Does friction contribute to formability improvement using servo press?

Servo press forming machines are advanced forming systems that are capable of imparting interrupted punch motion, resulting in enhanced room temperature formability. The exact mechanism of the formability improvement is not yet established. The contribution of interrupted motion in the ductility improvement has been studied through stress relaxation phenomena in uniaxial tensile (UT) tests. However, the reason for improved formability observed when employing servo press is complicated due to the additional contribution from frictional effects. In the present work, an attempt is made to decouple the friction effect on formability improvement numerically. The improved formability is studied using a hole expansion test (HET). The limit of forming during hole expansion is modeled using the Hosford-Coulomb (HC) damage criteria, which is implemented as a user subroutine in a commercial explicit finite element (FE) software. Only the contribution of stress relaxation is accounted for in the evolution of the damage variable during interrupted loading. Therefore, the difference between simulation and experimental hole expansion ratio (HER) can be used to decouple the friction effect from the overall formability improvement during hole expansion. The improvement in HER due to stress relaxation and friction effect is different. The study showed that the model effectively captures the hole expansion deformation process in both monotonic and interrupted loading conditions. Compared to stress relaxation, friction effect played a major role during interrupted HET.


Introduction
Dual-phase (DP) steels, owing to their superior mechanical properties, are widely used in automotive applications [1][2][3]. However, poor stretch-flangeability is a concern associated with many advanced high strength steels (AHSS) including DP steels [4][5][6]. Stretch-flangeability, also referred to as edge formability, plays an important role in many sheet forming operations including bending and flanging. The hole expansion test (HET) is commonly used to measure the edge formability. In a standard HET, the blank with a center hole (CH) is expanded using a conical punch. The test is continuously monitored till the appearance of a through thickness crack, and the final diameter of the hole at fracture is measured. The ratio of the change in the hole diameter to the initial hole diameter is referred to as hole expansion ratio (HER). A large value of HER suggests higher edge formability. Generally, the CH of a standard HET † Kali PRASAD and Aishwary GUPTA contributed equally to this work. * Corresponding author: Kali  Damage state variable : Lode angle parameter specimen is prepared by punching process as it closely resembles the industrial practice. However, researchers [7][8][9] have also investigated other traditional machining processes like drilling, boring, wire cut electric discharge machining (W-EDM), and laser machining. It was experimentally shown that specimen prepared through W-EDM process yields the highest HER [8]. This trend was attributed to relatively very few defects with hole edge preparation from W-EDM process compared to those of other machining processes. It has been shown that in addition to the hole edge, strain path also plays a significant role in HER [10]. The influence of strain path on edge formability can be tested by varying the punch geometry [10,11]. A conical punch would induce a uniaxial stress state, whereas a flat bottom punch subjects the specimen to a plane strain state in the vicinity of failure [11]. A hemispherical punch induces a complex and continuously varying strain path (stress state). Since plane strain state leads to early failure, it is expected that the HER estimated using a flatbottomed punch would be less than that obtained by a conical punch, as reported by Pathak et al. [11], using dual phase and complex phase steels. Furthermore, the HER was also found to depend on several metallurgical factors such as material microstructure [1,12], non-metallic inclusions [13], heat treatment condition [14,15], alloy chemistry, [5] and relative strengths of the individual phases [4,15]. Since the stress state is nearly uniaxial when deformed using a conical punch [3,8,11], attempts have been made to correlate uniaxial tensile (UT) properties with HER [16,17]. It has been shown that yield strength (YS), ultimate tensile strength (UTS), uniform elongation (UE), etc., have a positive correlation with HER [17,18]. Fang et al. [14] investigated the hole expansion behavior of C-Mn steels with different heat treatment conditions and concluded that HER increases with the ratio of YS/UTS. Recent studies indicate that HER correlates better with fracture toughness than tensile properties [6]. High HER is obtained in materials exhibiting higher fracture toughness. HER can be potentially improved if the onset of fracture can be delayed during plastic deformation. It has been shown that stress relaxation during plastic deformation delays the onset of fracture. This has been experimentally verified in many materials [19][20][21][22][23][24]. The improvement is primarily attributed to the combined effect of dislocation annihilation and homogenization of internal stress [25][26][27]. During HET, intermittent stopping of the conical punch during deformation is expected to delay the onset of fracture due to stress relaxation and improve HER.
