shape. This technique is commonly used to obtain stepped shafts as well as forging preforms [101]. Figure 12 shows a schematic diagram of the cross-wedge rolling process. The sequence of forming is determined by the shape of the rolls (Figure 12a). Wedge rolls typically consist of three zones, named the knifing zone, stretching zone, and sizing zone (Figure 12b). In the knifing zone, the tools cut the material to the required depth while gradually decreasing the diameter. In the stretching zone, the diameter is further reduced to the desired width. Finally, the undesired curvatures generated during the previous phases are eliminated in the sizing zone [102].

Cross-wedge rolling is a manufacturing process that offers several benefits over traditional techniques, such as high efficiency, better utilization of materials, increased product strength, lower energy consumption, easy automation, and environmental friendliness [103]. Despite these advantages, the process has not been extensively adopted in the industry due to the difficulty of designing forming tools that maintain process stability while preventing defects. Some papers in the literature use numerical simulations for a better understanding of the cross-wedge rolling process. Table 8 includes the main rolling parameters used in the numerical simulations, whereas Table 9 summarizes the key characteristics of the numerical models applied in the numerical simulations performed in the papers analyzed in this section.

Wang et al. [104] simulated the deformation mechanism of the hot cross-wedge rolling process of AISI 5140 stainless steel shafts in DEFORM-3D. Simulations were performed using a thermomechanical model coupled to a microstructural model. Using the information included in Table 8 and Table 9, the authors were able to predict the dynamic recrystallized grain size of austenite (Figure 13). This figure shows that the grain size in the axial head of the workpiece remained in the range 107–116 µm after rolling due to low deformation, whereas in the central region it was fined down to 20–30 µm. Values of the distribution of effective strain fields, effective strain rate, temperature, and microstructure of the material were obtained.

Bartnicki and Pater [105] used MSC SuperForm 2002 software to simulate the manufacturing process of a hollow shaft made of steel 45 through cold cross-wedge rolling. The objective of this study was to enhance the cross-wedge rolling technology design. The simulation results for wall thickness and rolling load were compared with experimental tests and found to be in good agreement. This work established the relationships between the stability of the process, the strain rate, and the wall thickness of the formed parts. In a later publication [101], the authors simulated a three-roll cold cross-wedge rolling process. They found that the stability of this process, when using hollow shafts, was better than that of the traditional two-roll methods.

Pater [102] developed a new two-stage concept for designing tools for the hot cross-wedge rolling process of 20MnCr5 steel. The optimal process variant was selected after studying several cases and was then subjected to detailed analysis using three-dimensional FEM models, both mechanical and thermomechanical, developed in MARC/AutoForge V2.3. Results were obtained for the distribution of strain, strain rate, mean stress, and rolling load. According to the author, the proposed model could predict forming process instabilities such as uncontrolled slipping and core necking.

Kache et al. [106] used FORGE 2009 software to develop a warm cross-wedge rolling process to obtain axisymmetric parts with area reduction for 38MnVS6 micro-alloyed steel. The objective was to carry out this new process using FEM to take advantage of the benefits of warm rolling and the cost savings offered by numerical simulation. The warm rolling simulations were performed for temperatures of 850 °C and 950 °C and hot rolling for 1250 °C. In warm conditions, the simulated horizontal rolling force was three times higher than in hot conditions. The FEM simulations were verified by performing experimental tests on downsized workpieces. Results confirmed the feasibility of warm cross-wedge rolling and the ability of the model to predict forces with an assumed error of up to 24%. According to the authors, the error in the simulations was caused by the material model used as input data.

Huang et al. [107] used cross-wedge rolling under warm and hot conditions to investigate the manufacturing process of AISI 4140 steel bolts. They compared simulated and experimental conditions by using DEFORM-3D software for initial temperatures of 650, 700, 1000, and 1050 °C. When comparing the results for the four simulated temperatures, a tension-compression alternation was identified, changing four times per rotation. According to the authors, tension-compression alternation could result in the propagation of micro-cracks and eventually in the formation of cavities. Thus, since the highest stress values corresponded to the warm rolling conditions, they concluded that for these conditions there was a higher risk of micro-cavity formation. On the other hand, they used the normalized Cockcroft–Latham criterion to analyze the fracture tendency. According to the obtained results, the samples simulated for warm cross-wedge rolling were more prone to breaking than those modeled by hot cross-wedge rolling. After analysis, it was found that warm cross-wedge rolling produced rolling force and torque over three times higher than those produced by hot cross-wedge rolling. These results are in agreement with those observed by Kache et al. [106].

