A Review of Friction and Lubricant in Metal Forming

Abstract

Friction conditions, along with the flow behaviors of materials, significantly impact plastic deformation during metal forming. Extensive practical research on friction has thus been conducted, and industrial development has been remarkable. However, it has been continuously pointed out in academia that an in-depth analysis of friction laws and phenomena linked to the metal forming simulation (MFS) has not yet reached a sufficient state from an engineering perspective. Despite the significant impact of friction on the MFSs, its importance has been underestimated, and the related studies have been relatively limited. A few researchers on metal forming emphasized the inadequacy of the constant shear friction law (CSFL) and the constant friction coefficient for the Coulomb friction law (CFL). Yet, most researchers still use the CFL with a constant friction coefficient or the CSFL. Many researchers have related the friction coefficient to the yield criterion, and they believe that the friction coefficient cannot exceed a certain value (for example, 0.577). It has also been believed that the sticking condition is the same as the friction factor of unity, even though displacement and traction cannot be prescribed simultaneously in natural phenomena. Despite many researchers’ innovative academic and industrial contributions, friction phenomena in metal forming remain in an incomplete state of confusion. This study reviewed and synthesized research on friction phenomena during metal forming. The main review topics include friction phenomena, friction modeling and friction laws, friction-related issues, lubrication regime change (LRC), lubricants, and tribometers, with their application spaces limited to metal forming. This review synthesizes existing research related to friction in metal forming, proceeding in a problem-identifying and solution-oriented manner.

1. Introduction

1.1. Significance of Friction in Metal Forming

Metal forming is a technology that uses dies to directly apply heat and force to materials to produce the desired shapes of mechanical or automobile parts at low costs. Understanding and efficiently utilizing the natural phenomena occurring at material–die interfaces under elevated temperature and high pressure is thus fundamental. Engineering activities during the research and development of metal forming processes are significantly important in the metal forming industry [1,2,3,4,5,6]. Traditionally, the success of the metal forming process is determined by the competitiveness of the price and quality of its products. Nowadays, environmental friendliness in parts manufacturing is becoming important. Several trial-and-error experiments were inevitable while developing a metal forming process that satisfies the comprehensive requirements even three decades ago. The trial-and-error experienced during the development of metal forming processes is the direct cause of increasing costs, delayed responses to market demands, increased defect rates, and environmental destruction due to excessive energy use. These persistent problems are being overcome by simulation technology in metal forming.

Most research papers related to metal forming emphasize the importance of friction phenomena and materials’ flow behaviors. Friction has a direct impact on both the macroscopic and microscopic structures of the product [7], as well as on the wear of the die and the dimensional accuracy of the product. Additionally, it is crucial from the standpoint of economically selecting equipment [8], because it is sensitive to forming loads. Many studies on flow behaviors are reaching a quantitative level. These results are largely attributed to the advancement of inverse analysis technology that integrates advanced material testing techniques and CAE technology [9,10,11,12,13,14,15,16].

There are some difficulties in obtaining flow stresses at large strains at room temperature, which pose challenges to analyzing metal forming processes that are locally exposed to large strains, such as automatic multi-stage cold forging [17] and the burnishing process for preloading [18]. However, a slight difference in flow stress at large strains does not lead to a significant difference in plastic deformation, which strongly depends on the mass conservation law. In other words, the mass conservation law, which has a high accuracy in finite element (FE) predictions, significantly contributes to the stability of the flow, thus playing a role in compensating for the errors of the flow function.

On the contrary, the law of friction directly affects the deformation of materials and the wear of dies without any control device for material deformation. Nevertheless, research on friction in metal forming has been limited because the tribological phenomenon during metal forming occurs under extreme conditions of elevated temperature or high pressure, and the fundamental cause, lubrication, experiences drastic changes, represented by the LRC [19]. The impact of friction on the metal forming process is significant [20,21]. Generally, friction has an adverse effect [22]. The fundamental cause of die wear, which is directly related to productivity, is friction [23,24,25,26,27]. Lubricants are thus used to control friction during metal forming.

However, friction does not only have adverse effects. Friction enables rolling. Many bulk metal forming (BMF) processes, including forging, are characterized by inducing appropriate deformation, or metal flow, within the material, which improves the material’s microstructure. Without friction in the forging process, controlling the metal flow lines (grain or fiber flow lines) becomes difficult [28]. Of course, friction increases the forming load and dominates die wear [29,30]. On the contrary, the increased forming load may promote the sound formation of metal flow lines and cause a locally high strain rate. These have a significant impact on metal forming, whether it is good or bad. Friction also significantly impacts the changes in the microstructure of metal forming products, particularly the material near the material–die interface [7,31].

The CFL has been used in the strip rolling process, where the friction phenomenon is significant [32,33], since there are substantial pressure differences at each contact point, and the impact of the friction hill [34] due to friction on the process is considerable. Using the CSFL, it may be impossible to describe the material’s motion accurately, including the sticking phenomenon [32] occurring at the material–die interface.

1.2. Main Factors Affecting Friction

Suh and Sin [35] presented the new postulate that the friction conditions are affected by the sliding distance and the environment because of the deformation of surface asperities, the plowing by wear particles and hard asperities, and the sticking of the flat portions of the sliding material–die interface. They also revealed four or six stages of the friction regime depending on the sliding conditions.

Works on friction emphasize that the friction phenomenon has a deep functional relationship with pressure [27,36,37], temperature [27,38,39], relative velocity [36,38,39,40,41,42,43], surface expansion [44] or surface strain, the contact area ratio [15,45], lubricants [46,47,48,49], surface coating [50], material and die material properties [27], the oxide layer [51,52,53], surface flow pattern [36,44], surface roughness [42,54,55,56,57,58,59,60], the metal forming process [8], and the like.

From a macroscopic perspective, many factors influence friction, but the essential factors among these are pressure (normal stress) and the relative velocity of the material to die. The mathematical model of the friction phenomenon must include the relative velocity. This relative velocity determines the direction of the friction stress, and accordingly, the material–die interface is divided into the slipping region, sticking region, and transition region.

A function that connects these three regions with a single formula, known as a smoothing function [61,62], is used effectively. This smoothing function must be treated as part of the friction model in problem-solving, specifically in process simulation. However, many research papers have not done so. As a result, the sticking region and the transition region have been improperly assumed or ignored. These inevitably lead to misunderstandings about lubrication phenomena in metal forming, which are difficult to verify experimentally. Despite this, the development of lubrication engineering continues, and the lubricants that control it have been evolving in a direction that considers health and the environment. The environment in hot forging factories has thus improved significantly, and there is also a general trend of decreasing friction coefficient values in the case of cold forging.

The history of friction parallels that of human tool use. Around 450 years ago, Leonardo da Vinci was known to have quantitatively studied friction for the first time in human history [63]. Early researchers recognized friction as a result of plowing. Tabor [64] developed the early adhesion friction model, considering the increase in the real contact area ratio as the friction stress is increased to the point at which gross sliding occurs.

1.3. Reality and Issues of Friction

A CSFL was used in a few application papers related to simulation technology for BMF processes. For instance, there are cases where it is asserted that the CSFL adequately reflects actual phenomena [65,66], which is contrary to the tribological phenomena affected by many changing factors during plastic deformation, described in Section 1.2. There are numerous review papers, but they do not provide answers for the contradictory issues. The uncertainties of the friction model and some issues, including the coupling of friction and flow behavior, have never been resolved. CAE software explains friction phenomena from the perspective of its unique friction laws that are favorable for obtaining the economically advantageous convergence of the software [67].

The most significant misunderstanding about friction lies in the insufficient treatment of the effect of relative velocity in the formulation of friction stress. Relative velocity determines the direction of the friction stress. In addition, the friction smoothing function [61,68,69] used instead of the relative velocity ensures no slipping between the material and die, leading to a reduced friction stress when the pressure at the material–die inter-face is very high, enabling a solution that satisfies the yield criterion without requiring special numerical techniques [70,71], even when using a significant friction coefficient with the CFL [72,73].

The simplest problem arises from the dead metal phenomenon occurring in cylinder compression. Just because the pressure is high in the dead metal area (or sticking occurs) does not mean that the yield shear stress acts as the friction stress. The dead metal area corresponds to a necessary boundary condition where displacement or velocity is constrained to zero. Therefore, the friction stress at the sticking region satisfying the yield criterion is a secondary result of the process simulation. Many researchers point out the problem of CFL, as it has considerable friction stress at high pressures, but this is a result of overlooking the role of the smoothing function and the sticking phenomenon that occurs under high pressure. This kind of reasoning can be observed in many papers, but it is invalid [74].

Many researchers [31,65,66] argue that using the CSFL [75] is a way to resolve the issues of CFL caused by the occurrence of high pressure at the material–die interface [76]. However, the CSFL inevitably leads to critical results that overestimate friction stress at material–die interfaces under low pressure [70].

If Siebel [75] developed the CSFL to compensate for the fact that friction is not high under high pressure, the motivation for this can be criticized. The CSFL is also seen as being aimed at scholars who rely on analytical approaches based on upper-bound methods [19]. If the general friction law (GFL) originated from addressing the weaknesses of this CSFL [56,77], it would also be subject to criticism, because it may lead to a lower friction on average.

The CFL is more complex than the CSFL because it requires the continuous improvement of the unknown stress information during the application process [78]. To solve this problem associated with the CFL, using the penalty method [33,79] allows for approximating the normal stress as a velocity function, which can resolve these issues. However, this approach has a computation time disadvantage due to the stiffness matrix’s asymmetry. During an MFS being used in a traditional way, the problem of the friction stress’s dependence on the pressure can be overcome in an engineering manner by intermittently or iteratively improving the pressure at the material–die interface [78].

Wilson [19] pointed out that researchers view friction not as a tribological phenomenon but rather from the perspective of researchers’ convenience. He argued that researchers hold certain beliefs about the constant friction coefficient and factor. According to the research by Hatzenbichler et al. [67], the analysis results of different commercial software for the same ring compression problem are significantly different. In the comparative studies of commercial software conducted by Zhang et al. [80] on simple T-shape compression under the same conditions, distinct differences were observed in terms of deformation shape. Such theoretical and practical friction-related uncertainties persist even today, many years later. Notably, there is even confusion about the definition of the friction law. It is also proposed that this issue may have affected the software and that the related engineering analysis software may have solidified this problem.

Friction is a very complex problem with many influencing factors. New technologies are being actively integrated to solve these issues. Kchaou [81] studied tribology at elevated temperatures, that is, hot forming tribology, and presented the framework of the coupling between experimental design and machine learning to overcome various limitations concerned with the interdisciplinary character of hot temperature tribology. Different scenarios are discussed to develop new collaboration methods among the design of experiments, numerical development, and machine learning algorithms.

1.4. MFS and Friction Identification

Currently, various MFS technologies with accuracy are leading the development of metal forming processes toward a direction that eliminates trial production. Even 10 years ago, calculation time was the most critical factor in evaluating an MFS software [82,83]. At that time, the simulation results often served as simple assistants rather than companions for the designers. Recently, the accuracy of the MFS technology has become essential to realize the dream of metal forming process design engineers. However, the importance of the accuracy of the simulation has endlessly grown [84,85]. Factors affecting the FE predictions include the material model, numerical model of the process, mesh quality, material properties represented by flow behaviors, tribological features, press models, and the like. Among them, the flow behaviors [86,87,88] and tribological features [70,89,90] have been the hottest concerns for a long time.

For two decades, flow characterization technology has steadily improved with the development of material testing equipment and the advance in the inverse analysis technology [9,10,11,12,13,14,15,16] to characterize the flow behaviors of the materials. On the other hand, research on friction in metal forming is inactive compared to its importance. It seems likely that the friction problem in steel forging has not been greatly highlighted, because the dominance of flow behaviors of the steels with significant elongation and a high strain hardening capability on the material’s plastic deformation is clear against tribological conditions.

However, the effect of flow behavior on plastic deformation is a little weakened in the case of low strain hardening materials like aluminum alloys but that of friction on macroscopic phenomena becomes strong. Low strain hardening of the material evokes friction, significantly affecting the plastic deformation of the material. This feature also appears in the hot forging of aluminum alloys [91]. As a result, from the perspective of a forging process design engineer accustomed to steels with a moderate strain hardening capability, the friction while forging low strain hardening materials like aluminum alloys must be a big troublemaker.