The first systematic study of interrupted HET was recently reported by Prasad et al. [9]. As expected, the HER evaluated during interrupted HET was much higher than the corresponding monotonic values. Interestingly, the strain increment to onset of fracture during HET in the above study was much higher than the ductility improvement observed during UT test at similar conditions. Since a conical punch was used, the difference was not due to any change in strain path ① . The authors attributed the difference to the effect of friction on the interrupted behavior. The friction coefficient between punch and blank is influenced by several factors such as contact pressure, sliding velocity, material grade, lubrication, and temperature [29]. The magnitude of the friction coefficient influences the metal flow over the punch, determining the quality of the component being formed. The concept of friction coefficient is strictly applicable to Coloumb's adhesion friction law that assumes a linear relation between friction (shear) force and normal force. In typical sheet forming applications, the normal force is high, and the linearity of Coloumb's law is not adhered. Yet, most numerical simulations assume a constancy of friction coefficient. A common example on the role of friction in sheet forming can be demonstrated using a hemispherical punch test, where the location of failure is shifted from the pole due to friction [30][31][32]. The hemispherical punch test is used to establish the forming limit diagram (FLD), an accepted failure criterion to evaluate the formability [33]. Kim et al. [34] stamped AHSS by varying the lubrication condition and determined the critical interface pressure and temperature that leads to failure. In an interesting study, Stembalski et al. [35] estimated the friction coefficient to be inversely proportional to normal pressure and sliding velocity. Similar observations were reported in Refs. [36,37]. Furthermore, analytical equations were proposed in this study to account for the influence of normal pressure and sliding velocity while estimating the ① Changing the strain path during deformation can enhance the failure limit [28]. friction coefficient. The aforesaid studies clearly show the importance of friction condition while forming sheet metal components.
In servo press forming, interrupted motion has been frequently employed to increase formability and decrease springback. During the interrupted motion, both normal pressure and velocity decrease. As discussed earlier, friction coefficient is sensitive to normal pressure and velocity [36,37]. Therefore, it is likely that the friction coefficient varies during the interrupted motion, which influences the sheet formability. In addition to the friction effect, other components such as stress relaxation and change in strain path possibly play a crucial role in affecting the formability. Therefore, elucidating the underlying mechanism and decoupling and quantifying individual component contributions are essential for developing robust process models for sheet metal forming applications employing servo presses. Except for a recent work by Prasad et al. [9], no comprehensive investigations in this direction have been conducted.
In the present work, it is attempted to analyze the HER improvement during interrupted loading in the framework of continuum damage theory. The reason for using continuum damage model to analyze the intermittent HET is multifold, while the limiting strain of localized neck formation through forming limit diagram is commonly utilized to analyze sheet forming process, the failure in HET is by fracture, and the fracture strain exceeds the limiting strain. Besides, the edge cracking is strongly influenced by the edge preparation process (drilling, punching, etc.), which prevents the proposition of a correlation between the localized necking strain under the same strain path.
Extensive research has been done in the field of ductile fracture modeling, and various fracture models have been proposed. Earlier studies by McClintock [38], Rice and Tracy [39], Gurson [40], and Needleman and Tvergaard [41] focused primarily on the influence of hydrostatic stress on the void growth to predict ductile fracture. Based on this understanding, for tensile dominated loading, various triaxiality ( )  based empirical formulations were proposed to model ductile damage [39,[43][44][45]. Chung et al. [45] used stress triaxiality based ductile fracture criterion to predict HER for three grades of AHSS. Butcher et al. [46] used Gurson-Tvergaard-Needleman (GTN) based damage model to describe the material behavior for DP600 steel. Barnwal et al. [47] used triaxiality based ductile fracture criteria proposed by Rice and Tracey [39] to predict the onset of fracture in HET. It has been shown that damage models based only on triaxiality could not completely capture the damage behavior under shear dominant loading [48]. A more general fracture criteria was introduced by Xue and Wierzbicki [49] with the third stress invariant in the weighing function. Xue [44] established the influence of Lode angle parameter on the damage evolution of material and proposed a damage model based on both stress triaxiality and Lode angle parameter. Furthermore, recent fracture models, such as the model proposed by Bai and Wierzbiki [50], shear stress based modified Mohr-Coulomb model [51], and Hosford-Coulomb (HC) model [52] included the influence of stress triaxiality and Lode angle parameter in the numerical formulations. Recent comparative studies on ductile fracture models have shown that HC model shows better predictive capability for various loading conditions [53].
In the present work, analysis of HET using the HC damage model is extended to analyze the interrupted HET. The objective of the present work is to quantify the role of friction in HET. In the absence of friction effect, it is expected that the increment in fracture strain due to stress relaxation would be similar to that of UT test. Therefore, the contribution of fracture strain increment during HET is obtained by extrapolating the ductility improvement measured under uniaxial tension. The evolution of damage parameters is fit to model the ductility improvement during stress relaxation. The damage model thus obtained is used to predict the HER. The above methodology predicts only the contribution of viscoplastic effect on the HER improvement. The difference between the trend in experiment and simulation gives the role of friction in interrupted HER.

