Failure analysis of AISI 430 stainless steel sheet under stretching and bending conditions

Ferritic stainless steels have been widely used to substitute austenitic steels because of their lower cost and higher deep drawing capacity. However, failures, such as shear fracture, have been observed in parts with small radii during the drawing process in applications where both bending and stretching occurs. Conventional techniques, such as failure criterion consideration, finite element method, and forming limit diagram, are unable to predict this type of fracture, which has hampered the development of new processes and products. To overcome this limitation, herein, we investigated the effects of tool radius, direction, and test speed on the formability to fracture of an AISI 430 stainless steel sheet under bending and stretching conditions. An equipment was used to perform the draw-bend fracture (DBF) test based on the bending under tension test. The DBF test efficiently reproduced the tensile, mixed, and shear fractures. Furthermore, we determined the fracture limit strain on the outer surface, thickness, and sidewall of the sample, as well as the coefficient of friction. The data showed that the radius to thickness ratio related to the process parameters had a direct impact on the experimental results. In addition, the results can be utilized for setting design guidelines and failure prevention.


Introduction
Ferritic stainless steels (FSSs) are metallic alloys based on Fe-Cr with the < Cr > content of 10.5-27%, which exhibit high corrosion resistance and possess long shelf life [1,2]. In recent years, FSSs have been widely used in various industries (e.g., civil, home appliances, food, pharmaceutical, chemical, bioengineering, and automotive) as an alternative to austenitic stainless steels, mainly because of their lower cost and excellent ability for deep drawing. In addition, they have a lower thermal expansion, higher thermal conductivity, and higher resistance to oxidation at high temperatures, and are not hardened by heat treatment, but by cold working [3,4].
However, one of the considerable challenges of these and other metal alloys is the successful stamping of parts, which requires having an excellent surface appearance, minimal material consumption, high productivity, low tool wear, and absence of fractures. Over the years, researchers around the world have devoted efforts to primarily mitigate failures during the sheet metal forming (SMF) process, as this has been the main barrier for the wide-scale use of new materials and materials with specific properties [5][6][7][8][9][10][11]. Since it is addressing the use of new materials and materials with specific properties, we could highlight the fact that material characterization is very important, such as the scanning electron microscopy (SEM) and advanced manufacturing tools. For these reasons, advanced SEM equipment has been used in the microanalysis of the properties of materials in general [12].
It can be seen in Fig. 1a [13] also observed this type of failure during a deep drawing operation. This type of failure contrasts the crystalline texture of the material. Many studies have emphasized the excellent deep drawing capacity of the AISI 430 steel owing to the cold rolling texture in the center of the sheet [14,15]. Damborg [16] revealed that most conventional low-C alloys typically fail in response to stretching in regions under plane strain or with soft radii. This type of failure is characterized as a shear fracture because it occurs in parts with small radii under low load with little or no apparent necking, in contrast to a tensile fracture. To date, this behavior has only been investigated in advanced high strength steels (AHSSs), but not in FSSs. Among the materials investigated is the DP590 steel, which has formability similar to that of the AISI 430 steel. Several authors have performed a series of studies on the shear fracture in AHSSs and concluded that a critical R∕t for each test material exists between the safe and fail zones occurring during stretch and bending [7,9,[17][18][19]. Ghosh and Hecker [20] observed that the bending on deformation has a positive influence on SMF processes. However, Baudelet and Ragab's [21] experiment investigated the out-of-plane and in-plane stretch, and concluded that it is found that the former method produces higher limit strains than the latter under identical degrees of strain biaxiality and initial sheet thickness. Cheong [22] explained that the superposition of stretching and bending can lead to a nonlinear deformation path. Currently available techniques, such as failure criterion consideration, finite element analysis (FEA), and FLD, are based on localized strain before fracture; therefore, they are unable to predict the shear fracture in parts with small radii because there is almost no necking. Supporting these arguments, it can be seen in Fig. 1a, b that the fracture limit strain on the outer surface of the example part in the plane region (point P3) was above FLD 0 ; on the other hand, in the regions with small radii (points P4 and P5), fracture limit strain was below FLD 0 , i.e., the fracture occurred in a region initially considered safe to stamp the part, denoting that the FLD method was ineffective in predicting faults in regions with small radii of parts.