Bulzak et al. [108] performed a comparative analysis of warm and hot cross-wedge rolling of ball-shaped DIN C45 steel pins using Simufact Forming 15 software. Simulations were performed for temperatures of 650, 800, and 1000 °C. The validation of the FEM model relied on comparing the forces obtained from simulation and experimental tests. Warm rolling at 650 °C recorded forces up to 80 kN, whereas hot rolling at 1000 °C presented a maximum force of 35 kN. The difference in force values between warm and hot rolling was smaller than those observed in [106,107]. Once validated, strain and stress were analyzed. The simulation results are shown in Figure 14. As can be seen in Figure 14a, the highest values of the strain were reached when laminating at 650 °C. When increasing temperatures, a reduction of the strain values was observed, being hardly noticeable when using values of 800 and 1000 °C. The largest strain values were obtained on the surface of the spherical region of the workpiece, with the values decreasing in the radial direction as they approached the center. Figure 14b shows the distribution of the normalized Cockcroft–Latham fracture criterion. According to these results, the value of the fracture criterion increased at the corners of the sphere area at 650 °C. However, the most exposed area to fracture was the central one of the pin ends. The stress was analyzed at six points distributed along the length of the part. On the one hand, it was observed that the stress decreased when the forming temperature increased. On the other hand, it was observed that stress presented positive and negative values. As discussed in [107], the authors highlighted the risk that this can pose because this favors the propagation of micro-cracks. From the simulation results, it was concluded that, despite the advantages of warm cross-wedge rolling, the nature of the stress during cross-wedge rolling is more advantageous during hot rolling due to the lower amplitude of the stress changes.

Bulzak et al. [22]. investigated the influence of hot cross-wedge rolling on the development of internal cracks in DIN C45 steel parts. For this purpose, Simufact Forming 16 software was used to simulate by FEM different cross-wedge rolling schemes using different configurations and number of tools: flat wedges, roll wedges, roll wedge-concave segments, and two concave wedges. From the results, it was found that the degree of damage increased by increasing the ovalization of the laminated parts. Specifically, the highest degree of damage occurred during the flat wedge rolling method, while the lowest degree of damage occurred when using concave wedges. By using this setup, it was possible to reduce the damage by half.

Table 8. Main rolling parameters used in the numerical simulations of the cross-wedge rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	Forming Angle [°]	Spreading Angle [°]	Friction Coef.	ø0 Workpiece [mm]	øf Workpiece [mm]
[104]	AISI 5140	HR	1000	10	28	6	1.0	22	17
[105]	DIN C45	CR; HR	RT; 1100	N/S	20; 45	6; 9	1.0	30	18
[101]	CP Pb	CR	RT	9.5	20–40	12–18	1.0	30	17–24
[102]	20MnCr5 steel	HR	1050	N/S	22.5–45	5; 7	0.5; 1.0	22–60	14–40
[106]	38MnVS6 steel	WR; HR	850–1250	N/S	25; 30	5–9	0.8	42	30
[107]	42CrMo steel	WR; HR	650–1050	10	36	7.34	0.9	30	N/S
[108]	DIN C45	WR; HR	650–1000	N/S	30	10	0.7	29	N/S
[22]	DIN C45	HR	1150	7.5–16.8	15	10	0.8	33	22