Materials and methods
DP600 steel with thickness = 2.6 mm obtained from ArcelorMittal ① was studied. The UT and HET results ① Certain commercial equipment, instruments, software, or materials are identified to describe a procedure or concept adequately. Such identification is not intended to imply recommendation, endorsement, or implication by National Institute of Standards and Technology (NIST) that the equipment, instruments, software, or materials identified are necessarily the best available for the purpose.
are reported in Ref. [9]. The procedure and key results are summarized briefly for completeness. The uniaxial and stress relaxation tests were performed at a strain rate of 0.042 s −1 ② . Two stress relaxation tests were performed by interrupting the deformation at 50% and 70% of UTS for a period of 60 s. Additional tensile tests were performed in specimens with different specimen geometries (Section 3.4). These experiments are performed to calibrate the fracture model.
The HET were conducted as per the International Standardization Organization (ISO) 16630:2017 standard [54] (Fig. 1). HET experiments were conducted in monotonic mode and interrupted loading condition. In the interrupted HET, the punch motion was interrupted for a duration of 60 s after reaching a certain pre-defined depth. During the holding period, the specimen was not unloaded. Detailed experimental procedure is mentioned in our recent work [9]. A schematic illustration of punch displacement for the two loading modes is schematically shown in Fig. 2. The test was continuously monitored through a camera, and the appearance of through thickness crack was used to stop the test. The HER value is calculated by Eq. (1): where f d and i d are the average final and initial hole diameters of the test specimen, respectively. Equivalent failure strain eq ( )  during HET was estimated analytically by Eq. (2) following Butcher et al. [55]: where inner d and outer d refer to the inner and outer diameters at failure, respectively, i t is the initial sheet thickness, and edge t is the sheet thickness around the circumference at failure. c  and t  are circumferential and thickness strains, respectively. In this work, one ② Corresponding to average strain rate during hole expansion test.
| https://mc03.manuscriptcentral.com/friction  monotonic HET and two interrupted HETs were performed (50% and 70% of the monotonic punch displacement), and each interrupted HET was performed for 60 s holding time.
3 Constitutive modeling and calibration of ductile damage model