Accordingly, experimental methods have been the most effective in the reproduction, characterization, and analysis of this type of failure. Recently, in an attempt to improve failure predictability, the crack opening modes were associated with the formability limits. Failure by fracture is characterized by the fracture forming limit line (FFL), originally proposed by Embury and Duncan [23] for cracks opening by tension (mode I of fracture mechanics [24]) and by the shear fracture forming limit line (SFFL), recently disclosed by Isik et al. [25] for cracks opening by in-plane shear (mode II of fracture mechanics). The mechanics and physics behind these two fracture lines are comprehensively discussed by Martins et al. [26]. Despite the above-mentioned advances in the characterization of fracture in a metallic sheet, the determination of the SFFL (mode II) and of the transition between the FFL and the SFFL (mixed-mode, consisting of crack opening by a combination of Fig. 1 a Fractures in an example part with AISI 430 steel; b forming limit diagram (FLD). RD-0° is the sheet rolling direction tension and in-plane shear) is still an open research topic. As with occur FLD, these methods cannot predict crack opening in regions with overlapping bending and stretching. However, there are few mechanical tests capable of reliably reproducing the shear fracture of metal sheets with small radii of the punch and die in a laboratory environment. Over the years, several experimental tests have been developed to evaluate the formability of metal sheets under bending and stretching conditions, such as the angular stretch-bend test (ASBT) [27], modified Duncan-Shabel apparatus [17], bending under tension (BUT) test [28,29], and use of the stretch forming simulator (SFS) [18]. Among these, the latter two tests have been widely used because they accurately simulate the plastic strain mechanics of the sheet over the tool. Shih et al. [19] used the SFS and conducted the modified-BUT tests to reproduce shear fractures and observed that both fracture limit curves converged satisfactorily.
Originally, the BUT test was developed to simulate the contact and the deformation of a sheet metal at the die radius. During the test, a metallic strip is forced to slide over a fixed or free cylindrical pin using two independently controlled hydraulic actuators arranged at a 90° angle [28,29]. The strip is subjected to the combined effect of the bending, unbending, and stretching efforts [30]. Subsequently, Sung et al. [7] modified the BUT test to operate it at a wide range of stamping speeds and R∕t ratios, ensuring high consistency and reproducibility of results. This test became known as the tensile draw-bend fracture (DBF) test. The mechanics and physics behind these and other formability tests are comprehensively discussed by Trzepiecinski and Lemu [31] and Schell and Groche [32]. In addition, the authors discussed applications relevant to SMF processes, machines, and process integration.
However, to date, no study has been conducted to analyze the failures in AISI 430 steel under bending and stretching conditions, causing a gap in our knowledge related to the formability of this material. Therefore, the present work aimed to reproduce, characterize, and analyze the failures presented by this material under these deformation conditions. For this, an equipment based on the BUT test was used to perform the DBF test. In addition, process parameters, such as pin radius, direction, and test speed, were varied to investigate their effects on the fracture limit deformation on the outer surface, thickness, and sidewall of the sheet, as well as on the coefficient of friction value. The results were plotted as a function of the R∕t ratio and then compared, analyzed, and discussed.

Material and sample preparation
The material investigated was an FSS AISI 430 sheet with an initial thickness, t 0 , of 0.8 mm, and the bending pins were made of the AISI O1 steel. The steel sheet under the as-received condition was cold-rolled, annealed, and pickled, with a slight skin pass. Its chemical composition is shown in Table 1.