Various studies [92] on tribo-test methods were conducted to identify tribological phenomena and to obtain parameters such as the friction coefficient. They include the standard or non-standard ring compression tests [9,15,44,93,94,95,96,97,98,99,100], various upsetting or cylinder compression tests [11,12,14,15,101,102,103,104,105,106], forward and/or backward extrusion tests [11,12,74,107,108,109,110,111,112,113,114,115,116], double cup and/or spike forging tests [117,118,119,120], contact area tests [16], ball-on-disk friction tests [27,121], pin-on-disk testing [38,122], sheet tensile or strip stretch tests [123,124,125], sheet rotational tests [126], deep drawing tests [127], strip drawing friction tests [115], Erichsen tests [128], ball ironing tests [129], two-disk apparatus tests [130], flat die friction tests or sheet metal friction tests [40,131], and inverse analysis methods or combined numerical and experimental approaches [9,10,11,12,13,14,15]. A few researchers [57,132,133,134,135,136,137,138,139] relied on analytical solutions to model or identify friction.

The most fundamental reason it is challenging to elucidate friction phenomena is that the lubrication at the material–die interface and the flow and heat transfer phenomena of materials are highly coupled [140]. Particularly, temperature directly impacts the lubricants and the flow stress of the material near the material–die interface, which leads to the need to separate the influences of lubrication and temperature in macroscopic phenomena. This issue is not easy, so the two elements, lubrication and flow, which are significantly affected by temperature, maintain a deep coupling state. Moreover, in metal forming, lubrication phenomena occur under extremely high temperature and pressure conditions, making experimental elucidation challenging.

Many studies have simultaneously used experiments and the finite element method (FEM) [44]. Various friction models have been proposed [68,141] to describe the tribological features at the material–die interface. Among them, the representative ones are the CFL [73], CSFL [75], GFL [56,77], hybrid friction law (HFL) [142,143], and modified friction models [68]. A few researchers [41,65,66,67,69,70,141,143,144] have conducted comparative studies of friction models.

1.5. Background and Purpose of the Paper

Beynon [145] summarized many aspects of the tribology in hot metal forming and emphasized that uncertainty prevails in the description of friction and heat transfer at the material–die interface. Tan [141] reviewed vast works on friction laws in BMF, including the CFL, CSFL, GFL, absolute constant stress law, and empirical friction law. Dohda et al. [53] reviewed the effects of temperature, coating, and lubrication on the tribological features in hot metal forming and tribometers for different metal forming processes at elevated temperatures. Nielsen and Bay [146] reviewed the most important contributions during the last 80 years, covering experimental techniques, analytical methods, and numerical approaches, with an emphasis on the significance of the real contact area. Seshacharyulu et al. [147] summarized the role of friction, applicability of friction laws, different tests to measure friction coefficients, and friction models in sheet metal forming (SMF).

Ahmad et al. [148] reviewed the influence of vegetable oil and surface texture on the tribological behavior in metal forming. Trzepiecinski and Lemu [90] summarized recent advancements and trends in friction testing for SMF and classified the tribological tests into direct and indirect measurement tests of the friction coefficient. Lee et al. [63] reviewed the works on the friction models by categorizing them into the boundary and mixed lubrication conditions, focusing on the contact models, to provide understanding and insight into the friction modeling and simulation.

Yang et al. [149] reviewed transient tribological phenomena, the interaction between friction and wear, and friction modeling techniques for transient behaviors for metal forming applications. They established interactive friction modeling for different application scenarios, including lubricated conditions, dry sliding conditions (metal-on-metal contact), and coated systems. Aiman et al. [150] reviewed the studies on the tribological analyses and the potential use of bio-lubricants in metal forming. Bay et al. [151] introduced a few solutions achieved through the activities under the name of ‘SHETRIB—New environmentally benign SMF tribology systems’ to develop new, environmentally friendly tribology systems for metal forming to substitute hazardous lubricants such as chlorinated paraffin oils.

This study reviews research on friction to identify the confusion of results and the causes and emphasizes the characteristics of friction evolution in metal forming. By synthesizing various research results, we recognize the friction phenomenon in metal forming and its mainstream research, identify the gap between industrial sites and researchers, and diagnose its causes. Notably, by presenting the pitfalls that researchers can easily fall into along with their grounds, we suggest a sound development of friction models and application software and a research direction centered on the essence of friction evolving during metal forming rather than being researcher-centered.

1.6. Review Methodology and List of Contents

The contents of major review papers related to tribological phenomena in metal forming were first reviewed, and a few root papers with many citations and the keywords were determined. As shown in

Table 1. Topics, keywords, and key contents.

Topics	Keywords	Section
Tribological issues in metal forming	Tribological issues, friction, lubrication	Section 2.1
Various factors affecting friction	Pressure, relative velocity, lubricant, temperature, strain rate, surface expansion	Section 2.2
Lubrication regime and LRC	Thick film, thin film, mixed, boundary lubrication, LRC	Section 2.3
Identification of friction laws	Nominal friction stress, friction law, Coulomb, constant shear, general, hybrid, smoothing function	Section 3.1
Modified friction laws for the varying friction condition during metal forming	Generalized friction coefficient parameter, critical surface strain, LRC, state variable effect	Section 3.2
Issues regarding the CFL and sticking phenomenon	Penalty scheme, sticking, CFL	Section 3.3
Ring compression test	Ring compression, friction calibration, sticking, friction hill	Section 3.4
Similarity and difference between CFL and CSFL	CFL, CoCSFL	Section 3.5
Critique of traditional friction laws with a constant friction coefficient or factor	Constant friction coefficient, constant friction factor, LRC, forward extrusion,	Section 3.6
HFL and state variabilization of the friction coefficient and factor	Hybrid friction law (HFL), friction coefficient	Section 3.7
Typical examples of LRC	Critical surface strain, cold forward and backward extrusion, hot forging, low strain hardening	Section 3.8
Lubricants	Lubricant	Section 4
Friction test and acquisition of tribological parameters	Friction test, tribometer	Section 5

, searches were mainly conducted through Google Scholar using the keywords. As a result, it was concluded that research on friction models used for CAE purposes, particularly in the field of BMF, is insufficient. Therefore, based on the authors’ experiences and analyses of the related literature, a forward-looking review of friction phenomena from the perspective of MFS was conducted. Topics, keywords, and key contents are summarized in Table 1.

2. Metal Forming Focusing on Tribology

2.1. Tribological Issues in Metal Forming

The quality of forged parts, including metal flow lines (grain flows), plastic deformation, and the like, is determined by the process design and conditions, such as die geometry and the tribological and thermal conditions at the material–die interface besides the initial material’s conditions. Many of these process parameters are well known and controllable, but it is difficult to predict the influence of friction on the process and product quality [109].

Friction conditions change during metal forming. Metal forming is an extreme lubrication application at an elevated temperature and high pressure. As a result, the LRC [130] occurs rapidly and partially. Challen and Oxley [152] presented the theoretical friction coefficients and wear rates from the analytical solutions of the plastic deformation of a soft asperity by a hard one. They proposed three different lubrication regimes depending on the interfacial conditions. The change in lubrication state during metal forming is significant, and LRCs are a characteristic of the lubrication phenomenon that occurs in metal forming.

The change in friction conditions during SMF is minor compared to BMF. Since the SMF generally falls under net-shape manufacturing, where the product is used as the final product, it is necessary to minimize friction in metal forming. The success of many sheet forming operations depends critically on the friction of the sheet on the blank-holder [36]. The CFL is generally applied to analyze SMF processes.

Many studies on friction, which is influenced by many factors, have been conducted in the field of SMF. Westeneng [135] presented a new contact model and a friction model and revealed that the friction coefficient is dependent on the nominal pressure, bulk strain, hardness, asperity height distribution of the material, roughness parameters of the tool, and boundary lubricant. Hol et al. [153] proposed a physics-based friction model to account for surface topography changes and friction evolution in the boundary lubrication region during SMF. They found that the friction coefficient varies spatially and temporally and depends on local process conditions such as contact pressure and the plastic deformation of the material. Shisode et al. [139] combined the flattening and asperity plowing models to create a multi-scale friction model for the boundary lubrication regime during the cup drawing process of zinc-coated steel sheets.

Different friction or lubrication regimes may occur depending on the lubricant and the surface topographies of the die and material. The boundary lubrication regime commonly prevails in SMF, particularly when a low amount of lubricant is utilized. In this case, the normal and tangential loads at the material–die interface are transmitted due to asperities. The friction coefficient in this regime depends on the micro-scale asperity interactions between the contacting surfaces. During SMF, the friction coefficient varies due to the evolution of the contact conditions at the material–die interface. Karupannasamy et al. [60] presented a multi-scale friction model to describe the friction in SMF, based on the asperity flattening and plowing mechanisms for pure plastic contact conditions. They indicated that friction depends on the surface topography of the contacting surfaces. The multi-scale friction model was further developed considering loading and reloading conditions for asperity deformation [126].

BMF is primarily used to manufacture parts intended for load support and power transmission. Therefore, the internal metal flow lines and the product’s strength are more important than the appearance. In BMF, where large plastic deformations are required, like in forging [91], or where the relative motion between the material and die is significant, like in drawing [154], a sudden change in friction conditions known as LRC arises.

Friction is both an object to be eliminated and a valuable factor for enhancing product quality. In forging, where healthy plastic deformation within the material is sought, friction is directly related to the integrity of metal flow. However, friction has a detrimental effect on the increase in wear and load. Mechanically, rolling is a process that maintains the resistance of the rolls against the sheet through friction, making friction essential [70]. Zhang and Yang [28] emphasized the importance of friction in achieving and facilitating sound metal flow, especially in a local loading process. Han and Hua [20] stressed the importance of friction in cold rotary forging because it not only determines the stable forming but also significantly affects the metal flow and the material’s forming limit. Alexandrov et al. [21] studied the correlation between the integrated strain rate intensity factor and the evolution of material properties near the material–die interfaces in metal forming processes. Razali et al. [7] revealed that the lubrication phenomenon in hot forging directly affects the metal flow lines and that the forged product’s microstructure directly influences their formation. In net-shape manufacturing, like coining, friction can also cause surface defects. Zhong et al. [22] presented a novel radial friction work model to predict the tendency of flash lines, which is one of the decisive defects occurring in coining.

The BMF process involves many processes sensitive to friction. These include forward and backward extrusion, ring compression, and the metal forming of materials with low strain hardening. Even in the case of an automatic multi-stage cold forging where lubrication film treatment is not possible between stages, friction remains essential [17]. The drawing process can also be considered another extreme process in terms of friction. The extreme friction occurs between the material and the die. Extreme contact and friction occur at localized points, especially from the perspective of the die. These promote the phase change in the lubricant, leading to accelerated localized wear. Accordingly, an LRC due to temperature can occur during the drawing process [154].

In hot forging, the weak friction may be unfavorable for quality control through the metal flow lines. However, since BMF is carried out at elevated temperatures and pressures, the wear life of the die has become a primary concern in most processes. In the case of cold forging, wear-resistant coatings for the die and lubrication coatings for the materials are essential. In hot forging, the use of lubricants is necessary to reduce friction, and it is also crucial to have wear-resistant treatment on dies. In BMF with low strain hardening, the influence of friction increases. It is not an exaggeration to say that BMF, especially forging, is a battle between the material’s strain hardening and friction. When the flow of materials is bidirectional or the area reduction ratio in extrusion is large, the effect of friction increases. One example is shown in Figure 1, which depicts a hot aluminum forging [91] that has a small wall thickness relative to its volume. In the forging of such products, the deformation shape is greatly influenced by friction, and there is a significant difference in the degree of damage to the lubrication film across different surface areas. In this process, especially when the strain hardening is slight, the probability of the LRC occurring increases.

In BMF, including forging and extrusion, the material–die interface experiences a significant LRC. The surface of materials in cold forging is mostly coated with a solid lubricant. The coating of the lubricant on the material’s surface can also be used in the hot forging of aluminum alloys [91]. The state of lubrication changes rapidly depending on the degree of damage to the coated surface. If the coating remains intact in the thick film lubrication regime, the friction stress is bound to be very small (for example, friction coefficient: 0.016 [19]) because it blocks the direct contact between the material and die. Direct contact between the material and die causes micro-bonding, fracture, separation of asperities, and debris creation, which increases friction stress and generates friction heat, ultimately accelerating adhesive wear [155]. It thus creates a vicious cycle. A tribological phenomenon is thus significantly affected by the destruction of the coated lubricant film, especially in cold forging. Friction cannot be simply described by the traditional friction laws [19]. Recent research works [17,91,154] revealed that friction is greatly affected by the tribological conditions at the material–die interface. Notably, as Wilson [19] emphasized, the CFL with a constant friction coefficient is helpful to predict metal flow only in the boundary lubrication regime. Yang et al. [115] tried to find the best tribological conditions in a rotary draw bending process, which are much affected by lubricant types and tube/tool materials.