Plasticity and hardening model
In the present work, von Mises yield criteria is used along with associative flow rule and isotropic hardening, as given in Eqs. (3) and (4). The strain hardening behavior is modeled using the combined Swift and Voce hardening law, as given in Eq. (5).
where 2 J is the second invariant of deviatoric stress tensor, 1

Ductile fracture modeling
An arbitrary stress state can be represented using stress triaxiality ( ) and Lode angle parameter ( ). Both  and  control the effect of stress state on void evolution. Stress triaxiality is the ratio of mean stress to the hydrostatic stress (Eq. (6)). The parameter controls the micro-void growth during ductile fracture. A lower value of stress triaxiality prevents the void growth, thus postponing the fracture.  is a function of the third invariant of the stress deviator (Eq. (7)). It is used to distinguish between the different shear stress states in three dimensions. The parameter accounts for the shape change of voids, which is dependent on the specific shear stress state.

HC ductile fracture model
Mohr and Marcadet [52] proposed HC fracture criterion to model ductile fracture initiation for advanced high strength sheets under proportional loading. HC fracture model assumes localization of deformation in a narrow zone, and the localization criterion can be given as where HF  is the Hosford stress, with proportional to the maximum shear stress on the deviatoric plane, which refer to cohesion and frictional terms, respectively [52]. For a = 1, the HC model reduces to Mohr-Coulomb model [52]. Using  -dependent functions, Eqs. (10)-(12) can be obtained.
Using Eqs. (10)- (12) in Eq. (8), HF  can be written as Strain at onset of fracture ( f  ) is formulated by taking the inverse of hardening law A damage variable c ( ) D is introduced as a state variable given by Eq. (14): where d denotes the equivalent plastic strain increment. c D = 1 marks the onset of fracture, and the corresponding f  is referred to as fracture strain.
The use of damage variable ensures strain path dependence on the onset of fracture.
For damage modeling in case of stress relaxation, the fracture model is split into two parts, i.e., the onset of fracture surface before the stress relaxation point and the onset of fracture surface after the stress relaxation point. Two sets of parameters (a, b, c) need to be calibrated to denote the fracture surface with and without relaxation. Let r  be the equivalent plastic strain, at which relaxation occurs. Then, for while for  greater than or equal to r  , the fracture parameters will switch to relaxation fracture parameters r r ( , , a b and r ). c Therefore, r  may also be referred to as switching strain for the fracture surface. Thus, the evolution of c D is modified according to Eq. (15) in case of stress relaxation.

Fracture tests and HC model calibration
To obtain the material parameters of the HC model, experimental fracture tests were performed over a wide range of stress states. The specimen geometries were chosen such that they provide a wide range of stress states. Four types of specimen geometry were chosen as UT, notch specimen (NT), CH, and in-plane shear specimen (SH), as shown in Fig. 3. Specimens were cut along the rolling direction of the sheet using W-EDM. Specimens were tested using a 100 kN universal tensile testing machine (Z100, Zwick/Roell) equipped with a video extensometer at a strain rate of 0.042 s −1 . All the experiments were repeated three times for statistical significance. For calibration of HC damage parameters, finite element (FE) simulations were carried out for each specimen geometry. It is assumed that the location of the onset of fracture coincides with the location of the highest equivalent plastic strain in each specimen geometry. Thus, the critical element is selected at the location of the highest equivalent plastic strain, as shown in Fig. 4.    Table 1. The ductility improvement due to stress relaxation is estimated from interrupted UT tests. The forcedisplacement data of the stress relaxation tests interrupted as 50% and 70% of UTS strain are shown in Fig. 6. The observed trend is in line with Refs. [19][20][21] on ductility improvement due to stress relaxation.
For the calibration of fracture parameters in case of relaxation, a similar approach is used with the help   of the obtained m m , , a b and m c and plastic strain at the relaxation point, at which the model switches to r r , , a b and r .
c Then, similar to the monotonic case, the minimization problem is solved such that the predicted fracture strain according to the modified HC model coincides with the experimental fracture point for each of the relaxation cases, i.e., relaxation at 50% and 70% UTS. For the estimation of relaxation HC parameters for HET, the fracture strains for 50% and 70% punch travel relaxation points are estimated using the empirical equation of ductility improvement (Eq. (18)), which has been reported in the work of Prasad et al. [9].
where r  is the ratio of relaxation strain to monotonic strain,  is the strain rate, t is the relaxation time, and  is the strain at the beginning of relaxation. The strain at the start of relaxation was estimated using monotonic HET simulation for 50% and 70% punch travel. The HC model parameters were then calibrated following the procedure explained in Section 3.4. It is to be noted that for simplicity, only parameter b in HC model is modified, and parameters a and c are assumed to be invariant during stress relaxation. The parameters a and c primarily control the shape of the fracture surface, whereas parameter b primarily controls the position of fracture surface in the z direction. The assumption of shape of fracture surface being the same in case of relaxation has been taken for simplicity. As the primary focus of this study is towards the application in HET, this assumption is acceptable as the loading history in case of HET is very similar to the uniaxial case. The calibrated relaxation model parameters for the modified HC model with stress relaxation point, at which the monotonic model parameters are switched, are given in Table 2. r  for the HET is extrapolated using Eq. (18). The three-simensional (3D) fracture surfaces for different HC model parameters are shown in Fig. 7.