The surface roughness (Ra) of the samples and bending pins were measured with a portable rugosimeter, model Rugosurf 20 (Tesa SA, Renens, Switzerland), and the average results were 0.051 ± 0.010 μm and 0.270 ± 0.048 μm, respectively. Their hardness was determined using a Vickers microhardness tester, model HMV-2 T (Shimadzu, Kyoto, and Japan), with a load of 4.9 N, and the average results were 158 ± 6 HV and 746 ± 11 HV, respectively.
The tensile mechanical properties of the AISI 430 steel were determined using a universal testing machine, model DL30000 (Instron/Emic, Massachusetts, USA). Three tensile samples, with geometry according to ASTM E8/E8M [33], were cut in three different directions (0°, 45°, and 90°) in relation to the original direction of sheet rolling using the wire electrical discharge machining, model EURO-EW1 (Eurostec, Caxias do Sul, Brazil). The mechanical properties of the AISI 430 steel sheet are listed in Table 2. The procedures described by Banabic et al. [34] were used to determine coefficients n , r , r b , and Δr. The samples used in the conformability tests were cut on a mechanical guillotine according to the geometry shown in

Draw-bend fracture (DBF) test
As shown in Fig. 3a, an apparatus based on the BUT test has been used in the execution of the DBF test. Its limit characteristics consist of the application of a force of 44.5 kN, a speed of 75 mm/s, and displacement of the hydraulic actuators by up to 250 mm. In addition, conformability tests may be performed using free and fixed pins. As can be seen in Fig. 3b, a fixed pin tool holder was specially designed to vary the radius of the bending pin in this study. The top surface of the pins was positioned at the intersection of the lines of action of the two hydraulic cylinders to maintain tangency at an angle of 90°.
As illustrated in Fig. 4, the data acquisition system (DAS) of the equipment used USB interface devices (Loadstar Sensors, Fremont, USA), which were responsible for the acquisition and signal processing of load and displacement sensors. This equipment still contained a meter and torque sensor, but it was used only in applications with free pins. With the aid of an application software (SensorVUE), these data were stored simultaneously on a computer. Subsequently, the graphs were plotted for an analysis of the results. Puroshit [36] used a DAS similar to that shown in Fig. 4. Table 3 lists the operational parameters adopted in the DBF test. In each new test, the tribo-surfaces were cleaned with acetone, and then a mineral oil-based lubricant was applied abundantly using a silicone oil brush. Three samples were tested under each test condition to ensure the repeatability of the results. Figure 3c shows that the DBF test is based on the operating principle of the BUT test, wherein four components of forces are assumed during the sliding of the sheet under the bending pin's radius of curvature. Several authors [29,38,39] have explained that the force required to pull the strip ( F 1 ) around where F 1 is the frontal or pulling force, F 2 is the back or restraining force, F b is the bending force, and F f is the frictional force.

Constitutive equations and fractured samples analysis
The fracture limit strain on the outer surface of the sheet ( 1f ), which occurred in the direction perpendicular to the fracture of the samples during the DBF test, was obtained by Eq. (2), as described by Silva et al. [40]. For CGA, we used the Zürich procedure n.5 presented by Parniere and Sanz [41], which evolved into the position-dependent method described by norm ISO 12004-2 [42]. To improve the accuracy of measuring the strains imposed on the samples, a digital microscope (up to 1000 × magnification and 2.0 MP resolution) and Image-Pro Plus software (version 6.0) were used to capture images and measure the deterministic grid of secant circles (before and after the experimental tests), respectively.
where l major is the length of the major axis of the ellipses resulting from the longitudinal deformation of the secant circles.
The thickness reduction ( R f ) and fracture limit strain in the sheet thickness ( 3f ) were determined using Eqs. (3) and (4), respectively, as reported by Nielsen and Martins [43]. The fracture thickness ( t f ) was obtained using a digital micrometer with 0.001 mm precision, and its value corresponded to the average of five measurements at different locations in the fractured section of the sample. Since the deformation was compressive, its result was plotted in modulus for comparison purposes.