Friction is a significant cause of wear that determines the productivity of metal forming processes [27]. Wang et al. [26] investigated the effect of load on wear behavior at elevated temperatures. They found that, at higher loads, the oxide layer more easily transmits the load to the counter surface, resulting in greater wear loss. Torres et al. [25] suggested that higher applied loads at elevated temperatures can increase the wear rate of steel. Stott and Jordan [23] showed that the anti-wear layers during sliding between metal contact surfaces at higher loads at 550 and 600 °C were broken down, leading to enhanced wear damage.

The frictional heat generated at the material–die interface produces welded debris from adhesive wear, increases friction, and reduces the operation efficiency [91]. Lee et al. [91] emphasized that traditional friction laws with constant friction parameters sometimes lead to entirely different results from the actual material flow. Raj et al. [156] used the varying friction at the roller–bead interface for the thermo-mechanical FEAs of the micro-rolling process. They reported that the interfacial friction caused irregular surface stress–strain and rolling load.

Navarrete et al. [157] studied the die wear in the coining process. They explored the combined experiments and simulations of the material’s elastoplastic deformation, friction effects, and die material wear rate, contributing to the productivity.

2.2. Various Factors Affecting Friction

Many factors significantly affect friction in metal forming. Summarizing related studies, they include the pressure [27], temperature [27,39], relative velocity [39,42,43], surface expansion [44] or surface strain, contact area ratio [16,45], lubricants [46,47,49,157], surface coating [50], the material properties of the material and die [27], oxide layer [51,52,53], surface flow pattern [44], surface roughness [42,54,55,56,58,59], and metal forming process [8].

• Pressure: Among the main factors affecting friction, the one with the most significant impact on friction is pressure. In an extreme case, without pressure there is no friction. Due to the complexity of mechanics in metal forming, however, pressure has become the most controversial topic. When friction stress is given (i.e., traction prescribed), the displacement or velocity in that direction should remain unknown. Despite this, there is a tendency to equate the friction factor of 1 with sticking (velocity boundary) [69]. This misunderstanding provides fertile ground for many controversial theoretical developments related to friction in metal forming.
• Relative velocity: It has at least a right to determine the direction of friction stress. When the right vanishes, the friction stress becomes unknown and has nothing to do with the frictional law. Svoboda and Jopek [43] experimentally showed the significant dependence of the friction coefficient on the strain rate (which is almost proportional to the relative velocity) at elevated temperatures. They showed that the friction coefficient decreases when the relative velocity is incredible.
• Lubricant: Kahhal et al. [47] determined the friction coefficient for each graphite, mica, and glass powder lubricant during the hot metal forming of an alloy steel. The results showed that the predicted friction coefficient for dry samples was 0.62, whereas the friction coefficients obtained using graphite, mica, and glass powder were 0.46, 0.29, and 0.18, respectively. Özakın and Erdil [49] conducted a study to replace synthetic-based lubricants for cold metal forming with vegetable lubricants. They reported that the performance of the new lubricant did not meet that of the existing synthetic lubricants in terms of the roughness transfer ratio. However, such attempts and advancements will continue in response to the modern demand for green manufacturing with a healthy shopfloor [158,159].
• Temperature, strain rate: Mirahmadi et al. [39] investigated the effect of the temperature and strain rate of the Ti6Al4V alloy on the friction factor using the isothermal compression test at elevated temperatures. It was found that they significantly affect the friction factor in a specific temperature range. Sheng et al. [27] experimentally showed the significant effect of temperature and test load on the friction coefficient. They revealed that the oxide layers favor obtaining a low friction coefficient under all the experimental conditions. The average friction coefficient decreased with the increasing load at 300 °C in a ball-on-disk tribometer, whereas it increased with the increase in applied load at 500 °C.
• Surface expansion, surface flow pattern including sliding velocity: Noh et al. [44] revealed through ring compression tests of perfectly plastic materials and FEM that the factors determining low and high friction include surface expansion and other surface flow patterns, such as sliding velocity and so on.
• Coating: Patil et al. [48] utilized the tin coating layer as a solid lubricant in cold steel tube drawing. They argued that an optimal thickness of the coating layer exists from the perspective of friction, specifically in drawing load. Using the Reynolds equation, Patri and Cheng [54,55] researched friction in thin or thick film lubrication regimes.
• Surface roughness: Zhang et al. [58] conducted a numerical analysis on the effect of surface roughness on friction under 2D planar conditions, while Mahrenholtz et al. [59] did so in a 3D space. Sigvant et al. [42] studied the effect of tool roughness and strain rate on the material flow in SMF, focusing on the CAE application.
• Various factors: Zhang et al. [58] studied the microscopic friction model to reveal the effect of roughness, pressure, adhesive friction coefficient, and relative velocity on the friction stress at the unlubricated material–die interface. They also studied the local friction model relying on the dimensionless lubrication number employed for calculating the varying friction coefficient with dynamic lubricant viscosity, relative velocity, contact pressure, and surface roughness. In the microscopic friction model, the friction coefficient increased with the velocity, while in the local friction model, the friction coefficient decreased with the velocity.

2.3. Lubrication Regime and LRC

In metal forming, the material–die interface changes as the process proceeds. With the progress of the process, pressure changes, temperature changes, and lubricant changes, surface roughness and surface expansion change. All other factors also change with the plastic deformation of the material. Therefore, friction must change. In some cases, friction changes drastically or discontinuously. This phenomenon is referred to as LRC [130,160].

Wilson [160] explained the changes in friction conditions during BMF as four LRCs [160,161,162,163]. Thick, thin [54,55], mixed [19,153,160,161,164,165,166], and boundary lubrication regimes [153] are fundamental. Other researchers have utilized this theory. Zhang et al. [58] studied microscopic and local friction models to investigate the factors affecting friction stresses. They explained the change in friction in SMF using three lubrication regimes, involving boundary, hydrodynamic, and mixed lubrication regimes.

Zhang et al. [167] showed that the magnitude of friction and surface roughness changes significantly upon reaching a 45% reduction in ring height during the cold ring compression of AISI 1045 steel. Almost the same phenomenon occurred during a cold ring compression process of aluminum alloy 5052, as reported by Zhang et al. [168]. Most friction models for the mixed and hydrodynamic lubrication regimes were based on the assumption of a Newtonian fluid, and their corresponding viscous shear stresses were calculated [165] for the friction stresses.

Hossen et al. [169] conducted experimental and numerical works to establish the variable friction coefficient model to improve the pulling force estimation for the split-sleeve cold expansion process. They reported an LRC during the process in which the friction coefficient changed from 0.03 to 0.13. The friction coefficient was influenced by fundamental contact area changes due to softer surface deformation. Leu [45] revealed that the friction coefficient experiences a sharp jump when the contact area ratio, which is sensitive to pressure and friction stress at the boundary, approaches 1.0. This fact helps explain the LRC and the critical surface strain [91]. He insisted that a friction coefficient μ in dry friction is a function of the dimensionless stress ratio and contact area ratio. However, Wang et al. [16] conducted tests on the contact between the specimen of ultrathin pure titanium sheets and rigid dies during a rubber mat forming experiment, showing that the friction coefficient decreases with normal pressure. This result contrasts with Leu’s findings, which claimed a significant increase in the friction coefficient due to the rise in the actual contact area dependent on the interfacial pressure. Darendeliler et al. [170] employed a variable friction coefficient model relating the parameters of a sheet metal drawing process to its local lubrication conditions for FE analyses of the process. The experiments validated FE predictions.

3. Friction Modeling and Solving

3.1. Identification of Friction Laws

As shown in Figure 2a, the friction stress ${\sigma }_t$ at the material–die interface can be treated as a function [70] of the normal stress ${\sigma }_n$ and acts in the direction that prevents relative motion between the two objects. Friction laws relate the friction stress to the pressure (negative of normal stress), as summarized in

Table 2. Friction laws.

Friction Law	Formulation	Features	Reference
CFL	Equation (1)	Constant friction coefficient Large normal stress issue	[73]
CSFL	Equation (2)	Constant friction factor Low normal stress issue	[75]
GFL	Equation (3)	Improved CSFL, contact area ratio	[77]
HFL	Equation (4)	Combined CFL and CSFL	[12,143,171]
Neumaier friction law	Equation (8)	Modified CSFL	[141]
Norton friction law	Equation (9)	Viscous friction model	[172]
IFUM friction law	Equation (10)	Mixed friction law of CFL and CSFL	[68]
Bernhardt friction law	Equation (11)	Avoid the normal stress issue in the CFL	[173]
Coulomb–Amonton friction law	Equation (12)	Avoid the normal stress issue in the CFL	[174]
Hol et al.’s FL	None	Not a closed-form function model considering the surface texture change	[175]

. There are two traditional friction laws for the functional relationship between the friction stress and normal stress or yield shear stress (k). Karman [73] developed a CFL (${\tau }_{cf}=-{\mu \sigma }_n,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} {\tau }_{cf}$ is referred to as nominal Coulomb friction stress from now on) to describe the friction behaviors between the workpiece and die in a strip rolling process. Since the nominal Coulomb friction stress can exceed the yield shear stress, they were hesitant to apply to CFL, owing to the possibly overestimated friction stress when no scheme to deviate from this matter was employed. Hence, Siebel [75] developed a CSFL (${\tau }_f=mk$, where ${ \tau }_f$ is the nominal constant shear friction stress, and m is the friction factor) for simplicity.

Notably, the two traditional friction laws with the constant friction coefficient and factor have some problems in describing the varying friction conditions during metal forming, where the interfacial condition changes significantly. Wanheim and Bay [77] thus presented the GFL (${\tau }_f=f\mathsf{\alpha }k$), where f and α are the friction factor and contact area ratio, respectively. This friction model was intended to describe the valid friction stress for the normal pressure range. As seen in Figure 2b [9], the curve representing the GFL is similar to the curve representing the CFL in low pressure conditions (p/flow stress is not less than 1.5). It converges towards the curve representing constant friction for high pressures (p/flow stress is greater than 3 to 4).

For numerical simplicity, a practical HFL [12,143,171], a combination of the CFL and CSFL, sometimes called Coulomb–Tresca friction, was employed, as shown in Figure 2b. The HFL at low friction follows the traditional CFL and the CSFL at high friction (with a hybrid friction factor $m^{\prime }$, defined in Figure 2b). When $m^{\prime }=m$, the HFL is biased on the CSFL side, like the GFL. Otherwise, when $m^{\prime }>m$, it is biased on the CFL side. The former may underestimate the friction. The latter stresses the CFL and the hybrid friction factor $m^{\prime }$, near unity [176], to control the friction stress at a high pressure for numerical efficiency or better convergence, particularly in the three-dimensional case.

From the standpoint of problem-solving, the CFL, CSFL, GFL, and HFL are expressed by the following equations, respectively:

$${\sigma }_t=-\mu {\sigma }_n\mathrm{g}\left(v_t\right)$$

(1)

$${\sigma }_t=mk\mathrm{g}\left(v_t\right)$$

(2)

$${\sigma }_t=f\alpha k\mathrm{g}\left(v_t\right)$$

(3)

$${\sigma }_t=\left\{\begin{array}{c}-\mu {\sigma }_{n}g\left(v_t\right) if \left|\mu {\sigma }_n\right|\le m^{\prime }k\\ m^{\prime }k if \left|\mu {\sigma }_n\right|>m^{\prime }k \end{array}\right.$$

(4)

where the function $\mathrm{g}\left(v_t\right)$, called the smoothing function, reflects the effect of the relative velocity $v_t-{\overline{v}}_t$ on friction stress ${\sigma }_t$, and the following function [61] is widely used:

$$\mathrm{g}\left(v_t\right)=-{\displaystyle \frac{2}{\pi }}{\tan }^{-1}\left({\displaystyle \frac{v_t-{\overline{v}}_t}{a}}\right)$$

(5)

where a is a positive constant but is significantly small compared to |${\overline{v}}_t$|.

The smoothing function in Equation (4) plays a vital role in FEA. If the absolute value of $v_t-{\overline{v}}_t$ in the sliding material–die interface is significantly less than a, the smoothing function converges to 1. Moreover, the sign of the function matches the relative velocity $v_t-{\overline{v}}_t$. However, among the material–die interfaces, there exists a partially sticking region where the following should be satisfied:

$$v_t-{\overline{v}}_t=0$$

(6)

The actual boundary conditions are formulated as one of the Equations (1)–(4), but there exist different boundary conditions that must be satisfied at a specific material–die interface. This problem is complex from a numerical analysis perspective, because the contact region must be divided into sliding and sticking regions, and the boundaries change during the analysis. This issue makes it difficult to achieve precise and efficient analysis. These problems can be solved numerically using the smoothing function in Equation (5). For the smoothing function to be zero, Equation (6) must hold. The reverse is also true. Therefore, during the process of finding convergence, if the nominal Coulomb friction stress at a point is greater than k, the absolute value of the smoothing function should decrease to meet the yield criterion. If the value of a is sufficiently small compared to the relative velocity, Equation (5) will approximately represent the sticking condition within an acceptable range at this point. On the other hand, the following function was used instead of the smoothing function in Equation (5) [69]:

$$\mathrm{g}(v_t)={\displaystyle \frac{v_t-{\overline{v}}_t}{\sqrt{(v_t-{\overline{v}}_t{)}^2+{v_0}^2}}}$$

(7)

where $v_0$ is a constant for solving numerical problems when the relative velocity is 0.