FE simulation
FE simulation of the specimens (Fig. 3) were performed with ABAQUS\Explicit 6.14 software. Three-dimensional continuum elements (C3D8R) were used to mesh the sheet specimens. Classical  (Fig. 8). The effective plastic strain, continuum damage variable stress triaxiality ( ), and Lode angle parameter (  ) are defined as state variables in the VUMAT. The onset of fracture for each FE simulation is assumed when the damage variable reaches unity. The mesh size has been chosen based on a mesh sensitivity analysis; an element size of 0.1 mm was used near the critical region. Around ten through thickness elements have been chosen for each specimen geometry. FE simulation of HET was similar to that of UT specimen, with the difference only in the boundary conditions. In the case of HET simulation, the blank edges were completely constrained. The punch was restricted to move only in the vertical direction with a punch velocity of 10 mm/min. A friction coefficient of 0.2 was assumed to model the interaction at the tool blank interface.

FE simulation of the fracture tests
As explained in Section 3.4, FE simulations for various specimen geometries were performed with the calibrated fracture model for monotonic case. The experimental and simulated force-displacement curves obtained from the fracture tests are shown in Figs. 9(a)-9(d). The onsets of fracture point are shown with red marks in Fig. 9. The representative plots show an excellent match between experimental and simulated data. This confirms the accuracy of the calibrated model for the investigated stress states. For UT specimen, FE simulations were performed with the modified HC model for stress relaxation at 50% and 70% UTS strain. The parameters of HC model were switched at the point of relaxation, as explained in Section 3.3. Figure 10 shows the onset of fracture for the three cases of UT testing.