The fracture limit strain on the sample wall due to stretching was determined by the ratio between the maximum length and initial of the notch, L max ∕L 0 . L max was obtained by an LVDT sensor (KTM series 275, 0.05% accuracy) attached to the front hydraulic cylinder rod of the apparatus shown in Fig. 3a. On the other hand, L 0 was obtained by a digital caliper (Mitutoyo 0-200 mm, 0.01 mm accuracy). The morphology of the fracture surfaces was analyzed using a scanning electron microscope, model JSM-6510LV (Jeol, Tokyo, Japan) with an acceleration of 20 kV.
The COF (µ) was calculated using Eq. (5). This equation has been used for decades [38,44] in the tribology of SMF with cylindrical tools because it considers the geometric parameters of the tribological pair in the constitutive equation, such as the sheet thickness and tool radius.
The bending force ( F b ) was determined using Eq. (6), as introduced by Swift [45]. The sample width ( w ) was assumed to be the notch width because the localized deformation was concentrated in this region.   Figure 5a shows the schematic representation of the deep drawing process and Fig. 5b-d show the different types of fractures that were reproduced during the DBF test: tensile, mixed, and shear fractures. These fractures were designated as types 1, 2, and 3, respectively. A condition of plane strain was assumed during the experimental tests of the samples.

Characterization of fracture types
The experimental results showed that as radius/thickness ( R∕t) ratio revealed, the fracture tends to move from the curvature region to the flat region of the sheet. Figure 5a, b show that type 1 is similar to the fracture that occurs in the sidewall of stamped parts. Generally, this failure is referred to as ductile fracture and is observed in metallic materials in a conventional tensile test, as the fracture occurred when the sample-resistant section was reduced by localized deformation or necking. Dieter [46] explained that before the ductile fracture, the necking introduces a triaxial stress state, and a hydrostatic component of stress acts at the center of the notch region until the sample reached its minimum dimensions (Fig. 6a) and, consequently, its final rupture. A commonly accepted interpretation is that a tensile fracture originates from the initiation, growth, and coalescence of microscopic voids during plastic deformation [47]. This void nucleation generally occurs at the interfaces of inclusions and second-phase particles, and the dissociation of these interfaces is the dominant mechanism in void nucleation. These behaviors were confirmed by an interrupted tensile test, as shown in Fig. 7a. The fracture propagated from the center to the edges of the notch (Fig. 7b), and the morphological aspect of the fracture surface (Fig. 7c) was similar to that of the sample subjected to the DBF test (Fig. 6b). Both fracture surfaces were characterized by the presence of equiaxed dimples, which, from a macroscopic perspective, are normal to the direction of the applied force ( F 1 ).
In contrast, Fig. 5a, d show that type 3 tends to occur under the tool radius and is similar to brittle fracture, which, by definition, occurs with low energy absorption (i.e., with little or no macroscopic plastic deformation) under stresses lower than those corresponding to the generalized yielding, and with a significantly high crack-propagation velocity [46,47]. Furthermore, the fracture started at the edge of the sample and propagated perpendicularly to its longitudinal axis while its minimum dimensions (Fig. 8a) were still much larger than those of type 1 (Fig. 6a). Supporting these results, Martínez-Donaire et al. [48] demonstrated that the fracture initiation site was shifted closer to the sheet free edge as the punch radius decreased and with low plastic strain. Figure 8b clearly shows that the fracture surface morphology contains sheared dimples or highly elongated, characteristics typical of those obtained in a shear fracture. The shear fracture mechanism during an SMF operation is not by pure shear as seen in sheet metal cutting, which would result in a flat surface practically without any significant topographic relief, but by sliding between crystallographic planes. The presence of cleavage facets on the fracture surface (Fig. 8b) supported these arguments, since they occurred due to the separation of the crystallographic planes by rupturing atomic bonds [49].