In addition to the traditional friction laws, many researchers have proposed friction laws, which, broadly speaking, can be seen as extreme cases or applications of the traditional friction laws [141]. Neumaier’s friction model [68] is aimed at improving the CSFL as follows:

$${\sigma }_t={\displaystyle \frac{2}{\pi }}mk\left(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\left|{\sigma }_n\right|}{{\sigma }_Y}}\right)\right)\mathrm{a}\mathrm{c}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{v_t-{\overline{v}}_t}{C}}\right)$$

(8)

where C adjusts the effect of the relative velocity on the friction stress. m, k, ${\sigma }_n$, and ${\mathsf{\sigma }}_Y$ are the friction factor, yield shear stress, normal stress, and the flow stress, respectively.

There is a friction law, derived from the CFL and CSFL, which particularly considers the effect of relative velocity. The viscous friction model, the Norton friction model [172], is defined as follows:

$${\sigma }_t=-\alpha K{\left({\left(v_t-{\overline{v}}_t\right)}^2+v_0^2\right)}^{{\displaystyle \frac{pf-1}{2}}}\left(v_t-{\overline{v}}_t\right)$$

(9)

where α is the viscous friction coefficient, and K is regarded as a material constant, which is considered a function of the flow stress and effective strain rate. pf is the sensitivity to the sliding velocity.

The IFUM (Institute of Metal Forming and Metal Forming Machines) friction model [68] is defined as follows:

$${\sigma }_t=\left[0.3\left(1-{\displaystyle \frac{\overline{\sigma }}{{\sigma }_Y}}\right){\sigma }_n+mk{\displaystyle \frac{\overline{\sigma }}{{\sigma }_Y}}\left(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\left|{\sigma }_n\right|}{{\sigma }_Y}}\right)\right)\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{v_t-{\overline{v}}_t}{C}}\right)}^2\right)$$

(10)

where C is a friction control parameter at a large relative velocity. The higher it is, the higher the friction stress.

Hol et al. [175] proposed the friction model for large-scale forming simulations of SMF processes based on the micro-scale surface changes due to the combination of normal loading and stretching with the adhesion and plowing effect between contacting asperities.

Notably, Bernhardt [173] aimed to solve the problem of high normal stress by correlating the friction coefficient itself with the normal stress at the contact boundary as follows:

$${\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{{\sigma }_Y^2}{{\sigma }_n^2}}-{\left({\displaystyle \frac{1-2v}{1-v}}\right)}^2\right)}$$

(11)

It should be noted that many researchers mentioned the limit of the friction coefficient. For example, Leu [45] argued that it cannot exceed 0.577 in practical applications, relating the friction coefficient to the yield criterion, although this is controversial. A few researchers studied friction and the laws of friction based on this theory.

Due to the normal stress issue in the CFL, many modified models have been proposed, and the following Coulomb–Amonton friction model [174] is also a typical example:

$${\displaystyle \frac{{\sigma }_t}{k}}=\sqrt[n]{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}{\left({\displaystyle \frac{{\mu \sigma }_n}{k}}\right)}^n}$$

(12)

Luo et al. [41] conducted a comparative study on the friction laws in the hot forging of a Ti-6Al-4V turbine blade. They indicated the low friction problem of the CSFL. They also criticized the CFL, stating that the CFL is limited in wide application due to the possibly overestimated friction stress. This criticism is a result of ignoring the role of the smoothing function. They also acknowledged the realism of the IFUM friction model while emphasizing the importance of velocity in friction. In the friction evaluation study conducted by Murillo-Marrodan et al. [69], which simulated a skew rolling process, the IFUM friction model predicted inappropriate results. In contrast, they concluded that the Norton friction model was the most suitable. However, it should be noted that the evaluation should be dependent on the features provided by the specific commercial software and the techniques it is based on.

3.2. Modified Friction Laws for the Varying Friction Condition During Metal Forming

From the standpoint of metal forming, the traditional CFL has two problems. The simple linearity between pressure and friction stress is a fundamental problem. The change in lubrication regime at the material–die interface due to extreme tribological states during metal forming is also significant. Therefore, the friction coefficient is bound to change during metal forming.

Wilson and Cazeault [177] revealed that the friction coefficient during wax-lubricated strip drawing was almost zero in the thick film lubrication regime, while it rose to 0.16 in the boundary lubrication regime. They also showed that the friction coefficient changed drastically around the die semi-angle from 15 to 20 degrees. This fact implies that the LRC causes a drastic change in the friction condition and die wear. This fact emphasizes that the friction coefficient is a function of the tribological parameters [19].

Lee et al. [91] treated the friction coefficient in Equation (1) as a function of state variables, including the pressure, the material’s surface strain, and temperature, to enhance the traditional CFL from the LRC perspective. For convenience, the following separable variable function, consisting of the influence of each state variable, that is, the weight, was used:

$$\mu ={\mu }_0{W}_E\left(\epsilon \right)W_T\left(T\right)W_p\left(p\right)W_{\alpha }\left(\alpha \right)$$

(13)

where ${\mu }_0$ is the friction constant. $W_E\left(\epsilon \right)$, $W_T\left(T\right)$, $W_p\left(p\right)$, and $W_{\alpha }\left(\alpha \right)$ are weighting functions of the surface strain, temperature, pressure, and contact area ratio, respectively. Each weighting function can be defined as a piecewise linear function, as shown in Figure 3 for $W_E\left(\epsilon \right)$, which can be identified by an inverse analysis technique [91].

Likewise, the friction factor can also be considered as a function of the state variables as follows [178]:

$$m=m_0{W}_E\left(\epsilon \right)W_T\left(T\right)W_p\left(p\right)W_{\alpha }\left(\alpha \right)$$

(14)

where $m_0$ is the friction constant.

3.3. Issues Regarding the CFL and Sticking Phenomenon

From the perspective of the FE prediction of the metal forming process, the CFL remains a highly controversial topic. In mechanics, sticking (essential boundary condition) and friction conditions (natural boundary condition or mixed boundary condition) cannot be imposed simultaneously. However, in many research papers, the conditions of m = 1 and sticking are described as being the same. In engineering, the sticking region and $v_t$ and ${\overline{v}}_t$ in Equation (5) cannot completely match due to a kind of penalty method. In this region, the Coulomb friction stress is determined by Equation (1), and the nominal Coulomb friction stress plays a kind of penalty constant function that only determines the slight difference between $v_t$ and ${\overline{v}}_t$. In contrast, the friction stress itself is an unknown constant. It is noteworthy that the calculated penalty constant defined at the sticking material–die interface, which varies from position to position, and the related actual friction stress are as follows, respectively:

$$K={\displaystyle \frac{\pi \left\{{\alpha }^2+{\left(v_t-{\overline{v}}_t\right)}^2\right\}}{2a}}{\tau }_{cf}=-{\displaystyle \frac{\pi \mu {\sigma }_n\left\{a^2+{\left(v_t-{\overline{v}}_t\right)}^2\right\}}{2a}}$$

(15)

$${\sigma }_t={\displaystyle \frac{2aK}{\pi \left(a^2+{\left(v_t-{\overline{v}}_t\right)}^2\right)}}\mathrm{g}(v_t)$$

(16)

In conclusion, the friction coefficient does not need to impose additional constraints other than being dependent on state variables. The yield criterion is applied to all particles of the material, so there is no need to mention additional constraints specifically at the boundaries. When a convergent solution is obtained in metal forming, the result satisfies the yield criterion for all particles. Therefore, the imposition of additional conditions on the friction coefficient is assessed as originating from, not considering, the sticking phenomenon or the dependence on relative velocity as factors affecting the frictional force.

In addition to the case that the nominal Coulomb friction stress exceeds the yield shear stress, the change in the direction of friction stress at a single point must be accompanied by a transitional region. Within this region, a sticking zone exists. For the convenience of theoretical development, many idealized friction models ignore this transition region [141], but this neglect can lead to misunderstandings about unrealistic friction models and friction coefficients. Therefore, the problem of determining this sticking region and its peak is significant in analyzing metal forming processes [141]. The smoothing function of Equation (5), which is aligned with this purpose, is natural from the perspective of mathematical modeling of friction phenomena and is very important. In practice, such considerations must be made when developing a significant friction model.

Notably, in implementing this numerically, considering the stable acquisition of the converged solution and computational efficiency, the imposition of constraints regarding the maximum friction stress is a separate issue. Empirically, acquiring a converged solution is not problematic in the simulation of two-dimensional plane strain and axisymmetric metal forming processes, as seen in the example in this subsection.

In the case of three dimensions, however, it may be necessary to impose conditions on the friction stress regarding the economic acquisition of convergence. The HFL centered on the friction coefficient is a representative method for this. The inadequacy of a constant friction coefficient (which fails to reflect the functional relationship of a real contact area ratio and pressure, etc.) can also be improved from the perspective of numerical analysis, meaning functions of state variables can easily resolve it and the like [91]. It is noteworthy that the smoothing function has high nonlinearity, and as a result, many integration points must be used in simulating a complicated forging process like a crankshaft forging process, which can lead to a significant increase in computation time and deteriorate the convergence.

The shear stress at the sticking interface in the plastic region should satisfy both the equation of the equilibrium and yield criterion. However, the friction stress at the material–die interface in the plastic region where slipping occurs must satisfy the equation of the equilibrium, yield criterion, and friction law. Since the CFL in Equation (1) also deals with the sticking condition, there is no mathematical problem in expressing all types of contact stresses, including friction stress in the sliding material–die interface and tangential stress in a sticking region. Equation (5) thus effectively prevents the friction stress in the CFL from being greater than the yield shear stress (k). When the nominal Coulomb friction stress exceeds k, $\left|v_t-{\overline{v}}_t\right|$ should be small enough to meet the yield criterion. Thus, sticking in the material–die interface should occur when the nominal Coulomb friction stress is sufficiently great. It should be noted here that the tangential stress in a sticking condition generally does not reach k [70].

In addition to the two friction zones discussed above, a real problem involves the transition zone. This transition zone is also approximately satisfiable using the same logic. The solution obtained in this way, especially the scope of the transition zone, may depend on the value of a [179]. However, if the value is appropriate, it does not pose an engineering problem, indicating that understanding the friction models and their numerical techniques is required before using commercial software, including the above perspective [71].

The above fact implies that the separation of the smoothing function in the friction law for FEM is impossible. Due to the extreme example of a contact condition known as sticking, which commonly occurs during metal forming, an integrated understanding and representation that takes the sticking phenomenon into account is required. For instance, sticking also occurs in ring compression tests, and the friction hill in rolling is a representative example. The material around the sticking region moves away from the center of the sticking region. This flow phenomenon of the material occurs because the die exerts friction stress towards the center of the sticking region on the material. As a result, the pressure in the sticking region generally increases, and due to the increased nominal Coulomb friction stress, the smoothing function should approach zero. This means that the relative velocity between the material and die is close to zero, which eventually imposes an approximate sticking condition.

In the transition zone, friction varies, resulting in continuous changes in the velocity field. When the velocity difference exceeds the threshold, the smoothing function of the arctangent function rapidly converges to unity. In other words, the friction stress converges with the nominal Coulomb friction stress. Meanwhile, just as it is difficult to ascertain friction in the transition zone, scientific determination regarding the a-value in Equation (5) is not easy. The related literature [68,71] recommends values that are sufficiently smaller than the velocity of the die. In conclusion, the upper limit constraint of 0.577 for the friction coefficient μ lacks theoretical justification and does not reflect a comprehensive view of the contact problems in metal forming. Therefore, this point cannot serve as a motivation for developing a new friction law.

On the other hand, realistically speaking, there are not even numerical problems in solving the FE equations under the CFL of Equation (1) in 2D (axisymmetric, plane strain). Ghawi et al. [71] incorporated the CFL at all material–die interfaces, effectively capturing friction hill phenomena and significant pressure variations. They analyzed the friction smoothing function of Equation (1), focusing on the a-value to address the unique challenges of high-pressure friction.

However, the situation may be different in the case of 3D, because the direction of the coordinate axis t, where the frictional force acts, changes during the convergence process of the solutions, and g$\left(v_t\right)$ is a highly nonlinear function. Theoretically, there are no problems, but considering the convergence challenges and computational inefficiencies that may arise when it is sufficiently small, the HFL is useful.