HET results
To comprehend the hole expansion deformation process, FE simulation of HET was performed. Figure 11(a) shows the distribution of stress triaxiality for HET specimen. The stress triaxility values near the hole edge are in the range of ~0.33-0.37, which nearly corresponds to the uniaxial stress state condition.  From this distribution, it is concluded that the deformation in HET is primarily concentrated near the hole edge, and it nearly deforms in uniaxial stress condition. Figures 12(a)-12(f) show the comparison of experimental and simulated deformed HET specimens as well as contours of damage state variable (SDV2). It can be seen that the damage variable has a maximum value at the outer hole edge, where the onset of fracture occurs. Moreover, to evaluate the stress state at the hole edge, three elements viz. outer, middle, and inner edge along the through thickness direction were chosen. The major ( 1  ) and minor strains ( 2  ) corresponding to these respective elements were estimated and superposed in the strain path corresponding to the uniaxial stress state for isotropic      ), as shown in Fig. 13(a).
It is observed that the outer and middle edge deforms nearly in an uniaxial stress state. However, the inner edge deviates from the uniaxial stress state. This deviation is possibly due to the compressive stress and friction condition between the sheet and conical punch. Figure 13(b) shows the major ( 1  ) and minor strains ( 2  ) in the outer edge element. In Fig. 13(b), the respective "×" symbol refers to the major and minor strains for monotonic and interrupted loading conditions at fracture, estimated using experiment and FE analysis. The fracture points shifts when the specimen was subjected to interrupted loading. Figure 14 shows the evolution of the HC damage variables with punch displacement in monotonic and interrupted HET. The fracture is assumed to initiate when damage variable reaches to unity. It is to be noted that during monotonic HET, the damage variable monotonically increases with punch displacement, and the specimen fails at comparatively less failure  strain. However, in interrupted HET, the specimens underwent larger failure strain before the initiation of fracture, which is manifested by delay in the saturation of damage variable.
The HERs were estimated by Eq. (1). Figures 15(a) and 15(b) show the comparison of experimental and simulated HER values and corresponding fracture strains for monotonic loading condition. It is observed that experimental HER and fracture strain are higher than those of predicted through FE analysis. This difference is attributed to the definition of failure or fracture in experiment and FE analysis. In FE analysis, the failure is considered when the damage variable saturates to unity. The FE analysis accounts only for the initiation of fracture, and the evolution of fracture was not taken into consideration, whereas in experiment, the HER values were estimated once the through thickness crack appears. This accounts for both damage initiation and propagation. Additionally, uncertainty associated with the detection of through thickness crack also poses an experimental challenge to accurately estimate the HER and fracture strain. Due to this, the experimental values were higher than those of predicted data.
To further understand the effect of interrupted loading in HET, the experimental and simulated HER values and their corresponding fracture strains are shown in Figs. 16(a) and 16(b), respectively. On comparing with those of monotonic loading condition (Figs. 15(a) and 15(b)), it is observed that interrupted HET has resulted into higher values of HER and fracture strains. This improvement was found to depend on punch travel. Moreover, the increment in www.Springer.com/journal/40544 | Friction HER and corresponding fracture strain was much higher in experiment compared to that in the simulated value. Since the interrupted HET experiments were performed without unloading the specimens, as explained earlier in Section 2, the samples were subjected to stress relaxation phenomena. In addition to stress relaxation, elastic recovery during relaxation also alters the contact stresses and contact areas between punch and blank. This influences the mechanical behavior of the specimen during interrupted HET by changing the pressure-dependent friction coefficient [59,60]. However, the simulated HET accounts only for the stress relaxation effect, which is due to the viscoplastic effect of the material. Therefore, the difference in trend between the experimental and predicted values will give the net contribution of improvement predominantly due to friction effect.
The contributions of stress relaxation and friction effect in interrupted HET are quantified using a parameter (   for simulation captures only the effect of stress relaxation. The difference in the slope is attributed purely due to friction effect. It is important to note that the contribution of friction effect increases with the punch displacement at interruption. This explains the formability improvement in interrupted HET when the interruption was performed at higher punch travel.
In order to account this friction effect in simulation, the evolution of damage variable should be made a function of  , i.e., presently

Conclusions
The present study investigates the effect of monotonic and interrupted loading on the hole expansion deformation behavior of DP600 steel. A comprehensive FE analysis of HET was performed, and HC ductile damage model was implemented in the FE model. It is observed that the FE model effectively captures the hole expansion deformation process in both monotonic and interrupted loading conditions. Compared to those of monotonic loading condition, higher values of HER and fracture strains were observed in interrupted HET. The overall improvement in HER was primarily due to the two concurring effects, namely stress relaxation and friction effect. The friction was found to play a major role compared to stress relaxation in improving HER during interrupted HET.