A comparative analysis between Figs. 8b and 9b evidence that the failure that occurred in the example part is a shear fracture, as both surfaces presented the same fracture mechanisms, i.e., sheared dimples and cleavage facets. This also indicated that the DBF test efficiently reproduced, in a laboratory environment, a fracture commonly observed in industrial practices, especially in the region of small radii of the punch.
On the other hand, type 2 is called mixed because it occurs during the transition between a shear and tensile fracture. Figure 5a, c show that this type of fracture tends to occur on the tangent point line between the side wall and punch radius. Sung et al. [7] explained that, in industrial practices, this type of fracture usually starts close to the edge of the sample like a type 3, but tends to propagate at a similar angle to type 1. However, it is unlikely to occur frequently as the R∕t ratio is quite high in most SMF applications. Shih and Shi [18] emphasized that this type of fracture is significantly sensitive to the restraining condition ( F 2 ) or the tension level applied during the bending. Figure 10 shows the effect of the ratio R∕t on the fracture limit strain on the outer surface ( 1f ) of the samples subjected to different directions and test speeds. In a first analysis, the results indicated that 1f increased significantly with increasing R∕t ratio. This behavior was due to the increase in the yield stress of the sample by strain hardening. In recent experimental studies, several researchers have observed the same tendency for other materials [50][51][52][53]. In general, the results showed that 1f , in the RD direction, reached its maximum values for a critical R∕t ratio, approximately equal to 13.0 and 9.0 for the velocity of 2.5 and 25 mm/s, respectively. On the other hand, in the TD direction, the critical R∕t ratio was approximately 10.0 for both speeds. Most likely, the differences observed can be attributed to the balance between the combined effect of bending and stretching on the stress gradient during plastic deformation of the metallic strip.

Fracture limit strain at outer surface ( " 1f )
Additionally, 1f decreased in both directions as the test speed increased. It is well known that in SMF processes the increase in velocity has a significative effect on the strain rate and, consequently, on the material properties. At room temperature, the work hardening of the material increases with increasing strain rate due to a more  [54] demonstrated, through an analytical prediction of the behavior of metal sheets under stretching and bending, that 1f tends to be higher for materials with a high hardening coefficient ( n ); however, for a higher R∕t ratio, this value tends to decrease, as stretching dominates the plastic deformation process. This reported behavior can be clearly seen in Fig. 10a, b, approximately from R∕t>12.0. At lower speeds, the TD direction (~ 9.0 < R∕t<12.0) exhibited the highest possible value of 1f . These different behaviors can be attributed to the heterogeneity of the material properties. Supporting these arguments, Table 2 shows that the TD sample showed a higher anisotropy coefficient (~ 17%) than the RD sample, indicating that the sheet plane surface deform more until necking and fracture occurred.
A more careful analysis of the results showed that as R∕t increased, 1f tends to move from the curvature region to the flat region of the sheet, agreeing with the discussion in the previous section. Therefore, the sheet fracture mechanism changed from the shear fracture in the punch radius to the tensile fracture in the punch sidewall. This behavior has also been observed in studies of other materials [7,19,50]. Supporting these arguments, the macrographic images of the fractured samples ( Fig. 5b-d) and the results shown in Fig. 10 demonstrated that type 1, type 2, and type 3 occurred when R∕t>6.0, 4.5 < R∕t<6.0, and R∕t<4.5, respectively.
A comparative analysis between Figs. 1a and 10 revealed that the magnitudes of the deformations were very similar, indicating again that the DBF test effectively reproduced an SMF industrial operation. Therefore, these results could be used as design guidelines and failure prevention criteria in the development of processes and products from FSS AISI 430. In addition, they can be used in the SMF processes to avoid problems related to conformability in the parts' radii. Figure 11 shows the effect of the ratio R∕t on the fracture limit strain upon the thickness ( 3f ) of samples subjected to different directions and test speeds. From this figure, it is clear that 3f increases with an increase in the ratio R∕t , i.e., the sheet thickness decreases. In this regard, Yoshida et al. [54] explained that during stretching and bending, the yield strength of the sheet ( S y ) increases with the outer surface strain 1f because of its cold work hardening; however, the thickness of the sheet decreases. Since the sample is in a plane strain state, this behavior obeys the law of volume constancy.