Sticking occurs partially in processes such as ring compression, forward and backward extrusion, and rolling, and it is essential to note that this phenomenon is unknown. Therefore, in a strict sense, Equations (1)–(4) are neither stress boundary conditions nor velocity or displacement boundary conditions. They are boundary conditions that predict both the friction stress and sticking condition simultaneously in a problem-friendly manner. Such a problem-friendly friction model is essential for efficient and accurate practical MFS. Since all friction laws are based on the concern that the CFL can exceed the yield shear stress in the sticking or high-pressure region, the smoothing function should be considered when considering the friction law.

3.4. Ring Compression Test

Kunogi [180] first utilized the ring compression test, conceptually depicted in Figure 4a, as a simple and fast method for comparing lubricant properties in forging. The ring compression test has been widely employed to measure friction in BMF and characterize the material’s flow behaviors because it is sensitive to friction conditions [181,182,183,184,185,186,187,188,189,190,191,192,193].

Male and Cockcroft [182] first established the friction calibration curves (FCCs) for the ring compression test based on experimental and theoretical analysis. For the ring compression, some theoretical analyses were carried out to elaborate FCCs [184]. The type of material and the forming condition may influence the FCCs. Avitzur [186] created the FCCs using a compensatory method through an analytical approach. Sofuoglu and Rasty [185] experimentally and numerically revealed the difference in FCCs between black and white plasticines. Zhu et al. [189] considered the heat transfer in constructing the FCCs using an FEM. Zhang et al. [193] studied the influence of forming conditions on the FCCs using an FEM. Hwang et al. [33] constructed the FCCs using a penalty method that focuses on the accuracy of friction stress, using both the CFL and CSFL.

Figure 4b shows typical FCCs for the material with a flow function of $\overline{\sigma }=50.3{\left(1+\overline{\epsilon }/0.05\right)}^{0.26}$. Note that the initial ratio of the outer diameter to the inner diameter to the height of the ring is 6:3:2, as shown in Figure 4a.

Comparing the slopes of the curves in Figure 4 shows that the CFL and CSFL produce nearly the same results, especially for a smaller reduction and lower friction. However, as the friction and reduction increase, the difference in the slope of the FCCs increases because the CFL reflects the normal stress variation at high friction and high reduction, while the CSFL does not. Joun et al. [70] revealed almost the same results for the viscoplastic material.

Apparently, the FCCs obtained using the ring compression tests imply the friction characteristics. However, the FCCs do not always provide accurate or reasonable information about the frictional features. The pattern of the FCCs of the CFL and CSFL is similar, and the friction coefficient and factor that correspond with each other can thus exist. Based on this fact, it has been frequently believed that the CFL and CSFL are equivalent. Notably, this may not be true from the standpoint of forging. The normal stress at the material–die interface does not significantly change during a ring compression test compared with the typical forging. The friction stress does not significantly change during the ring compression test, depending on the location, even though the CFL is used. Thus, the two FCCs should be similar to each other. On the contrary, the friction at the material–die interface during forging can be significantly different from position to position. The CFL and its variants are thus better than the CSFL.

Figure 5 shows the FE predictions of a ring compression process [194] and compares the results analyzed under the same conditions using two friction coefficients, namely Case (a) 0.1 (low friction) and Case (b) 0.5 (high friction). First, the ratio of the friction stress to the absolute value of the normal stress was checked. In Case (a), the ratio is 0.1 at all nodes, while in Case (b), the values vary depending on the location. As seen in Figure 5a, since all nodes in Case (a) move outward, the smoothing function becomes 1.0. Therefore, the smoothing function in Equation (1) only determines the direction of friction.

On the contrary, the motion pattern of the material in Case (b) is entirely different from that of Case (a). As seen in Figure 5b, there is a friction hill, and the direction of the friction stress changes around this point. Additionally, the value remains below 0.5 in a very wide region. Regions that are not 0.5 belong to the sticking region or the influence zone of the sticking centered around the friction hill or are part of the transition region. In this sticking region and transition zone, the smoothing function determines the direction and magnitude of the unknown friction stress. It is similar to calculating the unknown friction stress using a penalty method [33,79]. Here, the friction coefficient and the normal stress merely contribute to the penalty method’s penalty constant.

Mittelman et al. [98] stated that the ring compression test is inappropriate when the friction coefficient is above 0.45. As shown in Figure 5, although the FE simulation was made assuming a friction coefficient of 0.5, a sticking region occurred due to high pressure, and the calculated friction coefficient in the sticking region cannot be considered high. In other words, on a general level, high friction does not occur in metal forming. The reason is that the sticking phenomenon results from plastic deformation and the increase in contact pressure. However, Mittelman et al. [98] and many other researchers [195] have confused the condition of the unity friction factor with sticking or the limiting friction coefficient of 0.577. The mechanical phenomenon of sticking is not a simple issue [7]. Sticking in the extrusion or forging with a flat-face die may occur thermomechanically regardless of the friction conditions [7,144].

There are claims that it is an advantage to evaluate friction without load and flow function information, but obtaining precise FCCs requires results that are based on refined flow functions. The drawback is that, since the strain increments during the tests are not large, we can only obtain friction information limited to low strain from the ring compression. Since the change in pressure during ring compression is not large compared to forging, the CFL and the CSFL must be similar. Therefore, arguing the justification of the CSFL through ring compression is unreasonable.

3.5. Similarity and Difference Between CFL and CSFL

The comparisons between the CFL and CSFL were given by several researchers [70,141,144]. Tan [141] compared several friction laws (CFL, CSFL, GFL, and the pressure index exponent friction law) using cylinder upsetting tests for an A6082 alloy lubricated with different lubricants. He concluded that calibration curves of the contact area ratio are more sensitive to friction at the material–die interface than those of the pressure at the material–die interface. It was also shown that all the friction laws produced outcomes consistent with the tests. They insisted that the upsetting of materials with significant strain hardening is influenced by the dead metal at the center and the resulting small relative motion, as well as the effect of the arctangent function, implying that the upsetting experiments on materials with significant strain hardening are not suitable for friction evaluation.

Bašić [144] compared the CFL and CSFL in the context of the extrusion process using FVM. They confirmed that the two friction laws regarding extrusion pressure are clearly different. In other words, a continuous increase in the friction factor leads to a continuous rise in extrusion stress. In contrast, the friction coefficient increases the extrusion stress up to the friction coefficient of 0.3, and any further increase in the friction coefficient does not affect extrusion stress.

Murillo-Marrodan et al. [69] compared the velocities and power consumptions in a skew rolling process obtained using four friction models (CSFL, Norton, Neumaier, IFUM), excluding the CFL because of a hard change in friction stress in their CFL [67].

Joun et al. [70] studied the effects of the friction law on the material’s plastic deformation and forming loads in strip rolling, ring gear forging, multi-step extrusion, and pipe shrinkage and expansion. In the case of a low aspect ratio, contact ratio, and friction value, the CFL and the CSFL predicted nearly the same FE predictions. Friction in BMF is less dominant when the material is well-lubricated and the plastic deformation is not too severe. In this case, including cold forging, the two traditional laws of Coulomb friction and constant shear friction yield similar results.

Ring compression tests often come up when discussing the CFL (with the friction coefficient) or CSFL (with the friction factor). The ring compression test at room temperature convinces researchers that the two friction laws are similar. Ultimately, it leads to an underestimation of Coulomb friction and daringly dismisses the opinions of lubrication experts that various factors influence friction. The complacency of commercial software developers and users is solidifying this phenomenon.

It is unreasonable to compare the CFL and CSFL directly. After all, neither is perfect, and they have different backgrounds. In terms of friction, they show similarities in simple problems. However, when the deformation is complex and there are significant changes in contact pressure at the material–die interface, they inevitably lead to large differences. Without even borrowing Wilson’s argument, it is clear that, under any conditions, the CSFL where the difference in friction stress is slight will have many problems. The CFL assumes that the normal stress and friction stress have a linear relationship. In this regard, the CFL is often deemed to be more flexible than the CSFL, which is only influenced by the material’s flow behavior. Although there is a growing preference for this in practice, the use of modified models reflecting the characteristics of the process is unavoidable in practical applications.

Many researchers have attempted to compare and evaluate these two types of friction. For example, Balasundar and Raghu [66] compared the CFL and CSFL through the numerical analysis of an ECAP process and reported contradicting results. They evaluated the effect of the CFL and CSFL on the deformation pattern, strain distribution, and load requirement during the ECAP process. They concluded that the CSFL is better than the CFL in the numerical analysis of the ECAP process. However, it may be an exaggeration to generalize this. The CSFL fits better in the ECAP process because the interfacial damage during ECAP can be neglected. It can be determined that this is not due to the CSFL fitting well, but rather that the thick lubrication regime state is maintained during the process, which keeps the friction coefficient low.

Zhang et al. [80] found that the friction coefficient is related to the friction factor as follows:

$$m=2.3\mu$$

(17)

which is almost the same as

$$m=2.4\mu (\mathrm{o}\mathrm{n} \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e})$$

(18)

The above relationship was obtained by Joun et al. [70] under the conditions of $\overline{\sigma }=66.0{\dot{\overline{\epsilon }}}^{0.195}$ and tool velocity of −200 mm/s using the rigid-viscoplastic FEM.

The similarity of the two friction laws means there is a small variation in the pressure distribution in the testing method compared to a complex metal forming process. Therefore, meaningful comparison results cannot be obtained from ring compression tests where the pressure does not change much. Joun et al. [70] reported that, when two friction laws equivalent in the context of a ring compression test were employed to simulate a ring gear process ($\overline{\sigma }=66.0{\dot{\overline{\epsilon }}}^{0.195}$, punch velocity = −500 mm/s) using a viscoplastic FEM, the CFL (friction coefficient = 0.1) exhibited around a 30% difference in the forming load compared to the CSFL (friction factor = 0.2). The difference comes from the fundamental difference in plastic flow caused by the two friction laws and friction constants that are equal in ring compression tests. Figure 6 shows that the CSFL in the center of hot forging generally predicts a milder plastic flow pattern than CFL. Empirically, the actual plastic flow pattern is closer to the CFL [196]. The constant friction law diverges from the CFL as the process becomes complex and the pressure varies significantly from position to position, making it distinct from the actual phenomenon. Zhang and Yang [143] attempted to determine the relationship between the friction coefficient and friction factor in the Coulomb–Tresca friction model, but they could not establish a quantitative relationship between them.

3.6. Critique of Traditional Friction Laws with a Constant Friction Coefficient or Factor

Wilson [1] studied the LRC during bulk metal forming. He presented that the lubrication state at the material–die interface in BMF starts from a thick film lubrication regime and evolves towards a boundary lubrication regime via thin film and mixed lubrication regimes. Wilson emphasized that the CSFL can satisfy only the thick film lubrication regime, which seldom occurs during bulk metal forming. Nonetheless, the literature survey shows that many researchers and engineers rely on the CSFL.

The controversy regarding the friction law persists. Many variables influence it, and the yield criterion complicates the issue in simulating metal forming processes. For instance, although many researchers currently use CFL, Jain and Foster [140] claimed that the CFL applies only when the pressure is low, unlike in metal forming. In addition, by emphasizing that the friction factor must change according to the forming conditions, they implied that there is an inadequacy of the CSFL in the metal forming process, where changes in friction conditions, especially pressure and temperature, and deformation are inevitable.

Behrens and Schafshall [65] presented the CSFL with the adaptive friction factor based on the neural network and modified ring compression test. They presented that the CSFL is better than the CFL under high-pressure conditions. However, there is a limitation in comparing the two friction laws because the pressure difference in ring compression is not significant. Thus, this claim is likely to be controversial. The increase in the actual contact area ratio determines the phenomenon of reducing the friction coefficient at high pressure. As seen in Figure 5b, it is judged to be the effect of sticking that occurs at high pressure, namely, the phenomenon of the reduced friction coefficient. In other words, it is considered a natural phenomenon where friction stress decreases due to the reduction in asperities and the occurrence of sticking. This phenomenon can be resolved by the smoothing function in Equation (5). If a convergent solution has been obtained, the yield criterion should be automatically satisfied. For this to happen, the friction stress must be less than k. It is a phenomenon akin to sticking, where the value of the smoothing function decreases so that the friction stress reaches a suitable convergence value regardless of the magnitude of the pressure. In the past, when only the yield criterion was considered without sufficient experience accumulated regarding such issues, it was judged that the friction stress must be less than k [197]. They ignored the relative velocity of the material to the die. They considered the magnitude of the friction stress as the nominal friction stress, that is, mk (nominal constant shear friction stress) or $-{\mu \sigma }_n$ (nominal Coulomb friction stress).

A specific confusing situation related to friction can be found in the research results of Hatzenbichler et al. [67]. They conducted the benchmark between selected commercial software based on a model for ring compression tests. The same physical and numerical influence parameters were used for all simulations. They compared friction features in terms of plastic deformation during the ring compression using the commercial software but found significant differences in the FE predictions, showing a significant divergence in the predictions, depending on the software used.