N/S: not specified; CP: commercially pure; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Table 8. Main rolling parameters used in the numerical simulations of the cross-wedge rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	Forming Angle [°]	Spreading Angle [°]	Friction Coef.	ø0 Workpiece [mm]	øf Workpiece [mm]
[104]	AISI 5140	HR	1000	10	28	6	1.0	22	17
[105]	DIN C45	CR; HR	RT; 1100	N/S	20; 45	6; 9	1.0	30	18
[101]	CP Pb	CR	RT	9.5	20–40	12–18	1.0	30	17–24
[102]	20MnCr5 steel	HR	1050	N/S	22.5–45	5; 7	0.5; 1.0	22–60	14–40
[106]	38MnVS6 steel	WR; HR	850–1250	N/S	25; 30	5–9	0.8	42	30
[107]	42CrMo steel	WR; HR	650–1050	10	36	7.34	0.9	30	N/S
[108]	DIN C45	WR; HR	650–1000	N/S	30	10	0.7	29	N/S
[22]	DIN C45	HR	1150	7.5–16.8	15	10	0.8	33	22

N/S: not specified; CP: commercially pure; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Table 9. Main characteristics of the numerical models applied in the numerical simulations of the cross-wedge rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[104]	FEM	Im	La	3D	Tr	Th-Me	DEFORM	Tetra	N/S	N/S
[105]	FEM	N/S	La	3D	Tr	Th-Me	MSC SuperForm	Hexa	N/S	N/S
[101]	FEM	N/S	La	3D	Tr	Th-Me	MSC SuperForm	Hexa	N/S	N/S
[102]	FEM	N/S	La	3D	Tr	Th-Me	MARC/AutoForge	Hexa	N/S	N/S
[106]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	N/S	2
[107]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	100,000	N/S
[108]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[22]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; Im: implicit; La: Lagrangian; Tr: transient; Th-Me: thermomechanical.

Table 9. Main characteristics of the numerical models applied in the numerical simulations of the cross-wedge rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[104]	FEM	Im	La	3D	Tr	Th-Me	DEFORM	Tetra	N/S	N/S
[105]	FEM	N/S	La	3D	Tr	Th-Me	MSC SuperForm	Hexa	N/S	N/S
[101]	FEM	N/S	La	3D	Tr	Th-Me	MSC SuperForm	Hexa	N/S	N/S
[102]	FEM	N/S	La	3D	Tr	Th-Me	MARC/AutoForge	Hexa	N/S	N/S
[106]	FEM	N/S	La	3D	Tr	Th-Me	FORGE	N/S	N/S	2
[107]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	N/S	100,000	N/S
[108]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[22]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	N/S	N/S	N/S

N/S: not specified; FEM: finite element method; Im: implicit; La: Lagrangian; Tr: transient; Th-Me: thermomechanical.

A comparison of rolling parameters used in the different papers can be made from the information included in Table 8. Thus, it is observed that the most studied materials are structural steels, processed by hot or warm rolling at temperatures between 650 and 1250 °C. These processes have been studied using workpieces with diameters between 22 and 60 mm, rolling speeds within the range of 7.5 and 16.8 rpm, forming angles from 15 to 45°, and spreading angles that range from 5 to 18°. Another variable taken into account is the friction coefficient, the values studied being between 0.5 and 1.

As regards the main characteristics of the numerical models used in Table 9, it is noted that the 3D FEM thermomechanical models have been developed using different simulation programs, including DEFORM, MSC SuperForm, MARC/AutoForge, and Simufact Forming. Lagrangian definition has been used in all cases, and in [104], an implicit method has been employed. Due to the nature of the process studied, simulations in the transient regime have been carried out in all the articles. Finally, as far as the mesh is concerned, mainly hexahedral elements have been used.

2.5. Skew Rolling

Skew rolling is considered one of the most effective ways for manufacturing small-sized ball bearings [109] that involves passing a semi-finished bar product through helically shaped rollers [110], Figure 15. The bar is continuously passed through the rollers to produce the desired shape and size of the ball bearings [111].

Skew rolling has several advantages over traditional manufacturing methods. It enables continuous production, eliminating the need for tools to stop and start for billet placement and product removal. It also minimizes lost motion in the tools and results in minimal material loss, high precision in the formed products, and favorable structural positioning. Additionally, the process is relatively easy to automate, which contributes to its high efficiency [112]. Skew rolling has been studied by different researchers in recent years and is an attractive method for producing semi-finished ball products in an industrial setting. Table 10 and Table 11 summarized the main rolling parameters and characteristics of the numerical simulations performed in the papers analyzed in this section.