Fracture limit strain of thickness ( " 3f )
A more careful analysis of the results showed that 3f tends to stabilize more quickly in e TD (~ R∕t = 7.5 ) than in the RD (~ R∕t = 13.0 ). This behavior was explained by the mechanical properties of the material (Table 2); the higher the normal anisotropy coefficient (r), the higher is the material's resistance to thinning, allowing the surface and wall of the strip to deform more until the fracture. The results also showed that when the R∕t ratio was very low (≤ 1.5), type 3 occurred almost instantly, as there was practically no apparent necking in the sample section ( 3f < 7%), indicating that the thickness strain ceased even before the sample reached uniform elongation. However, this behavior differed from that of type 1 because when R∕t was high the ductility of the sheet still remained quite significant ( 3f > 30%), even after reaching the uniform elongation, ceasing only with the sample fracture. Increasing the test speed had a more significant effect on the RD, most likely due to its lower normal anisotropy coefficient; consequently, the thickness reduction tended to increase.
In summary, these results suggested that for predicting a fracture in SMF processes, the FLD is valid only for failures induced by necking on flat surfaces and under approximately proportional loading conditions. In this regard, Baudelet and Ragab [21] emphasized that nonlinear deformation trajectories and out-ofplane deformations are excluded from FLDs, which explains the unpredictability of the fracture of AISI 430 steel shown in Fig. 1. Figure 12 shows the effect of the R∕t ratio on the limit wall stretch of the samples subjected to different test directions and speeds. It is clearly noted that L max ∕L 0 increased significantly with increasing R∕t because of the lower strain hardening of the material as the bending severity decreased. According to the bending theory [34,46], the larger the radius of curvature, the smaller the decrease in sheet thickness in the bending region and, consequently, the higher the capacity of the sheet to stretch or deform longitudinally. Several researchers [18][19][20][21][54][55][56][57] have investigated the effect of the radius of curvature on the limit wall stretch of other materials and observed the same trend.

Limit wall stretch ( L max ∕L 0 )
Notably, it is known that during pure bending, the convex side of the bend (outer surface) experiences tension and the concave side experiences compression. Due to this non-uniform status through the thickness, a strain gradient is generated with a neutral layer (NL) where neither tension nor compression is observed. Generally, more severe bending provides a stronger strain gradient [22,30,46]. In sheet metal forming, the bending mode is one of plane strain since the width of the sheet is generally much larger than the sheet thickness. However, the difference between the different stress states along the material thickness causes a displacement of the NL. In this regard, Ma and Welo [58] explained that for most metallic materials during bending, the NL moves away from the center of the sheet as the Fig. 10 Effect of R∕t ratio on the fracture limit strain of outer surface ( 1f ) at different test velocities. a RD-0°; b TD-90°F ig. 11 Effect of the R∕t ratio on the fracture limit strain of thickness ( 3f ) at different test velocities. a RD-0°; b TD-90°2 radius of curvature increases, causing changes in the strain gradient. Cheong [22] emphasized that the strain gradient has a critical impact on strain instability, and even though the outmost layer reaches the critical strain limit from FLD, the inner layers that are in moderate tension or compression provide stability and serve to mitigate necking. In this context, Tharrett and Stoughton [50] proposed an empirical concave side rule (CSR), which stated that strain instability occurs only when all of the material layers in the sheet have exceeded the forming limit strain. As a result, in forming operations with in-plane tension and out-of-plane bending, the initiation of necking is delayed until the innermost layer has reached the limit strain. Figure 11 confirms this behavior as the R∕t ratio increased. Therefore, under these conditions, the strip wall tends to reach higher levels of stretch, as can be seen in Fig. 12a, b.