Joun et al. [76] studied the multi-step forward cold extrusion process in Figure 7, which experiences the LRC, and solved this tribological phenomenon using the friction coefficient function of the effective strain of the material at the material–die interface. They simulated the process with two friction conditions to visualize this phenomenon by the comparison of the FE predictions. The analysis information is as follows:

-

Flow stress: $\mathsf{\sigma }=50.3{\left(1+20\epsilon \right)}^{0.26}\mathrm{M}\mathrm{P}\mathrm{a}$;

-

Punch’s speed: 1 mm/s.

-

Friction conditions:

Case A. Constant friction coefficient = 0.1;

Case B. Piecewise linear friction coefficient function defined by the following sample points (effective strain, friction coefficient): (0, 0.01); (0.3, 0.03); (0.5, 0.05); (1.0, 0.1); and (2.0, 0.2).

The FE predictions of the effective strain and grid distortion of the upper material at the final stroke are shown in Figure 8, indicating that the material at the entrance in Case A experienced plastic deformation. Notably, the upper material in Case B remained plastically undeformed, even though the friction coefficient exceeded 0.18 at the third step.

It was well known that the lubricated film coated on the material is not damaged much within the container during the multi-step cold forward extrusion process, even though the material’s surface near the exit is severely scratched. This tribological phenomenology implies that the LRC occurs in the multi-step cold forward extrusion process with a considerable total reduction in area, emphasizing that the magnitude of the friction coefficient must change and reflect the LRC as the state of the material–die interface changes.

The traditional CFL problem described above is less likely to occur when using a CSFL. However, in this case, if the length of the extrusion container is large, the inaccuracy of the load increases, making it impossible to express the unavoidable high friction due to lubricant damage and excessive surface expansion at the outlet. Above all, the issue of friction stress being determined almost independently of the significantly varying pressure at different locations is a fundamental defect of the CSFL.

The GFL to solve such problems can be used to address the excessive friction issues mentioned above; however, because it is a modification of the CSFL, it cannot avoid the fundamental problem. Therefore, a simplified friction model developed based on a single phenomenon has limitations in reflecting the actual situation. An extreme example is the LRC [17,91,139,154] commonly occurring in BMF. It is noteworthy that research on friction during metal forming is needed for the forging of various difficult to forge materials, such as aluminum and magnesium alloys, as well as the need for the scientific forging process to be developed in accordance with net-zero requirements.

3.7. HFL and State Variabilization of the Friction Coefficient and Factor

As seen in Figure 2, the GFL is a refinement that addresses the excessive friction stress predicted by the CSFL at low pressures. While there are clear improvements, there are issues related to not allowing friction stress beyond a certain level determined by the real friction factor. In brief, the GFL is deemed to be the CSFL. However, numerous examples emphasize its invalidity. Additionally, there are problems with obtaining the actual contact area ratio.

Many researchers have tried to deal with the practical problems of the traditional CFL. Notably, most works are controversial because they are based on the existence of the maximum friction coefficient, the equivalence of the sticking condition to the maximum friction factor (m = 1), or the equation of μ = 0.577m. However, it is not without practical issues related to the CFL under high pressure and high friction.

The direction of the frictional force aligns with the direction of the object’s relative velocity from the target’s perspective. Since the line of sliding in a two-dimensional case is fixed, and only its sign should thus be determined by the smoothing function, the problem of relative velocity related to friction is very simple. However, the situation differs in three-dimensional cases. The sliding direction is a function of the varying nodal velocities during the convergence process. The friction thus has a relatively large impact on the solution convergence. The magnitude of the friction stress follows the same scheme as the two-dimensional case. Except for cases where the penalty method [33,79] is used to approximate the friction stress as a function of the velocity field, the load or stress information calculated at the previous iteration or analysis step should be utilized to calculate the current friction stress. This scheme may decelerate the solution convergence because it can hardly employ the change rate of friction stress for calculating the improved solution. Therefore, numerical problems may arise from the solution, and depending on the technique, this can lead to non-negligible differences in results [67].

A few studies were based on the HFL, or Coulomb–Tresca friction law, where the CFL is applied when the nominal Coulomb friction stress cannot exceed the yield shear stress. Contrarily, an HFL is used at extremely high pressure, which may cause sticking. This method has strengths in terms of the convergence of the solution and computation time in simulating complicated three-dimensional processes when obtaining FE predictions. However, like all other methods, it has some disadvantages in accurately predicting the sticking region. Fortunately, the occurrence of the sticking region is influenced by the die shape and friction. The HFL thus started from believing in the CFL.

Orowan [32] used the CFL but employed static friction (assuming the friction stress is k) when the pressure and friction coefficient exceed the yield shear stress at a high pressure. He was thus the first to use a type of HFL. In the cold-bond-rolling process of double-layer sheets by Tzou et al. [176], the CFL was used, but when high pressure occurs at the material–die interface, the shear stress of the k value was utilized. Huang and Tzou [198] employed a special HFL of CFL (outer side) and sticking condition (inner side) to solve the problem that the nominal Coulomb friction stress near the central axis exceeds k in the compression forming of a rotating disk. The works of Tzou et al. and Huang and Tzou are based on a type of extreme HFL first utilized by Orowan [32]. Unlike them, Joun et al. [76] used the HFL to achieve a stable solution convergence and economical calculation in practical applications.

Notably, unlike Orowan [32] and Joun et al. [76], Zhang and Ou [171], Gavrus et al. [12], and Ghassemali et al. [13] used the HFL to address problems at low pressure with a focus on the CSFL like the GFL. Zhang and Yang [143] compared an HFL that simultaneously uses the CFL and CSFL with traditional friction models. Zhang and Ou [171] investigated the relationship between the friction coefficient and friction factor in an HFL using ring compression tests in dry and lubricated environments. Gavrus et al. [12] determined the friction parameters of the HFL using an inverse analysis method with the experimental load of a forward extrusion process. Ghassemali et al. [13] reported that an HFL was required to be implemented for better fitting with the experimental results of a progressive micro-forming process compared to the CFL and CSFL. They used the following relation between the friction coefficient and friction factor to determine the friction coefficient in the HFL, considering its upper limit.

$$\mu =0.577m$$

(19)

Note that Equation (19) is based on von Mises’ yield criterion [45], which is a little different from Zhang et al. [80] and Joun et al. [70], as shown in Equations (17) and (18), respectively.

Although the form of the general and HFLs is the same, there is a difference depending on what is placed at the center of the concept. The GFL is a type of HFL that introduces the contact area ratio to achieve results similar to the CFL at low pressures, that is, to solve the problem of the CSFL at low pressure. However, the GFL was inherently designed to solve the inaccuracy of the CSFL at low pressures. The problems it seeks to address are thus different from those of the HFL. As seen in Figure 2 [171], the HFL centered on a CSFL underestimates the friction stress. On the contrary, in the HFL focused on the CFL, the hypothetical friction factor is intended to regulate excessive friction stress. The hypothetical friction factor is usually determined to be less than 1.0, considering computational efficiency with better solution convergence.

From the perspective of an analytical method and FEM, the CSFL is advantageous because the friction stress is given as a base value. The HFL is fundamentally based on the CFL and supported by the strategy of preventing the friction stress from exceeding the yield shear stress. This strategy helps the CFL overcome its matter of inefficiency. The traditional CFL’s inaccuracy arises from the assumed constant value of the friction coefficient. This issue can thus be resolved by expressing the friction coefficient as a function of the friction-influencing factors, like in Equations (13) or (14).

3.8. Typical Examples of LRC

3.8.1. Cold Forward and Backward Extrusion of an A6061 Alloy

Wagener and Wolf [8] emphasized that the assumption of constant friction in metal forming was prevalent, and they highlighted that friction varies depending on the process, lubrication conditions, and the specific areas of the material–die interface through experiments with various materials and cold processes. The forward and backward extrusion process is known to be sensitive to friction. The left side of Figure 9 shows the experiments of the axisymmetric forward and backward extrusion process of aluminum alloy A6061 [199]. The main points in this process are the two shape variables at the top and bottom.

The best size for the shape variable on the bottom side obtained using the initial flow curve in Figure 10 and constant friction coefficients (μ = 0.0, 0.025, 0.05, 0.075, 0.1, 0.15, and 0.2) or friction factors (m = 0.0, 0.05, 0.1, 0.15, 0.2, 0.3, and 0.4) was a maximum of 0.57 mm [199] at the frictionless condition. It is significantly less than the experimental value of 2.5 mm, indicating that it is impossible to obtain engineering-significant results for this process using traditional CFL and CSFL.

Thus, the piecewise linear continuous function shown in Figure 11 was regarded as the friction coefficient, and it was optimized along with the flow curve in Figure 10. The flow curve up to point F′, corresponding to the fracture point in the tensile test, as shown in Figure 10, was obtained using the combined elastoplastic FEM and tensile test method [200,201], and the flow curve thereafter was determined using optimization techniques along with the friction coefficient function. The optimal flow curve and friction coefficient function, obtained by using the difference between the experimental values and predicted values of the two shape variables as the objective function, are shown in Figure 10 and Figure 11, respectively.

The FE predictions in Figure 9 were obtained using the optimal flow curve and friction conditions, indicating that the difference between experimental and FE-predicted values is negligible from an engineering perspective. The friction coefficient function in Figure 11 reveals the LRC, showing that a sudden increase in the friction coefficient occurs when the effective strain of the material at the contact surface reaches 0.8. This fact means that the GFL, which depends directly on the actual contact ratio that changes gradually according to the magnitude of the pressure and the like, cannot explain this phenomenon.

On the other hand, based on the aforementioned content, it can be seen that strong friction occurs at the lower internal interface, owing to an LRC, as the effective strain increases, as shown in Figure 9. The relatively strong friction in the lower internal interface to the outer interface suppresses the flow of the material. It is thus believed that the bending phenomenon of the lower cross-section, which could not be predicted with a constant friction coefficient, has been predicted using the optimal friction conditions.

3.8.2. Hot Forging of an A4032 Alloy Piston

Lee et al. [91] experimentally and numerically studied a hot forging process of an A4032 alloy piston. They employed verified flow information to reveal the tribological phenomenon. They tried to find appropriate solutions using various constant friction coefficients [91] and friction constants [178]. However, as shown in Figure 12 (when the friction coefficient is 0.2), all the FE predictions exhibited significant differences from the experiments, particularly in the region highlighted by the red dashed box.

On the other hand, when using the friction coefficient function shown in Figure 13, obtained through optimization, which changes rapidly from the moment the material’s surface strain reaches 1.5, similar numerical results to the experimental results could be obtained, as shown in Figure 14.

The effective strain of the material differs from position to position, as can be seen in Figure 12b and Figure 14a. It is interesting to note that the effective strain pattern is similar to the color of the scratched surface in Figure 12a. This fact means that the friction coefficient changes rapidly when the surface strain reaches around 1.5, as shown in Figure 13, implying that the LRC from a thick film to boundary lubrication schemes occurred in the regions where the material surface’s color became bright. Almost the same phenomenon can occur during the cold or hot forging of low strain hardening materials, particularly when the bulk material flows through narrower die gaps.

4. Lubricants

The history of metal forming is inseparably linked to the history of lubricants, because lubricants effectively alleviate the friction and remove the heat generated during metal forming. Lubricants play a crucial role in reducing the friction between metal and dies in plastic deformation, improving process efficiency, and maintaining the quality of metals. Many researchers have conducted studies and experiments to understand the characteristics of lubricants [202,203,204,205,206,207,208,209,210,211,212,213].