In articles published in 2013 and 2016, Pater et al. [113,114,115] developed rolls with wedge-shaped flanges to produce 100Cr6 bearing steel balls by hot skew rolling. To test the effectiveness of the designed rolls, 3D thermomechanical FEM simulations were conducted using Simufact Forming versions 11 and 12. The simulations helped to analyze various factors such as the distribution of effective strain, temperature, material failure criterion, loads, and torques. The degree of roll wear was also studied. The results obtained were used to evaluate the feasibility of designing rolls with wedge-shaped flanges.

Cao et al. [116] used Simufact Forming to simulate the cold skew rolling process of 100Cr6 bearing steel balls to analyze the occurrence of internal defects. The simulation results provided relevant information about the deformation characteristics in the skew rolling process. Specifically, the laws of evolution and distribution of strain, stress, and damage based on the Lemaitre relative damage model were analyzed. The study also established the origins of central loosening and cavity defects during the skew rolling process, which can serve as a basis for future research on the deformation mechanism and process optimization.

Huo et al. [111] developed multi-axial constitutive equations implemented in DEFORM-3D to predict the microstructure evolution of 100Cr6 bearing steel balls during warm skew rolling at 750 °C. Simulations were carried out using mainly the parameters given in Table 10 and Table 11. Using these data, it was predicted the distribution of normalized dislocation density, carbide phase transformation fraction, and carbide spheroidization fraction. Skew rolling experiments validated the simulation results for 30 mm diameter specimens, showing a good match between predicted and experimental data. This study concluded that FEM simulations were highly accurate for predictions of the microstructural evolution of bearing steel balls produced through warm skew rolling.

The study conducted by Bulzak et al. [117] used FORGE NxT to simulate the warm skew rolling process of manufacturing 100Cr6 steel bearing balls at 750 °C. The numerical model was validated by comparing predicted torque and radial force values with experimental ones, and a good agreement was observed. However, a slight overestimation occurred in the steady state and was attributed to differences in the ball separation nature between simulation and experiments. Figure 16a shows the simulated temperature distribution, which matched well with experimental values. It was particularly accurate in connector areas where rapid temperature rise occurred due to plastic deformation work transformation into heat. Figure 16b displays the effective strain distribution, showing high deformation in ball connector areas and a deformation gradient in the axial direction toward the center of the balls.

Recently, Liu et al. discussed in [47] the advantages and disadvantages of the single-side variable lead method and the double-side variable lead method used in skew rolling for the fabrication of 26 mm diameter bearing steel balls. Moreover, 3D FEM numerical simulations were performed using Simufact Forming 16 and validated comparing with experimental rolling operations. Values of the strain and deformation distributions as well as the evolution of temperature and rolling force parameters were obtained and analyzed. According to the simulations, the double-side variable lead method had a higher forming accuracy and was more suitable for the fabrication of bearing steel balls. After a statistical study, the authors determined that, using this method, the radius deviation of the rolled balls could be controlled to 0.1 mm and the ovality was less than 1%.

Table 10. Main rolling parameters used in the numerical simulations of the skew rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	Feed Angle [°]	Friction Coef.	ø Billet [mm]	ø Balls [mm]
[115]	100Cr6 steel	HR	1150	60	3	1.0	33	33
[114]	100Cr6 steel	HR	1100	60	3	1.0	33	33
[113]	100Cr6 steel	HR	1150	60	6	1.0	30	30
[116]	100Cr6 steel	CR	RT	60	2	0.2	7.1	7.4
[111]	100Cr6 steel	WR	750	110	2.5	0.7	30	30
[117]	100Cr6 steel	WR	750	15	8	0.8	43	44.4
[47]	100Cr6 steel	HR	1050	60	N/S	0.8	25	26

N/S: not specified; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Table 10. Main rolling parameters used in the numerical simulations of the skew rolling process.