Additionally, the magnitude of L max ∕L 0 decreased with an increase in the test speed. As discussed, increasing the strain rate caused the yield strength of the material to increase due to strain hardening and, as result, the sheet ductility decreased. Fractures occur when all fibers close to the outer surface reach the limit strain ( 1f ) [52]. At high speed, both directions (RD and TD) showed very similar behavior; however, at low speed, the TD sample (Fig. 10b) exhibited a higher L max ∕L 0 value, most likely, due to the high normal anisotropy coefficient in this direction, which increased the resistance to thinning and, consequently, their stretching capacity.
Therefore, the experimental results showed that the limit strain of the sheet metal ( limit ) under bending and stretching is assumed to be the superposition of the bending limit deformation ( bend ) and stretch ( stretch ). This suggests that the smaller the contribution of bend in the plastic deformation process-this occurs with the displacement of the NL out of the plane as the ratio R∕t increases-the more limit approaches stretch and, consequently, of FLD 0 . Considering this, the industrial practice has produced thicker parts and with large bend radius to ensure success in SMF operations. However, these practices directly impact the other parameters; for example, an increase in the force required to deform a part, an increase in production costs, and an increase in the friction and wear process between the tribo-surfaces. Experimental analysis is a very powerful tool in this context, as it allows a more curated analysis of the physical phenomena involved in the plastic deformation process of a metallic sheet, which can determine more assertive solutions in developing processes and products. Figure 13 shows the effect of the R∕t ratio on the COF of the samples subjected to different test directions and speeds. We noticed that the COF increased with a decrease in the pin radius; however, this behavior changed from a critical R∕t ratio, from which the COF increased with an increase in the pin radius. Nanayakkara et al. [38] observed a similar behavior for a galvanized steel sheet subjected to the BUT test and concluded that from a critical radius, the tribo-system changed from a mixed lubrication regime to a hydrodynamic lubrication regime. Andreasen et al. [59] explained that this change may be related to the varying contact pressure and sheet roughness due to the plastic strain by stretching.

Coefficient of friction (COF)
Kim et al. [60] demonstrated that the contact pressure increases as the radius of the pin decreases; moreover, it is not uniformly distributed on the contact surface. According to the Stribeck curve, when the contact pressure increases ( p ), the lubricant viscosity ( ) or speed ( v ) decreases [61]. Under these conditions, the ability of the lubricant to separate the contact surfaces and stabilize the COF is diminished [62]. Several authors [63][64][65] have reported that at higher pressures, the lubricating film can be expelled from the friction zone or break, increasing the interaction between the asperities of the tribo-surfaces and, consequently, the friction resistance. Once the sheet surface is softer (~ 4.8 times) than the bending pin, an increase in this interaction produces deep galling on the sheet surface and with the material transfer, as shown in Fig. 14a. This suggests that the high level of friction and galling may have contributed to the occurrence of shear fracture. Figure 14b also supports these arguments by showcasing the different micro-effects of friction and wear mechanisms that governed the interface of the tribo-contact.
Still according to the Stribeck curve, the COF in fluidlubricated contacts is a nonlinear function with the Hersey number ( v∕p ) [61]. From a critical value of this number, the COF tends to increase continuously, suggesting that significant changes occur at the contact interface, which explains the increase in the COF from the R∕t ratio critical. In SMF operations, the lubricating film is very thin; therefore, the lubrication regime is said to be micro-hydrodynamic [63]. This lubrication regime is very sensitive to topographical changes at the contact interface. Recent studies have shown that the surface roughness and COF increase as the relative elongation of the sheet increases [66,67]. This increase in surface roughness is explained by a theory about the formation of Lüders bands during the plastic deformation of steels with low carbon [68][69][70]. Makhkamov [71] emphasized that in addition to the plastic deformation altering the surface roughness, it also eliminates most of the elastic effects on contact and opens a new surface by the action of dislocation sliding.