Hersey [202] established the relationship among friction, viscosity, speed, and pressure in horizontal journal bearings, laying the foundation of fluid lubrication theory and systematized lubrication conditions using the dimensionless parameter known as the Hersey number. Bowden and Tabor [203] argued that thin metallic films formed on surfaces during bearing operation are a key lubrication mechanism. Their research revealed that the composition and properties of bearing metals determine the formation and stability of these films and high friction and wear behavior, contributing to the development of the theory of boundary lubrication. Green [204] theoretically analyzed the junction model of unlubricated metal contacts and explained that the actual contact surface consists of microscopic contact points. He also demonstrated that friction between metals is primarily governed by processes such as the formation, shearing, and breaking of these microjunctions. Montgomery [205] experimentally demonstrated that the chemical reaction of lubricants on aluminum contact surfaces significantly affects wear behavior. In particular, polar additives form a protective film to reduce wear, whereas certain chemical species can promote corrosion and adhesive wear through surface reactions. Nayak [206] modeled surface roughness as a Gaussian random process and proposed a method to probabilistically predict the peak density, curvature, and distribution. Wilson [207] demonstrated that, at the onset of the extrusion of ductile metals such as aluminum, the fluid lubrication film collapses instantaneously due to surface velocity differences and pressure gradients. Booker and Huebner [208] presented an engineering approach for analyzing Reynolds’ equation using the finite element method, demonstrating that lubrication problems with complex boundary conditions and geometries can also be solved numerically. Patir and Cheng [55] proposed the Average Flow Model to analyze lubrication between rough surfaces, quantifying the effective effects of surface roughness using flow factors and enabling practical lubrication analysis considering surface roughness directionality and contact conditions. Through analytical methods, Challen and Oxley [209] elucidated how local micro-asperity contact transitions to full contact in material–die interfaces. Wilson and Wang [210] demonstrated that, in simple stretch forming, fluid lubrication behavior transitions from boundary lubrication to fluid lubrication depending on the lubrication state, as described by the Reynolds equation, and that the wedge effect primarily governs the formation of the lubricant film. Black et al. [211] experimentally and theoretically analyzed the deformation behavior that occurs when a hard wedge slides over a ductile surface. Wilson [212] described the effects of surface roughness, contact pressure, and the chemical properties of lubricants on the friction coefficient and shear behavior. Bay et al. [213] presented a model to predict friction behavior in cold forging processes for aluminum, steel, and stainless steel.

Lubricants reduce friction, decrease process load, and improve surface quality and tool life. To study lubricants, it is necessary to understand the frictional relationship between materials and dies. A few researchers focused on studying friction to establish these tribological relationships. Notably, the materials with great elongation and strain hardening capabilities, involving most forgeable steels, tend to transfer the plastic deformation to their neighborhood. The dominance of flow behaviors during steel forging is thus evident.

In contrast, for materials with a low strain hardening capability, such as aluminum alloys, the influence of flow behavior on the macroscopic phenomena of the process is somewhat reduced, but the effect of friction increases sharply. The low strain hardening capability of the material induces friction, which significantly affects the material’s plastic deformation. Therefore, from the perspective of engineers accustomed to steel with pronounced work hardening, friction during the forging of low work hardening materials like aluminum alloys is a major concern. Studies have typically focused on forging steel and aluminum alloys for lubricants. Consequently, this study conducted a literature review limited to research on these two metals. Research on lubricants for hot forging under various conditions has been conducted for a long time [214,215,216,217,218,219,220,221,222,223]. Godfrey [214] experimentally demonstrated that Tricresyl Phosphate acts as a lubricant by forming a protective film (tribofilm) on the steel surface through the generation of phosphate-based compounds. Bay et al. [114] simulated the extreme tribological conditions of cold forging and clarified the correlation between friction and lubricant performance with temperature, surface expansion, and sliding distance. In particular, they showed that phosphate coating combined with MoS2 exhibits excellent lubrication.

Kim et al. [215] experimentally studied the effects of the lubricants containing various additives on the friction and wear characteristics of steel surfaces under boundary lubrication conditions. Johnson et al. [216] elucidated the formation mechanism of solid lubricant films generated when phosphate ester vapors react with steel surfaces in high-temperature environments. Bay et al. [224] systematically summarized experimental methodologies for evaluating lubricant performance in SMF processes. They analyzed how lubricant performance varies depending on the process’s friction conditions, temperature, and pressure. Farias et al. [217] revealed that zinc phosphate coatings, when combined with lubricants, can reduce steel friction and enhance wear resistance. Byon et al. [225] emphasized that the optimal combination of coating materials and lubricants in the wire drawing process stabilizes the forming force and improves surface quality. Bay, N. [218] tried to overcome the limitations of conventional zinc phosphate coating-based lubrication systems and introduce new lubrication systems to enhance process efficiency and environmental sustainability. Dubois et al. [219] revealed the effects of interactions among the temperature, materials, and lubricants on friction and surface quality, showing that graphite and MoS2 lubricants perform well under high-temperature conditions. Wang et al. [220] conducted research on a new lubricating coating that could replace zinc phosphate coatings. Zhang and Li [221] demonstrated that manganese phosphate treatment effectively improves the friction and wear characteristics of steel piston materials under boundary lubrication conditions. Lorenz et al. [222] utilized the double cup extrusion test to experimentally measure the friction coefficients to compare and analyze the friction behavior of various lubricants for cold forging and, through this, evaluated the performance of the lubricants. Kim et al. [120] assessed the friction characteristics of dry-coated and forming oils in cold forging processes using double cup and spike forging tests. Fleming [223] evaluated lubricant performance under boundary lubrication conditions in cold rotary forging processes, suggesting optimal lubrication strategies to improve the forming quality of high-strength martensitic steels.

Although the research history is short compared to steel, studies on lubricants for applying aluminum alloy [226,227,228,229,230,231,232,233,234,235,236,237] in plastic deformation are very actively conducted. Bay [226] presented standard lubrication strategies for steel and aluminum in cold forging and evaluated the effects of phosphate, calcium aluminate, and fluoride in lubrication for the cold forging of aluminum. Manisekar and Narayanasamy [227] experimentally analyzed the effect of friction conditions on barreling phenomena through the cold forging experiments of aluminum square billets under asymmetric friction conditions. Malayappan and Narayanasamy [228] experimentally analyzed the effect of asymmetric friction conditions on the flat die surface during the cold forging process of aluminum cylindrical billets, revealing that the friction coefficient significantly impacts the degree of barreling. Petrov [229] presented a systematic approach for selecting lubricants in the hot isothermal forging of aluminum alloys and argued that graphite-based lubricants are optimal under most conditions. Baskaran and Narayanasamy [230] experimentally analyzed the effect of lubricants on barreling in the cold forging process of aluminum billets and found that the friction coefficient significantly influences the degree of barreling. Narayanasamy et al. [231] reported that lubricants affect friction conditions, controlling the deformation of metals, which plays a crucial role in barreling phenomena. Sagisaka et al. [232] evaluated the performance of environmentally friendly lubricants in aluminum cold forging using spline extrusion-based friction tests. Feyzullahoğlu et al. [233] assessed the effects of forging and heat treatment on the friction and wear characteristics of aluminum–silicon and aluminum–lead-based bearing alloys. Medea et al. [234] experimentally analyzed how the performance of lubricants changes with the temperature during the cold forging process of aluminum alloy A1050. Priyadarshini et al. [235] used cylindrical specimens of aluminum 6082 alloy to evaluate the impact of friction conditions (dry, grease, and mineral oil) on barreling. Yang et al. [236] investigated the effects of various contact conditions (temperature, load, and sliding speed) on the friction coefficient and lubricant failure in aluminum forming processes. Schell et al. [237] conducted flat die strip drawing tests over a sheet temperature range of 20 °C to 425 °C to analyze the performance of various types of lubricants and their mixtures and proposed improvement measures.

5. Friction Test and Acquisition of Tribological Parameters

Trzepiecinski and Lemu [90] summarized the methods for describing friction conditions in conventional and incremental SMFs, focusing on the main disadvantages and limitations of modeling the friction phenomena in specific areas of the material. They emphasized that there are direct and indirect test methods. Isogawa and Ito [92] analyzed the tribo-test methods using various forging processes, including ring compression, extrusion, spike forging, etc. Dohda et al. [53] reviewed the works on tribological features and tribometers at elevated temperatures, focusing mainly on the experimental works.

Ring compression test: Dudkiewicz et al. [100] conducted hot ring compression tests to obtain the friction coefficient for the hot forging of stainless steel using various lubricants. As a result, the friction coefficient ranged from 0.19 to 0.32, depending on the material and lubricant. Hill and Turner [99] conducted ring compression tests on various materials under different conditions (temperature) and lubricants. Due to the issue of shear force exceeding at high pressures, they used the CSFL. They relied on experimental data and inverse analysis techniques using finite element methods to identify the friction condition. They showed that temperature variation in the experimental parameter matrix played a more significant role in determining deformation than the lubrication agent.

DePierre and Gurney [93] developed the method for utilizing the ring compression tests to quantitatively reveal the friction factors and flow curves of the test material. Gzyl et al. [95] examined the performance of various lubricants for aluminum alloy A5083 at 200 °C using ring compression tests. The combined FEM and experiment method was used to identify friction factors. Gomez et al. [15] conducted a large-scale ring compression test and identified the friction using an inverse analysis technique. They revealed that the inner diameter reduction is highly sensitive to the friction coefficient. Zhang et al. [97] developed the incremental ring compression test to identify the friction condition in the cold-rolling process of a spline or thread shaft. The magnitude of the friction coefficient or factor obtained by this test exhibited a notable difference from the magnitude by a traditional ring compression test. Rao and Xu [94] evaluated the interfacial friction factor and the material’s flow behavior using the FCCs in ring compression testing and the neural network approach. FE predictions for different ring geometries and friction factors were made, and acceptable results were obtained with the maximum error of about 5% for both the friction factor and flow information.

Generally, the advantage of ring compression is that it allows friction conditions to be determined without using flow information. However, Mittelman et al. [98] utilized the inverse analysis technique, a combination of ring compression testing and finite element method, to elucidate friction phenomena according to temperature, strain rate, and friction conditions, emphasizing the importance of simultaneous analysis of flow stress and friction conditions.

Obtaining the friction coefficient or friction factor from ring compression tests is fundamentally based on the assumption that this value remains constant at the material–die interface. However, according to Wilson et al. [238], this value varies with the radius. In the ring compression test, the material’s deformation is insignificant, and the normal stress at the material–die interface is so small that it cannot be compared to conventional forging. Many researchers have appreciated the ring compression test for its methodological simplicity. However, the limitations due to significantly smaller deformations and pressures compared to the forging process are evident. Therefore, various experimental approaches have been attempted to obtain friction information.

• Complementary or non-standard ring compression test: Since a pressure higher than the flow stress is applied in ring compression testing, there are limitations in evaluating the friction state of exceptional processes (e.g., forward extrusion with low extrusion ratio) where experiments are conducted at a pressure lower than this. Petersen et al. [9] used a new complementary ring-test geometry to reduce the increase in ring compression load due to barreling. In this model, normal stresses over some areas of the material–die interface are lower than the material’s flow stress. They used inverse analysis technology to identify the friction condition shown in Figure 2, focusing on the GFL. Sanodiya and Choudhary [96] identified friction from the compression tests of non-standard (or differently shaped) specimens that can replace standard ring compression test specimens. It was confirmed that the results obtained from the non-standard specimens were similar to those obtained from the existing specimens.
• Warm and hot upsetting sliding test: Soranansri et al. [106] used the warm and hot upsetting sliding test to identify the friction coefficient and friction factor during aluminum forming processes at elevated temperatures. The experiment was conducted for the A6082-T6 alloy using the AISI H13 tool steel under dry contact conditions at 400 °C. It was experimentally found that the friction coefficient was 0.57, and the friction factor ranged between 0.76 and 0.90, depending on the flow behavior.
• Upsetting or cylinder compression: In cylinder compression tests, the deformation shape, namely the barrel shape, is sensitive to friction. Utilizing this point, numerous studies have been conducted to evaluate friction. However, the compressive load does not sensitively change to friction. Friction increases the material’s resistance to deformation, which leads to an increase in the compressive load. However, friction reduces the rate of increase in the area of the interfacial surface directly related to the compressive load. Therefore, the sensitivity of friction to the compressive load decreases [15,105]. Ebrahimi and Najafizadeh [101] presented the method for calculating the friction factor using cold and hot cylinder compression tests. They used only the measured barreling of the compressed cylinder with the upper-bound solutions to calculate the friction factor since the barreling is dependent on the friction and the initial height-to-diameter ratio.

Azushima et al. [103] conducted lubricated cylinder upset tests using a specimen of commercially pure aluminum and a liquid lubricant to obtain the friction coefficient at the material–die interface. They revealed that friction coefficients depend on the reduction in height and the position at the material–die interface. Li and Ma [14] conducted experimental upsetting to determine the flow behavior and friction performance for a range of press amounts from the original specimen to half its height in adhesive friction. Lin and Chen [102] calculated the variation in a friction coefficient during the upsetting process of a mild steel using the experimental load information. They relied on the axisymmetric elastothermoplastic FEM, an inverse analysis technique based on the optimization method, the Levenberg–Marquardt method, and the like.

• Theoretical (numerical or analytical) methods: Sun et al. [239] presented a novel method to calculate the interfacial friction stress in friction stir welding without determining the friction factor first. The interfacial stress was described by a function of temperature and calculated by a three-dimensional computational fluid dynamics (CFD) model. Based on slip line analysis, Challen and Oxley [132] used two-dimensional rigid cylindrical asperities to identify the CFL during material removal occurring in moving contact between abrasive or polishing grits and softer workpieces. Their approach contributed to the works of Ma et al. [136] and Hol et al. [153] for their multi-scale friction models. Ma et al. [136] presented a multi-asperity macro-scale friction model for the aluminum extrusion process by adapting a wedge-shaped single asperity model to determine the friction force at each sheet material–die contact patch from which the overall friction coefficient was calculated.