Refs.	Material	Condition	Workpiece Temp. [°C]	Rev. [rpm]	Feed Angle [°]	Friction Coef.	ø Billet [mm]	ø Balls [mm]
[115]	100Cr6 steel	HR	1150	60	3	1.0	33	33
[114]	100Cr6 steel	HR	1100	60	3	1.0	33	33
[113]	100Cr6 steel	HR	1150	60	6	1.0	30	30
[116]	100Cr6 steel	CR	RT	60	2	0.2	7.1	7.4
[111]	100Cr6 steel	WR	750	110	2.5	0.7	30	30
[117]	100Cr6 steel	WR	750	15	8	0.8	43	44.4
[47]	100Cr6 steel	HR	1050	60	N/S	0.8	25	26

N/S: not specified; HR: hot rolling; WR: warm rolling; CR: cold rolling; RT: room temperature.

Table 11. Main characteristics of the numerical models applied in the numerical simulations of the skew rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[115]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	N/S	N/S	N/S
[114]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[113]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[116]	FEM	Im	La	3D	Tr	Th-Me	Simufact Forming	Hexa	1920	N/S
[111]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	100,000	N/S
[117]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.25
[47]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	30,012	N/S

N/S: not specified; FEM: finite element method; Im: implicit; La: Lagrangian; Tr: transient; Th-Me: thermomechanical.

Table 11. Main characteristics of the numerical models applied in the numerical simulations of the skew rolling process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[115]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	N/S	N/S	N/S
[114]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[113]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	N/S	N/S
[116]	FEM	Im	La	3D	Tr	Th-Me	Simufact Forming	Hexa	1920	N/S
[111]	FEM	N/S	La	3D	Tr	Th-Me	DEFORM	Tetra	100,000	N/S
[117]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.25
[47]	FEM	N/S	La	3D	Tr	Th-Me	Simufact Forming	Hexa	30,012	N/S

N/S: not specified; FEM: finite element method; Im: implicit; La: Lagrangian; Tr: transient; Th-Me: thermomechanical.

An analysis of the information provided in Table 10 shows that 100Cr6 steel has been used in all the studies discussed in this section. In most cases, warm or hot rolling conditions have been studied, using temperatures of 750 °C in warm rolling and temperatures ranging from 1050 to 1150 °C in hot rolling. Only in [116] cold skew rolling process has been studied, observing non-homogeneous deformations in the balls and fractures in the center of the balls. As far as experimental parameters are concerned, Table 10 shows that the processes studied in the literature have used values of 15, 60, and 110 rpm for the roll speed and feed angles between 2 and 8°, while the values of the friction coefficient used depend on the simulation model employed. As regards the diameter of the rolled balls, values between 7.4 and 44.4 mm were used, being the diameters of the billets the same or slightly smaller.

As can be seen in Table 11, numerical simulations have been performed using 3D thermomechanical FEM models, being Simufact Forming the most used program. DEFORM and FORGE NxT have been used only in [111,117], respectively. Implicit analysis has only been reported in [116], and its use is not indicated in the rest of the papers under consideration. On the other hand, in all the articles, Lagrangian definition is used in the meshing operations, using indistinctly tetrahedral and hexahedral elements, whose size, 1.25 mm, is only indicated in [117]. The number of elements used varies between 1920 and 100,000 depending on the size of the model and the desired accuracy. Finally, due to the nature of the process studied, all simulations have been carried out in a transient regime.

2.6. Tube Piercing

Tube piercing, also known as skew rolling piercing, is a hot rolling process used in the manufacture of seamless tubes. In this process, a cylindrical billet heated to over 70% of the melting temperature of the material is placed between two rolls, and a piercing plug begins to pierce and stretch the billet along its entire length [118]. Simultaneously, two large disks, referred to as Dieschers, govern the geometry of the pierced billet to enhance geometrical accuracy [119]. A schematic representation of the parts involved in tube piercing is shown in Figure 17.

Tube piercing is a complex process that involves exposing the material to extreme conditions, including ultra-high strain rates, high temperatures, and high process speed [120]. To maximize the lifespan of the tooling system and minimize energy consumption, it is essential to identify the best process parameters [121]. Although the process is complex, numerical simulations are used to understand the thermomechanical phenomena associated with billet behavior. Table 12 and Table 13 summarized the main rolling parameters and characteristics of the numerical simulations performed in the papers analyzed in this section.