However, the contribution of roughness to the tribo-system cannot be analyzed in isolation. It is well known that a higher surface roughness guarantees enhanced lubrication due to the presence of more valleys; however, this is true only to a certain extent, as varying the contact pressure affects the adhesion mechanisms and plastic strain of sheet asperities. Since sheet asperities are softer than tool asperities, at high pressures, they tend to undergo a greater degree of flattening and, consequently, adhesive forces, frictional resistance, and surface wear tend to increase.
In addition, several researchers [38,72] have demonstrated that the bending force is significant in friction analysis and, therefore, cannot be ignored. Equation (6) shows that F b decreases with increasing pin radius. According to Eq. (1), the force required to pull the strip ( F 1 ) over the pin increases as F b decreases, resulting in an increase in the normal load on the sheet surface that causes a greater degree of flattening of its asperities. Equation (5) also supports these arguments.
The COFs obtained showed the same tendencies as other researchers [60,63] observed in their experiments, COF decreased as the test speed increased. At higher speeds, it was observed that the RD and TD samples presented very similar behaviors, suggesting that the COF became more independent of the contact pressure under this condition. The Stribeck curve also supports this argument. This was because the lubrication regime at the contact interface became more similar to micro-hydrodynamic lubrication, where the load was carried more by the lubricating film.
However, at lower speeds, the COF in the RD direction was higher than that in the TD direction, most likely due to ridging of the AISI 430 steel sheet under plastic  [73] explained that FSS exhibits ridges parallel to the rolling direction when subjected to drawing or deep drawing operations. Generally, the ridges have a depth in the range of 20-50 μm. Luiz and Rodrigues [39,67] explained that, although the ridges are considered superficial defects that negatively affect the visual aspect of parts, the depth and width of the ridges are relevant for thin sheets, because the distribution of the lubricant and the efforts are not uniform at the contact interface, which can cause different lubrication regimes in the same tribo-system and, as result, the friction resistance tends to increase.

Conclusions
The main conclusions obtained from the analysis of the experimental results are summarized below: • The DBF test was able to reliably reproduce, under a plane strain state, three different types of fractures in an AISI 430 sheet steel: type 1 (necking failure in the punch sidewall), type 2 (failure in the tangent point between the punch radius and sidewall), and type 3 (failure in the punch radius). • The fracture limit strain on the outer surface increased with increasing R∕t and, simultaneously, shifted and changed its typical characteristics. The tensile fracture occurred when R∕t>6.0, mixed fracture when 4.5 < R∕t <6.0, and the shear fracture when R∕t<4. 5. The results showed still that in the RD direction, fracture limit strain on the outer surface reached its maximum values for a critical R∕t ratio, approximately equal to 13.1 and 9.0 for the velocity of 2.5 and 25 mm/s, respectively. On the other hand, in the TD direction, the critical R∕t ratio was approximately 10.3 for both speeds. • The fracture limit strain at the thickness increased with increasing R∕t ; however, beyond a critical point in their relationship, it exhibited a tendency to stabilize. This indicated that the FLD was valid only for neckinginduced failures on flat surfaces and under approximately proportional loading conditions. • The wall stretch limit increased with increasing R∕t , most likely due to lesser work hardening of the material. The increase in sheet resistance to thinning in the TD provided a more enhanced stretch than the RD. • The COF increased with the decrease in the pin radius; however, this behavior changed from a critical R∕t ratio, where the COF increased with increasing pin radius. Parameters such as the variation of contact pressure, bending force, lubrication regime, roughness, and sheet surface ridging had a direct impact on the COF.
• Finally, these results can be used as design guidelines and failure prevention criteria in the development of processes and products from AISI 430 steel. In addition, they can be used to avoid failures related to conformability in parts' radii.