Mishra et al. [137] proposed an analytical model for an ellipsoidal and elliptical paraboloid-shaped rigid asperity plowing through a softer substrate, which behaved like a perfectly plastic material. The model considered the asperity orientation relative to the sliding direction. Greenwood and Williamson [57] presented a model for calculating the asperity deformation under the normal load using an elastic Hertz contact theory without considering strain hardening during plastic deformation. Shisode et al. [138] presented a semi-analytical normal load contact model for the coating and substrate materials, coating thickness, and measured surface topography of the rough surface.

Sutcliffe [133] predicted the asperity deformation to examine the crushing of surface asperities by a frictionless die under conditions of bulk deformation of the underlying material. However, the slip-line method for the simplified plane strain problem was insufficient for realistic surface topography. Westeneng [135] adapted Sutcliffe’s plane strain approach [133] to derive a contact model for arbitrarily shaped asperities and height distribution. Wilson and Sheu [134] proposed the plane strain model to investigate the wedge-shaped asperity flattening in the presence of normal load and bulk strain using an upper-bound method.

• Various extrusion tests: Nakamura et al. [107] established a friction testing method with a combined forward rod–backward can extrusion, obtained the theoretical FCCs, and evaluated a friction coefficient and friction factor along the container wall and the conical die surface in the forward rod extrusion without information about the forming forces and the flow stress of the material. Hsu and Huang [108] investigated the friction distribution in a combined forward and backward extrusion process. They developed an inverse analysis technique to determine the friction coefficient of the lubricated material–die interface where grease or different lubricants were applied. Buschhausen et al. [109] proposed a friction test, based on a double backward extrusion process, to obtain a friction factor from the most significant difference in extruded cup heights. The FCCs were constructed.

Sofuoglu and Gedikli [111] developed an open-die backward extrusion test to determine the friction coefficient. They calculated the FCCs constructed by the FEAs of the processes with different normalized specimens’ aspect ratios and die geometries. Gariety et al. [110] used the double cup backward extrusion test for the performance evaluation of various lubricants. They constructed the FCCs, i.e., cup height ratio vs. punch stroke, for different friction factor values. Ito et al. [112] proposed a new tribosimulator with a high surface expansion condition under high normal stress, a type of backward extrusion test with a taper punch. It is thus named the taper cup test.

Takahashi et al. [113] studied the effect of production rate on the performance of a double-layer-type, environmentally friendly lubricant in the cold forging of an aluminum alloy using the tribological test of combined forward and backward can extrusion. The lubrication performance was investigated numerically and experimentally, revealing that an increase in the production rate decreases the friction coefficient. Kačmarčik et al. [74] proposed a friction-sensitive double backward extrusion method where the backward extrusion cylinder is backward extruded by a special punch with an opening through the center along with the related FCCs. The final height of the specimen determined the friction coefficient.

Gavrus et al. [11] proposed the friction test method based on a special upsetting test and an optimal direct extrusion test to identify the material’s flow behavior and friction coefficient directly from the load–stroke curves using an inverse analysis technique. Bay et al. [114] presented a new friction and lubrication test for cold forging. The cylindrical billet was backward extruded, and then, at the end of the extrusion process, the punch was rotated while the die was kept stationary. The test allows for controlled variation in the surface expansion range of 0–2000%, the tool temperature in the range of 20–270 °C, and the sliding length between 0 and infinity. Zhang et al. [80] presented the T-shape compression, a new friction testing method, composed of a cylinder’s combined compression and extrusion between a flat punch and a V-grooved die. The experiments and FE predictions showed that the stroke–load curve and the height of the extruded part are both sensitive to friction. The experiments provided the FCCs.

Gavrus et al. [12] conducted the experimental upsetting and extrusion tests of an A5083 aluminum alloy to characterize the corresponding flow behaviors and to identify friction coefficients using only the recorded load–stroke curves. The friction parameters were found by a two-step inverse analysis method applied to a finite element model of the extrusion process. In contrast, the real flow behavior of the material was characterized by a compression test of a special dumbbell specimen. Chen et al. [104] established quantitative correlations among the friction coefficient or factor, flow curve of the material, forming load, and die geometry in the forward extrusion with a conical die.

• Contact area test: Wang et al. [16] conducted the contact test. They proposed a friction coefficient model based on real contact area change, which accurately predicted friction changes with normal pressure and material properties compared to the classical CFL.
• Double cup and spike forging tests: Kim et al. [120] created FCCs to determine the friction factor through finite element analysis in double cup and spike forging tests. They emphasized that different tribo-testing methods yielded different friction factors under the lubricating condition of dry-in-place coating (by the water-based lubricants) with forming oil. Xu and Rao [117] conducted FE simulations of the spike forging processes to reveal the effects of different geometric parameters, processing variables, and interfacial conditions on the instantaneous spike height under cold and hot forming conditions. Hu et al. [118] developed an optimized spike forging test method using FEM and optimization techniques. Hu et al. [119] studied the effect of tooling surface on the friction during cold forging of an aluminum alloy using a ring-with-boss compression test and an optimum spike forging test. They emphasized the friction anisotropy, which hindered the friction factor from being determined directly with flat ground platens.
• Ball or pin-on-disk friction test: Sheng et al. [27] used the ball-on-disk tribometer to reveal the effect of temperature and test load on the friction coefficient. Wang et al. [121] studied the load-dependence of tribological behaviors of the sodium carbonate coating on stainless steel using the ball-on-disk friction test at elevated temperatures. Wang et al. [122] studied an experimental method of pin-on-disk testing for developing a pressure-dependent variable friction model for DP780 AHSS sheet sliding against the DC53 cold-work tool, aiming at more accurately predicting the springback. The developed model was successfully applied to the forming and springback of U-shaped bending under tension, focusing on the accurate springback prediction. Grüebler and Hora [38] conducted the pin-on-disk test of the stainless steel sheet to reveal the effect of temperature and velocity dependence on friction. They identified the friction coefficient using the combined FEM and experiment method. Friction tests using different temperatures showed a change in the friction regime.
• Sheet strip stretch or rotation test: Karadogan and Hatipoglu [125] easily and roughly calculated the stress and the friction coefficient at the material–die interface from the strain information measured by an optical strain measurement system, assuming the flow behavior of the sheet material through a sheet strip stretch test. Karupannasamy et al. [126] conducted the rotational friction test to measure friction under loading/reloading for SMF processes.
• Deep drawing test: Hu and Vollertsen [127] presented the friction test method for deep drawing applications to study the size effects. The friction coefficient function was derived from the results of strip drawing investigations to describe the friction behavior in the whole deep drawing process.
• Erichsen test: Giuliano [128] employed the Erichsen test to identify the friction coefficient of the CFL by coupling experimental and numerical results.
• Simulative twist compression test: Yang et al. [115] employed the simulative twist compression test to obtain the friction conditions for a rotary draw bending process where friction influences wrinkling, wall thickness variation, and cross-section deformation. Ma et al. [116] identified the significant influential factors affecting rules and mechanisms on the friction coefficient using the high-temperature twist compression test combined with the design of experiments. They also presented a pressure- and temperature-related dynamic friction model, which is better than the CFL at predicting the defect in metal forming.
• Ball ironing test: Sae-eaw and Aue-u-lan [129] proposed a ball ironing test as a simulative tribo-test to evaluate the lubricants of the thick sheet ironing process, focusing on a large surface expansion, large pressure, and high relative velocity. Finite element modeling and statistical analysis were employed to determine the maximum load and the specimen’s final height. According to the results, the maximum load is very sensitive to the friction, which is used as an indicator to evaluate and approximate the friction coefficient with FCCs.
• Strip tensile or drawing test: To evaluate friction under conditions, Duncan et al. [123] developed the tensile strip test that simulates the stretching of a sheet over the punch corner radius in forming a shallow stamping in a typical draw die. Hao et al. [124] developed two physical models or friction simulators based on stretching a strip around a pin to characterize SMF friction. They determined the effects of strain, stretching speed, lubrication, pin radius, and wrap angle on the friction coefficient. Trzepieciński and Fejkiel [240] studied the effect of sheet deformation on the change in the surface roughness and friction coefficient and the correlation between the surface roughness and frictional conditions of the tested sheets in the strip drawing test. According to their friction and wear behavior, Kondratiuk and Kuhn [241] evaluated the hot dip aluminum–silicon and electroplated zinc alloy coatings for hot forming applications. The friction coefficient was revealed in hot strip drawing experiments. Additionally, wear characteristics were evaluated in hot forming tests.
• Inverse analysis technique: Szeliga et al. [10] applied the inverse analysis to identify friction and rheological models in metal forming.
• Flat die friction test—sheet metal friction test: Lee et al. [131] studied the effect of surface roughness and lubricants using a sheet metal friction tester, i.e., a flat die friction tester. They formulated the friction coefficient as a function of surface roughness. Han and Kim [40] studied the effect of contact pressure on friction, focusing on the SMF of high-strength materials. They conducted the flat-type friction test with a high-strength bare steel sheet under various contact pressures, revealing that the effect of contact pressure on the friction behavior of the steel sheet is significant, especially on HSS stamping with a wide range of contact pressures.
• Scuffing test: Schipper and De Gee [130] studied the relationship between a lubrication mode diagram for concentrated contacts and the ‘IRG transition’ diagram. The scuffing test was conducted using a two-disk apparatus, revealing that three distinct lubrication regimes could be observed, including elastohydrodynamic lubrication, mixed lubrication, and boundary lubrication.

6. Conclusions

This review of the numerous studies regarding the factors affecting friction and evolving under severely elevated temperatures and high pressures during plastic deformation in metal forming is by no means insufficient for experiencing the important lubrication phenomena along with the history of many practical academic advancements. However, there has been a lack of studies on how to utilize the research works and, particularly, on the correlation of the relative velocity of the material to the die and sticking phenomena. For instance, while many studies emphasized the problem of using the CFL at high pressure, they overlooked the effect of high pressure on the relative velocity of the material to the die. It should be noted that this relative velocity determines the direction of the friction stress and is intrinsically related to the presence of sticking. In MFSs, higher-order relative velocity smoothing functions, such as the arctangent, are used to reflect this characteristic. A review of various papers has emphasized that such realistic problems are not being addressed in depth in academic publications.

During metal forming, friction at the material–die interface is influenced by various factors. Various experimental and analytical studies have been conducted on the characteristics of friction in metal forming carried out at elevated temperatures or high pressures. However, these studies were not smoothly connected to the MFS technology that has become mainstream over the past 30 years. The reason is that, due to the nature of friction, it exhibits high nonlinearity and is linked to most state variables, directly affecting the convergence of the solution, making it difficult to separate it into a separate user routine. Above all, since the 2000s, the development of friction routines has been led by a small group of researchers centered around software companies, which fundamentally causes a mismatch between the diversity of friction and the technological development environment.

In metal forming, the characteristics of friction are represented by friction evolution and LRC. Based on the LRC, Wilson argued that the CSFL is unreasonable and that a constant friction coefficient is unrealistic. Through several typical examples, it was demonstrated that treating the friction coefficient as a function of state variables, including pressure, effective strain, temperature, and surface expansion, is a general and practical method to account for the various types of friction occurring in different processes, including the LRC.

Various testing methods have been developed to determine the friction coefficients and factors. Recently, friction tests utilizing AI have also been attempted. The most widely used is the ring compression test. This testing method is more economical than other tests, and it also ensures experimental simplicity and clarity of results. However, it was emphasized that it has the disadvantage of not being able to represent the large pressure variations and various friction conditions that occur during forging. Particularly, it was stressed that one should avoid discussing the validity of a specific friction model or giving results from comparative studies of friction models more significance than the constraints imposed by using the ring compression test as the basis.

Friction phenomena are very complex, and the dispersion of experimental results is also quite large since the material–die interface in metal forming processes is subjected to extreme conditions of elevated temperatures and high pressure. As a result, this review study revealed that the laws of friction and related information are under confusion. In the past, friction was not a key concern. During MFSs, friction was neglected in core aspects because simulation technology was significantly affected by numerical issues and the material’s flow behaviors. In many applied studies, friction was treated merely as a simple assumption.

With the advancement of measurement technologies regarding flow behaviors and macroscopic phenomena, the accuracy of related engineering analyses has improved, and as friction and wear become critical in the new age of new material technologies and net-zero, the significance of friction is increasingly recognized. In the future, the academic and technical development related to friction is expected to advance continuously alongside the sophistication of MFS technology, that is, through considering the evolution of tribological features at the material–die interface to achieve its high accuracy.