Pater et al. compared in [122] the tube piercing process of 100Cr6 medium carbon steel tubes using a two-roll mill and a three-roll mill. FORGE NxT 1.1 software was used to create a FEM model for simulation. Based on the simulation results, it was observed that the tube piercing process in the three-roll mill was more favorable. By using this rolling mill, the predicted process duration was reduced by 30%, and energy consumption was reduced by 24%. When analyzing the predicted strain by simulation, higher values were obtained for the two-roll mill. According to the authors, this was due to the longer duration of the process. Similarly, the temperature was also affected by the duration of rolling, with higher values obtained when using the tree-roll mill. The Cockroft–Latham damage criterion was also evaluated, whose results are provided in Figure 18. The information presented in this figure showed a higher probability of material failure in the case of two-roll piercing. Additionally, a smaller ovalization of the cross-section was achieved.

Topa et al. [121] conducted 3D numerical simulations of the tube piercing process using an arbitrary Lagrangian–Eulerian formulation that was created in LS-DYNA software. The study was focused on examining the effect of different variables such as workpiece speed, process temperature, and plug size on the maximum stress levels of the produced pipes. Initially, the authors conducted a mesh sensitivity analysis, which determined that the optimal finite element size was 2 mm. It was concluded that the maximum stress increased with workpiece speed and decreased with increasing workpiece temperature and plug diameter.

Derazkola et al. [123,124] studied the impact of rolls-billet and plug-billet friction coefficients on the tube piercing process for manufacturing Cr13 super-martensitic stainless steel pipes. A 3D FEM model was developed in FORGE NxT 3.2 to analyze several factors such as plastic deformation, strain rate, temperature evolution, and surface twisting. The results indicated that the plastic strain and strain rate increased with the rolls-billet friction coefficient, whereas they decreased with increasing plug-billet friction coefficient. However, the temperature evolution remained unaffected by these coefficients.

The study developed by Derazkola et al. was extended in [125]. In this paper, the effects of tool-workpiece material behavior on geometric deviation and material fracture were analyzed by FEM using the data contained in Table 12 and Table 13. Comparison of the simulated results with experimental data revealed that the mean experimental diameters were between 206 and 208 mm, while the simulated diameters were 210 mm. The heterogeneity of the deformation was attributed to variations in the stress flow of the material during rolling. Moreover, it was determined that heat transfer interactions between the part and the tooling resulted in the formation of precipitates that induced a local increase in stress, leading to deviations in different axes of the tube.

Table 12. Main rolling parameters used in the numerical simulations of the tube piercing process.

Refs.	Material	Condition	Ring Temp. [°C]	Rev. [rpm]	Speed [mm·s−1]	Roll Friction Coef.	Plug Friction Coef.	Feed Angle [°]	ømax Plug [mm]	ø0 Ring [mm]
[122]	100Cr6 steel	HR	1180	60	N/S	0.95	0.10	8	34	60
[121]	N/S	CR	0–20	N/S	5–30	N/S	N/S	6–12	27–37	45
[123]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.0–0.15	12	N/S	202
[124]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.06–0.15	12	N/S	202
[125]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.1–0.3	10	N/S	202

N/S: not specified; HR: hot rolling; CR: cold rolling.

Table 12. Main rolling parameters used in the numerical simulations of the tube piercing process.

Refs.	Material	Condition	Ring Temp. [°C]	Rev. [rpm]	Speed [mm·s−1]	Roll Friction Coef.	Plug Friction Coef.	Feed Angle [°]	ømax Plug [mm]	ø0 Ring [mm]
[122]	100Cr6 steel	HR	1180	60	N/S	0.95	0.10	8	34	60
[121]	N/S	CR	0–20	N/S	5–30	N/S	N/S	6–12	27–37	45
[123]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.0–0.15	12	N/S	202
[124]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.06–0.15	12	N/S	202
[125]	SS Cr13	HR	1250	111	N/S	0.10–0.60	0.1–0.3	10	N/S	202

N/S: not specified; HR: hot rolling; CR: cold rolling.

Table 13. Main characteristics of the numerical models applied in the numerical simulations of the tube piercing process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[122]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	N/S	N/S	N/S
[121]	FEM	N/S	ALE	3D	Tr	Th-Me	LS-DYNA	Tetra	54,612	2.0
[123]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.5
[124]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.5
[125]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	N/S

N/S: not specified; FEM: finite element method; La: Lagrangian; ALE: arbitrary Lagrangian–Eulerian; Tr: transient; Th-Me: thermomechanical.

Table 13. Main characteristics of the numerical models applied in the numerical simulations of the tube piercing process.

Refs.	Method	Solution	Definition	Discretization	Regime	Analysis	Software	Element	Mesh Size	Element Size [mm]
[122]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	N/S	N/S	N/S
[121]	FEM	N/S	ALE	3D	Tr	Th-Me	LS-DYNA	Tetra	54,612	2.0
[123]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.5
[124]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	1.5
[125]	FEM	N/S	La	3D	Tr	Th-Me	FORGE NxT	Tetra	N/S	N/S

N/S: not specified; FEM: finite element method; La: Lagrangian; ALE: arbitrary Lagrangian–Eulerian; Tr: transient; Th-Me: thermomechanical.

As indicated in Table 12 in the articles discussed in this section, the manufacture of seamless carbon steel and stainless steel tubes has been simulated. The tube piercing process is fundamentally a hot rolling process, so most of the discussed papers simulations have been carried out for temperatures of 1180 and 1250 °C. Only in [121] has the cold rolling of seamless tubes been studied. Regarding the parameters of the simulated processes, it should be noted that there are papers in which not all values are provided. However, it can be observed roll speed values of 60 and 111 rpm; workpiece feed rates of 5, 15, and 30 mm·s⁻¹; feed angles between 8 and 12°; initial billet diameter, with dimensions ranging from 45 to 202 mm and plug diameters of 27 and 37 mm. The values of the friction coefficients are different and depend on the tool and the friction criterion applied.

Table 13 shows that numerical simulations have been performed using 3D thermomechanical finite element models obtained using mainly FORGE NxT, whereas LS-DYNA has only been used in [121]. In all articles, the Lagrangian definition is used for meshing operations, except in [121], where an arbitrary Lagrangian–Eulerian definition was applied. In those articles where information on mesh size is given, tetrahedral finite elements with a minimum mesh size of 1.5 and 2.0 mm have been used. Finally, it should be noted that the articles analyzed do not specify the use of explicit or implicit models, and in all cases, the simulations have been carried out for a transient regime.

3. Outlook

Numerical simulation of rolling processes has become an essential tool for achieving a better understanding of the behavior of materials during the forming process. This increased understanding of the effect of rolling on part properties in turn allows the optimization of rolling processes and facilitates the development and implementation of alternative techniques. Because of these benefits, there are an increasing number of studies dealing with the numerical simulation of rolling processes.

The literature search carried out in this review article has provided a visualization of the trend followed by different authors in recent years, as well as identifying those areas that require further research.

As it has been observed from the literature review, most of the publications are mainly focused on the simulation of different hot rolling processes due to the interest in the thermomechanical complexities involved. However, the works focused on the simulation of cold rolling continue to be a challenge due to the distortions that the workpieces may suffer due to residual stresses. Information provided by simulating cold rolling processes could be used to improve the quality of the products, preventing distortions such as springback. This would make it possible to obtain near-net components without the need for post-processing operations to compensate for distortions.

Although there is a smaller number of works oriented to the study of warm rolling processes, its simulation could improve the understanding of the thermomechanical phenomena experienced by the materials in this temperature regime, which would allow the development of new operations and rolling schedules that would improve the performance of the manufactured products at a lower energy cost.

As regards the materials studied, most of the articles consulted focus on the use of steel and stainless steel. Although these materials have excellent mechanical properties, there are currently numerous alternative alloys with specific applications of major industrial interest. In the aerospace sector, for example, titanium alloys and nickel-based superalloys are commonly used in ring-shaped parts for propulsion systems. It would be of great interest to further study alternative alloys to broaden the available knowledge and improve their applicability.

On the other hand, based on the information included in Table 1, in recent years most of the research work has focused on flat rolling and, to a lesser extent, on ring rolling. However, few studies have covered other complex processes such as cross-wedge rolling, skew rolling, and tube piercing. Similarly, there are few current publications of interest focused on the manufacture of long products by shape rolling.

With regard to the size of rolled products, especially flat products and long products obtained by flat rolling and shape rolling, it has been observed that in most cases the simulations are focused on the study of parts of reduced dimensions. This is mainly due to the computing power of personal computers. Mainly, in flat rolling, there are few studies that deal with the simulation of slabs or sheets of industrial scale dimensions. In articles analyzing large parts, simplifications such as the use of 2D models are usually adopted. Another strategy employed is to study only a single pass of the rolling schedule. This significantly reduces the depth of knowledge that can be acquired from simulation-based studies of these processes.

It should be noted that many articles do not provide sufficient data on the rolling parameters and characteristics of the numerical simulations carried out. Consequently, the methodology used can be difficult to reproduce, and consequently, the results provided may be difficult or impossible to contrast. In addition, the lack of key data in the research articles makes comparison with other publications considerably more difficult.

Moreover, it has been observed that in some articles the simulations have been performed using material characterization data obtained from other bibliographic sources or using experimental conditions that do not faithfully represent the process conditions. To the extent that material characterization data are obtained under conditions close to those of the process, the quality of the simulations will be improved. In the same way, it is equally important to have reliable characterization data of the real process to be simulated, since the validation of the simulations depends on them.

Finally, it should be noted that a current trend, beyond the scope of this article, is the use of numerical simulation to study hybrid and complex processes. Thus, for example, in [29] the combined rolling-extrusion process was simulated, in [37,38] equal-channel angular rolling, in [49] radial shear rolling, in [126] asymmetric rolling, and in [46] hollow embossing rolling. The literature review of these studies could be dealt with in future publications.

4. Summary and Conclusions

In this paper, a literature review has been performed in which some of the most relevant research works of the last decades regarding the numerical simulation of rolling processes have been discussed, with the aim to serve as a reference for future research works on this topic. Based on the revision performed, the following conclusions have been reached:

• FEM is used in almost all cases as a predictive tool due to its accuracy and its ability to deal with the complexity of the problems proposed. To apply FEM, most authors choose to use proprietary programs that integrate pre- and post-processing tools, as well as solvers, in a single environment. Some of the most used options include ABAQUS, DEFORM, FORGE, LS-DYNA, MSC Marc and Simufact Forming.
• Steels are the most studied alloys in the consulted papers. Additionally, articles have been found focused on the study of aluminum alloys, titanium alloys, and superalloys, among other materials.
• Numerical simulation of rolling processes proves to be flexible and adaptable to the requirements of the study, as well as to the limitations of the available resources. Thus, it is possible to find studies using 2D or 3D models; mechanical, thermal, or thermomechanical models, which can be coupled with microstructural models; transient or stationary studies; small or large workpieces; one-pass or multi-pass simulations; models focused only on the workpiece and those that also contemplate the evolution of the rolls. Increasing the complexity of the model leads to an increase in computing time, which may result in some simulations being unfeasible, especially when using low-performance computers.
• In almost all the studies discussed in this review, validation of the simulated models has been performed. In most cases, similar results to experimental values have been obtained. However, some cases have been found in which deviations from the experimental results have been observed. The authors attribute these discrepancies to an incorrect modeling of the material or of the rolling parameters.
• Numerical simulations of rolling processes offer the possibility of predicting rolling loads, strain, strain rate, stress, and temperature distributions, among other results. In the case of using a microstructural model, it is also possible to predict the evolution of the grain size and the phases present in the alloy. Thus, the understanding of the phenomena occurring during rolling can be improved.
• Most of the publications consulted in the literature focused on the numerical simulation of hot and cold rolling. Only a few articles discussed the simulation of warm rolling due to their limited application. Regarding the types of rolling, many studies were found focused on the simulation of flat and shape rolling, while the number of articles on other types of rolling was more limited.
• Different strategies are used to reduce model complexity and computational times. For this purpose, small specimens and a limited number of passes are the most common strategies used. These simplifications provide limited knowledge of the rolling process. In cases requiring a broader and deeper knowledge of the studied topic, it would be necessary to perform simulations as similar as possible to the real case.
• Despite the growing number of publications on the numerical simulation of rolling processes, shortcomings have been identified that can be covered by future work.